Notebook: Roger B. Farquhar, 1852-1853

Proposition 3 Theorem If four quantities are in proportion they will be in proportion when taken alternately, We have by hypothesis M:N::P:Q hense by prop 1st M x Q = N x P divide each into P x Q and we have P/M = Q/N, then by def 3rd M:P::N:Q Q, E, D

Proposition of Theorem If there be four proportional quantities and four other proportional quantities having the antecedents the same in both the consequents will be proportional, We have by hypothesis M:N::P:Q and M:R:P:S and by def 4th P/M = S/R and P/M = Q/N Hense Q/N = S/R and by def 3rd N:Q ::R:S, Q, E, D

We have by hypothesis M:P::N:Q and M:P ::R:S hense as above N:Q::R:S,

Proposition 5th Theorem If four quantities are in proportion they will be in proportion when taken inversely, We have by hypothesis M:N::P:Q hense prop 1st M x Q = N x P divide each by N x Q and we have M/N = P/Q or N:M::Q:P Q, E, D.

Proposition 6 Theorem If four quantities are in proportion they will be in proportion by composition or division, We have by hypothesis M:N::P:Q, Hense by def 4th N/M = Q/P annex each side to 1 by the sign ± and we have 1 ± N/M = 1 ± Q/P or (M±N)/M = (P±Q)/P M:M ± N::P:P ± Q or M ± N:M::P ± Q:P, Again prop 5th M/N = P/Q annex ± 1 to each side and we have M/N = ± 1 = P/Q ± 1 or (M ± N)/N = (P ± Q)/Q hense M ± N:N::P ± Q:Q, Q, E, D

Proposition 7 Theorem Equimultiples of any two quantities have the same ratio as the quantities themselves We have by the hypothesis N/M multiply Nums and Denomi by M and we have N/M = MN/MM hense M:N::MM:MN or MM:MN::M:N, Q, E, D

Proposition 8th Theorem Of four proportional quantities if there be taken any equimultiples of the two consequents the four resulting quantities will be proportional, We have by hypothesis M:N::P:Q hense N/M = Q/P multiple each side by N/M & we have NN/MM = NQ/MP or MM:NN::MP:NQ, Q, E, D

Proposition 9 Thoerem 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 will have the same ratio

We have by hypothesis M:N::P:Q and M:P::M:N By def 4th N/M = Q/P and by alternation M/M = N/P annex these to the former by the sign ± and we have (N ± M)/M = (Q ± M)/P or M:N ± M::P:Q ± M or M:P::N ± M:Q ± N,

Proposition 10 Theorem If any number of quantities are proportional any one antecedent will be to its consequent as the sum of all the antecedents to the sum of the consequents, We have by hypothesis M:N::P:Q:: R:S Then M/M = N/N we also have M:P::N:Q by alternation Then P/M = Q/N we also have M:R::N:S by alternation Then R/M = S/N Now add all these together observing that they all have a common denominator and we have (M + P + R)/M = (N + Q + S)/N {or M:N + P + R::N:N + Q + S} {M:N::M + P + R:N + Q + S} Q, E, D

Proposition 11 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, We have by hypothesis M + N multiple Numerator and Denominator of the fraction N/M by 1 ± 1/M & we have N/M = (N ± N/M)/(M ± M/M) or M:N::M ± M/M:N ± N/M Q, E, D,

Proposition 12 Theorem If four quantities are proportional their squares or cubes will also be proportional. We have by hypothesis M:N::P:Q N/M = Q/P or (N^2)/(M^2) = (Q^2)/(P^2) or (N^3)/(M^3)=(Q^3)/(P^3) or (N^m)/(M^m)=(Q^m)/(P^m) M^2:N^2::P^2:Q or M^3:N^3::P^3:Q^3 or M^m:N^m::P^m:Q^m, Q, E, D,

Proposition 13 Theorem If there be two sets of proportional quantities the product of the corresponding terms will be proportional, We have by hypothesis M:N::P:Q and R:S::Y:V N/M=Q/P and S/R = V/Y then multiply the fractions and we have (N x S)/(M x R) = (Q x V)/(P x Y) hense M x R:N x S::P x Y:Q x V. Quad, Erat, Demonstration R B. Farquhar.

Finished 6th mo 6th 1853

This page is blank