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143

ΑΝ

INTRODUCTION

TO THE

Mathematicks.

H

PART II.

PROEM.

AVING formerly wrote a small 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 have not feen that Tract, or any other of that Kind; I thought it convenient to accommodate the young Geometer with the firft 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, Subtraction, Multiplication, Dibifion, Involution, and Evolution, as in common Arithmetick but differently performed; and therefore fome call it Algebraick Arithmetick. Others call it arithmetick in Specie, because all the Quantities concerned in any Queftion, remain in their fubftituted Letters (howfoever managed by Addition, Subtraction, or Multiplication, &c.) without being deftroyed or changed into others, as Figures in common Arithmetick are.

Mr Harriot called it Logistica Speciola,, or Specious Compur CHAP.

tation.

CHAP. I.

Concerning the Method of Noting down Duantities; and Tracing their Steps, &c.

THE

Sect. 1. Of Motation.

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 small 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 ftands for a taken three times, or three times a, and 7b ftands 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 b is 1b, &c.

The Signs by which Quantities are chiefly managed, are the fame, and have the fame Signification, with thofe 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.

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SInvolution.

այ the Sign of Evolution, or extracting Roots.

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

Those 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, or abdo, &c. they are called Compound whole Quantities.

And

And when Quantities are expreffed or fet down like Vulgar a+b a b + de

a

Fractions, Thus- or

d

or

called Fractional or broken Quantities.

b6

&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 Queftion, may ftand in any order at Pleasure, viz. the moft convenient for the next Operation. As a+b-d may ftand thus b-d+a, 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 understood to have the Sign + before it. As a ista, 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.

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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 ftands.

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

THE Method of tracing the Steps, ufed in bringing the Quan tities concerned in any Question to an Equation, is beft 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 useful 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 ftand,

Thus 12 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 againft 3 1+232 +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 second 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,

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1+232+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 1a9

266

I-234 a―b=9—6=3 the Diff. between 9 and 6.

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

Thus 1a9

266

1 x 23a x b or a b≈ 9×6=54 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 reprefent; 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 thefe 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 these five Axioms perfectly by Heart.

Thefe Things being premifed, 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.

СНАР.

CHA P. II.

Concerning the Sir Principal Rules of Algebraick Arithmetick, of whole Duantities.

Sect. 1. Addition of whole Duantities.

AD

DDITION of whole Quantities admits of three
Cafes.

Cafe 1. If the Quantities are like, and have like Signs; add the Co-efficients or prefixt Numbers together, and to their Sum adjoin the Quantities with the fame Sign.

Exam. 1. Exam. 2. Exam. 3. Exam. 4.

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156
36

-7bc

-8bc

24 186

-15bc

Thus Exam. 5. Exam. 6. Exam. 7. 13a+56 39-5b6ab12 224+ 762a7b3ab +24 1+235a+1265a-1269 ab +36

The Reafon of thefe Additions is evident from the Work of Common Arithmetick. For fuppofe a, to represent one Crown, to which if I add one Crown, the Sum will be two Crowns, or 2a, as in Exam. 1.

Or if we fuppofea, to represent the Want or Debt of one Crown, to which if another Want or Debt of one Crown be added, the Sum muft needs be the Want or Debt of two Crowns, or 2a; as in Example 2. And fo for all the rest.

Cafe 2. If the Quantities are alike, and have unlike Signs; fubtract the Co-efficients from each other, and to their Difference join the Quantities with the Sign of the greater.

|Exam.8. Exam.9. Exam. 10.|Exam. 11.

+59
-3a+za

5 a

7 bc
6 be

9 abd +7abd

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Exam. 13.

7a-5b8ab7bc +15

2-5a+76 +12ab+7bc-24

1+23 24+26 4ab9

The

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