A Short Account of the History of Mathematics · W. W. Rouse Ball
(b) The Conoids and Spheroids contains forty propositions on quadrics of revolution (sent to Dositheus in Alexandria) mostly concerned with an investigation of their volumes.
(c) Archimedes also wrote a treatise on certain semi-regular polyhedrons, that is, solids contained by regular but dissimilar polygons. This is lost, but references to it are given by Pappus.
(iii) On arithmetic Archimedes wrote two papers. One (addressed to Zeuxippus) was on the principles of numeration; this is now lost. The other (addressed to Gelon) was called Yamm'iths (the sand-reckoner), and in this he meets an objection which had been urged against his first paper.
The object of the first paper had been to suggest a convenient system by which numbers of any magnitude could be represented; and it would seem that some philosophers at Syracuse had doubted whether the system was practicable. Archimedes says people talk of the sand on the Sicilian shore as something beyond the power of calculation, but he can estimate it; and, further, he will illustrate the power of his method by finding a superior limit to the number of grains of sand which would fill the whole universe, i.e. a sphere whose centre is the earth, and radius the distance of the sun. He begins by saying that in ordinary Greek nomenclature it was only possible to express numbers from 1 up to 10^8 : these are expressed in what he says he may call units of the first order. If 10^8 be termed a unit of the second order, any number from 10^8 to 10^16 can be expressed as so many units of the second order plus so many units of the first order. If 10^16 be a unit of the third order any number up to 10^24 can be then expressed, and so on. Assuming that 10,000 grains of sand occupy a sphere whose radius is not less than 1/80 th of a finger-breadth, and that the diameter of the universe is not greater than 10^10 stadia, he finds that the number of grains of sand required to fill the solar universe is less than 10^51 .
Probably this system of numeration was suggested merely as a scientific curiosity. The Greek system of numeration with which we are acquainted had been only recently introduced, most likely at Alexandria, and was sufficient for all the purposes for which the Greeks then required numbers; and Archimedes used that system in all his papers. On the other hand, it has been conjectured that Archimedes and Apollonius had some symbolism based on the decimal system for their own investigations, and it is possible that it was the one here sketched out. The units suggested by Archimedes form a geometrical progression, having 10^8 for the radix. He incidentally adds that it will be convenient to remember that the product of the m th and n th terms of a geometrical progression, whose first term is unity, is equal to the (m + n) th term of the series, that is, that r^m × r^n = r^m+n .
To these two arithmetical papers I may add the following celebrated problem which he sent to the Alexandrian mathematicians. The sun had a herd of bulls and cows, all of which were either white, grey, dun, or piebald: the number of piebald bulls was less than the number of white bulls by 5/6 ths of the number of grey bulls, it was less than the number of grey bulls by 9/20 ths of the number of dun bulls, and it was less than the number of dun bulls by 13/42 nds of the number of white bulls; the number of white cows was 7/12 ths of the number of grey cattle (bulls and cows), the number of grey cows was 9/20 ths of the number of dun cattle, the number of dun cows was 11/30 ths of the number of piebald cattle, and the number of piebald cows was 13/42 nds of the number of white cattle. The problem was to find the composition of the herd. The problem is indeterminate, but the solution in lowest integers is
white bulls, 10,366,482; white cows, 7,206,360;
grey bulls, 7,460,514; grey cows, 4,893,246;
dun bulls, 7,358,060; dun cows, 3,515,820;
piebald bulls, 4,149,387; piebald cows, 5,439,213.
In the classical solution, attributed to Archimedes, these numbers are multiplied by 80.
Nesselmann believes, from internal evidence, that the problem has been falsely attributed to Archimedes. It certainly is unlike his extant work, but it was attributed to him among the ancients, and is generally thought to be genuine, though possibly it has come down to us in a modified form. It is in verse, and a later copyist has added the additional conditions that the sum of the white and grey bulls shall be a square number, and the sum of the piebald and dun bulls a triangular number.
It is perhaps worthy of note that in the enunciation the fractions are represented as a sum of fractions whose numerators are unity: thus Archimedes wrote 1/7 + 1/6 instead of 13/42 , in the same way as Ahmes would have done.
