II 



-1) DlSCOVKItY. 



1NVKKSK, IXVKKSION. 



of all UM countries in which Greece had colonied. So far 



from it. that when in the sixteenth century Murolioo lived and wrote, 

 the (allowing epigram was made upon him 



T* qaoque Zttcla tallt, Maurolrce, m sit In ano 

 CUra 8/iaeoslo SteeU at* teoe. 



The moat difficult question of all U undoubtedly what degree of 

 merit belong* to a discovery, and the Battlement of the question where 

 in the list it place* its author. The public in general judges by 

 utility; whereas it it notorious that many discoveries show more 

 power of mind than others of much greater value to the world. The 

 rule of utility is a good one for mankind in general ; but it must be 

 taken with modification by the historian of science. Who has most 

 benefited his species, and who has shown himself most above his 

 species in intellect, are two different questions. The merit of the 

 inventor, and his genius, are not comparable quantities. The merit is 

 determined l>y the study, the personal risk or inconvenience which it 

 was requisite to undergo, the patience and perseverance which must 

 have been shown, and the goodness of the motive which appears to 

 have actuated. The genius is the greater the leas the pain and labour, 

 and is wholly independent of moral consideration!). A patient school- 

 boy who multiplies one number by another with time and care, has 

 more merit than the wonderful youths who have sometimes appeared, 

 and who can do it in their heads ; but the first has far less mental 

 power, in this one line, than the second. All these things are plnin on 

 the first statement ; but they are far from receiving due attention, 

 and will so remain until the history of discovery is written without too 

 luiu-h deference to popular reputation. 



We may mention, as a thing to be guarded against, the disposition 

 to depredate a discovery because it is not something more than it 

 pretends to be, which is frequently combined with a wish to judge of 

 its merit by an arbitrary A priori standard of what it ought to have 

 been. Sturm's theorem in a very pretty instance. Before it appeared, 

 a purely theoretical and strictly certain method was eagerly nought 

 after, and any such, however difficult, would have been held a great 

 gain. The object U at lost attained, but in a manner which is trouhlc- 

 aome to use. To look at the way in which gome writers now mention 

 it. one would suppose they had entirely forgotten how many investiga- 

 tors of the first order had given up the subject without producing any 

 method at aU. 



Since the preceding article was published, many confirmations have 

 arisen of our assertion that for the most part discoveries are made by 

 more than one person when the right time arrives. The eighth 

 satellite of Saturn was discovered on one and the s.une day in England 

 and America ; and several of the smaller planets, those specks of 

 which there are hundreds, perhaps thousands, in our system, have been 

 twice announced within a few days by observers of whom the second 

 was ignorant of the success of his predecessor. But the most 

 remarkable case of simultaneous investigation which ever occurred 

 was the solution of the inverse problem of gravitation by Leverrier 

 and Adams, which resulted in the discovery of NEITUNE by Qalle, at 

 Berlin, on information furnished by Leverrier. It is not our intention 

 to enter on this celebrated case : we shall only make a remark on the 

 di/Ktiwion which it produced. [ASTEROIDS.] 



It soon became apparent that in the scientific world, both of 

 England and France, the word publtfation retains its old meaning, 

 though in common language it now means circulation by help of the 

 press. To publish is, in science as in law, to make known to others, 

 no matter how. Some little disposition was shown by a few to con- 

 found publication, the genus, with printing, the species ; but without 

 producing any general assent. And a reference was made to H'art'm/'* 

 rule, a phrase of which we gave no explanation in the preceding article. 

 We proceed to supply this omission, because we have since heard it 

 cited as the rule to which men of science appeal in matters of priority 

 They follow, it i Raid, the rule laid down by Waring, that tin- firsl 

 publisher is to be held the discoverer : and it is frequently insistec 

 that this word publication is used in the modern and restricted sense. 



Waring (' Medit. Analyt,' pref. p. ii, 1 77H) Kinking about a chum of 

 John Bernoulli to something which had been shown both by Newton 

 and Leibnitr., m having been found by himself before he had seen 

 their writing*, speaks as follows 



"Bid MOM potMt MM Intli in HIM pirtc, U mlhl mnptr dicendn* a 

 intftitor, qoi prlmtn rrulgtTlt, rcl nltem cam mid commnnlcarlt, vix ruin 

 larraiatur allquii, dlfrniu mathcmaticl nomine, qul de inn Ingenin mult 

 crifrtorum prionua reperU baud delciit." 



That u to say, Waring held the inventor to be the first who ervlyatc 

 -tells to all, no matter how or at least tells to his own friends. An 

 Li truth even these conditions are not so much Waring's rule, 



117.. _._ . I. r .\ t i - . . . 



ii' >i, .-" i if nil 11 n CM U IK n I III'', (- 



Waring* example* of the way in which discovery may be prove* 

 inder his rule. That rule is implied in his opening dictum, that n 



under MM .-, * n.*v i mv a implied in ms opening nictum, mat n 

 man mar be his own witneas ; that is, evidence distinct from his own 

 and sufficient without it, must be produced. And this U commo 

 ewe. A man must prove that he is the discoverer : it is not enotig 

 to prove that he is either the discoverer, or a nelf deceiver or a rogue. 

