The fourth book indicates 'where and how the figures are to bend.' It is, in point of fact, 'an application of the science of geometrical projection to the drawing of the human body expressed by lines and plane surfaces, and represented under different aspects and in different positions.' He declares in his preface that he intends to write nothing about the inward parts of the body, and at the beginning of the fourth book says: 'But how to describe the limbs, and how wonderfully they fit into each other, is known to those who occupy themselves with anatomy, and I leave it to them to speak of these things.' He himself is content with briefly pointing out the limits within which the body can be bent, and how the joints become enlarged when they are stretched and in action. In the first book he gives figures of bodies varying from six to nine, and even ten, heads in stature, though the latter proportions are only treated as supposititious cases, and not as actually occurring conditions. Thus he represents a pair of robust peasants, male and female, in whom he makes the foot one-sixth, the head one-seventh, and the hand one-tenth, of the entire stature.

He then gives another pair of figures, also male and female, of a less robust and more slender form, in whom the head is one-eighth, the hand one-tenth, and the foot, in the male, one-sixth, and in the female one-seventh, of the entire stature. 'The vertical and horizontal lines into which he divides the head,' says Topinard, 'merit special attention. He established his first horizontal line to orient the head in profile, and drew it so as to pass by the lower part of the lobule of the ear and the lower part of the nose. Amongst the other lines are two called slanting - the one a tangent to the chin and to the two lips, the other a tangent to the frontal eminences, to the glabella and to the nose. At the point of meeting of this line with the horizontal line above mentioned is an angle which the authors of the "Crania Ethnica" have described as a sort of facial angle which preceded that of Camper. It is a fact that on a figure of a negro given by Durer, in which the two lines are represented, the angle is more acute than amongst Europeans, and the forehead therefore rendered more retiring.' Very different opinions seem to have been held respecting the value of Durer's work ; Michael Angelo is said to have thought but little of it, whilst Hogarth, in the book from which I have already quoted, says: 'Albert Durer, who drew mathematically, never so much as deviated into grace, which he must sometimes have done in copying the life, if he had not been fettered with his own impracticable rules of proportion.' On the other hand, Francisco Pacheco, the master of the great Velasquez, in his book on painting, recommends that the female figure should be studied from Durer's drawings, instead of from the living model.

Passing to other Germans, Bergmuller, who published in 1723 a book entitled 'Anthropometrica' Lichtensteger and Zeising, all devised canons which were more or less fantastic and artificial. The last-named author published his 'Lebre von der Proportionen' in 1854, the details of which rested upon the following proposition : Proportion is a fundamental necessity for beauty of form: if the division of a whole consisting of unequal parts is to appear proportional, the relation of the unequal parts to one another must be the same as the relation of the parts to the whole; that is, the smaller parts must be related to the greater, as the greater to the whole. From this rule he deduced his so-called 'Goldenen Schnitt' as a canon of ideal beauty in the division of all structures. This section consisted in a line so divided that the smaller part bore the same portion to the larger as that did to the whole.

Schadow, who was sculptor at the court of the King of Prussia, published in 1884 his work on proportions, entitled 'Polycletus,' a name which was that of one of the earliest devisers of a canon, the author of the celebrated 'Doryphorus.' Of Schadow so great an authority as Quetelet had a high opinion within certain limitations. 'We find him,' he says, an artist before all things: that which unceasingly occupied him was grace, was the elegance of forms, much more than the law of proportions and of stature, and he is correct up to a certain point.' In his system he describes the face as the portion between the upper part of the orbit and the lower part of the chin, and he states that this distance is in a full-grown man five inches. He divides this space into six parts, the first extending to midway between the orbit and the lower limit of the nose, the second to the last-named point, the third to the angle of the mouth, the fourth midway from this to the chin, and the last to the point of the chin itself. The foot of a man of five foot six inches in stature should be ten inches - this is the same length as the ulna, and both are, therefore, double the length of the face according to his definition of that region.

In the female the face is four and a half inches, and the foot nine; whilst in the child the head is six and a half inches, and the foot five and a half. One of the most interesting attempts to solve the question of proportions is that of Carus, the celebrated Dresden physiologist, who published a work called ' Die Proportions-Lehre der Menschlichen Gestalt' in 1854. His views are well expressed in a letter to Quetelet, which the latter quotes. He says: 'I have considered the proportions of man as an object of morphology, and I have tried to find in consequence physical laws to fix that which we may call the canon, or, according to the expression of architects, when they are dealing with the column, the module, of our organization.' Having then given an account of the progress of his ideas, and having stated that the statuary Kictochel had made a figure from his directions, he proceeds: 'It is twenty years and more since I repeated in several places in my writings if anyone wishes to find the true key to our proportion he must set out with the vertebral column, which is, so to speak, the true organic ell divided into twenty-four inches (free vertebras). When the ovum of a mammal is opened at the commencement of its formation there is found, as the first model of the future animal, the germinal area grooved in the middle with a line, which becomes the vertebral column at a later period.