References

[Archimedes (Heath)] The Works of Archimedes. Heath T L (ed), with a supplement The Methodof Archimedes recently discovered by Heiberg, New York, Dover Publications

[Aristotle, Metaphysics (Ross)] Aristotle, Metaphysics. Translated Ross W D, http://www.classicallibrary.org/aristotle/metaphysics/index.htm

[Aristotle, Physics] Aristotle, Physics [http://classics.mit.edu/Aristotle/physics.html][Aristotle, Physics (Stanford)] Aristotle, Physics [http://classics.mit.edu/Aristotle/physics.html][Arthur, Fictions] Arthur R T W, Four phases in Leibniz’s early thought on infinitesimals. [http://

www.humanities.mcmaster.ca/∼rarthur/articles.arthur.htm][Arthur, Syncategorematic] Arthur R T W, Leibniz’s syncategorematic infinitesimals, smooth

infinitesimal analysis, and Newton’s proposition 6. [http://www.humanities.mcmaster.ca/∼rarthur/articles.arthur.htm]

[Arthur, Law of continuity] Arthur R T W, A complete denial of continuity? Leibniz’s law ofcontinuity. [http://www.humanities.mcmaster.ca/∼rarthur/articles.arthur.htm]

[Leibniz, Continuum] The Labyrinth of the Continuum: Writings on the Continuum Problem1672–1686 (2001). Ed., sel. & transl. Arthur R T W. Yale University Press, New Haven

[Arthur, Newton] Arthur R T W, Newton’s fluxional proof of the vector addition of motive forces.[http://www.humanities.mcmaster.ca/∼rarthur/articles.arthur.htm]

[Berkeley, Analyst] Berkeley G (1734) The Analyst; or a discourse addressed to an infidel math-ematician. London. [http://www.maths.tcd.ie/∼dwilkins/Berkeley/ETextsTCD.html]

[Bernoulli, Daniel, Hydromechanica] Bernoulli D (1738) Hydrodynamica, sive de viribus etmotibus fluidorum commentarii. Opus academicum ab auctore, dum Petropoli ageret, con-gestum. Johann Reinhold Dulsecker, Straßburg

[Bernoulli, Daniel] Bemoulli D (1726) Examen principiorum Mechanicae. Comm. Petrop. To. I[Bernoulli, Daniel , Wikipedia] Daniel Bernoulli [http://en.wikipedia.org/wiki/Daniel Bernoulli][Bernoulli, Letter to Euler] Bernoulli J. letter from November 6, 1737 to Euler. In: Opera Omnia,

II, 1, Preface[Bernoulli 1691–92] Bernoulli J (1924) Vorlesung uber das Rechnen mit Differentialen.

Schafheitlin P (ed) Die Differentialrechnung von Johann Bernoulli aus dem Jahre 1691/92.In: Oswalds Klassiker der exakten Wissenschaft. Akademische Verlagsgesellschaft, Leipzig

[Johann Bernoulli, Response to Taylor] Taylor versus continental mathematicians [http://www-groups.dcs.st-and.ac.uk/∼history/Extras/Taylor continental.html]

[Bertrand] Bertrand J (1873) Comptes Rendus 77:849[Bohlmann] Bohlmann G (1899) Ubersicht uber die wichtigsten Lehrbucher der Infinitesimal-

Rechnung von Euler bis auf die heutige Zeit. Jahresberichte der Deutschen Mathematiker-Vereinigung 6(2):91

[Bohr 1] Bohr N (1913) On the Constitutions of Atoms and Molecules. Phil. Mag: 26, 1, 476, 875[Bohr Correspondence] Bohr N (1918) On the Quantum Theory of Line Spectra. D. Kgl. Danske

Vidensk. Selsk. Skrifter, Naturvidensk, og Mathem. Afd. 8. Raekke, IV.1, 1–3[Bohr 2] Bohr N (1914) On the effect of electric and magnetic fields on spectral lines. Phil. Mag:

27, 506

313

314 References

[Bohr 3] Bohr N (1915) On the series spectrum of the hydrogen and the structure of the atom.Phil. Mag: 29, 332

[Bohr 4] Bohr N (1915) On the quantum theory of radiation and the structure of the atom. Phil.Mag: 30, 394

[Bohr, Nobel] Bohr N (1922) The structure of the atom. Nobel Lecture[Bos] Bos H J M, Lectures in the history of mathematics. Providence RI, American Math.

Soc., 1997[Bos, Leibniz] Bos H J M (1974–75) Differentials, Higher-Order Differentials and the Derivative

in the Leibnizian Calculus. Archive for the History of the Exact Sciences, 14, 1–90[Bradley, D’Antonio, Sandifer] Bradley R E, D’Antonio L A and Sandifer C E (eds) Euler at 300:

An Appreciation. Mathematical Association of America[Bradley, Sandifer] Bradley R E, Sandifer C E (eds) Leonhard Euler 5. Life, Work and Legacy.

Elsevier, Amsterdam[Brumbaugh] Brumbaugh Z (2000) The integration theory of Gottfried Wilhelm Leibniz. Rutgers.

[http://www.math.rutgers.edu/∼cherlin/History/Papers2000/brumbau.html][Cajori] Cajori F (1952) A History of Mathematical Notations: vol. II, Notation Mainly in Higher

Mathematics. Open Court Publishing Co., Chicago[Cantor] Cantor G (1885) Uber die verschiedenen Standpunkte in bezug auf das aktuelle Un-

endliche. Ztschr. fur Philos und philos Kritik 88:224–233; Cantor (1890) Gesammelte Ab-handlungen zur Lehre vom Transfiniten. I. Abtheilung. Pfeffer C E M, Halle a S

[Cassirer] Cassirer E (1999) Das Erkenntnisproblem in der Philosophie und Wissenschaft derneueren Zeit, Bd. II. Hamburg

[Cavalieri] Cavalieri B (1653) Geometria indivisibilibus continuorum nova quadam ratione pro-mota. Bonnoniae [http://diglib.hab.de/wdb.php?dir=drucke/8-9-geom-1]

[Chandrasekhar] Chandrasekhar S (1995) Newton’s Principia for the Common Reader. Oxford[Chatelet, Institutions] Chatelet E du (1740) Institutions de physique. Paris[Chatelet, Naturlehre] Chatelet E du (1743) Naturlehre an ihren Sohn, erster Theil nach der

zweyten franz. Ausgabe ubersetzt von Wolf Balthasar Adolf von Steinwehr. RengerischeBuchhandlung, Halle/Leipzig

[Child] The early mathematical manuscripts of Leibniz (1920) translated from the Latin textspublished by Carl Immanuel Gerhardt with critical and historical notes by Child J M. Theopen court publishing company, Chicago, London

[Condon] Condon E U (1960) Sixty Years of Quantum Physics. Read before the Philosophical So-ciety of Washington December 2, 1960. [http://www.philsoc.org/MeetingArchive.html#1500]

[Condorcet, Eulogy] Condorcet A de (1783, 1787) Eulogy to Mr. Euler. In: History of the RoyalAcademy of Sciences 1783. Paris [http://www-groups.dcs.stand.ac.uk/∼history/Extras/Eu-ler elogium.html]

[Condorcet, Letters] Condorcet A de (1787, 1823), Preface to Letters of Euler on different sub-jects in Natural Philosophy addressed to a German Princess. Edinburgh. Quoted from: Lettresde M. Euler a une princesse d’Allemagne sur differentes questions de physique et de philoso-phie. Nouvelle edition avec des additions par MM. le marquis de Condorcet et de La Croix.Royez, Paris

[Couturat, Cassirer] Couturat L (1903) Le systeme de Leibniz d’apres M. Cassirer. In: Revue demetaphysique et de morale 83:11. [http://facdephilo.univ-lyon3.fr/rev.htm]

[Couturat, Opuscules] Couturat L (1903) Opuscules et fragments inedits de Leibniz. Felix Al-can, Paris

[d’Alembert, Traite] d’Alembert J L R (1743) Traite de Dynamique, dans lequel les loix del’equilibre & du mouvement des corps sont reduites au plus petit nombre possible, &demontrees d’une maniere nouvelle, & ou l’on donne un Principe general pour trouer le Mou-vement de plusieurs Corps qui agissent les uns sur les autres d’une maniere quelconque. Paris

[d’Alembert, Encyclopedie] L’Encyclopedie ou Dictionnaire raisonne des sciences, des arts et desmatiers (1751–1780)

[Descartes, Principia] Descartes R (1644) Principia philosophiae. Amsterdam [http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?dir=desca princ 099 la 1905; step=textonly]

References 315

[Descartes, Principles (Mahoney)] Descartes R (1995) The Principles of Philosophy, translated byGeorge MacDonald Ross

[Descartes, Principles (Ross)] Descartes R (1998–99) The Principles of Philosophy, translated byGeorge MacDonald Ross

[Dijksterhuis] Dijksterhuis E J (1956) Die Mechanisierung des Weltbildes. Berlin[Djerassi, Calculus] Djerassi C (2004) Calculus. [http://www.djerassi.com/calculus/calculus.html][Duchesneau] Duchesneau D M (1998) Leibniz’s Theoretical Shift in the Phoranomus and Dy-

namica de Potentia on the Foundation of the Calculus. Perspectives on Science 6: 1 & 2:77–109. The MIT Press

[Dunham] Dunham W (1999) Euler: The Master of Us All. Washington (DC)[Dunham, Euler] Dunham W (ed) (2007) Genius of Euler: Reflections on his Life and Work.

Mathematical Association of America[Ehrenfest 1917] Ehrenfest P (1917) Proc. Amsterdam Acad 20:200[Ehrenfest 1920] Ehrenfest P (1920) Welche Rolle spielt die Dimensionalitat des Raumes in den

Grundgesetzen der Physik? Ann Physik 61:440[Ehrenfest] Ehrenfest P (1959) Collected scientific papers. Klein M J (ed) North-Holland Publish-

ing Company, Amsterdam[Einstein, Bewegte] Einstein A (1905) Zu Elektrodynamik bewegter Korper. Ann Phys 17:

891–921[Einstein, Heuristisch] Einstein A (1905) Uber einen die Erzeugung und Verwandlung des Lichts

betreffenden heuristischen Gesichtspunkt. Ann. Phys. 17: 132[Einstein, Absorption] Einstein A (1906) Zur Theorie der Lichterzeugung und Absorption. Ann.

Phys. 20: 199[Einstein, Warme] Einstein A (1906) Die Plancksche Theorie der Strahlung und die Theorie der

spezifischen Warme. Ann. Phys. 22: 180[Einstein, Allgemeine Relat] Einstein A (1916) Die Grundlage der allgemeinen Rela-

tivitatstheorie. Ann. Phys. 49: 769–822[Enders 2004] Enders P, Suisky D (2004) Uber das Auswahlproblem in der klassischen Mechanik

und in der Quantenmechanik. In: Nova Acta Leopoldina Supplementum 18:13–17. Halle[Enders 2005] Enders P, Suisky D (2005) Quantization as selection problem. Int J Theor Phys

44(2):161–194[Euclid, Elements] Euclid’s elements of geometry (2007). The Greek text of J. L. Heiberg

(1883–1885) from Euclidis Elementa, edited et Latine interpretatus est I. L. Heiberg, in aed-ibus G. G. Teubneri, 1883–1885, edited, and provided with a modern English translation byRichard Fitzpartick

[Euclid (Heath)] The thirteen books of Euclid’s Elements (1908). Translated from the text ofHeiberg with introduction and commentary by T. L. Heath. University Press, Cambridge

The works of Euler will be cited according to Leonardi Euleri Opera Omnia sub auspiciis Soci-etatis Scientarium Naturalium Helveticae, Zurich und Basel 1911–1986 or by Enestrom index.

