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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 i 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 rediscovered 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.

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Another branch of Physics originated at this time was that connected with Electricity and Magnetism. Although there had been a few previous observations on the subject by Cardan, Mercator, and Porta, it may be said to have commenced with the work of William Gilbert (1540-1603), physician in ordinary to Queen Elizabeth. His experiments were pub lished in 1600.

The necessity of an experimental foundation for science was in the course of this period advocated with considerable effect by Francis Bacon (1561-1626) in his Novum Organum, published in 1620. Here he laid down the principles which should guide those making experiments in any branch of Physics, and gave rules by which the results of induction could be tested. Bacon's book appealed to men of education, and helped to secure recognition for the proposition that experiment and observation are necessary preludes to the formation of scientific theories. For practical purposes, however, it was of but little use. Bacon thought that investigations could be made by rule, and did not realise that the creation of scientific hypotheses was impossible without imagination. The book had more influence among philosophers and men of letters. than among scientific students.

Towards the middle of the seventeenth century the progress of scientific learning received a great stimulus, especially in England and France, from the foundation of academies or societies, created for the purpose of encouraging scientific investigations and providing a common meeting-place where those engaged in it could interchange ideas. Some account of these associations and of the part which they played in the history of science will be found in another section of this chapter.

Great as was the advance in knowledge made during the first half of the seventeenth century, that from 1660 to 1730 was even more marked. In the branches of Pure Mathematics previously mentioned it will suffice to say that Algebra and Trigonometry became more analytical, and that Newton's discovery of the binomial theorem and his work on the theory of equations were especially notable. Towards the end of the period the extension of Trigonometry to imaginary quantities was made by Abraham Demoivre of London (1667-1754), whose name is associated with the fundamental theorem on the subject. No new developments of Pure Geometry took place during this period; but the classical methods were applied to various problems with extraordinary ingenuity by Newton in the first book of the Principia. The methods of Analytical Geometry were also developed and it became a familiar tool in the hands of mathematicians.

A novel and potent instrument of research was developed in the infinitesimal calculus. This method of analysis, expressed in the notation of fluxions and fluents, was used by Newton (1642-1727) in or before 1666; but no account of it was published until 1692, though its

Invention of the Infinitesimal Calculus 717

general outline was known by his friends and pupils long anterior to that year. The notation of the fluxional calculus is for most purposes less convenient than that of the differential calculus. The latter notation was invented by Leibniz (1646-1716), probably in 1675, and was published in 1684.

The idea of a fluxion or differential coefficient, as treated in this period, is simple. When two quantities for instance, the radius of a sphere and its volume are so related that a change in one causes a change in the other, the one is said to be a function of the other. The ratio of the rates at which they change is termed the differential coefficient or fluxion of the one with regard to the other, and the process by which this ratio is determined is known as differentiation. Knowing the differential coefficient and one set of corresponding values of the two quantities, we are able by summation to determine the relation between. them; but often the process is difficult. If however we can reverse the process of differentiation, we can obtain this result directly. This process of reversal is termed integration, and was first employed by Newton and Leibniz. It was at once seen that problems connected with the quadrature of curves, and the determination of volumes (which were soluble by summation, as had been shown by the employment of indivisibles) were reducible to integration. In Mechanics also, by integration, velocities could be deduced from known accelerations, and distances traversed from known velocities. In short, wherever things change according to known laws, here was a possible method of finding the relation between them. It is true that, when we try to express observed phenomena in the language of the calculus, we usually obtain an equation involving the variables, and their differential coefficients- and possibly the solution may be beyond our powers. Even so, the method is often fruitful and its use marked a real advance in thought and power.

With the various applications-important though they were of the calculus to Geometry and Mechanics we need not concern ourselves, but one application is sufficiently important to demand a word in passing. This was the discovery in 1712 by Brook Taylor (1685-1731) of the well-known theorem by which a function of a single variable can be expanded in powers of it. It was published in 1715, though no satisfactory proof was given at the time.

The ideas of the infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. There is no doubt that the differential notation is due to Leibniz; but an acute controversy arose as to whether the general idea of the calculus was taken by him from a manuscript by Newton, to which it was supposed he had had access, or whether it was discovered independently. During the eighteenth century the prevalent opinion was against Leibniz; but to-day the majority of judges think it more likely that the inventions were independent. The controversy was complicated by bitter personalities. It was natural,

though unfortunate, that British mathematicians were thus led to confine themselves to the methods used by Newton. The consequence was that, shortly after Newton's death, the British school fell out of touch with the great continental mathematicians of the eighteenth century, and it was not until about 1820, when the value of analytical methods was recognised, that Newton's countrymen again took any considerable share in the development of Mathematics.

