741 



SQUARE ROOT. 



SQUARE ROOT. 



290225296 - 10000= = 190225296 

 290225296 - 17000 2 = 1225296 

 290225296 17030 2 = 204396 

 290225296-17036 2 = 



whence the given number is the square of 17036. 



The rationale of the approximate extraction is as follows : Suppose 

 we wish to find, to four places of decimals, the square root of 1'74 ; 

 that is, to find a fraction a, such that a* shall be less than 1'74, but 

 that (a + '0001)* shall be greater than 1'74. Give this fraction the 

 denominator 1 ,00,00,00,00, which requires that its numerator shall be 

 1,74,00,00,00. This numerator is found, by the integer rule, to lie 

 between 13190 3 and 13191*, or 173976100 and 1740C2481. We have, 

 then, 



(1-8190)' = 173976100 

 (1-3191)* = 1-74002481 



which satisfy the conditions. 



The common process of contraction, explained in books of arithmetic, 

 has the following rule : When the number of places in the divisor 

 baa so much increased as to exceed by 2 or more the number of places 

 yet remaining to be found, instead of proceeding with the complete 

 operation, leave the remainder unaugmented by any new period, strike 

 one figure off the divisor, and proceed as in contracted division. If 

 R be the remainder, a the part of the root found, 6 that remaining to 

 b found, we have 6(2d + i) = B. If a be very large compared with b, 

 2o5 = R nearly, or 6=B-=-2o nearly; now 2a is the divisor in the rule. 

 The fact i that 6 must lie between 



2a 



Processes of this sort are often best shown, as to mere operation, by 

 an instance in which the numerical computation gives no trouble. 

 The following is a complete instance of the rule, exhibited in finding 

 the sqxiare root of 4444-444444, Ac. : 



4444 444444, Ac. (66 666666666667 

 36 



126J844 

 756 



1326)8844 

 7956 



13326)88844 



nws 



133326)888844 

 799956 



1333326)8888844 

 7999956 



13333326)88888844 

 7L':iy99 56 



13,3,8,3,3,8,2)8888888 

 7999899 



BMHfl 



MM) 1 



88889 

 80000 



The given number in, in fact, 4444), which is the square of 66J. 



Let the number be y, the integer part of its root a, and write under 

 one another in the first column, a, 1, a. Proceed to form the second, 

 third, &c., columns, each from the preceding, in the following way : 

 If p, q, r, be in one column (found), and p', q' )' in the next (to be 

 found), then 



N p 2 

 1. g' is , and is always integer. 



2. / is the integer part of 



3. p' is }V p. 



a 4- j> 



~~ 



Thus, for the second column we have (21 16)-f-l, or 5, for the second 

 row ; the integer of (4 + 4)-=-5, or 1, for the third row; and 1 x 5 4, 

 or 1, for the first row. In the third column we have (21 1)-=-5, or 4, 

 for the second row ; the integer of (4 + 1)^4, or 1, for the third row ; 

 and 1 x 41, or 3, for the first row. In the fourth column we have 

 (21 9)-=-4, or 3, for the second row ; the integer of (4 + 3)-^3, or 2, for 

 the third row; and 2x33, or 3, for the first row, and so on. This 

 process must, after a time, begin to repeat itself ; as soon as this 

 happens, the last row shows the integer of the square root, and th 

 succession of denominators of the continued fraction, which mustba 

 repeated as often as is necessary to give the required degree of accuracy. 

 Thus, 



1_1__1_1_1_1_1 1_ 1 1 1 



+ 1+ 1+ 2+ 1+ 1+ 8+ 1+ 1+ 2+ 1+ 1 + ' &c - 



If we proceed with the continued fraction as in the article cited, we 

 have for the successive approximations to its value 



1134^ ^21L12I??i 4 il 769 



I 2 5 7 12 103 115 218 551 769 1320' : 



and 4 only differs from \/2l in the 8th decimal place. The use 



of this method is, not to extract the square root of any number which 

 accidentally occurs, but to find convenient approximations, if possible, 

 for square roots which frequently occur, and as to which it is therefore 

 worth examination whether there may not be some common fraction 

 which will serve the purpose as well as a decimal of considerable accu- 

 racy. For instance, we take V2 and V 3, which represent the diagonals 

 of a square and a cube having a unit for their sides. The processes 

 are as follows : 



V2 = 



~ 5T 2 + Ac. V3 - 1 + , j. ;_ i ^ ^ , Ac. 



1111, Ac. 

 2121, Ac. 

 1212, Ac. 



1 1 1 1 



IT -IT IT 



The successive approximations to the fractions are 



1 2 5 12 29 70 169 408 



2 5 12 29 70 169 408 985' * c ' 

 1 2 8 j[ 11 SO 41 112 153 418 571 



T 3 4 11 15 41 56 153 209 571 780' & ' 



Thus we immediately see a convenient mode of finding the diagonal 

 of a square, derived from 



29 99 100 1 



V2 = 1 + 70 = 



70 



70 



-, nearly, 



the excess of which above the truth is less than the 11830th part of 

 the side. Thus, to find the diagonal of a square of 769'23 feet, we 

 have 



76923 

 769 23 



70)76153 77 

 1087-911 

 which is too great by only about 6-hundredths of a foot. 



