410 



APPROXIMATION. 



APPROXIMATION. 



4-0 



or an approximate value of h : so that the new value of x obtained 

 rom the step just made is 



""TV 



Vith this new valne of x we may recommence the process, and find a 

 lew correction ; and so on. 

 Resuming the example, we find putting a = 21, 



fa = a 3 - 2a 5 = '061, 



inverse process, and, if of sufficiently frequent occurrence, a name is 

 given to it, and the rule for finding it is put into words, and arranged 

 in its most systematic form. Thus the process of squaring/ or multi- 

 plying a number by itself, is known when multiplication is known, and 

 the question is easily answered, what is the square of 2} or any other 

 number, or what result* from the process of Hjuaring employed upon 

 the number 2J ? From this arise such questions as the following 

 The result of squaring is found to be 50 ; what was the numfar 

 employed .' This can only be answered approximately; that is, no 

 number squared can give exactly 50, though one can be found, the 

 square of which is as near 50 as we please. This operation occiirs 

 sufficiently often to receive the name of the extraction of the square 

 root, and the rule for approximating to it is well known. We can now 

 carry the generalisation a step farther, for the result of the last is to 

 put a new process into our hands which we may consider as direct, since 

 the means of performing it in all eases, approximately at least, have been 

 found. We may now ask, what is the result of the process denoted by 



any number being substituted instead of x : but the inverse question 

 namely, suppose the above process to have been performed, and the 

 result to be 20 ; what number was employed ? presents itself and 

 requires new investigations. . Neither the direct nor inverse process in 

 this case has received a name ; and it is evident that, name as many as 

 we may, each addition will give new processes, require new inverse 

 processes, and so on ad infinitum. 



Previous to entering upon the process of approximation, it is neces- 

 sary to inquire into the effect which a small change in the number 

 employed would produce upon the result. We say a tmall change, 

 because the changes of any magnitude require formulae of great in- 

 tricacy, compared with small changes. The consideration of the effect 

 of such changes is, among other things, the object of the DIFFERENTIAL 

 CALCULUS ; into which we can here enter no further than to state, 

 that in connexion with every process it discovers others, which we 

 shall here call by the names of the first derived process, the second 

 derived process, &c. ; the two first of which are indispensable, the 

 first for obtaining the approximation, the second for ascertaining the 

 degree of accuracy to which the approximation has been carried. 

 These derived processes (as we here call them) are the first and second 

 differential coefficients. [DIFFERENTIAL CALCULUS.] 



Let fx represent the required process or FUNCTION. Let /' x and 

 f'x represent ita first and second derived functions. We suppose 

 this notation known to the reader; but any one who has studied 

 algebra may be prepared to follow us by reading the first thirteen 

 pages of the treatise of the Society for the Diffusion of Useful Know- 

 ledge, entitled ' Elementary Illustrations of the Differential and Integral 

 Calculus.' If the operations which fx indicates to have been performed 

 upon x, be successively performed on a and a + h, giving fa and / 

 (a + h), it may be proved that 



/"(+) 



2 



(A), 



where 8 is a fraction less than unity, or Bh is less than h. This rule 

 only admits of exception where fx is such that either /" x becomes 

 very large, or f'x very small, for some value of x lying between a and 

 a + h ; and since in approximations A is a very small quantity, this 

 will rarely happen, and when it does happen, the results of an attempt 

 to approximate will soon point it out. Let ug now suppose that we wish 

 to find x in such a way that fx= 0. Every case may be easily reduced 

 to this : for example, to solve x 1 = 7 is to find or approximate to a 

 value of x, which makes z' 7 = 0. The first step is to find by trial 

 some value of x which will very nearly satisfy the proposed condition, 

 that is to find a, so that fa shall be small. No general rule can be 

 given for this part of the process, which is, however, easily done in 

 most cases. To carry an example with us, let us suppose it required 

 to solve the equation : 



a? - Zx = 5 



or to make 



a* Zx 5=0 



Here/x is x* Zx 5, and by 'the rules of the differential calculus, 

 f'x is 3.Z 3 2, and /" x is 6x. We soon find that there is a rool 



between 2 and 21, for if x=2, then x 3 2x is 4, and if x= 21, it is 

 S'061 ; the first less than 5, the latter greater, but not much. We 



therefore take 21 as the approximate value of x found by trial. 



Returning now to equation (A), let us suppose a the approximate 



value increased by h, in such a way that a + k shall be the real value 



of x required, or / (a + A) = 0. This gives 



h= 



(a + Bh) ( B ) 



in which h is not, strictly speaking, determined, because it occurs on 

 the second as well as the first side. But h is small, because a is nearly 

 the value required, and therefore we may approximate to the value 

 A from (B) by rejecting the small term 



from the denominator of the fraction, which gives 



ARTS ASD sci. err. VOL. i. 



. 



/'* 



A = 



061 



= '0054 nearly, 



nearly. 



Prying this value in x 3 1x 5, we find it '005, nearly ; less than the 

 tenth part of its preceding value. With 2'0946 for a, the process 

 must be now repeated. 

 The degree of approximation thus obtained may be estimated as 



follows, though we can only very briefly explain it to those who have 

 no more practice in the differential calculus than we have hitherto 

 supposed. Resuming the correct equation (B), we see that, if we call 

 fa, as obtained, a small quantity of the first order, (fa)- of the second, 

 and so on, then h will be of the same order as fa, unless f'a, be also of 



,hat order, which is one of the excepted cases. Hence, in rejecting Oh, 

 we reject only quantities of the first order from the term /" (a -t- Oh), 

 or of the second from 4 h f" (a + 8/i), or of the third order from the 

 whole fraction, since fa is itself of the first order. This will appear 



rom the development of the second side of (B) by common division. 

 Thus rejecting Oh, and developing 



fa + 4 h f" a 

 is far as terms of the second order, we have 



f'a, \ f'a 2 * C 'J 



n which, if on the second side we write ^ for h, which rejects 



;erms of the second order only, we still reject terms of the third order 

 only in the value of A. Hence 



and its ratio to its preceding value !, is 



1 _ a f" a ? a 

 2 (/'a) 3 ' 



whence * a ^-- represents roughly the greatest part of itself, b'y 



which the correction &~ may be erroneous, the sign indicating whether 



/ a 



it is too small or too great. In the preceding example, where a = 21, 

 and where 



fa = a' 2a 5 = '061, 



fa = So- 2 = 11-23, 



f'a = Ba = 12-6, 



the preceding fraction is roughly , so that the correction '0055 



32 



may possibly be one thirty-second of itself too great, or about '0002 

 too great. 



This method does not appear to be of much use for the second 

 approximation ; but becomes more powerful at every succeeding step. 

 Whatever number of correct decimal places is obtained at the end of 

 any one of the successive approximations, it is, roughly speaking, 

 doubled by the next ; since the second term of the preceding develop- 

 ment of h, being 



l(fW (fa) *' 



is of the same order as the square of h, or of the same order as 



7~" 



In treating the various articles, DIVISION, SQUARE ROOT, &c., EQUA- 

 TION, we shall show that principles analogoxis to the preceding have 

 been adopted in the rules for approximating. 



Various methods of approximation are found in the Hindoo Algebra; 

 but, as far as we can find, Vieta is the first who generalised the main 

 principle so far as to connect the approximate solution of equations 

 with the particular cases of division and the square root, which were 

 known before Hutton, in his ' History of Algebra ' (see his Tracts), 

 attributes this extension to StevinuH, but on searching the works of 

 the Intter, we cannot find anything which, in our opinion, justifies tho 



