duct of the Means.

And the former Product is the latter, which proves, in some measure, our Rule.

3. Find whether and be Commenfurable, as allo their mutual Proportion.

The greatest Common Divisor of b" andc"b)" is=

יה

by

which each being Divided, the Quotients are and; But

I and are Rational Quantities, and equal to 1 and c refpectively; confequently the given Surds are Commensurable, and

::I

IF

CHAP. II.

Multiplication of Simple Surds.

Kule.

F the given Surds be not of the fame kind; that is, if they have different Indices: reduce them to one and the least kind; that is, to their least common Index, (by the 1st Propofition of the laft Chap.) then Multiply them by one another, without their Indices; and laftly annex the faid leaft common Index to the Product. So this new Root fhall be the Product fought.

a" 2"

Again 213 × 3]3 is (by the 1st Proposition of the last Chaptery

is (by the 1ft Propofition of the laft Chapter)

ft

And univerfallyxis (by the 1st Propofition of the laft = X× -= -.

Chapter) =

Scholia.

1

1. When a Surd is to be Multiplyed by a rational Quantity, it will be fometimes convenient to Connex them with the Sign X, fo the Product of a and may be writ thus 4 x b or thus, a"√b.

2. When two Rational Quantities are joyn'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 joyn'd together with the Sign × is the Product required. So a x xbd] = (for a × b\" is =

x xis = ac xbd (for ax

s=c"d"; But a′′b\"

ly ac x bd").

a"b" and c X

bl X "da"
a"c" bd" which is manifeft-

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

alfo a rais'd to the mth Power is

You have Examples enough of this kind, in Part V.

CHAP.

CHA P. IV.

Division of Simple Surds.

Rule.

F the Surds be of the fame kind, Divide the Dividend by the Divifor, without their Indices, and over the Quotient place the common Index, and this new Root is the Quotient fought; But if they be not of the fame kind, firft reduce them to the fame kind (by Chap. 2. Propofition 1.) and then proceed in manner aforefaid.

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

Allo abj→ cal

And univerfally is (by Propofition 1. Chapter 2.)

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 Diwifor: and the two Quotients connected together with (or fome rimes without) the Sign x is the Quotient defired.

CHAP. V. and Vi.

Addition and Subftration of Simple Surds.

D' Ivide the given Surds by their greateft common Divifor, and if the Quotients be Rational (that is to fay, if the given Surds are Commenfurable) Multiply the Sum of the faid Quotients by the faid greatest common Divifor, and the Product shall be the Sum of the Surds propos'd; or Multiply the difference of the faid Rational Quotients by their faid greateft common Divifor, and the Product shall be the difference of the two, Surds propos'd.

Examples norem

1. Let it be required to add 8 to √32. Their greatest common Divifor is 8.

By which each of the given Surds being Divided, the Quotients arer and √4, which are Rational Numbers, and equal to 1 and 2.

The Sum of which Quotients is 3, which being Multiplyed by the greatest common Divifor (8) the Product is 3× √√√9, X√872, which is 832. and

2. I demand the Sum of

b

C

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

C8

and

I

I

4 whofe greatest common Divifor is By which each of them being Divided, the Quotients will be

b2

b and c, which are Rational Quantities, and equal to ¿? and c.

Μ

The

The Sum of which Quotients is ba+c2, which being Mul

?

3. I demand the Sum of 6"" and be

Their greateft 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 d+e; which being Multiplyed by the

greatest common Divisor, the Product is d+ex

is=bd|= +bc"|".

which

This laft Example being Univerfal, will ferve to demonstrate our

Rule.

Suppose

Then bď| is = 4 ×

bd|is ax =ad.
Exa

And bis = axc = ac.

But ad aed+c xa Sum of the two given Quantities.

Subftitute for a, and you'll have d+ex b

of the two given Quantities. W.W.D.

In like manner if it were required to Subftract

the Remainder will be found (by our Rule) to be Or if it were required to find the Value of

other Terms, it will (by our Rule) be found =

Sum

from bơ"|

in

WL