A Short Account of the History of Mathematics · W. W. Rouse Ball
For authorities, see footnote above on p. . All dates given hereafter are to be taken as anno domini unless the contrary is expressly stated.
I concluded the last chapter by stating that the first school of Alexandria may be said to have come to an end at about the same time as the country lost its nominal independence. But, although the schools at Alexandria suffered from the disturbances which affected the whole Roman world in the transition, in fact if not in name, from a republic to an empire, there was no break of continuity; the teaching in the university was never abandoned; and as soon as order was again established, students began once more to flock to Alexandria. This time of confusion was, however, contemporaneous with a change in the prevalent views of philosophy which thenceforward were mostly neo-platonic or neo-pythagorean, and it therefore fitly marks the commencement of a new period. These mystical opinions reacted on the mathematical school, and this may partially account for the paucity of good work.
Though Greek influence was still predominant and the Greek language always used, Alexandria now became the intellectual centre for most of the Mediterranean nations which were subject to Rome. It should be added, however, that the direct connection with it of many of the mathematicians of this time is at least doubtful, but their knowledge was ultimately obtained from the Alexandrian teachers, and they are usually described as of the second Alexandrian school. Such mathematics as were taught at Rome were derived from Greek sources, and we may therefore conveniently consider their extent in connection with this chapter.
There is no doubt that throughout the first century after Christ geometry continued to be that subject in science to which most attention was devoted. But by this time it was evident that the geometry of Archimedes and Apollonius was not capable of much further extension; and such geometrical treatises as were produced consisted mostly of commentaries on the writings of the great mathematicians of a preceding age. In this century the only original works of any ability of which we know anything were two by Serenus and one by Menelaus.
Serenus. Menelaus. Those by Serenus of Antissa or of Antinoe, circ. 70, are on the plane sections of the cone and cylinder, in the course of which he lays down the fundamental proposition of transversals. That by Menelaus of Alexandria, circ. 98, is on spherical trigonometry, investigated in the Euclidean method. The fundamental theorem on which the subject is based is the relation between the six segments of the sides of a spherical triangle, formed by the arc of a great circle which cuts them [book iii, prop. 1]. Menelaus also wrote on the calculation of chords, that is, on plane trigonometry; this is lost.
Nicomachus. Towards the close of this century, circ. 100, a Jew, Nicomachus, of Gerasa, published an Arithmetic, which (or rather the Latin translation of it) remained for a thousand years a standard authority on the subject. Geometrical demonstrations are here abandoned, and the work is a mere classification of the results then known, with numerical illustrations: the evidence for the truth of the propositions enunciated, for I cannot call them proofs, being in general an induction from numerical instances. The object of the book is the study of the properties of numbers, and particularly of their ratios. Nicomachus commences with the usual distinctions between even, odd, prime, and perfect numbers; he next discusses fractions in a somewhat clumsy manner; he then turns to polygonal and to solid numbers; and finally treats of ratio, proportion, and the progressions. Arithmetic of this kind is usually termed Boethian, and the work of Boethius on it was a recognised text-book in the middle ages.
Theon. Another text-book on arithmetic on much the same lines as that of Nicomachus was produced by Theon of Smyrna, circ. 130. It formed the first book of his work on mathematics, written with the view of facilitating the study of Plato's writings.
Thymaridas. Another mathematician, reckoned by some writers as of about the same date as Theon, was Thymaridas, who is worthy of notice from the fact that he is the earliest known writer who explicitly enunciates an algebraical theorem. He states that, if the sum of any number of quantities be given, and also the sum of every pair which contains one of them, then this quantity is equal to one (n - 2) th part of the difference between the sum of these pairs and the first given sum. Thus, if
x1 + x2 + + xn = S, and if x1 + x2 = s2, x1 + x3 = s3, , and x1 + xn = sn, then x1 = (s2 + s3 + + sn - S)/(n - 2).
He does not seem to have used a symbol to denote the unknown quantity, but he always represents it by the same word, which is an approximation to symbolism.
Ptolemy.
About the same time as these writers Ptolemy of Alexandria, who died in 168, produced his great work on astronomy, which will preserve his name as long as the history of science endures. This treatise is usually known as the Almagest: the name is derived from the Arabic title al midschisti, which is said to be a corruption of meg'isth [majhmatik'h] s'untaxis. The work is founded on the writings of Hipparchus, and, though it did not sensibly advance the theory of the subject, it presents the views of the older writer with a completeness and elegance which will always make it a standard treatise. We gather from it that Ptolemy made observations at Alexandria from the years 125 to 150; he, however, was but an indifferent practical astronomer, and the observations of Hipparchus are generally more accurate than those of his expounder.
