A Short Account of the History of Mathematics · W. W. Rouse Ball
The history of these schools has been discussed by G. Loria in his Le Scienze Esatte nell' Antica Grecia, Modena, 1893--1900; by Cantor, chaps. v--viii; by G. J. Allman in his Greek Geometry from Thales to Euclid, Dublin, 1889; by J. Gow, in his Greek Mathematics, Cambridge, 1884; by C. A. Bretschneider in his Die Geometrie und die Geometer vor Eukleides, Leipzig, 1870; and partially by H. Hankel in his posthumous Geschichte der Mathematik, Leipzig, 1874.
With the foundation of the Ionian and Pythagorean schools we emerge from the region of antiquarian research and conjecture into the light of history. The materials at our disposal for estimating the knowledge of the philosophers of these schools previous to about the year 430 b.c. are, however, very scanty Not only have all but fragments of the different mathematical treatises then written been lost, but we possess no copy of the history of mathematics written about 325 b.c. by Eudemus (who was a pupil of Aristotle). Luckily Proclus, who about 450 a.d. wrote a commentary on the earlier part of Euclid's Elements, was familiar with Eudemus's work, and freely utilised it in his historical references. We have also a fragment of the General View of Mathematics written by Geminus about 50 b.c., in which the methods of proof used by the early Greek geometricians are compared with those current at a later date. In addition to these general statements we have biographies of a few of the leading mathematicians, and some scattered notes in various writers in which allusions are made to the lives and works of others. The original authorities are criticised and discussed at length in the works mentioned in the footnote to the heading of the chapter.
Thales.
The founder of the earliest Greek school of mathematics and philosophy was Thales, one of the seven sages of Greece, who was born about 640 b.c. at Miletus, and died in the same town about 550 b.c. The materials for an account of his life consist of little more than a few anecdotes which have been handed down by tradition.
During the early part of his life Thales was engaged partly in commerce and partly in public affairs; and to judge by two stories that have been preserved, he was then as distinguished for shrewdness in business and readiness in resource as he was subsequently celebrated in science. It is said that once when transporting some salt which was loaded on mules, one of the animals slipping in a stream got its load wet and so caused some of the salt to be dissolved, and finding its burden thus lightened it rolled over at the next ford to which it came; to break it of this trick Thales loaded it with rags and sponges which, by absorbing the water, made the load heavier and soon effectually cured it of its troublesome habit. At another time, according to Aristotle, when there was a prospect of an unusually abundant crop of olives Thales got possession of all the olive-presses of the district; and, having thus "cornered" them, he was able to make his own terms for lending them out, or buying the olives, and thus realized a large sum. These tales may be apocryphal, but it is certain that he must have had considerable reputation as a man of affairs and as a good engineer, since he was employed to construct an embankment so as to divert the river Halys in such a way as to permit of the construction of a ford.
Probably it was as a merchant that Thales first went to Egypt, but during his leisure there he studied astronomy and geometry. He was middle-aged when he returned to Miletus; he seems then to have abandoned business and public life, and to have devoted himself to the study of philosophy and science--subjects which in the Ionian, Pythagorean, and perhaps also the Athenian schools, were closely connected: his views on philosophy do not here concern us. He continued to live at Miletus till his death circ. 550 b.c.
We cannot form any exact idea as to how Thales presented his geometrical teaching. We infer, however, from Proclus that it consisted of a number of isolated propositions which were not arranged in a logical sequence, but that the proofs were deductive, so that the theorems were not a mere statement of an induction from a large number of special instances, as probably was the case with the Egyptian geometricians. The deductive character which he thus gave to the science is his chief claim to distinction.
The following comprise the chief propositions that can now with reasonable probability be attributed to him; they are concerned with the geometry of angles and straight lines.
(i) The angles at the base of an isosceles triangle are equal (Euc. i, 5). Proclus seems to imply that this was proved by taking another exactly equal isosceles triangle, turning it over, and then superposing it on the first--a sort of experimental demonstration.
(ii) If two straight lines cut one another, the vertically opposite angles are equal (Euc. i, 15). Thales may have regarded this as obvious, for Proclus adds that Euclid was the first to give a strict proof of it.
(iii) A triangle is determined if its base and base angles be given (cf. Euc. i, 26). Apparently this was applied to find the distance of a ship at sea--the base being a tower, and the base angles being obtained by observation.
