(a) Proportion

The theory of proportions had a great fascination for Renaissance artists. Their canons were not only intended as a means of artistic workmanship, they were meant to achieve harmony. Proportions in painting, sculpture, and architecture were like harmony in music and gave intense delight.
Proportion is not only found in numbers and measurements but also in sounds, weights, time, and position, and whatever power there may be.
The Roman architect Vitruvius had transmitted some data of a Greek canon for the proportions of the human figure and these were revived in Renaissance time. A drawing by Leonardo, now in the Academy at Venice, was reproduced in an edition of Vitruvius' book, published in 1511, in order to illustrate the statement that a well-made human body with arms outstretched and feet together can be inscribed in a square; while the same body spread-eagled occupies a circle described around the navel. The proportions of the human body are here related to the most perfect geometric figures and may be said to be integrated into the spherical cosmos. Leonardo endeavoured to verify and elaborate Vitruvius' mathematical formulae in order to put them on a scientific basis by empirical observations, and for this purpose he collected data from living models.
Geometry is infinite because every continuous quantity is divisible to infinity in one direction or the other. But the discontinuous quantity commences in unity and increases to infinity, and as it has been said the continuous quantity increases to infinity and decreases to infinity. And if you give me a line of twenty braccia I will tell you how to make one of twenty-one.
Every part of the whole must be in proportion to the whole … I would have the same thing understood as applying to all animals and plants.
From painting which serves the eye, the noblest sense, arises harmony of proportions; just as many different voices joined together and singing simultaneously produce a harmonious proportion which gives such satisfaction to the sense of hearing that listeners remain spellbound with admiration as if half alive. But the effect of the beautiful proportion of an angelic face in painting is much greater, for these proportions produce a harmonious concord which reaches the eye simultaneously, just as a chord in music affects the ear; and if this beautiful harmony be shown to the lover of her whose beauty is portrayed, he will without doubt remain spellbound in admiration and in a joy without parallel and superior to all other sensations.
The painter in his harmonious proportions makes the component parts react simultaneously so that they can be seen at one and the same time both together and separately; together, by viewing the design of the composition as a whole; and separately by viewing the design of its component parts.
Vitruvius, the architect, says in his work on architecture that the measurements of the human body are distributed by nature as follows: 4 fingers make 1 palm; 4 palms make 1 foot; 6 palms make 1 cubit; 4 cubits make a man's height; and 4 cubits make one pace; and 24 palms make a man; and these measures he used in buildings.
If you open your legs so much as to decrease your height by 1/14 and spread and raise your arms so that your middle fingers are on a level with the top of your head, you must know that the navel will be the centre of a circle of which the outspread limbs touch the circumference; and the space between the legs will form an equilateral triangle.
The span of a man's outspread arms is equal to his height.
From the roots of the hair to the bottom of the chin is the tenth part of a man's height; from the bottom of the chin to the crown of the head is the eighth of the man's height; from the top of the breast to the crown of the head is the sixth of the man; from the top of the breast to the roots of the hair is the seventh part of the whole height; from the nipples to the crown of the head is a fourth part of the man. The maximum width of the shoulders is the fourth part of the height; from the elbow to the tip of the middle finger is the fifth part; from the elbow to the end of the shoulder is the eighth part. The complete hand is the tenth part. The penis begins at the centre of the man. The foot is the seventh part of the man. From the sole of the foot to just below the knee is the fourth part of the man. From below the knee to where the penis begins is the fourth part of the man.
The distance between the chin and the nose and that between the eyebrows and the beginning of the hair is equal to the height of the ear and is a third of the face.
The length of the foot from the end of the toes to the heel goes twice into that from the heel to the knee, that is, where the leg-bone joins the thigh bone. The hand to the wrist goes four times into the distance from the tip of the longest finger to the shoulder-joint.
A man's width across the hips is equal to the distance from the top of the hip to the bottom of the buttock, when he stands equally balanced on both feet; and there is the same distance from the top of the hip to the armpit. The waist, or narrower part above the hips, will be half-way between the armpits and the bottom of the buttock.
Every man at three years is half the full height he will grow to at last.
There is a great difference in the length between the joints in men and boys. In man the distance from the shoulder joint to the elbow, and from the elbow to the tip of the thumb, and from one shoulder to the other, is in each instance two heads, while in a boy it is only one head; because Nature forms for us the size which is the home of the intellect before forming what contains the vital elements.
Remember to be very careful in giving your figures limbs that they should appear to be in proportion to the size of the body and agree with the age. Thus a youth has limbs that are not very muscular nor strongly veined, and the surface is delicate and round and tender in colour. In man the limbs are sinewy and muscular; while in old men the surface is wrinkled, rugged and knotty, and the veins very prominent.
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