numbers are related? Please look carefully at
| Simple Number |
Square for the Distance |
Cube for the Time | |||||
| 1 | 1 × | 1 = | 1 | 1 × | 1 × | 1 = | 1 |
| 2 | 2 × | 2 = | 4 | 2 × | 2 × | 2 = | 8 |
| 3 | 3 × | 3 = | 9 | 3 × | 3 × | 3 = | 27 |
| 4 | 4 × | 4 = | 16 | 4 × | 4 × | 4 = | 64 |
| 10 | 10 × | 10 = | 100 | 10 × | 10 × | 10 = | 1000 |
this little table. Four is called the square of two, because if one square is twice as long and twice as broad as another, it is four times the first in size. We are almost reminded of the Red Queen's argument with Alice, "Five times as warm and five times as cold—just as I'm five times as rich as you are and five times as clever." Poor Alice gave it up like a riddle with no answer: but we must not give it up here, we must go even a step further. When one thing is twice as long and twice as broad and twice as high, we go beyond the square to the cube.
If we multiply 2 × 2 × 2, we get 8, which is the cube of 2: and when the distance is the square of a number, the time is the cube of the same number.
None of the planets fits these numbers exactly: there is no planet at 9 times the Earth's distance, but Saturn is at 10 times—just a little bigger than 9—and we can therefore see from the table that it will take rather more than 27 years to go round the Sun (29½ years as a matter of fact). But we need not trouble about the exact figures if we understand the principle of this great Third Law of Kepler.
On that law Sir Isaac Newton and others went to work, and soon saw that it told them the Law of