(iv) The sides of equiangular triangles are proportionals (Euc. vi, 4, or perhaps rather Euc. vi, 2). This is said to have been used by Thales when in Egypt to find the height of a pyramid. In a dialogue given by Plutarch, the speaker, addressing Thales, says, "Placing your stick at the end of the shadow of the pyramid, you made by the sun's rays two triangles, and so proved that the [height of the] pyramid was to the [length of the] stick as the shadow of the pyramid to the shadow of the stick." It would seem that the theorem was unknown to the Egyptians, and we are told that the king Amasis, who was present, was astonished at this application of abstract science.
(v) A circle is bisected by any diameter. This may have been enunciated by Thales, but it must have been recognised as an obvious fact from the earliest times.
(vi) The angle subtended by a diameter of a circle at any point in the circumference is a right angle (Euc. iii, 31). This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a right-angled triangle in a circle he sacrificed an ox to the immortal gods. It has been conjectured that he may have come to this conclusion by noting that the diagonals of a rectangle are equal and bisect one another, and that therefore a rectangle can be inscribed in a circle. If so, and if he went on to apply proposition (i), he would have discovered that the sum of the angles of a right-angled triangle is equal to two right angles, a fact with which it is believed that he was acquainted. It has been remarked that the shape of the tiles used in paving floors may have suggested these results.
On the whole it seems unlikely that he knew how to draw a perpendicular from a point to a line; but if he possessed this knowledge, it is possible he was also aware, as suggested by some modern commentators, that the sum of the angles of any triangle is equal to two right angles. As far as equilateral and right-angled triangles are concerned, we know from Eudemus that the first geometers proved the general property separately for three species of triangles, and it is not unlikely that they proved it thus. The area about a point can be filled by the angles of six equilateral triangles or tiles, hence the proposition is true for an equilateral triangle. Again, any two equal right-angled triangles can be placed in juxtaposition so as to form a rectangle, the sum of whose angles is four right angles; hence the proposition is true for a right-angled triangle. Lastly, any triangle can be split into the sum of two right-angled triangles by drawing a perpendicular from the biggest angle on the opposite side, and therefore again the proposition is true. The first of these proofs is evidently included in the last, but there is nothing improbable in the suggestion that the early Greek geometers continued to teach the first proposition in the form above given.
Thales wrote on astronomy, and among his contemporaries was more famous as an astronomer than as a geometrician. A story runs that one night, when walking out, he was looking so intently at the stars that he tumbled into a ditch, on which an old woman exclaimed, "How can you tell what is going on in the sky when you can't see what is lying at your own feet?"--an anecdote which was often quoted to illustrate the unpractical character of philosophers.
Without going into astronomical details, it may be mentioned that he taught that a year contained about 365 days, and not (as is said to have been previously reckoned) twelve months of thirty days each. It is said that his predecessors occasionally intercalated a month to keep the seasons in their customary places, and if so they must have realized that the year contains, on the average, more than 360 days. There is some reason to think that he believed the earth to be a disc-like body floating on water. He predicted a solar eclipse which took place at or about the time he foretold; the actual date was either May 28, 585 b.c., or September 30, 609 b.c. But though this prophecy and its fulfilment gave extraordinary prestige to his teaching, and secured him the name of one of the seven sages of Greece, it is most likely that he only made use of one of the Egyptian or Chaldaean registers which stated that solar eclipses recur at intervals of about 18 years 11 days.
Among the pupils of Thales were Anaximander, Anaximenes, Mamercus, and Mandryatus. Of the three mentioned last we know next to nothing. Anaximander was born in 611 b.c., and died in 545 b.c., and succeeded Thales as head of the school at Miletus. According to Suidas he wrote a treatise on geometry in which, tradition says, he paid particular attention to the properties of spheres, and dwelt at length on the philosophical ideas involved in the conception of infinity in space and time. He constructed terrestrial and celestial globes.
