(μάθημα or μάθησις) properly signifies the discipline of the mind acquired in studying a science, rather than its subject matter; and is a term descriptive of any species of knowledge which tends to improve the mental faculties. It is employed, however, in a restricted sense, to denote the science whose object is the discovery of the abstract relations of number and magnitude, and their application, by means of observed laws, to the explanation of natural phenomena, and the improvement of the useful arts.

Although in every department of mathematics results are obtained by strictly logical deduction from a few first principles explicitly assumed, yet the science may be regarded as consisting of two distinct branches, according to the nature of the evidence on which the truth of its first principles is admitted. In the one, which constitutes pure mathematics, the first principles require no special inductive process to convince us of their truth, and scarcely, indeed, demand the evidence of our senses. Whether they are notions inherent in the mind, or deductions from our constant and earliest experience, they are universally allowed to be self-evident, in the sense that we cannot conceive them to have been otherwise than they are. Thus pure mathematics is founded upon definitions of necessary truth, and is pre-eminently the most certain of all the sciences.

Mixed mathematics denotes the application of pure mathematics to natural objects; and presupposes some knowledge of their properties derived from the senses, or of general laws obtained by induction from a sufficient number of observations. The logical, or strictly mathematical processes of deduction in the pure and mixed mathematics are identical; the difference between the two sciences being, that in the one, the first principles are self-evident, while in the other, laws and facts, which are not necessarily self-evident, but derived from observation, are admitted, along with the axioms and definitions of pure mathematics, as fundamental principles of the science. Again, in mixed mathematics we are often presented with conclusions derived from hypotheses regarding the constitution of nature. If we ask whether these are correct conclusions from the assumed hypotheses, our investigations must be conducted on principles purely mathematical; but if we inquire whether the conclusions have a real existence in nature, we must appeal to observation; and our reasoning becomes inductive. It is from this mixture of mathematical deduction with experimental processes that the mixed sciences derive their name, and not from any difference between their mathematics and those of pure science.

Thus the evidence for the conclusions of mixed mathematical science is inferior to that obtained in pure mathematics, only in so far as the utmost confidence in the truth of laws arising from even the highest degree of probability attained by inductive reasoning, must always fall short of the conviction which follows from the perception of necessary truth. If, however, we estimate the rank of a science by the perfection of its methods,—the power it confers of pursuing the most lengthened and complex processes of reasoning with perfect certainty of the correctness of the results—its consequently perpetually extending and cumulative character—the almost hopeless difficulty of the problems it has solved—the remoteness of the conclusions to which it has conducted the inquirer into natural laws—the number and importance of the truths with which it has enriched mankind—or, in short, the almost incredible degree in which it has aided and amplified the reasoning faculties—we must accord both to pure and mixed mathematics the first place among the various departments of human knowledge.

The different branches of mathematical science may be classified as follows:

1. Pure mathematics consists of the following branches: 1. Arithmetic, which may be subdivided into pure arithmetic, or the ordinary science of numbers, and of numerical calculations conducted by means of the common arithmetical notation; and algebra, or the methods of calculation by means of general symbols, so far as the processes are restricted to actual numerical operations. (See ARITHMETIC.) 2. Geometry, which investigates the properties of figures in the manner of Euclid's Elements, without the aid of algebraical processes. (See GEOMETRY, and CONIC SECTIONS.) 3. Algebra, or the calculus of operations, differing from arithmetical algebra in the more extended definition of its operations, and their more universal applications. (See ALGEBRA.) 4. The application of algebra to geometry, including coordinate geometry. (See ANALYTICAL GEOMETRY.) 5. The differential and integral calculus, which include the general theory of limits, or those methods of operation in which the limits of ratios are employed in calculation, and denoted by specific algebraical symbols. (See FLUXIONS.) II. Mixed mathematics includes the application of pure mathematics to mechanics, astronomy, optics, heat, electricity, &c. (See Astronomy, &c.) The theory of probabilities is usually included under this head, although it has perhaps equal claims to rank under pure mathematics.

(For the history of mathematical science the reader is referred to the articles Algebra and Geometry, and to the Preliminary Dissertations.)