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article on the me of the word infinite in the exact sciences : the ideas 

 >1Trf ...^ true or false, are thoM which will, in the first instance, 

 present themselves to the mind ; and thow who object to one method 

 of expression will embody the same thought* in another. 



The other extreme in the scale of quantity is the perfect absence of 

 all magnitude, expressed in the word " nothing " or the technical term 

 "tero." It is necessary to treat the two together in mathematical 

 reasoning, since all difficulties which belong to the one term belong 

 equally to the other. We have also to consider the words " infinitely 

 small " as well aa " infinitely great.'' 



There are three distinct methods of proceeding in regard to the 

 employment of these terms in mathematical reasoning. Firstly, we 

 have those who would use the words " infinite " or " nothing " in tlu-ir 

 absolute sense, relying upon the reality of the conception which they 

 hare of the things signified by them. Secondly, there are others who 

 would entirely banish the use of the words, because in their absolute 

 sense they do not represent assignable magnitudes. Thirdly, others 

 admit the use of the words, guarding them by definitions which point 

 out the processes in the expression of the results of which they may be 



To the first it is answered that the absolute use of <*> and (the 

 i,.t),-t,iAt irM symbols of infinite magnitude and absence of all mag- 



nitude) in the same manner as symbols of definite quantity, is extremely 

 liable to lead to error ; which was never entirely avoided by the advo- 

 cates of this system, except by abandoning their theory, and applying 

 in practice the maxims alluded to under the third of the preceding 

 heads. The absurdity of an absolute and unrestricted use of these 

 terminal symbols may be very easily shown, if it be maintained that 

 they are to be used precisely as other symbols. For example, it will 

 readily be conceded that if a times x be y. and if b times x be y, then 

 a and 6 must be equal. Now twice an infinite magnitude is infinite, 

 'and three times an infinite magnitude is infinite, therefore 2 and 3 

 must be asserted to be equal. The advocates of the unrestricted use 

 of co and avoid such results by a method of selection which amounts 

 to keeping within the definition presently to be noticed. 



To the second of the throe sects above mentioned it may readily be 

 conceded that they have a right to refuse any branch of mathematical 

 reasoning, so far as themselves only are concerned. But we deny that 

 the code of mathematical controversy contains any such axiom as that 

 " mathematics is the science of assignable magnitudes only," by which 

 to chum the submission of an opponent. The general rule is, that 

 mathematical demonstration exists wherever there in logical deduction 

 from universally obvious maiim. with respect to magnitude. Nor 

 does the word " universally " here mean that such maxims must have 

 been obvious to every individual of the human race. If so there 

 would be no such thing as mathematical demonstration : for there 

 have beenr found instances in which persons have denied that the sum 

 of all the parts makes up the whole. It would not be very easy to lay 

 down a rule by which it should be determined what fraction of dissent 

 is fatal to an axiom, but the following appears to us to be the practice. 

 When any individual who has been successful in advancing the mathe- 

 matical sciences, and whose talent and originality are widely known, 

 disputes what is usually received as a first principle, it is customary 

 for subsequent writers on the same subject to preserve his objections, 

 and place them before the reader. If two or three such persons unite 

 in an objection, the fact of there being a majority of the same class on 

 the other side would not save the principle attacked from being con- 

 sidered as dubious. All differences which affect results are very soon 

 settled ; but those which only array one mode of attaining a certain 

 conclusion against another, depending as they i In tr the most part 

 upon the manner in which fundamental and indefinable terms are 

 understood, are generally perpetuated from one age to another. Now 

 it is a proposition which is very rarely disputed, that the science of 

 mathematics is conversant with more than assignable magnitude, and 

 that the notion of infinity, though requiring to be used with care, is 

 one with regard to which sound and obvious maiims of reasoning may 

 be laid down. 



We proceed to state these principles, that is, to enunciate the method 

 followed by the third of the sects mentioned. If we look at the man- 

 ner in which we derive the notion of infinity, we shall not find any one 

 who imagines that he absolutely grasps infinite space, time, or number, 

 by one single and independent conception of his mind. To space, 

 space may be added, to this again space may be added, and so on with- 

 out limit, until the space thus accumulated is greater than any definite 

 space [IxDEfiMTK] which was named at the outset of the process. 

 From thence comes the notion of infinite : we cannot imagine the 

 greatest possible space, because any space, however great, being dis- 

 tinctly conceived, we can as distinctly conceive a greater. Consequently 

 the phrase " space is infinite," whatever more it may imply, certainly 

 may be allowed to stand for an abbreviation of the preceding two 

 sentences, And in like manner, if we see a conclusion which we can 

 nearly attain by the use of a large magnitude, more nearly by the use 

 of a larger, and so on without limit, that is to say, as nearly as you 

 please, if we may use a magnitude as large as we please, but which is 

 never absolutely attained by any magnitude however great then such 

 conclusion may be mid, for abbreviation, to be absolutely true when 

 the magnitude is infinite. It may appear to some as if the conclusion, 

 under the preceding circumstances, is really true when the magnitude 



in infinite : this may or may not be the case, but the mathematical use 

 of the word infinite doe* not require the question to be raised. The 

 convention under which that term is introduced demands tl. 

 preceding conditions shall be fulfilled, and excludes the word wh< 

 they are not fulfilled : those who think that the fulfilment <>f the con- 

 ditions makes that which we call a convention a necessary consequence, 

 meet on common ground with those who would reject the . 

 notion of infinity. The former are allowed their own words, and their 

 own result, together with their own method of arriving at it ; the 

 latter are not required to use the word infinite, except as an abbrevia- 

 tion : to the mere collocation of the letters which compose that word 

 they can hardly object, and the conditions of it* introduction are 

 precise and intelligible. We shall now give a few instances of the 

 development of propositions in which the word infinite appears. 



