The product of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b. For aa, which means the square on the line a, we write a².
§ 21. The first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows:—
| Prop. | 1. | a (b + c + d + ... ) = ab + ac + ad + ... |
| ” | 2. | ab + ac = a² if b + c = a. |
| ” | 3. | a (a + b) = a² + ab. |
| ” | 4. | (a + b)² = a² + 2ab + b². |
| ” | 5. | (a + b)(a − b) + b² = a². |
| ” | 6. | (a + b)(a − b) + b² = a². |
| ” | 7. | a² + (a − b)² = 2a (a − b) + b². |
| ” | 8. | 4(a + b)a + b² = (2a + b)². |
| ” | 9. | (a + b)² + (a − b)² = 2a² + 2b². |
| ” | 10. | (a + b)² + (a − b)² = 2a² + 2b². |
It will be seen that 5 and 6, and also 9 and 10, are identical. In Euclid’s statement they do not look the same, the figures being arranged differently.
If the letters a, b, c, ... denoted numbers, it follows from algebra that each of these formulae is true. But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers. To prove them we have to discover the laws which rule the operations introduced, viz. addition and multiplication of segments. This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.
§ 22. In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups and then add these groups. But this also holds for the sum of segments and for the sum of rectangles, as a little consideration shows. That the sum of rectangles has always a meaning follows from the Props. 43-45 in the first book. These laws about addition are reducible to the two—
a + b = b + a
(1),
a + (b + c) = a + b + c
(2);