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" 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 which the first of the others has to the last. NB This is usually... "
Euclid's plane geometry, books iii.-vi., practically applied; or, Gradations ... - Seite 217
von Euclides - 1861
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The First Six Books of the Elements of Euclid: With Notes

Euclid - 1833 - 183 Seiten
...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
...PROPOSITION XXIII. THBOR. — If there be any number of magnitudes, and as many See N. others, whifh, taken two and two in a cross order, have the same ratio, the Jirsl shall have to the last of the Jir si magnitudes, the same ratio which the Jirsl has to the last...
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The Elements of Euclid: viz. the first six books, together with the eleventh ...

Robert Simson - 1835 - 513 Seiten
...the number of magnitudes. Therefore, " if there be any number," &c. QED PROP. XXIII. THEOR. See N. If there be any number of magnitudes, and as many...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...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1835 - 316 Seiten
...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...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...
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The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...

Euclid - 1835 - 513 Seiten
...D is less than F. Therefore, " if there be three," &c. QED PROP. XXII. THEOR. ABCABC DEFDEF See N. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio ; the first will have to the last of the first magnitudes the same ratio...
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An Elementary Treatise on Algebra, Theoretical and Practical ...

James Ryan, Robert Adrain - 1835 - 361 Seiten
...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 ihe first rank of magnitudes, the same ratio which the first of the others has lo the last. NB This...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1836 - 471 Seiten
...: 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 Element of Geometry

John Playfair - 1836 - 114 Seiten
...and C : D : : E : F ; then A : D : : E : H. Because A, B, C, are three quantities, and F, G, H, three others, which taken two and two in a cross order have the same ratio, by the , first case, A : C : : F : H, but C : D : : E : F, wherefore again by the first case, A : D...
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The Teacher's Assistant in the "Course of Mathematics Adapted to the Method ...

1836 - 472 Seiten
...third, the fourth is greater than the sixth ; if equal, equal; and if less, less. XXII. If there beany number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first will have to the last of the first magnitudes, the same ratio...
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Elements of Plane Geometry According to Euclid

Andrew Bell - 1837 - 240 Seiten
...E; therefore, by the first case, since C ^ A, F -^ D, that is, D *£. F. PROPOSITION XXn. 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...
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