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. Principia. Liber I. Selections. English
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.

    15/200 (page 9)
    LEMMA VI. ]f any arc A C B, given in position^ is subtended by its chord A B, and in any point A, in the middle of a con- tirmed curvature, is touched by a right line A D, produced both ways ; then, if the points A and B approach one ano¬ ther and meet; I say that the angle BAD, contained be¬ tween the chord and the Uingeni, will be diminished in¬ definitely, and will ultimately vanish,—(Fig. 3.) For, if that angle does not vanish, the arc A C B will contain with the tangent A D an angle equal to a rectilinear angle; and, therefore, the curvature at the point A will not be continued. Which is against the supposition. LEMMA VII. The same things being supposed, I say, that the idtimate ra¬ tio of the arc, the chord, and the tangent, to each other, is the ratio of equality. For, while the point B approaches towards the point A, let A B and A D be considered as produ¬ ced to tlie remote points b and d, and let b d be drawn parallel to the secant B D. Let the arc K c h be al¬ ways similar to the arc A C B. Then, supposing the points A and B to coincide, the angle d A b will van¬ ish, by the preceding Lemma; and, therefore, the right lines A A d, which are always finite, and the intermediate arc A c b will coincide, and become B