1 . When z is infinite, A is equal to B. This may be said in abbre- 

 viation of the following : When z is great, A is nearly equal to B ; A 

 may be made as nearly equal to B as you please if we may take z as 

 great as we please ; but no value of ;, however grr .: , will make A 

 absolutely equal to B. 



2. A finite quantity x, divided by an infinite quantity, is nothing. 

 If f be divided by a comparatively great quantity, the quotient is 

 small ; this quotient may be made as small as you please, if we may 

 take the divisor as great as we please ; but no divisor, however great, 

 w ill make the quotient absolutely equal to nothing. 



3. Kvery circle is a regular polygon of an infinitely great number of 

 infinitely small sides. An inscribed polygon of a large number of 

 small sides nearly coincides with the circle ; a polygon may be made to 

 coincide with the circle as nearly as you please, if its number of sides 

 may be as great as we please, and their lengths as small as we please ; 

 but no polygon, however great its number of sides, and however email 

 the sides, can absolutely coincide with the circle. 



4. When x is infinite A and B ore both infinite ; but A is infinitely 

 greater than B. This may be said when if z increase, A and B both 

 increase, so that A and B may both be mode greater than any quantity 

 you name, provided we may make x as great as we please : provided 

 also that A increases faster than B, so that when you name any number, 

 however great, we, being allowed to make x as great as we please, can 

 make A contain B more than that number of times. 



.">. When x=a, z is infinite. This may be said when z is great if x 

 be nearly equal to a, and may be made as great as you please, if 

 be made as nearly equal to a as we please: provided that, li- 

 near x may be to a, z has still on assignable value. 



The preceding instances are sufficient to show what is meant when 

 the terms " infinity," " infinite," or " infinitely great," appear : we now 

 proceed to the correlatives " nothing," " infinitely small," " evanescent," 

 tc. The independent use of the term " infinitely small," as laid down 

 by some writers, is yet more difficult than that of infinitely great If 

 A be an assignable magnitude, x is said to be infinitely small when it is 

 so small that it is absolutely incomparable to A, so that \ + ,r and A 

 may be taken as equal. Now, unless x be absolutely equal to nothing, 

 this cannot be ; so that the infinitely small quantity, as thus defined, 

 can have no magnitude whatever. Here we seem to rest, not iu an 

 absurd, but in a useless conclusion : for what possible benefit can arise 

 from inventing a new word to stand for the n"t/iin;i l>y which two 

 equal magnitudes differ. A little further -consideration of the term 

 " nothing" will here be necessary. 



There is one process of arithmetic which yiel In an absolute 

 namely, subtraction. From a take a, and nothing remains. Con- 

 sequently, in considering the idea of the absence of :ill m.i "lutude, we 

 u MI. illy refer it to the result of that process by which it is directly and 

 unambiguously obtained. But from no other process of ariti 

 does this idea arise, except by the same train of ideas which lead- 

 the use of the word infinite. We cannot, for example, obtain the 

 quotient" nothing" liy dividing one finite magnitude by another; we 

 can make the result small, smaller, as small as we please, In 

 absolutely nothing. When therefore we consider an e M ;i at ion made by 

 addition and subtraction of terms, the absolute result may be used 

 without recervation : tint-. -.!./ + 3 = o, and 1x + 3 - a 

 written for each other without any particular examination 

 symbol 0. But in any other case we can on], i- the limit 



towards which we approach by an interminable succession of diminu- 

 tions, no one of which is ever final, corresponding to the 

 succession of augr y which we attain the notion of it, 



In strict analogy therefore with our former proceeding (mutatis 

 mutandis, we repeat our words) if we see a conclusion which we can 

 nearly attain l>y the use of a small magnitude, more nearly by the use 

 of a smaller, and so on without limit, that is to ray, as nearly as you 

 please by the use of a magnitude aa small as we please, but v. I 

 never absolutely attained by any magnitude however small; then 

 such conclusion may be said, for abbreviation, to be absolutely true 

 he magnitude is nothing. The sentences immediately following 

 the first occurrence of the preceding words may now be repeated, 

 only changing " infinite " and " infinity " into " nothing." 



But in the meanwhile the term infinitely small does not appear, and 

 its use seems to be superseded by i " nothing." And 



it is true that " nothing," introduced under the preceding conditions, 

 might supply the place of an infinitely small quantity. But since 

 there is an absolute use of the term " nothing," derived from ( 



