Now, the three great problems of antiquity which engaged the attention and wonderment of geometricians throughout the Middle Ages, were "the Squaring of the Circle," "the Duplication of the Cube," and lastly, "the Trisection of an Angle," even Euclid being unable to show how to do it; and yet it will be seen that the diagonal of one of the subsidiary figures in the tri-subdivision, together with the diagonal of the whole figure, actually trisect the angle at the corner of the rectangle. It is true that it only showed them how to trisect one kind of angle, but it was that particular angle which was so dear to them as symbolising their craft, and which was created by the Equilateral Triangle. All these unique properties place the figure far above that of a square for practical work, because even when the diagonal of a square is given, it is impossible to find the exact length of any of its sides or _vice versa_; whereas in the Vesica rectangle the diagonal is exactly double its shorter side, and upon any length of line which may be taken on the tracing-board as a base for elevation, an Equilateral Triangle will be found whose sides are of course all equal and therefore known, as they are equal to the base, and whose line joining apex to centre of base is a true Plumb line, forming at its foot the perfect right angle, so important in the laying of every stone of a building. In the volume referred to I have given a skeleton plan upon such a scale of subdivision that a tracing-board, of 5 feet by 8 feet, would be divided up into over one million parts, and, as all these subdivisions are perfect representations of the original Vesica figure with all its properties, the design of the largest building, with the minutest detail, could be drafted with absolute accuracy. There are many other curious properties of this Figure, but they are difficult to explain without diagrams. I will, however, give one more example of its creative power. The problem of describing a Pentagon must have puzzled architects considerably in those early times, but this was again easily accomplished by means of the Vesica. Albrecht Duerer, the great designer and engraver, who lived at the end of the fifteenth century, refers to the Vesica in his works (_Dureri Institutune Geometricarum_, lib. ii. p. 56) in a way which shows that it was as commonly known in his time as the Circle, Square, and Triangle. His instructions for forming a Pentagon are: "Designa circino invariato tres piscium vesicas" (describe with unchanged compasses three vesicae piscium). Three similar circles are described with centres at the angles of an Equilateral Triangle, forming the three Vesicae, by means of which the Pentagon is drawn, and from which also we get a beautiful form of arch very common in the thirteenth century (_vide_ illustrations in _Magister Mathesios_). This is also the method used in that old manuscript of the fifteenth century named "Geometria deutsch." In this old MS.