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A B C D

For, if one of the extremes, as A, be the greatest, then, because B is less than A, D is less than C ([14]), and because C is less than A, D is less than B ([18]); therefore D is the least. But, because A:B::C:D, dividendo, A-B:B:: C-D:D; and because in this proportion B is greater than D, A-B is greater than C-D ([18]. Cor.) Therefore, if B+D be added to each, the sum of A and D will be greater than the sum of C and B. If one of the means, as B, should be greatest, then invertendo B:A::D:Č; and hence, as before, the sum of B and C, that is, of the greatest and least, is greater than the sum of A and D.

Cor. 3. If three magnitudes be proportionals, the sum of the extremes will be greater than twice the mean, and therefore half the sum greater than the mean.

For, if three magnitudes be propor tionals, and if the mean be greater than one of the extremes, it must be less than the other; or again, if it be less than one, it must be greater than the other ([14] or [18]). Therefore, the two extremes are the greatest and least. Half the sum of two magnitudes, being as much greater than the one as it is less than the other (I. ax. 9.), is called an arithmetical mean between the two. It appears therefore, that the arithmetical mean between two magnitudes is greater than the geometrical mean.

PROP. [21].

*

If four magnitudes A, B, C, D be proportionals, they shall also be proportionals when taken conjointly; that is, the sum of the first and second shall be to the second, as the sum of the third and fourth to the fourth; or componendo A+B:B::C+D:D.

For the measures M, N remaining as in the preceding proposition, that is, being contained in A and C respectively 7 times, and in B and D respectively 4 times, are contained in A+B and C+D respectively 7 +4 or 11 times; and, therefore, 11:4 will be the ratio of A+ B to B, and also of C+D to D. Therefore A+B: B::C+D:D.

Therefore, &c.

The case is here excepted in which the magnitudes are equal to one another; for in that case it is manifest that the arithmetical mean is equal to the geometrical mean,

Cor. Hence, invertendo and componendo, (after the order of Cor. 1. in the preceding proposition,) if four magnitudes A, B, C, D be proportionals, the first shall be to the sum of the first and second, as the third to the sum of the third and fourth; or, A:A +B::C:C+D.*

PROP. [22].

If one magnitude be to another as a magnitude taken from the first to a magnitude taken from the other, the remainder shall be to the remainder in the same ratio.

Let A, B be any two magnitudes, from which respectively let there be taken the magnitudes A', B', which have to one another the same ratio which A has to B; the remainders A-A' and B-B' shall be to one another in the same ratio.

For, because A: B:: A': B', alternando, A: A':: B: B'; therefore, dividendo, A-A': A':: B-B': B'; and again, alternando, AA': B-B′ : : A' : B', that is ([12]) :: A: B. Therefore, &c.

Cor. 1. If there be any number of magnitudes, A, B, C, D, &c., in geometrical progression, the differences A-B, B-C, C-D, &c. will form a geometrical progression, in which the successive terms have the same ratio with the successive terms of the former.

For B is to C as A to B, C to D as B to C, and so on; therefore, A-B is to B-C as A to B, B-C to C~D as B to C, i. e. as A to B, i. e. as A~B to B~C ([12]); and so on.

Cor. 2. And conversely, any number of magnitudes A, B, C, D, &c. in geometrical progression, may be considered as the differences of other magnitudes, A', B', C', D', E', &c. forming a geometrical progression, in which the first term A' is to A as A to A-B, and the successive terms have the same ratio with the successive terms of the former.

short and obvious, to prefer that which is the most It is usual, when two demonstrations are equally elementary, or approaches nearest to first principles. Now, it may be observed of this corollary, the corre the present section, as the theorem miscendo, and sponding corollary of Prop. [28], and other parts of Prop, [25], that, stated, as is here the case, of commensurable proportionals, they may, at least, as obviously and briefly, be referred to def. [7] as they may he derived from other propositions. The connected demonstrations have in this case been preferred with a view to the next Section, where nearly all the propositions of this Section will be re-stated, and such as have been referred to def. [7], will have to be reconsidered with reference to a more general definition. A similar consideration will be found to have directed the arrangement of the propositions.