[Euler Archive] The Euler Archive. [http://www.eulersociety.org] [http://math.dartmouth.edu/∼euler/]

[Euler E015/016] Euler L (1736) Mechanica sive motus scientia analytice exposita. In: OperaOmnia II, 1

[Euler E015/016 (Wolfers)] Leonhard Euler’s Mechanik oder analytische Darstellung der Wis-senschaft von der Bewegung (1848) Wolfers J Ph (ed) C. A. Koch’s Verlagsbuchhandlung,Greifswald

[Euler E017] Euler L (1738) Rechenkunst, written 1734 presented 1738[Euler E044] Euler L (1740) De infinitis curvis eiusdem generis seu methodus inveniendi aequa-

tiones pro infinitis curvis eiusdem generis, presented 1734, published 1740[Euler E065] Euler L (1744) Methodus inveniendi lineas curvas maximi minimive proprietate

gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti, written 1743, pre-sented 1744. In: Opera Omnia I, 24

[Euler E081] Euler L (1746) Gedancken von den Elementen der Corper, written 1746, presented1746. In: Opera Omnia II, 2

316 References

[Euler E101] Euler L (1748), Introductio in analysin infinitorum, volume 1, written 1745, pre-sented 1748. In: Opera Omnia I, 8

[Euler E102] Euler L (1748) Introductio in analysin infinitorum, volume 2, written 1745, pre-sented 1748. In: Opera Omnia I, 9

[Euler E149] Euler L (1748) Reflexions sur l’espace et le tems, written 1748, presented 1748. In:Opera Omnia III,

[Euler E177] Euler L (1750) Decouverte d’un nouveau principe de mecanique, written 1750, pre-sented 1750. In: Opera Omnia II, 5

[Euler E181] Euler L (1750) Recherches sur l’origine des forces, written 1750, presented 1750.In: Opera Omnia III, 2

[Euler E212] Euler L (1755) Institutiones calculi differentialis cum eius usu in analysi finitorumac doctrina serierum, volume 1, written 1748, presented 1755. In: Opera Omnia I, 10

[Euler E212 (Michelsen)] Euler L (1790) Vollstandige Anleitung zur Differential-Rechnung. Ausdem Lat. ubers. und mit Anm. und Zus. begl. von J. A. C. Michelsen. Lagarde undFriedrich, Berlin

[Euler E289] Euler L (1765) Theoria motus corporum solidorum seu rigidorum, written 1760,presented 1765. In: Opera Omnia II, 3 and 4

[Euler E289 (Wolfers)] Leonhard Euler’s Theorie der Bewegung fester und starrer Korper (1853)Wolfers J Ph (ed) C. A. Koch’s Verlagsbuchhandlung, Greifswald

[Euler E342] Euler L (1763) Institutionum calculi integralis, vol. 1, published 1763. In: OperaOmnia I, 10

[Euler E343] Euler L (1768) Lettres a une princesse d’Allemagne sur divers sujets de physique &de philosophie, written 1760, presented 1768. In: Opera Omnia III, 11 and 12

[Euler 344] Euler L (1768) Lettres a une princesse d’Allemagne sur divers sujets de physique &de philosophie, written 1760, presented 1768. In: Opera Omnia III, 11 and 12

[Euler E366] Euler L (1763) Institutionum calculi integralis, vol. 2, published 1763 and 1769. In:Opera Omnia I, 11

[Euler E385] Euler L (1770) Institutionum calculi integralis, vol. 3, published 1770. In: OperaOmnia I, 12

[Euler E387] Euler L (1770) Vollstandige Anleitung zur Algebra, vol 1, written 1767, pre-sented 1770, in: Opera Omnia I, 1

[Euler E388] Euler L (1770) Vollstandige Anleitung zur Algebra, vol 2, written 1767, presented1770, in: Opera Omnia I, 1

[Euler E417] Euler L (1771) Lettres a une princess d’Allemagne sur divers sujets de physique &de philosophie, written 1762, presented 1771. In: Opera Omnia III, 11 and 12

[Euler E814] Euler L (1862) Institutionum calculi differentialis[Euler E842] Euler L (1862) Anleitung zur Naturlehre, written 1750, presented 1862. In: Opera

Omnia III, 1[Euler E145] Euler L (1748) Recherches sur les plus grands et plus petits qui se trouvent dans les

actions des forces, written 1748, presented 1748[Euler E146] Euler L (1748) Reflexions sur quelques loix generales de la nature qui s’observent

dans les effets des forces quelconques, written 1748, presented 1748[Euler E176] Euler L (1750) Expose concernant l’examen de la lettre de M. de Leibnitz alleguee

par M. le Professeur Koenig, dans le mois de mars, 1751 des Actes de Leipzig, a l’occasiondu principe de la moindre action, presented 1750

[Euler E181] Euler L (1750) Recherches sur l’origine des force, written 1750, presented 1750[Euler E182] Euler L (1750) Lettre de M. Euler a M. Merain, presented 1750[Euler E186] Euler L (1753) Dissertation sur le principe de la moindre action, avec l’examen des

objections de M. le Professeur Koenig faites contre ce principe[Euler E197] Euler L (1753) Harmonie entre les principes generaux de repos et de mouvement de

M. de Maupertuis, presented 1751. In: Opera Omnia III, 5[Euler E198] Euler L (1753) Sur le principe de la moindre action, presented 1751[Euler E199] Euler L (1753) Examen de la dissertation de M. le Professeur Koenig, inseree dans

les actes de Leipzig, pour le mois de mars 1751[Euler E200] Euler L (1753) Essay d’une demonstration metaphysique du principe general de

l’equilibre, written 1753, presented 1751

References 317

[Euler, Correspondence with scholars] Euler L (1963) Correspondence with scholars (Pisma kucenym). Moscow/Leningrad, Izdat. Akad. Nauk SSSR

[Euler E015/016 (Stackel)] Stackel P (1912) Vorwort des Hausgebers, written in 1912. In: OperaOmnia VI, 1

[Euler 1727] Euler L (1727) Calculus differentialis. Manuscript, 30 pages. Archive of the Peters-burg Academy of Sciences, f 136, op 1, Nr 183

[Eves] Eves H (1900) An Introduction to the History of Mathematics. 6th ed, Saunders CollegePublishing, Philadelphia

[Falckenberg] Falckenberg R (1892) Geschichte der neueren Philosophie. Leipzig[Fellmann (Row)] Fellmann E A (1995) Leonhard Euler. Rowohlt, Reinbek[Fellmann (Birk)] Fellmann E A (2007) Leonhard Euler. Birkhauser, Basel[Feynman] Feynman R P (1948) Space-Time Approach to Non-Relativistic Quantum Mechanics.

Rev Mod Phys 20: 367[Fontenelle] Fontenelle B B (1700) In: Histoire de l’Academie Royale des Sciences, Paris[Friedman] Friedman M (1992) Kant and the Exact Sciences. Cambridge[Galileo, Discorsi] Galilei G (1665) Mathematical Discourses and Demonstartions. Thomas

Salusbury’s English translation. ARCHIMEDES. Project [http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?dir=galil disco 069 en 1665;step=textonly]

[Gehler] Gehler J S T (1787) Physikalisches Worterbuch oder Versuch einer Erklarung dervornehmsten Begriffe und Kunstworter der Naturlehre mit kurzen Nachrichten von derGeschichte der Erfindungen und Beschreibungen der Werkzeuge begleitet in alphabetischerOrdnung. Leipzig [http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/archim/dict/hw?step=list;id=d007;max=100]

[Gerhardt, Historia] Gerhardt C I (1846) Historia et Origo Calculi Differentialis a G. G. Leibnitioconscripta. Hannover

[Gerhardt, Leibniz] Gerhardt C I (1848) Die Entdeckung der Differentialrechnung durch Leib-niz. Halle

[Gerhardt, Geschichte] Gerhardt C I (1855) Geschichte der hoheren Analysis, Erste Abtheilung,Die Entdeckung der Hoheren Analysis. Halle

[Guicciardini, Hermann] Guicciardini N (1996) An Episode in the History of Dynamics: JakobHermann’s Proof (1716–1717) of Proposition 1, Book 1, of Newton’s Principia. HistoriaMathematica 23: 2: 167–181

[Guicciardini, Reading] Guicciardini N (1999) Reading the Principia. Cambridge University PressCambridge

[Guicciardini, Contextualism] Guicciardini N (2003) Conceptualism and contextualism in therecent historiography of Newton’s Principia. Historia Mathematica 30: 4: 407–431

[Guicciardini (Gray)] Gray (2000) Reading the Principia. The Debate on Newton’s MathematicalMethods for Natural Philosophy from 1687 to 1736 by Niccolo Guicciardini. Reviewed byGray J. [http://www.maa.org/reviews/readnewton.html]

[Hagengruber, 2008] Emilie du Chatelet between Leibniz and Newton (2008) Hagengruber R(ed). Springer, New York (in print)

[Hagengruber, 2007] Emilie du Chatelet und die deutsche Aufklarung (2007) Hagengruber R andHecht H (eds). Olms-Verlag

[Hagengruber, Naturlehre] Emilie du Chatelet: Naturlehre an ihren Sohn (2008). Eingeleitet vonHagengruber R und Hecht H (eds). Hildesheim, Olms-Verlag

[Hagengruber, Metaphysik] Hagengruber R (1999) Eine Metaphysik in Briefen. E. du Chatelet anP. L. M. de Maupertuis. In: Hecht H (ed) Pierre Louis Moreau de Maupertuis (1698–1759).Spitz-Verlag, Berlin

[Hamilton 1] Hamilton W R (1834) On a general method in dynamics. In: Philosophical Transac-tions of the Royal Society part II for 1834: 247–308. Dublin

[Hamilton 2] Hamilton W R (1835) On a general method in dynamics. In: Philosophical Transac-tions of the Royal Society, part II for 1835: 95–114. Dublin

[Heisenberg 1925] Heisenberg W (1925) Uber quantentheoretische Umdeutung kinematischerund mechanischer Beziehungen. Z Physik 33: 879–893

[Heisenberg B J 1926] Born M, Heisenberg W, Jordan P (1926) Zur Quantenmechanik II. ZPhysik 35: 557–615

318 References

[Heisenberg, Anschaulich] Heisenberg W (1927) Uber den anschaulichen Inhalt der quantenthe-oretischen Kinematik und Mechanik. Z Physik 43: 172–198

[Heisenberg, Quantentheorie] Heisenberg W (1929) Die Entwicklung der Quantentheorie. Natur-wissenschaften 17: 490–496

[Helmholtz, Vorlesungen] Helmholtz H von (1911) Vorlesungen uber die Dynamik discreterMassenpunkte. Leipzig

[Helmholtz, Kraft] Helmholtz H von (1847) Uber die Erhaltung der Kraft, eine physikalische Ab-handlung. Berlin

[Hermann, Phoronomia] Hermann J (1716) Phoronomia sive de viribus et motibus solidorum etfluidorum libri II. Amsterdam

[Hermann, Responsio] Hermann J (1700) Responsio ad Clarissimi Viri Bernhardt Nieuwentijtconsiderationes secundas circa calculi differentialis principia editas. Basel

[Hoppe] Hoppe E (1928) Zur Geschichte der Infinitesimalrechnung bis Leibniz und Newton.Jahresberichte der Deutschen Mathematiker-Vereinigung 37: 148–187

[Huygens, Traite] Hyugens (1690) Traite de la lumiere. Leiden[Jammer, Mass] Jammer M (1961) Concepts of mass in classical and modern physics. Harvard

University Press, Cambridge, Massachusetts[Jammer, Force] Jammer M (1957) Concepts of force. A study in the foundation of dynamics.

