The so-called forty-seventh problem of Euclid was undoubtedly regarded by the ancient mystics as a divinely appointed Canon of Universal harmony, to the fundamental mathematical truth of which might be traced most of the laws of symmetry and order, which reign throughout the vast expanse of Cosmos. The precise nature of the various connections must be established by analogies, rather than by concrete examples, because the peculiar geometrical figure, in question, was probably deemed one of the sacred mysteries by the ancients and peculiarly one which might not be hewn, cut, carved or marked in any manner by which it might become legible or intelligible to the profane.
Its delineation, graphically, we probably first owe to the Arab geometers of the early days of our present era, among whom no scruple could have prevailed. The only truly ancient example the writer has ever seen is on a little clay Tessera from an Egyptian tomb forty centuries old, upon which it is rudely, but unmistakably, present.
The psychology of geometrical science to the sympathetic student is as thrilling as that of music or color to specialists therein.
There are marvels in Mathematics, indeed, but in Geometry, mathematics’ are applied and every point, line, superficies and solid, in its relations and potentialities bespeaks a subtle something which Finite Mind instinctively recognizes as the production of Infinite Mind.
We would be utterly unable to realize, much less formulate the notion of “Law,” pervading Cosmos, if all facts and forces did not present unswerving, uncompromising obedience to the geometrical verities which are at the root of all of that which we term “Manifestation ” – visualized existence.
Of the limited volume of these fundamental geometrical laws-and there are surprisingly few to account for the splendors of the Universe they present to our vision, the fact that the hypotenuse of a right-angle of three to four, invariably and to the ultra-microscopic fraction, measures five, as we have seen, is one of the most perfect proofs of pre-determined plan. The resulting triangle of 3-4-5 is not the only example of such a correlation between exact quantities for Geometry is full of them, but it is the simplest and most basic and calls attention to countless other remarkable factors to the cosmic whole, to which its intimate relationship may be demonstrated.
It would naturally be a physical impossibility to cover the enormous ground available for discussion, within present limits, but some slight notion of the tremendous influence of Geometry in general and this notable Pythagorean proposition, in particular, upon the philosophies and theologies of the World, may be obtained by deciphering a few ancient occultisms concerning it.
In order to get our bearing, we are compelled to formulate an ascertained law of development in all of the latter.
Their unique basis is the recognizable and ascertainable phenomena of the universe, the definition of which constitutes what modern man terms “Science,” but which the ancient mind referred more directly to Deity, as an arbitrary power.
All such definition was necessarily confined to terms of time, space, number and proportion, which can only be expressed in those of Mathematics or Geometry, the most stupendous of all contemplations being the observation of the vast cosmic machine from an astronomical standpoint.
That which struck the ancient mind with the greatest wonder was, self-evidently, the fact that if all of the wonders of the Universe resolved themselves into mathematical expressions and the latter into geometrical figures, often simple ones at that, which in turn melted into the simple fundamentals of the latter science, then in the latter must be found the dynamic concentrations which most closely approach a conception of Divinity.
A further step along the line of developing rational occultism was gained by the employment of identical characters to express both letters and numbers. That which we now call the decimal system was worked by “the power of nine” or “three times three.”
The letters of the Hebrew alphabet and the Greek letters derived from an analogous source, counted from Aleph or Alpha, 1 to Teth or Theta, 9 and from Yod or Iota, 10, to Peh or Pi, 80. Beginning with R, 200 (in Greek, 100) the Hebrew letters are one digital value higher than the corresponding Greek letters. It must also be remarked that the Hebrew letter Vau or 6, corresponded to the now missing Digamma in Greek, once placed between Eta and Zeta.
With this equipment we can now understand the significance of the ancient Hebrew Deity names, such as Al Shdi, Biblically translated “God Almighty,” (El Shaddai) or 1+30+300+4+10 -345, or 3, 4 and 5, in decimal places. Also the corresponding name Ahih Ashr Ahih, “I am that I am,” amounting to 543, the same figures reversed, and their addition 345+543-888, the basis of the Greek word .1esous or 10+8±200+70±400+200, the true meaning of which, along a special line of demonstration, is the Solar Logos, or World creating, sustaining and preserving manifestations of Deity, a symbolism still dimly preserved in Masonry under the veil of a twenty-four inch gauge.
The Egyptian name of Ales, a Son, which was that of Moses, is equal to 345, while the word “Messiah” is really files Jah, or “Son of God,” a Cabalism on the Pythagorean triangle, for the word Jah is equivalent to 16 or the D Y sum of 3+4=7 and 4+5=9, the two chief cosmic numbers derived from this triangle . 3+4-, 5 equaling 12, the Hebrews took the ineffable name JHVH, three times repeated and permuted the letters into HJH (3), HVVH (4), VJHJH (5), reading “He that was, He that is and He that will be” or “is to come,” thus paraphrasing the declaration of Horns in the “Book of the Dead .” “I am yesterday, today and tomorrow,” and the inscription over the Shrine of Isis, “I Isis am all that ever has been, now is or ever will be and no mortal has ever lifted my veil.”
We also know that the secret supreme being of the Egyptians was HU-HI, the Mongolian dual principle, HO and HI, which assimilated to the Sanskrit FOHAT or “Cosmic Energy” and ICHCHA, “Human Will.”
It is relatively easy, now, to follow the Egyptian philosophy of the 47th problem, which attributed the square of 3 to Asar (Osiris), the square of 4 to Ishah (Isis) and the square of 5 to Chr (Horns) . Let us remark, in passing, that Chr is the Sanskrit word for “Light,” ECH, reversed and is the origin of the tradition that upon “discovering” the 47th problem, Pythagoras joyfully uttered the word Eureka. Greek mythology is full of stolen Oriental sacred words thus metamorphosed.
Asar, which is 3X54,162, is Ash Ra, the Solar Fire, Ishah (a name still preserved in the Arabic Ayesha), “The Female,” meaning terrestrial “Nature, 4X54 or 216, which is the sum of the cubes of 3, 4 and 5, and Chr the Male principle or fructifying spirit of Nature, is 2X54, or 108. All of these numbers are multiples of 9 or 3X3, and this last number, especially, is the one defined by Hindu Astrology as the full term of human life, and thus represents humanity in the abstract. Egyptian philosophy regarded the great cosmic process of generation and regeneration as resultant upon the infinitely reiterated passages of the Male principle through the Female principle, Father and Son being, spiritually, a Unit (“I and my Father are One”).
The number 270 (9X30) represents the number, in days, of human gestation and is the sum of 162 (Father) and 108 (Son). Thus, merely supplying the natural philosophical postulate that the Father passing his vital principle through the Mother, continues to live in the Son, we secure a Pythagorean triangle of 162-216-270.
If the 47th problem be exhibited, as it frequently is, divided into alternating black and white squares. symbolic, among other things, of the “Dual principle,” the division will be 26 (10+5+6 +5 ; JHVH) of one color and 24 (8±8±8) of the other, while the measure round-about the whole figure will be 36 spaces, giving the Pythagorean sacred number of the Sun, upon which the Sage of Samos swore his pupils to, secrecy. The 9+16+25=50 squares thus gave rise to the old Greek Cabalistic name of the Sun, “Damnamenos of the fifty faces.” The Hebrews celebrated each fiftieth year as a Jubilee, calling it the “Power of the Number Twelve” (3+4+5). In present conclusion, it is regrettable that we have only just begun to enter upon the mere fringe of this wonderful subject, concerning which, if anything may most truly be said that “the World itself could not contain all the books which might be written concerning it.