Commentary on Proposition V.1¶
Back to Proposition V.1.
De Morgan remarks of V. 1—6 that they are “simple propositions of concrete arithmetic, covered in language which makes them unintelligible to modern ears. The first, for instance, states no more than that ten acres and ten roods make ten times as much as one acre and one rood.” One aim therefore of notes on these as well as the other propositions of Book V. should be to make their purport clearer to the learner by setting them side by side with the same truths expressed in the much shorter and more familiar modern (algebraical) notation. In doing so, we shall express magnitudes by the first letters of the alphabet, $a$, $b$, $c$ etc., adopting small instead of capital letters so as to avoid confusion with Euclid’s lettering; and we shall use the small letters $m$, $n$, $p$ etc. to represent integral numbers. Thus $ma$ will always mean $m$ times $a$ or the $m^{\text{th}}$ multiple of $a$ (counting $1.a$ as the first, $2.a$ as the second multiple, and so on).
Prop. 1 then asserts that, if $ma$, $mb$, $mc$ etc. be any equimultiples of $a$, $b$, $c$ etc., then
$$ma + mb + mc + \dots = m(a + b + c + \dots).$$