The work is divided into thirteen books. In the first book Ptolemy discusses various preliminary matters; treats of trigonometry, plane or spherical; gives a table of chords, that is, of natural sines (which is substantially correct and is probably taken from the lost work of Hipparchus); and explains the obliquity of the ecliptic; in this book he uses degrees, minutes, and seconds as measures of angles. The second book is devoted chiefly to phenomena depending on the spherical form of the earth: he remarks that the explanations would be much simplified if the earth were supposed to rotate on its axis once a day, but states that this hypothesis is inconsistent with known facts. In the third book he explains the motion of the sun round the earth by means of excentrics and epicycles: and in the fourth and fifth books he treats the motion of the moon in a similar way. The sixth book is devoted to the theory of eclipses; and in it he gives 3° 8' 30" , that is 317/120 , as the approximate value of pi , which is equivalent to taking it equal to 3.1416. The seventh and eighth books contain a catalogue (probably copied from Hipparchus) of 1028 fixed stars determined by indicating those, three or more, that appear to be in a plane passing through the observer's eye: and in another work Ptolemy added a list of annual sidereal phenomena. The remaining books are given up to the theory of the planets.
This work is a splendid testimony to the ability of its author. It became at once the standard authority on astronomy, and remained so till Copernicus and Kepler shewed that the sun and not the earth must be regarded as the centre of the solar system.
The idea of excentrics and epicycles on which the theories of Hipparchus and Ptolemy are based has been often ridiculed in modern times. No doubt at a later time, when more accurate observations had been made, the necessity of introducing epicycle on epicycle in order to bring the theory into accordance with the facts made it very complicated. But De Morgan has acutely observed that in so far as the ancient astronomers supposed that it was necessary to resolve every celestial motion into a series of uniform circular motions they erred greatly, but that, if the hypothesis be regarded as a convenient way of expressing known facts, it is not only legitimate but convenient. The theory suffices to describe either the angular motion of the heavenly bodies or their change in distance. The ancient astronomers were concerned only with the former question, and it fairly met their needs; for the latter question it is less convenient. In fact it was as good a theory as for their purposes and with their instruments and knowledge it was possible to frame, and corresponds to the expression of a given function as a sum of sines or cosines, a method which is of frequent use in modern analysis.
In spite of the trouble taken by Delambre it is almost impossible to separate the results due to Hipparchus from those due to Ptolemy. But Delambre and De Morgan agree in thinking that the observations quoted, the fundamental ideas, and the explanation of the apparent solar motion are due to Hipparchus; while all the detailed explanations and calculations of the lunar and planetary motions are due to Ptolemy.
The Almagest shews that Ptolemy was a geometrician of the first rank, though it is with the application of geometry to astronomy that he is chiefly concerned. He was also the author of numerous other treatises. Amongst these is one on pure geometry in which he proposed to cancel Euclid's postulate on parallel lines, and to prove it in the following manner. Let the straight line EFGH meet the two straight lines AB and CD so as to make the sum of the angles BFG and FGD equal to two right angles. It is required to prove that AB and CD are parallel. If possible let them not be parallel, then they will meet when produced say at M (or N ). But the angle AFG is the supplement of BFG , and is therefore equal to FGD : similarly the angle FGC is equal to the angle BFG . Hence the sum of the angles AFG and FGC is equal to two right angles, and the lines BA and DC will therefore if produced meet at N (or M ). But two straight lines cannot enclose a space, therefore AB and CD cannot meet when produced, that is, they are parallel. Conversely, if AB and CD be parallel, then AF and CG are not less parallel than FB and GD ; and therefore whatever be the sum of the angles AFG and FGC such also must be the sum of the angles FGD and BFG . But the sum of the four angles is equal to four right angles, and therefore the sum of the angles BFG and FGD must be equal to two right angles.
Ptolemy wrote another work to shew that there could not be more than three dimensions in space: he also discussed orthographic and stereographic projections with special reference to the construction of sun-dials. He wrote on geography, and stated that the length of one degree of latitude is 500 stadia. A book on sound is sometimes attributed to him, but on doubtful authority.
