A Short Account of the History of Mathematics · W. W. Rouse Ball
The history of the Alexandrian Schools is discussed by G. Loria in his Le Scienze Esatte nell' Antica Grecia, Modena, 1893--1900; by Cantor, chaps. xii--xxiii; and by J. Gow in his History of Greek Mathematics, Cambridge, 1884. The subject of Greek algebra is treated by E. H. F. Nesselmann in his Die Algebra der Griechen, Berlin, 1842; see also L. Matthiessen, Grundzüge der antiken und modernen Algebra der litteralen Gleichungen, Leipzig, 1878. The Greek treatment of the conic sections forms the subject of Die Lehre von den Kegelschnitten in Altertum, by H. G. Zeuthen, Copenhagen, 1886. The materials for the history of these schools have been subjected to a searching criticism by S. P. Tannery, and most of his papers are collected in his Géométrie Grecque, Paris, 1887.
The earliest attempt to found a university, as we understand the word, was made at Alexandria. Richly endowed, supplied with lecture rooms, libraries, museums, laboratories, and gardens, it became at once the intellectual metropolis of the Greek race, and remained so for a thousand years. It was particularly fortunate in producing within the first century of its existence three of the greatest mathematicians of antiquity--Euclid, Archimedes, and Apollonius. They laid down the lines on which mathematics subsequently developed, and treated it as a subject distinct from philosophy: hence the foundation of the Alexandrian Schools is rightly taken as the commencement of a new era. Thenceforward, until the destruction of the city by the Arabs in 641 a.d., the history of mathematics centres more or less round that of Alexandria; for this reason the Alexandrian Schools are commonly taken to include all Greek mathematicians of their time.
The city and university of Alexandria were created under the following circumstances. Alexander the Great had ascended the throne of Macedonia in 336 b.c. at the early age of twenty, and by 332 b.c. he had conquered or subdued Greece, Asia Minor, and Egypt. Following the plan he adopted whenever a commanding site had been left unoccupied, he founded a new city on the Mediterranean near one mouth of the Nile; and he himself sketched out the ground-plan, and arranged for drafts of Greeks, Egyptians, and Jews to be sent to occupy it. The city was intended to be the most magnificent in the world, and, the better to secure this, its erection was left in the hands of Dinocrates, the architect of the temple of Diana at Ephesus.
After Alexander's death in 323 b.c. his empire was divided, and Egypt fell to the lot of Ptolemy, who chose Alexandria as the capital of his kingdom. A short period of confusion followed, but as soon as Ptolemy was settled on the throne, say about 306 b.c., he determined to attract, so far as he was able, learned men of all sorts to his new city; and he at once began the erection of the university buildings on a piece of ground immediately adjoining his palace. The university was ready to be opened somewhere about 300 b.c., and Ptolemy, who wished to secure for its staff the most eminent philosophers of the time, naturally turned to Athens to find them. The great library which was the central feature of the scheme was placed under Demetrius Phalereus, a distinguished Athenian, and so rapidly did it grow that within forty years it (together with the Egyptian annexe) possessed about 600,000 rolls. The mathematical department was placed under Euclid, who was thus the first, as he was one of the most famous, of the mathematicians of the Alexandrian school.
It happens that contemporaneously with the foundation of this school the information on which our history is based becomes more ample and certain. Many of the works of the Alexandrian mathematicians are still extant; and we have besides an invaluable treatise by Pappus, described below, in which their best-known treatises are collated, discussed, and criticized. It curiously turns out that just as we begin to be able to speak with confidence on the subject-matter which was taught, we find that our information as to the personality of the teachers becomes vague; and we know very little of the lives of the mathematicians mentioned in this and the next chapter, even the dates at which they lived being frequently in doubt.
