given Refolvend, and the Homogeneal Power (viz. the like Power) of the Root thus taken, may be lefs either in Excefs, or Defect. Which Difference being reduced, or depreffed lower, becomes fo prepared, that by plain Divifion (comparatively) there will arife fuch Quotient Figures as will both correct and increase the first Root to three Places of Figures at least, fometimes to four, or five Places of Figures; according as the faid first Difference happens to be more or lefs (of which you may have observed Inftances): But yet there will be a Remainder left, and perhaps an Excefs or Defect in the Root fo increased, viz. in the laft Figure of it.

Now to rectify the faid Excess or Defect in the Root, and to discover whether the given Refolvend be a true Figurate Number, or not: That is, whether it have a true Root of it's kind; it will be neceffary to make a fecond Operation; by taking the Root so increased, and proceeding with it and the given Refolvend, in all refpects as in the firft Work (like to the third Example of extracting the Cube Root); I fay, if the given Refolvend have a true Root, it will appear at this fecond Operation, and all the aforefaid Differences, &c. will be vanished; provided the Root required is not to have more than three (or four) Places of Figures in it.

But if the Root be to have more than three Figures in it; or, that the given Refolvend prove to be a Surd Number; then there will be a Difference as before; which will afford Quotient Figures to rectify and increase the Root last taken, to three Times as many Places of Figures, as it had at the Beginning of that second Operation. As you may fee in the aforefaid Example 3. of the Cube Root; wherein that Root is increased to twelve Places of Figures at two Operations; which if it were to be extracted the Old (and ftill Common) way, it would require at least forty times the Number of Figures I have here used.

Again, if there chance to be a Miftake committed in any Operation performed by the Method here laid down, that Miftake will not deftroy the precedent Work, but will be rectified in the next Operation, although it were not difcovered before. And thus you may proceed on to a third Operation, which will afford 27 Places of Figures in the Root, &c. with very little Trouble, if compared with former Methods.

The brief Account, which I have here given (by Way of Explaining the Nature of this Method of extracting Roots) being well confidered and compared with the feveral Operations of the foregoing Examples, muit needs help the Learner to form fuch an Idea of it, that he cannot (I prefume) but understand how to

proceed

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 inserted the whole Business of Interest and Annuities; but finding that it would require too large a Difcourfe, to fhew the Grounds and Reafons of the feveral Theorems ufeful therein, I have therefore referved that Work for the Close of the next Part. Neither indeed can the raifing of those Theorems be fo 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 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 small fmattering of Algebra. I fhall 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 hafty in paffing from one Rule to another, and then he will find it very ealy to be attained.

AN

A N

143

INTRODUCTION

TO THЕ

Mathematicks.

H

PART II,

PROEM.

AVING formerly wrote a small Tract of Algebra, perhaps it may feem (to fame) 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 bere fully handled at large.

The Principal Rules are addition, Subtraction, Multiplication, Division, Involution, and Evolution, as in common Arithmetick, but differently performed; and therefore fome call it Algebraick Arithimerick. Others call it Arithmetick in Specie, because all the Quantities concerned in any Queftion, remain in their fubftituted Letters (howfoever y 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 Computation.

С НА Р.

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

Sect. 1. Of otation.

THE Method of noting down Letters for Quantities, is various, according to every one's Fancy; but I thall here follow the fame as in my former Tract, and reprefent 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 represented 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 76 ftands for seven times b, &c.

All Numbers thus prefixt to any Quantity, are called Coefficients or Fellow-Factors; becaufe 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 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. WJ

the

Involution.

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 7ld. &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 ab-de, &c. they are called Compound whole Quantities.

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

Fractions, Thus .. or

d

or

called Fractional or broken Quantities.

b

&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 a-d+b, ord+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 underflood 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 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 ftands.

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

HE Method of tracing the Steps, ufed in bringing the Quantities concerned in any Queftion 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 stand, Thus la

26

Firft 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+2a + b (the third Step) fet down their Sum, viz. a+b. Then againit 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 b=6. Then it will be,

Thus a <=9 26 =6

1+23a+6=9+615 being the Sum of 9 and 6.