853 



INDIGO-BLUE. 



INDULGENCE. 



854 



Indinic Acid is formed in combination with potash, yielding black 

 crystals, by the continued action of the alkali upon indin ; the indinate 

 of potash is readily decomposed by water, and still more easily by 

 acids ; alcohol dissolves a little of it, but by long washing decomposes 

 it. In the ah- it becomes of a light rose colour. Its composition 

 appears to be (C 3 .H,KN S 0,). 



.Chlorine and Uromine Derivatives of Indin are produced by the 

 action of heat on the corresponding derivatives of isathyd. 



ffydrindin (C M H M N 4 0, + 2Aq.) appears to contain the elements of 

 indin plus hydrogen. It occurs in small white prismatic crystals, and 

 is prepared by acting upon indin with an alcoholic solution of potash. 

 The re-action also gives rise to Jtai-indin, 'a body forming pale yellow 

 acicular crystals. 



INDIGO-BLUE. [INDIGO.] 



INDIGO-WHITE. [INDIGO.] 



INDIGOGENE. [INDIGO.] 



INDIGOTIC ACID. [INDIGO.] 



INDIGOTIC GROUP. According to Gerhard's system of classifi- 

 cation of organic compounds, this group, belonging to the BENZOIC 

 SERIES, consists of a number of bodies derived exclusively from indigo, 

 under which the individual members of this group will be found 

 described. Indigotic compounds possess a certain relation to salicylic, 

 kinonic, and phenylic compounds. Thus salicylic acid, nitrosalicylie 

 acid, perchlorokinonpicric acid, and aniline, can be obtained from 

 indigotic compounds; but when once thus broken up, no indigotic 

 body has yet been reproduced from these derivatives. 



INDIGOTIN. [INDIGO.] 



INDIN-. [INDIOO.] 



INDINIC ACID. [INDIGO.] 



INDUCEMENT. [PLEADING.] 



INDUCTION (txayurf,), as denned by Archbishop Whately, is " a 

 land of argument which infers respecting a whole class what has 

 been ascertained respecting one or more individuals of that class." 

 According to Sir William Hamilton the word has been employed to 

 designate three very different operations : 1. The objective process of 

 investigating particular facts, as preparatory to Induction, which he 

 observes is manifestly not a process qf reasoning of any kind ; 2. A 

 material illation of a universal from a singular, as warranted either by 

 the general analogy of nature or the special presumptions afforded by 

 the object matter of any real science; 3. A formal illation of a 

 universal from the individual, as legitimated solely by the laws of 

 thought and abstracted from the conditions of any particular matter. 

 The second of these operations is the inductive method of Bacon, 

 which proceeds by means of rejections and conclusions, so as to arrive 

 at those axioms or general laws from which we may infer by way of 

 synthesis other particulars unknown to us, and perhaps placed beyond 

 reach of direct examination. (' Nov. Org.,' ' Aph.,' c. iii., c. v.) 

 Aristotle's definition coincides with the third, and induction " is an 

 inference drawn from all particulars." (' Prior Analy.,' ii., c. xxiii.) 

 The second and third processes are improperly confounded by most 

 writers on logic, and treated as one simple and purely logical operation. 

 But the second is not a logical process at all ; since the conclusion is 

 not necessarily inferrible from the premise, for the some of the ante- 

 cedent does not necessarily legitimate the all of the conclusion, 

 notwithstanding that the procedure may be warranted by the material 

 problem of the science, or the fundamental principles of the human 

 understanding. The third alone is properly an induction of logic; for 

 logic does not consider things, but the general forms of thought un.ler 

 which 'the mind conceives them; and the logical inference is not 

 determined by any relation of causality between the premise and 

 conclusion, but by the subjective relation of reason and consequence as 

 involved in the thought. The inductive process is exactly the reverse 

 of the deductive ; for while the latter proceeds from the whole to the 

 part, the former ascends from the part to the whole : since it is only 

 under the character of a constituted or containing whole, or as a con- 

 stituent and contained part, that anything can become the term of 

 logical argumentation. Of these two processes, Sir William Hamilton 

 gives the following figures : 



Induction. 

 X T z are A. 

 X T z are whole B. 

 .'. whole B is A. 



or, 



A contains x T z. 

 X Y z contains B. 

 .'. A contains u. 



Deduction. 

 Bis A. 



X T z are under B. 

 .'. X Y z are A. 



or, 



A contains B. 

 B contains x Y z. 

 .'. A contains x Y z. 



This confusion of material and logical induction led Gillies and 

 others to insist on the sameness of the Baconian and Aristotelian 

 induction ; while Campbell and Dugald Stewart, who totally mistook 

 the value of all logical inference, yet rightly maintained their diffe- 

 rence. 



