These centres are centres of gravity, and they are symmetrically distributed around the centre of gravity of the original triangle.
The figure formed by joining them must therefore be symmetrical, and, as in this case, it is a triangle, it must be always equilateral.
No. LXXIII.—TRY THIS EXPERIMENT
There can be no better instance of how the eye may be deceived than is so strikingly afforded in these very curious diagrams:—
The square which obviously contains sixty-four small squares, is to be cut into four parts, as is shown by the thicker lines. When these four pieces are quite simply put together, as shown in the second figure, there seem to be sixty-five squares instead of sixty-four.
This phenomena is due to the fact that the edges of the four pieces, which lie along the diagonal A B, do not exactly coincide in direction. In reality they include a very narrow diamond, not easily detected, whose area is just equal to that of one of the sixty-four small squares.
FIGURES IN SWARMS
Very curious are the results when the nine digits in reverse order are multiplied by 9 and its multiples up to 81. Thus:—
| 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | × | 9 | = | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | |
| × | 18 | = | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 8 | |||||||||
| × | 27 | = | 2 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 7 | |||||||||
| × | 36 | = | 3 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 6 | |||||||||
| × | 45 | = | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | |||||||||
| × | 54 | = | 5 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | |||||||||
| × | 63 | = | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | |||||||||
| × | 72 | = | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | |||||||||
| × | 81 | = | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | |||||||||