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will express, at once, the ratio of the first to the second, and of the third to the fourth (1. Cor. 3.),
The same conclusion may be stated in other words, as follows:
If four magnitudes be proportionals, and if A, B, C, D, represent those magnitudes numerically, that is, if A and B represent the numbers of times the unit of their kind is contained in the two first, and if C and D represent the numbers of times the unit of their kind is contained in the two last, the quotient or A
fraction shall be equal to and conversely.
that cited by the word dividendo : di
viding 1 by each side, B
A A B tiplying into each, CD' the theorems cited by the words invertendo and alternando respectively. And with equal readiness the rest may be derived from the same equation. As they are all however sufficiently conmainder of this scholium will be confined sidered in their several places, the reto the explanation of certain rules commonly practised in the treatment of proportions, which suppose the terms A, B, C, D to be numbers, and are at once deА С
rivable from the equation B D
Rule. 1. If four magnitudes, which are numerically represented by A, B, C, D, be proportionals, the product of the extremes will be equal to the product of the means, i. e. if A: B::CD, AXD =Bx C: and conversely if Ax D=BxC the magnitudes represented by A, B, C, D will be proportionals. А С
For, since is the founda
This equation=D tion of the theory of proportion as it is treated in Arithmetic or Algebra, (see Arithmetic, art. 127 and 128.) and
leads with great facility to all the the theorems of the foregoing section. Thus,
It is shown in the treatise on Arithmetic (art, 61.) that the same number can have but one set of prime factors: from this it follows, that if a dbxe (as is the case when
Arith. art, 127.), and if a be prime to b, the other factor d on the first side must contain b, and therefore must be of the form Xb, where l is some whole number multiplying the whole number b; and hence it is evident that c is likewise of the form Xa, where I is the same whole number multiplying the whole number a. Therefore
В D' multiplying each side by B×D, AxD-BX C: and hence, conversely, dividing each side by B×D, BD,. e. A, B, C, D are proportionals.
7 both sides by M,
by dividing and d by their greatest common factor (see Prop. 5. Scholium); and hence likewise if
Or the Rule may be thus stated: Expunge all common factors, except
Ax A' Cx C'
i. e. if AB::C: D
AXA': BX B':: Cx C': Dx D'. Again, if there be a third proportion A": B" :: C": D", the terms of this being multiplied into those of the preceding, A× A'× A" : B× B' × B" : : Cx C'×C": DX D'×D"; and so on, if there be any number of proportions.
When this Rule is applied, the resulting proportion is said to be compounded
of the others, and hence the rule is
called, "compounding the proportions." We may observe that the compound proportion commonly admits of reduction by Rule 2., on account of the same term or terms occurring in more than one of the component proportions.
If, for example, A: B :: C: D
the proportion which is compounded of these is Ax B'x A" : BxAxC':: Cx DX B': DXC'XD": which is reducible by Rule 2. to A" : B :: C: D".
The terms are, however, seldom (or never) so intermixed as in this example. The end which is usually proposed in the compounding of proportions is to obtain the ratio of one magnitude K to another L by means of a number of intermediate magnitudes; in order to which, K is made the antecedent of the leading ratio of the first proportion, and L the consequent of the like ratio in the last.
Thus if K: BC :D
and B: B':: C': D' and B': L:: C":D" by compounding the proportions, KxB XB: BxB'XL:: CxC'x C": Dx D' xD", and hence by Rule 2, K:L:: CX C'x C": DxD'xD".
SECTION 3.-The General Theory of Proportion.
The foregoing theorems have been established upon the supposition that the magnitudes spoken of are commensurable. This, however, is not always the case with magnitudes: there are some (examples will appear in a future page) which have no common measure, and which are therefore said to be incommensurable.
In the present section, the similar relations of such magnitudes will be briefly considered; a new definition will be laid down, comprehending that already given (def. ) of the proportion of commensurable magnitudes, at the same time that it does not require that the magnitudes which satisfy it shall be commensurable; and to this new definition the be shown to apply equally as to the theorems of the preceding section will
In the first place, then, it is evident that, incommensurable magnitudes having no common part, their ratio can never be exactly expressed by numbers. Numbers which shall serve to compare two such may nevertheless be obtained magnitudes A and B to any required degree of accuracy.
Let B be divided into any large number of equal parts, a million for example: then A will contain a certain number of these parts with an excess which is less than one of them, less, that is, than a millionth part of B; so that if we take no account of this excess in our estimate of their relative magnitude, we shall commit an error of less than one-millionth. And it is plain that, in this manner, by dividing B into a still greater number of equal parts, the error of our estimate may be made as small as we please.
