may be a fourth proportional:...... Euclid does not demonstrate it, nor does he shew how to find the fourth proportional, before the 12th Proposition of the 6th Book.... "
The following demonstration is given by Austin in his Examination of the first six books of Euclid's Elements.
Let AE be to EB as CF is to FD: AB shall be to BE as CD is to DF.
For, because AE is to EB as CF is to FD, therefore, alternately, AE is to CF as EB is to FD. [V. 16.
And as one of the antecedents is to its consequent so is the sum of the antecedents to the sum of the consequents; [V. 12.
therefore as EB is to FD so are AE and EB together to CF and FD together, that is, AB is to CD as EB is to FD.
Therefore, alternately, AB is to EB as CD is to FD. [V. 16.
V. 25. The first step in the demonstration of this proposition is "take AG equal to E and CH equal to F"; and here a reference is sometimes given to I. 3. But the magnitudes in the proposition are not necessarily straight lines, so that this reference to I. 3 should not be given; it must however be assumed that we can perform on the magnitudes considered, an operation similar to that which is performed on straight lines in I. 3. Since the fifth Book of the Elements treats of magnitudes generally, and not merely of lengths, areas, and angles, there is no reference made in it to any proposition of the first four Books.
Simson adds four propositions relating to compound ratio, which he distinguishes by the letters F, G, H, K; it seems however unnecessary to reproduce them as they are now rarely read and never required.
THE SIXTH BOOK.
The sixth Book of the Elements consists of the application of the theory of proportion to establish properties of geometrical figures,
VI. Def. I. Eor an important remark bearing on the first definition, see the note on VI. 5.
VI. Def. 1. The second definition is useless, for Euclid makes no mention of reciprocal figures.
