basis found exactly applicable to the work of the fifteenth century, since which time mathematical proportions have been generally employed." And after referring to numerous scale drawings of Churches, windows, doors, and arches, he points out that every student of Church architecture must pronounce those of the untraceried and traceried first point to be the most beautiful of all, those of the Norman to be a degree less so, and those of the perpendicular and debased to be far inferior to either, and in that analysis we find that the Equilateral Triangle was used almost exclusively for determining one order (the Gothic), the Square for another (the Norman), and the Square diagonally divided for the other (the debased). Now let me try to describe the wonderful properties of the Vesica Piscis, so that you may understand the mystery which shrouded it in the minds of those Mediaeval builders. The rectangle formed by the length and breadth of this figure, in the simplest form, has several extraordinary properties; it may be cut into three equal parts by straight lines parallel to the shorter side, and these parts will all be precisely and geometrically similar to each other and to the whole figure,--strangely applicable to the symbolism attached at that time to the Trinity in Unity,--and the subdivision may be proceeded with indefinitely without making any change in form. However often the operation is performed, the parts remain identical with the original figure, having all its extraordinary properties, the Equilateral Triangle appearing everywhere, whereas no other rectangle can have this curious property. It may also be cut into four equal parts by straight lines parallel to its sides, and again each of these parts will be true Vesicas, exactly similar to each other, and to the whole, and of course the Equilateral Triangle is again everywhere. Again, if two out of the tri-subdivisions mentioned above be taken, the form of these together is exactly similar, geometrically, to half the original figure, and again the Equilateral Triangle is ubiquitous on every base line. Again, the diagonal of the rectangle is exactly double the length of its shorter side, which characteristic is absolutely _unique_, and greatly increases its usefulness for plotting out designs; and this property of course holds good for all the rectangles formed by the original figure and for the other species of subdivision. But perhaps its most mysterious property (though not of any practical use) to those who had studied Geometry, and to whom this figure was the symbol of the Divine Trinity in Unity, so dear to them, was the fact that it actually put into their hands the means of _trisecting_ the Right Angle.