MST Topics in History of Mathematics

Transcription

1 MST Topics in History of Mathematics Euclid s Elements, the Works of Archimedes, and the Nine Chapters of Mathematical Art Paul Yiu Department of Mathematics Florida Atlantic University Summer 2017 June 28A

2 Greek geometry before Euclid Chronology -586 Babylonian captivity -585 Thales of Miletus; deductive geometry -580 Birth of Pythagoras -540 Pythagorean arithmetic and geometry -430 Elements of 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 s Elements -225 Apollonius: Conics -212 Death of Archimedes +75 Works of Heron of Alexandria +250 Diophantus: Arithmetica +320 Pappus: Mathematical Collections +485 Death of Proclus

3 2 Thales of Miletus Theorem (Thales). The angle inscribed in semicircle is a right angle.

4 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. (Euclid I.15) 4. If two triangles are such that two angles and a side of one are equal respectively to two angles and a side of the other, then the triangles are congruent. (Euclid I.26)

5 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. D A E B LetABC be a triangle, and through A letde be drawn parallel tobc. Now since BC,DE are parallel, and the alternative angles are equal, the angle DAB is equal to the angle ABC, andeac is equal to ACB. LetBAC be added to both. Then the angles DAB, BAC,CAE, that is, the angles DAB, 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. C

6 5 Iamblichus: On the Pythagorean Life (Thomas, ): It is related of Hippasus that he was a Pythagoreans, and that, owing to his being the first to publish and describe 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).

7 6 Euclid, Elements X, Scholium i The Pythagoreans were the first to make inquiry into commensurability, having first discovered it as a result of their observation of numbers; for though the unit is a common measure of all numbers they 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 divisible ad infinitum and do not leave some part which, being the least possible, will not admit of further division, but that remainder can be divided ad infinitum so as to give an infinite number of parts, of which each can be divided ad 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 Pythagoreans who made public the investigation of these matters perished in a shipwreck.

8 7 Democritus Archimedes, Method:... in case of those theorems concerning the cone and pyramid of 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.

9 8 Duplication of the cube Theon of Smyrna (Thomas 257): In his work entitled Platonicus Eratosthenes says that, when the god announced to the Delians by oracle that to get rid of a plague they must construct an altar double of the existing one, their craftsmen fell into great perplexity in trying to find how 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 mathematics and their contempt for geometry.

10 9 Eutocius: Commentary on Archimedes Sphere and Cylinder, II (Thomas ): It became a subject of inquiry among geometers in what manner one might double the given solid while it remained the same shape, and this problem was called the 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 proportionals could be found in 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 with an oracle, fell into the same quandary, and sent over to ask the geometers who were with Plato in the Academy to find what they sought. When these men applied themselves diligently and sought to 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.

11 10 Nicomedes solution Given: Two lines AD and DC at right angles. To construct: Two continued mean proportionals betweenad andcd. M A D F H B E C K T Z Construction: Complete the rectangle ADCB and take the midpoints E,F ofbc and AB. Join DF to meetcb produced ath. Construct EZ perpendicular tobc so thatcz = AF. Join HZ and construct the parallel through C. On this parallel construct a point T so that ZT meetbc produced atk with TK = CZ. Join KD and produce it to meetba produced atm. Then AD : AM = AM : CK = CK : CD, andam, CK are two continued mean proportionals between AD andcd.

12 11 MAA Focus June/July 2014, 27. x andy are two continuous mean proportionals between1and2: A x 1 1 Y 1 B 1 C y X 1 Z

13 12 Cubic root of 2 by paper folding B. Casselman, If Euclid had been Japanese, Notices of AMS, 54 (2007) A paper square ABCD is divided into three strips of equal area by the parallel lines PQ andrs. The square is then folded so that C falls on AD ands falls onpq (as C in the second diagram). Then AC C D = 3 2. A B A B P Q P S C R S R D C D C

14 13 Trisection of an angle Archimedes [Book of Lemmas, Proposition 8] Given angle AOB with OA = OB (contained in a circle, centero), construct a line through A such that the intercept between the circle and the line BO has the same length as the radius of the circle. Then A OC = 1 3 AOB. A A C O B

