3. Suppose it was required to Extract the Biquadrat-Root of 6612111747853987761. :x+y:x:xx:x:x+y:x:x+y:=x+4x3y+6xxyy Operation. +4xy3 +y4. Canon. Div. Div. 6612111747853987761 (50000 + 700 + 9 = 50709 the (Root required. Here y=700. 521295372000000= 4x3 Here x = 50700. 4691658348000000=4x3 1249258140000 6xxyy 4692907753987761 Ablat. o Remainder. Here y=9. Part. IV. If the given Power hath not fuch a Root as is required, you may notwithstanding find a Root nearer the truth, than any affigned in the following manner. Suppose it was required to find the Square-Root of 2 nearly. Divifor 2=2x) I Operation. 2 (1+.4+.01 +.004 +.0002 + &c. = (1.4142+ is the Square-Root of 2 nearly. I XX Refol. The common Method of Extracting the Square-Root, CubeRoot, &c. of Numbers, is only an Abridgement of the foregoing Method, and is thus perform'd, viz. Place the firft Point always over the Figure which is in the Place of the Units. Place alfo a Point over every other Figure, denoted by the Denominator of the Index of the Root to be Extracted; that is, if it be the Square-Root, Cube-Root, &c. that is to be Extracted, point over every 2d. 3d. &c. Figures refpectively to the Left-hand, and if there be any Decimals in the given Number, to the Right-hand of the Figure, which is in the Place of Units. And as many Points as there are over the Places of whole Numbers, fo many Places of whole Numbers must be in the Root; and the reft are Decimals. Again, any Binomial, as x+y, being Involved according to the Denominator (1 being the Numerator) of the Index of the Root to be Extracted, produces your Canon for Evolving. Then, 1ft. By the Table of Powers in Page 42, or otherwise, find the greatest Power that is contained in the firft Period towards the Left-hand; viz. the greateft Square, if it be the SquareRoot; the greatest Cube, if it be the Cube-Root; &c. that is to be Extracted; then having plac'd the Root x in the place affigned for it, which is likewife call'd the Root; Subtract the faid Power from the faid Period. و After the Remainder place the Figures in the next Period, and call that Number your Refolvend; call alfo the Value of the Co-efficient of y in the fecond Term of your Canon your Divifor. Then ask how oft the Divifor is contained in the Refolvend, omitting all the Figures in the last mention'd Period but the firft; the Answer or Number y fet in the Root next after the Value of x. Then find the Ablativum thus: Place the Figures which are the Value of the fecond Term of your Canon, fo as the laft of them may be under the first of the laft mention'd Period, and the Values of the 3d, 4th, 5th, c. Terms one, two, three, &c. Places refpectively, more to the Right-hand, than thofe of the 2d. Term; and the Sum of the 2d, 3d, 4th, 5th, &c. Terms, plac'd as aforesaid, is the Ablativum, which take from the Refolvend. But here Note, that if the Ablativum thus found fhould be greater than the Refolvend, then the Value of y is too great, and must be made lefs. After the Remainder place the Figures in the next Period, and call that Number your Refolvend, and call the Figures plac'd in the Root x; by which find the Value of the next y, in like manner, as before directed; and fo proceed 'till you have done with all your Periods: And if afterwards there is a Remainder, place Cyphers after it, in order to find as many Decimal Figures as you please. Example. Let it be required to Extract the Square-Root of 6968, nearly. Operation. Div. 1668 = 2x)124400 Refolvend. 116809 Ablat. = 2xy +yy. The other Figures of the Root to the 12th may be found by Divifion; thus, 166949.0) 784975,0 (47018 117179 148 15 Whence the Square-Root of 6968 is nearly 83.474547018. CHAP. CHA P. III. Evolution Mixt of Numbers, or the Method of Extracting the Roots of Adfected Equations; which Method is generally call'd the umeral Gregefis. 1. LET it be required to find one of the Affirmative Values of a in this Equation, viz. a3-171.91an † 7905.6a =71460. Suppofe x+y=a; then, aaa Canon. xxx + 3xxy+3xyy + y3 343.82xy-171.91yy {= 7905.6a 171.91aa- 171.91xx +7905. 6a = 7905.6x+7905. by 71460. First, I fuppofe a 20; then a3 — 171. 91aa (=8000-68764 + 158112) 97348, which is more than 71460; therefore a 20. = Again, I fuppofe a = 10; then a3—171.91aa+7905.6a = 62865 which is lefs than 71460; therefore a 10: That is, a is 20, and 10; confequently 10 is the first Member of one of the Values of a; that is 10= X. Operation. 71460 (10+1+.9+.01=11.91 =4 +79056+7905.6x 2 62865 =x3 171.91 xx Here x 10 |