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. 1784 Excerpt: ... the diameter GE falls wholly within the curve b. For, if any point O be assumed in it, it is evident that OD is greater than OF, Cor. Hence, the two legs of the curve continually diverge from the axis. PROPOSITION III, A straight line bisecting the angle formed by two lines drawnfrom the fame point in the curve the ...
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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. 1784 Excerpt: ... the diameter GE falls wholly within the curve b. For, if any point O be assumed in it, it is evident that OD is greater than OF, Cor. Hence, the two legs of the curve continually diverge from the axis. PROPOSITION III, A straight line bisecting the angle formed by two lines drawnfrom the fame point in the curve the one io the focusi and the other perpendicular to the direffrix, is a tangent to the parabola in that point. Fig. 2. The straight line GE, bisecting the angle DGF, is a tangent to the parabola in G. For, let H be any other point in GE, from which let there be drawn HF and HD, also HA perpendicu-. lar to AB, Then, because GE bisects the vertical angle of the isosceles triangle GDF, it will also bisect the base DF at right angles. Hence the triangles % 4-HED, HEF are equal in every respecta. Thus, HF 3.0.19.1. is equal to HD, and therefore greater than HAb, 0 Consequently, the point H is without the curve c. Cor, Cor. 1. Hence the method of drawing a tangent from any point in the curve. Cor. 2. If a straight line be drawn from the focus, to any point in the directrix, the perpendicular which bisects it will touch the parabola; also, every perpendicular to it, which cuts the curve, will be nearer to the focus than to the point in the directrix. Cor. 3. The parabola is concave towards the axis. PROPOSITION IV. Every straight line, drawn through the focus of a-parabola, except the axis, meets the curve in two points. Let FQ be a line passing through the focus, FD Fig-3perpendicular to it, DA, DE each equal to DF; AG, EH parallel to CF, and intersected by FQ, in G, H a; c 2.C.16.1. and let the points AF be joined. Then, because DA is equal to DF, the angles DAF, DFA are equal; and these being taken from the right angles DAG, DFG3 the remainders, the an...
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