A Short Account of the History of Mathematics · W. W. Rouse Ball
Except a few isolated papers, Fermat published nothing in his lifetime, and gave no systematic exposition of his methods. Some of the most striking of his results were found after his death on loose sheets of paper or written in the margins of works which he had read and annotated, and are unaccompanied by any proof. It is thus somewhat difficult to estimate the character of his investigations. He was constitutionally modest and retiring, and does not seem to have intended his papers to be published. It is probable that he revised his notes as occasion required, and that his published works represent the final form of his researches, and therefore cannot be dated much earlier than 1660. I shall consider separately (i) his investigations in the theory of numbers; (ii) his use in geometry of analysis and of infinitesimals; and (iii) his method of treating questions of probability.
(i) The theory of numbers appears to have been the favourite study of Fermat. He prepared an edition of Diophantus, and the notes and comments thereon contain numerous theorems of considerable elegance. Most of the proofs of Fermat are lost, and it is possible that some of them were not rigorous--an induction by analogy and the intuition of genius sufficing to lead him to correct results. The following examples will illustrate these investigations.
(a) If p be a prime and a be prime to p , then a^p-1-1 is divisible by p , that is, a^p-1-1equiv 0 pmodp . A proof of this, first given by Euler, is well known. A more general theorem is that a^phi(n)-1equiv 0 pmodn where a is prime to n , and phi(n) is the number of integers less than n and prime to it.
(b) An odd prime can be expressed as the difference of two square integers in one and only one way. Fermat's proof is as follows. Let n be the prime, and suppose it equal to x^2-y^2 , that is, to (x+y)(x-y) . Now, by hypothesis, the only integral factors of n are n and unity, hence x+y=n and x-y=1 . Solving these equations we get x=1/2(n+1) and y=1/2(n-1) .
(c) He gave a proof of the statement made by Diophantus that the sum of the squares of two integers cannot be of the form 4n-1 ; and he added a corollary which I take to mean that it is impossible that the product of a square and a prime of the form 4n-1 [even if multiplied by a number prime to the latter], can be either a square or the sum of two squares. For example, 44 is a multiple of 11 (which is of the form 4 × 3 - 1 ) by 4 , hence it cannot be expressed as the sum of two squares. He also stated that a number of the form a^2+b^2 , where a is prime to b , cannot be divided by a prime of the form 4n-1 .
(d) Every prime of the form 4n+1 is expressible, and that in one way only, as the sum of two squares. This problem was first solved by Euler, who shewed that a number of the form 2^m(4n+1) can be always expressed as the sum of two squares.
(e) If a, b, c, be integers, such that a^2+b^2= c^2 , then ab cannot be a square. Lagrange gave a solution of this.
(f) The determination of a number x such that x^2n+1 may be a square, where n is a given integer which is not a square. Lagrange gave a solution of this.
(g) There is only one integral solution of the equation x^2+2= y^3 ; and there are only two integral solutions of the equation x^2+4=y^3 . The required solutions are evidently for the first equation x=5 , and for the second equation x=2 and x=11 . This question was issued as a challenge to the English mathematicians Wallis and Digby.
(h) No integral values of x, y, z can be found to satisfy the equation x^n+y^n=z^n, if n be an integer greater than 2. This proposition has acquired extraordinary celebrity from the fact that no general demonstration of it has been given, but there is no reason to doubt that it is true.
Probably Fermat discovered its truth first for the case n=3 , and then for the case n=4 . His proof for the former of these cases is lost, but that for the latter is extant, and a similar proof for the case of n=3 can be given. These proofs depend upon shewing that, if three integral values of x, y, z can be found which satisfy the equation, then it will be possible to find three other and smaller integers which also satisfy it: in this way, finally, we shew that the equation must be satisfied by three values which obviously do not satisfy it. Thus no integral solution is possible. This method is inapplicable to the general case.
Fermat's discovery of the general theorem was made later. A proof can be given on the assumption that a number can be resolved into the product of powers of primes in one and only one way. It is possible that Fermat's argument rested on some such supposition, but this is an unsupported conjecture. The assumption is true of real integers, but is not necessarily true for algebraic integers--an algebraic integer being defined as a root of an algebraic equation x^n+a1x^n-1 + ... + an = 0 , whose coefficients, a , are arithmetical integers; for instance, a + b sqrt( -m), where a, b, and m are arithmetical integers, is an algebraic integer. Thus, admitting the use of these generalised integers, 21 can be expressed in three ways as the product of primes, namely, of 3 and 7 , or of 4+sqrt(-5) and 4 - sqrt(-5) , or of 1 + 2sqrt(-5) and 1-2sqrt(-5) ; and similarly there are values of n for which Fermat's equation leads to expressions which can be factorised in more than one way.
In 1823 Legendre obtained a proof for the case of n = 5 ; in 1832 Dirichlet gave one for n=14 , and in 1840 Lamé and Lebesgue gave proofs for n = 7 . In 1849 Kummer proved the truth of the theorem for all numbers which satisfy certain Bernoullian conditions. The only numbers less than 100 which do not do so are 37, 59, 67, and the theorem can, by other arguments, be proved for these three cases. It has now been extended far beyond 100, and no exception to it has yet been found. The following extracts, from a letter now in the university library at Leyden, will give an idea of Fermat's methods; the letter is undated, but it would appear that, at the time Fermat wrote it, he had proved the proposition (h) above only for the case when n= 3 .