Pappus. Ptolemy had shewn not only that geometry could be applied to astronomy, but had indicated how new methods of analysis like trigonometry might be thence developed. He found however no successors to take up the work he had commenced so brilliantly, and we must look forward 150 years before we find another geometrician of any eminence. That geometrician was Pappus who lived and taught at Alexandria about the end of the third century. We know that he had numerous pupils, and it is probable that he temporarily revived an interest in the study of geometry.
Pappus wrote several books, but the only one which has come down to us is his Sunagwg'h, a collection of mathematical papers arranged in eight books of which the first and part of the second have been lost. This collection was intended to be a synopsis of Greek mathematics together with comments and additional propositions by the editor. A careful comparison of various extant works with the account given of them in this book shews that it is trustworthy, and we rely largely on it for our knowledge of other works now lost. It is not arranged chronologically, but all the treatises on the same subject are grouped together, and it is most likely that it gives roughly the order in which the classical authors were read at Alexandria. Probably the first book, which is now lost, was on arithmetic. The next four books deal with geometry exclusive of conic sections; the sixth with astronomy including, as subsidiary subjects, optics and trigonometry; the seventh with analysis, conics, and porisms; and the eighth with mechanics.
The last two books contain a good deal of original work by Pappus; at the same time it should be remarked that in two or three cases he has been detected in appropriating proofs from earlier authors, and it is possible he may have done this in other cases.
Subject to this suspicion we may say that Pappus's best work is in geometry. He discovered the directrix in the conic sections, but he investigated only a few isolated properties: the earliest comprehensive account was given by Newton and Boscovich. As an illustration of his power I may mention that he solved [book vii, prop. 107] the problem to inscribe in a given circle a triangle whose sides produced shall pass through three collinear points. This question was in the eighteenth century generalised by Cramer by supposing the three given points to be anywhere; and was considered a difficult problem.
, Paris, 1904, vol. xi, pp. 219--220. It was sent in 1742 as a challenge to Castillon, and in 1776 he published a solution. Lagrange, Euler, Lhulier, Fuss, and Lexell also gave solutions in 1780. A few years later the problem was set to a Neapolitan lad A. Giordano, who was only 16 but who had shewn marked mathematical ability, and he extended it to the case of a polygon of n sides which pass through n given points, and gave a solution both simple and elegant. Poncelet extended it to conics of any species and subject to other restrictions.
In mechanics Pappus shewed that the centre of mass of a triangular lamina is the same as that of an inscribed triangular lamina whose vertices divide each of the sides of the original triangle in the same ratio. He also discovered the two theorems on the surface and volume of a solid of revolution which are still quoted in text-books under his name: these are that the volume generated by the revolution of a curve about an axis is equal to the product of the area of the curve and the length of the path described by its centre of mass; and the surface is equal to the product of the perimeter of the curve and the length of the path described by its centre of mass.
The problems above mentioned are but samples of many brilliant but isolated theorems which were enunciated by Pappus. His work as a whole and his comments shew that he was a geometrician of power; but it was his misfortune to live at a time when but little interest was taken in geometry, and when the subject, as then treated, had been practically exhausted.
Possibly a small tract on multiplication and division of sexagesimal fractions, which would seem to have been written about this time, is due to Pappus.
Throughout the second and third centuries, that is, from the time of Nicomachus, interest in geometry had steadily decreased, and more and more attention had been paid to the theory of numbers, though the results were in no way commensurate with the time devoted to the subject. It will be remembered that Euclid used lines as symbols for any magnitudes, and investigated a number of theorems about numbers in a strictly scientific manner, but he confined himself to cases where a geometrical representation was possible. There are indications in the works of Archimedes that he was prepared to carry the subject much further: he introduced numbers into his geometrical discussions and divided lines by lines, but he was fully occupied by other researches and had no time to devote to arithmetic. Hero abandoned the geometrical representation of numbers, but he, Nicomachus, and other later writers on arithmetic did not succeed in creating any other symbolism for numbers in general, and thus when they enunciated a theorem they were content to verify it by a large number of numerical examples. They doubtless knew how to solve a quadratic equation with numerical coefficients--for, as pointed out above, geometrical solutions of the equations ax^2 - bx + c = 0 and ax^2 + bx -c = 0 are given in Euc. vi, 28 and 29--but probably this represented their highest attainment.
