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" If there be any number of magnitudes, and as many others, which, taken two and two in 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 usually... "
Euclid's plane geometry, books iii.-vi., practically applied; or, Gradations ... - Seite 217
von Euclides - 1861
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The Elements of Plane and Solid Geometry

Henry William Watson - 1871 - 285 Seiten
...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 others, which taken two and two in order have the same ratio, the first shall have to the last of the first set of magnitudes the same...
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Elements of geometry, containing books i. to vi.and portions of books xi ...

Euclides, James Hamblin Smith - 1872 - 349 Seiten
...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
...; that is, 1. D is less than F. Therefore, if there be three, &c. QED PROPOSITION 22. — Theorem. If there be any number of magnitudes, and as many others, which taken two and two in order, have the same ratio ; the first shall have to the last of the first magnitudes, the same ratio...
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Euclid, Book V. Proved Algebraically, So Far as it Relates to Commensurable ...

Euclid, Lewis Carroll - 1874 - 62 Seiten
...> 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
...G' G H' &c. ThenA or_ E - F E H' F G G H' therefore A : D :: E : H. PROPOSITION XXIIL THEOREM. — If there be any number of magnitudes, and as many...in a cross 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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Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with ...

Robert Potts - 1876 - 403 Seiten
...first; that is, 1) is less than F. Therefore, if there be three, Sic. QED, PROPOSITION XXII. THEOKEM. If there be any number of magnitudes, and as many others, which taken two and two in order, have the same ratio ; the first shall have to the last of the first magnitudes, the same ratio...
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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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The Elements of Euclid for the Use of Schools and Colleges: With Notes, an ...

Isaac Todhunter - 1880 - 400 Seiten
...is greater oD; t is, D is less than F. Wherefore, if there be three &c. QED PROPOSITION 22. THEOREM. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall fiave to the last of the first magnitudes...
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The Elements of geometry; or, The first six books, with the eleventh and ...

Euclides - 1881
...* ex aquali," or * ex ayuo," that it, from equality of distance U'ff. \9). PROP. XXIII. THEOREM. y there be any number of magnitudes, and as many others,...in a cross order, have the same ratio ; the first shatt havt to the last of the first magnitudes the same ratio which the first hat to the last of the...
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