Euclid.
by A. De Morgan in Smith's Dictionary of Greek and Roman Biography, London, 1849; the article on Irrational Quantity by A. De Morgan in the Penny Cyclopaedia, London, 1839; Litterargeschichtliche Studien über Euklid, by J. L. Heiberg, Leipzig, 1882; and above all Euclid's Elements, translated with an introduction and commentary by T. L. Heath, 3 volumes, Cambridge, 1908. The latest complete edition of all Euclid's works is that by J. L. Heiberg and H. Menge, Leipzig, 1883--96. --This century produced three of the greatest mathematicians of antiquity, namely Euclid, Archimedes, and Apollonius. The earliest of these was Euclid. Of his life we know next to nothing, save that he was of Greek descent, and was born about 330 b.c.; he died about 275 b.c. It would appear that he was well acquainted with the Platonic geometry, but he does not seem to have read Aristotle's works; and these facts are supposed to strengthen the tradition that he was educated at Athens. Whatever may have been his previous training and career, he proved a most successful teacher when settled at Alexandria. He impressed his own individuality on the teaching of the new university to such an extent that to his successors and almost to his contemporaries the name Euclid meant (as it does to us) the book or books he wrote, and not the man himself. Some of the medieval writers went so far as to deny his existence, and with the ingenuity of philologists they explained that the term was only a corruption of >ukli a key, and dis geometry. The former word was presumably derived from kle'is. I can only explain the meaning assigned to dis by the conjecture that as the Pythagoreans said that the number two symbolized a line, possibly a schoolman may have thought that it could be taken as indicative of geometry.
From the meagre notices of Euclid which have come down to us we find that the saying that there is no royal road in geometry was attributed to Euclid as well as to Menaechmus; but it is an epigrammatic remark which has had many imitators. According to tradition, Euclid was noticeable for his gentleness and modesty. Of his teaching, an anecdote has been preserved. Stobaeus, who is a somewhat doubtful authority, tells us that, when a lad who had just begun geometry asked, "What do I gain by learning all this stuff?" Euclid insisted that knowledge was worth acquiring for its own sake, but made his slave give the boy some coppers, "since," said he, "he must make a profit out of what he learns."
Euclid was the author of several works, but his reputation rests mainly on his Elements. This treatise contains a systematic exposition of the leading propositions of elementary metrical geometry (exclusive of conic sections) and of the theory of numbers. It was at once adopted by the Greeks as the standard text-book on the elements of pure mathematics, and it is probable that it was written for that purpose and not as a philosophical attempt to shew that the results of geometry and arithmetic are necessary truths.
The modern text is founded on an edition or commentary prepared by Theon, the father of Hypatia (circ. 380 a.d.). There is at the Vatican a copy (circ. 1000 a.d.) of an older text, and we have besides quotations from the work and references to it by numerous writers of various dates. From these sources we gather that the definitions, axioms, and postulates were rearranged and slightly altered by subsequent editors, but that the propositions themselves are substantially as Euclid wrote them.
As to the matter of the work. The geometrical part is to a large extent a compilation from the works of previous writers. Thus the substance of books i and ii (except perhaps the treatment of parallels) is probably due to Pythagoras; that of book iii to Hippocrates; that of book v to Eudoxus; and the bulk of books iv, vi, xi, and xii to the later Pythagorean or Athenian schools. But this material was rearranged, obvious deductions were omitted (for instance, the proposition that the perpendiculars from the angular points of a triangle on the opposite sides meet in a point was cut out), and in some cases new proofs substituted. Book X, which deals with irrational magnitudes, may be founded on the lost book of Theaetetus; but probably much of it is original, for Proclus says that while Euclid arranged the propositions of Eudoxus he completed many of those of Theaetetus. The whole was presented as a complete and consistent body of theorems.
The form in which the propositions are presented, consisting of enunciation, statement, construction, proof, and conclusion, is due to Euclid: so also is the synthetical character of the work, each proof being written out as a logically correct train of reasoning but without any clue to the method by which it was obtained.
The defects of Euclid's Elements as a text-book of geometry have been often stated; the most prominent are these. (i) The definitions and axioms contain many assumptions which are not obvious, and in particular the postulate or axiom about parallel lines is not self-evident. (ii) No explanation is given as to the reason why the proofs take the form in which they are presented, that is, the synthetical proof is given but not the analysis by which it was obtained. (iii) There is no attempt made to generalize the results arrived at; for instance, the idea of an angle is never extended so as to cover the case where it is equal to or greater than two right angles: the second half of the thirty-third proposition in the sixth book, as now printed, appears to be an exception, but it is due to Theon and not to Euclid. (iv) The principle of superposition as a method of proof might be used more frequently with advantage. (v) The classification is imperfect. And (vi) the work is unnecessarily long and verbose. Some of those objections do not apply to certain of the recent school editions of the Elements.
