gregory, j. - optica promota

1James Gregory's Optica Promota

§0.1. The Translator's Introduction to Gregory's Optica Promota.

We will not give a complete history of the short but productive academic life of JamesGregory (1637 - 1675). Here we relate only some details relevant to the Optica Promota 1itself. First, we note that each section in the translation consists of a collection of relatedtheorems and problems, presented in their original order, prefixed by a synopsis, andfollowed by references and notes - to which the original Latin text is finally adjoined. Itmay be remarked that the amount of published material on James Gregory isdisappointingly small, and most of that is not readily available. Thus, modern optics texts,even those which present some sort of cursory historical overview, usually omit Gregoryaltogether. It is hoped that anyone who succeeds in reading this translation andcommentary will become convinced that Gregory did make the initial fundamentalmental leap needed to elucidate image construction; also in his work will be found thebeginnings of Huygens' Principle, as well as a number of the tricks used in producing raydiagrams, such as the use of auxiliary axes, etc. In this translator's opinion, the book is awonderful piece of work, and the various parts of it eventually are orchestrated togetherin a masterful manner: indeed, the material is raised to a high level of completeness fromvery meagre beginnings, under the influence of a mind of amazing capabilities. The lackof appreciation of Gregory's achievements in optics may well lie with Gregory himself, ashe did not 'spell out' his methods explicitly, but left his readers to ponder over the raydiagrams. Thus, previous recent commentaries on the Promota, such as the few pages inchapter VII of H.W.Turnbull's Tercentenary Memorial Volume2, and Antoni Malet in hisPrinceton University doctoral thesis (1989)3, do not bring out the significance of the raydiagrams. Whiteside was later to cast a desultory eye over the work, and to expand on itsweaknesses rather than its strengths, which are obscured by Gregory's style ofpresentation. This impasse has been overcome by the present writer, and a version of thework is presented in translation with the diagrams explained in detail in the majority ofcases, while others have been left for the reader following the lead given. Ours is the firstgeneration liberated from much drudgery by the computer, and a drawing package hasallowed the ray diagrams to be examined in an interactive way, to the point that theyseem obvious and natural Also, by making use of the modern technique of applyingABCD transfer matrices, adapted here for lenses and mirrors with ellipsoidal orhyperbolic surfaces, a rigorous theory can be developed. Ray diagrams like allgeometrical demonstrations suffer from the defect of presenting only one situation,whereas algebraic methods are quite general in application, although they do show whatis going on with the light rays, which is probably why they still find a place in text books.

An interesting article by A.D.C. Simpson in the Journal of the History of Astronomy4

sheds some light of its own on the origins of the Optica Promota, apart from whatGregory tells us himself in his preamble. From Simpson's researches we learn that theyouthful Gregory who hailed from the northern climes of Aberdeen in Scotland wasgoing off to do a grand tour of Europe, a popular activity for well-heeled young men inBritain at the time - though Gregory's tour was of a more serious academic nature thanthat of most young men of the time. During his London sojourn, he was to have his littlebook on optics published, on which he appears to have spent several years in thecomposition, and a start made on a mirror for his telescope. It was Gregory's chagrinwhile in London to discover that a work on optics, including refraction and reflection by


lenses and mirrors with conoidal surfaces, had already been published by Descartes as anillustration of his method of reasoning many years previously. Indeed, the law ofrefraction as we know it was first published in the Dioptique 5in 1637 , and Descartes hadapplied the law to the ellipse and the hyperbola for rays parallel to the axis in much thesame way as Gregory, in an effort to explain the imaging properties of lenses.

The modern reader may wonder at this lapse in scholarship, but one must recall theseturbulent times - the Civil War in the 1640's had cast a long shadow over the academicworld in Britain - in which the Gregory family had become embroiled (to the extent thatGregory's eldest brother was murdered in a feud), and the difficulty in obtaining books inwhat was then a remote part of the country. We may point out that the Anderson family,well-off merchants and influential in the N.E. of Scotland at the time, to whom JamesGregory was maternally related, had already produced Alexander Anderson, anoutstanding mathematician who became a professor of mathematics in Paris, and whoedited the unpublished posthumous works of Vieta ~ 1617. It was possibly due to thisgreat-uncle that the library at Aberdeen had the optics works used by Gregory6. We alsolearn from Simpson that copies of Mersenne's Harmonica and Cogitata physico-mathematica were acquired by the same library in 1635. The first of these works,although one of the first books on musical theory, also contained designs for reflectingtelescopes originally due to Cavalieri. The interested person can access this informationin an article by Arotti Bonaventura Cavalieri, Marin Mersenne, and the ReflectingTelescope in Isis7. Only in 1644 did Mersenne bring some order into his published workswith the final version of his Cogitata and Geometriae Universiae, the latter containingsome unpublished work by Warner and Hobbes on optics. These early ideas on reflectingtelescope design used a concave mirror with a hole at the vertex, through which lightpassed to a further concave or convex mirror, and Gregory may have been aware of themvia Mersenne - if he was interested in the theory of music! However, he may have feltthat his own work was far superior - as he had produced working designs for mirrors andlenses. Also, the presence of Warner's work is of considerable interest, as he was presentwhen Thomas Harriot first discovered the law of refraction around 1600, and thus hiswritings may show Harriot's influence: for example, the ray diagram demonstratingrefraction presented by Warner is different from that developed by Descartes andsubsequently used by Hobbes and Mersenne. It is of course a fruitless exercise to indulgein speculation at this late date.

Gregory fared even worse when an attempt was made to fabricate the reflectingtelescope than with his book, which initially was beyond the craftsmanship of the time.Thus, all of Gregory's hopes were dashed. However, the seeds were sown: Newtonproduced a simplified version of the telescope, and a little later Robert Hooke succeededin producing a reflecting telescope according to Gregory's design. To this day, theaspheric lenses and mirrors are still hard to make. Nevertheless, we are left with the raydiagrams, which portray imaging devices for paraxial rays (without spherical aberration).It will, however, greatly aid the reader's understanding and appreciation of Gregory'swork, if we consider first the passage of rays through these surfaces from a modern pointof view. This work is set out in the next section.


Notes & References for §0.1.

1. Optica Promota seu abdita radiorum reflexorum & refractionum mysteria, geometrice enucleata; cuiappendix, subtilissimorum astronomiae problematon resolutionem exhibens. Autore Jacobo Gregorio,Aberdonensi. Londoni 1663.

2. James Gregory Tercentenary Memorial Volume. H.W. Turnbull. Bell (1939)

3. Studies on James Gregorie (1638 -1675). Antoni Malet. Princeton University doctoral thesis (1989).

4. James Gregory and the reflecting telescope. A.D.C. Simpson. Journal of the History of Astronomy(JHA), xxiii (1992), pp.77-92.

5. Discourse on Method, Optics, Geometry, and Meteorology Rene Descartes. An English translation andforward by Paul J. Olscamp. The Library of Liberal Arts. Bobbs-Merrill Co. (1965).

6. In ref. 4, Simpson informs us that the Marishal College library acquired its volume of Risner's OpticaeThesaurus... in 1613. The source is unknown, but correspond to the time when Alexander Anderson wasactive.

7. Bonaventura Cavalieri, Marin Mersenne, and the Reflecting Telescope. P. E. Ariotti. Isis 66 (1975), pp.303-321.

§0.2 Gregory's Optics from a Modern Viewpoint.The material presented here has been transcribed into modern language and published

in The European Journal of Physics in Feb. 2006 with the title 'ABCD transfer matricesand paraxial ray tracing for elliptic and hyperbolic lenses and mirrors' by the presentwriter. It can be viewed at stacks.org/EJP/27/393. Thus, we are in a good position tobegin examining Gregory's work.

§0.3. The Method Used to Present the Material:The work is divided into sections. A section of the work may be a single theorem, or a

group of related theorems. These have been given a section number not present in theoriginal, for reference purposes. The purest may not like this intrusion, but for most of usit is a useful addition. From my experiences with other translations, a satisfactoryapproach is one in which:(i) a synopsis of a section of the work is first given in modern terminology;(ii) a translation of the section is given in modern English. I try to avoid using ponderousLatin style sentences; the aim is to reproduce the meaning the original writer had in mind,which may or may not be a literal translation of the Latin text. Very occasionally thereare sentences which I have found obscure, where I have had to take an educated guess atwhat Gregory meant. May I offer an apology for any of these which are incorrect. I haveused simple present, past, and future tenses of verbs in line with modern math/physicstext usage rather than subjunctives etc in a slavish manner.(iii) notes and comments as referenced in (ii);(iv) the original Latin text for the whole section is lumped together.

The translation is hence several approaches rolled into one. You are at liberty to readonly the synoptic material in (i); or you can read sections (i) - (iii); while a Latinorientated person may wish to read the last section also.


In general, diagrams are labelled according to the Theorem or Problem they relate to.There are of course no such labels in the original text. These diagrams are set out in thetranslation section and not repeated in section (iv). Occasionally there is a need forextended explanatory diagrams in section (i), relating Gregory's diagrams to those wehave already discussed above in §0. There is no table of contents in the original, whichwe have added rather fully to enable you to navigate the text, and there is no index. Wecommence with the title page, followed by the Preface and Definitions. A dedication toCharles II is not included. The original work is set out rather in the form of Euclid'sGeometry. Gregory was a top-notch mathematician, and I have had to labour greatly toproduce a mathematical scheme that agrees with his geometrical approach in this firstsection. Very occasionally he has made arithmetical errors that I have corrected. Hisknowledge of the physiological optics of the eye was of course rudimentary and oftenwrong.


OPTICA PROMOTA,( The Advancement of Optics)

Seu(or)

Abdita radiorum reflexorum & refractorum( The mysteries of the reflection and refraction of rays are resolved,)

MYSTERIA,Geometrice Enucleata ;(elucidated with the aid of geometry;)

Cui subnectitur(to which is added an appendix )

APPENDIX,Subtilissimorum Astronomiae

Problematum resolutionemexhibens.

(demonstrating the resolution of the most subtle astronomical problems.)

__________________________________

Authore Jacobo Gregorio,Aberdonensi Scoto.

_________________________________________________________________________________

LONDINI.Excudebat F. Hayes, pro S. Thomas, ad Insigne

Episcopi, in Coemeterio Paulino, 1663.


§0.4. To Mathematical Readers.

It is a superfluous task in your work on optics - that outstanding light of the learned! -to extol the wonders of optics For optics offers a number of aids to vision - the keenest ofthe senses. The use of special lenses affords the clearest possible vision for those peoplewho are short or long-sighted, excepting eye defects caused by disease. The telescopeenables us to look skywards to view the starry heavens - for most of us an unknown andincredible place. Again, the microscope enlarges the appearance of the smallest objectfor our careful examination. Thus optics supplies a useful handle to all the naturalsciences, allowing things either to be made larger or to be brought nearer for distinctviewing.

Nevertheless when considered among all the mathematical sciences, nothing of anygreat consequence (as far as I know) has been handed down to us from antiquity 1 : forwith due regard for things from the olden times, and with the exclusion of the optics ofEuclid, Alhazen [or Ibn Al-Hasan, etc, (965 - 1038)] and Witelo [Vitellius (1220 -1280)]2, very few of the other obscure and long-winded authors appear to have offeredmuch by way of improvement. But in recent times following Galileo3- a man for all timemost worthy of our praise - many have striven at great cost to themselves to garner intheir works some of the bountiful harvest of ideas left by that Mercurial star. I includemyself among such people, urged on by a certain youthful ardour and boldness thatarose from my readings of the discovery of the elliptic inequality [i.e. Kepler's Laws] . Tothis end, I have equipped myself with the necessary mathematical means to facilitate theoptical speculations set out here, that terminate with the design of new telescopes ofparticular geometries4.

From the analogies indicated in the first proposition of this small tract, I haverevealed a hypothesis for the measurement of natural refractions. Clearly I was unaware- on account of the lack of new Mathematical books in the otherwise renowned AberdeenUniversity library - [Gregory's brother David was the librarian there] that the same hasalready been found by Descartes. Following this I made use of a few Lemmas selectedfrom the Commentary on Archimedes by Rivault5, to 26 of these I have added problems.Subsequently, I was in difficulties for a long space of time, deprived of all hope ofprogress [i.e. in understanding image formation by lenses and mirrors]; but with thecontinuing encouragement and help of my brother David Gregory, a man of no meanability in mathematics - to whom I am truly indebted, if in whatever of these sciences Ishould excel. Finally I have produced a series of diagrams for consideration, and I havetaken the image [formed by a lens] to be nothing more than the divergence of the raysfrom the individual visible points coming from a single surface [of a source or previousimage]. I have added Corollaries to the individual problems considered to aid theexamination of these images.

I have added Theorems 26 and 27 concerning pencils and the determination of thecones of rays. Theorems 28, 29, 30, 31, 32, 34, 35, 36 account for the position of theimage, and 33 for seeing a distinct image. Theorems 37, 38, 39, 40, 41, 42, 43 take careof the position of the apices of the pencils and cones of rays. Finally, from thispresentation the most general propositions are readily shown by geometricaldemonstration: all of reflection [by conoid mirrors], and the understanding of refraction[by conoid surfaces], as with a denser transparent medium so with the less dense case


considered together. From these theoretical considerations there finally emerge newkinds of optical instruments, some already known about, and some unknown - especiallythe telescope.

Finally I have added an Appendix on Astronomy; in which all that can be desired to beknown about parallax, ellipses, and apparent diameters can be found from observation,with the help of the optical devices I have designed. As for a disgrace to the art ofAstronomy, namely the parallax of the sun [ i.e. a reliable estimate of the earth/sundistance], I demonstrate different ways of finding this [by using the transits of Mercuryand Venus]. And finally, I show all the inequalities of the planets which I have examinedclosely many times. However, I have abandoned any attempt at explaining the moon'smotion, on account of so much irregularity.[This had originally been discovered byGalileo.] For from these irregular motions which we may cast up in the course of ourlate night studies, I can say that I have struggled in vain with the foremost mathematics togenerate an orbit. So one must relish even the attempt, but file it away in the drawer ofour slavish trials!

Yours most observantly, J. G.

Notes & References for §0.7.Gregory had apparently just finished reading Descartes' First discussion in the Dioptique when he

amended his manuscript, as there is a faint echo of that philosopher's line of thought in the openingparagraph of the O.P.1. Gregory fails to mention Ptolemy's Optics, which is a treasure house of classical optics. See Ptolémé.L'Optique de Claude Ptolémé.. Albert Lejeune. Louvin University.(1956). Also, A.M. Smith : Ptolemy'sTheory of visual perception. An English translation of the Optics... Transactions of the AmericanPhilosophical Society 86, Pt 2. (1996).

2. Opticae Thesaurus Alhazeni Arabis......item Vitellonis....Federico Risnero. Basile (1572). Available inlibraries, from the Johnson Reprint Corporation (1972).

3. Siderius Nuncius. Galileo. (1610).

4. Such as in Kepler's Optics, translated by W. Donahue, Green Lion Press, 2000.

5. Archimedes: Opera quae exstant graece et latine novis demon....; Rivault.(Paris, 1615).

§0.5. Definitions.

1. Rays are the lines along which the fiery corpuscles run that arise from lightproducing bodies.

2. A body shining [from diffuse reflected light] is opaque and unpolished, reflectingsome of the fiery corpuscles, and absorbing others.

3. The colour conveyed into the eye is the tincture of the fiery corpuscles emerging fromthe radiating material.

4. Vision shall be from the reception of these corpuscles reflected from a shining body,and conveying the colour into the eye.

5. The rays of a single point are those which have been reflected from one point of ashining body.


6. Parallel rays are those which are always equally distant each to the other amongstthemselves.

7. Diverging rays are those which concur in a point when produced in both directions:those rays produced in the opposite direction to the motion form the ray-bearing cone- the apex of the cone is the point of concurrence of the rays.

8. Converging rays are those which concur in a point in the direction of the motionwhen produced in both directions; these rays are called a pencil, and the point ofconcurrence the apex of the pencil.It is to be noted however, that the terms parallelism, divergence, and convergence areto be applied only to rays coming from a single point.

9. The image produced by a lens or mirror is a likeness of the radiating body, arisingeither from the divergence or convergence of rays from single points of the shiningbody, to or from the individual points of the one image surface.

10. An image before the eye [i.e. a real image], arises from the apices of the light bearingcones from single radiating points of matter, brought together in a single surface.

11. The image behind the eye [i.e. a virtual image] arises from the apices of the pencils ofsingle radiating points of matter, brought together in a single structure [behind thelens or mirror]; although it is not properly said to be an image, nevertheless in opticsit is scarcely less effective than the [real] image before the eye.

12. An object visible to the eye is produced either by a luminous body, or from a previousreal or virtual image [produced by a lens or mirror]; nevertheless more often the termis used for a radiating body, especially so when the talk is about the image.

13. The diameter of a visible object is the distance between any two extreme andopposite points.

14. The centre of visibility is the mid-point of the diameter.15. The angle of vision is the angle taken by the two rays from the extreme visible points,

intersecting in the centre of the surface of the eye.16. The sections of conics are the circle, ellipse, parabola, hyperbola, and straight lines.17. The line of the conic section, or the circumference, is the common section of the plane

cutting the surface of the cone.18. A right line is said to be perpendicular to the circumference of a conic section when it

cuts the tangent at the point of contact at right angles.19. Conical surfaces are the surfaces of figures generated by the revolution of sections of

the cone about individual axes.20. A mirror is an opaque polished body fashioned from one conical surface.21. A lens is a transparent polished body fashioned from two conical surfaces with a

common axis.22. The axis of a lens or mirror is the same as the axis of the conical surfaces from

which it is comprised.23. The vertices of mirrors or lenses are the same as the vertices of the conical surfaces

from which the mirror or lens is comprised.24. The incident vertex is that in which the rays are incident, the vertex of the emergent

rays is that from which they emerge; from which it appears that these verticescoincide for mirrors, while with lenses they are truly separated.

25. The thickness of the lens is the difference in length of the vertices.


26. The diameter of the lens or mirror is the distance between two of their extreme andopposite points; and hence the centre of the lens or mirror is the mid point of thediameter.

27. The angle containing the rays is the angle taken from the centre of the visible and thediameter of the lens or the mirror.

28. The axis of the pencil, or of the ray bearing cone, is that ray which is at right anglesto the surface of the mirror or lens.

29. The nearest image is that which is seen immediately by the eye; the second image thatby means of which the first image is seen by the eye; and thus in succession [for asequence of lenses or lenses with a mirror]: But the furthest image is the first imageprojected, and therefore brought to the eye by all the remaining intermediate images.

30. The rays of the first, second, third, and the final image, etc., are those which convergeto the first, second image, etc. ; or which diverge from the same.

31. The incident vertex of the first image, of the second, third, etc., is the vertex ofincidence of that lens or mirror in which the rays are incident, of the first image, ofthe second, etc.; and in the same way the vertex of these emergent is the vertex of theemergent lens, or mirror, from which these emerge.

32. The telescope is an optical device with lenses or mirrors placed together, having acommon axis, allowing the accurate vision of things far away.

33. The microscope is an optical device with lenses or mirrors placed together, having acommon axis, allowing the accurate vision of things nearby.

34. The icoscope is an optical device with lenses or mirrors, having a common axis,depicting visible images in a plane [i.e. a projecting telescope or microscope].

35. Relating to the first, second, third, etc., image, is the mirror or lens, projecting thefirst, second, third, etc., image.

36. Long - sighted [usually elderly] people are those who see far away objects distinctly,while nearby things are indistinct.

37. Myopic [i.e. short - sighted] are those who see far away things indistinctly, nearbythings distinctly.

§0.6. Postulates.

1. Light rays are weakened by their distance from the source;2. Rays cannot be weakened by reflection alone from mirrors ; otherwise corpuscles

would be destroyed, which cannot happen;3. Rays transmitted are weakened by refraction, because a great fraction of the

corpuscles are reflected backwards at the common interface of differenttransparencies.

4. The rays coming from remote visible objects are considered to be parallel.5. From a given axis and foci of a conic section, the section itself is given.

§0.7. Lectoribus Mathematicis.

Supervacaneum esset (praeclara Literarum lumina) de mirandis Optices Encomiis apud vospanegyricum instituere; visum enim sensuum pulcherrimum adjuvat; myopibus & presbytis (absque;Optices ope quodammodo caecis) clarissimum praebet visum; oculos humanos ab hisce terrestribus, adcaelestia majoribus nostris incognita, & incredibilia elevat; & ad perfectam minutissimorum corporum


visionem rursus detrudit: Unde ansam praebet, omnes scientiae naturales, & amplificandi, & promovendi.Nihilominus inter universas scientias Mathematicas, nulla ab antiquis adeo neglecta, & a paucioribusexculta: Omissis enim Euclide, Alhazeno, & Vitellione, authoribus etiam perobscuris & prolixis; eveteribus nullus aliquid momenti ( quantum scio) nobis reliquit: E neotericis autem, post Galilaeumsidereum illum Mercurium, & omni aevo laudibus celebrandum, multi, & scriptis, & sumptibus immensis,in hac messe laudabiliter sudarunt: inter quos ego, juvenili quodam ardore instigatus, & ex inventioneinaequalitatis ellipticae audaciam nactus, hisce speculationibus Opticis, & praecipuae geometricaeTelescopiorum demonstrationi, pro facultate ingenii me accinxi; & ex analogiis in prima huius tractatulipropositione declaratis, inveni primam hujus Opticae partem, de genuina refractionum hypothesi &mensura; nescius scilicet (propter inopiam novorum librorum Mathematicorum, in alias inclytaeBibliothecae Aberdoneinsi) haec eadem a Cartesio fuisse inventa: & ope aliquot Lemmatum, e RivaltiCommentariis in Archemedem desumptorum, Problemata ad 26 huius adjunxi: ubi diu haesi omni speprogrediendi orbatus; sed continuis hortatibus, & auxiliis fratris mei Davidis Gregorii in Mathematicis nonparum versati (cui, si quid in hisce scientiis praestitero, me illud debere non inficias ibo) animatus, tandemincidi in seriam imaginis considerationem, & deprehendi imaginem nihil aliud esse, quam radiorum,singulorum visibilis punctorum, a singulis aliis punctis in una superficie existentibus, divergentiam: quoperspecto, singulis problematibus addidi sua Corollaria; & adjunxi 26, 27, pencillorum, & conorumradiorum determinationi; 28, 29, 30, 31, 32, 34, 35, 36, imaginis loco, & visioni distinctae, 33 ; 37, 38, 39,40, 41, 42, 43, apicum, penicillorum & conorum radio[so]rum, loco inservientes. Quibus praemissis,usque; ad finem patent facillima & universalissimae propositiones geometricae demonstratae, totamCatoptricorum, & Dioptricorum doctrinam, tam mediante diaphano densiore, quam rariorecomprehendentes: e quibus emergunt infinita machinamenta Optica diversa, partim ante cognita, partimincognita. Deinde appendicem astronomicam addo; in qua, omnia desiderata de parallaxium, eclipsium, &diametrorum apparentiuim observationibus, ope machinarum opticarum perficio: & Astronomorumopprobrium, nempe solis parallaxim diversis modis invenire doceo: Et denique; omnes planetaruminaequalitates scrutari multipliciter demonstro; Lunam tantum propter maxime irregularem motum vixtactam relinquens. His inquam molitus sum praeclara Mathematica propagare: quod si in nostrislucubrationibus humanius quid acciderit, conatum saltem fovete; & servorum vestrorum catalogo inserite.

Vestri Observantissimum, J. G.

[1]Definitiones.

1. Radii sunt linea, in quibus discurrunt corpuscula ignea, è corporibus lucidis ortum habentia.2. Materia radians, est corpus opacum, & impolitum, corpuscula ignea reflectens, & aliquem illis

praebens ingressum.3. Color in oculum delatus, est tinctura corpusculorum igneorum, è materiâ radiante emergentium.4. Visio sit receptione horum corpusculorum a materia radiante reflexorum, & colorem in oculum

deferentium.5. Radii unius puncti, sunt qui ab uno materiae radiantis puncto reflectuntur.6. Radii paralleli, sunt qui aequaliter semper a se invicem distant.7. Radii divergentes, sunt qui utrinque producti concurrunt in punctum, ad partes motus contrarias; &

vocantur hi radii conus radiorum & punctum concursus apex coni.8. Radii convergentes, sunt qui utrinque producti concurrunt ad partes motus, & vocantur penicillum,

& punctum concursus apex penicilli.Notandum tamen, parallelismum, divergentiam, & convergentiam, applicari tantum unius puncti radiis.

9. Imago est similitudo materiae radiantis, orsa ex divergentiâ, vel convergentiâ radiorum, singulorummateriae radiantis punctorum, a punctis singulis, vel ad puncta singula unius superficiei.

10. Imago ante oculum, oritur ex apicibus conorum radiosorum, singulorum materiae radiantispunctorum, in una superficie congregatis.

11. Imago post oculum, oritur ex apicibus pencillorum, singulorum materiae radiantis punctorum,[2]

in una superficie congregatis; qua non propriè dicitur imago, sed tamen in Opticis, vix minorem habetefficaciam , quàm imago ante oculum.


12. Visibile est, sive materie radians, sive imago ante, sive post oculam; saepius tamen usurpatur, promateriâ radiante, praesertim ubi sermo etiam est de imagine.

13. Diameter visibilis, est distantia inter qualibet duo puncta extrema, & opposita.14. Centrum visibilis, est diametri punctum medium.15. Angulus visorius,, est anguli comprehensus a duobus radiis, extremorum visibilis punctorum, in

centro oculi, se invicem secantibus.16. Sectiones Conica sunt, circulus, ellipsis, parabola, hyperbola, linea recta.17. Sectiones conicae linea, seu circumferentia, est communis sectio plani secantis cum superficie coni.18. Linea recta dicitur esse circumferentiae sectionis conicae perpendicularis; cum secat contingentem

in puncto contactus ad angulos rectos.19. Superficies conicae, sunt superficies figurarum, ex revolutione sectorum conicarum circa proprios

axes, genitarum.20. Speculum est corpus opacum, & politum, ab una superficie conica à comprehensum.21. Lens est corpus diaphanum, & politum, a duobus superficiebus conicis, communem axem

habentibus, comprehensum.22. Axis lentis, vel speculum. est idem cum axe superficierum conicarum, a quibus comprehenditur.23. Vertices lentis, vel speculi, sunt eadem cum verticibus superficierum conicarum, a quibus

comprehenditur lens vel speculum.24. Vertex incidentiae, est illa, in quam incidunt radii: vertex emersionis, illa, ex qua emergunt: Unde

patet, in speculis hae vertices esse unam; in lentibus vero diversas.25. Crassities lentis, est verticum distantia.26. Diameter lentis, vel speculi, est distantia inter duo ipsius puncta extrema & opposita; & proinde

centrum lentis, vel speculi, est diametri punctum medium.27. Angulus radiosus, est angulus comprehensus a centro visibilis, & diametro lentis, vel speculi.

[3]28. Axis penicilli, vel coni radiosi, est ille radius, qui ad superficiem speculi, vel lentis, est rectus.29. Imago prima, est illa, quae immediatè ab oculo videtur; imago secundo, illa quae mediante imagino

prima ab oculo videtur; & sic deinceps: Imago autem ultima, est imago visibilis primò projecta, & ideomediantibus omnibus reliquis, in oculum allata.

30. Radii imaginis primis, secundae, tertae, ultimae, &c., sunt illi, qui ad imaginem primam, secundam,&c., convergunt, vel ab iisdem divergunt.

31. Vertex incidentiae, imaginis primae, secundae, tertiae, &c., est vertex incidentiae, illius lentis, velspeculi, in quam incidunt radii, imaginis primae, secundae, &c.; & eodem modo vertex earum emersionis,est vertex emersionis lentis, vel speculi, ex quà emergunt earum radii.

32. Telescopium, est machina optica, ex lentibus, vel speculi, communem axem habentibus, composita ;longinquorum efficiens accuratem visionem.

33. Microscopium, est machina optica, ex lentibus, vel speculis, communem axem habentibus,composita; propinquorum efficiens accuratam visionem.

34. Icoscopium, est machina optica, ex lentibus, vel speculis, composita; visibilium imagines in planodepingens.

35. Pertinens ad imaginem primam, secundam, tertiam, &c., est speculum, vel lens, projiciensimaginem primam, secundem, tertiam, &c.

36. Presbyti sunt, qui remota distinctè videns; propinqua confuse.37. Myopes sunt, qui remota confusè vident ; propinqua distinctè.

Postulata.

1. Radii non debilitantur, solâ distantià a radiante.2. Radii non debilitantur, solâ reflectione a speculis ; alioquin corpus annihilaretur,quod fieri non potest.3. Radii debilitantur, sola refractione, quoniam eorum plerique in superficie diversorumdiaphanorum communi, ad contrarias partes reflectuntur.4. Radii visibilium longinquorum, sunt quo ad sensum paralleli.5. E datis, axi, & focis sectionis conicae, datur & ipsa sectio.


§1.1. Synopsis Proposition 1.

In the first proposition the rationale behind the refraction method is explained. There isto be a correspondence set up between reflection and refraction by surfaces derived fromconic sections. The former is already well known geometrically at this time, and Gregoryintends showing that the latter can also be derives geometrically for surfaces of the sameform. The opening premise is the refraction of a light ray towards or away from thenormal in entering a more/less dense medium He borrows heavily from Kepler's Optics inconsidering limiting situations where the degree of refraction is extreme. Thus, atransparent sphere with an infinite density will refract a parallel beam in the air throughthe centre of the sphere: in a less extreme situation the parallel beam can be refracted topass through a focus of an ellipsoid, which has an ellipse as cross-section. The far focusmust be used in order that physically meaningful angles of refraction occur. In a similarmanner, a parallel beam in the infinitely dense transparent medium can come to a focus inthe air outside a plane bounding the dense medium: in a less extreme situation the parallelbeam in the dense medium can be refracted by the surface of a hyperboloid through thefocus of the other branch; in this case the cross-section is a hyperbola. The case of theparaboloid corresponds to no refraction. Hence, lenses with parabolic cross-sections willnot occur in this work.

[5]

Optica Promota._______________________________________________________________________

§1.2. Proposition 1.

The Proposition shall be to examine: which pray shall be the surface whichmeasures refractions?1

It has been commonly observed by those involved in Mathematics, whatever the truthof what else they may say, that light rays passing through a less dense transparentmedium and incident obliquely on another denser medium, are refracted at the surface ofthe second medium, and bend towards the perpendicular, excited by the points ofincidence. Conversely, rays crossing the denser medium and incident obliquely on therarer medium, are refracted by the surface of the second medium, and bend away fromthe before-mentioned perpendicular. It is not for us to set forth here the origins of thisrefraction : indeed Alhazen, Vittellio, Kepler, and many others have discussed thesecauses at length; moreover, since the measurements of these effects as set forth by theseauthors are revealed to be less than reliable, we shall attempt on that account to exert alittle influence in the midst of all this confusion - which perhaps will be of some use tomathematicians. But concerning these things which are to be the subject of deeperconsiderations, it may be permitted to argue a little by analogy, before we approach thesubject with geometrical rigor.

[6]


AAAAAAA

B

A

A

A

A

A

B

D

E

[Figures 1and 2.]

It is clear enough from the elements of Optics that much of Reflection [Catoptrics] andRefraction [Dioptrics] have properties in common; therefore perhaps some commonproperty will remain in the measurement of both reflection and refraction. But all themystery of reflection that lies hidden in conic sections has been demonstrated - as will beshown in turn - and hence perhaps also a measure of refraction will be concealed therein.For the following cases considered, regular reflection cannot occur unless the reflectingsurface is a conic section, and perhaps there may not be a rule for refraction either unlessthe refracting surface is a conic section also.

If in truth the reflecting surface isthe concavity of a parabola, and theincident rays are parallel to the axis,then they are reflected into the focus. Itcan be asked therefore, for a givensurface of refraction: is it the case thatall the rays incident on this surfaceparallel to a certain specific line can berefracted into any one assigned point ?From the preceding it is probable - ifsuch a surface can be made - that itshall be a conic section. But our searchfor such a surface making use ofanalogies may be undertaken byexamining extreme situations. Thus, wemay consider the medium in which therays are incident to be the densestpossible2: in which case the refracted

rays will be perpendicular to the surface of the medium (see Kepler. Ast. Opt. fo. 113).The conic section is sought therefore for which all the perpendiculars are themselvesconcurrent in one point: the circle is such a section [Figure 1]. For the second case, theparallel rays are considered to be passing through the densest medium

[7][i.e. to the left of DE]. The bundle of rays [Gregory calls this the form or shape] leavesalong the same lines by which it enters; but the bundle enters with the lines perpendicularto the surface, as hitherto said, and so [tracing the rays backwards] the bundle of raysemerges from the densest medium turned away from the perpendicular lines3. Thereforethe conic section is sought to which all the perpendiculars are parallel: but the straightline is such a section [Figure 2].

Some parallel rays A, A, A, etc. lying in one plane therefore are supposed to berefracted by the surface of the densest medium, where the refracting surface is a circle; inwhich case the refracted rays concur to the centre of the circle B. The single point [i.e. thefocus] from an extreme point of view therefore is satisfactory. Also to be supposed, anumber of rays arising from a single point B advancing in the rare medium, are to berefracted by the densest medium - the refracting surface of which is taken as the straightline DE - in which case all the refracted lines emerge parallel in the densest medium.Conversely, if these parallel rays lying in a single plane of the densest medium areconsidered to be refracted into the more rare medium by the surface DE, (because the


same bundle of rays leaves which enter) all the rays A, A, A, etc., are concurrent in thepoint B; thus the converse situation is satisfactory from this extreme point of view. Iftruly, everything is examined carefully, then it will be seem - on account of theaforementioned reasons - that all the rays, either parallel or non-parallel, which areincident on the circular surface of the densest medium for refraction, are concurrent in thecentre of the circle. Now we ask: how does this come about? The answer is :- Well,however the line is drawn incident on the circle, (provided they are co-planar) an axiscan be drawn parallel to it, and without doubt the circle can be considered as a kind ofellipse, so that any diameter can be called the axis, from which it appears that the specialline sought is the axis of a conic section.

For the other case, from observation it will also be apparent that all the parallel rays ina single plane of the densest medium are not so much assigned to a single fixed point, butto any point you wish beyond the line DE, and the component parts concur in B. Also weask : how does this come about? The reply is :- Well, (supposing the straight line to bethe branch of a hyperbola) any point outside the densest medium can be accepted as thelocation of the focus, from which it can be seen that the focus is the required point ofconcurrence. But of the two foci of the hyperbola, either real or imaginary [depending onwhether we have a real hyperbola or this degenerate straight line case], it will be the pointof concurrence which stands furthest from the point of incidence of the rays, otherwisethe angle of refraction would be greater than a right angle, which cannot happen [i.e. thefocus of the far branch of the hyperbola is used].

From these pre-tests of the medium using extreme values, we may attempt to answerthe following questions :- By considering rays passing either from the rare medium intothe densest, or from the densest into the outermost rare medium,

[8]by necessity it follows that the rays from one medium incident on another of the samedensity, to be the mean between the two aforementioned extremes ; but in this case thereshall be no refraction. For the parabola, therefore, (which is the mean between the circleand the straight line) all the lines parallel to the axis and co-planer with it coincident on it,ought to be concurrent in the focus by refraction. These are incident from points at thegreatest distance, and so the focus shall be at an infinite distance from the vertex of theparabola. Therefore all the rays incident on the parabola, and drawn from theaforementioned imaginary focus, are parallel to the axis. If truly they are parallel to theaxis both before and after incidence, then in general they are free from refraction, as isthe proposition.

We may therefore conclude from the analogy that one is able to find a surface ofrefraction for all different kinds of transparent media, which shall be a conic section, inwhich coplanar parallel rays in one medium are refracted by another medium to concur ata point. Now when the rays are parallel in the denser medium and they concur in the rarermedium, then the surface of refraction approaches almost to the most obtuse ofhyperbolas, i.e. a straight line. On the contrary, when the rays are parallel in the lessdense medium and they concur in the denser medium then the surface of refractionapproaches almost to the most obtuse of ellipses, i.e. a circle. Truly from these discardedanalogous trifles we may come close to more reliable evidence for establishing thescientific origin of refraction.


§1.3. Notes on Proposition 1:1 The author is unaware of the now customary form of Snell's Law of refraction : heintends to relate the 'optical density' to the focal property of a conic section, which weshow below in modern terms for a medium of refractive index n. Initially he argues byanalogy, assuming that an unknown law of refraction can be applied to conic sections justas the law of reflection can be applied, which certainly is the case as reflecting surfacescan bring parallel rays to a focus. In this matter, he follows the lead of Kepler inconsidering extreme cases, a ploy still used in understanding new physical phenomena.

2 That is, consider an infinite refractive index. This at least has the effect of changing aparallel beam into a focused beam; and conversely, by making use of a circular cross-section and a plane surface respectively. See the James Gregory Tercentenary Volume.Page 454 onwards. See also, p. 127 of Kepler's Optics, translated by W. Donahue, GreenLion Press, 2000.

3 Thus showing the principle of reversibility of a light ray.

[5]§1.4. Propositio 1.

Propositum sit inquirere, quaenam sit superficies quae metitur refractiones.

Omnibus in Mathesi vel leviter veratis, vulgo notum est, radios luminosos per diaphanum rariustranseuntes, & in aliud diaphamum densius obliquè incidentes, refringi in superficie secundi diaphani, & adperpendiculares vergere ab incidentiae punctis excitatas ; & è contrario radios per diaphanam densiustranseuntes, & in aliud diaphanum tenuius obliq ; incidentes, refringi in superficie secundi diaphani, & apraedictis perpendicularibus divergere. Cujus refractionis causus & elementa non nostrum est hic explicare,abunde enim de his disputarunt Alkazanus, Vitellio, Keplerus, & mulit alii: Quoniamve: ob quae de earummensurâ ab authoribus profaeruntur minus solida videntur, paucula quaedam de hac re (Mathematicisforsan non inutilia) in medium adducere conabimur. De his autem quae altioris sunt considerationis, liceatpaululum analogicè disputare, priusquam ad αχριβειαν geometricam accedamus.

[6]Satis patet ex Opticis elementis, multa Catoptricae, & Dioptricae esse communia ; forsan igitur; & in

reflectionum, & in refractionum mensuris, aliquid commune haerebit : Totum autem reflectionemmysterium, in sectionibus conicus latere compertum est; (ut deinceps patebit) forte igitur & refractionummensura illic latebit. Secundo non fit regularis reflectio, nisi superficies reflectionis sit sectio conica ;fortassis ergo nec regularis refractio, nisi refractionis superficies sit sectio etiam conica.

Si vero superficies reflectionis sit concavitas parabolae, & radii incidentes axi paralleli, omnesreflectuntur in focum : Quaeritur ergo num possit dati superficies refractionis, ita ut omni radii in eamincidentes, speciali cuidam lineae paralleli, refringantur in unum aliquod punctum determinatum? expraedictis probabile est ( si talis detur) hanc superficiem esse conicam sectionem : ut autem talemsuperficiem analogicè inquiramur, ab extremis incipiatur ; & concipiamus medium in quod incidunt radiiesse densissimum ; radii refracti, ad superficiem medii perpendiculares erunt. ( Keplerus Ast. Opt. fo. 113)Quaeritur igitur sectio Conica, cuius omnes perpendiculares ad sui lineam, in unum punctum concurrant ?talis autem est circulus. Secundo concipiatur illud medium densissimum, per quod transeunt

[7] radii paralleli ; & quoniam eisdem lineis egreditur forma quibus ingreditur ; ingreditur autem forma, lineissuperficiei perpendicularibus, ut hactenus dictum ; ergo egrediuntur radii, sive forma, e medio densissimolineis perpendicularibus : Quaeritur igitur sectio conica cujus omnes perpendiculares ad sui lineam sint


paralleli ? talis autem est linea recta. Supponendo igitur, radios parallelos quotcunq; A, A, A, &c. in unoplano, refringi in superficie medii densissimi, & superficiem refractionis esse circulum ; radii refracticoncurrent in centrum circuli B: uni igitur ex extremis est satisfactum. Supponendo etiam radios quotcunq,;ex puncto B existente in medio raro prodeuntes, refringi in medio densissimo cujus superficies refractioniscomprehendetur linea recta DE, omnes lineae refractae evadent parallelae in medio densissimo. Si vero hiradii paralleli in uno plano medii densissimi, concipiantur in medii rarioris superficie DE refringi, (quoniam iisdem lineis egreditur forma quibus ingreditur) concurrent omnes radii A, A, A, &c. in punctumB; atq; ita altera ex extremis est satisfactum. Si vero, rem diligenter quis intueatur, videbit ( propterpraedictas rationes) omnes radios, sive parallelos, sive non parallelos, in superficiem refractionis circularemmedii densissimi incidentes, in circuli centrum concurrere. Quaeritur autem unde hoc proveniat ?Respondetur ; videtur hoc provenire ex eo, quod quomodocunq ducatur linea in circulum incidens,(dummodo cum circulo in eodem sit plano) axis ipsi parallelus duci possit ; concipiendo nimirum, circulumesse ellipseps speciem, quaevis illius diameter potest dici axis ; unde videtur ; axem sectionum conicarum,esse lineam specialem quaesitam. Animadverdenti quoque patebit, omnes radios, in uno plano mediidensissimi parallelos, non in unum tantum, sed in quodlibet punctum assignatum, extra lineam DE, adpartes B concurrere : Quaeritur etiam unde hoc proveniat ? Respondetur; hoc vedetur provenire ex eo quod,(supponens lineam rectam esse hyperbolam) quodlibet punctum extra ipsam possit sumi loco foci ; undevidetur focum esse punctum concursus quaesitum. E duobus autem focis, vel realibus, vel imaginariis, iserit punctum concursus, qui longissime a radiorum incidentia distat, alioquin angulus refractionis essetmajor recto, quod fieri non potest. Hisce de extremis praelibatis medium tentemus : Si autem radiosprovenientes e diaphano raro in densissimum, & e diaphano

[8]densissimo in rarum extrema concipiamus ; necessario sequitur radios ex uno diaphano, in aliud ejusdemdensitatis incidentes, esse medium inter praedicta duo extrema ; in hoc autem casu nulla sit refractio ; Inparabola ergo (quae media est inter circulum & lineam rectam) omnes lineae axi parallelae, & in eodemcum parabolâ plano, in ipsam incidentes, debent per refractionem concurrere in focum, a punctisincidentiae maximè remotum : at focus iste a vertice parabolae infinitè distat ; omnes igitur radii inparabolam incidentes, & ad praedictum focum imaginatium ducti, sunt axi paralleli : si vero & ante, & postincidentiam, sint axi paralleli, a refractione omnino sunt liberi, quod est propositum. Concludimus igituranalogicè pro omni diaphanorum diversitate, inverniri posse superficiem refractionis (quae sit sectioconica) in quâ lineae parallelae in plano unius diaphani, in altero refractae concurrant in punctum : quoautem densius fuerit diaphanum in quo radii sunt paralleli, & quo rarius diaphanum in quo concurrunt ; eopropius accedit superficies refractionis ad hyperbolarum obtusissimum, id est lineam rectam : & e contra;quo rarius fuerit diaphanum, in quo radii sunt paralleli, & quo densius fuerit diaphanum, in quo concurrunt,eo propius accedit superficies refractionis ad ellipsium obtussimam, id est circulum. Verùm relictis hisceanalogiae nugis, ad experientiae scientiarum originis certiora testimonia accedamus.

17JAMES GREGORY'S OPTICA PROMOTA

A B

C D

V

E

ON

R T

L

M

G

Prop.2-Figure 1.

§2. Synopsis Propositions 2 - 6: The Law of Refraction for DenseEllipsoids and Hyperboloids.

According to Gregory, for a ray with a given angle of incidence i at a plane boundaryseparating two media, the angle of deviation d of the ray as it passes from one medium tothe other is a measure of the refraction between the surfaces.

In Prop. 2, an experiment is described in which the angles labeled i and d aremeasured.

Prop. 3 shows how to construct an ellipse in which the incident ray is parallel to theprinciple axis at some point on the ellipse, with an angle i to the normal of the tangent atthat point, while the refracted ray at an angle r = i - d passes through the far focus. Notethat new ideas introduced are called Theorems, while applications of given material arecalled Problems.

Prop. 4 demonstrates that the ratio sin i : sin r reduces to the ratio axis length : inter-focal distance.

Prop. 5 extends the result of Prop. 4 to the case of many rays parallel to the axis,which all pass through the far focus of the dense ellipsoid. There are subsequently sometables for refraction through water, glass, etc, taken from the works of Witelo, Kercher,which he compares with his own measurements. Gregory appears to have conducted veryprecise experiments of his own. Of interest to the modern reader is the practice ofcomparing all the measurements to those of a particular angle, rather than taking anaverage, for the actual idea of a refractive index was not yet in use.

Prop. 6 introduces the other kind of refracting surface - related to the hyperbola. In theellipsoidal case, rays parallel to the axis are sent through the far focus by refraction at theelliptic interface. In this case, rays diverging in air from the far focus are rendered parallelon refraction at the hyperbolic interface of the dense medium.

[8, cont'd]§2. Prop.2.1 Prop. 2. Problem.

To find the refraction of any medium with air.

Let some plane ABCD be set up parallel tothe horizontal or close to it ; and let some otherpoint be fixed at a higher position E, andthrough E a perpendicular VEG may beconsidered to be drawn to the horizontal, andthe angle of incidence in the medium (of whichthe refraction of the angle in the medium isrequired) is GEL, which is measured with anastrolabe or quadrant. Finally the whole spacebetween the plane ABCD and the point E isfilled up a medium; the smooth surface of whichshall be accurately parallel to the horizontal atthe point E.

[9]


By viewing with the eye placed at E, a small body placed at L shining brightly willappear to be shining brightly at M. Therefore, by measuring the angle NEV, [or GEM],the difference between that angle and the first angle OEV, [or GEL] will give the angle ofrefraction NEO sought, coming together with the angle of incidence NEV in air at E : [thetask] which had to be accomplished. Anyone who wishes to find the angles of refractionby other means, may consult Witelo, Kepler, and the other dioptrics authorities.

§2. Prop.2.2. Note on Prop. 2.

The Law of Refraction as we know it, had first been established experimentally byThomas Harriot in the summer of 1601, but he had not communicated his discoverybeyond a close circle of friends that included Aylesbury and Warner. [vide J. Lohne,Essays on Thomas Harriot, J. Arch. Exact Sciences, p.275,(1979)]: the law was to berediscovered by Snell in 1624, but was not published by him. Descartes (1637) hadindependently discovered the law experimentally, while Fermat had applied his principleof least time to give the first theoretical explanation of the phenomena of refraction andreflection. Thus the scientific community, such as it was at the time, was familiar withSnell's Law when Gregory produced his book. Gregory however did not have the law ofrefraction as a ratio of sines, though he measured refraction with a ratio that can bereduced to this form for the case of conoidal surfaces. The method adopted by Gregory tomeasure the angle of refraction of a ray, say through a flat glass slab, appears to be asfollows:

1. A small light source at L is observed in air initially through a small opening at Ewith the eye placed at O.

2. The medium is placed in position with E on or very close to the smooth horizontalsurface, and again the image of the light from L is observed - now refracted at thesurface, and passing through E along EN. The observer considers the image to lie at M,which can be found using the parallax method, [by simultaneously viewing a small objectplaced outside the medium, and adjusting to give the same height ML, when there is norelative motion on moving the eye slightly]. Gregory regards the angle of deviation NEOas a measure of the refraction: the same experiment survives to this day, where onemeasures the true depth and the apparent depth of an object, from which the refractiveindex of the medium can be extracted.

§2. Prop.2.3. Prop. 2. Problema.

Refractiones cujuscunq; diaphani aere invenire.

Sit planum aliquod ABCD stabilitum, & horizonti parallelum, vel eo circiter; sitq; punctum aliquodfirmissimè stabilitum in sublimi positum E, & per punctum E concipiatur duci perpendicularis adhorizontem VEG, sitque angulus incidentiae in diaphano (cujus anguli refractio requiritur) GEL, qui faci èmensuratur astrolabio, vel quadrante: & in puncto L, figatur corpusculum resplendens, & tandem impleaturtotum illud spatium inter planum ABCD, & punctum E, diaphano optimè polito, cujus superficies ad Epunctum, horizonti sit exquisitè parallela, &

[9] aspicienti per punctum E, apparebit resplendens in M; explorato igitur angulo NEV, seu GEM; differentiainter illum, & priorem angulum OEV, seu GEL dabit angulum refractionis quaesitum NEO, competentem


EN

AB

L

D

C

P

R

MO

T

Prop.3-Figure 1.

angulo incidentiae NEV in aere, quod faciendum erat. Qui alios, refractionum angulos inveniendi, modosdesiderat Vitellionem, Keplerum, alioque dioptrices auctores consulat.

§2. Prop.3.1. Prop. 3. Problem.

With two acute angles given, [i.e. i and d with i > d ] to find an ellipse such that theline parallel to the axis, incident on this ellipse, shall make an angle with the tangent,equal to the complement of the larger angle, and the line from the point of incidence tothe focus at the greater distance shall make an equal angle with the axis to the smallerangle.

Let the two angles be given, ABC the larger and DBE the smaller, and the tangent lineRBP of the ellipse shall be foundperpendicular to CB at the point B, the lineAB shall be parallel to the axis of thisellipse, and the line BE shall cross throughthe further focus. Through any point of theline EB [i.e. the actual size of the ellipse isnot important], without doubt E, EO isdrawn parallel to the line AB, to which theother line CB is produced in O, and theangle OBL is made equal to the angle OBE,and MN shall be equal to the sum of EB and

BL. This shall be equal to the axis of the ellipse sought, with the positions of the foci at Land E and the axes MN. The ellipse MBN can be described which necessarily will crossthrough the point B.

Conversely, since LB and BE together are equal to the axis MN (by the converse of[Prop.] 48, Book 3, Apollonius), and since the angles OBE and OBL are equal, if they aretaken from the right angles RBO and OBP,

[10] then the equal angles EBP and RBL are left; and therefore the line RBP is made to touchthe ellipse in the point B ( by the converse of [Prop.] 52, Book 3, Apollonius). With ABparallel to the axis MN, the angle RBA is the complement of the given larger angle ABC;and because the lines AD and MN are parallel, the angle to the further focus BEO isequal to the given smaller angle DBE, as required.

§2. Prop.3.2. Note on Prop. 3.Angle ABC is the angle of incidence i, while Gregory has taken the angle of deviation

DBE or d, as a measure of the refraction by the medium. Thus, the experimentalprocedure of Prop. 2 for measuring refraction is adopted for the curved surfaces of thelenses to be subsequently discussed.


EN

AB

L

D

C

P

R

O

T

i

rd

dii

r

r

Prop.4 -Fig.1.

r

§2. Prop.3.3. Prop.3. Problema.

Datis duobus angulis, non obtusis, invenire ellipsin, ut linea axi parallela, in eam incidens, efficiat cumtangente angulum, equalem complemento majoris, & recta a puncta incidentiae ad focum maximedistantem, efficiat cum axe angulum aequalem minori.

