A History of Mathematics/Modern Europe/Vieta to Descartes

by Florian Cajori

 

VIETA TO DESCARTES.

 

The ecclesiastical power, which in the ignorant ages was an unmixed benefit, in more enlightened ages became a serious evil. Thus, in France, during the reigns preceding that of Henry IV., the theological spirit predominated. This is painfully shown by the massacres of Vassy and of St. Bartholomew. Being engaged in religious disputes, people had no leisure for science and for secular literature. Hence, down to the time of Henry IV., the French "had not put forth a single work, the destruction of which would now be a loss to Europe." In England, on the other hand, no religious wars were waged. The people were comparatively indifferent about religious strifes; they concentrated their ability upon secular matters, and acquired, in the sixteenth century, a literature which is immortalised by the genius of Shakespeare and Spenser. This great literary age in England was followed by a great scientific age. At the close of the sixteenth century, the shackles of ecclesiastical authority were thrown off by France. The ascension of Henry IV. to the throne was followed in 1598 by the Edict of Nantes, granting freedom of worship to the Huguenots, and thereby terminating religious wars. The genius of the French nation now began to blossom. Cardinal Richelieu, during the reign of Louis XIII., pursued the broad policy of not favouring the opinions of any sect, but of promoting the interests of the nation. His age was remarkable for the progress of knowledge. It produced that great secular literature, the counterpart of which was found in England in the sixteenth century. The seventeenth century was made illustrious also by the great French mathematicians, Roberval, Descartes, Desargues, Fermat, and Pascal.

More gloomy is the picture in Germany. The great changes which revolutionised the world in the sixteenth century, and which led England to national greatness, led Germany to degradation. The first effects of the Reformation there were salutary. At the close of the fifteenth and during the sixteenth century, Germany had been conspicuous for her scientific pursuits. She had been the leader in astronomy and trigonometry. Algebra also, excepting for the discoveries in cubic equations, was, before the time of Vieta, in a more advanced state there than elsewhere. But at the beginning of the seventeenth century, when the sun of science began to rise in France, it set in Germany. Theologic disputes and religious strife ensued. The Thirty Years' War (1618–1648) proved ruinous. The German empire was shattered, and became a mere lax confederation of petty despotisms. Commerce was destroyed; national feeling died out. Art disappeared, and in literature there was only a slavish imitation of French artificiality. Nor did Germany recover from this low state for 200 years; for in 1756 began another struggle, the Seven Years' War, which turned Prussia into a wasted land. Thus it followed that at the beginning of the seventeenth century, the great Kepler was the only German mathematician of eminence, and that in the interval of 200 years between Kepler and Gauss, there arose no great mathematician in Germany excepting Leibniz.

Up to the seventeenth century, mathematics was cultivated but little in Great Britain. During the sixteenth century, she brought forth no mathematician comparable with Vieta, Stifel, or Tartaglia. But with the time of Recorde, the English became conspicuous for numerical skill. The first important arithmetical work of English authorship was published in Latin in 1522 by Cuthbert Tonstall (1474–1559). He had studied at Oxford, Cambridge, and Padua, and drew freely from the works of Pacioli and Regiomontanus. Reprints of his arithmetic appeared in England and France. After Recorde the higher branches of mathematics began to be studied. Later, Scotland brought forth Napier, the inventor of logarithms. The instantaneous appreciation of their value is doubtless the result of superiority in calculation. In Italy, and especially in France, geometry, which for a long time had been an almost stationary science, began to be studied with success. Galileo, Torricelli, Roberval, Fermat, Desargues, Pascal, Descartes, and the English Wallis are the great revolutioners of this science. Theoretical mechanics began to be studied. The foundations were laid by Fermat and Pascal for the theory of numbers and the theory of probability.