On the other hand, the propositions in Euclid are arranged so as to form a chain of geometrical reasoning, proceeding from certain almost obvious assumptions by easy steps to results of considerable complexity. The demonstrations are rigorous, often elegant, and not too difficult for a beginner. Lastly, nearly all the elementary metrical (as opposed to the graphical) properties of space are investigated, while the fact that for two thousand years it was the usual text-book on the subject raises a strong presumption that it is not unsuitable for the purpose.
On the Continent rather more than a century ago, Euclid was generally superseded by other text-books. In England determined efforts have lately been made with the same purpose, and numerous other works on elementary geometry have been produced in the last decade. The change is too recent to enable us to say definitely what its effect may be. But as far as I can judge, boys who have learnt their geometry on the new system know more facts, but have missed the mental and logical training which was inseparable from a judicious study of Euclid's treatise.
I do not think that all the objections above stated can fairly be urged against Euclid himself. He published a collection of problems, generally known as the Dedom'ena or Data. This contains 95 illustrations of the kind of deductions which frequently have to be made in analysis; such as that, if one of the data of the problem under consideration be that one angle of some triangle in the figure is constant, then it is legitimate to conclude that the ratio of the area of the rectangle under the sides containing the angle to the area of the triangle is known [prop. 66]. Pappus says that the work was written for those "who wish to acquire the power of solving problems." It is in fact a gradual series of exercises in geometrical analysis. In short the Elements gave the principal results, and were intended to serve as a training in the science of reasoning, while the Data were intended to develop originality.
Euclid also wrote a work called Per'i Diair'esewn or De Divisionibus, known to us only through an Arabic translation which may be itself imperfect. This is a collection of 36 problems on the division of areas into parts which bear to one another a given ratio. It is not unlikely that this was only one of several such collections of examples--possibly including the Fallacies and the Porisms--but even by itself it shews that the value of exercises and riders was fully recognized by Euclid.
I may here add a suggestion made by De Morgan, whose comments on Euclid's writings were notably ingenious and informing. From internal evidence he thought it likely that the Elements were written towards the close of Euclid's life, and that their present form represents the first draft of the proposed work, which, with the exception of the tenth book, Euclid did not live to revise. This opinion is generally discredited, and there is no extrinsic evidence to support it.
The geometrical parts of the Elements are so well known that I need do no more than allude to them. Euclid admitted only those constructions which could be made by the use of a ruler and compasses. He also excluded practical work and hypothetical constructions. The first four books and book vi deal with plane geometry; the theory of proportion (of any magnitudes) is discussed in book v; and books xi and xii treat of solid geometry. On the hypothesis that the Elements are the first draft of Euclid's proposed work, it is possible that book xiii is a sort of appendix containing some additional propositions which would have been put ultimately in one or other of the earlier books. Thus, as mentioned above, the first five propositions which deal with a line cut in golden section might be added to the second book. The next seven propositions are concerned with the relations between certain incommensurable lines in plane figures (such as the radius of a circle and the sides of an inscribed regular triangle, pentagon, hexagon, and decagon) which are treated by the methods of the tenth book and as an illustration of them. Constructions of the five regular solids are discussed in the last six propositions, and it seems probable that Euclid and his contemporaries attached great importance to this group of problems. Bretschneider inclined to think that the thirteenth book is a summary of part of the lost work of Aristaeus: but the illustrations of the methods of the tenth book are due most probably to Theaetetus.
Books vii, viii, ix, and x of the Elements are given up to the theory of numbers. The mere art of calculation or logistik'h was taught to boys when quite young, it was stigmatized by Plato as childish, and never received much attention from Greek mathematicians; nor was it regarded as forming part of a course of mathematics. We do not know how it was taught, but the abacus certainly played a prominent part in it. The scientific treatment of numbers was called >arijmhtik'h, which I have here generally translated as the science of numbers. It had special reference to ratio, proportion, and the theory of numbers. It is with this alone that most of the extant Greek works deal.
