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IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. NB This is...
Euclid's plane geometry, books iii.-vi., practically applied; or, Gradations ... - Side 214
av Euclides - 1861
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## Euclid's Elements of geometry [book 1-6, 11,12] with explanatory notes ...

Euclides - 1845
...be the number of magnitudes. Therefore, if there be any number, &c. QED PROPOSITION XXIII. THEOREM. If there be any number of magnitudes, and as many others, which taken two and two in a cross order, have the same ratio ; the Jirsl shall have to the last of the first magnitudes the same...
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## Euclid in Paragraphs: The Elements of Euclid: Containing the First Six Books ...

Euclid - 1845 - 199 sider
...number of magnitudes. Therefore, if there be any number, &c. QED ABCD EFGH PROPOSITION XXIII. THEOR. — If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, have the same ratio ; the first shall have to the last of the first magnitudes, the...
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## The First Six, and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclid, James Thomson - 1845 - 352 sider
...the number of magnitudes. Therefore, if there be any numberof magnitudes, &c. PROP. XXIII. THEOR. — If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, have the same ratio ; the first has to the last of the first magnitudes the same ratio...
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## The Elements of Euclid, the parts read in the University of Cambridge [book ...

Euclides - 1846
...greater than D, that is, D is less than F. Wherefore, If there be three %c. QED PROP. XXII. THEOS. If there be any number of magnitudes, and as many...which, taken two and two in order, have the same ratio, then the first shall have to the last of the first magnitudes the same ratio which the first of the...
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## Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's ...

Dennis M'Curdy - 1846 - 138 sider
...be greater than Recite (a) def. 10, 5 ; (b) p. 15, 5; (c) Note 3, page 32; (d) p. A of b. 5. 22 Th. If there be any number of magnitudes, and as many...which, taken two and two in order, have the same ratio: then the first is to the last of one rank, as the first is to the last of the other rank. G, H, in...
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## Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1846 - 317 sider
...the same manner is the demonstration extended to any number of magnitudes. PROP. XXIII. THEOR. iff there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio ; thefirst will have to the last of the first magnitudes the same...
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## The definitions, postulates, axioms, and enunciations of the propositions of ...

Euclides - 1848
...others. • NB This is usually cited by the words "ex tequali," or "ex aguo." PROP. XXIII. THEOREM. If there be any number of magnitudes, and as many others, which taken two and two, in a cross order, have the same ratio ; the first shall have to the last of the first magnitudes the same...
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## Elements of Geometry: Containing the First Six Books of Euclid : with a ...

John Playfair - 1849 - 317 sider
...and B : A : : F : E, therefore, by this first case, since CyA, F7 D, that is, D/F. PROP. XXII. THEOR. If there be any number of magnitudes, and as many others, which, taken two ana two in order, have the same ratio ; the first will have to the last of the first magnitudes, the...
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## The geometry, by T. S. Davies. Conic sections, by Stephen Fenwick

...less than F. Therefore, if there be three, etc. QED c • EUCLID S ELEMENTS. PROPOSITION XXII. THEOR. If there be any number of magnitudes, and as many...the first magnitudes, the same ratio which the first of tke others has to the last. NB This is usually cited by the words " ex sequali," or " ex sequo."....
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## The first six books of the Elements of Euclid, with numerous exercises

Euclides - 1853 - 147 sider
...number of magnitudes. Therefore, if there be any number, <fec. QED PROPOSITION XXIII. — THEOREM. If there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio ; the, first sliall have to the last of the first magnitudes the...
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