We shall first consider the improvements made in the art of calculating. The nations of antiquity experimented thousands of years upon numeral notations before they happened to strike upon the so-called "Arabic notation." In the simple expedient of the cipher, which was introduced by the Hindoos about the fifth or sixth century after Christ, mathematics received one of the most powerful impulses. It would seem that after the "Arabic notation" was once thoroughly understood, decimal fractions would occur at once as an obvious extension of it. But "it is curious to think how much science had attempted in physical research and how deeply numbers had been pondered, before it was perceived that the all-powerful simplicity of the 'Arabic notation' was as valuable and as manageable in an infinitely descending as in an infinitely ascending progression."^[28] Simple as decimal fractions appear to us, the invention of them is not the result of one mind or even of one age. They came into use by almost imperceptible degrees. The first mathematicians identified with their history did not perceive their true nature and importance, and failed to invent a suitable notation. The idea of decimal fractions makes its first appearance in methods for approximating to the square roots of numbers. Thus John of Seville, presumably in imitation of Hindoo rules, adds $\scriptstyle {2n}$ ciphers to the number, then finds the square root, and takes this as the numerator of a fraction whose denominator is 1 followed by $\scriptstyle {n}$ ciphers. The same method was followed by Cardan, but it failed to be generally adopted even by his Italian contemporaries; for otherwise it would certainly have been at least mentioned by Cataldi (died 1626) in a work devoted exclusively to the extraction of roots. Cataldi finds the square root by means of continued fractions—a method ingenious and novel, but for practical purposes inferior to Cardan's. Orontius Finaeus (died 1555) in France, and William Buckley (died about 1550) in England extracted the square root in the same way as Cardan and John of Seville. The invention of decimals is frequently attributed to Regiomontanus, on the ground that instead of placing the sinus totus, in trigonometry, equal to a multiple of 60, like the Greeks, he put it $\scriptstyle {=100,000}$. But here the trigonometrical lines were expressed in integers, and not in fractions. Though he adopted a decimal division of the radius, he and his successors did not apply the idea outside of trigonometry and, indeed, had no notion whatever of decimal fractions. To Simon Stevin of Bruges in Belgium (1548–1620), a man who did a great deal of work in most diverse fields of science, we owe the first systematic treatment of decimal fractions. In his La Disme (1585) he describes in very express terms the advantages, not only of decimal fractions, but also of the decimal division in systems of weights and measures. Stevin applied the new fractions "to all the operations of ordinary arithmetic."^[25] What he lacked was a suitable notation. In place of our decimal point, he used a cipher; to each place in the fraction was attached the corresponding index. Thus, in his notation, the number 5.912 would be $\scriptstyle {{\overset {\mathsf {0}}{5}}{\overset {\mathsf {1}}{9}}{\overset {\mathsf {2}}{1}}{\overset {\mathsf {3}}{2}}}$ or $\scriptstyle {5\bigcirc \!\!\!\!{\mathsf {0}}\,9\bigcirc \!\!\!\!{\mathsf {1}}\,1\bigcirc \!\!\!\!{\mathsf {2}}\,2\bigcirc \!\!\!\!{\mathsf {3}}\,}$. These indices, though cumbrous in practice, are of interest, because they are the germ of an important innovation. To Stevin belongs the honour of inventing our present mode of designating powers and also of introducing fractional exponents into algebra. Strictly speaking, this had been done much earlier by Oresme, but it remained wholly unnoticed. Not even Stevin's innovations were immediately appreciated or at once accepted, but, unlike Oresme's, they remained a secure possession. No improvement was made in the notation of decimals till the beginning of the seventeenth century. After Stevin, decimals were used by Joost Bürgi, a Swiss by birth, who prepared a manuscript on arithmetic soon after 1592, and by Johann Hartmann Beyer, who assumes the invention as his own. In 1603, he published at Frankfurt on the Main a Logistica Decimalis. With Bürgi, a zero placed underneath the digit in unit's place answers as sign of separation. Beyer's notation resembles Stevin's. The decimal point, says Peacock, is due to Napier, who in 1617 published his Rabdologia, containing a treatise on decimals, wherein the decimal point is used in one or two instances. In the English translation of Napier's Mirifici logarithmorum canonis descriptio, executed by Edward Wright in 1616, and corrected by the author, the decimal point occurs in the tables. There is no mention of decimals in English arithmetics between 1619 and 1631. Oughtred in 1631 designates the fraction .56 thus, $\scriptstyle {0|{\underline {56}}}$. Albert Girard, a pupil of Stevin, in 1629 uses the point on one occasion. John Wallis in 1657 writes $\scriptstyle {12|{\underline {345}}}$, but afterwards in his algebra adopts the usual point. De Morgan says that "to the first quarter of the eighteenth century we must refer not only the complete and final victory of the decimal point, but also that of the now universal method of performing the operations of division and extraction of the square root."^[27] We have dwelt at some length on the progress of the decimal notation, because "the history of language…is of the highest order of interest, as well as utility: its suggestions are the best lesson for the future which a reflecting mind can have."^[27]

The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms. The invention of logarithms in the first quarter of the seventeenth century was admirably timed, for Kepler was then examining planetary orbits, and Galileo had just turned the telescope to the stars. During the Renaissance German mathematicians had constructed trigonometrical tables of great accuracy, but this greater precision enormously increased the work of the calculator. It is no exaggeration to say that the invention of logarithms "by shortening the labours doubled the life of the astronomer." Logarithms were invented by John Napier, Baron of Merchiston, in Scotland (1550–1617). It is one of the greatest curiosities of the history of science that Napier constructed logarithms before exponents were used. To be sure, Stifel and Stevin made some attempts to denote powers by indices, but this notation was not generally known,—not even to Harriot, whose algebra appeared long after Napier's death. That logarithms flow naturally from the exponential symbol was not observed until much later. It was Euler who first considered logarithms as being indices of powers. What, then, was Napier's line of thought?

Let AB be a definite line, DE a line extending from D indefinitely. Imagine two points starting at the same moment; the one moving from A toward B, the other from D toward E. Let the velocity during the first moment be the same for both: let that of the point on line DE be uniform; but the velocity of the point on AB decreasing in such a way that when it arrives at any point C, its velocity is proportional to the remaining distance BC. While the first point moves over a distance AC, the second one moves over a distance DF. Napier calls DF the logarithm of BC.

Napier's process is so unique and so different from all other modes of presenting the subject that there cannot be the shadow of a doubt that this invention is entirely his own; it is the result of unaided, isolated speculation. He first sought the logarithms only of sines; the line AB was the sine of 90° and was taken $\scriptstyle {=10^{7}}$; BC was the sine of the arc, and DF its logarithm. We notice that as the motion proceeds, BC decreases in geometrical progression, while DF increases in arithmetical progression. Let $\scriptstyle {AB=a=10^{7}}$, let $\scriptstyle {x=DF}$, $\scriptstyle {y=BC}$, then $\scriptstyle {AC=a-y}$. The velocity of the point C is $\scriptstyle {{d(a-y) \over {dt}}=y}$; this gives — nat. log $\scriptstyle {y=t+c}$. When $\scriptstyle {t=0}$, then $\scriptstyle {y=a}$ and $\scriptstyle {c=-{\text{nat. log }}a}$. Again, let $\scriptstyle {{dx \over dt}=a}$ be the velocity of the point F, then $\scriptstyle {x=at}$. Substituting for t and c their values and remembering that $\scriptstyle {a=10^{7}}$ and that by definition $\scriptstyle {x={\text{Nap. log }}y}$, we get

$\scriptstyle {{\text{Nap. log }}y=10^{7}{\text{ nat. log }}{10^{7} \over y}}$.

