In every arithmetical triangle each cell is equal to its reciprocal.
For in the second base, JB, it is evident that the two reciprocal cells, J, B, are equal to each other and to G.
In the third base, QKC, it is also obvious that the reciprocals, Q, C, are equal to each other and to G.
In the fourth base it is obvious that the extremes, W, D, are again equal to each other and to G.
And those between, R, L, are obviously equal since R = Q + K and L = K + C. But Q + K = K + C by what has just been shown. Therefore, etc.
Similarly it can be shown for all the other bases that reciprocals are equal, because the extremes are always equal to G and the rest can always be considered as the sum of cells in the preceding base which are themselves reciprocals.
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