Harvard University Press, Cambridge, Massachusetts[Jesseph, Berkeley] Jesseph D M (1993) Berkeley’s Philosophy of Mathematics. Series: (SCF)

Science and Its Conceptual Foundations series. The University of Chicago Press, Chicago[Jesseph, Leibniz] Jesseph D M (1998) Leibniz on the Foundation of the Calculus. The Ques-

tion of the Reality of Infinitesimal Magnitudes. Perspectives on Science 6: 1 & 2: 6–40. TheMIT Press

[Jushkevich, Euler] Jushkevich A (1983) Euler’s unpublished manuscript Calculus Differentialis.In: Euler-Gedenkband des Kantons Basel. Birkhauser, Basel, S. 161–170

[Kant, KdrV] Kant I (1781) Kritik der reinen Vernunft. In: Kant’s gesammelte Schriften Bd. III,Kant’s Werke Bd. III, Berlin 1904

[Kant, Meta Anfangsgrunde] Kant I (1786) Metaphysische Anfangsgrunde der Naturwis-senschaft. In: Kant’s gesammelte Schriften Bd. IV, Kant’s Werke Bd. IV, Berlin 1903

[Kastner, Anfangsgrunde] Kastner A G (1766) Anfangsgrunde der hoheren Mechanik. Gottingen[Keisler, Calculus] Keisler H J (1986) Elementary Calculus: An Infinitesimal Approach (2d ed)

Prindle, Weber and Schmidt Publishers, Boston[Keisler] Keisler H J (2000) Elementary Calculus. An Approach Using Infinitesimals. Uni-

versity of Wisconsin (made from [Keisler, Calculus]) [https://www.math.wisc.edu/∼keisler/calc.html]

[Keynes] Keynes J M (1947) Newton, the Man. [http://www-history.mcs.st-and.ac.uk/Extras/Keynes Newton.html]

[Keynes, Reagle] Reagle Jr. J M (1996) At the Fulcrum of time. [http://reagle.org/joseph/1996/stuff/newton.txt]

[Klein, Elementarmathematik] Klein F (1933) Elementarmathematik vom hoheren Standpunktaus, Erster Band, Arithmetik, Algebra, Analysis. In: Die Grundlehren der mathematischenWissenschaften in Einzeldarstellungen, Band XIV. Springer, Berlin

[Klein, Arithmetization] Klein F (1895) Uber Arithmetisierung der Mathematik. In: Nachrichtenvon der Konigl. Gesellschaft der Wissenschaften zu Gottingen

[Knobloch, Rigorous] Knobloch E (2002) Leibniz’s Rigorous Foundation of Infinitesimal Geom-etry by means of Riemannian Sums. Synthese 133: 59–73

[Knobloch, Notizbucher] Knobloch E (1989) Leonhard Eulers Mathematische Notizbucher. An-nals of Science 46: 277–302

[Knobloch Parmentier, Leibniz] Leibniz G W (2004) Quadrature arithmetique du cercle, del’ellipse et de l’hyperbole et la Trigonometrie sans tables trigonometriques qui en est lecorollaire. Introduction, traduction et note de Marc Parmentier. Texte latin edite par EberhardKnobloch. Paris

[Kowalewski] Newtons Abhandlung uber die Quadratur der Kurven (1704). Aus dem Lateinis-chen ubersetzt und herausgegeben von Kowalewski G (ed). Leipzig

References 319

[Krantz] Isaac Newton by James Gleick. Reviewed by Krantz S G (2003) Notices of the AMS 50:11: 1404–06 [http://www.ams.org/notices/200311/rev-krantz.pdf]

[Lacroix] Lacroix S F (1828) Traite elementaire de calcul differentiel et de calcul integral. Paris[Lagrange, Mecanique] Lagrange P S de (1788) Mecanique analytique. Paris 1788[Lagrange, Works 11] Lagrange P S de (1888), Oevres de Lagrange. Serret J–A et Darboux G

(eds) Tome onzieme. Mecanique analytique. Tome Premier. Paris[Lagrange, Works 12] Lagrange P S de (1888), Oevres de Lagrange. Serret J–A et Darboux G

(eds) Tome douzieme. Mecanique analytique. Tome Second. Paris[Lagrange, Fonctions] Lagrange P S de (1806) Lecons sur le calcul des fonctions. Nouvelle

edition, revue, corrigee et augmentee par l’Auteur. Paris 1806[Lagrange, Works 10] Lagrange P S de (1884), Oevres de Lagrange. Serret J (ed) Tome dixieme.

Lecons sur le calcul des fonctions. Nouvelle edition, revue, corrigee et augmentee parl’Auteur. Paris

[Landau/Lifschitz, Quantum] Landau L D and Lifschitz E M (1966) Quantum Mechanics, Non-Relativistic Theory. Volume 3 of a Course of Theoretical Physics. Authorized Translation fromthe Russian by J. B. Sykes and J. S. Bell. London Paris, Pergamon Press

[Laugwitz, Nonstandard] Laugwitz D (1973) Ein Weg zur Nonstandard-Analysis. Jahresberichteder Deutschen Mathematiker-Vereinigung 75: 66–93

[Laugwitz, Zahlen] Laugwitz D (1986) Zahlen and Kontinuum: Eine Einfuhrung in die In-finitesimalmathematik, Lehrbucher and Monographien zur Didaktik der Mathematik 5. Bibli-ographisches Institut, Mannheim

Leibniz. Collected Papers[Leibniz (Pertz)] Leibnizens gesammelte Werke aus den Handschriften der Koniglichen Biblio-

thek zu Hannover (1858). Pertz G H (ed). Schmidt H W, Halle[Leibniz A] Gottfried Wilhelm Leibniz. Samtliche Schriften und Briefe (1923 ff.). Hrsg. von

der Preußischen Akademie der Wissenschaften, weitergefuhrt von der Akademie der Wis-senschaften der DDR in Zusammenarbeit mit dem Leibniz-Archiv der NiedersachsischenLandesbibliothek Hannover und der Leibniz-Forschungsstelle der Westfalischen Wilhelms-Universitat Munster. Darmstadt (later Leipzig, Berlin)

Leibniz. Mathematical Papers[Leibniz, Mathematische Schriften] Leibniz G W (1971) Mathematische Schriften, heraus-

gegeben von C. I. Gerhardt. Georg Olms Verlag, Hildesheim New York[Leibniz, GM] Leibniz G W (1971) Mathematische Schriften, herausgegeben von C. I. Gerhardt,

Georg Olms Verlag, Hildesheim New YorkLeibniz. Philosophical papers[Leibniz GP] Die philosophischen Schriften von Gottfried Wilhelm Leibniz (1875–90) Gerhardt

C I (ed) Weidman, Berlin. Reprint: (1960–1979) Olms, Hildesheim[Leibniz (Herring)] Gottfried Wilhelm Leibniz. Philosophische Schriften Bd. 4. Schriften zur

Logik und zur philosophsichen Grundlegung von Mathematik und Naturwissenschaft. Her-ring (ed) Suhrkamp

[Leibniz, Monadology (Latta)] Leibniz G W (1999) Monadology. Translated by Latta R,[http://www.rbjones.com/rbjpub/philos/classics/leibniz/monad.htm; http://en.wikisource.org/wiki/Monadology/Latta translation]

[Leibniz (Child)] The early mathematical manuscripts of Leibniz (1920) translated from the Latintexts published by Carl Immanuel Gerhardt with critical and historical notes by Child J M.The open court publishing company, Chicago, London

[Leibniz, Homogeneity] Leibniz G W, Symbolismus memorabilis calculi angebraici et infinitesi-malis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcenden-tali. In: [Leibniz GM V, p. 377]

[Leibniz, Historia] Leibniz G W. Historia et origo calculi differentialis. In: [Leibniz GM V, p. 392][Leibniz, Hypothesis] Leibniz G W (1670–71) Hypothesis physica nova. Theoria motus abstracti

and Theoria motus concreti. London In: [Leibniz GP IV, p. 177][Leibniz, Elementa] Leibniz G W (1680) Elementa calculi novi pro differentiis et summis, tangen-

tibus et quadraturis, maximis et minimis, dimensionibus linearum, superficierum, solidorum,aliisque communem calclulum transcendentibus. In: [Gerhardt, Historia, p. 32], [Child, p. 136]

320 References

[Leibniz, Cum] Leibniz G W. Cum prodiisset. In: [Gerhardt, Historia, p. 39], [Child, p. 145, an-swer to Nieuwentijt]

[Leibniz, Nova methodus] Leibniz G W (1684) Nova methodus pro maximis et minimis, itemquetangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculigenus (Acta Erudit Lips An 1684). In: [Leibniz GM V, p. 222]

[Leibniz, Brevis] Leibniz G W (1686) Brevis demonstratio erroris memorabilis Cartesii et alio-rum circa legem naturalem, secundum quam volunt a Deo eandem semper quantitatem motusconservari; qua et in re mechanica abutuntur. In: [Leibniz, GM VI, p. 117]

[Leibniz, Pacidius] Leibniz G W. Pacidius Philalethi. In: [Couturat, Opuscules, pp. 549–627][Leibniz, Phoranomus] Leibniz G W (1689). Phoranomus seu de potentia et legibus naturae (Pref-

ace). In: [Couturat, Opuscules, pp. 590–593] and [Leıbniz, Phoranomus (Robinet)][Leibniz, Phoranomus (Robinet)] Leibniz G W (1689, 1992). Phoranomus seu de potentia et leg-

ibus naturae. Robinet (ed) Olschki, Firenze[Leibniz, Nouveaux Essais] Leibniz G W (1996) Nouveaux essais sur l’entendement humain. In:

Philosophische Schriften, Band 3.1 und 3.2 Engelhardt W von, Holz H H (eds) Suhrkamp,Frankfurt a M

[Leibniz, Specimen] Leibniz G W (1982) Specimen Dynamicum. Dosch H G (ed) Felix Meiner,Hamburg

[Leibniz, Theodizee] Leibniz G W (1996) Die Theodizee. In: Philosophische Schriften, Band 2.1Herring H (ed) Suhrkamp, Frankfurt a M

[Leibniz, Initia] Leibniz G W (1966) Initia rerum mathematicarum metaphysica. In: Schriftenzur Logik und zur philosophischen Grundlegung von Mathematik und Naturwissenschaften.Philosophische Schriften, Band 4. Herring H (ed) Suhrkamp

[Leibniz, Monadologie] Leibniz G W (1982) Vernunftprinzipien der Natur und der Gnade, Mon-adologie. Herring H (ed) Felix Meiner, Hamburg

[Leibniz, Definitiones] Leibniz G W (1679) Definitiones cogitationesque metaphysicae. In: [Leib-niz A VI 4b1, Nr. 267]

[Leibniz, De ipsa] Leibniz G W (1698, 1966) De ipsa natura sive de vi insita actionibusque creatu-rarum (1698). In: Schriften zur Logik und zur philosophischen Grundlegung von Mathematikund Naturwissenschaften. Hamburg 1966

[Leibniz (1712)] Leibniz G W (1712) Observatio quod rationes sive proportiones non habeantlocum circa quantities nihilo minors, et de vero sensu methodi infinitesimales. In: Acta Erudit[Leibniz GM V, p. 387]

[Leibniz Clarke] Leibniz G W (1716) Briefwechsel mit Clarke. In: Die philosophischen Schriftenvon G W Leibniz. Gerhardt C I (ed). [Leibniz GP, VII, 353]

[Leibniz Clarke (Alexander)] The Leibniz-Clarke correspondence (1976) Alexander H G (ed)[Leibniz, Responsio] Leibniz G W (1695) Responsio ad nonnullas difficultates a Dn. Bernardo

Niewentiit circa methodum differentialem seu infinitesimalem motas. Acta Eruditorum [Leib-niz (Pertz)]

[Lichtenberg, Sudelbucher] Lichtenberg G W. Sudelbucher. Heft B 28[Mach, Mechanik] Mach E (1921) Die Mechanik in ihrer Entwicklung. Brockhaus, Leipzig[Mach, Mass] Mach E (1868) Uber die Definition der Masse. In: Carl’s Repetitorium der Experi-

mentalphysik 4: 335[Maltese] Maltese G (2000) On the relativity of motion in Leonhard Euler’s science. Archive for

History of Exact Sciences 54: 4:319–348[Maupertuis, Repos] Maupertuis P L M de (1740, 1768) Lois du repos de corps. In: Œuvres de

Maupertuis 4:45–64. Bruyset, Lyon[Maupertuis, Accord] Maupertuis P L M de (1744, 1768) Accord de differentes Loix de la Nature

qui avoient jusqu’ici paru incompatibles. In: Œuvres de Maupertuis 4:3–28. Bruyset, Lyon[Maupertuis, Mouvement] Maupertuis P L M de (1746, 1768) Les lois du mouvement et du repos

deduites d’un principe metaphysique. In: Œuvres de Maupertuis 4:31–42. Bruyset, Lyon[Maupertuis, Examen] Maupertuis P L M de (1746, 1768) Examen philosophique de la preuve de

l’Existence de Dieu. In: Œuvres de Maupertuis 4:389–424. Bruyset, Lyon[Maxwell, Electromagnetism] Maxwell J C (1865) A Dynamical Theory of the Electromagnetic

Field. Philosophical Transactions of the Royal Society of London 155:459–512

References 321

[Maxwell, Treatise] Maxwell J C (1873) A treatise on electricity and magnetism. Clarendon Press,Oxford

[Meli] Bertolini Meli D (1993) Equivalence and Priority: Newton versus Leibniz. Including Leib-niz’s Unpublished Manuscripts on the Principia. Clarendon Press, Oxford

[Meyer] Meyers Conversations-Lexikon. 1885–1892 [http://www.retrobibliothek.de][Minkowski, Space and time] Minkowski H (1908) Space and Time. In: Hendrik A. Lorentz, Al-

bert Einstein, Hermann Minkowski, and Hermann Weyl (1952) The Principle of Relativity: ACollection of Original Memoirs on the Special and General Theory of Relativity. Dover, NewYork

[Minkowski, Raum und Zeit] Minkowski H (1909) Raum und Zeit. Vortrag, gehalten auf der 80.Naturforscherversammlung zu Koln am 21. September 1908. In: Jahresberichte der deutschenMathematiker-Vereinigung, Gutzmer A (ed) 18: 75–88. Teubner, Leipzig

[Newton, Principia] Newton I (1687) Philosophiae naturalis principia mathematica. London[Newton, Principia 1713] Newton I (1713) Philosophiae naturalis principia mathematica.

