Popular Science Monthly/Volume 38/November 1890/My Class in Geometry
By GEORGE ILES.
A VIVID recollection of my boyhood is the general disfavor with which my school-fellows used to open Euclid. It was in vain the teacher said that geometry underlies not only architecture and engineering, but navigation and astronomy. As we never had any illustration of this alleged underlying to make the fact stick in our minds, but were strictly kept to theorem and problem, Euclid remained for most of us the driest and dreariest lesson of the day. This was not the case with me, for geometry happened to be my favorite study, and the easy triumph of leading the class in it was mine. As years of active life succeeded my school-days I could not help observing a good many examples of the truths set forth in the lines and figures I had conned as a boy; examples which, had they been presented at school, would certainly have somewhat diminished Euclid's unpopularity. In fullness of time it fell to my lot to be concerned in the instruction of three boys—one of fourteen, the second twelve, the third a few months younger. In thinking over how I might make attractive what had once been my best-enjoyed lessons, I took up my ink-stained Euclid—Playfair's edition. A glance at its pages dispossessed me of all notion of going systematically through the propositions—they took on at that moment a particularly rigid look, as if their connection with the world of fact and life was of the remotest. Why, I thought, not take a hint from the new mode of studying physics and chemistry? If a boy gets a better idea of a wheel and axle from a real wheel and axle than from a picture, or more clearly understands the chief characteristic of oxygen when he sees wood and iron burned in it than when he only hears about its combustive energy, why not give him geometry embodied in a fact before stating it in abstract principle? Deciding to try what could be done in putting book and blackboard last instead of first, I made a beginning. Taking the boys for a walk, I drew their attention to the shape of the lot on which their house stood. Its depth was nearly thrice its width, and a low fence surrounded it. As we went along the road, a suburban one near Montreal, we noticed the shapes of other fenced lots and fields. Counting our paces and noting their number, we walked around two of the latter. This established the fact that both fields were square, and that while the area of one was an acre and a half, that of the other was ten. When we returned home the boys were asked to make drawings of the house-lot and of the two square fields, showing to a scale how they differed in size. This task accomplished, they drew a diagram of the house-lot as it would be if square instead of oblong. With a foot-rule passed around the diagram it was soon clear to them that, if the four sides of the lot were equal, some fencing could be saved. The next question was whether any other form of lot having straight sides could be inclosed with as little fence as a square. Rectangles, triangles, and polygons were drawn in considerable variety and number and their areas calculated, only to confirm a suspicion the boys had entertained from the first—that of lots of practicable form square ones need least fencing. In comparing their notes of the number of paces taken in walking around the two square fields, a fact of some interest came out. While the larger field contained nearly seven times as much land as the other, it only needed about two and a half times the length of fencing to surround it. Taking a drawing of the larger inclosure, I divided it into four equal parts by two lines drawn at right angles to each other. It only needed a moment for the boys to 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 wealth of heat and light because the immensity of its bulk has, comparatively speaking, so little surface to radiate from.
To make the law concerned in all this 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 that of each separate cube. With twenty-seven blocks built into a cube, they found that structure 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: Observation and comparison brought the boys to the rule which told exactly what proportion of surface remained exposed. They wrote, "Like solids vary in surface as the square, and in contents as the cube of their like dimensions." They were glad to note that the first half of their new rule was nothing but their old one of the farms and fields over again.
As the law at which we had now arrived is one of the most important in geometry, I took pains to illustrate it in a variety of ways. Taking a long, narrow vial of clear glass, nearly filled with water and corked, I passed it around, requesting each of the boys to shake it smartly, hold it upright, and observe which of the bubbles came to the surface first. All three declared that the biggest did, but it was a little while before they could be made to discern why. They had to be reminded of the cinders and the building-blocks before they saw that a small bubble's comparatively large surface retarded its motion through the water. The next day we visited Montreal's wharves, and, pacing alongside several vessels, jotted down their length. In response to questions, the boys showed their mastery of the principle which decides that the larger a ship the less is its surface in proportion to tonnage. Going aboard an Allan liner, of five thousand tons burden, we descended to the engine-room; we next visited a steamer of somewhat less than one thousand tons, and inspected her engines—engines having proportionately to power much larger moving surfaces to be retarded by friction than those we had seen a few minutes before. On being reminded of their experiments with the vial, the boys were pleasantly surprised to find that the largest bubble and the ocean racer come first to their respective ports by virtue of their identical quality of bigness, by reason of the economies which dwell with size. As we walked homeward, the youngest of our party espied a street-vender with a supply of gaudy toy-balloons. One of them bought, I dare say the little fellow's mind was pretty confident that there was no Euclid in that plaything. It proved otherwise. That evening he calculated how much the lifting power of his balloon would gain on its surface were its dimensions increased one thousand or ten thousand fold—step by step approaching the conclusion that, if air-ships are ever to be manageable in the face of adverse winds, they must be made vastly larger than any balloons as yet put together.
