...OAv OB, be proportional, and the following statement hold good:— If there be four magnitudes, and any equimultiples whatsoever of the first and third...equimultiples whatsoever of the second and fourth being taken, if the equimultiples of the first and third together exceed, or are equal to, or are less...

...Or, to bring it still nearer to the language of Euclid's definition: — The first of four magnitades is said to have the same ratio to the second, which...equimultiples whatsoever of the first and third being taken, the second is contained as often in the equimultiple of the first, as the fourth is contained in the...

...any integers, m,*l. mA. or m/ll : na, : : m/f, : na,. That i«, if the first of four magnitudes has the same ratio to the second which the third has to the fourth ; then any equimultiples whatever of the first and third shall have the game ratio to any equimultiples...

...into three, and also into five equal parts. 6. When is the first of four magnitudes said to have the the same ratio to the second which the third has to the fourth ? Prove that triangles which have the same altitude are to one another as their bases. Show also that...

...two magnitudes of the same kind to one another, in respect of quantity, is called their ratio. XXX. The first of four magnitudes is said to have the same ratio to the second, which the third has to the fouitl', when any equimultiples whatsoever of the first and third i being taken, ai;d any equimultiples...

...rectilineal figures. Explain homologous, alternando, ex sequali. When is the first of four magnitudes said to have the same ratio to the second which the third has to the fourth ? 7. In a right angled triangle, if a perpendicular be drawn from the right angle to the base, the...

...triangle, pentagon, and hexagon. 7. Give Euclid's definition of ratio. When is the first of four magnitudes said to have the same ratio to the second which the third has to the fourth ? 8. The sides about the equal angles of equiangular triangles are proportional. If a straight line...

...together. [V. Definition 5. Wherefore, if any number &c. Q.EJ>. PROPOSITION 13. THEOREM. If the first have the same ratio to the second which the third has to the fourth, but the third to the fourth a greater ratio than the fifth to the si.cth, thefirst shall have to ths...

...they cannot be said to be of the same kind, and so cannot have any ratio to each other. T. The Brst of four magnitudes is said to have the same ratio...of the first be less than that of the second, the mu'tiple of the third is also less than tlint of the fourth : or, if the multiple of the first be equal...

...17 QED 272. DBF. Euclid's test of a proportion is as follows :-•" The first of four magnitudes iä said to have the same ratio to the second which the...equimultiples whatsoever of the second and fourth ; (l ± P] a : \ q/ : 0 ± -) V q/ :b±Р-b \ b : : a : b, Q/ ~T~ — Cz ч Ч : : a : b. "If the multiple...