(iv) On mechanics the extant works of Archimedes are two in number, namely, (a) his Mechanics, and (c) his Hydrostatics.
(a) The Mechanics is a work on statics with special reference to the equilibrium of plane laminas and to properties of their centres of gravity; it consists of twenty-five propositions in two books. In the first part of book i, most of the elementary properties of the centre of gravity are proved [props. 1--8]; and in the remainder of book i, [props. 9--15] and in book ii the centres of gravity of a variety of plane areas, such as parallelograms, triangles, trapeziums, and parabolic areas are determined.
As an illustration of the influence of Archimedes on the history of mathematics, I may mention that the science of statics rested on his theory of the lever until 1586, when Stevinus published his treatise on statics.
His reasoning is sufficiently illustrated by an outline of his proof for the case of two weights, P and Q , placed at their centres of gravity, A and B , on a weightless bar AB . He wants to shew that the centre of gravity of P and Q is at a point O on the bar such that P.OA = Q.OB .
On the line AB (produced if necessary) take points H and K , so that HB = BK = AO ; and a point L so that LA = OB . It follows that LH will be bisected at A , HK at B , and LK at O ; also LH : HK = AH : HB = OB : AO = P : Q . Hence, by a previous proposition, we may consider that the effect of P is the same as that of a heavy uniform bar LH of weight P , and the effect of Q is the same as that of a similar heavy uniform bar HK of weight Q . Hence the effect of the weights is the same as that of a heavy uniform bar LK . But the centre of gravity of such a bar is at its middle point O .
(b) Archimedes also wrote a treatise on levers and perhaps, on all the mechanical machines. The book is lost, but we know from Pappus that it contained a discussion of how a given weight could be moved with a given power. It was in this work probably that Archimedes discussed the theory of a certain compound pulley consisting of three or more simple pulleys which he had invented, and which was used in some public works in Syracuse. It is well known that he boasted that, if he had but a fixed fulcrum, he could move the whole earth; and a commentator of later date says that he added he would do it by using a compound pulley.
(c) His work on floating bodies contains nineteen propositions in two books, and was the first attempt to apply mathematical reasoning to hydrostatics. The story of the manner in which his attention was directed to the subject is told by Vitruvius. Hiero, the king of Syracuse, had given some gold to a goldsmith to make into a crown. The crown was delivered, made up, and of the proper weight, but it was suspected that the workman had appropriated some of the gold, replacing it by an equal weight of silver. Archimedes was thereupon consulted. Shortly afterwards, when in the public baths, he noticed that his body was pressed upwards by a force which increased the more completely he was immersed in the water. Recognising the value of the observation, he rushed out, just as he was, and ran home through the streets, shouting e<'urhka, e<'urhka, "I have found it, I have found it." There (to follow a later account) on making accurate experiments he found that when equal weights of gold and silver were weighed in water they no longer appeared equal: each seemed lighter than before by the weight of the water it displaced, and as the silver was more bulky than the gold its weight was more diminished. Hence, if on a balance he weighed the crown against an equal weight of gold and then immersed the whole in water, the gold would outweigh the crown if any silver had been used in its construction. Tradition says that the goldsmith was found to be fraudulent.
Archimedes began the work by proving that the surface of a fluid at rest is spherical, the centre of the sphere being at the centre of the earth. He then proved that the pressure of the fluid on a body, wholly or partially immersed, is equal to the weight of the fluid displaced; and thence found the position of equilibrium of a floating body, which he illustrated by spherical segments and paraboloids of revolution floating on a fluid. Some of the latter problems involve geometrical reasoning of considerable complexity.
The following is a fair specimen of the questions considered. A solid in the shape of a paraboloid of revolution of height h and latus rectum 4a floats in water, with its vertex immersed and its base wholly above the surface. If equilibrium be possible when the axis is not vertical, then the density of the body must be less than (h-3a)^2/h^3 [book ii, prop. 4]. When it is recollected that Archimedes was unacquainted with trigonometry or analytical geometry, the fact that he could discover and prove a proposition such as that just quoted will serve as an illustration of his powers of analysis.
