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Now if these Numbers be prefixt to the aforesaid Letters, all the Terms will be compleated with their respective Unciæ, and will fand thus ; a tabt 2105b? + 35a0b3 + 35a3b4 +121a*bs t7ab +67

But that the business of finding tbese Uncie, may be render'd yet more easy for Practice, it will be convenient to consider what Series or Progression the Unsiæ of each Term do make from the aforesaid Additions. Thus

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The Uncic of the first Terms, is only a Series of Units, whose Sum is every where the Unciæ of the second Term.

The Oncie of the second Term, is a Series of Numbers in Arithmetick Progreffion; whose Sum is every where the Onciæ of the next fuperiour Power in the third Term, and may be found by Theorem 3. Chap. I. Part VIII.

6 + 1 X6 That is in the seventh Power it will be

21 the Uncie of the third Term.

The rest of the Onciæ are a Compounded Series, whose res. pective Sams may be obtained from the Unciæ of their precedent Terms.

X5 =35, then 35 X 4

= 35.


Thus 21 %

X 2

Again 35 X 3


21 ; alfoz & 2

7, and 7 xs


From hence may be deduc'd this general Rule,



Mule. If the Index of the first Letter of any Term be Makiplyed into its own Uncia, and that Product be divided by the Number of Terms to that place, the Quotient will be the Uncie of the next succeeding Term forward.

That is, by the help of those Indices that belong to the several Powers of the first or leading Letter only, i (as a) the true Uncia of every Term may be easily found.

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7 X 6


3. Again

Examples. Let it be required to compleat all the Terms of the aforesaid feveral Powers, viti al + ab to atba + ab + ab + abs Tabs + b with their proper Uncid.

1. The Index of a? (the firft Term) to wit 7 will be the Oncia of the second Term. Thus 47 + 71°b.

2. Then half the second Terms Index into its Uncie, Viz. 21, will be the third Terms Uncie. Thus 'a? + 7ab +21asba will be the three first Terms. 21 XS

35 is the Unciæ of the fourth Term,

-3 Then it will be a? + 7266 +213562 + 35a*b.

35 X 4 4.

= 35 will be Oncie of the fifth Term. Then it will be a? + 74b + 2146+ 35a4b3 + 35a3b4, 35 X 3

21 is the Unciæ of the sixth Term. 5

21 X 2 6. Also =7 is the Unciæ of the feventh Teri.


XI 7

= 1 is the Uncie of the eighth Term.

7 Wherefore all the Terms, when compleated, stand

Thus, ał +7436 + 21a9b2 + 35a4b3 + 35ab* † 21a*b + ab + b7.

In like manner, if it be required to complear all the Terms of a + ab + aab? t-a3bi t-ab4 tabs t bo ( which are the Terms of the sixth Power of a tb, without their Unciæ) it may be done thus,


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s. Again

7. And 7 X 1

ift..s is the Unciæ ofa"; and 6 of ab.
2di§. 6*5=,

15 is the Uncie of the third Term,

Third," 3dly.


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15x4=20 is the Uncie of the fourth Term.


20 X 3 = is is the Uncie of the fifth Term. 19 X 2

= 6 is the Uncia of the fixth Term.

bebly. =r is the Uncie of the seventh Term.

Therefore all the Terms when Compleated, stand thus,

+ 6asb + 1520be + 2043b3 + 15a2b4 + 6abs +66. Now here it may be further observed, that the Unciæ do only increase until the Indices of the two Letters become equal, or change Places; and the rest of the Unciæ will return, or des crease in the same Order. That is, wherever the Indices of the Letters are alike, there the Uncie will be alike.

And therefore one needs to find the Unciæ (as before) but to half the Numbers of Terms in any Power.

Corollary. If n be equal to any whole Number; then the several Terms of the mob Power of á fb will be=am tomam-sbtmx bi tmx

63 &C. 3


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Х 2

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Any 3, 4, 5, 6o. Simple Quantities connected by the Signs to,

-, may be involved in like manner with the * foregoing. But bere ought rather to be In. That is by volved to any required Power, by the same Multiplication. Power of a Binomial, in order to know the Nature and Composition of Powers, and thence the manner of Evolving them. Thus Suppose it were required to lovolve s tobi, to the third

The third Power of the Binomial a fab is a? + 3 aab t zabb

+ b3, which is your Canon for Involving.

g= a (i. e. if. a)

& TAB And therefore g} = a3 Orthus: put gth=a, zghb = 3abb

358h= zaab and othand fogth-iža-i,



? gth-i

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hero willbo g +3gh+3ghthaan


a_ 3a i +3ai-;; hora restore gth fora, and

- 32 Porvor of

gi: +38gh + 3ghh + b3 =23
388i" 6ghi -- 3hbi
3hhi = 3aab

Here a (i. e. 2d. a)
+3gii + 3hii = 3abb
+ ,

8th. And 2d. 6

63 g+ 3h

+ 3hb
3i8g - 6bi

+ b

Is the Cube of g t by + zii +3hii

-i, which was required. is.

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Involution of Dumbers. ANy absolute Number being firft reduc'd into its leveral Mem

bers (which are the several Significant Figures in the given absolute Number with their due Number of Cyphers after each of them, and before such of them as are Decimals, along with the Point or Decimal Character) may be Involved to any required Power, by the help of the same Power of a Binomial (as in the foregoing Example in Specie, which shews and demonftrates the manner of doing it) always observing to begin with the greatest Members of the said Number.

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Examples. 1 Let it be required to find the Square of 5709. 1. The Square of the Binomial a+b, is aa'+ 2ab + bb, which is your Canon for Involving. 2. 5700 5000 + 700 + 9

sooo a (i. e. ift. a)

700 = b (i.e. iftib)
(5000 X 5000 ) 25000000 =
(2 X 5000 x 700 7000000 2ab

(700 X 700 =) (5700.X 57.00 =) 32490000 = S .

Here'a (i. e. 2d. a) (2 X 5700 X9=) 102600 = 2ab

55700. And b(i.c. (9x9 =)

81 bb

2d. b) 9. 32592681 The Square required. G


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.490000 = bb

2. What is the Cube of 463?

The Cube of the Binomial a +b, is qua + 3aab + zabb
+ bbb Canon.
463 =400 + 60+ 3.


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99252847 the Cube of 463 Answer.

Scholium. The Square Cube, &c. of any Number with Cyphers after it to the place of Units inclusive, will have twice thrice, &c. selpectively the said Number of Cyphers after it; wherefore in Involving, or Evolving Numbers; the following Table will be aflifting.


The Reot.


8 9 Square

16 25

38 49 64 81 Cube

64 | 125 216

343 512 729 Biquadrai.

16 811 2361, 625 | 1296 2401 4096 6560 sth Power. 32 1243 | 1024 3125 1976 | 16807 i 32768 | 59049

&c. 3. What is the Square of 1 . 217

1.21 is =1.7.2+ioi

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