Development of Geometry.

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oblivion mainly owing to the fact that the system of Analytical Geometry introduced by Descartes was far more powerful as a method of research.

The Cartesian system of Analytical Geometry was expounded by Descartes in the tract on Geometry appended to his Discours. In effect, Descartes asserted that the position of 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 coordinates of a number of points which form a curve, of which the equation f(x, y)=0 expressed 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 coordinates; 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) coordinates 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 recognised 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 coordinates, and that the points in which two curves intersect can be determined by finding the roots common to their two equations.

We need not describe the details of Descartes' work. His great reputation ensured appreciation of his investigations, and an edition of this tract with notes by Beaune and a commentary by van Schooten, issued in 1659, became a standard text-book; henceforth the subject was familiar to mathematicians. It should perhaps be added that it is probable that the principles of Analytical Geometry had been worked out independently by Pierre de Fermat of Toulouse (1601-65) at least as early as by Descartes; but, as they were not then published, we need not discuss this point further.

More than one writer at this time concerned himself with the division of quantities, such as areas and volumes, into infinitesimals, and with the summation of such infinitesimals, thus escaping the long and tedious method of exhaustions used by the Greeks. In this connexion we should in particular mention the names of Kepler, Cavalieri, and somewhat later that of Fermat. The most important exposition of the subject was that given by Wallis in 1656, in which he applied it to determine the quadrature of a curve whose equation could be expressed in the form

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Development of Mechanics.

y=Zaam. These investigations foreshadowed the introduction of the infinitesimal calculus by Newton and Leibniz towards the end of the seventeenth century.

Before leaving the subject of Pure Mathematics, we must in passing mention the theory of numbers and that of probabilities. The former, under the stimulus of the writings of one of the greatest mathematicians, Fermat, attracted considerable attention. The latter was created by Pascal (1623-62) and Fermat.

Pure Mathematics are a useful if not necessary instrument of research; but the general reader takes more interest in the history of their application than in their own-in results rather than in methods. We turn now to consider the development of applied science during this period.

As before, we begin with Mechanics. Simon Stevinus of Bruges (1548–1620), who died at the Hague, used, though he did not explicitly enunciate, the triangle of forces, which he treated as the fundamental theorem of Statics (1586). A similar position was taken up by Galileo (1564-1642). A year or two later the last-mentioned mathematician laid the foundations of the science of Dynamics. In 1589, when professor at Pisa, he made experiments from the leaning tower there on the rate at which bodies of different weights would fall. It was at once apparent that the generally accepted assertion of Aristotle was incorrect, and that, save for the resistance of the air, all bodies fell at the same rate, and through distances proportional to the square of the time which had elapsed from the instant when they were allowed to drop. Of this Galileo gave a public demonstration; but, though his Aristotelian colleagues could not explain the result, many of them preferred to assert that there must be some mistake rather than admit the possibility that Aristotle was wrong. The ridicule cast by Galileo on this argument caused friction, and in 1591 he was obliged to resign his chair. His writings at this time show that he had already formed correct ideas of momentum and centrifugal force. He had proved that the path of a projectile was a parabola, and was aware that the pendulum was isochronous. The last fact he discovered by noticing that the great bronze lamp hanging from the roof of the cathedral at Pisa performed its oscillations, whether large or small, in equal times. He nowhere stated the laws of motion in a definite form; but probably he was acquainted generally with the principles of the first two laws as enunciated by Newton. His astronomical work was accomplished shortly after he left Pisa, and to this reference is made below. Towards the end of his life he again took up the subject of Mechanics, and a book by him, published in 1638, has been described as a masterpiece of popular exposition of its principles. In it he describes his pendulum experiments, and the theory of impact. A year or two later he invented a pendulum clock, though the fact was not generally known at the time. Mechanics were discussed by Descartes in 1644; but he did not substantially advance the theory. The correct

Development of Astronomy.

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theory of impact was given by Wren and Wallis. The next marked development of the subject took place under the influence of Huygens and Newton and is referred to later.

The most striking achievement of this period in the eyes of an ordinary citizen of the time was the establishment of the Copernican system of Astronomy. We have already alluded to the publication by Copernicus of his hypothesis. The next stage in its development was due to Kepler (1571-1630). He served under Tycho Brahe, one of the most skilful observers of his time, and making use of Brahe's observations succeeded, after many and laborious efforts, in reducing the planetary motions to three comparatively simple laws. 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. These laws pointed to the fact that the sun and not the earth should be regarded as the centre of the solar system. We may add that Kepler attempted to explain why these motions took place by a hypothesis which is somewhat like Descartes' theory of vortices described below. He also suggested that the tides were due to the attraction of the

moon.

