docs/examples/kernel/davinci0.txt

The Project Gutenberg EBook of The Notebooks of Leonardo Da Vinci, Complete
by Leonardo Da Vinci
(#3 in our series by Leonardo Da Vinci)

Copyright laws are changing all over the world. Be sure to check the
copyright laws for your country before downloading or redistributing
this or any other Project Gutenberg eBook.

This header should be the first thing seen when viewing this Project
Gutenberg file.  Please do not remove it.  Do not change or edit the
header without written permission.

Please read the "legal small print," and other information about the
eBook and Project Gutenberg at the bottom of this file.  Included is
important information about your specific rights and restrictions in
how the file may be used.  You can also find out about how to make a
donation to Project Gutenberg, and how to get involved.


**Welcome To The World of Free Plain Vanilla Electronic Texts**

**eBooks Readable By Both Humans and By Computers, Since 1971**

*****These eBooks Were Prepared By Thousands of Volunteers!*****


Title: The Notebooks of Leonardo Da Vinci, Complete

Author: Leonardo Da Vinci

Release Date: Jan, 2004  [EBook #5000]
[Yes, we are more than one year ahead of schedule]
[Most recently updated June 26, 2002]

Edition: 10

Language: English

Character set encoding: ASCII

*** START OF THE PROJECT GUTENBERG EBOOK, THE NOTEBOOKS OF LEONARDO DA
    VINCI, COMPLETE ***




This eBook was produced by Charles Aldarondo and the Distributed
Proofreaders team.



The Notebooks of Leonardo Da Vinci

Volume 1

Translated by Jean Paul Richter

1888






PREFACE.





A singular fatality has ruled the destiny of nearly all the most
famous of Leonardo da Vinci's works. Two of the three most important
were never completed, obstacles having arisen during his life-time,
which obliged him to leave them unfinished; namely the Sforza
Monument and the Wall-painting of the Battle of Anghiari, while the
third--the picture of the Last Supper at Milan--has suffered
irremediable injury from decay and the repeated restorations to
which it was recklessly subjected during the XVIIth and XVIIIth
centuries. Nevertheless, no other picture of the Renaissance has
become so wellknown and popular through copies of every description.

Vasari says, and rightly, in his Life of Leonardo, "that he laboured
much more by his word than in fact or by deed", and the biographer
evidently had in his mind the numerous works in Manuscript which
have been preserved to this day. To us, now, it seems almost
inexplicable that these valuable and interesting original texts
should have remained so long unpublished, and indeed forgotten. It
is certain that during the XVIth and XVIIth centuries their
exceptional value was highly appreciated. This is proved not merely
by the prices which they commanded, but also by the exceptional
interest which has been attached to the change of ownership of
merely a few pages of Manuscript.

That, notwithstanding this eagerness to possess the Manuscripts,
their contents remained a mystery, can only be accounted for by the
many and great difficulties attending the task of deciphering them.
The handwriting is so peculiar that it requires considerable
practice to read even a few detached phrases, much more to solve
with any certainty the numerous difficulties of alternative
readings, and to master the sense as a connected whole. Vasari
observes with reference to Leonardos writing: "he wrote backwards,
in rude characters, and with the left hand, so that any one who is
not practised in reading them, cannot understand them". The aid of a
mirror in reading reversed handwriting appears to me available only
for a first experimental reading. Speaking from my own experience,
the persistent use of it is too fatiguing and inconvenient to be
practically advisable, considering the enormous mass of Manuscripts
to be deciphered. And as, after all, Leonardo's handwriting runs
backwards just as all Oriental character runs backwards--that is
to say from right to left--the difficulty of reading direct from the
writing is not insuperable. This obvious peculiarity in the writing
is not, however, by any means the only obstacle in the way of
mastering the text. Leonardo made use of an orthography peculiar to
himself; he had a fashion of amalgamating several short words into
one long one, or, again, he would quite arbitrarily divide a long
word into two separate halves; added to this there is no punctuation
whatever to regulate the division and construction of the sentences,
nor are there any accents--and the reader may imagine that such
difficulties were almost sufficient to make the task seem a
desperate one to a beginner. It is therefore not surprising that the
good intentions of some of Leonardo s most reverent admirers should
have failed.

Leonardos literary labours in various departments both of Art and of
Science were those essentially of an enquirer, hence the analytical
method is that which he employs in arguing out his investigations
and dissertations. The vast structure of his scientific theories is
consequently built up of numerous separate researches, and it is
much to be lamented that he should never have collated and arranged
them. His love for detailed research--as it seems to me--was the
reason that in almost all the Manuscripts, the different paragraphs
appear to us to be in utter confusion; on one and the same page,
observations on the most dissimilar subjects follow each other
without any connection. A page, for instance, will begin with some
principles of astronomy, or the motion of the earth; then come the
laws of sound, and finally some precepts as to colour. Another page
will begin with his investigations on the structure of the
intestines, and end with philosophical remarks as to the relations
of poetry to painting; and so forth.

Leonardo himself lamented this confusion, and for that reason I do
not think that the publication of the texts in the order in which
they occur in the originals would at all fulfil his intentions. No
reader could find his way through such a labyrinth; Leonardo himself
could not have done it.

Added to this, more than half of the five thousand manuscript pages
which now remain to us, are written on loose leaves, and at present
arranged in a manner which has no justification beyond the fancy of
the collector who first brought them together to make volumes of
more or less extent. Nay, even in the volumes, the pages of which
were numbered by Leonardo himself, their order, so far as the
connection of the texts was concerned, was obviously a matter of
indifference to him. The only point he seems to have kept in view,
when first writing down his notes, was that each observation should
be complete to the end on the page on which it was begun. The
exceptions to this rule are extremely few, and it is certainly
noteworthy that we find in such cases, in bound volumes with his
numbered pages, the written observations: "turn over", "This is the
continuation of the previous page", and the like. Is not this
sufficient to prove that it was only in quite exceptional cases that
the writer intended the consecutive pages to remain connected, when
he should, at last, carry out the often planned arrangement of his
writings?

What this final arrangement was to be, Leonardo has in most cases
indicated with considerable completeness. In other cases this
authoritative clue is wanting, but the difficulties arising from
this are not insuperable; for, as the subject of the separate
paragraphs is always distinct and well defined in itself, it is
quite possible to construct a well-planned whole, out of the
scattered materials of his scientific system, and I may venture to
state that I have devoted especial care and thought to the due
execution of this responsible task.

The beginning of Leonardo's literary labours dates from about his
thirty-seventh year, and he seems to have carried them on without
any serious interruption till his death. Thus the Manuscripts that
remain represent a period of about thirty years. Within this space
of time his handwriting altered so little that it is impossible to
judge from it of the date of any particular text. The exact dates,
indeed, can only be assigned to certain note-books in which the year
is incidentally indicated, and in which the order of the leaves has
not been altered since Leonardo used them. The assistance these
afford for a chronological arrangement of the Manuscripts is
generally self evident. By this clue I have assigned to the original
Manuscripts now scattered through England, Italy and France, the
order of their production, as in many matters of detail it is highly
important to be able to verify the time and place at which certain
observations were made and registered. For this purpose the
Bibliography of the Manuscripts given at the end of Vol. II, may be
regarded as an Index, not far short of complete, of all Leonardo s
literary works now extant. The consecutive numbers (from 1 to 1566)
at the head of each passage in this work, indicate their logical
sequence with reference to the subjects; while the letters and
figures to the left of each paragraph refer to the original
Manuscript and number of the page, on which that particular passage
is to be found. Thus the reader, by referring to the List of
Manuscripts at the beginning of Volume I, and to the Bibliography at
the end of Volume II, can, in every instance, easily ascertain, not
merely the period to which the passage belongs, but also exactly
where it stood in the original document. Thus, too, by following the
sequence of the numbers in the Bibliographical index, the reader may
reconstruct the original order of the Manuscripts and recompose the
various texts to be found on the original sheets--so much of it,
that is to say, as by its subject-matter came within the scope of
this work. It may, however, be here observed that Leonardo s
Manuscripts contain, besides the passages here printed, a great
number of notes and dissertations on Mechanics, Physics, and some
other subjects, many of which could only be satisfactorily dealt
with by specialists. I have given as complete a review of these
writings as seemed necessary in the Bibliographical notes.

In 1651, Raphael Trichet Dufresne, of Paris, published a selection
from Leonardo's writings on painting, and this treatise became so
popular that it has since been reprinted about two-and-twenty times,
and in six different languages. But none of these editions were
derived from the original texts, which were supposed to have been
lost, but from early copies, in which Leonardo's text had been more
or less mutilated, and which were all fragmentary. The oldest and on
the whole the best copy of Leonardo's essays and precepts on
Painting is in the Vatican Library; this has been twice printed,
first by Manzi, in 1817, and secondly by Ludwig, in 1882. Still,
this ancient copy, and the published editions of it, contain much
for which it would be rash to hold Leonardo responsible, and some
portions--such as the very important rules for the proportions of
the human figure--are wholly wanting; on the other hand they contain
passages which, if they are genuine, cannot now be verified from any
original Manuscript extant. These copies, at any rate neither give
us the original order of the texts, as written by Leonardo, nor do
they afford any substitute, by connecting them on a rational scheme;
indeed, in their chaotic confusion they are anything rather than
satisfactory reading. The fault, no doubt, rests with the compiler
of the Vatican copy, which would seem to be the source whence all
the published and extensively known texts were derived; for, instead
of arranging the passages himself, he was satisfied with recording a
suggestion for a final arrangement of them into eight distinct
parts, without attempting to carry out his scheme. Under the
mistaken idea that this plan of distribution might be that, not of
the compiler, but of Leonardo himself, the various editors, down to
the present day, have very injudiciously continued to adopt this
order--or rather disorder.

