1. t**+2 = xx + 2xy + 9, is your Canon for Extracting the Square- Roor. 2dly. The * greatest Member of the SquareRoot of 4624 is 60 (for 70 x 70 = 4900); * See the Tablo therefore * = 60, and xx = 3600, which ca. in Page 42. x ta ken from 4624, leaves 1024 for a Resolvend, and the Coefficient of , in the second Term of your Canon is 2x 120 for your Divisor; by which Dividing the Resolvend, wiz. * 1024, the Quocient is 8' = y; therefore 2xy + y = 1084 is the Ablativum, which taken from the Resolvend, leaves o, Whence the Square Root of 4624 is 60 +8 = 68.. a Operation, 1024 Refolvend, Div. - 120 = 2x 960247 Here y=$: 64 yy 2. Ler it be required to Extract the Cube Root of 99252847. *+,x* tex*+ 3x3y + 3x99 + you Canon. Operation. 9925284? (400 +60+3=463 the Roor required: 64000000 = x3 {Here * = 400. See the Tabla Refol. 35252847 in Page 42. * Didi. 48000o = 3x2 28800000 = 3x*y. Here y = 60. Here you may les that by Dividing the thereby found, would be but the 33336000 Ablativum. Ablativum so produc'd, would excud ihs Refolvend, wherefore y = 60. 1916847 Refolvend. Divifor 634800 = 3x Here x = 460, 1904400 70, 19044003**y {Here y=3. ..12420 3xyy 27 NY 1916847. Ablativum. o Remainder. 3. Suppose it was rrequired to Extract the Biquadrat Root of 6652111747853987761. *+,xx+yx *tyxx ty=** + 4xy + ba'yo + 4xy? t ya Canon. Operation. 6612111747853987761 (soooo + 700 + 9 = 62500000 0000 0000060 = ** (50709 the Root required, 362111747853987761 Refolvend. Here x 50000. Div. 500000000000000=4* 350000000000000000=4*y Here y 700. 7350000000000000 = $x*yy 686000000000004.%y3 240100000000 yyyy 357418840100000000 Ablativum. 4692907753987761 Refolvend. 521295372000000 =4*. Here'x'=50700 4691658348000000 = 4x*y. Here y = 9. 1249258140000 = 6x*y? 14784676093437 4692907753987761' Ablativum. Remainder. If the given Power hath not such a Root as is required, you may notwithstanding find a Root nearer the truth,' than any atligned in the following manner. Suppose it was required to find the Square Root of a very Operation. 2 (1+ition +.004 :0902 + c, =1:4142 + near, - Divifor 2 33x) ! Refolvendo 8 = 2xy.' Here y=4 96 Ablativum. 04 Refolvendo ; (And 3.1. : 0281 Ablativume i Div. 2.82 =2x) 0119 Rcfolvend. OL128 = 2XY Here x 1.41.16 i odoo 16 = 9 And yes, 0043 011296 Ablativum. Div. 2 - 818 = 2x) = 600604 Resolveud. 0005656 2XY SHere 1.414. 00000004 99 Andy. 0002. 00056564 Ablativum.' cobo3836 Remainder. The common Method of Extracting the Square Root, CubeRoot, &c. of Numbers, is only an Abridgement of the fore. going Method, and is thus perform'd, viz. Place the first Point always over the Figure, which is in the Place of Units, Place also 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 Roor, &c. that is to be Extracted, Point over the 2d. 3d. &c. Figure respectively to the Left Hand ; and if there be any Decimals in the given Number, to the Right Hand of the said Figure, which is in the place of Units. And as many Points as there are over the places of whole Numbers, so many places of whole Numbers must be in the Root, and the rest are Decimals. Again any Binomial as x + y, being Involved into it self, as often as there are Units in the Denominator (1 being the Numefator) of the Index of the Root to be Extracted, produces your Canon for Evolving. Then. ; ift. By the Table of Powers in Page 42 Por otherwise), find the greatest Power that is containd in the first Period towards the Left Hand; viz. the greatest Square, if it be the Square Root the greatest Cube, if it be the Cube-Roor, &c. tbar is to be Extracted ; then having plac'd the Root = x in the place assign'd for it, which is likewile call'd the Roots Subftract the said Power from the said Period. Atter the Remainder, place the Figures in the next Period, and call that Number your Resolvend call also the Value of the ; Coefficient of y in the second Term of your Canon, your Divisor. Then ask how oft the Divisor is contained in the Resolvend omitting all the figures in the laft Period but the first, the Number = y set in the Roor next after the Value of * ; then find the Ablativum thus. Place the Figures which are the value of the 24. Term of your Canon, fo as the last of them may be under the 18 of the laft mention'd Period, and the Values of the 3d. 4th. 5th. &c. Terms 1, 2, 3, &c. Places respectively more to the Right Hand, than those of the 2d. Term; and the Sum of the Values of the 2d, 3d, 4th, seh, &c. Terms plac'd as aforesaid, is the faid Ablativum, which rake from your Resolvend. But here Note, that if the Ablativum thus found fould be greater than the Re. folvend, then the Value of , is too Great, and muft be made Lels. After the Remainder, place the Figures in the next Period, and call thar Number your Resolvend, and call the Figures placa in the Root *; by which find the Value of the next), in like manner as before directed, and so proceed till you have done with all your Period; and if afterwards there is a Remainder, place Cyphers atter it, in order to find as many Deeimal Figures as you please. Examples. Let it be required to Extract the Square. Root of 6968, very 48 IXY 489 Ablatioun . 79.00 Refolvendo Div. 166 – 3x) 664 = 2xy 16 = y 6656 Ablativum. yy * Here I go on with the 667776 Operation as it ought' to practis'd. 8347425 The other Figures of the Roor to the 11th Fig. Inclusive, may 117179 148 16 Whence the Square Root of 6968 is very near=83. 474547 018. CH A P. III. Evolution wziet of Numbers, or the Method of Ex tracting the Roots of Adfeaed Æquations; which Values of a. 4 Canon. a' =** + 3*°; +3879 +»! }= 1860990. + And a * |