exactly with some points on the plan or section, these are really of no more importance than other points which might just as well have been taken.  The theorist draws our attention to those points in the building which correspond with his geometry, and leaves on one side those which do not.  Now it may certainly be assumed that any builders intending to lay out a building on the basis of a geometrical figure would have done so with precise exactitude, and that they would have selected the most obviously important points of the plan or section for the geometrical spacing.  In illustration of this point, I have given (Fig. 25) a skeleton diagram of a Roman arch, supposed to be set out on a geometrical figure.  The center of the circle is on the intersection of lines connecting the outer projection of the main cornice with the perpendiculars from those points on the ground line.  This point at the intersection is also the center of the circle of the archway itself.  But the upper part of the imaginary circle beyond cuts the middle of the attic cornice.  If the arch were to be regarded as set out in reference to this circle, it should certainly have given the most important line—­the top line, of the upper cornice, not an inferior and less important line; and that is pretty much the case with all these proportion theories (except in regard to Greek Doric temples); they are right as to one or two points of the building, but break down when you attempt to apply them further.  It is exceedingly probable that many of these apparent geometric coincidences really arise, quite naturally, from the employment of some fixed measure of division in setting out buildings.  Thus, if an apartment of somewhere about 30 feet by 25 feet is to be set out, the builder employing a foot measure naturally sets out exactly 30 feet one way and 25 feet the other way.  It is easier and simpler to do so than to take chance fractional measurements.  Then comes your geometrical theorist, and observes that “the apartment is planned precisely in the proportion of six to five.”  So it is, but it is only the philosophy of the measuring-tape, after all.  Secondly, it is a question whether the value of this geometrical basis is so great as has sometimes been argued, seeing that the results of it in most cases cannot be judged by the eye.  If, for instance, the room we are in were nearly in the proportion of seven in length to five in width, I doubt whether any of us here could tell by looking at it whether it were truly so or not, or even, if it were a foot out one way or the other, in which direction the excess lay; and if this be the case, the advantage of such a geometrical basis must be rather imaginary than real.

[Illustration:  Figs. 26 through 28]