It will be noticed that the mechanical investigations of Archimedes were concerned with statics. It may be added that though the Greeks attacked a few problems in dynamics, they did it with but indifferent success: some of their remarks were acute, but they did not sufficiently realise that the fundamental facts on which the theory must be based can be established only by carefully devised observations and experiments. It was not until the time of Galileo and Newton that this was done.
(v) We know, both from occasional references in his works and from remarks by other writers, that Archimedes was largely occupied in astronomical observations. He wrote a book, Per'i Sfeiropoi'ias, on the construction of a celestial sphere, which is lost; and he constructed a sphere of the stars, and an orrery. These, after the capture of Syracuse, were taken by Marcellus to Rome, and were preserved as curiosities for at least two or three hundred years.
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This mere catalogue of his works will show how wonderful were his achievements; but no one who has not actually read some of his writings can form a just appreciation of his extraordinary ability. This will be still further increased if we recollect that the only principles used by Archimedes, in addition to those contained in Euclid's Elements and Conic sections, are that of all lines like ACB , ADB , ... connecting two points A and B , the straight line is the shortest, and of the curved lines, the inner one ADB is shorter than the outer one ACB ; together with two similar statements for space of three dimensions.
In the old and medieval world Archimedes was reckoned as the first of mathematicians, but possibly the best tribute to his fame is the fact that those writers who have spoken most highly of his work and ability are those who have been themselves the most distinguished men of their own generation.
Apollonius. The third great mathematician of this century was Apollonius of Perga, who is chiefly celebrated for having produced a systematic treatise on the conic sections which not only included all that was previously known about them, but immensely extended the knowledge of these curves. This work was accepted at once as the standard text-book on the subject, and completely superseded the previous treatises of Menaechmus, Aristaeus, and Euclid which until that time had been in general use.
We know very little of Apollonius himself. He was born about 260 b.c., and died about 200 b.c. He studied in Alexandria for many years, and probably lectured there; he is represented by Pappus as "vain, jealous of the reputation of others, and ready to seize every opportunity to depreciate them." It is curious that while we know next to nothing of his life, or of that of his contemporary Eratosthenes, yet their nicknames, which were respectively epsilon and beta, have come down to us. Dr. Gow has ingeniously suggested that the lecture rooms at Alexandria were numbered, and that they always used the rooms numbered 5 and 2 respectively.
Apollonius spent some years at Pergamum in Pamphylia, where a university had been recently established and endowed in imitation of that at Alexandria. There he met Eudemus and Attalus, to whom he subsequently sent each book of his conics as it came out with an explanatory note. He returned to Alexandria, and lived there till his death, which was nearly contemporaneous with that of Archimedes.
In his great work on conic sections Apollonius so thoroughly investigated the properties of these curves that he left but little for his successors to add. But his proofs are long and involved, and I think most readers will be content to accept a short analysis of his work, and the assurance that his demonstrations are valid. Dr. Zeuthen believes that many of the properties enunciated were obtained in the first instance by the use of co-ordinate geometry, and that the demonstrations were translated subsequently into geometrical form. If this be so, we must suppose that the classical writers were familiar with some branches of analytical geometry--Dr. Zeuthen says the use of orthogonal and oblique co-ordinates, and of transformations depending on abridged notation--that this knowledge was confined to a limited school, and was finally lost. This is a mere conjecture and is unsupported by any direct evidence, but it has been accepted by some writers as affording an explanation of the extent and arrangement of the work.
The treatise contained about four hundred propositions, and was divided into eight books; we have the Greek text of the first four of these, and we also possess copies of the commentaries by Pappus and Eutocius on the whole work. In the ninth century an Arabic translation was made of the first seven books, which were the only ones then extant; we have two manuscripts of this version. The eighth book is lost.
In the letter to Eudemus which accompanied the first book Apollonius says that he undertook the work at the request of Naucrates, a geometrician who had been staying with him at Alexandria, and, though he had given some of his friends a rough draft of it, he had preferred to revise it carefully before sending it to Pergamum. In the note which accompanied the next book, he asks Eudemus to read it and communicate it to others who can understand it, and in particular to Philonides, a certain geometrician whom the author had met at Ephesus.
