Some Contemporaries of Descartes, Fermat, Pascal and Huygens

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 Cogita 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.

The preface to the Cogitata contains a statement (possibly due to Fermat) that, in order that may be prime, the only values of p, not greater than 257, which are possible are 1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257; the number 67 is probably a misprint for 61. With this correction the statement appears to be true, and it has been verified for all except twenty-one values of p, namely 71, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241, and 257. Of these values, Mersenne asserted that p = 127 and p = 257 make prime, and that the other nineteen values make a composite number. It has been asserted that the statement has been verified when p = 89 and 127, but these verifications rest on long numerical calculations made by single computators and not published; until these demonstrations have been confirmed we may say that twenty-one cases still await verification or require further investigation. The factors of when p = 89 are not known, the calculation merely showing that the number could not be prime. It is most likely that these results are particular cases of some general theorem on the subject which remains to be discovered.

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 , where 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 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 rectifications 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

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, x² + y² = z², x² = u² + v², 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

he proved this by writing the equation of a hyperbola in the form

to which Wallis's method of quadrature could be applied. The same series had been independently discovered by Saint-Vincent.

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

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.

Tchirnhausen

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

Ole 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.

René Descartes (1596 - 1650)

We may consider Descartes as the first of the modern school of mathematics. René Descartes was born near Tours on March 31, 1596, and died at Stockholm on February 11, 1650; thus he was a contemporary of Galileo and Desargues. His father, who, as the name implies, was of good family, was accustomed to spend half the year at Rennes when the local parliament, in which he held a commission as councillor, was in session, and the rest of the time on his family estate of Les Cartes at La Haye. René, the second of a family of two sons and one daughter, was sent at the age of eight years to the Jesuit School at La Flêche, and of the admirable discipline and education there given he speaks most highly. On account of his delicate health he was permitted to lie in bed till late in the mornings; this was a custom which he always followed, and when he visited Pascal in 1647 he told him that the only way to do good work in mathematics and to preserve his health was never to allow anyone to make him get up in the morning before he felt inclined to do so; an opinion which I chronicle for the benefit of any schoolboy into whose hands this work may fall.

On leaving school in 1612 Descartes went to Paris to be introduced to the world of fashion. Here, through the medium of the Jesuits, he made the acquaintance of Mydorge, and renewed his schoolboy friendship with Mersenne, and together with them he devoted the two years of 1615 and 1616 to the study of mathematics. At that time a man of position usually entered either the army or the church; Descartes chose the former profession, and in 1617 joined the army of Prince Maurice of Orange, then at Breda. Walking through the streets there he saw a placard in Dutch which excited his curiosity, and stopping the first passer, asked him to translate it into either French or Latin. The stranger, who happened to be Isaac Beeckman, the head of the Dutch College at Dort, offered to do so if Descartes would answer it; the placard being, in fact, a challenge to all the world to solve a certain geometrical problem. Descartes worked it out within a few hours, and a warm friendship between him and Beeckman was the result. This unexpected test of his mathematical attainments made the uncongenial life of the army distasteful to him, but under family influence and tradition he remained a soldier, and was persuaded at the commencement of the Thirty Years' War to volunteer under Count de Bucquoy in the army of Bavaria. He continued all this time to occupy his leisure with mathematical studies, and was accustomed to date the first ideas of his new philosophy and of his analytical geometry from three dreams which he experienced on the night of November 10, 1619, at Neuberg, when campaigning on the Danube. He regarded this as the critical day of his life, and one which determined his whole future.

He resigned his commission in the spring of 1621, and spent the next five years in travel, during most of which time he continued to study pure mathematics. In 1626 we find him settled at Paris, ``a little well-built figure, modestly clad in green taffety, and only wearing sword and feather in token of his quality as a gentleman.'' During the first two years there he interested himself in general society, and spent his leisure in the construction of optical instruments; but these pursuits were merely the relaxations of one who failed to find in philosophy that theory of the universe which he was convinced finally awaited him.