In discussing Euclid's arrangement of the subject, we must therefore bear in mind that those who attended his lectures were already familiar with the art of calculation. The system of numeration adopted by the Greeks is described later, but it was so clumsy that it rendered the scientific treatment of numbers much more difficult than that of geometry; hence Euclid commenced his mathematical course with plane geometry. At the same time it must be observed that the results of the second book, though geometrical in form, are capable of expression in algebraical language, and the fact that numbers could be represented by lines was probably insisted on at an early stage, and illustrated by concrete examples. This graphical method of using lines to represent numbers possesses the obvious advantage of leading to proofs which are true for all numbers, rational or irrational. It will be noticed that among other propositions in the second book we get geometrical proofs of the distributive and commutative laws, of rules for multiplication, and finally geometrical solutions of the equations a(a-x)=x^2 , that is x^2+ax-a^2=0 (Euc. ii, 11), and x^2-ab=0 (Euc. ii, 14): the solution of the first of these equations is given in the form sqrtsmash[b]a^2+(1/2a)^2-1/2a . The solutions of the equations ax^2-bx+c=0 and ax^2+bx-c=0 are given later in Euc. vi, 28 and vi, 29; the cases when a=1 can be deduced from the identities proved in Euc. ii, 5 and 6, but it is doubtful if Euclid recognized this.
The results of the fifth book, in which the theory of proportion is considered, apply to any magnitudes, and therefore are true of numbers as well as of geometrical magnitudes. In the opinion of many writers this is the most satisfactory way of treating the theory of proportion on a scientific basis; and it was used by Euclid as the foundation on which he built the theory of numbers. The theory of proportion given in this book is believed to be due to Eudoxus. The treatment of the same subject in the seventh book is less elegant, and is supposed to be a reproduction of the Pythagorean teaching. This double discussion of proportion is, as far as it goes, in favour of the conjecture that Euclid did not live to revise the work.
In books vii, viii, and ix Euclid discusses the theory of rational numbers. He commences the seventh book with some definitions founded on the Pythagorean notation. In propositions 1 to 3 he shews that if, in the usual process for finding the greatest common measure of two numbers, the last divisor be unity, the numbers must be prime; and he thence deduces the rule for finding their G.C.M. Propositions 4 to 22 include the theory of fractions, which he bases on the theory of proportion; among other results he shews that ab = ba [prop. 16]. In propositions 23 to 34 he treats of prime numbers, giving many of the theorems in modern text-books on algebra. In propositions 35 to 41 he discusses the least common multiple of numbers, and some miscellaneous problems.
The eighth book is chiefly devoted to numbers in continued proportion, that is, in a geometrical progression; and the cases where one or more is a product, square, or cube are specially considered.
In the ninth book Euclid continues the discussion of geometrical progressions, and in proposition 35 he enunciates the rule for the summation of a series of n terms, though the proof is given only for the case where n is equal to 4. He also develops the theory of primes, shews that the number of primes is infinite [prop. 20], and discusses the properties of odd and even numbers. He concludes by shewing that a number of the form 2^n-1 (2^n-1) , where 2^n-1 is a prime, is a "perfect" number [prop. 36].
In the tenth book Euclid deals with certain irrational magnitudes; and, since the Greeks possessed no symbolism for surds, he was forced to adopt a geometrical representation. Propositions 1 to 21 deal generally with incommensurable magnitudes. The rest of the book, namely, propositions 22 to 117, is devoted to the discussion of every possible variety of lines which can be represented by sqrt(sqrta ± sqrtb) , where a and b denote commensurable lines. There are twenty-five species of such lines, and that Euclid could detect and classify them all is in the opinion of so competent an authority as Nesselmann the most striking illustration of his genius. No further advance in the theory of incommensurable magnitudes was made until the subject was taken up by Leonardo and Cardan after an interval of more than a thousand years.