Anaximander is alleged to have introduced the use of the style or gnomon into Greece. This, in principle, consisted only of a stick stuck upright in a horizontal piece of ground. It was originally used as a sun-dial, in which case it was placed at the centre of three concentric circles, so that every two hours the end of its shadow passed from one circle to another. Such sun-dials have been found at Pompeii and Tusculum. It is said that he employed these styles to determine his meridian (presumably by marking the lines of shadow cast by the style at sunrise and sunset on the same day, and taking the plane bisecting the angle so formed); and thence, by observing the time of year when the noon-altitude of the sun was greatest and least, he got the solstices; thence, by taking half the sum of the noon-altitudes of the sun at the two solstices, he found the inclination of the equator to the horizon (which determined the altitude of the place), and, by taking half their difference, he found the inclination of the ecliptic to the equator. There seems good reason to think that he did actually determine the latitude of Sparta, but it is more doubtful whether he really made the rest of these astronomical deductions.
We need not here concern ourselves further with the successors of Thales. The school he established continued to flourish till about 400 b.c., but, as time went on, its members occupied themselves more and more with philosophy and less with mathematics. We know very little of the mathematicians comprised in it, but they would seem to have devoted most of their attention to astronomy. They exercised but slight influence on the further advance of Greek mathematics, which was made almost entirely under the influence of the Pythagoreans, who not only immensely developed the science of geometry, but created a science of numbers. If Thales was the first to direct general attention to geometry, it was Pythagoras, says Proclus, quoting from Eudemus, who "changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom and investigated its theorems in an intellectual manner"; and it is accordingly to Pythagoras that we must now direct attention.
Pythagoras.
Pythagoras was born at Samos about 569 b.c., perhaps of Tyrian parents, and died in 500 b.c. He was thus a contemporary of Thales. The details of his life are somewhat doubtful, but the following account is, I think, substantially correct. He studied first under Pherecydes of Syros, and then under Anaximander; by the latter he was recommended to go to Thebes, and there or at Memphis he spent some years. After leaving Egypt he travelled in Asia Minor, and then settled at Samos, where he gave lectures but without much success. About 529 b.c. he migrated to Sicily with his mother, and with a single disciple who seems to have been the sole fruit of his labours at Samos. Thence he went to Tarentum, but very shortly moved to Croton, a Dorian colony in the south of Italy. Here the schools that he opened were crowded with enthusiastic audiences; citizens of all ranks, especially those of the upper classes, attended, and even the women broke a law which forbade their going to public meetings and flocked to hear him. Amongst his most attentive auditors was Theano, the young and beautiful daughter of his host Milo, whom, in spite of the disparity of their ages, he married. She wrote a biography of her husband, but unfortunately it is lost.
Pythagoras divided those who attended his lectures into two classes, whom we may term probationers and Pythagoreans. The majority were probationers, but it was only to the Pythagoreans that his chief discoveries were revealed. The latter formed a brotherhood with all things in common, holding the same philosophical and political beliefs, engaged in the same pursuits, and bound by oath not to reveal the teaching or secrets of the school; their food was simple; their discipline severe; and their mode of life arranged to encourage self-command, temperance, purity, and obedience. This strict discipline and secret organisation gave the society a temporary supremacy in the state which brought on it the hatred of various classes; and, finally, instigated by his political opponents, the mob murdered Pythagoras and many of his followers.
Though the political influence of the Pythagoreans was thus destroyed, they seem to have re-established themselves at once as a philosophical and mathematical society, with Tarentum as their headquarters, and they continued to flourish for more than a hundred years.
Pythagoras himself did not publish any books; the assumption of his school was that all their knowledge was held in common and veiled from the outside world, and, further, that the glory of any fresh discovery must be referred back to their founder. Thus Hippasus (circ. 470 b.c.) is said to have been drowned for violating his oath by publicly boasting that he had added the dodecahedron to the number of regular solids enumerated by Pythagoras. Gradually, as the society became more scattered, this custom was abandoned, and treatises containing the substance of their teaching and doctrines were written. The first book of the kind was composed, about 370 b.c., by Philolaus, and we are told that Plato secured a copy of it. We may say that during the early part of the fifth century before Christ the Pythagoreans were considerably in advance of their contemporaries, but by the end of that time their more prominent discoveries and doctrines had become known to the outside world, and the centre of intellectual activity was transferred to Athens.