I, like other enquirers, had given up the original Manuscript of the
Trattato della Pittura for lost, till, in the beginning of 1880, I
was enabled, by the liberality of Lord Ashburnham, to inspect his
Manuscripts, and was so happy as to discover among them the original
text of the best-known portion of the Trattato in his magnificent
library at Ashburnham Place. Though this discovery was of a fragment
only--but a considerable fragment--inciting me to further search,
it gave the key to the mystery which had so long enveloped the first
origin of all the known copies of the Trattato. The extensive
researches I was subsequently enabled to prosecute, and the results
of which are combined in this work, were only rendered possible by
the unrestricted permission granted me to investigate all the
Manuscripts by Leonardo dispersed throughout Europe, and to
reproduce the highly important original sketches they contain, by
the process of "photogravure". Her Majesty the Queen graciously
accorded me special permission to copy for publication the
Manuscripts at the Royal Library at Windsor. The Commission Centrale
Administrative de l'Institut de France, Paris, gave me, in the most
liberal manner, in answer to an application from Sir Frederic
Leighton, P. R. A., Corresponding member of the Institut, free
permission to work for several months in their private collection at
deciphering the Manuscripts preserved there. The same favour which
Lord Ashburnham had already granted me was extended to me by the
Earl of Leicester, the Marchese Trivulsi, and the Curators of the
Ambrosian Library at Milan, by the Conte Manzoni at Rome and by
other private owners of Manuscripts of Leonardo's; as also by the
Directors of the Louvre at Paris; the Accademia at Venice; the
Uffizi at Florence; the Royal Library at Turin; and the British
Museum, and the South Kensington Museum. I am also greatly indebted
to the Librarians of these various collections for much assistance
in my labours; and more particularly to Monsieur Louis Lalanne, of
the Institut de France, the Abbate Ceriani, of the Ambrosian
Library, Mr. Maude Thompson, Keeper of Manuscripts at the British
Museum, Mr. Holmes, the Queens Librarian at Windsor, the Revd Vere
Bayne, Librarian of Christ Church College at Oxford, and the Revd A.
Napier, Librarian to the Earl of Leicester at Holkham Hall.

In correcting the Italian text for the press, I have had the
advantage of valuable advice from the Commendatore Giov. Morelli,
Senatore del Regno, and from Signor Gustavo Frizzoni, of Milan. The
translation, under many difficulties, of the Italian text into
English, is mainly due to Mrs. R. C. Bell; while the rendering of
several of the most puzzling and important passages, particularly in
the second half of Vol. I, I owe to the indefatigable interest taken
in this work by Mr. E. J. Poynter R. A. Finally I must express my
thanks to Mr. Alfred Marks, of Long Ditton, who has most kindly
assisted me throughout in the revision of the proof sheets.

The notes and dissertations on the texts on Architecture in Vol. II
I owe to my friend Baron Henri de Geymuller, of Paris.

I may further mention with regard to the illustrations, that the
negatives for the production of the "photo-gravures" by Monsieur
Dujardin of Paris were all taken direct from the originals.

It is scarcely necessary to add that most of the drawings here
reproduced in facsimile have never been published before. As I am
now, on the termination of a work of several years' duration, in a
position to review the general tenour of Leonardos writings, I may
perhaps be permitted to add a word as to my own estimate of the
value of their contents. I have already shown that it is due to
nothing but a fortuitous succession of unfortunate circumstances,
that we should not, long since, have known Leonardo, not merely as a
Painter, but as an Author, a Philosopher, and a Naturalist. There
can be no doubt that in more than one department his principles and
discoveries were infinitely more in accord with the teachings of
modern science, than with the views of his contemporaries. For this
reason his extraordinary gifts and merits are far more likely to be
appreciated in our own time than they could have been during the
preceding centuries. He has been unjustly accused of having
squandered his powers, by beginning a variety of studies and then,
having hardly begun, throwing them aside. The truth is that the
labours of three centuries have hardly sufficed for the elucidation
of some of the problems which occupied his mighty mind.

Alexander von Humboldt has borne witness that "he was the first to
start on the road towards the point where all the impressions of our
senses converge in the idea of the Unity of Nature" Nay, yet more
may be said. The very words which are inscribed on the monument of
Alexander von Humboldt himself, at Berlin, are perhaps the most
appropriate in which we can sum up our estimate of Leonardo's
genius:

"Majestati naturae par ingenium."

LONDON, April 1883.

F. P. R.







CONTENTS OF VOLUME I.





PROLEGOMENA AND GENERAL INTRODUCTION TO THE BOOK ON PAINTING

Clavis Sigillorum and Index of Manuscripts.--The author's intention
to publish his MSS. (1).--The preparation of the MSS. for
publication (2).--Admonition to readers (3).--The disorder in the
MSS. (4).--Suggestions for the arrangement of MSS. treating of
particular subjects (5--8).--General introductions to the book on
painting (9--13).--The plan of the book on painting (14--17).--The
use of the book on painting (18).--Necessity of theoretical
knowledge (19, 20).--The function of the eye (21--23).--Variability
of the eye (24).--Focus of sight (25).--Differences of perception by
one eye and by both eyes (26--29).--The comparative size of the
image depends on the amount of light (30--39).

II.

LINEAR PERSPECTIVE

General remarks on perspective (40--41).--The elements of
perspective:--of the point (42--46).--Of the line (47--48).--The
nature of the outline (49).--Definition of perspective (50).--The
perception of the object depends on the direction of the eye
(51).--Experimental proof of the existence of the pyramid of sight
(52--55).--The relations of the distance point to the vanishing
point (55--56).--How to measure the pyramid of vision (57).--The
production of the pyramid of vision (58--64).--Proof by experiment
(65--66).--General conclusions (67).--That the contrary is
impossible (68).--A parallel case (69).--The function of the eye, as
explained by the camera obscura (70--71).--The practice of
perspective (72--73).--Refraction of the rays falling upon the eye
(74--75).--The inversion of the images (76).--The intersection of
the rays (77--82).--Demonstration of perspective by means of a
vertical glass plane (83--85.)--The angle of sight varies with the
distance (86--88).--Opposite pyramids in juxtaposition (89).--On
simple and complex perspective (90).--The proper distance of objects
from the eye (91--92).--The relative size of objects with regard to
their distance from the eye (93--98).--The apparent size of objects
denned by calculation (99--106).--On natural perspective (107--109).

III.

SIX BOOKS ON LIGHT AND SHADE

GENERAL INTRODUCTION.--Prolegomena (110).--Scheme of the books on
light and shade (111).--Different principles and plans of treatment
(112--116).--Different sorts of light (117--118).--Definition of
the nature of shadows (119--122).--Of the various kinds of shadows
(123--125).--Of the various kinds of light (126--127).--General
remarks (128--129).--FIRST BOOK ON LIGHT AND SHADE.--On the nature
of light (130--131).--The difference between light and lustre
(132--135).--The relations of luminous to illuminated bodies (136).
--Experiments on the relation of light and shadow within a room
(137--140).--Light and shadow with regard to the position of the
eye (141--145).--The law of the incidence of light
(146--147).--SECOND BOOK ON LIGHT AND SHADE.--Gradations of strength
in the shadows (148--149).--On the intensity of shadows as dependent
on the distance from the light (150--152).--On the proportion of
light and shadow (153--157).--THIRD BOOK ON LIGHT AND
SHADE.--Definition of derived shadow (158--159).--Different sorts of
derived shadows (160--162).--On the relation of derived and primary
shadow (163--165).--On the shape of derived shadows (166--174).--On
the relative intensity of derived shadows (175--179).--Shadow as
produced by two lights of different size (180--181).--The effect of
light at different distances (182).--Further complications in the
derived shadows (183--187).--FOURTH BOOK ON LIGHT AND SHADE.--On the
shape of cast shadows (188--191).--On the outlines of cast shadows
(192--195).--On the relative size of cast shadows (196.
197).--Effects on cast shadows by the tone of the back ground
(198).--A disputed proposition (199).--On the relative depth of
cast shadows (200--202).--FIFTH BOOK ON LIGHT AND
SHADE.--Principles of reflection (203. 204).--On reverberation
(205).--Reflection on water (206. 207).--Experiments with the mirror
(208--210).--Appendix:--On shadows in movement (211--212).--SIXTH
BOOK ON LIGHT AND SHADE.--The effect of rays passing through holes
(213. 214).--On gradation of shadows (215. 216).--On relative
proportion of light and shadows (216--221).

IV.

