IXVitl.rTION" AND EVOLUTION. 



INVOLUTION AXP EVOLUTION. 



m 



The UMwer 1 41421356:2878 is comet to the last place inclusive. 

 (The contracted diruiun follows the thick line.) The rule by which to 

 judge of the extent to which the full process chmilil be continued is u 

 followi : Carry it on until the last column but one has at least two 

 more figures than the number of root figures remaining to be fouml. 



Such is the method which miut place its author among those valuable 

 tarantors who find out simple adaptations which have been overlooked 

 by their predecessors. It is not a little remarkable that this, the most 

 important facilitation which the solution of numerical equations has 

 received since the time of Viete, and which is, \ckn tuotm, a very 

 obvious extension of the extraction of roots, should have only preceded 

 by a few years the most important addition to the method of ascer- 

 taining the number of roots which baa been made since Des Cartes, and 

 which is also, when known, an equally simple result of the process of 

 finding the highest common factor of two algebraical expressions. 

 [STCRM'S THEOREM.] 



Two of the most remarkable applications of this method are the 

 solution of equations of the second degree, which U made as simple as 

 the extraction of the square root, and the extraction of the cube root, 

 which is reduced from an impracticably complicated process to one of 

 perfectly easy performance. 



AM an example of the first, required the solution of 



24T-r:r=2. 



Since the root is less than unity, the preparation for decimals is 

 made at the outset. 



10 

 14 

 24 

 14 

 880 



16 

 



16 



200(78077643 

 168 

 "3200 

 8168 

 320000 

 288498 

 81602 

 28864 



4122 



or x= '78077643, which is correct with the exception of the last place. 

 The extraction of the square root, say of 10, is done by solving the 

 equation * +0x= 10; but it will be found that the solution of any 

 equation of the form & + or= 4 may be performed by the same rule as 

 the extraction of the square root. We shall show this, beginning with 

 Homer's rule, and changing to the other after a few steps. Let the 

 equation be 



2 



4 



60 



63 



660 



661 



M 



10(2-81662479 

 8 



200 

 189 



1100 

 661 



6626)43900 

 WM 



66326)414400 

 8979M 



In the extraction of roots, the method of pointing and bringing 

 down the periods as they are wanted may be followed. The following 

 is the process for the extraction of the cube root of 205692449327 ; it 

 being remembered that the question is the solution of an equation of 

 the farm * + Oz* + 0* = a : - 



1 







6 



10 



160 



169 



108 



17700 



177M 





 25 



7600 

 1431 



- . : 



im 



MMM44MS7(MOt 



125 



80692 

 80879 



"81844!i:c.'7 



813449827 



63109 

 1044 



The following process is the extraction of the cube root of 1'808, and 

 will serve as an example of the complete process, omitting only the 

 first column, which, with the exception of the unit at the bead, is 

 blank. And this U also the type of the solution of any cubic equation 

 whatsoever ; the oiily difference being that the beads of the first and 

 second working columns are ciphers in the extraction of the simple 

 root, and significant in all other < 



00 

 a 10 

 620 

 e80 



84 



/36i 



A 362 

 i 3630 

 j 3688 



'/- 3646 



I 86540 

 m 36542 

 n 36544 



o865ti:<> 

 p 365463 

 q 866466 

 r 3654690 

 < 8654699 

 t 3654708 



H 8654717 

 xwv 



The preparation for decimals makes the answer ten times too great ; 

 BO that the cube root of 1-808 is 1-218. . .002, of which only the last 

 figure 2 cannot be depended upon. The preceding contains every 

 figure which need be written down, all the connecting operations being 

 those which are usually performed mentally, and one only is required 

 for each figure. 



The vertical lines in the example show that part of the operation 

 in which the contraction takes place, and the point at which the 

 contraction becomes simple contracted division is marked by a thick 

 horizontal line. 



To enable the beginner to examine the process, we have placed 

 a letter in every line of the first working column, by which the 

 parts of the second column which are connected with it may be 

 traced; while a letter doubled in the second column shows a mul- 

 tiplicand the product of which by the root figure is found as marked 

 in the third column. The letters under the last line of the first 

 column mark the figures cut off in the several contractions, and their 

 results in the other columns are traced in the same way : the same 

 for the letters under the second column. 



One simplification might be made after the learner has practised a 

 number of examples conducted as above. In the second working 

 column, certain lines namely, the second 6, the second e, the second 

 A, &c. are not used except to be added to the next Hue. Hence, each 

 of the lines on which a letter is doubled might be formed by adding 

 the first, third, and fourth preceding lines, and the effect would be to 

 omit some of the lines and some of the most simple additions. The 

 second column, beginning from pp inclusive, is a specimen, and 

 changing the line in which ciphers are annexed (and the ciphers should 

 always be annexed to mark the step) would be 



