# 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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### Contents

 CHAPTER 9 CHAPTER II 38 CHAPTER III 56 CHAPTER IV 73 CHAPTER V 93
 CHAPTER VI 137 THE METHOD OF NEWTON 170 CHAPTER VIII 209 NOTE A 247

### Popular passages

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...