It is found, for example, (by methods which will be noticed hereafter) that, if the diameter of a circle be divided into 7 equal parts, the circumference will contain not quite 22 of those parts; if, again, the diameter be divided into 113 equal parts, the circumference will contain not quite 355 of those parts: if into 10,000,000, the circumference will con
tain not quite 31,415,927; and so on. Therefore the ratio of the circumference to the diameter is expressed by the ratio 227 nearly; more nearly by 355: 133; still more nearly by 31,415,927 10,000,000; and so on: nor is there any limit to the accuracy of this approach, although there should (as is really the case) be no two numbers by which it can be expressed exactly.
This consideration brings us directly to the only case in which, consistently with the view already taken of the subject of equal ratios, two magnitudes may be said to be similarly related (or in the same ratio) to two others of the same kind respectively, with which they are incommensurable. The ratios of the former to the latter, each to each, must admit of being approximately represented by the same numbers, to how great an extent soever the degree of approximation may be carried in other words, any like parts whatsoever of the two latter magnitudes, however minute they may be taken, must be contained in the two former, each in each, the same number of times, with corresponding* remainders less than the parts.
It has been already observed that this obtains with regard to the proportionals of def. . When four magnitudes are proportionals by that definition, which supposes the first two and second two to be commensurable, there are, indeed, some like parts of the second and fourth which are contained in the first and third the same number of times without remainders; viz. the greatest common measures of the first two and second two, and any like parts of the greatest common measures: it is easy to perceive, however (and the same has been demonstrated at large in Prop. ), that any other like parts of the
second and fourth will be contained in the first and third the same number of
cond which the third has to the fourth, when any like parts whatsoever of the second and fourth are contained in the first and third the same number of times exactly, or the same number of times with corresponding remainders less than the parts.
For example: let ABCD, EFGH be two rectangles having the same altitude, and let AB, EF be their bases. Let the base E F be divided into any number of equal parts E f, &c., and the rectangle E F G H into as many equal rectangles, Efg H, &c. by lines drawn
through the points of division parallel to EH. Then if A b, &c. be taken equal to Ef, and if straight lines be drawn through the points b, &c. parallel to AD; the base AB, and the rectangle ABCD, will contain, the one a certain number of parts equal to Ef, and the other the same number of rectangles equal to Efg H, either exactly, or with and Efg H. And this will always be the corresponding remainders less than Ef case, whatsoever be the number of parts into which E F is divided. Therefore, according to def. 7., the two rectangles and their two bases are proportionals.
tudes contain any part of the second a Def. 8. If the first of four magnigreater number of times, with or without a remainder, than the third contains the like part of the fourth, the first is said to have to the second a greater ratio than the third has to the fourth : also, in this case, the third is said to have to the fourth a less ratio than the first has to the second.
As from Prop.  with regard to commensurable proportionals, so from the terms of our new general definitions 7. and 8. with regard to the proportionals described in def. 7. it is at once evident that of four magnitudes, A, B, C, D, the first A cannot be said to have to the second B the same ratio which the third C has to the fourth Def. 7. The first of four magnitudes D, according to def. 7, and at the same
times, with corresponding less remainders. The following, therefore, is to be considered as the general test of two magnitudes A and C, having the same ratio to two others B and D, of the same kind with the former two respectively.
is said to have the same ratio to the se
By the word " corresponding" here used, it is merely intended to point out the fact of there being two remainders, i. e. a remainder in the comparison of the two first magnitudes, and a remainder corresponding to it in the comparison of the two last. And in the same sense the word is to be understood in subsequent passages on the same subject.
time a greater or a less ratio than C has to D, according to def. 8.: much less can A be said to have to B at the same time both a greater and a less ratio than C has to D.*
* See note at Prop. [9.]
We proceed to the properties of this more general description of proportionals, which will be found the same with those already demonstrated in the preceding Section of commensurable proportionals. They will be considered accordingly in the same order, and will have the same numbers affixed to them. It will be observed, also, that they are stated in the same words, with the exception of Prop. 9., which is little more than another form of expressing def. 7., and its corollaries, which again express the same thing in different terms, the 2d. of them being, in fact, Euclid's celebrated definition of proportionals.
Next, let this be the case with four magnitudes, A, B, C, D, whatsoever numbers be substituted for m and n: A, B, C, D shall be proportionals.
For, if A and C contain exactly m of the nth parts into which B and D are divided, the four A, B, C, D are commensurable proportionals, according to def. 7; and, therefore, also (by Prop.  of the last Section, as has been already observed) proportionals according to def. 7. Again, if A contain more than m, as m' and a remainder, that is more than m' and less than m'+1 parts
if equal equal, and if less less, whatsoever values may be given to m and n: and, conversely, if this be the case with four magnitudes, they shall be proportionals."