Je ne m'en servis au commencement que pour demontrer les propositions negatives, comme par exemple, qu'il n'y a aucu nombre moindre de l'unité qu'un multiple de 3 qui soit composé d'un quarré et du triple d'un autre quarré. Qu'il n'y a aucun triangle rectangle de nombres dont l'aire soit un nombre quarré. La preuve se fait par >apagwg'hn t'hn e>is >ad'unaton en cette manière. S'il y auoit aucun triangle rectangle en nombres entiers, qui eust son aire esgale à un quarré, il y auroit un autre triangle moindre que celuy la qui auroit la mesme proprieté. S'il y en auoit un second moindre que le premier qui eust la mesme proprieté il y en auroit par un pareil raisonnement un troisieme moindre que ce second qui auroit la mesme proprieté et enfin un quatrieme, un cinquieme etc. a l'infini en descendant. Or est il qu'estant donné un nombre il n'y en a point infinis en descendant moindres que celuy la, j'entens parler tousjours des nombres entiers. D'ou on conclud qu'il est donc impossible qu'il y ait aucun triangle rectangle dont l'aire soit quarré. Vide foliu post sequens.
Je fus longtemps sans pouuoir appliquer ma methode aux questions affirmatiues, parce que le tour et le biais pour y venir est beaucoup plus malaisé que celuy dont je me sers aux negatives. De sorte que lors qu'il me falut demonstrer que tout nombre premier qui surpasse de l'unité un multiple de 4, est composé de deux quarrez je me treuuay en belle peine. Mais enfin une meditation diverses fois reiterée me donna les lumieres qui me manquoient. Et les questions affirmatiues passerent par ma methode a l'ayde de quelques nouueaux principes qu'il y fallust joindre par necessité. Ce progres de mon raisonnement en ces questions affirmatives estoit tel. Si un nombre premier pris a discretion qui surpasse de l'unité un multiple de 4 n'est point composé de deux quarrez il y aura un nombre premier de mesme nature moindre que le donné; et ensuite un troisieme encore moindre, etc. en descendant a l'infini jusques a ce que uous arriviez au nombre 5, qui est le moindre de tous ceux de cette nature, lequel il s'en suivroit n'estre pas composé de deux quarrez, ce qu'il est pourtant d'ou on doit inferer par la deduction a l'impossible que tous ceux de cette nature sont par consequent composez de 2 quarrez.
Il y a infinies questions de cette espece. Mais il y en a quelques autres qui demandent de nouveaux principes pour y appliquer la descente, et la recherche en est quelques fois si mal aisée, qu'on n'y peut venir qu'auec une peine extreme. Telle est la question suiuante que Bachet sur Diophante avoüe n'avoir jamais peu demonstrer, sur le suject de laquelle M ^r . Descartes fait dans une de ses lettres la mesme declaration, jusques la qu'il confesse qu'il la juge si difficile, qu'il ne voit point de voye pour la resoudre. Tout nombre est quarré, ou composé de deux, de trois, ou de quatre quarréz. Je l'ay enfin rangée sous ma methode et je demonstre que si un nombre donné n'estoit point de cette nature il y en auroit un moindre qui ne le seroit pas non plus, puis un troisieme moindre que le second &c. a l'infini, d'ou l'on infere que tous les nombres sont de cette nature.
J'ay ensuite consideré certaines questions qui bien que negatives ne restent pas de receuoir tres-grande difficulté, la methode pour y pratiquer la descente estant tout a fait diuerse des precedentes comme il sera aisé d'esprouuer. Telles sont les suiuantes. Il n'y a aucun cube diuisible en deux cubes. Il n'y a qu'un seul quarré en entiers qui augmenté du binaire fasse un cube, ledit quarré est 25. II n'y a que deux quarrez en entiers lesquels augmentés de 4 fassent cube, lesdits quarrez sont 4 et 121.
Apres auoir couru toutes ces questions la plupart de diuerses (sic) nature et de differente façon de demonstrer, j'ay passé a l'inuention des regles generales pour resoudre les equations simples et doubles de Diophante. On propose par exemple 2 quarr. + 7957 esgaux a un quarré (hoc est 2xx + 7967proptoquadr. ) J'ay une regle generale pour resoudre cette equation si elle est possible, ou decouvrir son impossibilité. Et ainsi en tons les cas et en tous nombres tant des quarrez que des unitez. On propose cette equation double 2x + 3 et 3x + 5 esgaux chaucon a un quarré. Bachet se glorifie en ses commentaires sur Diophante d'auoir trouvé une regle en deux cas particuliers. Je la donne generale en toute sorte de cas. Et determine par regle si elle est possible ou non.
Voila sommairement le conte de mes recherches sur le suject des nombres. Je ne l'ay escrit que parce que j'apprehende que le loisir d'estendre et de mettre au long toutes ces demonstrations et ces méthodes me manquera. En tout cas cette indication seruira aux sçauants pour trouver d'eux mesmes ce que je n'estens point, principalement si Mr. de Carcaui et Frénicle leur font part de quelques demonstrations par la descente que je leur ay enuoyees sur le suject de quelques propositions negatiues. Et peut estre la posterité me scaura gré de luy avoir fait connoistre que les anciens n'ont pas tout sceu, et cette relation pourra passer dans l'esprit de ceux qui viendront apres moy pour traditio lampadis ad filios, comme parle le grand Chancelier d'Angleterre, suiuant le sentiment et la deuise duquel j'adjousteray, multi pertransibunt et augebitur scientia.
