u 



INTEGRATION, FINITE. 



INTELLECT. 



81(1 



which accompanies the UM of dirergent series, which in tlie most 

 important m ahuiuatical question now under discussion. If we ware 

 to judge nf tli. iu:ur.' by the put, we ihouM prophesy that divergent 

 Htin would one dy Uke their undisputed place miunp; well under- 

 stood object* of analysis, as negative quantities and their logarithms, 

 imaginary quantities and their exponentials, infinitely small quantities 

 with their different orden, discontinuous solutions of differential 

 equations, Ac., have successively done, each under a fire of objections 

 which has well served the progress of science, by the defensive 

 Tescsrrhre which 'it hss rendered necessary. U is fortunate for ana- 

 lysis that so many of those who find difficulties propose the entire 

 rejection of the symbols or methods in which the difficulties exist : 

 the proposition excite* those who are against any rejection to efforts 

 which they perhaps would not make, if they bid only to meet the 

 doubts of allies, instead of the attacks of opponents. That the sym- 

 bolic expressions of which we are speaking will never vanish out of 

 remembrance, we may confidently predict : of all the point* of diffi- 

 culty of which we have spoken, it may be said, in the words of 

 Horace 



" Naturam expellu run*, tamen Mque recurret ; " 



they will come, and will demand explanation until they get it. They 

 will conquer by numbers, as Fontenelle said the symbol of infinity had 

 done. And it is to be hoped and expected that no difficulty will be 

 completely resolved, without the appearance of a successor, to excite 

 new efforts, and be the stimulating cause of further progress. We 

 should be sorry to think we had arrived at the " last impossibilities * 

 of pure mathematics." A very valuable accession to the literature of 

 definite integrals has been recently made by Mr. Bierens de Haan, who 

 has filled the fourth volume of the ' Transactions of the Royal Academy 

 of Sciences of Amsterdam ' with a 'i- 8 ' f definite integrals, with their 

 values, aqd references to the authors who have given them. The list 

 occupies 550 quarto pages. 



INTEGRATION, KINITE. By this term is meant the summation 

 of any number of terms of a series which follows a regular law ; and 

 just as INTEGRATION was reduced in a preceding article to the deter- 

 mination of a function from its differential coefficient, so finite inte- 

 gration or summation may be reduced to the determination of a function 

 from its difference. [DIFFERENCE.] 



First, let there be a function of x, $x, and let x successively become 

 x + Ar, x + 2&r ....... up to z + (n 1 )&x, so that n different values 



are given to x. It is required to sum the series 



jx + 4>(x + Ax) + #* + 2 A*) + . . . + <*(x + H^lAx) 

 Let *= rAjr, and let 4>(t>A*) be called ifr. Then the series becomes 



This sum is a function of , and such, that if be changed into 

 + l, one more term if( + n) will be added : consequently it must be 

 the function which has f (*) for its difference. If, then, we denote 

 the preceding sum by X4>(v + n), we find 



or A and 2 express operations which are inverse to one another. 

 Remark that the symbol Za does not denote the sum of a number of 

 terms up to a inclusive, but up to a exclusive : thus 



1+2 + 3+ + (-!) + is X( + l) 



All that precedes has no reference to the term with which we begin : 



thus 4+5+ +n and 1+2 + 8 + 4+5 + +n arc equally denoted 



by 2(n + 1). This symbol U therefore indefinite, but it will be found 

 that the process by which it is to be determined give* an indefinite 

 result. 



Suppose, for Instance, we have ascertained that 4 ("' + ) is the 

 function whose difference is (n + 1), which will be found to be the 

 case; or 



} { B * + j = + !. 



It is equally true that | ( : +) + C has n + 1 for its difference, where 

 c may be anything whatever, provided that it do not change when n 

 changes. Hence 



The inverse method of difference*, o/ that of finite integration, is 

 'ounded upon the preceding principles and notation : but no far as the 

 nere summation of simple series is concerned, the following rules will 

 j sufficient : 



1. Let a be the first term of a series of terms, a, 4, c, ic. Form 

 the successive differences of a [DIFFERENCE], which will all vanish 

 after a certain point in every instance to which this rule applies. 

