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· Def. XXI. COMPOSITION of RATIOS.
In ariy Number of Quantities, and in any Order, whatever; the Ratio of the first to the last is equal to that which is compounded of all the intermediate Ratios:
If A, B, C, and D, represent as many Quantities; whatever is the Ratio of A to B, of B to C, and of C to D; the Ratio of A to D, is equal to the Ratio compounded of them all.
Thus,if A is to B as 3 to 5, B to Cas 5 to8, andCto D as to 6.
C 8 5
i.e. Multiplyail the Antecedents into one another; 3 x 5 x8=120; anddivide theSum, by that of all the Consequents; 5 X 8 x 6=240, then is A to D, as 120 to 240, or as 3 to 6; i.e. as I to 2, or ifs.
* This last Definition is generally given in the fixth Book, for which I cannot conceive a reafon. The Doctrine of Proportion is a distinct fubject, and therefore, all which relates to it Thould be together, in one Book ; which, according to Euclid, is the fifth : to which in the use and application of it, in the sixth and other Books,) we may refer on all Occasions.
Dr. Barrow, indeed, gives nearly the fame thing, in both. In Def. 20. B. 5. he says, “ any number of Magnitudes being put; the proportion, of the first to the last, is compounded out of the proportion of the first to the second, the second to the third, and the third to the fourth, &c. till the proportion arise.”
The proportion, which is between the first and the last, will, and muft neceffarily, arile, by the composition of all the intermediate Ratios, which is evident from the manner of compounding them; nor is it possible to be otherwise ; feeing that, each of the intermediate Quantities is both Antecedent and Confeqüent. Therefore, the Ratio emerging at last, seeing, it depends entirely on the first and lait, muít ever be equal to the Ratio: of the firit to the last; let the intermediate Quantities be ever so many, and in what Proportion foever.
According to Euclid, there are fix various ways of reasoning, from Proportion, including ordinate and inordinate Ratio in that of Equality; which, in effect, is all one. Inordinate Ratio differs from ordinate, in nothing but the disposal of the last Quantity; for, whether it be considered and compared as an Antecedent or as a Confiquent, by placing it first or last; the Ratio, in either case, being the same, it will still be as the first to the last, so is the first to the last; but not, one Quantity, (added) to the other, as in the former case, where the Quantities, added, are both Antecedents, or both Consequents.
But there is another way of reasoning by Proportion, distinct from the rest, which is not in Euclid; viz. mixed Retio, (Def. XVIII.) that is, comparing the sun of the Antęcedent and Consequent with their difference. In respect of Converse Ratio, it differs in nothing from inverse, which includes all Converse Ratios whatever; fince, al! Ratios, that are analogous, are inversely so, or conversely, which is the same thing. Nor do I see any Reason, why the 15th Definition, of Compounded Ratio, has not a converse, as well as the 16th of Divided Ratio ; seeing that, one holds good as well as the other; and is as neceflary to be known, it þeing as frequently made use of as the other.
That the manner, and the order, of the several changes may be seen at one view, I have given them over again; with the converse of each, in the following abstract; keeping the Same Ratio throughout the whole. In which five Cafes, viz. alternating, inverting, compounding, dividing and mixing, with their converse, together with equality, ordinate and inordinate, is contained all the variety in which the Terms of analogous, Ratios, may be changed and still be analogous; and which, answers almost every purpose, of reasoning from Proportion.
If A:B::C:D, directly, as 3 to 2; thus, 21:14:: 9:6. Then, A:C::B:D, alternately.
21: 9::14:6. Also, B:A::D:C, inversely.
14:21:: 6:9. And, A + B:A, or B,::C+D:C, or D.
35:21, or 14, :: 15 :9, or 6. § by Composition. Aor BCorD,:C+D. 21, or 14, : 35 :: 9, or 6,: 15
7:21, or 14,:: 3:9, or 6, Also; A, or B, : A-B::C, or D:C-D
conversely, 21, or 14: 7 :: 9, or 6:3 Again, A+B:A-B::C+D:C-D, mixing; 35: 7:15: 3. or, A-B: A+B::C-D:C+D,conversely, 7:35:: 3:15.