(ii) I next proceed to mention Fermat's use in geometry of analysis and of infinitesimals. It would seem from his correspondence that he had thought out the principles of analytical geometry for himself before reading Descartes's Géométrie, and had realised that from the equation, or, as he calls it, the "specific property," of a curve all its properties could be deduced. His extant papers on geometry deal, however, mainly with the application of infinitesimals to the determination of the tangents to curves, to the quadrature of curves, and to questions of maxima and minima; probably these papers are a revision of his original manuscripts (which he destroyed), and were written about 1663, but there is no doubt that he was in possession of the general idea of his method for finding maxima and minima as early as 1628 or 1629.
He obtained the subtangent to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were x and x-e ; but there is nothing to indicate that he was aware that the process was general, and, though in the course of his work he used the principle, it is probable that he never separated it, so to speak, from the symbols of the particular problem he was considering. The first definite statement of the method was due to Barrow, and was published in 1669.
Fermat also obtained the areas of parabolas and hyperbolas of any order, and determined the centres of mass of a few simple laminae and of a paraboloid of revolution. As an example of his method of solving these questions I will quote his solution of the problem to find the area between the parabola y^3 = px^2 , the axis of x , and the line x=a . He says that, if the several ordinates at the points for which x is equal to a , a(1-e) , a(1-e)^2 , ... be drawn, then the area will be split into a number of little rectangles whose areas are respectively ae(pa^2)^1/3, ae(1-e) pa^2(1-e)^2^1/3, . The sum of these is p^1/3 a^5/3 e / {1-(1-e)^5/3} ; and by a subsidiary proposition (for he was not acquainted with the binomial theorem) he finds the limit of this, when e vanishes, to be 3/5 p^1/3 a^5/3 . The theorems last mentioned were published only after his death; and probably they were not written till after he had read the works of Cavalieri and Wallis.
Kepler had remarked that the values of a function immediately adjacent to and on either side of a maximum (or minimum) value must be equal. Fermat applied this principle to a few examples. Thus, to find the maximum value of x(a-x) , his method is essentially equivalent to taking a consecutive value of x , namely x-e where e is very small, and putting x(a-x) = (x-e)(a-x+e) . Simplifying, and ultimately putting e=0 , we get x = 1/2 a . This value of x makes the given expression a maximum.
(iii) Fermat must share with Pascal the honour of having founded the theory of probabilities. I have already mentioned the problem proposed to Pascal, and which he communicated to Fermat, and have there given Pascal's solution. Fermat's solution depends on the theory of combinations, and will be sufficiently illustrated by the following example, the substance of which is taken from a letter dated August 24, 1654, which occurs in the correspondence with Pascal. Fermat discusses the case of two players, A and B, where A wants two points to win and B three points. Then the game will be certainly decided in the course of four trials. Take the letters a and b, and write down all the combinations that can be formed of four letters. These combinations are 16 in number, namely, aaaa, aaab, aaba, aabb; abaa, abab, abba, abbb; baaa, baab, baba, babb; bbaa, bbab, bbba, bbbb. Now every combination in which a occurs twice or oftener represents a case favourable to A, and every combination in which b occurs three times or oftener represents a case favourable to B. Thus, on counting them, it will be found that there are 11 cases favourable to A, and 5 cases favourable to B; and, since these cases are all equally likely, A's chance of winning the game is to B's chance as 11 is to 5.
The only other problem on this subject which, as far as I know, attracted the attention of Fermat was also proposed to him by Pascal, and was as follows. A person undertakes to throw a six with a die in eight throws; supposing him to have made three throws without success, what portion of the stake should he be allowed to take on condition of giving up his fourth throw? Fermat's reasoning is as follows. The chance of success is 1/6 , so that he should be allowed to take 1/6 of the stake on condition of giving up his throw. But, if we wish to estimate the value of the fourth throw before any throw is made, then the first throw is worth 1/6 of the stake; the second is worth 1/6 of what remains, that is, 5/36 of the stake; the third throw is worth 1/6 of what now remains, that is, 25/216 of the stake; the fourth throw is worth 1/6 of what now remains, that is, 125/1296 of the stake.
Fermat does not seem to have carried the matter much further, but his correspondence with Pascal shows that his views on the fundamental principles of the subject were accurate: those of Pascal were not altogether correct.
Fermat's reputation is quite unique in the history of science. The problems on numbers which he had proposed long defied all efforts to solve them, and many of them yielded only to the skill of Euler. One still remains unsolved. This extraordinary achievement has overshadowed his other work, but in fact it is all of the highest order of excellence, and we can only regret that he thought fit to write so little.
Huygens. Christian Huygens was born at the Hague on April 14, 1629, and died in the same town on June 8, 1695. He generally wrote his name as Hugens, but I follow the usual custom in spelling it as above: it is also sometimes written as Huyghens. His life was uneventful, and there is little more to record in it than a statement of his various memoirs and researches.
