Contact with the mathematician Luca Pacioli in Milan had a profound effect on Leonardo. His subsequent studies of proportions in nature indicate an increasingly scientific approach.

In the highly finished study of a dog’s head on the right sheet, Leonardo has exploited the expressive potential of red chalk to model the head in light and shade, using the white of the paper to define the highlights. The linear technique beautifully conveys the wiry texture of the dog’s coat and the complex shadows of its muzzle.

Once the profile had been captured, Leonardo proceeded to establish the proportions of the animal’s head. One can imagine his delight in discovering that the distance from the tip of the dog’s snout to its brow is exactly equal to the distance from its brow to its ear, and that the lowest point of snout is on a line with the lowest point of the ear.

On the opposite page, Leonardo creates designs, titled “fine works of mesh” which were related to designs for theatrical costumes inspired by shells and the patterns of the feathers of a birds’ wing.

In Leonardo's words
No human investigation can be called true science without passing through mathematical tests; and if you say that the sciences which begin and end in the mind contain truth, this cannot be conceded, and must be denied for many reasons. First and foremost because in such mental discourses, experience does not come in, without which nothing reveals itself with certainty.