173 



CONVERGENT. 



CONVERSE. 



174 



the prisoner committed shall not bo conveyed out of her Majesty's 

 dominions within two months from the time of his committal, any of 

 her Majesty's judges, on application made to them, and after notice of 

 such application has been sent to the secretary of state (or to the acting 

 governor in a colony), may order such person to be discharged, unless 

 good cause shall be shown to the contrary. The act is to extend to all 

 her Majesty's present or future possessions, and to continue in force 

 during the continuance of the convention. 



The act relating to America (c. 76) is similar in its nature and 

 purposes to the one relating to France ; but the crimes specified in- 

 clude, in addition, piracy, arson, and robbery, and do not include 

 fraudulent bankruptcy. "Difficulties, however, have arisen as to the 

 jurisdiction of either country as to crimes committed at sea. A 

 murder may be committed on an English subject on board an American 

 vessel (or vice" versa), and in whichever country it arrives it is unde- 

 cided whether either country has jurisdiction over the case. 



In 1841 a case occurred of a fraudulent French bankrupt who had 

 escaped to England, and the French government demanded that he 

 should be given up under the Convention Treaty. He was arrested 

 and taken to prison ; but before the surrender could take place he 

 applied for writ of habeas corpus, on the ground that fraudulent bank- 

 ruptcy was an offence unknown to the law of England, and that there- 

 fore it was contrary to law to arrest him or keep him in custody on 

 such a charge. The warrant of commitment did not specify that the 

 prisoner should be given up on requisition duly made according to the 

 act, but the words were, " until he shall be delivered by due course of 

 law." In consequence of the defective application of the Convention 

 Treaty in this particular case the prisoner was discharged. 



CONVERGENT, CONVERGENCY, DIVERGENT, DIVER- 

 GENCY. When a series of numbers proceeding without end, has 

 terms which diminish in such a manner that no number whatsoever of 

 them added together will be as great as a certain given number, the 

 series is called convergent. But when such a number can be added 

 together as will surpass any given number, however great, the series is 

 called divergent. Thus, of the two following series 

 111 111 



the first is convergent, for no number of its terms, however great, will 

 amount to 2 : the second is divergent, and the sum of its terms may be 

 made to exceed any number. By going a mile, then half a mile, then 

 a quarter of a mile, Ac., two miles could never be completed : but by 

 going a mile, then half a mile, then one-third of a mile, Ac., a hundred 

 million of miles, or any greater number, could be surpassed. 



The subject of the convergency of series is one of fundamental 

 importance in the whole of the mathematics ; but it is only recently 

 that it has been treated, in works on algebra, in the manner which its 

 importance requires. Algebraical writers once imagined that a series, 

 however obtained, is safe and fit for use, whether convergent or diver- 

 gent. This opens a question which has caused much discussion, and 

 on which we cuinot here enter. We shall state the results of investi- 

 gation on the subject, together with the references to sources of 

 information. 



1. Series of increasing terms are certainly divergent. 



2. Series of decreasing positive terms are divergent, unless the terms 

 diminish without limit. 



8. Of series of positive terms which diminish without limit, a test 

 of convergency or divergency may frequently be given as follows. Let 

 a, b, c, d, e, Ac., be the terms of the series : form the new series 

 bed e f . 



a' V ? d' ? &C < A): 



then, if there ever arrive a term of the series (A), from and after which 

 all the terms are not only less than unity, but tend towards a limit 

 which is less than unity, the series is certainly convergent : but if the 

 terms aforesaid become greater than unity, and continue so from and 

 after a given term, the series is certainly divergent : and if the limit in 

 the first case be not less th in unity, but unity itself, the series may be 

 either convergent or divergent, and each particular case must be 

 examined by itself. Instances of both sorts can be given. 



4. Series of the form a + bx + ex' + dx" + ex* + Ac., can always be 

 made convergent by giving a sufficiently small value to x, except only 

 In the case where the terms in the series (A) increase without limit 

 from and after any term. If they do not increase without limit, let 

 L be the limit ; then the preceding series is convergent whenever "Lx 

 is leas than unity, is referable to the preceding case when L.c is equal 

 to unity, and divergent when L.r is greater than unity. But if L = 0, 

 thj preceding is always convergent. 



6. Series whose terms are alternately positive and negative, are 

 always convergent when the terms diminish without limit, and the 

 error committed by taking any number of terms to stand for the 

 whole value, is never so great as the first term thus rejected. For 



instance, if the answer to a question bel - + 5 T + Ac., then 1 



2t 3 4 



is not wrong by -,1 - is not wrong by -, 1 - + - is not wrong by 



mm 9mm 



2, and so on. The result* are alternately too great and too small. 



6. When such a series as the last has its terms not diminishing 

 without limit, but towards a finite limit, the sum of any number of 

 terms, increased by half the limit, is never wrong by so much as the 

 first-rejected term differs from the limit. 



