proceed in extracting the Root out of any fingle Power, how high foever 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 eafier performed: As will appear in the next Part.

I did intend to have here inferted the whole Bufinefs of Intereft and Annuities; but finding that it would require too large a Difcourfe, to fhew the Grounds and Reasons of the feveral Theorems ufeful therein, I have therefore referved that Work for the Clofe of the next Part. Neither indeed can the raifing of thofe Theorems be fo well delivered in Words, as by an Algebraick Way of arguing; which renders them not only much fhorter, but alfo plainer and easier to be understood."

I have alfo omitted that Rule in Arithmetick, ufually called the Rule of Pofition, or Rule of Falfe: Because all fuch Questions, as can be answered by that gueffing Rule, are much better done by any one who hath but a very fmall fmattering of Algebra. I fhall therefore conclude this Part of Numerical Arithmetick; and proceed to that of Algebraick Arithmetick, wherein I would advife the young Learner not to be too hafty in paffing from one Rule to another, and then he will find it very eafy to be attained.

AN

143

AN

INTRODUCTION

TO THE

Mathematicks.

H

PART II.

PROEM.

AVING formerly wrote a fmall Tract of Algebra, perhaps it may feem (to fome) very improper to write again upon the fame Subject; but only (as the ufual Cuftom is) to

have referred my Reader to that Tract. However, becaufe the following Parts of this Treatife are managed by an Algebraick Method of arguing; which may fall into the Hands of those who bave not feen that Tract, 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 fo he may not be at a Lofs as he proceeds farther on: Befides, what I formerly wrote was only a Compendium of that which is here fully handled at large.

The Principal Rules are Addition, Subtracion, Multiplication, Divifion, Involution, and Evolution, as in common Arithmetick but differently performed; and therefore fome call ic Algebraick Arithmetick. Others call it Arithmetick in Specie, because all the Quantities concerned in any Question, remain in their fubftituted Letters (howfoever 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 Speciofa,, or Specious Compu

tation.

CHAP.

Concerning the ethod of Noting down Quantities ; and Tracing their Steps, &c.

ΤΗ

Sect. 1. Of Dotation.

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

e. u. or y.

The given Quantities are reprefented by the fmall Confonants, b. c. d. f. g. &c.

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

When any Quantity (either given or fought) is taken more than once, you must prefix it's Number to it; as 3a stands for a taken three times, or three times a, and 76 ftands for feven 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 fuppofed or understood to have an Unit prefixed to it; as a is 1a, or b is 1b, &c.

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

G

Involution.

Viz. Ս the Sign of Evolution, or extracting Roots.

Irrationality, or Sign of a Surd Root. All Quantities that are expreffed by Numbers only (as in Vulgar Arithmetick) are called Abfolute Numbers.

Thofe Quantities that are reprefented by fingle 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 reprefented by different or unlike Letters, are connected together by the Signs (or); as a+b, a−b, os abde, &c. they are called Compound whole Quantities.

And

And when Quantities are expreffed or fet down like Vulgar rab+de, &c. they are

a

Fractions, Thus, or

a+b
d

or

called Fractional or broken Quantities.

b. с

c

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

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

TH

HE Method of tracing the Steps, ufed in bringing the Quantities concerned in any Question to an Equation, is best performed by regiftring the feveral Operations with Figures and Signs placed in the Margin of the Work, according as the feveral Operations require; being very ufeful in long and tedious Operations. For Inftance: If it be required to fet down, and register the Sum of the two Quantities, a and b, the Work will stand, Thus|1|2

25

First fet down the propofed Quantities, a and b, over-against the Figures 1, 2, in the fmall Column, (which are here called Steps) and against 3 1+23+6 (the third Step) fet down their Sum, viz. a + b. Then against that third Step, fet down 1+2 in the Margin; which denotes that the Quantities against the first and fecond Steps are added together, and that thofe in the third Step are their Sum. To illuftrate this in Numbers, fuppofe a 9 and 66. Then it will be,

1+234 +6=9+6=15 being the Sum of 9 and 6.

U

Again,

Again, If it were required to fet down the Difference of the fame two Quantities; then it will be,

Thus 19

266

I-234-6=9-6=3 the Diff. between 9 and 6.

Or if it were required to fet down their Product. Then it will be,

Thus a=9 266.

1 x 234 x b or ab9x654 the Prod. of 9 into 6.

&c.

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

=

Arioms.

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

Note, I advife the Learner to get thefe five Axioms perfectly by Heart.

Thefe Things being premised, and a perfect Knowledge of the Signs and their Significations being gained, the young Algebraift may proceed to the following Rules. But firft 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 fhould 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.