 | John Playfair - Euclid's Elements - 1855 - 340 pages
...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... | |
 | Euclides - 1855 - 270 pages
...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... | |
 | Euclides - 1855 - 230 pages
...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... | |
 | Euclid - 1859 - 150 pages
...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... | |
 | Euclides - 1860 - 288 pages
...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... | |
 | Eucleides - 1860 - 396 pages
...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... | |
 | 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... | |
 | 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... | |
 | 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... | |
 | 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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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... 