Cambridge[Newton, Principia 1726] Newton I (1726) Philosophiae naturalis principia mathematica. London[Newton, Principia (Chatelet)] Newton I (1759) Principes mathematiques de la philosophie na-

turelle. Translated by Emilie du Chatelet. Paris[Newton (Collins), Commercium] Commercium Epistolicum D. Johannis Collins & aliorum, De

Analysi promota (1712) London[Newton, Account] An Account of the Book entituled Commercium Epistolicum D. Johannis

Collini & aliorum, De Analysi promota, published by order of the Royal Society, in rela-tion to the Disoute between Mr. Leibnitz and Dr. Keill, about the Right of Invention of theMethod of Fluxions, by some call’d the Differential Method (1714). Phil Trans of the RoyalSoc. No. 342, January and February 1714, 342:173–224

[Newton, Math] The Mathematical Papers of Isaac Newton. Whiteside D T (ed). Cambridge Uni-versity Press, Cambridge

[Newton, Opticks] Newton I (1704) Opticks. London[Newton, De motu] Newton I (1684) De motu corporum in gyrium. [http://www.answers.com/

topic/de-motu-corporum-in-gyrum][Newton, De gravitatione] Newton I (1670) De gravitatione et aequipondo fluidorum. Allen W B

(tr) [http://www.msu.edu/∼allenwi/translations/De Gravitatione et Aequipondio Fluidorumtranslation.htm]

[Newton, Serium] Newton (1671) De Methodis serium et fluxium (On the method of series andfluxions) but only published 1736

[Newton, Quadrature]1 Newton (1704) De quadratura curvarum (On the quadrature of curves) In:Opticks [Newton, Opticks]

[Newton, Quadrature (Harris)] Newton I (1710) On the quadrature of curves. Harris J (tr)In: Harris J (1710) Lexicon Technicum. Vol 2. [http://www.maths.tcd.ie/pub/HistMath/People/Newton/Quadratura/]

[Newton, Quadrature (Kowalewski)] Newtons Abhandlung uber die Quadratur der Kurven(1704). Aus dem Lateinischen ubersetzt und herausgegeben von G. Kowalewski, Leipzig 1908

[Newton, Method of Fluxions] Newton I (1740) La Methode des Fluxions et de suites in-finies. Paris

[Newton, Notebook] Newton I (Notebook) Questiones quædam Philosophiæ. Source: Add. Ms.3996, Cambridge University Library, Cambridge. [http://www.newtonproject.sussex.ac.uk/texts/viewtext.php?id=THEM00092&mode=normalized]

1 “The Introductio ad Quadraturum Curvarum is the introduction that Newton wrote to one oftwo mathematical treatises appended to the first edition of his Opticks, published in 1704. Thesemathematical treatises were republished in 1711, in Analysis per Quantitatum Series, Fluxiones,ac Differentias, cum Enumeratione Linearum Tertii Ordinis, edited by William Jones. The Latintext available here is taken from this edition of 1711. Also available is a translation into Englishmade by John Harris and published in the second volume of his Lexicon Technicum, published in1710.” [http://www.maths.tcd.ie/pub/ HistMath/People/Newton/ Quadratura/]

322 References

[Nick, Gegenmodelle] Nick K R (2001) Kontinentale Gegenmodelle zu Newtons Gravitations-theorie (Dissertation), Franfurt a M. [http://deposit.ddb.de/cgi-bin/dokserv?idn=963471651&dok var=d1&dok ext=pdf.gz&filename=963471651.pdf.gz]

[Nieuwentijt, Analysis] Nieuwentijt B (1695) Analysis infinitorum seu curvilineorum proprietatesex polygonorum natura deductae. Wolters, Amsterdam

[Nieuwentijt, Considerationes] Nieuwentijt B (1696) Considerationes secundae circa calculidifferentialis principia et responsio ad virum nobilissimum G. G. Leibnitum. Wolters,Amsterdam

[Park] Park D (1990). The How and the Why: An Essay on the Origins and Development ofPhysical Theory. Princeton University Press, Princeton

[Pauli] Pauli W (1926) Uber das Wasserstoffspektrum vom Standpunkt der neuen Quanten-mechanik. Z Phys 36:336–363

[Pemberton] Pemberton H (1728). A view of Sir Isaac Newton’s Philosophy. Palmer, London[Pyenson] Pyenson L (1977) Hermann Minkowski and Einstein’s Special Theory of Relativity:

With an appendix of Minkowski’s ‘Funktiontheorie’ manuscript. Arch History Exact Sci17(1): 71–95

[Phili] Phili C (2001) Has flux’s concept ancient roots? An attempt at an approach.[http://galileo.fcien.edu.uy/numero 24.htm]

[Planck 1900] Planck M (1900) Uber eine Verbesserung der Wienschen Spektralgleichung. Ver-handlungen der Deutschen Physikalischen Gesellschaft 2:202–204

[Planck 1906] Planck M (1906) Vorlesungen uber die Theorie der Warmestrahlung. Johann Am-brosius Barth, Leipzig

[Planck, Vorlesungen] Planck M (1913) Vorlesungen uber die Theorie der Warmestrahlung (2nded). Johann Ambrosius Barth, Leipzig

[Planck 1913 (Masius)] Planck M (1913) Vorlesungen uber die Theorie der Warmestrahlung (2nded). Translated from 2d ed. by Morton Masius as The theory of heat radiation (Philadelphia:Blakiston, 1913, reprinted 1988 with introduction by Allan A. Needell)

[Planck SelbstBiographie] Planck M (1948) Wissenschaftliche Selbstbiographie. Johann Ambro-sius Barth, Leipzig

[Plutarch] Plutarch, Life of Marcellus. Dryden J (tr) [http://www.math.nyu.edu/∼crorres/Archimedes/Lever/LeverQuotes.html]

[Proclus] Proclus (1788) The Philosophical and Mathematical Commentaries of Proclus on thefirst book of Euclid’s Elements. Taylor T (tr) London

[Redondo] Redondo F A G (2007) Constants, units, measures and dimensions in Leonhard Euler’smechanics. In: Euler reconsidered, ed. by Roger Baker, Kendrick Press

[Reichenbach, Space and Time] Reichenbach H (1928) Philosophie der Raum-Zeit-Lehre. Thephilosophy of space and time (tr) Dover Publications, New York 1958

[Reichenbach] Reichenbach H (1979) Die Bewegungslehre von Newton, Leibniz und Huygens.In: Gesammelte Werke Kamlah A (ed) Bd. 3. Braunschweig/Wiesbaden

[Reichenberger] Reichenberger A (2008) “mv2” – eine philosophisch Ketzerei? Emilie duChatelet’s Verteidigung des Leibnizschen Kraftmaßes. In. Hagengruber R and Hecht H (eds)Emilie du Chatelet und die deutsche Aufklarung. Olms, Hildesheim (in print)

[Robinson] Robinson A (1966) Non-standard Analysis. North-Holland Publishing Co, Amster-dam. Reprinted by Princeton University Press, Princeton, 1996

[Sandifer] Sandifer C E (2003) Euler’s life-long plan for mechanics. Euler Society Meeting 2003[http://www.southernct.edu/∼sandifer/Ed/History/Preprints/Preprints.htm]

[Sandifer, Early] Sandifer C E (2007) The Early Mathematics of Leonhard Euler. MathematicalAssociation of America

[Sandifer, How] Sandifer C E (2007) How Euler Did It. Mathematical Association of America[Sandifer, Bradley, D’Antonio] Euler at 300: An Appreciation. Bradley R E, D’Antonio L A and

Sandifer C E (eds) Mathematical Association of America[Sandifer, Bradley] Leonhard Euler 5. Life, Work and Legacy. Bradley R E and Sandifer C E (eds)

Elsevier, Amsterdam[Schmieden Laugwitz] Schmieden C, Laugwitz D (1958) Eine Erweiterung der Infinitesimalrech-

nung. Math Zeitschrift 69:1–39

References 323

[Schrodinger, Quantenbahnen] Schrodinger E (1923) Uber eine bemerkenswerte Eigenschaft derQuantenbahnen. Z Physik 12:13

[Schrodinger, First Announcement] Schrodinger E (1926) Quantisierung als Eigenwertproblem.Ann Physik 79:361–376

[Schrodinger, Second Announcement] Schrodinger E (1926) Quantisierung als Eigenwertprob-lem. Ann Physik 79:489–527

[Schrodinger, Heisenberg] Schrodinger E (1926) Uber das Verhaltnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen. Ann Physik 79:734–756

[Schrodinger, Third Announcement] Schrodinger E (1926) Quantisierung als Eigenwertproblem(Dritte Mitteilung: Storungstheorie, mit Anwendungen auf den Starkeffekt der Balmerlinien).Ann Physik 80:437–490

[Schrodinger, Fourth Announcement] Schrodinger E (1926) Quantisierung als Eigenwertproblem(Vierte Mitteilung). Ann Physik 81:109–139

[Schrodinger Nobel Lecture] Schrodinger E (1933), Nobel Lecture. [http://nobelprize.org/physics/laureates/1933/]

[Schrodinger, Naturwissenschaften] Schrodinger E (1935) Die gegenwartige Situation in derQuantenmechanik. Naturwissenschaften 23 (48): 807

[Simonyi] Simonyi K (1990) Kulturgeschichte der Physik. Von den Anfangen bis heute. Berlin[Smolin] Smolin L (2005) The case of background independence. [arXiv:heph-th/0507235v1 25

Jul 2005][Snobelen, Newton] Snobelen S D (2008) Isaac Newton. Theology, Prophecy, Science and Reli-

gion. [http://www.isaac.newton.org][Sommerfeld, Mechanik] Sommerfeld A (1994) Vorlesungen uber theoretische Physik, Band 1:

Mechanik. Harri Deutsch, Frankfurt[Suisky 2001] Suisky D, Enders E (2001) Leibniz’s Foundation of Mechanics and the Devel-

opment in 18th Century Mechanics Initiated by Euler. In: Proc. VII International LeibnizCongress. Poser H (ed) Berlin

[Suisky 2005] Suisky D, Enders P (2005) Dynamical derivation of Lorentz transformation. An-nual Meeting of German Physical Society, Berlin 2005. [http://old.dpg-tagungen.de/archive/2005/gr18.pdf]

[Suisky 2006] Suisky D (2006) On the derivation of Lorentz transformation using ordering re-lations. Annual Meeting of German Physical Society, Dortmund 2006. [http://www.dpg-tagungen.de/archive/2006/dortmund/akphil10.pdf]

[Suisky 2008] Suisky D (2008) Leonhard Euler and Emilie du Chatelet. In: Emilie du Chateletund die deutsche Aufklarung. Hagengruber R, Hecht H (eds). Olms-Verlag

[Szabo] Szabo I (1977) Der philosophische Streit um das wahre Kraftmaß im 17. und 18. Jahrhun-dert. In: Szabo I (1977) Geschichte der mechanischen Prinzipien und ihrer wichtigsten An-wendungen. Birkhauser, Basel

[Taylor, Methodus] Taylor B (1715) Methodus Incrementorum Directa & Inverse. London[Tercent Basel] The Euler Tercentenary. Basel 2007. [http://www.euler-2007.ch/en/] and the links

therein[Tercent Berlin] The Euler Tercentenary. Berlin 2007. [http://euler.bbaw.de/][Tercent StPeter] The Euler Tercentenary. St. Petersburg 2007. [http://www.pdmi.ras.ru/EIMI/