It is evident from this formula that Napier's logarithms are not the same as the natural logarithms. Napier's logarithms increase as the number itself decreases. He took the logarithm of $\scriptstyle {\sin {90}=0}$; i.e. the logarithm of $\scriptstyle {10^{7}=0}$. The logarithm of $\scriptstyle {\sin {\alpha }}$ increased from zero as $\scriptstyle {\alpha }$ decreased from 90°. Napier's genesis of logarithms from the conception of two flowing points reminds us of Newton's doctrine of fluxions. The relation between geometric and arithmetical progressions, so skilfully utilised by Napier, had been observed by Archimedes, Stifel, and others. Napier did not determine the base to his system of logarithms. The notion of a "base" in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system, but such a base would not reproduce accurately all of Napier's figures, owing to slight inaccuracies in the calculation of the tables. Napier's great invention was given to the world in 1614 in a work entitled Mirifici logarithmorum canonis descriptio. In it he explained the nature of his logarithms, and gave a logarithmic table of the natural sines of a quadrant from minute to minute.

Henry Briggs (1556–1631), in Napier's time professor of geometry at Gresham College, London, and afterwards professor at Oxford, was so struck with admiration of Napier's book, that he left his studies in London to do homage to the Scottish philosopher. Briggs was delayed in his journey, and Napier complained to a common friend, "Ah, John, Mr. Briggs will not come." At that very moment knocks were heard at the gate, and Briggs was brought into the lord's chamber. Almost one-quarter of an hour was spent, each beholding the other without speaking a word. At last Briggs began: "My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help in astronomy, viz. the logarithms; but, my lord, being by you found out, I wonder nobody found it out before, when now known it is so easy."^[28] Briggs suggested to Napier the advantage that would result from retaining zero for the logarithm of the whole sine, but choosing 10,000,000,000 for the logarithm of the 10th part of that same sine, i.e. of 5° 44′ 22″. Napier said that he had already thought of the change, and he pointed out a slight improvement on Briggs' idea; viz. that zero should be the logarithm of 1, and 10,000,000,000 that of the whole sine, thereby making the characteristic of numbers greater than unity positive and not negative, as suggested by Briggs. Briggs admitted this to be more convenient. The invention of "Briggian logarithms" occurred, therefore, to Briggs and Napier independently. The great practical advantage of the new system was that its fundamental progression was accommodated to the base, 10, of our numerical scale. Briggs devoted all his energies to the construction of tables upon the new plan. Napier died in 1617, with the satisfaction of having found in Briggs an able friend to bring to completion his unfinished plans. In 1624 Briggs published his Arithmetica logarithmica, containing the logarithms to 14 places of numbers, from 1 to 20,000 and from 90,000 to 100,000. The gap from 20,000 to 90,000 was filled up by that illustrious successor of Napier and Briggs, Adrian Vlacq of Gouda in Holland. He published in 1628 a table of logarithms from 1 to 100,000, of which 70,000 were calculated by himself. The first publication of Briggian logarithms of trigonometric functions was made in 1620 by Gunter, a colleague of Briggs, who found the logarithmic sines and tangents for every minute to seven places. Gunter was the inventor of the words cosine and cotangent. Briggs devoted the last years of his life to calculating more extensive Briggian logarithms of trigonometric functions, but he died in 1631, leaving his work unfinished. It was carried on by the English Henry Gellibrand, and then published by Vlacq at his own expense. Briggs divided a degree into 100 parts, but owing to the publication by Vlacq of trigonometrical tables constructed on the old sexagesimal division, Briggs' innovation remained unrecognised. Briggs and Vlacq published four fundamental works, the results of which "have never been superseded by any subsequent calculations."

The first logarithms upon the natural base e were published by John Speidell in his New Logarithmes (London, 1619), which contains the natural logarithms of sines, tangents, and secants.

The only possible rival of John Napier in the invention of logarithms was the Swiss Justus Byrgius (Joost Bürgi). He published a rude table of logarithms six years after the appearance of the Canon Mirificus, but it appears that he conceived the idea and constructed that table as early, if not earlier, than Napier did his. But he neglected to have the results published until Napier's logarithms were known and admired throughout Europe.

Among the various inventions of Napier to assist the memory of the student or calculator, is "Napier's rule of circular parts" for the solution of spherical right triangles. It is, perhaps, "the happiest example of artificial memory that is known."