PERSPECTIVE OF DISAPPEARANCE

Definition (222. 223).--An illustration by experiment (224).--A
guiding rule (225).---An experiment (226).--On indistinctness at
short distances (227--231).--On indistinctness at great distances
(232--234).--The importance of light and shade in the Prospettiva
de' perdimenti (235--239).--The effect of light or dark backgrounds
on the apparent size of objects (240--250).--Propositions on
Prospettiva de' perdimenti from MS. C. (250--262).

V.

THEORY OF COLOURS

The reciprocal effects of colours on objects placed opposite each
other (263--271).--Combination of different colours in cast shadows
(272).--The effect of colours in the camera obscura (273. 274).--On
the colours of derived shadows (275. 276).--On the nature of colours
(277. 278).--On gradations in the depth of colours (279. 280).--On
the reflection of colours (281--283).--On the use of dark and light
colours in painting (284--286).--On the colours of the rainbow
(287--288).

VI.

PERSPECTIVE OF COLOUR AND AERIAL PERSPECTIVE

General rules (289--291).--An exceptional case (292).--An experiment
(293).--The practice of the Prospettiva de' colori (294).--The rules
of aerial perspective (295--297).--On the relative density of the
atmosphere (298--299).--On the colour of the atmosphere (300--307).

VII.

ON THE PROPORTIONS AND ON THE MOVEMENTS OF THE HUMAN FIGURE

Preliminary observations (308. 309).--Proportions of the head and
face (310--318).--Proportions of the head seen in front
(319--321).--Proportions of the foot (322--323).--Relative
proportions of the hand and foot (324).--Relative proportions of
the foot and of the face (325--327).--Proportions of the leg
(328--331).--On the central point of the whole body (332).--The
relative proportions of the torso and of the whole figure
(333).--The relative proportions of the head and of the torso
(334).--The relative proportions of the torso and of the leg (335.
336).--The relative proportions of the torso and of the foot
(337).--The proportions of the whole figure (338--341).--The torso
from the front and back (342).--Vitruvius' scheme of proportions
(343).--The arm and head (344).--Proportions of the arm
(345--349).--The movement of the arm (350--354).--The movement of
the torso (355--361).--The proportions vary at different ages
(362--367).--The movement of the human figure (368--375).--Of
walking up and down (375--379).--On the human body in action
(380--388).--On hair falling down in curls (389).--On draperies

(390--392).

VIII.

BOTANY FOR PAINTERS, AND ELEMENTS OF LANDSCAPE PAINTING

Classification of trees (393).--The relative thickness of the
branches to the trunk (394--396).--The law of proportion in the
growth of the branches (397--402).--The direction of growth
(403--407).--The forms of trees (408--411).--The insertion of the
leaves (412--419).--Light on branches and leaves (420--422).--The
proportions of light and shade in a leaf (423--426).--Of the
transparency of leaves (427--429).--The gradations of shade and
colour in leaves (430--434).--A classification of trees according to
their colours (435).--The proportions of light and shade in trees
(436--440).--The distribution of light and shade with reference to
the position of the spectator (441--443).--The effects of morning
light (444--448).--The effects of midday light (449).--The
appearance of trees in the distance (450--451).--The cast shadow of
trees (452. 453).--Light and shade on groups of trees
(454--457).--On the treatment of light for landscapes
(458--464).--On the treatment of light for views of towns
(465--469).--The effect of wind on trees (470--473).--Light and
shade on clouds (474--477).--On images reflected in water (478).--Of
rainbows and rain (479. 480).--Of flower seeds (481).

IX.

THE PRACTICE OF PAINTING

I. MORAL PRECEPTS FOR THE STUDENT OF PAINTING.--How to ascertain the
dispositions for an artistic career (482).--The course of
instruction for an artist (483--485).--The study of the antique
(486. 487).--The necessity of anatomical knowledge (488. 489).--How
to acquire practice (490).--Industry and thoroughness the first
conditions (491--493.)--The artist's private life and choice of
company (493. 494).--The distribution of time for studying (495--
497).--On the productive power of minor artists (498--501).--A
caution against one-sided study (502).--How to acquire universality
(503--506).--Useful games and exercises (507. 508).--II. THE
ARTIST'S STUDIO.--INSTRUMENTS AND HELPS FOR THE APPLICATION OF
PERSPECTIVE.--ON JUDGING OF A PICTURE.--On the size of the studio
(509).--On the construction of windows (510--512).--On the best
light for painting (513--520).--On various helps in preparing a
picture (521--530).--On the management of works (531. 532).--On the
limitations of painting (533--535).--On the choice of a position
(536. 537).--The apparent size of figures in a picture (538.
539).--The right position of the artist, when painting and of the
spectator (540--547).--III. THE PRACTICAL METHODS OF LIGHT AND SHADE
AND AERIAL PERSPECTIVE.--Gradations of light and shade (548).--On
the choice of light for a picture (549--554).--The distribution of
light and shade (555--559).--The juxtaposition of light and shade
(560. 561).--On the lighting of the background (562--565).--On the
lighting of white objects (566).--The methods of aerial perspective
(567--570).--IV. OF PORTRAIT AND FIGURE PAINTING.--Of sketching
figures and portraits (571. 572).--The position of the head
(573).--Of the light on the face (574--576).--General suggestions
for historical pictures (577--581).--How to represent the
differences of age and sex (582. 583).--Of representing the emotions
(584).--Of representing imaginary animals (585).--The selection of
forms (586--591).--How to pose figures (592).--Of appropriate
gestures (593--600).--V. SUGGESTIONS FOR COMPOSITIONS.--Of painting
battle-pieces (601--603).--Of depicting night-scenes (604).--Of
depicting a tempest (605. 606).--Of representing the deluge
(607--609).--Of depicting natural phenomena (610. 611).--VI. THE
ARTIST'S MATERIALS.--Of chalk and paper (612--617).--On the
preparation and use of colours (618--627).--Of preparing the panel
(628).--The preparation of oils (629--634).--On varnishes (635--
637).--On chemical _materials (638--650).--VII. PHILOSOPHY AND
HISTORY OF THE ART OF PAINTING.--The relation of art and nature
(651. 652).--Painting is superior to poetry (653. 654).--Painting is
superior to sculpture (655. 656).--Aphorisms (657--659).--On the
history of painting (660. 661).--The painter's scope (662).

X.

STUDIES AND SKETCHES FOR PICTURES AND DECORATIONS

On pictures of the Madonna (663).--Bernardo di Bandino's portrait
(664).--Notes on the Last Supper (665--668).--On the battle of
Anghiari (669).--Allegorical representations referring to the duke
of Milan (670--673).--Allegorical representations
(674--678).--Arrangement of a picture (679).--List of drawings
(680).--Mottoes and Emblems (681--702).






The author's intention to publish his MSS.

1.

How by a certain machine many may stay some time under water. And
how and wherefore I do not describe my method of remaining under
water and how long I can remain without eating. And I do not publish
nor divulge these, by reason of the evil nature of men, who would
use them for assassinations at the bottom of the sea by destroying
ships, and sinking them, together with the men in them. Nevertheless
I will impart others, which are not dangerous because the mouth of
the tube through which you breathe is above the water, supported on
air sacks or cork.

[Footnote: The leaf on which this passage is written, is headed with
the words _Casi_ 39, and most of these cases begin with the word
'_Come_', like the two here given, which are the 26th and 27th. 7.
_Sughero_. In the Codex Antlanticus 377a; 1170a there is a sketch,
drawn with the pen, representing a man with a tube in his mouth, and
at the farther end of the tube a disk. By the tube the word
'_Channa_' is written, and by the disk the word '_sughero_'.]

The preparation of the MSS. for publication.

2.

When you put together the science of the motions of water, remember
to include under each proposition its application and use, in order
that this science may not be useless.--

[Footnote: A comparatively small portion of Leonardo's notes on
water-power was published at Bologna in 1828, under the title: "_Del
moto e misura dell'Acqua, di L. da Vinci_".]

Admonition to readers.

3.

Let no man who is not a Mathematician read the elements of my work.

The disorder in the MSS.

4.

Begun at Florence, in the house of Piero di Braccio Martelli, on the
22nd day of March 1508. And this is to be a collection without
order, taken from many papers which I have copied here, hoping to
arrange them later each in its place, according to the subjects of
which they may treat. But I believe that before I am at the end of
this [task] I shall have to repeat the same things several times;
for which, O reader! do not blame me, for the subjects are many and
memory cannot retain them [all] and say: 'I will not write this
because I wrote it before.' And if I wished to avoid falling into
this fault, it would be necessary in every case when I wanted to
copy [a passage] that, not to repeat myself, I should read over all
that had gone before; and all the more since the intervals are long
between one time of writing and the next.

[Footnote: 1. In the history of Florence in the early part of the
XVIth century _Piero di Braccio Martelli_ is frequently mentioned as
_Commissario della Signoria_. He was famous for his learning and at
his death left four books on Mathematics ready for the press; comp.
LITTA, _Famiglie celebri Italiane_, _Famiglia Martelli di
Firenze_.--In the Official Catalogue of MSS. in the Brit. Mus., New
Series Vol. I., where this passage is printed, _Barto_ has been
wrongly given for Braccio.

