 | Mathematics - 1835 - 684 pages
...to the second, and of the third to the fourth be expressed by the same terms ; the first is said to have to the second the same ratio which the third has to the fourth ; and the four magnitudes are called proportionals. For example, let ABC D, and EFGH be two rectangles,... | |
 | Mathematics - 1836 - 488 pages
...the sixth ; the first lias also to the second a greater ratio than the fifth has to the sixth. XIV. If the first have to the second the same ratio which the third has to the fourth, and if the first be greater than the third, the second shall be greater than the fourth ; if equal,... | |
 | Mathematics - 1836 - 366 pages
...arithmetical, and the last three in harmonical progression • it is required to prove that the first has to the second the same ratio which the third has to the fourth. 45. Extract the fourth root of m? (m2— 3«2) + «2(rc2— 3w2) + 4 (m — n) (m + ri) mn */ _ \.... | |
 | John Playfair - Geometry - 1836 - 148 pages
...soon whatever be the number of magnitudes. Therefore, If, &c. QED PROP. XVIII. THEOR. If the first has to the second the same ratio which the third has to the' fourth; and the fifth tri the second, the same ratio which the sixth has to the fourth ; the first and fifth... | |
 | Euclid, James Thomson - Geometry - 1837 - 410 pages
...B, so are A, C, E together, to B, D, F together : wherefore, if any number, &c. PROP. XIIT. THEOR. IF the first have to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth ; the first will also have... | |
 | Andrew Bell - Euclid's Elements - 1837 - 290 pages
...second the same ratio which the third, together with the fourth, has to the fourth, the first shall have to the second the same ratio which the third has to the fourth. If A + B : B : : C + D : D, then by division A : B : : C : D. Take mA and nB any equimultiples of A... | |
 | John Playfair - Euclid's Elements - 1837 - 332 pages
...equimultiples of B, and of B+D+F ; therefore (def. 5. 5.) A : B : : A+C+E : B+D+F. PROP. XIII. THEOR. If the first have to the second the same ratio which the third has tothe fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first... | |
 | Robert Simson - Geometry - 1838 - 434 pages
...PROP. A. THEOR. G_ K c— TJ B DEF C_ H— B DEF If therefore two magniIF the first of four magnitudes have to the second the same ratio which the third...the fourth; then, if the first be greater than the second, the third is also greater than the fourth ; and if equal, equal ; if less, less.* Take any... | |
 | Euclides - 1840 - 192 pages
...B, and of B + D + F; therefore, A : B : : A + C + E .• B + D + F (v. Def. 5). PROP. XIII. THEOR. If the first have to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first has also to... | |
 | Oliver Byrne - Mathematics - 1841 - 144 pages
...when m — n = 1. PROP. A. THEO. If the first of the four magnitudes has the same ratio to the second, which the third has to the fourth, then, if the first be greater than the second, the third is also greater than the fourth ; and if equal, equal ; if less, less. Let • :... | |
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THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the... 
