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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 Elements: Or, Second Lessons in Geometry,in the Order of Simson's ... - Page 83
by Dennis M'Curdy - 1846 - 138 pages
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An elementary course of practical mathematics, Part 1

James Elliot - 1860 - 252 pages
...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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Euclid's plane geometry, books iii.-vi., practically applied; or, Gradations ...

Euclides - 1861 - 464 pages
...to the last of the first magnitudes, the same ratio which the first has to the last of the others. ' If there be any number of magnitudes, and as many others, which taken two and two, have the same ratio, they shall have the same ratio by equality. "— EUCLID. CON. Pst. 1, V. DEM....
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The Propositions of the Fifth Book of Euclid Proved Algebraically: with an ...

George Sturton Ward - Geometry, Algebraic - 1862 - 104 pages
...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 school edition. Euclid's Elements of geometry, the first six books, by R ...

Euclides - 1864 - 448 pages
...than F. Therefore, if there be three, &c. QED PROPOSITION XXII. THEOREM. If there be any number nf 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 has to the...
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The College Euclid: Comprising the First Six and the Parts of the Eleventh ...

Euclides - 1865 - 402 pages
...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 - Euclid's Elements - 1867 - 424 pages
...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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The Elements of Euclid for the Use of Schools and Colleges: Comprising the ...

Isaac Todhunter - Euclid's Elements - 1867 - 426 pages
...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.

Euclid - 1868 - 138 pages
...то irX^боc, avvSvo \afi▀avofiiva Kai ┐v тф avry Xˇy^* кaÝ Siţaov 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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Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with ...

Robert Potts - 1868 - 434 pages
...greater than D, by case first ; PROPOSITION XXII. THEOREM. If there be any number of magnitudes, and a3 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 has to the...
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The Elements of Plane and Solid Geometry

Henry William Watson - Geometry - 1871 - 320 pages
...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...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 ratio as the first has to...
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