Proposition: 5.12: Sum of Components of Equal Ratios

(Proposition 12 from Book 5 of Euclid's “Elements”)

If there are any number of magnitudes whatsoever (which are) proportional then as one of the leading (magnitudes is) to one of the following, so will all of the leading (magnitudes) be to all of the following.

Let there be any number of magnitudes whatsoever, $A$, $B$, $C$, $D$, $E$, $F$, (which are) proportional, (so that) as $A$ (is) to $B$, so $C$ (is) to $D$, and $E$ to $F$.
I say that as $A$ is to $B$, so $A$, $C$, $E$ (are) to $B$, $D$, $F$.

fig12e

Modern Formulation

In modern notation, this proposition reads that if \[\frac\alpha{\alpha'}=\frac\beta{\beta'}=\frac\gamma{\gamma'}=\ldots.\] then \[\frac\alpha{\alpha'}=\frac{\alpha+\beta+\gamma+\cdots}{\alpha'+\beta'+\gamma'+\cdots},\]

for all positive real numbers.

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10
Sections: 11