Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 75
third, a pentagonal; the fourth, an hexagonal pyramid; and so
of the rest.
CHAPTER XI.
The generation of solid numbers.
As linear numbers are those which proceeding from unity
extend to infinity, such as 1. 2. 3. 4. 5. 6, etc. and as superficial
numbers are formed from the addition of these, in a similar
manner solid are generated from the junction of superficial
numbers. Thus, triangular pyramids are formed from the
addition of triangular numbers; square pyramids from the
addition of squares ; pentagonal, from the addition of pentagons;
and so of the rest.
The first triangle therefore, in power or capacity, is unity,
and unity is also the first pyramid. But the second triangle is
3; and this added to 1 the first triangle, forms the second pyramid
4. If to these the third triangle 6 is added, the third pyramid
10 will be generated. And if to these, the fourth triangle
10 is added, the fourth pyramid 20 will be formed. In all the
rest likewise, there will be the same mode of conjunction.
Triangles.
13 6 10 15 21 28 36 45 55
Pyramids from Triangles.
1 4 10 20 35 56 84 120 165 220
Jn this conjunction therefore, it is necessary that the last of the
conjoined numbers should always be as it were the base of the
pyramid. For it is found to be broader than all the rest; and
all the numbers that are conjoined prior to it are necessarilv
less, till we arrive at unity, which in a certain respect obtains
the place of a point and a vertex.