The first four books deal with the elements of the subject, and of these the first three are founded on Euclid's previous work (which was itself based on the earlier treatises by Menaechmus and Aristaeus). Heracleides asserts that much of the matter in these books was stolen from an unpublished work of Archimedes, but a critical examination by Heiberg has shown that this is improbable.
Apollonius begins by defining a cone on a circular base. He then investigates the different plane sections of it, and shows that they are divisible into three kinds of curves which he calls ellipses, parabolas, and hyperbolas. He proves the proposition that, if A , A' be the vertices of a conic, and if P be any point on it, and PM the perpendicular drawn from P on AA' , then (in the usual notation) the ratio MP^2: AM ctrdot MA' is constant in an ellipse or hyperbola, and the ratio MP^2: AM is constant in a parabola. These are the characteristic properties on which almost all the rest of the work is based. He next shows that, if A be the vertex, l the latus rectum, and if AM and MP be the abscissa and ordinate of any point on a conic (see above figure), then MP^2 is less than, equal to, or greater than l ctrdot AM according as the conic is an ellipse, parabola, or hyperbola; hence the names which he gave to the curves and by which they are still known.
He had no idea of the directrix, and was not aware that the parabola had a focus, but, with the exception of the propositions which involve these, his first three books contain most of the propositions which are found in modern text-books. In the fourth book he develops the theory of lines cut harmonically, and treats of the points of intersection of systems of conics. In the fifth book he commences with the theory of maxima and minima; applies it to find the centre of curvature at any point of a conic, and the evolute of the curve; and discusses the number of normals which can be drawn from a point to a conic. In the sixth book he treats of similar conics. The seventh and eighth books were given up to a discussion of conjugate diameters; the latter of these was conjecturally restored by E. Halley in his edition of 1710.
The verbose explanations make the book repulsive to most modern readers; but the arrangement and reasoning are unexceptional, and it has been not unfitly described as the crown of Greek geometry. It is the work on which the reputation of Apollonius rests, and it earned for him the name of "the great geometrician."
Besides this immense treatise he wrote numerous shorter works; of course the books were written in Greek, but they are usually referred to by their Latin titles: those about which we now know anything are enumerated below. He was the author of a work on the problem "given two co-planar straight lines Aa and Bb , drawn through fixed points A and B ; to draw a line Oab from a given point O outside them cutting them in a and b , so that Aa shall be to Bb in a given ratio." He reduced the question to seventy-seven separate cases and gave an appropriate solution, with the aid of conics, for each case; this was published by E. Halley (translated from an Arabic copy) in 1706. He also wrote a treatise De Sectione Spatii (restored by E. Halley in 1706) on the same problem under the condition that the rectangle Aa ctrdot Bb was given. He wrote another entitled De Sectione Determinata (restored by R. Simson in 1749), dealing with problems such as to find a point P in a given straight line AB , so that PA^2 shall be to PB in a given ratio. He wrote another De Tactionibus (restored by Vieta in 1600) on the construction of a circle which shall touch three given circles. Another work was his De Inclinationibus (restored by M. Ghetaldi in 1607) on the problem to draw a line so that the intercept between two given lines, or the circumferences of two given circles, shall be of a given length. He was also the author of a treatise in three books on plane loci, De Locis Planis (restored by Fermat in 1637, and by R. Simson in 1746), and of another on the regular solids. And, lastly, he wrote a treatise on unclassed incommensurables, being a commentary on the tenth book of Euclid. It is believed that in one or more of the lost books he used the method of conical projections.
Besides these geometrical works he wrote on the methods of arithmetical calculation. All that we know of this is derived from some remarks of Pappus. Friedlein thinks that it was merely a sort of ready-reckoner. It seems, however, more probable that Apollonius here suggested a system of numeration similar to that proposed by Archimedes, but proceeding by tetrads instead of octads, and described a notation for it. It will be noticed that our modern notation goes by hexads, a million = 10^6 , a billion = 10^12 , a trillion = 10^18 , etc. It is not impossible that Apollonius also pointed out that a decimal system of notation, involving only nine symbols, would facilitate numerical multiplications.