Ordinate Ratio of equality. IF A: B::C:D; and, if B:E::D: F; then, A: E::C: F. 21:14::9:6; and, 14:42:: 6:18; then 21:42:: 9:18.
Inordinate Ratio of equality. IfA: B::C:D; and, if B: E::F:C; then, A: E::F:D. 21:14:: 9:6; and, 14:42:: 3:9; then, 21:42:: 3: 6.
Thus, having shewn all the variety of changes, and, that the Ratios emerging from each, in numbers, are equal or analogous; there remains, one might reasonable suppose, nothing more to be done. For, after this knowledge once obtained, of what use can it be to perplex the Student, with a tedious Demonstration of what, in Numbers, is clear and manifeft; and which, I am persuaded, needs no further Demonstration. Nor do I see, excepting Alternate Ratio, (which is, also, sufficiently evident, in Numbers) that any of the rest require Demonstration ; or are made more manifeft by it. For, the Ratios of all commensurable Quantities may be expressed by Numbers; and it is sufficiently manifest, that, by Analagy, it must also hold true in respect of Quantities, which are incommensurable.
It is evident, that, fince A:B::C:D, whether the Ratio of A to B be commensurable or incommenfurable, the Ratio of C to D being the same, as A to B, they will still be analogous in every change. E. g. Suppose the Ratio of A to B and of C to D be as the Side
of a Square to the Diagonal; or, as the Diameter of a Circle is to its Circumference, &c.
Now, it is manifest, that the side of one Square is to its Diagonal, as the side of any other Square is to its Dia. gonal; consequently, their Ratios are equal, notwithstanding they cannot be expresled in Numbers; for 3:5:: 9:15, i. e. 3:5:: 3:5; consequently, 3:3::5:5, &c. Therefore, since the Side of one Square, is to its Diagonal, as the Side of any other Square to its Diagonal; consequently, as Side is to Side, so is Diagonal to Diagonal; also, as the side added to the Diagonal, is to either Side or Diagonal; fo is any other Side added to its Diagonal, to either ; likewise as one Diagonal is to the Side, fo is the other Diagonal to the Side ; or, as the Diagonal, less the Side, to either, so is the Diagonal, less the Side, to either; and thus it must ever be through all the variety of Changes that can be, if they are ordered and changed after the same manner, on both sides.
In what I have advanced, on the Subject of Proportion, is contained the essence of the whole fifth Book of Euclid and if it be clearly understood, the Doctrine of Proportion is already acquired; and consequently, all that can be said more of it, by way of Demonstration, is, in a great measure, fuperfluous. However, not to leave the whole Doctrine of Proportion without a Foundation, least it should be censured by the scrupulous, I have demonstrated or more fully illustrated, fome of the most effential, on which the whole depends ; some of the others I have made Axioms, for being self evident they require no Demonstration.
AXIOMS, or self-evident PROPOSITIONS, Axi I. One Quantity is to any other Quantity, as all the
Parts of one, is to all the parts of the other. Ama b.ç.
Let A and D be two Quantities, 'and, let A be
divided, any how, into Parts, a, b, c; also let D е f
be divided any how, in d, e, and f.
Then, A :B:: a +c:dite + f.."
11.. Equimultiples, * or equal Parts, of equal Quantitjes B В
are equal. 1816
Let A and B be equal Quantitics ; then, if C and D'be equal twice or thrice, &c. of A and B,
respectively; 2 A=2B, or 3A=3B, i. e. C=D. с D
Also, if A and B be divided into two or more
equal Parts, a'a, bb,&c; a == b or A=}B, &c.
III. Any Multiple of the whole is equal to the same А 6 Multiple of all its Parts, taken together, i
Let B be taken any Multiple of A; then, whatb
ever Múltiple B is of A, it is manifest, that it a 6
contains the same Multiple of a and by the Parts of A and B ; a, a, a, and b, b, b.
IV.s. The seventh Proposition of Euclide,
Equal Quantities have the fame Ratio to any third Quantity, or to equal Quantities.
For, if A and B'be equal Quantities, they have the same Ratio to a third Quantity, C, or to equal Quantities, D and E; feeing that, either may
be taken for the other.
By Equimultiples is meant, that the Quantities are taken, or multiplied, an equal number of times.