§ 18. To compare areas of other figures, Euclid shows first, in Prop. 42, how to draw a parallelogram which is equal in area to a given triangle, and has one of its angles equal to a given angle. If the given angle is right, then the problem is solved to draw a “rectangle” equal in area to a given triangle.
Next this parallelogram is transformed into another parallelogram, which has one of its sides equal to a given straight line, whilst its angles remain unaltered. This may be done by aid of the theorem in
Prop. 43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.
Thus the problem (Prop. 44) is solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).
As every polygon can be divided into a number of triangles, we can now construct a parallelogram having a given angle, say a right angle, and being equal in area to a given polygon. For each of the triangles into which the polygon has been divided, a parallelogram may be constructed, having one side equal to a given straight line and one angle equal to a given angle. If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length. This is done in Prop. 45.
Herewith a means is found to compare areas of different polygons. We need only construct two rectangles equal in area to the given polygons, and having each one side of given length. By comparing the unequal sides we are enabled to judge whether the areas are equal, or which is the greater. Euclid does not state this consequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.
Prop. 46 is: To describe a square on a given straight line.
§ 19. The first book concludes with one of the most important theorems in the whole of geometry, and one which has been celebrated since the earliest times. It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his name. If we call that side in a right-angled triangle which is opposite the right angle the hypotenuse, we may state it as follows:—
Theorem of Pythagoras (Prop. 47).—In every right-angled triangle the square on the hypotenuse is equal to the sum of the squares of the other sides.
And conversely—