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 of deducing from symbols which imply operations on magnitude, and which are to be used in a given manner, the consequences of the fundamental definitions. The leading idea of this science is operation or process, just as 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, &c., or the science would be useless in mathematics, commonly so called; but it is not the leas true a system of methods of operation, based upon general definitions, and conducted by strict logic, may be made to apply either to math meta) or geometry, according to the manner in which the generalities of the definition are afterward., 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 less mixed up with arithmetic, both in common algebra and in its application to geometry. We may describe this division as follows: 1. Pure Arithmetic, subdivided into particular and universal:' the former, 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 numerical processes. The science which treats of the peculiar relations of numbers, and subdivides them into classes possessing distinct pro perties, is called the theory of numbers, and is an extension which fre quently requires a higher algebra.
2. Pure Geometry, which investigates the properties of figures in the manner of Euclid, that is, with restrictions which confine the student to the straight line and circle as the means of operation and the bound arise of figure. [Osousana.] This science includes solid geometry, 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 line 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 includes trigonometry, and all those parts of geometry in which problems are numerically solved, and the method of Euclid is abandoned. Thus 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. Differential and Integral 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 of a limit, and even propositions which belong to the differential calculus in everything but form, are contained in the elements of Euclid, and in the application of arithmetic to geometry. The calculus of eUerences and the calculus of variations are 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 mixed 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 tho sciences which have matter for their subject, to mechanics, optics, &e. But considering that in all these subjects a few simple principles are the groundwork of tho whole deduction, they might be explained as intended to answer two distinct questions : first, what are the consequences of such and such assumptions upon the constitution of matter I secondly, are these con sequences found to be true of matter as it exists, and are the assump tions therefore to be also regarded as true I In the reply to the first question, the science is wholly mathematical ; to the second, wholly 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 mathematic, 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 cau 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 eecond, 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.