750375v
[Commentary:
This large folded sheet (Add MS
f. ) is one of the few that Harriot dated: 12 December 1591. It is concerned with the piling of bullets, for which two questions naturally arise: 1. Given the number of bullets and the required shape of the base, how many bullets must be placed at each level, and how many levels? 2. Given a pile of a certain shape, how many bullets does it contain?
(a) Square and oblong numbers:
The numbers 1, 2, 3, 4, ... down the key diagonal indicate the length of the shorter side of the base, and therefore also the height of the pyramid.
The numbers 1, 2, 3, 4, ... along the top indicate the length of the longer side of the base.
Thus for an oblong pyramid on a 3 x 7 base, move right from 3 in the key diagonal and down from 7 on the top line. The entries in the appropriate square show that there are 21 bullets in the base and 38 in the pyramid as a whole.
In particular, the square pyramidal numbers 1, 5, 14, 30, 55, 91 ... appear in the squares immediately to the right of the key diagonal; thus a square pyramid with base 3 x 3 has 9 bullets in its base and 14 bullets in
(b) Triangular numbers:
The triangular pyramidal numbers 1, 4, 10, 20, 35, ... appear as the lower entries in the squares below and to the left of the key diagonal, to which they are joined by short lines. Thus a triangular pyramid with three levels contains 6 bullets in its base and 10 bullets in total. As Harriot points out in his note f. , the triangular numbers 1, 3, 6, 10, 15, ... may be read off as the differences of the oblong numbers in the previous row. Conversely, the entire table can be generated by successive addition from the triangular and square numbers. ]
(a) Square and oblong numbers:
The numbers 1, 2, 3, 4, ... down the key diagonal indicate the length of the shorter side of the base, and therefore also the height of the pyramid.
The numbers 1, 2, 3, 4, ... along the top indicate the length of the longer side of the base.
Thus for an oblong pyramid on a 3 x 7 base, move right from 3 in the key diagonal and down from 7 on the top line. The entries in the appropriate square show that there are 21 bullets in the base and 38 in the pyramid as a whole.
In particular, the square pyramidal numbers 1, 5, 14, 30, 55, 91 ... appear in the squares immediately to the right of the key diagonal; thus a square pyramid with base 3 x 3 has 9 bullets in its base and 14 bullets in
(b) Triangular numbers:
The triangular pyramidal numbers 1, 4, 10, 20, 35, ... appear as the lower entries in the squares below and to the left of the key diagonal, to which they are joined by short lines. Thus a triangular pyramid with three levels contains 6 bullets in its base and 10 bullets in total. As Harriot points out in his note f. , the triangular numbers 1, 3, 6, 10, 15, ... may be read off as the differences of the oblong numbers in the previous row. Conversely, the entire table can be generated by successive addition from the triangular and square numbers. ]
These are those speciall
groundplats upon the which
may be orderly piled bullets:
The triangle: the square: and
the
groundplats upon the which
may be orderly piled bullets:
The triangle: the square: and
the
Concerning piling there are two
questions: one which the nomber
of bullets to be piled being given
with the forme of the groundplat,
to know how many must be placed
in every rounde, with how many
roundes in the sayd
questions: one which the nomber
of bullets to be piled being given
with the forme of the groundplat,
to know how many must be placed
in every rounde, with how many
roundes in the sayd