=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._