In 1628 Cardinal de Berulle, the founder of the Oratorians, met Descartes, and was so much impressed by his conversation that he urged on him the duty of devoting his life to the examination of truth. Descartes agreed, and the better to secure himself from interruption moved to Holland, then at the height of his power. There for twenty years he lived, giving up all his time to philosophy and mathematics. Science, he says, may be compared to a tree; metaphysics is the root, physics is the trunk, and the three chief branches are mechanics, medicine, and morals, these forming the three applications of our knowledge, namely, to the external world, to the human body, and to the conduct of life.

He spend the first four years, 1629 to 1633, of his stay in Holland in writing Le Monde, which embodies an attempt to give a physical theory of the universe; but finding that its publication was likely to bring on him the hostility of the church, and having no desire to pose as a martyr, he abandoned it: the incomplete manuscript was published in 1664. He then devoted himself to composing a treatise on universal science; this was published at Leyden in 1637 under the title Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences, and was accompanied with three appendices (which possibly were not issued till 1638) entitled La Dioptrique, Les Météores, and La Géométrie; it is from the last of these that the invention of analytical geometry dates. In 1641 he published a work called Meditationes, in which he explained at some length his views on philosophy as sketched out in the Discours. In 1644 he issued the Principia Philosophiae, the greater part of which was devoted to physical science, especially the laws of motion and the theory of vortices. In 1647 he received a pension from the French court in honour of his discoveries. He went to Sweden on the invitation of the Queen in 1649, and died a few months later of inflammation of the lungs.

In appearance, Descartes was a small man with large head, projecting brow, prominent nose, and black hair coming down to his eyebrows. His voice was feeble. In disposition he was cold and selfish. Considering the range of his studies he was by no means widely read, and he despised both learning and art unless something tangible could be extracted therefrom. He never married, and left no descendants, though he had one illegitimate daughter, who died young.

As to his philosophical theories, it will be sufficient to say that he discussed the same problems which have been debated for the last two thousand years, and probably will be debated with equal zeal two thousand years hence. It is hardly necessary to say that the problems themselves are of importance and interest, but from the nature of the case no solution ever offered is capable either of rigid proof or of disproof; all that can be effected is to make one explanation more probable than another, and whenever a philosopher like Descartes believes that he has at last finally settled a question it has been possible for his successors to point out the fallacy in his assumptions. I have read somewhere that philosophy has always been chiefly engaged with the inter-relations of God, Nature, and Man. The earliest philosophers were Greeks who occupied themselves mainly with the relations between God and Nature, and dealt with Man separately. The Christian Church was so absorbed in the relation of God to Man as entirely to neglect Nature. Finally, modern philosophers concern themselves chiefly with the relations between Man and Nature. Whether this is a correct historical generalization of the views which have been successively prevalent I do not care to discuss here, but the statement as to the scope of modern philosophy marks the limitations of Descartes's writings.

Descartes's chief contributions to mathematics were his analytical geometry and his theory of vortices, and it is on his researches in connection with the former of these subjects that his mathematical reputation rests.

Analytical geometry does not consist merely (as is sometimes loosely said) in the application of algebra to geometry; that had been done by Archimedes and many others, and had become the usual method of procedure in the works of the mathematicians of the sixteenth century. The great advance made by Descartes was that he saw that a point in a plane could be completely determined if its distances, say x and y, from two fixed lines drawn at right angles in the plane were given, with the convention familiar to us as to the interpretation of positive and negative values; and that though an equation f(x,y) = 0 was indeterminate and could be satisfied by an infinite number of values of x and y, yet these values of x and y determined the co-ordinates of a number of points which form a curve, of which the equation f(x,y) = 0 expresses some geometrical property, that is, a property true of the curve at every point on it. Descartes asserted that a point in space could be similarly determined by three co-ordinates, but he confined his attention to plane curves.

It was at once seen that in order to investigate the properties of a curve it was sufficient to select, as a definition, any characteristic geometrical property, and to express it by means of an equation between the (current) co-ordinates of any point on the curve, that is, to translate the definition into the language of analytical geometry. The equation so obtained contains implicitly every property of the curve, and any particular property can be deduced from it by ordinary algebra without troubling about the geometry of the figure. This may have been dimly recognized or foreshadowed by earlier writers, but Descartes went further and pointed out the very important facts that two or more curves can be referred to one and the same system of co-ordinates, and that the points in which two curves intersect can be determined by finding the roots common to their two equations. I need not go further into details, for nearly everyone to whom the above is intelligible will have read analytical geometry, and is able to appreciate the value of its invention.