The most brilliant conquest in algebra during the sixteenth century had been the solution of cubic and bi-quadratic equations. All attempts at solving algebraically equations of higher degrees remaining fruitless, a new line of inquiry—the properties of equations and their roots—was gradually opened up. We have seen that Vieta had attained a partial knowledge of the relations between roots and coefficients. Peletarius, a Frenchman, had observed as early as 1558, that the root of an equation is a divisor of the last term. One who extended the theory of equations somewhat further than Vieta, was Albert Girard (1590–1634), a Flemish mathematician. Like Vieta, this ingenious author applied algebra to geometry, and was the first who understood the use of negative roots in the solution of geometric problems. He spoke of imaginary quantities; inferred by induction that every equation has as many roots as there are units in the number expressing its degree; and first showed how to express the sums of their powers in terms of the coefficients. Another algebraist of considerable power was the English Thomas Harriot (1560–1621). He accompanied the first colony sent out by Sir Walter Raleigh to Virginia. After having surveyed that country he returned to England. As a mathematician, he was the boast of his country. He brought the theory of equations under one comprehensive point of view by grasping that truth in its full extent to which Vieta and Girard only approximated; viz. that in an equation in its simplest form, the coefficient of the second term with its sign changed is equal to the sum of the roots; the coefficient of the third is equal to the sum of the products of every two of the roots; etc. He was the first to decompose equations into their simple factors; but, since he failed to recognise imaginary and even negative roots, he failed also to prove that every equation could be thus decomposed. Harriot made some changes in algebraic notation, adopting small letters of the alphabet in place of the capitals used by Vieta. The symbols of inequality $\scriptstyle {>}$ and $\scriptstyle {<}$ were introduced by him. Harriot's work, Artis Analyticæ praxis, was published in 1631, ten years after his death. William Oughtred (1574–1660) contributed vastly to the propagation of mathematical knowledge in England by his treatises, which were long used in the universities. He introduced $\scriptstyle {\times }$ as symbol of multiplication, and $\scriptstyle {::}$ as that of proportion. By him ratio was expressed by only one dot. In the eighteenth century Christian Wolf secured the general adoption of the dot as a symbol of multiplication, and the sign for ratio was thereupon changed to two dots. Oughtred's ministerial duties left him but little time for the pursuit of mathematics during daytime, and evenings his economical wife denied him the use of a light.

Algebra was now in a state of sufficient perfection to enable Descartes to take that important step which forms one of the grand epochs in the history of mathematics,—the application of algebraic analysis to define the nature and investigate the properties of algebraic curves.

In geometry, the determination of the areas of curvilinear figures was diligently studied at this period. Paul Guldin (1577–1643), a Swiss mathematician of considerable note, rediscovered the following theorem, published in his Centrobaryca, which has been named after him, though first found in the Mathematical Collections of Pappus: The volume of a solid of revolution is equal to the area of the generating figure, multiplied by the circumference described by the centre of gravity. We shall see that this method excels that of Kepler and Cavalieri in following a more exact and natural course; but it has the disadvantage of necessitating the determination of the centre of gravity, which in itself may be a more difficult problem than the original one of finding the volume. Guldin made some attempts to prove his theorem, but Cavalieri pointed out the weakness of his demonstration.

Johannes Kepler (1671–1630) was a native of Würtemberg and imbibed Copernican principles while at the University of Tübingen. His pursuit of science was repeatedly interrupted by war, religious persecution, pecuniary embarrassments, frequent changes of residence, and family troubles. In 1600 he became for one year assistant to the Danish astronomer, Tycho Brahe, in the observatory near Prague. The relation between the two great astronomers was not always of an agreeable character. Kepler's publications are voluminous. His first attempt to explain the solar system was made in 1596, when he thought he had discovered a curious relation between the five regular solids and the number and distance of the planets. The publication of this pseudo-discovery brought him much fame. Maturer reflection and intercourse with Tycho Brahe and Galileo led him to investigations and results more worthy of his genius—"Kepler's laws." He enriched pure mathematics as well as astronomy. It is not strange that he was interested in the mathematical science which had done him so much service; for "if the Greeks had not cultivated conic sections, Kepler could not have superseded Ptolemy."^[11] The Greeks never dreamed that these curves would ever be of practical use; Aristæus and Apollonius studied them merely to satisfy their intellectual cravings after the ideal; yet the conic sections assisted Kepler in tracing the march of the planets in their elliptic orbits. Kepler made also extended use of logarithms and decimal fractions, and was enthusiastic in diffusing a knowledge of them. At one time, while purchasing wine, he was struck by the inaccuracy of the ordinary modes of determining the contents of kegs. This led him to the study of the volumes of solids of revolution and to the publication of the Stereometria Doliorum in 1615. In it he deals first with the solids known to Archimedes and then takes up others. Kepler introduced a new idea into geometry; namely, that of infinitely great and infinitely small quantities. Greek mathematicians always shunned this notion, but with it modern mathematicians have completely revolutionised the science. In comparing rectilinear figures, the method of superposition was employed by the ancients, but in comparing rectilinear and curvilinear figures with each other, this method failed because no addition or subtraction of rectilinear figures could ever produce curvilinear ones. To meet this case, they devised the Method of Exhaustion, which was long and difficult; it was purely synthetical, and in general required that the conclusion should be known at the outset. The new notion of infinity led gradually to the invention of methods immeasurably more powerful. Kepler conceived the circle to be composed of an infinite number of triangles having their common vertices at the centre, and their bases in the circumference; and the sphere to consist of an infinite number of pyramids. He applied conceptions of this kind to the determination of the areas and volumes of figures generated by curves revolving about any line as axis, but succeeded in solving only a few of the simplest out of the 84 problems which he proposed for investigation in his Stereometria.

