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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 ... - Σελίδα 83
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...firft ; and therefore D is lefs than F. Therefore if there be three, &c. Q._E. D. PROP. XXII. THEOR, TF there be any number of magnitudes, and as many others, which taken two and two in order have the fame ratio ; the firft fhall have to the laft of the firft magnitudes the fame ratio which the firft...

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...Therefsre, i( there be any number, &c. Q^ED BC FG D. H. PROP. Book V. PROP. XXIII. THEO R. SeeK. TF there be any number of magnitudes, and as many others, which taken two and two, in a crofs order, have the fame ratio ; the firft mall have to the laft of the firft magnitudes the fame...

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...of magnitudes. Theiefore, if there be any number, &c. Q^ED K a P IUO P. v. PROP. XXIII. THEO R. K. IF there be any number of magnitudes, and as many others, which, taken two and two, in a crofs order, have the fame ratio ; the firft mail have to the laft of the firft magnitudes the fame...

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...if there be any number, &c. Q;_, ED • Book V. SeeN. a. if- Jkn ft 4ii. J. PROP. XXIII. THEO R. TF there be any number of magnitudes, and as many ,*, others, which, taken two and two, in a crofs order, have the fame ratio ; the firft lhall have to the laft of the firft magnitudes the fame...

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...the demonftration extended to any number of magnitudes. Therefore, Scc. Q^ ED PROP. XXIII. THEO R. IF there be any number of magnitudes, and as many others, which, taken two and two, in a crofs order, have the fame ratio; the firft (hall have to the laft of the firft magnitudes the fame...

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...л . /Yj _ V -'HI ^n<^ ^° 4I>' v/batever be the number of magnitudes. PROPOSITION XXIIL— THEOREM. If there be any number of magnitudes, and as many others, which taken two and two inordinately, have the fame ratio ; then, by perturberate equality (Def. XX.), the firft mall have...

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...as E to H ; and fo on, whatever be the number of magnitudes. Wherefore, &c. Q^ED PROP. XXIII. THEOR. IF there be any number of magnitudes, and as many others, which, taken two and two, in a crofs order, have the fame ratio ; the firll mail have to the laft of the firft magnitudes the fame...

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...therefore, by the firft cafe, fince C > A, F > D, that is, D < F. Therefore. . f RO P. ^XXII. THEO R. IF there be any number of magnitudes, and as many...others, which, taken two and two in order, have the fame ratio ; the firft will have to the laft of the firft magnitudes, the fame ratio which the firft...

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...F>D, that is, D<F. Therefore, &c. QED But A : B : : : : D : E, and : E : De, and, PROP. XXII. THEOK. IF there be any number of magnitudes, and as many...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 which the first of the others...

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...whatever be the number of magnitudes. Therefore, if there be any number, &c. 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 first shall have to the last of the first magnitudes the same...




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