In the last proposition of the tenth book [prop. 117] the side and diagonal of a square are proved to be incommensurable. The proof is so short and easy that I may quote it. If possible let the side be to the diagonal in a commensurable ratio, namely, that of two integers, a and b . Suppose this ratio reduced to its lowest terms so that a and b have no common divisor other than unity, that is, they are prime to one another. Then (by Euc. i, 47) b^2 = 2a^2 ; therefore b^2 is an even number; therefore b is an even number; hence, since a is prime to b , a must be an odd number. Again, since it has been shewn that b is an even number, b may be represented by 2n ; therefore (2n)^2 = 2a^2 ; therefore a^2 = 2n^2 ; therefore a^2 is an even number; therefore a is an even number. Thus the same number a must be both odd and even, which is absurd; therefore the side and diagonal are incommensurable. Hankel believes that this proof was due to Pythagoras, and this is not unlikely. This proposition is also proved in another way in Euc. x, 9, and for this and other reasons it is now usually believed to be an interpolation by some commentator on the Elements.
In addition to the Elements and the two collections of riders above mentioned (which are extant) Euclid wrote the following books on geometry: (i) an elementary treatise on conic sections in four books; (ii) a book on surface loci, probably confined to curves on the cone and cylinder; (iii) a collection of geometrical fallacies, which were to be used as exercises in the detection of errors; and (iv) a treatise on porisms arranged in three books. All of these are lost, but the work on porisms was discussed at such length by Pappus, that some writers have thought it possible to restore it. In particular, Chasles in 1860 published what he considered to be substantially a reproduction of it. In this will be found the conceptions of cross ratios and projection, and those ideas of modern geometry which were used so extensively by Chasles and other writers of the nineteenth century. It should be realized, however, that the statements of the classical writers concerning this book are either very brief or have come to us only in a mutilated form, and De Morgan frankly says that he found them unintelligible, an opinion in which most of those who read them will, I think, concur.
Euclid published a book on optics, treated geometrically, which contains 61 propositions founded on 12 assumptions. It commences with the assumption that objects are seen by rays emitted from the eye in straight lines, "for if light proceeded from the object we should not, as we often do, fail to perceive a needle on the floor." A work called Catoptrica is also attributed to him by some of the older writers; the text is corrupt and the authorship doubtful; it consists of 31 propositions dealing with reflexions in plane, convex, and concave mirrors. The geometry of both books is Euclidean in form.
Euclid has been credited with an ingenious demonstration of the principle of the lever, but its authenticity is doubtful. He also wrote the Phaenomena, a treatise on geometrical astronomy. It contains references to the work of Autolycus and to some book on spherical geometry by an unknown writer. Pappus asserts that Euclid also composed a book on the elements of music: this may refer to the Sectio Canonis, which is by Euclid, and deals with musical intervals.
To these works I may add the following little problem, which occurs in the Palatine Anthology and is attributed by tradition to Euclid. "A mule and a donkey were going to market laden with wheat. The mule said, 'If you gave me one measure I should carry twice as much as you, but if I gave you one we should bear equal burdens.' Tell me, learned geometrician, what were their burdens." It is impossible to say whether the question is due to Euclid, but there is nothing improbable in the suggestion.
It will be noticed that Euclid dealt only with magnitudes, and did not concern himself with their numerical measures, but it would seem from the works of Aristarchus and Archimedes that this was not the case with all the Greek mathematicians of that time. As one of the works of the former is extant it will serve as another illustration of Greek mathematics of this period.
Aristarchus. Aristarchus of Samos, born in 310 b.c. and died in 250 b.c., was an astronomer rather than a mathematician. He asserted, at any rate as a working hypothesis, that the sun was the centre of the universe, and that the earth revolved round the sun. This view, in spite of the simple explanation it afforded of various phenomena, was generally rejected by his contemporaries. But his propositions on the measurement of the sizes and distances of the sun and moon were accurate in principle, and his results were accepted by Archimedes in his Yamm'iths, mentioned below, as approximately correct. There are 19 theorems, of which I select the seventh as a typical illustration, because it shews the way in which the Greeks evaded the difficulty of finding the numerical value of surds.
Aristarchus observed the angular distance between the moon when dichotomized and the sun, and found it to be twenty-nine thirtieths of a right angle. It is actually about 89° 21' , but of course his instruments were of the roughest description. He then proceeded to shew that the distance of the sun is greater than eighteen and less than twenty times the distance of the moon in the following manner.
