| 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... | |
| 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... | |
| 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.... | |
| 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... | |
| 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... | |
| 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... | |
| 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... | |
| 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... | |
| 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.... | |
| 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... | |
| |