14774
[Commentary:
At the beginning of this set of sheets Harriot has written: 'Waste papers of figurate nombers'.
They are waste only in the sense that they contain rough working. At the same time,
they show Harriot attempting something highly original, namely,
finding formulae for sequences of figurate numbers.
In modern terms, we would say he is fitting third-, fourth- or fifth-degree polynomials
to numerical sequences.
At the top is the sequence 1, 5, 14, 30, 55, ... of sums of squares (or of square-pyramidal numbers, see Add MS 6782 f. ). Just below that, Harriot has written the polynomial , which, it seems, is his first attempt to find a formula for the numbers in the sequence, multiplied by 24 (that is, 24, 120, 336, ...). Putting gives , as required. Putting , however, gives , which is too large. This calculation can be seen displayed vertically just below the formula. Harriot notes that falls short of this total by 8.
Similar trial and error calculations appear on this and several pages that ]
At the top is the sequence 1, 5, 14, 30, 55, ... of sums of squares (or of square-pyramidal numbers, see Add MS 6782 f. ). Just below that, Harriot has written the polynomial , which, it seems, is his first attempt to find a formula for the numbers in the sequence, multiplied by 24 (that is, 24, 120, 336, ...). Putting gives , as required. Putting , however, gives , which is too large. This calculation can be seen displayed vertically just below the formula. Harriot notes that falls short of this total by 8.
Similar trial and error calculations appear on this and several pages that ]
Waste papers.
of figurate
of figurate