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CHAPTER III
GEOMETRIC SYMBOLS OF PROGRESS BY PULSATION

"When He made a decree for the rain, and a way for the lightning, He saw Wisdom; and to man He said, The fear of the Lord is Wisdom."
Job xxviii. 26—28.

While such Seers as Moses, Isaiah, and Odin were struggling to make the masses attend to the Genesis of things, or process of becoming, to give them some sort of faith in the very existence of any Laws of moral development by reaction, certain thinkers of less actively social and philanthropic temperament retired into solitude to study those Laws in detail, and to impart their knowledge to a few chosen pupils. Many of them studied the Laws of mental development by the method of mathematical analysis. The vulgar called such students Wizards and other harsh names. The pursuit of mathematical Logic was often forbidden; the books of "Grammarye" were burned; and sometimes the writers and readers of them. At other times Wizards were held in high honour. Is it not possible that the famous breast-plate of Aaron, which was consulted in mental difficulties, may have had inscribed on it a selection of the Natural and Geometric Symbols of orderly mental sequence?

In modern times it has been assumed that the study of mathematical Logic must have been "mystical" or "fanciful." But in the middle of last century the Science was revived by a group of writers, among the foremost of whom may be mentioned Babbage, De Morgan, Gratry, and Boole. These philosophers have reclaimed us from bondage to the ignorant dictatorship of the opponents of "Grammarye," by proving that all the more important thought-processes can be illustrated Algebraically. Whatever can be stated in Algebraic symbols may legitimately be expressed, so far as possible, by mechanical action, or in diagrams. The practical possibilities of Algebraic notation are, it is true, far wider than those of mechanical or pictorial representation; but, when a truth has once been expressed Algebraically, it can no longer be considered fanciful to illustrate, by motion or by diagram, as much of it as is capable of such illustration. We are therefore free to teach the elements of the Higher Logic, to those unacquainted with the notation of Algebra, by the same methods as were used of old in schools of Free-Masonry and of Prophecy. Messrs. Benjamin Betts and Howard Hinton are creating a simple system of representation, by the use of which the study of Logic by diagrams will some day be carried much further than has ever yet been possible. In this Chapter I propose to give some idea of the elements of the Science which teaches geometrically the laws of mental Pulsation.

A stone, escaping from a sling, exhibits tangential motion, overcoming the counteracting force of the string. According to modern definition, tangential momentum is not, properly speaking, a Force.

"Let knowledge grow from more to more, yet more of reverence in us dwell;" reverence for the Light vouchsafed to those who had not our advantages in the way of technical knowledge. To the thinker of the far back ages, these were the facts presented as a basis for speculation:—

If he dropped a stone from his hand, something made it fall at once, straight towards the earth. If he put it in a string and whirled it, something prevented its either falling as far as the string would allow, or yielding quite freely to the pull of the string. If it escaped from the string, something made it fly, not straight to the earth, but off towards the distant horizon. What were these rival somethings? Were they warring dæmons? And was his string a magic implement which altered the balance of power between these unseen personalities?

If the thinker was a Hebrew Prophet, his doctrine of Unity provided him with an answer. The various phenomena were not the work of rival personalities, but various manifestations of the Dunamis or Power exerted by The Unity Who speaks through diversities. And when his confidence in that Unity had given him skill to conquer an opponent, both more richly endowed by Nature than himself and more amply provided with material and mechanical appliances (of learning ?), he went back to his sheepfold and sang under the silent stars: "Oh! how love I Thy Law; all the day long is my study in it." The knowledge of Science possessed by the ancient Hebrew was no doubt very elementary; but perhaps it does his more fortunate European successor no harm to reflect that, so far as it went, it was sound. Nor is it bad for the over-cultured, over-specialized, over-examined little victims of advanced education to have their hearts brought into sympathy with the emotions of the Sacred Past, while their fingers are handling its implements, and their intelligence is brought into contact with its problems. The practice of playing with such toys as the sling and stone, the sucking-valve, the old-fashioned rope-maker's wheel, and the bandalore, may be made a means of accustoming children's nerves to the feeling of Nature's opposing tendencies, and may prepare the organization for receiving knowledge, later on, into the conscious mind. By training the hand to trace out Nature's action, we train the unconscious brain to act spontaneously in accordance with Natural Law; and the unconscious mind, so trained, is the best teacher of the conscious mind.

Another interesting exercise is that of tracing the Pentagram. Number the angles of a regular Pentagon in order; and draw straight lines from 1 to 3, from 3 to 5, from 5 to 2, from 2 to 4, from 4 to 1. Repeat several times in succession. Choose the size of the Pentagon to suit the size of the hand. After some practice, the Pentagram should be drawn by freehand. The exercise may be varied by tracing the Heptagram, i.e. passing from one point to another of a regular Heptagon in this order—1, 4, 7, 3, 6, 2, 5, 1. Childish as the foregoing description may seem, the tracing of these figures gives a curious feeling of arriving at completeness, by a series of tentative Pulsations backwards and forwards; and in old days the idea of magic attached itself to the exercise. At a time when the ability to investigate the angles of the Pentagram represented a high degree of mathematical skill, we may imagine that some enthusiast, in a fit of that tender fun which is characteristic of scientific genius, conceived the idea of tracing the figure on his threshold, saying to his pupils, "No lying spirit can enter, nor can science degenerate into sorcery, if study is put under the safe guardianship of accurate mathematics. Very slight inaccuracies," he would add, "leave room for the entrance of any kind of treachery and deception."

