perceive how these lines of division, representing as they did so much new fencing, explained why the small field had proportionately to area so much longer a boundary than the large one.
A chess-board served as another illustration. Taking each of its sixty-four squares to represent a farm duly inclosed, it was easy to see how a farmer rich enough to buy the whole number, were he to combine them in one stretch of land, could dispense with an immense quantity of lumber or wire fencing. During a journey from Montreal to Quebec the boys had their attention directed to the disadvantageous way in which many of the farms had been divided into strips long and narrow. "Just like a row of chess squares run together," said one of the lads.
When a good many examples had impressed the lesson on their minds pretty thoroughly, I had them write under their drawings, taking care that the terms used were understood: "Like plane figures vary in boundary as their like linear dimensions; they vary in area as the square of their like linear dimensions." It proved, however, that while the boys knew this to be true of squares, they could not at first comprehend that it was equally true of other forms. They drew equilateral and other triangles and ascertained that they conformed to the rule, but I was taken aback a little when the eldest boy said, "It isn't so with circles, is it?" His doubt was duly removed, but the remark showed how easy it is to make words outrun ideas; how hard it is for a young mind to recognize new cases of a general law with which in other examples it is quite familiar.
One chilly evening the sitting-room in which my pupils and I sat was warmed by a grate-fire. Shaking out some small live coals, I bade the boys observe which of them turned black soonest. They were quick to see that the smallest did, but they were unable to tell why. They were reminded of the rule they had committed to paper, but to no purpose, until I broke a large glowing coal into a score of fragments which became black almost at once. Then one of them cried, "Why, smashing that coal gave it more surface!" This young fellow was studying the elements of astronomy at school, so I had him give us some account of how the planets differ from one another in size, how the moon compares with the earth in mass, and how vastly larger than any of its worlds is the sun. Explaining to him the theory of the solar system's fiery origin, I shall not soon forget his keen delight—in which the others presently shared—when it burst upon him that because the moon is much smaller than the earth it must be much colder; that, indeed, it is like a small cinder compared with a large one. It was easy to advance from this to understanding why Jupiter, with eleven times the diameter of the earth, still glows faintly in the sky; and then to note that the sun pours out its