_The Conduct of the Understanding, Sect. 7._ =427.= As an exercise of the reasoning faculty, pure mathematics is an admirable exercise, because it consists of _reasoning_ alone, and does not encumber the student with an exercise of _judgment_: and it is well to begin with learning one thing at a time, and to defer a combination of mental exercises to a later period.--WHATELY, R. _Annotations to Bacon’s Essays (Boston, 1873), Essay 1, p. 493._ =428.= It hath been an old remark, that Geometry is an excellent Logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived, by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth.--BERKELY, GEORGE. _The Analyst, 2; Works (London, 1898), Vol. 3, p. 10._ =429.= Suppose then I want to give myself a little training in the art of reasoning; suppose I want to get out of the region of conjecture and probability, free myself from the difficult task of weighing evidence, and putting instances together to arrive at general propositions, and simply desire to know how to deal with my general propositions when I get them, and how to deduce right inferences from them; it is clear that I shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true. For in all our thinking, if we come to erroneous conclusions, we come to them either by accepting false premises to start with--in which case our reasoning, however good, will not save us from error; or by reasoning badly, in which case the data we start from may be perfectly sound, and yet our conclusions may be false. But in the mathematical or pure sciences,--geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,--we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. When Plato wrote over the portal of his school. “Let no one ignorant of geometry enter here,” he did not mean that questions relating to lines and surfaces would be discussed by his disciples. On the contrary, the topics to which he directed their attention were some of the deepest problems,--social, political, moral,--on which the mind