[1428]. The intellectual habits of the Mathematicians are, in some respects, the same with those [of the Metaphysicians] we have been now considering; but, in other respects, they differ widely. Both are favourable to the improvement of the power of attention, but not in the same manner, nor in the same degree.
Those of the metaphysician give capacity of fixing the attention on the subjects of our consciousness, without being distracted by things external; but they afford little or no exercise to that species of attention which enables us to follow long processes of reasoning, and to keep in view all the various steps of an investigation till we arrive at the conclusion. In mathematics, such processes are much longer than in any other science; and hence the study of it is peculiarly calculated to strengthen the power of steady and concatenated thinking,—a power which, in all the pursuits of life, whether speculative or active, is one of the most valuable endowments we can possess. This command of attention, however, it may be proper to add, is to be acquired, not by the practice of modern methods, but by the study of Greek geometry, more particularly, by accustoming ourselves to pursue long trains of demonstration, without availing ourselves of the aid of any sensible diagrams; the thoughts being directed solely by those ideal delineations which the powers of conception and of memory enable us to form.—Stewart,Dugald.
Philosophy of the Human Mind, Part 3, chap. 1, sect.3.
[1429]. They [the Greeks] speculated and theorized under a lively persuasion that a Science of every part of nature was possible, and was a fit object for the exercise of a man’s best faculties; and they were speedily led to the conviction that such a science must clothe its conclusions in the language of mathematics. This conviction is eminently conspicuous in the writings of Plato.... Probably no succeeding step in the discovery of the Laws of Nature was of so much importance as the full adoption of this pervading conviction, that there must be Mathematical Laws of Nature, and that it is the business of Philosophy to discover these Laws. This conviction continues, through all the succeeding ages of the history of the science, to be the animating and supporting principle of scientific investigation and discovery.—Whewell,W.
History of the Inductive Sciences, Vol. 1, bk. 2, chap.3.
[1430]. For to pass by those Ancients, the wonderful Pythagoras, the sagacious Democritus, the divine Plato, the most subtle and very learned Aristotle, Men whom every Age has hitherto acknowledged as deservedly honored, as the greatest Philosophers, the Ring-leaders of Arts; in whose Judgments how much these Studies [mathematics] were esteemed, is abundantly proclaimed in History and confirmed by their famous Monuments, which are everywhere interspersed and bespangled with Mathematical Reasonings and Examples, as with so many Stars; and consequently anyone not in some Degree conversant in these Studies will in vain expect to understand, or unlock their hidden Meanings, without the Help of a Mathematical Key: For who can play well on Aristotle’s Instrument but with a Mathematical Quill; or not be altogether deaf to the Lessons of natural Philosophy, while ignorant of Geometry? Who void of (Geometry shall I say, or) Arithmetic can comprehend Plato’s Socrates lisping with Children concerning Square Numbers; or can conceive Plato himself treating not only of the Universe, but the Polity of Commonwealths regulated by the Laws of Geometry, and formed according to a Mathematical Plan?—Barrow,Isaac.
Mathematical Lectures (London, 1734), pp. 26-27.
And Reason now through number, time, and space
Darts the keen lustre of her serious eye;