notes on Optics or on Perspective as bear exclusively on Mathematics or Physics could not be included in the arrangement of the libro di pittura which is here presented to the reader.  They are however but few.].  But its axioms being laid down at great length, I shall abridge them to a conclusive brevity, arranging them on the method both of their natural order and of mathematical demonstration; sometimes by deduction of the effects from the causes, and sometimes arguing the causes from the effects; adding also to my own conclusions some which, though not included in them, may nevertheless be inferred from them.  Thus, if the Lord—­who is the light of all things—­vouchsafe to enlighten me, I will treat of Light; wherefore I will divide the present work into 3 Parts [Footnote:  10.  In the middle ages—­for instance, by ROGER BACON, by VITELLONE, with whose works Leonardo was certainly familiar, and by all the writers of the Renaissance Perspective and Optics were not regarded as distinct sciences.  Perspective, indeed, is in its widest application the science of seeing.  Although to Leonardo the two sciences were clearly separate, it is not so as to their names; thus we find axioms in Optics under the heading Perspective.  According to this arrangement of the materials for the theoretical portion of the libro di pittura propositions in Perspective and in Optics stand side by side or occur alternately.  Although this particular chapter deals only with Optics, it is not improbable that the words partiro la presente opera in 3 parti may refer to the same division into three sections which is spoken of in chapters 14 to 17.].

The plan of the book on Painting (14—­17).

14.

ON THE THREE BRANCHES OF PERSPECTIVE.

There are three branches of perspective; the first deals with the reasons of the (apparent) diminution of objects as they recede from the eye, and is known as Diminishing Perspective.—­The second contains the way in which colours vary as they recede from the eye.  The third and last is concerned with the explanation of how the objects [in a picture] ought to be less finished in proportion as they are remote (and the names are as follows): 

Linear Perspective.  The Perspective of Colour.  The Perspective of
Disappearance.

[Footnote:  13.  From the character of the handwriting I infer that this passage was written before the year 1490.].

15.

ON PAINTING AND PERSPECTIVE.

The divisions of Perspective are 3, as used in drawing; of these, the first includes the diminution in size of opaque objects; the second treats of the diminution and loss of outline in such opaque objects; the third, of the diminution and loss of colour at long distances.

[Footnote:  The division is here the same as in the previous chapter No. 14, and this is worthy of note when we connect it with the fact that a space of about 20 years must have intervened between the writing of the two passages.]