SQUARE ROOT, the name given to a number with reference to its square. Thus, 49 being the square of 7, 7 is the square root of 49. When an integer has no integer square root, it has no square root at all in finite terms : thus 2 has no square root. But since 1.4142136 multiplied by itself gives very nearly 2, or has a square very near to 2, it is customary to say that is very nearly the square root of 2 : more properly, the square root of something very near 2.
The extraction of the square root came into Europe with the Indian arithmetic, the method followed by Theon and other Greeks (which was substantially the same) having been forgotten. The earliest extensive treatise on the subject is that of P. A. Cataldi (Bologna, 1613), though the books of algebra and arithmetic had then been long in the habit of giving the rule. The process presently given for finding the square root of a number in a continued fraction was first given (in a lees easy rule) by the same Cataldi, who was thus the first who used continued fractions. This fact was pointed out by M. Libri, before which Lord 13rounker was generally considered as the first who used continued fractions.
The rule for the extraction of the square root is a tentative inverse process very much resembling division, and is contained under the general rule given in INVOLUTION ANT) EvoturtoN. The peculiar simplicity of this case, however, allows of a condensation of form, and makes the demonstration easy. The general rule just alluded to might be demonstrated on the same principle.
In order to turn the square of a into the square of a+ b, we must add to the former 2a b +1', or (2 a + b. This follows from An example will now show how the square root is extracted ; first roughly, afterwards more skilfully in the choice of trials. Let the number be 104713. The square of 100 being 10000, which is much The best method of making the trials depends upon the following circumstances : 1. A square number followed by an even number of ciphers, such as 16000000, is also a square number.
2. If b (2 a+ b) is to be found as near as possible to n, and if 2a be considerable compared with to, the value of I is near to that given by Taking the number 104713, and parting it into periods of two num tors each, we have 10, 47, 13, and 9,00,00 is the highest square belonging to a simple unit followed by ciphers, which cau be contained in it. Choose 300 for the first part of the root, and we have 14713 for the remainder. If b be the number of tens in the root, we have to make (2 x 300 +101) as near as we can to 14713, or 10L not being much compared with 600, we must try 10 b x 600=14713, or b = 14713+6000, whence 2 is the highest (perhaps too high, but that will be seen by the remainder). If 10 (600+101) is 12400, which, subtracted
from 14713, gives 2313. The part of the root now obtained is 320, and if c be the number of units, c (2 x 320 +c) must be made equal, or as near as can be, to 2313. Now c is very small compared with 640, and c x 640=2313 shows that c=3 at most, giving 3 x 643, or 1929, to be subtracted from 2313, which leaves 384. The process may be written thus: which, omitting superfluous ciphers, is the one commonly used. We do not intend to dwell on the common process, which is in all the books, but confine ourselves to the explanation, which is frequently omitted.
We now take a longer instance, at full length, followed by a state ment of its results :— 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 P74 ; that is, to find a fraction a, such that at shall be less than 114, but that (a+ shall be greater than 114. 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 and or 173976100 and 1740C2481. We have, then,— = 113976100 = 114002481 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 has 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 unaugmcnted by any new period, strike one figure off the divisor, and proceed as in contracted division. If It be the remainder, a the part of the root found, b that remaining to be found, we have 6(24+14=R. If a he very large compared with b, 2 ab= n nearly, or b= n+.2 a nearly; now 2a is the divisor In the rule. The fact is that b must lie between and R Ta 2a+ 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 square root of &c.
The given number is, in fact, 4444:, which is the square of 66g.
Wheu R is the remainder, and a the part of the root found, the remaining part of the root is the continued fraction [FRACTIONS, CONTINUED.j