In 1651 he published an essay in which he shewed the fallacy in a system of quadratures proposed by Grégoire de Saint-Vincent, who was well versed in the geometry of the Greeks, but had not grasped the essential points in the more modern methods. This essay was followed by tracts on the quadrature of the conics and the approximate rectification of the circle.
In 1654 his attention was directed to the improvement of the telescope. In conjunction with his brother he devised a new and better way of grinding and polishing lenses. As a result of these improvements he was able during the following two years, 1655 and 1656, to resolve numerous astronomical questions; as, for example, the nature of Saturn's appendage. His astronomical observations required some exact means of measuring time, and he was thus led in 1656 to invent the pendulum clock, as described in his tract Horologium, 1658. The time-pieces previously in use had been balance clocks.
In the year 1657 Huygens wrote a small work on the calculus of probabilities founded on the correspondence of Pascal and Fermat. He spent a couple of years in England about this time. His reputation was now so great that in 1665 Louis XIV. offered him a pension if he would live in Paris, which accordingly then became his place of residence.
In 1668 he sent to the Royal Society of London, in answer to a problem they had proposed, a memoir in which (simultaneously with Wallis and Wren) he proved by experiment that the momentum in a certain direction before the collision of two bodies is equal to the momentum in that direction after the collision. This was one of the points in mechanics on which Descartes had been mistaken.
The most important of Huygens's work was his Horologium Oscillatorium published at Paris in 1673. The first chapter is devoted to pendulum clocks. The second chapter contains a complete account of the descent of heavy bodies under their own weights in a vacuum, either vertically down or on smooth curves. Amongst other propositions he shews that the cycloid is tautochronous. In the third chapter he defines evolutes and involutes, proves some of their more elementary properties, and illustrates his methods by finding the evolutes of the cycloid and the parabola. These are the earliest instances in which the envelope of a moving line was determined. In the fourth chapter he solves the problem of the compound pendulum, and shews that the centres of oscillation and suspension are interchangeable. In the fifth and last chapter he discusses again the theory of clocks, points out that if the bob of the pendulum were, by means of cycloidal checks, made to oscillate in a cycloid the oscillations would be isochronous; and finishes by shewing that the centrifugal force on a body which moves round a circle of radius r with a uniform velocity v varies directly as v^2 and inversely as r . This work contains the first attempt to apply dynamics to bodies of finite size and not merely to particles.
In 1675 Huygens proposed to regulate the motion of watches by the use of the balance spring, in the theory of which he had been perhaps anticipated in a somewhat ambiguous and incomplete statement made by Hooke in 1658. Watches or portable clocks had been invented early in the sixteenth century, and by the end of that century were not very uncommon, but they were clumsy and unreliable, being driven by a main spring and regulated by a conical pulley and verge escapement; moreover, until 1687 they had only one hand. The first watch whose motion was regulated by a balance spring was made at Paris under Huygens's directions, and presented by him to Louis XIV.
The increasing intolerance of the Catholics led to his return to Holland in 1681, and after the revocation of the edict of Nantes he refused to hold any further communication with France. He now devoted himself to the construction of lenses of enormous focal length: of these three of focal lengths 123 feet, 180 feet, and 210 feet, were subsequently given by him to the Royal Society of London, in whose possession they still remain. It was about this time that he discovered the achromatic eye-piece (for a telescope) which is known by his name. In 1689 he came from Holland to England in order to make the acquaintance of Newton, whose Principia had been published in 1687. Huygens fully recognized the intellectual merits of the work, but seems to have deemed any theory incomplete which did not explain gravitation by mechanical means.
On his return in 1690 Huygens published his treatise on light in which the undulatory theory was expounded and explained. Most of this had been written as early as 1678. The general idea of the theory had been suggested by Robert Hooke in 1664, but he had not investigated its consequences in any detail. Only three ways have been suggested in which light can be produced mechanically. Either the eye may be supposed to send out something which, so to speak, feels the object (as the Greeks believed); or the object perceived may send out something which hits or affects the eye (as assumed in the emission theory); or there may be some medium between the eye and the object, and the object may cause some change in the form or condition of this intervening medium and thus affect the eye (as Hooke and Huygens supposed in the wave or undulatory theory). According to this last theory space is filled with an extremely rare ether, and light is caused by a series of waves or vibrations in this ether which are set in motion by the pulsations of the luminous body. From this hypothesis Huygens deduced the laws of reflexion and refraction, explained the phenomena of double refraction, and gave a construction for the extraordinary ray in biaxal crystals; while he found by experiment the chief phenomena of polarization.
The immense reputation and unrivalled powers of Newton led to disbelief in a theory which he rejected, and to the general adoption of Newton's emission theory. Within the present century crucial experiments have been devised which give different results according as one or the other theory is adopted; all these experiments agree with the results of the undulatory theory and differ from the results of the Newtonian theory; the latter is therefore untenable. Until, however, the theory of interference, suggested by Young, was worked out by Fresnel, the hypothesis of Huygens failed to account for all the facts, and even now the properties which, under it, have to be attributed to the intervening medium or ether involve difficulties of which we still seek a solution. Hence the problem as to how the effects of light are really produced cannot be said to be finally solved.
