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PART, is a Magnitude of a Magni
tude, the less of the greater, when
ibe lefser measures the greater.
II. But a Multiple is a Magnitude of a Magnitude, the greater of the lejer, when
the lesser measures the greater. fII. Ratio, is a certain mutual Habitude of Mag
nitudes of the same kind, according to Quantity. IV. Magnitudes are said to have Proportion to,
each other, which being inultiplied can exceed 1 one another. V. Magnitudes are said to be in the same Ratio,
the first to the second, and the third to the fourth, when the Equimultiples of the first and third, compared wiih the Equimultiples of the second and fourth, according to any Multipli-, cation whatsoever, are either both together greater, equal, or less, than the Equimultiples of the second and fourth, if those be taken that answer each other.
That is, if there be four Magnitudes, and you take any Equimultiples of the first and third, and also any Equimultiples of the second and fourth. And if the Multiple of the first be greater than the Multiple of the second, and also the Multiple of the third greater than the Multiple of the fourth: Or, if the Multiple of the first be equal to the Multiple of the second; and also the Multiple of the third equal to the Multiple of the fourth : Or, lastly, if the Multiple of the firft be less than the Multiple of the second, and also that of the third less than that of the fourth, and these Things happen according to every Multiplication whatsoever ; then the four Magnitudes are in the fame Ratio, the first to the second, as the third to the fourth.
VI. Magnitudes that have the same Proportion, are
Expounders usually lay down here that Definition which Euclid has given for Numbers only, in his seventh Book, viz. That
Magnitudes are said to be Proportionals, when the first is the same Equimultiple of the second, as the third is of the fourth, or the same Part, or Parts.
But this Definition appertains only to Numbers and commensurable Quantities; and so since it is not universal, Euclid did well to reject it in this Element, which treats of the Properties of all Proportionals; and to fubftitute another general one, agreeing to all Kinds of Magnitudes. In the mean time, Expounders very much endeavour to demonstrate the Definition here laid down by Euclid, by the usual received Definition of proportional Numbers; but this much easier flows from that, than that from this; which may be thus demonstated:
First, Let A, B, C, D, be four Magnitudes which are in the same Ratio, according to the Conditions that Magnitudes in the fame Ratio must have laid down in the fifth Definition. And let the first be a Multiple of the second. I say, the third is also the fame Multiple of the fourth. For Example: Let A
be equal to 5B. Then C shall be equal to 5D. Take any Number. For Example, 2, by which let 5 be multiplied, and the Product will
A: B: C: D be 10: And let 2A, 2C, be Equimultiples of the first and third Magnitudes A and C: 2A, 10B, 2C, 10D Also, let 10B and 10D be Equimultiples of the second and fourth Magnitudes B and D. Then (by Def. 5.) if 2A be equal to 10B, 2C shall be equal to 10D. But since A (from the Hypothesis) is five Times B, 2A shall be equal to 10B; and lo 2C equal to 10D, and C equal to 5D; that is, C will be five Times D. W. W. D.
Şecondly, Let A be any Part of B; then C will be the same Part of D. Fór because A is to B, as C is to D; and since A is fome Part of B; then B will be a Multiple of A: And so (by Cafe 1.) D will be the fame Multiple of C, and accordingly C shall be the fame Part of the Magnitude D, as A is of B. W.W.D.
Thirdly, Let A be equal to any Number of whatsoever Parts of B. I say, C is equal to the fame Number of the like Parts of D. For Example: Let A be a fourth Part of five Times B ; that is, let A be equal to B. I say, C is also equal to įD. For because A is equal to B, each of them being multiplied by 4, then 4A will be equal to 5B. And lo if the Équimultiples of the first
A: B:: C: D. and third, viz. 4A, 4C be alfumed ; as also the Equimultiples of the second and fourth, viz. 4A, 5B, 4C, 5D. 5B, 5D, and (by the Definition) if 4A is equal to 5B; then 4C is equal to sĎ. But 4A has been proved equal to 5B, and so 4C shall be equal to 5D, and C equal to D. W.W.D.
And univerfally, if A be equal to B, C will be equal to D, For let A and C be multiplied by m, and B and A: B:: C: D D by n. And because A is equal
mA, nB, mC, D to-B; mA shall be equal to nB; wherefore (by Def. 5.) mC will be equal to xD, and C equal to " D. W.W.D.
VII. 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; then the first to the second is said to have a greater Proportion than the
third to the fourth. VIII. Analogy is a Similitude of Proportions. IX. Analogy at least consists of three Termş. X. When thrée Magnitudes are Proportionals, the
first is said to have to the third, a Duplicate Ra
fio to what it has to the second. XI. But when four Magnitudes are Proportionals
, the first Mall have a triplicate Ratio to the fourth of what it has to the second; and so always one more in Order, as the Proportionals
hall be extended. XII. Homologous Magnitudes, or Magnitudes of a
like Ratio, are said to be such whosé Antecedents are to the Antecedents, and Consequents to the
Consequents. XIII. Alternate Ratio, is the comparing of the An.
tecedent with the Antecedent, and the Consequent
with the Consequent. XIV Inverse Ratio, is when the Consequent is
taken as the Antecedent, and so compared will
the Antecedent as a Consequent. XV. Compounded Rațio, is when the Antecedent
and Consequent taken both as one, is compared
to the Consequent itself. XVI. Divided Ratio, is when the Excess wherein
the Antecedent exceeds the Consequent, is com
pared with the Consequent. XVII. Converse Ratio, is when the Antecedent is
compared with the Excess, by which the Antece
dent exceeds the Consequent. XVIII. Ratio of Equality, is where there are taken
more than two Magnitudes in one Order, and a like Number of Magnitudes in another Order, comparing two to two being in the same Pro
as th: ibird to the fourth; then also shall the Equi-
second B as the third C hath to the fourth D; and let E and F, the Equimultiples of A and C, be any how taken; as also G, H, the Equimultiples of B and D. I say E' is to Gas F is to H.
For take K and L, any Equimultiples of E and F; and also M and N of G and H.
Then because E is the fame Multiple of A, as p is of C, and K, L are taken Equimultiples of E, K E A B Ġ Ñ F, K will be * the same
* 3 of this Multiple of A, as L is of L F C D HN C. For the fame Reason, M is the same Multiple of B, as N is of D. And since A is to B, as C is to D, and K and L are Equimultiples of A and C; and also M and N Equimultiples of B and D. If K exceeds M, then + L will exceed N; if equal, equal; or less, less.
of this And K, L are Equimultiples of E, F, and M, N, any other Equimultiples of GH, Therefore, as E is to G,
of Dof. 5.