Apollonius was interested in astronomy, and wrote a book on the stations and regressions of the planets of which Ptolemy made some use in writing the Almagest. He also wrote a treatise on the use and theory of the screw in statics.
This is a long list, but I should suppose that most of these works were short tracts on special points.
Like so many of his predecessors, he too gave a construction for finding two mean proportionals between two given lines, and thereby duplicating the cube. It was as follows. Let OA and OB be the given lines. Construct a rectangle OADB , of which they are adjacent sides. Bisect AB in C . Then, if with C as centre we can describe a circle cutting OA produced in a , and cutting OB produced in b , so that aDb shall be a straight line, the problem is effected. For it is easily shewn that 2pt plus 2pt minus 1pt
Oa Aa + CA^2 = Ca^2. Similarly Ob Bb + CB^2 = Cb^2. Hence Oa Aa = Ob Bb. That is, Oa : Ob = Bb : Aa.
But, by similar triangles,
BD : Bb = Oa : Ob = Aa : AD. Therefore Oa : Bb = Bb : Aa = Aa : OB,
that is, Bb and Oa are the two mean proportionals between OA and OB . It is impossible to construct the circle whose centre is C by Euclidean geometry, but Apollonius gave a mechanical way of describing it. This construction is quoted by several Arabic writers.
In one of the most brilliant passages of his Aperçu historique Chasles remarks that, while Archimedes and Apollonius were the most able geometricians of the old world, their works are distinguished by a contrast which runs through the whole subsequent history of geometry. Archimedes, in attacking the problem of the quadrature of curvilinear areas, established the principles of the geometry which rests on measurements; this naturally gave rise to the infinitesimal calculus, and in fact the method of exhaustions as used by Archimedes does not differ in principle from the method of limits as used by Newton. Apollonius, on the other hand, in investigating the properties of conic sections by means of transversals involving the ratio of rectilineal distances and of perspective, laid the foundations of the geometry of form and position.
Eratosthenes. Among the contemporaries of Archimedes and Apollonius I may mention Eratosthenes. Born at Cyrene in 275 b.c., he was educated at Alexandria--perhaps at the same time as Archimedes, of whom he was a personal friend--and Athens, and was at an early age entrusted with the care of the university library at Alexandria, a post which probably he occupied till his death. He was the Admirable Crichton of his age, and distinguished for his athletic, literary, and scientific attainments: he was also something of a poet. He lost his sight by ophthalmia, then as now a curse of the valley of the Nile, and, refusing to live when he was no longer able to read, he committed suicide in 194 b.c.
In science he was chiefly interested in astronomy and geodesy, and he constructed various astronomical instruments which were used for some centuries at the university. He suggested the calendar (now known as Julian), in which every fourth year contains 366 days; and he determined the obliquity of the ecliptic as 23° 51' 20" . He measured the length of a degree on the earth's surface, making it to be about 79 miles, which is too long by nearly 10 miles, and thence calculated the circumference of the earth to be 252,000 stadia. If we take the Olympic stadium of 2021/4 yards, this is equivalent to saying that the radius is about 4600 miles, but there was also an Egyptian stadium, and if he used this he estimated the radius as 3925 miles, which is very near the truth. The principle used in the determination is correct.
Of Eratosthenes's work in mathematics we have two extant illustrations: one in a description of an instrument to duplicate a cube, and the other in a rule he gave for constructing a table of prime numbers. The former is given in many books. The latter, called the "sieve of Eratosthenes," was as follows: write down all the numbers from 1 upwards; then every second number from 2 is a multiple of 2 and may be cancelled; every third number from 3 is a multiple of 3 and may be cancelled; every fifth number from 5 is a multiple of 5 and may be cancelled; and so on. It has been estimated that it would involve working for about 300 hours to thus find the primes in the numbers from 1 to 1,000,000. The labour of determining whether any particular number is a prime may be, however, much shortened by observing that if a number can be expressed as the product of two factors, one must be less and the other greater than the square root of the number, unless the number is the square of a prime, in which case the two factors are equal. Hence every composite number must be divisible by a prime which is not greater than its square root.