7. When series produced by algebraical development have their 

 terms alternately positive and negative, the error committed by 

 stopping at any term is never, in any case which the student will meet 

 with, so great as the first rejected term, even though the series become 

 afterwards one of continually increasing terms. If, then, such a series 

 have the first few terms rapidly diminishing, a close approximation 

 may be made by means of them to the real value of the expanded 

 function. For instance, in the series 1 - Zx + 2 . Zx- 2.3. 4.Z 3 -t- . . . . 

 (in which an attempt to calculate from the whole series would be 

 utterly futile, since, however small x may be, there must be terms of 

 every degree of magnitude) when x is small, an approximation may be 

 made to its value from the terms which decrease. Thus if x '1, in 

 which case the series is 



1 - 2 + 06 - 024 -t- 0120 - 00720 + &c., 



(and the first term which surpasses that preceding, is 2 . 3 ..... 11 a- 3 ) : 

 the aggregate of the terms up to 2 . 3 ____ 9.e 8 inclusive, will not differ 

 from the true value of the expression by so much as 2 . 3 ____ 10*", or 

 0036288. The proof of this curious proposition, in the vast number of 

 cases in which it is true, may be deduced from Lagrange's Theorem on 

 the Limits of Taylor's Series. (' Lib. Useful Knowledge,' Differential 

 Calculus, p. 78.) 



Series which are functions of x may be divided into 1. Those which 

 are sometimes convergent, and sometimes divergent, such as the deve- 

 lopment of (l+x) m . 2. Those which are always convergent at last, 

 but in which the appearance of divergency (increasing ttrms) may be 

 continued as long as we please, such as the development of c". 3. Series 

 which are always divergent, but to which a similar appearance of con- 

 vergency can be given, such asl + 2o; + 2.8x s + ____ , and the like. 

 4. Series which are always convergent or always divergent, and never 

 can be made to exhibit any symptom of approach to the other state, 

 such as 



1 / 1 \ , x 



-+[x* + -] + . . . and - - ; 

 x \ x?J 1 + a? 



- - - 

 ! + * 



The series which are always convergent, both in reality and appear- 

 ance, and upon which, therefore, an arithmetical algebraist would 

 reckon with most security, do, in fact, offer difficulties of a very 

 peculiar character. They are the only ones in which the usual alge- 

 braical generalisations would lead to absolute error (so far as has yet 

 appeared). 



On this subject generally, see Peacock's 'Algebra,' and 'Report, on 

 Analysis' ('Rep. Brit. Assoc ,' vol. ii.); Cauchy, ' Cours d' Analyse;' 

 Grunert, ' Supplemente zu Klugel'd Wb'rterbuche der Reinen Mathe- 

 matik,' in the article ' Couvergenz der Reinen ;' ' Encyc. Metrop.,' 

 article ' Calculus of Functions.' See also SERIES. 



CONVERSE. We here state the ordinary theory of the logical 

 converse. Converse, in logic, means a proposition which is formed 

 from another by interchanging the subject and predicate, thus : the 

 converse of " Every A is B " is " Every B is A." But care must be 

 taken to put the proposition in its usual logical form before conversion. 

 Thus the converse of " Every A has a B " is not " Every B has an A." 

 For the proposition first stated is 



-p [~ A ~| |~ is ~| Pa thing which has a B~| 

 "** |_ju6/ecJ [_cop,UaJ 



predicate. 





and the converse is " Every thing which has a B is an A." 



Of the four forms to which all assertions can be reduced, namely 

 (A) "Every A is B"; (E) " no A is B " ; (I) "some As are Bs"; 

 (0) " some As are not Bs ", the logical converses (so called) are those in 

 which the new subject appears with the same degree of generality of 

 assertion as the old one. Thus the converse of " Every A is B ", is 

 " Every B is A.' Consequently in the first and fourth forms, or the 

 general affirmative and the particular negative, the logical converse is 

 not necessarily true. Thus " Every A is B,' does not give " Every B 

 is A " necessarily, but only " some Bs are As'. The latter is called by 

 writers on logic conversion per accidens, a term which, as Dr. Wallis 

 has declined to explain it, we shall leave as we find it, adopting the 

 phrase diminished or limited conversion, and calling the first kind 

 simple conversion. The only other method of conversion which has a 

 definite name is that in which the subject and predicate are made 

 contradictory to the former ones, as when we convert the proposi- 

 tion, " All equilateral triangles are equiangular triangles," into " All 

 triangles not equiangular are triangles not equilateral." This is called 

 conversion by contra-position. Restricting ourselves to converses which 

 are necessarily as true as the direct propositions, we have the 

 following rules with respect to A, E, I, and O above. 



E and I are simply convertible. 



E and A are convertible by diminution. 



A and O are convertible by contra-position. 



. Nothing is more apt to make a beginner believe that " Every A is 

 B " yields " Every B is A," than the study of geometry without close 

 attention to the meaning of terms and the force of the parts of an 

 assertion. For as a majority of the earlier propositions have their 



