The Philosophy of Mathematics: With Special Reference to the Elements of Geometry and the Infinitesimal Method
J.B. Lippincott & Company, 1886 - Mathematics - 248 pages
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abscissa abstract number absurd algebra Analytical Geometry angle approach Archimedes axiom axis Cavalieri circle circumference circumscribed clear co-ordinates coincide curve curvilinear figure curvilinear space D'Alembert definition demonstration Descartes determinate difference differential calculus Duhamel dx² equation error evident exactly equal expression false finite frustum Hence hypothesis increment indefinitely indeterminate infinite number infinitely small quantities infinitesimal analysis infinitesimal method infinity inscribed lemma logical magnitudes mathematicians mathematics method of exhaustion method of indivisibles method of Leibnitz method of limits mind Newton number of sides objection obscure ordinate parabola parallel parallelograms Pascal Philosophy of Mathematics polygon precisely principle pyramid question rectangles rectilinear reductio ad absurdum regular polygon represent result right lines says Carnot Scholium subtangent supposed supposition surface symbol tangent line term thing tion triangle true truth ultimate ratios ultimately equal variable volume zero
Page 195 - But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In...
Page 195 - ... none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases.
Page 155 - If you had committed only one error, you would not have come at a true solution of the problem. But by virtue of a twofold mistake you arrive, though not at science, yet at truth. For science it cannot be called, when you proceed blindfold, and arrive at the truth not knowing how or by what means.
Page 195 - There is a limit which the velocity at the end of the motion may attain, but not exceed ; and this is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be.
Page 173 - Hence the ultimate sum of those evanescent parallelograms will in all parts coincide with the curvilinear figure. COR. 2. Much more will the rectilinear figure comprehended under the chords of the evanescent arcs ab, bc, cd, &c., ultimately coincide with the curvilinear figure.
Page 173 - M inscribed and circumscribed become ultimately equal one to the other, and much more will the intermediate curvilinear figure be ultimately equal to either. QED
Page 183 - ... ratios of quantities decreasing without limit do always converge, and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.
Page 172 - But this rectangle, because its breadth AB is supposed diminished in infinitum, becomes less than any given space. And therefore (by Lem. I) the...