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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 Trait. 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 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: Bifides, what I formerly wrote was only a Compendium of that which 'is here fully handled at large.

The Principal Rules are addition, Subtraaton, 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 Speciosan, or Specious Compistation,

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els bandes Rules are you and me and the

THccording to every pract, and reprefenman (23) and if

C H A P. I. Concerning the method of Noting down Duantities ;

and Tracing their Steps, &c.

Sect. 1. Of Dotation. THE Method of noting down Letters for Quantities, is various,

according to every one's Fancy; but I shall here follow the same as in my former Tract, and represent the Quantity sought (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. 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 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 ia, or bis 1b, &c.

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

Involution.
Miz. { w Sthe Sign of 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. 6. d. &c. or by several 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 -); as atb, a-b, or ab--ade, &c. they are called Compound whole Quantities.

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And when Quantities are expressed or fet down like Vulgar

atb . abt de Fractions, Thus īs

, &c. they are

di ba? called Fractional or broken Quantities.

The Sign wherewith Quancities are connected, always belongs to that Quantity which immediately follows it ; and therefore all the Quancities concerned in any Question, may stand in any order at Pleasure, viz. the most convenient for the next Operation. As a tb-d may stand thus b-dta, or thus a d+b,

- tatb &c. these being still the same, 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 ta, or b-distband, &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 Neceflity of prefixing it before that Quantity to which it belongs, wherever the Quantity stands.

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Again, If it were required to set down the Difference of the same two Quantities; then it will be, Thuslila = 9

216 = 6 I-213a - b=9-6=3 the Diff. between 9 and 6.

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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. 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 Algebraift 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.

CHĄ P.

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