It would seem then that, in spite of the time given to their study, arithmetic and algebra had not made any sensible advance since the time of Archimedes. The problems of this kind which excited most interest in the third century may be illustrated from a collection of questions, printed in the Palatine Anthology, which was made by Metrodorus at the beginning of the next century, about 310. Some of them are due to the editor, but some are of an anterior date, and they fairly illustrate the way in which arithmetic was leading up to algebraical methods. The following are typical examples. "Four pipes discharge into a cistern: one fills it in one day; another in two days; the third in three days; the fourth in four days: if all run together how soon will they fill the cistern?" "Demochares has lived a fourth of his life as a boy; a fifth as a youth; a third as a man; and has spent thirteen years in his dotage: how old is he?" "Make a crown of gold, copper, tin, and iron weighing 60 minae: gold and copper shall be two-thirds of it; gold and tin three-fourths of it; and gold and iron three-fifths of it: find the weights of the gold, copper, tin, and iron which are required." The last is a numerical illustration of Thymaridas's theorem quoted above.
It is believed that these problems were solved by rhetorical algebra, that is, by a process of algebraical reasoning expressed in words and without the use of any symbols. This, according to Nesselmann, is the first stage in the development of algebra, and we find it used both by Ahmes and by the earliest Arabian, Persian, and Italian algebraists: examples of its use in the solution of a geometrical problem and in the rule for the solution of a quadratic equation are given later. On this view then a rhetorical algebra had been gradually evolved by the Greeks, or was then in process of evolution. Its development was however very imperfect. Hankel, who is no unfriendly critic, says that the results attained as the net outcome of the work of six centuries on the theory of numbers are, whether we look at the form or the substance, unimportant or even childish, and are not in any way the commencement of a science.
In the midst of this decaying interest in geometry and these feeble attempts at algebraic arithmetic, a single algebraist of marked originality suddenly appeared who created what was practically a new science. This was Diophantus who introduced a system of abbreviations for those operations and quantities which constantly recur, though in using them he observed all the rules of grammatical syntax. The resulting science is called by Nesselmann syncopated algebra: it is a sort of shorthand. Broadly speaking, it may be said that European algebra did not advance beyond this stage until the close of the sixteenth century.
Modern algebra has progressed one stage further and is entirely symbolic; that is, it has a language of its own and a system of notation which has no obvious connection with the things represented, while the operations are performed according to certain rules which are distinct from the laws of grammatical construction.
Diophantus.
All that we know of Diophantus is that he lived at Alexandria, and that most likely he was not a Greek. Even the date of his career is uncertain; it cannot reasonably be put before the middle of the third century, and it seems probable that he was alive in the early years of the fourth century, that is, shortly after the death of Pappus. He was 84 when he died.
In the above sketch of the lines on which algebra has developed I credited Diophantus with the invention of syncopated algebra. This is a point on which opinions differ, and some writers believe that he only systematized the knowledge which was familiar to his contemporaries. In support of this latter opinion it may be stated that Cantor thinks that there are traces of the use of algebraic symbolism in Pappus, and Freidlein mentions a Greek papyrus in which the signs / and -.6ex are used for addition and subtraction respectively; but no other direct evidence for the non-originality of Diophantus has been produced, and no ancient author gives any sanction to this opinion.
Diophantus wrote a short essay on polygonal numbers; a treatise on algebra which has come down to us in a mutilated condition; and a work on porisms which is lost.
The Polygonal Numbers contains ten propositions, and was probably his earliest work. In this he reverts to the classical system by which numbers are represented by lines, a construction is (if necessary) made, and a strictly deductive proof follows: it may be noticed that in it he quotes propositions, such as Euc. ii, 3, and ii, 8, as referring to numbers and not to magnitudes.
His chief work is his Arithmetic. This is really a treatise on algebra; algebraic symbols are used, and the problems are treated analytically. Diophantus tacitly assumes, as is done in nearly all modern algebra, that the steps are reversible. He applies this algebra to find solutions (though frequently only particular ones) of several problems involving numbers. I propose to consider successively the notation, the methods of analysis employed, and the subject-matter of this work.