The third century before Christ, which opens with the career of Euclid and closes with the death of Apollonius, is the most brilliant era in the history of Greek mathematics. But the great mathematicians of that century were geometricians, and under their influence attention was directed almost solely to that branch of mathematics. With the methods they used, and to which their successors were by tradition confined, it was hardly possible to make any further great advance: to fill up a few details in a work that was completed in its essential parts was all that could be effected. It was not till after the lapse of nearly 1800 years that the genius of Descartes opened the way to any further progress in geometry, and I therefore pass over the numerous writers who followed Apollonius with but slight mention. Indeed it may be said roughly that during the next thousand years Pappus was the sole geometrician of great original ability; and during this long period almost the only other pure mathematicians of exceptional genius were Hipparchus and Ptolemy, who laid the foundations of trigonometry, and Diophantus, who laid those of algebra.
Early in the second century, circ. 180 b.c., we find the names of three mathematicians--Hypsicles, Nicomedes, and Diocles--who in their own day were famous.
Hypsicles. The first of these was Hypsicles, who added a fourteenth book to Euclid's Elements in which the regular solids were discussed. In another small work, entitled Risings, we find for the first time in Greek mathematics a right angle divided in the Babylonian manner into ninety degrees; possibly Eratosthenes may have previously estimated angles by the number of degrees they contain, but this is only a matter of conjecture.
Nicomedes. The second was Nicomedes, who invented the curve known as the conchoid or the shell-shaped curve. If from a fixed point S a line be drawn cutting a given fixed straight line in Q , and if P be taken on SQ so that the length QP is constant (say d ), then the locus of P is the conchoid. Its equation may be put in the form r = a sec theta ± d . It is easy with its aid to trisect a given angle or to duplicate a cube; and this no doubt was the cause of its invention.
Diocles. The third of these mathematicians was Diocles, the inventor of the curve known as the cissoid or the ivy-shaped curve, which, like the conchoid, was used to give a solution of the duplication problem. He defined it thus: let AOA' and BOB' be two fixed diameters of a circle at right angles to one another. Draw two chords QQ' and RR' parallel to BOB' and equidistant from it. Then the locus of the intersection of AR and QQ' will be the cissoid. Its equation can be expressed in the form y^2(2a-x) = x^3 . The curve may be used to duplicate the cube. For, if OA and OE be the two lines between which it is required to insert two geometrical means, and if, in the figure constructed as above, A'E cut the cissoid in P , and AP cut OB in D, we have OD^3 = OA^2 ctrdot OE . Thus OD is one of the means required, and the other mean can be found at once.
Diocles also solved (by the aid of conic sections) a problem which had been proposed by Archimedes, namely, to draw a plane which will divide a sphere into two parts whose volumes shall bear to one another a given ratio.
Perseus. Zenodorus. About a quarter of a century later, say about 150 b.c., Perseus investigated the various plane sections of the anchor-ring, and Zenodorus wrote a treatise on isoperimetrical figures. Part of the latter work has been preserved; one proposition which will serve to show the nature of the problems discussed is that "of segments of circles, having equal arcs, the semicircle is the greatest."
Towards the close of this century we find two mathematicians who, by turning their attention to new subjects, gave a fresh stimulus to the study of mathematics. These were Hipparchus and Hero.
Hipparchus.
Hipparchus was the most eminent of Greek astronomers--his chief predecessors being Eudoxus, Aristarchus, Archimedes, and Eratosthenes. Hipparchus is said to have been born about 160 b.c. at Nicaea in Bithynia; it is probable that he spent some years at Alexandria, but finally he took up his abode at Rhodes where he made most of his observations. Delambre has obtained an ingenious confirmation of the tradition which asserted that Hipparchus lived in the second century before Christ. Hipparchus in one place says that the longitude of a certain star eta Canis observed by him was exactly 90° , and it should be noted that he was an extremely careful observer. Now in 1750 it was 116° 4' 10" , and, as the first point of Aries regredes at the rate of 50.2" a year, the observation was made about 120 b.c.
