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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 Elements: Or, Second Lessons in Geometry,in the Order of Simson's ... - Page 83
by Dennis M'Curdy - 1846 - 138 pages
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An Elementary Treatise on Algebra, Theoretical and Practical ...

James Ryan, Robert Adrain - Algebra - 1824 - 542 pages
...truth of the proposition may be shown, whatever be the number of magnitudes. QED 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 6rst shall have to the last of the first magnitudes the...
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An Elementary Treatise on Algebra: Theoretical and Practical ...

James Ryan - Algebra - 1826 - 383 pages
...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 others, •which, take two and two in order, have the same ratio ; the first shall have to the last of the first rank...
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The Elements of Euclid: The Errors by which Theon, Or Others, Have Long ...

Robert Simson - Trigonometry - 1827 - 513 pages
...Therefore, if there be three, &c.. SeeN. * 4.'5. * 20. 5. •f- Constr. *5Def. 5. PROP. XXII. 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 theJirst has to the...
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The First Six Books of the Elements of Euclid, with a Commentary and ...

Euclid, Dionysius Lardner - Euclid's Elements - 1828 - 544 pages
...may be extended to series consisting of any number of magnitudes. PROPOSITION XXIII. THEOREM. (502) 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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A Treatise on Algebra

George Peacock - Algebra - 1830 - 732 pages
...perturbatfi. A similar demonstration will apply when there are more than three magnitudes in each set. 413. " If there be any number of magnitudes and as many others, which, taken two and two in a cross order, jlaye ^ same rati0; namely, if the first be to the second of the first set of magnitudes,...
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Elements of Geometry: Containing the First Six Books of Euclid : with a ...

John Playfair - Euclid's Elements - 1832 - 333 pages
...therefore, by the first case, since C> A, F>D, that is, D^_F. Therefore, &c. QED PROP. XXII. THEOR. Jf there be any number of magnitudes, and as many others, which, taken two and two in order, have (he same ratio; the Jirst will have to the lust of the first magnitudes, the same ratio which the first...
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The Mechanics' Magazine, Museum, Register, Journal, and Gazette, Volume 19

Industrial arts - 1833 - 496 pages
...1st case; but it seems tome impossible to make it shorter than in Euclid. II. The 23d Proposition. If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, Lave the same ratio, the first will have to the last of the first magnitudes the same...
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Iron: An Illustrated Weekly Journal for Iron and Steel ..., Volume 19

Perry Fairfax Nursey - Industrial arts - 1833 - 488 pages
...1st case ; but it seems tome impossible to make it shorter than in Euclid. II. The 23d Proposition. 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 First Six Books of the Elements of Euclid: With Notes

Euclid - Euclid's Elements - 1833 - 183 pages
...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 ...

Euclid - 1835 - 540 pages
...the number of magnitudes. Therefore, " if there be any number," &c. QED PROP. XXIII. THEOR. se 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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