First, as to the notation. Diophantus always employed a symbol to represent the unknown quantity in his equations, but as he had only one symbol he could not use more than one unknown at a time. The unknown quantity is called <o >arijm'os, and is represented by Mysigma' or Mysigma^oprime . It is usually printed as sigma . In the plural it is denoted by sigmasigma or sigmasigma^oiota . This symbol may be a corruption of alpha^rho , or perhaps it may be the final sigma of this word, or possibly it may stand for the word swr'os a heap. The square of the unknown is called d'unamis, and denoted by delta^upsilon : the cube k'ubos, and denoted by kappa^upsilon ; and so on up to the sixth power.
The coefficients of the unknown quantity and its powers are numbers, and a numerical coefficient is written immediately after the quantity it multiplies: thus sigma'alpha = x , and sigmasigma^oiota iotaalpha = sigmasigma iotaalpha = 11x . An absolute term is regarded as a certain number of units or mon'ades which are represented by mu^o : thus mu^oalpha = 1 , mu^o iotaalpha = 11 .
There is no sign for addition beyond juxtaposition. Subtraction is represented by , and this symbol affects all the symbols that follow it. Equality is represented by iota . Thus
^ overline ^omicron ^omicron iota represents (x^3 + 8x) - (5x^2 + 1) = x.
Diophantus also introduced a somewhat similar notation for fractions involving the unknown quantity, but into the details of this I need not here enter.
It will be noticed that all these symbols are mere abbreviations for words, and Diophantus reasons out his proofs, writing these abbreviations in the middle of his text. In most manuscripts there is a marginal summary in which the symbols alone are used and which is really symbolic algebra; but probably this is the addition of some scribe of later times.
This introduction of a contraction or a symbol instead of a word to represent an unknown quantity marks a greater advance than anyone not acquainted with the subject would imagine, and those who have never had the aid of some such abbreviated symbolism find it almost impossible to understand complicated algebraical processes. It is likely enough that it might have been introduced earlier, but for the unlucky system of numeration adopted by the Greeks by which they used all the letters of the alphabet to denote particular numbers and thus made it impossible to employ them to represent any number.
Next, as to the knowledge of algebraic methods shewn in the book. Diophantus commences with some definitions which include an explanation of his notation, and in giving the symbol for minus he states that a subtraction multiplied by a subtraction gives an addition; by this he means that the product of -b and -d in the expansion of (a-b)(c-d) is +bd , but in applying the rule he always takes care that the numbers a , b , c , d are so chosen that a is greater than b and c is greater than d .
The whole of the work itself, or at least as much as is now extant, is devoted to solving problems which lead to equations. It contains rules for solving a simple equation of the first degree and a binomial quadratic. Probably the rule for solving any quadratic equation was given in that part of the work which is now lost, but where the equation is of the form ax^2 + bx + c = 0 he seems to have multiplied by a and then "completed the square" in much the same way as is now done: when the roots are negative or irrational the equation is rejected as "impossible," and even when both roots are positive he never gives more than one, always taking the positive value of the square root. Diophantus solves one cubic equation, namely, x^3 + x = 4x^2 + 4 [book vi, prob. 19].
The greater part of the work is however given up to indeterminate equations between two or three variables. When the equation is between two variables, then, if it be of the first degree, he assumes a suitable value for one variable and solves the equation for the other. Most of his equations are of the form y^2 = Ax^2 + Bx + C . Whenever A or C is equal to zero, he is able to solve the equation completely. When this is not the case, then, if A = a^2 , he assumes y = ax+m ; if C = c^2 , he assumes y = mx+c ; and lastly, if the equation can be put in the form y^2 = (ax ± b)^2 + c^2 , he assumes y = mx : where in each case m has some particular numerical value suitable to the problem under consideration. A few particular equations of a higher order occur, but in these he generally alters the problem so as to enable him to reduce the equation to one of the above forms.
The simultaneous indeterminate equations involving three variables, or "double equations" as he calls them, which he considers are of the forms y^2 = Ax^2 + Bx + C and z^2 = ax^2 + bx + c . If A and a both vanish, he solves the equations in one of two ways. It will be enough to give one of his methods which is as follows: he subtracts and thus gets an equation of the form y^2 - z^2 = mx + n ; hence, if y ± z = lambda , then y mp z = (mx + n) / lambda ; and solving he finds y and z . His treatment of "double equations" of a higher order lacks generality and depends on the particular numerical conditions of the problem.
Lastly, as to the matter of the book. The problems he attacks and the analysis he uses are so various that they cannot be described concisely and I have therefore selected five typical problems to illustrate his methods. What seems to strike his critics most is the ingenuity with which he selects as his unknown some quantity which leads to equations such as he can solve, and the artifices by which he finds numerical solutions of his equations.
