2. If a3+ a be equal to 1860990, 'tis required to find one of the Values of a. Suppofe x+ya, then Canon. a2 = x3 + 3xxy + 3xyy +y3 {=1860990 +a=x+ g Confequently 100+20+3=123a; that is, one of the Values of a. If any more of the Roots or Values of a, in the propos'd Equation be required; and that you can find the first Member (ft. x) of any fuch Root, then you may proceed to Extract that Root by the fame Canon: But if you can't readily find any fuch first Member, Reduce all the Terms of the Equation to one Side, making them equal to o; next divide each part of this Equation, by a -the known Root; afterwards tranfpofe the Abfolute Number in the Quotients; and then frame a Canon for this last Equation as hath been already taught, in order to Evolve the Abfolute Number therein by that Canon. But I find that a3 a 18609900, Divided by a 123 yields in the Quotients, aa+123 a + 151300; the Abfolute Number in which being Tranfpos'd, will give this Quadratick Equation, aa 123 a=15130, which Equation hath no Root, either Affirmative or Negative; wherefore no other Root of, or in the propos'd Equation can be found by this Method but 123; notwithstanding the above Quadratick, and confequently the propos'd Equation hath two Imaginary Roots, that is to fay, Impoffible Roots, which are found by Part X. to be 123-45391 and 123+√45391 2 2 which along with 123 are the three Roots or Values of a in the propos'd Cubick Equation, CHA P. IV. Evolution of Fractions. FIrft prepare the Fractions propos'd to be Evolved, viz. by reducing Compound Fractions to fimple ones, as alfo Mixt Fractions to Fractions of the fame Denomination, and by reducing them to their leaft Terms, by Se&t. 4th, 1st, and 3ɗ. Chap. II. Part. II Then If the Numerator and Denominator have each of them fuch a Root as is required, Evolve them, and their respective Roots will be the Numerator and Denominator of the new Fraction required. Examples. 1. Thus the Square-Root of 1 of 38 of 4=88 is = 8 a 2. Alfo the Cube-Root of 1+ a3 + zaab +zabb + b3 baab2bbb a3 —zaab+3abb—b3 a3 − zaab + zabb — 63 will be found to be a+b a-b 3. The Cube-Root of 1296 216 6 that is of is Sometimes it fo falls out, that the Numerator may have fuch a Roo as is required, when the Denominator hath not, or the Denominator may have fuch a Root, when the Numerator hath not. In fuch Cafes the Roots may be fet down as follow; viz. The Square-Root of 40 27c3d3 But when neither the Numerator, nor the Denominator (being in their least Terms) have just fuch a Root as is required; prefix the Řadical Sign of the Root to the Fraction; thus the Square-Root of 9 is: And the Cube-Root of 2 is Again, If the Fractional Numbers given can be reduc'd to Decimals of the fame Value, you may reduce them to fuch, and then Evolve. Examples. 1. Thus the Cube-Root of 1296 1296 750 750 is found to be 1.2, by Reducing to a Decimal of the fame Value (which is done by Dividing the Numerator, with a fufficient Number of Decimal Cyphers plac'd after it by the Denominator 'till nothing remains), and then Extracting the Cube-Root of that Decimal Fraction, that is of 1.728. 2. Also if 3y3-23y+1443713, that is 7.875: then one of the Values of y is 5. 3. Again, If as + 3 a3 = 5157 +, 'tis required to find one of the Values of a. First reduce the Fractions in this Equation to Decimals of the fame Value, and the faid Equation will become as +.75a3 =5157.625. Secondly, Suppofe x+y=a; then, a2 = x2 + 5x + y + 10x3 y2+10x23y3 +5xy+ + y2 } = +.75a3=+•75x3+2.25x2y +2.25xy' +.75y3 5157.625. Operation. 5157.625 (5+.5=5.5=a Answer. But if the Fractions in the propos'd Equation cannot be reduc'd to Decimals of the fame Value (you may notwithstanding reduce them near the Truth, and then proceed to Extraction; or) Reduce the Equation to a common Denominator, and then Multiply each Part of that Equation by the common Denominator, Er. thus, |