Let S be the sun, E the earth, and M the moon. Then when the moon is dichotomized, that is, when the bright part which we see is exactly a half-circle, the angle between MS and ME is a right angle. With E as centre, and radii ES and EM describe circles, as in the figure below. Draw EA perpendicular to ES . Draw EF bisecting the angle AES , and EG bisecting the angle AEF , as in the figure. Let EM (produced) cut AF in H . The angle AEM is by hypothesis 1/30 th of a right angle. Hence we have
angle AEG : angle AEH = 14 rt. : 130 rt. = 15 : 2, AG : AH [= AEG : AEH] > 15 : 2 alpha .
Again FG^2 : AG^2 = EF^2 : EA^2 (Euc. vi, 3) = 2 : 1 (Euc. i, 47),
FG^2 : AG^2 & > 49 : 25, FG : AG & > 7 : 5, AF : AG & > 12 : 5, AE : AG & > 12 : 5. beta
Compounding the ratios ( alpha ) and ( beta ), we have AE : AH > 18 : 1. But the triangles EMS and EAH are similar, ES : EM > 18 : 1.
I will leave the second half of the proposition to amuse any reader who may care to prove it: the analysis is straightforward. In a somewhat similar way Aristarchus found the ratio of the radii of the sun, earth, and moon.
We know very little of Conon and Dositheus, the immediate successors of Euclid at Alexandria, or of their contemporaries Zeuxippus and Nicoteles, who most likely also lectured there, except that Archimedes, who was a student at Alexandria probably shortly after Euclid's death, had a high opinion of their ability and corresponded with the three first mentioned. Their work and reputation has been completely overshadowed by that of Archimedes.
Archimedes. Archimedes, who probably was related to the royal family at Syracuse, was born there in 287 b.c. and died in 212 b.c. He went to the university of Alexandria and attended the lectures of Conon, but, as soon as he had finished his studies, returned to Sicily where he passed the remainder of his life. He took no part in public affairs, but his mechanical ingenuity was astonishing, and, on any difficulties which could be overcome by material means arising, his advice was generally asked by the government.
Archimedes, like Plato, held that it was undesirable for a philosopher to seek to apply the results of science to any practical use; but in fact he did introduce a large number of new inventions. The stories of the detection of the fraudulent goldsmith and of the use of burning-glasses to destroy the ships of the Roman blockading squadron will recur to most readers. Perhaps it is not as well known that Hiero, who had built a ship so large that he could not launch it off the slips, applied to Archimedes. The difficulty was overcome by means of an apparatus of cogwheels worked by an endless screw, but we are not told exactly how the machine was used. It is said that it was on this occasion, in acknowledging the compliments of Hiero, that Archimedes made the well-known remark that had he but a fixed fulcrum he could move the earth.
Most mathematicians are aware that the Archimedean screw was another of his inventions. It consists of a tube, open at both ends, and bent into the form of a spiral like a corkscrew. If one end be immersed in water, and the axis of the instrument (i.e. the axis of the cylinder on the surface of which the tube lies) be inclined to the vertical at a sufficiently big angle, and the instrument turned round it, the water will flow along the tube and out at the other end. In order that it may work, the inclination of the axis of the instrument to the vertical must be greater than the pitch of the screw. It was used in Egypt to drain the fields after an inundation of the Nile, and was also frequently applied to take water out of the hold of a ship.
The story that Archimedes set fire to the Roman ships by means of burning-glasses and concave mirrors is not mentioned till some centuries after his death, and is generally rejected. The mirror of Archimedes is said to have been made in the form of a hexagon surrounded by rings of polygons; and Buffon in 1747 contrived, by the use of a single composite mirror made on this model, to set fire to wood at a distance of 150 feet, and to melt lead at a distance of 140 feet. This was in April and as far north as Paris, so in a Sicilian summer the use of several such mirrors might be a serious annoyance to a blockading fleet, if the ships were sufficiently near. It is perhaps worth mentioning that a similar device is said to have been used in the defence of Constantinople in 514 a.d., and is alluded to by writers who either were present at the siege or obtained their information from those who were engaged in it.
