A Short Account of the History of Mathematics · W. W. Rouse Ball
See footnote to chapter xii.
The closing years of the renaissance were marked by a revival of interest in nearly all branches of mathematics and science. As far as pure mathematics is concerned we have already seen that during the last half of the sixteenth century there had been a great advance in algebra, theory of equations, and trigonometry; and we shall shortly see (in the second section of this chapter) that in the early part of the seventeenth century some new processes in geometry were invented. If, however, we turn to applied mathematics it is impossible not to be struck by the fact that even as late as the middle or end of the sixteenth century no marked progress in the theory had been made from the time of Archimedes. Statics (of solids) and hydrostatics remained in much the state in which he had left them, while dynamics as a science did not exist. It was Stevinus who gave the first impulse to the renewed study of statics, and Galileo who laid the foundation of dynamics; and to their works the first section of this chapter is devoted.
Stevinus.
Simon Stevinus was born at Bruges in 1548, and died at the Hague in 1620. We know very little of his life save that he was originally a merchant's clerk at Antwerp, and at a later period of his life was the friend of Prince Maurice of Orange, by whom he was made quartermaster-general of the Dutch army.
To his contemporaries he was best known for his works on fortifications and military engineering, and the principles he laid down are said to be in accordance with those which are now usually accepted. To the general populace he was also well known on account of his invention of a carriage which was propelled by sails; this ran on the sea-shore, carried twenty-eight people, and easily outstripped horses galloping by the side; his model of it was destroyed in 1802 by the French when they invaded Holland. It was chiefly owing to the influence of Stevinus that the Dutch and French began a proper system of book-keeping in the national accounts.
I have already alluded to the introduction in his Arithmetic, published in 1585, of exponents to mark the power to which quantities were raised; for instance, he wrote 3x^2 - 5x + 1 as 3 circledigit2 -5 circledigit1 + 1 circledigit0 . His notation for decimal fractions was of a similar character. He further suggested the use of fractional (but not negative) exponents. In the same book he likewise suggested a decimal system of weights and measures.
He also published a geometry which is ingenious though it does not contain many results which were not previously known; in it some theorems on perspective are enunciated.
It is, however, on his Statics and Hydrostatics, published (in Flemish) at Leyden in 1586, that his fame rests. In this work he enunciates the triangle of forces--a theorem which some think was first propounded by Leonardo da Vinci. Stevinus regards this as the fundamental proposition of the subject. Previous to the publication of his work the science of statics had rested on the theory of the lever, but subsequently it became usual to commence by proving the possibility of representing forces by straight lines, and thus reducing many theorems to geometrical propositions, and in particular to obtaining in this way a proof of the parallelogram (which is equivalent to the triangle) of forces. Stevinus is not clear in his arrangement of the various propositions or in their logical sequence, and the new treatment of the subject was not definitely established before the appearance in 1687 of Varignon's work on mechanics. Stevinus also found the force which must be exerted along the line of greatest slope to support a given weight on an inclined plane--a problem the solution of which had been long in dispute. He further distinguishes between stable and unstable equilibrium. In hydrostatics he discusses the question of the pressure which a fluid can exercise, and explains the so-called hydrostatic paradox.
His method of finding the resolved part of a force in a given direction, as illustrated by the case of a weight resting on an inclined plane, is a good specimen of his work and is worth quoting.
He takes a wedge ABC whose base AC is horizontal [and whose sides BA , BC are in the ratio of 2 to 1]. A thread connecting a number of small equal equidistant weights is placed over the wedge as indicated in the figure below (which I reproduce from his demonstration) so that the number of these weights on BA is to the number on BC in the same proportion as BA is to BC . This is always possible if the dimensions of the wedge be properly chosen, and he places four weights resting on BA and two on BC ; but we may replace these weights by a heavy uniform chain TSLVT without altering his argument. He says in effect, that experience shews that such a chain will remain at rest; if not, we could obtain perpetual motion. Thus the effect in the direction BA of the weight of the part TS of the chain must balance the effect in the direction BC of the weight of the part TV of the chain. Of course BC may be vertical, and if so the above statement is equivalent to saying that the effect in the direction BA of the weight of the chain on it is diminished in the proportion of BC to BA ; in other words, if a weight W rests on an inclined plane of inclination alpha the component of W down the line of greatest slope is W sin alpha .