Leibniz was a man of extraordinary versatility; and, in addition to his diplomatic activity, played a prominent part in the literary and philosophical history of his time. Mathematics were not his main interest, and he produced very little mathematical work of importance besides his papers on the calculus; his reputation in this subject rests largely on the attention which he drew to it. In 1686 and 1694 he wrote papers on the principles of the new calculus. In these, his statements of the objects and methods of the infinitesimal calculus are somewhat obscure, and his attempt to place the subject on a metaphysical basis did not tend to clearness; but the fact that all the results of modern Mathematics are expressed in the language invented by him has proved the best monument of his work.

Newton elaborated the calculus more completely than Leibniz, but his methods were buried in note-books inaccessible to all save a few friends; and the general adoption of Leibniz' notation was largely due to the fact that, through a text-book published in 1696 by the Marquis de L'Hospital of Paris, it was at once made known to all interested in the subject. It was also regularly used by Peter Varignon (1654-1722), the most eminent French mathematician of the time, and by the brothers James Bernoulli (1654–1705) and John Bernoulli (1667–1748) —men of remarkable ability who applied the new calculus to solve numerous problems. The Bernoullis were the most prominent continental teachers of this period and their influence was exceptionally potent. The accounts at first given of Newton's method of fluxions were less complete; and more than a generation passed after the production of L'Hospital's work, before Colson in 1736, and Maclaurin in 1742, published systematic expositions of the fluxional method.

We turn next to the subject of Mechanics, which was placed on a scientific basis through the researches of Newton. The investigations by Galileo on the fall of heavy bodies, and the theory of pendulums, were completed by Huygens (1629-95) in his Horologium Oscillatorium. published at Paris in 1673. In this work he determined the centrifugal force on a body moving in a circle with uniform velocity; he also considered the motion of bodies of finite size and not merely of particles. Newton's investigations on Mechanics are included in his Principia. It will suffice here to say that he based the subject on three laws of motion, and he applied the principles to the statics and dynamics of rigid bodies and fluids; probably he carried the investigations as far as was possible

Newton's theory of gravitation

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with the analysis at his command. He distinguished between mass and weight, and this was an important point. He also created the theory of attractions, which will be more naturally noted in connexion with his theory of gravitation.

The fundamental principles of Newton's theory of gravitation seem to have occurred to him shortly after he had taken his degree at Cambridge. His reasoning at this time, 1666, appears to have been as follows. He knew that gravity extended to the tops of the highest hills; and he conjectured that it might extend as far as the moon, and be the force which retained it in its orbit about the earth. This hypothesis he verified by the following argument. If a stone is allowed to fall near the surface of the earth, the attraction of the earth causes it to move through sixteen feet in one second. Now Newton, as also other mathematicians, had suspected from Kepler's law that the attraction of the earth on a body would be found to decrease as the body was removed further away from the earth, inversely as the square of the distance from the centre of the earth. He knew the radius of the earth and the distance of the moon, and therefore on this hypothesis could find the magnitude of the earth's attraction at the distance of the moon. Further, assuming that the moon moved in a circle, he could calculate the force that was necessary to retain it in its orbit. In 1666, his estimate of the radius of the earth was inaccurate, and, when he made the calculation, he found that this force was rather greater than the earth's attraction on the moon. This discrepancy did not shake his faith in the belief that gravity extended to the moon and varied inversely as the square of the distance; but he conjectured that some other forcesuch, for example, as Descartes' vortices-acted on the moon as well as gravity.

In 1679 Newton repeated his calculations on the lunar orbit; and, using a correct value of the radius of the earth, he found the verification of his former hypothesis was complete. He then proceeded to the general theory of the motion of a particle under a centripetal force - that is, one directed to a fixed point- and showed that the vector to the particle would sweep over equal areas in equal times. He also proved that, if a particle describes an ellipse under a centripetal force to a focus, the law must be that of the inverse square of the distance from the focus; and, conversely, that the orbit of a particle projected under the influence of such a force would be a conic. In 1684 Halley asked Newton what the orbit of a planet would be, if the law of attraction were that of the inverse square, as was commonly suspected to be approximately the case. Newton asserted that it was an ellipse, and sent the demonstration which he had discovered in 1679. Halley, at once recognising the importance of the communication, induced Newton to undertake the investigation of the whole problem of gravitation, and to publish his results.

It would seem that Newton had long believed that every particle of