Commentary on Proposition II.3¶
Back to Proposition II.3.
If we leave out of account the contents of Book II. itself and merely look to the applicability of propositions to general use, this proposition and the preceding are, as already indicated, of great importance, and particularly so to the semi-algebraical method just described, which seems to have found its first exponents in Heron and Pappus. Thus the proposition that the difference of the squares on two straight lines is equal to the rectangle contained by the sum and the difference of the straight lines, which is generally given as equivalent to II. 5, 6, can be proved by means of II. 1, 2, 3, as shown by Lardner. For suppose the given straight lines are AB, BC, the latter being measured along BA.
Then, by II. 2, the square on AB is equal to the sum of the rectangles AB, BC and AB, AC.
By II. 3, the rectangle AB, BC is equal to the sum of the square on BC and the rectangle AC, CB.
Therefore the square on AB is equal to the square BC together with the sum of the rectangles AC, AB and AC, CB.
But, by II. 1, the sum of the latter rectangles is equal to the rectangle contained by AC and the sum of AB, BC, i.e. the rectangle contained by the sum and difference of AB, BC.
Hence the square on AB is equal to the square on BC and the rectangle contained by the sum and difference of AB, BC; that is, the difference of the squares on AB, BC is equal to the rectangle contained by the sum and difference of AB, BC.