Though it is impossible to separate precisely the discoveries of Pythagoras himself from those of his school of a later date, we know from Proclus that it was Pythagoras who gave geometry that rigorous character of deduction which it still bears, and made it the foundation of a liberal education; and there is reason to believe that he was the first to arrange the leading propositions of the subject in a logical order. It was also, according to Aristoxenus, the glory of his school that they raised arithmetic above the needs of merchants. It was their boast that they sought knowledge and not wealth, or in the language of one of their maxims, "a figure and a step forwards, not a figure to gain three oboli."
Pythagoras was primarily a moral reformer and philosopher, but his system of morality and philosophy was built on a mathematical foundation. His mathematical researches were, however, designed to lead up to a system of philosophy whose exposition was the main object of his teaching. The Pythagoreans began by dividing the mathematical subjects with which they dealt into four divisions: numbers absolute or arithmetic, numbers applied or music, magnitudes at rest or geometry, and magnitudes in motion or astronomy. This "quadrivium" was long considered as constituting the necessary and sufficient course of study for a liberal education. Even in the case of geometry and arithmetic (which are founded on inferences unconsciously made and common to all men) the Pythagorean presentation was involved with philosophy; and there is no doubt that their teaching of the sciences of astronomy, mechanics, and music (which can rest safely only on the results of conscious observation and experiment) was intermingled with metaphysics even more closely. It will be convenient to begin by describing their treatment of geometry and arithmetic.
First, as to their geometry. Pythagoras probably knew and taught the substance of what is contained in the first two books of Euclid about parallels, triangles, and parallelograms, and was acquainted with a few other isolated theorems including some elementary propositions on irrational magnitudes; but it is suspected that many of his proofs were not rigorous, and in particular that the converse of a theorem was sometimes assumed without a proof. It is hardly necessary to say that we are unable to reproduce the whole body of Pythagorean teaching on this subject, but we gather from the notes of Proclus on Euclid, and from a few stray remarks in other writers, that it included the following propositions, most of which are on the geometry of areas.
(i) It commenced with a number of definitions, which probably were rather statements connecting mathematical ideas with philosophy than explanations of the terms used. One has been preserved in the definition of a point as unity having position.
(ii) The sum of the angles of a triangle was shown to be equal to two right angles (Euc. i, 32); and in the proof, which has been preserved, the results of the propositions Euc. i, 13 and the first part of Euc. i, 29 are quoted. The demonstration is substantially the same as that in Euclid, and it is most likely that the proofs there given of the two propositions last mentioned are also due to Pythagoras himself.
(iii) Pythagoras certainly proved the properties of right-angled triangles which are given in Euc. i, 47 and i, 48. We know that the proofs of these propositions which are found in Euclid were of Euclid's own invention; and a good deal of curiosity has been excited to discover what was the demonstration which was originally offered by Pythagoras of the first of these theorems. It has been conjectured that not improbably it may have been one of the two following.
( alpha ) Any square ABCD can be split up, as in Euc. ii, 4, into two squares BK and DK and two equal rectangles AK and CK : that is, it is equal to the square on FK , the square on EK , and four times the triangle AEF . But, if points be taken, G on BC , H on CD , and E on DA , so that BG , CH , and DE are each equal to AF , it can be easily shown that EFGH is a square, and that the triangles AEF , BFG , CGH , and DHE are equal: thus the square ABCD is also equal to the square on EF and four times the triangle AEF . Hence the square on EF is equal to the sum of the squares on FK and EK .
( beta ) Let ABC be a right-angled triangle, A being the right angle. Draw AD perpendicular to BC . The triangles ABC and DBA are similar,
BC : AB = AB : BD. Similarly BC : AC = AC : DC. Hence AB^2 + AC^2 = BC(BD + DC) = BC^2.
This proof requires a knowledge of the results of Euc. ii, 2, vi, 4, and vi, 17, with all of which Pythagoras was acquainted.
(iv) Pythagoras is credited by some writers with the discovery of the theorems Euc. i, 44, and i, 45, and with giving a solution of the problem Euc. ii, 14. It is said that on the discovery of the necessary construction for the problem last mentioned he sacrificed an ox, but as his school had all things in common the liberality was less striking than it seems at first. The Pythagoreans of a later date were aware of the extension given in Euc. vi, 25, and Allman thinks that Pythagoras himself was acquainted with it, but this must be regarded as doubtful. It will be noticed that Euc. ii, 14 provides a geometrical solution of the equation x^2 = ab .