15 14 Trisection of an angle To triangle an angle AOB, pass a line through O such that the intercept between the parallel and the perpendicular at A to the line OB is 2 OA. Then this line is a trisector of angle AOB. A E M D O B

16 15 Angle trisection with the use of conics To trisect an angle AOB, construct the hyperbola with focus A, directrix OM, and eccentricity2, to intersect the arc AB at C. Then AOB = 3 AOC. C K B M P A O

17 16 The quadratrix of Hippias A horizontal line HK (with initial position AB) falls vertically, and a radius OP (with initial position OA) rotates about O, both uniformly and arrive atoc at the same time. The locus of the intersection Q = HK OP is the quadratrix. A B P H Q K O C

18 17 Trisection of an angle by the quadratrix B P K Q O To trisect angleaob, letob intersect the quadratrix atp. Trisect the segment OP atk. Construct the parallel through K tooa to intersect the quadratrix at Q. ThenOQ is a trisector of angle AOB. A

19 18 Angle trisection by paper folding B. Casselman, If Euclid had been Japanese, Notices of AMS, 54 (2007) A M B A M B P P Q P Q R R S R D S D C D C

20 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 that a rectilinear figure is equal to a circular area.

21 20 Pappus on the quadratrix For the squaring of the circle a certain line was used by Dionstratus 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 center C, while BHT is a quadratrix generated in the aforesaid manner, it is proved that the ratio of the arc DEB towards the straight line BC is the same as that ofbc towards the straight line CT. arc BED : AB = AB : CT. B A E H C Construct a length b such that CT : BC = BC : b. Then the rectangle with sides b andbc is equal to the quadrant BED. 2 T D

22 21 Incommensurables Aristotle, Prior Analytics For all who argue per impossibile infer by syllogism a false conclusion, and prove the original conclusion hypothetically when something impossible follows from a contradictory assumption, as, for example, that the diagonal [of a square] is incommensurable [with the side] because odd numbers are equal to even if it is assumed to be commensurate. It is inferred by syllogism that odd numbers are equal to even, and proved hypothetically that the diagonal is commensurate, since a false conclusion follows from the contradictory assumption.

23 22 Incommensurability of the diagonal and side of a square If the diagonal d and the side s of a square have a (unit) common measure, these are numbers satisfying d 2 = 2s 2. d 2 is an even number. Therefore, d is an even number. Since d andsdo not have common measure, s is an odd number. Writing d = 2m, we have (2m) 2 = 2s 2, 4m 2 = 2s 2, ands 2 = 2m 2. This shows that is s 2 an even number. Therefore, s is also an even number. But s cannot be both odd and even. This contradiction shows that the diagonal and the side of a square are incommensurable.

24 23 Euclid s Elements There is no royal road to geometry. Euclid to Ptolemy and Alexander

25 24 Proclus on Euclid s Elements It is a difficult task in any science to select and arrange properly the elements out of which all others matters are produced and into which they can be resolved. Of those who have attempted it some 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 general many ways of constructing elementary expositions have been individually invented....

26 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 bits makes knowledge of it difficult to attain. Judged by all these criteria, you will find Euclid s introduction superior to others.

27 26 Euclid s Elements Books I VI VII IX X XI XIII Subject Plane geometry Number theory Theory of irrational constructible quantities Solid geometry Book I II III IV V VI Total Definitions Common notions 5 Postulates 5 Propositions Book VII VIII IX X XI XII XIII Total Definitions Propositions

28 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 line makes the adjacent angles equal to one another, each of the equal angles is [a] right [angle], and the straight line standing on the other is called a perpendicular to that on which it stands. Definition (I.23). Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

29 28 The common notions (1) Things which equal the same thing also equal one another. a = b and b = c = a = c. (2) If equals are added to equals, then the wholes are equal. a = b and c = d = a+c = b+d. (3) If equals are subtracted from equals, then the remainders are equal. a = b and c = d = c a = d b. (4) Things which coincide with one another equal one another. (5) The whole is greater than the part.

30 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 makes the interior angles on the same side less 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.

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