of the first A to the last D, will be expressed by an antecedent which is the product of all the antecedents, and a consequent which is the product of all the consequents." Hence, also, a numerical ratio is frequently said to be compounded of two or of any other number of numerical ratios, when its antecedent is the product of all their antecedents, and its consequent the product of all their consequents: for the magnitudes whose ratio it denotes will in such a case have to one another a ratio which is compounded of the ratios expressed by those others.

In a geometrical progression A, B, C, D, &c. of commensurable magnitudes, the successive terms have a common numerical ratio, e. g. 5:7; therefore, the ratio of A to C, i. e. the duplicate of A to B, is 5 x 57 x 7, the ratio of A to D, i. e. the triplicate of A to B, is 5 x 5 X 5:7 x 7 x 7; and so on.

PROP. [28].

If there be two fixed magnitudes, A and B, which are the limits of two others, P and Q, (that is to which P and Q, by increasing together or by diminishing together, may be made to approach more nearly than by any the same given difference), and if Pbe to Q always in the same given ratio of C to D, A shall be

to B in the same ratio.

First, let P and Q approach to A and B respectively by a continual increase, so that P and Q can never equal, much less exceed, A and B, but may be made to approach to A and B more nearly than by any the same given difference. And let a magnitude B' be taken such that A: B':: C: D. Then, if B' is not equal to B, it must either be less than Bor greater than B. First, let it be supposed less, as by any difference b. Then, because P: Q::C: D, and A: B'::C: D, ([12]) A: B':: P: Q; but A is always greater than P; therefore, B' is always greater than Q [(18)]. Wherefore, because Q is always less than B', which is less than B by b, Q cannot approach to B within the difference b, which is against the supposition. Therefore, B' cannot be less than B.

Again, if B' be supposed greater than B, take A' such that A': B'::A: B. Then, because B is less than B', A is less than A' ([18]), as by some difference a. And, because A': B'::A; B, and P: QAB, ([12]) A': B'::P:Q:

• See note at prop. [23],

but B' is always greater than Q, because it is supposed to be greater than B, which is greater than Q; therefore A' is always greater than P ([18]). Wherefore, because P is always less than A', which is less than A by a, P cannot approach to A within the difference a, which is against the supposition. Therefore B' cannot be greater than B.

Therefore, in this case, B' cannot but be equal to B; that is, A: B::C: D. And the other case, in which P and Q approach to A and B respectively, by a continual decrease, may be demonstrated after the same manner; indeed in the same words, if the word " greater" be everywhere substituted for "less," and "less" for " greater."

Therefore, &c.

This proposition will be found of very extensive application in Geometry. By help of it, the lengths of plane curves, and the areas bounded by them, the curved surfaces of solids, and the contents they envelope, may in many instances be brought into comparison with little greater difficulty than right lines, rectilineal areas, and solids bounded by planes. This will be exemplified in subcases which suppose the magnitudes sequent parts of the present treatise in compared to be similar, or of the same form; but the use of the proposition is by regarded as one of the first steps to what no means confined to these. It may be is called the higher Geometry, and in this view, likewise, is well worth the attention of the student.

General Scholium

On the proportion of commensurable magnitudes.

It was shown in the first proposition of this section ([9].) that if four magnitudes be proportionals, and if any mea

tain number of times in the first, a like

sure of the second be contained a cer

measure of the fourth shall be contained

the same number of times in the third. Hence it follows, that any terms expressing the ratio of the first to the third to the fourth. But no terms can second, express also the ratio of the express the ratio of two magnitudes except the lowest, and such as are equimultiples of the lowest terms; that is, except m, n, if m, n are the lowest terms, and xm, 1x n, where l is a number multiplying the lowest terms (6. Cor. 1.). And (Arith. art.80.). There

m

1 x m

n 1 x n

fore, if four magnitudes be proportionals,

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 C

fraction shall be equal to

versely.

A C

--

D

: and con

is the founda

This equation B 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 theorems of the foregoing section. Thus,

с

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 xd=bXc a (as is the case when = Arith. art 127.), and b d if a be prime to b, the other factor d on the first side must contain b, and therefore must be of the form 1Xb, where I is some whole number multiplying the whole number b; and hence it is evident that e is likewise of the form Xa, where I is the same whole number multiplying the whole number a. Therefore

A+B C+D

=

Ꭰ

adding 1 to each side, B which is the theorem cited by the Latin word componendo; again subtracting A-B C-D, 1 from each, which is B

D

that cited by the word dividendo: di

viding 1 by each side,

B

tiplying C

into each,

B D

=

and mul

Α

で

=

L

LX M

D

C

dividing both

LxM' multiplying

A C

B

Or if

Kx M

a

both sides by M,

equimultiples of a and b, that is isis is reducible to by dividing and d by their greatest common factor (see Prop. 5. Scholium); and hence likewise if

Kx M
LxM D' L D'

C K C

=

for

Or the Rule may be thus stated: Expunge all common factors, except

they be common only to the two extremes, or to the two means. For, in all other cases, if, as before, Kx M and LX M be the terms having a common factor, and if P and Q be the terms not having a common factor, by multiplying

extremes and means, K X PX M = LXQXM, and dividing each side by M, Kx PLX Q.

Of these three cases, the one which occurs most frequently is the last, viz. that in which the common factor is found in the terms of the same ratio.

Rule 3. If there be two or more proportions, the products of the corresponding antecedents and consequents shall constitute a proportion.

A C A' C B D' BD multiplying together the quantities upon the left hand, and also those upon the right of the two equations, CX C'

Ax A

=

BxB' DX D'

i. e. if A: B::C: D
and A': B'::C': D'

A× A': B× B' :: CxC': DxD'. Again, if there be a third proportion A": B" :: C":D", the terms of this being multiplied into those of the preceding, Ax A' × A" : B× B'x B" :: Cx

C'xC": DxDxD"; 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, ABC: D
and B': A :D : C'
and A": C':: B' : D",

the proportion which is compounded of

these is Ax B'x A" : BxAX C':: Cx DXB: DxC'x D": 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.

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.

lations of such magnitudes will be briefly In the present section, the similar reconsidered; a new definition will be laid down, comprehending that already given (def. [7]) 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 theorems of the preceding section will be shown to apply equally as to the

former.

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 may nevertheless be obtained which shall serve to compare two such 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 22:7 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. [7]. 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. [9]), that any other like parts of the second and fourth will be contained in the first and third the same number of 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.

Def. 7. The first of four magnitudes 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. c. 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.

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, Eƒg 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 corresponding remainders less than Eƒ and Efg H. And this will always be the 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.

Def. 8. If the first of four magnitudes contain any part of the second a greater number of times, with or without a remainder, than the third conis said to have to the second a greater tains the like part of the fourth, the first 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. [9] 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 D, according to def. 7, and at the same 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.]

E

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 n th 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. 9] 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 n 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 equimultiples of A, B, C, D; and of the equimultiples of two magnitudes, one will be greater than, or equal to, or less than the cther, according as the corresponding magnitude is greater than, or equal to, or less than the corresponding magnitude of the other; and conversely (ax. 1, 2, 3, 4).*

"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 multiple 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. [7] 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 uc test essentially different from that which is here adopted. The greatest geometers in dwelling upon this part of the Elements have ever found 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, than the doctrine of proportionals." Euclid has, indeed, left little in this respect, as in others, to be