Stevinus was somewhat dogmatic in his statements, and allowed no one to differ from his conclusions, "and those," says he, in one place, "who cannot see this, may the Author of nature have pity upon their unfortunate eyes, for the fault is not in the thing, but in the sight which we are unable to give them."
Galileo. Just as the modern treatment of statics originates with Stevinus, so the foundation of the science of dynamics is due to Galileo. Galileo Galilei was born at Pisa on February 18, 1564, and died near Florence on January 8, 1642. His father, a poor descendant of an old and noble Florentine house, was himself a fair mathematician and a good musician. Galileo was educated at the monastery of Vallombrosa, where his literary ability and mechanical ingenuity attracted considerable attention. He was persuaded to become a novitiate of the order in 1579, but his father, who had other views, at once removed him, and sent him in 1581 to the university of Pisa to study medicine. It was there that he noticed that the great bronze lamp, hanging from the roof of the cathedral, performed its oscillations in equal times, and independently of whether the oscillations were large or small--a fact which he verified by counting his pulse. He had been hitherto kept in ignorance of mathematics, but one day, by chance hearing a lecture on geometry (by Ricci), he was so fascinated by the science that thenceforward he devoted his leisure to its study, and finally got leave to discontinue his medical studies. He left the university in 1585, and almost immediately commenced his original researches.
He published in 1586 an account of the hydrostatic balance, and in 1588 an essay on the centre of gravity in solids; these were not printed till later. The fame of these works secured for him in 1589 the appointment to the mathematical chair at Pisa--the stipend, as was then the case with most professorships, being very small. During the next three years he carried on, from the leaning tower, that series of experiments on falling bodies which established the first principles of dynamics. Unfortunately, the manner in which he promulgated his discoveries, and the ridicule he threw on those who opposed him, gave not unnatural offence, and in 1591 he was obliged to resign his position.
At this time he seems to have been much hampered by want of money. Influence was, however, exerted on his behalf with the Venetian senate, and he was appointed professor at Padua, a chair which he held for eighteen years, 1592--1610. His lectures there seem to have been chiefly on mechanics and hydrostatics, and the substance of them is contained in his treatise on mechanics, which was published in 1612. In these lectures he repeated his Pisan experiments, and demonstrated that falling bodies did not (as was then commonly believed) descend with velocities proportional, amongst other things, to their weights. He further shewed that, if it were assumed that they descended with a uniformly accelerated motion, it was possible to deduce the relations connecting velocity, space, and time which did actually exist. At a later date, by observing the times of descent of bodies sliding down inclined planes, he shewed that this hypothesis was true. He also proved that the path of a projectile is a parabola, and in doing so implicitly used the principles laid down in the first two laws of motion as enunciated by Newton. He gave an accurate definition of momentum which some writers have thought may be taken to imply a recognition of the truth of the third law of motion. The laws of motion are, however, nowhere enunciated in a precise and definite form, and Galileo must be regarded rather as preparing the way for Newton than as being himself the creator of the science of dynamics.
In statics he laid down the principle that in machines what was gained in power was lost in speed, and in the same ratio. In the statics of solids he found the force which can support a given weight on an inclined plane; in hydrostatics he propounded the more elementary theorems on pressure and on floating bodies; while among hydrostatical instruments he used, and perhaps invented, the thermometer, though in a somewhat imperfect form.
It is, however, as an astronomer that most people regard Galileo, and though, strictly speaking, his astronomical researches lie outside the subject-matter of this book, it may be interesting to give the leading facts. It was in the spring of 1609 that Galileo heard that a tube containing lenses had been made by an optician, Hans Lippershey, of Middleburg, which served to magnify objects seen through it. This gave him the clue, and he constructed a telescope of that kind which still bears his name, and of which an ordinary opera-glass is an example. Within a few months he had produced instruments which were capable of magnifying thirty-two diameters, and within a year he had made and published observations on the solar spots, the lunar mountains, Jupiter's satellites, the phases of Venus, and Saturn's ring. The discovery of the microscope followed naturally from that of the telescope. Honours and emoluments were showered on him, and he was enabled in 1610 to give up his professorship and retire to Florence. In 1611 he paid a temporary visit to Rome, and exhibited in the gardens of the Quirinal the new worlds revealed by the telescope.
