MST Topics inHistory of Mathematics

Euclid’s Elements,the Works of Archimedes,

and the Nine Chapters of Mathematical Art

Paul Yiu

Department of MathematicsFlorida Atlantic University

Summer 2017

June 28A

Greek geometry before Euclid

Chronology

-586 Babylonian captivity-585 Thales of Miletus; deductive geometry-580 Birth of Pythagoras-540 Pythagorean arithmetic and geometry-430 Elementsof Hippocrates of Chios-427 Birth of Plato-420 Incommensurables-399 Death of Socrates-360 Eudoxus on proportion and exhaustion-347 Death of Plato-335 Eudemus:History of Geometry-332 Alexandria founded-323 Death of Alexander-322 Death of Aristotle-300 Euclid’sElements-225 Apollonius:Conics-212 Death of Archimedes+75 Works of Heron of Alexandria

+250 Diophantus:Arithmetica+320 Pappus:Mathematical Collections+485 Death of Proclus

2

Thales of Miletus

Theorem (Thales). The angle inscribed in semicircle is a right angle.

3

Other theorems attributed to Thales

Eudemus’History (ca -320):

1. A circle is bisected by a diameter.

2. The base angle of an isosceles triangle are equal. (Euclid I.5)

3. The pairs of vertical angles formed by two intersecting lines are equal. (Eu-clid I.15)

4. If two triangles are such thattwo angles and a side of oneare equal respectively totwo angles and a side of the other,then the triangles are congruent. (Euclid I.26)

4

Pythagoras of Samos

Proclus, on Euclid I: (p.298)

Eudemus the Peripatetic ascribes to the Pythagoreans the discovery of this theorem,that any triangle has internal angles equal to two right angles.He says they proved the theorem in question after this fashion.

A

B C

D E

LetABC be a triangle,and throughA letDE be drawn parallel toBC.Now sinceBC, DE are parallel,and the alternative angles are equal,the angle DAB is equal to the angleABC,andEAC is equal toACB.LetBAC be added to both.Then the anglesDAB, BAC, CAE, that is,the anglesDAB, BAE, that is, two right angles,are equal to the three angles of the triangle.Therefore the three angles of the triangle are equal to two right angles.Such is the proof of the Pythagoreans.

5

Iamblichus:On the Pythagorean Life(Thomas, 222–225):

It is related of Hippasus that he was a Pythagoreans,and that, owing to his being the first to publish anddescribe the sphere from the twelve pentagons,he perished at sea for his impiety,but he received credit for the discovery,though really it all belonged to HIM(for in this way they refer to Pythagoras,and they do not call him by his name).

6

Euclid, Elements X, Scholium i

The Pythagoreans were the first to make inquiry intocommensurability,having first discovered it as a result of their observation ofnumbers;for though the unit is a common measure of all numbersthey could not find a common measure of all magnitudes.The reason is that all numbers, of whatsoever kind,however they be divided leave some least part which will not suffer further division;but all magnitudes are divisiblead infinitumand do not leave some part which,being the least possible, will not admit of further division,but that remainder can be dividedad infinitumso as to give an infinite number of parts,of which each can be dividedad infinitum;and in sum, magnitude partakes in division of the principle of the infinite,but in its entirety of the principle of the finite,while number in division partakes of the finite,but in its entirety of the infinite.. . .There is a legend that the first of the Pythagoreanswho made public the investigation of these mattersperished in a shipwreck.

7

Democritus

Archimedes,Method:

. . . in case of those theorems concerning the cone and pyramidof which Eudoxus first discovered the proof,the theorem that the cone is the third part of the cylinder,and the pyramind of the prism,having the same base and equal height,no small share of the credit should be given to Democritus,who was the first to make the assertion with regard to the said figure,though without proof.

8

Duplication of the cube

Theon of Smyrna (Thomas 257):

In his work entitledPlatonicusEratosthenes says that,when the god announced to the Delians by oracle that to get ridof a plaguethey must construct an altar double of the existing one,their craftsmen fell into great perplexity in trying to findhow a solid could be made double of another solid,and they went to ask Plato about it.He told them that the god had given this oracle,not because he wanted an altar of double the size,but he wished, in setting this task before them,to reproach the Greeks for their neglect of mathematicsand their contempt for geometry.

