152 



ANALYSIS. 



or agreement of tilings in certain respects. The 

 knowledge which n-sis merely on this relation is 

 l. The conclusion deduced from the 



Miuihirity of tilings in certain respects, that they are 

 similar, also, in other respects, i> railed, in logic, an 

 analogical conclusion, ami amounts only to a proba- 

 bility. This reasoning is applied to the explanation 

 i if authors (analocfia interpretations), and particularly 

 10 the interpretation of the Holy Scripture*, in which 

 c-niisisiriicy of doctrine is taken for granted (analo- 

 fia /</'/) It is also used in the application of the 

 laws, to form a judgment, in any particular case, by 

 a comparison of former decisions in similar cases. 

 In practical medicine, it is used in the application of 

 remedies A great part of the principles of experi- 

 mental philosophy are established by inferring a 

 further uniformity from that which lias been already 

 settled. In grammar, by analogy is meant a confor- 

 mity in the organization of words. In mathematics, 

 it is Uie similitude of certain proportions. Newton 

 yives analogy the second place amongst his laws of 

 philosophizing, and may be said to have established 

 Mime of the most cliaracteristie parts of his system, 

 as arising out of the doctrine of gravitation, on its 

 sober and patient use. In fact, analogical reasoning 

 is essential in inductive philosophy, though it must 

 be used with caution. The history of philosophy 

 shows innumerable instances of the wildest errors, as 

 well as of the sublimest discoveries arising from its 

 application. The modern philosophy of Germany 

 has suffered much in point of correctness and clear- 

 ness, from several bold speculators, led away by 

 fancied analogies between the moral and physical 

 world; though it cannot be denied, that much of 

 Uie progress of that nation in philosophical investi- 

 gations is due to the use of the same instrument. 



ANALYSIS, in philosophy ; the mode of resolving a 



compound idea into its simple parts, in order to con- 



sider them more distinctly, and arrive at a more pre- 



cise knowledge of the whole. It is opposed to syn- 



thesis, by which we combine and class our percep- 



( inns, and contrive expressions for our thoughts, so 



as to represent their several divisions, classes, and 



relations. Analysis is regressive, searching into 



principles ; synthesis is progressive, carrying forward 



acknowledged truths to their application. Analysis, 



in mathematics, is, in the widest sense, the expres- 



sion and developement of the functions of quantities 



by calculation. There are two ways of representing 



the relations between quantities, to wit, by construc- 



tion, and by calculation. Pure geometry determines 



all magnitudes by construction, i. e. by the mental 



drawing of" lines, whose intersections give the pro- 



posed quantities ; analysis, on the contrary, makes 



use of symbolical formulas, called equations, to ex- 



press relations. In this widest extent of the idea ol 



analysis, algebra, assisted by literal arithmetic, ap- 



pears as the first part of the system. Analysis, in a 



narrower sense, is distinguished from algebra, inas- 



much as it considers quantities in a different point ol 



view. While algebra speaks of the known and 



unknown, analysis treats of the unchanging or con- 



stant, and of the clianging or variable. The alge- 



braic equation, a? -f- a x b = 0, for example, seeks 



an expression for the unknown x by means of the 



known a and b ; but the analytical equation, y 2 = 



a x, expresses the law of the formation of the varia- 



ble y, by means of the variable x, together with the 



constant a In its application to geometry, analysis 



seeks by calculation the geometrical magnitudes for 



an assumed or undetermined unit. The analysis ol 



the ancients was exhibited only in geometry, anc 



made use only of geometrical assistance, whereby it 



is distinguished from Uie analysis of the moderns 



which, a* before said, extends to all measurable ob- 



inl expresses in equations the mutual depen- 

 Icnce of magnitudes. Hut analysis and algebra 

 cseiuhlc each other in this, that both, as ii shown 

 nore fully in Uie article on algebra, reason in a 

 .anguage, into the expressions of which certain con- 

 litions are translated, and then, according to the 

 rules of Uie language, are treated more fully, in 

 order to arrive at the result. An;. lysis, when con- 

 idered in this light, appears to IK- tin- widest extent 

 of the province of this language. Analysis, in the 

 more limited sens,., is divided into lower and higher, 

 Jie bounds of which run very much into one anoUier, 

 l>ecause many branches of learning are accessible in 

 both ways. While we comprise in lower analysis, 

 besides arithmetic and algebra, the doctrines ol 

 functions, of series, combinations, logarithms, and 

 curves, we comprehend in the higher the differential 

 and integral calculus, which arc also included in tin- 

 name infinitesimal calculus ; the first of which tin- 

 French consider as belonging, in a wider sense, to 

 the theorie desfonctions analytiqucs. A good account 

 of the ancient analysis is given by Pappus of Alex- 

 andria, a mathematician of the 4th century, in his 

 Collection of Geometrical Problems,* in which there 

 is also a list of the analytical writings of the ancients. 

 What progress was made after the destruction of the 

 Roman empire, particularly by the Arabians, in 

 algebraical, and, as interwoven with them, in ana- 

 lytical inquiries, has been related in the article on 

 algebra. Newton and Leibnitz (q. v.) invented the 

 above-mentioned infinitesimal calculus. After them, 

 Euler and the brothers Bernouilli (q. v.) laboured 

 with splendid success for the further improvement of 

 mathematical analysis ;-and, in later times, d'Aiem- 

 bert, Laplace, Lagrange, &c. have raised it still 

 higher. Hindenburg (q. v.) is the inventor of the 

 analysis of combinations. We have not room here 

 to go into detail with respect to Uie other analytical 

 doctrines. Euler's Introductio in Analysin Infinito- 

 rumft Lausanne, 1748, 2 vols. (new ed., Leyden, 

 1797) still continues one of the most important works, 

 in regard to the analysis of finite quantities. In close 

 connexion with this stands Uie same author's ///.v//- 

 tutiones Calculi differ entialis, Petersburg, 1755, 4to. 

 Lagrange's Theorie des Fonctions Analytiques (new 

 ed., Paris, 1813, 4to.) is, on account of the depth of 

 its views and its many valuable applications to geo- 

 metry and mechanics, a valuable work for the study 

 of the connexion between Uie analysis of finite quan- 

 tities, and the so named (though, indeed, here con- 

 sidered in a very different light) calculation of 

 infinities. As this work cannot oe understood with- 

 out a good acquaintance with general- and very ab- 

 stract calculations, we would connect with it the 

 same author's Lemons sur le Calcul des Fonctions 

 (new ed., Paris, 1806). Arbogast's Calcul des Deri- 

 vations, Strasburg, 1800, 4to, is new in its views of 

 the analysis of finite quantities. The most excellent 

 of the old works on the integral calculus is Euler's 

 Institutiones Calculi Integralis, Petersburg, 1768 

 1770, 3 vols., 4to. The present state of the integral 

 calculus, after the improvements of the French 

 analysts, may be learned from Lacroix's Trails du 

 Calcul differentiel et du Calcul integral, Paris, 1797 

 and seq., 3 vols., 4-to. (There has since appeared a 

 new edition.) For beginners, we recommend Pas- 

 quich's Mathematical Analysis, Leipsic, 1791, and, 

 for more advanced students, the same author's Ele~ 



* There is a Latin translation of it by Commandinus: . 

 Mathemat. Collatiniifs, Comment ariis illustrate, Bonn, 

 1059, folio. The Greek, text is not published. 



t It has this title on acconnt of the application which i 

 here made of the idea of the infinite, and its coiicexioo 

 with the higher analysis. 