Later on it came to be believed that drawing a Pentagram on the threshold of a study prevented Satan from entering, provided it were drawn with sufficient accuracy. If at any point the junction were imperfect, the devil entered through the gap! The process by which the Pentagram degenerated from being the symbol of a scientific accuracy which shields from temptation to duplicity, into being a magic weapon of defence against a personal devil, is typical of all such degradations.

We now pass from the Science which teaches Natural Law by training the muscles, to that which addresses itself more directly to the intellect, and through the intellect to the soul.

The reform in the teaching of mathematics, now in agitation, depends essentially on getting teachers to understand that the chalk in the lecturer's hand becomes, at a given moment in the lesson, a Revealer, independent of (and, for the moment, superior to) the man who holds it; a Teacher of Teachers, King of Kings, and Lord of Lords. Yet how easily this essential doctrine of mathematics slips over into the slavish dogmas which ignorant people connect with the so-called doctrine of Transubstantiation! And how wisely did the English Church decree that the "transubstantiated" bread shall be eaten at once, not preserved as sacred! We do not wish the children to attach superstitious ideas to the chalk (or bread) when the demonstration is over; but if 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, 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 winding pathway, would naturally be represented to the early Geometrician (who lived where snakes were common and corkscrew wires not yet invented) by a snake coiled round a branch and hanging freely. The Geometry of the snake-coil is a more advanced stage of the Logic of the Divining-rod. In the fable of the rods that were shown to Pharaoh, the rods of both parties turned into snakes before that of the good prophet could swallow those of the bad magicians.

The Pentagram and the Magic Squares and other puzzles used by ancient "Wizards" illustrate the same principle, though less clearly. The principle itself was well known to the writer of the narrative of Abraham's calling. Abraham lived at a time when children were recklessly sacrificed to the gods whenever the parent felt prompted to do so. First he saw there was something wrong about that; children were meant for something better than to be killed. Next, it occurred to him to think that he might be wrong after all; ought he not to hold himself ready to give his son to God if God willed? He prepared to obey—if God should will. Then he was led to see that God wills to accept the offering of our children in a different way; we are to offer them to The Unity; but so as to hold them in trust for the establishment of human society. Thus he was led gradually up the spiral of revelation; at each step giving up something which had seemed (though it had not really been) the essence of some previous revelation.

How Superstition has fastened on that standard narration of spiral progress, of which Abraham is the hero, it is hardly necessary to point out.

The whirling storm-wind forms an accurate diagram for illustrating the relation of the ex-centric to humanity. Having drawn the spiral, with tangents running in various directions, one tells the pupil that the spiral itself represents the methods of the orthodox Ecclesia, the constituted authorities, the recognized teachers. The tangents represent various paths taken by ex-centrics, non-conformists, men who have received the first personal inspiration, the first birth into a new world; who have cared, and dared, to go off on lines of their own, each investigating as he is impelled by his own genius.

That is very well for a time; the youth of genius should find out for himself what is the direction in which he is impelled to go. But before he offers his gift on the National Altar, before he can really contribute to the general store of Knowledge, he must pull himself into line with general progress. How shall he do this?

The advice usually given is, to conform to the general custom, submit to the orthodox authorities. But there is a better way. The normal to any tangent leads towards the centre of equipoise, of balance, of calm progress; the spot where is concentrated the maximum of force with the minimum expenditure of force in mere useless motion.

But how can any man find certainly the true normal to his own tangential direction, the path which will guide his own peculiar genius towards the centre of progress and of power?

The direction given by the diagram is emphatic and unmistakable. He should look out for the brother ex-centric who most especially "offends" him, who is going on what seems to him to be most obviously and certainly the wrong road, because it leads in the opposite direction to what he is sure is, for him, the path of inspiration. Let him unify with that brother, call out to that brother to meet him and join hands. When these two meet, they will be in the centre of calm force. This conversion into the normal is the second birth of genius, the Regeneration which converts mere inspiration into Revelation; for inspiration is at best partial; Revelation comes at the moment of the combination of opposites.

As the spiral shows, all inspirations are inadequate; and nearly all are, in themselves, actually misleading.

A dim suggestion makes itself felt as we think of the spiral wire. The whole coil might, if long enough, be treated as if it were a wire, and wrapped in its turn round and round into a larger coil. We leave this consideration to the imagination of any reader who cares to pursue it. For none have more need of patience and reticence than those who are teaching the art of generalizing mathematical method, who are transferring to the analysis of ethics and history that knowledge of the human mind which is gained by mathematical investigation. All the various rays of light shed by our various studies came originally form One Source, and are meant to be re-united; but not till the appointed time is come. The danger of premature synthesis is that it leads to the vicious habit of what is called reasoning by analogy; jumping to conclusions about outer things and historical events. True analogy is of use as throwing light on Laws of Thought; it teaches, not Truth in itself but the nature of the relation of the human consciousness to Truth. True Logic is not dogmatic about facts; it clears the ground of false conclusions, and leaves The Eternal Pulsation to speak in silence to the soul.

We now go back for a moment to the Tree-Symbolism. In the Hebrew Scriptures knowledge is spoken of as a Tree. In the Odin religion we are taught that: Its root is the knowledge of earth and its crown is the knowledge of Heaven.

  1. This detailed account of the use of these symbols for educational purposes will be found in "The Preparation of the Child for Science." (Oxford : Clarendon Press).