I call arithmetical triangle a figure constructed as follows:
From any point, G, I draw two lines perpendicular to each other, G-AK, G-I in each of which I take as many equal and contiguous parts as I please, beginning with G, which I number 1, 2, 3, 4, etc., and these numbers are the exponents of the sections of the lines.
Next I connect the points of the first section in each of the two lines by another line, which is the base of the resulting triangle.
In the same way I connect the two points of the second section by another line, making a second triangle of which it is the base.
And in this way connecting all the points of section with the same exponent, I construct as many triangles and bases as there are exponents.
Through each of the points of section and parallel to the sides I draw lines whose intersections make little squares which I call cells.
Cells between two parallels drawn from left to right are called cells of the same parallel row, as, for example, cells G, B, C, D, E, F, H, I, or J, K, L, M, N, O, P.
Those between two lines are drawn from top to bottom are claeed cells of the same perpendicular row, as for example, cells G, J, Q, W, AB, AF, AI, AK, or B, K, R, X, AC, AG, AJ.
Those cut diagonally by the same base are called cells of the same base, as for example, W, R, L, D or Q, K, C.
Cells of the same base equidistant from its extremities are called reciprocals, as, for example, R, L and N, AC, because the parallel exponent of one is the same as the perpendicular exponent of the other, as is apparent in the above example, where R is in the second perpendicular row and in the fourth parallel row and its reciprocal, L, is in the second parallel row and in the fourth parallel row, reciprocally. It is very easy to demonstrate that cells with exponents reciprocally the same are in the same base and are equidistant from its extremities.
It is also very easy to demonstrate that the perpendicular exponent of any cell when added to is parallel exponent exceeds by unity the exponent of its base.
For example, cell Y is in the third perpendicular row and in the fourth parallel row and in the sixth base, and the exponents of rows 3 and 4, added together, exceed by unity the exponent of base 6, a property which follows from the fact that the two sides of the triangle have the same number of parts; but this is understood rather than demonstrated.
Of the same kind is the observation that each base has one more cell than the preceding base, and that each has as many cells as its exponent has units; thus the second base, BJ, has two cells, the third, CKQ, has three, etc .
Now the numbers assigned to each cell are found by the following method:
The number of the first cell, which is at the right angle, is arbitrary; but that number having been assigned, all the rest are determined, and for this reason it is called the generator of the triangle. Each of the others is specified by a single rule as follows:
The number of each cell is equal to the sum of the numbers of the perpendicular and parallel cells immediately preceding. Thus cell S, that is, the number of cell S, equals the sum of cell R and cell L, and similarly with the rest
Whence several consequences are drawn. The most important follow, wherein I consider triangles generated by unity, but what is said of them will hold for all others.
Next :: (1st Consequence)