Besides these works Huygens took part in most of the controversies and challenges which then played so large a part in the mathematical world, and wrote several minor tracts. In one of these he investigated the form and properties of the catenary. In another he stated in general terms the rule for finding maxima and minima of which Fermat had made use, and shewed that the subtangent of an algebraical curve f(x, y) = 0 was equal to yfy/fx where fy is the derived function of f(x, y) regarded as a function of y . In some posthumous works, issued at Leyden in 1703, he further shewed how from the focal lengths of the component lenses the magnifying power of a telescope could be determined; and explained some of the phenomena connected with haloes and parhelia.
I should add that almost all his demonstrations, like those of Newton, are rigidly geometrical, and he would seem to have made no use of the differential or fluxional calculus, though he admitted the validity of the methods used therein. Thus, even when first written, his works were expressed in an archaic language, and perhaps received less attention than their intrinsic merits deserved.
I have now traced the development of mathematics for a period which we may take roughly as dating from 1635 to 1675, under the influence of Descartes, Cavalieri, Pascal, Wallis, Fermat, and Huygens. The life of Newton partly overlaps this period; his works and influence are considered in the next chapter.
I may dismiss the remaining mathematicians of this time with comparatively slight notice. The most eminent of them are Bachet, Barrow, Brouncker, Collins, De la Hire, de Laloubère, Frénicle, James Gregory, Hooke, Hudde, Nicholas Mercator, Mersenne, Pell, Roberval, Roemer, Rolle, Saint-Vincent, Sluze, Torricelli, Tschirnhausen, van Schooten, Viviani, and Wren. In the following notes I have arranged the above-mentioned mathematicians so that as far as possible their chief contributions shall come in chronological order.
Bachet. Claude Gaspard Bachet de Méziriac was born at Bourg in 1581, and died in 1638. He wrote the Problèmes plaisants, of which the first edition was issued in 1612, a second and enlarged edition was brought out in 1624; this contains an interesting collection of arithmetical tricks and questions, many of which are quoted in my Mathematical Recreations and Essays. He also wrote Les éléments arithmétiques, which exists in manuscript; and a translation of the Arithmetic of Diophantus. Bachet was the earliest writer who discussed the solution of indeterminate equations by means of continued fractions.
Mersenne. Marin Mersenne, born in 1588 and died at Paris in 1648, was a Franciscan friar, who made it his business to be acquainted and correspond with the French mathematicians of that date and many of their foreign contemporaries. In 1634 he published a translation of Galileo's mechanics; in 1644 he issued his Cogitata Physico-Mathematica, by which he is best known, containing an account of some experiments in physics; he also wrote a synopsis of mathematics, which was printed in 1664.
In the preface to the Cogitata a statement is made about perfect numbers, which implies that the only values of p not greater than 257 which make N prime, where N = 2^p - 1 , are 1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257: all prime values of N are known as Mersenne's Numbers. Some years ago I gave reasons for thinking that 67 was a misprint for 61. Until 1911, no error in this corrected statement was established, and it was gradually verified for all except sixteen values of p . In 1911, however, it was proved that N was prime when p = 89 , and three years later that it was prime when p = 107 : two facts at variance with Mersenne's statement. The prime or composite character of N now remains unknown for only ten values of p , namely, 139, 149, 157, 167, 193, 199, 227, 229, 241, and 257. We may safely say that the methods used to-day in establishing the known results for many of the higher values of p could not have been employed by Mersenne. It would be interesting to discover how he reached his conclusions, which are true if p does not exceed 88. Some recent writers conjecture that his statement was the result of a guess, intelligent though erroneous, as to the possible forms of p : I find it difficult to accept this opinion, but further discussion of the problem would be out of place here.
The theory of perfect numbers depends directly on that of Mersenne's Numbers. It is probable that all perfect numbers are included in the formula 2^p-1(2^p-1) , where 2^p-1 is a prime. Euclid proved that any number of this form is perfect; Euler shewed that the formula includes all even perfect numbers; and there is reason to believe--though a rigid demonstration is wanting--that an odd number cannot be perfect. If we assume that the last of these statements is true, then every perfect number is of the above form. Thus, if p = 2, 3, 5, 7, 13, 17, 19, 31, 61, then, by Mersenne's rule, the corresponding values of 2^p-1 are prime; they are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951; and the corresponding perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, and 2658455991569831744654692615953842176.
Roberval. Gilles Personier (de) Roberval, born at Roberval in 1602 and died at Paris in 1675, described himself from the place of his birth as de Roberval, a seigniorial title to which he had no right. He discussed the nature of the tangents to curves, solved some of the easier questions connected with the cycloid, generalized Archimedes's theorems on the spiral, wrote on mechanics, and on the method of indivisibles, which he rendered more precise and logical. He was a professor in the university of Paris, and in correspondence with nearly all the leading mathematicians of his time.
Van Schooten. Frans van Schooten, to whom we owe an edition of Vieta's works, succeeded his father (who had taught mathematics to Huygens, Hudde, and Sluze) as professor at Leyden in 1646. He brought out in 1659 a Latin translation of Descartes's Géométrie, and in 1657 a collection of mathematical exercises in which he recommended the use of co-ordinates in space of three dimensions. He died in 1661.
