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56 THE NATURE OF MATHEMATICS still more accurate law of the motions of any number of bodies under any forces. Let us now try to think clearly of what we mean by such a rule, or, as it is usually called, a " scientific " or " natural law," and why it plays an important part in the arrangement of our knowledge in such a convenient way that we can at once, so to speak, lay our hand on any particular fact the need of which is shown by practical or theoretical circumstances. For this purpose, we will see how Galileo, in a work published in 1638, attacked the problem of a falling body. Consider a body falling freely to the earth : Galileo tried to find out, not why it fell but how it fell, that is to say, in what mathematical form the distance fallen through and the velocity attained depends on the time taken in falling and the space fallen through. Freely falling bodies are followed with more difficulty by the eye the farther they have fallen ; their impact on the hand receiving them is, in like measure, sharper; the sound of their striking louder. The velocity accordingly increases with the time elapsed and the space traversed. Thus, the modern inquirer would ask : What function is the number (v) representing the velocity of those (s and t} representing the distance fallen through and the time of falling 1 Galileo asked, in his primitive way : Is v proportional to s, is v proportional to 1 1 Tims he made assumptions, and then ascertained by actual trial the correctness or otherwise of these assumptions. One of Galileo's assumptions was, thus, that the velocity acquired in the descent is proportional to the time of the descent. That is to say, if a body falls once, and then falls again during twice as long an interval of time as it first fell, it will attain in the second instance double the velocity it acquired in the first. To find by experiment whether or not this assumption accorded with observed facts ; as it was difficult to prove by any direct means that the velocity acquired was proportional to the time of descent, but easier to investigate by what law the distance increased with the time, Galileo deduced from his assumption the relation that obtained between the distance and the time. This very important deduction he effected as follows. On the straight line OA, let the abscissae OE, OC, OQ, and so on, represent in length various lengths of time elapsed from
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