proceed in extracting the Roof out of any fingle Power, how high soever it be; without the Help of an Algebraick Theorem. Not, but when that comes to be once understood; the Work will be much readier and easier performed: As will appear in the next Part.

I did intend to have here inserted the whole Business of Interest and Annuities; but finding that it would require too large a Difcourse, to shew the Grounds and Reasons of the several Theorems ufeful therein, I have therefore reserved that Work for the Close of the next Part. Neither indeed can the raising of those Theorems be so well delivered in Words, as by an Algebraick Way of arguing; which renders them not only much shorter, but also plainer and easier to be understood.

I have also omitted that Rule in Arithmetick, usually called the Rule of Position, or Rule of False: Because all fuch Questions, as can be answered by that guessing Rule, are much better done by any one who hath but a very small smattering of Algebra. I shall therefore conclude this Part of Numerical Arithmetick; and proceed to that of Algebraick Arithmetick, wherein I would advise the young Learner not to be too hasty in paffing from one Rule to another, and then he will find it very easy to be attained.

AN

143

AN

INTRODUCTION

AVING formerly wrote a small Tract of Algebra, perhaps it may seem (to fome) very improper to write again upon the fame Subject; but only (as the usual Custom is) to have referred my Reader to that Tract. However, because the following Parts of this Treatise are managed by an Algebraick Method of arguing; which may fall into the Hands of those who have not seen that Traft, or any other of that Kind; I thought it convenient to accommodate the young Geometer with the first Elements, or Principal Rules, by which all Operations in this Art are performed; that so he may not be at a Loss as he proceeds farther on: Befides, what I formerly wrote was only a Compendium of that which is bere fully handled at large.

The Principal Rules are Addition, Subtracion, Multiplication, Division, Involution, and Evolution, as in common Arithmetick but differently performed; and therefore some call it Algebraick Arithmetick. Others call it Arithmetick in Specie, because all the Quantities concerned in any Question, remain in their substituted Letters (howsoever managed by Addition, Subtraction, or Multiplication, &c.) without being destroyed or changed into others, as Figures in common Arithmetick are.

Mr Harriot called it Logistica Speciela,, or Specious Compu

tation.

CHAP. I.

Concerning the Method of Moting down Duantities;

and Tracing their Steps, &c.

Sect. 1. Of Motation.

THE Method of noting down Letters for Quantities, is various, according to every one's Fancy; but I shall here follow the fame as in my former Tract, and represent the Quantity fought (be it Line or Number, &c.) by the small (a,) and if more Quantities than one are fought, by the other small Vowels,

e. u. or y.

The given Quantities are represented by the small Confonants, b. c. d. f. g. &c.

And for Distinction fake, mark the Points or Ends of Lines in all Schemes, with the capital or great Letters, viz. A. В. C. D. &c.

When any Quantity (either given or fought) is taken more than once, you must prefix it's Number to it; as za stands for a taken three times, or three times a, and 76 stands for seven times b, &c.

All Numbers thus prefixt to any Quantity, are called Coefficients or Fellow-Factors; because they multiply the Quantity; and if any Quantity be without a Co-efficient, it is always fupposed or understood to have an Unit prefixed to it; as a is 1a, or bis 1b, &c.

The Signs by which Quantities are chiefly managed, are the fame, and have the fame Signification, with those in the first Part, page 5. which I here prefume the Reader to be very well acquainted with. To them must be here added these three more;

Viz.

{}

the Sign of

{

Involution.
Evolution, or extracting Roots.
Irrationality, or Sign of a Surd Root.

All Quantities that are expressed by Numbers only (as in Vulgar Arithmetick) are called Absolute Numbers.

Those Quantities that are represented by single Letters, as, a. b. c. d. &c. or by feveral Letters that are immediately joined together; as ab. cd. or 7bd. &c. are called Simple or Single whole Quantities.

But when different Quantities represented by different or unlike Letters, are connected together by the Signs (or); 35 a+b, a-b, or ab-dc, &c. they are called Compound whole Quantities.

And

And when Quantities are expressed or fet down like Vulgar

a

or

ab+dc
b-c

a+b Fractions, Thus, or d' called Fractional or broken Quantities.

&c. they are

The Sign wherewith Quantities are connected, always belongs to that Quantity which immediately follows it; and therefore all the Quantities concerned in any Question, may stand in any order at Pleasure, viz. the most convenient for the next Operation. As a+b-d-may stand thus b-d+a, or thus ad+b, or-d+a+b &c. these being ftill the fame, tho' differently placed.

That Quantity which hath no Sign before it (as generally the leading Quantity hath not) is always understood to have the Sign + before it. As a is +a, or b-d is +b-d, &c. for the Sign + is the Affirmative Sign, and therefore all leading or Positive Quantities are understood to have it, as well as those that are to be added.

But the Sign - being the Negative Sign, or Sign of Defect, there is a Neceffity of prefixing it before that Quantity to which it belongs, wherever the Quantity stands.

Sect. 2. Of tracing the Steps used in bringing Duantities to an Equation.

THE Method of tracing the Steps, ufed in bringing the Quantities concerned in any Question to an Equation, is best performed by registring the several Operations with Figures and Signs placed in the Margin of the Work, according as the several Operations require; being very useful in long and tedious Operations.

25

For Instance: If it be required to set down, and register the Sum of the two Quantities, a and b, the Work will stand, Thus 1/2 First set down the proposed Quantities, a and b, over-against the Figures 1, 2, in the small Co- lumn, (which are here called Steps) and against 3 +2/32+6 (the third Step) set down their Sum, viz. a +6. Then against that third Step, set down 1+2 in the Margin; which denotes that the Quantities against the first and second Steps are added together, and that those in the third Step are their Sum. To illustrate this in Numbers, suppose a = 9 and 6=6.

Then it will be,

Thus 1 a

26

=

= 6

1+23+6=9+6= 15 being the Sum of 9 and 6.

U

Again, Again, If it were required to set down the Difference of the

same two Quantities; then it will be,

Thus 1/a = 9

1-23a-b=9-6=3 the Diff. between 9 and 6.

Or if it were required to set down their Product.

Then it will be,

Thus 1/a = 9

266

Ix23 axbor ab = 9 x 6 = 54 the Prod. of 9 into 6.

Note, Letters fet or joined immediately together (like a Word) fignify the Rectangle or Product of those Quantities they represent; as in the last Example, wherein ab=54 is the Product of a=9 and b = 6. छ.

Axioms.

1. If equal Quantities be added to equal Quantities, the Sum of these Quantities will be equal.

2. If equal Quantities be taken from equal Quantities, the Quantities remaining will be equal.

3. If equal Quantities be multiplied with equal Quantities, their Products will be equal.

4. If equal Quantities be divided by equal Quantities, their Quotients will be equal.

5. Those Quantities, that are equal to one and the fame Thing, are equal to one another.

Note, I advise the Learner to get these five Axioms perfectly by Heart.

These Things being premised, and a perfect Knowledge of the Signs and their Significations being gained, the young Algebraist may proceed to the following Rules. But first I must make bold to advise him here, (as I have formerly done) that he be very ready in one Rule before he undertakes the next.

That is, He should be expert in Addition, before he meddles with Subtraction; and in Subtraction, before he undertakes Multiplication, &c. because they have a Dependency one upon another.

CHAP.