Descartes's Géométrie is divided into three books: the first two of these treat of analytical geometry, and the third includes an analysis of the algebra then current. It is somewhat difficult to follow the reasoning, but the obscurity was intentional. ``Je n'ai rien omis.'' says he, ``qu'à dessein ... j'avois prévu que certaines gens qui se vantent de sçavoir tout n'auroient par manqué de dire que je n'avois rien écrit qu'ils n'eussent sçu auparavant, si je me fusse rendu assez intelligible pour eux.''

The first book commences with an explanation of the principles of analytical geometry, and contains a discussion of a certain problem which had been propounded by Pappus in the seventh book of his and of which some particular cases had been considered by Euclid and Apollonius. The general theorem had baffled previous geometricians, and it was in the attempt to solve it that Descartes was led to the invention of analytical geometry. The full enunciation of the problem is rather involved, but the most important case is to find the locus of a point such that the product of the perpendiculars on m given straight lines shall be in a constant ratio to the product of the perpendiculars on n other given straight lines. The ancients had solved this geometrically for the case m = 1, n = 1, and the case m = 1, n = 2. Pappus had further stated that, if m = n = 2, the locus is a conic, but he gave no proof; Descartes also failed to prove this by pure geometry, but he shewed that the curve is represented by an equation of the second degree, that is, a conic; subsequently Newton gave an elegant solution of the problem by pure geometry.

In the second book Descartes divides curves into two classes, namely, geometrical and mechanical curves. He defines geometrical curves as those which can be generated by the intersection of two lines each moving parallel to one co-ordinate axis with ``commensurable'' velocities; by which terms he means that dy/dx is an algebraical function, as, for example, is the case in the ellipse and the cissoid. He calls a curve mechanical when the ratio of the velocities of these lines is ``incommensurable''; by which term he means that dy/dx is a trancendental function, as, for example, is the case in the cycloid and the quadratrix. Descartes confined his discussion to geometrical curves, and did not treat of the theory of mechanical curves. The classification into algebraical and transcendental curves now usual is due to Newton.

Descartes also paid particular attention to the theory of the tangents to curves - as perhaps might be inferred from his system of classification just alluded to. The then current definition of a tangent at a point was a straight line through the point such that between it and the curve no other straight line could be drawn, that is, the straight line of closest contact. Descartes proposed to substitute for this a statement equivalent to the assertion that the tangent is the limiting position of the secant; Fermat, and at a later date Maclaurin and Lagrange, adopted this definition. Barrow, followed by Newton and Leibnitz, considered a curve as the limit of an inscribed polygon when the sides become indefinitely small, and stated that the side of the polygon when produced became in the limit a tangent to the curve. Roberval, on the other hand, defined a tangent at a point as the direction of motion at that instant of a point which was describing the curve. The results are the same whichever definition is selected, but the controversy as to which definition was the correct one was none the less lively. In his letters Descartes illustrated his theory by giving the general rule for drawing tangents and normals to a roulette.

The method used by Descartes to find the tangent or normal at any point of a given curve was substantially as follows. He determined the centre and radius of a circle which should cut the curve in two consecutive points there. The tangent to the circle at that point will be the required tangent to the curve. In modern text-books it is usual to express the condition that two of the points in which a straight line (such as y = mx + c) cuts the curve shall coincide with the given point: this enables us to determine m and c, and thus the equation of the tangent there is determined. Descartes, however, did not venture to do this, but selecting a circle as the simplest curve and one to which he knew how to draw a tangent, he so fixed his circle as to make it touch the given curve at the point in question, and thus reduced the problem to drawing a tangent to a circle. I should note in passing that he only applied this method to curves which are symmetrical about an axis, and he took the centre of the circle on the axis.

