PROPOSITION I. THEOREM.

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

X

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=M×Q.

MP

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

NxP
M

NXP (Prop. I.): hence, Q':
; but Q
M
sequently, Q-Q' and the four quantities are proportional; that
is, MN: P: Q.

PROPOSITION III. THEOREM.

=

; 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, Mx Q=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

Q S

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

hence

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.

D

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 Nx P=M×Q;

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
(MN) x P (P±Q) × M.

But M±N and P, may be considered the two extremes, and PQ 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. NxM, 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=NxP; hence, m. Mxn. Q=n. N×m. 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.

For, since And since Add

:

M N P: Q, and let also
M: P: : mn, then will
M: P: N±m : Q±n.
M:N::P:Q, MxQ=NxP.
M: Pm : n, M×n=Pxm
MxQMxn=Nx P±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

MN: P: Q: R: S, &c. then will
MN: M+P+R: N+Q+ S

MN: P: Q, we have M×Q=NxP
MNR: S, we have MxS=NxR
MXN MXN

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

PROPOSITION, XI. THEOREM,

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

M N

Let M and N be any two magnitudes, and. and be like

m

m

parts of each: then will

Let

then will

and

m

N

For, it is obvious that Mx (N) =Nx (M±
(M+M) since

m

N.M

Consequently, the four quan

m

each is cqual to M.N±

tities are proportional (Prop. II.).

For,

or, and

MN:

M
M±± :N±

If four quantities are proportional, their squares or cubes will also be proportional.

Let
and

then will
For since

and

or

N

m

therefore,

PROPOSITION XII. THEOREM.

M: NP: Q,

M2: N2: : P2 : Q2
M3: N3 P3 :

Q3

therefore, M2: N° :: P2: Q°

and

M3: N3:: P3 : Q3

MxQ=Nx P, since M: N:: P: Q

M2 × Q2=N2 × P2,
M3 x Q3N3x P3,

Cor. In the same way it may be shown that like powers or roots of proportional quantities are proportionals.

by squaring both members,
by cubing both members;

If there be two sets of proportional quantities, the products of the corresponding terms will be proportional.

PROPOSITION XIII. THEOREM.

MN: P: Q

R: ST : V

MXR NXS :: P×T: Q× V.
MxQ=NxP

RxV=SxT, we shall have
MxQxRxV=NxPxSxT

MXRxQxV=N×S×P×T
MXR NXS :: PXT: Q× V.