Commentary on Proposition VII.6

Back to Proposition VII.6.

[Folio 0311 | p. 305]

If

$$a = \frac{m}{n}b, \text{ and } c = \frac{m}{n}d,$$

then

$$a + c = \frac{m}{n}(b + d).$$

More generally, if

$$a = \frac{m}{n}b,\ c = \frac{m}{n}d,\ e = \frac{m}{n}f, \dots$$

then

$$(a + c + e + g + \dots) = \frac{m}{n}(b + d + f + h + \dots).$$

[Folio 0312 | p. 306]

In Euclid’s proposition $m < n$, but the generality of the result is of course not affected. This proposition and the last are complementary to V. 1, which proves the corresponding result with multiple substituted for “part” or “parts.”