But whatever be the truth as to this story, there is no doubt that Archimedes devised the catapults which kept the Romans, who were then besieging Syracuse, at bay for a considerable time. These were constructed so that the range could be made either short or long at pleasure, and so that they could be discharged through a small loophole without exposing the artillery-men to the fire of the enemy. So effective did they prove that the siege was turned into a blockade, and three years elapsed before the town was taken.
Archimedes was killed during the sack of the city which followed its capture, in spite of the orders, given by the consul Marcellus who was in command of the Romans, that his house and life should be spared. It is said that a soldier entered his study while he was regarding a geometrical diagram drawn in sand on the floor, which was the usual way of drawing figures in classical times. Archimedes told him to get off the diagram, and not spoil it. The soldier, feeling insulted at having orders given to him and ignorant of who the old man was, killed him. According to another and more probable account, the cupidity of the troops was excited by seeing his instruments, constructed of polished brass which they supposed to be made of gold.
The Romans erected a splendid tomb to Archimedes, on which was engraved (in accordance with a wish he had expressed) the figure of a sphere inscribed in a cylinder, in commemoration of the proof he had given that the volume of a sphere was equal to two-thirds that of the circumscribing right cylinder, and its surface to four times the area of a great circle. Cicero gives a charming account of his efforts (which were successful) to rediscover the tomb in 75 b.c.
It is difficult to explain in a concise form the works or discoveries of Archimedes, partly because he wrote on nearly all the mathematical subjects then known, and partly because his writings are contained in a series of disconnected monographs. Thus, while Euclid aimed at producing systematic treatises which could be understood by all students who had attained a certain level of education, Archimedes wrote a number of brilliant essays addressed chiefly to the most educated mathematicians of the day. The work for which he is perhaps now best known is his treatment of the mechanics of solids and fluids; but he and his contemporaries esteemed his geometrical discoveries of the quadrature of a parabolic area and of a spherical surface, and his rule for finding the volume of a sphere as more remarkable; while at a somewhat later time his numerous mechanical inventions excited most attention.
(i) On plane geometry the extant works of Archimedes are three in number, namely, (a) the Measure of the Circle, (b) the Quadrature of the Parabola, and (c) one on Spirals.
(a) The Measure of the Circle contains three propositions. In the first proposition Archimedes proves that the area is the same as that of a right-angled triangle whose sides are equal respectively to the radius a and the circumference of the circle, i.e. the area is equal to 1/2a (2pi a) . In the second proposition he shows that pi a^2 : (2a)^2 = 11 : 14 very nearly; and next, in the third proposition, that pi is less than 31/7 and greater than 310/71 . These theorems are of course proved geometrically. To demonstrate the two latter propositions, he inscribes in and circumscribes about a circle regular polygons of ninety-six sides, calculates their perimeters, and then assumes the circumference of the circle to lie between them: this leads to the result 6336 / 20171/4 < pi < 14688 / 46731/2 , from which he deduces the limits given above. It would seem from the proof that he had some (at present unknown) method of extracting the square roots of numbers approximately. The table which he formed of the numerical values of the chords of a circle is essentially a table of natural sines, and may have suggested the subsequent work on these lines of Hipparchus and Ptolemy.
(b) The Quadrature of the Parabola contains twenty-four propositions. Archimedes begins this work, which was sent to Dositheus, by establishing some properties of conics [props. 1--5]. He then states correctly the area cut off from a parabola by any chord, and gives a proof which rests on a preliminary mechanical experiment of the ratio of areas which balance when suspended from the arms of a lever [props. 6--17]; and, lastly, he gives a geometrical demonstration of this result [props. 18--24]. The latter is, of course, based on the method of exhaustions, but for brevity I will, in quoting it, use the method of limits.
Let the area of the parabola (see figure above) be bounded by the chord PQ . Draw VM the diameter to the chord PQ , then (by a previous proposition), V is more remote from PQ than any other point in the arc PVQ . Let the area of the triangle PVQ be denoted by triangle . In the segments bounded by VP and VQ inscribe triangles in the same way as the triangle PVQ was inscribed in the given segment. Each of these triangles is (by a previous proposition of his) equal to 1/8triangle , and their sum is therefore 1/4triangle . Similarly in the four segments left inscribe triangles; their sum will be 1/16triangle . Proceeding in this way the area of the given segment is shown to be equal to the limit of + 4 + 16 + + 4^n + , when n is indefinitely large.