Other points of mathematical interest in Kepler's works are (1) the statement of the earliest problem of inverse tangents; (2) an investigation which amounts to the evaluation of the definite integral $\scriptstyle {\int \limits _{~~0}^{~~\phi }\sin \phi d\phi =1-\cos \phi }$; (3) the assertion that the circumference of an ellipse, whose axes are $\scriptstyle {2a}$ and $\scriptstyle {2b}$, is nearly $\scriptstyle {\pi (a+b)}$; (4) a passage from which it has been inferred that Kepler knew the variation of a function near its maximum value to disappear; (5) the assumption of the principle of continuity (which differentiates modern from ancient geometry), when he shows that a parabola has a focus at infinity, that lines radiating from this "cæcus focus" are parallel and have no other point at infinity.

The Stereometria led Cavalieri, an Italian Jesuit, to the consideration of infinitely small quantities. Bonaventura Cavalieri (1598–1647), a pupil of Galileo and professor at Bologna, is celebrated for his Geometria indivisibilibus continuorum nova quadam ratione promota, 1635. This work expounds his method of Indivisibles, which occupies an intermediate place between the method of exhaustion of the Greeks and the methods of Newton and Leibniz. He considers lines as composed of an infinite number of points, surfaces as composed of an infinite number of lines, and solids of an infinite number of planes. The relative magnitude of two solids or surfaces could then be found simply by the summation of series of planes or lines. For example, he finds the sum of the squares of all lines making up a triangle equal to one-third the sum of the squares of all lines of a parallelogram of equal base and altitude; for if in a triangle, the first line at the apex be 1, then the second is 2, the third is 3, and so on; and the sum of their squares is

$\scriptstyle {1^{2}+2^{2}+3^{2}+\cdots +n^{2}=n(n+1)(2n+1)\div 6}$.

In the parallelogram, each of the lines is n and their number is n; hence the total sum of their squares is $\scriptstyle {n^{2}}$. The ratio between the two sums is therefore

$\scriptstyle {n(n+1)(2n+1)\div 6n^{3}={\frac {1}{3}}}$,

since n is infinite. From this he concludes that the pyramid or cone is respectively $\scriptstyle {\frac {1}{3}}$ of a prism or cylinder of equal base and altitude, since the polygons or circles composing the former decrease from the base to the apex in the same way as the squares of the lines parallel to the base in a triangle decrease from base to apex. By the Method of Indivisibles, Cavalieri solved the majority of the problems proposed by Kepler. Though expeditious and yielding correct results, Cavalieri's method lacks a scientific foundation. If a line has absolutely no width, then no number, however great, of lines can ever make up an area; if a plane has no thickness whatever, then even an infinite number of planes cannot form a solid. The reason why this method led to correct conclusions is that one area is to another area in the same ratio as the sum of the series of lines in the one is to the sum of the series of lines in the other. Though unscientific, Cavalieri's method was used for fifty years as a sort of integral calculus. It yielded solutions to some difficult problems. Guldin made a severe attack on Cavalieri and his method. The latter published in 1647, after the death of Guldin, a treatise entitled Exercitationes geometricœ sex, in which he replied to the objections of his opponent and attempted to give a clearer explanation of his method. Guldin had never been able to demonstrate the theorem named after him, except by metaphysical reasoning, but Cavalieri proved it by the method of indivisibles. A revised edition of the Geometry of Indivisibles appeared in 1653.

There is an important curve, not known to the ancients, which now began to be studied with great zeal. Roberval gave it the name of "trochoid," Pascal the name of "roulette," Galileo the name of "cycloid." The invention of this curve seems to be due to Galileo, who valued it for the graceful form it would give to arches in architecture. He ascertained its area by weighing paper figures of the cycloid against that of the generating circle, and found thereby the first area to be nearly but not exactly thrice the latter. A mathematical determination was made by his pupil, Evangelista Torricelli (1608–1647), who is more widely known as a physicist than as a mathematician.

By the Method of Indivisibles he demonstrated its area to be triple that of the revolving circle, and published his solution. This same quadrature had been effected a few years earlier by Roberval in France, but his solution was not known to the Italians. Roberval, being a man of irritable and violent disposition, unjustly accused the mild and amiable Torricelli of stealing the proof. This accusation of plagiarism created so much chagrin with Torricelli that it is considered to have been the cause of his early death. Vincenzo Viviani, another prominent pupil of Galileo, determined the tangent to the cycloid. This was accomplished in France by Descartes and Fermat.

