The Elements of Euclid, containing the first six books, with a selection of geometrical problems. To which is added the parts of the eleventh and ... read at the universities. By J. Martin - Tapa blanda

Sinopsis

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. 1874 Excerpt: ... 1. IfKbe greater than G, L is greater than H; and if equal, equal; if less, less (V. Def. 5); but K, L are any equimultiples whatever of E, F (constr.), and G, H any whatever of B, D; therefore 2. As E is to B,soisF to D (V. Def. 5).1 And in the same way the other case is demonstrated. Proposition 5.--Theorem. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other; the remainder shall be the same multiple of the remainder, that the whole is of the whole. Let the magnitude AB be the same multiple of CD, that AE taken from the first, is of CF taken from the other. Then the remainder EB shall be the same multiple of the remainder FD, that the whole AB is of the whole CD. G A E B C F D Construction. Take A G the same multiple of FD, that AE is of CF. Demonstration. Then AE is the same multiple of CF, that EG is of CD (V. 1); but AE, by the hypothesis, is the same multiple of CF, that AB is of CD; therefore EG is the same multiple of CD that AB is of CD; wherefore 1. EG is equal to AB (V. Ax. 1); take from each of them the common magnitude AE; and 2. The remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF, that AG is of FD, (constr.), and that AG has been proved equal to EB, therefore 3. AE is the same multiple of CF, that EB is of FD; but AE is the same multiple of CF that AB is of CD (hyp.); therefore 4. EB is the same multiple of FD, that AB is of CD. Therefore, if one magnitude, &c. Q.E.D. Proposition 6.--Theorem. If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two; the remainders are either equal to these others, or equimultiples of them. Let the two magnitudes A B, CD be equimultiples of...

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