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27610268676757812,5 (381078125 + 8
272 38 )
4102 + 10 3805 3810)
7 2667245 38107) 3096226 +
8 3048592 381078 ) 4763 4757
38107805 3810781) 95269528 +
76215622 38107812) 1905390612,5
5 1905390612,5 381078125
(0) Having found the Square Root of the given Resolvend, 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 Eirst Root 7.. + 2
72.) 18161 +25 +25 144
725 2 Divisor 745) 3761
(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, 325 725 * 725 725 725 725, will be found = 1452 20537353515725 the given Resolvend.
Sect. 7. To Ertract the Root of the seventh Power. Having pointed the given Resolvend, as it's Index denotes, viz.
into Periods of seven Figures, seek out such a Number of the seventh Power, by the Table of Powers, as comes nearest to the first Period of the Resolvend; whether it be greater or lefser, calling it's respective Root more than Just, 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 Resolvend, and that Number of the seventh Power (found by the Table of Powers) by subtracting the leffer from the greater.
Next find the Sursolid or fifth Power of that Root with it's an. nexed Cyphers (which you may also do by the Table of Powers) and multiply that Sursolid Number with 7, the Index of the given Resolvend ; that Product must be a Divisor, by which the foresaid Difference must be divided, that so it may be depressed to a Square, to be pointed, &c. as before in the Cube, &c. then make the first Root, without it's Cyphers, a Divisor; working with it and the new Resolvend (as before) only here you must increase, or diminith the Divisor with thrice the Quotient Figure *.
Example. What is the second Surfolid Root, or that of the seventh Power,
of 382986553955078125 the Resolvend pointed. -2187 the nearest of the seventh 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 Divisor, by which the aforesaid Dif. ference must be divided; which contracted may stand thus, 1701) 16428655 (9658,23 &c. First Root
300 + 3 x 20 = + 60
300 1 Divisor 360) 9658 (+25 60+3x05=+75 72
325 = the true Root required. 2 Divisor
(before, 283 the Remainder to be rejedted, as
That is, by cwice adding or subtracting the triple Quotient Figure, as was done with the double Quotient Figure for the Root of the fifth Power, page 136 ; and ebe lingle Quotient Figure for thic Cube Root, page 132.
Hence I have found 325 to be the true Root required, that is, the true Root of the seventh Power,
I think it needless to proceed farther; viz. to insert 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 single 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 single Side or Root. That is, whether it be More, or Less than Just, &c.
Yet methinks I hear the young Learner say, it is possible to follow the Directions and Examples, as they are here laid down; but still here is not the Reason why they are so, and so, performed ; and why there should be a Remainder left after the Root is found; viz. when the given Resolvend hath a true Root of it's Kind.
Ic is true, the Reasons of these are not here laid down ; neither indeed can they be rendered so plain and intelligible by Words, as by an Algebraick Process, from whence the Theorems or Rules here given, had their first Invention ; as shall be thewed in the next Part, when I come to treat of resolving compounded or adfected Æquations ; however, take this short 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 still common) Methods of extracting Roots, which require the first single Side or Root of the first Period (in any Resolvend) to be taken exactly true, and then by involving, and other tedious Ways of ordering it, there is formed a Divisor ; which helps to grope out by Trials a second Figure in the Root. And so proceed on from Point to Point; fill 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 spoils the whole Process, which must then he wholly begun anew, or at least from that part of it where the Error first entered.
But the Nature and Design of the Method which I have here laid down is quite otherwise; it being fo contrived, as to gradually lelsen the Difference betwixt any proposed 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 insignificant.
Therefore when any Number is propofed to have it's Root extracted; it is here required to take the next neareft Root of the first Period in the Resolvend; that so the Difference betwixt the given Resolvend, and the Homogeneal Power (viz. the like Power of the Root thus taken, may be less either in Excess, or Defect. Which Difference being reduced, or depressed lower becomes so prepared, that by plain Division (comparatively) there will arise such Quotient Figures as will both correct and increase the first Root to three Places of Figures at least, sometimes to four, or five Places of Figures ; according as the said firft Difference happens to be more or less (of which you may have observed Instances): But yet there will be a Remainder left, and perhaps an Excess or Defect in the Root so increased, viz. in the last Figure of it.
Now to rectify the said Excess or Defect in the Root, and to discover whether the given Resolvend be a true Figurate Number, or not: That is, whether it have a true Root of it's kind; it will be necessary to make a second Operation; by taking the Root so increased, and proceeding with it and the given Resolvend, in all respects as in the firft Work (like to the third Example of extracting the Cube Root); I say, if the given Resolvend have a true Root, it will appear at this second Operation, and all the aforesaid Differences, &c. will be vanquished; provided the Root required is not to have more than three (or four) Places of Fi
gures in it.
But if the Root be to have more than three Figures in it; or, that the given Resolvend 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 see in the aforesaid 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 fill Common) way, it would require at least forty times the Number of Figures I have here used.
Again, if there chance to be a Mistake committed in any Operation performed by the Method here laid down, that Mistake will not destroy the precedent Work, but will be rectified in the next Operation, although it were not discovered 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 considered and compared with the several Operations of the foregoing Examples, must needs help the Learner to form such an Idea of it, that he cannot (I presume) but understand how to
proceed proceed in extracting the Root out of any single Power, how high soever 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 easier performed: As will appear in the next Part.
I did intend to have here inserted the whole Business of Intereft and Annuities; but finding that it would require too large a Dircourse, to shew the Grounds and Reasons of the several Theorems useful therein, I have therefore reserved that Work for the Ciofe of the next Part. Neither indeed can the raising of those Theorems be so 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 also omitted that Rule in Arithmetick, usually called the Rule of Position, or Rule of Falfe: Because all such Questions, as can be answered by that guessing Rule, are much better done by any one who hath but a very small fmattering of Algebra. I shall 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 hasty in passing from one Rule to another, and then he will find it very easy to be attained.