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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

## Elements of geometry, containing books i. to vi.and portions of books xi ...

Euclides, James Hamblin Smith - Geometry - 1872 - 376 pages
...is greater than FV 9. Similarly the other cases may bo proved. Q. nn PROPOSITION XXI. (Eucl. v. 22.) If there be any number of magnitudes, and as many others, which have the same ratio taken two and two in order, the first must have to the last of the first magnitudes...

## The Elements of Euclid, containing the first six books, with a selection of ...

Euclides - 1874 - 342 pages
...be the number of magnitudes. Therefore, if there be any number, &o. QED PROPOSITION 23. — Theorem. 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...

## Elements of Euclid Adapted to Modern Methods in Geometry

Euclid, James Bryce, David Munn (F.R.S.E.) - Geometry - 1874 - 234 pages
...proportion will be B+A:BA::D + C:DG PHOP. XI.— THEOREM. (Euc. V. 22, 23). Tf there be three or more magnitudes, and as many others which, taken two and two in order, have the same ratio, then by equality, the first IMS to the last of the first rank the same ratio that the first has to the last...

## Euclid, Book V. Proved Algebraically, So Far as it Relates to Commensurable ...

Euclid, Lewis Carroll - Euclid's Elements - 1874 - 80 pages
...> l ; С .-. , xn« by с, а>с. Similarly, if А=С, а=с; and if А<С, а< с. QED PROP. XXII. If there be any number of magnitudes, and as many others which, taken two and two, have the same ratio : the first of the first rank shall have to the last the same ratio as the first...

## Euclid, book v. proved algebraically, so far as it relates to commensurable ...

Charles Lutwidge Dodgson - 1874 - 96 pages
....•. , substituting, - > 1 ; QED by c, a>c. Similarly, if A=C, a — c; and if A<C, a< c. PROP. XXII. If there be any number of magnitudes, and as many others which, taken two and two, have the same ratio : the first of the first rank shall have to the last the same ratio as the first...

## Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with ...

Robert Potts - Geometry, Plane - 1876 - 446 pages
...first; that is, 1) is less than F. Therefore, if there be three, Sic. QED, PROPOSITION XXII. THEOKEM. If there be any number of magnitudes, and as many...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 has to the...

## First Public Examination in Literis Graecis Et Latinis

University of Oxford - Greek language - 1879 - 412 pages
...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...

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

Isaac Todhunter - Euclid's Elements - 1880 - 426 pages
...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...