629 



MATHEMATICS. 



MATHEMATICS. 



530 



their distance from tlie common axis. Hence, if the radius of the 

 whole cylinder be r, and that of any one of the internal cylinders be x ; 

 also, if f' represent the force of cohesion between any two particles in 



F' 



the outer circumference, we have r : F' : : x : x. The last term ex- 

 presses the like force in the circumference of the cylinder, whose 



F' 

 radius is x, and the momentum of cohesion is a?. But as all the 



particles in that circumference exert the same power, and the number 



p' 

 of particles is proportional to x, it follows that a? will represent the 



F' 



sum of all the forces in the latter circumference, and a? dx will 



represent the sum in a hollow cylinder whose thickness, in the direction 

 of the radius, is infinitely small and equal to dx. Then, by the rules 

 of integration, we have for the strength of the whole cylinder the 



p' F' 



expression 7- x 4 , which, between x and x = r, becomes -r r 3 ; and 



hence the whole strength of any cylinder varies as the cube of the 

 radius. 



But the area of any section of a cylinder whose radius is r being 

 =r 2 r (r being the half circumference of a circle when the radius = 1) 

 it is evident that F' i a ir will represent the lateral cohesion by which all 

 the particles resist being separated by the force of detrusion. There- 

 fore the ratio between the force of detrusion and that of torsion will be 



r 3 

 as r* v to - ; or as 4* to r. 



And since the strength under the latter strain depends on the radius, 

 it is evident that a hollow cylinder must be stronger than a solid one ; 

 the areas of the material in the transverse sections being equal to each 

 other. 



The preceding reasoning is entirely founded on the supposition 

 that the sections of the various solids considered had but a small 

 importance upon their resistance to the various forces considered ; but 

 this is far from being the case in practice, and especially in the case of 

 cast and wrought iron, the resistances of the upper and of the lower 

 fibres of a rectangular beam differ to such an extent as to have rendered 

 it necessary to carry out a long series of investigations into the real 

 forms of solids of those particular materials presenting the greatest 

 resistance. Mr. Hodgkinson, after Tredgold and Barlow in our own 

 country, and Messrs. Morin and Love, after Navier, Coriolis, and Vicat, 

 in France, have studied this complicated question ; and the results of 

 their inquiries, so far as the practical arts of construction are con- 

 cerned, have been already alluded to under GIBDER, in which article 

 also will be found enumerated the principal authors upon the subject 

 it may be desirable for the student to consult. Very useful tables of 

 the resistances of materials to efforts of compression, and of extension, 

 may be found in Can's 'Synopsis of Practical Philosophy,' 1843; 

 Beardmore's ' Hydraulic, &c. Tables,' 1852 ; Moseley's ' Engineering and 

 Architecture,' 1843; Willis'* edition of Barlow, on 'Materials and 

 Construction,' 1851 ; and Claudel's ' Formules a 1'usage des Ingenieurs,' 

 1854. 



MATHEMATICS (juifcjo-u, or ndBrtna), a came given in the first 

 instance to a branch of knowledge, not as descriptive of its subject- 

 matter, but of the methods and consequences of learning it. The word 

 /u0ij<m, and the Latin disciplina by which it has been rendered, have 

 been the origin of the vernacular terms mal/temutics and discipline, the 

 meanings of which have long since separated. The properties of space 

 and number, the subject-matters of the ful8ri<ru, have usurped the 

 name ; so that anything which relates to them, however learnt, is called 

 mathematics : the Latin word, on the contrary, still retains the signi- 

 fication of a corrective process ; and, in speaking of any branch of 

 knowledge, is applied when power of mind is derived from the methods 

 of learning it, as well as knowledge from the results. 



The original use of the word mathematics cannot be gathered, so far 

 as we can find, from any express contemporary authority ; a few pass- 

 ages in which the term is used without explanation, as one of notoriety, 

 being all that can be cited, and mostly from Plato. Later writers, as 

 lor instance Anatolius (cited by Heilbronner), A.D. 270, give the deri- 

 vation above alluded to. But before the time of Anatolius the 

 meaning of the word had been extended : thus the book of Sextus 

 Empiricus "against the mathematicians" is, as Vossius remarks, 

 directed as much against grammarians and musicians as against arith- 

 meticians and geometers. And John Tzetzes, in the 12th century, 

 includes under the fiato/i/MTa nearly what the universities afterwards 

 called by the name of arts ; calling grammar, rhetoric, and philosophy, 

 the disciplines (futSfi^ara), and arithmetic, music, geometry, and astro- 

 nomy, arts (TX(<) included under philosophy. 



Tlio distinction between the old and new meaning of mathematics is 

 most requisite to be kept in mind, because arguments are frequently 

 urged for and against mathematics, in which the discipline is con- 

 founded with the communication of facts and processes about space 

 and number ; and because it is our intention in the present article, cou- 

 fining ourselves to the most important view of the science, as well as to 

 the etymological meaning of its Dame, to offer a few remarks on tho 

 diicijttine called mathematics. 



