metre. subdivided as to extension, i. e., as to size or shape, that these also can be measured and compared. It is important to observe, however, that it is not necessary actually to measure them, as a preliminary step to recognising that they are in proportion. In other words, it is not necessary to determine what the ratio between them is, but merely that it exists. The same principle applies here as in rhythm. To experience the effects of this, we do not need to decide what the metre is-whether initial or terminal, iambic or trochaic-only that there is a But while this is true, the metre must be capable of being analysed; and we must be conscious that it is so, although, perhaps, we ourselves do not care to go through with the analytic process. In the same way, the impression which the mind receives of proportion is due to measurements of which, if it choose, it may become conscious as distinguished from those of which, as measurements, it must forever remain unconscious. This fact is noteworthy, because, so far as it can be recognised, it enables one to perceive why proportion in the arts of sight, is not, as has been almost universally supposed, the analogue of harmony in the arts of sound. As will be shown in Chapter XVII., harmony is produced in these arts whenever the number of vibrations per second determining the pitch of one tone sustains a certain ratio to the number of vibrations per second determining the pitch of another tone. But only the investigations of science have been able to discover that this is the reason for the effect. The mind cannot count the vibrations. It is not conscious of them; but only of an agreeable thrill or glow in case they coalesce, as they do when they sustain to one another the required harmonic ratio. Now if we go upon the supposition that the measure 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. 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 1:11, and in the second that which corresponds to 1+1:1+1+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 I: I 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 I 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 45, 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 45, 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 FIG. 59.-LINES SUBDIVIDED TO INDICATE PROPORTION. 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 |