Cor. 2. (Euc. v. def. 5.) Or, again, "if A, B, C, D be proportionals, and if ʼn A be greater than m B, n C shall likewise be greater than m D, if equal equal, and if less less, whatsoever values may be given to m and n: and, conversely, if this be the case with four magnitudes, they shall be proportionals."
For n A, m B, n C, m D are equimul
"The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multi
ple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth."Euc. v. def. 5.
This definition of proportionals has been sometimes found fault with as too abstruse and recondite for beginners; which would not perhaps have been the case, had its connexion with the more obvious but
contined view of def.  been always pointed out. For, we have seen that a general theory of propor tion, which shall embrace indifferently all magnitudes, whether commensurable or otherwise, admits of no test essentially different from that which is upon this part of the Elements have ever found here adopted. The greatest geometers in dwelling cause to admire the profoundness and sagacity of their author. Witness the energetic testimony of Barrow, "That there is nothing in the whole body of the Elements of a more subtile invention, nothing more solidly established, or more accurately handled, indeed, left little in this respect, as in others, to be than the doctrine of proportionals." Euclid has,
Cor. 3. (Euc. v. def. 7.) In like manner, it may be shown, that, if there be four magnitudes A, B, C, D, and if a multiple of A, as m A, can be found which is greater than a multiple of B as n B, while the corresponding multiple of C, viz. m C, is not greater than the corresponding multiple of D, viz. n D, A has to B a greater ratio than C has to D. For, A will be greater than of B, but C will not be greater than of D; that is, the nth part of B
will be contained in A m times or more with a remainder, but the nth part of D will not be contained in C so much as m times with a remainder. (See def. 8.)
PROP. 10. (Euc. v. 7.)
Equal magnitudes have the same ratio to the same magnitude: and the same has the same ratio to equal magnitudes.
For, any the same part of the same magnitude will be contained the same number of times in equal magnitudes with corresponding less remainders. And, again, any like parts of equal magnitudes, being equal to one another (ax. 2), will be contained the same number of times with the same remainder in the same magnitude.
Cor. If a ratio which is compounded of two ratios be a ratio of equality, one of these must be the reciprocal of the other. (See  Cor.)
PROP. 11. (Euc. v. 8.)
Of two unequal magnitudes, the greater has a greater ratio to the same magnitude: and the same magnitude has a greater ratio to the lesser of the two.
The first is evident: for, of any the same magnitude a part may be found less than the difference of two unequal magnitudes, which part must evidently be contained a greater number of times in the greater than in the lesser of the two.
In the second place, therefore, let A and B be any two magnitudes of which A is the greater, and let C be any third magnitude: C shall have to the greater A a less ratio than the same C has to the other B.
done by succeeding writers, but to follow his steps as closely as they are able: the principles, the theorems, the demonstrations they are in search of are all to be found in this one masterpiece, and for the most part under the simplest form. From this it will, of course, be easily understood how large a portion of the present Section is borrowed from the 5th Book of the Elements,
Let D be the difference of A and B: then, whether B and D be or be not both of them greater than C, multiples m B, m D may be taken of them which are both greater than C. And, because C is less than m B, let multiples of C be taken, as 2 C, 3 C, &c. until a multiple be found, as p+1.C, which is the first greater than m B. Then, because m B is not less than the preceding multiple p C, and because m Dis greater than C, the two m B, m D together are greater than p C and C together, that is, than p+1.C. But, because A is equal to B and D together, it is evident that m A is equal to m B and m D together. Therefore m A is greater than p+1.C, and m B less than p+1. C. Therefore the p+1th part of A is contained in C, not so much as m times, and the p+1th part of B is contained in C, m times with a remainder. Therefore (def. 8.) C has a less ratio to A than it has to B, or a greater ratio to B than it has to A. (See also 9. Cor. 3.) Therefore, &c.
Cor. 1. (Euc. v. 9.) Magnitudes which have the same ratio to the same magnitude are equal to one another: as likewise those to which the same magnitude has the same ratio.
Cor. 2. A ratio which is compounded of two ratios, one of which is the reciprocal of the other, is a ratio of equality: for, A having to C a ratio which is compounded of the ratios of A to B and of B to C, if the latter be the same with the ratio of B to A, A must be equal to C.
Cor. 3. (Euc. v.10.) If one of two magnitudes have a greater ratio to the same magnitude than the other has, the first must be greater than the other and if the same magnitude have a greater ratio to one of two magnitudes than it has to the other, the first must be less than the other.
PROP. 12. (Euc. v. 11.)
Magnitudes A, B and C, D, which have the same ratio with the same magnitudes P, Q, have the same ratio with one another.
For any part of B will be contained in A exactly or with a remainder, as often as a like part of Q is contained in P exactly or with a remainder, because AB::P: Q, that is, as often as a like part of D, is contained in C exactly, or with a remainder, because C:D :: P: Q. Therefore, &c.