We see clearly from the concluding sentence of section 49, where the author directly addresses the painter, that he must certainly have intended to include the elements of mathematics in his Book on the art of Painting.  They are therefore here placed at the beginning.  In section 50 the theory of the “Pyramid of Sight” is distinctly and expressly put forward as the fundamental principle of linear perspective, and sections 52 to 57 treat of it fully.  This theory of sight can scarcely be traced to any author of antiquity.  Such passages as occur in Euclid for instance, may, it is true, have proved suggestive to the painters of the Renaissance, but it would be rash to say any thing decisive on this point.

Leon Battista Alberti treats of the “Pyramid of Sight” at some length in his first Book of Painting; but his explanation differs widely from Leonardo’s in the details.  Leonardo, like Alberti, may have borrowed the broad lines of his theory from some views commonly accepted among painters at the time; but he certainly worked out its application in a perfectly original manner.

The axioms as to the perception of the pyramid of rays are followed by explanations of its origin, and proofs of its universal application (58—­69).  The author recurs to the subject with endless variations; it is evidently of fundamental importance in his artistic theory and practice.  It is unnecessary to discuss how far this theory has any scientific value at the present day; so much as this, at any rate, seems certain:  that from the artist’s point of view it may still claim to be of immense practical utility.

According to Leonardo, on one hand, the laws of perspective are an inalienable condition of the existence of objects in space; on the other hand, by a natural law, the eye, whatever it sees and wherever it turns, is subjected to the perception of the pyramid of rays in the form of a minute target.  Thus it sees objects in perspective independently of the will of the spectator, since the eye receives the images by means of the pyramid of rays “just as a magnet attracts iron”.

In connection with this we have the function of the eye explained by the Camera obscura, and this is all the more interesting and important because no writer previous to Leonardo had treated of this subject_ (70—­73). Subsequent passages, of no less special interest, betray his knowledge of refraction and of the inversion of the image in the camera and in the eye (74—­82).

From the principle of the transmission of the image to the eye and to the camera obscura he deduces the means of producing an artificial construction of the pyramid of rays or—­which is the same thing—­of the image.  The fundamental axioms as to the angle of sight and the vanishing point are thus presented in a manner which is as complete as it is simple and intelligible (86—­89).

Leonardo distinguishes between simple and complex perspective (90, 91). The last sections treat of the apparent size of objects at various distances and of the way to estimate it (92—­109).