So much, then, we may say with Bacon "concerning the several classes of Idols and their equipage, all of which must be renounced and put away with a fixed and solemn determination, and the understanding thoroughly freed and cleansed; the entrance into the kingdom of man, founded on the sciences, being not much other than the kingdom of heaven, whereinto none may enter except as a little child." It may perhaps be urged that the result of such a survey as we have taken of the obstacles to clear thought is to leave the mind dazed and discouraged, partly because the suggestions made for the conquest of these obstacles, though easily formulated in theory are difficult and sometimes impossible in practice, and partly because the general if not expressed tendency of our analysis is (it may be said) in the direction of that Pyrrhonic skepticism which "doomed men to perpetual darkness." To the former objection I have only to reply that it is one to which all discussions of the principles and problems of conduct are necessarily open. "If to do were as easy as to know what were good to do, chapels had been churches, and poor men's cottages princes' palaces."[27] None the less, to state as lucidly as we can what were good to do under certain circumstances is properly regarded as part of the business of ethics. The other point is touched upon by Bacon himself in words which it would be impertinent to seek to better: "It will also be thought that by forbidding men to pronounce and set down principles as established until they have duly arrived through the intermediate steps at the highest generalities, I maintain a sort of suspension of the judgment, and bring it to what the Greeks call acatalepsia—a denial of the capacity of the mind to comprehend truth. But in reality that which I meditate and propound is not acatalepsia, but eucatalepsia; not denial of the capacity to understand, but provision for understanding truly; for I do not take away authority from the senses, but supply them with helps; I do not slight the understanding, but govern it. And better surely it is that we should know all that we need to know, and yet think our knowledge imperfect, than that we should think our knowledge perfect, and yet not know anything we need to know."
MATHEMATICS FOR CHILDREN.
By M. LAISANT.
Except with persons having specially favorable surroundings, I believe that the vast majority of parents have a feeling of dread at the thought of putting their children to the study of mathematics. They know that the child must learn something about it in order to pass his examinations; but with this knowledge goes an apprehension of loading his mind with those ideas which are so complicated and hard to acquire, and we put off the dreaded moment of setting him to work as late as possible.
While I believe it is wise to spare the child all useless overwork, I am persuaded also that the best way of sparing him is not to shrink from initiating him into hard work, if that can be done in a rational way.
I regard all the sciences as, at least to a certain extent, experimental, and, notwithstanding the views of those who would regard the mathematical sciences as a series of operations in pure logic, resting upon strictly ideal conceptions, I believe that we may affirm that there does not exist a mathematical idea that can enter our brain without the previous contemplation of the outer world and the facts it offers to our observation. This affirmation, the discussion of which now would carry us too far, may help to a clear idea of the way we should try to convey the first mathematical ideas to the mind of the child.
The outer world is the first thing the child should be taught to regard and concerning which he should be given as much information as possible—information which he will have no trouble in storing, we may well believe, and from this outer world the first mathematical notions should be borrowed; to these should succeed later an abstraction, which is less complicated than it seems.
Our primary teaching of arithmetic now follows in the tracks of that of grammar, as we might as well say that the teaching of grammar follows in the tracks of that of arithmetic. That is, in either case we teach the child a number of abstract and confusing definitions which he can not comprehend, imposing on him a series of rules to follow under the pretext of giving him a good practical direction, and we force him to learn and memorize these rules whether they are good for anything or not.
When the child has grown older he is given two or three short lessons a week in science, nine tenths of which, with his fleeting memory, he forgets before the next week's lessons come on. He can not relish anything that is taught him in that way, and it would be vastly better to give him no scientific ideas at all than to scatter them around in such a way, for all teachers agree that a fresh pupil is more easily dealt with and can be taught more satisfactorily and thoroughly than one who has been mistaught.