Page:Popular Science Monthly Volume 73.djvu/325

 not regarded as mathematics, and, in the other, it is supposed to comprise all that is generally included under the term mathematics. With such a wide range of usage among eminent authorities it is evident that an acceptable definition is hopeless.

These instances appear sufficient to emphasize the fact that the terms arithmetic, algebra and geometry have no definite meanings in mathematical literature. They may be compared with the names of the constellations, which attract the attention of the amateur but are not generally taken very seriously by the professional astronomer since their boundaries are not defined with clearness. Just as it may be difficult to establish a connection between the figures represented by the names of some of the constellations and the arrangement of the brighter stars in them, so it is difficult to see much connection between the meaning of the terms arithmetic, algebra and geometry, and some of the subjects classed under these heads. In a growing science it is very desirable to have some elastic terms—terms to which we assign broader and perhaps even different meanings as our knowledge advances. In fact, the term mathematics is itself preeminently one whose meaning is a matter of slow development, even if we accept such brief definitions as mathematics is the science of saving thought, or "mathematics is the science of drawing necessary conclusions."

The fact that many things which appear unrelated when studied superficially exhibit the most intimate connections when viewed from a higher standpoint has doubtless been a potent cause of the variety of usage as regards general terms of classification. There are no natural lines of division in mathematics. In fact, one of the most attractive phases in the development of mathematics is the discovery of the relations existing between what was supposed to be unrelated. In other words, the unifying of mathematical truths is one of the chief concerns of many of the workers in this domain. Although the elements of arithmetic, algebra and geometry appear sufficiently distinct to the beginner, the marks of distinction vanish one by one as one proceeds in following up the ideas starting from these centers, as is evidenced by the term analytic geometry, since analysis and algebra were synonyms for Newton, Euler and Lagrange.

Notwithstanding the fact that there are no natural lines of division in mathematics, classification is essential and need not be entirely artificial; for, marks of differences which are only superficial are, nevertheless, worthy of note and frequently furnish convenient centers for groups of very closely related ideas. Both subject-matter and method offer many such superficial marks of difference which are utilized for the sake of classification. As we go away from these centers we naturally reach facts which seem equally closely related to more than one center, and in such cases it is necessary to have either