2. _addi 22 di marzo 1508_. The Christian era was computed in
Florence at that time from the Incarnation (Lady day, March 25th).
Hence this should be 1509 by our reckoning.

3. _racolto tratto di molte carte le quali io ho qui copiate_. We
must suppose that Leonardo means that he has copied out his own MSS.
and not those of others. The first thirteen leaves of the MS. in the
Brit. Mus. are a fair copy of some notes on physics.]

Suggestions for the arrangement of MSS treating of particular
subjects.(5-8).

5.

Of digging a canal. Put this in the Book of useful inventions and in
proving them bring forward the propositions already proved. And this
is the proper order; since if you wished to show the usefulness of
any plan you would be obliged again to devise new machines to prove
its utility and thus would confuse the order of the forty Books and
also the order of the diagrams; that is to say you would have to mix
up practice with theory, which would produce a confused and
incoherent work.

6.

I am not to blame for putting forward, in the course of my work on
science, any general rule derived from a previous conclusion.

7.

The Book of the science of Mechanics must precede the Book of useful
inventions.--Have your books on anatomy bound! [Footnote: 4. The
numerous notes on anatomy written on loose leaves and now in the
Royal collection at Windsor can best be classified in four Books,
corresponding to the different character and size of the paper. When
Leonardo speaks of '_li tua libri di notomia_', he probably means
the MSS. which still exist; if this hypothesis is correct the
present condition of these leaves might seem to prove that he only
carried out his purpose with one of the Books on anatomy. A borrowed
book on Anatomy is mentioned in F.O.]

8.

The order of your book must proceed on this plan: first simple
beams, then (those) supported from below, then suspended in part,
then wholly [suspended]. Then beams as supporting other weights
[Footnote: 4. Leonardo's notes on Mechanics are extraordinarily
numerous; but, for the reasons assigned in my introduction, they
have not been included in the present work.].

General introductions to the book on Painting (9-13).

9.

INTRODUCTION.

Seeing that I can find no subject specially useful or
pleasing--since the men who have come before me have taken for their
own every useful or necessary theme--I must do like one who, being
poor, comes last to the fair, and can find no other way of providing
himself than by taking all the things already seen by other buyers,
and not taken but refused by reason of their lesser value. I, then,
will load my humble pack with this despised and rejected
merchandise, the refuse of so many buyers; and will go about to
distribute it, not indeed in great cities, but in the poorer towns,
taking such a price as the wares I offer may be worth. [Footnote: It
need hardly be pointed out that there is in this 'Proemio' a covert
irony. In the second and third prefaces, Leonardo characterises his
rivals and opponents more closely. His protest is directed against
Neo-latinism as professed by most of the humanists of his time; its
futility is now no longer questioned.]

10.

INTRODUCTION.

I know that many will call this useless work [Footnote: 3. questa
essere opera inutile. By opera we must here understand libro di
pittura and particularly the treatise on Perspective.]; and they
will be those of whom Demetrius [Footnote: 4. Demetrio. "With regard
to the passage attributed to Demetrius", Dr. H. MULLER STRUBING
writes, "I know not what to make of it. It is certainly not
Demetrius Phalereus that is meant and it can hardly be Demetrius
Poliorcetes. Who then can it be--for the name is a very common one?
It may be a clerical error for Demades and the maxim is quite in the
spirit of his writings I have not however been able to find any
corresponding passage either in the 'Fragments' (C. MULLER, _Orat.
Att._, II. 441) nor in the Supplements collected by DIETZ (_Rhein.
Mus._, vol. 29, p. 108)."

The same passage occurs as a simple Memorandum in the MS. Tr. 57,
apparently as a note for this '_Proemio_' thus affording some data
as to the time where these introductions were written.] declared
that he took no more account of the wind that came out their mouth
in words, than of that they expelled from their lower parts: men who
desire nothing but material riches and are absolutely devoid of that
of wisdom, which is the food and the only true riches of the mind.
For so much more worthy as the soul is than the body, so much more
noble are the possessions of the soul than those of the body. And
often, when I see one of these men take this work in his hand, I
wonder that he does not put it to his nose, like a monkey, or ask me
if it is something good to eat.

[Footnote: In the original, the Proemio di prospettiva cioe
dell'uffitio dell'occhio (see No. 21) stands between this and the
preceding one, No. 9.]

INTRODUCTION.

I am fully concious that, not being a literary man, certain
presumptuous persons will think that they may reasonably blame me;
alleging that I am not a man of letters. Foolish folks! do they not
know that I might retort as Marius did to the Roman Patricians
[Footnote 21: _Come Mario disse ai patriti Romani_. "I am unable to
find the words here attributed by Leonardo to Marius, either in
Plutarch's Life of Marius or in the Apophthegmata (_Moralia_,
p.202). Nor do they occur in the writings of Valerius Maximus (who
frequently mentions Marius) nor in Velleius Paterculus (II, 11 to
43), Dio Cassius, Aulus Gellius, or Macrobius. Professor E.
MENDELSON of Dorpat, the editor of Herodian, assures me that no such
passage is the found in that author" (communication from Dr. MULLER
STRUBING). Leonardo evidently meant to allude to some well known
incident in Roman history and the mention of Marius is the result
probably of some confusion. We may perhaps read, for Marius,
Menenius Agrippa, though in that case it is true we must alter
Patriti to Plebei. The change is a serious one. but it would render
the passage perfectly clear.] by saying: That they, who deck
themselves out in the labours of others will not allow me my own.
They will say that I, having no literary skill, cannot properly
express that which I desire to treat of [Footnote 26: _le mie cose
.... che d'altra parola_. This can hardly be reconciled with Mons.
RAVAISSON'S estimate of L. da Vinci's learning. "_Leonard de Vinci
etait un admirateur et un disciple des anciens, aussi bien dans
l'art que dans la science et il tenait a passer pour tel meme aux
yeux de la posterite._" _Gaz. des Beaux arts. Oct. 1877.]; but they
do not know that my subjects are to be dealt with by experience
rather than by words [Footnote 28: See Footnote 26]; and
[experience] has been the mistress of those who wrote well. And so,
as mistress, I will cite her in all cases.

11.

Though I may not, like them, be able to quote other authors, I shall
rely on that which is much greater and more worthy:--on experience,
the mistress of their Masters. They go about puffed up and pompous,
dressed and decorated with [the fruits], not of their own labours,
but of those of others. And they will not allow me my own. They will
scorn me as an inventor; but how much more might they--who are not
inventors but vaunters and declaimers of the works of others--be
blamed.

INTRODUCTION.

And those men who are inventors and interpreters between Nature and
Man, as compared with boasters and declaimers of the works of
others, must be regarded and not otherwise esteemed than as the
object in front of a mirror, when compared with its image seen in
the mirror. For the first is something in itself, and the other
nothingness.--Folks little indebted to Nature, since it is only by
chance that they wear the human form and without it I might class
them with the herds of beasts.

12.

Many will think they may reasonably blame me by alleging that my
proofs are opposed to the authority of certain men held in the
highest reverence by their inexperienced judgments; not considering
that my works are the issue of pure and simple experience, who is
the one true mistress. These rules are sufficient to enable you to
know the true from the false--and this aids men to look only for
things that are possible and with due moderation--and not to wrap
yourself in ignorance, a thing which can have no good result, so
that in despair you would give yourself up to melancholy.

13.

Among all the studies of natural causes and reasons Light chiefly
delights the beholder; and among the great features of Mathematics
the certainty of its demonstrations is what preeminently (tends to)
elevate the mind of the investigator. Perspective, therefore, must
be preferred to all the discourses and systems of human learning. In
this branch [of science] the beam of light is explained on those
methods of demonstration which form the glory not so much of
Mathematics as of Physics and are graced with the flowers of both
[Footnote: 5. Such of Leonardo's 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.]

16.

THE DISCOURSE ON PAINTING.

Perspective, as bearing on drawing, is divided into three principal
sections; of which the first treats of the diminution in the size of
bodies at different distances. The second part is that which treats
of the diminution in colour in these objects. The third [deals with]
the diminished distinctness of the forms and outlines displayed by
the objects at various distances.

17.

ON THE SECTIONS OF [THE BOOK ON] PAINTING.

The first thing in painting is that the objects it represents should
appear in relief, and that the grounds surrounding them at different
distances shall appear within the vertical plane of the foreground
of the picture by means of the 3 branches of Perspective, which are:
the diminution in the distinctness of the forms of the objects, the
diminution in their magnitude; and the diminution in their colour.
And of these 3 classes of Perspective the first results from [the
structure of] the eye, while the other two are caused by the
atmosphere which intervenes between the eye and the objects seen by
it. The second essential in painting is appropriate action and a due
variety in the figures, so that the men may not all look like
brothers, &c.

[Footnote: This and the two foregoing chapters must have been
written in 1513 to 1516. They undoubtedly indicate the scheme which
Leonardo wished to carry out in arranging his researches on
Perspective as applied to Painting. This is important because it is
an evidence against the supposition of H. LUDWIG and others, that
Leonardo had collected his principles of Perspective in one book so
early as before 1500; a Book which, according to the hypothesis,
must have been lost at a very early period, or destroyed possibly,
by the French (!) in 1500 (see H. LUDWIG. L. da Vinci: _Das Buch van
der Malerei_. Vienna 1882 III, 7 and 8).]