Except for a short commentary on a poem of Aratus dealing with astronomy all his works are lost, but Ptolemy's great treatise, the Almagest, described below, was founded on the observations and writings of Hipparchus, and from the notes there given we infer that the chief discoveries of Hipparchus were as follows. He determined the duration of the year to within six minutes of its true value. He calculated the inclination of the ecliptic and equator as 23° 51' ; it was actually at that time 23° 46' . He estimated the annual precession of the equinoxes as 59" ; it is 50.2" . He stated the lunar parallax as 57' , which is nearly correct. He worked out the eccentricity of the solar orbit as 1/24 ; it is very approximately 1/30 . He determined the perigee and mean motion of the sun and of the moon, and he calculated the extent of the shifting of the plane of the moon's motion. Finally he obtained the synodic periods of the five planets then known. I leave the details of his observations and calculations to writers who deal specially with astronomy such as Delambre; but it may be fairly said that this work placed the subject for the first time on a scientific basis.
To account for the lunar motion Hipparchus supposed the moon to move with uniform velocity in a circle, the earth occupying a position near (but not at) the centre of this circle. This is equivalent to saying that the orbit is an epicycle of the first order. The longitude of the moon obtained on this hypothesis is correct to the first order of small quantities for a few revolutions. To make it correct for any length of time Hipparchus further supposed that the apse line moved forward about 3° a month, thus giving a correction for eviction. He explained the motion of the sun in a similar manner. This theory accounted for all the facts which could be determined with the instruments then in use, and in particular enabled him to calculate the details of eclipses with considerable accuracy.
He commenced a series of planetary observations to enable his successors to frame a theory to account for their motions; and with great perspicacity he predicted that to do this it would be necessary to introduce epicycles of a higher order, that is, to introduce three or more circles the centre of each successive one moving uniformly along the circumference of the preceding one.
He also formed a list of 1080 of the fixed stars. It is said that the sudden appearance in the heavens of a new and brilliant star called his attention to the need of such a catalogue; and the appearance of such a star during his lifetime is confirmed by Chinese records.
No further advance in the theory of astronomy was made until the time of Copernicus, though the principles laid down by Hipparchus were extended and worked out in detail by Ptolemy.
Investigations such as these naturally led to trigonometry, and Hipparchus must be credited with the invention of that subject. It is known that in plane trigonometry he constructed a table of chords of arcs, which is practically the same as one of natural sines; and that in spherical trigonometry he had some method of solving triangles: but his works are lost, and we can give no details. It is believed, however, that the elegant theorem, printed as Euc. vi, d, and generally known as Ptolemy's Theorem, is due to Hipparchus and was copied from him by Ptolemy. It contains implicitly the addition formulae for sin(A ± B) and cos(A ± B) ; and Carnot showed how the whole of elementary plane trigonometry could be deduced from it.
I ought also to add that Hipparchus was the first to indicate the position of a place on the earth by means of its latitude and longitude.
Hero. The second of these mathematicians was Hero of Alexandria, who placed engineering and land-surveying on a scientific basis. He was a pupil of Ctesibus, who invented several ingenious machines, and is alluded to as if he were a mathematician of note. It is not likely that Hero flourished before 80 b.c., but the precise period at which he lived is uncertain.
In pure mathematics Hero's principal and most characteristic work consists of (i) some elementary geometry, with applications to the determination of the areas of fields of given shapes; (ii) propositions on finding the volumes of certain solids, with applications to theatres, baths, banquet-halls, and so on; (iii) a rule to find the height of an inaccessible object; and (iv) tables of weights and measures. He invented a solution of the duplication problem which is practically the same as that which Apollonius had already discovered. Some commentators think that he knew how to solve a quadratic equation even when the coefficients were not numerical; but this is doubtful. He proved the formula that the area of a triangle is equal to {s(s-a) (s-b) (s-c)}^1/2 , where s is the semiperimeter, and a , b , c , the lengths of the sides, and gave as an illustration a triangle whose sides were in the ratio 13:14:15. He seems to have been acquainted with the trigonometry of Hipparchus, and the values of cot 2pi/n are computed for various values of n , but he nowhere quotes a formula or expressly uses the value of the sine; it is probable that like the later Greeks he regarded trigonometry as forming an introduction to, and being an integral part of, astronomy.