For, if such a progression be taken, A', B', C', &c. in which A' is to A as A to AB, and A' to B' as A to B; then because A': B' :: A: B, convertendo, A': A'B' :: A: A~B; but A': A:: A: A-B; therefore ([12]) A': A B'A': A; and because in this proportion the first term is the same with the third, the second is equal to the fourth ([18)], that is, A'B' is equal to A. But (by Cor. 1.), A'~B', B'~C', C'~D', &c. form a progression in which the successive terms, as A'-B', B'C', have the same ratio with A', B', that is, with A, B ([12]). Therefore, also, B'C' is equal to B ([18]), and hence again C'-D' is equal to C, and so on. In fact, the progressions A, B, C, D, &c. and A'-B', B'C', C'-D', D'E', &c. having the same first terms, and the same common ratio of their terms, cannot but be identical.

PROP. [23].

If one magnitude be to another as a third magnitude of the same kind to a fourth, the sum of the first and third shall be to the sum of the second and fourth in the same ratio.

Let A, B be any two magnitudes, tó which respectively let there be added the magnitudes A', B', which have the same ratio to one another which A has to B: the wholes A+A' and B+B' shall be to one another in the same ratio.

For, because A: B:: A': B', alternando, A: A':: B: B'; therefore componendo, A+A: A'::B+B': B'; and again, alternando, A+A': B+B'::A': B', that is, ([12.]) :: A: B. Therefore, &c.

Cor. 1. Hence, if there be any number of magnitudes of the same kind antecedents, and as many consequents, and if every antecedent have the same ratio to its consequent, the sum of all the antecedents shall have the same ratio to the sum of all the consequents.

Cor. 2. If the ratio of A' to B' be not the same with the ratio of A to B, the ratio of A+A' to B+B' will not be the same with the ratio of A to B ; but less, if A' be to B' in a less ratio, or greater, if A' be to B' in a greater ratio.

For if A' have to B' a less ratio than A has to B, A' must be less than a magnitude P, which has to B' the same ratio

It is here assumed that to two given magnitudes of the same kind and a third there is some magnitude which is a fourth proportional: a truth obvious enough in the case of commensurable proportion here supposed; for if there be taken a common measure of the first two magnitudes and a part which is contained in the third as often as that

as A to B ([11] Cor. 3.); and therefore A+A' must have a less ratio ([11]) to B+B' than A+P has to B+B'; but A+P has to B+B' the same ratio as A to B, by the proposition: therefore A+A' has to B+B' a less ratio than A has to B ([12] Cor. 2.): and the other case in which A' has to B' a greater ratio than A to B, admits of a similar demonstration.

Cor. 3. Hence, if the ratio of A' to B' be not the same with the ratio of A to B, the ratio of A+ A' to B+B' will lie between the ratios of A to B, and of A' to B'; that is, it will be greater than the lesser of the two, and less than the greater of the two.

PROP. [24].

If there be three magnitudes of the same kind A, B, C, and other three A', B', C', which, taken two and two in order, have the same ratio, viz. A to B the same ratio as A' to B', and B to C the same ratio as B' to C'; then ex æquali in proportione directâ (or ex æquo) the first shall be to the third of the first magnitudes, as the first to the third of the others; or, as it may be more briefly stated,

if A: B::A': B'

and BC::B' : C', then ex æquali, A: C::A': C'.

Let 3 4 be the common ratio of A to B and of A' to B', and 5: 7 the common ratio of B to C and of B' to C'. Then, if B be divided into 4 x 5 or 20 equal parts, one of these parts will be contained 3 x 5, or 15 times in A, and 4x7 or 28 times in C, because 3: 4, or 3×5: 4×5 (1. Cor. 2) is the ratio of A to B, and 5: 7, or 4 × 5: 4×7 is the ratio of B to C. And for the like reason, if B' be likewise divided into 20 parts, one of these parts will be contained 3 x 5 or 15 times in A', and 4 × 7 or 28 times in C'. Therefore, 15: 28 is the ratio of A to C, and also of A' to C'; and, consequently A: C: A':: C'.

Therefore, &c.

Cor. 1. The same may be stated of any number of magnitudes A, B, C, D, and A', B', C', D': that is,

if A: BA': B' and B C::B': C' and CD::C': D' then, ex æquali, A : D::C' : D'.

For by the first two proportions A: C

common measure is contained in the first, a multiple containing this part as often as that common measure is contained in the second will be the fourth proportional required. The case of general proportion will be noticed at prop. 23. of the next section.

:: A': C', and from this combined with the third, A: D:: A': D'; and so on for any number of proportions.

This may be stated in the following words: "ratios, which are compounded of any number of equal ratios in the same order, are equal to one another." (def. 12.)

