The Operative Masons had always formed their Right-Angled Triangles by means of mundane measures of 3, 4, and 5 units to each side respectively, as was done by the Harpedonaptae of Egypt 5000 years ago, and 2500 years later by Pythagoras, and this same method continues to be used to this day; but to those of a religious turn of mind, who had only lately become conversant with Euclid, and looked upon Geometry not only as the height of all learning, but, as they progressed in the knowledge of its bearing on the Science of building, actually made it synonymous with Tectonic Art (the old MSS. which have come down to us from that time _invariably_ state that "at the head of all the Sciences stands _Geometry which is Masonry_"), there must have come a wave of wonderful enthusiasm when they first discovered that the Geometrical way of creating a Right Angle, as given in Euclid I. ii., was by means of an Equilateral Triangle, by joining the Apex with the centre of the base. This Equilateral Triangle was the earliest symbol we know of the Divine _Logos_ in connection with that wonderful figure the Vesica Piscis; and as the Bible declared that the Universe was created by the Logos (the Word), so the Square which represented the Universe was naturally created by means of the Equilateral Triangle. A great mystery this must have appeared to those who, like the Hellenic philosophers, postulated that everything on Earth has its counterpart in Heaven, and who, in their religious mysticism, were always looking for signs of the transcendental in their temporal surroundings. But in what awe and reverence must they have held Geometry, when they further found that the Equilateral Triangle, representing the Logos, was itself generated, as shown in the _first_ Problem of Euclid, upon which the whole Science of Geometry was therefore based, by the intersection of two Circles! These two Circles were held by the Greeks, at the beginning of our Era, to represent the Past and Future Eternities generating the Logos; but the whole figure (Euclid I. i.) was at the time we are now dealing with looked upon by Mediaeval Architects as representing the Three Divine _personae_, and that part, or _cavity_, of the figure which is bounded by the Arcs of the two circles, and which takes to itself one-third of each of the two generating circles (making its perifera exactly equal with that remaining to each of the two circles, all three therefore being _co-equal_), and in which the Equilateral Triangle is formed (_vide_ frontispiece), was naturally held by the Mediaeval Architects, and indeed from earliest times, as the most sacred Christian Emblem--namely, that of _Regeneration_ or "New Birth."