Notice that the unit figure of the cube is always the same as the number we are cubing; so that the cube of 4 ends in 4; the cube of 6 ends in 6; and so on for the five numbers given. This holds good without exception.

Therefore, any cube whose unit figure is 1, 4, 5, 6, or 9 has 1, 4, 5, 6, or 9 as the unit figure of the root.

There are still four numbers to consider, which we may group like this:

Here we see that the cube of 2 is 8, while the cube of 8 ends in 2; the cube of 3 ends in 7, while the cube of 7 ends in 3.

Therefore, any cube whose unit figure is 8 has 2 as the unit figure of the root; any cube whose unit figure is 2 has 8 as the unit figure of the root.

The same may be said of 7 and 3. When one is the unit figure of the cube, the other is the unit figure of the root. Keeping these facts in mind, we can readily write the unit figure of the root of any cube. And remembering the cubes of the numbers 1 to 9, we can as easily determine the first figure of the root, and, if the cube is of only two periods, the whole root.

For example, 592,704 is a perfect cube. By inspection, the first figure of the root we see is 8; and as the cube ends in 4, its, root must also, being 84. Take the cube 300,763. The first figure of the root is evidently 6; and as the cube ends in 3, its root must end in 7, making 67.

In any cube of more than two periods, we can apply this method to determine the first and last figures of the root, and use the regular rule for the intermediate figures. The cube 130,323,843 contains three periods and three figures in the root. The first and last figures of the root are 5 and 7 respectively, leaving us to determine the second figure in the regular way, which shows the root to be 507.

This method can only be used when the number we operate upon is known to be a perfect cube. It very often happens that from the nature of the work we know a certain number to be a cube whose root we must find, and in all such cases the method I have tried to explain is very convenient.-N. E. Journal.

DEPARTMENT OF PEDAGOGY.

Conducted by ARNOLD TOMPKINS, Chair of Pedagogy, University of Illinois, at Champaign.]

PEDAGOGICAL BOOKS AND EMERSON'S "CIRCLES."

It is usually a sign of narrowness when a man swears by any man's system of thought; whether it be in science, politics, philosophy, or pedagogy. One man and one system can present only one aspect of the whole truth. One should accept gratefully Spencer's system of philosophy, but with the understanding that he speaks from one standpoint and can present only one hemisphere of the truth When one becomes a pronounced disciple of Hegel he is apt to become blinded to other valid aspects of the truth. In becoming strictly an adherent of Froebel or Herbart one loses the many-sided truth and many-sided interest in the problem of pedagogy. Sectarianism in thought cannot lead to the inner court of the temple of truth. To gain the greatest height and the broadest horizon one's pedagogical books and reading should be of the most diverse character.

But what I now wish to insist on is the fact that pedagogical books are not limited to those so labeled. As a rule the best books to help the teacher are not those written with that distinct purpose. And the sooner the teacher learns to read vital books from the pedagogical point of view the better. The purpose of this article is to encourage such practice. It seems that when a course of reading is made out for the teachers it is expected that something will be chosen from books directly addressed to the teacher. This may be well, but it is not the only thing, and not always the best thing, that can be done. The New York School Journal, in criticising the action of the Indiana Reading Circle Board in adopting certain books not labeled teachers' books seems to think it not well for teachers to browse on common territory with other people; forgetting that the Indiana teacher has the power of assimilating the common food of the mind into pedagogical bone and muscle. The teacher would be in a bad plight indeed were he prohibited from the inspiration and guidance received from non-professional books. In fact there comes a time in one's professional growth when

pedagogical books give the least pedagogical help, as compared with other books; just as the minister who keeps helpfully close to the heart and life of his people must turn from his books of theological speculation to those which spring from the inner life of the soul, and are equally good for all classes and all professions. Pedagogical books, in the strict sense, are to train us so that we may give pedagogical interpretation to all things and to all books. They belong to the preparatory school of the profession; and, while they are never to be put aside, the teacher must rise to the universal point of view where his teaching power is reinforced from all quarters; to where even the universe is given its pedagogical interpretation. Such is the inevitable growth of the thinking teacher. So true is this that the reverse is also true. When one comes to ripe knowledge in any line of study, as that of science, sociology, religion, or philosophy, he arrives at the pedagogical point of view and is inspired with a pedagogical interest. Prof. Dewey, Dean of the Department of Philosophy in Chicago University, and who is doing such great service in the interest of philosophy, claims, if I interpret him correctly, that his chief interest in philosophy is a pedagogical one. For him it is a question of life and the development of man. The sociologist's whole problem is that of adjusting the forces of society so as to bring out the full life and true character of the individual. The scientist's problem is, how to get the most and best life out of the study of nature. His problem is one of life for the individual; and in this he has the true pedagogical motive. The preacher and the teacher are both seeking the ways of salvation for the man. No system of thought o aspect of truth can be pressed to its ultimate meaning without becoming pedagogical.

