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One frosty evening I sat with three young pupils in a room 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, until I broke a large glowing coal into a score of fragments, which almost at once turned black. Then one of them cried, “Why, smashing that coal gave it more surface!” This young scholar was studying the elements of astronomy that year, 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 volume, and how vastly larger than any of its worlds is the sun. Explaining to him the fiery origin of the solar system, I shall not soon forget his 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 cooler; 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 by its own light, and then to comprehending that the sun pours out its wealth of heat and light because the immensity of its bulk means a comparatively small surface to radiate from.

Cube built of 27 cubes of 9 times more surface.

To make the law concerned in these examples definite and clear, I took eight blocks, each an inch cube, and had the boys tell me how much surface each had—six square inches. Building the eight blocks into one cube, they then counted the square inches of its surface—twenty-four: four times as many as those of each separate cube. With twenty-seven blocks built into a cube, that structure was found to have a surface of fifty-four square inches—nine times that of each component block. As the blocks underwent the building process, a portion of their surfaces came into contact, and thus hidden could not count in the outer surfaces of the large cubes. The outer surfaces of these large cubes I then painted white; when each was separated into its eight or twenty-seven blocks, we saw in unpainted wood how surfaces were increased by this separation into the original small cubes. Observation and comparison brought the boys to the rule involved in these simple experiments. They wrote: Solids of the same form vary in surface as the square, and in contents as the cube, of their like dimensions.

This elementary law I traced that year in a variety of illustrations presented in “A Class in Geometry,” published by A. S. Barnes & Co., New York. Our excursions, since extended, are here given as an example of the knitting value of a pervasive rule kept constantly in mind.