It consists of Ten Points inscribed pyramid-like (from one to four) within its three sides, and it symbolizes the Universe in the famous Pythagorean Decad. The upper single point is a Monad, and represents a Unit-Point, which is the Unity whence all proceeds. All is of the same essence with it. While the ten points within the equilateral Triangle represent the phenomenal world, the three sides enclosing the pyramid of points are the barriers of noumenal Matter, or Substance, that separate it from the world of Thought.
Pythagoras considered a point to correspond in proportion to unity; a line to 2; a superfice to 3; a solid to 4; and he defined a point as a monad having position, and the beginning of all things; a line was thought to correspond with duality, because it was produced by the first motion from indivisible nature, and formed the junction of two points. A superfice was compared to the number three because it is the first of all causes that are found in figures; for a circle, which is the principal of all round figures, comprises a triad, in centre—space—circumference. But a triangle, which is the first of all rectilineal figures, is included in a ternary, and receives its form according to that number; and was considered by [pg 676]the Pythagoreans to be the author of all sublunary things. The four points at the base of the Pythagorean triangle correspond with a solid or cube, which combines the principles of length, breadth, and thickness, for no solid can have less than four extreme boundary points.[1059]
It is argued that “the human mind cannot conceive an indivisible unit short of the annihilation of the idea with its subject.” This is an error, as the Pythagoreans have proved, and a number of Seers before them, although there is a special training needed for the conception, and although the profane mind can hardly grasp it. But there are such things as “Meta-mathematics” and “Meta-geometry.” Even Mathematics pure and simple proceed from the universal to the particular, from the mathematical indivisible point to solid figures. The teaching originated in India, and was taught in Europe by Pythagoras, who, throwing a veil over the Circle and the Point—which no living man can define except as incomprehensible abstractions—laid the origin of the differentiated cosmic Matter in the base of the Triangle. Thus the latter became the earliest of geometrical figures. The author of New Aspects of Life, dealing with the Kabalistic Mysteries, objects to the objectivization, so to speak, of the Pythagorean conception and the use of the equilateral triangle, and calls it a “misnomer.” His argument that a solid equilateral body—
One whose base, as well as each of its sides, form equal triangles—must have four co-equal sides or surfaces, while a triangular plane will as necessarily possess five,[1060]
—demonstrates on the contrary the grandeur of the conception in all its Esoteric application to the idea of the pregenesis, and the genesis of Kosmos. Granted, that an ideal Triangle, depicted by mathematical, imaginary lines,
Can have no sides at all, being simply a phantom of the mind to which, if sides be imputed, these must be the sides of the object it constructively represents.[1061]
But in such case most of the scientific hypotheses are no better than “phantoms of the mind”; they are unverifiable, except on inference, and have been adopted merely to answer scientific necessities. Furthermore, the ideal Triangle—“as the abstract idea of a triangular body, and, therefore, as the type of an abstract idea”—accomplished and carried out to perfection the double symbolism intended. As an emblem applicable to the objective idea, the simple triangle became a solid. When repeated in stone, facing the four cardinal points, it [pg 677] assumed the shape of the Pyramid—the symbol of the phenomenal merging into the noumenal Universe of thought, at the apex of the four triangles; and, as an “imaginary figure constructed of three mathematical lines,” it symbolized the subjective spheres—these lines “enclosing a mathematical space—which is equal to nothing enclosing nothing.” And this because, to the senses and the untrained consciousness of the Profane and the Scientist, everything beyond the line of differentiated Matter—i.e., outside of, and beyond the realm of even the most Spiritual Substance—has to remain for ever equal to nothing. It is the Ain Suph—the No Thing.
Yet these “phantoms of the mind” are in truth no greater abstractions than the abstract ideas in general as to evolution and physical development—e.g., Gravity, Matter, Force, etc.—on which the exact Sciences are based. Our most eminent Chemists and Physicists are earnestly pursuing the not hopeless attempt of finally tracing to its hiding-place the Protyle, or the basic line of the Pythagorean Triangle. The latter is, as we have said, the grandest conception imaginable, for it symbolizes both the ideal and the visible universes.[1062] For if
The possible unit is only a possibility as an actuality of nature, as an individual of any kind, [and as] every individual natural object is capable of division, and by division loses its unity, or ceases to be a unit,[1063]
this is true only of the realm of exact Science in a world as deceptive as it is illusive. In the realm of Esoteric Science the Unit divided ad infinitum, instead of losing its unity, approaches with every division the planes of the only eternal Reality. The eye of the Seer can follow it and behold it in all its pregenetic glory. This same idea of the reality of the subjective, and the unreality of the objective Universe, is found at the bottom of the Pythagorean and Platonic Teachings—limited to the Elect alone; for Porphyry, speaking of the Monad and the Duad, says that the former only was considered substantial and real, “that most simple Being, the cause of all unity and the measure of all things.”