Saint-Vincent. Grégoire de Saint-Vincent, a Jesuit, born at Bruges in 1584 and died at Ghent in 1667, discovered the expansion of log (1+x) in ascending powers of x . Although a circle-squarer he is worthy of mention for the numerous theorems of interest which he discovered in his search after the impossible, and Montucla ingeniously remarks that "no one ever squared the circle with so much ability or (except for his principal object) with so much success." He wrote two books on the subject, one published in 1647 and the other in 1668, which cover some two or three thousand closely printed pages; the fallacy in the quadrature was pointed out by Huygens. In the former work he used indivisibles. An earlier work entitled Theoremata Mathematica, published in 1624, contains a clear account of the method of exhaustions, which is applied to several quadratures, notably that of the hyperbola.
Torricelli. Evangelista Torricelli, born at Faenza on Oct. 15, 1608, and died at Florence in 1647, wrote on the quadrature of the cycloid and conics; the rectification of the logarithmic spiral; the theory of the barometer; the value of gravity found by observing the motion of two weights connected by a string passing over a fixed pulley; the theory of projectiles; and the motion of fluids.
Hudde. Johann Hudde, burgomaster of Amsterdam, was born there in 1633, and died in the same town in 1704. He wrote two tracts in 1659: one was on the reduction of equations which have equal roots; in the other he stated what is equivalent to the proposition that if f(x, y) = 0 be the algebraical equation of a curve, then the subtangent is -ypartial f/partial y Big/ partial f/partial x ; but being ignorant of the notation of the calculus his enunciation is involved.
Frénicle. Bernard Frénicle de Bessy, born in Paris circ. 1605 and died in 1670, wrote numerous papers on combinations and on the theory of numbers, also on magic squares. It may be interesting to add that he challenged Huygens to solve the following system of equations in integers, x^2+y^2=z^2 , x^2=u^2+v^2 , x-y=u-v . A solution was given by M. Pépin in 1880.
De Laloubère. Antoine de Laloubère, a Jesuit, born in Languedoc in 1600 and died at Toulouse in 1664, is chiefly celebrated for an incorrect solution of Pascal's problems on the cycloid, which he gave in 1660, but he has a better claim to distinction in having been the first mathematician to study the properties of the helix.
N. Mercator. Nicholas Mercator (sometimes known as Kauffmann) was born in Holstein about 1620, but resided most of his life in England. He went to France in 1683, where he designed and constructed the fountains at Versailles, but the payment agreed on was refused unless he would turn Catholic; he died of vexation and poverty in Paris in 1687. He wrote a treatise on logarithms entitled Logarithmo-technica, published in 1668, and discovered the series (1+x) = x - 12 x^2 + 13 x^3 - 14 x^4 + ; he proved this by writing the equation of a hyperbola in the form y = 11+x = 1 - x + x^2 - x^3 + , to which Wallis's method of quadrature could be applied. The same series had been independently discovered by Saint-Vincent.
Barrow. Isaac Barrow was born in London in 1630, and died at Cambridge in 1677. He went to school first at Charterhouse (where he was so troublesome that his father was heard to pray that if it pleased God to take any of his children he could best spare Isaac), and subsequently to Felstead. He completed his education at Trinity College, Cambridge; after taking his degree in 1648, he was elected to a fellowship in 1649; he then resided for a few years in college, but in 1655 he was driven out by the persecution of the Independents. He spent the next four years in the East of Europe, and after many adventures returned to England in 1659. He was ordained the next year, and appointed to the professorship of Greek at Cambridge. In 1662 he was made professor of geometry at Gresham College, and in 1663 was selected as the first occupier of the Lucasian chair at Cambridge. He resigned the latter to his pupil Newton in 1669, whose superior abilities he recognized and frankly acknowledged. For the remainder of his life he devoted himself to the study of divinity. He was appointed master of Trinity College in 1672, and held the post until his death.
He is described as "low in stature, lean, and of a pale complexion," slovenly in his dress, and an inveterate smoker. He was noted for his strength and courage, and once when travelling in the East he saved the ship by his own prowess from capture by pirates. A ready and caustic wit made him a favourite of Charles II., and induced the courtiers to respect even if they did not appreciate him. He wrote with a sustained and somewhat stately eloquence, and with his blameless life and scrupulous conscientiousness was an impressive personage of the time.
His earliest work was a complete edition of the Elements of Euclid, which he issued in Latin in 1655, and in English in 1660; in 1657 he published an edition of the Data. His lectures, delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones Mathematicae; these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year, and suggest the analysis by which Archimedes was led to his chief results. In 1669 he issued his Lectiones Opticae et Geometricae. It is said in the preface that Newton revised and corrected these lectures, adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. This, which is his most important work in mathematics, was republished with a few minor alterations in 1674. In 1675 he published an edition with numerous comments of the first four books of the Conics of Apollonius, and of the extant works of Archimedes and Theodosius.
In the optical lectures many problems connected with the reflexion and refraction of light are treated with ingenuity. The geometrical focus of a point seen by reflexion or refraction is defined; and it is explained that the image of an object is the locus of the geometrical foci of every point on it. Barrow also worked out a few of the easier properties of thin lenses, and considerably simplified the Cartesian explanation of the rainbow.