The use of the book on Painting.

18.

These rules are of use only in correcting the figures; since every
man makes some mistakes in his first compositions and he who knows
them not, cannot amend them. But you, knowing your errors, will
correct your works and where you find mistakes amend them, and
remember never to fall into them again. But if you try to apply
these rules in composition you will never make an end, and will
produce confusion in your works.

These rules will enable you to have a free and sound judgment; since
good judgment is born of clear understanding, and a clear
understanding comes of reasons derived from sound rules, and sound
rules are the issue of sound experience--the common mother of all
the sciences and arts. Hence, bearing in mind the precepts of my
rules, you will be able, merely by your amended judgment, to
criticise and recognise every thing that is out of proportion in a
work, whether in the perspective or in the figures or any thing
else.

Necessity of theoretical knowledge (19. 20).

19.

OF THE MISTAKES MADE BY THOSE WHO PRACTISE WITHOUT KNOWLEDGE.

Those who are in love with practice without knowledge are like the
sailor who gets into a ship without rudder or compass and who never
can be certain whether he is going. Practice must always be founded
on sound theory, and to this Perspective is the guide and the
gateway; and without this nothing can be done well in the matter of
drawing.

20.

The painter who draws merely by practice and by eye, without any
reason, is like a mirror which copies every thing placed in front of
it without being conscious of their existence.

The function of the eye (21-23).

21.

INTRODUCTION TO PERSPECTIVE:--THAT IS OF THE FUNCTION OF THE EYE.

Behold here O reader! a thing concerning which we cannot trust our
forefathers, the ancients, who tried to define what the Soul and
Life are--which are beyond proof, whereas those things, which can at
any time be clearly known and proved by experience, remained for
many ages unknown or falsely understood. The eye, whose function we
so certainly know by experience, has, down to my own time, been
defined by an infinite number of authors as one thing; but I find,
by experience, that it is quite another. [Footnote 13: Compare the
note to No. 70.]

[Footnote: In section 13 we already find it indicated that the study
of Perspective and of Optics is to be based on that of the functions
of the eye. Leonardo also refers to the science of the eye, in his
astronomical researches, for instance in MS. F 25b '_Ordine del
provare la terra essere una stella: Imprima difinisce l'occhio'_,
&c. Compare also MS. E 15b and F 60b. The principles of astronomical
perspective.]

22.

Here [in the eye] forms, here colours, here the character of every
part of the universe are concentrated to a point; and that point is
so marvellous a thing ... Oh! marvellous, O stupendous Necessity--by
thy laws thou dost compel every effect to be the direct result of
its cause, by the shortest path. These [indeed] are miracles;...

In so small a space it can be reproduced and rearranged in its whole
expanse. Describe in your anatomy what proportion there is between
the diameters of all the images in the eye and the distance from
them of the crystalline lens.

23.

OF THE 10 ATTRIBUTES OF THE EYE, ALL CONCERNED IN PAINTING.

Painting is concerned with all the 10 attributes of sight; which
are:--Darkness, Light, Solidity and Colour, Form and Position,
Distance and Propinquity, Motion and Rest. This little work of mine
will be a tissue [of the studies] of these attributes, reminding the
painter of the rules and methods by which he should use his art to
imitate all the works of Nature which adorn the world.

24.

ON PAINTING.

Variability of the eye.

1st. The pupil of the eye contracts, in proportion to the increase
of light which is reflected in it. 2nd. The pupil of the eye expands
in proportion to the diminution in the day light, or any other
light, that is reflected in it. 3rd. [Footnote: 8. The subject of
this third proposition we find fully discussed in MS. G. 44a.]. The
eye perceives and recognises the objects of its vision with greater
intensity in proportion as the pupil is more widely dilated; and
this can be proved by the case of nocturnal animals, such as cats,
and certain birds--as the owl and others--in which the pupil varies
in a high degree from large to small, &c., when in the dark or in
the light. 4th. The eye [out of doors] in an illuminated atmosphere
sees darkness behind the windows of houses which [nevertheless] are
light. 5th. All colours when placed in the shade appear of an equal
degree of darkness, among themselves. 6th. But all colours when
placed in a full light, never vary from their true and essential
hue.

25.

OF THE EYE.

Focus of sight.

If the eye is required to look at an object placed too near to it,
it cannot judge of it well--as happens to a man who tries to see the
tip of his nose. Hence, as a general rule, Nature teaches us that an
object can never be seen perfectly unless the space between it and
the eye is equal, at least, to the length of the face.

Differences of perception by one eye and by both eyes (26-29).

26.

OF THE EYE.

When both eyes direct the pyramid of sight to an object, that object
becomes clearly seen and comprehended by the eyes.

27.

Objects seen by one and the same eye appear sometimes large, and
sometimes small.

28.

The motion of a spectator who sees an object at rest often makes it
seem as though the object at rest had acquired the motion of the
moving body, while the moving person appears to be at rest.

ON PAINTING.

Objects in relief, when seen from a short distance with one eye,
look like a perfect picture. If you look with the eye _a_, _b_ at
the spot _c_, this point _c_ will appear to be at _d_, _f_, and if
you look at it with the eye _g_, _h_ will appear to be at _m_. A
picture can never contain in itself both aspects.

29.

Let the object in relief _t_ be seen by both eyes; if you will look
at the object with the right eye _m_, keeping the left eye _n_ shut,
the object will appear, or fill up the space, at _a_; and if you
shut the right eye and open the left, the object (will occupy the)
space _b_; and if you open both eyes, the object will no longer
appear at _a_ or _b_, but at _e_, _r_, _f_. Why will not a picture
seen by both eyes produce the effect of relief, as [real] relief
does when seen by both eyes; and why should a picture seen with one
eye give the same effect of relief as real relief would under the
same conditions of light and shade?

[Footnote: In the sketch, _m_ is the left eye and _n_ the right,
while the text reverses this lettering. We must therefore suppose
that the face in which the eyes _m_ and _n_ are placed is opposite
to the spectator.]

30.

The comparative size of the image depends on the amount of light
(30-39).

The eye will hold and retain in itself the image of a luminous body
better than that of a shaded object. The reason is that the eye is
in itself perfectly dark and since two things that are alike cannot
be distinguished, therefore the night, and other dark objects cannot
be seen or recognised by the eye. Light is totally contrary and
gives more distinctness, and counteracts and differs from the usual
darkness of the eye, hence it leaves the impression of its image.

31.

Every object we see will appear larger at midnight than at midday,
and larger in the morning than at midday.

This happens because the pupil of the eye is much smaller at midday
than at any other time.

32.

The pupil which is largest will see objects the largest. This is
evident when we look at luminous bodies, and particularly at those
in the sky. When the eye comes out of darkness and suddenly looks up
at these bodies, they at first appear larger and then diminish; and
if you were to look at those bodies through a small opening, you
would see them smaller still, because a smaller part of the pupil
would exercise its function.

[Footnote: 9. _buso_ in the Lomb. dialect is the same as _buco_.]

33.

When the eye, coming out of darkness suddenly sees a luminous body,
it will appear much larger at first sight than after long looking at
it. The illuminated object will look larger and more brilliant, when
seen with two eyes than with only one. A luminous object will appear
smaller in size, when the eye sees it through a smaller opening. A
luminous body of an oval form will appear rounder in proportion as
it is farther from the eye.

34.

Why when the eye has just seen the light, does the half light look
dark to it, and in the same way if it turns from the darkness the
half light look very bright?

35.

ON PAINTING.

If the eye, when [out of doors] in the luminous atmosphere, sees a
place in shadow, this will look very much darker than it really is.
This happens only because the eye when out in the air contracts the
pupil in proportion as the atmosphere reflected in it is more
luminous. And the more the pupil contracts, the less luminous do the
objects appear that it sees. But as soon as the eye enters into a
shady place the darkness of the shadow suddenly seems to diminish.
This occurs because the greater the darkness into which the pupil
goes the more its size increases, and this increase makes the
darkness seem less.

[Footnote 14: _La luce entrera_. _Luce_ occurs here in the sense of
pupil of the eye as in no 51: C. A. 84b; 245a; I--5; and in many
other places.]

36.

ON PERSPECTIVE.

The eye which turns from a white object in the light of the sun and
goes into a less fully lighted place will see everything as dark.
And this happens either because the pupils of the eyes which have
rested on this brilliantly lighted white object have contracted so
much that, given at first a certain extent of surface, they will
have lost more than 3/4 of their size; and, lacking in size, they
are also deficient in [seeing] power. Though you might say to me: A
little bird (then) coming down would see comparatively little, and
from the smallness of his pupils the white might seem black! To this
I should reply that here we must have regard to the proportion of
the mass of that portion of the brain which is given up to the sense
of sight and to nothing else. Or--to return--this pupil in Man
dilates and contracts according to the brightness or darkness of
(surrounding) objects; and since it takes some time to dilate and
contract, it cannot see immediately on going out of the light and
into the shade, nor, in the same way, out of the shade into the
light, and this very thing has already deceived me in painting an
eye, and from that I learnt it.

