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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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The Elements of Euclid for the Use of Schools and Colleges: With Notes, an ...

Isaac Todhunter - 1880 - 400 sider
...is greater oD; t is, D is less than F. Wherefore, if there be three &c. QED PROPOSITION 22. THEOREM. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall fiave to the last of the first magnitudes...
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The Elements of geometry; or, The first six books, with the eleventh and ...

Euclides - 1881
...* ex aquali," or * ex ayuo," that it, from equality of distance U'ff. \9). PROP. XXIII. THEOREM. y 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 shatt havt to the last of the first magnitudes the same...
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The Elements of Euclid for the Use of Schools and Colleges: Comprising the ...

Euclid, Isaac Todhunter - 1883 - 400 sider
...is, D is less than F. AB PE C F AB D B B I EUCLID'S ELEMENTS DE I GKMHL 1ST PROPOSITION 22. THEOREM. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes...
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The Elements of Plane and Solid Geometry

Henry William Watson - 1884
...times the sum A+C+E is equal to m times the sum B + D + F, therefore ± is equal to , PROPOSITION 12. If there be any number of magnitudes, and as many...taken two and two in order have the same ratio, the Jirst shall have to the last of the first set of magnitudes the same ratio as the first has to the...
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The Elements of Euclid, books i. to vi., with deductions, appendices and ...

Euclides - 1884
...BV 13 Now C:B = E:D; .-. E: F is less than E:D; F. 13 D is less than FF 10 PROPOSITION 22. THEOREM. If there be any number of magnitudes, and as many others, which taken two and two in direct order, have the same ratio ; tJie first shall have to the last of the first magnitudes the same...
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The Harpur Euclid: An Edition of Euclid's Elements

Edward Mann Langley, W. Seys Phillips - 1890 - 515 sider
...[IST PART. and the theorem can be easily extended to any number of magnitudes. Hence generally : — 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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Ellen: Or, Whisperings of an Old Pine, Volum 1

Joseph Battell - 1903
...which the first of the others has to the last.' " See Proposition XX. PROPOSITION XXIII. ' If there are 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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Books 3-9

Euclid - 1908
...P." He also gives the extension of the proposition to any number of magnitudes, enunciating it thus : "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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The Thirteen Books of Euclid's Elements

Euclid
...He also gives the extension of the proposition to any number of magnitudes, enunciating it thus : " 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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Pass school, group A (-D).

Oxford univ, exam. papers, 2nd publ. exam - 1884
...rectilineal figure and equal to another given rectilineal figure. 8. Inscribe a square in a given circle. 9. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in a cross order, the first shall have to the last of the first...
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