It would seem that Galileo had always believed in the Copernican system, but was afraid of promulgating it on account of the ridicule it excited. The existence of Jupiter's satellites seemed, however, to make its truth almost certain, and he now boldly preached it. The orthodox party resented his action, and in February 1616 the Inquisition declared that to suppose the sun the centre of the solar system was false, and opposed to Holy Scripture. The edict of March 5, 1616, which carried this into effect, has never been repealed, though it has been long tacitly ignored. It is well known that towards the middle of the seventeenth century the Jesuits evaded it by treating the theory as an hypothesis from which, though false, certain results would follow.
In January 1632 Galileo published his dialogues on the system of the world, in which in clear and forcible language he expounded the Copernican theory. In these, apparently through jealousy of Kepler's fame, he does not so much as mention Kepler's laws (the first two of which had been published in 1609, and the third in 1619); he rejects Kepler's hypothesis that the tides are caused by the attraction of the moon, and tries to explain their existence (which he alleges is a confirmation of the Copernican hypothesis) by the statement that different parts of the earth rotate with different velocities. He was more successful in showing that mechanical principles would account for the fact that a stone thrown straight up falls again to the place from which it was thrown--a fact which previously had been one of the chief difficulties in the way of any theory which supposed the earth to be in motion.
The publication of this book was approved by the papal censor, but substantially was contrary to the edict of 1616. Galileo was summoned to Rome, forced to recant, do penance, and was released only on promise of obedience. The documents recently printed show that he was threatened with the torture, but probably there was no intention of carrying the threat into effect.
When released he again took up his work on mechanics, and by 1636 had finished a book which was published under the title Discorsi intorno a due nuove scienze at Leyden in 1638. In 1637 he lost his sight, but with the aid of pupils he continued his experiments on mechanics and hydrostatics, and in particular on the possibility of using a pendulum to regulate a clock, and on the theory of impact.
An anecdote of this time has been preserved which, though probably not authentic, is sufficiently interesting to bear repetition. According to one version of the story, Galileo was interviewed by some members of a Florentine guild who wanted their pumps altered so as to raise water to a height which was greater than thirty feet; and thereupon he remarked that it might be desirable to first find out why the water rose at all. A bystander intervened and said there was no difficulty about that, because nature abhorred a vacuum. Yes, said Galileo, but apparently it is only a vacuum which is less than thirty feet. His favourite pupil Torricelli was present, and thus had his attention directed to the question, which he subsequently elucidated.
Galileo's work may, I think, be fairly summed up by saying that his researches on mechanics are deserving of high praise, and that they are memorable for clearly enunciating the fact that science must be founded on laws obtained by experiment; his astronomical observations and his deductions therefrom were also excellent, and were expounded with a literary skill which leaves nothing to be desired; but though he produced some of the evidence which placed the Copernican theory on a satisfactory basis, he did not himself make any special advance in the theory of astronomy.
Francis Bacon. The necessity of an experimental foundation for science was also advocated with considerable effect by Galileo's contemporary Francis Bacon (Lord Verulam), who was born at London on Jan. 22, 1561, and died on April 9, 1626. He was educated at Trinity College, Cambridge. His career in politics and at the bar culminated in his becoming Lord Chancellor, with the title of Lord Verulam. The story of his subsequent degradation for accepting bribes is well known.
His chief work is the Novum Organum, published in 1620, in which he lays down the principles which should guide those who are making experiments on which they propose to found a theory of any branch of physics or applied mathematics. He gave rules by which the results of induction could be tested, hasty generalisations avoided, and experiments used to check one another. The influence of this treatise in the eighteenth century was great, but it is probable that during the preceding century it was little read, and the remark repeated by several French writers that Bacon and Descartes are the creators of modern philosophy rests on a misapprehension of Bacon's influence on his contemporaries; any detailed account of this book belongs, however, to the history of scientific ideas rather than to that of mathematics.