Sint dati duo anguli, ABC major, DBE minor, sitque invenienda ellipsis tangens lineam RBP ad CBperpendicularem in puncto B, cujus axis rectae AB sit parallelus, & linea BE per focum maxime distantemtranseat. Per punctum quodlibet lineae EB, nimirum E, ducatur lineae AB parallela, EO, quae utrinqueproducatur : producatur CB in O, & fit angulus OBL aequalis angulo OBE, fitque MN aequalis EB & BLsimul; quam dico esse axem ellipseos quaesitae, positis focis L & E: axe MN, & focis L, E, describaturellipsis MBN quae necessario transibit per punctum B; quoniam LB, BE simul sint aequales axi MN (perconversum [Prop.] 48, lib 3, Apoll.), & quoniam anguli OBE, OBL sunt aequales, si a rectis RBO, OBPauferantur,

[10]relinquuntur anguli EBP, RBL aequales; tangit igitur linea RBP ellipsin in puncto B ( per conversum[Prop.] 52, lib 3, Apoll.); facitq; cum AB, axi MN parallela, angula RBA aequalem complemento angulidati majoris ABC; & ob parallelisimum linearum AD, MN, angulus ad focum remotiorem BEO, aequalisest angulo dato minori DBE, quod erat faciendum.

§2. Prop.4.1. Prop. 4. Theorem.

With the same situation, I say that the sine of the difference of the given angles shall be tothe sine of the larger angle, as the separation the foci to the ellipse axis.

For the line EB [see Prop.3 -Fig.1] may be produced to T, and BT made equal to BL,and TL drawn. Therefore the angles BTL and BLT are equal, and also the angle LBE isequal to the sum of both, and therefore EBO, or half the angle EBL, is equal to the angleBTL, therefore the triangles EBO are ETL are similar. But the angle BOL is equal to thelarger given [i] ABC, on account of the parallel lines AB and MO, and the angle BEO isequal to the smaller given angle DBE [d]. But BEO and OBE added together are equal tothe angle BOL, and therefore the angle OBE, or LTE is equal to that angle, and this is thedifference of the given angles ABC and DBE; and also the angle TLM is equal to theangle BOL, or to the given larger angle ABC.

[11]As a consequence we conclude the sine of the difference of the angles given, that is thesine of the angle LTE is to the sine of the larger angle given TLM, as the separation ofthe foci LE is to the length of the axis of the ellipse TE. Q.E.D.

§2. Prop.4.2. Note on Prop. 4.

Gregory has independently discovered aform of the familiar law of refractionsini/sinr = n, where i is the angle of incidenceABC, r the angle of refraction OBE, and n theindex of refraction of the medium relative toair. For the angle of deviation d used byGregory is given by d = i - r, while n is related


to the eccentricity e of the ellipse of major diameter 2a by n = 1/e. Thus, sin i/sin(i - d) =2a/2ae, or sin i/sin r = 1/e, where e is the eccentricity of the ellipse in modern terms,though this particular terminology was not in use at the time. Indeed, the focus/directrixproperty of conic sections was not discovered until the beginning of the 19th century bythe two Belgian mathematicians Quatelet and Dandelin - see e.g. Eves: An Introduction tothe History of Mathematics, p. 169.

§2. Prop.4.3. Prop. 4. Theorema.

Iisdem positis, dico sinum differentiae angulorum datorum, esse ad sinum anguli majoris, ut focorumdistantia, ad axem ellipseos.

Producator enim linea EB in T; fitq BT aequalis BL, & ducatur TL ; erunt igitur anguli BTL, BLT aequales,& LBE aequales ambobus simul, ergo & EBO semissis anguli EBL, aequalis erit anglulo BTL, triangulaigitur EBO, ETL sunt equiangula ; est autem angulus BOL aequalis majori dato ABC, ob parallelismumlinearum AB, MO, estque angulus BEO aequalis minori angulo dato DBE: BEO autem & OBE suntaequales angulo BOL; igitur angulus OBE, vel illi aequalis LTE, est differentia angulorum datorum ABC,DBE; est quoq; angulus TLM aequalis angulo BOL, vel majori dato ABC. Concludimus

[11]ergo, sinum differentae angulorum datorum, hoc est angluli LTE, esse ad sinum anguli majoris datinimirum TLM, ut distantia focorum LE, ad axem ellipseos TE, quod erat demonstrandum.


For the same situation, if the larger angle were the angle of incidence of some ray fromthe rare to the dense transparent medium, and the smaller angle agreeing by refractionwith the said angle of incidence, and the ellipse found that forms the surface of refractionfrom the same rare medium into the dense. I say that all the rays parallel to the axis ofthe ellipse, and incident on the ellipse, are refracted at the points of incidence, and areconcurrent at the focus: moreover this ellipse may be called the ellipse of theaforementioned dense medium.

From the above analogous discussion, it is seen the parallel rays in one plane of therare medium meet in one point of the dense medium, only if the surface of refraction shallbe a certain ellipse with fixed measurements, which is appropriate for the density of themedium. Since indeed the circle gathers together the parallel rays from the rare mediuminto a single point of the densest medium; the parabola certainly gathers together theparallel rays from one medium into an imaginary point of another medium of the samedensity, standing apart at an infinite separation; therefore it follows for the ellipse, whichis intermediate between these figures, that the parallel rays in the plane of the raremedium are gathered together in a single point of the medium of intermediate density.

Therefore with these things touched on, we may undertake the experimentaldemonstration of this Theorem; and we may suppose that it is truth, in order that it maybe revealed to be absurd (if an absurdity should lie hidden within it). Witelo observed therefraction of water [i.e. the angle of deviation], agreeing with an angle of incidence in air300: to be 7030'. From this observation by the proceeding theory, we find the dimensionsof the ellipse thus shall be as 50000, the sine of the angle of incidence 300, to 38268, thesine of the angle of refraction 220:30': [or more conveniently] 10000 the axis of theellipse, to 7654 the separation of the foci. Hence by the same Theorem, we can compute


the rest of the angles of refraction: as 10000 - the axis of the ellipse is to 7654 - theseparation of the foci, as 17365 the sine of the angle of incidence 100, is to 13291 the sineof the angle of refraction: 70:38', which taken from the angle of incidence 100 leaves theangle of refraction

[12] 20:22' equal to the difference from the observed angle, 20:15', except for 7'. In this waywe have computed the rest that follow. But these disagreements [in values] need notdisturb the reader, for these reasons. Indeed from the first, from the observations ofWitelo, it appears that it was enough to have observed only to [the nearest] half degree.And (as Kepler observes in Astron. Opt. fol. p.116.), he is sure from his own trials onmastering refraction, by putting his hand to it, he might reduce these to order through theequality of the second increments [Gregory, however, does not involve himself inKepler's fitting scheme]: All of these observations indeed increase with increments of 30'[by this statement, Gregory appears to mean Witelo's angles are measured to within a halfor even a quarter of a degree: Gregory's own data in Table 5.3, shows accuracy to thenearest minute]._____________________________________________________________________The refraction observations from Witelo; and our calculated discrepancies of these.

The observed refractions from Arthansius Kercher, & our method following calculated with differences.

[13]But in the work of Kercher from the differences of the observations [Prop.5 -Table 2], thereis not even the shade of order: thus it shall be beyond all doubt, that these observationsare in error. And not without wonder, it will seem (if anyone should consider the thingmore accurately) that certain differences had crept in, with so subtle an enquiry; where atrivial error in the base of the calculations is multiplied in those

Refraction bywater from air.

The refraction of glassfrom air.

The refraction of glassfrom water.

Angles. ofincidencein air. Obs. Calcul. Diff. Obs. Calcul. Diff. Obs. Calcul. Diff.

0 0 0 ' 0 ' 0 '10 2 15 2 22 + 7 03 00 03 20 + 20 00 30 01 12 + 42 1020 4 30 4 50 + 20 06 30 06 48 + 18 01 30 02 27 + 57 2030 7 30 7 30 0 0 10 30 10 30 00 00 03 00 03 50 + 50 3040 11 0 10 32 - 28 15 00 14 35 - 25 05 00 05 28 + 28 4050 15 0 14 6 - 54 20 00 19 14 - 46 07 30 07 30 00 00 5060 19 30 18 29 - 61 25 30 24 40 - 50 10 30 10 12 - 18 6070 24 0 24 1 - 29 31 30 31 08 - 22 14 00 14 02 + 02 7080 30 0 31 5 + 65 38 00 38 58 + 58 18 00 19 34 94 80 An

gles

of i

ncid

ence

in w

ate r

[Prop.5 -Table 1]

Refraction of water Refraction of wine Refraction of oil Refraction of glassObs. Calc. Diff. Obs. Calc. Diff

.Obs. Calc. Diff

.Obs. Calc. Diff

.0 ' 0 ' ' 0 ' 0 ' ' 0 ' 0 ' ' 0 ' 0 ' '

2 20 2 12 + 8 2 30 2 28 - 2 2 50 2 33 - 17 3 10 3 27 + 174 38 4 56 + 18 4 45 5 3 + 10 5 10 5 12 + 2 6 40 7 1 + 217 40 7 40 0 0 7 50 7 50 0 0 8 4 8 4 0 0 10 50 10 50 0 011 9 10 45 - 24 11 4 10 59 - 5 11 50 11 18 - 32 15 8 15 2 - 615 6 14 24 - 42 15 10 14 41 - 29 16 10 15 5 - 65 20 12 19 48 - 2419 40 18 50 - 50 19 50 19 12 -38 20 20 19 42 - 38 25 50 25 20 - 3024 49 24 26 - 23 24 50 24 50 0 0 25 12 25 25 + 13 31 10 31 54 + 44A

ngle

s of i

ncid

ence

in a

ir

30 4 31 33 + 89 30 10 32 0 +110 30 54 32 38 + 104 38 10 39 43 + 93


Refraction of spring water from our own Observationswith the Calculation of their Differences.

Ang. ofincidence

in air.

Refract.observed.

Refractioncalculated.

Differences

0 ' 0 ' 0 ' '

13 28 03 28 03 25 - 0326 48 05 48 07 03 + 1541 50 11 50 11 50 00 0059 12 19 12 19 07 - 0571 20 26 20 26 05 - 15

[Prop.5 -Table 3]

proceeding. But the truth of this Theorem hasbeen apparent many times from our differenttrials: just as it will be evident from this singleexample, which from our angles of refractionwe have measured for spring water in ouryouth; where reliably by our first problem, andfinely enough (from the size of theinstrument), for the angles of incidence inwater 100, 200, 300, 400, 450. [These angles donot correspond with those in Prop.5 -Table 3.]If truly these observations made with great

care do not give satisfaction: Come mathematicians! And from more subtle observationsconfirm this most beautiful speculation of refraction.

CorollaryIt follows from this theorem, the angle between the line of refraction with the axis of

the dense ellipse, (see the angle BEO in the figure above), shall always be the angle ofrefraction, meeting the angle of incidence ABC. But now we can explain the restgeometrically from the aforementioned analogy with the help of theorems.


Iisdem positis; si angulus major fuerit angulus incidentiae alicujus Radii diaphano raro in densum, &angulus minor refractio dicto angulo incidentiae competens, fueritq, ellipsis inventa, superficiesrefractionis ex eodem diaphano raro in densum ; dico omnes radios axi ellipseos parallelos,& in ellipsimincidentes, in punctis incidentiae Refringi, & in focum concurrere :Vocetur autem haec ellipsis, ellipsisdensitatis praedictorum diaphanorum.

Ex superiore discursu Analogico, videtur radios parallelos in uno plano diaphani rari, congregari inunum punctum diaphani densioris, si modo, superficies refractionis fit ellipsis certa cujusdam dimensionis,quae diaphanorum densitati conveniat. Quoniam enim circulus congregat Radios Parallelos e medio raro inunum punctum medii densissimi ; Parabola vero congregat radios parallelos, ex uno diaphano in punctumimaginarium, alterius diaphani ejusdem densitatis, infinite distans ; Sequitur igitur ellipsim, quae media estinter hasce figuras, radios parallelos in plano diaphani rari congregare in unum punctum diaphanimediocriter densiosis. His igitur praelibatis, ad demonstrationem hujus Theorematis experimentalemaccedamus; & supponamus verum esse, ut absurdum (si quod lateat) patefiat. Observavit Vitelliorefractionem aquae, competentem angulo incidentiae in aere 300: esse 7030', & hac observatione perpraecedens Theorem, ita inveniemus ellipseos dimensiones, fit ut 50000: sinus anguli incidentiae 300: ad38268 sinum anguli refracti 220:30': 10000 axis ellipseos, ad 7654 distantia focorum. Et eodemTheoremate reliquas refractiones ita Computemus; ut 10000 axis ellipseos ad 7654 the distantiam focorum,ita 17365 sinum anguli incidentiae 100, ad 13291 sinum anguli refracti: 70:38': qui ablatus ab anguloincidentiae 100: relinquet angulum refractionis [12] 20:22': differentem ab observatione, 20:15', nisi 7':atque ita sequentia Computavimus. Sed lectorem ne moveat haec discrepantia, ob has rationes: Primo enim,satis apparet observationibus Vitellionis eum observasse tantummodo ad graduum semisses; Et (ut notatKeplerus in Astron. Opt. fol. 116.), certum est suis ab experientia captis refractionibus, manum admovisse,ut in ordinem illas, per secundorum incrementorum aequalitem, redigeret: [13] Omnia enim harumobservationum incrementa surgunt per differentiis 30'. In Kercheri autem observationum differentiis nullaest vel ordinis umbra; ita ut extra omne dubium sit, illius observationes esse fallaces. Nec sane mirum,videbitur (si quis rem accuratius intueatur) differentias quasdam in tam subtili disquisitione irrepsisse; ubilevis error in calculationum radice, in processu multiplicatur. Huius autem Theorematis veritas, pervariaexperimenta nobis multoties emicuit: veluti ex hoc unico, quod ad aquam fontanam habuimus, apparebit ;ubi per primum nostrum problema fideliter, & (pro instrumenti magnitudine) satis subtiliter, ad angulosincidentiae in aqua 100, 200, 300, 400, 450 : tales juvenibus refractiones. Si vero curiosis ingeniis hae


observationes non satisfecerint; agite Mathematici, & subtilioribus observationibus, hanc pulcherrimamDioptrices speculationem confirmate.

CorollariumSequitur ex hoc theoremate, angulum lineam refractionis cum axe ellipseos densitatis, (v.g. in

superiore figura angulum BEO), semper esse angulum refractionis, Competentem angulo incidentiae ABC.Restat autem nunc ut quod superest , ex praedicta analogia, ope theorematis, geometrice demonstremus.

[14]§2. Prop.6.1. Prop. 6. Theorem.

If the ratio of the length of the axis to the inter-focal distance of a dense ellipsebetween two media is thus as the ratio of the inter- focal distance to the length of its axisof a hyperbola , and if the hyperbola is the surface of refraction of the rays arising fromits own outer focus in the rarer medium, then all the given rays are refracted parallel bythe surface of the denser medium. This hyperbola is called the dense hyperbola of thegiven media.

The ratio is AC to BD, the axis length of the dense ellipse between any two mediaAKC to the separation of the foci, and this is thus as EH to LN, the separation of the fociof the hyperbola to the axis length of the same. The hyperbola ILY is the surface ofrefraction. The ray EI is incident on a branch of the hyperbola coming from the far focus,and passing from the rarer to the denser medium at

[15]the point I. The ray EI is refracted through Z, thus QIZ is the angle of refraction [ i.e. thedeviation d]. I say that IZ is parallel to the axis of the hyperbola ENLH. For the line IT is

drawn through the point I, tangent to the hyperbola ILY at the point I. A perpendicularline IM is drawn from the point I, and EIM is the angle of incidence, which is equal tothe angle GDF from the focus D of the ellipse [i.e. equal to the angle of incidence for theellipse: see the note following], and the line BG is drawn equal in length to the axis AC;from the preceding corollary it is apparent that the angle GBD is equal to the angle ofrefraction QIZ [i.e. the angle of deviation d], meeting the angle of incidence GDF, orEIM: and thus IO is made equal to the line IH, and OH is joined; and EO is equal to thelength of the axis of the hyperbola, and IT dividing the angle HIO in two equal parts is

B C

SK

DF A

G

HLE N T

IZ

O

QM

Y

Prop. 6 - Figure 1.


perpendicular to OH (Apol. 3.51 & 3.48). But IT is perpendicular to IM, and therefore IMand OH are parallel, and hence the angle EIM is equal to the angle IOH: but EIM is equalto the angle GDF, and therefore IOH is equal to the same GDF. Therefore in the [similar]triangles EOH, BDG the two sides HE, EO are proportional to the two sides GB and BD,[thus, the inter-focal distance for the ellipse is proportional to the vertex separation of thehyperbola; while the vertex separation of the ellipse is proportional to the inter-focaldistance for the hyperbola: conjugate conics], and the angle GBD is equal to the angleHEO, i.e. the angle QIZ is equal to the angle QEH, IZ and EH are therefore parallel.Q.e.d.

Scholium.It is also possible to find the hyperbola by trial first, and then the ellipse is deduced

from this by a geometrical demonstration. But we have composed these theorems in thesame order in which they were found by us.

And from these surfaces the angles of refraction are measured in the clearestanalogous manner which you can show - through tests with which you can steadfastlyagree, and by geometrical demonstrations in three dimensions that you can prove. It nowremains that we may explain optical devices, before [demonstrated by others] throughapproximation, now shown with geometric precision by us. Since truly, up to this pointwe have said so much about the surfaces of refraction, it is necessary that we shouldpresent some other Lemmas, with the help of which (without doubt of such kinds thatconstitute all the machinery of optics) one is allowed to proceed from surfaces to solids.

§2. Prop.6.2. Notes on Prop. 6.

From Prop. 6 - Fig.2, it is seen that the triangle associated with the ellipse, BDG, issimilar to the corresponding triangle EOH for the hyperbola. Hence if e, where 0 < e < 1,is the eccentricity of the ellipse with major diameter 2a, then AC = BK + KD = BG = 2a ,

B C

SK

DF A

G

E N T

ZO

q

M

HL

I

y

ir d

r

r

ri d

ird

i

d

iir

Prop. 6 - Figure 2.


and BD = 2ae ; similarly, for the hyperbola with eccentricity e' > 1, NL = EI - IH = EO=2a', and EH = 2a'e'.

[14]§2. Prop.6.3. Prop. 6. Theoremata.

Si fuerit; ut axis ellipseos densitatis duorum diaphanorum, ad distantiam focorum ejusdem,ita Distantiafocorum Hyperbolae, ad sui axem; fueritq: hyperbola superficies refractionis Radiorum ex foco suoexteriore, in diaphano rariore existente : Omnes dicti radii, per refractionem in superficie densiorisdiaphani , ad Parallelismum reducentur. Vocetur autem has hyperbola, hyperbola Densitatis praedictorumDiaphanorum.

Sit ut AC axis Ellipseos densatatis duorum diaphanorum quorumcunq ; AKC; ad BD, distantiam focorum;ita EH, distantiam focorum hyperbolae, ad LN, axem ejusdem: sitq hyperbola ILY superficies refractionis,in quam incidat Radius EI ex foco exteriore E, in diaphano rariore existente, in punctum I, existens insuperficie diaphani densioris, & refrangatur linea EI in Z, ita ait angulus QIZ, sit angulus refractionis : DicoIZ esse parallelam axi hyperbolae ENLH. Ducatur enim per

[15]punctum I, linea IT, tangens Hyperbolam ILY in puncto I, & a puncto I ducatur linea IT perpendicularis,IM, eritq; EIM angulus incidentiae, cui fiat equalis ex foco ellipseos D, angulus GDF, & ducta recta BG,axi AC aequali; ex praecedente corollario evidens est, angulum GBD esse aequalem angulo refractionisQIZ, competentem angulo incidentiae GDF, seu EIM: fiat itaq; IO aequalis rectae IH, & jungatur OH;eritque EO aequalis axi, hyperbolae NL, & IT dividens angulum HIO bifarium (Ap.3.51, Ap.3.48), estperpendicularis ad OH; sed & perpendicularis est ad IM; igitur IM, OH sunt parallelae, & angulus EIMaequalis angulo IOH: sed EIM, est aequalis angulo GDF, igitur & IOH eidem GDF est aequalis. Intriangulis igitur EOH, BDG duo latera HE, EO sunt proportionalia duobus lateribus GB, BD, & angulusHOE, aequalis angulo GDB, ergo & angulus GBD, est aequalis angulo HEO, hoc est angulus QIZ, aequalisangulo QEH, parallelae igitur sunt IZ & EH , quod erat demonstrandum.

Scholium.Poterat etiam & hyperbola densitatis per experientiam primo inveniri, & ellipsis demonstratione

Geometrica ex ea deduci; Nos autem, haec scripsimus eadem Methodo, qua a nobis reperta sunt.Atque hisce de superficiebus, quae Refractiones metiuntur per analogiam clarissime Monstratis, per

experientias firmiter probatis, & per demonstrationem Geometricam solide confirmatis : Restat nunc utMachinas Opticas, ante per approximationem, nunc Geometrice demonstremus. Quoniam vero, hactenus desuperficiebus refractionum tantum, loquuti sumus; oportet ut aliquot Lemmata praemittamus, quorum ope asuperficiebus ad solida (qualia nimirum sunt omnia Machinamenta Optica) liceat Progredi.


[16]

§3. Synopsis of Propositions 7 - 17: The Generation of Spheroidal andConoidal Surfaces from Conic Sections and the Behaviour of RaysRefracted through them.

The first of these Propositions show how the refracting surfaces are generated fromconic sections. We are to regard the conoid either as a paraboloid or the branch of ahyperboloid, while the spheroid is an ellipsoid with an axis of symmetry. The refractionof a set of parallel rays along the optic axis for individual surfaces is then considered. Ingeneral, parallel rays are refracted through the far focal point by these surfaces. Thesituations considered include parallel rays incident on a dense spheroid, parallel rays in adense hyperboloid incident at a less dense interface, the ray reversed cases of these, andthe cases where the less and more dense media are interchanged, and their ray reversedcases. The reflection by the inner surface of a parabola is also considered.

Prop. 7 shows the axial symmetry of spheroidal and conoidal surfaces.Prop. 8 asserts that all sections perpendicular to the axis of the cone are circles.Prop. 9 asserts that a plane tangential to a conoid or spheroid is perpendicular to a

plane through the point of contact and the axis of the conoid or spheroid.Prop. 10 asserts that a ray incident on a refracting or reflecting conoidal or spheroidal

surface in a plane containing the axis, is refracted or reflected in that plane by the conicsection which generated the conoid or spheroid.

Prop. 11 demonstrates the focusing property of a parabola for the reflection of raysparallel to the axis, and locates the position of the focus at one quarter of the length of thelatus rectum from the vertex within the parabola.

Prop. 12 asserts that an ellipse or hyperboloid of a given kind, i. e. one with a knownaxis to inter-focal separation ratio, can be constructed from the known positions of avertex and focus from proportion.

Prop. 13 asserts that equally spaced parallel rays incident at some angle on the planeinterface between two media are refracted and sent out parallel and equally spaced at adifferent angle in the second medium.

Prop. 14 asserts that parallel rays incident along the axis of a spheroid in a less densemedium are refracted through the far focal point of the ellipse of cross-section. Also,parallel rays incident along the axis of a hyperboloid in the denser medium are refractedthrough the focal point of the far branch of the hyperboloid of cross-section in the rarermedium.

Prop. 15 is the converse of Prop. 14; which follows by reversing the directions of therays.

Prop. 16 considers the case where the dense medium lies on the outside of thesurface. We are to consider refraction by a hollow spheroidal surface and by ahyperboloidal surface with the denser medium filling the space between the branches.

Prop. 17 is the converse of Prop. 16 with the rays reversed.


A

BC

D

E

F

G H

I

K M

L

Prop. 7- Figure 1.

§3. Prop.7.1. Prop. 7. Lemma.

All sections of a conoid or spheroid cut through the axes return the same section of thecone from which they are generated, while the axis of the section is the same as that ofthe conoid or spheroid.

Let ABCD be any conoid orspheroid you wish of which the axisis AC, which is cut by a plane tomake the section ABCD. I say thatthe section EAFC shall be the sameconic section, from which theconoid or spheroid ABCD wasgenerated. I also say that AC is theaxis of this conic section. The conicsection ABCD may be plainly seen,from which the figure has beengenerated by rotation. Indeed ifABCD itself is rotated, so that it

may fall in the place of the section AECF, then either the whole section will agree inevery part, or it will be different from AECF itself. If it agrees, then both AECF andABCD are the same, and as we assert the axis AC of both is the same. If it should differin some part then it follows that there is something is in the figure which does not agreewith the rotation of the conic section ABCD, which is absurd. Therefore, there will be nodisagreements, and therefore they are equal; and they have a common axis AC, whichhad to be shown.

§3. Prop.7.2. Prop. 7. Lemma.

Omnis conois, vel sphaerois, per axes secta, reddit eandem coni sectionem, ex qua estgenita. Sectionis autem axis, idem est cum axe conoidis, vel Sphaeroidis.

Sit Conois, Sphaerois quaelibet ABCD, cujus axis AC, per quem plano secetur, fiatq;sectio EAFC : Dico sectionem EAFC , esse eandem sectionem Conicam, ex qua estgenerita Conois vel sphaerois ABCD: cujus etiam sectionis conicae dico AC esse axem.Intelligatur ABCD sectio conica, ex cujus revolutione genita est figura: etenim si ipsaABCD revolvatur, quod incidat in locum sectionis AECF, aut tota toti conveniet, aut abipsa AECF discrepabit: si conveniat, eadem est AECF, ac ipsa ABCD, & utriusq; idemest axis AC, ut intendimus : si discrepet aliqua parte, sequetur aliquid esse in figura, quoda sectionis conicae ABCD revolutione, non est constitutum, quod est absurdum. Non ergodiscrepabunt, & igitur equales sunt, habentq; axem communem AC, quoddemonstrandum erat.

[17]


§3. Prop.8.1. Prop. 8. A Lemma.

A plane section perpendicular to the axis of a conoid or spheroid is a circle having itscentre on the axis.

The conoid or the spheroid is cut by a plane perpendicular to the axis AC, meeting theaxis in the point K, and the section is GLHI, which I say is a circle [Prop.7 - Fig.1]Furthermore, the plane through the axis AKC is extended making the section EAFC,meeting the plane section perpendicular to the axis in the points I and M. The lines KIand KM are drawn to these points from the point K perpendicular to the axis AKC.Hence each section is connected in turn: for indeed the circle BEDF is made from therotation of the section BAFC in constructing the figure, the radii of which are CB, CE,CD, and CF at right angles to the axis, and hence the axis is connected with each of thesein turn. Therefore (Apol. 1.21) each of the three equal radii CB, CE and CD has indeedthe same ratio to its own parallel line from the three lines KG, KI and KH. Thus it isapparent (as CB, CE and CD are equal) that KG, KI and KH are equal. Therefore thecircle is GMHLI, the centre K of which lies on the axes of the conoid or spheroid (Apol.3.9). Q.e.d.

§3. Prop.8.2. Prop. 8. Lemma.Conoide vel spheroide, secta plano ad axem perpendiculari, sectione fit Circulus,Centrum habens in axe.

Dividatur conois, vel Sphaerois, plano ad axem AC perpendiculari, occurrente inpuncto K, & sit sectio GLHI, quam dico esse circulum: Etenim per axem AKC planumagatur faciens sectionum EAFC, occurrentem secanti plano perpendiculari ad axem, inpunctis I, M, ad quae a puncto K ducantur lineae KI, KM, qui ad axem AKC suntperpendiculares ; & proinde ordinatim applicatae: cum enim ex revolutione sectionisBAFC, in constitutione figurae, factus sit circulus BEDF, cujus radii sunt CB, CE, CD,CF, [ Ap. 1.21] ad axem recti, & proinde ordinatim quoque applicati; igitur unusquisquetrium Radiorum aequalium CB, CE, CD, [3.9] habet ad quamlibet sibi parallelam lineame tribus KG, KI, KH, eandem rationem, unde patet (cum CB, CE, CD, sint aequales) KG,KI, KH esse aequales; igitur circulus est GMHLI, cujus centrum K, in axe conoidis velsphaeroidis; quod demonstrandum erat.

§3. Prop.9.1. Prop. 9. A Lemma.

If a plane touches a conoid or spheroid, and another plane is drawn through the pointof contact and the axis. I say that these two planes mutually cut each other at rightangles.

ABCD is the conoid or spheroid to which the plane HIGF is a tangent at the point E.The plane BEALD is produced through the point E and the axis of the figure, cutting theplane HIGF in the line IF. I say that the planes HIGF and BEALD cut each othermutually at right angles. The plane EMLN is drawn through the point E, perpendicularto the axis AC, and if it is produced then it cuts the plane HIGF in the line GH . Since HGtouches the circle EMLN, it is perpendicular to the diameter EL, by Prop. 8. Because the


A

BC

D

MLE

FG

H

N

I

Prop. 9 - Figure 1.

planes BAD and EML cut eachother mutually at right angles(Apol. 3.18), and HG lying in theplane EMLN is

[18]

perpendicular to the commonsection EL, then HG will beperpendicular to the plane ABD.Therefore (Apol. 11.18) all theplanes drawn through HG, (out ofwhich number is the plane HIGF)are normal to the plane BAD.

Q.e.d.

Scholium.These three Lemmas are explained in almost the same way, generated by rotation in

all the solids.§3. Prop.9.2. Prop. 9. Lemma.

Si Conoidem, vel Sphaeroidem, tangat planum, & per punctum contactus, & axem,ducatur aliud planum ; Dico haec duo plana se invicem normaliter secare.

Sit Conois vel spherois ABD, quam tangat planum HIGF in puncto E, perque punctumE, & axem figurae producatur planum BEALD, donec secet planum HIGF in recta IF.Dico plana HIGF, BEALD, se invicem normaliter secare. Per punctum E ducatur planumEMLN, perpendiculare ad axem AC, & producatur, donec secet planum HIGF in rectaGH; & quoniam HG tangit circulum EMLN, erit ad ipsius diametrium ELperpendicularis ; & quia plana BAD, EML se invicem secant normaliter, & HG in planoEMLN, est

[18] perpendicularis ad communem sectionem EL; erit HG perpendicularis ad plano ABD:ergo & omnia plana per HG ducta, (e quorum numero est planum HIGF) ad planumBAD; erunt normalia ; quod erat demonstrandum.

Scholium.Haec tria Lemmata, eodem fere modo demonstrantur, in omnibus solidis ex

circumvolutione genitis.


A

B CD

EM

GH

F

L

Prop. 10 - Figure 1.


If the surface of a conoid or spheroid is also the surface of a dense medium forrefraction, or of a polished medium for reflection, and the line of incidence of a ray liesin the same plane as the axis of the conoid or spheroid, then the conic section from whichthe conoid or spheroid has been generated will always be the refracting or reflectingsurface for the ray.

Let ABC be the conoid or spheroid,either dense or with a polished surface.The axis AD and the line of the incidentML are coplanar, meeting the denserefracting medium or polished reflectingsurface at the point L. I say that thesurface for reflection or refraction of theincident ray ML is the same conicsection by which the conoid or spheroidwas described previously. The planeFEHG is drawn, touching the sameconoid or spheroid in the point L, and itis apparent from the laws of optics that the plane is the surface for reflection or refractionfor that incident ray ML cutting the perpendicular plane FEHG in the point L. Thustruly if the incident ray ML is there, and if the plane BAFLHC is drawn through the axisAD and the incident ray, then by Prop. 9 it shall be perpendicular to the plane FEHG,crossing through the point L. Hence the surface is the required surface of reflection orrefraction for the line of incidence ML, according to Prop. 7. Q.E.D.


Si superficies densi, aut politi, fuerit superficies Conoidis, aut Sphaeroidis, fueritquelinea incidentiae in eodem plano cum axe Conoidis, vel Spheroidis : sectio Conica ex quagenita est Conois, vel Sphaerois ; semper erit superficies Refractionis, vel Reflectionis.

Sit Conois, vel Sphaerois ABC, densum vel politum; cujus axis AD , sitque linea MLin eodem cum axe plano, L: Dico superficiem reflectionis, vel refractionis, lineaeincidentiae ML, esse eandem sectionem conicam ex qua describitur Conois vel Sphaerois.Ducatur planum FEHG, tangens Conoid vel Sphaeroid in puncto L: & patet ex doctrinaopticorum, illud planum esse superficiem reflectionis, vel refractionis, lineae incidentiaeML, quod perpendiculariter secat planum FEHG, in puncto L, ita ut linea incidentiaeML, in eo existat: Si vero per axem AD,

[19]& lineam incidentiae ML, ducatur planum BAFLHC; erit illud perpendiculare ad planumFEHG, & transibit per punctú L; ideoque erit superficiem reflectionis, vel refractionislineae incidentiae ML; est autem sectio conica, ex qua describitur Conois vel Sphaerois :quod erat demonstrandum.


A

B

C

D EM

G

H

F

L

R

N

Prop.11-Figure 1.


If a straight line is tangent to a parabola, and from the contact point a single line isdrawn within the parabola parallel to the axis. Another line is drawn from the samecontact point, making an angle with the tangent equal to the angle of the first line withthe tangent. This second line cuts the axis of the parabola within the parabola such thatthe distance between the vertex and the point of intersection is always equal to a quarterof the latus rectum. The point of intersection is called the focus of the parabola.

This is a most beautifuldeduction, which I know wasfirst come upon by Witelo, butsince he has overlooked thepleasing corollary, we ourselves- and perhaps in an easiermanner - shall demonstrate thistheorem itself otherwise. LetBACE be the parabola withaxis LAM, and the line LCN isa tangent to the parabola at thepoint C, from which CD is

drawn parallel to the axis, and the angle DCN is made equal to the angle LCF. I say thatAF is equal to a quarter of the latus rectum or focal chord R [drawn to the right of thefigure]. Let GC and AG [the text uses ME] be drawn symmetrically perpendicular andparallel to the axis, with CH perpendicular to the tangent line. These lines will be in theproportion R : GC : : GC : GA (= AL) , and HG : GC :: GC : GL; [ as ∆'s CGH and LGCare similar] hence GL : GA :: R : HG; but GL is twice the length GA, therefore R istwice the length HG;

[20] and since LM, CD are parallel; the angle NLM [the text has ELC] will be equal to theangle DCN, that is FCL; therefore FL and FC are equal: also the angles DCH, CHF, FCHare equal; hence, FH and FC are equal, and consequently FL is equal to FH. If thereforethe halves of LH and LG are taken from each other, i.e. giving LF and LA , then AF willbe the difference of LF and LA. Since FH is half of the latus rectum R, then AF is thefourth part of the latus rectum [i. e. half of (R - R/2)]. (Apol. 5.15). Q.e.d. The sametheorem can also be easily shown for the two remaining cases, which we omit for thesake of brevity.

Corollary 1.It follows from this Theorem that: DC + CF is equal to MA + AF; indeed DC + CF isequal to MG + FH, that is MF + GH; but GH is double FA itself; hence DC + CF = MF +2FA = MA + AF, which is the proposition. From this corollary the easiest way ofdescribing the parabola in the plane is given, that Kepler touched on in Ast. Opt.


A

B

CD

E

MG HFL

N

a2a

2ay

R


-x 1 x 1

1

Corollary 2.From this Theorem the second [corollary] follows, all the rays parallel to the axis are

reflected in the focus of the parabola, if the mode of reflection should be the concavesurface of a parabola.

[20]

§3. Prop.11.2.Notes on Theorem 11.

The first proportionality written inthe form GC2 = R.GA resembles themodern standard equation for theparabola be y2 = 4ax with origin at A(0, 0), and C is the point with co-ordinates (x1, y1) as in Prop. 10 -Fig.2, where we have marked in thevarious equal angles and lengths. Ifwe let the point G move to F, thenthe equation becomes (R/2)2 = R.FA,giving FA = R/4 = a.We may note the confusion of lettering in the text and diagram: it would appear that onehad been changed but not the other. Note that the first corollary translates straight into thelanguage of equal path lengths of all the rays: hence the focusing property of theparabola.


Si parabolam recta linea tangat, & a tactu ducatur una recta intra parabolam, axiparallelae, alia vero ad easdem partes, faciens angulum cum contingente, equalemprioris angulo cum contingente ; dico hanc lineam secundam, secare axem parabolaeintra parabolam, ita ut linea inter verticem & intersectionem; semper aequalis sitquadranti lateris Recti. Dicitur autem intersectionis punctum, focus Parabolae.

Hanc Pulcherrimam conclusionem, primus quod sciam invenit Vitellio, sed quoniamjucundam omisit Corollarum, nos aliter, & forsan facilius, hanc ipsam demonstrabimus.Sit Parabola BACE, cujus axis LAM, tangatque eam linea LCN, in puncto C, a quoducatur axi parallela CD, fiatque angulo DCN, aequalis LCF, dico AF esse aequalemquadranti lateris recti R. Sint ordinatim applicatae, GC, ME perpendicularis adcontingentem CH, eruntque hae lineae proportiones R:GC :: GC:GA = AL, &HG:GC::GC:GL; ergo GL:GA::R:HG; sed Gl est dupla lineae GA, ergo R est dupla lineaHG;

[20] & quoniam LM, CD sunt parallelae; erit angulus ELC , aequalis angulo DCN, hoc estFCL; ergo FL, FC sunt aequales: sunt etiam anguli DCH, CHF, FCH aequales; ergo, FH& FC sunt aequales, & consequenter FL & FH: si igitur duplorum LH, LG sumantur


ABC

D

G M N H


dimedia LF, LA; erit AF differentia dimediorum, semissis GH differentiae duplorum ;est autem GH semissis lateris recti R; ergo & AF est quanta pars lateris recti, quod eratdemonstrandum. Eadem etiam facilitate demonstrabitur hoc Theorema in duobus Reliquiscasibus, quos brevitatis gratia praetermittimus.

Corollary 1.Sequiter ex hoc Theoremate: DC + CF esse aequales MA + AF; sunt enim DC + CFaequales MG + FH, hoc est MF + GH; est autem GH dupla ipsius FA; ergo DC + CF =MF + 2FA = MA + AF, quod est propositum. Ex hoc corollario, datur facillimus modusdescribendi parabolam in plano, quem attigit Keplerus in Ast. Opt.

Corollarium 2.Ex hoc Theoremate sequiter secundo, omnes radios axi parallelos reflecti in focumparabolae, si modo superficies reflectionis fuerit concavitas surface parabolae.

[21]


Given the position of a single focus and vertex for a given kind of ellipse or hyperbola[i. e., one with a known axis to inter-focal separation ratio], the ellipse or hyperbola canbe found.

Let A be the vertex and B the focus of theellipse ALD, for which the ratio of theseparation of the foci to the axis is givenas E to GH. The axis length and the focalseparation of the ellipse ALD is sought.From GH, MN equal to E itself is takenaway [i. e. the ratios 1 - E/GH, 2NH/GH,NH/GH, and also 1 - NG/GH or GN/GH,are also known]. Thus, as GM is equal toNH, and the ratio GN to GH is as AB toAD, and AC is equal to BD, I say that[the unknown] AD is the axis, and C andB are the foci of the ellipse ALD.Conversely:If GN:GH::AB:AD, thenGN:AB::GH:AD ; & asGH:AD::NH:BD; [for GN/GH - 1 = AB/AD - 1, so NH/GH = BD/AD]GH:AD::2NH:2BD; [hence 1 - 2NH/GH = 1 - 2BD/AD, etc, leading to....]GH:AD::MN = E:CB; i. e. E:GH::CB:AD; (Apol. 5.19). Q.e.d.


Also one can proceed in the same way, if A is the focus of a hyperbola, B the vertex; Eto GH, the given ratio of the axes to the separation of the foci, and AD is the separation ofthe foci, and CB the axis. [This is the case of a conjugate ellipse and hyperbola. In thistheorem, everything follows by proportion from the given ellipse or hyperbola.]

[21]

§3. Prop.12.2. Prop. 12. Problem.Ex datis positione, uno foco, una vertice, cum ellipseos, vel hyperbolae specie ; Ellipsemaut Hyperbolam invenire.

Sit A vertex ellipseos ALD, B focus , sitque ut E ad GH; ita distantia focorum ellipseosALD, ad ipsius axem; quaeritur illius axis, & focorum distantia. Ex GH auferatur MN,aequalis ipsae E ; ita, ut GM sit aequalis NH; fiatque ut GN ad GH, ita AB, ad AD; sitq;AC aequalis ipsi BD: Dico AD esse axem ellipseos ALD, & C, B, illius focos . Quoniamenim ut GN:GH::AB:AD; erit utGN:AB::GH:AD; & utGH:AD::NH:BD;GH:AD::2NH:2BD;GH:AD::MN = E:CB; hoc est E:GH::CB:AD;quod erat ostendendum. Eodem etiam modo esset praecedendum, si A esset focusHyperbolae, B vertex; E ad GH, ratio axeos ad distantium focorum, essetque ADdistantia focorum, & CB axis.

[22]


Equally spaced parallel rays in one medium,[ on refraction] by another medium ofdiffering density are to be sent out equally spaced..Let the common surfaces of the mediums of different density be plane. Therefore theparallel rays incident on the plane surface are chosen with equal angles of incidenceeverywhere; therefore all of the angles of refraction of these are equal, which results inparallel rays of refraction too since they are made parallel by the surface of refraction, asmay be shown by the converse of Prop. 10, Book 11. Elements. Q. E. D.

ScholiumEven though this problem is demonstrated most generally here, in the following

however we shall make use of only one case ; indeed we always draw such a planethrough any given point, to which the parallel rays are normal [i.e. lie in a perpendicularplane] ; this special case is used more for the sake of convenience than necessity, as willbecome apparent to the knowing Reader with what follows.

[22]


A

E

R R R R R

M


R

L

E

R RR R

MN

A



Radios parallelos in uno Diaphano, per aliud diversae densitatis, aequi distantes mittere.

Sit communis superficies Diaphanorum diverarum densitatum plana, radii igiturparalleli, in superficiem planam incidentes, undique sortiuntur aequales incidentiaeangulos, ac proinde aequales refractiones, omnes ergo eorum anguli refracti suntaequales, qui cum fiant in superficiebus refractionum parallelis efficiunt quoque radiosrefractos parallelos, ut patet per conversum Prop. 10, libri 11. Elementi. Quod eratostendendum.

ScholiumHoc Problema etiamsi hic generalissime demonstretur, in sequentibus tamen unum

tantum illius casum usurpamus; semper enim Ducimus tale planum per punctum aliquoddatum, cui normales sunt radii paralleli ; quo casu magis ob commoditatem, quemnecessitatem utimur, ut intelligenti Lectori ex sequentibus patebit.


The parallel rays in one medium are gathered together ina given single point by a conoidal lens composed of amedium of different density; with the vertex of the conoidor spheroid given too, it is moreover in order that the linedrawn through the point of convergence and the givenvertex shall be parallel to the given rays.

In the first place the rays R, R, .. etc. are parallel in therarer medium, and they are to converge to the point E ofthe denser medium. The vertex of the spheroid is L with Ethe further focus, by Prop. 12. An ellipse LMA is madefrom the denser transparent medium, and with the axisLEA continuing parallel to the given rays, the spheroidLMA is made from the ellipse by revolution, by Prop. 10.I say all the rays parallel to the axis of the Spheroid (fromwhich a number are RR etc.) which are incident on thespheroid, are

[23]refracted at the surface of the spheroid and concur atthe focus E.

For let one of the parallel rays RM be incident atthe point M in the same plane as the axis, as it isparallel to that plane, it follows that the ellipse LMAwill be the surface of refraction of the ray RM (whichhas been generated by rotation of the spheroidLMA); RM therefore will be refracted in the point E,by Prop. 5. Q.E.D.


In the second case, let R, R,...etc. be parallel rays in the denser medium, and theyconverge to the given point E of the rarer medium. L shall be the vertex of the conoid,and it is to be made from the hyperbola NLM of the dense transparent medium withvertex L and further focus F, by Prop. 12. From the axis EAL continuing parallel to thegiven rays, the hyperbolic conoid is made by rotation of the hyperbola NEM. If the densematerial is as before, I say that all the rays parallel among themselves, are refracted in thesurface itself, and emerge into the rarer medium to concur at the focus E.

[24]The ray RM is one of the parallel rays incident in the point M, which therefore is in thesame plane as the axis. Hence the hyperbola NLM, (from which the conoid has generatedby rotation, by Prop. 10) will be the surface of refraction for the ray RM; which will berefracted in the point E; because of that, the ray coming from the point E and incident inM is refracted in MN, (Prop. 5); as is shown by Prop. 9, Book 10 of Witelo. Q.E.D.

CorollaryFrom this it appears to be sufficient, according to visual perception, that parallel rays

RR, etc., are to be gathered together in one point near E, : even if LE may be differentgeometrically from the length of the ray; since if there shall be no sensible differenceamong the causes, neither will there be any sensible differences among the effects.


Radios parallelos in uno Diaphano, in unicum punctum datum Diaphani diversaedensatatis congregare ; Data quoque Conoidis, vel Sphaeroidis vertice ; oportet autem,ut linea, ducta per punctum congregationis, & verticem datam, sit radiis datis parallela.

Sint primo radii in Diaphano rariore paralleli, R, R, .. &c., congregandi in Diaphamdensioris punctum E ; sitque Sphaeroidis vertex L; foco remotiore E, & vertice L,fiathorum diaphanorum ellipsis densitatis LMA ; & manente axe LEA, radiis datis parallelo,ex ellipseos circumvolutione fiat Sphaerois LMA, cujus materia sit ex praedicto diaphanodenso : dico omnes radios axi Sphaeroidis parallelos, (e quorum numero sunt R R &c.) inSphaeroidem incidentes,

[23]in superficie Sphaeroidis refringi, & in focum E concurrere : Sit enim Radius RM, unus eparallelis, incidens in punctum M, qui erit in eodem plano cum axe, ex eo quod sit illiparallelus; erit igitur Ellipsis LMA (ex cujus circumvolutione genita est sphaerois LMA )superficies refractionis radii RM; refringetur igitur RM in focum E; quod ostendendumerat.

Secundo; Sint radii in diaphano densiore paralleli RR &c. congregandi in diaphanirarioris punctum datum E : Sitq Conoidis vertex L; foco remotiore E, & vertice L, fiathorum diaphanorum Hyperbole densitatis NLM ; & manente axe EAL radiis datisParallelo, ex circumductione hyperbolae NEM, fiat Conois hyperbolica, cujus materia sifuerit ex praedicto denso; dico omnes radios in ipsa parallelos, in ipsius superficierefringi, & in focum E, in diaphano Rariore existentem, concurrere : Sit enim radius RM,

[24]


unus e parallelis incidens in punctum M, qui propterea erit in eodem cum axe plano ; eritigitur hyperbola NLM, (ex cujus circumductione genita est Conois) superficiesrefractionis Radii RM; qui refringetur in punctum E, ex eo quod radius puncto Eegrediens, & incidens in M refringatur in MR; ut patet per Prop. 9, lib. 10 Vitellionis.quod erat ostendendum.

CorollariumEx hoc satis apparet, Radios Parallelos RR, &c., congragari in unum punctum prope

E, quo ad sensum: etiamsi LE non aequi distet radii &c. geometrice; Quoniam si non sitsensibilis differentia inter causas, nec erit sensibilis differentia inter effectus.

[24 cont'd]


The rays diverging from a single point of one medium are restored to parallelism inanother medium of different density. The vertex of the required conoid or spheroid is alsogiven.

This problem is the converse of the antecedent, and is solved in the same way; asshown by Prep. 9, book 10 of Witelo.

CorollaryFrom this too it appears to be sufficient that the rays coming from any point near E,

(on account of the reasoning reported above) are perceptibly restored to parallelism bythe conoid or spheroid NLM.

[24 cont'd]


Radios ex unico puncto unius diaphani; provenientes, ad Parallelismum in alio diaphanodiversae densitatis reducere ;data quoque Conoidis vel Sphaeroidis vertice.

Problema hoc, conversum est antecedentis, eodemque modo solvitur; ut Patet Prop. 9,lib.10, Vitellionis.

Corollarium.Ex hoc quoque satis apparet radios, e puncto aliquo prope E provenientes, (propter

rationem superius allatam) sensibiliter ad parallelismum Reduci Conoide vel SpheroideNLM.

[25]


N

R

P

H

B

L

M

AAA A

G


P

LM

N

H

B

Z GR

A A A A A



Parallel rays in one medium shalldiverge in a medium of another density.Such rays can be traced back to some givenpoint, for which the vertex of the requiredconoid or spheroid has also been given. It isalso the case that the line drawn throughthe vertex and the given point is parallel tothe given rays.

The parallel rays in the rarer medium A,A, etc., are to be refracted thus so that theyappear to diverge from the point N: with thevertex M of the spheroid given too {prop.12}. The dense ellipse LMR [i.e. on theoutside] is made of this transparentmedium, N is the more removed focus, andM is the vertex. The axis NM of this ellipseremaining fixed with respect to the parallelrays A, A, etc. The spheroid LMR isdescribed by the rotation of this ellipse, composed of the rarer medium; indeed, the spacewhich is situated around the spheroid is composed of the denser medium. I say that all therays A, A, etc, incident on the surface of the spheroid, are refracted by the surface itself,and diverge from the focus N. Indeed from the rays A, A, etc., let one ray AL be incidenton the spheroid at the point L, and this ray is produced to P, and the line HLG is drawntouching the spheroid in the point L, crossing the plane of the ellipse through the axis ofthe spheroid and the point L. The ellipse is the surface of refraction RML. The line NLBis drawn from the point N through the point of contact. Therefore, from prop. 10, if thespheroid LMR is made from the denser medium and is enclosed by the rarer medium,then the angle of incidence

[26]HLP agree with the angle of refractionNLA from prop. 9 [i.e. the ray PL isrefracted along LN]. Hence, in the presentcase for the equivalent angle of incidenceALG the equivalent angle of refractionPLB agrees [i.e. in the sense that the rayhas bent through this angle]. Therefore theray AL is refracted at B, and appears todiverge from the point N. Q.e.d.

Secondly the parallel rays A A A, etc. inthe denser medium can be refracted thus,in order that they appear to diverge fromthe point N. Given too the vertex M of theconoid, N the more distant focus, and Mthe vertex. The dense hyperbola LMR is


made of this transparent medium, and by the rotation of which, with the axis NMremaining parallel to the given rays, the hyperbolic conoid RML is generated. Thus as thespace outside the conoid (where assuredly the parallel rays A, A, etc are present) is madefrom the denser medium, truly the cone itself consists of the rarer medium. I say that allthe rays A, A, etc. are refracted in the surface of the conoid RML, and diverge from thepoint N. Indeed the ray AL is one of these, for which the surface of refraction is thehyperbola RML itself, from which the conoid is described by Prop. 10. Through the pointof incidence L, the line HLG is drawn touching the hyperbola, and AL is produced to P,and from the point N, through L the line NLB is drawn. If therefore the conoid RML iscomposed of the denser medium, and is present in the rarer medium, from which it is nowcomposed by Prop. 6, then the line PL is refracted in N. Therefore for the equivalentsituation with the angle of incidence PLG, the refraction NLA is in agreement, and theline AL is refracted in B. Q.e.d.