 Then the sum of Uu> n terms is 



but a being any whole number less than , 3 (n + 1) may stand for a + 

 (a-t- 1) + ..... + n. Consequently c in the one must be taken in a 

 manner corresponding to a in the other. If n were equal to , tin; 

 series would be reduced to one term a, and 4 ( n 1 + n) + c would become 

 4 (a + o) + C. Dftunuiuo c so that these shall be equal : we have 

 then to make 



4 (o' + o + c, c*= -4 (cf-a) 



Tbew * th word, of Mr. A. Y. Vofl, of Lrlptlc, who pnbllnhcd in thl> 

 eoutry irwit on UM resolution of U kinds of equstiom, printed at Ltlptlc In 

 is own 



- 1 -l - 2 



* a + o~ A a + * 5 o 



EXAMPLE : 1 + 8 + 27 + 64 + 125 + 

 First diff. 7 19 87 61 



Second diff. 12 18 24 

 Third diff. 6 6 



Fourth diff. 



Here a = 1, A a = 7, A 1 o = 12, A'o = 6, A* a = 0, 

 A* a = 0, Ac. 



and the sum required U 



+ +n-2 3- + -a g ~ 



It may be convenient to give the reduction of the preceding formula 

 in the oases where all after the second differences vanish, and the same 

 for the third. Let a', a", a'", &.C., be the differences of a ; when o"' = 

 0, u" = 0, &c., the sum in one-sixth of 



o" ' + (a' -o') 3 n 5 f (6 a - 3 a' + 2 a') H. 

 When a"= 0, a*= 0, Ac., the sum is one twenty-fourth of 



a"' n* + P n s + <J n 1 + R n 

 where p = 4o"-6o'" 



q = 12 a' -12 a" + 11 a"' 



R = 24 a - 12 a' + 8 a" - 6 a'" 



2. Let there be a number of terms in uniformly increasing progres- 

 sion, such as 4, 44, 6. 54, ic. ; and let a series be formed by multiplying 

 a number of terms from the beginning, then the game number from 

 the second, and so on, as in 



3.4.5 + 4.5.6 + 5.6.7 + + 12.13.14 



To find the sum of this series, put an additional factor at the end of 

 the last term and at the beginning of the first term ; subtract the 

 latter from the former, and divide by the common difference of the 

 successive factors taken one more time than there are factors in each 

 term. Thus the sum of the preceding is 12.13.14.15 2.3.4.5 divided 

 by 1 taken 4 times. Again 



1.2 + 2.3 + 8.4 + 4.5 + 6. 



is 5.6.7 0.1.2 divided by 1 taken 8 times ; or 70 ; as may easily be 

 verified. Also 



1.3 + 3.5 T 5.7 -r 7.9 



is 7.9.11 ( 1) 1.3 divided by 2 taken 3 times; or 118. 



3. Let the series consist of reciprocals of terms like the preceding : 



To sum this series, strike off a factor from the end of the first term 

 and the beginning of the last term ; subtract the second from the first, 

 and divide by the common difference of the successive factors taken 

 one tiino leu than there are factors in each denominator. Thus the 

 auui of the preceding terms is 



jj-j j 3 ^ . divided by 1 taken twice 



jfjj 



57 



nce ; 



Similarly j~<j + jfjj + 5 



3 

 or =, aa may easily be verified. 



INTKU.KCT (inidltctiu), that which perceives and understands, 

 comprehending all the cognitive powers of the mind, in contradistinc- 

 tion to the active powers or the will. " The internal and immanent 

 acts of the reasonable soul (besides those of common sense, phantasy, 

 memory, passion and appetite, common to man and inferior animals) 

 are intellect and will, and the proper acts of these ore intellection, 

 deliberation, and determination, or decision." (Male's ' Origin <>i 

 kind.') In the Aristotelian philosophy the intellect (rain) first uorkn 

 on the phantasms, and discerns by a spontaneous energy what in many 

 is one, what in things diitximiUr ix similar and the same (ri> oi tf nomv 

 TOVTO t yovt fmurror). By this means it attains to a new kind of )>er- 

 oeptiona (ifti)), more comprehensive than those of sense ; and each of 

 these general ideas subsists entire in each individual of an infinite 