37.

Experiment [showing] the dilatation and contraction of the pupil,
from the motion of the sun and other luminaries. In proportion as
the sky is darker the stars appear of larger size, and if you were
to light up the medium these stars would look smaller; and this
difference arises solely from the pupil which dilates and contracts
with the amount of light in the medium which is interposed between
the eye and the luminous body. Let the experiment be made, by
placing a candle above your head at the same time that you look at a
star; then gradually lower the candle till it is on a level with the
ray that comes from the star to the eye, and then you will see the
star diminish so much that you will almost lose sight of it.

[Footnote: No reference is made in the text to the letters on the
accompanying diagram.]

38.

The pupil of the eye, in the open air, changes in size with every
degree of motion from the sun; and at every degree of its changes
one and the same object seen by it will appear of a different size;
although most frequently the relative scale of surrounding objects
does not allow us to detect these variations in any single object we
may look at.

39.

The eye--which sees all objects reversed--retains the images for
some time. This conclusion is proved by the results; because, the
eye having gazed at light retains some impression of it. After
looking (at it) there remain in the eye images of intense
brightness, that make any less brilliant spot seem dark until the
eye has lost the last trace of the impression of the stronger light.

_II.

Linear Perspective.

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).

General remarks on perspective (40-41).

40.

ON PAINTING.

Perspective is the best guide to the art of Painting.

[Footnote: 40. Compare 53, 2.]

41.

The art of perspective is of such a nature as to make what is flat
appear in relief and what is in relief flat.

The elements of perspective--Of the Point (42-46).

42.

All the problems of perspective are made clear by the five terms of
mathematicians, which are:--the point, the line, the angle, the
superficies and the solid. The point is unique of its kind. And the
point has neither height, breadth, length, nor depth, whence it is
to be regarded as indivisible and as having no dimensions in space.
The line is of three kinds, straight, curved and sinuous and it has
neither breadth, height, nor depth. Hence it is indivisible,
excepting in its length, and its ends are two points. The angle is
the junction of two lines in a point.

43.

A point is not part of a line.

44.

OF THE NATURAL POINT.

The smallest natural point is larger than all mathematical points,
and this is proved because the natural point has continuity, and any
thing that is continuous is infinitely divisible; but the
mathematical point is indivisible because it has no size.

[Footnote: This definition was inserted by Leonardo on a MS. copy on
parchment of the well-known _"Trattato d'Architettura civile e
militare"_ &c. by FRANCESCO DI GIORGIO; opposite a passage where the
author says: _'In prima he da sapere che punto e quella parie della
quale he nulla--Linia he luncheza senza apieza; &c.]

45.

1, The superficies is a limitation of the body. 2, and the
limitation of a body is no part of that body. 4, and the limitation
of one body is that which begins another. 3, that which is not part
of any body is nothing. Nothing is that which fills no space.

If one single point placed in a circle may be the starting point of
an infinite number of lines, and the termination of an infinite
number of lines, there must be an infinite number of points
separable from this point, and these when reunited become one again;
whence it follows that the part may be equal to the whole.

46.

The point, being indivisible, occupies no space. That which occupies
no space is nothing. The limiting surface of one thing is the
beginning of another. 2. That which is no part of any body is called
nothing. 1. That which has no limitations, has no form. The
limitations of two conterminous bodies are interchangeably the
surface of each. All the surfaces of a body are not parts of that
body.

Of the line (47-48).

47.

DEFINITION OF THE NATURE OF THE LINE.

The line has in itself neither matter nor substance and may rather
be called an imaginary idea than a real object; and this being its
nature it occupies no space. Therefore an infinite number of lines
may be conceived of as intersecting each other at a point, which has
no dimensions and is only of the thickness (if thickness it may be
called) of one single line.

HOW WE MAY CONCLUDE THAT A SUPERFICIES TERMINATES IN A POINT?

An angular surface is reduced to a point where it terminates in an
angle. Or, if the sides of that angle are produced in a straight
line, then--beyond that angle--another surface is generated,
smaller, or equal to, or larger than the first.

48.

OF DRAWING OUTLINE.

Consider with the greatest care the form of the outlines of every
object, and the character of their undulations. And these
undulations must be separately studied, as to whether the curves are
composed of arched convexities or angular concavities.

49.

The nature of the outline.

The boundaries of bodies are the least of all things. The
proposition is proved to be true, because the boundary of a thing is
a surface, which is not part of the body contained within that
surface; nor is it part of the air surrounding that body, but is the
medium interposted between the air and the body, as is proved in its
place. But the lateral boundaries of these bodies is the line
forming the boundary of the surface, which line is of invisible
thickness. Wherefore O painter! do not surround your bodies with
lines, and above all when representing objects smaller than nature;
for not only will their external outlines become indistinct, but
their parts will be invisible from distance.

50.

Definition of Perspective.

[Drawing is based upon perspective, which is nothing else than a
thorough knowledge of the function of the eye. And this function
simply consists in receiving in a pyramid the forms and colours of
all the objects placed before it. I say in a pyramid, because there
is no object so small that it will not be larger than the spot where
these pyramids are received into the eye. Therefore, if you extend
the lines from the edges of each body as they converge you will
bring them to a single point, and necessarily the said lines must
form a pyramid.]

[Perspective is nothing more than a rational demonstration applied
to the consideration of how objects in front of the eye transmit
their image to it, by means of a pyramid of lines. The _Pyramid_ is
the name I apply to the lines which, starting from the surface and
edges of each object, converge from a distance and meet in a single
point.]

[Perspective is a rational demonstration, by which we may
practically and clearly understand how objects transmit their own
image, by lines forming a Pyramid (centred) in the eye.]

Perspective is a rational demonstration by which experience confirms
that every object sends its image to the eye by a pyramid of lines;
and bodies of equal size will result in a pyramid of larger or
smaller size, according to the difference in their distance, one
from the other. By a pyramid of lines I mean those which start from
the surface and edges of bodies, and, converging from a distance
meet in a single point. A point is said to be that which [having no
dimensions] cannot be divided, and this point placed in the eye
receives all the points of the cone.

[Footnote: 50. 1-5. Compare with this the Proem. No. 21. The
paragraphs placed in brackets: lines 1-9, 10-14, and 17--20, are
evidently mere sketches and, as such, were cancelled by the writer;
but they serve as a commentary on the final paragraph, lines 22-29.]

51.

IN WHAT WAY THE EYE SEES OBJECTS PLACED IN FRONT OF IT.

The perception of the object depends on the direction of the eye.

Supposing that the ball figured above is the ball of the eye and let
the small portion of the ball which is cut off by the line _s t_ be
the pupil and all the objects mirrored on the centre of the face of
the eye, by means of the pupil, pass on at once and enter the pupil,
passing through the crystalline humour, which does not interfere in
the pupil with the things seen by means of the light. And the pupil
having received the objects, by means of the light, immediately
refers them and transmits them to the intellect by the line _a b_.
And you must know that the pupil transmits nothing perfectly to the
intellect or common sense excepting when the objects presented to it
by means of light, reach it by the line _a b;_ as, for instance, by
the line _b c_. For although the lines _m n_ and _f g_ may be seen
by the pupil they are not perfectly taken in, because they do not
coincide with the line _a b_. And the proof is this: If the eye,
shown above, wants to count the letters placed in front, the eye
will be obliged to turn from letter to letter, because it cannot
discern them unless they lie in the line _a b;_ as, for instance, in
the line _a c_. All visible objects reach the eye by the lines of a
pyramid, and the point of the pyramid is the apex and centre of it,
in the centre of the pupil, as figured above.

[Footnote: 51. In this problem the eye is conceived of as fixed and
immovable; this is plain from line 11.]

Experimental proof of the existence of the pyramid of sight (52-55).

52.

Perspective is a rational demonstration, confirmed by experience,
that all objects transmit their image to the eye by a pyramid of
lines.

By a pyramid of lines I understand those lines which start from the
edges of the surface of bodies, and converging from a distance, meet
in a single point; and this point, in the present instance, I will
show to be situated in the eye which is the universal judge of all
objects. By a point I mean that which cannot be divided into parts;
therefore this point, which is situated in the eye, being
indivisible, no body is seen by the eye, that is not larger than
this point. This being the case it is inevitable that the lines
which come from the object to the point must form a pyramid. And if
any man seeks to prove that the sense of sight does not reside in
this point, but rather in the black spot which is visible in the
middle of the pupil, I might reply to him that a small object could
never diminish at any distance, as it might be a grain of millet or
of oats or of some similar thing, and that object, if it were larger
than the said [black] spot would never be seen as a whole; as may be
seen in the diagram below. Let _a_. be the seat of sight, _b e_ the
lines which reach the eye. Let _e d_ be the grains of millet within
these lines. You plainly see that these will never diminish by
distance, and that the body _m n_ could not be entirely covered by
it. Therefore you must confess that the eye contains within itself
one single indivisible point _a_, to which all the points converge
of the pyramid of lines starting from an object, as is shown below.
Let _a_. _b_. be the eye; in the centre of it is the point above
mentioned. If the line _e f_ is to enter as an image into so small
an opening in the eye, you must confess that the smaller object
cannot enter into what is smaller than itself unless it is
diminished, and by diminishing it must take the form of a pyramid.

