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Geometry ; the elements of Euclid and Legendre simplified and arranged to exclude from geomtrical reasoning the reductio ad absurdum. with the ... geometry and trigonometry | |c By Law - Softcover
Synopsis
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. 1867 Excerpt: ... ABCD be a parallelogram, and EG, HK the parallelograms about the diagonal AC; the parallelograms EG, HK are similar to the whole parallelogram, and to one another. Because DC, GF are parallels, the angles ADC, AGF are (I.16) equal. For the same reason, because BC, EF are parallels, the angles ABC, AEF are equal; and (I.15, cor. 1) each of the angles BCD, EFG is equal to the opposite angle DAB, and therefore they are equal to one another; wherefore, in the parallelograms, the angle ABC is equal to AEF, and BAC common to the two triangles BAC, EAF; therefore (V. 3, cor.) as AB:BC:: AE: EF. And (IV. 5), because the opposite sides of parallelograms are equal to one another, AB: AD:: AE: AG; and DC: BO:: GF: EF; and also CD: DA:: FG: GA. Therefore the sides of the parallelograms BD, EG about the equal angles are proportionals; the parallelograms are, therefore (V. def. 1), similar to one another. In the same manner it would be shown that the parallelogram BD is similar to HK Therefore each of the parallelograms EG, HK is similar to BD. But (V. 14, cor.) rectilineal figures which are similar to the same figure, are similar to one another; therefore the parallelogram EG is similar to HK. Wherefore, etc. Scho. Hence, GF: FE:: FH: FK Therefore the sides of the parallelograms GK and EH, about the equal angles at F, are reciprocally proportional; and (V. 12) these parallelograms are equivalent; a conclusion which agrees with the eighth corollary to the fifteenth proposition of the first book. Prop. XVI.--Prob.--To describe a rectilineal figure which shall be similar to one given rectilineal figure, and equivalent to one another. Let ABC and D be given rectilineal figures. It is required to describe a figure similar to ABC, and equal to D. Upon the straight line BC de...
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- PublisherRareBooksClub.com
- Publication date2012
- ISBN 10 1236331265
- ISBN 13 9781236331267
- BindingPaperback
- LanguageEnglish
- Number of pages80
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