I select the following as characteristic examples.
(i) Find four numbers, the sum of every arrangement three at a time being given; say 22, 24, 27, and 20 [book i, prob. 17].
Let x be the sum of all four numbers; hence the numbers are x-22 , x-24 , x-27 , and x-20 .
x = (x-22) + (x-24) + (x-27) + (x-20). x = 31. the numbers are 9, 7, 4, and 11.
(ii) Divide a number, such as 13 which is the sum of two squares 4 and 9, into two other squares [book ii, prob. 10].
He says that since the given squares are 2^2 and 3^2 he will take (x + 2)^2 and (mx - 3)^2 as the required squares, and will assume m = 2 .
(x + 2)^2 + (2x - 3)^2 = 13. x = 8/5. the required squares are 324/25 and 1/25.
(iii) Find two squares such that the sum of the product and either is a square [book ii, prob. 29].
Let x^2 and y^2 be the numbers. Then x^2y^2 + y^2 and x^2y^2 + x^2 are squares. The first will be a square if x^2 + 1 be a square, which he assumes may be taken equal to (x - 2)^2 , hence x = 3/4 . He has now to make 9 (y^2+ 1)/16 a square, to do this he assumes that 9y^2 + 9 = (3y - 4)^2 , hence y = 7/24 . Therefore the squares required are 9/16 and 49/576 .
It will be recollected that Diophantus had only one symbol for an unknown quantity; and in this example he begins by calling the unknowns x^2 and 1 , but as soon as he has found x he then replaces the 1 by the symbol for the unknown quantity, and finds it in its turn.
(iv) To find a [rational] right-angled triangle such that the line bisecting an acute angle is rational [book vi, prob. 18].
His solution is as follows. Let ABC be the triangle of which C is the right-angle. Let the bisector AD = 5x , and let DC = 3x , hence AC = 4x . Next let BC be a multiple of 3 , say 3, therefore BD = 3-3x , hence AB = 4 - 4x (by Euc. vi, 3). Hence (4-4x)^2 = 3^2 + (4x)^2 (Euc. i, 47), therefore x = 7/32 . Multiplying by 32 we get for the sides of the triangle 28, 96, and 100; and for the bisector 35.
(v) A man buys x measures of wine, some at 8 drachmae a measure, the rest at 5 . He pays for them a square number of drachmae, such that, if 60 be added to it, the resulting number is x^2 . Find the number he bought at each price [book v, prob. 33].
The price paid was x^2 - 60 , hence 8x > x^2-60 and 5x < x^2-60 . From this it follows that x must be greater than 11 and less than 12.
Again x^2-60 is to be a square; suppose it is equal to (x-m)^2 then x = (m^2 + 60)/2m , we have therefore
11 < m^2+602m < 12; 19 < m < 21.
Diophantus therefore assumes that m is equal to 20, which gives him x = 111/2 ; and makes the total cost, i.e. x^2-60 , equal to 721/4 drachmae.
He has next to divide this cost into two parts which shall give the cost of the 8 drachmae measures and the 5 drachmae measures respectively. Let these parts be y and z .
Then 15z + 18(7214 - z) = 12. Therefore z = 5 × 7912, and y = 8 × 5912.
Therefore the number of 5 drachmae measures was 79/12 , and of 8 drachmae measures was 59/12 .
From the enunciation of this problem it would seem that the wine was of a poor quality, and Tannery ingeniously suggested that the prices mentioned for such a wine are higher than were usual until after the end of the second century. He therefore rejected the view which was formerly held that Diophantus lived in that century, but he did not seem to be aware that De Morgan had previously shewn that this opinion was untenable. Tannery inclined to think that Diophantus lived half a century earlier than I have supposed.
I mentioned that Diophantus wrote a third work entitled Porisms. The book is lost, but we have the enunciations of some of the propositions, and though we cannot tell whether they were rigorously proved by Diophantus they confirm our opinion of his ability and sagacity. It has been suggested that some of the theorems which he assumes in his arithmetic were proved in the porisms. Among the more striking of these results are the statements that the difference of the cubes of two numbers can be always expressed as the sum of the cubes of two other numbers; that no number of the form 4n-1 can be expressed as the sum of two squares; and that no number of the form 8n-1 (or possibly 24n + 7 ) can be expressed as the sum of three squares: to these we may perhaps add the proposition that any number can be expressed as a square or as the sum of two or three or four squares.
