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Commentary on Proposition II.7¶

An interesting variation of the form of this proposition may be obtained by regarding AB, BC as two given straight lines of which AB is the greater, and AC as the difference between the two straight lines. Thus the proposition shows that the squares on two straight lines are together equal to twice the rectangle contained by them and the square on their difference. That is, the square on the difference of two straight lines is equal to the sum of the squares on the straight lines diminished by twice the rectangle contained by them. In other words, just as II. 4 is the geometrical equivalent of the identity [ (a + b)^2 = a^2 + b^2 + 2ab, ] so II. 7 proves that [ (a - b)^2 = a^2 + b^2 - 2ab. ] The addition and subtraction of these formulae give the algebraical equivalent of the propositions II. 9, 10 and II. 8 respectively; and we have accordingly a suggestion of alternative methods of proving those propositions.