The Elements of Geometry [Euclid Book 1-3] in General Terms, with Notes &C. &C. Also a Variety of Problems & Theorems. [Ed. by J. Luby. With] the Elem - Softcover

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1821 edition. Excerpt: ... A VARIETY OS THE MOST IMPORTANT DEDUC1BLE FROM THE SECOND & THIRD BOOKS. PP&B ANALYTICALLY. PROP. 1. THEOR. The difference between the hypothenuse BC and rum of the sides BA,AC of a right angled triangle ABC it equal to tmo radii of the inscribed circle. Fig. 14, ANALYSIS, Suppose it to be the case;. then since BC with FG, GE is = to BA,AC together, BC with FA, AE is j= to BA,AC; for it is evident that FA,AE are respectively ac to Geji'gj since each of the. angles F, A, E, is a right angle,".: BC is'=: to BF and CE together, if this can, be. proved the. proposition is evident. Join BG; then since in the triangles BGD,BGF the sides BG and GD are respectively = to BG and GF, and the angles BDG,BFG are = as being right and the angles DBG,FBG of the same affection, the triangles are =5 in every way,.-, BD is = to BF; for tie same reason CD is = to CE,.. BC is = to the sum of BF and CE,,-. &c. PROP. 2, PROB. To produce a given right line AB so that the rectangle under the whole and-produced part may he equul ta a given quantity. Fig-15. ANAlYSls. Suppose it done, and that AC is the line produced. Jet F be sr to a side of the Q of the given quantity; then it is evident from. prop. 37, b. 3, that CA shalJ be = to the secant drawn from the extremity of the tangent (= to F,) through the centre of the circle, the diameter of which is = to the given line. Describe a circle of which the diameter is the givea line AB, draw to it a tangent IA--s to the given quantity, and from its extremity I draw through the centre the secant ID, it is evidently = to the whole produced line,.-. make BC == to GI, and the rect. ACB is = to the given quantity. Cor. 1. By being given the tangent and internal part of the secant passing in any way, we can find the secant....

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