Cor. 2. By help of this proposition another property of proportionals may be demonstrated, which is commonly cited by the word miscendo :* viz.

If four magnitudes A, B, C, D be proportionals, the sum of the first and second will be to their difference, as the sum of the third and fourth to their difference.

For, since A:B::C:D,

componendo A+B:B::C+D:D. And again, because invertendo B: A:: D: C, convertendo B: A-B:: D: C-D. Therefore, ex æquali A+B : A~B : : C+D: C-D, which is the property in question.

PROP. [25].

If two proportions have the same consequents, the sum of the first antecedents shall be to their consequent, as the sum of the second antecedents to their consequent; that is, if A: B:: C: D, and if A': B: C': D, then A+ A' : B : : C+C': D.

Because by the first proportion A: B :: C:D, and, by the second proportion, invertendo, B: A':: D: C'; ex æquali, A: A::C: C': hence, componendo, A+A': A': C+C': C', and on account of the second proportion, viz. A' : B :: C': D, ex æquali, A+ A': B :: C+ C': D, which is the property in question. Therefore, &c.

Cor. 1. The same may be stated of any number of proportions having the same consequents: that is, the sum of all the first antecedents shall be to their consequent, as the sum of all the second antecedents to their consequent.

Cor. 2. In like manner also it may shown (by" dividendo" instead of" componendo"), that if two proportions have the same consequents, the difference of the first antecedents shall be to their consequent as the difference of the second antecedents to their consequent.

Cor. 3. Hence, if there be any number of magnitudes of the same kind A, B, C, D, E, F, and as many others, A', B', C', D', E', F'; and if the ratios of the first to the second, of the second to the third, of the third to the fourth, and so on, be Sometimes also, and more appropriately, by the

English words" by sum and difference."

respectively the same in the two series, any two combinations by sum and difference of the magnitudes of the first series, e.g. A+C-E and B - C + D, shall be to one another as two similar combinations of the corresponding magnitudes of the second series, viz. A' + C'E' and B' - C' + D'.

For, because A: F:: A': F (ex æquali); and again, C: F:: C': F', by the proposition A+C: F:: A'+C': F'; but EF E': F'; therefore, by the preceding corollary, A+ C E: F:: A'+ C'E': F. In the same manner, it may be shown that B − C + D:F:: B'C' + D': F'; therefore, invertendo and ex æquali A+C-E: B-C + D :: A' + C ~ E' : B' —C' + D'. And a similar demonstration may be applied to any other combinations by sum and difference.

PROP. [26].

If there be three magnitudes of the same kind A, B, C, and other three A', B', C', which taken two and two, but in a cross order, have the same ratio, viz. A to B the same ratio as B' to C', and B to C the same ratio as A' to B'; then, ex æquali in proportione perturbatâ (or ex æquo perturbato) the first shall be to the third of the first magnitudes, as the first to the third of the others; or, as it may be more briefly stated,

if A: B: B' : C' and BC: A': B' then ex æquo perturbato, A: C: : A' : C',

For, if 3: 4 be the common ratio of A to B, and of B' to C', and 5 : 7 the common ratio of B to C and of A' to B', it may be shown, exactly in the same manner as in the demonstration of Prop. 24, that the ratio of A to C is 3x5 : 4 x 7, that is 15:28, and in like manner that the ratio of A' to C' is 5x3: 7×4, that is again 15: 28. Therefore, 15: 28 is the ratio of A to C, and also of A' to C', and consequently A: C:: A': C'. Therefore, &c.

Cor. The same may be stated of any number of magnitudes A, B, C, D, and A', B', C', D'; that is, if A: B:: C: D

and B:C:: B': C' and C:D:: A': B'.

then er æquo perturbato A:D :: A': D'; for by the two first proportions, A : C :: B': D', and from this combined with the third proportion, A: D: : A' : D', and so on for any number of proportions. This may be stated in the following words: "Ratios which are compounded of any number of equal ratios, but in a

reverse order, are equal to one another." of them will have to one another a ratio (Def. 12.)

PROP. [27].

Ratios which are compounded of the same ratios, in whatsoever orders, are the same with one another.

The case of ratios which are compounded of the same ratios in the same order is that of [24]. Cor. 1. The case, again, of ratios, which are compounded of the same ratios in a reverse order, is that of [26]. Cor. 1.