Corollary.Hence too it is clear enough to the senses that if parallel rays A, A, etc., diverge fromsome point near N, then the line NM shall not be geometrically parallel to the rays A Aetc.

[25]§3. Prop.16.2. Prop. 16. Problema.

Radios parallelos in uno diaphano, ad divergentiam, in alio diaphano aliteriusdensitatis, ab aliquo puncto dato reducere;data quoque Conoidis vel Sphaeroidis vertice:Oportet autem, ut linea per verticem, & punctum datum ducta, sit radiis datis parallela.

Sint radii Paralleli in Diaphano rariore A, A etc., ita refringendi, ut appareantdivergere ex puncto N; data quoque Sphaeroidis vertice M, foco remotiore N, & verticeM, fiat ellipsis densatatis horum diaphanorum LMR, & ejus axe NM immobili manente,radiis A, A, &c. parallelo, ex ejus circumvolutione describatur Sphaerois LMR constansex diaphano rariore; spatium vero in quo est Sphaerois, constet ex diaphano densiore:Dico omnes radios A, A, &c., in Sphaeroidis superficiem incidentes, in ipsius superficiesrefringi, & a foco N divergere: Sit enim e radius A, A, &c.,unus AL, incidens inSpheroidis superficiem in punctum L, & producatur in P, ducaturque contingensSphaeroidem in puncto L, linea HLG, in plano ellipseos, per axem Sphaeroidis, &punctum L, transeuntis, quae ellipsis est superficies refractionis, sitque RML; & ducaturex puncto N, per punctum contactus L, linea NLB: Si igitur Sphaerois LMR, essetdiaphanum densius, & includeretur diaphano rariore, ex quo nunc constat, tunc anguloincidentia.

[26]HLP, competeret refractio NLA; ergo & aequali angulo incidentiae ALG competitaequalis refractio PLB; refringitur ergo radius AL in B, divergens a puncto N; quodostendendum erat.

Sint secundo radii paralleli in diaphano densiore A A &c. ita refringendi, ut appareantdivergere e puncto N; Data quoque Conoidis vertice M: foco remotiore N, & vertice M,fiat hyperbola densitatis horum diaphanorum LMR, ex cujus circumvolutione, axe NM


radiis datis parallelo manente, fiat Conois Hyperbolica RML ; ita ut spatium extraConoidem, (ubi nimirum sunt radii paralleli A A &c.) constet ex diaphano densiore, ipsavero Conois ex diaphano rariore. Dico omnes radios A A &c. refringi in superficieConoidis RML, & divergi a puncto N. Sit enim ex illis unus, radius AL, cujus superficiesrefractionis est ipsa hyperbole RML, ex qua describitur Conois ; & per punctumincidentiae L, ducatur tangens hyperbolam Recta HLG , & producatur AL in P, & apuncto N, per L ducatur recta NLB; Si igitur Conois RML constatet ex diaphanodensiore, & existeret in diaphano rariore, ex quo nunc constat ; tunc refrangeretur radiusPL, in N; & angulo incidentiae PLG,competeret refractio NLA; aequali igitur anguloincidentiae ALH, competit aequalis refractio PLB ; refringitur igitur radius AL in B;quoderat ostendendum..

Corollarium.Hinc quoque satis patet radios A A &c. Parallelos, divergi ab aliquo puncto circiter N;quo ad sensum; etiamsi linea NM nonsit radiis A A &c., geometrice parallela.

[27]§3. Prop.17.1. Prop. 17. Problem.

The rays in one medium converging to a single given point become parallel in anothertransparent medium of different density; with the vertex of the required Conoid orSpheroid given too.This problem is the converse of the preceding too, and is solved in the same way; as isclear from Prop. 9, Book 10, Witelo.

Corollary.From this also it is apparent, that rays converging to some other point near N are sensiblyreduced to being parallel for the Conoid or Spheroid LMR.

Scholium.

Up to this point we have talked only about a single refraction, which happens at thesurface of a conoid or spheroid; indeed now we are to talk about the two-fold refractionof lenses (one refraction happens in the incidence of the rays, the other in the emergence),which are composed of conoidal or spheroidal frustrums. The same conclusions are to beshown always, for both mirrors and lenses - in order that the wondrous harmony mayappear between Catoptrics and Dioptrics.

[27]


Radios in uno diaphano,ad unicum punctum datum convergentes, ad parallelismium inalio diaphano diversae densatatis reducere ; data quoq; Conoidis, vel Sphaeroidisvertice.Hoc problema est conversum quoque antecedentis, eodemque modo solvitur ; ut patet exProp. 9, lib. 10, Vitellionis.


Corollarium.

Ex hoc etiam evidens est, radios ad punctum aliquod prope N convergentes, adparallelismum reduci quo ad sensum, Conoide, vel Sphaeroide LMR.

Scholium.Huc usque loquati sumus de unica tantum refractione, quae fit in superficie Conoidis,

Sphaeroidis; nunc vero loquimur de duplice refractione lentium (una fit in radiorumincidentia, altera in radiorum emersione ), quae ex frustis conoideon, vel sphaeroideoncomponuntur, easdem semper conclusiones demonstrando, & in speculis, & in lentibus ;ut appareat admiranda Harmonia, inter Catoptricam, & Dioptricam.


§4. Synopsis of Propositions 18 - 27: Image formation by mirrors andlenses with spheroidal and hyperboloidal surfaces.

Gregory pursues the analogy between reflection and refraction in his investigation ofthe imaging properties of mirrors and lenses with surfaces derived from conic sections. Asequence of propositions of increasing complexity is established, relating object andimage positions for special cases. These follow from the above theorems and also verifiedfrom products of the transfer matrices M1 - M8 introduced in §0.4.

Prop. 18 considers the focusing or meeting of parallel rays at a point in the samemedium by : (a) a concave reflecting paraboloid; (b) a solid lens set in air with a planeand a convex refracting hyperboloidal surface; and (c) a hollow refracting lens set in adense medium with a plane and a spheroidal refracting surface. The initial rays areparallel to the line joining the vertex to the point of concurrence or focal point of the rays- i. e. the optical axis.

Prop. 19 is the converse of Prop. 18, where the rays are reversed in direction.Prop. 20 considers the divergence of parallel rays from a special point, i. e. the focus,

in the same medium by: (a) a convex reflecting paraboloid; (b) a solid lens set in air witha plane and refracting convex hyperboloidal surfaces; and (c) a hollow lens set in a densemedium with a plane and a spheroidal refracting surface. The initial rays are againparallel to the optical axis.

Prop. 21 is the converse of Prop. 20, on reversing the rays.Prop. 22 considers the convergence to a focal point of rays diverging from another

focal point in the same medium on the optic axis by means of : (a) a concave reflectingellipsoid where the foci of the ellipsoid are used; (b) a solid lens set in air with equalconvex hyperboloidal refracting surfaces, where the far focal point of each hyperbola isused; and (c) a hollow lens set in a dense medium with equal spheroidal refractingsurfaces, where the far focal point of each ellipsoid of cross-section is used .

Prop. 23 is the converse of Prop. 22, on interchanging convex and concave surfacesfor both conoidal and spheroidal surfaces.

Prop. 24 considers the convergence to a focal point of rays of the rays converging toanother focal point in the same medium on the optic axis by means of : (a) a convexreflecting hyperboloid where the foci of the hyperboloid are used; (b) a solid lens set inair with unequal concave hyperboloidal refracting surfaces, where the unequal far focalpoint of each hyperbola are used; and (c) a hollow lens set in a dense medium withunequal hyperboloid refracting surfaces, where the far focal point of each hyperboloid ofcross-section are used .

Prop. 25 is the converse of Prop. 22, on exchanging convex spheroidal surfaces forconcave conoidal surfaces of unequal focal lengths.

Prop. 26 considers the reflection of a ray incident at some angle to the axis from thevertex of a concave mirror of revolution, together with a ray incident at a different angleto the axis. The rays and the axis are coplanar. The angle formed from the two rays ofincidence is equal to the angle formed from the two rays of reflection.

Prop. 27 is a comparable theorem for refraction, forming a vital part of Gregory'stheory of image formation by lenses with conoidal or spheroidal surfaces. Two rays atdifferent angles to the axis are incident at the vertex of a lens, one ray on either side ofthe axis. These rays are refracted into the lens at known angles, and they proceed to the


R R R RL

B

EM

A

C


RR

RR

O E

RR

RR

A O E



A

other surface. Two other rays are drawn from the vertex of emergence, each parallel to anincident ray. Within the lens, there are two sets of parallel rays. The theorem contendsthat the angle between the incident rays is equal to the angle between the emergentrefracted rays. We shall discuss this theorem in more depth in the synopsis to Prop. 44.


Parallel rays are to meet at a single point of the same medium, with the vertices of therequired lens or mirror also given: nevertheless the line drawn through the vertices ofthe lens or mirror and the point of concurrence is parallel to the given rays.

Reflection.Ler the parallel rays R, R, etc. meet in the singlepoint E: a concave parabolic mirror BAMC ismade with vertex A, focus E, and vertex A. I saythat all the rays R, R, etc. falling upon this mirrorare reflected through its focus E.

For let M be the point of incidence of the rayRM, and

[28]since the axis of the mirror AE, and the ray RMby supposition are parallel, then they are in thesame plane, and the parabola BAC from whichthe mirror BAMC has been generated byrotation, will be the reflecting surface. Hencethe ray RM is reflected in the focus E. SeeProp.10, Cor. 2, and Prop. 11. Q.e.d.

Refraction.Let the parallel rays R, R, etc. be broughttogether in a single point E, and let thevertices of incidence and emergence of thelens be A and O. The rays R, R, etc. are ledwith the help of the plane surface of thelens drawn through A, to which the rays R,R are normal, and within the lens they aresent parallel. [Beyond the lens] they meetat the focal point E, by means of refractionby the [solid] conoid, (Prop. 18 - Fig. 2) or[hollow] spheroid (Prop. 18 - Fig. 3), thevertex of which is O. See also Prop.13 &14. Q.e.d.


Corollary.From this it also follows that parallel rays meet near E, even if the rays are not parallel

to the line AE.[29]


Radios parallelos, in unicum punctum datum ejusdem diaphani congregare; datisquoque lentis vel speculi verticibus : opertet tamen, ut linea ducta per vertices lentis, velspeculi, & punctum concursi sit radiis datis parallela.

Catoptrice.Sint radii paralleli R R &c., in unam punctum E congregari, sitque vertex speculi A : focoE, & vertice A, fiat speculum parabolicum concavum BAMC; dico omnes radios R R &c.in hoc speculum incidentes reflecti in ipsius focum E. Sit enim M punctum incidentiaeradii RM, &

[28]quoniam AE axis speculi , & Radius RM, ex suppositione sunt parallelae;erunt in eodemplano ; & parabola BAC ex cujus circumvolutione est genitum speculum BAMC, eritsuperficies reflectionis ; radius igitur RM reflectetur in focum E, quod erat estendendum.

Dioptrice.Sint radii R R &c. paralleli in unum punctum E congregandi, sitque Lentis vertexincidentiae A, emersionis O. Radii R R &c. ope planae lentis superficiei per A ductae,cui sunt normales radii RR, intra lentem paralleli mittantur, in E punctum congregentur,ope conoidis, vel sphaeroidis, cujus vertex O; quod erat faciendum.

Corollarium.Ex hoc etiam sequitur, radios parallelos congregati in unam punctum circiter E,

etiamsi radii non sint geometrice paralleli rectae AE.[29]


To show that rays diverging from a single given point can be made parallel in thesame medium. The vertex of the required lens or mirror is also given, and it is necessarythat the vertices of the lens and the point of divergence are in the same straight line.

This problem is the converse of the preceding, and it is brought about by the samemethod; as it appears from Prop. 9, book 10 of Witelo. [Note: The proposition referred toin Witelo does not discuss lenses but gives the plane refraction case, which is applicableat the vertex of the lens.]

Corollary.From this it also follows that rays diverging from some point near E can be made

parallel with the help of the lens, or of the mirror mentioned above.


L

AAA

B

P R

N

H

M

G

A


AA

AA

L OB

AA

AA

L OB




Radios ex unico puncto dato divergentes, ad parallelismum reducere, in eodemdiaphano; datis quoque lentis vel speculi verticibus : Oportet autem, ut vertices lentis, &punctum divergentia, sint in eadem recta linea.

Hoc Problema est conversum antecedentis, eodemque modo perficitur ; ut patet ex Prop.9, lib. 10, Vitellionis.

Corollarium.Ex hoc etiam sequitur, radios divergentes ab aliquo puncto circiter E, fieri parallelos

ope lentis, vel speculi supradicti.


Parallel rays are made todiverge from some given point inthe same medium, where thevertices of the required lens ormirror are given too. It is necessarythat the line drawn through thevertex and the point of divergenceis parallel to the given rays.

Reflection.The parallel rays A, A, etc. are thusto be reflected, in order that theyappear to be diverging from thepoint B, and the vertex L of the mirror is given. From the vertex L and the focus B theparabolic convex mirror MLR is described, intersecting the parallel rays A,A, etc. Therays which are falling on the surface MLR, I say, are reflected by the surface,

[30]and diverge from the point B. Also a single rayAM incident on the point M, as it is parallel to theaxis of the conoid, has the parabolic surfaceMLR for reflection (from which the conoid isdescribed). (See Prop 10).The line HMG is drawnthrough M tangent to the parabola MLR, and theray AM is produced in P, from B BN is drawnthrough the point M. The angle BMG is equal tothe angle PMH, that is angle NMH is equal toangle AMG, and therefore AM is reflected intoN. (See Prop 11). Q.e.d.

Refraction.Let the parallel rays A, A, etc. be refracted thus inorder that they diverge from the point B by the


lens, of which the incident vertex of is L and of emergence O. The rays A, A, etc., withthe help of the plane surface of the lens, are drawn through L (to which the rays arethemselves the normals). They are sent parallel into the lens, and travel parallel within thelens, and finally are refracted to diverge from point B, with the aid of the solidhyperboloid (Fig. 2) or the hollow spheroid (Fig. 3), the vertex of which is O. (FollowingProp's 17&16). Q.e.d.

Corollary.From this it follows also that parallel rays diverge from some point around B, even if

they shall not be geometrically parallel to BL themselves.


Radios parallelos, ad divergentiam in eodem diaphano a quocunque puncto datoreducere, datis quoque lentis, vel speculi verticibus : Oportet tamen, ut recta per vertices,& divergentiae punctum ducta, sit radiis datis parallela.

Catoptrice.Sint radii paralleli AA &c. ita reflectendi, ut appareant divergi e puncto B, sitque dataspeculi vertex L. Vertice L, & foco B, describatur speculum parabolicum convexumMLR, intercipiens radios parallelos AA &c., quos radios in superficiem MLR incidentes,dico in ipsa reflecti, &

[30]divergi a puncto B. Sit enim ex illis, unus AM incidens in punctum M, qui, quoniam axiConoidis est parallelus, habebit parabolam MLR (ex qua describitur Conois) superficiemreflectionis : ducatur per M linea HMG tangens parabolam MLR in M, & producuturradius AM in P, & ex B ducatur BN, per punctum M; & erit angulus BMG, aequalisangulo PMH, hoc est NMH, aequalis angulo AMG; reflectitur igitur AM in N: quod eratostendendum.

Dioptrice.Sint radii paralleli AA &c. ita refringendi, ut divergant ex puncto B, lente cujus vertexincidentiae L, emersionis autem O. Radii AA &c., ope planae lentis superficiei, per Lductae (cui normales sunt ipsi radii) intra lentem paralleli, mittantur, & intra lentemparalleli, & B puncto divergantur, ope Conoidis, vel Sphaeroidis, cujus vertex O: quoderat faciendum.

Corollarium.Ex hoc etiam sequitur, radios parallelos divergi ex aliquo puncto circiter B, etiamsi

non sint geometrice paralleli ipsi BL.

[31]


HL

G

M A B N



The rays converging to one given point can be made parallel in the same medium, thevertices of the required lens or mirror are given too. It is necessary that the vertices ofthe lens and the point of convergence are in the same straight line.

This problem is the converse of the preceding, and is solved by the same method, as isclear from Prop. 9. book 10, Witello.

Corollary.From this it follows too, that the rays converging to some point around B will also be

made parallel with the help of the above mentioned lens or mirror.[31]


Radios ad unicum punctum datum convergentes, ad parallelismum in eodem diaphanoreducere; datis quoque lentis, vel speculi, verticibus : oportet tamen ut vertices lentis, &punctum convergentiae sint in eadem recta linea.

Hoc Problema est conversum antecedentis, eodemque modo solvitur, ut patet ex Prop. 9.lib. 10, Vitellionis.

Corollarium.Ex hoc quoq sequitur radios convergentes ad aliquod punctum circiter B, fieri etiam

parallelos, ope lentis, vel speculi supradicti.


The rays diverging from one given point converge to some other point of the samemedium ; with the vertices of the required lens or mirror given too : nevertheless it isnecessary that the points ofdivergence and concurrence,and the vertices of the lens ormirror are in the same straightline.

Reflection.Let the rays diverging frompoint A converge in point B,with the mirror of which thevertex is M. From the foci A, B

[32]and the vertex M, the concaveelliptical mirror MLN is drawn,in the surface of which the ray AL is incident in the point L: I say that the ray AL isreflected in the point L and passes through the focus B. For since AL and the axis of thespheroid are in the same plane; the ellipse LMN (from which the spheroid has beengenerated) will be the surface of reflection of the ray AL, and the angle of the ray AL


A M O B

A A O BM



with the contact will be equal to the angle BL from the same contact with the ellipseMLN; therefore the ray AL is reflected in B [Prop. 10]. Q.e.d.

Refraction.Let the rays diverging from the [focal]point A be made to converge to the [otherfocal] point B, by means of a lens forwhich the vertices of incidence andemergence are M and O. The raysdiverging from A are sent parallel withinthe lens with the help of the conoid[hyperboloidal surface: Prop. 22 - Fig. 2]or the spheroid [spheroidal surface : Prop.22 - Fig. 3], of which the vertex is M.Subsequently they are made to convergein the point B with the help of the conoidor spheroid with vertex O. [Prop's 15 &14.] Q.e.d.

Corollary.From this result it appears that if a radiating point is near A then the rays from that

point converge to some point around B by means of the lens or mirror mentioned above.

§4. Prop.22.2. Note.

We may note the use of the matrix methods introduced initially in the introductorysection to verify these theorems independently of the geometrical presentation. Thus, forexample, the lens in Prop. 22 - Fig. 2 can be represented by the matrices M6 and M5 for athin lens, giving the transfer matrix :

+−=

+−

+−= 1

12

011

11

01

11

0165M

)n(an)n(ann

)n(aM . The focal length of the lens is thus

a(n + 1)/2, while the focal lengths A & B associated with the hyperboloidal surfaces areat a distance a(n + 1) from their associated vertex. The effect of this transfer matrix is tochange the sign of the slope of an incident ray from A that passes through B. Similarly,the lens in Prop. 22 - Fig. 3 can be represented by the matrices M1 and M2, resulting in atransfer matrix:

+−=

+−

+−= 1

12

01

12

011

1

0121M

)n(ann

)n(an

n)n(anM with the same effect on the rays. The

focal length associated with this lens is a(n + 1)/2n.


G

L

HM

A

B

N

P

Q


§4. Prop. 22.3. Prop. 22. Problema.

Radios ab unico puncto dato divergentes, in aliud punctum ejusdem diaphanicongregare ; datis quoque lentis vel speculi verticibus: oportet tamen, ut punctumdivergentiae, punctum concursus, & vertices lentis vel speculi, sint in eadem rectae linea.

Catoptrice.Sint radii e puncto A divergentes, in punctum B congregati, speculo cujus vertex M.Focis A, B

[32]& vertex M, describatur speculum ellipticum Concavum MLN, in cujus superficiem,incidat radius AL, in puncto L: Dico radium AL in puncto L reflecti , & in focum Btendere. Quoniam enim AL, & axis Sphaeroidis sunt in eodem plano; erit ellipsis LMN(ex qua genita est Sphaerois) superficies reflectionis radii AL, & angulus radii AL cumcontingente, aequalis erit angulo BL cum eadem contingente ellipsem MLN; reflecteturigitur radius AL in B ; quod erat ostendendum.

Dioptrice.Sint radii e puncto A divergentes, in punctum B congregandi, lente, cujus vertexincidentiae M, emersionis O, radii ex A divergentes intra lentem paralleli mittantur, opeConoidis, vel Sphaeroidis, cujus vertex M, & intra lentem paralleli, in punctum Bcongregentur, ope Conoidis vel Sphaeroidis cujus vertex O; quod erat faciendum.

Corollarium.Ex hoc patet, si punctum radians fuerit prope A, ipsius radios congregari, in aliquod

punctum circiter B, ope lentis vel speculi supradicti.

[33]§4. Prop. 23.1. Prop. 23. Problem.

The rays that converge to one given point are made to diverge from another point inthe same medium. The vertices of the required lens or mirror are given too. It is alsonecessary for the points of convergence and divergence to be collinear with the vertices.

Reflection.The elliptic convex mirror MLN isdescribed with vertex M, foci A & B, andvertex M. Let the rays that converge tothe point A diverge from the point B . Isay that all the rays incident on thesurface of the mirror converge to thesame focus A, and are reflected by it todiverge from the other focus B. For letthe ray P converge to the focus A, whichtherefore will be in the same plane as theaxis of the mirror, and the surface ofreflection of the ray PL will be the ellipse


AAB OM

AM OB



MLN (from which the spheroid has been generated), by Prop. 10. This ray is produced toA, and BQ is drawn through L from B, and the straight line HLG is a tangent to theellipse at the point L. Therefore it is clear the angle ALG is equal to the angle BLH: fromthe vertex L the former will equal PLH and the latter QLG. The ray PL is thereforereflected into LQ. Q.e.d.

[34] Refraction.

The rays that converge to the point Aare made to diverge from the point B.This is achieved by a lens with vertexM for the incident rays and vertex Ofor the rays that emerge. The rays thatconverge to the [far focal point of thefirst surface] A are rendered parallelby the conoidal [Prop. 23 - Fig. 2] orspheroidal surface [Prop. 23 - Fig. 3]with vertex M, and sent parallel withinthe lens, subsequently the rays divergefrom the point B [the far focal pointassociated with the second surface], bymeans of the conoidal or spheroidalsurface with vertex O. (Prop's 17 &16.) Q.e.d.

Corollary.Thus it also follows, that rays

converging to some point near A willalso diverge from some point around B, with the help of the lens or mirror mentionedabove.

§4. Prop. 23.2. Note.

The diverging lens in Prop. 23 - Fig. 2 can be represented by the matrices M7 and M8 fora thin lens, giving the transfer matrix :

+=

+

+= 1

12

011

11

01

11

0178M

)n(an)n(ann

)n(aM . The focal length of the lens is thus

- a(n + 1)/2, while the focal lengths A & B associated with the hyperboloidal surfaces areat a distance a(n + 1) from their associated vertex. The effect of this transfer matrix is tochange the sign of the slope of an incident ray going towards A that appears to comefrom B. Similarly, the lens in Prop. 23 - Fig. 3 can be represented by the matrices M3 andM4, resulting in a transfer matrix:

+=

+

+= 1

12

01

12

011

1

0134M

)n(ann

)n(an

n)n(anM with the same change of direction of the

rays. The focal length associated with this lens is - a(n + 1)/2n.


P

L

MB

H

G

R

A


[33]§4. Prop. 23.2. Prop. 23. Problema.Radios ad unicum punctum datum convergentes, in eodem diaphano ab alio puncto datodivergere ; datis ; datis quoque lentis, vel speculi verticibus : Opertet tamen, ut punctumconvergentiae, punctum divergentiae, & vertices lentis, vel speculi, sint in eadem rectalinea.

Catoptrice.Sint radii ad punctum A convergentes, a puncto B divergendi, speculi cujus vertex M:focis A, B, & vertice M, describatur speculum ellipticum convexum MLN. Dico omnesRadios ad focum convergentes, & in speculi superficiem incidentes, in ipsa reflecti, & afoco B divergere. Sit enim ad focum A convergens radius PL qui igitur erit in eo planocum axe speculi, eritque Ellipsis MLN (ex qua genita est Sphaerois), superficiesReflectionis radii PL, qui producetur in A; & a B, per L, ducatur BQ, tangatque ellipsimLMN recta HLG in punctum L. Quoniam igitur angulus ALG , est aequalis angulo BLH;erit & priori ad verticem PLH, aequalis posteriori ad verticem QLG; reflectetur igitur PLin Q: quod ostendendum erat.

[34] Dioptrice.

Sint radii ad punctum A convergentes, a puncto B diverergendi, lente cujus vertexincidentiae M, emersionis O. Radii ad punctum A convergentes intra lentem parallelimittantur, ope conoidis vel sphaeroidis, cujus vertex M, & intra lentem paralleli, apuncto B divergentur, ope Conoidis vel Sphaeroidis cujus vertex O; quod erat faciendum.

Corollarium.Hinc etiam sequitur, radios convergentes ad aliquod punctum circiter A, etiam divergi abaliquo puncto circiter B, ope lentis vel speculi supradicti.

§4. Prop. 24.1. Prop. 24. Problem.The rays converging to one given point are made to converge to another given point of

the same medium with the help of a mirror orlens, the vertices of which are given too. It isnecessary morover that the two points ofconvergence are collinear with the verticesof the lens or mirror.

Reflection.A convex hyperboloidal mirror LMR isdescribed with foci A and B and vertex M.

[35] Let the rays that converge to some point Abe made to converge at another point B bymeans of the mirror with vertex M. But onlyif the vertex M is nearer to the point ofconvergence A than B. If in fact the vertexM is nearer to the point of convergence B,then a concave hyperboloid mirror isdescribed with foci A and B and vertex M.


A BOM

A BOM



[35]I say that all the rays converging to the focus A are reflected in the surface of the mirrorLMR, and are concurrent at the focus B. For let the ray PL be incident on the focus A,which therefore is in the same plane as the axis of the mirror. Hence the hyperbola LMRis the surface of reflection for the ray PL (from the revolution of which the mirror hasbeen generated) , by Prop. 10. The ray PL is produced to A, and from the point L the lineLB is drawn. Let the line HLG touch the hyperbola at the point L, and hence the angleALG i.e. HLP, is equal to the angle GLB. Therefore PL is reflected through the point B.Q.e.d.This conclusion can be shown for a hyperbolic concave mirror by a similar argument.

[36]

Refraction.Let the rays that converge to the point A be madeto converge in another point B, by the lens withincident vertex M and emergent vertex O. ByProp. 17, the rays converging to A are sentparallel on entering the lens, by means of theconoidal [Prop. 24 - Fig. 2] or spheroidal [Prop.24 - Fig. 3] surface with vertex M; within thelens the rays are parallel. Subsequently by Prop.14, the rays converge to the point B, by means ofthe second conoidal or spheroidal surface withvertex O. Q.e.d.

Corollary.Thus it follows, the rays converging to anotherpoint near A, also are to congregate in one point

around B, with the aid of the lens or the mirror mentioned above.

§4. Prop. 24.2. Note.The diverging lens in Prop. 24 - Fig. 2 can be represented by the matrices M5 and M7 fora thin lens, giving the transfer matrix :

−+

=

+

+−= 1

21

11

11

011

111

01

121

0175M

aa)n(n)n(nan

)n(aM . The focal length of the lens

is thus

−

+−

21

11

11

aa)n( , where a2 > a1, while the focal lengths A & B associated with the

hyperboloidal surfaces are at distances a1(n + 1) and a2(n + 1) from their associatedvertices M & O. The effect of this transfer matrix is to change the slope of an incident raygoing towards the focus A so that it appears to come from B. Similarly, the lens in Prop.24 - Fig. 3 can be represented by the matrices M6 and M8, resulting in the transfer matrix:


−+

=

+

+−= 1

21

11

11

01

111

011

121

0186M

aa)n(nn

)n(an)n(naM with the same change of

direction of the rays. The focal length associated with this lens is

−

+−

21

11

11

aa)n(n .

§4. Prop. 24.3. Prop. 24. Problema.Radios ad unicum punctum datum convergentes, in aliud punctum datum ejusdem

diaphani congregare ; datis quoque lentis, vel speculi verticibus : Opertet tamen, utpunctum convergentiae, punctum concursus, & vertices lentis, vel speculi, sint in eademrecta linea.

Catoptrice. Sint radii convergentes ad aliud punctum A, ad aliud punctum B congregandi, speculocujus vertex M. Focis A, & B, & vertice M, describatur speculum Hyperbolicumconvexum LMR; si modo vertex M, propior fuerit puncto convergentiae A: Si vero vertexM, propior fuerit puncto concursus B; focis A & B, & vertice M, describatur speculumhyperbolicum concavum.

[35]Dico omnes radios ad focum A convergentes, in superficie speculi LMR reflecti, & infocum B, congregari; sit enim radius PL ad focum A vergens, qui igitur est in eodemplano cum axe speculi, hyperbole igitur LMR (ex cujus revolutione genitum estspeculum), est superficies reflectionis radii PL, qui radius producetur in A; & a puncto Lducatur recta LB, tangatque Hyperbolam, recta HLG, in puncto L. angulus igitur angulusALG , hoc est HLP, est aequalis angulo GLB; reflectetur igitur PL in B: quod eratostendendum . Eodem quoque modo, demonstratur haec conclusio, in speculohyperbolico concavo.

Dioptrice.Sint radii ad punctum A convergentes, in aliud punctum B congregandi,

[36]lente cujus vertex incidentiae M, emersionis vero O. Radii ad A convergentes intralentem paralleli mittantur, ope conoidis vel sphaeroidis, cujus vertex M, & intra lentemparalleli, in punctum B congregentur , ope conoidis vel sphaeroidis, cujus vertex O; quoderat faciendum.

Corollarium.Hinc sequitur, radios convergentes ad aliquod punctum prope A, etiam congregari inunum puncto circiter B, ope lentis vel speculi supradicti.

§4. Prop. 25.1. Prop. 25. Problem.

The rays diverging from one given point are to diverge from another given point. Withthe vertices of the required lens or mirror given too: nevertheless it is necessary that thepoints of divergence and the vertices of the lens or mirror are collinear.

This problem is the converse of the antecedent, and is solved by the same method, asis shown from Prop. 9, Book 10, Witelo.


Corollary.From this problem it has also been clear that the rays diverging from another point

near B, can also to be made to diverge from one point near A, with the aid of the lens orthe mirror mentioned above.

[37]

Some Obvious Properties:1.

Through the determination of the properties which attend the resolution of theseproblems, it is clear in the case of refraction with small parts of conoids and spheroidsthus joined together to make lenses, that the axis of the lens is the common axis of bothparts, and that the vertices of the lens are the vertices of the parts.

2.Secondly, it is clear that for the resolution of all these problems for refraction and

corollaries, that the rays outside the lens are either parallel, or are converging to/diverging from a point, while within the lens they are always parallel.


Radios ab unico puncto dato divergentes, ab alio puncto duco divergere ; datis quoq;lentis, vel speculi verticibus : opertet tamen, ut divergentiae puncta, & vertices lentis, velspeculi, sint in eadem recta linea.

Hoc Problema est conversum antecedentis, eodemque modo solvitur, ut patet ex Prop.9. lib. 10, Vitellionis.

Corollarium.

Ex hoc problemate, etiam manifestum est ; radios divergentes ab uno puncto, circiterB, etiam divergi ab unico puncto circiter A, ope lentis vel speculi supradicti.

Manifestum 1.Per limitationes,quae resolutioni horum problematum inserviunt,

[37] Manifestum est in Dioptricis, Conoideon, & Sphaeroideon portiunculas ; ita in lentibusconnexas esse; ut axis lentis sit communis axis utriusq; portiunculae, & vertices lentisportiuncularum vertices.

Manifestum 2.Secundo, Manifestum est in omnibus horum problematum resolutionibus Diopticis, &Corollariis; radios extra lentem vel parallelos, vel ad unum punctum convergentes, vel abunico puncto divergentes; intra lentem semper esse parallelos.


L

B P

R

M

QCA


§4. Prop. 26.1. Prop. 26. Theorem.

If two rays are incident on the vertex of a mirror coplaner with the axis, by which theyare reflected, then the angle of incidence is equal to the angle of reflection for the rays.

Let the two rays BM & CM beincident in the same plane of themirror RML with vertex M, and therays are reflected at M to Q and P. Isay the angle BMC formed from theincident rays added together is equalto the angle PMQ formed from thereflected rays added together. Letthe axis of the mirror be AM, andsince BM & CM are coplaner withthe axis, & for these rays there isone and the same surface ofreflection, namely the figure RML,from which the mirror arises by rotation, from Prop. 10.

[38]The angle BMA is equal to the angle AMQ; & likewise the angle CMA is equal to theangle AMP; therefore BMA & CMA likewise added together without doubt BMC, isequal to the angle AMP. Likewise the angle AMQ , that is, to the angle PMQ; which itwas necessary to show.

§4. Prop. 26.2. Prop. 26. Theorema.

Si duo Radii, in eodem cum axe plano, incidant in speculi verticum, & ab eareflectantur; erit angulus a radiis incidentibus , aequalis angulo a radiis repercussis.

Sint duo radii BM, CM, in eodem cum axe plano, incidentes, in speculi RML, verticemM, & reflectentur in Q, & P : Dico angulum BMC, a radiis incidentibus comprehensum,esse aequalem angulo PMQ a radiis reflexis comprehenso. Sit speculi axis AM, &quoniam BM, CM sunt in eodem cum axe plano, erit illis una, & eadem superficiesreflectionis, figura nempe RML, ex cujus revolutione genitum est speculum ; eritq

[38]angulus BMA, aequalis angulo AMQ; & CMA, aequalis angulo AMP;igitur BMA &CMA simul, nimirum BMC, aequalis erit angulo AMP, & angulo AMQ simul, id estangulo PMQ; quod demonstrare oportuit.


AB

O

M

I

K

L

C

G H

N

E F

AB

P Q


§4. Prop. 27.1. Prop. 27. Theorem.Let there be two rays co-planar with the axis

outside the lens which concur and cross at the vertexof the lens. Indeed, there shall be two other rays,which are coplanar with the former rays, which areeach parallel to a previous ray crossing within thelens, and meeting at the other vertex of the lens, andemerging from that vertex. The angle subtended bythe two rays before entering will be equal to theangle subtended by the two final rays uponemerging.

Let the two rays AL & BL outside the lens incidentat L, and in the same plane as the axis of this lens, berefracted as the rays LM and LN in crossing the lensPMNQL . Let there be two other rays crossingparallel to the previous rays within the lens: GCparallel to LN, HC parallel to LM, and with all ofthese four rays arising from the one kind of

refracting surface. AL, BL are incident at the one vertex of the lens L, while HC & GCare refracted by the other vertex of the lens C to give the emerging rays CF & CE. Iassert that the angle ALB is equal to the angle ECF. For the lens PMNQL is drawn, withthe axis ILCK, through which (it is evident all of these rays are in the same plane as theaxis, by supposition) the surface of common refraction is drawn PMCNQHLG; indeedthis is the plane from which the lens has been generated by rotation, by Prop. 10. MLC& LCH will be the equal angles of incidence of the parallel rays ML & HC within thelens (plainly by supposing ML to be refracted in LB; for if BL is refracted in LM, thenconversely

[39]ML is refracted in LB). Therefore the refracted angles are equal too, which angles takenfrom or added to the same equal angles of incidence MLC & LCH, gives the equalangles ILB & ECK. In the same way, the equality of the angles ALI, KCF will beapproved. Therefore the whole angle ALB is equal to the whole angle ECF, which had tobe shown.

§4. Prop. 27.2. Note.At first sight, this theorem and the previous seem a little unusual. However, they form

part of the plan for locating images formed be reflection or refraction by lenses andmirrors. It is usual to draw the object with some degree of asymmetry, and hence theangles of incidence are from different sides of the axis, and of different sizes. As Gregoryhas pointed put, he intends to consider parallel rays within the lens, and it remains at thisstage for Gregory to connect up the rays that have been abruptly terminated at the in-going and out-going surfaces of the lens, using auxiliary axes.


§4. Prop. 27.2. Prop. 27. Theorema.Si duo radii in eodem cum axe plano,extra lentem, concurrant in vertice lentis, eam

transeuntes; duo vero alii, in eodem plano cum prioribus, intra lentem prioribus etiam intransitu paralleli, concurrans in altera lentis vertice, ex illa egredientes : anguluscomprehensus a duobus primis ante ingressum, aequalis erit angulo comprehenso aduobus postrimis post egressum.

Sint duo radii extra lentis AL, BL, in eodem tamen plano cum illius axe, lentem PMNQLtranseuntes, refracti in LN & LM; sintq; duo alii, intra lentem prioribus in transituParalleli, nempe GC ipsi LN, & HC ipsi LM ; omnibus hisce quatuor radiis in unorefractionis superficie existentibus; incidantq; AL, BL in unam lentis verticem L,egrediantur vero in HC, GC ex alia lentis vertice C, in CF, CE. Dico angulum ALB esseaequalem angulo ECF. Ducatur enim lens PMNQL, axis ILCK, per quem (quoniamomnes hi radii sunt in eodem plano cum axe ex suppositione) ducatur communissuperficies refractionis, PMCNQHLG, planum nempe, ex cujus revolutione grenita estlens ; eruntq; MLC, LCH anguli incidentiae radiorum parallelorum ML, HC (supposendonimium ML refringi in LB; nam si BL refringatur in LM, & e converso,

[39]ML refringetur in LB) aequales : & ideo refractiones quoq aequales, quae ab aequalibusincidentiae angulis MLC, LCH subtractae, vel iisdem additae, efficiunt angules ILB,ECK aequales. Eodem modo probabitur aequalitas angulorum ALI, KCF ; Totus igiturangulus ALB, est aequalis toti angulo ECF; quod demonstrare oportuit.


§5. Synopsis of Propositions 28 - 36: A miscellany of theorems relatingto the eye in optics, the intensity or power in a cone of rays, versions ofAlhazen's problem, image viewing, etc.

Prop. 28: The convergence or turning of the axes of the eyes is taken as a measure ofthe distance of an object from the viewer.

Prop. 29: As the previous theorem, but for single eye vision. Note §5.29.2 viewsthese theorems from a modern perspective.

Prop. 30: It is impossible for the eye to form an image from converging rays.Prop. 31: For an object subtending an angle at some distance from the eye along the

axis, the tangent of the half-angle varies inversely as the distance from the eye.Prop. 32: The previous theorem is extended to include lenses and mirrors. A

corollary identifies the equality of the angle of vision with the angle subtended by therays.

Prop. 32: The radiant power within a cone of rays from a point source is inproportion to the square of the radius of the base of the intercepting circle.

Prop. 33: The radiating powers within cones of rays proceeding from a point sourceare as the squares of the chords of the semi-angles of the bases of their radiating cones.Rays are made to converge to equal areas for comparison without weakening usingmirrors or lenses. The bases of the radiant cones are equidistant from the source. A firstcorollary extends the result to a radiating body, while a second corollary justifies the useof lenses or mirrors. We should note that this theorem is not the inverse square law, butmerely a statement about the amount of radiation proceeding into solid angles. However,the theorem is used when Gregory comes to consider lenses and mirrors of differingdiameters in telescopes, regarding their light-capturing abilities.

Prop. 34: To find the point of reflection from a given regular smooth surface when apoint is given on the incidence ray and also a point on the reflected ray. This is thefamous problem of Alhazen: Gregory's geometrical solution seems almost trivial;however, one has to find the point L where the ellipse and the curve have a commontangent. One can only presume that this be done by trial and error in a graphical way, andso cannot be considered as a proper solution. The fact that Gregory was to spend moretime on the problem when he became acquainted with analytical methods, as detailed onp. 437 of H. W. Turnbull's James Gregory Tercentenary Memorial Volume, wouldindicate that he was not convinced that this theorem actually solved the problem!

Prop. 35: For the given surface of a transparent medium, the point of refraction is tobe found where a point is given on the incident ray and another point is given on therefracted ray. This theorem is the refractive equivalent of the above mirror theorem, anddeserves the same comment, for both theorems are of the form of the minimum transittime for the ray connecting the given points - examples of Fermat's Principle. Suchtheorems in general require calculus to effect a solution, which was not available toGregory at this time.

Prop. 35: Tracing the rays from a point to a reflecting or refracting surface to the eye,and locating the position of the image, if it exists.



If some point is observed by a person with normal eyesight, then the position of thatpoint or its particular distance from the observer, is always estimated according to theturning of the axes of the eyes.

Let an observer's eyes be situated at the points M and N, and let A be any point in theirfield of view. If both the eyes are healthy and functioning properly, and there is nothingof interest to catch the observer's attention, then the axes of the eyes MO and NL arealways parallel in their natural resting position. But if both eye look at some point A at afinite distance, then the ocular axes MO and NL are turned a little towards MA and NA,

(for the clearest vision can only occur when the eyes are correctly orientated about theiraxes, which is in good agreement with experience, this indeed then is the most favouredexplanation). Again if the person's eyes should be looking at some nearby point H closerthan A, then the axes of the eyes MA and NA are turned more to give the directions MHand NH. It is hence possible to estimate the distance of an object from the viewer, a skillsharpened by the everyday experience of rotating the eyes about their axes.

Corollary.Thus it follows that the point observed in the field of view always lies at the

concurrence of the axes of the eyes.

[40]


Si ambobus oculis sanis, punctum quodlibet aspiciatur ; distantia illius ab oculissingulis, hoc est ipsius locus, semper aestimatur, secundum contortionem axiumoculorum.

Sit punctum quodlibet visibile A, oculorum centra MN. Si oculi M, N, nullo vitiolaborent, & oculatus cogitabundus haereat nihil intuens; tunc oculorum axes MO, NLsemper erunt parallel; nempe in situ suo naturali ; si vero ambo oculi intueantur punctumaliquod A e longinquo, paululum contorquentur axes oculares MO, NL in MA, NA,(quoniam in oculorum axibus solummodo sit visio distinctissima, ut experientia satisconstat, ratione vero firmissime demonstratur:), & si ambo oculi intueantur punctumaliquod propinquius H, adhunc magis contorquentur axes oculorum MA, NA, in MH &

A H

M

N

O

L



NH; atq; quotidiana experientia, majoris vel minoris contorsinis, natura est edocta, ut dedistantia seu loco visibilis, conjecturam facere possit.

Corollarium.Hinc sequitur, locum visibilis duobus oculis visu, semper esse in concursu axium

ocularium.[40]


If a point should be observed with only one eye, then the distance of the point from theeye, or the location of the point itself, is always estimated following the application of theeye to the field of view of that point.

A robust and healthy eye looking straight ahead and intercepting the parallel rays froma distant point is able to depict that object most clearly from the collected rays on theretina of the eye. If however a nearby point at some other angle is also shining and thesingle eye keeps its former position, then it is impossible for this point to be depictedclearly on the retina of the eye, as the ancient writers on optics show ad nauseam. Thereason for this happening, which is innate to nature, according to these ancient writers, isthat it is due to the fluids and moveable humors of the eye. These by some manner appearable to regulate nearby beams, in order that some may be depicted distinctly on the retinaof the eye. From the change of these humors, large or small, the mind is able to discern(taught by day to day experience), whither a radiating point is at a small or large distancefrom the eye. Thus an estimate can be made about the position of the point. In the samemanner that it can be said for short-sighted people, whose eyes can only view nearbyobjects.

Corollary 1.Thus it follows in the first place, that the position of a point seen with single eye

vision is always to be estimated from the point from which the rays emerge.

Corollary 2.Thus it follows secondly, if many points are sending out diverging rays in the same

manner as a single point but at different places, then no conclusion is drawn by the eye:regarding the location nor is there perfect vision.

Corollary 3.Thus it follows thirdly, if the rays of a single point are sent parallel to the eye, the

apparent location of that point is infinitely distant from the eye.

Corollary 4.Thus it follows fourthly, if the rays from a single point are to converge to another

point beyond the eye, it is not possible to assign the location of this point by looking at it(if anyone wanted to do so) , since the rays meet past the eye. An image of this kindgenerated by many points can conveniently be called an image behind the eye.


[40]

§5. Prop. 29.2. Note.Theorems 28 and 29 are a foray by Gregory into the worlds of binocular and

monocular vision . The accommodation of the eye was not understood at the time ofGregory's account, and the humors within the eye were thought to be involved in someunknown way. Although Gregory was correct in assigning a role to the converging of theeyes using binocular vision, the twisting of the eyes is limited to objects less than some200 metres away, the aim being to avoid double vision by superposing the images ratherthan to estimate distance. A number of contributing factors need to be considered indistance estimation, such as the degree of accommodation of the lens needed to form asharp image slightly different views and other visual clues. Stereoscopic vision arisesfrom the slightly different images presented by each eye for interpretation by the brain.

Predatory animals have evolved binocular vision to give them a beneficial 3-D view ofthe world, while animals preyed upon have evolved eyes placed on the sides of the headthat afford a much greater field of view for their safety, which gives them essentiallymonocular vision. Monocular vision can still use the cue of the degree of focusingrequired for a sharp image to judge distance, and also makes use of the other visual cuesavailable for binocular vision : the relative motion induced in the image by moving thehead around with respect to the far distance; the loss of definition or texture of distantobjects and their increasing dimness and smallness, slowness of movement across thevisual field, etc.


Si unico oculo, punctum quodlibet Asspiciatur; distantia puncti ab oculo, seu ipsiuslocus, semper aestimatur, secundum applicationem oculi, ad visionem illius puncti.

Oculus enim sanus, & bene valens, naturalem suum situm retinens, si puncti elonginquo radios intercipiat parallelos, his radiis collectis fortissime pingitur punctumvisibile longinquum in oculi retina. Si vero punctum radians, appropinquet, oculopristinum situm servante, impossibile est ut distincte pingitur radians in oculi retina, ut adnauseam demonstrant Scriptores Optici. Qua de causa, naturae insitum est, ut humoresoculi fluidos, & mobiles, aliquo modo disponat ad radiantia propinqua, ut distinctepingantur in oculi retina ; ex quorum humorum mutatione, magna vel parva, dignoscitnatura ( quotidianis experientiis edocta) num parvo, vel magno Intervallo, distet radiansab oculi ; atq; ita de illius loco conjecturam facit: eodem modo de Myopibus estdicendum, qui oculum habent, ad radiantia propinqua, a natura fabricatum.

Corollarium 1.Hinc sequitur primo, locum puncti visibilis uno oculi visi, semper aestimari in puncto,

ex quo proveniunt radii illius puncti visibilis.

Corollarium 2.Hinc sequitur secundo, si unius puncti radii e diversis punctis divergantur, nullam dari,

nec locum, nec visionem perfectam.


A

D C

B

E


Corollarium 3.Hinc sequitur tertio, si unius puncti radii in oculum paralleli mittantur, locum illius

puncti apparentem, infinite distare ab oculo.[41]

Corollarium 4.Hinc sequitur quarto, si radii unius puncti, ad aliud punctum post oculum

convergantur, nullum posse huic puncto locum assignari, nisi ( si quis voluerit) postoculum in radiorum concursu; Unde imago ex talibus punctis conflata, Commode vocaripotest, imago post oculum.


If the rays from an object point converge to an image point situated behind the eye,then it is impossible to form a distinct image.

Indeed the whole image is produced by the eyes, in order that it may view eitherremote distinct objects, which radiate almost parallel rays of light, or nearby objectswhich send out diverging rays. But the converging rays (which have been produced by amirror or lens in a manner not often found in nature) cannot be depicted clearly on theretina of the eye; since the crystalline humor gathers these rays to a point within thevitreous humour, and sends the diverging rays to the retina. A blurred image arises fromthese disordered rays, as the works of Kepler have shown.


Convergentibus unius puncti radiis, ad punctum situm post oculum, impossibile estfieri distinctam visionem.

Omnis enim oculis fabricatus est, ut aut remota distincta videat, quae radiant quasiparallele, aut propinqua, quae divergentes mittunt radios ; radiis autem convergentibus(qui ab arte, & non a natura ortum habent) nullius oculi retina distincte pingitur ; quoniamchrystallinus humor hos radios in punctum congregat, in humore vitreo, & disgregatos adretinam mittit, ex qua disgregatione oritur confusa visio : ut videri est apud Keplerum.


Consider the centre of the eye receding from or approaching in a straight line towardsan object in the field of view. The ratio of the tangent of the semi-angle of the object asseen from the first eye-object distance to the tangent of the semi-angle from the secondobject-distance varies reciprocally as the first eye-object distance to the second eye-object distance.

Let the line ABrepresent thevisible objectwhich is bisected


by C, and from C the normal CE is erected to the line AB, on which the centre of the eyeat the first position E is located ; and on the same line CE is located the position of thecentre of the eye for the second position D. I say, therefore, that the ratio CD to CE thusvaries reciprocally as the tangent of the angle CEB to the tangent of the angle CDB. Also(with the ray proceeding from the position CB) the ratio CD to CE varies directly as thetangent of the complement of angle CDB to the tangent of the complement of angle CEB.

[42]For the tangents of arcs are in reciprocal proportion to the tangents of the complements ofthe same arcs. Hence, as the ratio CD to CE, so the tangent of the angle CEB to thetangent of the angle CDB. Qed.


Si centrum oculi, directea visibili recedat, vel ad illiud accedat ; erit ut tangenssemianguli visorii unius stationis, ad tangentuem semianguli visorii alterius stationis ; itareciproce, distantia centri ocularis, a centro visibilis unius stationis, ad distantiam centriocularis, a centro visibilis alterius stationis.

Sit visibile recta AB , quae bisecetur in C, & a C erigatur ad rectam AB, normalis CE,in qua collocetur centrum oculare primae stationis E; & in eadem linea CE colloceturcentrum oculare secundae stationis D: Dico igitur ut CD, ad CE, ita reciproce tangensanguli CEB, ad tangentem anguli CDB. Erit enim ut CD, ad CE, ita (posito CB radio)tangens complem. anguli CDB, ad tangentem complementi anguli CEB;

[42]tangentes autem arcum sunt in reciproca proportione tangentium, complementum,eorumdem arcuum; erit igitur ut CD ad CE, ita tangens anguli CEB, ad tangentem anguliCDB ; quod demonstrare oportuit.


If any mirror or lens recedes from or proceeds towards an object point in a straightline, then the ratio of the tangent of the semi-angle of the lens for the first objectdistance, to the tangent of the same semi-angle of the second object distance thus variesreciprocally as the first object-lens distance to the second object-lens distance.

This theorem is demonstrated by the same method, from the preceding proposition.