53.

PERSPECTIVE.

Perspective comes in where judgment fails [as to the distance] in
objects which diminish. The eye can never be a true judge for
determining with exactitude how near one object is to another which
is equal to it [in size], if the top of that other is on the level
of the eye which sees them on that side, excepting by means of the
vertical plane which is the standard and guide of perspective. Let
_n_ be the eye, _e f_ the vertical plane above mentioned. Let _a b c
d_ be the three divisions, one below the other; if the lines _a n_
and _c n_ are of a given length and the eye _n_ is in the centre,
then _a b_ will look as large as _b c. c d_ is lower and farther off
from _n_, therefore it will look smaller. And the same effect will
appear in the three divisions of a face when the eye of the painter
who is drawing it is on a level with the eye of the person he is
painting.

54.

TO PROVE HOW OBJECTS REACH THE EYE.

If you look at the sun or some other luminous body and then shut
your eyes you will see it again inside your eye for a long time.
This is evidence that images enter into the eye.

The relations of the distance points to the vanishing point (55-56).

55.

ELEMENTS OF PERSPECTIVE.

All objects transmit their image to the eye in pyramids, and the
nearer to the eye these pyramids are intersected the smaller will
the image appear of the objects which cause them. Therefore, you may
intersect the pyramid with a vertical plane [Footnote 4: _Pariete_.
Compare the definitions in 85, 2-5, 6-27. These lines refer
exclusively to the third diagram. For the better understanding of
this it should be observed that _c s_ must be regarded as
representing the section or profile of a square plane, placed
horizontally (comp. lines 11, 14, 17) for which the word _pianura_
is subsequently employed (20, 22). Lines 6-13 contain certain
preliminary observations to guide the reader in understanding the
diagram; the last three seem to have been added as a supplement.
Leonardo's mistake in writing _t denota_ (line 6) for _f denota_ has
been rectified.] which reaches the base of the pyramid as is shown
in the plane _a n_.

The eye _f_ and the eye _t_ are one and the same thing; but the eye
_f_ marks the distance, that is to say how far you are standing from
the object; and the eye _t_ shows you the direction of it; that is
whether you are opposite, or on one side, or at an angle to the
object you are looking at. And remember that the eye _f_ and the eye
_t_ must always be kept on the same level. For example if you raise
or lower the eye from the distance point _f_ you must do the same
with the direction point _t_. And if the point _f_ shows how far the
eye is distant from the square plane but does not show on which side
it is placed--and, if in the same way, the point _t_ show _s_ the
direction and not the distance, in order to ascertain both you must
use both points and they will be one and the same thing. If the eye
_f_ could see a perfect square of which all the sides were equal to
the distance between _s_ and _c_, and if at the nearest end of the
side towards the eye a pole were placed, or some other straight
object, set up by a perpendicular line as shown at _r s_--then, I
say, that if you were to look at the side of the square that is
nearest to you it will appear at the bottom of the vertical plane _r
s_, and then look at the farther side and it would appear to you at
the height of the point _n_ on the vertical plane. Thus, by this
example, you can understand that if the eye is above a number of
objects all placed on the same level, one beyond another, the more
remote they are the higher they will seem, up to the level of the
eye, but no higher; because objects placed upon the level on which
your feet stand, so long as it is flat--even if it be extended into
infinity--would never be seen above the eye; since the eye has in
itself the point towards which all the cones tend and converge which
convey the images of the objects to the eye. And this point always
coincides with the point of diminution which is the extreme of all
we can see. And from the base line of the first pyramid as far as
the diminishing point

[Footnote: The two diagrams above the chapter are explained by the
first five lines. They have, however, more letters than are referred
to in the text, a circumstance we frequently find occasion to
remark.]

56.

there are only bases without pyramids which constantly diminish up
to this point. And from the first base where the vertical plane is
placed towards the point in the eye there will be only pyramids
without bases; as shown in the example given above. Now, let _a b_
be the said vertical plane and _r_ the point of the pyramid
terminating in the eye, and _n_ the point of diminution which is
always in a straight line opposite the eye and always moves as the
eye moves--just as when a rod is moved its shadow moves, and moves
with it, precisely as the shadow moves with a body. And each point
is the apex of a pyramid, all having a common base with the
intervening vertical plane. But although their bases are equal their
angles are not equal, because the diminishing point is the
termination of a smaller angle than that of the eye. If you ask me:
"By what practical experience can you show me these points?" I
reply--so far as concerns the diminishing point which moves with you
--when you walk by a ploughed field look at the straight furrows
which come down with their ends to the path where you are walking,
and you will see that each pair of furrows will look as though they
tried to get nearer and meet at the [farther] end.

[Footnote: For the easier understanding of the diagram and of its
connection with the preceding I may here remark that the square
plane shown above in profile by the line _c s_ is here indicated by
_e d o p_. According to lines 1, 3 _a b_ must be imagined as a plane
of glass placed perpendicularly at _o p_.]

57.

How to measure the pyramid of vision.

As regards the point in the eye; it is made more intelligible by
this: If you look into the eye of another person you will see your
own image. Now imagine 2 lines starting from your ears and going to
the ears of that image which you see in the other man's eye; you
will understand that these lines converge in such a way that they
would meet in a point a little way beyond your own image mirrored in
the eye. And if you want to measure the diminution of the pyramid in
the air which occupies the space between the object seen and the
eye, you must do it according to the diagram figured below. Let _m
n_ be a tower, and _e f_ a, rod, which you must move backwards and
forwards till its ends correspond with those of the tower [Footnote
9: _I sua stremi .. della storre_ (its ends ... of the tower) this
is the case at _e f_.]; then bring it nearer to the eye, at _c d_
and you will see that the image of the tower seems smaller, as at _r
o_. Then [again] bring it closer to the eye and you will see the rod
project far beyond the image of the tower from _a_ to _b_ and from
_t_ to _b_, and so you will discern that, a little farther within,
the lines must converge in a point.

The Production of pyramid of Vision (58-60).

58.

PERSPECTIVE.

The instant the atmosphere is illuminated it will be filled with an
infinite number of images which are produced by the various bodies
and colours assembled in it. And the eye is the target, a loadstone,
of these images.

59.

The whole surface of opaque bodies displays its whole image in all
the illuminated atmosphere which surrounds them on all sides.

60.

That the atmosphere attracts to itself, like a loadstone, all the
images of the objects that exist in it, and not their forms merely
but their nature may be clearly seen by the sun, which is a hot and
luminous body. All the atmosphere, which is the all-pervading
matter, absorbs light and heat, and reflects in itself the image of
the source of that heat and splendour and, in each minutest portion,
does the same. The Northpole does the same as the loadstone shows;
and the moon and the other planets, without suffering any
diminution, do the same. Among terrestrial things musk does the same
and other perfumes.

61.

All bodies together, and each by itself, give off to the surrounding
air an infinite number of images which are all-pervading and each
complete, each conveying the nature, colour and form of the body
which produces it.

It can clearly be shown that all bodies are, by their images,
all-pervading in the surrounding atmosphere, and each complete in
itself as to substance form and colour; this is seen by the images
of the various bodies which are reproduced in one single perforation
through which they transmit the objects by lines which intersect and
cause reversed pyramids, from the objects, so that they are upside
down on the dark plane where they are first reflected. The reason of
this is--

[Footnote: The diagram intended to illustrate the statement (Pl. II
No. i) occurs in the original between lines 3 and 4. The three
circles must be understood to represent three luminous bodies which
transmit their images through perforations in a wall into a dark
chamber, according to a law which is more fully explained in 75?81.
So far as concerns the present passage the diagram is only intended
to explain that the images of the three bodies may be made to
coalesce at any given spot. In the circles are written,
giallo--yellow, biacho--white, rosso--red.

The text breaks off at line 8. The paragraph No.40 follows here in
the original MS.]

62.

Every point is the termination of an infinite number of lines, which
diverge to form a base, and immediately, from the base the same
lines converge to a pyramid [imaging] both the colour and form. No
sooner is a form created or compounded than suddenly infinite lines
and angles are produced from it; and these lines, distributing
themselves and intersecting each other in the air, give rise to an
infinite number of angles opposite to each other. Given a base, each
opposite angle, will form a triangle having a form and proportion
equal to the larger angle; and if the base goes twice into each of
the 2 lines of the pyramid the smaller triangle will do the same.

63.

Every body in light and shade fills the surrounding air with
infinite images of itself; and these, by infinite pyramids diffused
in the air, represent this body throughout space and on every side.
Each pyramid that is composed of a long assemblage of rays includes
within itself an infinite number of pyramids and each has the same
power as all, and all as each. A circle of equidistant pyramids of
vision will give to their object angles of equal size; and an eye at
each point will see the object of the same size. The body of the
atmosphere is full of infinite pyramids composed of radiating
straight lines, which are produced from the surface of the bodies in
light and shade, existing in the air; and the farther they are from
the object which produces them the more acute they become and
although in their distribution they intersect and cross they never
mingle together, but pass through all the surrounding air,
independently converging, spreading, and diffused. And they are all
of equal power [and value]; all equal to each, and each equal to
all. By these the images of objects are transmitted through all
space and in every direction, and each pyramid, in itself, includes,
in each minutest part, the whole form of the body causing it.

