Corollary 3
In every arithmetic triangle, each cell is equal to the sum of all those of the preceding perpendicular rank, comprising the cells from its parallel rank to the first, inclusively.
Consider any cell C: I assert that it is equal to B + ψ + σ, which are the cells of the preceding perpendicular rank, from the parallel rank of the cell C to the first parallel rank.
C equals B + θ, θ equals ψ + π, π equals σ. Hence C equals B + ψ + σ.
Corollary 4
In every arithmetic triangle, each cell diminished by unity is equal to the sum of all those which are included between its perpendicular rank and its parallel rank, exclusively.
Consider any cell ξ: I assert that ξ - G equals R + θ + ψ + φ + λ + π + σ + G, which are all the numbers included between the ranks ξωCBA and the rank ξSμ, exclusively.
For ξ equals λ + R + ω, ω equals π + θ + C, C equals σ + ψ + B, B equals G + φ + A, A equals G. Hence ξ equals λ + R + π + θ + σ + ψ + G + φ + G.
Pascal makes a note. I have said in the statement: each cell diminished by unity, because unity is the generator; but if it were another number, it would be necessary to say: each cell diminished by generating number.
Corollary 12
In every arithmetical triangle, of two contiguous cells in the same base the upper is to the lower as the number of cells from the upper to the top of the base is to the number of cells from the lower to the bottom of the base, inclusive.
This one I shall just reformulate in modern notations: C(n, m) / C(n, m + 1) = (n - m) / (m + 1). Additional features of the triangle are listed on a separate page where I'll make more references to Pascal's Treatise.
The last of the nineteen corollaries is followed by a remark: Thence many other proportions may be drawn that I have passed over, because they may be easily deduced, and those who would like to apply themselves to it will perhaps find some, more elegant than these which I could present.
References
- H. Eves, Great Moments in Mathematics After 1650, MAA, 1983
- Great Books of the Western World, v 33, Encyclopaedia Britannica, Inc., 1952.
- J. A. Paulos, Beyond Numeracy, Vintage Books, 1992
- D. E. Smith, History of Mathematics, Dover, 1968
- D. E. Smith, A Source Book in Mathematics, Dover, 1959
Pascal's Triangle and the Binomial Coefficients
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Copyright © 1996-2012 Alexander Bogomolny