Corollary.From these propositions, if the eye, lens, or mirror, should recede from or advance

towards the shining point source in a straight line, then the ratio will be as the the tangentof the visual semi-angle of the first station to the tangent of the visual semi-angle of thesecond position. This will be thus as the first tangent of the semi-angle of the rays to thesecond tangent of the semi-angle of the rays.


CB

G F

A

D E

H I


§5. Prop. 32.2. Prop. 32. Theorema.Si speculum quodlibet, vel lens, directa a puncto radioso recedat, vel ad illud accedat;

erit ut tangens semianguli lentis, vel speculi, e puncto radioso unius stationis, adtangentem semianguli ejusdem alterius stationis ; ita reciproce distantia puncti a centrolentis unius stationis, ad distantiam puncti a centro lentis alterius stationis.

Hoc Theorema demonstratur eodem modo, quo Antecedens.

Corollarium.Ex hisce, si oculus, lens vel speculum, directe a radioso recedat, vel ad illud accedat ;

erit ut tangens semianguli visorii, unius stationis, ad tangentem semianguli visorii ,alterius stationis ; ita tangens semianguli radiosi, ejusdem prioris stationis, ad tangentemsemiangluli radiosi, alterius stationis.


The radiating powers within cones of rays, either for illumination or burning, (wherethe rays are made to converge to equal areas for comparison, without being weakened),are as the squares of the chords of the semi-angles of the bases of their radiating cones.

Let the point A be radiating equally on every side toinfinity, and let there be two cones of radiation DAE,BAC, of which the rays are gathered together in equalintervals GF and HI, without being weakened. I say thatthe strength of the illumination or conflagration1 in GF,compared to the strength of the illumination orconflagration in HI, is in the duplicate, i. e. square, ratioof the chords of half the angle BAC to the chord of halfthe angle DAE. Indeed as often as the base area of thecone DAE is contained in the base area of the coneBAC, so an equal number of times is the radiant powerof the rays of the cone DAE contained in the radiantpower of the rays of the cone BAC. In the same way,whatever the proportion is between the bases, so it is thesame with the radiant powers. That is, with the radiantcones intercepted by ideal lenses DE and BC radiatinginto equal areas GF and HI ; but the bases are equal tothe circles of which the radii are chords of the semi-angles of the radiations, as shown byArchimedes Book 1, Sphere & Cylinder,Prop. 40. The base areas therefore are as thesquare ratio of these chords2. The radiant powers therefore, either for burning, or forillumination, are in the same ratio. Q.e.d. This theorem can be demonstrated in the sameway, even if the areas HI and GF are reduced to bare points.


Corollary 1.

It follows from this theorem, if an extended radiating body is located in the vicinity ofA, and its rays converge in the equal areas HI and GF, then the strengths of all these raysfor combustion or illumination are in the square ratio of the preceding chords. For theradiating body A can be divided up into its radiating points, and thus because these areproportional magnitudes. The square of the chord of the semi-angle DAE is to the squareof the chord of the semi-angle BAC in the same ratio as the power of the body Aradiating into the area HI, (for illumination or conflagration) is to the power of the bodyA radiating into the area GF, (for we suppose that the cones of all the radiating pointsconverge in HI,

[44] with the rays having equal radiating angles: and we suppose the same for the conesgathered together to GF). Thus the ratio of the powers for one of the first points to one ofthe second points is indeed as the square of the chord of the half-angle DAE to the squareof the chord of the half-angle BAC. Thus, this shall be the ratio for the power ofillumination or burning of the whole radiating body A, for all of the first points summedtogether radiating into the first area HI, to the power of all the second points summedsimilarly radiating into the second area GF.

Corollary 2.

Thus it follows secondly (if the radiating body should be at A, and with the help ofmirrors or lenses DE and BC, the rays of single radiating points incident upon the lensesor mirrors converge to points of the segment HI, and likewise to points of the segmentGF.) The ratio of the radiant powers for illumination or burning applied to the segmentsHI and GF shall be as the squares of the chords of the semi-angles DAE and BACrespectively. Indeed all these radiant magnitudes are in proportion to the squares of thechords of the semi-angles DAE and BAC, illuminating points in the one surface HI andpoints of the other surface GF. Whenever points of the first and second areas are presentin equal amounts, the ratio of the power for a point of the first area to a point of thesecond area is as the square of the chords of half the angles DAE and BAC. The ratio forillumination or burning will be the same on adding the radiant powers for all the firstpoints in HI, and similarly for all the second points in the segment GF.

§5. Prop. 33.2. Notes.

Gregory compares the radiant powers using burning glasses of differing sizes,converging the rays to equal areas. The theory of heat was almost non-existent at thetime, and so when Gregory talks about burning we assume he means the heating effect ofradiation.



Vires Conorum radiosorum, in illustrando, vel comburendo (radiis nimirum in spatiaaequalia absq; debilitatione congregatis) sunt in duplicata ratione Chordarum, suorumsemiangulorum radiosorum.

Sit punctum, undiq; & aequaliter in infinitum radians A, sintq; coni duo radiosi DAE,BAC, quorumradii congregentur in spatia aequalis GF & HI, absq; debilitatione: Dicovim illustrationis, vel conflagrationis in GF, ad vim illustrationis, vel conflagrationis inHI, esse in duplicata ratione chordae, semissis anguli BAC, ad chordam semissis anguliDAE. Quoties enim continetur basis coni DAE, in base coni BAC, toties etiam contineturefficacia radiorum coni DAE, in efficacia radiorum coni BAC; & eodem modo, quaeproportio est inter bases, eadem est & inter vires, conis scilicet in aequalia spatiacongregatis ; Bases autem sunt aequales circulis, quorum radii snt chordaesemiangulorum radiosorum, ut demonstrat Archimedes Lib. 1, de sphae. & Cylind. Prop.40. Bases igitur sunt in duplicata ratione earum chordarum;vires ergo, sive incomburendo, sive in illustrando, sunt in eadem rarione; quod demonstrare oportuit.Eodem modo poterit hoc Theorema demonstrare, etiam spatia HI & GF sint mera puncta.

Corollarium 1.Ex hocTheoremate sequitur, si corpus radiosum fuerit in A, & ejus radii congregentur

in spatia aequalia HI, GF; eorum vires in comburendo, vel illustrando, esse in duplicataratione praedictarum chordarum : Dividatur enim corpus radiosum A, in sua punctaradiantia; quoniam itaq; hae sunt magnitudines proportionales; nempe quadratum chordaesemissis anguli DAE; quadratum chordae semissis anguli BAC; vis in illustrando; velcomburendo uniuscujusq; puncti corporis radiosi A, in spatia HI; & vis in illustrando, velcomburendo unius cujusq; puncti copporis radiosi A in spatio GF (supponimus enimomnium punctorum conos radiosos in HI,

[44] congregatos, aequales habere angulos radiosos; idem supponimus in conis ad GFcongregatis) : erit ut una antecedentium, nempe quadratum chordae semissis anguli DAE;ad unam consequentium, nempe quadratum chordae semissis anguli BAC; ita omnesantecedentes, nempe vires in illustrando vel comburendo totius corporis radiosi A, inspatio HI; ad omnes consequentes, nempe vires in illustrando, vel comburendo, totiuscorporis radiosi A, in spatio GF.

Corollarium 2.Hinc sequitur secundo (si corpus radiosum fuerit in A, & ope speculorum, vel lentium

DE, BC, congregentur radii singulorum radiosi punctorum, in lentes, vel speculaincidentes, in puncta spatii HI, & in puncta spatii GF) Vim in illustrando, velcomburendo spatii HI, esse ad vim in illustrando, vel comburendo spatii GF, utquadratum chordae semissis anguli DAE, ad quadratum chordae semissis anguli BAC;sunt enim omnes hae magnitudines proportionales, quadratum chordae semissis anguliDAE, quadratum chordae semissis anguli BAC, illustratio unius-cujusq; puncti in spatioHI, & illustratio unius-cujusq puncti in spatio GF: Cumq; antecedentes, & consequentessint numero aequales, propter aequalitatem spatiorum HI, GF; erit ut una antecedentium,nempe quadratum chordae semissis anguli DAE; ad unam consequentium, nempe


A BC

D

E

P

QM L N


quadratum chordae semissis anguli BAC; ita omnes antecedentes, nempe tota illustratio,vel confligratio in spatio HI; ad omnes consequentes, nempe totam illustrationem, velconflagrationem, in spatio GF.

§5. Prop. 34.1. Prop. 34. Problem.For a given regular smooth surface, to find the point of reflection from the surface

when an incident ray passes through a given point, and the reflected ray passes throughanother given point.

Let ACB be the given smoothregular surface; and let the points D,E be given; from the foci D, E aspheroid PLQ is described, touchingthe surface ACB in the point L: I saythat L is the point sought. Through Lis drawn the plane MLN, andtouching the other surface; &through the points DEL

[45]the other plane is drawn, cutting thesame spheroid through the axis; andwith a common section making theellipse LPQ, from which thespheroid is generated, cutting the plane MLN normally in the line LMN too, whichtouches the ellipse PLQ in the point L. Therefore the ellipse PLQ and the line LMN are inthe sought plane of reflection, and the lines drawn from the foci of the ellipse make theangles DLM, ELN equal: evidently the angles of incidence and reflection. L is thereforethe point of incidence Q.e.d.

§5. Prop. 34.2. Prop. 34. Problema.Data regulari politi superficie, & datis, uno puncto in linea incidentiae, & altero in

linea reflectionis ; punctum reflectionis invenire.

Sit data regularis politi superficies ACB ; sintq; data puncta D, E; focis D, Edescribatur sphaerois PLQ, tangens superficiem ACB in puncto L: Dico L esse punctumquaesitum. Per L ducatur planum MLN, utramq; superficiem tangens; & per puncta DEL

[45]ducatur aliud planum, secans sphaeroidem per axem; & communi sectione faciensellipsen LPQ, ex qua genera est sphaerois, secans quoq; planum MLN normaliter, inlinea LMN, quae ellipsen PLQ tangit in puncto L. Sunt igitur ellipsis PLQ, & linea LMN,in Plano reflectionis quaesito; & lineae ex ellipseos focis ductae, faciunt angulos DLM,ELN, nimirum incidentiae, & reflectionis, aequales; est igitur L punctum incidentiae,quod ostendendum erat.


A B

D

E

P

Q

M L N



For the given surface of a transparent medium, the point of refraction is to be foundwhere a point is given on the incident ray and another point is given on the refracted ray.

Let the surface of regular density be AB, and letthe given point on the line of incidence be E, and onthe line of refraction : from the more distant focusE, a spheroid QLP of the dense medium isdescribed, touching the dense surface in the point L,thus as the line DL is parallel to the axis of thespheroid: I say that L is the sought point ofrefraction. Through L is drawn the plane MLNtangent to the other surface; and through the pointsD, L, and P another plane is drawn cutting the axisof the spheroid and making a common ellipticsection with the dense medium normal to the planeMLN, which therefore is the surface of refraction.Therefore the ray DL is parallel to the axis QP,incident in that less dense mediun and refracted

through E in the denser medium. In the opposite direction, the ray EL passing through thepoint L is refracted through D. Therefore the point L is the point of refraction sought,Q.e.d.


Datis regulari densi superficie, & dats, uno puncto in linea incidentiae, & altero inlinea refractionis ; punctum refractionis invenire.

Sit regularis densi superficies AB, sitq; datum punctum in linea incidentiae E, & inlinea refractionis D : foco remotiore E, describatur sphaerois densitatis QLP, tangenssuperficiem densi AB in puncto L, ita ut recta DL sit axi sphaeroidis QP parallela: Dico Lesse punctum refractionis quaesitum. Per L ducatur planum MLN utramq; superficiemtangens ; perq; puncta D, L, P ducatur aliud planum, quod transibit per axemsphaeroidis,faciens communem sectionem ellipsem densitatis, ad Planum MLN rectam,quae igitur est superficies refractionis ; & ideo radius DL axi QP parallelos, in eamincidens, refringetur in E; & e contra EL ex puncto L refrangitur in D. Punctum igitur Lest punctum refractionis quaesitum, quod ostendendum erat.


For a given regular smooth reflecting surface, or the smooth surface of a refractingmedium, and with a given visible point as the object and the eye ; it is required to find thelocation of the visible image of the point formed either by reflection or refraction(provided the image point is distinct ).


A

B

D

E

F

L


Let the regular polished surface be EDF, A shall be the eye, and B the visible pointobject. Thus with the point B positioned in theline of an incidence ray, and with individualpoints of the pupil of the eye A presentseparately in the lines of reflection, the imagepoint of the reflected rays can be found. All thelines of reflection are drawn from the points ofthe pupil through their points of reflection,which concurr in L, and - provided this happens- the location of the image point B appears tothe eye. If however the rays do not concurr inone point, then nothing distinct will be seen,and no point that determines the position of theimage of the visible point B is given.

[47]

All of which are apparent from the corollaries to Prop. 29 of this work: in the same way,the locus of the image can be found for refraction.


Data regulari, densi, aut politi superficie ; & datis, puncto visibili, & oculo ; locumimaginis puncti visibilis reflexum, vel refractum (si modo detur distinctus) invenire.

Sit regularis politi superficies EDF, sit oculus A, & punctum visibile B: positis itaq;puncto B in linea incidentiae, & singulis punctis pupillae oculi A seorsum in lineisreflectionum, inveniantur puncta reflectionum : & a punctis pupilae, per sua reflectionumpuncta, ducantur omnes lineae reflectionum, in quarum concursu nempe L, (si modoconcurrant) erit locus apparens imaginis puncti B.

[47]Si vero in uno puncto non concurrant, nullus dabitur, distinctus, & determinatus locusimaginis, puncti visibilis B. Quae omnia patent ex corollariis ad 29 Prop. hujus : eodemquoque modo, inveniendus est locus imaginis in Dioptricis.


§6. Synopsis of Propositions 37 - 43:

The following six theorems prepare the way for non-axial rays to be used in paraxialray image formation. Prop. 37 and Prop. 38 deal with parallel rays incident at a smallangle to the optical axis of a hyperboloid and a spheroid where they are refracted to afocus at some point a little off the optical axis and nearly in the focal plane, while Prop.39 deals with the similar case of reflection for a parabolic mirror. Prop. 40 and Prop. 41are concerned with the almost equality of the angles subtended at the far focus and apoint close to it by two nearby points near the vertex of a hyperboloid or of spheroid.Prop. 42 and Prop. 43 are concerned with convex hyperboloidal and concave spheroidalmirrors, where a point in one focal plane near the focus is reflected to a point in the otherfocal plane.

Until this point, only axial rays have been used in the Promata in image formation,in a rather restricted sense. This was the stage in the development, indicated in thepreface, where Gregory felt at a loss to know how to proceed further, and he admits tohaving received some help from his brother David, also a mathematician. The use ofparaxial rays was of course a great step forward, when one considers the muted responseto lenses adopted by writers on optics at the time. Consider, for example, the writings ofMersenne (The version of his Cogitata & Geometriae published in 1644 represent hiscollected works on mathematics and physics apart from the letters), which set out whatwas known at the time about optics. The only major advance not known to Gregory wasthe law of refraction in a more general form than he had found himself. This is set out forexample by Mersenne in his Ballistica, p. 79, following Descartes' Dioptique. Mersennealso includes two extra optical works at the end of his Universae Geometriae, followingmore of his own thoughts on optics: the first is a short previously unpublished tract byWarner, who gives the law of refraction using a different kind of diagram; and Hobbes'sideas are presented, which again follow Descartes. Warner had of course been presentwhen Thomas Harriot had made the original deduction of the law of refraction fromexperiments performed around 1600 at Scion House - which Harriot only circulatedamongst his friends and never published. Gregory professes his unfortunate ignorance ofDescartes' work in the preface to the Promoto. We should also mention the sterling workdone by Fermat in deriving the laws of both reflection and refraction from a least timeprinciple at this time.

Prop. 37: If a visible object sends parallel rays through a hyperboloidal surface into adense medium, then the apices of the pencils of rays from individual paraxial pointsources lie very close to a single plane.

We are to understand the visible object to be either an extended distant object, such asthe canopy of the stars, or the intermediate image formed by a previous lens. The idea ofdividing an extended object up into a number of point sources is used. Gregory does notpursue the obvious question of relating the paraxial angle to the distance of the point ofconvergence of the paraxial rays from the focal plane, which we consider briefly in §6.Prop. 37.2. .

Prop. 38: If a light source sends parallel rays into a spheroid of a denser medium ,then the apices of the pencils of individual point sources are almost coplanar.


Prop. 39: If a light source sends parallel rays to the concave surface of a parabolicmirror, then the apices of the pencils from individual point sources are almost coplanar.

Prop. 40: From one focus of an ellipse, a short line is drawn at right angles to theaxis, and two lines are drawn from some point on this line. One line goes to the vertex ofthe ellipse and the other is truly perpendicular to the circumference of the ellipse. Also,from the preceeding focus of the ellipse, another line is drawn meeting the perpendicularin the circumference. In this case, the angle between the first two lines is almost equal tothe angle between the third line and the axis of the ellipse.

Prop. 41: A short line is drawn at right angles to the axis of a hyperbola from onefocus. Two lines are drawn from some point on this line: one line goes to the vertex ofthe hyperbola and the other is truly perpendicular to the circumference of the hyperbola.From the preceeding focus of the hyperbola, another line is drawn meeting theperpendicular in the circumference. In this case, the angle between the first two lines isalmost equal to the angle between the third line and the axis of the hyperbola.

Prop. 42: If rays are sent from a planar object present in the focal plane of a concaveelliptic mirror (of which the normal is the axis of the mirror) to the mirror surface, thenthe apices of the cones formed by the reflected rays from individual points are almost co-planar.

Prop. 43: If rays are sent from a planar object present in the focal plane of a convexhyperboloidal mirror (of which the normal is the axis of the mirror) to the mirror surface,then the apices of the cones formed by the reflected rays from individual points arealmost co-planar.


LM

NH

OR

Q

C

F G P

T


A B


If a visible object sends parallel rays into a dense medium through a hyperboloidalsurface, then the apices of the pencils of rays from individual paraxial point sources arealmost coplanar.

Let ACB be the medium with the hyperboloidal or conoidal surface, with the axis GCproduced to the exterior focus M, and through M is drawn the focal plane LMN normalto GM. I say that the apices of all the pencils coming from the converging parallel rays in

the conoid lie very close to the focalplane LMN. For let the axis of one suchpencil GRH be incident normally on thesurface of the conoid at the point R. Aplane is drawn through the axis of theconoid and the axis of the pencil (whichlie in the same plane), giving a commonhyperbolic section ACRB with theconoid, from which the conoid itself isgenerated. Now in the plane containingthe line LMN and the point R, a lineQRO is drawn tangent to the hyperbola,and it crosses to meet the axis in O.Another ray PR is drawn parallel to theaxis of the hyperbola, and this ray is

refracted in the exterior focus of the hyperbola M, which is the apex of the pencil of raysparallel to the axis. A line FC is drawn to meet the vertex of the hyperbola,

[48]which is parallel to GH itself, which is refracted to the apex H of the pencil of rays whichare all parallel to GH. Since the angles of incidence GRP and GCF are equal, then boththe angles of refraction RMG and GHC are also equal. The triangles GCH and GRM aresimilar, hence:GR : GM : : GC : GH.Again, from the similar triangles GRO and GMT:GR : GM : : GO : GT.Therefore GC : GO : : GH : GT.

If the point R is near the vertex of the hyperbola C, (concerning the pencils of whichwe have so much to say) GC will be nearly equal to GO: (as those who have been wellversed in conics are fully aware) and therefore GT will be itself approximately equal toGH; therefore the apex of the pencil H approaches as near as possible to the plane surfaceLMTN. Q.e.d.


L

M

N

H

O

Q

A

C

F

G

P

T


y

x

(an,0)ααn

α

(-a,0)

R

O'

αn

Z

α R'H'

B


α∆(n - 1)a2 − ∆

AO

(x , y )1 1

y1

1a /x2

B

CD

E

§6. Prop. 37.2. Notes.

I. Initially, we show that for points on theleft- hand branch of the hyperbola, close tothe vertex D in Fig. 2, the y co-ordinate canbe found in terms of some sagittal distance∆ > 0 associated with the point B, which hasco-ordinates (x1, y1). BE is the tangent at B,while A has the x co-ordinate n2x1,where x1 = -a - ∆. Now, from the right-angledsimilar triangles ABC & BCE, BC2 = AC.CE,and on inserting the various lengths in termsof co-ordinates, keeping only first orderterms in ∆ , we obtain : ∆a).n(y 2122

1 −= .We may observe that y1 ~ ∆1/2, which is an order of magnitude greater than ∆ , a resultgiving the extent of the focusing action of the surface for paraxial rays . This result canalso be obtained by direct substitution into the equation for the hyperbola.

II. If the point R with co-ordinates (x1, y1) is located near the x axis with origin at O' infig. 3 on the left-hand branch ofthe regular hyperbolax2/a2 - y2/b2 = 1, (reverting toGregory's notation), then theequation of the tangent QRO isgiven by:xx1/a2 - yy1/b2 = 1. This line cutsthe x-axis at the point O (a2/x1, 0),and the lengthOC = a (1 + a/x1), wherex1 < 0. Also, the normal GH tothe hyperbola through R (x1, y1)has the equation:y - y1 = α.(x - x1), ora2 x/x1 + b2y/y1 = a2 + b2 ingeneral. From the latter equation,when y = 0 we have xG = (1 +b2/a2). x1 = n2.x1, the x co-ordinateof G.

According to §0.2 Fig. 1(c), there is a constant path difference between any ray RMparallel to the axis and the equivalent ray CM from the same wave front through the apexC: a requisite for focusing. We now consider the paraxial ray case, and in particular thepath difference (p.d.) for the rays arriving at H that start from the wave front ZC in fig. 3.One ray GH lies along the auxiliary axis, while another lies along CH : the p.d. = (n.ZR+ RH) - CH. We now proceed to evaluate these quantities.


Applying the sine rule to triangle ACH, we find that ∆12

11

21

−−−

++== nn

naxn a)n(CH

Now, ZR = GR - GZ ~ (GO - GC) where cos α ~ 1.)n(a)n()a/(a)a(nx/axnGO 111 222

12

12 ++−=−−+=+−= ∆∆∆ .

GC = -n2.x1 - a = (n2-1)a +n2∆.Hence ZR = a2/x1 + a ~ ∆, a result we might have guessed.Again, RH = GH - GR ~ GH - GO. From the sine rule applied to triangle GHO:

GH/n = GC/(n - 1); hence, [ ] ∆∆ 132

1 1−−

++=−+= nn

nn a)n(na)a(nGH ,

where GM = an(n + 1).Hence, RH = GH - GO

)nn(a)n()n(a)n(a)n(n nnn 11111 2

122

13

+−++=+−−−++=−−∆∆∆ .

The above p.d. becomes:

.)n()nn(a)n(na)n(CHRHZR.n nnn ∆∆∆ ∆ 1111 2

112

−=+−++++−+−=−+−−

Hence, to the first order appriximation considered, the path difference is proportional to∆, and it follows that for a point x1, with say ∆ < λ/4 of C, for a given wavelength λ,reasonable focusing of the beam will occur at H. Following part I of this note, anequivalent variation for y can be established. The distance HT of the paraxial focal pointH from the focal plane NM is given by HT = CO.GT/GO from Prop. 37 ; CO = ∆ ; GT =an(n + 1) + O (∆) ; GO = (n2 - 1)a + O (∆) : giving HT = ∆

∆

∆11212

1−++−

+n

n)n(a)n(

)n(an ~ ,

placing further constraints on the magnitude of ∆ for focusing of paraxial rays in theoriginal plane.

Finally, we note that the observable quantity α is related to ∆ according to :

)n(ax)n(

y ~12

2

1121

−−= ∆α

Similar arguments can be given for the two following theorems.


Si visibile mittat radios parallelos, per Conoidem densitatis ; erunt apicespencilllorum, singulorum visibilis punctorum, in uno quam proxime plano.

Sit Conois densitatis ACB, cujus axis GC producetur ad focum exteriorem M; & perM ducatur planum LMN, cui normalis sit GM; dico omnes apices pencillorum, ex radiisin Conoide parallelis, genitorum, cadere quam proxime in planum LMN. Sit enim uniuspencili axis GRH, normaliter cadens in conoidis superficiem in puncto R, & per axemconoidis, & axem pencilli (qui sunt in eodem plano) ducatur planum, faciens cumconoide, communem sectionem hyperbolam ACRB, ex qua generatur, cum plano veroLMN, rectam LMN; & per punctum R, ducatur una linea tangens hyperbolam, nempe


LM

N

H

OR

Q

A B

C

F

G

P

T

KI


QRO, axi occurrens in O, alia vero PR, axi hyperbolae parallela, quae refringitur infocum hyperbolae exteriorem M, seu apicem pencilli radiorum

[48]axi parallelorum ; ducaturque ad vertice hyperbolae linea FC, ipsi GH parallela, qui ideorefringetur H, apicem pencilli radiorum, ipsi GH parallelorum: & quoniam angulusincidentiae GRP, est aequalis angulo incidentiae GCF, erit & angulus refractionis RMGangulo refractionis GHC aequalis {cor. 14. Hujus}: eruntq; triangla GCH, GRM similia,& ut

GR : GM : : GC : GH;Et ob similitudinem triangulorum GRO, GMT, ut

GR : GM : : GO : GT; ergo ut GC : GO : : GH : GT;Si vero punctum R non multum recedat a vertice Hyperbolae C, (de quibus pencilllis nostantum loquimur) GC ferme aequalis erit ipsi GO : (ut satis norunt qui in conicis versatisunt) ergo & GT, quam proxime aequalis erit ipsi GH; apex igitur pencilli H, quamproxime incidet in superficiem planum LMTN; quod demonstrare oportuit.

[49]§6. Prop. 38.1. Prop. 38. Theorem.

If a light source sends parallel rays into a spheroid of a denser medium , then theapices of the pencils of individual point sources are almost coplanar.

Let the dense spheroid be ARCB, the axisGC of which is produced to the further focusM; through M is drawn the plane LMTNwhich is normal to the axis CGM. I say thatall the apices of the pencils arising from theparallel rays incident on the spheroid fallclose to the plane LMTN. Let the axis of onepencil be FR, incident normally on thesurface of the spheroid at the point R, and aplane is drawn through both the main axis ofthe spheroid and the axis of the pencil,making a common elliptic section ACRBwith the spheroid, from which the spheroidis generated. With the normal plane LMTNthere is associated the line LMTN. Throughthe point R a line QRO is drawn tangent to the ellipse cutting the axis in O. Another lineKR, which is parallel to the axis of the ellipse, is refracted through the focus M: the apexof the pencil of rays parallel to the axis. A line IC is drawn through the vertex of theellipse C parallel to the ray FR itself, which therefore is refracted through H, the apex ofthe pencil of rays parallel to FR. Since the angles of incidence ICP and FRK are equal, itfollows that both the angles of refraction RHC and RMC are equal [Note: we would nowcall these latter angles the angles of deviation rather than refraction]. The triangles GCH,GRM are similar:

[50]


hence GR : GM : : GO : GH;And from the similitude of the triangles GRO & GMT:

GR : GM : : GO : GT;therefore: GC : GO : : GH : GT.If indeed the point R is near the vertex of the ellipse C, (concerning the pencils of whichwe have so much to say), then GC will be nearly equal to GH itself, and therefore GTwill be itself approximately equal to GH. Therefore the apex of the pencil H approachesas near as possible to the plane surface LMTN. Qed.

[49]


Si visibile mittat radios parallelos, in sphaeroidem densitatis ; erunt apicespencillorum, singulorum visibilis punctorum, in uno quam proxime plano.

Sit Sphaerois densitatis ARCB, cujus axis GC producetur ad focum remotiorem M; &per M ducatur planum LMTN, cui normalis sit axis CGM; dico omnes apicespencillorum, ex radiis in sphaeroidem parallele incidentibus, ortorum, cadere quamproxime proxime in planum LMTN. Sit enim unius pencili axis FR, normaliter incidensin sphaeroidis superficiem in puncto R, & per axem sphaeroidis & pencilli ducaturplanum, faciens cum sphaeroide communem sectionem ellipsen ACRB, ex qua generatur,cum plano vero LMTN, rectam LMTN; & per punctum R, ducatur una linea QROtangens ellipsem, & axi occurrens in O, alia vero KR, axi ellipseos parallela, quae igiturrefringitur in focum M, seu apicem pencilli radiorumaxi parallelorum ; ducaturq; per verticem ellipseos C, recta IC, ipsi FR, parallela, quaeideo refringetur in H, apicem pencilli, Radiorum ipsi FR parallelorum: et quoniamangulus incidentiae ICP, est aequalis angulo incidentiae FRK, erit & angulus refractionisRHC angulo refractionis RMC aequalis {cor. 14. Hujus}: eruntq; triangla GCH, GRMsimilia, & ut

[50]GR : GM : : GO : GT

Et ob similitudenem triangulorum GRO, GMT, erit utGR : GM : : GO : GT;

ergo ut GC : GO : : GH : GT;Si vero punctum R non recedat multum a vertice ellipseos C, (de quibus pencillis nostantum loquimur) GC ferme aequalis erit ipsi GH : apex igitur pencilli H, quam proximeincidit in superficiem planum LMTN; quod demonstrandum erat.


If a light source sends parallel rays to the concave surface of a parabolic mirror, thenthe apices of the pencils from individual point sources are almost coplanar.

Let ACB be a concave parabolic mirror with axis GC and focus M. Through the focus Ma plane is drawn normal to the axis GMC. I say that all apices of pencils arising from the


incident parallel rays fall quite near the plane LMTN drawn through the focus. For let GRbe the axis of one pencil, incident normally on the concavity of the mirror surface at thepoint R, and through the axis of the mirror a plane is drawn, making a common parabolicsection ACRB with the mirror, from the rotation of which section the mirror is generated.Indeed with the plane LMTN, the line LMTN is also generated; & through the point R aline QRO is drawn tangent to the parabola, meeting the axis in O. Also, to the same pointR, another line PR is drawn parallel to the axis GC, which therefore is reflected in thefocus M of the parabola, or the apex of the pencil of the rays parallel to the axis. The lineFC is drawn through the vertex C of the mirror, parallel to GR, which therefore isreflected in H, which is the apex of the pencil of the rays which are parallel to GR. Thus,since the angles of incidence FCG and GRP are equal, both the angles of reflection GRMand GCH are also equal. The triangles GCH and GRM are similar, and

[51]

GR : GM : : GC : GH;From the simililarity of the triangles GRO & GMT:

GR : GM : : GO : GT;therefore GC : GO : : GH : GT.

If the point R in near the vertex of the parabola C, (concerning the pencils of which wehave so much to say) GC will be nearly equal to GO: and therefore GH is approximatelyequal to GT. Therefore the apex of the pencil H approaches as near as possible to theplane surface LMTN. Which was to be show.


Si visibile mittat radios parallelos, in speculum conum parabolicum, erunt apicespencillorum, singulorum visibilis punctorum, in uno quam proxime plano.

Sit speculum cavum parabolicum ARCB, cujus axis GC, focus M; & per focum Mducatur planum cui perpendicularis sit axis GMC; dico apices omnium pencillorum, exradiis parallele incidentibus ortorum, cadere quam proxime proxime in planum LMTN,per focum ductum. Sit enim unius pencili axis GR, perpendiculariter incidens in cavemspeculi superficiem in puncto R, per quem, & axem speculi ducatur planum, faciens

AR

CO

L M N

P

T

H

F G

BQ



A

C

G

TM

O

Q

R


communem sectionem cum speculo parabolam ACRB, cujus revolutione generaturspeculum; cum plano vero LMTN, rectam LMTN; & per punctum R, ducatur rectatangens parabolam, & axi occurrens in O, nempe QRO; & ad idem punctum R, ducaturalia recta PR, axi GC parallela, quae igitur reflectetur in focum parabolam M, seu apicempencilli radiorum axi parallelorum ; ducaturq; ad verticem speculi C, recta FC, ipsi GRparallela, quae ideo reflectetur in H, apicem pencilli radiorum, ipsi GR parallelorum:quoniam itaq; angulus incidentiae FCG, est aequalis angulo incidentiae GRP, erit quoqueangulus reflexus GRM aequalis angulo reflexo GCH {cor. 18. Hujus}. Triangla igiturGCH, GRM sunt similia, & ut

[51]GR : GM : : GC : GH;

Et ob similitudenem triangulorum GRO, GMT, erit utGR : GM : : GO : GT;

ergo ut GC : GO : : GH : GT;Si vero punctum R non multum recedat a vertice parabola C, (de quibus pencillis nostantum loquimur) GC propendicum aequalis erit ipsi GO : erit ergo & GH fere aequalisipsi GT, apex igitur pencilli H, quam proxime incidit in planum LMTN; quoddemonstrandum erat.

[52]§6. Prop. 40.1. Prop. 40. Lemma.

A short line is drawn at right angles to the axis of an ellipse from one focus. Two linesare drawn from some point on this line: one line goes to the far vertex of the ellipse andthe other is truly perpendicular to the circumference of the ellipse. Also, from thepreceeding focus of the ellipse , another line is drawn meeting the perpendicular in thecircumference. In this case, the angle between the first two lines is almost equal to theangle between the third line and the axis of the ellipse.

ARC is the ellipse, and from thefocus M a line MT is drawnperpendicular to the axis AC. From apoint T, the line TC is drawn to thevertex C. Also from T to thecircumference of the ellipse, theperpendicular line TR is drawn to thetangent QRO with point of contact R.The line MR is drawn from the focus Mto the point R. I say that the angles RMC& RTC are almost equal. The axis AC isproduced so that it meets the tangentQRO. If therefore from the diameter OT,a circle is described then it will passthrough the points R & M, as the anglesTMO & RTO are right. Therefore, theangles RTO and RMO are equal, lying inthe same arc of the circle. But the angle


M

ARC

T

O

Q


RTC does not differ much from the angle RTO, for if the point R of the contact line ROlies near the vertex C, (which always happen on account of the shortness of the line MT),it meets the axis almost at C. And with the line OC being short, then the differencebetween the angles RMC & RTC, i.e. the angle CTO, is diminished. Therefore, if theline MT is made very short, then the angles RTC and RMC will be almost equal. Qed.

[52]§6. Prop. 40.2. Prop. 40. Lemma.

Si ex uno ellipseos foco, ducatur rectula, normalis axi ; & ab aliquo rectulae puncto,ducantur duae rectae, una ad elllipseos verticem, altera vero ellipseos circumferentiaeperpendicularis ; a foco etiam ellipseos praedicto, ducantur alia, occurensperpendiculari, in circumferentia : erit angulus comprehensus a duabus primis rectis,ferme aequalis angulo comprehenso, a tertia, & axe ellipseos.

Sit ellipsis ARC, ex cujus foco M, ducatur recta MT, axi AC perpendicularis; & apuncto T ad ellipseos verticem C, ducatur recta TC ; & ab eodem puncto T, ad ellipseoscircumferentiam, ducatur contingenti QRO, perpendicularis in puncto contactus TR; & afoco M, ad punctum R, ducatur recta MR. Dico angulos RMC, RTC fere aequales esse.Producatur axis AC, ut concurrat cum tangente QRO in O;si igitur diametro OT,describatur circulus, transibit per puncta R, M; quoniam anguli TMO, RTO sunt recti;anguli igitur RTO, RMO in eodem circuli portionis sunt aequales; angulus autem RTCnon multum differt ab angulo RTO, quoniam si R punctum, cadat prope C, (quod semperevenit propter brevitatem lineae MT), linea contactus RO, concurret cum axe, fere in C.Et pro parvitate rectae OC, diminuitur angulus CTO, differentia inter angulos RMC &RTC: quare si recta MT fuerit brevis; erunt anguli RTC, RMC ferme aequales; quoddemonstrandum erat.

[53]

§6. Prop. 41.1. Prop. 41. Lemma.

A short line is drawn at right angles to the axis of a hyperbola from one focus. Twolines are drawn from some point on this line: one line goes to the vertex of thehyperbola and the other is truly perpendicular to the circumference of the hyperbola.From the preceeding focus of the hyperbola, another line is drawn meeting theperpendicular in the circumference. In this case, the angle between the first two lines isalmost equal to the angle between the third line and the axis of the hyperbola.

Let ARC be the hyperbola, from the farfocus M of which the short line MT isdrawn perpendicular to the axis AC. Theline TC is drawn From the point T to thevertex C of the hyperbola, and also from Tanother is drawn to the point of contact Rperpendicular to the tangent QRO. The lineMR is drawn from the focus M to the pointR. I say that the angles CMR & CTR arealmost equal. The tangent QRO is produced


A

C

B

P

M

F K

L T N

OR

Q

G

H


so that it meets the axis in O. A circle can be described on the diameter TO passingthrough the points R & M, since the angles TMO & RTO are right. Angles OMR & OTRare equal, lying on the same arc of the circle. However the angle RTC does not differmuch from the angle RTO, for if the point R of the contact line RO lies near the vertex C,(which always happens on account of the shortness of the line MT), it meets the axisalmost at C. And if the line OC is made small in length, then the difference between theangles RMC & RTC, i.e, the angle CTO , is diminished. Therefore, if the line MT ismade very short, then the angles RTC & RMC are almost equal. Qed.

[53]§6. Prop. 41.2. Prop. 41. Lemma.

Si ex uno hyperbolae foco, ducatur rectula, normalis axi ; & ab aliquo rectulaepuncto, ducantur duae rectae, una ad hyperbolae verticem, altera vero circumferentiaeperpendicularis ; a foco etiam hyperbolae praedicto, ducantur alia, occurensperpendiculari, in circumferentia : erit angulus comprehensus a duabus primis rectis,ferme aequalis angulo comprehenso, a tertia, & axe hyperbolae.

Sit hyperbola ARC, ex cujus foco M, ducatur recta MT, axi AC perpendicularis; & apuncto T ad hyperbolae verticem C, ducatur recta TC ; & ab eodem puncto T, ducaturaltera ad punctum contactus R; tangenti ORQ perpendicularis TR ; & a foco M, adpunctum R, ducatur MR. Dico angulos CMR, CTR fere aequales esse. Producaturtangens QRO ut concurrat cum axe in O; & diametro TO, ducatur circulus, qui transibitper puncta R, & M; quoniam anguli TMO, RTO sunt recti; eruntque anguli OMR, OTR,in eodem circuli portion ; inter se aequales ; angulus autem RTC non differt multum abangulo RTO, quoniam si R punctum, cadat prope verticem C, (quod semper evenitpropter brevitatem lineae MT), tangens QRO, concurret cum axe, ferme in vertice C. Etpro brevitate rectae OC, diminuitur angulus CTO, differentia inter angulos RMC & RTC:quare si MT fuerit breviuscula; erunt anguli RTC, RMC aequales fere; quoddemonstrandum erat.

[54]


If rays are sent from aplanar object present inthe focal plane of aconcave elliptic mirror(of which the normal isthe axis of the mirror) tothe mirror surface, thenthe apices of the conesformed by the reflectedrays from the individualpoints are almost co-planar.


Let ACB be the surface of the elliptic concave mirror, PC is the axis and the foci are Pand M. Let the plane surface object FPK send rays from the focal plane through P, towhich the axis PC is perpendicular.

[55]Another plane LMT is drawn through the other focal plane M N. I declare that the apicesof the pencils of the single points on the radiating surface lie approximately in the planeLMTN. In order to show this, let the axis of one of the pencils be FR, incident normallyon the mirror at the point R. A plane is drawn through this line and the axis of the mirrorgiving the ellipse ACRB as a common section with the mirror, from which the mirror canbe generated. From the corresponding planes the lines FPK and LMTN are taken, and theline QRO is drawn which is a tangent to the ellipse at the point R, which meets the axis atO. The line PR is drawn from the focus P to the same point R, which is reflected throughthe other focus M, which is the apex [i.e. meeting point] of the pencil of rays divergingfrom P , by Prop. 22. The line FC is drawn from the point F to the vertex of the mirror,where it is reflected through H, the apex of the pencil of the diverging rays from the pointF. And so the angles CFR and CPR are equal; (for they shall not be very different, as wehave shown above), according to the cor. of Prop. 22. Therefore the angles of incidencePRF and PCF are equal; and also the angles of reflection GRM, GCH are equal.Therefore the triangles GRM and GCH are similar.And as

GR : GM : : GC : GH;And from the similitude of the triangles GRO and GMT,

GR : GM : : GO : GT;therefore as GC : GO : : GH : GT. Now if the point R is actually very near the vertex ofthe ellipse C (about the pencils of have so much to say at present) then GC will be nearlyequal to GO itself, and therefore GH will be itself approximately equal to GT. Hence theapex or image point H of the pencil of rays diverging from F approaches closely to theplane surface LMTN. Qed.


Si superficies plana, in foco speculi cavi elliptici (cui normalis sit axis speculi) mittatradios in speculum; erunt apices pencillorum, singulorum superficiei puncturom, in unoquam proxime plano.

Sit speculum ellipticum concavum, ACB cujus axis PC, foci P, M; sitq; in foco P,superficis plana radians, FPK

[55] cui perpendicularis sit axis PC ; per alterem focum M, ducatur aliud planum prioriparallelum, LMTN. Dico apices pencillorum singulorum superficieis radiantispunctorum, cadere quam proxime in planum LNTN. Sit enim unius pencilli axis FR,perpendiculariter incidens in speculum, in puncto R; per quem, & axem speculi, ducaturplanum, faciens commonem sectionem cum speculo, ellipsem ACRB, ex qua generaturspeculum;cum planis vero, rectas lineas FPK, LMTN; & per punctum R, ducatur rectaQRO, tangens ellipsem, & occurens axi in O; & ad idem punctum R, ducatur a foco P,


A

C

B

P

M

FK

L T N

O

R

Q

G

H

V

S


recta PR, quae ideo reflectetur in alterum focum M, seu apicem pencilli radiorum , exfoco P divergentium ; ducaturq; ad verticem speculi, ex puncto F, recta FC, quaereflectetur in H, apicem pencilli, radiorum ex puncto F divergentium. Sint itaque anguliCFR, CPR aequales; (parum enim differunt ut demonstrimus) ; eruntq; igitur anguliincidentiae PRF, PCF aequales : & ideo anguli reflexi FRV, PCS, seu illis ad verticemMRG, HCG, aequales erunt ; quare triangula GRM, GCH sunt similia.Et ut

GR : GM : : GC : GH;Et ob similitudinem triangulorum GRO, GMT, ut

GR : GM : : GO : GT;ergo ut GC : GO : : GH : GT; Si vero punctum R, non recedatmultum a vertice ellipseos C ; (de quibus pencillis nos tantum loquimur) GC fermeaequalis erit rectae GO ; erit ergo & GH , quam proxime aequalis ipsi GT; Incidit igiturapex pencilli radiorum, ex F divergentium, nempe H, in superficiem planam, LMTN sere;quod demonstrantium erat.


If rays are sent from a planar object present in the focal plane of a convex hyperbolicmirror (of which the normal shall be the axis of the mirror) to the mirror, then the apicesof pencils of the reflected rays of the individual points are almost coplanar.

[56, 57]Let ACB be the surface of the

convex hyperbolic mirror, the axis isPCG, and the foci are P and M. Let theplanar object FPK sent rays from thefocus P, which is perpendicular to theaxis PC. Through the other focus Manother plane LMTN is drawn parallelto the first. I declare that the apices orimage points of the pencils of theindividual object points on the surfaceare near the plane surface LMTN. Forlet the axis of one of the cones of raysFR be incident normally on the mirrorat the point R. Through R and the axisof the mirror, a plane is drawn makingas a common section with the mirrorthe hyperbola ACRB, from which themirror is generated. Indeed, the

intersection of this plane with the planes KPF and LMN gives the lines KPF and LMN.The line QRO is drawn through the point R tangent to the hyperbola, meeting the axis inO. The line PR is drawn from the focus P to the same point R, which is thus reflected


from the other focus M, or from the apex or image point of the cone of rays divergingfrom the point P, by Prop. 24. The line FC is next drawn from the point F to the vertex ofthe mirror which is then reflected from H, the apex or image point of the cone ofdiverging rays from the point F, by cor. Prop. 24. Therefore the angles CFR and CPR areequal; (which indeed differ little, as we have explained); and therefore the angles ofincidence PRF and PCF are equal: and also the angles of reflection FRV and PCS areequal. Therefore the triangles GRM, GCH are similar.And as

GR : GM : : GC : GH;And from the similitude of the triangles GRO, GMT:

GR : GM : : GO : GT;therefore GC : GO : : GH : GT. If the point R is near the vertex of the Hyperbola C (about which cones of rays we say somuch) then GC will be nearly equal to GO, and therefore GH will itself beapproximately equal to GT. Therefore the apex H of the pencil of rays of reflection fromthe point F is not far from the plane surface LMTN. Qed.

Scholium.In the corollories to our catoptic and dioptic problems (which have been resolved

geometrically to a large extent by the use of conic sections through the foci), we havesaid these same problem can be adequately resolved according to our visual sense forcertain other points a little distance from the foci. But from these theorems justdemonstrated, it is evident that all these points almost lie in a plane perpendicular to theaxis of the lens or mirror.

§6. Prop. 43.2. Prop. 43. Theorema.Si superficies plana, in foco speculi hyperbolici convexi (cui normalis sit axis speculi)mittat radios in speculum; erunt apices pencillorum, singulorum superficiei puncturom,in uno quam proxime plano.

[56, 57]Sit speculum hyperbolicum convexum; cujus axis PCG, foci P, M; sitq; in foco P,

superficis plana radians, FPK cui perpendicularis sit axis PC ; per alterem focum M,ducatur aliud planum, priori parallelum LMTN. Dico apices pencillorum singulorumsuperficieis radiantis punctorum, non procul abesse a superficie plana LNTN. Sit enimunius Coni radiosi axis FR, perpendiculariter incidens in specului superficiem in punctoR; per quem, & axem speculi, ducatur planum, faciens commonem sectionem cumspeculo, hyperbolam ACRB, ex qua generatur speculum;cum planis vero, rectas lineasFPK, LMN & per punctum R, ducatur recta QRO, tangens Hyperbolam, & axi occurensin O; ducaturq; a foco P, ad idem punctum R, recta PR, quae ideo reflectetur a foco M,seu apice coni radiorum puncti P reflexorum, ducatur deinde ad verticem speculi, expuncto F, recta FC, quae reflectetur ab H, apice coni radiorum reflexorum puncto F. Sintitaque anguli CFR, CPR aequales; (qui parum enim differunt ut demonstravimus) {22 &Cor.22. Hijus}; eruntq; igitur anguli incidentiae PRF, PCF aequales : & ideo angulireflexi GRM, GCH, aequales erunt ;quare triangula GRM, GCM sunt similia.Et ut

GR : GM : : GC : GH;


Et ob similitudinem triangulorum GRO, GMT, utGR : GM : : GO : GT;

ergo ut GC : GO : : GH : GT; Si vero punctum R, non recedatmultum a vertice Hyperbolae ; (de quibus conis radiosis nos tantum loquimur) rectae GCferme aequalis erit rectae GO ; erit ergo & GH , quam proxime aequalis ipsi GT; apexigitur coni radiorum reflexorum puncti F, nempe H, non procul distantia plano LMTN;quod demonstrantium erat.

Scholium.In Corollariis ad problemata nostra catoptrica, & dioptrica, diximus eadem problema,

(quae geometrice tantum resolvuntur per focos sectionum conicarum) quo ad sensumetiam resolvi, per alia quaedam puncta, a focis paululum distantia. Ex his autemTheorematibus, nuper demonstratis, paret omnia illa puncta esse in uno quam proximeplano, cui perpendicularis est axis lentis, vel speculi.


§7. Synopsis of Propositions 44 - 51:This section is probably the most important in the book, and deals with image

formation by lenses and mirrors of all kinds with conoidal surfaces.

Prop. 44: If rays from individual points of any visible object are rendered parallel bya lens or mirror, and these parallel rays are viewed with a normal relaxed eye, then avisible image of these points is always seen to appear at infinity. The visible image isseen with the same angle of vision which the object subtends at the incident vertex of thelens or mirror.A detailed explanation of the ray diagrams for this proposition and the next are givenbelow.

Prop. 45: Rays from the individual points of some real image are made to convergeto other points by means of a lens or mirror. If the eye is placed between the lens ormirror and the points of convergence of the new image, then from the view-point of thatangle of vision, for which the apices of the pencils of rays from the extreme points of theobject appear at the centre of the eye, a confused image is always observed. However,the rays from the end points of the image subtend the same angle at the vertex ofemergence as the rays from the object subtend at the vertex of incidence.

Prop. 46: Consider the same arrangement, but with the eye now placed beyond thepoints of concurrence of the points of the object at the apex of their pencil. An imagewill be seen for people with myopic eyes which is useful in determining their bounds ofdistinct vision. However, for long-sighted people the image is still confused. The imageof the object also appears inverted, and at that angle corresponding to where the apicesof the pencils of the extreme points appear from the centre of the eye.

This theorem gives a number of extra examples of image formation, and also includessome information regarding long and short sighed people. The myopic eye by its naturecan form an image from rays with a greater degree of divergence than the normal eye,and attention is drawn to this fact. An analysis of the refraction ray diagrams is presentedvia ray diagrams.

Prop. 47: Rays from the individual points of some visible image are made to divergefrom other points by a lens or mirror. The image of any of these visible points will appearin the apex of the radiant cone of the eye receiving these rays. In accordance with theprevious theorem for myopic eyes the bounds of distinct vision can be determined, whilea confused image will always appear for long-sighted eyes. And the visible image willappear from that angle of vision, from which the apices of the cones of rays of theextreme visible points, appear with the same angle from the vertex of emergence, withwhich the visible object appears from the vertex of incidence.

This theorem is complementary to the previous one in that the corresponding diagramscorrespond to the opposite situation: concave and concave mirrors are interchanged, asare dense and rare mediums for lenses of the same geometry, so that the object andimage are interchanged in each case. An analysis of the reflection ray diagrams ispresented via ray diagrams.

Prop. 48: To measure the size or diameter of the pupil.


Prop. 49: Parallel rays are not weakened on traversing any distance, and equallyilluminate the same object placed in the same way at any point along the path .

Prop. 50: If the object distance from the incident vertex to the object distance fromcentre of the eye [placed before the lens or mirror] is in the same ratio as the imagedistance from emergent vertex to the image distance from centre of the eye [placed afterthe lens or mirror] , then the image appears with the same angle of vision as the objectappears to the naked eye, when viewed through a lens or mirror.

Prop. 51: With the same positions of object and image, I say that the image appearsequally illuminated with the eye at O, as the object appears with the eye at L, providedthe rays illuminate the whole pupil of the eye.


Extended Note on Prop. 44:A detailed discussion of the imaging processes in the various lenses presented in Prop.

44 is given using both Gregory's scheme and modern methods. The cases of the denseconvex-planar and plano-concave lenses are set out initially, followed by imageformation for less dense lenses in a denser medium, and for reflection by a parabolicmirror.