The following is the manner in which he solved the problem to find the area of a triangle ABC the length of whose sides are a , b , c . Let s be the semiperimeter of the triangle. Let the inscribed circle touch the sides in D , E , F , and let O be its centre. On BC produced take H so that CH=AF , therefore BH = s . Draw OK at right angles to OB , and CK at right angles to BC ; let them meet in K . The area ABC or triangle is equal to the sum of the areas OBC , OCA , OAB = 1/2ar + 1/2br + 1/2cr = sr , that is, is equal to BH ctrdot OD . He then shews that the angle OAF = angle CBK ; hence the triangles OAF and CBK are similar.
BC:CK &= AF:OF = CH:OD, BC:CH &= CK:OD = CL:LD, BH:CH &= CD:LD, BH^2:CH BH = CD BD:LD BD = CD BD:OD^2.
Hence = BH OD = CH BH CD BD^12 = (s-a)s(s-c)(s-b)^12.
In applied mathematics Hero discussed the centre of gravity, the five simple machines, and the problem of moving a given weight with a given power; and in one place he suggested a way in which the power of a catapult could be tripled. He also wrote on the theory of hydraulic machines. He described a theodolite and cyclometer, and pointed out various problems in surveying for which they would be useful. But the most interesting of his smaller works are his Pneumatik'a and A>ut'omata, containing descriptions of about 100 small machines and mechanical toys, many of which are ingenious. In the former there is an account of a small stationary steam-engine which is of the form now known as Avery's patent: it was in common use in Scotland at the beginning of this century, but is not so economical as the form introduced by Watt. There is also an account of a double forcing pump to be used as a fire-engine. It is probable that in the hands of Hero these instruments never got beyond models. It is only recently that general attention has been directed to his discoveries, though Arago had alluded to them in his éloge on Watt.
All this is very different from the classical geometry and arithmetic of Euclid, or the mechanics of Archimedes. Hero did nothing to extend a knowledge of abstract mathematics; he learnt all that the text-books of the day could teach him, but he was interested in science only on account of its practical applications, and so long as his results were true he cared nothing for the logical accuracy of the process by which he arrived at them. Thus, in finding the area of a triangle, he took the square root of the product of four lines. The classical Greek geometricians permitted the use of the square and the cube of a line because these could be represented geometrically, but a figure of four dimensions is inconceivable, and certainly they would have rejected a proof which involved such a conception.
The successors of Hipparchus and Hero did not avail themselves of the opportunity thus opened of investigating new subjects, but fell back on the well-worn subject of geometry. Amongst the more eminent of these later geometricians were Theodosius and Dionysodorus, both of whom flourished about 50 b.c.
Theodosius. Theodosius was the author of a complete treatise on the geometry of the sphere, and of two works on astronomy.
Dionysodorus. Dionysodorus is known to us only by his solution
, second edition, Zürich, 1873, p. 101. of the problem to divide a hemisphere by a plane parallel to its base into two parts, whose volumes shall be in a given ratio. Like the solution by Diocles of the similar problem for a sphere above alluded to, it was effected by the aid of conic sections. Pliny says that Dionysodorus determined the length of the radius of the earth approximately as 42,000 stadia, which, if we take the Olympic stadium of 2021/4 yards, is a little less than 5000 miles; we do not know how it was obtained. This may be compared with the result given by Eratosthenes and mentioned above.
The administration of Egypt was definitely undertaken by Rome in 30 b.c. The closing years of the dynasty of the Ptolemies and the earlier years of the Roman occupation of the country were marked by much disorder, civil and political. The studies of the university were naturally interrupted, and it is customary to take this time as the close of the first Alexandrian school.
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End of part 7 of 28
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