In France, where geometry began to be cultivated with greatest success, Roberval, Fermat, Pascal, employed the Method of Indivisibles and made new improvements in it. Giles Persone de Roberval (1602–1675), for forty years professor of mathematics at the College of France in Paris, claimed for himself the invention of the Method of Indivisibles. Since his complete works were not published until after his death, it is difficult to settle questions of priority. Montucla and Chasles are of the opinion that he invented the method independent of and earlier than the Italian geometer, though the work of the latter was published much earlier than Roberval's. Marie finds it difficult to believe that the Frenchman borrowed nothing whatever from the Italian, for both could not have hit independently upon the word Indivisibles, which is applicable to infinitely small quantities, as conceived by Cavalieri, but not as conceived by Roberval. Roberval and Pascal improved the rational basis of the Method of Indivisibles, by considering an area as made up of an indefinite number of rectangles instead of lines, and a solid as composed of indefinitely small solids instead of surfaces. Roberval applied the method to the finding of areas, volumes, and centres of gravity. He effected the quadrature of a parabola of any degree $\scriptstyle {y^{m}=a^{m-1}x}$, and also of a parabola $\scriptstyle {y^{m}=a^{m-n}x^{n}}$. We have already mentioned his quadrature of the cycloid. Roberval is best known for his method of drawing tangents. He was the first to apply motion to the resolution of this important problem. His method is allied to Newton's principle of fluxions. Archimedes conceived his spiral to be generated by a double motion. This idea Roberval extended to all curves. Plane curves, as for instance the conic sections, may be generated by a point acted upon by two forces, and are the resultant of two motions. If at any point of the curve the resultant be resolved into its components, then the diagonal of the parallelogram determined by them is the tangent to the curve at that point. The greatest difficulty connected with this ingenious method consisted in resolving the resultant into components having the proper lengths and directions. Roberval did not always succeed in doing this, yet his new idea was a great step in advance. He broke off from the ancient definition of a tangent as a straight line having only one point in common with a curve,—a definition not valid for curves of higher degrees, nor apt even in curves of the second degree to bring out the properties of tangents and the parts they may be made to play in the generation of the curves. The subject of tangents received special attention also from Fermat, Descartes, and Barrow, and reached its highest development after the invention of the differential calculus. Fermat and Descartes defined tangents as secants whose two points of intersection with the curve coincide; Barrow considered a curve a polygon, and called one of its sides produced a tangent.

A profound scholar in all branches of learning and a mathematician of exceptional powers was Pierre de Fermat (1601–1665). He studied law at Toulouse, and in 1631 was made councillor for the parliament of Toulouse. His leisure time was mostly devoted to mathematics, which he studied with irresistible passion. Unlike Descartes and Pascal, he led a quiet and unaggressive life. Fermat has left the impress of his genius upon all branches of mathematics then known. A great contribution to geometry was his De maximis et minimis. About twenty years earlier, Kepler had first observed that the increment of a variable, as, for instance, the ordinate of a curve, is evanescent for values very near a maximum or a minimum value of the variable. Developing this idea, Fermat obtained his rule for maxima and minima. He substituted $\scriptstyle {x+e}$ for x in the given function of x and then equated to each other the two consecutive values of the function and divided the equation by e. If e be taken 0, then the roots of this equation are the values of x, making the function a maximum or a minimum. Fermat was in possession of this rule in 1629. The main difference between it and the rule of the differential calculus is that it introduces the indefinite quantity e instead of the infinitely small dx. Fermat made it the basis for his method of drawing tangents.

Owing to a want of explicitness in statement, Fermat's method of maxima and minima, and of tangents, was severely attacked by his great contemporary, Descartes, who could never be brought to render due justice to his merit. In the ensuing dispute, Fermat found two zealous defenders in Roberval and Pascal, the father; while Midorge, Desargues, and Hardy supported Descartes.

Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems."

A contemporary mathematician, whose genius excelled even that of the great Fermat, was Blaise Pascal (1623–1662). He was born at Clermont in Auvergne. In 1626 his father retired to Paris, where he devoted himself to teaching his son, for he would not trust his education to others. Blaise Pascal's genius for geometry showed itself when he was but twelve years old. His father was well skilled in mathematics, but did not wish his son to study it until he was perfectly acquainted with Latin and Greek. All mathematical books were hidden out of his sight. The boy once asked his father what mathematics treated of, and was answered, in general, "that it was the method of making figures with exactness, and of finding out what proportions they relatively had to one another." He was at the same time forbidden to talk any more about it, or ever to think of it. But his genius could not submit to be confined within these bounds. Starting with the bare fact that mathematics taught the means of making figures infallibly exact, he employed his thoughts about it and with a piece of charcoal drew figures upon the tiles of the pavement, trying the methods of drawing, for example, an exact circle or equilateral triangle. He gave names of his own to these figures and then formed axioms, and, in short, came to make perfect demonstrations. In this way he arrived unaided at the theorem that the sum of the three angles of a triangle is equal to two right angles. His father caught him in the act of studying this theorem, and was so astonished at the sublimity and force of his genius as to weep for joy. The father now gave him Euclid's Elements, which he, without assistance, mastered easily. His regular studies being languages, the boy employed only his hours of amusement on the study of geometry, yet he had so ready and lively a penetration that, at the age of sixteen, he wrote a treatise upon conics, which passed for such a surprising effort of genius, that it was said nothing equal to it in strength had been produced since the time of Archimedes. Descartes refused to believe that it was written by one so young as Pascal. This treatise was never published, and is now lost. Leibniz saw it in Paris and reported on a portion of its contents. The precocious youth made vast progress in all the sciences, but the constant application at so tender an age greatly impaired his health. Yet he continued working, and at nineteen invented his famous machine for performing arithmetical operations mechanically. This continued strain from overwork resulted in a permanent indisposition, and he would sometimes say that from the time he was eighteen, he never passed a day free from pain. At the age of twenty-four he resolved to lay aside the study of the human sciences and to consecrate his talents to religion. His Provincial Letters against the Jesuits are celebrated. But at times he returned to the favourite study of his youth. Being kept awake one night by a toothache, some thoughts undesignedly came into his head concerning the roulette or cycloid; one idea followed another; and he thus discovered properties of this curve even to demonstration. A correspondence between him and Fermat on certain problems was the beginning of the theory of probability. Pascal's illness increased, and he died at Paris at the early age of thirty-nine years.^[30] By him the answer to the objection to Cavalieri's Method of Indivisibles was put in the clearest form. Like Roberval, he explained "the sum of right lines" to mean "the sum of infinitely small rectangles." Pascal greatly advanced the knowledge of the cycloid. He determined the area of a section produced by any line parallel to the base; the volume generated by it revolving around its base or around the axis; and, finally, the centres of gravity of these volumes, and also of half these volumes cut by planes of symmetry. Before publishing his results, he sent, in 1658, to all mathematicians that famous challenge offering prizes for the first two solutions of these problems. Only Wallis and A. La Louère competed for them. The latter was quite unequal to the task; the former, being pressed for time, made numerous mistakes: neither got a prize. Pascal then published his own solutions, which produced a great sensation among scientific men. Wallis, too, published his, with the errors corrected. Though not competing for the prizes, Huygens, Wren, and Format solved some of the questions. The chief discoveries of Christopher Wren (1632–1723), the celebrated architect of St. Paul's Cathedral in London, were the rectification of a cycloidal arc and the determination of its centre of gravity. Fermat found the area generated by an arc of the cycloid. Huygens invented the cycloidal pendulum.