A: Dense convex-planar lens:

h

P

F1C1 F2 C2O

α

A

Q

S'α

RT

glassair

'α

A

B F

L

E

D

C

G

D

Fig.44 A

Fig.44 C

A

B F

L

E

D

C

G

DF

1 C2

Fig.44 B

p

q

r

sxx

First we note that the scheme presented here for image formation by conoidal surfacesanticipates the discussions presented in modern optics text such as Fundamentals ofOptics, Jenkins and White [McGraw-Hill, 1976] - where single convex and concavespherical surfaces are considered in turn (Chapter III ). Thus, in the modern case, raysentering a spherical convex glass surface from a focal point on the axis are refracted andsent through the glass as rays parallel to the axis, while rays sent from all off-axis pointssuffer from spherical aberration. In the hyperbolic case, rays parallel to the optic axis arerefracted without spherical aberration whatever their height, though of course there is stillaberration for rays refracted parallel to an auxiliary axes.

Figure 44A shows the initial diagram of the plano-convex lens in Prop. 44, rotated by900 for convenience. The curved surface is a hyperbola of revolution, and in this firstexample an object AB lies in the focal plane of the left-hand branch of the hyperbola (not


shown), and placed slightly asymmetric about the optic axis. Such objects are alwaysshown much larger in diagrams for convenience and the angles are much larger than fortrue paraxial ray tracing. Gregory has already established in Proposition 27 the manner inwhich rays from A & B may be traced through such a lens.

Within the lens medium, rays parallel to some auxiliary axis only are to be considered,according to the principle that parallel paraxial rays within the lens are refracted either toor from the same point in the focal plane through which the auxiliary axis passes - theactual focus used depending on whether the refracting surface makes the rays converge ordiverge. Thus, if B is such a point in Fig. 44B, where the vertical dotted line indicates thefocal plane, then the ray refracted at the nearest vertex of the lens D is parallel to theauxiliary axis within the lens. This axis is shown dashed and labeled s within the lens.Conversely, a ray emerging at the same angle (these angles are marked with a dot) fromthe other vertex D and below the axis is also parallel to the auxiliary axis within the lens,labeled r, and this ray also originates from the point B, according to the parallel paraxialray theorem. A similar argument applies for the ray from A to D. There are hence twoparallelograms, one with a vertex at D (in) and the other at D (out). For Gregory, theangles of refraction at the left-hand vertex D for the rays coming from A & B could befound from a table of experimental results for the glass medium concerned. The anglesfor the emergent rays are equal to these incident rays. The total angle BDA subtended byAB at D is thus equal to that formed by the emergent rays at the right-hand vertex FDE.

It is a great simplification to consider the cusp of the hyperbolic curve as a nodal pointof the lens. Hence, the almost horizontal sides of these parallelograms are parallel to theauxiliary optic axes, marked with dashed lines in Figure 44B, passing through the cuspC2. In most diagrams, the closely spaced rays near the vertices of the lens are omitted, aspresumably they would not be resolved in the printed diagram. The rays drawn entering avertex are hence not the same as those drawn leaving the other vertex. See also Figure44C - note that Gregory does not explicitly show any auxiliary axes, and considers onlythe refracted rays through D to define the direction. Thus, the line marked r in themedium is parallel to the auxiliary axis q in Fig. 44B. Any other paraxial ray from B suchas p travelling through the lens from A is also parallel to the axes q. Conversely, each raycorresponding to the same auxiliary axis has come from the same point in the focal plane,such as A or B. An eye placed at L will see parallel rays from A or B, or from both. Theimage will appear to be at infinity.

Figure 44C shows the path of any ray through the hyperbolic surface using a paraxialray in modern notation, as set out in §0.4 for M6. The thin convex-plano lens is given by

the matrix product ,n

MM)n(an)n(an

po

=

=

+−

+− 1

0101

001

1

111

16 where Mpo is the

plane surface matrix for the ray going out of the medium, and the focal length of the lensF1A is equal to a(n + 1) as required.


B: Dense plano-concave lens:

B

A

L

E

F

C

G

DD

C2h

PF1

C1

F2O AQ

S 'α

R

Tα

glass air

α

L

AF

C

DD

C2

F1

Figure 44D Figure 44E

Figure 44F

The plano-concave lens that follows is concerned with rendering parallel rays thatconverge in the absence of the lens to image points A and B in the focal plane of the lens,as in Fig. 44D. AB thus lies in the focal plane of the lens where the curved surface isagain hyperbolic but now concave. The in-going rays shown are rendered parallel to theauxiliary axis through A or B. An eye placed at L will hence see an image of A or B atinfinity. Figure 44E shows a simplified version for the rays converging to A, where theequality of the angles at the vertices can be determined from a single parallelogram.

Figure 44F shows the path of any ray through the hyperbolic surface using a paraxialray in modern notation, as set out in §0.4 for M8. The thin convex-plano lens is given by

the matrix product ,nnMM

)n(a)n(anpi

=

=

++10101

001

1

11

17 where Mpi is the

plane surface matrix for the ray going into the medium. The focal length of the lens is-a(n + 1) as required.


A B

F

L

E

DC

G

C1

F1

D

C G

B A

LE

F

A B

F

L

E

DC G

DC G

B A

L

E

F

C1

Figure 44G Figure 44H

Figure 44I Figure 44J

C: Concave \ Convex Parabolic Mirrors:

These latter diagrams for reflection are left as exercises.


D: Less dense concave-planar and convex-planar lenses in denser medium:

Figure 44K shows the path of rays through the ellipsoidal surface for paraxial rays, asset out in §0.4 for M2. The thin convex-plano lens in the denser medium has the ABCD

matrix given by:

=

+−

+− 1

01010

01

1121

)n(an

)n(an

nn .

The focal length of the lens is a(n + 1)/n as required.Figure 44L shows the path of rays through the ellipsoidal surface for paraxial rays, as

set out in §0.4 for M3. The thin convex-plano lens in the denser medium has the ABCD

matrix given by:

=

++10101

001

1121

)n(an

)n(an

nn . The focal length of the diverging

lens is a(n + 1)/n as required.

End of Extended Note on Prop. 44.

[58]Prop. 44. Theorem.

If rays from individual points of a visible object are rendered parallel by a lens ormirror, and these parallel rays are viewed by a normal relaxed eye, then a visible imageof these points is always seen to appear at infinity. The image subtends the same anglewith the eye that the object subtends at the incident vertex of the lens or mirror.

F1 C2

B

A

L

E

F

C

G

DD

Figure 44KA

DD

C1

B

F2

Figure 44L


Let AB be any visible object, or radiating matter, either before or behind the eye [i.e.the rays may be diverging from an actual object AB, or converging from some previouslens or mirror to form a prior image AB]. In any case, the object lies in the focal plane ofthe present lens or mirror CDG, and the end points are named A and B. The rays fromindividual points are rendered parallel by CDG, the axis of which is co-planar with thepoints A & B, but perpendicular to the plane of the visible object. The rays AD & BD aredrawn from the end points A & B to the lens (or mirror) vertex of incidence D, and thereflected or refracted rays DF & DE are drawn from the vertex of emergence D.

I declare that an eye, placed at L in the same plane as the axis of the lens or mirror andintercepting the parallel rays produced by the lens or mirror from AB, shall alwayscapture an image of that object through an angle of vision equal to the angle ADB. A rayGL is drawn to the eye from B parallel to DE. An equivalent ray, namely CL, is drawnfrom A, parallel to DF. These four lines, namely CL, DF, GL, DE, form a parallelogram,with equal opposite angles EDF &CGL; but the angle EDF equals the angle ADB, andtherefore the angle CLG is equal to the angle ADB. But the eye at L captures the objectAB viewed through the angle of vision CLG, as the extreme points A & B emit rays thatgive the angle CLG in the centre of the eye L, which therefore is equal to the angle ADB.This is the same angle by which the object can be seen from the vertex of incidence of thelens or mirror. The image appears at an infinite distance, and is distinct to a normalrelaxed eye, since each point appears to radiates parallel rays. QED.

For Reflections. Diverging Rays. Converging Rays.

D

C G

B A

LE

F

A B

F

L

E

DC G


For Refraction with a denser intermediate medium.Diverging Rays. Converging Rays.

For Refraction with a less dense intermediate medium.[60]

Diverging Rays. Converging Rays.

Further Comment on Prop. 44.In the cases of the concave mirror and the convex hyperbolic lens, A and B are the end

points of a radiating object, or of two discrete radiating points A and B, which lie in thefocal plane, close to, but distinct from the focus of a lens or mirror on the optic axis.

A B

FL

E

D

C GD

B A

LE F

C G

D

D

AB

FLE

D

C GD

BA

LE F

C G

D

D


Hence, rays from A and B reflected or refracted by the vertex are sent out in a parallelbeam at some angle to the axis. An eye capable of focusing a parallel beam will hence seethe image at this angle. The entire range of angles, or the angle into which the rays areradiated, is the same as the angle subtended by the object from the vertex of incidence.

[58]

Prop. 44. Theorema.

Si cujuscunque visibilis, singulorum punctorum radii, ad parallelismum reducantur :oculo radios parallelos recipienti, semper videbitur visibilis imago, eodem angulovisorio, quo videtur ex vertice incidentiae lentis, vel speculi. Apparetq; imago infinitedistans, & presbytis distincta.Sit visibile quodlibet AB, sive materia radians, sive imago ante, sive post oculum ;dummodo sit planum ; cujus extrema puncta A, & B: Reducantur singulorum punctorumradii, ad parallelismum, lente vel speculo CDG, cujus axis sit in plano cum punctis A, Bplano visibilis perpendicularis : & ab extremis punctis A, B, ad verticem incidentiae D,ducantur radii AD, BD ; & ad verticem emersionis D, radii refracti AD, BD ; quireflectentur vel refrangentur in DF, DE. Dico oculum, radios visibilis AB parallelos,intercipientem, semper comprehendere illius imaginem, per angulum visorium, aequalemangulo ADB. Sit enim centrum oculi L, in eodem plano cum axe lentis, vel speculi, &punctis extremis AB ; ad quod ducatur radius puncti B, nempe GL, ipsi DE parallelus ;idem radius puncti A, nempe CL, ipsi DF parallelos ; igitur quatuor istae rectae, nimitumCL, DF, GL, DE, elliciunt parallelogrammum, cujus anguli oppositi, EDF, CGL, suntaequales : Est autem angulus EDF, aequalis angulo ADB ; ergo & angulus CLG, estaequalis angulo ADB, sed oculus in L, apprehendit imaginem visibilis AB, per angulumvisorium CLG ; quoniam puncta extrema AB, emittunt radios, facientes in centro oculi L,angulum CLG, qui igitur aequalis est angulo ADB, nimirum eodem angulo, quo videturvisibile, ex vertice incidentiae lentis, vel speculi. Quod demonstrandum erat. Apparetinfinite distans, & presbytis distincta, quoniam unumquodque punctum radiat parallele.

Prop. 45. Theorem.

Rays from the individual points of some real image are made to converge to otherpoints by means of a lens or mirror. If the eye is placed between the lens or mirror andthe points of convergence of the new image, then from the view-point of that angle ofvision, for which the apices of the pencils of rays from the extreme points of the objectappear at the centre of the eye, a confused image is always observed. However, the raysfrom the end points of the image subtend the same angle at the vertex of emergence as therays from the object subtend at the vertex of incidence.

Consider an object with end points A and B as in the previous theorem. The rays fromthe individual points of AB are either converging or diverging. They are made toconverge by a lens or mirror CDG, the axis of which lies in the same plane as the pointsAB, in the plane of the perpendicular object. The apices of the pencils of rays are M andN. The rays CLN and GLM are drawn from the extreme points A and B. I am saying thatwith the eye at L, an image of the object is apparent with the angle of vision MLN or


CLG. For since the ray CN extended to N comes from the extreme point A of the object;and by the same reason GM comes from the end B of the object. Therefore the angleCLG, taken from the rays of the extreme points of the object intersecting each other at thecentre of the eye, is the angle of vision, within which the image of the object AB isapparent. The image does not appear sharp but is confused, since the rays from individualpoints converge to different points behind the eye. The position of the image is indeedapparent in these points, if by some means it has been found. Let ADB be the angle ofvision of the object AB, from the vertex of incidence D, which is equal to the angleMDN of the apex of the pencils of the extreme visible points from the vertex D of theemergent ray ; as is apparent from Prop. 26 and Prop. 27 of this work, which it wasrequired to show.

Scholium.

But if the line ML produced does not fall on the lens or mirror CDG, then the image ofthis point of this point will not be seen by the lens or mirror, of which the apex of thepencil is M.


M N

B

L

B

D G

B

A

A

A

O

C

Object at infinite distance : Object before the eye : Reflection.

M N

B

L

D G

A

O

C

D

C G

B A

L

O

M N

Object behind the eye :


For Refraction with a denser intermediate medium.

Object at infinite distance :

M N

B

L

B

D

G

BA A A

O

C D

Object before the eye :

O

M N

B

L

D

G

A

CD

Object behind the eye:

DC G

L

O

M N

B A

D


For refraction with a less dense intermediate medium.

Object behind the eye.

D

C G

L

O

MN

BA

D

Object before the eye.

M N

B

L

D

G

A

C D

O

Object at an infinite distance.

M N

B

L

B

D

G

BAA A

O

C D


Notes on Prop. 45:

Finally we are presented with the fully-fledged method of ray tracing for lenses withhyperboloidal surfaces placed in a less dense medium, paraboloidal mirrors, and lenseswith ellipsoidal surfaces placed in a more dense medium. In order to appreciate themethod, we set out some of the ray diagrams in detail. The first figure in each pair of

diagrams is taken from Gregory's text, while the latter shows focal planes, auxiliary axes,cusp points, etc, as in Figures 44A & B.

O

M

NB

LD

G

A C

D

F1C1 F

2 C2O

DF

1 C1F2C2

OD

A

BM

N

Figure 45 A.

Figure 45 B.


M

NB

L

B

D

G

B

A

A

A

O

C

DD D2C

S1 S2

F'2

V2

D

C

G

LO

M

N

B

A

D

DM

N

B

A

DC 1 2C

F1

2F

Figure 45 C.

Figure 45 D.


M

N

B

B

B

A

A

AF1

C1 F2C2

O

C1OF'1 C2 O F'2

A

M

C2

O F'2

C1 O F'1

A N

Figure 45 E.

Figure 45 G.

Figure 45 F.


Prop. 45. Theorema.

Si cujuscunque visibilis, singulorum punctorum radii, ad alia puncta convergantur ;oculo inter lentem, vel speculum, & puncta concursuum posito, semper apparebit imagovisibilis confusa, & eo angulo visorio, quo apices pencillorum, extremorum visibilispunctorum, ex oculi centro. Apices autem pencillorum, extremorum visibilis punctorum,eodem angulo apparent ex vertice emersionis, quo visibile ex vertice incidentiae.

Sit visibile quodlibet AB, sive materia radians, sive imago ante, sive post oculum,dummodo sit planum , cuius extrema puncta A, B. Convergantur singulorum punctorumradii, ad alia puncta, lente vel speculo CDG, cujus axis sit in plano cum punctis AB,plano visibilis perpendicularis. Sintq; punctorum extremorum A, B, pencillorum apicesM, N ducantur radii, CLN, GLM. Dico oculo in L, imaginem visibilis AB apparere, cumangulo visorio MLN, seu CLG. Quoniam enim radius CN, tendit ad N, provenit abextremo visibilis puncto A; & ob eandem rationem GM provenit, ab extremo visibilispuncto B; angulus igitur CLG, comprehensus radiis, extremorum visibilis punctorum, seinvicem in centro oculi secantibus, est angulus visorius, quo apparet imago visibilis AB.Apparet autem confusa, quoniam singulorum punctorum radii, ad alia puncta post oculumconverguntur ; in quibus etiam punctis, est apparens imaginis locus, si modo ullus detur :estq; ADB, angulus visorius visibilis AB, ex vertice incidentiae D, aequalis MDN, anguloapicum pencillorum, extremorum visibilis punctorum ex vertice emersionis D; ut patetper hujus 27 & 26 Prop. Quae omnia demonstranda erant.

Scholium.Si autem recta ML producta, in lentem vel speculum CDG non incidat, tunc non videbiturimago istius puncti, per lentem vel speculum CDG, cujus apex pencilli est M.

Prop. 46. Theorem.

Consider the same arrangement, but with the eye now placed beyond the points ofconcurrence of the points of the object at the apex of their pencil. A clear image will beseen for people with myopic eyes which is useful in determining their bounds of distinctvision. However, for long-sighted people the image is still confused. The image of theobject also appears inverted, and at that angle corresponding to where the apices of thepencils of the extreme points appear from the centre of the eye.

In the above figures, let the centre of the eye be placed at O beyond the point ofconcurrence of the rays. Thus the rays coming from the points of an object AB to the eyeO first converge to the apex of their pencil, and then diverge from it as if from a fountain.The eye is applied at O in order to observe the image depicted on the retina by these raysdiverging from the apex of their pencil; and by applying the eye the position of the pointsis judged to be in the apex of its own pencil of rays - if truly the apex of the pencil, orrather of the cone of rays, may be different for the myopic eye for that interval, and forwhich the retina itself is accustomed to be clearly depicted by the rays. In this case theaformentioned myopic eye will see the image of the point distinctly.


But for long-sighted eyes, (since the image on the retina of these is only depicted clearlyby parallel rays) there is no clear vision formed by these diverging rays. Which in allrespects is made clear enough, by Prop. 29 of this work onwards. I say also that the imageof the object appears inverted : i.e. the image of the point A is apparent in the partcorresponding to B, and formed in the opposite direction, while the image of point B isapparent in the part of the object corresponding to A . For A appears at N, from the apexof its pencil, which is apparent in the part B, with the eye surely placed at O ; as can bereadily deduced from Prop. 26 and 27 of this work; and in the same way, the image ofpoint B is apparent in the parts A. But since the extremes points of the object alwaysappear at M and N - the image formed will always appear with the angle of vision NOM:that is, with the angle by which the apices of the pencils of the extreme points of theimage appear from the centre of the eye. QED.

Scholium.But if the line OM produced is not incident on the lens or mirror, then the image of thispencil with apex is M will not be seen through the lens or mirror CDG. Which is also tobe understood in the following theorem.

Prop. 46. Theorema.

Iisdem positis ; oculo post puncta concursuum, apparebit imago, cujuslibet visibilispuncti, in apice sui penicilli ; myopibus, in determinata sua distantia, distincta ;Presbytis autem semper confusa. Videbitur quoque visibilis imago, everso sitis, & eoangulo visorio, quo apices penicillorum, extremorum visibilis punctorum, ex oculi centroapparent.In Figuris superioribus, sit oculi centrum O, post puncta concursuum ; radii igiturcujuslibet puncti visibilis AB, oculum O ferentes, primo congregantur in apice suipenicilli, & tunc divergunt ab apice praedicta, tanquam a fonte; Oculus igitur O, seapplicat, ut pingatur ipsius retina, istis radiis, ex apice peniocilli divergentibus, ex quaapplicatione existimat videns locum puncti, esse in apice sui penicilli: si vero apexpenicilli, vel potius coni radiosi, eo intervallo ab oculo myopis distet, quo solet ipsiusretina a radiante distincte pingi ; in hoc inquam casu myops praedicti puncti imaginem,distincte videbit : Presbytis autem, (quoniam eorum retina radiis solummodo parallelisdistincte pingitur) hisce radiis divergentibus nulla sit distincta visio; Quae omnia, satispatent, per Prop. 29. hujus & ejus consectaria. Dico imaginem quoque visibilis AB,apparere everso situ ; hoc est imaginem puncti A, apparere ad partes B, & e contraimaginem puncti B, ad partes A. A enim apparet in N, apice sui penicilli, quae apparet adpartes B, oculo nimirum in O posito ; ut facile deducitur ex 26 & 27 Prop. hujus ;eodemq; modo imago puncti B apparet ad partes A : Quoniam autem extrema visibilispuncta apparent semper in M, N; apparebit imago visibilis, angulo visorio NOM, hoc estangulo, quo apices penicillorum, extremorum visibilis punctorum, ex oculi centroapparent. Quae omnia demonstrare opportuit.

Scholium.Si autem recta OM producta, in lentem vel speculum CDG non incidat, tunc nonvidebitur imago istius puncti, cujus apex pencilli est M, per lentem vel speculum CDG:Quod in sequente etiam intelligendum.


Prop. 47. Theorem.

Rays from the individual points of some visible image are made to diverge from otherpoints by a lens or mirror. The image of any of these visible points will appear in theapex of the radiant cone of the eye receiving these rays. in accordance, for myopic eyes,with determining the bounds of their distinct vision; but for long-sighted eyes it willalways appear confused. And the visible image will appear from that angle of vision,from which the apices of the cones of rays of the extreme visible points, appear with thesame angle from the vertex of emergence, with which the visible object appears from thevertex of incidence.

Let there be some image, [ formed by a previous lens or mirror] which is formed eitherbefore or behind the eye or actual object before the eye, AB. This object shall lie in aplane with end points A and B. The rays that diverge from the individual points of thisobject are made to converge to other points, by the action of the lens or mirror CDG. Theaxis of CDG lies in a plane perpendicular to the points A and B in the plane of vision.The eye therefore, receiving these rays on its application to the apices of the radiant conesfrom the individual points of the visible image, is able itself to judge where the points Aand B are to be seen, from the apices of the aforementioned visible points, according tocor.1, Prop. 29 of this work. Thus, since the rays from neighbouring individual points ofthe image are diverging, a visible image is apparent for short-sighted people, within thebounds of their distinct vision. Long-sighted people however will always see a confusedimage for this situation.

Let M and N be the apices of the emergent cones of rays from the points of extremevision. Also let the centre of the eye be placed at L, where the two rays CL and GL of theextreme visible points concur, as they diverge from the points M and N. The angle NLMor CLG will be the angle of vision, by which the image is seen by the eye with centre L.This indeed is the angle by which the apices of the radiant cones of the extreme visiblepoints appear from the centre of the eye. Also the angle ADB, which is indeed the anglesubtended by the object at the vertex of incidence, is equal to the angle NDM or RDS,without doubt the angle which the apices of the radiant cones of the extreme visiblepoints subtend from the vertex of emergence. All of which is apparent from Prop. 27 &26 of this work. QED.

Corollary.For these theorems, the image is always seen to appear for the eye in the same place, withthe help of a lens or mirror; and therefore the tangents of the visual semi-angle are in thereciprocal ratio of the direct distances from the eye.

Note on Prop. 47:As indicated in the synopsis, inverse cases are considered in this theorem. As we have

already examined the refraction diagrams in detail in Prop. 46, the interchange of moreand less dense mediums for the lenses has been left as an exercise for the reader. We do,however, examine the reflection case briefly.


D CG

BA

L

NM

AB S

R

Centre of curvature

DC G

BA

L

N M

AB SR

A BSR

D

L

C G

N M

A BSR

D

L

C G

N M

AB

L

S

R

MN

AB

L

S

R

MN

Reflection

Object at infinity.Object before the eye.


Centre of curvature


DC G

BA

L

N M

AB

SR

Object at infinite distance.

For Reflection.

A BSR

D

L

C G

N M


AB

L

S

R

MN



For Refraction with a more dense intermediate medium.Object at infinite distance.

R

MN

B

L

D

G

BA A

CD

f

S


DC G

L

R

M N

D

BA

s1

s2f1'

S

Object past the eye.

L

GC

D

SR

D

M N

B A

For Refraction with a less dense intermediate medium.Object at infinite distance.

R

MN

B

L

D

G

BA A

CD

f

S

Object past the eye.

DC G

L

R

MN

D

BA

s1

s2

S


L

GCD

SR

D

M N

B A


[66]Prop. 47. Theorema.

Si cujuscunq; visibilis, singulorum punctorum radii, ab aliis punctis divergantur ; oculohos radios recipienti, apparebit imago cujuslibet visibilis punctis in apice sui coniradiosi, myopibus, in determinatis suis distantiis distincta, presbytis autem semperconfusa. Et imago visibilis, apparebit eo angulo visorio, quo apices conorumradiosorum, extremorum visibilis punctorum, ex centro oculi apparent : apices autemconorum radiosorum, extremorum visibilis punctorum, eodem apparent angulo ex verticeemersionis, quo visibile ex vertice incidentia.

Sit visibile quodlibet AB; sive materia radians, sive imago ante, sive post oculum,dummodo sit planum; cujus extrema puncta A, B. Divergantur singulorum punctorumradii, ab aliis punctis, lente vel speculo CDG, cujus axis sit in plano, cum punctis A, B,plano visibilis perpendicularis : Oculus igitur, hos radios recipiens, ex sua applicatione adapices conorum radiosorum, singulorum visibilis punctorum, existimat se videre, punctavisibilis A, B, in praedictis conorum radiosorum apicibus {cor.1. 29 Hujus}. Quoniamitaque singulorum punctorum radii, a punctis propinquis diverguntur, apparebit imagovisibilis, myobibus, in determinatis suis distantiis distincta ; Presbytis vero semperconfusa. Sint M, N apices conorum radiosorum, extremorum visibilis punctorum; sitq;centrum oculi L, in quod concurrunt duo extremorum visibilis punctorum radii CL, GL, apunctis N, M divergentes, eritq; NLM, seu CLG, angulus visorius, quo videtur imagovisibilis ab oculo, cujus centrum L; angulus nimirum, quo apices conorum radiosorum,extremorum visibilis punctorum, ex centro oculi apparent. Estq; angulus ADB, angulusnimirum quo visibile apparet ex vertice incidentiae, aequalis angulo NDM, seu RDS,angulo nempe, quo apices conorum radiosorum, extremorum visibilis punctorum, exvertice emersionis apparent; ut patet per Prop. 27 & 26 hujus; quae omnia demonstrandaerant.

[70]Corollarium.

Ex his Theorematibus, patet imaginem visibilis ope lentium vel speculum, oculoapparere, semper in eadem loco ; & igitur tangentes semiangulorum visualium sunt inreciproca ratione distantiarum directarum ab oculo.

Prop. 48. Problem.

To measure the size or diameter of the pupil.

Let the plate ABCD be made of copper or some other metal, in which there shall bethe thinnest of cracks BC [i.e. a narrow slit] at right angles to the horizontal. The cylinderGMN is at right angles to the horizontal too, in order that a line drawn parallel to thehorizontal from the axis of that cylinder to the middle of the crack is perpendicular to theplate. Let GN be the radius of the base of the cylinder, and GT the distance between thecylinder axis and the centre of the crack; the width of the slit BC is measured and noted.The line CM is drawn parallel to the horizontal from the point C at the edge of the slit


tangent to the cylinder GMN at some point M. In the same way, from point B at the otheredge of the crack, BN is drawn in the same plane as CM, i. e. parallel to the horizontal,tangent to the cylinder at the point N. [See the diagram below]. The lines NB and MC areproduced from B and C and these intersect at R. In triangle GTB with a right angle at T,from the given GT and TB, it follows that both BG the angle BGM can be found. Againin triangle GBN, with the right angle at N, the angle BGN is found from the given sidesBG and GN; then from BGN if the angle BGT is taken away then the angle RGN is left.The length GR can be found in the isosceles triangle BRC, given the angle BRC and theperpendicular RT. The diameter of visibility is OP in the same plane as the lines NRBand MRC, as nothing can be seen through the crack beyond the extremities of thecylinder GMN. Therefore the lines NRB and MRC on being produced exactly determinethe diameter OP. For consider the isosceles triangle ORP, (it is necessary indeed that theaxis of the eye and the plane ABCD are perpendicular) with the given angle ORP, andwith the perpendicular from the vertex R to the base OP - which is the distance of the eyefrom the point R - both the base OP and the visual diameter can be found, which it wasnecessary to observe.

Scholium.But the pupil is not always the same size; as in strong light it is made smaller, while inweak light it is larger.

Note on Prop. 47:Gregory realises that the size of the entrance pupil of the eye itself is needed to

maximise the light admitted via lenses and mirrors, by matching up the angles subtendedby cones of rays, as shown in Prop. 50 and Prop. 51. The procedure adopted here is opento criticism, but we do not intend to explore this further!

[70]Prop. 48. Problema.

Quantitatem foraminis uveae, feu diametrum visualem, obfervari.

Sit ex aere, vel ex quovis alio metallo, lamina ABCD; in qua sit rimula tenuissima BC.Horizonti recta : sitq; cylindrus GMN, horizonti quoq; rectus ; ita ut linea, horizontiparallela, ab illius axe, ad rimulae medium ducta, sit laminae perpendicularis : sintq; basis

G

N

M

P

O

TB

C R


A M

L NB

C

B

C

semidiameter GN; & distantia inter axem, & rimulae medium GT ; & latitudo rimulaeBC, notae mensurae. Et ab extremo, rimulae puncto C, ducatur linea horizonti parallelaCM, cylindrum GMN tangens ad alteras partes, in puncto M. Eodem modo, a B, alteroextremo rimulae puncto, ducatur BN, in eodem plano, cum CM, horizonti parallelo,tangens cylindrum, in puncto N : & producantur lineae NB, MC, ad partes B, C; sitq;

[71]earum intersectio in R; & in triangulo GTB, rectangulo ad T, e datis, GT, TB, datur &BG, & angulus BGM: Et rursus in triangulo GBN, rectangulo ad N, e datis, BG, GN,dabitur angulus BGN, a quo, angulus BGT ablatus, relinquit angulum RGN; datur latusGR :dantur igitur in triangulo isosceli BRC, angulus BRC, & perpendicularis RT. Sitdiameter visualis OP, in eodem plano cum rectis NRB, MRC, nihil videns per rimulamCB, ultra extremitates cylinderi GMN: Lineae igitur NRB, MRC productae, diametrumvisualem OP exacte comprehendunt ; & in triangulo isosceli ORP, (oportet enim ut axisoculi, laminae ABCD sit perpendicularis) datis angulo ORP, & perpendiculari a vertice Rin basem OP, seu distantia oculi a puncto R; dabitur & basis OP, diameter visualisquaesita; quam observare oportuit.

Scholium.Foramen autem uveae, non est semper ejusdem quantitatis ; sed in forti luce

diminuitur, in debili autem ampliatur.

Prop. 49. Theorem.

Parallel rays are not weakened on traversing any distance, and equally illuminate thesame object placed in the same way at any point along the path .

Let the parallel rays be AMNL, of which the outerrays are AL and MN, and which illuminate the sameobject BC placed in the same way at different distancesfrom the source of radiation. I say that BC is alwaysilluminated equally, since indeed the outer rays AL andMN which are illuminating the edges of the surface ofthe object BC are equidistant among themselves.Therefore the same object placed in the same wayalways intercepts each and every ray between the outeredges of the object BC, and since these rays give thesame illumination, then indeed the strength of the rayscannot be diminished by distance alone.

Corollary 1.Thus it follows that the pupil of the eye is always equally illuminated by the raysemanating from individual points, whether they come from the vertex of incidence oremission of the lens. (Indeed for all these things concerning a lens, the rays from thesevertices are parallel, as we have shown following Prop. 25 of this work). Only a smallpart of the illumination removed, which is drawn out by the opacity of the lens; and this


exception is always to be invoked in Dioptrics [Refraction]. [Gregory's optics relies onparallel beams either entering or leaving lenses, or are parallel within the lens.]

Corollary 2.For from this theorem and the first corollary, (if the rays of individual points from anyobject are made parallel), and the eye can receive these rays, then not only is an image ofthe object formed by the eye for that angle of vision from the vertex of incidence (as wehave shown from Prop. 44 of this work), but also it is received with the sameillumination.

Prop. 49. Theorema.

Radii paralleli, non debilitati; in omni distantia, aequaliter illustrant idem objectum,eodem modo positum.

Sint radii paralleli AMNL; quorum extremi funt AL, MN, illustrantes, idem objectum B,C, eodem modo pofitum, fed in diverfis distantiis, a radiorum fonte. Dico BC, aequaliterfemper illustrati. Quoniam enim extremi radii, e quibus sunt AL, MN, extremitatessuperficiei, objecti BC illustrantes, inter se aequidistant, idem igitur objectum, eodemmodo pofitum, semper intercipiunt omnes igitur radii, inter extremos objectum BC,intercipientes, in objectum BC semper incidunt: iidem igitur radii eandem dantillustrandem ; sola enim distantia, radiorum vim non debilitat.

Corollarium 1.

Hinc sequitur, foramen uveae oculatis semper aequaliter illustrati, a singulis radiantispunctis ; sive fuerit in vertice incidentiae, sive emersionis ; (in omni enim lente, ejusdempuncti radii sunt paralleli, ut ad manifestum secundum Prop. 25 hujus diximus) demptasolummodo particula ista illustrationis, quae exhauritur opacitate lentis; quae exceptio inDioptricis semper est adhibenda.

Corollarium 2.

Ex hoc Theoremate, & praemisso Corollario, (si cujuscunque visibilis, singulorumpunctorum radii, ad parallelismum reducantur) oculum hos radios recipientem, non solumcomprehendere visibile, eo angulo visorio, quo comprehenditur ex vertice incidentiae, (utdemonstravimus ad 44 Prop. hujus) sed etiam eadem illustratione.


Prop. 50. Theorem.If the object distance from incident vertex to the object distance from centre of the eye isin the same ratio as the image distance from emergent vertex to the image distance fromcentre of the eye, then the image appears with the same angle of vision as the objectappears to the naked eye, when viewed through a lens or mirror.

Let AVB be any real object or image [formed by a previous lens or mirror] as youplease, with this object or image either before or after the eye. However, AVB should bein a plane normal to the axis of the lens or mirror. With the help of some mirror or lens,the rays from the object plane AVB appear in the image plane MIN. Through the planesAVB, MIN, and the axis of the lens of mirror, a plane is drawn making a common sectionwith the object line AVB and the image line MIN. The object AVB appears with theangle of vision ALB from some axial point L. The ratio is thus as VD, the distance of theobject from the vertex of incidence, to VL, the distance of the object to the centre of theeye placed at L, thus as VD, the distance of the image from the vertex of emergence, toIO, the distance of the image from the centre of the eye at O. The image IMN appearsfrom the axial point O at the visual angle MON. I say that the angle MON is equal to theangle ALB.Indeed, as VD : VL: : ID : IO; and onrearranging, as VD : ID : : VL : IO;but as VD : ID : : VA : IN;Therefore, VL : IO : : VA : IN.Therefore the triangles AVL and NIO are similar, having equal right angles AVL andNIO, and sides in proportion around these equal angles. Therefore the angles ALV andION are equal, and the angles VLB and MOI are equal. Hence the whole angles ALB andMON are equal. QED.

Prop. 50. Theorema.

Si fuerit ut distantia visibilis, a vertice incidentia, ad distantia visibilis, ab oculi centro ;ita distantia imaginis, a vertice emersionis, ad distantiam imaginis, ab oculi centro :eodem angulo visorio, apparebit imago, ope lentis, vel speculi ; quo apparet visibile,nudo oculo visum.

Sit visibile quodlibet AVB; sive materia radians, sive imago ante, sive post oculum,modo sit planum ; cui axis lentis, vel speculi sit normalis : ope speculi cujuslibet, vellentis, appareat imago, plani radiantis AVB, in plano MIN ; & per plana AVB, MIN, &axem lentis, vel speculi VID, ducatur planum, faciens cum visibili communemsectionem, rectam AVB; cum imagine vero, rectam MIN. Et a quovis axeos puncto L,appareat visibile AVB, angulo visorie ALB; sitque ut VD, distantia visibilis a verticeincidentiae; ad VL, distantiam visibilis ab oculi centro: Ita D, distantia imaginis a verticeemersionis; ad IO, distantiam imaginis ab oculi centro: Et a puncto axeos O, appareatimago MIN, angulo visorio MON. Dico angulum MON ; esse aequalem angulo ALB:


Quoniam, ut VD : VL: : ID : IO; erit & permutando, ut VD : ID : : VL : IO; ut autem VD : ID : : VA : IN;Igitur, ut VL : IO : : VA : IN; triangulaigitur AVL, NIO, habentia angulos AVL, NIO, aequales, utpote rectos, & latera circaaequales angulos proportionalia, sunt similia ; angulus igitur ALV, aequalis est anguloION. Eodem modo, demonstrabitur aequalitas, angulorum VLB, MOI: totus igiturangulus ALB, est aequalis toti angulo MON. Quod erat demonstrandum.

Prop. 51. Theorem.With object and image positions unchanged , I say that the image appears equallyilluminated with the eye at O, as the object appears with the eye at L, provided the wholepupil of the eye is illuminated by the rays.

The ratio of the diameters is indeed as VL to VD: i. e. the ratio of the diameter of thepupil at L, to be illuminated by the paraxial rays from V, to the diameter of the samecone of rays reaching the lens or mirror, which has the vertex at D. And the ratio IO toID is also as VL to VD. Thus, the ratio is the diameter of the pupil at O, illuminated bythe rays from V, to the base diameter of the same cone of rays at the lens or mirror.Therefore, if the diameters of the pupils are supposed equal at the two positions, then thediameters of the bases of the cones of rays from V are also equal. Therefore the baseareas of these cones of rays are equal: for since they have a common axis VD, they tooare equal, in short, each cone contains the power of one and the same cone of rays. Thepowers of the cones in illuminating or burning (for rays at equal distances passing intoequal pupils) are as the square ratio of the chords of their radiant semi-angles. Since theradiant cones have equal radiant angles, the chords of these half-angles are equal, and sothe squares of chords are equal also [for the case of equal distances]. Thus, theilluminations are equal. Indeed, this is so for the illumination into the pupil L from therays coming from V, and the illumination due to the rays coming from the point V intothe pupil at O. Therefore, in like manner, the eye at L sees the illumination from theobject point V, with which the eye at O sees the illuminated image of this. In the sameway too, this equality of the illumination can be shown for all the points of the visibleobject AVB. QED.

Scholium.But if VL to VD, or IO to ID has a larger proportion than the diameter of the pupil to

the diameter of the lens or mirror to be illuminated by the rays, then in this case it can besaid that the whole pupil is illuminated by the rays from the individual points. If the ratiois equal, then the result is exact. If however the ratio is less then only a part of the pupil isilluminated, and since it is sufficient to be warned, the demonstration is indeed clear [inall cases].

Comment:The point V is the common vertex of two cones, with a common axis VD. The pupil isthe base of the small cone located at L, while the lens or mirror is the base of the largecone: thus, each receives the same amount of radiation, while the intensities vary as the


squares of the diameters. The same reasoning applies for the image point I, the newposition of the eye at O, and the lens or mirror. A summation is carried out over all thepoints of the object, from A to B, and all the paraxial rays in the angle ALB, the angle ofvision, is assumed to enter the eye. If the ratio of the lengths is greater than the ratio ofthe diameters then the whole pupil and lens or mirror is bathed in illumination; and onlyin part if the ratio is less; the theorem is thus true only for the case of equality. Thetheorem is concerned with the possible loss of light intensity due to a miss-match of thesize of the lens or mirror used when an image is processed for viewing.

[74]

A BV

L

MN

O

D

I

A BV

L

M N

O

O

D

I

For Reflection: with the object before the eye. The object behind the eye.

ABV

L

M N

O

D

I

O

A BV

L

MN

OD

I


[75]

M

V AB

LD

O

D

N

O

I

A BV

L

D

O

D

NM

O

I

A BV

L

D

O

D

N MI

A BV

L

D

O

D

N MI

For Refraction with the intermediate medium denser.

Object before the eye. Object behind the eye.


A BV

L

D

O

D

N MI

A BV

L

D

O

D

N MI

For Refraction with the intermediate medium less dense.

M

Object before the eye.D

A BV

L

D

O

D

N

O

I

M

V ABL

O

D

N

O

I

Object after the eye.


[76]Prop. 51. Theorema.

Iisdem positis; dico imaginem, eodem modo apparere illustratam, oculo in O; quoapparet visibile, oculo in L : dummodo totum foramen uvea, radiis illustretur.

Est enim, ut VL, ad VD; ita diameter foraminis uveae, radiis puncti visibilis V, illustrati,in L ; ad diametrum , baseos coni radiorum eorundem, in lente vel speculo; atq; ut IO adID ; hoc est VL, ad VD; ita diameter foraminis uveae, radiis puncti visibilis V, illustratiin O; ad diametrum baseos coni, radiorum eorundem, in praedicto quoque speculo vellente : cum igitur diametri foraminum uveae supponantur aequales; aequales quoq; erunt,in lente vel speculo, diametri basium, conorum, radiorum puncti V; quibus, illustraturforamen uveae in L, & foramen uveae in O. Bases igitur horum conorum aequales erunt,cumq, eundem communem habeant axem VD, prorsus aequales erunt; vel potius unus, &idem conus radiosus; eruntq; vires conorum radiosorum, in illustrando, vel comburendo(radiis in spatia aequalia nempe in foramina uvearum aequalia reductis) in duplicataratione, chordarum, suorum semiangulorum radiosorum; cumque coni radiosi, aequaleshabeant angulos radiosos; erunt & chordae semissium horum angulorum aequales;

[77]adeoq; & quadrata chordaram aequalia; unde, illustrationes aequales erunt : Nempe,illustratio radiorum puncti V, in foramine uveae L; & illustratio radiorum puncti V, inforamine uveae O: oculus igitur in L, eodem modo videt punctum visibile V, illustratum ;quo oculus in O videt illius imaginem illustratam. Eodem quoque modo haec aequalitasillustrationis in omnibus punctis visibilis AVB manifesta fiet; quod erat demonstrandum.

Scholium.Si autem VL, ad VD; vel IO, ad ID; majorem habeat proportionem ; quam diameterforaminis uveae, ad diametrum lentis, vel speculi, radiis illustrati; in hoc inquam casu,totum foramen uveae, singulorum punctorum radiis illustrabitur; si aequalem, praecise ;Si vero minorem, totum non illustrabitur; quod admonuisse sufficiat, demonstratio enimest manifesta.

James Gregory's Optica Promota 118

Prop. 52. Problem.

With the distance and the angle of vision of any visible [object, or previously formedimage] given; to represent the image of that object in whatever given measure ; with thethickness of the lens given too.

Let AVB be any visible object whatever, either radiating matter, or an image before orafter the eye, provided it shall be plane; in which two points A, B are taken. Then theimage of the radiant plane, AVB, is to be found, in which the images of the points A, Bshall stand apart by the right distance R. In the line AB, any point whatever is taken, nearthe centre, namely V; from which is drawn VD, perpendicular to the plane AVB, equal tothe distance of the given visible object, which is produced indefinitely. And DD taken inthis, equal to the thickness of the lens; and it shall become, as AB to VD; thus, R to DI.And if the object is before the eye then the rays diverging from V ; or if past the eyeconverging to V; [in either case] diverging from the point I, with the help of the lens ormirror, with vertices DD {according to Prop's.23, 24 etc of this work}. And through I aplane is drawn, parallel to the plane AVB. I say, the images of the points A, B shall bedistant by the given right line R. Since indeed, for the object AVB,

[78]supposed a plane, the apices of all the cones of rays of these points converge in the planeMIN. Hence in the plane MIN, as is stated clear enough from the scholium to Prop. 43 ofthis work. [Note: The text in the translator's copy of the O. P. is very unclear on this page,due to type from the previous page showing through.] The apices of the cones of rays ofthe points A, B therefore shall be the points M, N; and the points M, I, N shall be in astraight line. Since A, V, B are in one right line, indeed the rays AD, BD are in the sameplane, for the rays AD, BD are in the same plane as the axis, and therefore the surface ofthese, either of reflection or refraction, is one and the same; therefore reflected orrefracted in one plane with the axis, as the plane cuts the plane of the image in the rightline MIN; and on account of the equality of the angles ADV, IDN; the triangles ADV,IDN are similar. And as:

DV: DI : : AV : IN;In the same manner, as DV: DI : : VB : IM;but it was, as DV : DI : : AB : R; therefore R, & MN are equal. But theimages of the points appear in the apices of the cones of their own rays, that is, in M andN, with the separation for R: which was to be done. Also by the same method, the images of visible objects at an infinite distance can berepresented in terms of the given measure. See the diagrams. Let ADB be the angle ofvision, of any visible object at infinite distance, that the right line AB subtends, and itshall be; as AB to R ; thus the perpendicular into the subtended, DV to DI ; & the rays,themselves parallel to DV, diverge from the point I by the lens or mirror, of which thevertices are D, D, by the same manner as previously stated.

Scholium.But if the right line AB is small; so that it cannot be projected conveniently into a line Requal to itself: the image of AB itself may be projected into a line less than R, whichimage, by the same method, and if it might be visible itself, again projected into anotherimage equal to R itself. Because if it still may not be possible to happen, the operation


will have to be repeated. Nevertheless, to be noted, if the right line R is too long withrespect to the smallness of the lens or mirror ; the image MN is confused and indistincttowards the extremities M and N ; in the middle truly around the axis I of the lens ormirror produced, it will always be most distinct, since the image of the point V of thevisible object is projected in I, to the geometrical figure.

If indeed, the rays of the visible points shall gather in points of the plane of theimage, and a white unpolished plane, the rays of the visible object, shining strongly, shallbe fixed in the plane of the image, most clearly depicted in the white plane. For onaccount of the roughness of the plane, the rays of the individual points of the visibleobject, are concurring into images or points of the white plane. From the points of thesame plane, rebounding from which, they strike the eyes of observers; & each visiblepoint will appear depicted to be considered, in that point of the plane, from which all ofthese rays are reflected ; that is, in the apex of their own radiant cone ; or rather thepencils. But this image, projected by one mirror or lens, always appears in the invertedsituation: as is apparent from Prop. 46 of this work.

But if the plane placed in the position of the image were polished ; being reflected, allthe angles of incidence of the rays of the visible object are equal to the angles ofreflection ; which therefore are nearly all reflected into the lens or mirror. And if the eye ,from the side of the lens or mirror, may consider the aforementioned image, it will see asmall part of that, in the surface of the polished plane. Which are all clear enough, andnot in need of demonstration.


Cujuslibet visibilis, distantia, & angulo visorio, datis; illius imaginem in quavis datamensura representare: data quoque crassitie lentis.

Sit visibile AVB, quodlibet, sive materia radians, sive imago ante, vel post oculum,dummodo sit planum; in quo, sumantur duo puncta A, B. Sit igitur invenienda imagoplani radiantis, AVB; in qua, punctorum A, B, imagines, distent recta data R. In rectaAB, sumatur punctum quodlibet, prope ipsius medium, nimirum V ; e quo ducatur, planoAVB perpendicularis, VD, aequalis distantiae visibilis datae, quae producatur ininfinitum: Et sumatur in ea, DD, aequalis lentis crassitiei ; fiatque, ut AB, ad VD ; ita R,ad DI. Et radii ab V divergentes, si visibile sit ante oculum ; vel ad V convergentes, sipost oculum divergantur a puncto I, ope lentis, vel speculi, cujus vertices DD ; & per I,ducatur planum, plano visibilis AVB, parallelum. Dico, imagines punctorum A, B, distentrecta data R. Quoniam enim, visibile AVB, supponatur planum , apices conarumradiosorum omnium, illorum puncti convergunt in plano MIN. Hunc in plano MIN, ait escholium ad Prop. 43 hujus satis paret. Sint igitur apices conorum radiosorum,punctorum A, B, puncti M, N; & erunt puncta M, I, N, in una linea recta; quoniam A, V,B, sunt in una recta, radii enim AD, BD, sunt in eodem plana, cum axe & ideo,superficies illorum, sive reflectionis, sive refractionis, una est & eadem; reflectunturigitur, vel refringuntur, in una plano cum axe, quod planum secat planum imaginis, inrecta MIN; & ob aequalitatem angulorum, ADV, IDN; triangula ADV, IDN, sunt similia,


et utDV: DI : : AV : IN;

Eodem modo, ut DV: DI : : VB : IM;erat autem, ut DV : DI : : AB : R; igitur R, & MN, sunt aequales. Imaginesautem punctorum A, B apparent in apicibus, conorum suorum radiosorum, hoc est in M,& N. distantibus, recta R: quod faciendum erat.Eodem etiam modo, possunt imagines visibilium, infinite distantium, in data mensurarepresentari, v. g. Sit angulus ADB, angulus visorius, alicujus visibilis, infinitedistantium, quem subtendat recta AB, fiatq; ut AB, ad R ; ita perpendicularis insubtendentem, DV, ad DI ; & radii, paralleli ipfi DV, divergantur a puncto I, lente velspeculo cujus vertices D, D , eodem modo, ut hactenus dictum.

Scholium.Si autem recta AB fuerit parva ; ut non poterit commode projici, in rectam aequalem ipsiR : projiciat ipsius A, B imagino, in rectam, minorem ipso R, quae imago, eodem modo,ac si esset ipsum visibile, rursus projiciatur, in aliam imaginem, aequalem ipsi R: Quodsi adhuc fieri non possit, reiteranda erit operatio. Notandum tamen si recta R, sit nimislonga respectu parvitatis lentis, vel speculi ; imaginem MN, fuerat confusam &indistinctam, versus extreminintes M & N; in medio vero I circiter axem; lentis velspeculis producantur; semper erat distinctissimo; quoniam imago puncti visibilis V,projictitur in I, ad figorem Geometricum.

[78]

Si vero; radii visibilis punctorum, congregentur in puncti plani imaginis, &planum album, & impolitum, radios visibilis, fortiter revibrans, figatur in plano imaginis;imago videbitur, clarissime depicta, in plano albo. Nam propter plani impolitiam, radiisingulorum visibilis punctorum, in imaginis, vel plani albi puncta concurrentes ; abiisdem plani punctis, undeque repercussi, oculos videntium feriunt; & unumquodquevisibilis punctum , apparebit intuenti depictum, in illi plani puncto, a quo, omnes illiusradii reflectuntur ; hoc est in apice sui coni radiosi ; vel potius penicilli: Apparet autemhaec imago, unico speculo vel lente projecta, semper situ everso : ut patet per Prop. 46hujus.

Sed; si planum, in loco imaginis positum, fuerit politum ; reflectentur, omnesvisibilis radii, ad angulos reflectionum, aequales angulis incidentibus ; qui proptereareflectentur pene omnes in lentem vel speculum. Et si oculus, a latere lentis vel speculi,praedictam imaginem intueatur ; videbit illius particulam, in plani politi fuperficie : quaeomnia satis manifesta sunt, nec ulla egent demonstratione.


[80]

A BV

L

MN

D

I

In Catoptrics: visibile ante oculum

AB V

L

M N

D

I

Visibile post oculum

M NI

L

A B

D

V

R

AB V

L

MN

D

I


[81]

In Dioptricis intermediate diaphano densiore

A BV

L

D

D

NM I

Visibilis ante oculum

M

V AB

L

D

D

NI

Visibilis post oculum

A BV

L

D

D

N MI

A BV

L

D

D

N MI


[82]

In Dioptricis intermediate diaphano rariore

A BV

L

D

D

NM I


A BV

L

D

D

N MI

M

V AB

L

DD

NI


AB V

L

D

D

N MI


Prop. 53. Problem.For a given visible image; projected with the help of some mirrors or lenses : and withthe distances from each other in turn of the lenses, mirrors, and image given ; to find theangle of vision of the visible image from its own vertex of incidence.

