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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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The Elements of Euclid for the Use of Schools and Colleges: Comprising the ...

Euclid, Isaac Todhunter - 1867 - 400 sider
...whatever be the number of magnitudes. Wherefore, if there be any number &c. QED PROPOSITION 23. THEOREM. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in a cross order, the first shall have to the last of the first...
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Manual of Euclid: Books IV, V, VI, and Propositions 1-21 of Book XI

Euclid - 1868 - 141 sider
...то irX^боc, avvSvo \afißavofiiva Kai ¿v тф avry Xóy^* кaí Siîaov iv Tif avru \oyil> iarai. If there be any number of magnitudes, and as many others, which, taken two and two, have the same ratio; they shall be in the same ratio by equality. Statement. — Let there be three...
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The Elements of Plane and Solid Geometry

Henry William Watson - 1871 - 285 sider
...true whatever be the number of equal ratios and whatever be the magnitudes considered. PROPOSITION 12. If there be any number of magnitudes, and as many...ratio, the first shall have to the last of the first set of magnitudes the same ratio as the first has to the last of the second set of magnitudes. Fig....
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Elements of geometry, containing books i. to vi.and portions of books xi ...

Euclides, James Hamblin Smith - 1872 - 349 sider
...is greater than FV 9. Similarly the other cases may bo proved. Q. nn PROPOSITION XXI. (Eucl. v. 22.) If there be any number of magnitudes, and as many others, which have the same ratio taken two and two in order, the first must have to the last of the first magnitudes...
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The Elements of Euclid, containing the first six books, with a selection of ...

Euclides - 1874
...be the number of magnitudes. Therefore, if there be any number, &o. QED PROPOSITION 23. — 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 Euclid [selections from book 1-6] adapted to modern methods in ...

Euclides - 1874
...proportion will be B+A:BA::D + C:DG PHOP. XI.— THEOREM. (Euc. V. 22, 23). Tf there be three or more magnitudes, and as many others which, taken two and two in order, have the same ratio, then by equality, the first IMS to the last of the first rank the same ratio that the first has to...
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Euclid, Book V. Proved Algebraically, So Far as it Relates to Commensurable ...

Euclid, Lewis Carroll - 1874 - 62 sider
...> l ; С .-. , xn« by с, а>с. Similarly, if А=С, а=с; and if А<С, а< с. QED PROP. XXII. If there be any number of magnitudes, and as many others which, taken two and two, have the same ratio : the first of the first rank shall have to the last the same ratio as the first...
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Euclid, book v. proved algebraically, so far as it relates to commensurable ...

Charles Lutwidge Dodgson - 1874
....•. , substituting, - > 1 ; QED by c, a>c. Similarly, if A=C, a — c; and if A<C, a< c. PROP. XXII. If there be any number of magnitudes, and as many others which, taken two and two, have the same ratio : the first of the first rank shall have to the last the same ratio as the first...
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The Elements of Euclid, with many additional propositions, and explanatory ...

Euclides - 1876
...follows that if A is > C, D is > F; and if equal, equal ; if less, less. PROPOSITION XXII. THEOREM. — If there be any number of magnitudes, and as many...and two in order, have the same ratio ; the first has to the last of the first magnitudes the same ratio which the first has to the last of the others....
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First Public Examination in Literis Graecis Et Latinis

University of Oxford - 1879
...rectilineal figure and equal to another given rectilineal figure. 8. Inscribe a square in a given circle. 9. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in a cross order, the first shall have to the last of the first...
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