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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 ... - Side 219
av Euclides - 1861
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The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago ...

Robert Simson - 1762 - 466 sider
...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...
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The Elements of Euclid, Viz: The Errors, by which Theon, Or Others, Have ...

Robert Simson - 1775 - 520 sider
...therefore D is lefs than F. Therefore, if there be three, &c. Q^ED A D B E PROP. XXII. THEO R.' TF there be any number of magnitudes, and as many others, which taken two and two in order have the fame ratio i the firft (hall have to the laft of the firft magnitudes the fame ratio...
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The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago ...

Robert Simson - 1781 - 466 sider
...therefore D is lefs than F. Therefore if there be three, &c. OED C F A E Sec N. PROP. 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 mail have to the laft of the firft magnitudes the fame ratio...
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The First Six Books: Together with the Eleventh and Twelfth

Euclid - 1781 - 520 sider
...firft ; and therefore D is lefs than F. Therefore, if there be three, &c. QJ£. D. PROP. 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 fhall have to the laft of the firft magnitudes the fame ratio...
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Elements of Geometry;: Containing the First Six Books of Euclid, with Two ...

Euclid, John Playfair - 1795 - 400 sider
...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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The Elements of Mathematical Analysis, Abridged: For the Use of Students ...

Nicolas Vilant - 1798 - 170 sider
...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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The Elements of Euclid: Viz. the First Six Books, with the Eleventh and ...

Alexander Ingram - 1799 - 351 sider
...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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Elements of Geometry;: Containing the First Six Books of Euclid, with a ...

Euclid, John Playfair - 1804 - 440 sider
...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...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1806 - 311 sider
...nC, mD, nE, feF. Book V. Next, let there be four magnitudes, A, B, C, D, and other four, E, F, G, H, which, taken two and two, in a cross order, have the same ratio, A : B : : G : H, B : C : : F : G, and A, B, C, D, C : D : : E : F ; then A : D : : E : H. For, since...
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The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago ...

Robert Simson - 1806 - 518 sider
...ex aquo per tur bate." M EF N First, Let there be three magnitudes A, B, C, and other three D, E, F, which, taken two and two, in a cross order, have the same ratio, that is, such that A is to B, as E to F ; and as B is to C, so is D to E : A is to C, as D to F. Take...
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