Let the ratio of A to D, therefore, be compounded of the ratios of A to B, of B to C, and of C to D, and let the ratio of A' to D' be compounded of the ratios of A' to B', of B' to C', and of C' to D', which are the same respectively with the ratios of which the ratio of A to D is compounded, but without regard to order; thus, let 3: 4 be the common ratio of A to B, and of B' to C', 5:7 the common ratio of B to C and of A' to B', and 11: 8 the common ratio of C to D and of C' to D'. Then it is evident, from the demonstration of Prop. [24], that the ratio of A to C is 3 x 5 : 4 x 7, and hence again, the ratio of A to D, 3 x 5 x 11: 4x7 × 8. And on the other hand, the ratio of A' to C' is 5x3:7 x 4, and hence the ratio of A' to D', 5 × 3 × 11

: 7×4 × 8; which is the same with 3x5x11:4x7x8, because 5×3×11 is the same with 3x5x11, viz. 165, and 7×4×8 is the same with 4x7x8, viz. 224.* Therefore 165: 224 is the ratio both of A to D, and of A' to D', and consequently A: D:: A': D'. But the ratio of A to D is compounded of the same ratios with that of A' to D', without regard to order.

Therefore, &c.

Cor. 1. If there be any number of ratios as those of A to B, of C to D, of E to F, &c., magnitudes which have to one another a ratio compounded of any two of these shall have the same ratio to one another with any other magnitudes

which have to one another a ratio compounded of the same two; and, in like manner, magnitudes which have to one another a ratio compounded of any three

of these shall have the same ratio to one

another with any other magnitudes which have to one another a ratio compounded of the same three; and so on.

Cor. 2. If the ratios of A to B, of C to D, of E to F, &c. be all equal to one another, magnitudes which have to one another a ratio compounded of any two

See Arithmetic, art. 23.

which is the same with the duplicate ratio of A to B: and in like manner, magnitudes which have to one another will have to one another a ratio which a ratio compounded of any three of them is the same with the triplicate ratio of A to B; and so on.

cate, or triplicate, &c. of the same ratio Cor. 3. Ratios which are the dupli

are the same with one another.

Cor. 4. In the composition of ratios any two which are reciprocals of one may be neglected, without affecting the resulting compound ratio. (See [10.] Cor.)

another

compounded with the ratio of C to B, that is, with the reciprocal of B to C, the ratio A to B will be obtained, which being compounded with the direct ratio B to C, reproduces the ratio of A to C: that is, if one ratio be compounded with obtained, which being compounded with the reciprocal of another, a ratio will be that other, will again produce the first. This compounding of the reciprocal is sometimes called subducting, or taking away the direct ratio, and the result is termed the remaining ratio.

Cor. 5. If the ratio of A to C be

It appears, therefore, from the proposition, that if two ratios be equal to one another (and therefore compounded of equal ratios having any order in the composition of each), and if any of the equal ratios be subducted or taken away ([15]), the remaining ratios will be equal to one another.

Scholium.

The demonstrations which have been given of Prop. [24], [26], and [27], as above stated of commensurable magnitudes, are derived from a property of numerical ratios; viz. that "if the ratio of A to B be any whatever, as 3:4, and, again, the ratio of B to C any whatever, mined from these by multiplying their as 57, the ratio of A to C will be deterantecedents for a new antecedent, and their consequents for a new consequent." See the demonstration of Prop. [24],

which is referred to in the demonstra

tions of Prop. [26] and [27]. It follows that a similar rule may be observed with regard to any number of ratios which are compounded according to Def. 12 ; viz. 66 In all cases in which the several numerical ratios of A to B, B to C, C to D, can be assigned, i. e. when the magnitudes A, B, C, D, are commensurable (8, Cor. 1.), the compound ratio, or that

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. 57; 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 5x5 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 P be 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 B or greater than B. First, let it be supposed less, as by any difference b. Then, because P:Q::C: D, and A : B: CD, ([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 And, because A: B'::A: B, and P: QA B, ([12]) A': B'::P:Q:

a.

* 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 subsequent parts of the present treatise in cases which suppose the magnitudes compared to be similar, or of the same form; but the use of the proposition is by no means confined to these. It may be regarded as one of the first steps to what 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 meatain number of times in the first, a like sure of the second be contained a cer

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, and 1 x m, 1x n, where I is a number except m, n, if m, n are the lowest terms, multiplying the lowest terms (6. Cor. 1.). 7 x m (Arith. art.80.). There

measure of the fourth shall be contained

m

And n l x n

fore, if four magnitudes be proportionals,

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