The Greeks certainly employed such a system, though there are not sufficient data for us to judge exactly on what principle it was worked out. In regard to the Parthenon, and some other Greek buildings, Mr. Watkiss Lloyd has worked out a very probable theory, which will be found stated in a paper in the "Transactions of the Institute of Architects."

Vitruvius gives elaborate directions for the proportioning of the size of all the details in the various orders; and though we may doubt whether his system is really a correct representation of the Greek one, we can have no doubt that some such system was employed by them. Various theorists have endeavored to show that the system has prevailed of proportioning the principal heights and widths of buildings in accordance with geometrical figures, triangles of various angles especially; and very probably this system has from time to time been applied, in Gothic as well as in classical buildings. This idea is open to two criticisms, however. First, the facts and measurements which have been adduced in support of it, especially in regard to Gothic buildings, are commonly found on investigation to be only approximately true. The diagram of the section of the building has nearly always, according to my experience, to be "coaxed" a little in order to fit the theory; or it is found that though the geometrical figure suggested corresponds 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.

Figs. 26 through 28

Having spoken of plan as the basis of design, I should wish to conclude this lecture by suggesting also, what has never to my knowledge been prominently brought forward, that the plan itself, apart from any consideration of what we may build up upon it, is actually a form of artistic thought, of architectural poetry, so to speak. If we take three such plans as those shown in Figs. 26, 27, and 28, typical forms respectively of the Egyptian, Greek, and Gothic plans, we certainly can distinguish a special imaginative feeling or tendency in each of them. In the Egyptian, which I have called the type of "mystery," the plan continually diminishes as we proceed inward. In the third great compartment the columns are planted thick and close, so as to leave no possibility of seeing through the building except along a single avenue of columns at a time. The gloom and mystery of a deep forest are in it, and the plan finally ends, still lessening as it goes, in the small and presumably sacred compartment to which all this series of colonnaded halls leads up. In the Greek plan there is neither climax nor anti-climax, only the picturesque feature of an exterior colonnade encircling the building and surrounding a single oblong compartment.

It is a rationalistic plan, aiming neither at mystery nor aspiration. In the plan of Rheims (Fig. 28) we have the plan of climax or aspiration; as in the Egyptian, we approach the sacred portion through a long avenue of piers; but instead of narrowing, the plan extends as we approach the shrine. I think it will be recognized, putting aside all considerations of the style of the superstructure on these plans, that each of them in itself represents a distinct artistic conception. So in the plan of the Pantheon (Fig. 29), this entrance through a colonnaded porch into a vast circular compartment is in itself a great architectural idea, independently of the manner in which it is built up.