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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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An Elementary Treatise on Algebra: Theoretical and Practical

James Ryan, Robert Adrain - 1824 - 516 sider
...of the proposition may be shown, whatever be the number of magnitudes. QED 3«J 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 same...
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An Elementary Treatise on Algebra: Theoretical and Practical

James Ryan, Robert Adrain - 1824 - 516 sider
...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 - 1826 - 383 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 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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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

Euclid, John Playfair - 1826 - 320 sider
...Therefore, &e. QED PROP. XXII. THEOR: If there be any number of magnitudes, and as many Others, whieh, taken two and two in order, have the same ratio ; the first will have to the last of the first magnitudes, the same ratio whieh the first of the others has to...
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The Elements of Euclid: The Errors by which Theon, Or Others, Have Long ...

Robert Simson - 1827 - 513 sider
...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 - 1828 - 324 sider
...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 - 1830 - 685 sider
...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 - 1832 - 333 sider
...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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Iron: An Illustrated Weekly Journal for Iron and Steel ..., Volum 19

Perry Fairfax Nursey - 1833
...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 Mechanics' Magazine, Museum, Register, Journal, and Gazette, Volum 19

1833
...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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