of the Means:

And the former Product is equal to the latter, which proves, in fome Measure, our Rule.

3. To find whether

their Mutual Proportion.

and c be commenfurable, as also

I

The greatest common Divifor of b and c is ",by which both of them being Divided, the Quotients are 1" and "", which are Rational Quantities, and equal to 1 and r refpectively; confequently the propos'd Surds are Commenfurable, and b' ਯਾ ::

c

I

I. C.

CHA P. III.

Multiplication of Simple Surds.

Kule.

F the Surds be not of the fame kind; that is, if they have IF different Indices; reduce them to one and the leaft kind ; viz. to their least common Index (by Prop. 1. of the laft Chap.) then, if the Surds be not Imaginary, Multiply them by one another without their Indices; and laftly, annex the leaft common Index to the Product. So this new Root fhall be the Product fought. But if the propos'd Surds be Imaginary; See the following Remark.

Examples
b4

So √2×No3=√6. Also bx c is belt. And Uni

I

= 87 ×

Again 2 × 33 is (by Prop. 1. of the last Chap.) =

Also a3× 2a is (by Prop. 1. of the laft Char.) as bijis atboys.

And

And Univerfally xxy is (by Prop. of the last Chap.) ==

1. When a Surd is to be Multiplyed by a Rational Quantity, it will be fometimes convenient to connex them with the Sign ×; fo the Product of a and b may be writ thus axb or thus a b.

2. When two Rational Quantities are join'd with the Sign x to two Surds of the fame kind. Multiply the Rational Part by the Rational, and the Surd Part by the Surd; and the two Products join'd together with the Sign x is the Product required.

So axinto cxdà is = ac × bd|": For a×b" is =

I

a"b| ", and c × d" is=c"d\" ; but a′′b\" × c′′d," = a"c"bd|"which

is manifeftly acx bd\

— bd)".

3. When any Surd is to be rais'd to any given Power; Multiply the Index of the Surd by the Quantity that denotes that Power:

Alfo a" rais'd to the mth Power is

*Here Note that the Square of

✔a is a; and

Univerfally, that a rais'd to the nth-Power is—a,

Therefore baxc√ - a is bcx-a-bca; Alfo by-ax-c√a is b c x — a = + bea; Likewife-b-ax-c√— a = + bc x-a=bca; Again-bax J—ca = √ bx √ -ax√ex√-a = √ b c - a ybc:

Likewise bax - √ca — √ b x√-ax — √ ex√ — a = √ bc x — a = — a jbc. Again as 1 — a = b c √ — A.

Note

CHA P. IV.

Divilion of Simple Surds.

Kule.

F the Surds be not of the fame kind, first reduce them to the fame kind (by Prop. 1. Chap. II.) Then, if neither of the Surds be Imaginary, Divide the Dividend by the Divifor, without their Indices, and Superior to the Quotient place the Common Index; and this new Surd is the Quotient fought: But if either of the Surds be Imaginary, fee the Remarks in the last and in this Chap.

So 20 Divided by 5, gives 4 for a Quotient.

-4 -4

Alfo abca ab

1

And Univerfally z y = (by Prop. 1. Chap. II.)

And Univerfally z÷÷y (by Prop. 1. Chap. II.) z✈

When the Dividend and Divifor are two Rational Quantities prefixt to two Surds of the fame kind; Divide the Rational Part of the Dividend by the Rational Part of the Divifor, as alfo the Surd Part of the Dividend by the Surd Part of the Divifor; and the two Quotients connexed together with (or fometimes without) the Sign x is the Quotient defired.

Se

I

a

d

Also ab xcd÷bx cd|" is=ax 1]=a.

aisa:

**Note, that a÷√-a is = √-a: Also a÷-√

Likewife a√-a is ——√—a: Also a÷—√—ais=√—a: Again bca-baca.

CHA P. V. and VI.

Addition and Subtraction of Simple Durds.

ule.

DIvide the propos'd Surds by their greatest common Divifor, and if the Quotients be Ratic al (that is to fay, If the propos'd Surds be Commenfurable) Multiply the Sum of the Quotients by the faid greatest common Divifor, and the Product fhall be the Sum of the Surds propos'd: Or Multiply the Dif ference of the Rational Quotients by the faid greatest common Divifor, and the Product fhall be the Difference of the two Surds propos'd.

Examples.

1. Let it be required to add 18 to

32.

Their greatest common Divifor is 8, by which each of the propos'd Surds being Divided, the Quotients are 1 and 4, which are Rational Numbers, and equal to 1 and 2; the Sum of which is 3, which being Multiplyed by the greatest common Divifor (8) produces 3√8 (= √9 × √8=√ 72) which is = √8+√32.

: And, these reduc'd to a common Denominator, are equal

which each of them being Divided, the Quotients will be

I'

2

by

and c4 which are Rational Quantities, and equal to b and c2: The Sum of which Quotients is b+c2, which, being Multiplyed

by the greatest common Divifor, will give b2+c3:x

n

3. I demand the Sum of bd""" and be?

cb3

Their greatest common Divifor is b", by which each of them being Divided, the Quotients are and which are Rational Quantities, and equal to d and c; the Sum of which is dc, which being Multiplyed by the greatest common Divisor, the Product is :d+c:xb", which is = Ed"j" +

I

I

This laft Example, being Universal, will ferve to Demon ftrate our Rule.

Suppose |= a; then b♫ is = a×

d = ad;

And be" is a × cac : But ad+ac=d+cx a is Sum of the two propos'd Quantities; confequently (Sub

I

flituting b for a) dxb is equal to the Sum of the two propos'd Quantities. W. W. D.

n

In like Manner, if it were required to Subtract from be the Remainder will be found (by our Rule) to be c—1xb.

b

I

I

bc"|" in o

I

Or, if it were required to find the Value of ther Terms, it will (by our Rule) be foundIcx b " bn. When the Surds are Incommenfurable, neither their Sum nor Difference can be expreft by any fingle Root; but they are to

be