2007/Euler300/ and http://www.pdmi.ras.ru/EIMI/2007/Euler300/index.html][Tercent MAA] The Year of Euler. The MAA is Celebrating the 300th Anniversary

of Euler’s Birth. [http://www.maa.org/euler/; http://mathdl.maa.org/mathDL/1//?nodeId=1567&pa=content&sa=viewDocument] and [http://mathdl.maa.org/mathDL/1//?pa=content&sa=viewDocument&nodeId=1380&bodyId=1595]

[Thiele 1999] Thiele R (1999) “Er rechnete, wie andere atmen.” Eulers Beitrage zum Funktions-begriff. [http://www.zib.de/Euler/1999/thiele.html]

[Thiele 2007] Thiele R (2007) The Rise of the Function Concept in Analysis. In: Euler reconsid-ered. Baker R (ed) Kendrick Press

[Thiele, Euler at 300] see [Bradley, D’Antonio, Sandifer][Time Line] [http://www.sciencetimeline.net/1651.htm]

324 References

[Tropfke] Tropfke J (1940) Geschichte der Elementar-Mathematik, Vierter Band, Ebene Geome-trie. De Gryuter, Berlin

[Truesdell] Truesdell C (1968) Essays in the History of Mechanics. Springer, New York[Varignon 1700] Varignon P (1700) Maniere generale de determiner les Forces, les Vitesses, les

Espaces & les Tems, une seule de ces quatre choses etant donnee dans toutes sortes de mou-vemens rectilignes variez a discretion. In: Histoire de l’Academie Royale des sciences avecles memoires de mathematique et de physique pour la meme annee tires des registres de cetteacademie - Annee 1700: 22–27

[Varignon, Mecanique] Varignon P (1725) Nouvelle mecanique[Verelst] Verelst K (2006) Zeno’s Paradoxes. A Cardinal Problem. I. On Zenonian Plurality. In:

Paradox: Logical, Cognitive and Communicative Aspects. Proceedings of the First Interna-tional Symposion of Cognition, Logic and Communication. University of Latvia Press, Riga.[arXiv:math.HO/0604639 v1 28 April 2006]

[Vermeulen] Vermeulen B (1985) Berkeley and Nieuwentijt of Infinitesimals. In: BerkeleyNewsletter 8. Second edition. Furlong E J, Berman D (eds) Trinity College, Dublin

[Voltaire, Lettres] Voltaire (1734) Lettres anglaises ou Lettres philosophiques. Paris[Voltaire, Elemens] Voltaire (1738) Elemens de la philosophie de Newton. Paris[Voltaire, Newton] Voltaire (1740) La metaphysique de Neuton ou parallele des sentimens de Neu-

ton et de Leibnitz. Amsterdam[Voltaire, Candide] Voltaire (1759) Candide ou l’Optimisme. Paris[Vorlander] Vorlander K (1902) Geschichte der Philosophie. Leipzig 1908[Weissenborn] Weissenborn H (1856) Die Prinzipien der hoheren Analysis in ihrer Entwicklung

von Leibniz bis auf Lagrange, als ein historisch-kritischer Beitrag zur Geschichte der Mathe-matik. Halle

[Westfall, Never] Westfall R S (1980) Never at Rest: a Biography of Isaac Newton. Cambridge,New York

[Westfall, Newton] Westfall R S (1999) Isaac Newton. Heidelberg[Weyl, Raum und Zeit] Weyl H (1921) Raum, Zeit und Materie. Springer, Berlin[Weyl, Group Theory] Weyl H (1928) Gruppentheorie und Quantenmechanik. Hirzel, Leipzig[Whittaker Watson] Whittaker E T, Watson G N (1952) A course of modern analysis. University

Press, Cambridge[Wilczek 2004a] Wilczek F (2004) Whence the Force of F = ma? I: Culture Shock. Physics Today

57 [http://www.physicstoday.org/pt/vol-57/iss-10/p11.html][Wilczek 2004b] Wilczek F (2004) Whence the Force of F = ma? II: Rationalizations. Physics

Today 57 N12[Wilczek 2005] Wilczek F (2005) Whence the Force of F = ma? III: Cultural Diversity. Physics

Today 58 N7[Windelband] Windelband W (1892) Geschichte der Philosophie. Freiburg i B[Wolff, Math Lexicon] Wolff Ch (1734) Mathematisches Lexicon. Leipzig[Zeuthen] Zeuthen H G (1886) Die Lehre von den Kegelschnitten. Kopenhagen[Zeuthen, Apollonius] Zeuthen H G (1886) Apollonius theorems on conic section demon-

strated with the rigor of Euclid. [http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAT2765]

Index

Absolutemotion, Newton, 235time and space, 3time and space, Euler, 129, 184time and space, Newton, 7

Absolute and relativemotion, Newton, 114, 171time and space, Euler, 16, 49, 129, 165, 184,

207, 252, 264–265time, space and motion, Newton, 66

Absorption and emissionEinstein 1905 light quantum, 1906 specific

heat, 287Heisenberg 1925, 295light, energetic spectrum of atoms and

molecular aggregates, 285Planck 1900, hypothesis of

quantum emission, Hypothese derQuantenemission, 289

Schrodinger 1926, 287Algebra

Algebra Euler E387, 93, 110, 205,207, 226

Algorithmcalculus of differences, Euler E212, 223classical and quantum mechanics,

Wilczek, 10differential calculus, Euler E212,

222, 223fluents and fluxions, Newton, 224Nova methodus, Leibniz, 229–230

Ambiguitydefinition of relative velocity, 241

Analysis infinitorum, Euler E101, E102, 107Analytical expression

function, 151Analytically demonstrated

Euler’s program for mechanics, E015/016,111, 125

Ancient prototypeArchimedes, lever, 21–22geometry, Euclid, Heron, 25–28Klein, Zeuthen, 23Lagrange, limits, 32Leibniz, Specimen dynamicum, 38Newton, Euclid versus Descartes, 26rigour, Euler E212, § 85, 31Taylor on Cavalieri, 77Voltaire, 12–14Zeuthen, Klein, 23–24

Anleitung zur Naturlehre, Euler E842, 1, 8,108, 124, 127, 172, 220, 235, 236, 241,252, 253

Annihilation and creationLeibniz, creation and annihilation of

monads, 201Leibniz, generation of lines, surfaces and

solids, 65Newton, generation of lines, surfaces and

solids, 65Apposition of parts

Cavalieri, 120Leibniz, arithmetical series, 29Newton, excluded the procedure of

apposition of infinitesimal parts, 47Taylor, Method of Increments, 32

Archimedesarchimedean quantities, 22, 24, 135, 170,

191, 193, 200interpretation by Euler, E015/016, lever in

vacuum, 172interpretation by Leibniz, Specimen

dynamicum, 38Aristotle

divisibility in one way, two ways and threeways, 27

point, line, surface, body, 27

325

326 Index

Arithmetical foundation, 91, 93,140, 202

Arithmetical and geometrical ratioscalculus, Euler E212, 5, 12, 31, 65, 69, 102,

104, 112, 138, 148, 153, 159, 179, 199,203, 205, 207, 211, 216, 223, 225, 230,231, 238, 290, 314

rules for arithmetical ratios between finiteand infinitesimal quantities, Euler E212,121–122, 218

rules for geometric ratios between infinitelysmall quantities, the force of thecalculus, 97, 229

Arithmetical operationsaddition, subtraction, multiplication and

division, Algebra E387/388, 73, 85, 104,120, 204, 208, 228

addition, subtraction, multiplication anddivision, Rechenkunst E017, 73, 105,106, 112, 319

Arithmetical progressiondiscrete and continuous variables, Euler

E015/016, E212, 292–293discretization of motion, Euler E015/016,

E212, 133energetic states in quantum mechanics,

118, 275general scheme of increments and

differences, Euler E212, 215mechanics, Euler E015/016, 11, 13, 17, 38,

107, 118, 119, 123, 142, 150, 152, 160,171, 175, 189

with positive and negative numbers, 94, 232without infinitesimal real number zero,

quantum mechanics, 97Arithmetization

arithmetization of the calculus, Leibniz,Johann Bernoulli, Euler E212, 198–202

arithmetization of mathematics, Klein,31–32, 102, 175, 200, 312, 314

arithmetization of mechanics, EulerE015/016, 117–118

arithmetization of mechanics, Lagrange, 118historical development, Euler, Lagrange,

Cauchy, Weierstraß [Klein], 10, 38, 197Arthur

differentials as fictions, 204Leibniz’s differential calculus and theory of

motion between 1672 and 1677, 56, 125Assignable

assignable gaps, 51, 136, 201Assignable and inassignable

greater than any assignable number, EulerE212, 203

Leibniz, 1–32, 33–64, 65–100, 117, 120less than any assignable number, Euler

E212, 91, 219, 220, 222, 225, 231, 232,233, 273

syncategorematic interpretation, Arthur, 58,121, 136, 141, 193, 201, 231

Assignment, 144, 150, 164, 220Auflosungskunst, the art of resolution

1D, 2D and 3D motions, along a straightline, within a plane and in space,respectively, Euler, 131, 154, 155

Background independence, 35Barrow

continual motion, 25and Newton, 26

Berkeleycriticism of the calculus, 67, 96, 100on Newton, 67, 102

Bernoulli, Danielproposals for equation of motion, 38

Bernoulli, JohannBernoulli’s rule in Euler’s representation,

117, 205, 206, 211, 216, 226, 227, 229Brachystochrone problem, 105

Bewegungskrafte, 272, 274, 282Body

change of the state, 18, 20, 41, 52, 53, 76,125, 128, 134, 150, 178, 179, 186, 193,254, 259, 261, 262, 263, 274, 296

elastic, 44, 111, 237, 264extension, 62, 277and ghosts, 162impact, 56, 59, 62, 73, 129, 147, 149, 152,

177, 178, 278impenetrability, 44, 137, 150, 161–176, 188,

189, 253, 254inertia, 44, 55, 61, 64, 122, 124, 151, 160,

173, 189, 191infinitely small, 118, 121, 141, 302, 314influence by God and ghosts, 19interaction, at distance, 260interaction, impact, 24, 41, 55, 56, 59, 60,

73, 129, 136, 147, 176, 177, 178, 235,236, 278

length, breadth, height, 27, 74, 164, 218mass, heavy, 24, 42, 77, 269mass, inert, 44, 55, 61, 76, 150, 189, 299mass point, 133, 238, 274, 278, 282, 298mobility, 161–176, 254preservation of state, 16, 20, 41, 42, 63, 145,

179, 182, 186, 191resistance, 20, 169, 276rigid, 238

Index 327

state, 18, 261, 274steadfastness, 124, 135, 161–176, 254and vacuum, 99, 168, 170

Bohrcomplementarity, 155and Heisenberg, 66

Boltzmann, 285, 293Born

and Heisenberg, 66–67Bos

Leibniz’s foundation of the calculus, 30,81–91, 92, 182

Both . . . andconfigurations, 21, 25, 30, 48, 250, 264,

275–281, 294, 296, 298, 300–304, 314paths, principle of least action, 188, 296paths, Schrodinger, 189, 280, 287, 297, 298rest and motion, 2, 3, 17, 18, 19, 21, 42,

45, 61, 64, 105, 107, 123–129, 146,154, 159, 162, 168, 171, 172, 178, 181,184, 187, 188, 190, 238, 244, 254, 261,271, 274

Brachystochrone problem, 105Breadth

line is without breadth, 27, 78Brevis demonstratio [Leibniz], 34, 38, 73Bundle of paths [Schrodinger], 189, 280, 290,

299, 302

Calculuscriticism, Berkeley, 100criticism, d’Alembert, 141criticism, Lagrange, 224, 228criticism, Nieuwentijt, 36, 37, 38, 67, 86,

92, 100, 153, 201, 205of differences and sums, Leibniz 1680, 4,

66, 85, 88, 199, 206, 233differentialis, Euler 1727, 205, 206,

207–216, 218, 220, 222differentials, 32, 96, 97, 136, 138, 139, 142,

148, 150, 152, 157, 159, 193, 197, 199,200, 201, 203, 205, 206, 207, 208, 213,214, 226, 230, 293, 314

evanescent quantities, 30, 91, 92, 95, 142,203, 213, 227, 275

finite increments, 52, 55, 98, 138, 139, 205,206, 208, 216, 218, 222, 223, 274, 292

first and ultimate ratios, 52, 105, 140fluents and fluxions, 4, 5, 26, 38, 51, 67, 68,

