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.

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 sk