The writings of Diophantus exercised no perceptible influence on Greek mathematics; but his Arithmetic, when translated into Arabic in the tenth century, influenced the Arabian school, and so indirectly affected the progress of European mathematics. An imperfect copy of the original work was discovered in 1462; it was translated into Latin and published by Xylander in 1575; the translation excited general interest, and by that time the European algebraists had, on the whole, advanced beyond the point at which Diophantus had left off.
Iamblichus. Iamblichus, circ. 350, to whom we owe a valuable work on the Pythagorean discoveries and doctrines, seems also to have studied the properties of numbers. He enunciated the theorem that if a number which is equal to the sum of three integers of the form 3n , 3n-1 , 3n-2 be taken, and if the separate digits of this number be added, and if the separate digits of the result be again added, and so on, then the final result will be 6: for instance, the sum of 54, 53, and 52 is 159, the sum of the separate digits of 159 is 15, the sum of the separate digits of 15 is 6. To any one confined to the usual Greek numerical notation this must have been a difficult result to prove: possibly it was reached empirically.
The names of two commentators will practically conclude the long roll of Alexandrian mathematicians.
Theon. The first of these is Theon of Alexandria, who flourished about 370. He was not a mathematician of special note, but we are indebted to him for an edition of Euclid's Elements and a commentary on the Almagest; the latter gives a great deal of miscellaneous information about the numerical methods used by the Greeks.
Hypatia. The other was Hypatia the daughter of Theon. She was more distinguished than her father, and was the last Alexandrian mathematician of any general reputation: she wrote a commentary on the Conics of Apollonius and possibly some other works, but none of her writings are now extant. She was murdered at the instigation of the Christians in 415.
The fate of Hypatia may serve to remind us that the Eastern Christians, as soon as they became the dominant party in the state, showed themselves bitterly hostile to all forms of learning. That very singleness of purpose which had at first so materially aided their progress developed into a one-sidedness which refused to see any good outside their own body; and all who did not actively assist them were persecuted. The final establishment of Christianity in the East marks the end of the Greek scientific schools, though nominally they continued to exist for two hundred years more.
The hostility of the Eastern church to Greek science is further illustrated by the fall of the later Athenian school. This school occupies but a small space in our history. Ever since Plato's time a certain number of professional mathematicians had lived at Athens; and about the year 420 this school again acquired considerable reputation, largely in consequence of the numerous students who after the murder of Hypatia migrated there from Alexandria. Its most celebrated members were Proclus, Damascius, and Eutocius.
Proclus. Proclus was born at Constantinople in February 412 and died at Athens on April 17, 485. He wrote a commentary on the first book of Euclid's Elements, which contains a great deal of valuable information on the history of Greek mathematics: he is verbose and dull, but luckily he has preserved for us quotations from other and better authorities. Proclus was succeeded as head of the school by Marinus, and Marinus by Isidorus.
Damascius. Eutocius. Two pupils of Isidorus, who in their turn subsequently lectured at Athens, may be mentioned in passing. One of these, Damascius of Damascus, circ. 490, is commonly said to have added to Euclid's Elements a fifteenth book on the inscription of one regular solid in another, but his authorship of this has been questioned by some writers. The other, Eutocius, circ. 510, wrote commentaries on the first four books of the Conics of Apollonius and on various works of Archimedes.
This later Athenian school was carried on under great difficulties owing to the opposition of the Christians. Proclus, for example, was repeatedly threatened with death because he was "a philosopher." His remark, "after all my body does not matter, it is the spirit that I shall take with me when I die," which he made to some students who had offered to defend him, has been often quoted. The Christians, after several ineffectual attempts, at last got a decree from Justinian in 529 that "heathen learning" should no longer be studied at Athens. That date therefore marks the end of the Athenian school.
The church at Alexandria was less influential, and the city was more remote from the centre of civil power. The schools there were thus suffered to continue, though their existence was of a precarious character. Under these conditions mathematics continued to be read in Egypt for another hundred years, but all interest in the study had gone.
