greater, equal, and less. Propositions 11, 12, 15 and 16 may be considered as introduced to shew that, if four quantities of the same kind be proportionals they will also he proportionals when taken alternately. The remaining propositions shew that magnitudes are proportional by composition, by division, and ex aequo.
In this division of the fifth Book propositions 13 and 14 are supposed to be placed immediately after proposition 10; and they might be taken in this order without any change in Euclid's demonstrations.
The propositions headed A, B, C, D, E were supplied by Simson.
V. 1, 2, 3, 5, 6. These are simple propositions of Arithmetic, though they are here expressed in terms which make them appear less familiar than they really are. For example, V. i "states no more than that ten acres and ten roods make ten times as much as one acre and one rood." De Morgan.
In V. 5 Simson has substituted another construction for that given by Euclid, because Euclid's construction assumes that we can divide a given straight line into any assigned number of equal parts, and this problem is not solved until VI. 9.
V. 18. This demonstration is Simson's. We will give here Euclid's demonstration.
Let AE be to EB as CF is to FD: AB shall be to BE as CD is to DF.
For, if not, AB will be to BE as CD is to some magnitude less than DF, or greater than DF.
First, suppose that AB is to BE as CD is to DG, which is less than DF.
Then, because AB is to BE as CD is to DG, therefore AE is to EB as CG is to GD. [V. 17.
But AE is to EB as CF is to FD, [Hypothesis.
therefore CG is to GD as CF is to FD. [V. 11.
But CG is greater than CF; [Hypothesis.
therefore GD is greater than FD. [V. 11.
But GD is less than FD; which is impossible.
In the same manner it may be shewn that AB is not to BE as CD is to a magnitude greater than DF.
Therefore AB is to BE as CD is to DF.
The objection urged by Simson against Euclid's demonstration is that "it depends upon this hypothesis, that to any three magnitudes, two of which, at least, are of the same kind, there
