The History of Mathematics: A Brief Course

The great watershed in the history of mathematics is the invention of the calcu- lus. It synthesized nearly all the algebra and ...

Chapter 16. The Calculus The infinite occurs in three forms in calculus: the derivative, the integral, and the power series. Int ...

464 16. THE CALCULUS FIGURE 1. Fermat's method of finding the subtangent. take up the story of calculus at the point where algeb ...

PRELUDE TO THE CALCULUS 465 FIGURE 2. The folium of Descartes. Descartes and Fermat con- sidered only the loop in this curve. ...

466 16. THE CALCULUS FIGURE 3. RobervaFs quadrature of the cycloid. volume was known. The first figure was then asserted to be e ...

PRELUDE TO THE CALCULUS 467 c R A G Ç Ï Ì FIGURE 4. Fermat's quadrature of a generalized hyperbola. that the ratio AHm : AGm = ...

468 16. THE CALCULUS published a work on geometry in 1668 in which he stated the equivalent of the formula given earlier (unbekn ...

NEWTON AND LEIBNIZ 469 Newton's first version of the calculus. Newton first developed the calculus in what we would call param ...

470 16. THE CALCULUS time approach nearer to each other than by any given difference, become ultimately equal. If you deny it, s ...

NEWTON AND LEIBNIZ 471 y Æ X Figure 5. Leibniz' proof of the fundamental theorem of calculus. the horizontal axis.^2 The secon ...

472 16. THE CALCULUS proof; they contain the basic results on differentiation and integration of elemen- tary functions, includi ...

NEWTON AND LEIBNIZ 473 des infiniment petits (Infinitesimal Analysis) and has ever since been known as L'Hospital's rule. Like ...

474 16. THE CALCULUS Most students of calculus know the Maclaurin series as a special case of the Taylor series. Its discoverer ...

BRANCHES AND ROOTS OF THE CALCULUS 475 were hidden for some time, and for a blissful century mathematicians and physicists ope ...

476 16. THE CALCULUS It appeared at a very early stage that finding an integrating factor is not in general possible, and both N ...

BRANCHES AND ROOTS OF THE CALCULUS 477 at all. Given that the independent variable is usually time, that property corre- spond ...

(^478) 16. THE CALCULUS discussed in Chapter 17, along with the development of real analysis in general. For the rest of the pre ...

BRANCHES AND ROOTS OF THE CALCULUS 479 Q FIGURE 6. Left: Fermat's principle. Choosing the point Ï so that the time of travel f ...

(^480) 16. THE CALCULUS himself, his brother Jakob, and by both Newton and Leibniz.^6 According to Wood- house (1810, p. 150), N ...

BRANCHES AND ROOTS OF THE CALCULUS 481 extrema into absolute (unconstrained) extrema. Euler had given an explanation of this p ...