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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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The First Six Books of the Elements of Euclid: With Notes

Euclid - 1833 - 183 sider
...of A is contained in B; (ii)Def.5. therefore A is to C as B to DO (11). PROP. XXXIV. THEOR. Fig. 29. If there be any number of magnitudes, and as many others which taken two by two are in the same ratio, and if their proportion be ordinate ; they are ex aequali in the same...
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The Elements of Euclid; viz. the first six books, together with the eleventh ...

Euclides - 1834
...less than F. Therefore, if there be three, &c. Q. E, D. AB DE Ill PROPOSITION XXII. See N. THEOR. — If there be any number of magnitudes, and as many...which, taken two and two in order, have the same ratio, theJirst shall have to the last of the Jirst magnitudes, the same ratio which the Jirsl has to Ihe...
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An Elementary Treatise on Algebra, Theoretical and Practical ...

James Ryan, Robert Adrain - 1835 - 361 sider
...greater than C, D will be greater than F ; if equal, equal ; and if less, less. PROP. XXII. THEOR. If there be any number of magnitudes, and as many...the same ratio ; the first shall have to the last of ihe first rank of magnitudes, the same ratio which the first of the others has lo the last. NB This...
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The Elements of Euclid: viz. the first six books, together with the eleventh ...

Robert Simson - 1835 - 513 sider
...first; consequently D is less than F. Therefore, " if there be three" &c. QED PROP. XXII. THEOR. See N. If there be any number of magnitudes, and as many...and two in 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...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1835 - 316 sider
...H. In the same manner is the demonstration extended to any number of magnitudes. PROP. XXIII. THEOR. 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 will have to the last of the first magnitudes the same...
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The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...

Euclid - 1835 - 513 sider
...the number of magnitudes. Therefore, " if there be any number," &c. QED PROP. XXIII. THEOR. s«e N. 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 will have to the last of the first magnitudes t/ie same...
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The Element of Geometry

John Playfair - 1836 - 114 sider
...therefore ^=^ , or A : B : : E : F. JJ r D r PROP. XVI. THEOR. If there be any number of quantities, and as many others, which taken two and two in order have the same ratio, the first shall have to the laist of the first quantities the same ratio which the first of the others has to the last. This is...
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The Teacher's Assistant in the "Course of Mathematics Adapted to the Method ...

1836 - 472 sider
...last of the first magnitudes, the same ratio which the first of the others has to the last. XXIII. 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 will have to the last of the first magnitudes the same...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1836 - 471 sider
...: E, therefore, by the first case, since C^A, F^^D, that is, Therefore, &c. QED PROP. XXII. THEOR. If there be any number of magnitudes, and as many others, which, taken two and tico in order, have the same ratio ; the first will have to the last of tlie jirst magnitudes, the...
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The First Six and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclid, James Thomson - 1837 - 390 sider
...the number of magnitudes. Therefore, if there be any number of magnitudes, &c. PROP. XXIII. THEOR. 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 has to the last of the first magnitudes the same ratio...
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