Manual of Euclid: Books IV, V, VI.

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Cassell, Petter, and Galpin, 1868
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Page 28 - The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth: or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal...
Page 38 - If the first be the same multiple of the second which the third is of the fourth, and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth.
Page 3 - IN a given circle to inscribe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle ; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF.
Page 72 - The sides about the equal angles of equiangular triangles are proportionals ; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or consequents of the ratios. Let ABC, DCE be equiangular triangles, having the angle ABC equal to the angle DCE, and the angle ACB to the angle DEC, and consequently * the angle BAC equal to the a 32.
Page 92 - CF ; but K has to M the ratio which is compounded of the ratios of the sides ; therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the sides. Wherefore equiangular parallelograms, &c.
Page 76 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Page 69 - DE, and between the same parallels DE, BC; but ADE is another triangle ; and equal magnitudes have the same ratio to the same magnitude; (v. 7.) therefore, as the triangle BDE is to the triangle ADE...
Page 71 - Now let BD be to DC, as BA to AC, and join AD ; the angle CAD is equal to the angle DAE. The same construction being made, because BD is to DC as BA to AC ; and also BD to DC, BA to AF (2.
Page 29 - When of the equimultiples of four magnitudes (taken as in the fifth definition), the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth ; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth...
Page 60 - D, and other four E, F, G, H, which, two and two, have the same ratio, viz. as A is to B, so is E to F; and as B...

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