Before leaving the subject of applied mathematics I may add a few words on the writings of Guldinus, Wright, and Snell.
Guldinus. Habakkuk Guldinus, born at St. Gall on June 12, 1577, and died at Grätz on Nov. 3, 1643, was of Jewish descent, but was brought up as a Protestant; he was converted to Roman Catholicism, and became a Jesuit, when he took the Christian name of Paul, and it was to him that the Jesuit colleges at Rome and Grätz owed their mathematical reputation. The two theorems known by the name of Pappus (to which I have alluded above) were published by Guldinus in the fourth book of his De Centro Gravitatis, Vienna, 1635--1642. Not only were the rules in question taken without acknowledgment from Pappus, but (according to Montucla) the proof of them given by Guldinus was faulty, though he was successful in applying them to the determination of the volumes and surfaces of certain solids. The theorems were, however, previously unknown, and their enunciation excited considerable interest.
Wright. I may here also refer to Edward Wright, who is worthy of mention for having put the art of navigation on a scientific basis. Wright was born in Norfolk about 1560, and died in 1615. He was educated at Caius College, Cambridge, of which society he was subsequently a fellow. He seems to have been a good sailor, and he had a special talent for the construction of instruments. About 1600 he was elected lecturer on mathematics by the East India Company; he then settled in London, and shortly afterwards was appointed mathematical tutor to Henry, Prince of Wales, the son of James I. His mechanical ability may be illustrated by an orrery of his construction by which it was possible to predict eclipses; it was shewn in the Tower as a curiosity as late as 1675.
In the maps in use before the time of Gerard Mercator a degree, whether of latitude or longitude, had been represented in all cases by the same length, and the course to be pursued by a vessel was marked on the map by a straight line joining the ports of arrival and departure. Mercator had seen that this led to considerable errors, and had realised that to make this method of tracing the course of a ship at all accurate the space assigned on the map to a degree of latitude ought gradually to increase as the latitude increased. Using this principle, he had empirically constructed some charts, which were published about 1560 or 1570. Wright set himself the problem to determine the theory on which such maps should be drawn, and succeeded in discovering the law of the scale of the maps, though his rule is strictly correct for small arcs only. The result was published in the second edition of Blundeville's Exercises.
In 1599 Wright published his Certain Errors in Navigation Detected and Corrected, in which he explained the theory and inserted a table of meridional parts. The reasoning shews considerable geometrical power. In the course of the work he gives the declinations of thirty-two stars, explains the phenomena of the dip, parallax, and refraction, and adds a table of magnetic declinations; he assumes the earth to be stationary. In the following year he published some maps constructed on his principle. In these the northernmost point of Australia is shewn; the latitude of London is taken to be 51° 32' .
Snell. A contemporary of Guldinus and Wright was Willebrod Snell, whose name is still well known through his discovery in 1619 of the law of refraction in optics. Snell was born at Leyden in 1581, occupied a chair of mathematics at the university there, and died there on Oct. 30, 1626. He was one of those infant prodigies who occasionally appear, and at the age of twelve he is said to have been acquainted with the standard mathematical works. I will here only add that in geodesy he laid down the principles for determining the length of the arc of a meridian from the measurement of any base line, and in spherical trigonometry he discovered the properties of the polar or supplemental triangle.
The close of the sixteenth century was marked not only by the attempt to found a theory of dynamics based on laws derived from experiment, but also by a revived interest in geometry. This was largely due to the influence of Kepler.
Kepler.
Johann Kepler, one of the founders of modern astronomy, was born of humble parents near Stuttgart on Dec. 27, 1571, and died at Ratisbon on Nov. 15, 1630. He was educated under Mästlin at Tübingen. In 1593 he was appointed professor at Grätz, where he made the acquaintance of a wealthy widow, whom he married, but found too late that he had purchased his freedom from pecuniary troubles at the expense of domestic happiness. In 1599 he accepted an appointment as assistant to Tycho Brahe, and in 1601 succeeded his master as astronomer to the emperor Rudolph II. But his career was dogged by bad luck: first his stipend was not paid; next his wife went mad and then died, and a second marriage in 1611 did not prove fortunate; while, to complete his discomfort, he was expelled from his chair, and narrowly escaped condemnation for heterodoxy. During this time he depended for his income on telling fortunes and casting horoscopes, for, as he says, "nature which has conferred upon every animal the means of existence has designed astrology as an adjunct and ally to astronomy." He seems, however, to have had no scruple in charging heavily for his services, and to the surprise of his contemporaries was found at his death to possess a considerable hoard of money. He died while on a journey to try and recover for the benefit of his children some of the arrears of his stipend.
