The Forty-seventh Problem of Euclid. Here is a symbol the sovereign importance of which has been recognised by almost every student of our mysteries. Hoffman wrote a book about it; Sydney Klein devoted a magnificent study to it which will be found published in the Transactions of the Lodge Quatuor Coronati under the title of "The Great Symbol"; Dr. Anderson used it on the title page of his Constitutions and therein described it "as the foundation of all Masonry if duly observed"; scholars have vied with each other in attempting to uncover all the riches stowed away among its lines and angles.
Most of these interpreters, it must be said, have shown considerable dissatisfaction with the account of the Problem as given in the Lecture. There it is said that it was discovered by Pythagoras and that he was so overjoyed by it that he sacrificed a hecatomb to celebrate his discovery. This has behind it the authority of Vitruvius but even so it is hardly credible and that for the following reasons: the Problem was known to the Egyptians long before Pythagoras, and it is not possible that Pythagoras, who forbade the killing of animals, should have sacrificed a herd of oxen so needlessly; also, the explanation that this Problem is to teach us to be lovers of the arts and sciences is not very convincing.
Those who would defend the Monitor here urge that while the three, four, five triangle may have been used before Pythagoras he may have been the first to understand the Problem as a whole; that his "hecatomb" may have been made of wax figures of oxen, as was sometimes the practice; and that the Problem is so important to mathematics that it may well stand as an emblem of all arts and sciences. Between these two views one may take his choice.
Whatever may be the attitude of our authorities to the Monitorial interpretation they are all agreed that the Symbol is of the greatest importance. Dionysius Lardner, in his edition of "Euclid," writes : "It is by the influence of this proposition and that which establishes the similitude of equilateral triangles (in the sixth book) that geometry has been brought under the dominion of algebra; and it is upon the same principle that the whole science of trigonometry is founded." The Encyclopedia Britannica calls it "one of the most important in the whole of geometry, and one which has been celebrated since the earliest times. . . . On this theorem almost all geometrical measurement depends. which cannot be directly obtained." On its Masonic uses, our interpreters have written with equal enthusiasm: thus one, Brother J. F. Thompson, says that "in it are concealed more ancient symbolism than all other symbols used by, or incident to, our order. . . . In it we find concealed the jewels of the Worshipful Master, the Senior, and Junior Wardens," and also, he might have added, the Apron, the Square, the Tau square, cross, etc.
The brother who wishes to experiment for himself can easily do so by drawing the triangle after the following fashion; lay out a base line four inches in length; at one end erect a vertical three inches high; connect the ends of these two lines and the figure is drawn; this is not the strictly scientific way of going about it but it will serve. The point of this procedure is that whenever the vertical is three and the base is four, the hypothenuse, or long side, will be five; and the angle at the juncture of the base and the vertical will always be a right angle. After this manner a man can always prove a right angle with no mathematical instruments whatever. What this meant to the ancient builders, before such instruments were devised, or had come into common use, is plain to be seen.
But our concern here is not with the Problem as a geometric theorem but with it as a Masonic symbol. What is its Masonic meaning? Many answers can be given to this, none exhaustive, but all valuable; of these I can suggest but two or three.
If we experiment with a group of numbers falling into the series corresponding to three, four, five, we will find that they will always bear the same relationship to each other. In other words, the Problem establishes a harmonious relationship among numbers apparently unrelated. Does not this suggest something of the secret of Masonry? We select a large group of men; they seem to have little in common; but through our teachings, and the application of our principle of brotherhood, we are able to unite them into a harmonious fraternity. The Problem is in this view a symbol of Brotherhood.
The Egyptians made the base line to represent Osiris, the male principle; the vertical, Isis, or female principle; the hypothenuse represented Horus, the product of the two. Suppose we follow such a method and let the base represent our earthly nature; the vertical our spiritual nature; by a harmonious adjustment of these two a complete, or perfect man, will result - the same meaning which we found in the Three Lesser Lights.
Along with these two readings of the symbol
we might place an historical interpretation. The ancient builders,
as has been repeatedly said, did not have algebra and trigonometry,
nor were they in possession of architectural tables or instruments
such as we have; nevertheless they were obliged to fashion right
angles in the erection of their buildings; how could they have
done this without the Forty-seventh Problem, a method so simple
that any Apprentice could use it? It is not too much to say that
there would have been no ancient Masonry without the three, four,
five triangle, or the principle embodied in it; therefore it has
for us a peculiar value in that it represents the skill of our
early brethren in surmounting their obstacles. Since this principle
is so essential to the exact sciences we may agree with our Ritual
in seeing in it a symbol of all the arts and sciences. Just as
a crown may serve as an emblem of all government so may this triangle
serve as an emblem of all science. And since Masonry undertakes
to make character building into an art or a science we may also
find in the triangle, as Dr. Anderson said, "the foundation
of all Masonry if duly observed."
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