Having thus described so far as they are known the canons adopted by ancient artists, we must now turn to the consideration of those of more modern times. Amongst the Italians, Giotto (1276-1336), is said to have written on the subject, but I am not aware that any remains of his writings are extant. Alberti (1398-1475) made a much more successful attempt than any other early modern to deal with the subject of proportion. In fact, Topinard says of his work that it is an essay in rational anthropometry, and a very remarkable attempt for the period. Alberti took the foot as his unit, and states that it is included six times in the body, in which he followed Yitruvius and, according to Winckelmanns, various of the ancient sculptors. The foot he divided into ten parts, and each of these again into ten minutes, each of which thus formed the six-hundredth part of the body. Alberti's figures were based upon a number of measurements of the body relating to its height, transverse and antero-posterior diameters, and reduced to averages. Leonardo da Vinci (1445-1520), in his 'Treatise on Painting,' often mentions a standard of measurement, but never seems. to have been satisfied with any.

He took the face for his starting-point, and says that in his first infancy man has the width of his shoulders equal to the length of his face, and to that portion of the arm which is between the shoulder and the elbow when the arm is bent. It is also equal to the distance between the middle finger of the hand and the fold of the elbow, and to the interval between the bend of the knee and that of the ankle. But when man has come to his full stature all these measurements double in length, except the face, which, as the whole head, undergoes little change, and so the man who at adult stature is of a well-proportioned figure should have ten faces' height, the size of his shoulders should l»e two faces, 'and so all the parts of which I have spoken are alike of two faces.' He also says: 'Divide the head into twelve degrees, and each degree into twelve minutes, and each minute into twelve seconds, and so on until you have found a measure equal to the smallest parts of your figure,' a statement upon which possibly is based Rossi's surely sarcastic story that Leonardo had divided the face into 248,832 parts. Michael Angelo (1474-1568) left a sheet of proportions of which a representation is given in Fig. 1. It represents a man standing in three-quarter face, the head being in profile.

Fig. 1. Drawing by Michel Angelo Buonarrotti, from the copper engraving by Giov. Fabbri (Choulant).

The right arm is only partly shown, and the right leg and foot are incompletely represented. The skin is not removed, but the muscles are clearly shown, and the position of the left trochanter major is marked by a star. On the right side of the figure is a divided scale for the whole body, together with a special one for the arm. On the left side is a smaller representation of the proportions of the human body, which shows the bony skull, the cervical vertebrae, the first rib, clavicle and upper part of the scapula. The corresponding proportions of the outstretched arm to those of the middle line of the body are shown by three quadrants. From the vertex to the sole of the foot is described a semicircle whose diameter is formed by the length of the body. Along the perpendicular line in the smaller figure are the words: testa, collo, peto (petto), soto peto (sotto petto), col corpo, natura, coscia, congiunta, gamba, congiata di piedi. On the horizontal line, spala (spalla), congionta, oso (osso) di sopra, congionto, oso di soto (osso di sotto), congionto, oso (osso) de la mano. Under the clavicle, inguiniatura sopra il petto. But Michael Angelo has stated his opinion that the artist must rely upon his own eye as the surest guide to correct proportions.

A curious statement made by Lamozzo respecting Michael Angelo seems to have a bearing upon his ideas as to proportion, but it is phrased in as enigmatical a style as the directions of the alchemists, and to me is at least as unintelligible. Lamozzo says: 'And because in this place there falleth out a certain precept of Michael Angelo much for our purpose, I will not conceale it, leaving the farther interpretation and understanding thereof to the judicious reader. It is reported, then, that Michael Angelo upon a time gave this observation to the painter Marcus de Sciena his scholler; that he should alwaies make a figure pyramidall, serpent-like, and multiplied by one, two and three. In which precept (in mine opinion) the whole mysterie of the arte consisteth.' (The quotation is as given by Hogarth.)

Fig. 3. Linear scheme of the proportions of the so-called Germanicus and of the Apoxyomenos (Langer).

Amongst other Italians who wrote about the canon of proportion may be mentioned Paggi (1554-1629), who in a work entitled 'Acus Nautica,' which was published in 1601, gave some tables of proportions from which it is believed those subsequently issued by Testelin were copied. Barbaro, in his 'Practica della Perspectiva,' gave a series of proportions which he proposed as intermediate between those of Durer, which he considered to be too minute, and those of Vitruvius, which, on the other hand, he thought too general. Barca of Milan (1620) issued a sheet containing the proportions of Jupiter, Hercules, Minerva and Venus.

The Germans, as might have been expected from a nation always anxious to reduce all possible matters to scientific rules, and filled with a genuine love of art, have supplied various works on our subject. Of these, perhaps the earliest, and certainly one of the most famous, is that of Albrecht Durer (1470-1528), who had a very high opinionof the science of proportion, bestowed much thought upon the subject, and eventually published a work concerning it. His opinion of the potentialities of the subject was, in fact, almost overstrained, if one may judge from his statement that 'by means of outward proportion one can indicate the natures of men which correspond to fire, air, water, and earth, for the power of art is supreme.' His first book was entitled, 'Instruction in the Measurement, with the Compass and Rule, of Lines, Surfaces and Solid Bodies, drawn up by Albrecht Durer, and printed for the use of all lovers of art, with appropriate diagrams, in 152.' This book contains a course of applied geometry in connection with Euclid's elements; in fact, Durer states from the commencement of it that his book will be useless to anyone who understands the geometry of 'the very acute Euclid, for it has been written only for the very young and for those who have no one to instruct them accurately.' This work was followed by his book on Proportion, which was published with the following title: 'Herein are comprised four books on human proportions, composed and printed by Albrecht Durer, of Nurnberg, for the use of all those who love this art, MDXXVIII.' In his system of measurement of the human body he adopts two plans, for in the first book he uses as a standard a fraction of the entire height, whilst in the second his scale is composed of six hundred parts, like that of Alberti, a proof, says Thausing, that he had some acquaintance with the, at that time unpublished, writings of the Florentine. In the third book the varying proportions of the figures given in the first two are changed according to definite rules, the scale being increased and diminished in all kinds of different ways, but always with a certain consistency.