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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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Elements of Geometry: Containing the First Six Books of Euclid with a ...

John Playfair - 1855 - 318 sider
...demonstration extended to any number of magnitudes. nF(11.5.). Again, PROP. XXIII THEOR. If there le any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio ; the first will have to the last of the first magnitudes the same...
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The Elements of geometry; or, The first six books, with the eleventh and ...

Euclides - 1855
...Therefore, if there be three, &c. QED PROP. XXII. THEOREM. If there be any number of magnitudes, andas many others, which taken two and two in order, have the same ratio ; (he first shall have to the last of the first magnitudes, the same ratio which the first has to the...
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The Elements of Euclid, with many additional propositions, and explanatory ...

Euclides - 1855
...3 KMHLN ABC_EFG lhen B ' C ' DF ' G ' H' therefore A : D :: E : H. PROPOSITION XXIII. THEOREM.—Jf there be any number of magnitudes, and as many others, which, taken tivo and two, in a cross order, have tlie same ratio; the first has to the last of the first magnitudes...
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Manual of Euclid: books iv., v., vi., by J.A. Galbraith and S. Haughton

Euclides - 1859
...laa rо irXqöOç, aiivSvo \apßavoptva Kai iv rif avrif \oyif Kai Stîffov iv ry avru Xóyy Éffrai. 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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Euclid's Elements of plane geometry [book 1-6] explicitly enunciated, by J ...

Euclides - 1860
...same manner the demonstration is extended to any number of magnitudes. 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 shall have to the last of the first magnitudes the same...
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The Elements of Euclid with Many Additional Propositions and Explanatory Notes

Eucleides - 1860
...any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio. If there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio. If magnitudes are equal . . If two magnitudes are unequal. If a...
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An elementary course of practical mathematics, Del 1

James Elliot - 1860
...greatest and consequently c the least, b + c > a + d. THEOREM X. If there be any Number of Quantities, and as many others, which, taken two and two in Order, have the same Ratios, tlie first shall have to the last ofthefirst Quantities the same Ratio which the first of the...
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The Propositions of the Fifth Book of Euclid Proved Algebraically: with an ...

George Sturton Ward - 1862 - 79 sider
...n : : p : z. PROPOSITION XXIII. " Ex jEQUALI IN PROPORTIONS PERTURBATA," OR " Ex jEQUO PERTURBATO." 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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The College Euclid: Comprising the First Six and the Parts of the Eleventh ...

Euclides - 1865
...whatever be the number of magnitudes. Therefore, if there be any number, &c. QED PROP. XXm.— THEOREM. // there be any number of magnitudes, and as many others, which taken two and two in a cross order, have the same ratio ; then the first shall have to the last of the first magnitudes...
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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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