Ch. XII. ther three Figures in Order.

Whence, as it is usual, I distinguish the given Cube into the faid refpective Members, by pointing the right-hand Figure firft, and then every third Figure, as 1728, and

fo proceed to the Extraction of the Root, as follows:

The only Particular here to be obferv'd, is the fame with what was obferv'd in Reference to the fquare Root, Example 3; namely, that having by the firft Operation found the two Figures 57 of the Root fought, both the faid Figures in the fecond Operation are to be efteem'd as A; and the third Figure 2 of the Root is to be esteem'd as E. And if the faid Root did confift of four Figures, and fo the Operation were again to be repeated, then all the three Figures already found are to be esteem'd as A, and the Fourth remaining to be found as E. And fo on, according to the Number of the Figures, which the Root confists of.

K 4

If,

Ch. XII.

If, after that the fquare or cube Root ~ has been Extracted out of any Number

Ch. XII.

12.

Root, what.

A furd propos'd, according to the fore-going Rules, there remains any Part of the faid Number; then the faid Number is not an exact Square or Cube, but fo much over as is denoted by what remains of the faid Number. And it often happens, that in mathematical Operations, there is Occafion to look on fuch Numbers or Quantities, as Squares or Cubes, which yet are not fo; and confequently, whofe Roots can't be expreft in Numbers, (whence they are call'd (†)furd Roots,) but are wont to be denoted thus, viz. √96,

2

or 6, or fimply 6, denotes the Root of the Number 6 taken as a Square. In

3

like manner, ✔c6, or 6, denotes the Root of the Number 6 taken as a Cube. And therefore, the Square of the furd

6, is 6; the Cube of the furdc6, is likewise 6; and fo of any other Surd.

(+) I have not met with (as I remember) any Writer, that gives an account of the Reason of this Name. It is well known, that Surdus in Latin denotes one Deaf. And therefore I can't think of a better Reason, why a furd Root fhould be fo called, than that because it can't be expreft in Numbers, and confequently can't be truly pronounced, or heard pronounced by any other, any more than by a Deaf Man. Much after the fame manner as we fay a thing is Abfurd, because it is fo Irrational, as it is not fit to be heard: Which puts me in Mind that a furd Root is likewife agreeable to this laft Obfervation, otherwife call'd an Irrational Root.

As

3

bical Frac

zཏ་ tion.

As a Fraction is fquar'd or cub'd, by Ch. XII. Squaring or Cubing its Numerator and Denominator; fo a quadratick or cubical How to 13. Fraction is refolv'd into its Root, by re- extract the folving its Numerator and Denominator Root of a quadrainto their respective Roots. Thus, the tick or cu8 is Square of is, the Cube of And on the other Hand, the Root of the Square, or of the Cube is. If the Root of the Fraction can't be extracted, or exprefs'd in Numbers, then 'tis call'd a furd Fraction; as the Root of taking it for a Square, is; the Root of taking it for a Cube, isc.

8 27

14.

and Cubes

of the nine

The Squares and Cubes of the nine Digits or fingle Figures ought to be fo of the well known, as that at the bare Sight of Squares them their respective Roots fhould be perfectly known. To which End the Digits. faid Digits are here set down, with their respective Squares and Cubes on their Side.

Thefe Squares and Cubes of the nine 15. Digits, make up one of thofe numbering Extraction Rods, which commonly go by the Name quare or

of the

of cube Root,

Ch. XII. of Naper's Bones, as being of good Use, is render'd together with the other Rods or Bones, more easy in Extracting the fquare or cube Root of and quick But the quickest by Naper's any Number given. Method of Extracting the faid Roots is and efpeci- by the Help of Logarithms, as fhall be ally by Logarithms. fhewn in the laft Chapter of this Trea

Bones,

I.

The known Quantities are to be diftinguish'd from the

tife. And thus much for the Extraction of the fquare and cubical Root.

CHAP. XIII.

of Equation, or the Method of fol ving Questions, by (what is call'd) the Rule of Algebra.

A

NY Question being propos'd, put A, or any other Vowel, to denote the Quantity fought, as if it were already known; and put Confonants to denote the Quantities given. Or any other fuch like Method may be made Ufe of, whereunknown. by the known Quantities may be diftinguifh'd from the unknown.

2.

Then frame the feveral Quantities, An Equa- whether given or fought, according to tion is to, the State of the Queftion propos'd; and be form'd. out of the said Quantities fo fram'd form an Equation; that is, compare the faid Quantities together, 'till you perceive

fome