The invention of the telescope at the beginning of the seventeenth century facilitated observations of the nearer planets. The earliest discoveries with its aid were made by Galileo. In the spring of 1609 he heard that an optician of Middelburg had made a tube containing lenses which served to magnify objects seen through it. This gave him the clue, and he constructed a telescope of the kind which still bears his name, and of which an ordinary opera-glass is an example. The instrument magnified three diameters-that is, made objects appear as though only at one-third of their real distance. Encouraged by this success, he constructed a larger instrument of thirty-two diameters' power which magnified an object more than a thousand times. Intense interest was excited by these discoveries. He placed one of his instruments on a church tower at Venice, and, to the amazement of the merchants, showed them their ships approaching the harbour hours before any details could be detected by the eye. Turning his instrument to the heavens, he saw the lunar mountains, Jupiter's satellites, the phases of Venus, Saturn's ring, and the solar spots; from the motion of the latter he concluded that the sun rotates on its axis. In 1611, he exhibited in the garden of the Quirinal the wonders of the new worlds revealed by the telescope.

At first honours were showered upon him; but theological opposition arose so soon as it was realised that the observations tended to confirm

the Copernican theory. If that theory were true, some of the statements in the Bible could not be literally exact. Accordingly it was argued that, while the telescope might be a trustworthy instrument for terrestrial objects, it was not fitted to explore the heavens. In February, 1616, the Inquisition settled the matter, and declared that to suppose the sun the centre of the solar system was false, and contrary to Holy Scripture; and they embodied this assertion in an edict of March 5, 1616, which has never been repealed. For the time, Galileo bowed to the storm. In 1632, however, he published some dialogues on the system of the world, in which he clearly expounded the Copernican theory, and showed that mechanical principles would account for the fact that a stone thrown straight up into the air would fall again to the place from which it was thrown-a fact which previously had been one of the chief difficulties in accepting the view that the earth was in motion. The book was approved by the papal censor before publication; but none the less Galileo was summoned to Rome, forced to recant and do penance, and released only on promise of obedience to the edict of 1616. The dramatic persecution of Galileo has concentrated public attention on his work. But it should be noted that other mathematicians were also using the telescope to good advantage. In England Harriot had a large telescope through which he observed the satellites of Jupiter in 1610. Kepler also made various observations, and suggested that the eye-glass should be a convex lens. The transit of Venus was observed by Jeremiah Horrocks in Lancashire in 1639.

The acceptance of the Copernican system brought into prominence the problem of explaining the cause of the planetary motions. Descartes suggested in his Principia that space was filled with ether moving in whirlpools of varying sizes and under varying physical conditions. He supposed that the sun was the centre of a vortex in which the planets are swept round. Each planet was again the centre of another vortex in which its moons are swept round. He explained gravity by the action of these vortices, and suggested that smaller vortices round the molecules of bodies would account for cohesion. This suggestion was widely accepted, and is interesting as a genuine attempt to explain the phenomena of the universe by mechanical laws. But Descartes' assumptions were arbitrary, and unsupported by 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 shown 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. It will be sufficient here to say that Newton considered the theory in detail, and showed that its consequences are not only inconsistent with each of Kepler's laws and with the fundamental laws of Mechanics, but also at variance with the laws of nature assumed by Descartes.

The invention of the telescope and the almost simultaneous invention

Development of Physics.

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of the microscope naturally attracted attention to the subject of Optics, and especially to the law of refraction. Kepler asserted in 1611 that for small angles of incidence the angle of incidence was proportional to the angle of refraction, and, applying this, he was able to give in general outline the theory of the telescope. The correct law of refraction was discovered by Willebrod Snell (1591-1626), professor of Mechanics at Leyden. It was stated again, and perhaps discovered independently, by Descartes in 1637. The latter gave a theoretical proof resting on inaccurate assumptions; but Fermat deduced the laws both of reflexion and refraction from the assumption that light travels from a point in one medium to a point in another in the least time, and that the velocity of light decreases as the density of the medium increases.

The view that the velocity of light was finite, so boldly assumed by Fermat, had originated in the seventeenth century. Galileo made experiments on the subject, but was unable to arrive at a definite result, though he and the leading physicists seem to have supposed that the view was correct. It was not until 1676 that it was proved. This was done by Olaus Römer (1644-1710), a young Dane then living in Paris, by observations of the eclipses of Jupiter's moons. The theories of physical optics current at this time will be considered later.

Hydrostatics also received considerable attention during the earlier years of the seventeenth century. Here too the earliest experiments seem to have been made by Galileo, who showed that the air has weight, estimated its pressure by the height of the water column it could sustain, and definitely refuted the Aristotelian view that a vacuum could not exist. He also described his experiments on various physical subjects, notably on fluids. These investigations fairly entitle him to be termed the founder of modern Physics.

Galileo's work was carried on by his pupil Evangelista Torricelli of Florence (1608-47) who constructed a barometer. The description given of it was vague, but it suggested ideas to Pascal which led not only to his barometric experiments, but to proofs of the more elementary propositions relating to the pressure exerted by fluids. Later investigations were facilitated by the invention of the air-pump by Otto von Guericke of Magdeburg (1602-86). In England the subject was taken up by Robert Boyle (1627-91). His name is associated with the law which he discovered that the pressure exercised by a given quantity of a gas is proportional to its density. The law was re-discovered independently fourteen years later by Edme Mariotte (1620-81) in France, who did a great deal to popularise physical investigations in France, and was one of the founders of the French Académie des Sciences. The beginnings of experimental investigations on Heat were also indebted to the labours of Galileo, who invented a thermometer, though of an imperfect type; but it was nearly a century later before the subject was taken up systematically.