(v) Pythagoras showed that the plane about a point could be completely filled by equilateral triangles, by squares, or by regular hexagons--results that must have been familiar wherever tiles of these shapes were in common use.
(vi) The Pythagoreans were said to have attempted the quadrature of the circle: they stated that the circle was the most perfect of all plane figures.
(vii) They knew that there were five regular solids inscribable in a sphere, which was itself, they said, the most perfect of all solids.
(viii) From their phraseology in the science of numbers and from other occasional remarks, it would seem that they were acquainted with the methods used in the second and fifth books of Euclid, and knew something of irrational magnitudes. In particular, there is reason to believe that Pythagoras proved that the side and the diagonal of a square were incommensurable, and that it was this discovery which led the early Greeks to banish the conceptions of number and measurement from their geometry. A proof of this proposition which may be that due to Pythagoras is given below.
Next, as to their theory of numbers. In this Pythagoras was chiefly concerned with four different groups of problems which dealt respectively with polygonal numbers, with ratio and proportion, with the factors of numbers, and with numbers in series; but many of his arithmetical inquiries, and in particular the questions on polygonal numbers and proportion, were treated by geometrical methods.
Pythagoras commenced his theory of arithmetic by dividing all numbers into even or odd: the odd numbers being termed gnomons. An odd number, such as 2n + 1 , was regarded as the difference of two square numbers (n+1)^2 and n^2 ; and the sum of the gnomons from 1 to 2n+1 was stated to be a square number, viz. (n+1)^2 , its square root was termed a side. Products of two numbers were called plane, and if a product had no exact square root it was termed an oblong. A product of three numbers was called a solid number, and, if the three numbers were equal, a cube. All this has obvious reference to geometry, and the opinion is confirmed by Aristotle's remark that when a gnomon is put round a square the figure remains a square though it is increased in dimensions. Thus, in the figure given above in which n is taken equal to 5, the gnomon AKC (containing 11 small squares) when put round the square AC (containing 5^2 small squares) makes a square HL (containing 6^2 small squares). It is possible that several of the numerical theorems due to Greek writers were discovered and proved by an analogous method: the abacus can be used for many of these demonstrations.
The numbers (2n^2+2n+1) , (2n^2+2n) , and (2n+1) possessed special importance as representing the hypotenuse and two sides of a right-angled triangle: Cantor thinks that Pythagoras knew this fact before discovering the geometrical proposition Euc. i, 47. A more general expression for such numbers is (m^2+n^2) , 2mn , and (m^2-n^2) , or multiples of them: it will be noticed that the result obtained by Pythagoras can be deduced from these expressions by assuming m=n+1 ; at a later time Archytas and Plato gave rules which are equivalent to taking n=1 ; Diophantus knew the general expressions.
After this preliminary discussion the Pythagoreans proceeded to the four special problems already alluded to. Pythagoras was himself acquainted with triangular numbers; polygonal numbers of a higher order were discussed by later members of the school. A triangular number represents the sum of a number of counters laid in rows on a plane; the bottom row containing n , and each succeeding row one less: it is therefore equal to the sum of the series n + (n-1) + (n-2) + + 2 + 1, that is, to 1/2 n(n+1) . Thus the triangular number corresponding to 4 is 10. This is the explanation of the language of Pythagoras in the well-known passage in Lucian where the merchant asks Pythagoras what he can teach him. Pythagoras replies "I will teach you how to count." Merchant, "I know that already." Pythagoras, "How do you count?" Merchant, "One, two, three, four--" Pythagoras, "Stop! what you take to be four is ten, a perfect triangle and our symbol." The Pythagoreans are, on somewhat doubtful authority, said to have classified numbers by comparing them with the sum of their integral subdivisors or factors, calling a number excessive, perfect, or defective, according as the sum of these subdivisors was greater than, equal to, or less than the number: the classification at first being restricted to even numbers. The third group of problems which they considered dealt with numbers which formed a proportion; presumably these were discussed with the aid of geometry as is done in the fifth book of Euclid. Lastly, the Pythagoreans were concerned with series of numbers in arithmetical, geometrical, harmonical, and musical progressions. The three progressions first mentioned are well known; four integers are said to be in musical progression when they are in the ratio a : 2ab/(a+b) : 1/2(a+b) : b , for example, 6, 8, 9, and 12 are in musical progression.
