By A Wolf
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Extra info for A History of Science, Technology, and Philosophy in the Eighteenth Century
Legendre: M im. de I'Acad. Roy. , Paris, 1786, pp. 7 fF. ” After he had himself found the solution, Johann, following the custom o f the time, challenged “ the most ingenious mathema ticians in the whole world” to solve the problem likewise. Newton sent a correct solution o f this problem to a friend on the day after it reached him. Leibniz, Jakob Bernoulli, and L ’Hopital also solved it, showing that the required curve was a cycloid. This result caused all the more surprise as Huygens had already recognized that a particle falling down a cycloidal path requires the same time to reach the lowest point o f the curve whatever its starting-point.
He therefore rejected Fermat’s explanation. But he showed that it was still possible to regard a ray o f light as following the path o f least action in travelling from a point A in one medium to a point B in the other, provided that such action was measured by multiplying the distance, travelled by the ray in each medium, by the velocity o f light therein. yg) IS a minimum, and he thence deduced that sin a sin jS 68 HISTORY OF SCIENCE, TECHNOLOGY, AND PHILOSOPHY Fermat, on the other hand, had assumed that /A G , CB \ I -----j was a minimum, MECHANICS 69 the time-integral o f the vis viva^ or mv^dt.
Euler was the first distinctly to conceive the notion o f a mathematical MATHEMATICS 53 function, with which the earlier chapters o f the Introductio deal. This conception has been well described as perhaps the most fundamental of all the creations o f modern mathematics. — Introductio, I, i, 4). , where a, b, c stand for cpnstants. Later he considers functions of more than one independent variable. , addition, subtraction, multipHcation, division, raising to powers, and extraction o f roots. Logarithmic or trigonometrical functions of the variable, and those which involve it as an exponent, are classed as transcendental.