In describing Galileo's work I alluded briefly to the three laws in astronomy that Kepler had discovered, and in connection with which his name will be always associated. I may further add that he suggested that the planets might be retained in their orbits by magnetic vortices, but this was little more than a crude conjecture. I have also already mentioned the prominent part he took in bringing logarithms into general use on the continent. These are familiar facts; but it is not known so generally that Kepler was also a geometrician and algebraist of considerable power, and that he, Desargues, and perhaps Galileo, may be considered as forming a connecting link between the mathematicians of the renaissance and those of modern times.
Kepler's work in geometry consists rather in certain general principles enunciated, and illustrated by a few cases, than in any systematic exposition of the subject. In a short chapter on conics inserted in his Paralipomena, published in 1604, he lays down what has been called the principle of continuity, and gives as an example the statement that a parabola is at once the limiting case of an ellipse and of a hyperbola; he illustrates the same doctrine by reference to the foci of conics (the word focus was introduced by him); and he also explains that parallel lines should be regarded as meeting at infinity. He introduced the use of the eccentric angle in discussing properties of the ellipse.
In his Stereometria, which was published in 1615, he determines the volumes of certain vessels and the areas of certain surfaces, by means of infinitesimals instead of by the long and tedious method of exhaustions. These investigations as well as those of 1604 arose from a dispute with a wine merchant as to the proper way of gauging the contents of a cask. This use of infinitesimals was objected to by Guldinus and other writers as inaccurate, but though the methods of Kepler are not altogether free from objection he was substantially correct, and by applying the law of continuity to infinitesimals he prepared the way for Cavalieri's method of indivisibles, and the infinitesimal calculus of Newton and Leibnitz.
Kepler's work on astronomy lies outside the scope of this book. I will mention only that it was founded on the observations of Tycho Brahe, whose assistant he was. His three laws of planetary motion were the result of many and laborious efforts to reduce the phenomena of the solar system to certain simple rules. The first two were published in 1609, and stated that the planets describe ellipses round the sun, the sun being in a focus; and that the line joining the sun to any planet sweeps over equal areas in equal times. The third was published in 1619, and stated that the squares of the periodic times of the planets are proportional to the cubes of the major axes of their orbits. The laws were deduced from observations on the motions of Mars and the earth, and were extended by analogy to the other planets. I ought to add that he attempted to explain why these motions took place by a hypothesis which is not very different from Descartes's theory of vortices. He suggested that the tides were caused by the attraction of the moon. Kepler also devoted considerable time to the elucidation of the theories of vision and refraction in optics.
While the conceptions of the geometry of the Greeks were being extended by Kepler, a Frenchman, whose works until recently were almost unknown, was inventing a new method of investigating the subject--a method which is now known as projective geometry. This was the discovery of Desargues, whom I put (with some hesitation) at the close of this period, and not among the mathematicians of modern times.
Desargues. Gérard Desargues, born at Lyons in 1593, and died in 1662, was by profession an engineer and architect, but he gave some courses of gratuitous lectures in Paris from 1626 to about 1630 which made a great impression upon his contemporaries. Both Descartes and Pascal had a high opinion of his work and abilities, and both made considerable use of the theorems he had enunciated.