9

Eutocius:Commentary on Archimedes’ Sphere and Cylinder, II(Thomas 257–259):

It became a subject of inquiry among geometersin what manner one might double the given solid while it remained the same shape,and this problem was calledthe duplication of the cube;for given a cube, they sought to double it.When all were for a long time at a loss,Hippocrates of Chios first conceived that,if two mean proportionalscould be foundin continued proportion between two straight lines,of which the greater was double the lesser,the cube would be doubled,so that the puzzle was by him turned into no less a puzzle.After a time, it is related, certain Delians,when attempting to double a certain altar in accordance withan oracle,fell into the same quandary,and sent over to ask the geometerswho were with Plato in the Academy to find what they sought.When these men applied themselves diligently and soughtto find two mean proportionals between two given straight lines,Archytas of Taras is said to have found them by the half-cylinders,and Eudoxus by the so-called curved lines;but it turned out that all their solutions were theoretical,and they could not give a practical construction and turn it to use,except to a certain small extent Menaechmus, and that with difficulty.An easy mechanical solution, was however, found by me,and by means of it I will find, not only two means to the given straight lines,but as many as may be enjoined.

10

Nicomedes’ solution

Given: Two linesAD andDC at right angles.To construct: Two continued mean proportionals betweenAD andCD.

A

B C

D

H

E

Z

K

M

T

F

Construction: Complete the rectangleADCB andtake the midpointsE, F of BC andAB.JoinDF to meetCB produced atH.ConstructEZ perpendicular toBC so thatCZ = AF .JoinHZ and construct the parallel throughC.On this parallel construct a pointT so thatZT meetBC produced atK with TK = CZ.JoinKD and produce it to meetBA produced atM .Then

AD : AM = AM : CK = CK : CD,

andAM , CK are two continued mean proportionals betweenAD andCD.

11

MAA Focus June/July 2014, 27.x andy are two continuous mean proportionals between1 and2:

y

x

1

1

1

1

1

A

BC X

Z

Y

12

Cubic root of 2 by paper folding

B. Casselman, If Euclid had been Japanese,Notices of AMS, 54 (2007) 626–628.

A paper squareABCD is divided intothree strips of equal area by the parallel linesPQ andRS.The square is then folded so thatC falls onAD andS falls onPQ

(asC ′ in the second diagram).Then AC′

C′D= 3

√2.

D C

BA

S

Q

R

P

D C

BA

R

P

C′

S′

13

Trisection of an angle

Archimedes [Book of Lemmas, Proposition 8]

Given angleAOB with OA = OB (contained in a circle, centerO),construct a line throughA such thatthe intercept between the circle and the lineBO

has the same length as the radius of the circle.Then∠A′OC = 1

3∠AOB.

B

A A′

O

C

14

Trisection of an angle

To triangle an angleAOB,pass a line throughO such thatthe intercept between the parallel and the perpendicular atA to the lineOB

is 2 ·OA.Then this line is a trisector of angleAOB.

O

A

B

E

D

M

15

Angle trisection with the use of conics

To trisect an angleAOB,construct thehyperbola with focusA, directrixOM , and eccentricity2,to intersect the arcAB atC.Then∠AOB = 3∠AOC.

AB

C

M

K

P

O

16

The quadratrix of Hippias

A horizontal lineHK (with initial positionAB) falls vertically,and a radiusOP (with initial positionOA) rotates aboutO,both uniformly and arrive atOC at the same time.The locus of the intersectionQ = HK ∩OP is thequadratrix .

O

Q

A

C

P

H

B

K

17

Trisection of an angle by the quadratrix

P

O

B

A

QK

To trisect angleAOB, letOB intersect the quadratrix atP . Trisect thesegmentOP atK. Construct the parallel throughK toOA to intersect the quadratrix atQ.

ThenOQ is a trisector of angleAOB.

18

Angle trisection by paper folding

B. Casselman, If Euclid had been Japanese,Notices of AMS, 54 (2007) 626–628.

D C

BA

S

Q

R

P

M

D

BA

R

P

C

M

D′

P ′

R′

S

Q

19

Quadrature of the circle

Proclus on Euclid I.45:It is my opinion that this proposition is what led the ancients

to attempt the squaring of the circle.For if a parallelogram can be found equal to any rectilinear figure,it is worth inquiring whether it is not possible to prove thata rectilinear figure is equal to a circular area.

20

Pappus on the quadratrix

For the squaring of the circle a certain line was used byDionstratus’ and Nicomedes and certain other more recent geometers,and it takes the name from its special property:for it called by them the quadratrix, . . . .