The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves, and this is sufficiently important to require a detailed notice, because it illustrates the way in which Barrow, Hudde, and Sluze were working on the lines suggested by Fermat towards the methods of the differential calculus. Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known; hence, if the length of the subtangent MT could be found (thus determining the point T ), then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn, he got a small triangle PQR (which he called the differential triangle, because its sides PR and PQ were the differences of the abscissae and ordinates of P and Q ), so that TM:MP=QR:RP. To find QR:RP he supposed that x,y were the co-ordinates of P , and x-e,y-a those of Q (Barrow actually used p for x and m for y , but I alter these to agree with the modern practice). Substituting the co-ordinates of Q in the equation of the curve, and neglecting the squares and higher powers of e and a as compared with their first powers, he obtained e:a . The ratio a/e was subsequently (in accordance with a suggestion made by Sluze) termed the angular coefficient of the tangent at the point.
Barrow applied this method to the curves (i) x^2(x^2+y^2)=r^2y^2 ; (ii) x^3+y^3=r^3 ; (iii) x^3+y^3=rxy , called la galande; (iv) y=(r-x)tan pi x/2r , the quadratrix; and (v) y = rtan pi x/2r . It will be sufficient here if I take as an illustration the simpler case of the parabola y^2=px . Using the notation given above, we have for the point P , y^2=px ; and for the point Q , (y-a)^2=p(x-e) . Subtracting we get 2ay-a^2=pe . But, if a be an infinitesimal quantity, a^2 must be infinitely smaller and therefore may be neglected when compared with the quantities 2ay and pe . Hence 2ay=pe , that is, e:a=2y:p . Therefore TM:y=e:a=2y:p . Hence TM=2y^2/p=2x . This is exactly the procedure of the differential calculus, except that there we have a rule by which we can get the ratio a/e or dy/dx directly without the labour of going through a calculation similar to the above for every separate case.
Brouncker. William, Viscount Brouncker, one of the founders of the Royal Society of London, born about 1620, and died on April 5, 1684, was among the most brilliant mathematicians of this time, and was in intimate relations with Wallis, Fermat, and other leading mathematicians. I mentioned above his curious reproduction of Brahmagupta's solution of a certain indeterminate equation. Brouncker proved that the area enclosed between the equilateral hyperbola xy=1 , the axis of x , and the ordinates x=1 and x=2 , is equal either to 11 . 2 + 13 . 4 + 15 . 6 + , or to 1 - 12 + 13 - 14 + He also worked out other similar expressions for different areas bounded by the hyberbola and straight lines. He wrote on the rectification of the parabola and of the cycloid. It is noticeable that he used infinite series to express quantities whose values he could not otherwise determine. In answer to a request of Wallis to attempt the quadrature of the circle he shewed that the ratio of the area of a circle to the area of the circumscribed square, that is, the ratio of 4 to pi , is equal to 11 + 1^22 + 3^22 + 5^22 + 7^22 + Continued fractions had been employed by Bombelli in 1572, and had been systematically used by Cataldi in his treatise on finding the square roots of numbers, published at Bologna in 1613. Their properties and theory were given by Huygens, 1703, and Euler, 1744.
James Gregory. James Gregory, born at Drumoak near Aberdeen in 1638, and died at Edinburgh in October 1675, was successively professor at St. Andrews and Edinburgh. In 1660 he published his Optica Promota, in which the reflecting telescope known by his name is described. In 1667 he issued his Vera Circuli et Hyperbolae Quadratura in which he shewed how the areas of the circle and hyperbola could be obtained in the form of infinite convergent series, and here (I believe for the first time) we find a distinction drawn between convergent and divergent series. This work contains a remarkable geometrical proposition to the effect that the ratio of the area of any arbitrary sector of a circle to that of the inscribed or circumscribed regular polygons is not expressible by a finite number of algebraical terms. Hence he inferred that the quadrature of a circle was impossible; this was accepted by Montucla, but it is not conclusive, for it is conceivable that some particular sector might be squared, and this particular sector might be the whole circle. This book contains also the earliest enunciation of the expansions in series of sin x , cos x , sin^-1x or arcsin x , and cos^-1x or arccos x . It was reprinted in 1668 with an appendix, Geometriae Pars, in which Gregory explained how the volumes of solids of revolution could be determined. In 1671, or perhaps earlier, he established the theorem that = - 13 ^3 + 15 ^5 - , the result being true only if theta lie between -1/4pi and 1/4pi . This is the theorem on which many of the subsequent calculations of approximations to the numeral value of pi have been based.
Wren. Sir Christopher Wren was born at Knoyle, Wiltshire, on October 20, 1632, and died in London on February 25, 1723. Wren's reputation as a mathematician has been overshadowed by his fame as an architect, but he was Savilian professor of astronomy at Oxford from 1661 to 1673, and for some time president of the Royal Society. Together with Wallis and Huygens he investigated the laws of collision of bodies; he also discovered the two systems of generating lines on the hyperboloid of one sheet, though it is probable that he confined his attention to a hyperboloid of revolution. Besides these he wrote papers on the resistance of fluids, and the motion of the pendulum. He was a friend of Newton and (like Huygens, Hooke, Halley, and others) had made attempts to shew that the force under which the planets move varies inversely as the square of the distance from the sun.