70–77, 80, 81, 85, 113, 115, 132, 198,199, 211, 224, 290

historical development [Klein], 10, 38, 197limits, d’Alembert, 35

“omnis vis calculi versatur”, Euler E212,§ 85, 218

subject of the calculus, Euler E212, § 85, 68,72, 225

sums and differences, 37, 98of zeros, Lagrange, 224, 226

Cantoron actual infinity, 97, 205on differentials, 79, 96, 97, 207on the interpretation of Leibniz, 65, 79, 97,

98, 205Cauchy, 28, 35, 65, 91, 102, 104, 272Causality

classical interpretation, 42, 112, 155, 192,251, 270, 292, 293, 296, 303, 305, 307

quantum mechanical interpretation, 112,288, 295, 298, 300, 309, 314

Cause and effectHelmholtz’s comment on Mayer, 32interpretation of Helmholtz, 238interpretation of Leibniz, 37, 186interpretation of Mayer, 32, 195

Cavalieri, 27, 37, 67, 69, 73, 77, 78, 85, 100,120, 197, 201, 204

Change of motionincrement of velocity, 184increment of velocity is independent

of velocity, 72, 145, 150–153, 252,265, 274

in terms of change of momentum, Newton’s2nd Law, 54

in terms of change of velocity, Euler,Varignon, 55

Change of position, 36, 77, 158, 245, 249Change of state

by an external cause, 146, 171, 178, 187,191, 274, 296

by forces, 28, 41, 55, 57, 64, 116, 161, 162,174, 224, 254, 262

independent of state, Euler, 240, 243,296, 305

independent of state, Helmholtz, 296of rest or uniform motion, 166–167

Change of velocityby an external cause, 296by forces, 137, 141, 146, 262, 263independent of velocity, Euler, 72, 131, 145,

190, 262, 282independent of velocity, Helmholtz, 145,

150–153Cheated by the senses

Chatelet on relative motion, 12, 16–17, 18,22, 25, 64, 118, 182, 185, 248, 251, 252,257, 258

328 Index

Kastner on relative motion, 145, 184, 185,186, 241–243, 251, 252, 253, 261

Cipherdifferent signs to avoid confusion, 224–225,

230, 231Circle

and polygon, 53–54, 84, 144Coeunt, 114, 135, 174, 193Commercium Epistolicum, 69Compraesentia

and choice of a unit, Leibniz, 48–49and measurement, Leibniz, 48and perception, Leibniz, 48

Condon, 12, 269, 285, 286, 290Configurations

and conservation of energy, 283and extension of a system, 278–281and potential energy, 280of a system, 30, 279, 280, 281, 300–304,

306, 310of a system and paths of bodies, 178

Conservationof center of gravity, 148, 224of direction, 2, 174, 187of energy, 80, 115, 237, 278, 283, 287, 288,

297, 299, 305, 309of living forces, 36, 44, 53, 73, 112, 148,

161, 180, 194, 224, 269, 277, 309of mass, 160of momentum, 114, 153, 160, 258

Constants and variables, 68, 83, 91, 224, 225Contingent

and necessary, Leibniz, 177Continuity

and discrete things, Leibniz, 197of motion, 94, 112, 136, 142, 200, 217

Continuity, principle ofcontinuity of motion, exclusion of

interruption and jumps, Euler, 67continuity of motion, exclusion of leaps,

Leibniz, 217, 289extension to series of discrete numbers by

Euler, 217general form, Leibniz, 81Leibniz, 31, 37, 67, 74, 94, 125, 136, 195,

196, 230, 232, 289Robinson on Leibniz’s principle, the transfer

principle, 230Continuous

bodies as continuous things, Euler, 164creation of lines by the motion of points,

Newton, 74creation of mathematical quantities,

Newton, 7

Continuumcomposition of the continuum, Leibniz, 58divisibility, 164divisibility in infinity, 164indeterminate parts, Leibniz, 120

Contradictionprinciple of contradiction, 177

Countableenergetic values, 112ungereimt, die Mittelorte zahlen zu wollen,

Euler, 112Couturat, 9, 11, 13, 36, 62, 112, 160–161, 180,

237, 269

Dangicourt, 43, 65, 91letter to Dangicourt, Leibniz, 43, 91

Dead forcesand living forces, 15, 21–22, 64, 82, 161measure of dead forces, 60and statics, 22

Decrease and increaseof quantities, unlimited, Euler, 209

Decrements and increments, 77, 224Descartes

external causes, 2laws of nature, 2, 198motion in straight direction, 2principles of motion, 247“quantum in se est”, 3, 7, 20, 129, 170relative motion, 3, 7, 151, 244uniform motion, 239

Differential quotient, 52, 151, 225, 273, 274,275, 276

Differentialsdifferentio-differentials, 84, 86, 96, 136,

138, 159, 199, 219, 272, 290evanescent quantities, 30, 95, 142, 203,

227, 275of first, second and higher order, 214hierarchy, 96magnitude, 273

Divisibilityof bodies, 162, 164of bodies in inifinity, Euler, 162, 164

Divisibleand really divided, Leibniz, 47

Division, 27, 73, 83, 84, 85, 94, 98, 104,120, 123, 142, 166, 200, 204, 273,274, 299

Dynamics and static, 123, 127

Effects and cause, 44, 45, 181, 187, 193Eigenvalues, 287, 295

and quantization, 207

Index 329

Einsteinhypothesis of light quanta, 286quantization as selection problem, 309theory of relativity, 45, 240, 243, 298theory of specific heat, 287, 293

Either . . . orconfigurations, 21, 25, 30, 48, 250, 264,

275–281, 296, 297, 298, 300–310, 314paths, Schrodinger, 287rest and motion, 2, 3, 17, 18, 19, 21, 42, 45,

61, 64, 105, 107, 123–129, 154, 156,159, 162, 165, 168, 171, 172, 178, 181,184, 187, 188, 190, 238, 247, 248, 249,254, 261, 271, 274

Eleganceof the theory of fluents and fluxions,

Newton, 70–72Elements of bodies

Gedancken von den Elementen der Corper,220, 241

Ellipseand parabola, Leibniz, 225

Energykinetic, 35, 115, 149, 276, 277, 278, 279,

280, 281, 297, 301, 303, 304potential, 280, 283, 295, 297, 301, 303, 308total, 276–280, 287, 294, 295, 296, 297,

301, 303Equilibrium

of the lever, 170, 191Errors

and forces and time, Newton, 53, 76, 154Eternal, eternellement, 127, 191Euclid

Axioms, 75Proclus, 25–26

Euleranswer to Daniel Bernoulli, 153–159augmentation to infinity without limit, 216bodies and ghosts, 162bodies, properties, 1, 19, 58, 124, 126,

132, 147, 150, 171, 173, 186, 217,257, 261

calculus of differences, 16, 31, 32, 38, 56,138, 139, 157, 159, 196, 199, 205, 206,215, 217, 222, 223, 233, 290–292, 314

confidence in arithmetics and algorithms,56, 140

confidence in reason in comparison tosenses, 61, 141

criticism of the theory of monads, 107deep insight of Leibniz, 9, 15definition of mass, 25, 41, 148, 161–162

differential calculus, 32, 92, 93, 98, 111,112, 138, 139, 140, 153, 159, 167, 192,195, 205–208, 214, 215, 216, 218, 222,223, 290, 312

equation of motion, 17, 38, 80, 106, 152,183, 184, 190, 198, 214, 236, 238, 240,252, 257–259, 261, 265, 274, 275, 277,279, 281, 296, 297, 305

foundation of the calculus, 4, 11, 25, 30,31, 47, 65–100, 102, 104, 105, 117, 136,140, 141, 142, 156, 182, 195–233, 238,269, 272, 273, 291

foundation of mechanics, 8, 9, 19, 34, 40,56, 140, 161, 168, 175, 180, 181, 194,224, 225, 235, 236, 254, 269

freedom of ghosts, 133ghosts and bodies, 3, 189harmony between Maupertuis’ principles of

rest and motion, 21impenetrability of bodies, 172, 173infinite quantities of different magnitude,

84, 94, 228limit, without limit, 93, 216, 263“mistake of those”, infinite numbers of

different magnitude, 93“Mittelorte”, no counting of intermediate

positions, 112necessarily true, 141, 177, 290, 312observers, onlookers, Zuschauer, 17, 107,

183, 240, 255–256, 257, 258, 259origin of forces, E181, 107, 135, 150, 175principle of contradiction, 177principle of sufficient reason, 61, 171, 177,

178, 186, 238, 269quantity, 2, 5, 18, 27, 30, 41, 44, 48, 51, 52,

54, 56, 57, 60, 62, 67, 68, 71, 73, 74,75–97, 121, 144, 159, 160, 165, 174,176, 198, 200, 202, 203, 204, 205, 206,210, 211, 213, 214, 217, 218, 219, 225,226, 227, 228, 229, 230, 240, 263, 265,270, 273, 275, 276, 278, 295, 299, 301,302, 312

relative motion, 3, 7, 12, 16, 17, 19, 22,25, 36, 43, 46, 49, 51, 58, 62, 64, 107,114–118, 124, 145, 151, 169, 171, 174,175, 181–185, 193, 238, 240, 241–263

reliability of reason, 20, 35, 66, 124, 170,177, 190, 195, 196, 198, 202, 309

rigour of the ancients, 21, 77, 196, 314“Standhaftigkeit”, steadfastness, inertia,

126, 162, 167, 173, 253theory of gravitation, 1, 260theory of propagation of light, 19, 109,

163–164, 264

330 Index

theory of propagation of sound, 110,162–163

“Wirksamkeit”, efficiency, 20, 156, 253,254, 269–283

“Zuschauer”, observers, 17, 107, 183, 240,255–256, 256, 257, 258

Eulogyto Euler, Condorcet, 17, 183

Evanescent quantitiesBernoulli, 203Euler, E212, § 87, 30, 275Leibniz, 95, 142, 203, 213, 227, 275relation: “the finite evanesces with respect to

infinite”, 213, 227relation: “the infinitely small evanesces with

respect to finite”, 72, 142Extension

of a body, 59and configurations, 314and energy of the system of bodies,

Helmholtz, 296, 300as a property of bodies, Descartes, 128,

162, 172of a system of bodies, 16, 56, 59, 77, 155,

165, 169, 257, 271, 277–281, 289

Fictitious quantitiesdifferentials, Leibniz, 35, 79, 89, 91, 96,

97, 142Figures

no figures, Euler E212, 199no figures, Lagrange, Mecanique analytique,

118, 167, 175, 236, 313only one geometric line, Newton, 302

Finiteinfinitesimal, finite and infinite quantities,

hierarchy of, 77, 83, 199, 224Fleiss

“mit Fleiss ausgeschlossen”, influence ofghosts is excluded, Euler, 1, 3, 43, 124

Fluents and fluxions, 4, 5, 26, 38, 51, 70–77,80, 81, 85, 113, 115, 132, 199, 211,224, 290

Fluents, fluxions and moments, Newton, 5, 34,66, 71, 72, 202

Fluxcontinuous, Newton, 67, 89, 207, 292lines, surfaces and solid, generated by, 27,

50, 57, 71, 73, 74, 78, 85, 89, 113, 164time, generated by, 25, 49

FluxionsMethod of Fluxions, 25, 26, 34, 37, 47, 71,

72, 75, 76, 77–81, 82, 89, 94, 95, 142,197, 199, 244

Forceactive, 14–15, 20, 25, 61, 64, 82, 99, 125,

144, 173, 179, 185, 285active and passive forces, 14, 61, 64, 125,

173, 179analytical representation, 141, 277dead, 13, 21, 22, 60, 64, 66, 99, 125, 146,

149, 169, 181, 185, 202derivative, 24, 55, 58, 60, 61, 64, 67, 82,

114, 127, 129, 161, 177at distance, 193due to impact, 62, 129, 152, 178due to impenetrability, 177, 254of inertia, 14, 36, 44, 61, 62, 64, 82, 104,

125, 127, 128, 167–172, 173, 181, 184,185, 186, 187, 189, 191

inherent, 61–64, 82, 128, 179, 185, 190living, 2, 5, 7, 12, 13, 15, 21–24, 35, 36, 44,