With the same figures, the points M, N shall be the images of the two given points,projected with the help of some number of mirrors or lenses. However it is necessary forthe right line M, N to cut the axis produced in I. And MI, IN, ID shall be given, and thedistances from each other in turn of the lenses, mirrors, and the image. Therefore, withthe sides NI & ID given in triangle NID; and the angle NDI given, or that equal to ADV.And it shall be as the distance of the image of AVB, from the vertex of its own emission;to the distance of the same AVB, from the vertex of its own incidence VD ; thus as thetangent of the angle ADV, to the tangent of the angle, by which AV is seen from thepreceding angle of emergence. By the same method, the angle is found, by which VBappears from the same vertex of emergence; which added, will make the angle fromwhich AB, or the second image, appears from its vertex of emergence ; which is equal tothe angle by which the third angle appears, from its own vertex of incidence : from whichgiven, gives the same between the pertinent parts to the second image and to the third ;which were given between the pertinent parts to the first and second parts ; therefore bythe same reasoning, the angle will be found, by which the fourth image appears from itsvertex of incidence , and at last the angle from which the visible object is seen from itsangle of incidence.

Prop. 53. Problema.

Data imagino visibilie ; ope quotcunque speculorum vel lentium projecta : & datislentium, vel speculum, & imaginum, a se invicem distantiis ; visibilis angulum visorium,ex vertice suae incidentiae invenire.

Sint in iisdem figures ; puncta M, N imagines duorum visibilis punctorum, opequotcunque speculorum, vel lentium, projectae: Oportet tamen rectam M, N axemproductum secare in I. Sintque; data, MI , IN, ID, & lentium vel speculorum, &imaginum, a se invicem distantiae. Datis igitur in triangulo rectangulo NID, lateribus,NI, & ID; datur & angulus NDI, seu illi aequalis ADV. Sitq; ut distantia imaginis AVB, avertice suae emersionis; ad distantiam ejusdem AVB, a vertice suae incidentiae VD ; itatangens anguli ADV, ad tangentem anguli, quo AV videtur ex praedicta verticeemersionis. Eodem modo, invenitur angulus, quo VB, apparet ex eadem verticeemersionis; qui additi, efficiunt angulum, quo AB, seu secunda imago, apparet ex verticesuae emersionis ; qui, aequalis est angulo, quo tertia imago apparet, ex vertice suaeincidenriae: quo dato, dantur eadem, inter pertinentia ad imaginem secundam, & adtertiam ; quae dabantur, inter pertinentia ad imaginem primam, & secundam ; eodemigitur ratiocinio, reperietur angulus, quo quarta imago, apparet ex vertice suaeincidentiae, quo invenimus angulum tertiae, ex vertice ipsius incidentiae; & tandemangulus, quo visibile videtur ex vertice suae incidentiae.


[83] cont'd.

Prop. 54. Problem.

To illuminate the image of a visible object depicted in a plane in any given ratio.

Let AVB be the visible object, the image of which is depicted in the plane, by a lens ormirror, the diameter of which is DE; it shall be illuminated in the ratio R to S. Thus thesquare of the chord of the half angle DVE, to the square of the chord of the half angleFVG shall be as R to S. And the lens or mirror, of which the diameter is DE, is producedon both sides to G & F ; thus in order that FVG shall become the angle of the lens ormirror from the point V. Which is accomplished and displayed. For the lens or mirrorwill enlarge the illumination of the image in the ratio R to S: as this is apparent byCorollary 2 Prop. 33 of this work.

Corollary.Hence the method follows, the increased strengths of burning are devised in the givenratio ; for the greatest strengths of devising these, either shall be burning with the help ofthe sun or fire, always shall be present in the image of the sun or fire & the illuminationto be increased in some ratio; and the burning increased in the same ratio: as by Prop. 33the proof of this is obvious enough.

Prop. 54. Problema.

Imaginem visibilis, in plano depictam, in quacunque ratione data, illustrare.

Sit visibile AVB, cujus imago in plani depicta, lente vel speculo, cujus diameter DE;sit illustranda, in ratione R ad S. Sit ut R ad S, ita quadratum chordae, semissis anguliDVE, ad quadratum chordae semissis anguli FVG. Et producatur lens vel speculum,cujus diameter DE, ex omni parte in G, & F ; ita ut FVG, fiat angulus diametri, lentis, velspeculi , ex puncto V. Factumq; erit quod proponitur. Lens enim vel speculum, ampliabitimaginis illustrationem, in ratione R ad S: ut patet per Corollarium 2 Prop. 33 hujus.

R

S

A V B

FD E

G


[84]Corollarium.

Hinc sequitur modus, intendendi vires machinatum comburentium, in ratione data ;harum enim machinarum vires maximae, sive comburant ope solis, sive ignis, semperexistunt in solis vel ignis imagine, & in quacunque ratione augmentatur illustratio ; ineadem ratione, augmentatur & ustio : ut per Prop. 33, & ejus consectarium [-ia in originaltext] satis patet.

Prop. 55. Problem.

To present a distinct image of any visible object for a long sighted person, with a giventhickness of lens , for any visual angle you please, with the illumination at that angle.

Let AVB be anything visible whatever, either radiating matter, or an image before orafter the eye, provided it shall be plane : the extreme points of which shall be A, B. Forthe distinct visible image AVB may be represented to the eye of the long sighted personthus in order that the outermost points A, B shall appear with a visual angle equal to theangle MNO, and the visible image AVB shall have the illumination of this angle. Anypoint V is taken in the line AB, near its middle ; from which the line VD is drawn,perpendicular to the plane of the visible object ; and from the chord AB is drawn the partof the circle containing the angle MNO, which shall cut the perpendicular in D: and DDshall become equal to the given thickness of the lens ; and the rays diverging from thepoint V, if the visible object AVB should be before the eye , or converging to the point V,if it should be past the eye, shall be reduced to being parallel [Prop. 19 & 21 of thiswork}, by the lens or mirror, the vertices of which are DD. And by the Corollary to Prop.43 of this work, the rays of the individual points AB of the visible object, with the help ofthe same lens or mirror, also shall be reduced to being parallel. The eye then, receivingthese rays, the visible image AB will always appear with the visual angle ADB, with theillumination of that angle and distinct for the long sighted person; which was to beestablished. {Prop. 44, Cor. 2; & Prop. 49.}

Scholium.But if, on account of an inaccessible distance or smallness of the visible object AB, the

angle ADB cannot be taken conveniently equal to the angle MNO ; the images of thegiven points A, B are projected into an equivalent line, which will be able to subtend theangle MNO conveniently enough ; and by the same way, with the preceding image, if itmight be the given visible object.


In Catoptrics: visibile ante oculum

M

N

L

A B

D

V

O

L

Visibile post oculum

ABV

D

In Dioptricis intermediate diaphano densiore

A BV

L

D

D



AB V

L

D

D


Prop. 55. Problema.

Cujuslibet visibilis imaginem, quovis angulo visorio, cum illius anguli illustratione,presbytis distinctam representare ; cum data lentis crassitie.

Sit AVB visibile quodlibet, sive materia radians, sive imago ante, sive post oculum,dummodo sit planum; cujus extrema puncta, sint A, B. Oculo Presbyti, repraesenteturimago visibilis AVB, distincta, ita ut puncta extrema A, B, appareant angulo visorio,aequali angulo MNO, & imago visibilis AVB habeat, illius anguli illustrationem. In lineaAB, sumatur punctum quodlibet V, prope ipsius medium ; a quo, ducatur recta VD, planovisibilis perpendicularis ; & chorda AB, ducatur circuli portio continens angulum MNO,quae secet perpendicularem VD, in D: fiatq; D D aequalis datae lentis crassidei ; & radii,divergentes a puncto V, si visibile AVB, fuerit ante oculum ; vel ad punctum V,convergentes, si fuerit post oculum ; ad parallelismum reducantur {19 & 21 Hujus}, lentevel speculo, cujus vertices D, D: & per Corollarium Prop. 43 hujus , radii, singulorum

In Dioptricis intermediate diaphano rariore

A BV

L

DD


V AB

L

D

D



A V B

RP

T S

O

visibilis AB punctorum, ope ejusdem lentis, vel speculi, etiam ad parallelismumreducentur; oculo igitur, radios hosce recipienti, semper apparebit imago visibilis AB,angulo visorio ADB, cum illius anguli illustratione: {44. Hujus Cor.2}& presbytisdistincta; quod faciendum erat. {49. Hujus.}

[85]Scholium.

Si autem, ob visibilis AB distantiam inaccessibilem, vel parvitatem, commode nonpoterit sumi angulus ADB, aequalis angulo MNO ; projiciantur imagines, punctorum A,B, in rectam, aequalem datae, quae satis commode, angulum MNO subtendere possit; &eodem modo, cum imagine procedendum; ac si esset visibile datum.

Prop. 56. Problem. To represented the distinct image of any visible object, for any angle of vision, with theillumination of that angle; for the myopic eye, the distance of the vision of which shouldbe known, with the thickness of the lens given too.

Let AVB be any visible object, or radiating matter, with the image either before orafter the eye, provided it shall be plane; of which the most distant parts shall be A, B. Forthe myopic eye, of which the distance of [distinct] vision is S, the image of the visibleobject AVB may be shown distinctly ; thus in order that the extreme points of this A, B isshown with the angle of vision AOB given; and the image of the visible object AVB shallhave the illumination of that angle. In the triangle AOB, OV shall be normal to the baseAB; and OP shall be equal to the right line S ; then, RPT is drawn parallel to the baseAB; and the images of the points A, B are projected into points separated by the distanceRT. In the diagrams of Prop. 52 of this work, MIN shall be equal to RPT. Thus as NIshall be equal to RP; and MI equal to PT itself, and IL equal to the right line PO ; this isfor the given right line S. I say, that the eye of the myopic person at L shall see thedistinct image MIN, with the angle of vision AOB, and with the illumination of thatangle. Since indeed, the triangles MLN and ROT are equal. & similarly; triangle MLN issimilar to triangle AOB; & as

AB : MN : : VO : IL ;but asAB : MN : : VD : ID ;then, asVD : VO : : ID : IL;

and it shall be with, as the distanceof the visible object from thevertex of incidence VD; to theseparation of the visible object tothe centre of the eye VO, thus thedistance of the image from thevertex of emergence ID ; to thedistance of the image from the centre of the eye. The image MN will appear at L with thesame angle and illumination, with which the visible object appears to the eye at O : also


A B

M

O

N

the image appears distinct, because IL is equal to the right line S, for the given distanceof [clear] vision: Which were to be established.

Scholium.In the two previous propositions, we have been talking about the general method of

putting together telescopes and microscopes, in which we have examined the distinctvision of the image, the enlargement and the illumination. But there remains anotherconsideration, namely the angle by which a part of the image seen is viewed, but if thewhole cannot be seen. First however, it is to be known, [regarding] the lenses andmirrors, not less than the spheres, the images of the visible object not arising at the foci,sensibly enough are to be projected; as the writers of optics explain in their own way withspheres. From which it happens, since the image of [those] related to the last image maybe projected through the related images remaining, then incident at last on the eye. Yetknowing the visible images, and the images relating to the last image, to be made fromthe same rays; where indeed the rays emerging from one point of the visible object areconcurrent in another point ; that point of concurrence is the image of the previous visiblepoint, and also where the rays from one point of the visible from one related point to thefinal image are emerging, they are concurrent in the other ; that point of concurrence is

the image of the first point pertaining to the finalimage; and in the same way so for all the points inthe visible, as [for those] pertaining to the finalimage. With these things already touched on, we saythe part of the image seen, to appear from that angleby which a part of the retina illuminated by the raysof the image related to the final image, shall appearfrom the centre of the eye. Let AB be the imagerelated to the final image, the rays of whichilluminate part of the retina MN, and let the centre ofthe eye be L, I say the part of the image seen toappear with the angle of vision MON. Since indeedthe rays, which arrive from the related to the finalimage, also come from the visible object: thereforethe same space MN is also illuminated by the raysfrom the visible object ; therefore the points MN are

the furthest points of the visible object depicted in the retina of the eye, which appearwith the angle of vision MON ; then the proposition is apparent. From this it follows (ifthe centre of the eye O is placed in the image related to the final image e. g. in the rightline AB, which is always able to happen, when the second image is before the eye) to fillthe whole part of the image seen belonging to the first image ; and since it belongs to thefirst image, it will be able to be made wider to please; also the angle of vision of the partof the image seen can be augmented to please : And thus. concerning the part of theimage seen, to have said this little should be sufficient.


A V B

RP

T S

O

Prop. 56. Problema.

Cujuslibet visibilis imaginem, quovis angulo visorio, cum illius anguli illustratione; oculomyopis, cujus visus distantia sit cognita ; distinctam representare ; data quoque lentiscrassitie.

Sit AVB visibile quodlibet, sive materia radians, sive imago ante, sive post oculum,dummodo sit planum; cujus extrema puncta, sint A, B. Oculo myopis, cujus visusdistantia, est recta S, repraesentetur imago visibilis AVB, distincta ; ita ut ejus extremapuncta A, B, appareant angulo visorio, AOB, dato ; & visibilis AVB, imago habeat illiusanguli illustrationem. Sit in triangulo AOB, OV normalis ad basem AB; sitq; OP,aequalis rectae S ; deinde, ducatur RPT, parallela basi AB, & imagines puncturom A, B,projiciantur in puncta, distantia per rectam RT. In figuris Prop. 52 hujus, sit MIN,aequalis RPT ; ita ut NI, sit aequalis RP; & MI, aequalis ipsi PT, & IL, aequalis rectaePO; hoc est rectae datae S. Dico, oculum myopis in L, videre imaginem MIN distinctam,cum angulo visorio AOB, & illius anguli illustratione. Quoniam enim, Triangula MLN,ROT, sunt aequalia,

[87]& similia; triangulum MLN, est simile triangulo AOB; & utAB : MN : : VO : IL ;ut autemAB : MN : : VD : ID ;igitur, utVD : VO : : ID : IL;

cumque sit, ut distantia visibilis avertice incidentiae, VD ; addistantiam visibilis, ab oculicentro, VO, ita distantia imaginis,a vertice emersionis, ID ; addistantiam imaginis, ab oculicentro : eodem angulo, & eademillustratione, apparebit imago MN, in L; quibus apparet visibile oculo in O : apparet etiamimago distincta ; quia IL, aequalis est rectae S, datae visus distantiae : Quae faciendaerant.

Scholium.

In prioribus duabus propositionibus, loquenti sumus de methode generali componenditelescopia, & microscopia; in quibus consideravimus imaginis distinctam visionem,augmentationem, & illustrationem : Restat autem aliud considerandum, nempe angulus,quo videtur pars imaginis visa; si modo tota non videatur. Primo tamen sciendum estlentes & specula conica, non minus quam sphaerica, visibilium in focis non existentium,imagines, satis sensibiliter projicere; ut in sphaericis suo modo demonstrant Opticaescriptores : unde evenit, quod imago pertinentis ad imaginem ultimam, projiciatur perpertinentia ad imagines reliquas, donec tandem in oculum incidat : Sciendum tamen


A B

M

O

N

imagines visibilis, & imagines pertinentis ad imaginem ultimam, ex iisdem confici radiis;ubi enim radii ab uno puncto visibilis emergentes, in aliud punctum concurrunt ; illudpunctum concursus, est imago prioris puncti visibilis; & ubi radii etiam visibilis ab unopuncto pertinentis

[88]ad imaginem ultimam provenientes, in aliudconcurrunt ; illud punctum concursus est imagoprioris puncti pertinentis ad imaginem ultimam; &eodem modo de omnibus punctis tam in visibilis,quam in pertinente ad imaginem ultimam. Hiscepraelibatis, dicimus partem imaginis visam, apparereeo angulo, quo pars retinae illustrata a radiis,imaginis, pertinentis ad imaginem ultimam, ex oculicentro apparet. Sit imago pertinentis ad imaginemultimam AB cujus radii illustrent partem retinae MN,sitque centrum oculi L. Dico portionem imaginisvisam apparere angulo visorio MON. Quoniam enimradii, qui proveniunt a pertinente ad imaginemultimam, etiam proveniunt a visibili: ideo idemspatium MN illustratur etiam a radiis visibilis ; igiturpuncta MN sunt extrema visibilis puncta in oculiretina depicta, quae apparent angulo visorio MON ;

unde patet propositum. Ex hoc sequitur (si oculi centrum O ponatur in imagine pertinentisad imaginem ultimam e. g. in recta AB, quod semper fieri potest, quando secunda imagoest ante oculum) Partem imaginis visam totum pertinens ad imaginem primam implere ;cumque pertinens ad imaginem primam, poterit dilatari ad libitum ; potest quoque &angulus visorius partis imaginis visae augmentari ad libitum : De portione itaq ; imaginisvisa haec pauca dixisse sufficiat.

Prop. 57. Problem.

For a given visual angle of the image, brought to the eye with the help of some mirrors orlenses; and with the distances by themselves of the lenses or mirrors, and of the eye,given in turn; to find the angle of vision of the visible object from its own vertex ofincidence.

Let MLN be the visual angle in the figures of Prop. 52 & 53 of this work, where theimage given is handled with the help of some lenses or mirrors; and the given angles shallbe ILN, ILM, & LI shall be the given distance from the first image; therefore in the rightangled triangle LIN, with the angle ILN & the side IL given, & the side IN given; and bythe same method IM is found: with the image of the visible object MN given; projectedwith the help of some lenses or mirrors: & with the distances of the lenses or mirrors & ofthe image themselves given in turn ; the angle of vision of the visible object may bediscovered from its own vertex of incidence, by Prop. 53 of this work : which was to beestablished.


Yet more straight forward ; as ID to IL, thus the tangent of the angle MLI, to thetangent of the angle MDI, or BDV; similarly too VDB is found; and thus with thedistances of the lenses or mirrors & of the image, themselves given in turn ; & for thevisual angle of the image, from the vertex of its own incidence, ADB; to be proceeding inthe same manner as in the solution of Prop. 53 of this work.

Prop. 57. Problema.

Dato imaginis angulo visorio, ope quotcunque speculorum, vel lentium, in oculum allato;& datis, lentium vel speculorum, imaginum, & oculi, a se invicem distantiis; visibilisangulum visorium, ex vertice sua incidentiae invenire.

Sit in figuris Prop. 52 & 53 hujus ; MLN, angulus visorius, quo comprehenditur imagoope quotcunque lentium vel speculorum, datus ; sintq; dati anguli ILN, ILM, & LI,distantia oculi ab imagine prima ; in triangulo igitur rectangulo LIN, datis, angulo ILN, &latere IL, datur & latus IN ; eodemq; modo reperitur IM: data igitur visibilis imagine,MN; ope quotcunque lentium, vel speculorum, projecta : & datis, lentium, velspeculorum, & imaginum, a se invicem, distantiis ; inveniatur, visibilis angulus visorius,ex vertice suae incidentiae per Prop. 53 hujus : quod erat faciendum.

Expeditius tamen ; ut ID ad IL, ita tangens anguli MLI, ad tangentem anguli MDI, seuBDV; similiter quoque reperitur VDB; datis itaq; lentium, vel speculorum, & imaginum ase invicem distantiis, & angulo visorio imaginis, ex vertice suae incidentiae, ADB;eodem modo procedendum ut in solutione Prop. 53 hujus.

Prop. 58. Problem.

With one focus and the position of the axis given for a kind of ellipse ; and with theposition given of a line cutting the axis : to find the ellipse of which the given line shall beperpendicular to the circumference.

Let AB be the straight line produced in either direction, cutting the axis with the givenposition FA, in the point A: and the ellipse shall be found, of which the focus is F, andthe axis [lies] in the line FA. The ratio of the separation of the foci to the axis of theellipse shall be as R to S ; thus, as the right line BA, incident on the ellipse at the point E,shall be perpendicular to the line of the tangent EM of the ellipse. As R to S thus FA is toFE ; and the angle AEL is equal to FEA ; the line EL is joined. I say that the points F, Lare the focal points; and FE, EL together are equal to the axis of the ellipse sought. Forwith the ellipse described to such an extent, the line BAE, dividing the angle LEF in twoequal parts, will be perpendicular to the tangent, surely ME, in the point E. Since theangles LEA, AEF are equal ; thus as FE will be to FA, so EL to LA; thus as FE to FA,


AB

F

LE

M

R S

that is S to R, hence FE, EL likewise, surely the axis of the ellipse, to FA, AD likewise,surely the separation of the foci: therefore the ellipse with foci F, L has been described,passing through the point E, the ellipse sought; which was to be shown.

Prop. 58. Problema.

Datis; specie ellipseos, uno foco, & positione axeos ; dataq; linea, axem, positionedatum, secante: ellipsim invenire, cujus circumferentiae, perpendicularis sit linea data.

Sit recta AB, utrinq; producta ; secansaxem, positione datum FA, in puncto A:sitque invenienda ellipsis, cujus focus F,& axis in linea FA ; ratio distantiaefocorum, ad axem ellipseos, ut R ad S ;ita, ut BA recta, incidens incircumferentiam ellipseos, in puncto E,sit rectae, EM ellipsim tangenti, inpuncto E, perpendicularis. Sit ut R ad S,ita FA, ad FE ; sitq ; angulus AEL,aequalis angulo FEA ; & jungatur rectaEL. Dico F, L, puncta, esse focos, & FE,EL, simul, esse aequales axi ellipseos

quaesitae. Descripta enim tali ellipsi, recta BAE dividens angulum LEF bifariam, eritperpendicularis ad contingentem in puncto E, nempe ME ; & quoniam anguli LEA, AEFsunt aequales; erit ut FE, ad FA, ita EL, ad LA; igitur ut FE ad FA, hoc est S ad R, ita FE,EL simul,

[90]nempe axis ellipseos, ad FA, AL simul, nempe FL distantiam focorum : est igitur ellipsis,focis F, L, descripta, & transiens per punctum E, ellipsis quaesita ; quod ostendendumerat.

Prop. 59. Problem.

To build a telescope from a single lens, with the aid of which far-sighted eyes mayperceive the angle of vision of a distant visible object a great deal enlarged, with distinctvision.

Let AEB be the angle of vision of a distant visible object; and let a telescope beconstructed from a single lens, with the help of which a relaxed eye can see the image ofa distant object distinctly, with an angle of vision AOB : AEB, AOB shall be twotriangles above the same base AB; and the right line EOL shall be drawn perpendicular tothe base AB, and cutting this at L in two equal parts. Then with the focus L, and with theaxial position LE, a dense ellipse may be described of that denser medium, from which


A BL

O

F C

P Q

MD

N

E

the lens is to be made, and [another] from air, in which the eye exists; thus as the rightline BE produced shall be the perpendicular of this circumference in F; & for the sameellipse, AE produced will be the perpendicular to the circumference in C; with the samefocus L too, and with the same axial position LE, an ellipse of the first density [i.e. air] isalso described ; thus as the right line BO produced will be the perpendicular of thiscircumference in M; & for the same ellipse, AO produced will be the perpendicular to thecircumference in N; & from the revolution of these ellipsis about the common axis EL, acertain lens shall be made, from the before mentioned denser medium, with a convex partFC, and a concave part MN. I say that a long-sighted eye [ i. e. relaxed eye] in theconcavity MN, sees a distinct distant visible object, of which the angle of vision is AEB,having the centre in the axis of the lens produced, through the [increased] angle of visionMON or AOB. For indeed the extreme points ofvisibility are in a single plane with the axis of thelens by hypothesis; because the centre of visibilityis on the axis of the lens produced; & therefore thesurface of common refraction is the same figure,from the rotation of which the lens has beengenerated, surely FC, MN; & EC, EF produceddeal exactly with the visible object, since the angleFEC is equal to this angle of vision; & thereforethe rays of the extreme points of the visible objectare incident normally on the common surface ofrefraction, that is the axes of the pencils of theextreme points of vision, are the rays CE, FE, &therefore the apices of the pencils, of the extremepoints of vision, are B, A , surely the common sections of the axis of the pencils of theextreme points of the visible object with the plane drawn through the focus L, to whichthe perpendicular is the axis of the lens. Then if AO, BO are produced within thecommon surface of refraction to the surface of the lens at P & Q ; & the rays are drawn ofthe extreme visible points incident at P & Q ; & these rays may be refracted with regardto the apices of their pencils A & B, and as they shall be straight lines to the commonsurface of refraction at M & N; but all the rays falling on the surface of refraction NDMitself, & extending to the points of the line ALB, shall be reduced to being parallel withthese rays directly incident on the surface, and entering without refraction; as is easilydeduced from Prop. 17 & 37 of this work : therefore all the rays of the extreme points ofthe visible object are reduced to being parallel with the lines OB, OA ; & therefore thevisible object is seen with an angle of vision equal to AOB itself ; but it appears, & to thefar sighted eye distinct, since the rays of the individual visible points are parallel afterleaving the lens ; which was to be shown.

Scholium.

This problem could also be resolved with the help of a rarer medium between the visibleobject and the eye ; but since it is of no more usefulness, we omit the demonstration ofthis, but it is brought about with the aid of a dense hyperbola, this being brought aboutwith the help of an ellipse. Also the same illustration is able to demonstrate this, for


A BL

O

F C

P QM D N

E

which in the preceding, indeed with the angle of vision always proportional. But Irelinquish the demonstration, for the sake of exercising the talent of the studious.

Note on Prop. 59. Problem: We are dealingwith paraxial rays incident on an ellipsoid.In modern terms, the refractive index ischosen as the inverse of the eccentricity:thus, rays parallel to the optical axisconverge on the more distant focal point L;while paraxial rays converge to a point inthis focal plane. As Gregory shows, a raythrough the near focus E and perpendicularto the surface can be used to locate thispoint, A or B. Hence the position of the firstimage AB is located. The same reasoning isapplied to the second ellipse, which isconsidered as an ellipse of air in a glassmedium: just as a parallel beam of paraxialrays converge for a dense ellipse, so a beamconverging to the far focus of a less denseellipse are rendered parallel to the ray

through the near focus.

Prop. 59. Problema.

Ex unica lente, telescopium fabricare, cujus ope presbyti comprehendant angulumvisorium visibilis longinqui quantumlibet auctum, cum distincta visione.

Sit angulus visorius visibilis longinqui AEB sitq; telesciopium ex unica lenteconstruendum, cujus ope presbytus videat, visibilis longinqui imaginem distinctam, cumangulo visorio AOB: sint duo triangula isoscelia AEB, AOB super eadem basi AB;ducaturq; recta EOL, in basem AB perpendicularis, & eam bifariam secans in L. Deindefoco L , axeos positione LE, describant ellipsis densitatis illius diaphani densioris, e quofabriqanda est lens, & aeris, in quo existat [existas in ms.] oculus ; ita ut recta BEproducta, illius circumferentiae sit perpendicularis in F; & in eadem ellipsi, AE productaerit perpendicularis circumferentiae in C; eodem quoq; foco L, & eadem axeos positioneLE, describatur ellipsis prioris etiam densitatis, ita ut recta BO producta, illiuscircumferentiae sit perpendicularis in M; & in eadem ellipsi, AO producta, eritperpendicularis circumferentiae in N ; & ex revolutione harum ellipsium circa


communem axem EL, fiat lens quaedam, ex diaphano densiore praedicto, convexa adpartes FC, & concavitate ad partes MN. Dico oculum presbyti in concavitate MN, viderevisibile longinquum, cujus angulus visorius AEB, habens centrum in axe lentis producto,distinctum, per angulum visorium MON seu AOB. Sunt enim extrema visibilis puncta inuno plano cum axe lentis ex hypothesi ; quoniam centrum visibilis est in axe lentisproducto ; & igitur communis earum superficies refractionis, est eadem figura ex cujusrevolutione genita est lens, nempe FC, MN, & EC, EF productae exacte comprehenduntvisibile, quoniam angulus FEC est aequalis ejus angulo visorio; & igitur radiiextremorum visibilis punctorum in superficiem refractionis communem normaliterincidentes, id est axes penicillorum, extremorum visibilis punctorum, sunt radii CE, FE,& igitur apeces penicillorum, extremorum visibilis punctorum, sunt B,A, communesnempe sectiones axium penicillorum extremorum visibilis punctorum cum plano per focoL ducto, cui perpendicularis est axis lentis. Deinde si AO, BO in communi refractionissuperficie producantur ad lentis fuperficiem in P, & Q, & ducantur radii extremorumvisibilis punctorum incidentes in P & Q ; refringentur hi radii, in apices suorumpenicillorum A & B; cumq; sint recti ad superficiem communem refractionis in M & N;irrefracte penetrabunt ad A, & B ; superficies autem refractionis NDM omnes radios inse incidentes, & ad puncta rectae ALB tendentes, reducit ad parallelismum illis radiisdirecte in superficiem MDN incidentibus, & irrefracte penetrantibus ; ut facile deduciturex Prop. 17, & 37 hujus : omnes igitur radii extremorum visibilis punctorum reducunturad parallelismum lineis OB, OA ; & igitur visibile videtur angulo visorio aequali ipsiAOB ; apparet autem, & presbytis distinctum, quoniam singulorum visibilis punctorumradii, post egressum e lente sunt paralleli ; quod ostendendum erat.

Scholium.Poterat etiam hoc Problema resolvi, ope diaphani rarioris inter visibile & oculum ; sed

quoniam nullius est utilitatis, demonstrationem ejus praetermittimus, perficitur autemope hyperbolae densitatis, sicuti haec perficitur ope ellipseos. Eadem etiam illustratiodemonstrari poterat hic, quae in prioribus, nempe angulo visorio semper proportionata.Sed studiosis exercendi ingenii gratia, demonstrationem relinquo.

Epilogue.From these few propositions explained here generally, there depends not only the

whole principle of the telescope, but also everything about catoptrics and dioptrics.Indeed it is shown when the image shall be made distinct, when confused; in which placethe image of the visible object shall arise, and how great a part of that may be seen by theeye; in what ratio the image shall be enlarged or diminished, made bright or dim; how theimage is to be made, how it is to be calculated. Now it remains for us to say a little aboutcertain kinds of telescope. These two particular optical instruments are so much in dailyuse: the telescope for distant viewing, and the microscope for the near; to which we mayadd a third, namely the iconoscope, for projecting images ofvisible objects, shown in Prop. 52 of this work. These instruments are of three kinds:without doubt from pure dioptrics, being recognised so much until now; or from purecatoptrics; or from a mixture, part from catoptrics and part dioptrics. Then in each one,to be able to generate countless kinds, indeed from 2, 3, 4, etc lenses in the first, mirrorsin the second, and from mirrors and lenses in the third. Within each of these kinds thereare still two subdivisions, the one shows the image situated erect, the other inverted; from


which that is always to be equally preferred with the others which represents the imagesituated true and erect. But of the first kind, indeed from pure dioptrics, this is the onlyproperty, as many lenses as you please can be continued thus in order that the image maybe enlarged as much as you please; but this has disadvantages. In the first place, theimagines themselves generally are not able to grow in separation in a manageable enoughmanner. Secondly, pertaining to the final image, it is not able to be enlarged enough withregard to the illuminating images, except by weakening the rays of the visible object bythe great width. Thirdly, the thickness of the individual lenses weaken the rays slightly.These indeed are the properties of the second kind [of telescope]: first because,pertaining to the final image, it can be dilated as you please, without any weakening ofthe rays; secondly, because the images themselves need not grow in separation; but withthis they have not a small disadvantage, as it will be hardly possible to continue beyondtwo mirrors. But the third golden kind has no disadvantage, and it is able to have all theproperties of the former kinds, if lenses and mirrors may be duely disposed; this is ifmirrors shall be applicable to the ultimate and penultimate images, and lenses applicableto the remainder of the images. And thus, for the sake of an example, we will describeone telescope of this most perfect kind . Let AFCE be a concave parabolic mirror mostcarefully polished, in the focus C of which is placed a small concave elliptical mirrorhaving a common focus, and a common axes with the concave parabolic mirror, and maybe fastened in this position; but it is necessary that the preceding focus of this ellipticmirror shall approach as close as possible to its vertex, and the other focus shall be thegreatest distance from this vertex; the other focus of this elliptic mirror shall be F, with acommon axis produced beyond the parabolic mirror, and in the vertex of the of theparabolic mirror a round hole MN is hollowed out, in which hole a tube is placed havingthe same axis as the mirrors, big enough to be receiving the reflected rays of the visibleobject from the concave elliptic mirror, and the tube is produced to L, which is close to Fitself, and a crystalline lens is attached at L, of which F is the outer focus, but planar inthe direction of the eye, having a common axis with the mirrors and tube. And this willbe best resolved telescope made for far vision, indeed remote visible objects will appearto the eye through the tube most distinctly enlarged, which is almost in the ratio of thedistances of the vertices from the common foci; and made clear in the same way bywhich vision may be may be explained for such a visual angle; only if the diameterrelated to the final image shall permit the pupil of the eye to be filled with rays; but howthis may be known we have explained in the scholium to Prop. 51 of this work. Butconcerning the magnification of all telescopes, microscopes, and iconoscopes, this rule isto be observed : (indeed it is geometrical, if the centre of vision shall be so, as the eye, onthe axis of the instrument, the image shall never depart sensibly from the truth) as thedistance of the first image from the vertex of its emergence, to the distance of the samefrom the centre of the eye, thus the tangent of the semi - angle of the first image seenfrom the centre of the eye, to the tangent of the semi - angle of the same first image, fromthe vertex of its emergence. Again, as the distance of the second image from the vertex ofits emergence, to the distance of the same from the vertex of incidence, thus the tangentof the semi - angle of the second image seen from the vertex of its emergence, to thetangent of the semi - angle of the second image from the vertex of its emergence; or thetangent of the semi - angle of the third image seen from the vertex of its incidence. Thusit may proceed to the tangent of the semi - angle of the ultimate image seen from the


vertex of its emergence or of the visible object from its own vertex of incidence, whichsemi - angle doubled gives the angle of the object seen from its own vertex of incidence.One may look at the demonstration in Prop. 53 & 57 of this work.

Concerning the mechanical construction of these mirrors and lenses, from othersattempted in vain; I say nothing, as I am less versatile with mechanics. Yet to be assertedboldly, the perfection of optics in lenses and mirrors is sought in vain. If however itshould be pleasing to someone, he will be able to apply the particular propositions of thislittle tract, although indeed imperfect. Indeed the portion of a sphere (although it may notconcentrate the parallel rays in one point) presents the position of the image in onespherical surface, to have the same centre as the the portion, which surface is not alwaysable to concur with other spherical surfaces, yet concurrence is required as it shouldappear satisfactory, at least to the perception. Also, other imperfections of the sphericallenses are, arising from the stated; just as with telescopes with much enlarging,concerning the final image, it will hardly be able to be widened beyond two or threesteps; as will be evident enough from working it out, then horrible obscurity appears. Butagainst hyperbolic lenses, it is only objected that nothing will be able to be most clearlyseen, except a visible point arising on the axis of the instrument. But this weakness (ifthus it is allowed to be called) is sufficiently revealed in the eye itself, yet not to beimputing nature, for which nothing is in vain, as all will be most suitably carried throughto the end. Nevertheless, with conical lenses and mirrors not permitted, it will be ratherwith spherical parts used in the place of spheroids and paraboloids in catopotrics; as withhyperboloids in dioptrics, with which, parts of spheres are less appropriate.

With these we go to the stars.


Epilogus.Ex hisce paucis Propositionibus generaliter hic demonstratis, pendent non solum tota

telescopiorum doctrina, sed etiam universa catoptrica , & dioptrica ; ostenditur enim,quando fiat visio distincta, quando confusa, in quo loco existat visibilis imago, & quantaillius pars ab oculo videatur, in qua ratione fiat imaginis augmentatio, vel diminutio,illustratio, vel obscuratio, quomodo facienda, quomodo calculanda: nunc restat, utpaucula quaedam de telescopiorum generibus dicamus. In hunc usq; diem, duae sunttantum machinae opticae praecipuae; nempe telescopia ad remota aspicienda, &microscopium ad propinqua; quibus nos tertiam adjicimus; nempe Icoscopium, adprojiciendas visibilium imagines, in Prop. 52 hujus demonstratum: atq; hae machinaesunt trium generum; nimirum vel ex puris dioptricis, hactenus tantum cognitis; vel expuris catoptricis; vel ex mixis, partim catoptricis, partim dioptrics : Deinde in unoquoq;genere infinitae possunt esse species, nempe, ex 2, 3, 4, &c. lentibus in primo; speculis insecundo; lentibus, & fpeculis in tertio : & in singulis illis speciebus sunt adhuc duaesubdivisiones, una repraesentat imaginem in situ erecti, altera in situ everso, e quibus illa

A

B

C

DE

F

L L


semper est preferenda caeteris paribus, quae repraesentat imaginem in situ vero, & erecto.Primi autem generis, nempe e puris dioptricis, haec est sola proprietas; quod positlentibus quotvis continuari, ita ut visibile quantum libet amplificetur; haec autem habetincommoda; primo ipsius species plerunq; in longitudinem non satis tractabilemexcrescunt; secundo pertinens ad imaginem ultimam non potest satis dilatari adillustrandas imagines,

[93]absque magna crassitie radios visibilis debilitante ; tertio singularum lentium crassitiesradios visibilis aliquantulum debilitant. Secundi vero generis hae sunt proprietates, primoquod pertinens ad imaginem ultimam possit ad libitum dilatari, absque ulla radiorumdebilitatione ; secundo quod ipsius species non excrescant in longitudinem; hoc autemhabent incommodum non exiguum, quod vix possint earum specula continuari ultra duo.Tertium autem genus aureum nulla habet incommoda, & omnes priorum generumproprietates habere potest; si lentes & specula rite disponantur; hoc est si pertinentia adimaginem ultimam, & penultimam sint specula, & pertinentia ad imagines reliquas sintlentes. Nos itaq; exempli gratia unum hujus perfectissimi generis telescopiumdescribemus: Sit ADCE speculum parabolicum concavum exquisitissime politum, incujus foco C, ponatur parvum speculum ellipticum concavum habens communem focum,& communem axem cum speculo parabolico concavo, & in hoc situ figatur ; oportetautem, ut hujus speculi elliptici focus praedictus quam proxime accedat ad ipsiusverticem, & alter quam longissime ab ea distet, sit focus ipsius alter F, in communi axeproducto extra speculum parabolicum, & in parabolici speculi vertice excavetur foramenrotundum MN, in quo foramine ponatur tubus eundem habens axem cum speculis, satisamplus ad recipiendos radios visibilis a speculo concavo elliptico reflexos, & producaturin L quam proxime ipsi F ; & figatur in L lens chrystallinae, cujus focus exterior F, planaautem ad partes oculi, habens commumem axem cum speculis, & tubo ; eritq; haecfabrica telescopium optimum presbytis destinatum : visibilia enim longinqua, oculoapparebunt, per tubum distinctissime ampliata, quam proxime in ratione distantiarumverticum a focis communibus; & illustrata eodem modo, quo illustraretur visibile taliangulo visum; si modo diameter pertinentis ad imaginem ultimam permittat uveam oculiradiis impleri; quomodo autem hoc sciatur docuimus in Scholio Prop. 51 hujus. Sed deomni Telescopiorum, Microscopiorum, & Icoscopiorum amplificatione, sit haec regulaobservanda:(geometrica enim est, si tam centrum visibilis, quam centrum oculi, sint inaxe machinae, nunquam sensibiliter a veritate aberrat) ut distantia imaginis primae, avertice suae emersionis, ad distantiam

[95]ejusdem ab oculi centro, ita tangens semianguli visorii imaginis primae ex oculi centro,ad tangentum semianguli visorii ejusdem imaginis primae, ex vertice suae emersionis, velsemianguli visorii imaginis secundae, ex vertice suae incidentiae: & rursus ut distantiaimaginis secundae a vertice suae emersionis, ad distantiam ejusdem a vertex incidentiae,ita tangentem semianguli visorii imaginis secundae e vertice suae emersionis, velsemianguli visorii imaginis tertiae ex vertice suae incidentiae; & ita progrediatur adsemiangulum visorium imaginis ultimae ex vertice suae emersionis seu visibilis exvertice suae incidentiae, qui semiangulus duplicatus dat angulum visibilis quaesitum exvertice suae incidentiae. Demonstrationem videre licet in Prop. 53 & 57 hujus.


De Mechanica horum speculorum, & lentium, ab aliis frustra tentata; ego inmechanicis minus versatus nihil dico: audacter tamen asseri, opticae perfectionem inlentibus & speculis sphaericis frustra quaeri. Si vero cui placeat poterit praecipuas hujustractatuli propositiones sphaericis applicare; etsi non adeo perfecte: Portio enim sphaerae(praeterquam quod radios parallelos in unum punctum non congreget) locum imaginispraebet in una superficie sphaerica, habere idem centrum cum portione;quae superficiesnon potest omnimodo concurrere cum alia superficie sphaerica : [tasis?] tamen concursusrequiritur, saltem ad sensum, ut ex praedictis satis apparet. Aliae etiam sunt lentiumsphaericorum imperfectiones, a dictis emergentes; veluti, quod in telescopiis multumamplificantibus, pertinens ad imaginem ultimam, vix poterit ultra duos vel tres gradusdilatari; ut computanti fatis patebit : unde provenit horrenda obscuritas. At contra, lenteshyperbolicas solum objicitur, quod nihil possit distinctissime videm, praeter punctumvisibilis, in axe machinae existens :Sed haec infirmitas (si ita appellare liceat) in ipsooculo est satis manifesta; non tamen naturae imputanda ; quae nihil frustra, sed omniaquam commodissime peragit. Nihilominus, lentibus, & speculis conicis non concessis;satius erit portionibus sphaericis uti loco sphaerideon, & conoideon parabolicatum ; incatoptrica; quam hyperbolicarum in dioptrica; cum quibus portiones sphaericae minusconveniunt.

His itur ad astra.

James Gregory's Optica Promota143

A

B

C

P

D F

IEG

A

B

C

P

DF

IE

G

Z

RO

[96]

Prop. 60. Problem

With the two given heights of a phenomenon around the pole, in a meridian with theheight of the pole given; to find the sum or difference of the parallaxes of the givenheights.

In this problem there are three different cases; thefirst case is where the meridian height is observedbetween the pole and the vertical ; in which ABC shallbe the quadrant of the circle of the meridian, P thepole, D & E the true positions of the phenomena,above & below the pole, and these equidistant from thepole. Truly, the apparent positions shall be F, G. Thearc PI is equal to the arc PF [PE in the text]. I say thatthe arc IG is the sum of the parallaxes sought. Sinceindeed, the arcs PD & PE are equal; and the arcs PI &PF are equal, & EI itself equal to DF; which DF is the

parallax of the superior observation, and EG the parallax of the inferior. Therefore the arcIG, the sum of these, is the sum of the parallaxes sought. Which it was required to find.

The second case is when the zenith falls between the pole and the higher observation;in which BZC shall be the meridian circle, P the pole, Z the zenith, D & E the truelocations of the phenomenon, above and below the pole; F & G the apparent locations.Let the arc PI be equal to the arc PF, which is smaller than the arc PG by IG itself. I saythat the arc IG is the differences of the parallaxes sought. Indeed the arc PD is equal tothe arc PE, from which, because D & E are the true locations of the phenomena.

Therefore DF will be the upperparallax of the observation, and asEG shall be the lower parallax of theobservation, IG will be the differenceof the parallaxes observed; which itwas required to find.

The third case, in which thephenomenon is observed in thezenith, is the easiest, since thedifference of the apparent distancesfrom the pole, surely RO, is equal,not only to the difference of the

parallaxes, but also has no parallax in the vertical location.

Note: The diagrams show AP as the invariant direction of the polar axis; AB is thedirection of the vertical initially, while AC is the vertical 12 hours later. The apparentheight or angle measured relative to the polar axis AP is always less due to atmosphericrefraction, while the dotted arcs show the motions of the true and apparent heights on thecelestial sphere during the 12 hour period.


S

D

E

M

A B

Prop. 60. Problema.

Datis duabus altitudinibus, Phaenomeni circumpolaris, in meridiano; cum altitudine poli : datarumaltitudinum, parallaxium summam, vel differentiam invenire.

In hoc Problemate, tres sunt casus diversi; primus casus, est quando altitudo meridiana, supra polum,observatur inter polum & verticem ; in quo sit ABC, quadrans meridiani circuli, P polus, D & E vera locaPhaenomeni, supra & infra polum, adeoq; a polo aequidistantia : Loca vero apparentia sint F, G, infra locavera; sitq; PI arcus, arcui PE, aequalis. Dico IG arcum esse summam parallaxium quaesitam. Quoniamenim, arcus PD, & PE, sunt aequales; & arcus PI, arcui PF; erit & EI ipsi DF aequalis; qui DF estparallaxis, superioris observationis; & EG, parallaxis inferioris; arcus ergo IG, eorum summa; est summaparallaxium quaesita. Quam invenire oportuit.

Secundus casus est, quando zenith cadit inter polum & altiorem observationem; in quo sit BZC, circulusmeridianus; P polus, Z zenith, D, & E, vera loca Phaenomeni, supra & infra polum, F, & G, locaapparentia; sit PI arcus, aequalis arcui PF; qui minor est arce PG, per ipsum IG. Dico arcum IG, essedifferentiam parallaxium quaesitam. Est enim arcus PD, aequalis arcui PE, ex eo quod D, & E sunt veraloca Phaenomeni; erit igitur, & DF

[97]parallaxis superioris observationis, aequalis EI; cumq; EG, sit parallaxis inferioris observationis, erit earumdifferentia IG, differentia parallaxium quaesita; quam invenire oportuit.

Tertius casus, in quo Phaenomenon observatur in zenith, est facillimus; quoniam differentiadistantiarum apparentium a polo, nimirum RO, aequatur, non solum differentiae parallaxium, sed etiam invertice nullam habet parallaxem.

Prop. 61. Problem.Given the longitude and latitude of two places on the terrestrial globe ; to find the

angles with the common azimuth which the meridian of the places makes and the distancebetween them.

First, if those places shall be placed on opposite ends of a diameter, it is agreed that allthe azimuth circles are common.

Secondly, if the places differ only with latitude, the meridian circle is the commonazimuth.

Thirdly, if they differ with so much longitude , of if they differ with latitude andlongitude; thus the proposition is investigated. ASBM shall be a hemisphere of the earthin which there are the locations D & E with different latitude and longitude. The meridiancircles SDM, SEM and common azimuth DE are drawn. In the spherical triangle DSE,the sides DS, ES are given, with the angle DSE taken from these; from which the anglesSDE & SED are found, with the length DE of the separation of the places, which it wasnecessary to find.

Prop. 61. Problema.Datis longitudinibus, & latitudinibus, duorum locorum,

in globo terreno; angulos, quos commune Azimuth, cumMeridianis locorum facit ; & locorum distantiam invenire.

Primo, si loca ista sint ex diametro opposita; constat,omnes circulos Azimuthales esse communes.

Secundo, si sola latitudine differant, meridianus circulusest azimuth commune.

Tertio, si longitudine tantum; vel longitudine, &latitudine differant ; sic investigatur propositum. Sit ASBMterrae Hemisphaerium, in quo, sunt loca D, & E,


R

D

E

M

A

BO P

NI

longitudine, & latitudine, differentia; ducantur meridiani circuli, SDM, SEM, & azimuth commune DE ; intriangulo sphaerico, DSE, dantur latera DS, ES, cum angulo ab iis comprehenso DSE ; ex quibus,inveniantur anguli SDE, & SED, cum latere DE locorum distantia ; quae invenire oportuit.

Prop. 62. Problem.To find the sum or difference for the observed parallaxes P from two altitudes of the

same phenomenon brought together at the same time in different locations with acommon azimuth.

Let the circle ABDE be the common azimuth of the locations, B and D shall be thevertical points [the local zeniths]. In the first case, there shall be some [astronomical]phenomenon between the verticals B and D, of which the true position shall be O; andfor the observer beyond B, truly it would seem to be at P. Also, for the observer beyondD, it would appear to be at R. Therefore the observed arcs BP and DR, if they should beadded together at the same time, make an arc that exceeds the arc BD, the distancebetween the [true] positions, by the sum of the parallaxes RP.

In the second case, the true location of any phenomenon shall be I, outside theintercepted arc BD, appearing at M for the observer beyond B, and appearing at N for theobserver truly beyond D. Therefore with the arcs DB, BM subtracted at the same timefrom the arc DN, the difference MN of the parallaxes sought will remain.

Prop. 62. Problema.Ex collatis, duabus altitudinibus, ejusdem Phaenomeni; ad idem tempus, in diversis locis, & eorum

azimuth communi, observatis P parallaxium summam, vel differentiam, invenire.

Sit circulus ABDE, azimuth commune locorum; puncta verticalia sint B, D: sitq; primo inter vertices B,D Phaemomenon aliquod, cujus verus locus O; observatori vero infra B,

[99]videatur in P; & observatori infra D, videatur in R; arcus ergo BP, DR, observati; si simul addantur,

efficiunt arcum, excedentem arcum BD, distantiamlocorum, per RP, summam parallaxium.

Secundo, extra arcum interceptum, BD, sit I, veruslocus, alicujus Phaenomeni; observatori infra B,apparentis in M ; observatori vero infra D, apparentisinferius in N. Subductis ergo arcubus, DB, BM, simul, exarcu DN, remanebit MN, differentia parallaxium quaesita.

Prop. 63. Theorem.The distance of the phenomena, from the centre of the earth, is to the radius of the

earth as the sine of the angle of the apparent distance to the vertical to the sine of theparallaxes.

Let A be the centre of the earth, the position of the observer on the surface B, of whichthe zenith is D; and let the distance of the phenomenon from the centre of the earth beAO. The separation DBO therefore is [the angle] appearing from the vertical, and BOAthe angle of the parallax. In the plane triangle BOA: as the distance AO of the


D

A

B

O

phenomenon from the centre of the earth, to BA, the radius of the earth ; thus the sine ofthe angle DBO, or the [angular] separation from the vertical, to the sine of the angleBOA, or the parallax, which was to be demonstrated.

Prop. 63. Theorema.Distantia Phaenomeni, a centro terrae, est ad semidiametrum terra ; ut sinus, distantiae appatentis a

vertice; ad sinum parallaxeos.

Sit centrum terrae A, locus observatoris in terrae superficieB; cujus zenith D ; sitque distantia phaenomeni O, a centroterrae AO, est igitur DBO distantia apparens a vertice, & BOA,parallaxis

[100]; & in triangulo rectilineo BOA; ut AO, distantia phaenomeni acentro terrae; ad BA, semidiametrum terrae ; ita sinus anguliDBO, seu distantiae apparentis a vertice ; ad sinum anguliBOA, seu parallaxeos. Quod demonstrandum erat.

Prop. 64. Theorem.

The sines of the parallaxes, to the sines of distances appearing from the vertical aredirectly proportionals.

