cess is called 'induction,' and is the scientific method or process with which Bacon's name is generally identified, though I need hardly say that it is a process as old as the human intellect itself. Bacon only insisted on its importance, and helped to formulate it as an instrument for the discovery of truth. On the other hand there are some subjects to be studied, in which you begin with the large, general, universal truth, and proceed afterwards to deduce from this a number of special and detailed inferences. Such subjects are said to be studied deductively. In the former the movement of the thoughts is from the perception of particulars to the recognition of the general law. In the latter it is from the statement of the general to the recognition of the particulars. One sees that his neighbour is dead, he remembers the death of his parents or friends, he reads the history of the past, and by putting these experiences together, he arrives inductively at the conclusion that All men are mortal. He accepts this proposition. He muses over it. He adds, I too am a man. And he concludes, I therefore am mortal. Here the process is deductive. And sometimes in learning he must use one process, and sometimes another. And it is a great part of the business of education so to train the faculties that whichever process we adopt we should use it rightly, that our generalizations shall be valid and sound generalizations, and that our inferences shall be true, not hasty and illegitimate inferences, from the facts which may come before us.

Now Arithmetic and Geometry considered as sciences Arithmetic afford examples of both these kinds of learning. If I and Mathematics work out a few problems by experimental and chance mainly but methods, and having seen how the answer comes out, not wholly arrive at the conclusion that one method is best, I have

deductive.

Mathematics

a training

in logic.

reached this result by the method of analysis or induction. But if I start from axioms and definitions, and afterwards apply these to the solution of problems, I am availing myself of the method of deduction. But the method of deduction is, after all, the characteristic mode of procedure in arithmetical as well as in all other departments of mathematical science. We shall see hereafter that the physical sciences furnish the best training in inductive reasoning, for there you have in fact no axioms or admitted truths to start from, and must in all cases begin by the observation of phenomena and the collocation of experience. But elementary truths about number and about space, which are respectively the bases of arithmetic and geometry, have the great advantage of being very simple and very evident. They lie quite outside the region of contingency or controversy, and they therefore furnish a better basis for purely deductive or synthetic logic than any other class of subjects in which the very data from which we proceed are often disputed, or at least disputable.

Take a geometrical axiom-an elementary truth. concerning the properties of space-"two straight lines cannot enclose a space;" or an arithmetical axiom, an elementary truth concerning the properties of number, "to multiply by two numbers successively is to multiply by their product," and we observe that the moment we state them we perceive their necessary truth; there is no room for debate or difference of opinion; to understand either statement is to accept it. And so with all other of the fundamental axioms of geometry and arithmetic. Whatever particular facts prove ultimately to be contained in these general or universal truths must be true. As far as we can be certain of anything we are certain of these.

Deduction, the Mathematical Process. 319

Suppose then I want to give to myself a little training in the art of reasoning; suppose I wish to get out of the region of conjecture or 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 premisses 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. could exercise itself. Plato and his followers tried to think out together conclusions respecting the being, the duty, and the destiny of man, and the relation in which he stood to the gods and to the unseen world. What

Arithmetic

had geometry to do with these things? Simply this: That a man whose mind had not undergone a rigorous training in systematic thinking, and in the art of drawing legitimate inferences from premisses, was unfitted to enter on the discussion of these high topics; and that the sort of logical discipline which he needed was most likely to be obtained from geometry-the only mathematical science which in Plato's time had been formulated and reduced to a system. And we in this country have long acted on the same principle. Our future lawyers, clergy, and statesmen are expected at the University to learn a good deal about curves, and angles, and numbers and proportions; not because these subjects have the smallest relation to the needs of their lives, but because in the very act of learning them, they are likely to acquire that habit of steadfast and accurate thinking, which is indispensable to success in all the pursuits of life.

What mathematics therefore are expected to do for the mathe the advanced student at the University, Arithmetic, if matics of the School, 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 in achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in a school course that the art of thinking-consecutively, closely, logically-can be effectually taught.

I proceed to offer some practical suggestions as to the manner in which this principle, if once recognised, should dominate the teaching of Arithmetic, and determine your methods.

notation.

You have first of all to take care that so much of Our our Arithmetical system, as is arbitrary and conventional, artificial shall be shewn to be so, and not confounded with that part of Arithmetic which is permanently true, and based on the properties of number. We have for example adopted the number ten as the basis of our numeration; but there is nothing in the science of numbers to suggest this. Twelve or eight, or indeed any other number, might have served the same purpose, though not with quite the same convenience. Again the Arabic notation adopts the device of place to show the different values of figures: e.g. In 643 the 6 is shewn to mean 6 tens of tens, and the 4 to mean 4 tens, by the place in which they stand. But convenient as this arrangement is, other devices might have been adopted, which would have fulfilled the same purpose; and the Roman mode of representing the same number by DCXLIII may be with advantage compared; and its inconvenience practically tested by trying to work a sum with it. Again the

F. L.

2 I