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Book V. When, of equimultiples, the multiple of the first exceeds the
multiple of the second, but the multiple of the third does not exceed the multiple of the fourth ; the first to the second is said to have a greater ratio than the third to the fourth.
fourth a triplicate ratio of what it has to the second; and
whose antecedents are to the antecedents and consequents to
XIII. Alternate ratio is the comparing the antecedent with the antecedent, and consequent with the consequent.
XIV. Inverse ratio is, when the confequentis taken as the antecedent, and compared with the antecedent as a consequent.
XV. Compounded ratio is, when the antecedent and consequent, taken as one, are compared with the consequent itself.
XVI. Divided ratio is, when the excess, by which the antecedent exceeds the consequent, is compared with the consequent.
nitudes in one order, and a like number of magnitudes in ano-
XIX. Ordinate proportion is, the ratio being, as in the last, as the antecedent is to the confequent, in the first order of magnitudes;
so is the antecedent to the consequent in the second order of
tudes, and others equal to them in number, taken two and
A X I 0 MS.
E fuldesmare caual?o each of the fame, or of equal magni
tudes, are equal to each other.
equimultiples are equal, are equal to each other.
PRO P. I. THEO R.
F there be any number of magnitudes, equimultiples of a like
number of magnitudes, each of each, whatever multiple any one of the former magnitudes is of its correspondent one, the same multiple are all the former magnitudes of all the latter.
Let AB, CD, be magnitudes, equimultiples of E, F, whatever multiple AB is of E, and CD of F, the fame multiple AB, CD, together, is of E, F, together.
For, let the magnitudes in AB, equal to E, be AG, GB; and the magnitudes in CD, equal to F, be CH, HD, then AG, CH, are equal to E, F; and BG, HD, likewise equal to E, F; therefore, as often as AB contains E, and CD, F, so often AB, CD, contains E, F: Wherefore, if there are, &c.
PRO P. II. THEO R.
the fourth; and if the fifth be the same multiple of the second, that the fixth is of the fourth; then shall the first, added to the fifth, be the same multiple of the second, that the third, added to the lixth, is of the fourth.
Let the first AB be the same multiple of the second C, that Book V, the third DE is of the fourth F; and let the fifth BG be the fame multiple of the second C, that the fixth EH is of the fourth F; then AG will be the same multiple of C that DH is of F.
For, because AB is the same multiple of C that DE is of F, there are as many magnitudes in AB equal to C, as in DE, equal to F. For the same reason, there are as many magnitudes in BG equal to C, as there are in EH, equal to F; therefore there are as many magnitudes in AG equal to C, as there are in DH equal to Fa Wherefore, &c.
PRO P. III. THEOR.
IF the first be the same multiple of the second, that the third is of
the fourth, and there be taken equimultiples of the first and third, then will the magnitudes po taken be equimultiples of the second and fourth.
Let the first A be the same multiple of the second B that the third C is of the fourth D; and let EF, GH, be equimultiples of A, C sthen EF is the fame multiple of B, that GH is of D. For, let the magnitudes in EF, equal to A, be FK, KE ; and the magnitudes in GH, equal to C, be HL, LG; then there are as many magnitudes equal to B in FE, as there are magnitudes equal to Ď in GH"; wherefore FE is the same multiple of B, ihat GH is of D. Wherefore, &c.
PRO P. IV, THE OR.
If the first have the same ratio to the second that the third has to
the fourth, then shall also the equimultiples of the first have the Jame ratio to the equimultiple of the second that the equimultiple of the third has to that of the fourth.
Let there be four magnitudes, A, B, C, D, such, that A is to B as C to D. Let E, F, be taken the same multiples of A, C; and G, H, the same multiples of B, D, then E is to G as F is to H. For, take K, L, any equimultiples of E, F, and M, N, any equimultiples of G, H; then K‘is the same multiple of A' that L is of Ca. For the same reason, M is the a 3: same multiple of B that N is of Do; but, because A is to B as S is to D, if K be equal to M, I will be equal to N; if great
b def, s.
4 Book V. er, greater, and, if less, less; but K, L, are equimultiples of VE, F, and M, N, of G, H; wherefore E is to Gas F is to
Hb; but it is proved, that, if K be equal to M, L is equal to N; but, if K is equal to M, M is equal to K, and N to L; if greater, greater, and, if less, lefs; wherefore G is to E as H is to F. Wherefore, if four magnitudes be proportional, they will also be inversely proportional. Wherefore, &c.
IF one magnitude be the same multiple of another magnitude,
that a part taken from the one is of a part taken from the other; then the residue of the one shall be the same multiple of the residue of the other that the whole is of the whole.
Let AB be the same multiple of CD, that a part taken away AE is of a part taken away CF; then the residue EB shall be the same multiple of the residue FD, that the whole AB is of the whole CL.
For, let BE be the same multiple of CG that AE is of CF; then, because AE is the same multiple of CF that BE is of CG, AE will be the same multiple of CF that AB is of GF"; but AE is the same multiple of CF that AB is of CD, and AB is
the fame multiple of GF that it is of CD; therefore GF is e, b Ax. z. qual to CD b; take CF, which is common, from both, there re
mains GC equal to FD; therefore AE is the same multiple of CF that EB is of FD. Wherefore, &c.
PRO P. VI. THEO R.
magnitudes, equimultiples of the same, be taken away, the residuc fall either be equal to these magnitudes or equimultiples of them.
For, first, let GB be equal to E ; if HD is not equal to F, let KC be equal to F; then AB is the fame multiple of E that KH is of F a; but AB, CD, are put equimultiples of E, F; therefore
HK is the same multiple of F that CD is of F; therefore KH is Asi zi equal to CD 6. Take CH, which is common, from both, there
remains KC equal to HD; but KC was put equal to F; therefore HD is equal to F; after the same manner it may be demonstrated that, if GB is any equimultiple of E, HD will be the like equimultiple of F. Wherefore, &c.