45, 55, 57–59, 62–67, 79, 82, 99, 110,152, 153, 161, 176, 180, 181, 185, 194,238, 264, 269, 275, 276, 282, 294, 312

minimal forces to avoid impenetrability,principle of least action, 13, 20, 101,126, 150, 165, 188, 189, 254, 297

moving, 7, 14, 24, 36, 41, 45, 59–60, 61, 62,63, 64, 79, 82, 84, 114, 132, 146, 152,154, 155, 157, 165, 171, 175, 185, 187,189, 274

passive, 14, 61, 64, 82, 86, 99, 125, 178, 179primitive, 57, 61, 99, 113, 125, 127, 128,

177, 185primitive and derivative, 61, 64, 99, 125resisting, 61, 179

Forces and motionconstant force model, 53, 137, 282

Forces, origin ofgeneration of forces by bodies, 20, 60,

188, 270Freedom

of ghosts and impenetrability of bodies, 44,118, 133, 137, 171, 172, 189, 254

Functionanalytical expression, 72, 77, 151, 214, 223,

227, 247, 300constants and variables, 72, 79, 83, 95, 208,

225, 231independent and dependent variables, Euler,

5, 30, 51, 74, 79, 86, 115, 138, 142, 144,145, 151, 154, 158, 197, 198, 199, 202,207, 208, 213, 218, 274, 290, 300

GenitumNewton, 94

Index 331

Helmholtzconfigurations of a system, 308differential quotient, 56, 151, 238, 281, 286,

290, 294energy conservation, 80, 275, 277, 287, 288,

297, 299, 305extension of a system, 288–289Leibniz’s living forces, 44, 93potential energy, 295, 297, 301, 303,

306, 308Homogeneity

mathematically defined, 76mechanically defined, 113symbolismus memorabilis, Leibniz,

48, 214Hypotheses, 103, 190

Imaginary numbers, 229, 230Impenetrability

of bodies and freedom of ghosts,Euler, 189

independent of size and shape, 253magnitude of impenetrability, equal for all

bodies, 160, 203Increase and decrease

of quantities, unlimited, Euler, 86Increments and decrements

determinate or of constant magnitude, 273discrete increments and decrements and

continuous variables, 291of a function, 94, 138, 161, 214, 218, 223,

224, 232, 274geometric ratio of the increment of function

and the increment of variable, 225indeterminate or variable, 99, 223, 224,

241, 242and indivisibles, 78, 100, 211relation (geometric ratio) between

increments/decrements of a function anda variable, 79, 100, 149, 165, 214, 221,222, 225, 231, 232, 240, 259

of variables, 214, 274Increment of velocity

is independent of velocity, Euler, 72, 131,145, 150–153, 190, 262

is independent of velocity,Helmholtz, 282

Indeterminacyin the choice of independent variables,

75, 193Indivisibles, 21, 27, 28, 47, 51, 58, 66, 67, 69,

73, 75, 77, 100, 120, 136, 197, 200, 201,204, 211

Infinite degrees, 205

Infinitely little quantityone infinitely little quantity, O, Newton, 5,

67, 75, 197Infinite quantities

of different magnitude, Euler E387, xivInfinitesimal

quantities of different magnitude, 83, 93, 96,228, 231

Infinitum continuum vel discretumLeibniz, 88

Infinityas limit, Lagrange, 54, 223, 236

Instantaneous actionof forces, 137

Instant of time, 91Integration constants

mathematical interpretation, Euler,Wirksamkeit, 100

mechanical interpretation, Helmholtz,“living forces”, “work” and “totalenergy”, 277

Jesseph, 39, 90, 99, 200

Kastneron Euler’s mechanics, 122–123on relative motion, 180

Kepler, 43, 61, 75, 78, 102, 130, 143, 159, 162,168, 171, 172, 197, 244, 270

Keynes“magic was forgotten”, 37on Newton, 37, 40

Kinematicsand dynamics, 272–275

Kleinarithmetization of mathematics, 10, 24,

31–32, 99, 102, 196–200, 207arithmetization of mechanics, 28–31, 141,

175, 176, 195“Bahnbrecher”, break through by Newton

and Leibniz, 10, 31calculus of differences (Taylor, Euler) and

differential calculus, 215differentials, “less than any finite quantity,

but different from zero”, 230on Euler, Lagrange, Cauchy and

Weierstraß, 272on Euler’s concept of function, “like a

ferment”, 32formal foundation of mathematics by

Leibniz, 31–32logical consistency, 28naive intuition versus logical

consistency, 272

332 Index

polygon and circle, abstract logical point ofview, 39

principle of continuity, 31proposals to made use of Hamilton’s

theory, 63

Labyrinthlabyrinth of the contiuum, 201

Lacroixdifferential calculus and calculus of

differences, 206Lagrange

algebraic foundation of calculus, 119calculus of zeros, 224, 226foundation of mechanics by variational

principles, 12, 148mechanique analytique, 103, 167, 175,

196, 199mechanics as for dimensional geometry, 144

Lambert, 105, 176Language, 9, 14, 15, 16, 39, 42, 81, 90, 93,

99, 103, 119, 128, 162, 177, 182, 190,198, 208

Least partsof bodies, 141, 164, 166

Leibnizanswer to Nieuwentijt, 38best of all possible worlds, criticism by

Voltaire and Mach, 4, 180body is an extended resisting thing, 59dead and living forces, 13, 21, 64, 66, 82,

99, 125, 146, 181, 185differentials as fictitious quantities, 96,

136, 205evanescent quantities, 142excitation of internal forces by external

action of forces, 63fictitious quantities, 35, 79, 89, 91, 96, 136generation of lines by motion of points, 201on geometry and metaphysical

principles, 57impressions of dead forces, 82“Leges metaphysicas extensionis legibus

addendo”, 6, 20, 180living forces, measure of, 2, 5, 7, 12,

93, 149mechanics and phenomena described by

geometry and metaphysical principles, 3,37–38, 42, 43, 56, 58

monads, 43, 58, 107, 133, 141, 165, 174,239, 249

motion and time do not exist aswholes, 56

perspectives, 63, 174, 242, 249

possible and real worlds, 4, 188principles, 201rules for differentiation, 90, 210Specimen dynamicum, 38, 230Theoria motus abstracti, 38, 55, 136vacuum is an extended thing without

resistance, 59, 154, 165, 168Lever

Archimedes, 21, 22, 23, 24, 25, 36, 127Euler, 24, 147, 168, 170, 190, 191

Lichtenbergon Euler’s theory of mass points, 13

Lightvelocity, Einstein, 51, 239, 243velocity, Euler, 261, 266

Light quantumEinstein, 286, 299Planck, 286

Limitabsence of limit, Euler, 178approach to a limit, Lagrange, 35, 54,

122, 141d’Alembert, 122, 140Euler, 141, 226increase without limit, 278Lagrange, 228, 239polygon and circle as a model, 53–54,

84, 144veritables limites, Lagrange, 54, 226

LineEuclid, 26, 28, 73, 201Heron, generation of a line, 25Leibniz, generation of a line, 25“Mittelorte”, no counting of intermediate

positions, Euler, 112Newton, generation of a line, 73, 120one, two, three and four points defining

different geometrical objects, 134and plane, 27and point, 27and surface, 27

Living forcesand dead forces, 13, 21, 22, 62, 66, 82, 99,

125, 146, 181, 185and dynamics, 22, 276measure of living forces, Leibniz, 5, 12, 13,

93, 149

Machacceleration and mass, 114criticism of absolute space and time, 40mass, definition, “gegenuberstehende

Korper”, 181Newton’s physical principles, 260

Index 333

Massacceleration and mass, 114, 160critcism of metaphysics, 140definition of mass, Newton, 161mass, definition, “gegenuberstehende

Korper”, Mach, 181“the only exception is Euler”, Jammer, 160operational definition of mass, Euler,

159–160Mass point

difference between mass point andgeometrical point, Euler, 144

Maupertuisanalysis and criticism of the mechanical

principles of Descartes, Newton,Huygens, Leibniz and Euler, 11, 15

Euler’s representation and comments, 12,15, 21, 128

harmony between principles for rest andmotion, Euler, 184

minimal forces and least action, Euler, 13principle of least action, 106, 162, 163, 184,

236, 254, 296principle for rest, 184principle for rest and motion, 21

Measurementarithematics, representation of results by

arithmetical progression, Galileo, 29arithmetical differences between the terms

of the series, Galileo, 157division of distances and time intervals into

equal parts, Euler, 208division of distances and time intervals into

equal parts, Galileo, 28unit, choice of the unit, Leibniz, 48unit and object, Leibniz, 48unit and object, simultaneously present and

perceived, Leibniz, 48–49Mechanics

analytical, 116, 167, 215Eulerian and non-Eulerian, 262, 299

Melion the model of motion, 136

Methoddirect and indirect, 152, 188

Method of increments, 32, 208,215, 216

Modellingby thought experiments, 188

Modelsof the body, Euler, 132of the body, infinitely small body,

Euler, 118of the lever, Archimedes, 21–22

physics of models, Schrodinger, 285–310reaction of Lichtenberg to Euler’s model of

mass points, 13of the world consisting of a finite number of

bodies, Euler, 190of the world consisting of moving bodies

and moving observers, Euler, 261of the world to demonstrate mechanical

laws, Euler, 115Momentary increments

and differentials, Nova methodus, Leibniz,5, 85, 197, 210

Momentsfluents, fluxions and moments, Newton, 202

Monadscomment and reinterpretation by Euler, 133criticism by Euler, 107, 141distinguishing parts of the plenum, 164–165internal state of successions, 133least indivisible elements, 141, 239perspectives, 249

Motionabsolute, 17, 46, 47, 115–116, 118,

129, 142, 151, 169, 171, 182, 183,235, 238–241, 242, 252, 253, 256,259, 260

along a straight line, in a plane, in space, 80analytical representation of motion, 125,

133, 141–145, 155, 258–259apparent, “scheinbar”, 241change of position, 36, 77, 245change of state of rest or uniform motion,

21, 41, 53, 128, 150, 178, 190,247, 263

complete knowledge of motion, E842,§§ 21–25, 144

generates motion, Leibniz, 5, 57, 168,247, 251

non-uniform motion, 18, 118, 121, 134,136, 142, 143, 145, 152, 166, 167,256, 257

relative, 3, 7, 12, 16, 17–19, 22–25, 36, 43,46, 47, 49–51, 58, 62, 64, 107, 114–116,124, 145, 151, 159, 169, 171, 174–176,181–182, 184–185, 193, 238–259,261, 263

and rest, 16, 64, 114, 116, 125in terms of change of momentum, Newton’s

2nd Law, 54in terms of change of velocity, Euler,

Varignon, 154and time being no wholes, 132, 143, 240true, “wahre”, 253, 256

334 Index

uniform motion, 16, 18, 29, 31, 36, 40,41, 45, 47, 53, 54, 59, 64, 87, 88, 96,113, 114, 118, 121, 127, 131, 134,142, 143, 146, 150, 152, 157, 165,166–167, 174, 178, 190, 208, 238, 239,257, 263

Necessaryand contingent, Leibniz, 56, 139necessarily true, “non solum verum, sed

etiam necessario verum”, Euler, 141truth, geometric truth, Euler, 190–191

Neither . . . norpaths, 280rest and motion, 172, 187, 190

Newtonaccount of Commercium, 69activation of inertia by impressed

forces, 179axioms, 3, 29, 35, 40, 41, 45, 48, 97,

131, 140, 149, 160, 167, 178, 188,220, 296

comment on Leibniz’s calculus, 35Commercium, 34, 52, 66, 69, 79, 98,

120, 197on Descartes and Euclid, 6, 25–28, 39, 73,

74, 102on Descartes, Westfall, 1–32, 39, 41, 42,

59–60, 99, 112, 115, 130, 178, 179, 186errors, times and forces, 52–53fluents and fluxions, 4, 5, 26, 38, 70–72,

80, 81, 85, 113, 115, 132, 198,199, 211

force of inertia, 14, 36, 42, 62, 64, 82, 160,168, 171, 173, 185, 186

generation of lines by motion of points, 201genitum, 94impressed moving force, 44, 45, 59, 60, 63,

