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The "physiocrats" and the "encyclopædists" of the French school of practical philosophers also deserve notice, for they were professedly inspired by science and seeking to apply it to human society. Even in so humble a pursuit as the attempt to overcome in time of famine the prejudices of the populace against the potato, Turgot and his fellows did good work for applied science. Nor should we forget the service to social science of Count Rumford, who for the first time grappled boldly with problems as far apart as the control of mendicity, of smoky chimneys, and of poverty. Much of Rumford's best work, though done in the nineteenth century, had its origin in the scientific spirit and achievements of the eighteenth.

As the century drew to its close, an English physician, Edward Jenner, by the use of the basic methods of inductive scientific research accurate observation, skillful experimentation, careful generalization and thorough verification-created a new science, preventive medicine, and conferred upon mankind the priceless blessings of vaccination. (See Appendix G.)

The nebular hypothesis of Laplace, through its central idea of natural development rather than sudden and special (artificial) creation of the solar system, was an important preparation of men's minds for theories of transformation or evolution. Hutton's Theory of the Earth enforced the same idea for the familiar earth, while the metamorphoses of the parts of the flower, pointed out by Goethe, helped to pave the way for acceptance of the idea of gradual modification of organs and even of organisms into others. To these matters we shall return in our discussion of Evolution in the final chapter.

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CHAPTER XV

MODERN TENDENCIES IN MATHEMATICAL SCIENCE

Mathematics is the queen of the sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank. Gauss.

Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar, we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshals a whole troop of completed operations in the execution of a new one. Mach.

The iron labor of conscious logical reasoning demands great perseverance and great caution; it moves on but slowly, and is rarely illuminated by brilliant flashes of genius. It knows little of that facility with which the most varied instances come thronging into the memory of the philologist or historian. Rather is it an essential condition of the methodical progress of mathematical reasoning that the mind should remain concentrated on a single point, undisturbed alike by collateral ideas on the one hand, and by wishes and hopes on

the other, and moving on steadily in the direction it has deliberately chosen. Helmholtz.

Nature herself exhibits to us measurable and observable quantities in definite mathematical dependence; the conception of a function is suggested by all the processes of nature where we observe natural phenomena varying according to distance or to time. Nearly all the "known" functions have presented themselves in the attempt to solve geometrical, mechanical, or physical problems. - Merz.

We have now reached a period of maturity in the evolution of mathematical science beyond which any attempt to follow its details would involve technical discussions outside the range of this work. The present chapter will be devoted to a general survey of modern tendencies in pure and applied mathematics, in mechanics, in mathematical physics and in astronomy. The most notable single fact in the centuries under discussion is the increasing specialization resulting from the great expansion of scientific knowledge. It is no longer possible for the individual scholar to command the range at once of philosophy, mathematics, physics, chemistry, and the natural sciences. It has even become more and more difficult to have a general knowledge of any one of these broad fields.

MATHEMATICS AND MECHANICS IN THE EIGHTEENTH CENTURY. -The invention of the infinitesimal calculus by Newton and Leibnitz was comparable in its relations and consequences with the discovery of a new world by Columbus two centuries earlier. As in that case the discovery was not an absolutely sudden one; other explorers had hoped or imagined, but only genius of that highest order which we call inspired, gained the complete revelation. The years next following the great discovery were naturally a period of eager and wide-ranging exploration, of optimistic self-confident pioneering. Such was the power of the new method, that one might rashly hope no secret of nature could long resist its attack. As circumnavigation of the globe was not long in following the discovery of America, so the cycle of mathematical knowledge might be completed. The parallel has failed. The calculus grew out of the insistent grappling by mathematicians with

problems which had defied the feebler tools of the earlier mathematics. One obstacle after another has been gradually surmounted by the invention of new and more powerful methods of ever increasing generality, just as increasingly powerful telescopes have revealed unnumbered new suns; and no boundary or limit to this evolutionary progress can be foreseen or imagined. On the other hand, as the new world has been gradually settled, civilized, and cultivated, so the fields of mathematics which were opened up in the eighteenth century have been critically examined in the nineteenth, with much revision of fundamentals.

The main features of eighteenth century mathematics were: - the working out of the differential and integral calculus into substantially the form they have ever since retained; the beginnings of differential equations as a natural outgrowth of integral calculus, and the beginnings of the calculus of variations; the systematic application of the new ideas to mechanics, and in particular to celestial mechanics. The century was also notable for important discoveries in astronomy and physics, including for example that of the aberration of light; a vigorous attack on "the problem of three bodies"; and the earlier telescopic work of the Herschels, culminating in the discovery of a new planet, Uranus. Among the leading mathematicians of the period were Maclaurin of Scotland, various members of the Swiss Bernoulli family, Euler also a native of Switzerland, Lagrange of Italy, and in France, Clairaut, d'Alembert, and Laplace. In spite of the unique supremacy of Newton, the absence of Britons from this list is notable. The bitter personal controversy between Newton's adherents and those of Leibnitz produced or aggravated an unfortunate division between the English and the continental mathematicians. For the former, persistence in Newton's inferior notation became a matter of national pride, and progress was correspondingly retarded. Of the mathematicians named above, the Bernoullis and Euler on the continent and Maclaurin in Scotland bore a leading part in the systematization of the calculus, while Lagrange and Laplace were preeminent in the development of analytical and celestial mechanics respectively.

Maclaurin's (1698-1746) Treatise of Fluxions (1742) was "the first logical and systematic exposition of the method of fluxions," and the applications to problems contained in it were characterized by Lagrange as the "masterpiece of geometry, comparable with the finest and most ingenious work of Archimedes." Maclaurin's point of view may be illustrated by the following passage:

Magnitudes were supposed to be generated by motion; and, by comparing the increments that were generated in any equal successive parts of the time, it was first determined whether the motion was uniform, accelerated, or retarded. . . . When the motion was accelerated, this increment was resolved into two parts; that which alone would have been generated if the motion had not been accelerated, but had continued uniform from the beginning of the time, and that which was generated in consequence of the continual acceleration of the motion during that time. The latter part was rejected, and the former only retained for measuring the motion at the beginning of the time. And in like manner, when the motion was retarded, . . . so that the motion at the time proposed was accurately measured, and the ratio of the fluxions always accurately represented. In the method of infinitesimals, the element, by which any quantity increases or decreases, is supposed to be infinitely small, and is generally expressed by two or more terms, some of which are infinitely less than the rest, which being neglected as of no importance, the remaining terms form what is called the difference of the proposed quantity. The terms that are neglected in this manner, as infinitely less than the other terms of the element, are the very same which arise in consequence of the acceleration, or retardation, of the generating motion, during the infinitely small time in which the element is generated. . . . The conclusions are accurately true, without even an infinitely small

error. ...

Daniel Bernoulli (1700-1782) made such good use of the new mathematical methods in attacking previously unsolved problems of mechanics, that he has been called the founder of mathematical physics. He recognized the importance of the principle of the conservation of force anticipated in part by Huygens.

Euler (1707-1783), while Swiss by birth, spent most of his life

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