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In the same manner, the ratio of any other two magnitudes C and D may be expressed by Px C' to QxC', P and Q being also integral numbers, and their ratio will be the same as that of P to Q.

2. If there be four magnitudes A, B, C, and D, having such B. D

values that is equal to- then A is said to have the same ratio A C'

to B, that C has to D, or the ratio of A to B is equal to the ratio of C to D. When four quantities have this relation to each other, they are said to be in proportion.

To indicate that the ratio of A to B is equal to the ratio of C to D, the quantities are usually written thus, A: B:: C: D, and read, A is to B as C is to D. The quantities which are compared together are called the terms of the proportion. The first and last terms are called the two extremes, and the second and third terms, the two means.

3. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order.

4. Three quantities are in proportion, when the first has the same ratio to the second, that the second has to the third; and then the middle term is said to be a mean proportional between the other two.

5. Magnitudes are said to be in proportion by inversion, or inversely, when the consequents are taken as antecedents, and the antecedents as consequents.

6. Magnitudes are in proportion by alternation, or alternately, when antecedent is compared with antecedent, and consequent with consequent.

7. Magnitudes are in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent.

8. Magnitudes are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent.

9. Equimultiples of two quantities are the products which arise from multiplying the quantities by the same number: thus, mx A, mx B, are equimultiples of A and B, the common multiplier being m.

10. Two quantities A and B are said to be reciprocally proportional, or inversely proportional, when one increases in the same ratio as the other diminishes. In such case, either of them is equal to a constant quantity divided by the other, and their product is constant.

PROPOSITION I. THEOREM.

When four quantities are in proportion, the product of the two extremes is equal to the product of the two means

Let A, B, C, D, be four quantities in proportion, and M : N :: P:Q be their numerical representatives; then will M × Q= N Q NxP; for since the quantities are in proportion there. fore N=Mx: or NxP=MxQ.

M P

Cor. If there are three proportional quantities (Def. 4.), the product of the extremes will be equal to the square of the

mean.

PROPOSITION II. THEOREM.

If the product of two quantities be equal to the product of two other quantities, two of them will be the extremes and the other two the means of a proportion.

Let MxQ=Nx P; then will M: N:: P: Q.

For, if P have not to Q the ratio which M has to N, let P have to Q', a number greater or less than Q, the same ratio that M has to N; that is, let M:N:: P: Q'; then MxQ'=

NxP

=

; but Q= M sequently, Q-Q' and the four quantities are proportional; that is, MN: P: Q.

N×P (Prop. I.): hence, Q':

PROPOSITION III. THEOREM.

NxP

M

; con

If four quantities are in proportion, they will be in proportion when taken alternately.

Let M, N, P, Q, be the numerical representatives of four quanties in proportion; so that

M:N:: P: Q, then will M: P::N: Q.

Since MN: P: Q, by supposition, MxQ=NxP; therefore, M and Q may be made the extremes, and N and P the means of a proportion (Prop. II.); hence, M: P::N: Q.

PROPOSITION IV. THEOREM.

Let

and

then will

For, by alternation

If there be four proportional quantities, and four other proportional quantities, having the antecedents the same in both, the consequents will be proportional.

and

[blocks in formation]

M: P::R; S, or

Q S

hence N=R; or N:Q::R: S.

Cor. If there be two sets of proportionals, having an antecedent and consequent of the first, equal to an antecedent and consequent of the second, the remaining terms will be proportional.

PROPOSITION V. THEOREM.

If four quantities be in proportion, they will be in proportion when taken inversely.

M:N::P:Q; then will
N:M::Q:P.

Let

For, from the first proportion we have MxQ=Nx P, or NxP=MxQ

But the products Nx P and M× Q are the products of the extremes and means of the four quantities N, M, Q, P, and these products being equal,

N:M::Q:P (Prop. II.).

PROPOSITION VI. THEOREM.

If four quantities are in proportion, they will be in proportion by composition, or division.

Let, as before, M, N, P, Q, be the numerical representatives of the four quantities, so that

M:N::P.Q; then will
M±N:M:: P±Q:P.

For, from the first proportion, we have
MxQ=NxP, or NxP=MxQ;

Add each of the members of the last equation to, or subtract it from M.P, and we shall have,

M.P±N.P=M.P+M.Q; or
(M±N) × P=-(P±Q) × M.

But M±N and P, may be considered the two extremes, and
P+Q and M, the two means of a proportion: hence,
M±N:M:: P+Q: P.

PROPOSITION VII. THEOREM.

Equimultiples of any two quantities, have the same ratio as the quantities themselves.

Let M and N be any two quantities, and m any integral number; then will

m. M: m. N::M:N. For

m. MxN=m. Nx M, since the quantities in each member are the same; therefore, the quantities are proportional (Prop. II.); or

m. M:m. N:: M: N.

PROPOSITION VIII. THEOREM.

Of four proportional quantities, if there be taken any equimultiples of the two antecedents, and any equimultiples of the two consequents, the four resulting quantities will be proportional.

Let M, N, P, Q, be the numerical representatives of four quantities in proportion; and let m and n be any numbers whatever, then will

m. M: n. N:: m. P : n. Q.

For, since M:N:: P: Q, we have MxQ=N× P; hence, m. Mxn. Q=n. Nxm. P, by multiplying both members of the equation by mxn. But m. M and n. Q, may be regarded as the two extremes, and n. N and m. P, as the means of a proportion; hence, m. M: n. N:: m. P : n. Q.

Let

For, since
And since

Therefore,

Of four proportional quantities, if the two consequents be either augmented or diminished by quantities which have the same ratio as the antecedents, the resulting quantities and the antecedents will be proportional.

or, hence

PROPOSITION IX. THEOREM.

M: NP: Q, and let also
M: Pm : n, then will
M : P : N±m : Q±n.
M:N::P: Q, MxQ=NxP.
M: P:: m : n, M×n=Pxm
MxQMxn=NxP±Pxm
Mx (Q±n)=Px (N±m):
M: P: N±m: Q±n (Prop. II.).

PROPOSITION X. THEOREM.

If any number of quantities are proportionals, any one antecedent will be to its consequent, as the sum of all the antecedents to the sum of the consequents.

Let

M :
:: P Q R S, &c. then will
MN: M+P+R: N+Q+S
M: NP: Q, we have MxQ=NxP
MN, R: S, we have MxS=NxR
MXN MXN

For, since
And since

Add
and we have, M.N+M.Q+M.S=M.N+N.P+N.R
Mx (N+Q+S)=Nx (M+P+R)
therefore, MN :: M+P+R: N+Q+S.

or

PROPOSITION XI. THEOREM.

If two magnitudes be each increased or diminished by like parts of each, the resulting quantities will have the same ratic as the magnitudes themselves.

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