Books Books 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 ## The Elements of Euclid, books i. to vi., with deductions, appendices and ...

Euclides - 1884 - 434 pages
...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... ## The Harpur Euclid: An Edition of Euclid's Elements

Edward Mann Langley, W. Seys Phillips - 1890 - 538 pages
...[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... ## Ellen: Or, Whisperings of an Old Pine, Volume 1

Joseph Battell - Force and energy - 1903 - 722 pages
...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... ## Books 3-9

Euclid - Mathematics, Greek - 1908 - 456 pages
...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... ## The Thirteen Books of Euclid's Elements

Euclid - 452 pages
...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... 