T A Y 



T A v 



be. Wo put down tables for the development of this 

 (hoctioa. derived from tlir preceding. tables,** far as the 

 tenth power of x: to be used as follow.-- : 



ut 



Scc. 

 Then 



=4>'a . D"' b + <"a . D*" 



a . b" 



where the derivatives of the powers of 6, which do not 

 mean the same as hitherto, are to be taken from the fol- 

 lowing table : 



D6=c 



!>=/, 



D6=104c 



4-G06<Y> + l.V 



D'6"=156</ 



D6 S = 156V 



I>6=Ar, D i ft i 



r> < 6''=216* ( g'+ 1056r/+70Ae" + 105c-'e 

 1 ) ' 6' = :r>6 1 /+2106V<?+ lOSfc 3 

 J)'6 a =356V+ 1056V, DA=:216V. 



IW = 2S6Vi + 1 G8bc g + 280&/+ 2 10r/+ 280ce 

 IVA'zoCA"^ -t-4206V/+2806V+ai06c i; <;+ 105c 4 



D6'=286e 



D t 6"=566 i ;+2106V s , 



D'6 1 = 366* A + 252&-A + 5046^ +' 315A/< + 378e i # 



+ 12GOr</4-280e 8 

 IV6' = 846Vi + 75Gb t cg + 12G06V/+ 18906e*/+ 25206^ 



' 1 i=12Ci s /+ 12G06Ve+ 1260A 



'= 106ro +45c/+ 120<?A 4- 21Q/7i + 126-' 

 D'6> = 456V 4- 3GOArft -f (W06eA 4- 12606/g- 4- 



+ 2520eeg-4- 1575r/'+2100<? 2 / 

 D6 4 = 1206 n & + 12G06VA + 25206V<r + 1 



+ :J7H06r<# + 12G006c</ + 28006C 3 + 3150cY 



+ (iSOOr'e' 



D6 5 = 2106'A + 25206V^ + 42006V/ 4- 9450AV/ 



+ 12G006're + 12C006cV + 94:V 5 

 = 252//t' + 31506V/ + 21006V? 1 4- 126006 ll c 8 c 



+ 472.V/V 



= 2 106 4 / + 2T)206 s c< + 3 1506V 

 D-6"= 1206V +e>306*c, D6=45c. 



Vi- shall conclude this article by recommending that 

 the process of derivation should be introduced, without 

 demonstration of course, into elementary books of nl: 

 as one of 1helx-.t ei simple al*. r rln:iie:il opera- 



tion. \Vi: iin' firmly of opinion that the arithmetician 

 and the analyst should be trained early in the performance 

 of opeiations in which numerous details, each very Dim- 

 ple in itself, follow one another in rapid MiccesMoii with 

 much sameness and some diversity. Kor this reason we 

 should recommend, in aiithmetic, Homer's process [I:\vn- 

 ON AM) Kvni.i rin\] ; and in algebra, Arbogast's de- 

 rivation. We proceed accordingly to di vest thn method 

 of the phraseology of the dill'ercnlial calculus, and to jiut 

 it before ttie eienientaiy student in algebra. 



The name of the process is il- its primary ob- 



ject the raising of any power of an expression of tin 

 P4-cr + er'+/a + ,4c. immediately, (hat is to say, by 

 writing down the rcsolt at once, without any but simple 

 mental pioceswn in passing from term to term. The rules 

 are as follows : 



1. Begin with that power of 6 which is to be rail 



2. To pass from the coefficient of one power of .r to 

 that of the next, multiply each letter by its exponent : 

 then diminish that exponent by a unit : then Introduce I In- 

 next letter. And if this last process increase an exponent, 

 owing to the letter newly introduced having been in the 

 term before, divide by the increased exponent. />' 

 member iierrr to operate on any Irttrr errfpt the last in 



in, or the last but one; upon the last always, upon 

 the last but one when it immediately precedes 'the kit 

 in the original series 6, r, e,f, tec. 



3. If 6 + rj-4-, Sic. be not an infinite series, but a finite 

 number of terms, operate as if the succeeding letters 



illy equal to : for instance, if g be the last letter, 

 drop every term in which ft should appear, as fast as it 

 arise*. 



For example, the fifth power of b + cx + ex* +/x*. 

 Begin with 6 s , derive from it 56V, the two first terms are 



To form the coefficient of x*, take 56V, and observe 

 tlwit b and r follow each other in the series, so that in the 

 next derivation there are two processes. First, use c or c 1 , 

 the last letter, which by the rule gives IrV or e : so that 

 derivation applied to the first power of a letter gives 

 merely a change of that letter into the next : hence 5//V 

 gives 56V. But 6 4 , which must also be used, gives 46V, 

 and 56V gives 5(46*c)c ; so that c becomes c*. and we must 

 therefore divide by the increased exponent 2, giving 

 106V. Hence the next term is 



(56V4-106V)**. 



In the next derivation 56V gives only 56 < / r , for 6 not im- 

 mediately preceding e in the series 6, c, e, kc. is not used. 

 But 106V gives 



106'(2ee) 



I0'36*c') ft 

 '^- , or 206"ce-f 106V. 



Next term (56 4 /+206ce+106 1 e > )i. 



In the next derivation 56*/ must be neglected entirely, 

 because / is the last letter, and 6 is not the one imme- 

 diately preceding. Also 206V* 1 gives 2()6 : V/aiid 206V- -I- 1 -! 

 or 106V ; while 106V 3 gives 306V" and 2X lOAf.r'-H. or 

 06r 4 . The whole value of (b+cx+eaf+fx 3 f is as follows, 

 and a little practice would enable any one to write it down 

 at once, without any intermediate operations : 



6 s -t-56Vx+(56V 



* + G06V (/+ 106V +206cy+306c" 

 +(306V/ + 306V/+ U06<--i/+ 2( )6-^' + 

 + (30b'fp + 306r */* + 606r<-'/+ 5/-' + 3 V *f/+ 1 Or V ) 



This process, so simple as compared with the actual 

 performance of the four multiplications, has hitherto lain 

 hid in works on the higher parts of the differential cal- 

 culus : it is time it should take its place in every system 

 of algebra which contains the binomial theorem, of which 

 it is the legitimate extension. 



TAYLOR, JOHN, I.I..1V, was born about the year 

 1T"H, at Shrewsbury, where his lather, according to some 

 writers, was a poor shoemaker, or, according to oth 

 barber. He received his early education at the grammar- 

 school of his native place, and afterwards went to Cnni- 

 biidge, where he entered St. John's College, of which he 

 became a fellow in 1730. The great reputation which he 

 soon acquired as one of the best Greek scholars in t Ik- 

 University, procured him tin- nllie.' <.!' librarian of the Uni- 

 versity library, which however he afterwards exchanged 

 I'or that of registrar of the rniversity. His first work of 

 importance was his edition of the Greek orator Lysias, 

 under the title ' I._\siae ( iratimic- et Kragmenta, Graced et 

 I.atim':: ad tidem codicurn MSS. recommit, notis critic-is, 

 interpretation!', cacter,M|iic apparatu necessario donavit 

 Joanni's Taylor,' London, 1731), 4to. The year after he 

 Allied at Cambridge an octa\o edition of the same orator 

 for the use of students, with shori noti-s and a useful index 

 of the language. The study of the Attic orators led him 



