Commentary on Proposition X.12

Back to Proposition X.12.

[Folio 0039 | p. 35]

We have merely to go through the process of compounding two ratios in numbers.

$$\begin{aligned} &\text{Suppose } & a,\, b &\text{ each } ⌢ c. \\ &\text{Therefore } & a : c &= m : n\text{, say,} & \text{[x. 5]} \\ & & c : b &= p : q\text{, say.} \\ &\text{Now } & m : n &= mp : np, \\ &\text{and } & p : q &= np : nq. \\ &\text{Therefore } & a : c &= mp : np, \\ & & c : b &= np : nq, \\ &\text{whence, \textit{ex aequali},} & a : b &= mp : nq, \\ &\text{so that } & a &⌢ b. & \text{[x. 6]} \end{aligned}$$