| John Playfair - 1849 - 317 Seiten
...and B : A : : F : E, therefore, by this first case, since CyA, F7 D, that is, D/F. PROP. XXII. THEOR. **If there be any number of magnitudes, and as many others, which, taken two** ana two in order, have the same ratio ; the first will have to the last of the first magnitudes, the... | |
| Royal Military Academy, Woolwich - 1853
...less than F. Therefore, if there be three, etc. QED c • EUCLID S ELEMENTS. PROPOSITION XXII. THEOR. **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... | |
| Euclides - 1853 - 147 Seiten
...therefore, d is less than f Therefore, if there be three magnitudes, &c. QED PROPOSITION XXII. — THEOREM. **If there be any number of magnitudes, a/nd as ma/ny others, which, taken two and two in** order, have the sa/те ratio ; the first shall have to the last of thefirst magnitudes the same ratio... | |
| John Playfair - 1855 - 318 Seiten
...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,...in a cross order, have the same ratio ; the first** will have to the last of the first magnitudes the same ratio which the first of the others has to the... | |
| Euclides - 1855
...the last of the others. First, let there be three magnitudes A, B and С, and other three D, E and F, **which taken two and two in a cross order have the same ratio** ; that is, such that A is to B, as E is to F ; and as B is to C, so is D to E. A is to С, as D is... | |
| 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... | |
| 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... | |
| Robert Potts - 1860 - 361 Seiten
...first; that is, JD is less than JP. Therefore, if there be three, &c. QED PROPOSITION XXII. 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... | |
| Euclides - 1860
...I> : C ; and, by inversion.(Prqp. 4), A:A + B::C:C + D. PROPOSITION XXII. THEOREM. (EUCLID tSXII.) **If there be any number of magnitudes, and as many others, which, taken two and two in** order, have the same ratio ; the first ehall have to the last of the first magnitudes the same ratio,... | |
| Eucleides - 1860
...ratio taken two and two, but in a cross order, then, if the first magnitude be greater than the third. **If there be 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... | |
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