The first three sections of Newton's Principia; with copious notes and illustrations, and a great variety of deductions and problems. Designed for the use of students / By the Rev. John Carr.

  • Newton, Isaac, 1642-1727.
Date:
1826
Catalogue details

Licence: Public Domain Mark

Credit: The first three sections of Newton's Principia; with copious notes and illustrations, and a great variety of deductions and problems. Designed for the use of students / By the Rev. John Carr. Source: Wellcome Collection.

    13/200 (page 7)
    will be sfrecater than the difference of the inscribed and circumscribed figures; but, because its breadth A F is diminished indefinitely, it will become less than any given rectangle. Which was to be demon¬ strated. Coj\ 1. Hence the ultimate sum of the evanescent parallelograms coincides in every part with the cur¬ vilinear figure. Cor, 2. Much more does the rectilinear figure, which is comprehended under the chords of the evanescent arcs a b, b c, c d, &c. ultimately coincide with the curvilinear figure. Cor. 3. As also the circumscribed rectilinear figure, which is comprehended under the tangents of the same arcs. Cor. 4. And, therefore, these ultimate figures (as to their perimeters « c E,) are not rectilinear, but curvilinear limits of rectilinear figures. LEMMA IV. If in two figures A a c E, P p r T, there are inscribed fas before) two series of parallelograms, an expial number in each; and, their breadths being diminished indefinitely, if the ultimate ratios of the jmrallelograms in one fgure to those in the other, each to each respectively, are the same; I say, that those two figures A a c E, P p r T, are to each other in that same ratio.—(Fig. 2.) For, as the parallelograms in one are severally to