Not far from home stood a large store, displaying a miscellaneous stock of groceries, fruits, dry goods, shoes, and so on. As we cast our eyes about its shelves, counters, and floor, we saw many kinds of packages—cans of fish, marmalade, and oil, glass jars of preserves and olives, boxes of rice and starch, large paper sacks of flour. Outside the door stood half a dozen empty barrels and packing-cases. It certainly seemed as if the cost of paper, glass, tin, and lumber for packages must be an important item in retailing. One after another the boys discovered that the store was giving them their old lesson in a new form. They saw that the larger a jar or box the less material it needed. On their return home they were gradually led up to finding that form as well as size is an element in economy. Just as farms square in shape need least fence, they found that a cubical package needs least material to make it, and that tins of cylindrical form require least metal when of equal breadth and height.
Our next lesson was one for lack of which not a few inventors and designers have wasted time and money. Taking the trio to Victoria Bridge, we asked its custodian the length of its central span. His reply was, three hundred and fifty-two feet. When I asked the boys how matters would be changed if the span were twice as large, they soon perceived that, while increased in strength by breadth and thickness, it would be heavier by added length as well. On our return we compared two boards differing in each of their three dimensions as one and two, serving to make manifest why it often happens that a design for a bridge or roof, admirable as a model, fails in the large dimensions of practical construction.
One day a roofer had to be called in to make needful repairs. We went with him to the roof, and found the gutter choked with mud. How had it got there? A glance at the roof, an iron one, showed it covered with dust which the next shower would add to the deposit in the gutter. Dust-particles are extremely small and fine, and did not this explain how the wind had been able to take hold of them and carry them far up into the air? Although the boys had considerably less pocket-money than they liked, they had still enough to enable them to observe that the smallest coins were most worn. When they came to think it over, they readily hit on the reason why.
Our next lessons were intended to bring out the relations which subsist between several of the principal forms of solids. Two series of models in wood were accordingly made. The first consisted of a cube having a base five inches square, and a wedge and pyramid of similar base and height. The second series comprised a cylinder, sphere, and cone, each five inches broad and high. Taking the first series, a moment's comparison of the sides of wedge and cube told that one contained half as much wood as the other; but that the pyramid contained a third as much as the cube was not evident. Weighing the pyramid and cube brought out their relation, but a more satisfactory demonstration was desirable, for what was to assure us that the two solids were of the same specific
gravity? Taking a clear glass jar of an accurately cylindrical interior, measuring seven and a half inches in width by ten in height, it was half filled with water, and a foot-rule was vertically attached to its side. The models, which were neatly varnished, and therefore impervious to water, were then successively immersed and their displacement of the water noted. This proved that the pyramid had a third the contents of the cube, that the same proportion subsisted between the cone and cylinder, and that the sphere had twice the contents of the cone. Dividing the wedge by ten parallel lines an equal distance apart, I asked how the area of the smallest triangle so laid off, and that of the next smallest, compared with the area of the large triangle formed by the whole side of the wedge. "As the square of their sides" was the answer. Dipping the wedge below the surface of the water in the jar, edge downward, it was observed to displace water as the square of its depth of immersion. Reversing the process, the wedge became a simple means of extracting the square root. Dividing the vertical play of its displacement into sixteen parts drawn along the jar's side, we divided the wedge into four parts by equidistant parallel lines. Then, for example, if we sought the square root of nine, we immersed the wedge with its edge downward until it had displaced water to line nine on the jar's side. On the wedge the water stood at line three, the square root of nine. In a similar way the cone was observed to displace water as the cube of its depth of immersion, and therefore could be impressed into the service of extracting the cube root. For this purpose its total play of displacement in a jar of five and a half inches interior diameter was divided into twenty-seven parts, and the cone was marked off into three sections. To find the cube root of eight, we lowered the cone apex downward, until the water-level was brought to eight on the jar's side; at that moment the liquid encircled the cone at section two, denoting the cube root of eight. The pyramid immersed in the larger jar acted equally well as a cube-root extractor. Measuring both cone and pyramid at each of their sectional divisions, the boys were required to ascertain the rule governing their increase of sectional area, and arrived at the old familiar law of squares—a law true not only of all solids converging regularly to a point, but of all forces divergent or radiant from a center, simply because it is a law of space through which such forces exert themselves.