The beginning of the seventeenth century witnessed also a revival of synthetic geometry. One who treated conics still by ancient methods, but who succeeded in greatly simplifying many prolix proofs of Apollonius, was Claude Mydorge in Paris (1585–1647), a friend of Descartes. But it remained for Girard Desargues (1593–1662) of Lyons, and for Pascal, to leave the beaten track and cut out fresh paths. They introduced the important method of Perspective. All conics on a cone with circular base appear circular to an eye at the apex. Hence Desargues and Pascal conceived the treatment of the conic sections as projections of circles. Two important and beautiful theorems were given by Desargues: The one is on the "involution of the six points," in which a transversal meets a conic and an inscribed quadrangle; the other is that, if the vertices of two triangles, situated either in space or in a plane, lie on three lines meeting in a point, then their sides meet in three points lying on a line; and conversely. This last theorem has been employed in recent times by Branchion, Sturm, Gergonne, and Poncelet. Poncelet made it the basis of his beautiful theory of homoligical figures. We owe to Desargues the theory of involution and of transversals; also the beautiful conception that the two extremities of a straight line may be considered as meeting at infinity, and that parallels differ from other pairs of lines only in having their points of intersection at infinity. Pascal greatly admired Desargues' results, saying (in his Essais pour les Coniques), "I wish to acknowledge that I owe the little that I have discovered on this subject, to his writings." Pascal's and Desargues' writings contained the fundamental ideas of modern synthetic geometry. In Pascal's wonderful work on conics, written at the age of sixteen and now lost, were given the theorem on the anharmonic ratio, first found in Pappus, and also that celebrated proposition on the mystic hexagon, known as "Pascal's theorem," viz. that the opposite sides of a hexagon inscribed in a conic intersect in three points which are collinear. This theorem formed the keystone to his theory. He himself said that from this alone he deduced over 400 corollaries, embracing the conics of Apollonius and many other results. Thus the genius of Desargues and Pascal uncovered several of the rich treasures of modern synthetic geometry; but owing to the absorbing interest taken in the analytical geometry of Descartes and later in the differential calculus, the subject was almost entirely neglected until the present century.

In the theory of numbers no new results of scientific value had been reached for over 1000 years, extending from the times of Diophantus and the Hindoos until the beginning of the seventeenth century. But the illustrious period we are now considering produced men who rescued this science from the realm of mysticism and superstition, in which it had been so long imprisoned; the properties of numbers began again to be studied scientifically. Not being in possession of the Hindoo indeterminate analysis, many beautiful results of the Brahmins had to be re-discovered by the Europeans. Thus a solution in integers of linear indeterminate equations was re-discovered by the Frenchman Bachet de Méziriac (1581–1638), who was the earliest noteworthy European Diophantist. In 1612 he published Problèmes plaisants et délectables qui se font par lea nombres, and in 1621 a Greek edition of Diophantus with notes. The father of the modern theory of numbers is Fermat. He was so uncommunicative in disposition, that he generally concealed his methods and made known his results only. In some cases later analysts have been greatly puzzled in the attempt of supplying the proofs. Fermat owned a copy of Bachet's Diophantus, in which he entered numerous marginal notes. In 1670 these notes were incorporated in a new edition of Diophantus, brought out by his son. Other theorems on numbers, due to Fermat, were published in his Opera varia (edited by his son) and in Wallis's Commercium epistolicum of 1658. Of the following theorems, the first seven are found in the marginal notes:—

(1) $\scriptstyle {x^{n}+y^{n}=z^{n}}$ is impossible for integral values of x, y, and z, when $\scriptstyle {n>2}$. Remark: "I have found for this a truly wonderful proof, but the margin is too small to hold it." Repeatedly was this theorem made the prize question of learned societies. It has given rise to investigations of great interest and difficulty on the part of Euler, Lagrange, Dirichlet, and Kummer.

(2) A prime of the form $\scriptstyle {4n+1}$ is only once the hypothenuse of a right triangle; its square is twice; its cube is three times, etc. Example: $\scriptstyle {5^{2}=3^{2}+4^{2}}$; $\scriptstyle {25^{2}=15^{2}+20^{2}=7^{2}+24^{2}}$; $\scriptstyle {125^{2}=75^{2}+100^{2}=35^{2}+120^{2}=44^{2}+117^{2}}$.

(3) A prime of the form $\scriptstyle {4n+1}$ can be expressed once, and only once, as the sum of two squares. Proved by Euler.

