_Über das System der rein mathematischen Wissenschaften, Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 1, p. 36._ =112.= Pure mathematics is not concerned with magnitude. It is merely the doctrine of notation of relatively ordered thought operations which have become mechanical.--NOVALIS. _Schriften (Berlin, 1901), Zweiter Teil, p. 282._ =113.= Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematics has for its function to develop the consequences involved in the definition of a group of mathematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics.--CHRYSTAL, GEORGE. _Encyclopedia Britannica (9th edition), Article “Mathematics.”_ =114.= The purely formal sciences, logic and mathematics, deal with those relations which are, or can be, independent of the particular content or the substance of objects. To mathematics in particular fall those relations between objects which involve the concepts of magnitude, of measure and of number.--HANKEL, HERMANN. _Theorie der Complexen Zahlensysteme, (Leipzig, 1867), p. 1._ =115.= _Quantity is that which is operated with according to fixed mutually consistent laws._ Both operator and operand must derive their meaning from the laws of operation. In the case of ordinary algebra these are the three laws already indicated [the commutative, associative, and distributive laws], in the algebra of quaternions the same save the law of commutation for multiplication and division, and so on. It may be questioned whether this definition is sufficient, and it may be objected that it is vague; but the reader will do well to reflect that any definition must include the linear algebras of Peirce, the algebra of logic, and others that may be easily imagined, although they have not yet been developed. This general definition of quantity enables us to see how operators may be treated as quantities, and thus to understand the rationale of the so called symbolical methods.--CHRYSTAL, GEORGE. _Encyclopedia Britannica (9th edition), Article “Mathematics.”_ =116.= Mathematics--in a strict sense--is the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations. --MURRAY, J. A. H. _A New English Dictionary._