The First Six Books of the Elements of Euclid, with a Commentary and Geometrical Exercises: To which are Annexed a Treatise on Solid Geometry, and a Short Essay on the Ancient Geometrical Analysis

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John Taylor, 1828 - Euclid's Elements - 324 pages
 

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Page 36 - If two triangles have two sides of the one respectively equal to two sides of the other, and the contained angles supplemental, the two triangles are equal.
Page 44 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 52 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Page 148 - ... be equimultiples, the one of the second, and the other of the fourth. Let A the first be the same multiple of B the second, that C the third is of D the fourth ; and of A, C let equimultiples EF, GH be taken.
Page 166 - ... If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; then if the first magnitude be greater than the third, the fourth shall be greater than the sixth: and if equal, equal; and if less, less.
Page 158 - N ; and if equal, equal ; and if less, less : but if G be greater than L, it has been shown that L HC K E M F N H is greater than M ; and if equal, equal; and if less, less: therefore, if G be greater than L, K is greater than N ; and if equal, equal ; and if less less : and G, K are any equimultiples whatever of A, E ; and L, N any whatever of B, F : therefore as A is to B, so is E to F (5.
Page 320 - If two angles of one triangle are equal to two angles of another triangle, the third angles are equal, and the triangles are mutually equiangular.
Page 153 - IF the first be to the second as the third to the fourth, and if the first be a multiple, or part of the second; the third is the same multiple, or the same part of the fourth...
Page 153 - THEOR. IF the first be the same multiple of the second, or the same part of it, that the third is of the fourth ; the first is to the second, as the third is to the fourth...
Page 155 - D (as in fig. 2 and 3), this magnitude can be multiplied, so as to become greater than D, whether it be AC, or CB. Let it be multiplied until it...

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