Logic Taught by Love. Mary Everest Boole

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СКАЧАТЬ there is to be any vital reform in method we must make young teachers realize that, for a few moments in each lesson, he and the chalk change places; that for those moments the chalk, not he, is the true intermediary (or mediator) between the Unseen Revealer and the class. We cannot continue to boycott in England all vital mathematical teaching, just because stupid people have talked grovelling nonsense about the doctrine which is its vital essence.

      The manner in which a problem that baffles us when treated on its own level can often be solved by bringing to bear on the solution truths of a higher order than that contemplated when the question was first propounded, is well illustrated by the famous 47th Proposition of Euclid. The question proposed for solution is this:—Is there any constant relation between the length of ​the hypothenuse of a right-angled triangle and the lengths of the sides? We are now so familiar with the solution, we have so mechanicalized the process by which the answer is arrived at, that the significance of both escapes us. But let us place ourselves, in imagination, back at the time when the question was as yet nsolved and was being eagerly investigated. In studying the earlier problems of Euclid, questions about lengths of lines are settled by striking circles with compasses (which is virtually a process of measuring); and questions of area, etc., by superposition. Everything is referred to certain axioms which act as a hurdle set up for the purpose of giving children the exercise of climbing over it. The formal Logic in the beginning of Euclid exercises a certain mental agility; but everything which is really found out, is found out by trusting to the evidence of our senses aided by some mechanical process.

      But when we attempt to find a relation between the hypothenuse and sides of a right-angled triangle, all modes of measurement fail to show any fixed relation, and appear even to show that none exists. Those who were satisfied that nothing was valid except the evidence of the recognized instruments probably asserted that the existence of any fixed relation was disproved.

      But there were true Free-Masons in those days, or rather there were Free Geometers, the founders of Free-Masonry; bold, untamable spirits, who dared invoke the All-Seeing Eye of the Great Unity to enlighten their blindness; and who well knew that rules limiting the play of the human intellect were made, chiefly, to be defied. They claimed the right to seek Truth outside the limits marked by orthodox compasses; they ​knew that, when we find our way stopped in the order of thought to which we have hitherto been confined, such experience is an indication that the time has come to investigate afresh the question of the relation between different orders of thought. By transferring the search for a relation between the hypothenuse and the sides to an order of dimensions higher than that involved in the original question, we find that there is a constant relation, one indeed of absolute equality, between the square on the hypothenuse and the squares on the sides.

      Let us think with sympathy of the orthodox Geometricians. They thought, of course, that they had exhausted all the possibilities, and satisfactorily proved that the constant relation sought had no existence. And behold, here come dreamers, who claim the right to overthrow all established boundaries of knowledge; to evade difficulties by a mere trick; and to solve the question, declared unsolvable, by reference to some extra-linear order of ideas! We can well imagine their disgust. Alas for human short-sightedness! the defenders of orthodox methods are forgotten; and "the dreamers, the derided, the mad, blind men who saw" Truth, because they persisted in ignoring the cobweb barriers raised by intellectual timidity—these heretics built the Temple dedicated by the Wise Man to The Great Unity; and they also founded the Geometry of the Future.

      The moral of Euclid is this:—As long as we are investigating relations with no reference to any higher order of ideas than is obviously involved in those relations, we could make each discovery by some empirical method; a new order of thought begins at ​the point where we introduce into our reasoning considerations derived from an order of thought higher than that whose relations we are investigating.

      Now the present condition of moral and religious reasoning is about on a level with that of mathematical reasoning at the time when a few bold spirits were proposing to look for an equation between lines in the region of non-linear surface; and the majority were expressing scepticism and indulging in sneers. The parallel is perhaps all the more accurate, because reasoning about lines as lines is in itself, and necessarily, in a sense illusory. There is no such thing in Nature as a line, except the edge of a surface (nor, indeed, can there be any surface except the boundary of a solid).

Thousands of years before any such definite conception dawned in mathematics as that to which the name "dimensions" attaches itself now, it must have occurred to thoughtful men to study the shadows cast by solid objects. The duration of the shadow, it would be observed, is not coeval with that of the substance from which it is projected; a passing cloud is sufficient to obliterate it. Nor is its form solely determined by that of the solid object, but partly by the position of the latter relatively to the Source of Light. Pondering on this would lead to experiments in shadow-producing. Of the exact course of the shadow-study carried on by wise men of old we have no accurate record; but whoever is engaged in prosecuting similar investigations now, is sometimes irresistibly made to feel that he is going over the same ground as has been trodden by some "inspired" writer of the olden time. To avoid controversy, I propose here only to indicate a simple ​method by which any person of average intellect can commence the shadow-study, and introduce children to it. The points of contact between us and the ancients, I shall (with one important exception) leave the reader to find out for himself. I will observe, however, that the shadow-study was a favourite amusement of George Boole; and it would appear from the Seventh Book of the Republic, that Socrates was familiar with it.^[1]

      Those who are only beginning the shadow-study can work most conveniently with a single light overhead. Later on, combined and crossed lights can be used, and in some cases it will be useful to have a movable light. Place on the table a sheet of white paper. Hold between the paper and the light a ring. Call attention to the fact that the same ring casts a circular or an oval shadow, or a straight line, according to the position in which it is held. Also that the same series of shadows is produced by an elliptical ring as by a circular one. Either can be made to cast a shadow resembling in shape the other. A straight line, however (a knitting-needle for instance), cannot be made to cast a curved shadow on a plane; its shadow is always a straight line, which becomes shorter as the needle is tilted up, till at last it resembles a mere dot.

      If a circular disk of card-board be held horizontal under the light, it can be made to cast a series of shadows resembling in turn each of the conic sections (circle, ellipse, hyperbola, and parabola), by altering the position of the paper on which the shadow is cast. The same series of forms may be produced by placing a ​lighted night-light in the bottom of a tall jar, and throwing the shadow of the rim of the jar on surfaces held in different positions.

      The best paper to use is that which is ruled in small squares (it can be procured at the shops which furnish educational apparatus). The paper may with advantage be laid on the table with its lines pointing to the cardinal points of the compass; so that a line of shadow can be described by stating, e.g., that it crosses so many squares from north to south, and so many from east to west.

      Take a corkscrew-wire, with rings sufficiently large to throw a distinct shadow. It is possible to hold it so that its shadow is a mere circle; in another position it makes a mere wavy line. An ordinary spiral wire is easily procured, and in practice is sufficient; but we shall gain more instruction about the play of Natural forces if we picture to ourselves what would be the effect of using a spiral whose rings are elliptical. I shall assume here that we are using an elliptical spiral. The wire itself will then represent the path of a planet in space; one of its shadows pictures the path of the planet round its sun or suns; another, the path of the whirling storm-wind, to which Jesus compared that of Inspiration.

      Let us now place our (elliptical) spiral in such a position that it casts no shadow except an ellipse, and, for convenience of reference, let us agree that the longer axis points north and south. Let us picture to ourselves a tribe of microscopic creatures, СКАЧАТЬ