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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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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1844 - 317 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...two and two, in a cross order, have the same ratio; thefirst will have to the last of the first magnitudes the same ratio which thefirst of the others...
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Elements of algebra, by A. Ingram and J. Trotter

Alexander Ingram - 1844
...= \ ... a : d : : e : h. bed fgh db PROP. XV. If there are any number of quantities more than two, and as many others, which, taken two and two in a cross order, are proportionals ; then ex cequo inversely are the extreme terms in the first rank, proportional to...
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The First Six, and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclid, James Thomson - 1845 - 352 Seiten
...demonstrated in the 22d proposition of this book. 20. Ex ceguo, inversely: when there are three or more magnitudes, and as many others, which taken two and two in a cross order, have the same ratio ; that is, when the first magnitude is to the second in the first rank, as the last but one is to the...
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Euclid's Elements of geometry [book 1-6, 11,12] with explanatory notes ...

Euclides - 1845
...first ; that is, D is less than F. 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 ; thefirst shall have to the last of thefirst magnitudes, the same ratio...
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Euclid in Paragraphs: The Elements of Euclid: Containing the First Six Books ...

Euclid - 1845 - 199 Seiten
...and therefore, D is less than F. Therefore, if there be three, &c. QBD PROPOSITION XXII. THEOB. — 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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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1846 - 317 Seiten
...the same manner is the demonstration extended to any number of magnitudes. PROP. XXIII. THEOR. iff there be any number of magnitudes, and as many others,...two and two, in a cross order, have the same ratio ; thefirst will have to the last of the first magnitudes the same ratio which theJirst of the others...
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The Elements of Euclid, the parts read in the University of Cambridge [book ...

Euclides - 1846
...greater than D, that is, D is less than F. Wherefore, If there be three %c. QED PROP. XXII. THEOS. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio, then the first shall have to the last of the first magnitudes the same...
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Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's ...

Dennis M'Curdy - 1846 - 138 Seiten
...be greater than Recite (a) def. 10, 5 ; (b) p. 15, 5; (c) Note 3, page 32; (d) p. A of b. 5. 22 Th. If there be any number of magnitudes, and as many others ; which, taken two and two in order, have the same ratio: then the first is to the last of one rank, as the first is to the last...
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A Treatise on Algebra: Containing the Latest Improvements. Adapted to the ...

Charles William Hackley - 1846 - 503 Seiten
...efg bcd=fgh' or ae , -3=5-; ie, a:d::e:h. XIV. If there be any number of quantities more than two, and as many others, which, taken two and two in a cross order, are proportionals (ex ;equali perturbatA), the first will have to the last of the first rank the same...
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The definitions, postulates, axioms, and enunciations of the propositions of ...

Euclides - 1848
...fourth shall be greater than the sixth; and if equal, equal; and if less, less. PROP. 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...
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