 For ourselves, we admit no right in any one to lay down a rule : a 



we should be pressed to state some rule, we can only say that 

 e hold by the rule in Chrononhotonthologoa : 



"Col call a coach 1 and let a coach be called 1 

 And M (V mm via calleth to tta callrr ! " 



Let the man who discovers be held the discoverer ; and let evidence 

 settle who that man is : and let evidence be that which makes 

 mowing and reflecting men believe ; we know no other definition, 

 ind if knowing and reflecting men be seriously divided in opinion, let 

 ; be held undecided who is the discoverer. The reason of this rule U 

 liat it always has been the rule, is, and always will be the rule. The 

 ttempt to lay down a law of assignment of discovery has not succeeded : 

 lie law is quoted with respect until a dispute arises, and then all the 

 acts of the case are discussed, which is tantamount to a down 

 refusal to obey the law. For it is to be observed that the rule of first 

 ublication is of no import except in its negative or prohibitory 

 haracter. That the first publisher, no other fact except first public.i- 

 ion being on the record, shall be held the discoverer, needs no 

 reduction to rule : it is mere conclusion of course. The asserted rule 

 ne:ins that first publication shall exclude from the discussion all other 

 onsiderations of fact, or shall be conclusive against them, if introduced, 

 'his it never has been, and never will be. 

 INVKNToliY. [ KM-.TTOR.] 



INVKIfSK. I N V K1ISION. Any two operations of algebra are said 

 o be inverse when one of them undoes, so to speak, the effect of the 

 ither; so that if both be successively performed upon the same 

 [iiantity, the result is that quantity itself. For instance, the operations 

 tnplied in 1 -far* and V (x - 1) are inverse to one another; for 



We need do no more than name addition and subtraction, multipli- 

 cation and division, raising of powers and extraction of roots, as pairs 

 of inverse operations. 



The operation of inversion is the solution of an equation and vice 

 vend. Let it be required to find the operation inverse to <f> f. Assume 

 ka;= y, and find x in terms of y say .r = if >/, then 4>(<f y) = y, or f and 

 IT are inverse operations. Thug if x*-2j=y,;r=I + Jg + 1, and 

 either of the two, 1 + V x + 1, or 1 /x + 1, is inverse to x*-ir. 



It thus appears that a function may have more than one inverse 

 'unction, and there ore functions which have on infinite number ; but 

 .here ig a distinction by which one may be separated from all the rest. 

 Let the Greek letters in this article be all functional symbols, or marks 

 of operations to be performed, and let them come before the subject 

 of operation, the quantity x, or y, &c., in the order in which they are 

 :o be performed. Thus aip.t denotes the result of performing the 

 operation f> upon x, and then the operation a upon <p.r. Now let 

 =z give z=if r, where $ x is an unambiguous operation, and if is, 

 generally speaking, ambiguous, or presenting several different forms. 

 Then <p and if are inverse operations, and f tf z = , and "we might suppose 

 at first that x if <t> x ; that is to say, we might imagine that if destroys 



as well as that $ destroys if. But since if is ambiguous, it may be 

 h.it only one or more of the forms of \f will satisfy x if <?> x, and not 

 all ; and that this will be the cage with one ig obvious, while we can 

 show that it cannot happen with more than one. For though the same 

 operation, performed on different functions, may produce the same 

 function, yet different operations, performed on the same function, 

 must produce different functions. If then a and /3 be different forms 

 of if, we have $a.x=x and <t>ftx=x; but we cannot have l*>th a 

 and /3 $ x = x, where a and $ are different, j>x having absolutely the 

 same form and value in both equations. 



From all the inverses of a function $>.-, then, we separate that one, 

 ox, which gives both ^o.x-x and tL$x = x, and call it the conrertible 

 inverse. Ite symbol is -', so that 4>~' .r means that operation which 

 satisfies both the equations 4> T >~'x=2 and q>~ l 4>x=x. [KMMNIM.] 



tible inverse of 



In the preceding example 1+ V( 

 z-2.r: forl+ 



convertible 

 But 1 - 



is the 



gives 1 -(x-l) or %x; and we call l-V(* + l)an inconvertible 

 inverse. 



, function which has more than one inverse in not only a 

 function of .r, but the mine function of other functions of .<-. I,ct a./ 

 be an inconvertible inverse of x; then a <p.r is not .r, let it be a .r. 

 Then <f a x being <t,f*fxltfx,attfmx\tfx,Ki that <f> x is the same 

 function of x which it is of x. Thus in the preceding example x 1 2 x 

 is the same function of 2 t which it is of x ; or 



a!-2a;=(2-x)-2(2-i). 



\V. have then this theorem : every function has as many different 

 forms as inverses, and all these forms can be made by writing different 

 id of ./ in the original function; and each inverse 

 of the function Is the convertible inverse to one of its forms, and an 

 inconvertible inveise to all the rest. Thus 1 V(*+ 1), which 

 inconvertible inverse to x* Zx, is the convertible inverse of (2 j;)' 

 2(2-1): for 



The way to make the convertible inverse of a given function find all 



