CHAPTER I

DEFINITIONS AND OBJECT OF MATHEMATICS

=101.= I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music.--SYLVESTER, J. J.

_Presidential Address to the British Association, Exeter British Association Report (1869); Collected Mathematical Papers, Vol. 2, p. 659._

=102.= ... all the sciences which have for their end investigations concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, _mathematics_. And a proof that it far surpasses in facility and importance the sciences which depend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides; ....

--DESCARTES.

_Rules for the Direction of the Mind, Philosophy of D. [Torrey] (New York, 1892), p. 72._

=103.= [Mathematics] has for its object the _indirect_ measurement of magnitudes, and it _purposes to determine magnitudes by each other, according to the precise relations which exist between them_.--COMTE.

_Positive Philosophy [Martineau], Bk. 1, chap. 1._

=104.= The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.--COMTE.

_Positive Philosophy [Martineau], Bk. 1, chap. 2._

=105.= Mathematics is the science of the connection of magnitudes. Magnitude is anything that can be put equal or unequal to another thing. Two things are equal when in every assertion each may be replaced by the other.--GRASSMANN, HERMANN.

_Stücke aus dem Lehrbuche der Arithmetik, Werke (Leipzig, 1904), Bd. 2, p. 298._

=106.= Mathematic is either Pure or Mixed: To Pure Mathematic belong those sciences which handle Quantity entirely severed from matter and from axioms of natural philosophy. These are two, Geometry and Arithmetic; the one handling quantity continued, the other dissevered.... Mixed Mathematic has for its subject some axioms and parts of natural philosophy, and considers quantity in so far as it assists to explain, demonstrate and actuate these.

--BACON, FRANCIS.

_De Augmentis, Bk. 3; Advancement of Learning, Bk. 2._

=107.= The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed _Mixed Mathematics_, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called _Pure Mathematics_.--WHEWELL, WILLIAM.

_The Philosophy of the Inductive Sciences, Part 1, Bk. 2, chap. I, sect. 4. (London, 1858)._

=108.= Higher Mathematics is the art of reasoning about numerical relations between natural phenomena; and the several sections of Higher Mathematics are different modes of viewing these relations.--MELLOR, J. W.

_Higher Mathematics for Students of Chemistry and Physics (New York, 1902), Prologue._

=109.= Number, place, and combination ... the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred.--SYLVESTER, J. J.

_Philosophical Magazine, Vol. 24 (1844), p. 285; Collected Mathematical Papers, Vol. 1, p. 91._

=110.= There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all three of them combined, every mathematical truth admits of being referred; these are the three cardinal notions, of Number, Space and Order.

Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space.--SYLVESTER, J. J.

_A Probationary Lecture on Geometry, York British Association Report (1844),