Having found the Square Root of the given Refolvend, I proceed to extract the Cube Root of that Square Root.

That is, of 381078125

343 the nearest Cube, it's Root is 700.

Then 700 x 3 = 2100) 38078125 (18161

(36)

Hence I find 725 to be the Square cubed Root required; as may easily be tried by involving it to the fixth Power. That is, 725 × 725 × 725 × 725 × 725 × 725, will be found = 145220537353515725 the given Refolvend.

Sect,

Sect. 7. To Extract the Root of the feventh Power.
HAving pointed the given Refolvend, as it's Index denotes, viz.

into Periods of feven Figures, feek out fuch a Number of the feventh Power, by the Table of Powers, as comes nearest to the first Period of the Refolvend; whether it be greater or leffer, calling it's refpective Root more than Juft, or less than Juft, annexing it's proper Number of Cyphers &c. as in the Cube and Surfolid.

Then find the Difference between the given Refolvend, and that Number of the feventh Power (found by the Table of Powers) by fubtracting the leffer from the greater.

Next find the Surfolid or fifth Power of that Root with it's annexed Cyphers (which you may alfo do by the Table of Powers) and multiply that Surfolid Number with 7, the Index of the given Refolvend; that Product must be a Divifor, by which the forefaid Difference must be divided, that so it may be depreffed to a Square, to be pointed, &c. as before in the Cube, &c. then make the first Root, without it's Cyphers, a Divifor; working with it and the new Refolvend (as before) only here you must increase, or diminifh the Divifor with thrice the Quotient Figure*.

Example. What is the fecond Surfolid Root, or that of the seventh Power,

of 382986553955078125 the Refolvend pointed. -2187 the neareft of the feventh Power.

164286553955078125 their Difference.

The firft Root is 300 being less than Juft, and the fifth Power of 300 is 2430000000000, which being multiplied with 7 is 17010000000000 for a Divifor, by which the aforefaid Dif ference must be divided; which contracted may ftand thus, 1701) 16428655 (9658,23 &c.

300

First Root +3×20=+ 60

1 Divifor

60+3x05=+75

2 Divifor

325 the true Root required.

(before,

283 the Remainder to be rejected, as

That is, by twice adding or fubtracting the triple Quotient Figure, as was done with the double Quotient Figure for the Root of the fifth Power, page 136; and the Single Quotient Figure for the Cube Root, page 131.

T2

Hence

"

Hence I have found 325 to be the true Root required, that is, the true Root of the feventh Power.

I think it needlefs to proceed farther; viz. to infert Examples of higher Powers. For if what is already done be well understood, it will be easy to conceive how to proceed in extracting the Root of any fingle Power how high foever it be (for the Method is general and alike in all Powers) due Regard being had to their Indices; and to the first fingle Side or Root. That is, whether it be More, or Lefs than Juft, &c.

Yet methinks I hear the young Learner fay, it is poffible to follow the Directions and Examples, as they are here laid down; but still here is not the Reafon why they are fo, and fo, performed; and why there fhould be a Remainder left after the Root is found; viz. when the given Refolvend hath a true Root of it's Kind.

It is true, the Reafons of thefe are not here laid down; neither indeed can they be rendered fo plain and intelligible by Words, as by an Algebraick Procefs, from whence the Theorems or Rules here given, had their firft Invention; as fhall be fhewed in the next Part, when I come to treat of refolving compounded or adfected Equations; however, take this fhort and general Account of this Method.

This, and all other of the new Methods of Converging Series (as they are called) are very different from the former (and ftill common) Methods of extracting Roots, which require the firft fingle Side or Root of the first Period (in any Refolvend) to be taken exactly true, and then by involving, and other tedious Ways of ordering it, there is formed a Divifor; which helps to grope out by Trials a fecond Figure in the Root. And fo proceed on from Point to Point; ftill repeating the whole Work for every fingle Figure that comes into the Root. And if by Chance there be a Miftake or Error committed in any one Figure (as it is poffible there may) it fpoils the whole Procefs, which must then be wholly begun anew, or at least from that Part of it where the Error firft entered.

But the Nature and Defign of the Method which I have here laid down is quite otherwife; it being fo contrived, as to gradually leffen the Difference betwixt any propofed Power, and the like Power of another Number affumed; viz. it leffens that Difference until it is either quite vanquished, or becomes fo infinitely small as to be infignificant.

-Therefore when any Number is proposed to have it's Root extracted; it is here required to take the next nearest Root of the firft Period in the Refolvend; that fo the Difference betwixt the

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 so prepared, that by plain Divifion (comparatively) there will arife fuch Quotient Figures as will both correct and increase the firft Root to three Places of Figures at leaft, fometimes to four, or five Places of Figures; according as the faid firft 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 Excefs or Defect in the Root, and to difcover 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 fo 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 vanquished; 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 laft 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, muft needs help the Learner to form such 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 inferted the whole Bufinefs of Intereft 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 Ciofe 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 fhorter, 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 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 easy to be attained.

AN