Page:The American Cyclopædia (1879) Volume XI.djvu/287

 MATHEMATICS 275 braic symbols. The geometrical question is solved by resolving the corresponding algebraic equation. Algebra being defined as the ordi- nary analysis, calculus is the transcendental analysis, and has various applications in the higher departments of the science. The best achievements of modern mathematics are due to it. To these branches of mathematics the 19th century has added another, the final form of which is as yet undetermined, but the es- sential characteristics of which are to be found in the "quaternions" of Sir W. Eowan Ham- ilton, in the Ausdehnungslehre of H. Grass- mann, and in the " Linear- Associative Alge- bra " of Prof. Peirce. The great character- istics of this new science are : 1, the introduc- tion of several units differing in quality ; and 2, the rigid distinction between the multiplier and the multiplicand, or between the thing which acts and the thing acted upon. In the mathematical sciences, as hitherto treated, xy is always equal to yx ; it is a matter of indif- ference which quantity we regard as multi- plier and which as multiplicand. In the new science the distinction must be always regard- ed; xy and yx are entirely different things. The second characteristic is really a result of the first. Thus, in geometry, as treated by Grassmann, we have four different units, viz., a point and three mutually perpendicular straight lines. From the combinations of these units all the truths of geometry are deduced. Prof. Peirce, in his work above mentioned, has endeavored to fix a priori the laws which must regulate this introduction of units, and has divided algebra, according to the number of units introduced, into single, double, triple, &c. We can enter into no further explanations of this branch of mathematics, but will re- mark that as the great event in the intellectual history of the 17th century was the inven- tion of the calculus, so perhaps future histo- rians will regard this as the great event in the history of the 19th century. Algebra and ge- ometry are usually, but not with strict accura- cy, regarded as types respectively of analytical and synthetical reasoning. The former has an artificial language. Symbols are operated upon according to certain general rules, while the mind dismisses altogether the conceptions of the things which the symbols represent, whether lines, angles, velocities, forces, or whatever else. The steps in the processes are merely applications of the rule. The elements are symbols, and the results are only equations. Geometrical reasoning, on the contrary, is con- cerning things as they are. It retains the con- ceptions of quantities. It apprehends the na- ture of the new truths which it introduces at every step. Analysis is therefore the more powerful instrument for the professed mathe- matician, but geometry is the more effective mode of exercising the reason, and is a more useful part of the gymnastics of education. Comte, who makes mathematics preeminent in the hierarchy of the positive sciences, intro- duces a peculiar classification. Abstract math- ematics, according to him, embraces ordinary analysis, or the calculus of direct functions, and transcendental analysis, or the calculus of indirect functions. The former includes arith- metic and algebra ; the latter, the differential and integral calculus and the calculus of varia- tions. Concrete mathematics embraces syn- thetic and analytic geometry, the former being either graphic or algebraic, and the latter be- ing distinguished according as its objects are of two or three dimensions. Comte includes also rational mechanics, or the laws of statics and dynamics, as a department of concrete mathematics. If the universe were immov- able, there would be only geometrical phenom- ena ; but motions are mechanical phenomena. As commonly explained, the mixed mathe- matics are the applications of abstract mathe- matical laws to the objects of nature and art. From the universality and variety of these ob- jects, no strict and comprehensive classifica- tion of them has been made. Matter in rest and matter in motion are the primary phenom- ena in space and time. The laws which rule the one and the forces which impel the other are the first objects of inquiry. Mechanics treats of both, and is divided into statics and dynamics, dealing respectively with the equi- librium and the action of forces. Astronomy, hydraulics, pneumatics, optics, and acoustics may be regarded as subdivisions of dynamics. Surveying, architecture, fortification, and navi- gation are among the principal applications of mathematics to the arts. The pure mathe- matics are merely formal sciences. They oc- cupy and discipline but do not fill the mind. Their entirely formal character will be best appreciated by one or two illustrations. It is a law of falling bodies that the spaces passed through by the falling body are proportional to the squares of the times during which it falls. It is a law of geometry that the areas of circles are proportional to the squares of their radii. The mathematical formula expressing one of these laws also expresses the other. Let A : a=5 2 : B 2, and we may consider A and a as representing either spaces described by a falling body or areas of circles, and B and 5 as representing either the times during which it falls or radii of circles. In either case the formula is true. Yet the space described by a falling body and the area of a circle, the time during which a body falls and the radius of a circle, are wholly disparate notions. When we see a person adding a column of numbers, no inspection of the column itself will tell us what the person who wrote it down intended to represent by those numbers. He may have had in his mind sums of money, or yards of cloth, or bushels of wheat. Whatever it was, the process of finding the sum is in all cases the same. Again, an engineer investigating a prob- lem in regard to bridge building, an actuary one in life insurance, a machinist one in mechan- ics, might all arrive at the same algebraical for-