2723

3 nn--nn Confequently the Difference of the Sums

24

205

PART XIV.

Of Polygonal Numbers,

Olygonal, or Multangular Numbers are the Sums or Aggregates of a Rank of Numbers in Arithmetical Progreffion continued from Unity, and are fo called, because they reprefent the Number of Points that are required to fill fuch regular Polygons at equal Distances or Lines drawn parallel to the Sides of the Figure; as

Num. in Arit. Whofe com. Sums added from 1. Polygon.
Progreffion. Differ.is

I, 2, 3, 4, &c.)

I

1, 3, 5, 7, &c.

2

1, 4, 7, 10, &c.

3

I, 5, 9, 13, &c. &c.

4

&c.

1,3,6, 10, 15, 21, &c Triang.
1,4, 9, 16, 25, 36, &c.Quadr.
1, 5, 12, 22, 35, 51, &c. Pentang.
1, 6, 15, 28, 45, 66, &c. Hexang.
&c.
&c.

Hence, by Infpection, these two Observations are evident;

1. That the common Difference will be always lefs by 2, than the Number of Angles.

2. That the Side of the Polygon is equal to the Number of Terms which compofe it.

There

L

Since every Polygon is the Aggregate of Numbers in -:whofe firft Term is 1; therefore (by Part VIII. Chap. 1. Step 17.) pis=

n n n

d+n. 2.E.D.

Corollary.

And becaufe d is given (by Obfervation 1.) if its Value be fubftituted in this general Theorem, we may deduce particular ones for each Polygon: As in

52 72

But, becaufen may not be always given, fince we find p≈ 72 d-n; n therefore will be found (by Part X.) = d-2+v:dd- -4d-1-4+8dp. And because d is given

2

2 d

(by Obfer. 1.) if we fubftitute its Value, we fhall have parricular Theorems in this Cafe likewile; viz. in a

Triang.

Thus having fhew'd how from the Side given to find the Polygon; or from the Polygon to find the Side; I fhall now gsve an univerfal Theorem for finding any Trigonal, Pyramidal, or other Number undermentioned: As

Here it is evident, that each figurate Number is the Aggreate of the preceding Series fo far, or of the preceding figurate Number and that above itself.

Obfervation II.

It is also evident, if b and c be fuppos'd equal to any two whole Numbers, that b figurate Number of c Order is equal to c figurate Number of b Order.

LEMM A.

If n be the Side of a figurate Number of fuch an Or

=

I X

22-1-022---12---2x 2-+-3, &c. i. e. (fince a

X

2

3

4

to the refpective figurate Numbers of that Order; And, if m ben1 = the Side of the figurate Number of the

next greater Order; I say that I, I × M, I X

m+o

1

m-|-1,

2

X

3

4

m-\-2 m-1-3, &c. fhall be equal to the respective figurate

Numbers of that Order whofe Side is m.

Demonstration.

It is evident (by the Nature of Multiplication) that i =

be