Mathematics Volume 2

Mathematics Volume 2

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1830 edition. Excerpt: ... curve and its tangent so as not to cut the curve," or, in other words, that a curve admits of only one tangent at the same point of it, may be regarded as an axiom. That such is the case in the circle has been shown in Book iii. Prop. 2.; and hence, with regard to other curves, generally, it may be illustrated as follows: --Conceive a circle having the same tangent with the curve at the point M, and sufficiently small to fall within the curve, as in the adjoined figure. Then, since no straight line can be drawn through the point M so near to the tangent as not to cut the circumference of this circle, and since the curve lies between the circumference of this circle and the tangent; much less can any straight line be drawn so near to the tangent as not to intercept a part of the curve between itself and the tangent, and, consequently, being produced, to cut the curve. Prop. 7. The direction C D and the plane of projection A B betiig given; the orthographic projection of any point P whatever may be found upon the plane A B. It may, perhaps, appear at first, that if the tangent lies in the vertical plane, the curve must likewise lie in that plane; this, however, is not ft necessary consequence; the tanrent MH may be the common section of the plane of the curve with the vertical plane, ind this is the case which is supposed in tbe corollary. For, the direction CD not being parallel to the plane AB (def-2.), the straight line which is drawn through P parallel to C D is not parallel to that plane; since, otherwise, the line thus drawn would be parallel to the common section of a plane passing through it with the plane A B (IV. 10.), and therefore, also (IV. 6.), C D would be parallel to the same common section, that is, to a straight line in the...
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Product details

  • Paperback | 280 pages
  • 189 x 246 x 15mm | 503g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236611209
  • 9781236611208