(4) A number composed of two cubes can be resolved into two other cubes in an infinite multiplicity of ways.

(5) Every number is either a triangular number or the sum of two or three triangular numbers; either a square or the sum of two, three, or four squares; either a pentagonal number or the sum of two, three, four, or five pentagonal numbers; similarly for polygonal numbers in general. The proof of this and other theorems is promised by Fermat in a future work which never appeared. This theorem is also given, with others, in a letter of 1637(?) addressed to Pater Mersenne.

(6) As many numbers as you please may be found, such that the square of each remains a square on the addition to or subtraction from it of the sum of all the numbers.

(7) $\scriptstyle {x^{4}+y^{4}=z^{4}}$ is impossible.

(8) In a letter of 1640 he gives the celebrated theorem generally known as "Fermat's theorem," which we state in Gauss's notation: If p is prime, and a is prime to p, then $\scriptstyle {a^{p-1}\equiv 1\!\!\!\!{\pmod {p}}}$. It was proved by Euler.

(9) Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula $\scriptstyle {2^{2^{n}}+1}$ = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example $\scriptstyle {2^{2^{5}}+1}$ = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors, but was unable to explain the method by which he made his marvellous mental computation.

(10) An odd prime number can be expressed as the difference of two squares in one, and only one, way. This theorem, given in the Relation, was used by Fermat for the decomposition of large numbers into prime factors.

(11) If the integers a, b, c represent the sides of a right triangle, then its area cannot be a square number. This was proved by Lagrange.

(12) Fermat's solution of $\scriptstyle {ax^{2}+1=y^{2}}$, where a is integral but not a square, has come down in only the broadest outline, as given in the Relation. He proposed the problem to the Frenchman, Bernhard Frenicle de Bessy, and in 1657 to all living mathematicians. In England, Wallis and Lord Brounker conjointly found a laborious solution, which was published in 1658, and also in 1668, in an algebraical work brought out by John Pell. Though Pell had no other connection with the problem, it went by the name of "Pell's problem." The first solution was given by the Hindoos.

We are not sure that Fermat subjected all his theorems to rigorous proof. His methods of proof were entirely lost until 1879, when a document was found buried among the manuscripts of Huygens in the library of Leyden, entitled Relation des découvertes en la science des nombres. It appears from it that he used an inductive method, called by him la descente infinie ou indefinie. He says that this was particularly applicable in proving the impossibility of certain relations, as, for instance. Theorem 11, given above, but that he succeeded in using the method also in proving affirmative statements. Thus he proved Theorem 3 by showing that if we suppose there be a prime $\scriptstyle {4n+1}$ which does not possess this property, then there will be a smaller prime of the form $\scriptstyle {4n+1}$ not possessing it; and a third one smaller than the second, not possessing it; and so on. Thus descending indefinitely, he arrives at the number 5, which is the smallest prime factor of the form $\scriptstyle {4n+1}$. From the above supposition it would follow that 5 is not the sum of two squares—a conclusion contrary to fact. Hence the supposition is false, and the theorem is established. Fermat applied this method of descent with success in a large number of theorems. By this method Euler, Legendre, Dirichlet, proved several of his enunciations and many other numerical propositions.

A correspondence between Pascal and Fermat relating to a certain game of chance was the germ of the theory of probabilities, which has since attained a vast growth. Chevalier de Méré proposed to Pascal the fundamental problem, to determine the probability which each player has, at any given stage of the game, of winning the game. Pascal and Fermat supposed that the players have equal chances of winning a single point.

The former communicated this problem to Fermat, who studied it with lively interest and solved it by the theory of combinations, a theory which was diligently studied both by him and Pascal. The calculus of probabilities engaged the attention also of Huygens. The most important theorem reached by him was that, if A has p chances of winning a sum a, and q chances of winning a sum b, then he may expect to win the sum $\scriptstyle {{ap+bq} \over {p+q}}$. The next great work on the theory of probability was the Ars conjectandi of Jakob Bernoulli.

Among the ancients, Archimedes was the only one who attained clear and correct notions on theoretical statics. He had acquired firm possession of the idea of pressure, which lies at the root of mechanical science. But his ideas slept nearly twenty centuries, until the time of Stevin and Galileo. Stevin determined accurately the force necessary to sustain a body on a plane inclined at any angle to the horizon. He was in possession of a complete doctrine of equilibrium. While Stevin investigated statics, Galileo pursued principally dynamics. Galileo was the first to abandon the Aristotelian idea that bodies descend more quickly in proportion as they are heavier; he established the first law of motion; determined the laws of falling bodies; and, having obtained a clear notion of acceleration and of the independence of different motions, was able to prove that projectiles move in parabolic curves. Up to his time it was believed that a cannon-ball moved forward at first in a straight line and then suddenly fell vertically to the ground. Galileo had an understanding of centrifugal forces, and gave a correct definition of momentum. Though he formulated the fundamental principle of statics, known as the parallelogram of forces, yet he did not fully recognise its scope. The principle of virtual velocities was partly conceived by Guido Ubaldo (died 1607), and afterwards more fully by Galileo.

Galileo is the founder of the science of dynamics. Among his contemporaries it was chiefly the novelties he detected in the sky that made him celebrated, but Lagrange claims that his astronomical discoveries required only a telescope and perseverance, while it took an extraordinary genius to discover laws from phenomena, which we see constantly and of which the true explanation escaped all earlier philosophers. The first contributor to the science of mechanics after Galileo was Descartes.