If ABCD is a square andBED the arc of a circle with centerC,whileBHT is a quadratrix generated in the aforesaid manner,it is proved that the ratio of the arcDEB towards the straight lineBC

is the same as that ofBC towards the straight lineCT .

arcBED : AB = AB : CT.

AB

C D

H

T

E

Construct a lengthb such thatCT : BC = BC : b.Then the rectangle with sidesb

2andBC is equal to the quadrantBED.

21

Incommensurables

Aristotle, Prior Analytics

For all who argueper impossibileinfer by syllogism a false conclusion,and prove the original conclusion hypotheticallywhen something impossible follows from a contradictory assumption,as, for example, thatthe diagonal[of a square]is incommensurable[with the side]because odd numbers are equal to evenif it is assumed to be commensurate.It is inferred by syllogism thatodd numbers are equal to even,and proved hypothetically that the diagonal is commensurate,since a false conclusion follows from the contradictory assumption.

22

Incommensurability of the diagonal and side of a square

If the diagonald and the sides of a square have a (unit) common measure,these are numbers satisfyingd2 = 2s2.d2 is an even number.Therefore,d is anevennumber.Sinced ands do not have common measure,s is an odd number.Writing d = 2m, we have(2m)2 = 2s2,4m2 = 2s2, ands2 = 2m2.This shows that iss2 an even number.Therefore,s is also anevennumber.But s cannot be both odd and even.This contradiction shows thatthe diagonal and the side of a square are incommensurable.

23

Euclid’s Elements

There is no royal road to geometry.

Euclid to Ptolemy and Alexander

24

Proclus on Euclid’s Elements

It is a difficult task in any science to select and arrange properlythe elements out of which all others matters are producedand into which they can be resolved.Of those who have attempted itsome have brought together more theorems,some less;some have used rather short demonstrations,others have extended their treatment to great lengths;some have avoided the reduction to impossibility,others proportion;some have devised defenses in advance against attacks upon the starting points;and in generalmany ways of constructing elementary expositions have been individually invented.. . .

25

Proclus on Euclid’s Elements

. . .Such a treatise ought to be free of everything superfluous,for that is a hindrance to learning;the selections chosen must all be coherent and conducive to the end proposed,in order to be of the greatest usefulness for knowledge;it must devote great attention both to clarity and to conciseness,for what lacks these qualities confuses our understanding;it ought to aim at the comprehension of its theorems in a general form,for dividing one’s subject too minutely and teaching it by bitsmakes knowledge of it difficult to attain.Judged by all these criteria,you will find Euclid’s introduction superior to others.

26

Euclid’s ElementsBooks SubjectI – VI Plane geometryVII – IX Number theoryX Theory of irrational constructible quantitiesXI–XIII Solid geometry

Book I II III IV V VI TotalDefinitions 23 2 10 7 18 5 65Common notions 5Postulates 5Propositions 48 14 37 16 25 33 173

Book VII VIII IX X XI XII XIII TotalDefinitions 22 16 28 66Propositions 39 27 36 115 39 18 18 292

27

The first definitions

Definitions.(I.1). A point is that which has no part.(I.2). A line is breadthless length.(I.3). The extremities of a line are points.(I.4). A straight line is a line which lies evenly with the points on itself.

Definition (I.10).When a straight line set up on a straight linemakes the adjacent angles equal to one an-other, each of the equal angles is [a]right[angle], and the straight line standing onthe other is called aperpendicular to thaton which it stands.

Definition (I.23). Parallel straight lines are straight lines which, being in the sameplane and being produced indefinitely in both directions, donot meet one anotherin either direction.

28

The common notions

(1) Things which equal the same thing also equal one another.a = b andb = c =⇒ a = c.

(2) If equals are added to equals, then the wholes are equal.a = b andc = d =⇒ a+ c = b+ d.

(3) If equals are subtracted from equals, then the remainders are equal.a = b andc = d =⇒ c− a = d− b.

(4) Things which coincide with one another equal one another.

(5) The whole is greater than the part.

29

The postulates

Postulate 1.To draw a straight line from any point to any point.Postulate 2.To produce a finite straight line continuously in a straight line.Postulate 3.To describe a circle with any center and distance.Postulate 4.That all right angles are equal to each other.

Postulate 5.That, if a straight line falling on two straight lines makesthe interior angles on the same sideless than two right angles,the two straight lines, if produced indefinitely,meet on that side on which are the angles less than the two right angles.