ni 



MATHEMATICS. 



MATHEMATICS. 



HI 



no case Involve any of the uncertainty arising from the previous intro- 

 duction of that which may be false ; to form the habit of applying the 

 attimtKti closely to difficulties which can only be conquered by thought, 

 and over which the victory is certain if the right means be used ; to 

 establish confidence in abstract reasoning by the exhibition of pro- 

 cesses whose results may be verified in many different ways ; to help 

 in mng' the student to acquire correct notions and habits of 

 generalisation ; to give caution in receiving that which at first sight 

 appears good reasoning; to instil a correct estimate of the powers of 

 the mind, by pointing out the enormous extent of the consequences 

 which may be developed out of a few of its most fundamental notions, 

 and its incapacity to imagine, much leu to attain, the boundary of 

 knowledge; to methodise the invention of the moans of expressing 

 thought, and to make apparent the advantages of system and analogy 

 in the formation of language and symbols ; to sharpen the power of 

 investigation,' and the faculty of suggesting new combinations of the 

 resources of thought; to enable the historical student to look at men 

 of different races, opinions, and habits, in those parts of their minds 

 where it might be supposed a priori that all would most nearly agree ; 

 and to give the luxury of pursuing a study in which self- interest cannot 

 lay down premises nor deduce conclusions. 



As instrument* in the investigation of nature and the advancement 

 of the arts it is the object of the mathematical sciences to give correct 

 habits of judgment and ready means of expression in matters involving 

 degree and magnitude of all kinds ; to teach the method of decomposing 

 phenomena, and ascending from the complicated forms of manifestation 

 to the simple law which regulates them ; to trace the necessary con- 

 sequences of any law, assumed on suspicion, in order to compare 

 those consequences with phenomena ; to construct hypothetical repre- 

 sentations of laws, or approximations to laws, which shall sufficiently 

 represent phenomena ; to convert processes of known accuracy, but 

 complicated operation, into others which make up in simplicity for a 

 certain amount of inaccuracy, and to devise means for judging of that 

 amount of inaccuracy, and confining it within given limits ; to ascertain 

 the most probable result of observations or experiments which are dis- 

 cordant with each other from errors of measurement or unknown 

 causes of disturbance; to point out the species of experiments 

 which should be made to obtain a particular sort of information, or to 

 decide between two laws which existing phenomena both indicate as of 

 nearly equal credibility ; to make all those investigations, which are 

 necessary for the calculation of results to be used in practice, OB in 

 nautical astronomy, application of force and machinery, and conduct 

 of money transactions ; in a word, though that word by itself would 

 have not presented a sufficiently precise idea, to find out truth in 

 every matter in which nature is to be investigated, or her powers and 

 those of the mind to be applied to the physical progress of the 

 human race, or their advancement in the knowledge of the material 

 creation. 



The main branches of mathematical science were formerly stated to 

 be arithmetic and geometry, springing out of the simple notions of 

 number and space. This is too limited a description. Unquestionably 

 the science of numbers, strictly and demonstratively treated, and that 

 of geometry, or the deduction of the elementary properties of figure 

 from definitions which are entirely exclusive of numerical considera- 

 tions, must be considered as the elementary foundations, but not as 

 the ultimate divisions, of mathematics. To* them we must add the 

 science of operation, or algebra in its widest sense, the method oi 

 deducing from symbols which imply operations on magnitude, and 

 which are to be used in a given manner, the consequences of the 

 f umUincnUl definitions. The leading idea of this science is operation 

 or process, ju*t a. number is that of arithmetic, and space and figure 

 of geometry : it is of a more abstract and refined character than the 

 latter two, only because it does not immediately address itself to 

 notions which are formed in the common routine of life. It is the 

 most exact of the exact sciences, according to the idea of their exact- 

 ness which is frequently entertained, being more nearly based upon 

 definition than either arithmetic or geometry. It is true that the 

 definitions must be such as to present results which admit of appli- 

 cation to number, space, force, time, 4c., or the science would bo 

 useless in mathematics, commonly so called ; but it is not the less true 

 that a system of methods of operation, based upon general definitions, 

 and conducted by strict logic, may be made to apply either to arith- 

 metic or geometry, according to the manner in which the generalities 

 of the definition are afterwards made specific. 



The common division of the mathematical sciences will not admit 

 of the threefold separation just described, the science of operation 

 being more or leas mixed up with arithmetic, both in common algebra 

 and in iU application to geometry. We may describe this division as 

 follows : 



1. Pnrt Arithmetic, subdivided into particular and unirertal : the 

 orm-r, the common science of numbers (integral and fractional) and 

 calculation ; the latter, the science of numbers with general symbols, 

 or the introduction to methods of operation, restricted to purely 



w. ban (applied the word particular, ai opposed to universal : almbra 

 WM MOKUmei called unircrnl arlUiinrtic, but the phraw ncrcr became mural, 

 owtef to IU being obTioui to IboM who Mudled alftbn, that arithmetic bow. 

 tvtr fsnera) Its svabob mar be, U <m! 7 . p. rt of *&,. 



numerical processes. The science which treat* of the peculiar relations 

 of numbers, and subdivides them into olimm possessing distinct pro- 

 perties, is called the theory of Humbert, and is an extension which fre- 

 quently requires a higher algebra. 



