=1625.= What mathematics, therefore are expected to do for the advanced student at the university, Arithmetic, if taught demonstratively, is capable of doing for the children even of the humblest school. It furnishes training in reasoning, and particularly in deductive reasoning. It is a discipline in closeness and continuity of thought. It reveals the nature of fallacies, and refuses to avail itself of unverified assumptions. It is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope; in which authority goes for nothing. In other departments of instruction you have a right to ask for the scholar’s confidence, and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards. But here you are justified in saying to your pupil “Believe nothing which you cannot understand. Take nothing for granted.” In short, the proper office of arithmetic is to serve as elementary training in logic. All through your work as teachers you will bear in mind the fundamental difference between knowing and thinking; and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing, or even facility of achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in the school course that the art of thinking--consecutively, closely, logically--can be effectually taught.--FITCH, J. G. _Lectures on Teaching (New York, 1906), pp. 292-293._ =1626.= Arithmetic and geometry, those wings on which the astronomer soars as high as heaven.--BOYLE, ROBERT. _Usefulness of Mathematics to Natural Philosophy; Works (London, 1772), Vol. 3, p. 429._ =1627.= Arithmetical symbols are written diagrams and geometrical figures are graphic formulas.--HILBERT, D. _Mathematical Problems; Bulletin American Mathematical Society, Vol. 8 (1902), p. 443._