SQUARK ROOT. 



SQUARE ROOT. 



740 



rbol will not, without great can, avoid fake raaaoning in making 

 connection between geometry and algebra. 



A square U divided by its diagonal into two isosceles right-angled 



triangle! : Its diagonals are equal, and bisect each other at right angles. 



The easiest way of drawing a square U to describe two circles on the 



n, and bisecting A c in i>, to make c f and c r equal to A D ; the 



figure A K r B is then a square. 



Of all similar figures, squares are those to the areas of which 

 reasoning is most easily applied. If three similar figures be described 



on tho sides of a right-angled triangle, the Bum of those on the sides in 

 equal to that on the hypothenuse. This proposition is learned with 

 reference to squares [HvroTHKXi'sF.] long before it can be proved with 

 reference to similar figures in general : the consequence is, tint the 

 general proposition is almost overlooked by the previous occuri. 

 the particular case. We have noted in the article just cited the Hindu 

 proof of this celebrated cane; the simplest properties of the square 

 may be made to give a more easy proof, founded on the same prin- 

 ciple ; it being remembered that the first four propositions of the 

 second book of Euclid do not require the last two of the first book. 

 The proof is as follows : Let A B, B c, be the sides of a right-angled 

 triangle, and on their sum describe a square ; make c K, r H, K L, each 

 equal to A B. It is easily proved that L B E H is the square on the 

 hypothenuse of the triangle ; and it is made by subtracting four times 

 the triangle from the whole square A F. But if four times the triangle 



oo small, we go on adding 100 to the square root, until no more bun- 

 reds can be added; all the while forming the squares by the rule, 



each square from the preceding. We then begin to add tens, forming 

 he squares also, until the addition of one more ten would bring the 



square past 104713 ; we then add units, until cither the square is 

 xactly 104713, or the nearest to it. Or, instead of continual 

 dditions, we might subtract every number, as we get it, from 104718 

 mtil no more subtractions can be made. Both modes arc exhibited in 

 he following : 



100 J = 10000 104713 



100 (2 x 100 + 100) = 80000 10000 



be subtracted by taking away the rectangles AM and n D, wehav. 



ii. 4 ) the sum of the squares on the sides, which is therefore the same 



as the square on the diagonal. 



SQUARE HOOT, the name given to a number with reference to its 

 square. Thus, 4H being the square of 7,7 ia 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 baa no square root. But since 1 '41 42136 

 multiplied by itself gives very nearly '2, or has a square very near to 2, 

 it in customary to say that 1*41 42186 is very nearly the square root of 

 3 : 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 iindini; 

 the square root of a number in a continued fraction was first given (iu 

 less easy rule) by the same Cataldi, who was thus the first who used 

 continued fractions. This fact was pointed out by H. Lilui. 

 which Lord Brounker 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 in contained under the 

 general rule given in INVOMTION AND Evoi.i TION. The i 

 aimplicity of this case, however, allows of a condensation of form, am 

 make* the demonstration easy. The general nde just alluded to might 

 be demonstrated on the same principle. 



In 'Tiler to turn the square of a into the square of a + i, we muff 

 add to the former 2 a 6 + *', or (2 a + 6) 4. This follows from 



(a + bf = a" + 2 a 6 + M. 



An example will now show how the square root is extracted ; first 

 roughly, afterwards more skilfully in the choice of triala. Let the 

 number be 104713. The square of 100 being 10000, which is much 



200' = 40000 

 100 (S x 200 + 100) = 60000 



300?= 90000 

 10 (2 x 300 + 10) - 6100 



S10 ; = 96100 

 10 (2 x 310 + 10) = 6300 



8202 = 102400 

 1 (2 x 320 + 1) - 641 



321 3 = 103041 

 1 (2 x 321 + 1 = 643 



322' = 103684 

 1 (2 x 322 +1) - 045 



323-' = 104329 

 1(2x323 + 1) = 



94713 a 

 10000 



64718 i 



14713 c 

 6100 



2313 e 

 041 



641 



1029 

 Ml 



824= = 104976 



384 K 



In the first column we feel our way, so to speak, by hundreds, by 

 tens, and by units, up to the result that 323 s is too small, and 324'-' 

 too great; so that we see that 104713 has no square root. In the 

 second column we go down from 104713, and subtracting the sq 

 already formed in the first column, we come to the result that 104713 

 is 384 more than 323', but less than 324'. The results of the second 

 column are 



94713 = 104713 - 100 2 



64713 = 104713 - 200 2 



14713 = 104713 SOO 2 



8613 = 104713 - 310' 



2313 = 104713 - 320" 

 1672 = 104713 321-' 

 1029 = 104718- 

 384 = 104713 - 323* 



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. It'6(2a + 4) is to be found as near as possible to R, and if 2 n '' 

 considerable compared with I, the value of i is near to that given )>y 

 I x 2 a B, or I = R+-2a. 



Taking the number 104713, and parting it into periods of two num- 

 bers each, we have 10, 47, 13, and 9,00,00 is the highest square 

 belonging to a simple unit followed by ciphers, which can be contained 

 in it. Choose 300 for the first part of the root, and we have 14713 for 

 the remainder. If I be the number of tens in the root, we have to make 

 10 6 (2 x 300 + 10 M a< near as we can to 14713, or 10 4 not being much 

 compared with 600, we must try 10 4 x BOO = 14713, or 4 = 14713-4-6000, 

 whence 2 is the highest (perhaps too high, lnit that will be Keen by 

 the remainder). If 4 = 2, 104 (600 + 106) is 12400, which, subtracted 

 from 14713, gives 2313. The part of the root now obtained is :;jn. 

 and if c be the number of units, c(2 x 320 + f) must be made equal, or 

 as near as can be, to 2313. Now c is very small compared with >i In, 

 and c x 640 = 2313 shows that c = 3 at most, giving 3 x 643, or 1929, 

 to be subtracted from 2313, w-hich leaves 384. The process may be 

 written thus : 



104713 (300 + 20 + 8 

 90000 



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 



We now take a longer instance, at full length, followed by a state- 

 ment of its results : 



