series, do not measure the same. Thus, the panels in the lower story in the Assyrian tower in Fig. 60, page 338, show that the whole length of each story sustains a certain definite relationship to the whole length of each other story. So, too, the ornamental divisions in the spire in Chichester Cathedral (Fig. 61, page 339) show that the whole spire sustains an exact relationship of 3:1 to the square part of the tower visible below it.

We are told by W. W. Lloyd in his "Memoir on the Systems of Proportion," published with Cockerill's "Temples of Ægina and Bassæ," page 64, that all the architectural quantities as made proportionate were estimated by the Greeks chiefly in two ways: by rectilin

FIG 62.-FIGURES WITH LINES SUBDIVIDED TO INDICATE PROPORTION.

See pages 103 and 341.

ear proportions, i. e., by divisions of one continuous straight line; and by rectangular proportions, i. e., by a comparison of length and breadth, height and width, etc., at right angles. We have considered the first of these ways. In considering the second, we can expect, of course, no change in principle. In case the lines to be compared form adjacent sides of a rectangle, the ratio between the lines must be recognisable in the degree in which it can be expressed in small numbers, 1: 2, 23, 34, etc. Or, if comparatively large numbers be necessitated, they can still be recognised in the degree in

which certain marks suggest them to the eye. Notice this Fig. 62, representing 3: 5, and 4: 7. As applied in actual construction also, observe Fig. 60, page 338; and the like horizontal or vertical divisions in Fig. 28, page 219, Fig. 32, page 225, and Fig. 33, page 226.

Of course this method of making lengths and breadths seem in proportion in the same figure can make them seem so in adjacent figures; in other words, it can make one figure as a whole seem in proportion to another figure. If, in such cases, the figures be rectangles, they may be similar in width, and then their relationships may be determined by the ratios of their heights, as in the first three rectangles at the left of Fig. 63. Or if the

10

FIG. 63. RECTANGLES IN PROPORTION.

See page 341.

rectangles be similar in height, their relationships may be determined by the ratios of their widths, as in the fourth, fifth, and sixth rectangles in the same figure. Or, if the rectangles be similar neither in width nor in height, their relationships may still be determined by the ratios, each to each, of both these respective dimensions, as in the seventh, eighth, and ninth rectangles in Fig. 63.

So far we have considered only straight lines and rectangular figures. Of course, there are other figures, and they form a vast majority, that are not composed of lines of this character. It is evident that to compare the measurements of these figures, especially when they differ for different reasons is extremely difficult; not only so. but that it is impossible, unless all can be shown to

be allied to some simpler figure which can serve as a standard of measurement. This simpler figure, which is

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FIG. 64. FIGURES RELATED BECAUSE INSCRIBABLE IN THE SAME SQUARE. See page 342.

just as essential to the determining of like space-dimensions in shape as a yardstick is to the determining of like

FIG. 65. FIGURES RELATED BECAUSE INSCRIBABLE IN THE SAME RECTANGLE. See page 342.

lengths, may be either actually outlined at the time of comparing the measurements or only ideally imagined.

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FIG. 66. RELATIONSHIP OF FIGURES AS INDICATED AND AS NOT INDICATED. See pages 342 and 343.

But whether actually outlined or not, on the principle that things equal to the same thing are equal to one anОоолл ХХ

FI3 67.-FIGURES RELATED BECAUSE INSCRIBABLE IN FIGURES IN PROPORTION. See page 343.

other, all other figures inscribed in this simpler figure and that touch all its sides can, for this reason, be recognised as related. See Figs. 64, 65, and 66.

It is well to observe, however, that the more complex figures cannot always be recognised as being related, in case the outlines of the simpler figures do not accom

FIG. 68.-CHÂTEAU DE RANDAU, VICHY, FRANCE.
See page 343.

pany them. The first three forms in Fig. 66, when they are separated from the rectangles in which, in the last three forms, they are shown to be inscribable, do not suggest any particular relationship to one another. Nor would the fifth and sixth, or the seventh and eighth forms in Fig. 67, page 342, were it not for the rectangles in the first and second, with which the figure shows them to be connected. Or, to indicate the practical bearings upon art of this remark, it is conceivable that the different triangles described by the pitch of the gable-windows, roofs, and turrets in Fig. 68, page 343, would all be found to be exactly inscribable in rectangles which, according to what was said on page 342, are in