I ought not to conclude this part of the history without any mention of Roman mathematics, for it was through Rome that mathematics first passed into the curriculum of medieval Europe, and in Rome all modern history has its origin. There is, however, very little to say on the subject. The chief study of the place was in fact the art of government, whether by law, by persuasion, or by those material means on which all government ultimately rests. There were, no doubt, professors who could teach the results of Greek science, but there was no demand for a school of mathematics. Italians who wished to learn more than the elements of the science went to Alexandria or to places which drew their inspiration from Alexandria.
The subject as taught in the mathematical schools at Rome seems to have been confined in arithmetic to the art of calculation (no doubt by the aid of the abacus) and perhaps some of the easier parts of the work of Nicomachus, and in geometry to a few practical rules; though some of the arts founded on a knowledge of mathematics (especially that of surveying) were carried to a high pitch of excellence. It would seem also that special attention was paid to the representation of numbers by signs. The manner of indicating numbers up to ten by the use of fingers must have been in practice from quite early times, but about the first century it had been developed by the Romans into a finger-symbolism by which numbers up to 10,000 or perhaps more could be represented: this would seem to have been taught in the Roman schools. It is described by Bede, and therefore would seem to have been known as far west as Britain; Jerome also alludes to it; its use has still survived in the Persian bazaars.
I am not acquainted with any Latin work on the principles of mechanics, but there were numerous books on the practical side of the subject which dealt elaborately with architectural and engineering problems. We may judge what they were like by the Mathematici Veteres, which is a collection of various short treatises on catapults, engines of war, &c.: and by the Kesto'i, written by Sextus Julius Africanus about the end of the second century, part of which is included in the Mathematici Veteres, which contains, amongst other things, rules for finding the breadth of a river when the opposite bank is occupied by an enemy, how to signal with a semaphore, &c.
In the sixth century Boethius published a geometry containing a few propositions from Euclid and an arithmetic founded on that of Nicomachus; and about the same time Cassiodorus discussed the foundation of a liberal education which, after the preliminary trivium of grammar, logic, and rhetoric, meant the quadrivium of arithmetic, geometry, music, and astronomy. These works were written at Rome in the closing years of the Athenian and Alexandrian schools, and I therefore mention them here, but as their only value lies in the fact that they became recognized text-books in medieval education I postpone their consideration to chapter viii.
Theoretical mathematics was in fact an exotic study at Rome; not only was the genius of the people essentially practical, but, alike during the building of their empire, while it lasted, and under the Goths, all the conditions were unfavourable to abstract science.
On the other hand, Alexandria was exceptionally well placed to be a centre of science. From the foundation of the city to its capture by the Mohammedans it was disturbed neither by foreign nor by civil war, save only for a few years when the rule of the Ptolemies gave way to that of Rome: it was wealthy, and its rulers took a pride in endowing the university: and lastly, just as in commerce it became the meeting-place of the east and the west, so it had the good fortune to be the dwelling-place alike of Greeks and of various Semitic people; the one race shewed a peculiar aptitude for geometry, the other for sciences which rest on measurement. Here too, however, as time went on the conditions gradually became more unfavourable, the endless discussions on theological dogmas and the increasing insecurity of the empire tending to divert men's thoughts into other channels.
The precarious existence and unfruitful history of the last two centuries of the second Alexandrian School need no record. In 632 Mohammed died, and within ten years his successors had subdued Syria, Palestine, Mesopotamia, Persia, and Egypt. The precise date on which Alexandria fell is doubtful, but the most reliable Arab historians give December 10, 641--a date which at any rate is correct within eighteen months.
With the fall of Alexandria the long history of Greek mathematics came to a conclusion. It seems probable that the greater part of the famous university library and museum had been destroyed by the Christians a hundred or two hundred years previously, and what remained was unvalued and neglected. Some two or three years after the first capture of Alexandria a serious revolt occurred in Egypt, which was ultimately put down with great severity. I see no reason to doubt the truth of the account that after the capture of the city the Mohammedans destroyed such university buildings and collections as were still left. It is said that, when the Arab commander ordered the library to be burnt, the Greeks made such energetic protests that he consented to refer the matter to the caliph Omar. The caliph returned the answer, "As to the books you have mentioned, if they contain what is agreeable with the book of God, the book of God is sufficient without them; and, if they contain what is contrary to the book of God, there is no need for them; so give orders for their destruction." The account goes on to say that they were burnt in the public baths of the city, and that it took six months to consume them all.
· · ·
End of part 8 of 28
Continue readingChapter 6: The Byzantine School
Starvard