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MATHEM ATICAL GEOGRAPHY MATHEMATICS. 



In regard to the soul, the materialist maintains that 

 matter produces in itself spiritual changes, or that 

 the soul is a consequence of the whole bodily organ- 

 ization, by which matter is refined and ennobled into 

 mind. Among the advocates of this doctrine we 

 may mention Priestley. This theory, however, does 

 not explain how matter can think, and how physical 

 motion can produce mental changes, which we do 

 not observe in so many organic beings ; how, in 

 particular, a notion of its own activity can originate. 

 Numerous auxiliary hypotheses, therefore, have been 

 devised, as that of the vibration of nerves by Hartley. 

 In decided opposition, however, to materialism, is 

 our consciousness of the identity and liberty of man, 

 which would be annihilated by it, because matter is 

 governed by the necessity of nature, and free will 

 therefore excluded. Materialism is a very ancient 

 view of nature, and the predominant one in the most 

 ancient Greek philosophy, poetry,and mythology, sur- 

 rounded, however, by all the graces in which the 

 poetical spirit of this imaginative people could array 

 it. 



MATHEMATICAL GEOGRAPHY is the ap- 

 plication of mathematics and astronomy to the 

 measurement of the earth. The ancients had made 

 no inconsiderable progress in this science. This 

 science starts from two principles : 1 . that the earth 

 is to be considered as a sphere ; and, 2. that the 

 points and circles, imagined on the heavens, corre- 

 spond with points and circles on the earth. See 

 Earth, Pole, Equator, Tropics, Meridians, Degree, 

 Latitude, &c. ; see, also, Geography. 



MATHEMATICS. If we call everything, which 

 we can represent to our mind as composed of homo- 

 geneous parts, a magnitude, mathematics, according 

 to the common definition, is the science of determin- 

 ing magnitudes, i. e. of measuring or calculating. 

 Every magnitude appears as a collection of homo- 

 geneous parts, and may be considered in this sole 

 respect ; but it also appears under a particular form 

 or extension in space, which originates from the 

 composition of the homogeneous parts, and to which 

 belong the notions of situation, proportion of parts, 

 &c. Not only all objects of the bodily world, but 

 also time, powers, motion, light, tones, &c., may be 

 represented and treated as mathematical magnitudes. 

 The science of mathematics has to do only with these 

 two properties of magnitudes, the quantity of the 

 homogeneous parts, which gives the numerical mag- 

 nitude, and the form, which gives the magnitude of 

 extension. This is one way, and the most common, 

 of representing the subject: there are others more 

 philosophical, but less adapted to the limited space 

 which can be allowed to so vast a subject, in a work 

 like the present. In investigating these two pro- 

 perties of magnitudes, the peculiar strictness of the 

 proofs of mathematics gives to its conclusions and 

 all its processes a certainty, clearness, and general 

 application, which satisfies the mind, and elevates 

 and enlarges the sphere of its activity.* (See Me- 



* As a branch of intellectual culture, mathematics has great 

 excellencies and great defects. Its certainty, the precision 

 of its signs never conveying more nor less than the meaning 

 intended, its completeness in itself, and independence of all 

 other branches, distinguish it from every other science, and 

 nothing accustoms the young mind more to precision and ex- 

 actness of thought and expression than the study of mathe- 

 matics. But, on the other hand, these very excellencies render 

 it liable to give a partial direction to the mind, to withdraw it 

 from, and unfit it for pursuits of a different character. Hence 

 so many great mathematicians have appeared to be wholly 

 unfitted for other studies. On the whole, however, its advan- 

 tages are so great that it can never be dispensed with in a 

 liberal education. Nothing expands and elevates the mind 

 more than the acquisition of a mathematical truth, a law which 

 >beyed throughout the universe. The study of the conic 

 sections affords fine illustration of this influence ; and there 

 s few instances in which there will be much danger of the 

 pupil being unduly absorbed in the study. 



thod, Mathematical.) According as a magnitude i* 

 considered merely in the respects above mentioned, 

 or in connexion with other circumstances, matht- 

 HIM tics are divided into pure and applied. Pure ma- 

 thematics are again divided into arithmetic (q. v.), 

 which considers the numerical quality of magnitudes, 

 and geometry (q. v.), which treats of magnitudes in 

 their relations to space. In the solution of their 

 problems, the common mode of numerical calculation, 

 and also algebra (q. v.), and analysis (q. v.), are 

 employed. To the applied mathematics belong the 

 application of arithmetic to political, commercial, 

 and similar calculations ; of geometry to surveying 

 (q. v.), levelling, &c. ; of pure mathematics to the 

 powers and effects, the gravity, the sound, &c., of 

 the dry, liquid, and aeriform bodies in a state of rest, 

 in equilibrium or in motion, in one word, its applica- 

 tion to the mechanic sciences, (see Mechanics, 

 Hydrodynamics, &c.) ; to the rays of light in the 

 optical sciences (see Optics, Dioptrics, Perspective, 

 &c.) ; to the position, magnitude, motion, path, &c., 

 of heavenly bodies in the astronomical sciences (see 

 Astronomy), with which the measurement and calcu- 

 lation of time (see Chronology) and the art of making 

 sun-dials (see Dial) are closely connected. The 

 name of applied mathematics has sometimes been 

 so extended as to embrace the application of the 

 science to architecture, navigation, the military art, 

 geography, natural philosophy, &c. ; but in these 

 connexions it may more conveniently be considered 

 as forming a part of the respective sciences and arts. 

 It is to be regretted that there is as yet no perfectly 

 satisfactory work, treating of the history of this 

 science, so noble in itself, and so vast in its applica- 

 tion : even Kastner and Montucla leave much to be 

 desired. The establishment of mathematics on a 

 scientific basis probably took place among the 

 Indians and Egyptians. The first developeinent 

 of the science we find among the Greeks, those 

 great teachers of Europe in almost all branches. 

 Thales, and more particularly Pythagoras, Plato, 

 Eudoxus, investigated mathematics with a scientific 

 spirit, and extended its domain. It appears that 

 geometry, in those ages, was more thoroughly cul- 

 tivated than arithmetic. The ancients, indeed, 

 understood by the latter something different from 

 that which we understand by it. In fact, we have 

 not a clear idea of the ancient arithmetic. Their 

 numerical calculation was limited and awkward, 

 sufficient ground for which might be found in 

 their imperfect way of writing numbers, if there was 

 no other reason. Euclid's famous Elements, a work 

 of unrivalled excellence, considering the time of its 

 origin, the ingenious discoveries of Archimedes, the 

 deep investigation of Apollonius of Perga, carried 

 the geometry of the ancients to a height which has 

 been the admiration of all subsequent times. Since 

 then it has been made to bear more on astronomy, 

 and has become more connected with arithmetic. 

 Among the Greek mathematicians are still mentioned 

 Eratosthenes, Conon, Nicomedes, Hipparchus, Nico- 

 machus, Ptolemy, Diophantus, Theon, Proclus, Euto- 

 cius, Pappus, and others. It is remarkable that the 

 Romans showed little disposition for mathematics ; 

 but the Arabians, who learned mathematics, like 

 almost all their science, from the Greeks, occupied 

 themselves much with it. Algebra and trigonometry 

 owe them important improvements. Through the 

 Arabians, mathematics found entrance into Spain, 

 where, under Alphonso of Castile, a lively zeal was 

 displayed for the cultivation of this science. After 

 this, it found a fertile soil in Italy ; and in the con- 

 vents a monk would sometimes follow out its paths, 

 without, however, adding to its territory. This was 

 reserved for later ages. Mathematics owes much to 





