Euclid's Elements of Geometry; Chiefly from the Text of Dr. Simson, with Explanatory Notes ...

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. 1847 edition. Excerpt: ... First, the triangle a mm is intersected by the transversals AD, B6,, m'A aB mp . w . therefore -j-- . =--. -r= 1; -tt- ., . -r-- = 1. Again, the same triangle is intersected by the transversals AC, C c, ., - m' mC a A . m'O aC mp' therefore --i- 1; Ts- .tt-. -r . = 1, in Co Am Oa Cm /> m, ., mp ra'n mp' m'ri whence is deduced -- -. ---. ---- . ----= I. pm' nm p m' n m Secondly. In a similar way since the triangle nn'b is intersected by the transversals BA, Aa; and by BC, Cc; four relations arise, from which may be deduced nm' n'p nm n'p' m'n" pn ' mn ' p n Thirdly. The triangle pp'c is intersected by the transversal CA, Aa; and by CB, Bb; other four relations are found from which there results mp n'p' m'p np pn1'p'm" pn 'p'm Cor. From these three relations, each involving eight of the segments of the transversa], a relation may be found which involves only six segments; multiplying these results together, is deduced mp' nm' pn' p'n ' m'p' n'm Three other forms may be deduced from the same three expressions, (I.) by multiplying (1) and (2) together and dividing by (3); (II.) by multiplying (1) and (3) together and dividing by (2); (III.) by multiplying (2) and (3) together and dividing by (1). PROPOSITION V. If any polygon be intersected by a transversal, the segments of the sides have to each other a relation similar to that of the segments of the sides of a triangle. Let ABCD be a polygon of four sides, and let its opposite sides AB, CD be intersected in 6, c by a transversal which meets AD, CB produced in d, b. Join AC intersecting the transversal db in O. Then the transversal dcO intersects the triangle ACD, .., Ad Dc CO, n . therefore Prop. I. And the transversal ba O intersects the triangle ABC, ......