have measured the antique statues accurately and painstakingly to find the secret of their charm.  Science, by showing that different sounds and different colours are produced by waves of different lengths, and that therefore different colours and sounds can be expressed in terms of numbers, has certainly opened the door to a new consideration of this subject of beauty in relation to mathematics.  And the result of such an inquiry, if it is being or has been carried on, will be of much interest.

But there is something chilling to the artist in an array of dead figures, for he has a consciousness that the life of the whole matter will never be captured by such mechanical means.

The question we are interested to ask here is:  are there particular sentiments connected with the different relations of quantities, their proportions, as we found there were in connection with different arrangements of lines and masses?  Have abstract proportions any significance in art, as we found abstract line and mass arrangements had?  It is a difficult thing to be definite about, and I can only give my own feeling on the matter; but I think in some degree they have.

Proportion can be considered from our two points of view of unity and variety.  In so far as the proportions of any picture or object resolve themselves into a simple, easily grasped unity of relationship, a sense of repose and sublimity is produced.  In so far as the variety of proportion in the different parts is assertive and prevents the eye grasping the arrangement as a simple whole, a sense of the lively restlessness of life and activity is produced.  In other words, as we found in line arrangements, unity makes for sublimity, while variety makes for the expression of life.  Of course the scale of the object will have something to do with this.  That is to say, the most sublimely proportioned dog-kennel could never give us the impression of sublimity produced by a great temple.  In pictures the scale of the work is not of so great importance, a painting or drawing having the power of giving the impression of great size on a small scale.

The proportion that is most easily grasped is the half—­two equal parts.  This is the most devoid of variety, and therefore of life, and is only used when an effect of great repose and aloofness from life is wanted; and even then, never without some variety in the minor parts to give vitality.  The third and the quarter, and in fact any equal proportions, are others that are easily grasped and partake in a lesser degree of the same qualities as the half.  So that equality of proportion should be avoided except on those rare occasions when effects remote from nature and life are desired.  Nature seems to abhor equalities, never making two things alike or the same proportion if she can help it.  All systems founded on equalities, as are so many modern systems of social reform, are man’s work, the products of a machine-made age.