Ill 



ANALYSIS. 



AVAMERTA. 



Sin 



m when we speak of the proportion of death* to birth*, the proportion 

 >f wages to prate, the proportion of oanriotioM to commitment*. *<-. 

 Sometime* aUo it U ueed for portion, u when we pek of a Urge pro- 

 portion, anal) proportion, fair proportion ; in this cae however, a 

 rmlio M meant, M the put U considered M bearing certain relation to 

 .!, |* : 



(On the subject of Analogy see ArUtotle'. Pottir, e. 81 ; Wriorif. 

 b. ii. c. 2; //iV. A. i. c. 1 ; Copleotone in the Appendix to Watt- 

 ly't Mutant : Whately'. Rhtl., part I, c. 2. a, 6 ; Mill 1 * 8y*tm of Logic. 

 oep.iii. .. 1-Ji 



ANA'LYSIS. a Greek word, ignifying literally tit aft of /,>/ 

 or mtyiivj ; Ha opposite u i<r*lkr*i, which in the act of putting together. 

 The modern meaning of the term analysis is the process by which fact.*, 

 reeuha, or reaeoningn are neparated into thrir simple and coin|ineut 

 part*, or by mean* of which a simple truth is obtained when given in 

 a more complicated form ; so that, in it* most general sense, the 

 greatest part of human knowledge consists in the results of analysis. 

 It is, however, for the most part applied in a more particular manner 

 to the methods employed in those branches of inquiry, which most 

 strikingly exhibit direct analysis; namely, mathematics mid imttintl 

 philosophy, particularly chemistry. By a very incorrect misnomer. 

 algebra, the differential calculus, Ac., have been called by the general 

 name of onalittit, in opposition, not to t;fnthtni, but to r/eomttry, in 

 which latter science synthetical methods are must tistially applied. 

 This perversion of the term prevails on the Continent to such an extent, 

 that it most always be taken for granted, that ' analyst' stand* for the 

 algebraical branches of p<ire matliematics. In this sense it is again 

 subdivided into ' algebraical analysis ' and ' infinitesimal analysis," the 

 latter including the fliixinnal ordifferential calculus. And by ' geometrical 

 analysis' is frequently understood the application of algebra to geometry. 

 It must, however, be remarked, that the exact sciences have appropri- 

 ated this word, simply because in these branches of knowledge the use 

 of analysis has been made most conspicuous. 



{'< -nfuu'ng ourselves to the primitive meaning of the term, it is obvious 

 that all discovery must be entirely either the work of analysis or of 

 accident ; and that, therefore, geometrical analysis must be as old as 

 geometry. Nevertheless, this does not appear from the earliest treatises. 

 The work of Euclid is strictly synthetical. Instead of taking the pro- 

 position asserted, and examining it by means of preceding propositions, 

 and in the mean time assuming it to be true, in order to ascertain 

 whether the results deduced from it agree or disagree with what has 

 been already proved, Euclid first enunciates the point which he means 

 to establish, and then proceeds to put together the considerations by 

 which it is demonstrated, leaving the learner nothing to do hut to 

 judge of the truth or falsehood of each argument as it arises, without 

 taking into consideration the methods by which the arguments them- 

 selveo were first obtained. This is the natural and proper method of 

 teaching what has already been discovered, for its own sake ; not only 

 because it neglects to introduce difficult and embarrassing considera- 

 tions, and allows of the subject being broken up into portions which 

 are easily learnt at one time, but because there is, in reality, no per- 

 fectly general and certain method of analysis which can be made 

 obvious to the beginner. In attempting the analysis of a new problem, 

 though the discoverer will naturally first try those methods which 

 have been successful in preceding cases, he has no means of assuring 

 himself beforehand which will be successful. The chemist is similarly 

 circumstanced. Let a new substance, or one supposed to be such, be 

 presented to him, from which he is required to find out whether it is 

 already known, or if not, of what it is composed. No effective analysis 

 can commence without requiring the resnlta of all his previous know- 

 ledge ; for he must have some method of recognising each and every 

 substance with which he is acquainted, previously to pronouncing 

 whether or not that under consideration is one of them. He must 

 tben^ proceed to trials of that substance with various others, and 

 nothing but the sagacity which arises from previous experience can 

 direct him in his choice of the methods to be employed. No general 

 rules of analysis can be laid down : that is, no processes which must 

 end in the discovery of the component parts required. The same 

 observations may be made on mathematical analysis. We give a 

 geometrical instance, with its result, and the synthetical form of the 

 proposition arising oat of it 



^ The aides of a triangle ABC are cut in D,B, and r, by a straight line 

 Six segment* are thus formed, AD and on, whose sum M the side AB , 

 AK and to, whose mini is the Hide AC ; and vr and re. whow difference 

 is the nde BC. It U required to investigate the relation which exists 

 between these x Momenta, if there be any relation. 