Of the Pythagorean treatment of the applied subjects of the quadrivium, and the philosophical theories founded on them, we know very little. It would seem that Pythagoras was much impressed by certain numerical relations which occur in nature. It has been suggested that he was acquainted with some of the simpler facts of crystallography. It is thought that he was aware that the notes sounded by a vibrating string depend on the length of the string, and in particular that lengths which gave a note, its fifth and its octave were in the ratio 2:3:4 , forming terms in a musical progression. It would seem, too, that he believed that the distances of the astrological planets from the earth were also in musical progression, and that the heavenly bodies in their motion through space gave out harmonious sounds: hence the phrase the harmony of the spheres. These and similar conclusions seem to have suggested to him that the explanation of the order and harmony of the universe was to be found in the science of numbers, and that numbers are to some extent the cause of form as well as essential to its accurate measurement. He accordingly proceeded to attribute particular properties to particular numbers and geometrical figures. For example, he taught that the cause of colour was to be sought in properties of the number five, that the explanation of fire was to be discovered in the nature of the pyramid, and so on. I should not have alluded to this were it not that the Pythagorean tradition strengthened, or perhaps was chiefly responsible for the tendency of Greek writers to found the study of nature on philosophical conjectures and not on experimental observation--a tendency to which the defects of Hellenic science must be largely attributed.
After the death of Pythagoras his teaching seems to have been carried on by Epicharmus and Hippasus, and subsequently by Philolaus (specially distinguished as an astronomer), Archippus, and Lysis. About a century after the murder of Pythagoras we find Archytas recognised as the head of the school.
Archytas. Archytas, circ. 400 b.c., was one of the most influential citizens of Tarentum, and was made governor of the city no less than seven times. His influence among his contemporaries was very great, and he used it with Dionysius on one occasion to save the life of Plato. He was noted for the attention he paid to the comfort and education of his slaves and of children in the city. He was drowned in a shipwreck near Tarentum, and his body washed on shore--a fit punishment, in the eyes of the more rigid Pythagoreans, for his having departed from the lines of study laid down by their founder. Several of the leaders of the Athenian school were among his pupils and friends, and it is believed that much of their work was due to his inspiration.
The Pythagoreans at first made no attempt to apply their knowledge to mechanics, but Archytas is said to have treated it with the aid of geometry. He is alleged to have invented and worked out the theory of the pulley, and is credited with the construction of a flying bird and some other ingenious mechanical toys. He introduced various mechanical devices for constructing curves and solving problems. These were objected to by Plato, who thought that they destroyed the value of geometry as an intellectual exercise, and later Greek geometricians confined themselves to the use of two species of instruments, namely, rulers and compasses. Archytas was also interested in astronomy; he taught that the earth was a sphere rotating round its axis in twenty-four hours, and round which the heavenly bodies moved.
Archytas was one of the first to give a solution of the problem to duplicate a cube, that is, to find the side of a cube whose volume is double that of a given cube. This was one of the most famous problems of antiquity. The construction given by Archytas is equivalent to the following. On the diameter OA of the base of a right circular cylinder describe a semicircle whose plane is perpendicular to the base of the cylinder. Let the plane containing this semicircle rotate round the generator through O , then the surface traced out by the semicircle will cut the cylinder in a tortuous curve. This curve will be cut by a right cone whose axis is OA and semivertical angle is (say) 60° in a point P , such that the projection of OP on the base of the cylinder will be to the radius of the cylinder in the ratio of the side of the required cube to that of the given cube. The proof given by Archytas is of course geometrical; it will be enough here to remark that in the course of it he shews himself acquainted with the results of the propositions Euc. iii, 18, Euc. iii, 35, and Euc. xi, 19. To shew analytically that the construction is correct, take OA as the axis of x , and the generator through O as axis of z , then, with the usual notation in polar co-ordinates, and if a be the radius of the cylinder, we have for the equation of the surface described by the semicircle, r = 2asintheta ; for that of the cylinder, rsintheta = 2acosphi ; and for that of the cone, sinthetacosphi = 1/2 . These three surfaces cut in a point such that sin^3theta = 1/2 , and, therefore, if rho be the projection of OP on the base of the cylinder, then rho^3 = (rsintheta)^3 = 2a^3 . Hence the volume of the cube whose side is rho is twice that of a cube whose side is a . I mention the problem and give the construction used by Archytas to illustrate how considerable was the knowledge of the Pythagorean school at the time.
