If p be and

Corollary III.

any Affirmative whole Number greater than to an * indefinite Number; then

18 + 2P -|- 3P -|- 4o -|- 5o ---- &c. to n Terms

*Indefinitely

great.

NB. Tho' the 2d Term of this Canon namely × no be

indefinitely less than the firft, viz.

yet it will be con

venient to have it inferted for the great Ufe we have of that Canon (which will appear in Book II.): And fince the third Term is indefinitely lefs than the fecond, and the 4th Term indefinitely less than the 3d, &c. it will be needlefs to infert the 3d, 4th, &c. Terms in it only by; and therefore the faid Canon will ftand after the moft convenient Manner for our Ufe, as we have above defign'd it.

PART

Of the Nature of Series, or of Appzorimations.

the Uncia of the nth Power of a Binomial or Refidual; and

m MIM m

if m ben-1; I fay that 1, m, X

X

I

2

&c. fhall be equal

to the Uncia of the mth Power of the Binomial or Refidual.

Demonftration.

It is evident by the Genefis of Powers [See Pages 35, 36 and

37] If 1, n, —— ×

-2 n

2—2 × 2—3, &c. be equal to the Uncia of the nth Power

3

x

of a Binomial or Refidual, that the Uncia of the 2--1th Power of a Binomial or Refidual will be equal to 1,n+1,

And that they will be fo in infinitum, fufficiently appears from the Nature of the Operation. 2. E. D.

12

Scholia.

If n be the Index of the Root, and m=n+1= 2 the Index of the Square; then, fince 1, 2 (1),

X 22-3 (0), &c. are to the Uncia of the Root

4

of a Binomial or Refidual, 1, m,

X

m m..

m MI

X

3

I

2

3

4

(by our Lemma) equal to the Uncia of the Square of a Bi

nomial or Refidual.

Again, if n be 2 the Index of the Square, and m=n the Index of the Cube; then, fince 1, 12 (= 2),

n X

x

= 3

N I

22 2

3

2

X

o), &c.) are equal to the Unciæ of

m

I

n

4

the Square of a Binomial, or Residual, 1, m, X

will be (by our Lemma) equal to the Uncia of the Cube of a Binomial or Refidual.

Again, fuppofing n = 3 the Index of the Cube; then,

fince (by what has been already said ) 1, 2,

equal to the Unciæ of the Cube of a Binomial or Residual,

to the Uncia of the 4th Power of a Binomial or Refidual.

any affirmative whole Number, the nth Binomial or Refidual ax is — a" -|- 12

Again, if n and m be equal to any affirmative whole Numbers, then ax" is (by what has been before faid)

The Truth of this will appear from Algebraic Multiplication as far as you are pleafed to continue the Operation; and that it will be fo in infinitum is manifest from what has been already faid.

Now, fince this Product is fuch as I have exprefs'd, it must be fo, altho' n and m were equal to any Numbers whatfoever; for this Multiplication does not diftinguish what Numbers they are equal to; but, on the contrary, being Symbols, are to be confider'd therein only as fuch, that is, as univerfal. Hence

this multiplied by itself is (=ax" ) = an

x\" is (= a = x * ) =

faid); and this multiplied by a → x" is (= ax