Logic Taught by Love. Mary Everest Boole
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, whose true destiny should be to proceed upwards in the direction of the coil as a whole, and who have a blind but irresistible impulse to do so. They have no mode of ascending except along the wire; and no mode of expressing statements about distance, except on the paper. They have a vague, dawning consciousness of movement in the up and down direction, and of distance from the paper. In some, this consciousness of up and down is very much more developed than in others. In some it is so weak that they believe it to be mere illusion; in others so strong that they fancy all other modes of movement are illusion. But (we suppose) none can make definite statements about motion, except by reference to the lines on the paper. Progress upwards, therefore, is, for all of them, non-statable except in so far as it is connected with motion across the surface of the paper.
A group of them have climbed round a half-coil, beginning at the northern extremity of the longer axis; and have now arrived at its southern extremity. Their actual progress has been upwards, and amounts to half the distance between two coils; their expressed and apparent progress is the length of the longer axis of the shadow-ellipse on the paper. Whether any individual will be most conscious of his actual or of his recorded progress will depend on the condition of his individual consciousness. The condition common to them all is this:—Such progress as is actual is not recorded; and that which is recorded and registered is not actual nor permanent.
Every part of the progress which is registered will have to be unmade soon after it is made. Only that which is not registered is permanent. The particular group of creatures under our consideration first made some progress in a direction partly southward, but partly also eastward, away from the longer axis. Already they have had to unmake their eastward progress and come back to the longer axis. They have still, however, been able to congratulate themselves on a considerable amount of progress southward; and the optimists among them no doubt set up a theory that, after all, true progress is southward, and the eastward motion was only an accidental concomitant of the southward. But when the extremity of the long axis is reached, a terrible conflict sets in; the upward path is beginning to turn towards the north-west. The southward progress, therefore, is being lost! A most dramatic novel could be woven of the conflict of opinion and feeling that would arise, as soon as the nature of the situation came to be realized; for all the tragedy of History is summed up in the prosaic fact that "the actual progress is not statable; the recorded progress will all have to be unmade."
Whether the spiral wire was actually used in ancient times to teach the true principle of development, will perhaps never be known. In this connection it may be well to notice that the Spiral of Ascent, the