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 AnalysisJohn Taylor, 30 Upper Gower Street, Bookseller and Publisher to the University: and sold, 1828 - Euclid's Elements - 324 pages |
From inside the book
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Page xv
... equimultiples of the first and third are equal respectively to other equimultiples of the second and fourth , but yet have all the other properties of proportional quantities , and therefore it is necessary that they should be brought ...
... equimultiples of the first and third are equal respectively to other equimultiples of the second and fourth , but yet have all the other properties of proportional quantities , and therefore it is necessary that they should be brought ...
Page 136
... equimultiples of two others a , b , when a and b measure A and B respectively the same number of times . Thus the length one foot and the number 36 are equimultiples of the length one inch and the number 3 ; for an inch measures a foot ...
... equimultiples of two others a , b , when a and b measure A and B respectively the same number of times . Thus the length one foot and the number 36 are equimultiples of the length one inch and the number 3 ; for an inch measures a foot ...
Page 137
... equimultiples whatsoever of the first and third be taken , and also any equimultiples whatsoever of the second and fourth be assumed ; if , according as the mul- tiple of the first is greater , equal to , or less than the multiple of ...
... equimultiples whatsoever of the first and third be taken , and also any equimultiples whatsoever of the second and fourth be assumed ; if , according as the mul- tiple of the first is greater , equal to , or less than the multiple of ...
Page 138
... equimultiples of their consequents . These conditions may also be expressed thus : If the antecedents of two ratios be equimultiples or equisubmultiples of their consequents the ratios are equal . If , however , neither term of the ...
... equimultiples of their consequents . These conditions may also be expressed thus : If the antecedents of two ratios be equimultiples or equisubmultiples of their consequents the ratios are equal . If , however , neither term of the ...
Page 139
... equal when the ratios of every pair of equi- multiples of their antecedents to every pair of equimultiples of their consequents are ratios of the same species of inequality . Thus if A and a be multiplied by the same number , the ...
... equal when the ratios of every pair of equi- multiples of their antecedents to every pair of equimultiples of their consequents are ratios of the same species of inequality . Thus if A and a be multiplied by the same number , the ...
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The First Six Books of the Elements of Euclid: With a Commentary and ... Dionysius Lardner No preview available - 2018 |
Common terms and phrases
A B and B C A B D altitude angles A B C arcs Book centre circumference circumscribed coincide conical surface constructed demonstration diagonal diameter difference draw equal angles equal hyp equal sides equi equiangular equilateral triangle equimultiples Euclid external angle extremities geometry given circle given line given point given right line Hence homologous sides hypotenuse inscribed intersect isosceles triangle less magnitudes multiple opposite parallel parallelogram parallelopiped pentagon perpendicular plane polygon prism problem produced PROPOSITION proved pyramid radii radius rectangle rectilinear figure respectively equal right line A B segments sides A B similar solid angle square of A B surface tangent THEOREM third tiples triangles A B C vertex
Popular passages
Page 16 - 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 24 - 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 22 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Page 104 - ... 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 107 - ... 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 107 - 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 187 - 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 107 - 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 107 - 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 107 - 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...