Wallis, Brouncker, Wren, and Boyle (the last-named being a chemist and physicist rather than a mathematician) were the leading philosophers who founded the Royal Society of London. The society arose from the self-styled "indivisible college" in London in 1645; most of its members moved to Oxford during the civil war, where Hooke, who was then an assistant in Boyle's laboratory, joined in their meetings; the society was formally constituted in London in 1660, and was incorporated on July 15, 1662. The French Academy was founded in 1666, and the Berlin Academy in 1700. The Accademia dei Lincei was founded in 1603, but was dissolved in 1630.
Hooke. Robert Hooke, born at Freshwater on July 18, 1635, and died in London on March 3, 1703, was educated at Westminster, and Christ Church, Oxford, and in 1665 became professor of geometry at Gresham College, a post which he occupied till his death. He is still known by the law which he discovered, that the tension exerted by a stretched string is (within certain limits) proportional to the extension, or, in other words, that the stress is proportional to the strain. He invented and discussed the conical pendulum, and was the first to state explicitly that the motions of the heavenly bodies were merely dynamical problems. He was as jealous as he was vain and irritable, and accused both Newton and Huygens of unfairly appropriating his results. Like Huygens, Wren, and Halley, he made efforts to find the law of force under which the planets move about the sun, and he believed the law to be that of the inverse square of the distance. He, like Huygens, discovered that the small oscillations of a coiled spiral spring were practically isochronous, and was thus led to recommend (possibly in 1658) the use of the balance spring in watches. He had a watch of this kind made in London in 1675; it was finished just three months later than a similar one made in Paris under the directions of Huygens.
Collins. John Collins, born near Oxford on March 5, 1625, and died in London on November 10, 1683, was a man of great natural ability, but of slight education. Being devoted to mathematics, he spent his spare time in correspondence with the leading mathematicians of the time, for whom he was always ready to do anything in his power, and he has been described--not inaptly--as the English Mersenne. To him we are indebted for much information on the details of the discoveries of the period.
Pell. Another mathematician who devoted a considerable part of his time to making known the discoveries of others, and to correspondence with leading mathematicians, was John Pell. Pell was born in Sussex on March 1, 1610, and died in London on December 10, 1685. He was educated at Trinity College, Cambridge; he occupied in succession the mathematical chairs at Amsterdam and Breda; he then entered the English diplomatic service; but finally settled in 1661 in London, where he spent the last twenty years of his life. His chief works were an edition, with considerable new matter, of the Algebra by Branker and Rhonius, London, 1668; and a table of square numbers, London, 1672.
Sluze. René François Walther de Sluze (Slusius), canon of Liége, born on July 7, 1622, and died on March 19, 1685, found for the subtangent of a curve f(x,y)=0 an expression which is equivalent to -ypartial f/partial y Big/ partial f/partial x ; he wrote numerous tracts, and in particular discussed at some length spirals and points of inflexion.
Viviani. Vincenzo Viviani, a pupil of Galileo and Torricelli, born at Florence on April 5, 1622, and died there on September 22, 1703, brought out in 1659 a restoration of the lost book of Apollonius on conic sections, and in 1701 a restoration of the work of Aristaeus. He explained in 1677 how an angle could be trisected by the aid of the equilateral hyperbola or the conchoid. In 1692 he proposed the problem to construct four windows in a hemispherical vault so that the remainder of the surface can be accurately determined; a celebrated problem, of which analytical solutions were given by Wallis, Leibnitz, David Gregory, and James Bernoulli.
Tschirnhausen. Ehrenfried Walther von Tschirnhausen was born at Kislingswalde on April 10, 1631, and died at Dresden on October 11, 1708. In 1682 he worked out the theory of caustics by reflexion, or, as they were usually called, catacaustics, and shewed that they were rectifiable. This was the second case in which the envelope of a moving line was determined. He constructed burning mirrors of great power. The transformation by which he removed certain intermediate terms from a given algebraical equation is well known; it was published in the Acta Eruditorum for 1683.
De la Hire. Philippe De la Hire (or Lahire), born in Paris on March 18, 1640, and died there on April 21, 1719, wrote on graphical methods, 1673; on the conic sections, 1685; a treatise on epicycloids, 1694; one on roulettes, 1702; and, lastly, another on conchoids, 1708. His works on conic sections and epicycloids were founded on the teaching of Desargues, whose favourite pupil he was. He also translated the essay of Moschopulus on magic squares, and collected many of the theorems on them which were previously known; this was published in 1705.
Roemer. Olof Roemer, born at Aarhuus on September 25, 1644, and died at Copenhagen on September 19, 1710, was the first to measure the velocity of light; this was done in 1675 by means of the eclipses of Jupiter's satellites. He brought the transit and mural circle into common use, the altazimuth having been previously generally employed, and it was on his recommendation that astronomical observations of stars were subsequently made in general on the meridian. He was also the first to introduce micrometers and reading microscopes into an observatory. He also deduced from the properties of epicycloids the form of the teeth in toothed-wheels best fitted to secure a uniform motion.
Rolle. Michel Rolle, born at Ambert on April 21, 1652, and died in Paris on November 8, 1719, wrote an algebra in 1689, which contains the theorem on the position of the roots of an equation which is known by his name. He published in 1696 a treatise on the solutions of equations, whether determinate or indeterminate, and he produced several other minor works. He taught that the differential calculus, which, as we shall see later, had been introduced towards the close of the seventeenth century, was nothing but a collection of ingenious fallacies.
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