2. Part iieumttry, which investigates the properties of figures in the 

 manner of Euclid, that is, with restrictions which confine the stu<l< ut 

 to the straight line and circle as the means of operation and the bound- 

 aries of figure. [OioMKTBY.l This science includes solid geoi; 



as far as figures bounded by planes, the properties of the sphere, cone, 

 and cylinder, and of their plane sections ; but it does not allow any 

 conic section, except the straight lino and circle, to be employed in the 

 solution of problems. 



3. Algebra, including the general calculus of operations (though this 

 is not an elementary branch), and all methods which can be established 

 without the aid of processes exclusively belonging to the differential 

 calculus. The distinction between it and universal arithmetic is an 

 extended use of operations, preceded by an extended definition of their 

 meaning. 



4. Application of Algebra to Geometry. This includes triyonomctry, 

 and all those parts of geometry in which problems ore numerically 

 solved, and the method of Euclid is abandoned. Thug it includes the 

 conic sections as commonly taught; and in its higher parts is an 

 application of the differential calculus, as well as of algebra. 



5. JX/erential and Integral Calculus. Under this term we include 

 the general theory of limits ; that is, all digested methods of operation, 

 in which the limits of ratios are used as algebraical quantities under 

 specific symbols. This distinction is necessary, since the notion ..f a 

 limit, and even propositions which belong to the differential calculus iu 

 everything but form, are contained in the elements of Euclid, and in 

 the application of arithmetic to geometry. The calculut of differences 

 and the calculut of rariationt ore usually placed under this head : the 

 former, in its elementary parts, might be referred to common algebra ; 

 the latter is an extension of the differential calculus. 



The division of the mathematical sciences into pure and miiced is 

 convenient in some respects, though liable to lead to mistake. By the 

 former term is understood arithmetic, geometry, and all the preceding 

 list ; by the latter, their application to the sciences which have matter 

 for their subject, to mechanics, optics, &c. But considering that in all 

 these subjects a few simple principles are the groundwork of the 

 whole deduction, they might be explained as intended to answer two 

 distinct questions : first, what ore the consequences of such and such 

 assumptions upon the constitution of matter ? secondly, ore these con- 

 sequences found to be true of matter as it exists, and arc the assump- 

 tions therefore to be also regarded as true ? In the reply to the first 

 question, the science is wholly mathematical ; to the second, win illy 

 experimental in its processes, and inductive in its reasonings ; and this 

 is the mixture from which the joint answer to both questions derives 

 its name, and not from any difference between its mathematics and 

 those of the pure sciences. Again, a science does not take the name 

 of mixed mathematics simply because it is possible to apply mathe- 

 matical aid in the furtherance of its legitimate conclusions : such a 

 use of terms would be trifling with distinctions, since it would bring 

 political economy, chemistry, geology, and almost every part of natural 

 knowledge, under the same head as mechanics and hydrostatics. The 

 words in question should be reserved to denote those branches of 

 inquiry in which few and simple axioms are mathematically shown to 

 be sufficient for the deduction, if not of all phenomena, at least of all 

 which are most prominent. Taking the leading ideas of the mixed 

 sciences instead of their technical names, wo may describe them as 

 relating to motion, pressure, resistance, cohesion, light, heat, sound, 

 electricity, and magnetism. As disciplines, it is their main object to 

 teach the true method of inquiry into the laws, and, so far as can be 

 known, the causes, of material phenomena ; as instruments, it is not 

 necessary to say one word about them. Two only have not been 

 mentioned : the first, astronomy, which belongs to more than one of 

 the preceding; the second, the theory of probabilities, of which, though 

 placed among the mixed sciences, it may be affirmed that it ought to 

 be called an application of mathematics to logic. 



The most important question connected with the mathematical 

 sciences is the manner in which they should bo taught as disciplines of 

 the mind. This concerns all who consider any branch of knowledge in 

 that light; and, as education spreads, this view of the subject will 

 become of more and more consequence. Vitally essential as these 

 sciences ore to the advancement of the arts of life, we feel, in regard to 

 this branch of their utility, in the situation of those who know that 

 they must and will be attended to, because their cultivation is neces- 

 sary to the supply of wants which all can feel, and the promotion of 

 interests which all can understand. It is not so with the first-men ' 

 object of their study, but rather the reverse ; for the wants of lifo 

 being as easily supplied by the results of on illogical as of a logical 

 system (provided only that vicious reasoning be not allowed to produce 

 absolute falsehood), the facilities which laxity of reasoning affords in 

 the mere attainment of results will always recommend it to those 

 whose main object it is to apply the fruits of calculation to the uses 

 of life. Such has been, and, we are afraid, will continue to bo, 

 the tendency of the great advance which the last century made in 

 application. 



All we should positively contend for is the necessity of making the 