Indeed, the sines of the angles of the distances appearing from the vertical always are to the sines of theparallaxes, as the distances of the phenomenon from the centre of the earth, to the radius of the earth. Thus,as the sine of the distance appearing from the vertical for one observer, to the sine of the same parallax, sothe sine of the distance appearing from the vertical for another observer, to the sine of their parallax. Bybeing exchanged: as the sine of the distance appearing from the vertical of the one observer, to the sine ofthe distance appearing from the vertical of another observer; thus the sine of the parallax of the oneobserver, to the sine of the parallax of the other observer, which was to be demonstrated.

Prop. 64. Theorema.

Sinus Parallaxium, sinubus diftantiarum apparentiam a vertice, sunt directe proportionales.

Sinus enim, distantiarum apparentium a vertice, semper sunt ad sinus parallaxium , ut distantiaPhenomeni, a centro terrae, ad semidiametrum terrae; ergo ut sinus, distantiae apparentis a vertice, uniusobservationis; ad sinum parallaxeos ejusdem; ita sinus, distantiae apparentis a vertice, alteriusobservationis; ad sinum suae parallaxeos: & permutando ; ut sinus distantiae apparentis a vertice uniusobservationis ; ad sinum distantiae apparentis a vertice alterius observationis; ita sinus parallaxeos uniusobservationis ; ad sinum parallaxeos alterius observationis ; quod demonstrandum erat.

Prop. 65. Problem.Given the sum or difference of two arcs, with the ratio of the sines, to find the arcs

themselves.

Let the angle ABC be the given sum of the arcs, and the ratio of the sines AB to BD;with the smaller AB as radius, the semicircle ACE is described; and the straight lines CD,CE and CA are drawn. The line EF is drawn parallel to CA. And in the triangle CBD, asCB (i. e. AB) will be to BD, thus the sine of the angle CDB will be to the sine of theangle BCD ; therefore it is clear that the angles BCD and BDC themselves are to be


A B

C

E

FD

sought. Indeed, the sum ofthese is the given angle CBA,and the given ratio of thesines AB to BD. It is cleartoo, that the angle at thecircumference CEA is equalto half the angle at the centre,CBA. Therefore the angleECF, for which the greater of the angles sought exceeds half the sum ECB, is half thedifference of the same angles sought. Therefore, with the radius [meaning the adjacentside] CE placed, on account of the right angles ACE and CEF, AC will be as the tangentof half the sum of the angles sought, and EF as the tangent of half the difference of thesame angles. Since the lines AC and EF are parallel: as AD to ED, the sum of the termsof the ratio of the sines to the difference of the same; thus, AC to EF, the tangent of halfthe sum of the given arcs to the tangent of half the differences. Which added to half thegiven sum reveals the greater angle ; or from the same subtracted to bring about thesmaller angle. Or, as ED to AD, the difference of the terms of the ratio of the sines to thesum of the same; thus EF to AC, the tangent of half the difference of the given arcs to thetangents of half the sum: from which the given arcs are found as before. Which was to beaccomplished.

Prop. 65. Problema.Data summa, vel differentia, duorum arcuum; cum sinuum ratione; ipsos arcus invenire.

Sit data arcuum summa, angulus ABC, & sinuum ratio, AB, ad BD; minore AB semidiametro,describatur semicirculus ACE;

[101]& ducantur lineae rectae CD, CE, CA; atq rectae CA parallela ducat EF. Eritq; in triangulo rectilineo,

CBD; ut CB, (id est AB) ad BD, ita sinus anguli CDB; ad sinum anguli BCD; manifestum est igitur istosangulos, nempe BCD, BDC, esse quaesitos; eorum enim summa, est angulus CBA,datus, & sinuum ratio,AB, ad BD data: manifestum quoq; est, angulum CEA in circumferentia, esse semissem anguli CBA, incentro; & igitur angulus ECF, quo major quaesitorum, excedit semissem summae ECB, est semissisdifferentiae eorundem angulorum quaesitorum. Posito igitur CE radio ; ob angulos ACE, CEF, rectos ; eritAC tangens semissis summae angulorum quaesitorum, & EF, tangens semissis differentiae eorundemangulorum ; & quia rectae AC, EF, sunt parallelae ; erit ut AD, summa terminorum rationis sinuum; ad ED,differentiam eorundem; ita AC, tangens semissis summae arcuum datae ; ad EF, tangentem semissisdifferentiae ; qua addita, ad semissem summae datae, producitur angulus major ; vel ab eadem subducta,efficitur angulus minor. Vel, ut ED, differentia terminorum rationis sinuum ; ad AD summam eorundem;ita EF, tangens semissis differentiae arcuum datae; ad AC, tangentem semissis summae : qua datainveniuntur arcus ut prius. Quod erat faciendum.


A B

C

E

FDα + β

β

α (α + β)/2

( β − α)/2

α < β

y - x y + x

A

BCL P

DEO

Commentary on Prop. 65.

In the diagram, BD = y + x,and AB = y - x, the sum anddifference of the arcs x = Rα and y = Rβ, for some radius R.Now, in triangle CBD,CB/BD = sin α/sin β , andy/x= AD/ED = AC/EF = tan(β+α)/2.tan(β-α)/2

Hence, (x + y)/x = BD/y = 1 + tan(half-sum)/tan(half-diff.), from which y can befound, and x, etc.

Prop. 66. Problem.For a given spherical triangle, and with the arc of a great circle given: from an angle of the spherical

triangle, any arc of a great circle is drawn you please to the opposite side. To find the segments of thegiven side and [ the length ] of the arc given, from the given ratio of the sines from the sides of onesegment, and the difference of the arcs.

Let ABC be the spherical triangle, [of which] allthe sides and angles are given; and DE shall be agiven great arc of the circle. In the spherical triangleABC, an arc AL is drawn to the side BC placedopposite; then EO, [an arc] equal to the arc AL, istaken from DE. The sine of the arc DO to the sineof the arc BL is the given ratio; and the arcs DOand BL are sought. A perpendicular arc AP is sentfrom the angle A to the side BC placed opposite,

which will be given with the segments BP and PC, since all the sides and all the anglesare given in the triangle ABC. Thus the sine of the arc AL may be put surely as 1 [i. e.an unknown amount], and the sine of the arc LP will be given in terms of the unknownnumber. Then the sine of the arc BL will be given in related numbers from the givendifference of these, the sine of the arc LP and the sine of the arc BP. And in the sameway ; from the sine of the given arc DE, and from the sine of the arc OE, surely 1 , thesine of the arc DO will be given by the difference of these in related numbers. And sincethe sine of the arc DO to the sine of the arc BL will be the given ratio, and hence anequation will be given, from the resolution of which the value of the root will becomeknown, without doubt the sine of the arc AL, or the arc EO ; from which being given, therest are easily given, which were to be found.


Dato triangulo sphaerico, & arcu circuli maximi ; ductoq; ab angulo, trianguli sphaerici in latusoppositum, arcu circuli maximi quolibet ; e data ratione sinuum, unius segmenti lateris, & differentiaarcuum; lateris & arcus dati segmenta invenire.

Sit triangulum sphaericum ABC, data habens, omnia latera, & omnes angulos; sitq; datus arcus maximicirculi DE, & in triangulo sphaerico ABC, demittatur arcus AL, in latus oppositum BC ; deinde arcui ALaequalis EO, auferatur a DE ; & e data ratione, sinus arcus DO, ad sinum arcus BL ; quaeritur, & arcus DO


A

BCL P

D

EO

a

b

c

x

y

l - xx

p

qr

l

z

αβγ

Fig. 66 - 1.

AB

C

a

b

c

α

β

γ

c

α β

90 - b 90 - a

Fig. 66 - 2.

& arcus BL. Ab angulo A, in latus oppositum BC, demittatur perpendicularis arcus AP; qui dabitur, cumsegmentis BP, PC; quoniam omnia latera, & omnes anguli dantur in triangulo sphaerico ABC. Ponaturitaque sinus arcus AL ;nempe 1 ; dabitur & sinus arcus LP, in numeris cossicis. Deinde datis, sinu arcusLP, & sinu arcus BP, dabitur & sinus arcus BL eorum differentiae, in numeris cossicis. Et eodem modo ; edatis, sinu arcus DE, & sinu arcus OE, nempe 1 ; dabitur & sinus arcus DO, eorum differentiae, innumeris cossicis. Cumq; sit data ratio, sinus arcus DO, ad sinum arcus BL; dabitur hinc aequatio, ex cujusresolutione, innotescet valor radicis, sinus nimirum arcus AL, vel arcus EO ; quo dato, facile dantur reliquo,quae invenienda erant.

Commentary on Problem 66.

We are given the angles A, B, C ; thesides a, b, c ; together with the arcDE ( = l ), and sin(l - x) / sin y = k. Thelengths of the arcs y (and a - y) and thelength of the arc LA, or EO (= x), are to befound.The basic formulae for the angles A, B, Cand the sides a, b, c of a spherical triangle arethe cosine rules for sides (I) and angles (II),and the sine rule (III):

cos a = cos b cos c + sin b sin c cos A (I), and similarly for cos b and cos c in a cyclicmanner ;

cos A = - cos B cos C + sin B sin C cos a (II), and similarly for cos B and cos C in acyclic manner ; andsin a : sin b : sin c = sin A : sin B : sin C (III).

In addition, Napier's rules for right - angledspherical triangles are of assistance in solvingthis problem: For any arc on the circle, such as c,the cosine of the arc is equal to the product of thecotangents of the adjacent angles and also to theproduct of the sines of the complements of theopposite arcs. Thus, e.g. in Fig. 66 - 2,

cos c = cot α cot β = sin(90 - b) sin(90 - a) = cos a cos b.Napier's rules applied to triangle APL give :

cos(90 - p) = sin p = sin b sin C ; cos x = cos z cos p ; sin z = sin x sin β ;while in triangle APB, cos(90 - q) = sin q = cot B tan p. Hence p and q are known.

Now, sin(l - x) / sin y is given, or sin(l - x) = k sin y ; also,from the arcs, y = q - z, and sin y = sin(q - z) = sin q cos z - cos q sin z.

Hence, k sin q cos z - k cos q sin z = sin l cos x - cos l sin x, giving on substitution fromabove: k sin q cos x /cos p - k cos q sin x sin β = sin l cos x - cos l sin x.��


A

BCL P

D

E

ON

M

Hence, k sin q/cos p = sin l , and k cos q sin β = cos l; giving sin β = cot l cos p cot q ;hence β is known, and as cos β = cot x tan p from triangle APL, it follows that cot x andhence x is also known.

Prop. 67. ProblemFor two given spherical triangles, and from one angle of each an arc has been drawn

to the opposite side equal for both : to find the segments of each side from the given ratioof the sine of the segment of the side of one triangle to the sine of the segment of the sideof the other triangle.

Let ABC, EDM be two spherical triangles with allthe sides and angles having been given. From oneangle of each, say A and E, equal arcs AL, EO aredropped to each to the opposite sides, certainly BC,DM. And ratio is given of the sine of the arc DO tothe sine of the arc BL. Each arc BL and DO issought. The sine of the arc AL, or the sine of the arcEO, ( indeed both are equal by hypothesis) is put as1 : then from the same method , by which in theabove problem, both the sine of the arc DO and thesine of the arc BL may be found in related numbers,whenever the ratio of these may be given from

hypothesis. Hence an equation will be given, from the resolution of which will give thesine of the arc EO, or of the arc AL, the value of the root ; and from this being found,the remainder will be easily given, which were to be found.

Scholium.By the same means, other problems are able to be solved, even if AL, EO are not

equal, if the ratio of their sines is given.


Datis duobus triangulis sphaericis, & ab uno utriusque angulo, in latus oppositum, ducto arcu inutroque triangulo aequalis : e data ratione sinus, segmenti lateris unius trianguli ; ad sinum, segmentilateris alterius trianguli ; utriusque lateris segmenta invenire.

Sint duo triangula sphaerica, ABC, EDM, data habentia, omnia latera, & onmes angulos ; & ab unoutriusq, angulo nempe A & E demittantur, in utriusque latus oppositum, nimirum BC, DM, arcus aequalesAL, EO: sitque data ratio, sinus arcus DO, ad sinuum arcus BL. Quaeritur uterque arcus, BL & DO.Ponatur sinus arcus AL, seu sinus arcus EO, (sunt enim ex hypothesi aequales) 1 : deinde, eodem modo,quo in superiore problemate, inveniatur & sinus arcus DO, & sinus arcus BL, in numeris cossicis : cumquedetur eorum ratio, ex hypothesi ; Dabitur hinc aequatio ; quae resoluta, dabit sinum arcus EO, seu arcusAL, valorem radicis ; & eo reperto, facile dabuntur reliqua, quae invenienda erant.

Scholium.Iisdem mediis, possunt resolvi utraq; haec problemata ; etiamsi AL, EO, non sint aequales, si detur ratio

eorum sinuum.


ARZ

P

L

N

IM

N

L

ZA

RIM

P

Prop. 68. Problem.From two given altitudes, the given azimuths of the phenomenon and the altitude of

the pole, to show clearly the parallaxes of each altitude.

Let PZR be the arc of a meridian, P the pole, and Z the zenith. The true parallel of thephenomenon shall be AIL. R shall be the first one of the apparent positions of thephenomenon, in a meridian of which the altitude shall be known ( and therefore itsdistance from the vertical, ZR, shall be known). Another of the apparent positions shallbe at N, the distance of which from the vertical ZN and the angle NZP of the azimuthcircle to the meridian are given. But the true positions shall be in the commonintersections of the given azimuths with the true parallel, surely A and L. Since indeed inthe spherical triangle NZP, the sides ZN and ZP are given, and the angle NZP, then thewhole triangle NZP is given [i. e. from the two arcs and the included angle]. Thereforethe spherical triangle NZP, and the arc of the great circle PR are given. The arc PL isdrawn from the angle P to the side ZN. The ratio: sine of the segment LN to sine of thearc RA is given; for the difference of the arcs PR and PL, as PL is equal to PA (byhypothesis). (Indeed it is the same ratio which is between the sines of the given arcs ZR,

ZN). Therefore from Prop. 66, the arcs LN and AR shall be give, the parallaxes surely ofthe altitudes sought.

[105]In the second case, both the apparent positions M and N shall be beyond the meridian ;

but the true position (as was stated hitherto) will be in the common intersection of thegiven azimuth with the true parallel, needless to say, in I and L. And since the arcs ZM, ZN,ZP, and the angles MZP, NZP are known, the two triangles ZPM, ZPN, are given. The equalarcs PI, PL are drawn to the sides of which, surely ZM, ZN from the opposite angles; andthe ratio of the sine of the arc IM, to the sine of the arc LN is given : (for it is the same,which the sine of the distance appearing from the vertex ZN) therefore both the arcs IMand LN are given, the parallaxes sought, from Prop. 67 ; which were to be found.

Prop. 68. Problema.Ex Datis duobus altitudinibus ; cum Azimuthis Phaenomeni ; & altitudine Poli ; utriusque altitudinis

parallaxes, enucleare.


A

M

ON

D

B

Sit PZR, arcus meridiani, P polus, Z zenith ; verus Phaenomeni parallelus, AIL. Sitque primo unusapparentium locorum phaenomeni R, in meridiano, cujus altitudo, (& igitur ipsius a vertice distantia, ZR)sit cognita ; alter apparentium locorum in N, cujus distantia a vertice ZN, & angulus circuli azimuth, cummeridiano NZP, sint data ; loca autem vera, erunt in communi intersectione , azimuthorum datorum, cumparallelo vero, nempe A, & L. Quoniam igitur in triangulo sphaerico, NZP, dantur latera ZN, ZP, &angulus NZP; datur & totum triangulum NZP: Datur igitur triangulum sphaericum NZP, & arcus circulimaximi PR ; duciturque arcus PL, ab angulo P, in latus ZN; & datur ratio, sinus segmenti LN, ad sinumarcus RA, differentiae arcuum PR, & PL, seu illi aequalis ( ex hypothesi) PA ; (est enim eadem ratio quaeest inter sinus arcuum datorum ZR, ZN) dabitur igitur arcus LN, & arcus AR, parallaxes nempe altitudinumquaesitae. {66. Hujus. }

[105]Secundo, sint ambo apparentia loca, M, & N extra meridianum; loca autem vera ( ut hactenus dictum )

erunt in communi intersectione azimuthorum datorum cum parallelo vero ; nimirum in I, & L. Et quoniamcognita sunt ZM, ZN, ZP, MZP, NZP; dabuntur duo triangula sphaerica ZPM, ZPN, in quorum lateranempe ZM, ZN, ab angulis oppositis ducuntur arcus aequales PI, PL: daturq; ratio sinus arcus IM, ad sinumarcus LN: (eadem enim est, quae sinus distantiae apperentis a vertice ZN, ) datur ideo & arcus IM, & arcusLN, parallaxes quaesitae { 67. Hujus.}; quae inveniendae erant.

Prop. 69. Theorem.The ratio of the sine of the parallax of one phenomenon to the sine of the parallax of

another phenomenon is composed from the reciprocal proportion of the distance from thecentre of the earth, and from the direct proportion of the sine of the apparent distancefrom the vertical.

Let A be the centre of the earth, B the position of observation on the surface of theearth, of which the zenith is D. O and M shall be two phenomena. I say the ratio of thesine of the parallax BOA, to the sine of the parallax BMA, to be composed from the ratioAM to AO; and from the ratio, the sine of the angle DBO, to the sine of the angle DBM.BO is produced in N, and AN is equal to AM itself. It is therefore clear that thephenomenon M arising at N has the parallax BNA. And as AN, that is AM, shall be toAO: so the sine of the parallax BOA shall be to the sine of the parallax BNA. And, as thesine of the angle DBO to the sine of the angle DBM, thus the sine of the parallax BNA, tothe sine of the parallax BMA [applying the sine rule to triangles BAM & BAN, notingthat AM = AN]. If then there shall be several quantities, namely the sine of the angleBOS, the sine of the angle BNA, and the sine of the angle BMA ; the ratio of the first,surely the sine of the parallax BOA, to the last, surely the sine of the parallax BMA, iscomposed from the middle ratio;

[106]surely the sine of the angle BOA, to the sine ofthe angle BMA, or AN to AO ; and the sine of theangle BNA, to the sine of the angle BMA, or thesine of the angle DBO, to the sine of the angleDBM ; which was to be shown1.

Corollary.From the demonstration of the theorem it is

readily deduced that the sine, either of thehorizontal parallax or of the apparent radius is inthe inverse proportion to the distance of the


AB

CE

F

D

C

E

L

phenomenon from the centre of the earth ; and therefore the sines of the parallax aredirectly proportional to the sines of the apparent diameters2.

Notes on Prop. 69.1 The relevant triangles are (1) ABO ; (2) BAN ; (3) BAM. From these we obtain in turn:sin BOA/ sin DBO = AB/ AO ; sin DBO/ sin ANB = AN/ AB ;sin BMA/ sin DBM = AB/ AM. From these it follows thatsin BOA/ sin BMA = (AN/ AO) . sin DBO/ sin DBM = (AN/ AO) . sin BNA/ sin AMB.

2 From triangle BAM, sin BMA/ AB = sin DBM/ AM = sin BAM/ BM. Now, AB is theradius of the earth RE; the angle ABM or θ corresponds to the apparent radius, as we arelooking down on the earth from a pole; BMA is the horizontal parallax α, and BM is theapparent radius RA. Hence:sin α / RE = sin θ / AM = sin BAM/ BM, giving sin α = (RE sin θ)/AM, from which theresults follow.

Prop. 69. Theorema.

Ratio sinus parallaxeos unius Phaenomeni, ad sinum parallaxeos alterius Phaenomeni, est composita,ex reciproca proportione, distantiarum a centro terrae, & directa proportiione sinuum, distantiarumapparentium a vertice.

Sit centrum terrae A, locus observatoris in terrae superficie B, cujus zenith D : Sintque duophaenomena, O & M. Dico rationem sinus parallaxeos BOA, ad sinum parallaxeos BMA; essecompositam, ex ratione AM, ad AO ; & ex ratione, sinus anguli DBO, ad sinum anguli DBM. ProducaturBO, in N; sitque AN, aequalis ipsi AM: Manifestum est igitur phaenomenon M, in N existens habereparallaxem BNA : eritque , ut AN, hoc est AM, ad AO, ita sinus parallaxeos BOA, ad sinum parallaxeosBNA ; &, ut sinus anguli DBO, ad sinum anguli DBM, ita sinus parallaxeos BNA, ad sinum parallaxeosBMA. Si igitur fuerint quotcunq; quantitates, nempe sinus anguli BOA, sinus anguli BNA, & sinus anguliBMA ; ratio primae, nempe sinus parallaxeos BOA ; ad ultimam, nempe sinum parallaxeos BMA ;componitur ex rationibus mediarum ; nempe , sinus anguli

[106]BOA, ad sinum anguli BMA, vel AN ad AO ; & sinus anguli BNA, ad sinum anguli BMA vel sinus anguliDBO, ad sinum anguli DBM ; quod demonstrandum erat.

Corollarium.Ex Theorematis demonstratione facile deducitur, sinus, sive parallexium horizontalium, sive

semidiametrorum apparentium, esse in reciproca proportione distantiarum phaenomeni a centro terrae ; &igitur sinus parallaxium, sunt directe proportionales sinubus semidiametrorum apparentium.

Prop. 70. Theorem.From two given longitudes and

latitudes of the motion of a givenphenomenon in a great circle, to findthe point of intersection and theangle of that great circle with theecliptic.

ACB shall be the ecliptic of whichF is the pole and A the first [principal


star] of Aries, from which two longitudes of the phenomenon AB, AC may beconsidered; and BD, CE shall be latitudes, and D, E shall be the positions of thephenomenon. Then the great circle is produced in which the phenomenon is considered tomove, and L shall be

[107]the point of intersection with the ecliptic. In the spherical triangle DEF, the sides DF, FEare given, together with the angle of interception DFE, and therefore the angle DEF isgiven. Then in the spherical triangle ECL, with the right angle at C, the side EC is given,and the angle LEC, and hence the side CL will be found, the distance of the point ofintersection from C, and the angle ELC, which it was required to find.

Prop. 70. Theorema.

Ex datis, duabus longtudinibus, & latitudinius Phaenomeni, in circulo maximo moventis ; punctumintersectionis ; & angulum, illius circuli maximi cum ecliptica, invenire.

Sit ACB ecliptica cuius polus F, A principium arietis ; a quo numerentur duae phaenomenilongitudines, AB, AC ; sintque ; latitudines BD, CE, & loca phaenomeni D, E. Deinde producatur circulusmaximus, in quo moveri supponitur phaenomenon;

[107]sitque punctum intersectionis cum ecliptica L. In triangulo sphaerico DEF, dantur latera DF, FE, una cumangulo intercepto DFE; & igitur datur angulus DEF: Deinde in triangulo sphaerico ECL, rectangulo ad C,datur latus EC, & angulus LEC, & proinde dabitur latus GL, distantia puncti intersectionis a C; & angulusELC ; quae invenire oportuit.

Prop. 71. Problem.

From three given longitudes and latitudes of a phenomenon, from the movements in asmall circle ; to find the distance of the small circle from its pole, and from which to findthe position of the pole, according to the longitude and latitude.

Let ACBL be the ecliptic, the pole of which is F; A shall be the first star of Aries,from which the three longitudes of the phenomenon AC, AB, AL are measured ; and thealtitudes shall be CD, BN, LM ; & the positions of the phenomenon D, N, M. Andbecause of course the complements of the latitudes DF, NF, MF are given, with theangles of interception DFN, MFN from the differences of the longitudes. Both the arcsDN, MN and the angles DNF, MNE, NMF are given ; therefore the angle MND will begiven. Then two arcs EO, IO can be raised, perpendicular to the arcs DN, NM; andcutting these in two, at E and I: and the point O shall be the common intersection ofthese; which by necessity will be the pole of the minor circle in which the phenomenon ismoving. The arc IE is drawn, and in the triangle INE, the sides IN, NE are given, and thecommon angle INE; and hence the angles

[108]NEI, NIE and the side IE are given. Then in the triangle OIE the angles OEI, OIE, surelyfrom the complements of the angles EIN, IEN, and from the side IE, the side OI will begiven. Then in the spherical triangle MIO, from the given right angle at I, with sides IO,IM ; the side OM will not be forgotten , the distance of the minor circle from its ownpole. The angle OMI taken from the angle FMI leaves the given angle FMO. Therefore in


A

B

C

D

E

F

O

MNI

L

triangle FMO, from the given sides FM, MO, with thecommon angle FMO, the side FO will be given, thecomplement of the latitude of the pole of the minorcircle, or its distance from the pole of the ecliptic; andthe angle MFO will be given, the difference of thislongitude from the longitude of the phenomenon M.Which were to be found.

Prop. 71. Problema.

Ex datis tribus longitudinibus, & latitudinius Phaenomeni, incirculo minore moventis ; distantiam circuli minoris a suo polo ;

& poli locum, quo ad longitudinem, & latitudinem, invenire.

Sit ACBL, ecliptica; cujus polus F : A principium arietis, a quo innumerentur tres phaenomenilongitudines, AC, AB, AL ; sintq; atitudines, CD, BN, LM ; & loca phaenomeni D, N, M. & quoniam,dantur, DF, NF, MF, complementa nimirum latitudinum, cum angulis interceptis, DFN, MFN, differentiislongitudinum; dabuntur & arcus DN, MN ; & anguli DNF, MNE, NMF ; dabitur igitur angulus MND.Deinde, erigantur duo arcus EO, IO, perpendiculares arcubus DN, NM ; & eos bifariam secantes, in E, & I:sitque eorum communis intersectio, punctum O; quod necessario erit polus circuli minoris, in quo moveturPhaenomenon ; ducanturq; arcus IE ; & in triangulo sphaerico INE, dantur latera IN, NE, & angulusinterceptus INE ; ac proinde, dabuntur anguli NEI, NIE, & latus IE. Tunc in triangulo OIE; e datis , angulis,angulis OEI, OIE, complementis nimium angulorum

[108]EIN, IEN, & latere IE, dabitur latus OI. Deinde in triangulo sphaerico MIO, rectangulo ad I, e datis , IO,

IM, lateribus ; non ignorabitur latus OM, distantia circuli minoris, a suo polo; & angulus OMI, qui ablatusab angulo FMI relinquit angulum FMO datum. In triangulo igitur FMO, e datis, lateribus FM, MO, cumangulo intercepto FMO ; dabitur latus FO, complemenum latitudinis poli circuli minoris, seu distantia ejusa polo eclipticae ; & angulus MFO, differentia illius longitudinis a longitudine Phaenomeni M. Quaeinvenienda erant.

Prop. 72. Problem.To observe precisely the apparent diameter of the sun, moon, or any star you please.

The image of that [source] may be projected from one lens or mirror into another inorder that it may result in a sensible enough size, and the rays of a star may be stronglyshining, representing the image of that in a white plane, as we have taught in Prop. 52 ofthis work, and from its notes. Then the diameter of the image, which is measured mostprecisely, from which was given projected with the help of some lenses or mirrors, andfrom the given distances themselves of the lenses, mirrors, and the image in turn, theangle of vision is found, of the visible phenomenon itself (except for stars) from its ownvertex of incidence, this is the diameter of the star appearing by Prop 53 of this work. Butif the image shall not appear clear and bright enough; it may be made clear in some givenratio, by Prop. 54 of this work; also in the observation of more obscure locations, a moredistinct image is seen there; and with a more enlarged image (with everything else), therethe observation shall be more exact. Also eclipses of all kinds are observed with greatprecision by the same method.


A B

C D

H

I

N

L

Prop. 72. Problema.

Diametrum apparentem, solis, lunae, vel stellae cujuslibet, exactissime observare.

Projiciatur illius imago, ab una lente, vel speculo, in aliam, ut tandem evadat in magnitudinem satissensibilem ; & in plano albo, radios sideris fortiter revibrante, repraesenterur ejus imago, ut in Prop. 52hujus, & ejus scholiis, docuimus. Deinde mensuretur quam exactissime imaginis diameter ; qua data, opequotcunque speculorum, vel lentium, projecta ; datisque, speculorum, vel lentium, & imaginum, a seinvicem distantiis ; inveniatur angulus visorius, visibilis (nempe sideris) ex vertice suae incidentiae, hoc estdiameter apparens sideris per Prop 53 hujus. Si autem imago, non sit satis clara, & lucida ; illustretur, inquacunque ratione data, per Prop 54 hujus ; quo etiam obscurior fuerit locus, in quo sit observatio, eodistinctius videtur imago ; & quo magis amplificatur imago (caeteris paribus) eo exactior sit observatio.Eodem etiam modo observantur quam exactissime omnium generum eclipses.

Prop. 73. Problem.

To observe the separation of two stars close to each other, and the angle of the greatcircle drawn through their centres from any desired vertical drawn through the centre ofone star.

The axis of an icoscope is to be made in the plane of that vertical, of which the angleis sought for the great circle drawn through the centres of the stars, so that it will be ableto be moved freely to and fro, and always keeping the axis precisely in the vertical plane.ABCD shall be the image plane of this icoscope, in which the right line HI is noted, thecommon intersection of the image plane and of the aforementioned [great] vertical circle,in which the axis of the iconoscope produced is incident in the point L. Then the axis isturned most carefully until the star touches the vertical circle, the centre of which shouldbe noted in the vertical circle ; and the icoscope is moved up and down until the centre ofthe image of that star can be seen in the point L. And being close to the same, the centreof the other star is noted in the image plane in the point N. Therefore with the image LN

given, projected with the help of some lenses ormirrors; and from the given image of the lenses ormirrors, the visible angle of vision is found from theseparations between themselves, from the vertex of itsincidence, which will be the separation of the stars, theimages of which are L, N. Then the angle HLN ismeasured most precisely, which will be the angle ofthe vertical circle (in which the axis of the icoscope ispresent) with the great circle, in which the centres ofthe stars are present. For the line HI is the commonintersection of the vertical with the plane of the image,

and the line LN is the common intersection of the great circle drawn through the centresof the stars with the image plane.

[110]And with the common intersection of the vertical circle and the great circle through thecentres of the stars; indeed the axis of the iconoscope is at right angles to the plane of theimage; the angle HLN shall be equal to the angle of incidence of the plane of the verticalcircle and of the great circle through the centres of the stars: this is the spherical angleunderstood from these that was required to be found.


S

L

O

I R

B

Comment on Prop. 73.

Although the construction details of the icoscope are not provided, we consider it toconsist of a telescope with a lens or lenses that projects the final image on to a screen: bymeans of which measurements can be made in lieu of cross-wires. We regard the imageplane ABCD as being tangential to the celestial sphere at a point on a great circle, such asthe point L. The azimuth and altitude angles of L and N are measured by the associatedtelescope, hence LN lies on a great circle.

Prop. 73. Problema.

Distantiam , duorum siderum, sibi invicem propinquorum , & angulum circuli maximi, per eorumcentra ducti, cum verticali quolibet, per centrum unius siderum ducto, observare.

Fiatur icoscopii axis, in plano illius verticalis, cujus angulus cum circulo maximo per centra siderumducto quaeritur ; ita ut sursum , & deorsum, libere moveri poterit ; semper tenens axem, quam exactissime,in plano verticalis. Sitque hujus icoscopii planum imaginis ABCD, in quo noterur recta HI, communisintersectio, plani imaginis, & circuli verticalis praedicti : in quam incidat, axis icoscopii productus, inpuncto L. Deinde diligentissime attendatur, donec sidus cujus centrum in circulo verticali notari debeat,circulum verticalem attingat : & moveatur icoscopium sursum & deorsum, donec cernatur centrumimaginis illius sideris, in puncto L. Et eodem instante, notetur in plano imaginis, centrum alterius sideris, inpuncto N, Data igitur imagine LN, ope quotcunque lentium, vel speculorum , projecta ; & datis, lentium,vel speculoram, & imaginum, a se invicem distantiis; inveniatur angulus visorius visibilis, ex vertice suaeincidentiae; qui erit distantia siderum, quorum imagines sunt L, N, Deinde mensuretur quam exactissimeangulus HLN, qui erit angulus circuli verticalis, (in quo existit axis icoscopii) cum circulo maximo, in quoexistunt siderum centra: Est enim recta HI, communis intersectio verticalis cum plano imaginis, & recta LNcommunis intersectio circuli maximi per centra siderum ducti, cum plano imaginis :

[110]cumque communis intersectio circuli verticalis, & circuli maximi per centra siderum ; nempe icoscopiiaxis, sit ad planum imaginis rectus ; erit angulus HLN, aequalis angulo inclinationis planorum, circuliverticalis, & circuli maximi per centra siderum ; hoc est angulo sphaerico, ab illis comprehenso. Queminvenire oportuit.

Prop. 74. Problem.

To investigate exactly the parallax of any desired planet .

SIB shall be the common azimuth of two locations, of which the vertices are S, B; &the common azimuth SIB is to be observed from locations on both sides, a jointoccurrence of the body of the planet, of which the true location is I, from a fixed star of

which the position is O. But the apparent position of the planet shall be the point L fromthe location with vertex B ; and the position of the planet appearing to be at the point R,


from the location with vertex S. Then by Prop. 73 of this work, from the locations ofwhich the vertices are S, B, the arcs LO, RO are found, and the angles RLO, LRO : thelength LR can be found in four ways, the sum of the parallaxes of the planet of which thetrue position is I. But given the ratio of the sines of the same parallaxes; indeed it is thesame as for the ratio: sine arc BL to sine arc SR, and therefore will give the parallaxes LI,IR separately, which had to be found.

Scholium.The parallaxes of Saturn itself can be found with the help of this problem, only if the

exact observations are used.

Prop. 74. Problema.

Parallaxem Planae cujuslibet, exactissime investigare.

Sit SIB azimuth commune duorum locorum, quorum vertices S, B ; & azimuth commune SIB,observetur, ex utroque loco, conjunctio corporalis planetae, cujus verus locus I, cum stella fixa, cujus locusO : Sit autem apparens locus planetae, ex loco, cujus vertex B, punctum L; & apparens locus planetae, exloco, cujus vertex S, punctum R. Deinde per Prop. 73 hujus, ex locis quorum vertices S, B, investigenturarcus LO, RO, & angulis RLO, LRO; quatuor modis potest inveniri latus LR, summa parallaxeon planetaecujus verus locus I. Datur autem

[111]ratio sinuum earundem parallaxium ; eadem enim est cum ratione, sinus arcus BL, ad sinum arcus SR ;dabuntur igitur & parallaxes LI, IR seorsum : quas investigare oportuit.

Scholium.

Ope hujus problematis, poterunt & ipsius Saturni parallaxes inveniri ; si modo obsevationes exactaeadhibeantur.

Prop. 75. Lemma.

If there are two series of magnitudes in arithmetical proportion, and if the first [term]of the first series is added to the first term of the second series : and the second term ofthe first, to the second term of the second; and thus henceforth ; or if subtracted from thesame ; the sum or difference will be also a series of arithmetical proportionals.

Let there be the same difference between the magnitudes A and BC which there isbetween the magnitudes D and EF, surely H ; and let there be the same differencebetween the magnitudes I and KL which there is between the magnitudes M and NO,surely R. In the first case [involving subtraction], I say that A - I is deficient from BC -KL or that it is exceeded by the same difference that D - M is deficient or is exceeded byEF - NO. For PC = H is taken from BC, & TL = R is taken from KL ; and BP = A , KT= I ; therefore there will be no difference between A - I & BP - KT : Therefore the totaldifference between A - I, & BC - KL is the same as that between PC & TL, or H & R.And in the same way the difference is shown between D - M, & EF - NO to be equal tothe difference between H & R.

In the second case [involving addition], I say that the difference between A + I & BC+ KL, likewise the difference between D + M & EF + NO is equal to H + R: for there isno difference between A + I & BP + KT ; therefore the total difference between A + I &BC + KL


[112] is PC + TL = H + R ; which it was necessary to show.

Comment on Prop. 75.

We may regard this result as showing the state of algebra at the time:

A B C D E F

I K T L M N V O

P Sa

i

a + h d + h

i + r m + r

d

mProp. 75. Lemma.

Si fuerint duae series magnitudinium arithmeticae proportionalium, & prima, primae seriei, addaturprimae, secundae: & secunda primae, secundae secundae serei; & sic deinceps: vel si ab isdemsubstrahantur; facta, vel relicta, erunt etiam series arithmeticae proportionalium.

Sit eadem differentia , inter magnitudines A, & BC; quae inter magnitudines D, & EF ; nempe H ; & siteadem differentia inter magnitudines I, & KL ; quae inter magnitudines M, & NO; nimirum R. Dico primoA - I deficere a BC - KL, vel eandem excedere, eadem differentia, qua D - M excedere, vel deficit ab EF -NO. Auferatur a BC, PC = H ; & a KL, TL = R ; eritque BP = A ;& KT = I ; ideo nulla erit differentia interA - I, & BP - KT : Tota igitur differentia inter A - I, & BC - KL; eadem est, quae inter PC, & TL, seu H, &R. Eodemq; modo ostenditur differentiam inter D - M, & EF - NO esse aequalem differentiae inter H & R.

Dico secundo, differentiam, inter A + I, & BC + KL; item & differentiam inter D + M, & EF + NO esseaequalem H + R: nulla enim est differentia inter A + I & BP + KT ; tota igitur differentia

[112]inter A + I & BC + KL est PC + TL = H + R ; quae demonstrari oportuit.

Prop. 76. Problem.From the mean motion of a given planet, together with three observations from the centreof its apparent motion; the eccentricity and the position of the aphelion is found, with thesupposition that the planet is moving in an eccentric circle.

With centre A, the circle 1 2 3 D is described , and in the same circumference, thefollowing intervals 1 2, 2 3 shall be the arcs of the given observations of the centralmotion; thus the first [point] 1, the second 2, and the third 3, represent the positions ofthe planet in the circle. Then from the centre of the apparent motion B, the lines B1, B2,B3 are drawn ; thus the motions to be observed appear as the angles 1B2 & 2B3. Fromthe given arcs 12, 23, surely for the central motions; and from the angles 1B2, 2B3 fromthe apparent motions, the eccentricity AB can be found, and the position of the aphelionAB. 2B is produced as far as the circumference D; & the line AL is drawn through thecentre A parallel to 2D ; & a perpendicular AC is sent from the centre A to the line 2D,which divides the line 2D in two equal parts at C; and AC will be the sine of the arc L2:then the lines 1D and 3D are joined: & in the triangles 1BD, 3BD, the angles to D are

A B C D E F

I K T L M N V O

P S


A

BC

D

L1

2

3

given ; viz. half of the mean motions, or of the arcs 12 and 23; & the angles at B, surelythe supplements of the observed angles 1B2, 2B3 to the two lines : therefore the anglesB1D, B3D are given, and hence the ratio BD to ID; and BD to D3 : therefore the ratio1D

to 3D is given : & from the given ratio of thechords 1D, 3D, & the sum of the arcs 1D3 ;(surely of the remaining motion to complete thewhole circle) the arcs 1D, 3D may be found, &the lines 1D, 3D, BD, in parts of which theradius AL is 1000000 : & the difference of thearc 1D + 12, or 23 + 3D, & the semicircle, willbe equal to twice L2 itself; which will hencegive the arc L2 & the sine of this arc AC, withthe sine of the complement CD; from which theline DB is taken, BC is left. Finally, from theright angled triangle ABC, with the given sidesAC, BC; the eccentricity AB can be found, &

the angle ABC, for the location of the aphelion. Which it was necessary to find.

Prop. 76. Problema.

Ex dato planeta motu medio, una cum tribus observationibus ex centro sui motus apparentis ; ejusexcentricitatem, & aphelii locatus enucleare: supposito cum un circulo excentrico moveri.

Centro A, describatur circulus 1 2 3 D, & in ejusdem peripheria, secundum observationum intervalla , sintarcus motus medii dati 1 2, 2 3; ita ut 1, locum primum ; 2 , secundum ; 3 , tertium planetae locum, incirculo repraesenter. Deinde, a

[113]centro motus apparentis B, ducantur rectae B 1, B 2, B 3 ; ira ut anguli 1 B 2, 2 B 3, sint motus apparentesobservati ; & ex datis arcubus 12, 23, nimirum motibus mediis; & angulis 1B2, 2B3, motibus apparentibus; quaeritur excentricitas AB, & apheli positio AB: Producatur 2B, usque ad peripheriam in D; & ducaturper centrum A, rectae a 2D parallela, AL ; & demittatur ex centro A, rectae 2D parallela, AL; & demittaturex centro A, in rectam 2D perpendicularis AC, quae dividet rectam 2D bifariam in C; eritque sinus arcus L2: deinde, jungantur lineae 1 D, 3 D: & in triangulis 1BD, 3BD, dantur anguli ad D ; viz. dimidia motuummediorum, seu arcuum 12, 23; & anguli ad B, nempe residua angulorum observatorum 1B2, 2B3, ad duosrectos : dabuntur igitur, & anguli B1D, B3D ac proinde ratio BD, ad ID; & BD, ad D3 : ergo datur ratio 1Dad 3D: & data ratione chordatum 1D, 3D, & arcuum summa, 1D3 ; (residuum nempe motuum mediorum adintegrum circulum) inveniantur arcus 1D, 3D; & rectae 1D, 3 D, BD, in partibus, quarum radius AL est1000000 :& differentia arcus 1 D + 1 2, vel 23 + 3D, & semicirculi, aequalis erit duplo ipsius L2, dabiturproinde arcus L2, & ejus sinus AC, cum sinu complementi CD; a quo, recta DB ablata, relinquitur BC: exdatis denique, in triangulo rectangulo ABC, lateribus AC, BC; inveniantur AB excentricitas, & angulusABC, aphelii positio, ab observationea , Quae invenire oportuit.

Prop. 77. Theorem.If a planet may be moved along the line of an ellipse equally around one of the focuses ofthe ellipse, and truly appearing around the other focus with an equal motion to the focus;from the centre of the position of an eccentric circle, of which the radius is equal to thetransverse axis of the ellipse. By supposing that the planet moves with the sameintermediate motion in the eccentric circle, the motion appearing in the eccentric circlewill be the arithmetic mean between the common motion about the centre [which is alsothe first focus] and the apparent motion in the ellipse [about the second focus] : with the


A

B

C

C

OO

S

M

P

G

centre of the motion always appearing from the same position, truly from either focus ofthe ellipse.

Let the ellipse be APOG, in thecircumference of which the planetmoves from P to O ; equally aroundthe focus M, and around the focus S.And let

[114] BCC [ the text has ABG ] be thecircle of which the centre is M, &radius MB, equal to the transverseaxis of the ellipse AG; and theplanet moves from B to C, with thesame intermediate motion, withwhich it moves from P to O: thus, asMPB and MOC shall be madestraight lines. The lines of the truemotion shall be drawn so in the

circle as in the ellipse: SP, SO, SB, SC. I say that the angles PSO, BSC, BMC, are inarithmetic proportion ; that is, there is the same difference between PSO and BSC, whichthere is between BSC and BMC. Since indeed AG is equal to MC ; and MO + OS will beequal to AG and MC; and MO taken from both sides; OC = OS; & OCS = OSC; & AMC- OCS = ASC; & ASC - OCS = ASO : therefore the three angles AMC, ASC, ASO, are

[115]in continued arithmetical proportion ; the common difference of which is the angle OCS.And by the same method it can be demonstrated that the three angles AMB, ASB, ASP,are in continued arithmetical proportion, & have as a common difference the angle PBS.Therefore if these two series in order in arithmetical proportion are themselves added orsubtracted in turn; the angles BMC, BSC, PSO emerge in arithmetical proportion ; quoddemonstrandum erat.

Corollary.

Hence the method of the Geometrician is clear ; finding the eccentricity, or theseparation of the foci, & the position of the aphelion of the ellipse, following thehypothesis CL : V: & in all the writings of the most learned Dr Seth Ward. If indeedthree observations are given of the ellipse, with the intermediate motions ; thearithmetical means are taken, among the mean motions, and the apparent motions of twoobservations ; which puts the arithmetical means in the place of the motion appearing inthe circle, and with these apparent motions, and with the same intermediate motions,which in the ellipse, the eccentricity of the circle can be found , which will be the sameas the separation of the foci of the ellipse ; from the radial position of the eccentricitywith the axis of the ellipse, and it will be the position of the aphelion from the line of themean motion in the eccentricity, with the same position of the aphelion, or of thetransverse axis, from the same line of the mean arc in the ellipse. I had decided to lift thehand from the table, but because concerning the inequalities of the planets, Dr. Seth Ward


will make public some most acute propositions, it has pleased to make this little tractknown with these, and to add certain concerning the same material from our own store.

Prop. 77. Theorema.

Si Planeta, in linea elliptica aequaliter moveatur circa unum ellipseos focorum ; apparenter vero circaalterum ; foco motus aequalis, centro circuli excentrici posito, cujus radius aequalis est transverso axiellipseos ; & supposito, planetam eodem motu medio moveri in circulo excentrico; erit motus apparens incirculo excentrico, media arithmetica, inter motum medium communem, & motum apparentem, in ellipsi :eodem semper posito, centro motus apparentis, nempe altero ellipseos foco.

Sit ellipsis APOG, in cujus peripheria, moveatur planeta, a P, ad O ; aequaliter circa focum M ; &apparenter circa focum S. Sitque

[114]circulus ABG eujus centrum M, & radius MB, aequalis axi transverso ellipseos AG; & moveatur planeta, aB, ad C, eodem motu medio, quo a P, ad O: ita, ut MPB, MOC, fiant lineae rectae : ducanturq; lineae verimotus, tam in circulo, quam in ellipsi SP, SO, SB, SC. Dico angulos PSO, BSC, BMC, esse arithmeticepropertionales ; hoc est eandem esse differentiam inter PSO, & BSC, quae est inter BSC, & BMC,Quoniam enim AG, est aequalis MC ; erit & MO + OS = AG = MC; & ablata MO utrinque; erit OC = OS;& OCS = OSC; & AMC - OCS = ASC; & ASC - OCS = ASO : tres igitur anguli AMC, ASC, ASO, sunt

[115]in continue proportiones arithmeticae; quorum communis differentia est angulus OCS. Eodemque mododemonstrabitur, tres angulos AMB, ASB, ASP, esse continue proportionales arithmeticae, & haberecommunem differentiam , angulum PBS, Igitur si hae duae series, arithmeticae proportionalium, ordine sibiinvicem addantur, vel a se invicem subtrahantur ; emergent anguli BMC, BSC, PSO arithmeticaeproportionales ; quod demonstrandum erat.

Corollarium.

Hinc patet modus Geometricus; inveniendi excentricitatem, seu focorum distantiam, & positionemaphelii elliptici, secundum Hypothesin CL : V: & in omni literatura doctissimi D. Sethi Wardi. Si enimdentur tres observationes in ellipsi,cum motibus mediis ; Sumatur media arithmetica, inter motum medium,& motum apparentem duarum observationum ; quae media arithmetica ponatur loco motus apparentis incirculo, & cum hisce motibus apparentibus, & motibus medibus iisdem, qui in ellipsi ; inveniatur circuliexcentricitas, quae eadem erit cum distantia focorum ellipseos ; a posito excentrici radio, axe ellipseos ;eritque positio aphelii a linea medii motus in excentrico; eadem cum positione aphelii, seu axeos transversi,ab eadem linea medii arcus in ellipsi. Decreveram manum de tabula tollere : sed quoniam de planetaruminaequalitatibus, acutissimas aliquot propositiones evulgabit D. Sethus Ward ; placuit hunc tractatulum illisnobilitare, & quasdam de eadem materia e nostra penu adjungere.

Prop. 78. Problem.From the given mean motion of a planet, with the position of the aphelion, and the

ratio of the axis to the separation of the foci of the ellipse given ; to find the apparentmotion of the planet for the eye in agreement with the sun.

Let AP be the transverse axis of the ellipse, A the aphelion, P the perihelion, F the focusof the mean motion, S the sun, or the focus of the apparent motion, Q the locus of theplanet, of which the mean motion from the aphelion is AFQ. The true motion ASQ fromthe aphelion is sought. FQ is produced to R, and QR is equal to QS; & SR is drawn.Therefore, since in the triangle SFR the two sides FS, FR are given, together with thecontained angle; we have the angles at S and R ; but FSR - FRS = ASQ, for the anglethat it was necessary to find. Because if the distance SQ between the sun and the planet is


A

F

S

P

[Q]

[R]

sought ; as in triangle FSQ, all the angles are given, with theside FS ; which will also give the side SQ.

Prop. 78. Problema.

E datis, Planetae motu medio, Aphelii loco, & ratione axeostransveri, ad distantiam focorum ellipseos; planetae motum apparenteminvenire pro oculo in sole constituto.

Sit AP axis transversus ellipseos, A aphelium, P perihelium, F focusmedii motus, S Sol, sive focus motus apparentis, Q locus planetae, cujusmedius motus ab aphelio AFQ. Quaeritur ipsius verus motus ab aphelioASQ ? Producatur FQ ad R, & sit

[116]QR = RS; & ducatur SR. Quoniam igitur in triangulo SFR dantur duo latera FS, FR , una cum angulocomprehenso ; habemus angulos ad S & R ; at FSR - FRS = ASR, angulo quem invenire oportuit. Quod siquaeratur SQ distantia inter solem, & planeram ; quoniam in triangulo FSQ,dantur omnes anguli, cumlatere FS ;dabitur etiam latus SR.


N

NS

1

2

Prop. 79. Problem.

To find the central location of the nodes in the ecliptic, together with the distances of theplanet from the sun for the same, from observations made on the earth.

Let the first location of the earth be 1, and from that position the planet is observed in itsnode N without the latitude; and from the second location with the earth in location 2, thesame planet is observed, at the same node N. Let the sun be at S, and the line SN shall bethe common intersection of the plane of the ecliptic with the plane of the orbit of theplanet. In the triangle 1S2, the sides 1S and 2S are given, and the angle 1S2; and hencethe angles S12, S21 are given, and the side 12. And as N2S can be found fromobservation, it will give the angle N21; and in the same way N12 will be given. In thetriangle N12, from the given 12, N12, N21, 2N will be found. Finally, in triangle N2Sfrom the given N2, 2S, N2S, NS2 will be found for the position of the node N, and SNthe nodal distance of the planet from the sun.

Scholium.From such observations of the planets about the nodes, the most reliable method is picked

of finding the mean motion of the planets,since they have the slowest motion at thenodes.

Prop. 79. Problema.

Loca centrica nodorum in ecliptica, una cumplanetae, in iisdem, a sole distantiis, exobservationibus in terra habitis investigare.

Sit primo terra in 1 ; & ex ea observetur planeta innodo suo N carens latitudine: & secundo, ex terra in2, observetur idem planera, in eodem nodo N; sitque;

S sol; & recta SN, communis intersectio plani eclipticae, & plani orbitae planerae :[117]

In triangulo 1S2, dantur latera 1S, S2, & angulus 1S2; & proinde dantur anguli S12, S21, & latus 12.Cumque detur N2S ex observatione, dabitur & angulus N21 ; eodemque modo dabitur N12. Et in trianguloN12, e datis 12, N12, N21, invenitur 2N. Denique in triangulo N2S e datis N2, 2S, N2S; inveniatur NS2positio nodi N, & SN distantia planetae in nodo N a sole S.