In the light of this what better thing could a teacher's reading club do than to take such a book as Herbert Spencer's First Principles or Carlyle's Sartor Resartus and extract its pedagogical signification. In fact what other signification have such books? One cannot interpret Emerson's essay on "Circles" without viewing it from the teacher's standpoint; and the teacher seeking professional help can find no better source.

For instance, how much might the teacher evolve

from this: "The life of man is a self evolving circle, which, from a ring imperceptibly small, rushes on all sides outward to new and larger circles, and that without end." Again, "The only sin is limitation." How helpfully inspiring and how pointed in guidance for the teacher to feel that all his complex duties and efforts are for the removal of limitation, of sin. Once more in the Essay: "Generalization is always a new influx of the divinity in the mind. Hence the thrill that attends it." When it is considered that the work of the teacher is to bring the pupil into unity with the divine mind through the phenomena of the world, he may see the significance of generalization in the study of those phenomena. Pedagogical books present the method of generalization, but here we have given the ultimate meaning of the process, the guiding and inspiring truth to the teacher always. And so at every point in this essay is the pedagogical writer outdone; and at the close of its study one may see the whole theory and practice of teaching symbolized by the circle.

George McDonald's Donald Grant has been spoken of as a pedagogical novel. This might well be said of Holland's Arthur Bonnicastle. But may not the same be said of any good novel? A novel must exhibit the development of a character, -an ideal development. Such is the process the teacher is living out with the pupil. Every poem is the teacher's text. To get at the motive in education in the close sympathy between man and the spirit of the world about him, the best thing the teacher can do is to lay aside his dreary professional treatise and turn to Wordsworth's "Intimations of Immortality."

So that a pedagogical library consists of all books the teacher can turn to pedagogical account; and these include all deep and true books on whatever phase of thought and life.

SCIENCE WORK IN THE GRADES.

Recently Dr. S. A. Forbes, Dean of the College of Science, University of Illinois, read a paper before the University High School Conference on the subject of "Science Study in the grades as a preparation for University Work." It is full

of suggestion to the general teacher, irrespective of university preparation, and while too long to insert in the Journal in full, the following portion, which gives the spirit of the discussion, may be read with great profit.

"I think we may safely say that our average elementary school, in central and southern Illinois at least, is now to all intents and purposes essentially a school of language and of mathematics, and if the work in these departments is not well done it is practically a school in the mechanism of language and the processes of mathematics. I need not say that from our point of view this is very wrong. The sensitive and curious mind of the little child should be opened towards every side on which real knowledge lies. It seems to me a pitiful and cruel thing to shut the outside door on life, and to close and blind the windows through which the light of nature streams, and then to force the boy or girl to work day after day for years on the contents of the inner closets of his own and other people's minds by the artificial light of the text-book and the routine teacher's drill. Open the inner doors of the house of the mind in due time, of course, and see also that the sky lights in the roof are free and clear, and invite the sunbeam and the cloud shine and the flower glow, and the glitter of the wave and the rustle of the leafy tree and the buzz of the bumbling beetle at the window sill into the room, and help the pathetic little weavers at their looms to weave all this carefully into the web of knowledge as it grows slowly under their clumsy but well intending fingers.

Indeed, for my own part, I believe that the whole life and teaching of the graded school should be saturated with nature study from the first day to the last; and I believe that in so treating it we should not narrow and sacrifice anything anywhere, but that, on the contrary, we should expand and vivify the work of the school throughout. Certainly we should in this way avoid that deadening of the sensibilities and darkening and narrowing of life of the child in the public school which, as I see the process going on in one after another of my young acquaintances, affects me as it would to see a cataract slowly stealing over the eyes of each,