The problem is therefore reduced to finding the sum of a geometrical series. This he effects as follows. Let A, B, C, ..., J, K be a series of magnitudes such that each is one-fourth of that which precedes it. Take magnitudes b, c, ..., k equal respectively to 1/3B, 1/3C, ..., 1/3K . Then B + b = 13A, C + c = 13B, , K + k = 13J. Hence (B + C + ... + K) + (b + c + ... + k) = 1/3(A + B + ... + J); but, by hypothesis, (b + c + ... + j + k) = 1/3(B + C + ... +J) + 1/3K; (B + C + + K) + 13K = 13A. A + B + C + + K = 43A - 13K. Hence the sum of these magnitudes exceeds four times the third of the largest of them by one-third of the smallest of them.
Returning now to the problem of the quadrature of the parabola A stands for Delta , and ultimately K is indefinitely small; therefore the area of the parabolic segment is four-thirds that of the triangle PVQ , or two-thirds that of a rectangle whose base is PQ and altitude the distance of V from PQ .
While discussing the question of quadratures it may be added that in the fifth and sixth propositions of his work on conoids and spheroids he determined the area of an ellipse.
(c) The work on Spirals contains twenty-eight propositions on the properties of the curve now known as the spiral of Archimedes. It was sent to Dositheus at Alexandria accompanied by a letter, from which it appears that Archimedes had previously sent a note of his results to Conon, who had died before he had been able to prove them. The spiral is defined by saying that the vectorial angle and radius vector both increase uniformly, hence its equation is r = ctheta . Archimedes finds most of its properties, and determines the area inclosed between the curve and two radii vectores. This he does (in effect) by saying, in the language of the infinitesimal calculus, that an element of area is > 1/2r^2 dtheta and < 1/2(r + dr)^2 dtheta : to effect the sum of the elementary areas he gives two lemmas in which he sums (geometrically) the series a^2 + (2a)^2 + (3a)^2 + ... + (na)^2 [prop. 10], and a + 2a + 3a + ... + na [prop. 11].
(d) In addition to these he wrote a small treatise on geometrical methods, and works on parallel lines, triangles, the properties of right-angled triangles, data, the heptagon inscribed in a circle, and systems of circles touching one another; possibly he wrote others too. These are all lost, but it is probable that fragments of four of the propositions in the last-mentioned work are preserved in a Latin translation from an Arabic manuscript entitled Lemmas of Archimedes.
(ii) On geometry of three dimensions the extant works of Archimedes are two in number, namely (a), the Sphere and Cylinder, and (b) Conoids and Spheroids.
(a) The Sphere and Cylinder contains sixty propositions arranged in two books. Archimedes sent this like so many of his works to Dositheus at Alexandria; but he seems to have played a practical joke on his friends there, for he purposely misstated some of his results "to deceive those vain geometricians who say they have found everything, but never give their proofs, and sometimes claim that they have discovered what is impossible." He regarded this work as his masterpiece. It is too long for me to give an analysis of its contents, but I remark in passing that in it he finds expressions for the surface and volume of a pyramid, of a cone, and of a sphere, as well as of the figures produced by the revolution of polygons inscribed in a circle about a diameter of the circle. There are several other propositions on areas and volumes of which perhaps the most striking is the tenth proposition of the second book, namely, that "of all spherical segments whose surfaces are equal the hemisphere has the greatest volume." In the second proposition of the second book he enunciates the remarkable theorem that a line of length a can be divided so that a - x : b = 4a^2 : 9x^2 (where b is a given length), only if b be less than 1/3a ; that is to say, the cubic equation x^3 - ax^2 + 4/9a^2b = 0 can have a real and positive root only if a be greater than 3b . This proposition was required to complete his solution of the problem to divide a given sphere by a plane so that the volumes of the segments should be in a given ratio. One very simple cubic equation occurs in the Arithmetic of Diophantus, but with that exception no such equation appears again in the history of European mathematics for more than a thousand years.
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End of part 6 of 28
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