of the square, Periods of whole Numbers in the given Resolvend. [4] Find the difference between the given Resolvend and the power coming nearest the first Period. [5] Whatever odd power is given, the next lowest odd power to that of the said root must be found with its annexed cyphers ; i. e. if the ninth power be given, find the 7th power of the root and cyphers: if the lith power be given, find the 9th, d'e. [6] Multiply the next lowest odd power by the Index of the given power, and let that product be a divisor to the difference between the given resolvend and power first found, which depresses it to a square. [7] Point this square into periods of two figures cach. [8] Then make the first root without ils cyplier a divisor, and ask how oft it may be found in the first period [9] If the divisor be less than just, you must multipyi the quotient figure by half the iudex, i e if the index be 11, multiply the quotient figure by 5; if the index be 9, multiply it by 4, d.e. and add it to the divisor; but if it be more than just, you must subtract it from the divisor, having a cypher annexed or supposed to be annexed to the divisor ; which sum or difference must be multiplied by the said quotient figure, and so continued to every pew figure in the quotient. [10] If the first root with its cyphers be more than just, the quotient must be subtracted from it; but if it be less than just, it must be added to it; and the sum or difference will be the root required. 3. If an even power be giver, and the square root of that power be extracted, reduce it to an odd power ; you must then proceed with that odd power as the foregoing rules direct. EXAMPLES. .6436343 3236343 The cobe of 20 is =8000 And 8000 X5 is=40000 Then 40000)323634530 Again 2 )80 3 + 3 +2=678 Lastly 20 +3 23 the sursolid N? Ist divisor=26 Root required 2 to be rejected. Note. This is a very expeditious way of extracting the roots of high powers, but it is not always exact, because (as Àr. Ward observes, for it was taken from him) there will be à remainder, and sometimes an Excess or Defect in the last Figure of the root, when the given resolvend or power hath a true root-as appears by the fifth example following, whose true root should not be 384.3, as it there stands, but 384. 2. What is the sursolid of 48 ? Ans. 254803968. 3. What is the sursolid root of 8153726976? Ans. 96. 4. What is the sursolid root of 254803968 ? Ans. 48. 5. What is the sursolid root of 2 Ans. 384.3. 8349416423424 OF THE SQUARE-CUBE ROOT. Q. WHAT is a Square Cube ? A. Any Number involved six times, produces a Square Cube. EXAMPLES. 1. What is the Square-Cube of 48 ? Aos. 12230590464. 2. What is the Square-cube root of 78275778969.6.? Ans. 96. 3. What is the Square-cube root of 12230590464? Ans. 48 4. What is the Square-cube root of 320617590659 4816? Answer 384. OF THE SECOND SURSOLID-KUUT. Q. WHAT is the Second Sursolid ? A. Any number involved seven times produces a second Sursolid. EXAMPLES. 1. What is the 2d sursolid of 96 ? A. 75144747810816. 2. What is the second sursolid root of 75144747810816? Answer 96. 3. What is the second sursolid root of 587068342272 ? Answer 48. 4. What is the 2d sursolid root of 12311715481324093447 Answer 38 1.42. OF THE SQUARE BIQUADRATE-ROOT. Q. Wh4T is a Square Biquadrate ? 1. Any number involved eight times, is a Biquadrate squared, or square biquadrate. EXAMPLES. 1 What is the squared Biquadrate of 48 ? Ans. 281.9230429056. 2 What is the square biquadrate Ans. 96. root of 721389578838336? 3 What is the square biquadrate. Ans. 48. root of 28179280429056 ? 4. What is the square biquadrate Ans. 384. root of 472769874482815188096? S OF THE CUBED.CUBE ROOT. Q. What is a Cubed-Cube ? A. Any number involved nine times is a Cubed: Cube. EXAMPLES. 1 What is the Cubed-Cube-Root Ans. 96. 2. of 692533995824480256 ? 2 What is the Cubed-Cube-Root Ans. 48. 09. of 1352605460594688 ? 3. What is the Cubed-Cube-Root Ans. 384. 5. of 18154.363180141255228864 ? OF THE SQUARE SURSOLID ROOT. Q. What is a Squared Sursolid? A. Any number involved ten times, produces a Squared Sursolid. EXAMPLES. 1. What is the Squared Sursolid Root Ans. 48. of 64925062108545024? 2. What is the Squared Sursolid Root Ans. 96. of 66483263599150104576 ? 3. What is the Squared Sursolid Root Ans. 384. 3. of 697437 54611742420055883776? OF THE THIRD SURSOLID ROOT. Q. WHAT is a Third Sursolid ? A. Any number involved eleven times, produces a third Sursolid. EXAMPLES. 1. What is the third sursolid root Ans. 23. of 952809757913927 ? 2. What is the third sursolid root Ans. 48. of 3116402981210161152 ? EXAMPLES. 3. What is the third Sursolid root? of} Ans. 96, of 6382393305518410039296 ? OF THE SQUARED SQUARE CUBE ROOT. Q. WHAT is a Squared Square Cube ? A. Any number involved twelve times, produces & Squared Square Cube. 1. What is the root of this Squared Square Cube 149587343098087735296 ? 2. What is the root of this Squared Square A. 96. Cube 612709757329767363772416? 3. What is the root of this Squared Square? A. 384. Cube 102795639440290,9029176039807836? A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. 1. PREP RE the given number for extraetion, by pointing off from the unity place, as the root required directs. 2. Find the first figure in the root by your own judgment, or by inspection into the table of powers. 3. Subtract it from the given number. 4. Augment the remainder by the next figure in the given number, that is by the first figure in the next point, and call this your dividend. 5. Involve the whole Root, last found, into the next inferior power to that which is given. 6. Multiply it by the index of the given power, and call this your divisor. 7. Find a quotient figure by common Division, and annex it to the root. 8. Involve all the root thus found, into the given power 9. Subtract this power (always) from as mans points of the given power as you have brought down, beginning at the lowest place. 10. To the remainder bring dowo the first figure of the next point for new dividend. 11' find a new divisor as before, and in like manner proceed till the work is ended. EXAMPLES. | What is the Cube Root of 115501303? 115501303.(487 48) 515 Dividend 110592 Subtrabend 6912(49093 Dividend 115501303 Subtrahend To 4 x 4 X 3 =48 Divisor 48 X 8 X 48 =110592 Subtrahend 48 X 48 X 3 =6912 Divisor 487 X 487 X 487 =115501303 Subtrahend. ? What is the Biquadrate root of 56249134561 ? 4 X 4 X 4 x 40-256 Divisor 48 X 48 X 48 x 48=5303416 Subtrahend 48 X 48 X 48 X 4=442368 Divisor 487 X 487 X 487 X 487=56249134561 Sub. Note. This General Rule I receive from my worthy Friend, William Montaine, E:. F.R. S and teacher of the Mathematics at Shad-Thames, |