Subtracting c² from both sides, and writing a for c, we get
(a − b)² = a² − 2ab + b².
In Euclid’s Elements this form of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.
§ 24. The remaining two theorems (Props. 12 and 13) connect the square on one side of a triangle with the sum of the squares on the other sides, in case that the angle between the latter is acute or obtuse. They are important theorems in trigonometry, where it is possible to include them in a single theorem.
§ 25. There are in the second book two problems, Props. 11 and 14.
If written in the above symbolic language, the former requires to find a line x such that a(a − x) = x². Prop. 11 contains, therefore, the solution of a quadratic equation, which we may write x² + ax = a². The solution is required later on in the construction of a regular decagon.
More important is the problem in the last proposition (Prop. 14). It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation
x² = ab.
In Book I., 42-45, it has been shown how a rectangle may be constructed equal in area to a given figure bounded by straight lines. By aid of the new proposition we may therefore now determine a line such that the square on that line is equal in area to any given rectilinear figure, or we can square any such figure.
As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.