Scholium.Ex talibus observationibus planetarum in nodis, colligitur modus certissimus inveniendi motus medios

planetarum ; quoniam nodi tardissimum habent motum.


N

S

T

5

E D

C B

5

Prop. 80. Problem.

To find the inclination of the orbit of a planet with the plane of the ecliptic.

If the orbit of the planet is NCE, theintersection of which with the ecliptic, orwith the nodal line NST from the earth inthe line of the nodes arising from T, theplanet 5 is observed at E, the latitude ofwhich is ETD, the angle composed fromthe line T5E in the plane of the orbit of theplanet, and the line TD, in the plane of theecliptic. Another plane is drawn parallel tothe plane of the latitude ETD, through thecentre of the sun S ; and both these planesare normal to the plane of the ecliptic ; andthe lines BS, CS are parallel to the linesDT, ET ; and therefore angle BSN = angleDTS, and angle ETD = angle CSB: butDTS and ETD are given from observation ;giving therefore BSN and CSB. Then inthe spherical triangle CBN with the rightangle at B, from the given sides CB, that isthe angle CSB, and BN that is the angleBSN, the angle CNB can be found, theangle to the inclination sought.

Prop. 80. Problema.

Orbis planetarii cum ecliptica inclinationem invenire.

Si orbita planetae NCE, cujus intersectio cum ecliptica, seu linea nodorum rectae NST a terra in lineanodorum existente T, obervetur planeta, in E, cujus latitudo ETD, angulus comprehensus a recta T5E inplano orbitae planetariae, & recta TD, in plano eclipticae. Plano latitudinis ETD parallelum, ducatur aliudplanum, per centrum solis S ; eruntque ambo haec plana, plano eclipticae normalia ; & rectae BS, CSparallelae, rectis DT, ET ; & ideo BSN = DTS, ETD = CSB: dantur autem DTS, ETD ex observatione ;dantur igitur BSN,CSB. Deinde in triangulo sphaerico CBN rectangulo ad B, ex datis lateribus CB, hoc estangulo CSB, & BN hoc est angulo BSN ; inveniatur angulus CNB, inclinatio quaesita.

[118]

Prop. 81. Problem.From the observation of the planet, in conjunction or apposition with the sun; setting

aside the second inequality of this: to find the central latitude and the distance of theplanet from the sun.


S

N

C

B

5 5

T

B

5

5

O

SN

B

T

The planet 5, of which the node is N,is observed from the earth T inconjunction or in apposition with the sun.The observed locus of the planet in itsown orbit shall be C, in the ecliptic B.The angle BSN is given, indeed theseparation of the node from the observedposition of the planet in the ecliptic;therefore in the spherical triangle BNCwith the right angle at B, from the givenside BN, and from the angle BNC, the arcNC is found, for the distance of the

planet in its orbit from the node N. Whenever the position of the node N is given, theposition of the planet in its orbit is given; and hence the second inequality of this is setaside. Then in the same spherical triangle BCN the arc BC is found, the central latitude ofthe planet; and in the right angled triangle TS5, from the given angle TS5 for the centrallatitude of the planet, the observed latitude of the planet is TS, or the complement of thisto the two lines at right angles, and from the side TS the distance of the earth to the sun,the length S5 is found, the distance of the planet from the sun.

Prop. 81. Problema.

Ex observatione planetae, in conjunctione, vel oppositione cum Sole; ejus inequalitatem secundamexuere: latitudinem centricam, & a sole dictantiam reperire.

Observetur planeta, cujus nodus N, in conjunctione, vel opposition cum sole S, ex terra T. Sitqueplanetae, locus in ecliptica

[119]observatus B, in sua orbita C. Datur angulus BSN, nempe distantia nodi a loco planetae in ecliptica

observato; & ideo in triangulo sphaerico BNC rectangulo ad B, e datis latere BN, angulo BNC; reperiturarcus NC, distantia planetae in sua orbita, a node N; cumque detur locus nodi N, datur & locus planetae insua orbita; & proinde exuitur ejus secunda inaequalitas. Deinde in eodem triangulo sphaerico BCNinveniatur arcus BC, latitudo planetae centrica ; & in triangulo rectilineo TS5, e datis angulis TS5 latitudineplanetae centrica, TS latitudine planetae observata, vel ipsius complemento ad duos rectos, & latere TSdistantia terrae a sole; inveniatur latus S5, distantia planetae a sole.

Prop. 82. Problem.From any one observation of a planet, the

second inequality of which is set aside, to findthe central latitude and the distance from thesun.

The planet 5 is observed from the earth T,the nodal line of which is NS, and with theapparent latitude 5TB. The plane of theapparent latitude is produced, until it cuts theplane of the orbit of the planet in the line 5Nand the plane of the ecliptic in the line BTN.From the centre of the earth T, the


perpendicular TO to the plane of the ecliptic may be raised all the way to the plane of theorbit of the planet at O; and the line SO is joined, which by necessity is in the plane of theorbit of the planet. In the triangle NST, the side ST is given from the theory of the earth[i.e. the earth: sun distance], the angle NST from the location of the known node, theangle NTS from observation; therefore the angle TNS is given, and the sides TN and NS.Then, for the given orbit of the planet with the inclination to the ecliptic, and the angleTSN, and the distance of the earth from the node; the angle OST will be given : and asTN to TS, thus the tangent of the angle OST, to the tangent of the angle ONT. Thereforethe angle ONT is given, and the angle 5TB becomes known from observation. Therefore,in the triangle 5NT, from all the given angles, with the side NT, and the side 5N is given;and in the triangle 5NB with the right angle at B, from the given angle 5NB and the side5N, the side NB becomes known. Then in the triangle BNS, from the given sides BN andNS, with the angle intercepted BNS, the side BS and the angle BSN are found for thecentral longitude of the planet in the ecliptic, computed from the line of the nodes. And inthe spherical right angled triangle in the above figure [Prop. 81] CBN, from the givenside BN, surely the angle BSN in this figure, and the angle BNC for the inclination of theorbit, the line BC is found. This is the angle BS5 [ in current diagram], for the centrallatitude, and the side CN [above] is the angle 5SN, the central longitude of the planet inits orbit, computed from the line of the nodes: and hence the second inequality of this isset aside. However in the triangle 5SB with the right angle at B, from BS and the angleBS5; the side S5 will not be disregarded, the distance of the planet from the sun.

Prop. 82. Problema.

Ex unica planetae observatione quacunque; ejus inequalitatem secundam exuere: latitudinemcentricam, & a sole distantiam invenire.

Observetur planeta 5, cujus linea nodorum NS, ex terra T, cum latitidine apparente TB. Producaturplanum latitudinis apparentis, donec secuerit planum orbis planetarii in recta 5N, & planum eclipticae inrecta BTN. E centro terrae T, erigatur plano

[120]eclipticae perpendicularis TO, usq; ad planum orbis planetatii in O : & jungatur SO recta, quae necessarioest in plano orbis planetarii. In triangulo NST, datur latus ST ex theoria terrae, angulus NST ex loco nodicognito, angulus NTS ex observatione; dantur igitur angulus TNS, latus TN, latus NS. Deinde datis, orbisplanetarii cum ecliptica inclinatione, & angulo TSN distantia terrae a nodo ; dabitur angulus OST : & ut TNad TS, ita tangens anguli OST, ad tangentem anguli ONT ; datur igitur angulus ONT, & innotescit angulus5TB observatione; ideoq; in triangulo 5NT, e datis omnibus angulis, cum latere NT, datur & latus 5N: & intriangulo 5NB rectangulo ad B, e datis angulo 5NB, & latere 5N, innotescit latus NB. Deinde in trianguloBNS, e datis lateribus BN, NS, cum angulo intercepto BNS, reperiuntur latus BS, & angulus B5N,longitudo centrica planetae in ecliptica, a linea nodorum computata; & in triangulo sphaerico rectangulo insuperioribus figuris CBN, e datis latere BN, nempe angulo BSN in hac figura ; & angulo BNC orbiuminclinatione, inveniatur BC latus, hoc est angulus BS5, latitudo centrica, & latus CN, hoc est angulus 5SN,longitudo planetae centrica in suo orbe, a linea nodorum computata :& proinde exuitur secunda ejusinaequalitas. Tandem in triangulo 5SB rectangulo ad B;e datis BS, BS5; non ignorabitur latus S5, distantiaplanetae a sole.


B5

5

2

S

1B

Prop. 83. Problem.

From two observations of the planet at the same point of the orbit, the distance of thisplanet from the sun, the central place in the ecliptic is found; and from which thelongitude and the latitude of the planet are found; from the assumed theory of the earth.

The earth shall be at 1 first, and theplanet 5 is observed from thisposition, from which theperpendicular 5B may be dropped tothe plane of the ecliptic; and theobserved place of this in the eclipticshall be B with latitude 51B; and afterone or more whole revolutions of theplanet, the planet 5 is again observedfrom the earth 2, and the sun shall beS. From the theory of the earth, all thesides and all the angles of the triangle1S2 are given; also, the angles S1Band S2B are given from observation;therefore in triangle 21B, given theangles 21B and 12B ; and hence theangle 2B1 is given : and moreovergiven the side 21, from which the side B1 is found; & in the triangle 1B5, with the rightangle at B, from the given side 1B, & from the angle B15, for the observed latitude; B5can be found. Then in triangle B1S, from B1,1S, B1S given, BS is given, & 1SB, thecentral position of the planet in the plane of the ecliptic, from the earth in position 1 iscomputed; & in triangle BS5 with the right angle at B, from the given distances B5 andBS, the angle BS5 is found for the central latitude of the planet, and its distance S5 fromthe sun.

Scholium.In this problem the position of the node may be given ; it is manifest that both themaximum inclination and the central position of the planet in its orbit are to be given, or ( if the maximum inclination shall be given) the position of the node and the centralposition of the planet in its orbit shall be given. In the right angled spherical triangle, thecentral latitude is one line, while the central longitude in the ecliptic is another line,computed from the line of the nodes; the central longitude of the planet in its orbitcomputed from the node is a third line, and the inclination of the orbit [to the ecliptic] isone angle. From which four, two are given and two are shown to be unknown. But fromthe two central places of the ecliptic and the two central latitudes, both the givenmaximum inclination and the position of the node in the ecliptic are given, by Prop. 70 ofthis work. And from three positions of the planet in its orbit, both the position of theaphelion and the kind of the ellipse of the planet are given by Coroll. 77 of this work.Therefore it is agreed, by applying the method many times, the inequalities of all theplanets are to be found.


P Q

CN

N

L

L

F S


Ex binis observationibus planetae, in eodem orbis puncto ; ejus distantiam a sole, locum centricum inecliptica, quo ad longitudinem, & latitudinem invenire; supposita tantum terrae theoria.

Sit primo terra in 1, & ex ea observetur planeta 5, e quo in eclipticae planum, demittatur perpendicularis 5B;eritq; locus illius observatus in ecliptica B, cum latitudine 51B; & post unam, vel plures planetaerevolutiones integras, rursus observetur planeta 5, ex terra 2, sitque sol S. Dantur ex theoria terrae, omnialatera, & omnes anguli, trianguli 1S2 ; dantur etiam anguli S1B, S2B, ex observatione; igitur in triangulo21B, dantur anguli 21B, 12B; & proinde angulus 2B1 : datur autem & latus 21, e quibus inveniatur latusB1; & in triangulo 1B5, rectangulo ad B, e datis latere 1B, & angulo B15, latitudine observata; reperiturB5. Deinde in triangulo B1S, e datis B1,1S, B1S, datur BS, & 1SB, locus planetae centricus in ecliptica, aterra in 1 computatus; & in triangulo BS5 rectangulo ad B, e datis B5, BS, innotescit angulus BS5, latitudoplanetae centrica; & latus S5, distantia illius a sole.

[122]

Scholium.

Manifestum est ( in hoc problemate daretur locus nodi) dari, & inclinationem maximam, & locum planetaecentricum in suo orbe; vel ( si daretur inclinatio maxima ) dari locum nodi, & locum planetae centricum insuo orbe: In triangulo enim sphaerico rectangulo, latitudo centrica est latus unum, longitudo centrica inecliptica, a linea nodorum computata, latus alterum ; longitudo planetae centrica in suo orbe, a nodocomputata, latis tertium , & orbium inclinatio, unus angilus. E quibus quatuor, duo dati, duos ignotossemper manifestant. Ex duobus autem locis centricis in ecliptica, & duabus latitudinibus centricis; datur &inclinatio maxima, & nodi locus in ecliptica, per 70 hujus. Et ex tribus planetae locis centricis in suo orbe,datur & aphelii positio; & species ellipseos planetariae, per Corol. 77 hujus. Constat igitur modusmultiplex, inveniendi omnes planetarum inaequalitates.

Prop. 84. Problem.

From two given central positions of the planet in its orbit, together with the centralmotion of this planet, and the distances from the sun; to find the position of the aphelionand the kind of ellipse.

Let PCQN be the ellipse of the planet,of which F is the focus of the centralmotion, or the sun S is the focus of thetrue motion, with PQ the transverse axis,and C and N shall be the positions of theplanet. From the given sides SC and SNin the triangle CSN, to wit the distancesof the planet from the sun ; and with theangle CSN taken from these and theknown positions of the planet from the

centre, the side CN and the angle NCS can be found. The lines FC and FN are drawn;and as NS + NF = CS + CF ; CS - SN will be equal to NF - CF and equal to LN: for FC= FL ; and in the isosceles triangle FLC, all the angles are given: truly LFC, from thecentral motion of the planet between the observations C and N and the angles FLC and


P Q

CD

N

F S

A

B

K

V

HN

M

L

S

YP

DT

O

R

FCL on account of their equality ; therefore the angle CLN is given. Hence in triangleCLN, from the angle CLN, and the given sides NC and NL, the angle LCN is found ;hence the angle FCN = FCL + LCN is also given. In triangle FCN, from all the givenangles, with the side CN given, the side CF will be given, and CF + CS = PQ. And thenin triangle FCS, from FC and CS given, the angle FCS is known from the found anglesFCN and NCS. [Hence] the distance FS between the foci, and the angle FSC for thelocation of the aphelion are found from the observation C.

Scholium.But if anyone should want the

resolution of this problem without thesupposed central motion: by anobservation from the earth D, and thus DNmay found by the same method, by whichCN was found in the [above] proposition.Let PQ 14 be the axis, and therefore FC =14 - CS, & FN = 14 - NS will be given;therefore in the quadrilateral FCSN, all thesides are given, together with the diameterCN, from which the diameter FS may befound. In the quadrilateral FDSN too, all the sides are given from observation: SD, SN,FD = 14 - DS, and FN = 14 - NS; together with the diameter DN, from which again FS issought. Therefore given the equation between FS first found, & FS found second, fromthe resolution of which everything sought shall be made clear.

But the most pretty of all the methods is the one set forth by Dr. Ward on page 50 ofAstronomiae Geometricae, which neither supposes the central motion, nor the sun to bethe focus of the ellipse; but however, the centre of the sun is fixed and the planets aremoving in elliptical lines [curves]. From which the origin of the central motion, if such isgiven, and the position of the sun within the ellipse, being supposed [different] from theother methods, is inferred by a geometrical demonstration. But because of the excellence

of this method, we ourselves willtry to produce another example ofthe use of this in accordance withcentral motion. Let there be fivepoints H, K, L, M, N on theellipse, which shall nowheredetermine parallel lines, and thelines NK, MH, are drawn cuttingeach other in T: while LV isdrawn parallel to the line MH,cutting NK in Y. Fromobservation, all the angles at [thesun] S are given by a line betweenS and the five points of the ellipse: hence the lines MN, NK, NL,and the angles MNS, KNS, MNT,NMS, HMS, NMT, NLY are


known. Then in the rectilinear triangle MNT, from all the given angles, with the sideMN, the sides NT, MT are also given ; and in the rectilinear triangle NLY, from all thegiven angles, (for MTN = LYN) with the side LN; the sides NY, LY are given; andNT.TK : NY.YK : : MT.TH : LY.YV [From Prop. 17, Appolonius, Book III; see e. g. :T. Heath, A History of Greek Mathematics, p. 153; Dover] ; therefore YV is given, andhence the whole length LV. The lines LV [text has LY] and MH are bisected in P and O,and the line APDOB is drawn, surely the diameter of the ellipse; to which the ordinatelines PV and OH are joined ; and to the same diameter the ordinate line KR is joined. Onaccount of the similarity of the triangles PDY and TDO: PY + TO : TY : : PY : DY ; andas the angle PYD is given, the lines PD, DO, DR, KR will be found. Therefore, let thelines be given by numbers: OH 10, PV 12, RK 8, PO 4, RP 2, with the angle KRB 60[degrees]. Let BO be put as 1 , and BP will be 1 + 4, but 100: 144 : : 1 .OA : (1 +4).PA [from OH2/PV2 = BO.OA/BP.PA]; therefore by the converse of Prop. 99, from theproportionalities of Gregory of St. Vincent, 100 is to 144 and is to the ratio 1 to 1 + 4and to the ratio PA to OA = PA + 4; therefore PA is to PA + 4 as

100144 to 1 + 4, or as

144 to 100 + 400; and therefore the difference of these, surely400 - 44 : 144 : : 4 : 576

400 - 44= PA ; hence PA + 4 = 1600 + 400

400 - 44= AO,

& RA = 664 - 800400 - 44

; and BO x OA = 1600 + 400 q

400 - 44: 100 : : BR x RA = 664q + 1384 - 4800

400 - 44: 64;

therefore400 - 44

= 102400 + 25600q 66400q + 318400 - 480000,400 - 44

and from the reduced eqn. 17q = 200 - 15 ; as from 1 = q 21700 - 45 = 3017 = BO is given.17

[ RA = AO - 6; BR = 6 + ]

r

[Note: The correct algebra gives 17x2 +15x -200 = 0, with the solution x = (- 15 + √13825)/34 = 3.017 as shown; the original text contains a number of errors -as shown below in the Latin text.]

And the conjugate diameters are easily given from these ; and from the conjugatediameters, and the angle of the conjugate axes of these by the example from page 50 ast.Geometricae, or with the aid of Prop. 72 and 73 for the ellipse from Gregory of St.Vincent. And finally from the above given, everything sought will be easily found, surelythe centre of the mean motion, (if it shall be in the nature of things) the position of thesun within the ellipse, and the eccentricity, or the distance of the centre of the sun fromthe centre of the mean motion, and thus the kind of ellipse of the planet.

Prop. 84. Problema.

Ex datis duobus locis centricis planetae in suo orbe, una cum ipsius motu medio, & a sole distantiis :positionem aphelii, & ellipseos planetaria speciem, invenire.


Sit PCQN ellipsis planetae,cujus focus medii motus F, focus veri motus, sive sol S, axis transversus PQ,loca planetae C, N: E datis in triangulo CSN lateribus SC, SN, distantiis nimirum planetae a sole ; & anguloab illis comprehenso CSN,

[123]e locis planetae centricis cognito ; inveniatur latus CN, & angulus NCS, Ducantur rectae FC, FN; &quoniam NS + NF = CS + CF ; erit CS - SN = NF - CF = LN: est enim FC = FL ; & in triangulo FLCisosceli, dantur omnes anguli, nempe LFC ; ex medio motu inter observationes planetae C, N, & anguliFLC, FCL propter eorum aequalitatem ; ergo datur angulus CLN, Deinde in triangulo CLN, e datis CLN,NC, NL, reperitur LCN ; datur ideo angulus FCN = FCL + LCN; & in triangulo FCN, e datis omnibusangulis , cum latere CN, datur latus CF: & CF + CS = PQ. Denique in triangulo FCS, e datis FC, CS [CFin text] & angulo FCS, ex angulis inventis FCN, NCS cognito ; inveniuntur FS distantia focurum; & FSCpositio aphelii, ab observatione C.

Scholium.Si autem quis desideret hujus Problematis resolutionem, motu medio non supposito: detur terria

observatio D, & proinde eodem modo invenietur DN, quo in proportione invenitur CN. Sit axis PQ14 &ideo dabuntur FC = 14 - CS, & FN = 14 - NS ; in quadrilatero igitur FCSN, dantur omnia latera, una cumdiametro CN, e quibus inveniatur duameter FS. In quadrilatero quoq; FDSN, dantur omnia latera SD, SNex observatione, FD = 14 - DS, FN = 14 - NS una cum diametro DN, e quibus rursus inquiratur FS :daturigitur aequatio inter FS primo inventam, & FS secundo inventam, cujus resolutio manifestae omniaquaesita.

Methodus autem omnium pulcherrima, est illa quam tradit D. Wardus ast. Geometriae, pag. 50, quaenec supponit motum medium, nec solem esse ellipseos focum , sed tantum , solis centrum immobile, &planetas moveri, in lineis ellipticis; Unde centrum motus

[124]medii, (si tale detur) & locum solis intra ellipsem, ab aliis supposita, demonstratione Geometrica concludit.Sed propter eximium hujus methodi, usum conabimur nos aliam illius praxem in medium afferre. Sintquinque puncta in ellipsi, H, K, L, M, N, quae nusquam terminent lineas parallelas, ducantur NK, MH,secantes sese in T: lineae autem MH ducatur parallela LV, secans NK in Y. Observationibus dantur omnesanguli ad S, & rectae inter S, & quinq; puncta ellipseos : & proinde cognoscuntur rectae MN, NK, NL, &anguli MNS, KNS, MNT, NMS, HMS, NMT, NLY. Deinde in triangulo rectilineo MNT, e datis omnibusangulis , cum latere MN, dantur etiam latera NT, MT; & in triangulo rectilineo NLY, e datis omnibusangulis, (est enim MTN = LYN) cum latere LN; dantur latera NY, LY ; at NT x TK : NY x YK :: MT xTH:LY x YV ; ergo datur YV, & proinde tota LV, Bisecentur rectae LY, MH, in P, & O, & ducatur rectaAPDOB, diameter nempe ellioseos, cui ordinatim applicantur PV, OH ; eidemq; diametro ordinatimapplicetur recta KR : & ob similitudinem triangulorum PDY, TDO, erit PY + TL : TY : : PY : DY ;cumque detur angulus PYD, non latebunt PD,DO, DR, KR. Sint igitur, in numeris datae, rectae, OH 10, PV12, RK 8, PO 4, RP2, cum angulo KRB 60. Ponatur BO 1 , eritq; BP 1 + 4, sed 100: 144 : : 1 x OA :1 + 4 x PA; igitur per conversum 99, de proportionalitatibus Greg. a s. Vincentio; 100 est ad 144 ; atradio 1 ad 1 + 4 est ad rationem PA ad OA

[125]= PA + 4; igitur PA est ad PA + 4, ut

100144 ad 1, + 4, vel 144 ad 100 + 400; & ideo horum

differentia, nempe


B

5

5

2

S

1B

400 - 44 : 144 : : 4: 576400 - 44

= PA ; proinde PA + = 1600 + 400

400 - 44= AO,

& RA = 664 - 800400 - 44

; at BC x OA = 1600 + 400 q

400 - 44: 100 : : BR x RA = 664q + 3184 - 4800

400 - 44: 64;

igitur400 - 44

= 102400 + 25600q 66400q + 318400 - 480000,400 - 44

& aequatione reducta 17q = 200 -90 ; ut de datur 1 = q 5425 - 45 = 1686 = BO.17

r

Atque; ex his facile dantur diametri conjugatae ; & ex diametris conjugatis, & earum angulo, axes conjugatiper praxem pag. 50 ast. Geometricae, vel ope 72, & 73 de ellipse Grego. as. vincentio. Et tandem expraedictis datis, facile innotescunt omnia quasita, centrum nempe motus medii, (si sit in rerum natura) locussolis intra ellipsem, & excentricita, seu distantia centrij solaris a centro medii motus, & ellipseosplanetariae species.

Prop. 85. Problem.

From the presumed theoretical positions of the earth and the planet, according to which the longitude,latitude, and the position of the planet from the earth are to becomputed.

The position is calculated from the given central motion of the earthitself, which shall be 1 in figure of Prop. 83 of this work [redrawnhere]: and in short by the same method the position of the planet in itsorbit, which shall be 5 ; and from the given distance from the node,and the inclination of the orbit, the angle BS5 may be found, thecentral latitude of the planet, and the position of the planet in theecliptic B; and with the position of the earth in the ecliptic given 1, theangle 1SB is given. Given also from the theories of the earth and theplanet SO, S5 and in right-angled triangle SB5, from 5S, BS5 given,BS is found. Then in triangle BS1 from BS and S1 given; and fromthe contained angle BS1, the side 1B is given, and the angle S1B, theposition of the planet in the ecliptic from the earth 1. And finally, as1B to SB, thus the tangent of the angle

[126]BS5, to the tangent of the angle B15, the angle of the latitude sought.

Prop. 85. Problema.

Suppositis Theoriis Terrae, & Planetae ; locum Planetae e Terra, quo ad longitudenem, & latitudinemsupputare.

Ex terrae mediis motibus datis supputetur ipsius locus, qui in figura 83 hujus sit 1 {78 hujus} : & eodemprorsus modo supputetur locus planetae in suo orbe, qui sit 5; & data a nodo distantia & inclinationeorbium, inveniatur BS5, latitudo planetae centrica, & locus illius in ecliptica B; cumq; detur locus terrae inecliptica 1, datur angulus 1SB. Dantur etiam ex theoriis terrae & planetae SO, S5 & in triangulo rectanguloSB5, e datis 5S, BS5 & in triangulo rectangulo BS5, e datis 5S, BS5 reperitur BS. Deinde in triangulo BS1e datisBS, S1; & angulo comprehenso BS1, datur latus 1B, & angulus S1B, locus planetae in ecliptica, eterra 1. Et tandem, ut 1B ad SB, ita tangens


PZ

O

L

MN

[126]anguli BS, ad tangentem anguli B15, latitudinis quaesitae.

Prop. 86. Problem.

From one eclipse of the moon ; to find the lunar parallax and the diameter of theshadow for the moon in transit : with the supposition of the theory of the sun's motiononly, and with the fixed positions of the stars.

It may be observed in aneclipse of the moon howprecisely the horns of themoon may become to theperpendicular, or arise in asingle azimuth angle: (butonly if it occurs), and at thatpoint of time, the separationof the horns is observed, theheight from one of the horns,and its declination and theright ascension according tothe fixed stars; and hence thedeclination and the rightascension of the mid-pointbetween the horns will befound. Let the distance of themid-point between the hornsfrom the [celestial] pole P bethe arc PL, and let the

complement of the apparent altitude, or the distance of the same apparent point from thevertical [zenith] Z be the arc ZL.

[127]And in the spherical triangle ZPL, from the three given sides, the angle ZLP may befound. And the centre of the shadow, or the point opposite the sun, shall be O ; and sincethe declination and the right ascension of this shall be given in the spherical triangle LPO[the following quantities] shall be given: the side PO, the distance of the centre of theshadow from the pole; the angle OPL, the difference of the right ascensions; and theobserved side LP; and therefore the side LO will be given ; and the angle OLP. Then O,from the centre of the shadow, may fall on the vertical circle ZL, and truly and moreevidently, appearing to cross the horns of the moon in the perpendicular arc OM. Andsince there are two circles on the sphere [i.e. celestial], truly the circle of the moon, andthe circle of the shadow, they mutually cut each other in the horns of the moon. The arc ofone circle from the pole, the perpendicular joining the arc of their greatest circle ofintersection, truly OM, divides the aforementioned arc in two in the point M ; M will bethe true position of this point ; L is the apparent position of this join, surely of the mid-point between the horns [recall that this point is not seen 'head-on', but at an angle, hencethe parallax] ; and the arc ML the points of the same parallax, which is found from the


resolution of the spherical triangle LMO with the right-angle at M, from the given angleMLO = ZLP + PLO, and the side LO. Then N shall be the true position of one of thehorns, and in the spherical triangle MNO with the right-angle at M [N in text], from thegiven MN, MO, NO is given, the semi-diameter of the shadow. But I must satisfy thetwo, from little geometrical tricks and our vision, that this parallax is the natural parallaxof the centre of that circle in the body of the moon, which is drawn through the horns ofthe moon; for this given parallax, the geometrical parallax of the centre of the moon, canbe found by Prop. 69 of this work, which in no circumstance differs sensibly from theparallax now found.

Scholium.But because it is difficult to find the right ascension of the first fixed stars, and theparallax of the moon will be found much more exact from the common azimuthfollowing Prop. 62, 63, and 64 of this work; rather it might be by the parallax of themoon thus found, and the converse of this problem, to find the right-ascension of thefixed stars. I was deciding to add certain things about the theory of the moon; but a stonesticks in the entrance, still unshaken by the astronomers, and entirely prohibiting ageometrical entrance: indeed the central lunar motions cannot be determined by ageometrical method up to this day. For these truly supposed, and from three positions ofthe moon in her orbit, from the second inequality of freedom, kinds of lunar ellipses canbe found, and the position of the apogee in each one: from observation, by Corollary 77of this work; if the motion of the apogee may be taken from both the central motion andthe true motion ; and remaining may be put in the place of both the central motion and thetrue motion ; by preceding in the same way as is taught in that Corollary. In short I avoidthe second inequality of the moon, since the astronomers themselves are still ignorantabout this. However our mind is, also with some firm reason, that the moon has someannual inequality brought together with her first inequality, from which she might easilybe set free, with the help of three lunar eclipses observed from one place of the ecliptic.

Prop. 86. Problema.

Ex unica eclipsi lunari ; parallaxem lunae, & diametrum umbrae in transitu lunae investigare :suppositis tantum solis theoria, & fixarum locis.

Observetur quam exactissime in eclipsi lunae, cum cornua lunae fiant ad perpendiculum, sive existant inuno Azimuth: (si modo accidat) & in eo temporis articulo, observetur distantia cornuum, altitudo uniuscornu, & illius declinatio, & ascensio recta, per stellas fixas; ac proinde non latebit, declinatio, & ascensiorecta, puncti medii inter cornus. Sit distantia puncti medii inter cornua a polo P, arcus PL , sitq;complementum altitudinis apparentis, seu distantia apparens ejusdem puncti a vertice Z, arcus

[127]ZL. Et in triangulo sphaerico ZPL, e datis tribus lateribus, inveniatur angulus ZLP. Sitq; centrum umbrae,seu oppositum solis O; & quoniam datur ipsius declinatio, & ascensio recta, dabuntur in triangulo sphaericoLPO, latus PO, distantia centri umbrae a polo, angulus OPL, differentia ascensionum rectarum , & latusLPobservatum; dabitur ergo latus LO; & angulus OLP. Deinde a centro umbrae O cadat in circulumverticalem ZL, & vere, & apparenter, per cornua lunae transeuntem, arcus perpendicularis OM: Et quoniamduo circuli in sphaera, nempe circulus lunae, & circulus umbrae, se mutuo secant in lunae cornubus ; arcusa polo unius circuli, perpendicularis ad arcum circuli maximi eorum intersectiones jungentem, nempe OM,bifariam dividit praedictum arcum in puncti M ; erit igitur M locus verus illius ; juncti, cujus L est locusapparens, nimirum puncti medii inter cornu ; & arcus ML ejusdem puncti parallaxis, qui reperitur exresolutione trianguli sphaerici LMO rectanguli ad M, e datis angulo MLO = ZLP + PLO, & latere LO.Deinde sit N verus locus unius cornu, & in triangulo sphaerico MNO rectangulo ad N, e datis MN, MO,


datur NO, semidiameter umbrae. Ut autem geometricis ingeniis aliquantulum satisfaciam, duo hancparallaxem esse genuinam parallaxem, centri illius circuli in corpore lunae, qui ducitur per illius cornua, &nostrum visum ;data autem hac parallaxi, poterit inveniri & geometrica parallaxis centri lunaris, per 69hujus, quae nunquam sensibiliter differt a parallaxi nunc inventa.

Scholium.Sed quoniam difficulter invenitur ascensio recta primae stellae fixae, & parallaxis lunae multo exactiusinveniri poterit, per commune azimuth secundum 62, 64, 65 hujus ; satius esset per parallaxem lunae itainventam, & hujus problematis conversum, ascensioinem rectam primae stellae fixae investigare.Statuebam de theoria lunae quaedam adjungere; sed in ipso vestibulo haeret saxum, ab astronomis adhucinconcussum, & ingressum geometricum omnino prohibens: motus enim medii lunares in hunc usque diem,non determinantur methodo geometrica. Illis vero suppositis, & datis tribus locis lunae insua orbita, asecondis inaequalitatibus liberatis, poterit inveniri species ellipsess lunaris, & positio apogaei

[128]in unaquaq: observatione, per Corollarium 77 hujus; si auferatur motus apogaei, & a motu medio, & motuvero ; & residua ponantur loco motus medii, & motus veri ; eodem modo procedendo ut in illo Corollariodocetur. A secundis lunae inaequalitatibus prorsus abstineo, quoniam adhuc ipsis astronomis suntincognitae. Nostra tamen mens est, aliqua etiam ratione stabilita; lunam habere aliquam inaequalitatemannuam, cum prima sua inaequalitate commissam; a qua facile liberaretur, ope trium eclipsium lunarium, inuno eclipticae loco observatarum.

Prop. 87. Problem.

The parallaxes of two planets are to be investigated from the conjunction of each body.

Let the true locations of the observations of the two planets be H and C, in the bodyconjunction: and the planet C is observed in the common azimuth of the two locations,the poles of which are A and B. Let the apparent positions of the planets C and H, fromthe position of which the pole is A be E and L ; and from the position of the pole B theapparent positions are D and G; and the arcs EL and DG of the angles CEL and GDCmay be observed at the same time, according to Prop. 73 of this work. But the arcs AEand BD are given, to be observed finely enough to spread out the four parts; and hencethe arcs AL and BG are given, and the angles HGD and HLE ; from which given theparallaxes CD, CE, HG, HL are to be found. But since it is impossible to find the arcsAE, AL; BD and BG with the same care that the rest of the arcs and angles arediscovered; we therefore reject these from the calculation : yet they reveal with enoughaccuracy the ratio of the sines of the arcs DC, CE, and GH, HL, by Prop. 64 of this work.But the proportion of their distances from the centre of the earth is given by the theoriesof the planets C and H. Therefore by Prop. 69 of this work, given the proportion of thesine of the arc CE, to the sine of the arc HL; and the sine of the arc DC, to the sine of thearc HG. And therefore the mutual proportions are given between the sines of theparallaxes CE, CD, HG, HL. Hence from one given all become known; the perpendiculararc LF is dropped from the point L to the common azimuth ACB: and in the sphericaltriangle ELF, with the right-angle at F, from the given side LE, and from the angle LEF;the side FE may be found, and the side FL and the angle FLE: therefore the angle FLHwill become known. The sine of the parallax CE may be put as 1 ; and from the givensines of the arcs FE and EC, the sine of the arc FC of the sum of these is formed: then inthe spherical triangle LFC, with the right-angle at F, from the given sines of the sides FC,FL, the sine of the side LC and of the angle FLC may be found: and thus from the givensines of the angle HLF and the angle CLF, the sine of their difference may be found,


A

B

C

D

E

L

H

N

G

F

surely of the angle HLC.The perpendicular arc CN isdropped from the point C onthe azimuth AHL: and thusin the spherical triangleLCN, with the right-angle atN ; from the given sine ofthe side LC and the sine ofthe angle CLN, the sines ofthe sides CN and LN may befound; but given the sine ofthe arc HN, without doubtthe difference of the arcsLH and LN. And again inthe spherical triangle HNC,with the right-angle at N ;from the given sines of thesides HN and NC, the sine ofthe side HC may be found.And thus by the samemethod the sine of the arcHC may be found from thegiven sines of the arcs DC and GH, in the fractional part of which the sine of the arc CEis 1 : therefore an equation will be given, between the sine of the arc HC found first,and the same sine found from the second calculation ; the resolution of which will givethe value of the root, surely the sine of the parallax CE.

Scholium.This prettiest of problems has a use, but perhaps a very laborious one, in the observationsof Venus or Mercury obscuring a little part of the sun : indeed from such the parallax ofthe sun will be able to be investigated. Up to the present we have talked about parallaxeswith respect to the globe of the earth; certain follow for parallaxes of the great orbit of theearth.

Prop. 87. Problem.

Ex duorum Planetarum conjunctione corporali; utriusq; planeta Parallaxes investigari.

Sint loca vera duorum Planetarum, in conjunctione corporali observatorum H, C: & observetur planeta C incommuni azimuth duorum locorum, quorum vertices A, B: sintq; planetarum C, H, e loco cujus vertex A,loca apparentia E, L; & e loco cujus vertex B, loca apparentia D, G; & eodem instante observentur per 73hujus arcus EL, DG, anguli CEL, GDC: Dantur autem arcus AE, BD, quadrantibus ad intentum satissubtiliter observari; & proinde dabuntur arcus AL, BG, & anguli HGD, HLF ; e quibus datis, inveniendaesint parallexes CD, CE, HG, HL : Quoniam autem impossibile est arcus AE, AL, BD, BG, eadem diligentiainvenire, qua reperiuntur arcus, & anguli reliqui; ideo eos e calculo rejicimus: Manifestant tamen satisaccurate, rationem sinuum , arcuum DC, CE, & GH, HL, per 64 hujus. Datur autem ex planetarum C, H,theoriis, proportio suarum distantiarum a centro terrae, & igitur per 69 hujus, datur proportio sinus arcusCE, ad sinum arcus HL; & sinus arcus DC, ad sinum arcus HG: & ideo dantur proportiones mutuae intersinus parallaxium CE, CD, HG, HL : & proinde uno dato innotescunt omnes, ex puncto L in azimuth


A

C D

L

MN

B

commune ACB, demittatur arcus perpendicularis LF: & in triangulo sphaerico ELF, rectangulo ad F, e [Ein original] datis latere LE, & angulo LEF; inveniantur latus FE, latus FL, & angulus FLE: innotescit igiturangulus FLH. Ponatur sinus parallaxi [parallaxeos in original] CE 1 ; & e datis sinubus arcuum FE, EC,inveniatur sinus arcus FC eorum summae: deinde in triangulo sphaerico LFC, rectangulo ad F, e datissinubus laterum FC, FL, inveniatur sinus lateris LC, & anguli FLC: e datis itaq; sinu anguli HLF, & sinuanguli CLF, inveniatur sinus eorum differentiae, nempe anguli HLC:

[130]& demittatur a puncto C in azimuth AHL, arcus perpendicularis CN : in triangulo itaq; sphaerico LCN,rectangulo ad N ; e datis, sinu lateris LC, sinu anguli CLN; inveniatur sinus laterum CN, LN; datur autemsinus arcus HN, differentiae nimirum arcuum LH, LN. Et tandem in triangulo sphaerico HNC, rectanguload N ; e datis sinubus laterum HN, NC, inveniatur sinus lateris HC. Et eodem prorsus modo inveniatursinus arcus HC, e datis sinubus arcuum DC, GH, in partibus, quarum sinus arcus CE 1 : dabitur ergoaequatio, inter sinum arcus HC primo inventum , & sinum eundem secundo inventum ; cujus resolutiodabit valorem radicis, nempe sinum parallexeos CE.

Scholium.

Hoc Problema pulcherrimum habet usum, sed forsan laboriosum, in observationibus Veneris, vel Mercuriiparticulam Solis obscurantis : ex talibus enim solis parallexis investigari poterit. Hactenus loquuti sumus deparallaxibus respectu globi terrestris: sequuntur quaedam de parallaxibus magni orbis terrae.

Prop. 88. Problem.

To investigate the parallax of fixed stars (by a sensing method).

Let ABCD be an arc of the ecliptic, in whichA may be assigned to a point in apposition tothe sun; and let the true position of the star beL, N appearing as the apparent location. Thedeclination is found from the meridianaltitude of the star; and the right-ascensionfrom any equal motion, or by Scholium 86 ofthis work, and hence the latitude itself will befound, which shall be NC ; and the position inthe ecliptic C. Then, in the spherical right-angled triangle ANC ; from the given sides AC and CN ; the side AN is found, and theangle CAN. Again, with the apposition of the sun holding, an observer at point D shallobserve the latitude MB, and the place in the ecliptic B of the same star (Of which thetrue position is L, but now appearing at M) and by the same way as before, the arc MDand the angle MDB of the same star are found. Finally in the spherical triangle ADL,from the given angles ADL and DAL ; and from the side AD,

[131]from the known motion of the sun, the side AL will be given, which taken from the arcAN, leaves the parallax LN, of the observation N ; and the side DL, which taken from thearc DM, leaves LM, the parallax of the observation M.


OT

AN

LB

HI

A

NL

HI

S

R

B

B

Prop. 88. Problema.Parallaxem stellae fixae (modo sensibilem) investigare.

Sit arcus eclipticae ABCD, in quo assignetur punctum soli oppositum A ; sitq; verus locus stellae L,apparens N; cujus loci apparentis, declinatio invenitur per altitudinem stellae meridianam; & ascensio recta,per motum aliquem aequabilem ; vel Scholium 86 hujus ; & proinde non latebit ipsius latitudo, quo sit NC ;& locus in ecliptica C. Deinde, in triangulo sphaerico rectangulo, ANC ; e datis lateribus AC, CN ;invenitur latus AN, & angulus CAN. Rursus opposito solis punctum D tenente, observetur ejusdem stellae(cujus verus locus L, nunc autem apparens locus M) latitudo MB, & locus in ecliptica B; & eodem modoquo ante, invenitur arcus MD, & angulus MDB. Tandem in triangulo sphaerico ADL, e datis angulis ADL,DAL ; & latere AD,

[131]per motum solis cognito, dabitur latus AL, quod ab arcu AN ablatum, relinquit LN parallaxem,observationis N ; & latus DL, quod ab arcu DM ablatum ; relinquit LM, parallexem observationis M.

Prop. 89. Theorem.The ratio, that can be observed of the sine of the parallax of a phenomenon to the sine

derived from another parallax of the same phenomenon at another time, is composedfrom the direct proportion of the sines of the apparent distances from the sun; from thedirect proportion of the distances of the earth from the sun ; and the reciprocalproportion of the distances of the phenomenon from the sun.

Let the sun be S, with the earth T for the first time of observation, and the phenomenonfor the same time A; the earth R for the time of the second observation, the phenomenon

B, with O opposite the sun. WithCentre S and radius SA the circleANL is described, and the rightlines SRTO, TA, SA, RB, SB aredrawn. I say that the ratio: the sineof the parallax TAS to the sine ofthe parallax RBS, is to becomposed from the ratio of thesine of the angle STA to the sine ofthe angle SRB; with the ratio TS toRS ; and from the ratio BS to AS.For the right line TN is drawnparallel to RB: and to theintersections of the lines RB and

TN with the circle ANL, surely L and N, the lines SL and SN are drawn; and to the linesTN and RB, the perpendicular SH is drawn. And (by Prop. 64 of this work) the sine ofthe angle TAS to the sine of the angle TNS shall thus be as the sine of the angle STA tothe sine of the angle STN: and as the sine of the angle TNS, surely ( putting NS for theradius) SH; to the sine of the angle RLS, surely (for the radius has put LS = NS) SI; thusTS to RS: and as the sine of the angle RLS the sine of the angle RBS; thus BS to LS =SA. If therefore there were as many as, surely, the sine of the angle TAS, the sine of theangle TNS, the sine of the angle RLS, the sine of the angle RBS; the ratio of the first tothe last, surely the sine of the parallax TAS, to the sine RBS, composed from the mean


ratio, which have been shown with the same ratio; the sine of the angle STA, to the sineof the angle SRL; TS to RS ; BS to AS: therefore the proposition is shown.

Comment on Prop. 89.From Theorems 63 and 64, we have the parallax of the observation, namely the

difference of the angles subtended between the vertical and the phenomenon, for anobserver located on the surface of the earth and the angle measured from the centre of theearth; from the theorems it is easily shown that the ratio of the sines of the parallaxes isthe same as the ratio of the sines of the angles to the vertical, for two observers atdifferent locations at the same time. In the present theorem, the sine rule applied totriangles TAS and TNS, noting SA = SN, gives sin TAS / sin TNS = sin STA / sin STN.Also, from the right-angled triangles and the parallel bases, sin TNS = HS/SN, andsin RLS = SI / SL, giving sin TNS / sin RLS = HS / SI = TS / RS.Also, sin RLS / sin RBS = BS / LS. Hence, if we consider sin TAS / sin TNS, andsin RLS / sin RBS:sin TAS / sin RBS = sin TNS / sin RLS × BS / LS × sin STA / sin SRL =TS /RS × BS / AS × sin STA / sin SRL.

The situation in the theorem might occur if the phenomenon were a comet, and thepositions of the comet A and B were observed at time intervals of 12 hours from the samelocation on earth, and TR is related to the diameter of the earth.

Prop. 89. Theorema.

Ratio, sinus parallaxeos Phaenomeni, ad sinum, alterino parallaxeos ejusdem Phaenomeni, alio temporeobservati; est composita ex directecta proportione sinuum, distantiarum apparentium a sole ; directaproportione distantiarum terrae a sole; & reciproca proportione distantiarum Phaenomeni a sole.

Sit sol S, terra tempore primae observationis T, phaenomenon eodem tempore A; terra temporesecundae observationis R, Phaenomenon B, solis oppositio O. Centro S & radio SA describatur circulusANL, & ducantur rectae SRTO, TA, SA, RB, SB. Dico rationem, sinus parallaxeos TAS, ad sinumparallaxeos RBS, esse compositam ; ex ratione sinus anguli STA, ad sinum anguli SRB; ratione TS ad RS ;& ratione BS, ad AS. Rectae RB, parallela ducatur TN: & ad intersectiones rectarum RB, TN, cum circuloANL, nempe L, N, ducantur rectae SL, SN; & ad rectas TN, RB, perpendicularis ducatur SH. Eritque (perhujus 64) ut sinus anguli TAS, ad sinum anguli TNS ; ita sinus anguli STA, ad sinum anguli STN: & utsinus anguli TNS, nempe ( positio NS ratio) SH; ad sinum anguli RLS. nempe (posito ratio LS = NS) SI;ita TS, ad RS: & ut sinus anguli RLS ad sinum anguli RBS; ita BS, ad LS = SA. Si igitur fuerintquotcunque quantitates, nempe sinus anguli TAS, sinus anguli TNS, sinus anguli RLS, sinus anguli RBS;ratio primae ad ultimam, nempe sinus parallaxeos TAS, ad sinum RBS, componentur ex rationibusmediarum , quae eadem demonstratae sunt cum rationibus ; sinus anguli STA, ad sinum anguli SRL; TS adTS ; BS ad AS: patet ergo propositum.

Prop. 90. Problem.From three given longitudes and latitudes of any phenomenon moving around the sun,

on setting aside the second inequality of this and the inclination of the orbit to theecliptic; for each observation of the phenomenon to find the distance from the sun withthe supposition however of the theory of the earth, and with the ratios of the distances ofthe phenomenon from the sun.


A

B

C

R

S

T

I

L

E

F

D

M

O

P

Q

A

I

E

LD

FM

C

R

S

T

O

P

Q

B

Let AB be an arc of the ecliptic, and AC shall be the orbit of the phenomenon: threelocations R, S, T of thephenomenon areobserved from the earthin its orbit [with referencepoints] O, P, Q. Becausethe positions of thephenomenon observedfrom the earth are given ;the same apparentdistances will also begiven, either from thedirection of the sun orfrom the directionopposite the sun, surelyIO, LP, and MQ; and thearcs IL and LM ; and the

angles OIA, PLA, and QMA; and by the preceding proposition the ratios of the sines ofthe parallaxes RO, SP, and TQ become known. The sine of the arc RO is put as 1 ;therefore the sines of the arcs SP, TQ will be given in terms of unknown numbers : butthe sines of the arcs IO, LP and MQ are given; therefore the sines of the arcs IR, LS, MTare found in terms of unknown numbers. And from the given sines of the arcs IR, LS,MT, with the sines of the angles RIF, SLD, TME ; the sines of the perpendicular arcs RE,SD, and TF can be found and the sines of the arcs ED and DF. Then from the given sinesof the arcs ER, DS and ED; the sine of the angle CAB is sought: by Prop. 70 of this work:and in the same way from the given sines of the arcs DS, FT and DF, the sine of the sameangle CAB is found: and it will give an equation between the sine of the angle CAB firstfound, and the second found, the solution of which equation will reveal all sought.

Scholium.But this work and labour are to find the proportions of the distances of the

phenomenon from the sun ; however they are acquired with enough probable conjecture,through the change of the diameter [of the orbit that appears], the slowing down orspeeding up of the central motion; yet everything most certainly has to be done from acombination of many observations: but our task has not been to set up non-geometricalmethods to provide more detailed explanations.

Comment on Prop. 90 : The problem of determining the orbit of a comet had to awaitthe coming of analytical methods as set out by Isaac Newton in the third book of thePrincipia, some 20 years later. It is interesting to note that Gregory seems to be hinting atsuch a development in the final sentence.

The End.


Prop. 90. Problema.

E datis tribus longitudinibus, & latitudinibus, Phaenomeni cujuslibet, circa solem agitati ; ipsiusinaequalitatem secundam exuere ; orbis inclinationem cum ecliptica, & in unaquaq; observatione,Phaenomeni a sole distantiam investigare : suppositis tantum , telluris theoria, & rationibus distantiarumPhaenomeni a Sole.

Sit arcus eclipticae AB, orbis Phaenomeni AC, tria loca Phaenomeni e terra observata O, P, Q, in suoorbe R, S, T. Quoniam loca phaenomeni e terra observata dantur ; dabuntur etiam ejusdem, vel a sole, velab oppositionibus solis, distantiae apparentes, nempe IO, LP, MQ; & arcus IL, LM ; & anguli OIA, PLA,QMA, & perProp. praecedentem innotescunt rationes sinuum, parallaxium RO, SP, TQ. Ponatur sinus arcusRO 1 ; dabantur igitur sinus arcuum SP, TQ in numeris cofficis :dantur autem sinus arcuum IO, LP, MQ ;deprehenduntur ergo sinus arcuum IR, LS, MT, in numeris cofficis. Et ex datis sinubus arcuum IR, LS,MT, cum sinuus angulorum RIF, SLD, TME ; inveniantur sinus arcuum perpendicularium RE, SD, TF, &sinus arcuum ED, DF. Deinde ex datis sinubus arcuum ER, DS, ED; quaeratur sinus anguli CAB: per 70hujus:& eodem modo e datis sinubus arcuum DS, FT, DF, inquiratur sinus ejusdem anguli CAB:dabiturque aequatio inter sinum anguli CAB primo inventum, & eundem secundo inventum, cujusaequationis resolutio manifestabit omnia quaesita.

Scholium.Proportiones autem distantiarum Phaenomeni a sole invenire, hic labor, hoc opus ; nihilominus

acquiruntur probabili satis conjectura, per diametri apparentis mutationem, motus centrici tarditatem velvelocitatem; omnium tamen certissime ex multarum observationum collatione : sed nostri non est institutimethodos ageometricas fusius explicare.

FINIS.

gregory, j. - optica promota

Documents