Some relation, wifl be thrown out of the question upon the slightest 

 conriderrtion : the sum of the six lines is not the same in every 



triangle, neither U their product. Leaving this unorganised nni. ,. 

 examination, we recollect, that if CB were parallel to in , th. 

 similar triangles ADE, ABC, would give a w.-ll-kinmu relation 1 

 AD, DB, AE, and EC. To try whether thin may help u 

 to DK, which gives the proportion 



AD : DO : : AE 

 or if we represent the lines by the number of units which they < 



AD X EC = AK X DO . . I l.i 



Because oc is parallel to DP, we h,ue 



UD : BD : : cr : BF, 

 .D x BF = BD x or . . (2.) 

 and the equations (1.) and (2.) multiplied together, and the result 

 divided by the common factor D, gives 



AD X tO X BF = AK X BD X CK . . 



whence the relation required between the six lines is as follow 

 them lie separated into two lota of three lines caeh. ii. 

 no two Hues which have a common extremity are 1 ,' mate 



lot ; then the product of the first three will be equal to the prod 

 tee second three. 



If instead of asking for the relation, if any exist. U-t 

 lines, the equation (8.) had been given, and it had been re,,, 

 detect whether it were true or false, the process would 

 similar ; and we should have found that the equation (3.) in true. 

 necessary consequence of the jiroposition . that a line drawn ]rallel to 

 one side of a triangle divides the other sides into proportion 



The synthetical form of the preceding process differs from it 

 less on the paper than would IKS the case in the mind of a rtii.lmt . who 

 had actually hit ujion the solution in the progress of it 

 l-'or. not Iveing able to tell the various steps by whieh 

 readers would endeavour to arrive at the same corn-In, i-.n. , 

 obliged to prompt him with a right guess, and thereby <:iv< him 

 synthetical description of that which was in our minds ai 

 process. _ It only remains, therefore, to make the demon-- 

 tlieln-.il in form, which, will now be readily seen. \\ : 

 stating the proportion to IK- proved, directing to draw co parallel to 

 DF, without giving any reason, and combining the steps of 1 1 , 

 demonstration. 



The geometrical analysis is generally ascribed to the school of I 

 but, in reality, as we have already observed, must be of a date a- 

 as geometrical reasoning itself. The use of FORISIIS, or pn.Mei, 

 also Loci] admitting an indefinite number of solutions, the establish - 

 ment of the properties of the CONIC SECTIONS, and the various efforts 

 made for the DUPLICATION of the cube and the THISK TIOX of tlv 

 all of which were the work of the school alreadv mentioned 

 certainly increased the power of the analyst, that is, made the 

 of discovery more obvious and more successful; but tin : 

 in the methods which entitles them to the exclusi\ e . 

 geometrical analysis. 



Tlw peculiar distinction between algebra and geometry is. that tin- 

 analytical method is pursued in the former from the commencement. 

 The solution of a problem consists in inquiring into the i 

 of the solution tappoted to befwtnd, by introducing at evci -\ 

 known truth, such as will produce a more simple consequence, and thus 

 reasoning backwards, so to speak, until at last the answer ii 

 directly produced in numliers, which was before implicitly inv..! 

 the conditions of the problem. The methods are more general than in 

 geometry, that is, a larger number of problems may be sol\ 

 process. The same observations apply still more strongly to the higher 

 parts of algebra, and the differential calculus. 



The solution of equations of the first four degrees, and the approxi- 

 mation to that of all higher degrees, render the olut.ion of a 

 vast number of common problems a matter of certainty. The Dilution 

 of differential equations, win-re that can lie don,-. ;. ; ,ii additional step 

 of even a more important character. Within the last century, i 

 matical analysis has made considerable approaches to a state whi - Ii 

 enables us to determine, almost immediately, whether a problem 

 be solved by such means as we possess, or not ; no small advantage, 

 when it is considered how much time was previously m 

 attempt to attain results whieh have sine, ,, to lie imi 



ANALYSIS, CHKMICAL. [CII,:MICAL ANALYSIS.] 



ANALYSIS, EUDIOMKTHIC. [GASOMETRIO ANALYSIS.] 



. \\AMKKTA or ANAMM.TA. the name- of a pm.s of plant, 



ing to the natural order Menitpermafta-, to whieh the plant 



yielding the Cooouhu Inrlicus of commerce is now referred. It ha* 



tin- following characters : flowers dioecious, calyx of six sepals in a 



dot, Lie series with two-close pressed brae' 



on separate flowers united into a central column, dilated at the 

 antli. n uniu, fons, covering tin- whol, clol.oseapeM.f die eolmiin. The 

 (lowers with pistils are not known, hut. the fruit is a one- to t hi -e. 

 drupe. The seed is globose, deeply excavated at the hihuii, albumen 

 fleshy, cotyl' ihin. diverging. The plant whieh yiel 



berrien of commerce is the only species of this genus. It is n 

 Climbing shrub, and is met with on the coasts of Malabar and the 

 Eastern: \\,.,\ A,, a mirta Corciiliu ; it possesses a powerful 



bitter poisonous principle, and is used for external applications only. 



