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P R O E M.
AVING formerly wrote a small Traet 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 so 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 bere fully bandled at large.
The Principal Rules are addition, Subtraction, spultiplication, Dibision, 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 fubftituted 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 Speciosas, or Specious Compu. tation,
CH A P. I. Concerning the goethod of Hoting down Duantities ;
and Tracing their Steps, & c.
Sect. 1. Of Notation.
, is various,
I 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, l. U. or y.
The given Quantities are represented by the small Consonants, b. c. d. f. 8. & 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 ii's Number to it; as za stands for a taken three times, or three times a, and 76 stands for seven times b, 06.
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 b is 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 presume the Reader to be very well acquainted with. To them must be here added these three more;
Involution. l'iz. w the 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. c. 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 a+b, 2-b, or ab-do, &c. they are called Compound whole Quantities
And when Quantities are expressed or set down like Vulgar
atb abtdc Fra&tions, Thus
&c. they are d
bcalled Fractional or broken Quantities.
The Sign where with Quantities are connected, always belongs to that Quantity which imrnediately 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 tb-d may ftand thus b-d + a, or thus adtb, ord 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 tbefore it. As a is ta, or bmd is +6d, &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 Necessity of prefixing it before that Quantity to which it belongs, wherever the Quantity stands.
Of tracing the Steps used in bringing
Duantities to an Equation. Method of tracing the Stepsused in bringing the Quantities concerned in any Question to an Equation, is best performed by regiftring 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.
For Instance: If it be required to set down, and register the Sum of the two Quantities, a and b, the Work will stand, Thus112 First set down the proposed Quantities, a and b, 215 over-against the Figures 1, 2, in the small Co
lumn, (which are here called Steps) and against 3 1+2/3/1 + b (the third Step) set down their Sum, viz. a to b. Then againit that third Step, fet down 1+2 in the Margin; which denotes that the Quanticies against the first and second Sceps are added together, and that those in the third Step are their Sum.
To illustrate this in Numbers, fuppose a=9 and b= 6. Then it will be, Thus i la
216 = 6 +232 +6=2+6=15 being the sum of 9 and 6.
Again, If it were required to set down the Difference of the fame two Quantities; then it will be, Thusila
216 = 6
Or if it were required to set down their Product.
216 = 6
Note, Letters set or joined immediately together (like a Iord) fignify the Rectangle or Product of those Quantities they represent; as in the last Example, wherein ab=54 is the Produkt 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. Those Quantities, that are equal to one and the same Thing, are equal to one another.
Note, I advise the Learner to get these five Axioms perfeélly by Heart.
These Things being premised, and a perfect Knowledge of the Signs and their Significations being gained, the young Algebrais! 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,
C H A P.
CH A P. II.
Concerning the Sir Principal Rules of algebraick
Arithmetick, of whole Quantities.
1 a 121 a
Cafe 1. If the Quantities are like, and have like Signs; add
15bc Thus! \Exam. 5. Exam. 6. Exam. 7.
13a+ 5632— 5 6 ab to 12
12 2a+ 7b2a-76 3 ab + 24 1+253 54 +126152-1261 9 ab + 36
The Reason of these Additions is evident from the Work of Common Arithmetick. For suppose 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 suppose -a, to represent the Want or Debt of one Crown, 10 which if another Want or Debt of one Crown be added, the Sum muft nuds be the Want or Debt of two Crowns, or - 2a; as in Example 2. And so for all the rest.
Case 2. If the Quantities are alike, and have unlike Signs; subtraat tbe 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. sa
- 3a + 3a 6 bc i+231 + 2a 1
70-56-8ab-7bc +15 2-5a +70+ 12 ab +7bc24 1+2 3 29 +261 4ab9