The obscure style deliberately adopted by Descartes diminished the circulation and immediate appreciation of these books; but a Latin translation of them, with explanatory notes, was prepared by F. de Beaune, and an edition of this, with a commentary by F. van Schooten, issued in 1659, was widely read.

The third book of the Géométrie contains an analysis of the algebra then current, and it has affected the language of the subject by fixing the custom of employing the letters at the beginning of the alphabet to denote known quantities, and those at the end of the alphabet to denote unknown quantities. [On the origin of the custom of using x to represent an unknown example, see a note by G. Eneström in the Bibliotheca Mathematica, 1885, p. 43.] Descartes further introduced the system of indices now in use; very likely it was original on his part, but I would here remind the reader that the suggestion had been made by previous writers, though it had not been generally adopted. It is doubtful whether or not Descartes recognized that his letters might represent any quantities, positive or negative, and that it was sufficient to prove a proposition for one general case. He was the earliest writer to realise the advantage to be obtained by taking all the terms of an equation to one side of it, though Stifel and Harriot had sometimes employed that form by choice. He realised the meaning of negative quantities and used them freely. In this book he made use of the rule for finding the limit to the number of positive and of negative roots of an algebraical equation, which is still known by his name; and introduced the method of indeterminate coefficients for the solution of equations. He believed that he had given a method by which algebraical equations of any order could be solved, but in this he was mistaken. It may also be mentioned that he enunciated the theorem, commonly attributed to Euler, on the relation between the numbers of faces, edges and angles of a polyhedron: this is in one of the papers published by Careil.

Of the two other appendices to the Discours one was devoted to optics. The chief interest of this consists in the statement given of the law of refraction. This appears to have been taken from Snell's work, though, unfortunately, it is enunciated in a way which might lead a reader to suppose that it is due to the researches of Descartes. Descartes would seem to have repeated Snell's experiments when in Paris in 1626 or 1627, and it is possible that he subsequently forgot how much he owed to the earlier investigations of Snell. A large part of the optics is devoted to determining the best shape for the lenses of a telescope, but the mechanical difficulties in grinding a surface of glass to a required form are so great as to render these investigations of little practical use. Descartes seems to have been doubtful whether to regard the rays of light as proceeding from the eye and so to speak touching the object, as the Greeks had done, or as proceeding from the object, and so affecting the eye; but, since he considered the velocity of light to be infinite, he did not deem the point particularly important.

The other appendix, on meteors, contains an explanation of numerous atmospheric phenomena, including the rainbow; the explanation of the latter is necessarily incomplete, since Descartes was unacquainted with the fact that the refractive index of a substance is different for lights of different colours.

Descartes's physical theory of the universe, embodying most of the results contained in his earlier and unpublished Le Monde, is given in his Principia, 1644, and rests on a metaphysical basis. He commences with a discussion on motion; and then lays down ten laws of nature, of which the first two are almost identical with the first two laws of motion as given by Newton; the remaining eight laws are inaccurate. He next proceeds to discuss the nature of matter which he regards as uniform in kind though there are three forms of it. He assumes that the matter of the universe must be in motion, and that the motion must result in a number of vortices. He states that the sun is the centre of an immense whirlpool of this matter, in which the planets float and are swept round like straws in a whirlpool of water. Each planet is supposed to be the centre of a secondary whirlpool by which its satellites are carried: these secondary whirlpools are supposed to produce variations of density in the surrounding medium which constitute the primary whirlpool, and so cause the planets to move in ellipses and not in circles. All these assumptions are arbitrary and unsupported by any investigation. It is not difficult to prove that on his hypothesis the sun would be in the centre of these ellipses, and not at a focus (as Kepler had shewn was the case), and that the weight of a body at every place on the surface of the earth except the equator would act in a direction which was not vertical; but it will be sufficient here to say that Newton in the second book of his Principia, 1687, considered the theory in detail, and shewed that its consequences are not only inconsistent with each of Kepler's laws and with the fundamental laws of mechanics, but are also at variance with the laws of nature assumed by Descartes. Still, in spite of its crudeness and its inherent defects, the theory of vortices marks a fresh era in astronomy, for it was an attempt to explain the phenomena of the whole universe by the same mechanical laws which experiment shews to be true on the earth.