76, 79, 82, 113, 129, 140, 147, 167, 169,181, 185, 213, 269

on infinite numbers of different magnitude,93–94

integrated program of science and religion,Snobelen, 40

on “men of recent times”, 3, 39, 74“more elegant”, only ”one infinitely small

quantity, O, 5phenomena and forces, 43, 177, 244polygon and circle, modelling of the action

of central forces, 130ratios, first and last, 93rules for differentiation, 5, 214very beginning of motion, 51–53, 66,

81, 146

Nieuwentijtcriticism of the calculus, 37rejection of higher order differentials, 86Verelst, interpretation of Nieuwentij’s

theory, 96Nonstandard analysis

extension principle, 231rules for infinitesimal, finite and infinite

quantities, 122standard part principle, 231, 314transfer principle, 231

Nothingzeros, Euler, 223

Observersand bodies, 245and different perspectives of the world,

Leibniz, 245distinction between bodies and observers,

Euler, 252–253distinction between substances, Descartes,

4, 17–20making use of light for communication,

Euler, 263–266moving relatively to each other, Euler, 247relatively to each other resting and moving

observers, Euler, 261resting observers, 246, 248, 249, 256, 263separation of bodies and observers,

245, 249taking into account the finite velocity of

light, retardation of the observation ofevents, Euler, 264

union of bodies and observers, Euler’srelativistic theory, 241–243

union of bodies and observers, Leibniz,248–249

Occupation of placesby bodies, exclusion principle, 191

Orderof arithmetical progressions, 138, 208, 292of numbers, 50, 66of successions and coexistences, time and

space, 50Origin of forces

avoiding penetration, 53, 254, 297change of the state, 41, 52, 53, 126,

145–150impenetrability, 172minimal forces, 152, 262

Parabolacomposition of motions, Galileo, 28–31and ellipse, Leibniz, 225

Index 335

Pembertonon Newton’s studies of Descartes and

Euclid, 6, 26, 38Perpetual motion

Leibniz on perpetual motion, 45, 55Planck

action parameter, 289action parameter and classical physics, 289entropy and probability, 289, 293, 295hypothesis of quantum emission, 289scientific autobiography, 290universal parameter, 293

Planeforces, geometrically related to planes,

Euler, 156planes and straight lines, Euler, 155planes and straight lines in spaces of

different dimension, Ehrenfest, 155stability of motion and trajectories in 3D

space, Ehrenfest, 155Points

and line, surface and solid, 26, 27mathematical and physical points,

Euler, 165Polygon

and circle, Lagrange, 54, 129and circle, Leibniz, 75and circle as a model for straight and

curved, 84and circle, Newton, 129, 136circumscribed and inscribed, 54, 129–130

Positionabsolute, 242of a body, 113, 131, 252limited space, 113, 115relative, 49, 56, 124, 241, 249, 276

Potentiaconservation, neither less nor more potency

in cause and effect, Leibniz, 44, 55,153, 181

Potential energy, 254, 278, 279, 280, 295, 297,301, 306, 308

Principlesof contradiction, 16, 182, 177, 312external and internal, Euler E289, 20,

132–135external, 20, 44, 46, 97, 118, 126, 127,

145–153, 173, 247, 252general and fundamental principle of

mechanics, Euler E177, 178, 192internal, 44, 46, 114, 117, 118, 123–129,

132–133, 147, 149, 150–152, 170, 173,247, 252, 254

of least action, 13, 21, 53, 103, 106, 107,125, 148, 162, 163, 168, 184, 188, 224,236, 296

of sufficient reason, 61, 171, 177, 178, 186,238, 269

Proclusthe origin of the concept of flux, 25, 26–27

Quantification and ordertime and space, 50–51

Quantitas motus, 60, 62Quantities of the same sort

analytical treatment, Euler, 253, 276, 277space and time, Euler, 33, 45

Quantitydeterminate, 205, 224, 225finite, 11, 52, 54, 67, 71, 74, 79, 85, 86, 91,

95, 96, 98, 136, 140, 144, 148, 193, 196,202, 205, 206, 211, 215, 216–220, 225,227, 231, 265, 302, 312, 314

indeterminate, 91, 159, 223, 295infinite, 54, 73, 84, 96, 140, 196, 202, 204,

216–220, 223, 227, 228, 302infinite of different magnitude, 83, 93, 204,

217, 228, 232infinitesimal, 216–220infinitesimal of different magnitude, 83, 93,

94, 96, 204, 217, 232measurement, 218, 302

Quantizationaction parameter, Planck, 125, 240, 286,

287, 289, 295Bohr and Sommerfeld, 112differential equation, 262, 275, 286, 305as eigenvalue problem, Schrodinger, 287,

295, 309of energy, 308of energy or frequencies, 290matrix mechanics, 305as selection problem, Einstein, 295–296,

301, 306, 309Quantum numbers, 301

Ratiosarithmetic, 96, 225, 229first, 92geometric, 68, 72, 73, 79, 97, 121, 138, 139,

143, 218, 219, 225, 228, 229, 275, 302ultimate, last, 51, 52, 93, 92, 105, 136, 140,

197, 292Reaction principle, 181, 237Reason

sufficient, principle of sufficient reason, 61,171, 177, 178, 186, 238, 269

336 Index

RechenkunstRechenkunst and algebra, Euler, 104

Reichenbachon Leibniz and Newton, 16, 45, 46, 49, 240

Restabsolute, 47, 48, 64, 169, 170, 238, 244, 250change of rest and change of motion,

122, 127and dead and living forces, 13, 21, 64, 66,

82, 99, 125, 146, 181as evanescent motion, Leibniz, 72, 114, 125,

142, 213and motion, 2–3, 17–19, 21, 42, 45, 61, 64,

105, 107, 116, 117, 118, 119, 121–127,145, 146, 154, 159, 162, 165, 167, 168,171, 172, 178, 181, 184, 187, 188, 190,238, 247–249, 254, 261, 271, 274

relative, 22, 47, 64, 123, 169, 181, 184, 238,249, 250, 252, 257, 265

Rigourof the Ancients, Euler, 226, 269, 270of the Ancients, Lagrange, 196of the Ancients, Leibniz, 22, 57, 59,

125, 269of the Ancients, Newton, 3, 4, 5, 39, 74,

77, 269of the Ancients, Taylor, 216arithmetic, 196geometric, 196, 226

Robinsonon Leibniz’s principle of continuity, 37non-standard analysis, 65

Russelon the “popular” and the “esoteric” Leibniz,

13, 36

Schrodingeramplitude equation, 293, 294, 305, 306amplitude function, 281, 294configuration space, but not pq-space, 294,

305, 306“full force of logical opposition”, 189,

297, 298on intuition and methods of transcendental

algebra, 103quantization of energy or frequencies, 294quantization as selection problem, 304–310on recommendations concerning Hamilton’s

theory by Klein, 63, 298relation to Heisenberg’s matrix

mechanics, 305on the relations between classical and

quantum mechanics, 151, 286, 297“selbsttatig aussondern”, 307

transfer of classical concepts into quantummechanics, 31

undulatory mechanics, 294, 299, 304wave equation, 287, 300, 306wave function, 267, 280, 287, 294, 296, 297,

300, 301, 303, 312Selection problem

in classical mechanics, 297energetic spectrum of molecular aggregates,

quantization as selection problem,Einstein, 304

in quantum mechanics, 298Senses and ratio

cheated by our senses, Chatelet, 248reliability of ratio, Euler, 127

Shadowspace separated from time, 144, 239, 267time and space separated from each other

are mere shadows, Minkowski, 267the union of space and time,

Minkowski, 144Smolin

background independence, 154Sound

propagation of sound and propagation oflight, 163

Space and timeabsolute, 245, 260absolute, Newton, 45, 239, 240, 258,

261–263Leibniz, 9, 16, 17, 35, 47–52, 48, 51–52,

188, 214and motion, 8, 16, 35, 40, 46, 49, 64, 116,

181, 236, 239, 241as orders, 49, 89quantification and order, 50–51quantities of the same sort, Euler, 48relational definition, Leibniz, 45, 240, 248relational theory, “newly come”, Leibniz,

50, 249, 251relative, Newton, 115spatiolum, Euler, Wolfers, 275

Spatiolumand tempusculum, 75, 224, 293“Zeittheilchen”, 275

Stadium, 49, 159, 242, 244, 246State

changed by an external cause, 2, 127, 135,146, 150, 151, 171, 178, 186, 187, 190,191, 274, 296

changed by forces, 126, 193, 194, 262change of the state, mutually and

simultaneously, 191

Index 337

change of velocity is independent ofvelocity, Euler, 72, 131, 145

change of velocity is independent ofvelocity, Helmholtz, 273, 274, 282

conservation of state and change of position,170, 187

of rest and motion, 2, 3, 18, 19, 21, 42,117, 121, 123–127, 132, 145, 146,190, 274

the smallest body can change the state of thelargest body, 203

Staticsancient prototype of science of

equilibrium, 59and dynamics, 22, 24, 118, 123, 127, 146

Sufficient reasonfor changes in nature, 61principle, 61, 171, 177, 178, 182, 186, 238,

269, 312Syncategorematic, 58, 121, 135, 136, 141, 193,

201, 231

Tayloron Cavalieri and the perfection of ancient

geometers, 77“Grenzubergang von unerhorter Kuhnheit”

[Klein], 32making use of infinitesimal quantities, 197method of increments, 216

Tempusculumand spatiolum, 75, 224, 275, 293“verschwindendes Zeittheilchen”,

Helmholtz, 275“Zeittheilchen”, 275

thought experimentsEuler’s models, 188

Timeabsolute, 3, 7, 36, 45, 46, 74, 76, 115, 129,

181, 184, 235, 238–240, 252, 258, 260,261, 263, 264, 265, 270

and motion, 47, 89, 239, 297relational definition, Leibniz, 45–46and space, 3, 5, 7, 12, 16, 25, 36, 43, 45,

46–49, 50, 59, 87, 89, 115, 119, 128,129, 132, 143, 165, 166, 174, 183, 184,207, 225, 235, 238–240, 252, 257, 260,265, 267, 274

tempusculum, Euler, Wolfers, 275generated by a continual flux, Newton, 25as order of successions, Leibniz, 9, 89Planck’s time defined by universal

parameters, 287–288in quantum mechanics, 302Zeittheilchen, Wolfers, Helmholtz, 275

Time and spaceand geometry, Euler, 132mechanics as a four dimensional geometry,

Lagrange, 144Transcreation, 51, 132, 201Transfer principle

Robinson on Leibniz’s principle ofcontinuity, the transfer principle,230–231

Translationand motion, 113, 121

TropfkeGeschichte der Elementarmathematik, 200

Truncationof a system of differential equations,

mathematical procedure and mechanicalinterpretation, 159, 208, 306

Truthcontingent and necessary, 177

Unassignable, 51, 90–91, 112, 136, 199–200,218

Uncertainty relation, 19Units, 48, 212, 264, 273

Vacuumbody and vacuum, 99vacuum as a part of a mechanical

model, 157vacuum and plenum, 12

Variables and constantsvariables and functions, 224

Varignoninfinitesimals and uniform motion, 225time, space, motion and forces, 34, 38, 56

Velocitychanged by an external cause, 127changed by forces, 133, 139, 141,

259, 261change of velocity is independent of

velocity, Euler, 72change of velocity is independent of

velocity, Helmholtz, 273as a fluent, 42, 76, 77,

79, 80relative velocity, 22, 247,

249, 256, 257, 258, 263,264, 265

Verelstinterpretation of Nieuwentijt’s

theory, 96Vermeulen, 96, 204Veteres

rigour, Cavalieri, 197

338 Index

rigour, Euler, 226rigour, Lagrange, 196rigour, Leibniz, 57rigour, Newton, 77rigour, Taylor, 196rigour, Voltaire, 180static, 270

Voltairecriticism of metaphysics, 9a Frenchman in London, 12on Newton and Leibniz, 8

Wallis, 38, 201Westfall

Newton’s relation to Descartes, 3Weyl

on the relations between mathematics andphysics, 236

Whittakertreatment of differential equations, 307–308

Wilczekalgorithms in classical and quantum

mechanics, 288missing algorithms in classical

mechanics, 35

Zeittheilchen“verschwindende Zeittheilchen”, Helmholtz,

274Wolfers, 275

Zeno, 42, 49, 93, 96, 159, 165, 242, 243, 244,246

Zerocalculus of zeros, 224, 226

Zeuthen, 23