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.

    12/200 (page 6)
    her of 'parallelograms A b, Be, C d, ^c. contained un¬ der equal bases A B, B C, CD, 4’e., and the sides B b, C c, D d, S^c, parallel to A a, the side of the figure ; and the parallelograms a K b 1, b L c m, c M d n, S^c. are completed. Then., if the breadth of those parallelograms is diminished^ and their number is augmented continuedhj ; I say^ that the ultimate ratios, icliich the inscribed figure A K b L c M d D, the circumscribed figure A a 1 b m c n d 0 E, and the curvilinear figure A a b c d E, have to each other, are ratios of equality.—(Fig. 1.) For the difference of the inscribed and circumscri¬ bed figure is the sum of the parallelograms K /, L m, M n, D o, that is (because of the equality of all their bases,) the rectangle under one of their bases K h, and the sum of their altitudes K a ; that is, the rec¬ tangle A B Z But this rectangle, because its breadth AB is diminished indefinitely, becomes less than any given rectangle. Therefore (by Lem. I.) the inscri¬ bed and circumscribed, and much more the inter¬ mediate curvilinear figure become ultimately equal. Which was to be demonstrated. LEMMA III. The same idtimafe ratios are also ratios of equality, when the breadths A B, B C, CD, S^c. of the parallelograms are unequal, and are all diminished indefinitely. For let A F be equal to the greatest breadth; and let the parallelogram F A <2 y’ be completed. This