Theodorus. Another Pythagorean of about the same date as Archytas was Theodorus of Cyrene, who is said to have proved geometrically that the numbers represented by sqrt(3) , sqrt(5) , sqrt(6) , sqrt(7) , sqrt(8) , sqrt(10) , sqrt(11) , sqrt(12) , sqrt(13) , sqrt(14) , sqrt(15) , and sqrt(17) are incommensurable with unity. Theaetetus was one of his pupils.
Perhaps Timaeus of Locri and Bryso of Heraclea should be mentioned as other distinguished Pythagoreans of this time. It is believed that Bryso attempted to find the area of a circle by inscribing and circumscribing squares, and finally obtained polygons between whose areas the area of the circle lay; but it is said that at some point he assumed that the area of the circle was the arithmetic mean between an inscribed and a circumscribed polygon.
It would be a mistake to suppose that Miletus and Tarentum were the only places where, in the fifth century, Greeks were engaged in laying a scientific foundation for the study of mathematics. These towns represented the centres of chief activity, but there were few cities or colonies of any importance where lectures on philosophy and geometry were not given. Among these smaller schools I may mention those at Chios, Elea, and Thrace.
The best known philosopher of the School of Chios was Oenopides, who was born about 500 b.c., and died about 430 b.c. He devoted himself chiefly to astronomy, but he had studied geometry in Egypt, and is credited with the solution of two problems, namely, to draw a straight line from a given external point perpendicular to a given straight line (Euc. i, 12), and at a given point to construct an angle equal to a given angle (Euc. i, 23).
Another important centre was at Elea in Italy. This was founded in Sicily by Xenophanes. He was followed by Parmenides, Zeno, and Melissus. The members of the Eleatic School were famous for the difficulties they raised in connection with questions that required the use of infinite series, such, for example, as the well-known paradox of Achilles and the tortoise, enunciated by Zeno, one of their most prominent members. Zeno was born in 495 b.c., and was executed at Elea in 435 b.c. in consequence of some conspiracy against the state; he was a pupil of Parmenides, with whom he visited Athens, circ. 455--450 b.c.
Zeno argued that if Achilles ran ten times as fast as a tortoise, yet if the tortoise had (say) 1000 yards start it could never be overtaken: for, when Achilles had gone the 1000 yards, the tortoise would still be 100 yards in front of him; by the time he had covered these 100 yards, it would still be 10 yards in front of him; and so on for ever: thus Achilles would get nearer and nearer to the tortoise, but never overtake it. The fallacy is usually explained by the argument that the time required to overtake the tortoise, can be divided into an infinite number of parts, as stated in the question, but these get smaller and smaller in geometrical progression, and the sum of them all is a finite time: after the lapse of that time Achilles would be in front of the tortoise. Probably Zeno would have replied that this argument rests on the assumption that space is infinitely divisible, which is the question under discussion: he himself asserted that magnitudes are not infinitely divisible.
These paradoxes made the Greeks look with suspicion on the use of infinitesimals, and ultimately led to the invention of the method of exhaustions.
The Atomistic School, having its headquarters in Thrace, was another important centre. This was founded by Leucippus, who was a pupil of Zeno. He was succeeded by Democritus and Epicurus. Its most famous mathematician was Democritus, born at Abdera in 460 b.c., and said to have died in 370 b.c., who, besides philosophical works, wrote on plane and solid geometry, incommensurable lines, perspective, and numbers. These works are all lost. From the Archimedean MS., discovered by Heiberg in 1906, it would seem that Democritus enunciated, but without a proof, the proposition that the volume of a pyramid is equal to one-third that of a prism of an equal base and of equal height.
But though several distinguished individual philosophers may be mentioned who, during the fifth century, lectured at different cities, they mostly seem to have drawn their inspiration from Tarentum, and towards the end of the century to have looked to Athens as the intellectual capital of the Greek world; and it is to the Athenian schools that we owe the next great advance in mathematics.
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End of part 4 of 28
Continue readingChapter 3: The Schools of Athens and Cyzicus
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