By Aristotle, induction and deduction are viewed as in certain 

 respect* similar in form ; but in others as diametrically opposed, the 

 latter being an analysis of the whole into its parts, by descending from 

 the more general to the more particular ; but the former descends by 

 a synthetical process from the parts to the whole. The logicians, who 



misapprehended the nature of induction, reduced it to a deductive 

 syllogism of the third form, and thereby overthrew the validity of all 

 deduction itself, since the latter is only possible by means of the 

 former, which legitimates the proposition from which its reasoning 

 proceeds. 



Again, the Aristotelian induction was drawn from all the particulars, 

 whereas the confusion which Sir W. Hamilton has pointed out gave 

 rise to a division of the inductive process into perfect and imperfect, 

 according as the enumeration of particulars is complete or incomplete. 

 The latter gives only a probable result, whereas the necessity of the 

 conclusion is essential to all logical inference, as its demonstrative 

 stringency depends upon the form of the illation, and not upon the 

 truth of the premises. It is proper to add, that no one ever knew the 

 distinction between the imperfect and perfect forms of the conclusion 

 better than Aristotle himself. 



INDUCTION (Mathematics). The method of induction, in the 

 sense in which the werd is used in natural philosophy, is not known 

 in pure mathematics. There certainly are instances in which a general 

 proposition is proved by a collection of the demonstrations of different 

 cases, which may remind the investigator of the inductive process, or 

 the collection of the general from the particular. Such instances 

 however must not be taken as permanent, for it usually happens that a 

 general demonstration is discovered" as soon as attention is turned to 

 the subject. 



There is however one particular method of proceeding which is 

 extremely common in mathematical reasoning, and to which we 

 propose to give the name of successive induction. It has the character 

 of induction as defined by the logicians, because it is really the 

 collection of a general truth from a demonstration which implies the 

 examination of every particular case ; but it adds to the necessary 

 character of induction that each case depends upon one or more of 

 those which precede. Substituting demonstration for observation, the 

 mathematical process is truly inductive. A couple of instances of the 

 method will enable the mathematical reader to recognise a mode of 

 investigation with which he is already familiar. 



Example 1. The sum of any number of successive odd numbers, 

 beginning from unity, is a square number, namely, the square of half 

 the even number which follows the last odd number. Let this 

 proposition be true in any one single instance ; that is, n being some 

 whole number, let 1, 3, 5, .... up to 2 + 1 put together give (n + I) 3 . 

 Then the next odd number being 2/JH-3, the sum of all the odd 

 numbers up to 2n + 3 will 'be ( + l)'-' + 2 + 3, or n 2 + 4ra + 4, or 

 (re + 2) a . But + 2 is. the half of the even number next following 

 '.in + 3: consequently, if the proposition be true of any one set of odd 

 numbers, it is true of one more. But it is true of the first odd 

 number 1, for this is the square of half the even number next 

 following. Consequently, being true of 1, it is true of 1 + 3; being 

 true of 1 + 3, it is true of 1 + 3 + 5 ; being true of 1 + 3 + 5, it is true 

 of 1 + 3 + 5 + 7, and so on ad injinitum. 



Example 2. The formula x* a" , n being a whole number, is 

 always algebraically divisible by x a. 



x* a," =*x* a* -* 



=x(x*~ l a-') + a'~ l (x-a) 



In this last expression the second term a*~ l (x a) is obviously 

 divisible by .ca : if then a"-' a" -'be divisible by x a, the whole 

 of the second side of the last equation will be divisible by x a, and 

 therefore x* a" will be divisible by x a. If then any one of the 

 succession 



xa, a? a?, x^a 3 , x*a 4 , &c. 



be divisible by xa, so is the next. But this is obviously true of the 

 first, therefore it is true of the second ; being true of the second, it is 

 true of the third ; and so on, ad injinitum. 



There are cases in which the successive induction only brings any 

 term within the general rule, when two, three, or more of the terms 

 immediately preceding are brought withjn it. Thus, in the applica- 

 tion of this method to the deduction of the well-known consequence of 



x+ - = 2 cos 6, namely, x* + = 2 cos. n 9, 

 x x* 



it can only be shown that any one case of this theorem is true, by 

 showing that the preceding two cases are true ; thus its truth, when 

 n=5 and =6, makes it necessarily follow when n = 7. In this case 

 the two first instances must be established (when n = 1 by hypothesis, 

 and when =2 by independent demonstration), which two establish 

 the third, the second and third establish the fourth, and so on, 



An instance of mathematical induction occurs in many equations of 

 differences, in every recurring series, &c. 



INDUCTION, ELECTRICAL. [ELECTRICITY, COMMON; GAL- 

 VANISM ; MAONETO-ELECTBICIT Y.] 



INDUCTION. [BENEFICE.] 



INDULGENCE is a power claimed by the Roman Catholic church, of 

 granting to contrite and confessed sinners remission of the penalty, or 

 part of the penalty, which they ought to suffer here or hereafter in 

 expiation of their sins. The indulgence does not remit the guilt, 

 "culpa," nor the eternal punishment awarded to the impenitent 

 sinner, but only the temporal penalty which the repentant sinner, 