While I was glad to use examples and models to instruct my pupils, I wished them to grasp certain geometrical relations through exercise of imagination. They had long known that the area of a parallelogram is the product of its base and height; they were now required to conceive that any triangle has half the area of a parallelogram of equal height and base. It was easy then to show them the very old way of ascertaining the area of a circle, the method which conceives it to be made up of an indefinitely great number of triangles whose bases become the circle's circumference, and whose altitude is the circle's radius. Rolling the cylindrical model once around on a sheet of paper, its circuit was marked off; this was made the base-line of a parallelogram having a height equal to half the cylinder's breadth; half that area was clearly equal to the surface of the circle forming the cylinder's section. Another method of proving the relation between the area of a circle and its circumference was followed by the boys with fair promptness. I asked them to imagine a circular disk to be made up by the contact of a great number of concentric rings. Supposing the disk to be a foot in diameter and each ring to be the millionth of a foot wide, I inquired, "How many rings would there be?" "Half as many, half a million." To the question, "What would be the size of the average ring's circumference?" "Half that of the whole circle." was the reply. They were thus brought to it that if a circle rolled around once is found to have 3·1416 lineal units for its circumference, its area must be ·7854, or one half of one half as much, expressed in superficial units of the same order.
A terrestrial globe was the text for our next lesson. Assuming its form to be spherical, shift its axis as we might, it was clear that its center remained at rest during rotation in all planes. A hint here as to why the calculations of the astronomer are less difficult than if the planets were of other than globular form, for each orb as affected by gravitation may be practically considered as condensed at its center. Turning from astronomy to navigation, we glanced at the principle of great-circle sailing. On the equator of our terrestrial globe we found the Gillolo Islands and Cape San Francisco. A ship's shortest course plainly lay along the equatorial line which joined them. When I asked which was the shortest route from San Francisco, California, to Figami Island, Japan, the boys concurred in the wrong answer, "Along the thirty-eighth parallel." Taking a brass semicircle equal in diameter to the globe's equator, and applying it so as to touch both places, the lads saw at once that the shortest route would take a ship somewhat toward the north for the first half of her voyage; that if two ports are to be joined by an arc, the largest circle of which that arc can form a part marks out the shortest track; and that this largest or great circle is practically no other than a new equator cutting the earth in a plane inclined to the geographical equator.
By this time about a year had elapsed since our little class in geometry had been formed, and its progress was very satisfactory. The eldest boy was now studying Euclid at a high school and earning high marks for his proficiency. In the lessons I have described, and in others which followed them, all three lads showed their interest by being constantly on the lookout for new illustrations. Let an instance or two of this suffice. One day they walked to an immense sugar-refinery some distance off, paced around it, estimated its height, and brought me their calculations as to its storage capacity in comparison with that of a small warehouse near by; calculations showing how much outer wall and roof were saved in the vast proportions of the refinery. At home an extension of the house was heated in the winter by a small stove; at a neighboring station of the street railway there was a much larger stove of the same pattern. Counting efficiency to depend on surface, one of the boys asked me if it would not be better to have two small stoves instead of that large one. He was perfectly conversant with the reason why steam-fitters make their heating-coils of small pipes, and why their radiators abound in knobs and ridges.
It may be no more than the effect of bias due to an individual preference for the study, but, in the light of its influence on these three young minds, I can not help thinking that geometry affords a most happy means of developing powers of observation and reasoning. When the boys came to study plants, minerals, and insects they found their knowledge of Euclid gave them a new and vital thread whereon to string what they learned. This was even more decidedly the case when they came to study the various modes of motion and certain principles of engineering science. Mr. W. G. Spencer, the father of Herbert Spencer, in an invaluable little book has shown how geometry can be taught so as to educe the noble faculty of invention. At the high school at Yonkers, New York, of which Mr. E. R. Shaw is principal, I have seen most original and beautiful solutions of Mr. Spencer's problems worked out by the pupils.