ments determining the effects of proportion are of the same nature as are those determining the effects of harmony, it is evident that we must suppose ourselves dealing with factors of which the mind is unconscious; and must remain ignorant until science has come into possession of certain data not yet discovered. Is it any wonder that those accepting this supposition who have tried to explain the effects, have either held that they cannot be explained at all, or have made attempts at explanation which may be said in a general way to have failed to prove convincing? Is it any wonder that, even when acknowledging that the Greeks once had a knowledge of the subject, very many in our own times, after seeking for this knowledge in wrong directions, have conceived of the subject as hidden in almost impenetrable mystery, as involving principles which it is wellnigh useless for present artists to attempt either to understand or to apply?

It is important to notice, too, that the effects of proportion, as interpreted here, must be ascribed to measurements that are apparent, but not necessarily actual. One cannot well judge of the relations between the measurements of the parts of a body, or between the measurements of these and the measurements of the whole, except so far as he looks at the body from a distance where all the parts can be compared together. But, as shown on page 102, certain measurements need to be actually different, in order, when seen from a distance, to seem to be alike. Effects of proportion, therefore, are not determined by actual measurements, but by what the measurements appear to be, after perspective and the methods associated with it have made them appear as they do. The principles underlying Greek

proportion cannot, therefore, be ascertained by merely measuring with a tape-line the different members of a Greek façade.

I

Once more, inasmuch as proportion, like rhythm, is based upon the requirements of composition, it is important to notice that fundamentally, measurements go together because they appear to be exactly alike, that is, as II; and that the mind accepts the ratios of certain small numbers that are not alike, like 1:2 or 2:3, because it is able to recognise in the first that which corresponds to 1:1+1, and in the second that which corresponds to I+II+I+I. Finally, connected with this, it is important to notice that as rhythm starts by putting together similar small parts such as feet and lines, and produces the general effect of the whole as a result of the combined effects of these parts, so does artistic proportion. For instance, the height of the front of the Parthenon is to its breadth as 9:14. But we need not consider the architect as aiming primarily at this proportion; or that it is any more than a secondary, though, of course, a necessary result of the relations, the one to the other, of the different separate measurements put together in order to form the whole. If we lose sight of this fact, we may never be able to the end of time to explain why the Greeks used such proportions, in their columns, as 5:81, or, in their façades, as 9:14.

In view of what has been said in previous chapters of this book, it is easy for us to recognise why the ratio of II should be characteristic of the measurements of the majority, perhaps, of art-products in the realm of sight. Everything that was said of the repetition of like forms. on pages 270 to 275 applies equally to like measurements. Whether we compare with one another like features, as

in columns, flutings, windows, mouldings, eyes, arms, legs; or unlike features, as in capitals, friezes, architraves, metophs, triglyphs, foreheads, noses, ears, chins, we find that is the fundamental proportion from which all other ratios are developed.

It is evident that other ratios can be developed from this in such ways as to make the fact of proportion apparent in only the degree in which the numbers representing the ratios are small. After 11, the next easiest to recognise is that of 1:2, as between the first of the upper and of the lower lines at the left of Fig. 58.

FIG. 58.-LINES IN PROPORTION.

See pages 337 and 338.

Nor is it difficult to recognise the relationship of 1:3. as between the second pair of lines in this figure, or of 2: 3, as between the third pair. But it is evident that as the numbers representing the ratios increase in value, these ratios become less recognisable; as, for instance, when they are as 4:5, or as 57, as between, respectively, the fourth and fifth pairs of lines in this Fig. 58. When, at last, we get to a relationship that can be expressed only by large numbers like 10: 11, or 15: 16, the mind is no longer able to recognise even its existence.

There is a way, however, in which one may be made to recognise it, even when represented by comparatively large numbers. This is when, in accordance with the elementary process in proportion of putting like with like, the wholes of the forms that are to be compared are measured off into like subdivisions. For instance, it is far more easy to recognise the relationship of 4:5, or at least that there is such a relationship, when it is

expressed as in Fig. 59, below, than when it is expressed as in lines like those in Fig. 58, page 337. Accordingly, like subdivisions when they are indicated as in Fig. 59

See page 338.

may show not only the relationship that each subdivision sustains to each other subdivision that measures the same

as itself, but the relationship also that whole series of subdivisions sustain to other series of them, which, as