ARTS AND SCI. DIV. VOL. V. 



In the time of Plato, which was probably that of the application of 

 words which imply " the discipline" to that one exercise of mind which 

 consists in making deductions by pure reasoning from the self-evident 

 properties of space and number, it is probable that such restriction of 

 the word was easily justifiable. At present we have, besides mathe- 

 matics, also physics, the study of antiquity, grammar, &c., which have 

 all been made disciplines, but not one of which was then entitled to 

 that appellation. Nevertheless it has happened that writers, misled 

 partly by the name of mathematics and partly by the pre-eminence of 

 mathematical reasoning in strictness and connection, have spoken as if 

 it were the only cultivator of the pure reasoning power. 



Much discussion has arisen upon the question whether those primary 

 propositions which, from our clear apprehension and willing admission 

 of them, are called self-evident, are notions inherent in the mind, or 

 deductions of early experience. Except to mention this controversy, 

 we have here nothing to do with it. The certainty of these proposi- 

 tions is all that we want, and this is conceded by both sides. The con- 

 sideration _ however of the fundamental supports of mathematical 

 reasoning is useful and interesting, and, as a safeguard, even necessary. 

 It is not long since a school of metaphysicians existed who imagined 

 that because all mathematical definitions are precise, therefore the 

 exact sciences are founded upon definition. It was not to them a 

 necessary result of the constitution of our faculties that the three 

 angles of every triangle make up the same amount, but a consequence 

 of definition, which might have been something else, upon different 

 suppositions. We can hardly undertake to explain what we do not 

 understand : if the opinion we have quoted have not the meaning we 

 have given to it, there is in it some confusion of terms. We recom- 

 mend every beginner in the subject to seek no knowledge about tho 

 character of fundamental propositions until he shall have become well 

 acquainted with then- consequences. He must take care to admit 

 nothing which .is not, or cannot be made, most evidently true; 

 and he will find that all axioms, as they are called, have the 

 highe.it sort of certainty, namely, that they cannot be imagined 

 otherwise. 



AVhatever may be the psychological hypothesis to which it is 

 referred, it is certain that there is a real distinction between a 

 mathematical fact and one of any other kind. If we say that an un- 

 supported bit of lead will fall to the ground, we state a fact of which 

 we are as certain, in the sense of reliance, as we are of the other pro- 

 position that two straight lines cannot inclose a space. But in the for- 

 mer proposition an exception, or even a permanent alteration of tho 

 law, is conceivable by the imagination : in the latter the mind would 

 feel sensible of absurdity if it attempted to construct the idea of an 

 inclosure bounded by only two straight lines. No distinctive phrases 

 can be too strong to express tho essential difference of these two asser- 

 tions ; but it is a misfortune that all terms which create a sufficient 

 distinction are linked to one or another theory of the human mind. 

 If the mathematical student can receive these terms as indicative of 

 the difference of species, without bending before an hypothesis about 

 the conformation of his own reason, he will do well to adopt them ; 

 if, on the other hand, he feel compelled to agree with any one 

 system of mental philosophy, he will neither impede nor advance his 

 mathematical career. 



The unavoidable certainty and definite character of mathematical 

 conclusions have obtained for mathematics the name of exact science : 

 but to this name it has not exclusive right. The laws under which wo 

 must think are the foundation of a science which has an equal claim 

 with mathematics to any epithet which indicates cither necessity or 

 precision. Accordingly logic and mathematics are separate branches 

 of exact science. There are but three things of which we cannot 

 divest ourselves so long as we imagine ourselves to retain both existence 

 and consciousness of existence : they are thought, space, and time. 

 With everything else there is a possibility of dispensing ; that is, the 

 imagination can conceive everything got rid of, and out of existence, 

 except its own consciousness in some kind of activity, and the space 

 and time without which it cannot conceive existence. The necessary 

 laws of thought are the subject matter of logic : the necessary pro- 

 perties of space and time are the subject matter of mathematics. 

 Number is an offspring of the notion of time ; enumeration is a succes- 

 sion in time ; in no other way can number be distinguished from 

 multitude. And geometry is, without need of illustration, the offspring 

 of the notion of space. 



There has long been something of schism between the cultivators of 

 logic and those of mathematics. To establish this fact, and b 

 speculate on the reasons of it, would require more space than we have 

 to give. The effect has been unfortunate. The mathematician dis- 

 penses with the analysis of the laws of thought ; the logicians avoid 

 that field in which the laws of thought are applied to necessary 

 matter. This state of things is perhaps mending, but until the im- 

 provement is very decided, neither mathematics nor logic will be what 

 they ought to be, as component parts of liberal education. 



The sciences of which wo speak may be considered either as dis- 

 ciplines of the mind, or as instruments in the investigation of nature and 

 the advancement of the arts. In the former point of view thuir object 

 is to strengthen the power of logical deduction by frequent examples ; 

 to give a view of the difference between reasoning on probable premises 

 and on certain ones, by the construction of > body of results which in 



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