In 1636 Desargues issued a work on perspective; but most of his researches were embodied in his Brouillon proiect on conics, published in 1639, a copy of which was discovered by Chasles in 1845. I take the following summary of it from C. Taylor's work on conics. Desargues commences with a statement of the doctrine of continuity as laid down by Kepler: thus the points at the opposite ends of a straight line are regarded as coincident, parallel lines are treated as meeting at a point at infinity, and parallel planes on a line at infinity, while a straight line may be considered as a circle whose centre is at infinity. The theory of involution of six points, with its special cases, is laid down, and the projective property of pencils in involution is established. The theory of polar lines is expounded, and its analogue in space suggested. A tangent is defined as the limiting case of a secant, and an asymptote as a tangent at infinity. Desargues shows that the lines which join four points in a plane determine three pairs of lines in involution on any transversal, and from any conic through the four points another pair of lines can be obtained which are in involution with any two of the former. He proves that the points of intersection of the diagonals and the two pairs of opposite sides of any quadrilateral inscribed in a conic are a conjugate triad with respect to the conic, and when one of the three points is at infinity its polar is a diameter; but he fails to explain the case in which the quadrilateral is a parallelogram, although he had formed the conception of a straight line which was wholly at infinity. The book, therefore, may be fairly said to contain the fundamental theorems on involution, homology, poles and polars, and perspective.
The influence exerted by the lectures of Desargues on Descartes, Pascal, and the French geometricians of the seventeenth century was considerable; but the subject of projective geometry soon fell into oblivion, chiefly because the analytical geometry of Descartes was so much more powerful as a method of proof or discovery.
The researches of Kepler and Desargues will serve to remind us that as the geometry of the Greeks was not capable of much further extension, mathematicians were now beginning to seek for new methods of investigation, and were extending the conceptions of geometry. The invention of analytical geometry and of the infinitesimal calculus temporarily diverted attention from pure geometry, but at the beginning of the last century there was a revival of interest in it, and since then it has been a favourite subject of study with many mathematicians.
Thus by the beginning of the seventeenth century we may say that the fundamental principles of arithmetic, algebra, theory of equations, and trigonometry had been laid down, and the outlines of the subjects as we know them had been traced. It must be, however, remembered that there were no good elementary text-books on these subjects; and a knowledge of them was therefore confined to those who could extract it from the ponderous treatises in which it lay buried. Though much of the modern algebraical and trigonometrical notation had been introduced, it was not familiar to mathematicians, nor was it even universally accepted; and it was not until the end of the seventeenth century that the language of these subjects was definitely fixed. Considering the absence of good text-books, I am inclined rather to admire the rapidity with which it came into universal use, than to cavil at the hesitation to trust to it alone which many writers showed.
If we turn to applied mathematics, we find, on the other hand, that the science of statics had made but little advance in the eighteen centuries that had elapsed since the time of Archimedes, while the foundations of dynamics were laid by Galileo only at the close of the sixteenth century. In fact, as we shall see later, it was not until the time of Newton that the science of mechanics was placed on a satisfactory basis. The fundamental conceptions of mechanics are difficult, but the ignorance of the principles of the subject shown by the mathematicians of this time is greater than would have been anticipated from their knowledge of pure mathematics.
With this exception, we may say that the principles of analytical geometry and of the infinitesimal calculus were needed before there was likely to be much further progress. The former was employed by Descartes in 1637, the latter was invented by Newton some thirty or forty years later, and their introduction may be taken as marking the commencement of the period of modern mathematics.
THIRD PERIODModern Mathematicsmodern
The history of modern mathematics begins with the invention of analytical geometry and the infinitesimal calculus. The mathematics is far more complex than that produced in either of the preceding periods; but, during the seventeenth and eighteenth centuries, it may be generally described as characterized by the development of analysis, and its application to the phenomena of nature.
I continue the chronological arrangement of the subject. Chapter xv
contains the history of the forty years from 1635 to 1675, and an account of the mathematical discoveries of Descartes, Cavalieri, Pascal, Wallis, Fermat, and Huygens. Chapter xvi is given up to a discussion of Newton's researches. Chapter xvii contains an account of the works of Leibnitz and his followers during the first half of the eighteenth century (including D'Alembert), and of the contemporary English school to the death of Maclaurin. The works of Euler, Lagrange, Laplace, and their contemporaries form the subject-matter of chapter xviii.
Lastly, in chapter xix I have added some notes on a few of the mathematicians of recent times; but I exclude all detailed reference to living writers, and partly because of this, partly for other reasons there given, the account of contemporary mathematics does not profess to cover the subject.
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