Sect. 2. Multiplication of Surd Quantities. Cafe 1. WHEN the Quantities are puie surds of the fame Kind; multiply them together, and to their Product prefix

Cafe 2. If Surd Quantities of the fame Kind (as before) are joined to rational Quantities, then multiply the rational into the rational; and the Surd into the Surd, and join their Products together.

EXAMPLES.

I d √ b c
236√ a

5 c d √ ba+da

zavca

15 vab 5√ d

1 x 23 3 db √ be a 15 eday beaa + d ca a 75 √ ab d

Sect. 6. Divifion of Surd Quantities.

Cafe 1. WHEN the Quantities are pure Surds of the fame Kind, and can be divided off, (viz. without leaving a Remainder) divide them, and to their Quotient prefix their radical

EXAMPLES.

I√ba√bcaa + d ca a✔ aa aa

Sign.

1÷23√ a √ ba+da

Cafe 2. If Surd Quantities, of the fame Kind, are joined to rational Quantities; then divide the rational by the rational, if it can be, and to their Quotient join the Quotient of the Surd divided by the Surd with it's firft radical Sign.

Note, If any Square be divided by it's Root, the Quotient will

be it's Root.

EXAMPLES.

bb +2bc+cca a aa—2bba a + b b b b √bb +2 b c + ccly as 2bba a+b+

1-23│√ a│√bb + 2 b c + c c √ √ aa—2bbaa+b+

Sect. 4. Involution of Surd Quantities.

Cafe 1.WHEN the Surds are not joined to rational Quantities; they are involved to the fame Height as their Index denotes, by only taking away their radical Sign,

Cafe 2. When the Surds are joined to rational Quantities; involve the rational Quantities to the fame Height as the Index of the Surd denotes; then multiply thofe involved Quantities into the Surd Quantities, after their radical 9ign is taken away, as before.

I

Ia:34bc3d: 34 aa+bb

2 a a ab c 27dddaa +27 d d dbb ddda a ab

The Reason of only taking away the radical Sign, as in Cafe 1. is cafily conceived, if you confider that any Root being involved into itfelf, produces a Square, &c. And from thence the Reafon of thofe Operations performed by the fecond Cafe may be thus ftated.

Suppose b√ a =x. Then vaper Axiom 4. and both Sides of the Equation being equally involved, it will be a= x x Then multiplying both Sides of the Equation into bb, iṭ becomes b bax per Axiom 3. Which was to be proved.

bb

x

Again, Let 5 dvcax: Then ca=

and ca

5 ď

xx

25 dd

Alfo from hence it will be eafy to deduce the Reason of multiplying Surd Quantities, according to both the Cafes.

Suppose { 16=%} Example 1. Cafe 1.

2√a=xS

I

3

bazz xx. per Axiom 2.

6√ ba=zx. which was to be proved.

For

3 Ꮟ d

36ď

from what is proved above.

5x3bd63bdbca = zx, &c. for the reft..

Divifion being the Converfe to Multiplication, needs no other Proof.

CHAP. V.

Concerning the Nature of Equations and how to prepare them for a solution.

WHEN any Problem or Question is proposed to be analytically refolved; it is very requifite that the true Design or Meaning thereof, be fully and clearly comprehended (in all it's Parts) that fo it may be truly abftracted from fuch ambiguous Words as Questions of this Kind are often difguifed with; otherwife it will be very difficult, if not impofiible, to ftate the Queftion right in it's fubftituted Letters, and ever to bring it to an Equation by fuch various Methods of ordering thofe Letters as the Nature of the Questions may require.

Now

Now the Knowledge of this difficult Part of the Work is only to be obtained by Practice, and a careful minding the Solution of fuch leading Questions as are in themfelves very eafy. And for that Reafon I have inferted a Collection of feveral Queftions; wherein there is great Variety.

Having got fo clear an Understanding of the Queftion propofed, as to place down all the Quantities concerned in their due Order, viz. all the fubftituted Letters, in fuch Order as their Nature requires; the next Thing must be to confider whether it be limited or not. That is, whether it admits of more Anfwers than one. And to discover that, obferve the two following Rules.

RULE I.

When the Number of the Quantities fought exceed the Number of the given Equations, the Question is capable of innumerable Answers.

EXAMPLE.

Suppose a Question were propofed thus; there are three fuch Numbers, that if the firft be added to the fecond, their Sum will And if the second be added to the third, their Sum will be 46. What are thofe Numbers ?

Let the three Numbers be reprefented by three Letters, thus, call the first, the second e, and the third y.

Then

sa+c=22 2e+y=46 +46

{

according to the Question.

Here the Number of Quantities fought are three; a, e, y, and the Number of the given Equations are but two. Therefore this Question is not limited, but admits of various Answers; because for any one of those three Letters you may take any Number at Pleasure, that is less than 22. Which with a little Confideration

will be very easy to conceive.

RULE 2.

When the Number of the given Equations (not depending upon one another) are just as many as the Number of the Quantities fought; then is the Queftion truly limited, viz. each Quantity fought bath but one fingle Value.

As for inftance, let the aforefaid Queftion be proposed thus. There are three Numbers (a, c, and y, as before) if the firft be added to the fecond, their Sum will be 22; if the fecond be added

to

J

to the third, their Sum will be 46; and if the first be added to the third, their Sum will be 36. What are the Numbers? That is, a +e = 22. e+y=46. and a+y=36. Now the Queftion is perfectly limited, each fingle Quantity having but one fingle Value, to wit a 6, e16, and y = 30.

N. B. If the Number of the given Equations exceeds the Number of the Quantities fought; they not only limit the Question, but oftentimes render it impoffible, by being proposed inconfiftent one to another.

Having truly ftated the Queftion in it's fubftituted Letters, and found it limited to one Anfwer (or at least so bounded as to have a certain determinate Number of Anfwers) then let all those subftituted Letters be fo ordered or compared together, either by adding, fubtracting, multiplying, or dividing them, &c. according as the Nature of the Queftion requires, until all the unknown Quantities except one, are caft off or vanifhed; but therein great Care must be taken to keep them to an exact Equality; and when that unknown Quantity, or fome Power of it (as Square, Cube, &c.) is found equal to thofe that are known; then the Question is faid to be brought to an Equation, and confequently to a Solution, viz. fitted for an Anfwer.

But no particular Rules can be prescribed for the cafting off, or getting away Quantities out of an Equation; that Part of the Art is only to be obtained by Care and Practice. And when that is done, it generally happens fo, that the unknown Quantity which is retained in the Equation, is fo mixed and entangled with thofe that are known; that it often requires fome Trouble and Skill to bring it (or it's Powers, &c.) to one Side of the Equation, and those that are known to the other Side; (ftill keeping them to a juft Equality) which the ingenious Mr Scooten in his Prin cipia Mathefeos Univerfalis, calls Reduction of Equations.

The Bufinefs of reducing Equations (as of most, if not all Algebraick Operations) is grounded and depends upon a right Application of the five Axioms propofed in Page 146, and therefore, if those Axioms be well understood, the Reafon of fuch Operations must needs appear very plain, and the Work be eafily performed; as in the following Sections.