64.

The body of the atmosphere is full of infinite radiating pyramids
produced by the objects existing in it. These intersect and cross
each other with independent convergence without interfering with
each other and pass through all the surrounding atmosphere; and are
of equal force and value--all being equal to each, each to all. And
by means of these, images of the body are transmitted everywhere and
on all sides, and each receives in itself every minutest portion of
the object that produces it.

Proof by experiment (65-66).

65.

PERSPECTIVE.

The air is filled with endless images of the objects distributed in
it; and all are represented in all, and all in one, and all in each,
whence it happens that if two mirrors are placed in such a manner as
to face each other exactly, the first will be reflected in the
second and the second in the first. The first being reflected in the
second takes to it the image of itself with all the images
represented in it, among which is the image of the second mirror,
and so, image within image, they go on to infinity in such a manner
as that each mirror has within it a mirror, each smaller than the
last and one inside the other. Thus, by this example, it is clearly
proved that every object sends its image to every spot whence the
object itself can be seen; and the converse: That the same object
may receive in itself all the images of the objects that are in
front of it. Hence the eye transmits through the atmosphere its own
image to all the objects that are in front of it and receives them
into itself, that is to say on its surface, whence they are taken in
by the common sense, which considers them and if they are pleasing
commits them to the memory. Whence I am of opinion: That the
invisible images in the eyes are produced towards the object, as the
image of the object to the eye. That the images of the objects must
be disseminated through the air. An instance may be seen in several
mirrors placed in a circle, which will reflect each other endlessly.
When one has reached the other it is returned to the object that
produced it, and thence--being diminished--it is returned again to
the object and then comes back once more, and this happens
endlessly. If you put a light between two flat mirrors with a
distance of 1 braccio between them you will see in each of them an
infinite number of lights, one smaller than another, to the last. If
at night you put a light between the walls of a room, all the parts
of that wall will be tinted with the image of that light. And they
will receive the light and the light will fall on them, mutually,
that is to say, when there is no obstacle to interrupt the
transmission of the images. This same example is seen in a greater
degree in the distribution of the solar rays which all together, and
each by itself, convey to the object the image of the body which
causes it. That each body by itself alone fills with its images the
atmosphere around it, and that the same air is able, at the same
time, to receive the images of the endless other objects which are
in it, this is clearly proved by these examples. And every object is
everywhere visible in the whole of the atmosphere, and the whole in
every smallest part of it; and all the objects in the whole, and all
in each smallest part; each in all and all in every part.

66.

The images of objects are all diffused through the atmosphere which
receives them; and all on every side in it. To prove this, let _a c
e_ be objects of which the images are admitted to a dark chamber by
the small holes _n p_ and thrown upon the plane _f i_ opposite to
these holes. As many images will be produced in the chamber on the
plane as the number of the said holes.

67.

General conclusions.

All objects project their whole image and likeness, diffused and
mingled in the whole of the atmosphere, opposite to themselves. The
image of every point of the bodily surface, exists in every part of
the atmosphere. All the images of the objects are in every part of
the atmosphere. The whole, and each part of the image of the
atmosphere is [reflected] in each point of the surface of the bodies
presented to it. Therefore both the part and the whole of the images
of the objects exist, both in the whole and in the parts of the
surface of these visible bodies. Whence we may evidently say that
the image of each object exists, as a whole and in every part, in
each part and in the whole interchangeably in every existing body.
As is seen in two mirrors placed opposite to each other.

68.

That the contrary is impossible.

It is impossible that the eye should project from itself, by visual
rays, the visual virtue, since, as soon as it opens, that front
portion [of the eye] which would give rise to this emanation would
have to go forth to the object and this it could not do without
time. And this being so, it could not travel so high as the sun in a
month's time when the eye wanted to see it. And if it could reach
the sun it would necessarily follow that it should perpetually
remain in a continuous line from the eye to the sun and should
always diverge in such a way as to form between the sun and the eye
the base and the apex of a pyramid. This being the case, if the eye
consisted of a million worlds, it would not prevent its being
consumed in the projection of its virtue; and if this virtue would
have to travel through the air as perfumes do, the winds would bent
it and carry it into another place. But we do [in fact] see the mass
of the sun with the same rapidity as [an object] at the distance of
a braccio, and the power of sight is not disturbed by the blowing of
the winds nor by any other accident.

[Footnote: The view here refuted by Leonardo was maintained among
others by Bramantino, Leonardo's Milanese contemporary. LOMAZZO
writes as follows in his Trattato dell' Arte della pittura &c.
(Milano 1584. Libr. V cp. XXI): Sovviemmi di aver gia letto in certi
scritti alcune cose di Bramantino milanese, celebratissimo pittore,
attenente alla prospettiva, le quali ho voluto riferire, e quasi
intessere in questo luogo, affinche sappiamo qual fosse l'opinione
di cosi chiaro e famoso pittore intorno alla prospettiva . . Scrive
Bramantino che la prospettiva e una cosa che contrafa il naturale, e
che cio si fa in tre modi

Circa il primo modo che si fa con ragione, per essere la cosa in
poche parole conclusa da Bramantino in maniera che giudico non
potersi dir meglio, contenendovi si tutta Parte del principio al
fine, io riferiro per appunto le proprie parole sue (cp. XXII, Prima
prospettiva di Bramantino). La prima prospettiva fa le cose di
punto, e l'altra non mai, e la terza piu appresso. Adunque la prima
si dimanda prospettiva, cioe ragione, la quale fa l'effetto dell'
occhio, facendo crescere e calare secondo gli effetti degli occhi.
Questo crescere e calare non procede della cosa propria, che in se
per esser lontana, ovvero vicina, per quello effetto non puo
crescere e sminuire, ma procede dagli effetti degli occhi, i quali
sono piccioli, e percio volendo vedere tanto gran cosa_, bisogna che
mandino fuora la virtu visiva, _la quale si dilata in tanta
larghezza, che piglia tutto quello che vuoi vedere, ed_ arrivando a
quella cosa la vede dove e: _e da lei agli occhi per quello circuito
fino all' occhio, e tutto quello termine e pieno di quella cosa_.

It is worthy of note that Leonardo had made his memorandum refuting
this view, at Milan in 1492]

69.

A parallel case.

Just as a stone flung into the water becomes the centre and cause of
many circles, and as sound diffuses itself in circles in the air: so
any object, placed in the luminous atmosphere, diffuses itself in
circles, and fills the surrounding air with infinite images of
itself. And is repeated, the whole every-where, and the whole in
every smallest part. This can be proved by experiment, since if you
shut a window that faces west and make a hole [Footnote: 6. Here the
text breaks off.] . .

[Footnote: Compare LIBRI, _Histoire des sciences mathematiques en
Italie_. Tome III, p. 43.]

The function of the eye as explained by the camera obscura (70. 71).

70.

If the object in front of the eye sends its image to the eye, the
eye, on the other hand, sends its image to the object, and no
portion whatever of the object is lost in the images it throws off,
for any reason either in the eye or the object. Therefore we may
rather believe it to be the nature and potency of our luminous
atmosphere which absorbs the images of the objects existing in it,
than the nature of the objects, to send their images through the
air. If the object opposite to the eye were to send its image to the
eye, the eye would have to do the same to the object, whence it
might seem that these images were an emanation. But, if so, it would
be necessary [to admit] that every object became rapidly smaller;
because each object appears by its images in the surrounding
atmosphere. That is: the whole object in the whole atmosphere, and
in each part; and all the objects in the whole atmosphere and all of
them in each part; speaking of that atmosphere which is able to
contain in itself the straight and radiating lines of the images
projected by the objects. From this it seems necessary to admit that
it is in the nature of the atmosphere, which subsists between the
objects, and which attracts the images of things to itself like a
loadstone, being placed between them.

PROVE HOW ALL OBJECTS, PLACED IN ONE POSITION, ARE ALL EVERYWHERE
AND ALL IN EACH PART.

I say that if the front of a building--or any open piazza or
field--which is illuminated by the sun has a dwelling opposite to
it, and if, in the front which does not face the sun, you make a
small round hole, all the illuminated objects will project their
images through that hole and be visible inside the dwelling on the
opposite wall which may be made white; and there, in fact, they will
be upside down, and if you make similar openings in several places
in the same wall you will have the same result from each. Hence the
images of the illuminated objects are all everywhere on this wall
and all in each minutest part of it. The reason, as we clearly know,
is that this hole must admit some light to the said dwelling, and
the light admitted by it is derived from one or many luminous
bodies. If these bodies are of various colours and shapes the rays
forming the images are of various colours and shapes, and so will
the representations be on the wall.

[Footnote: 70. 15--23. This section has already been published in the
"_Saggio delle Opere di Leonardo da Vinci_" Milan 1872, pp. 13, 14.
G. Govi observes upon it, that Leonardo is not to be regarded as the
inventor of the Camera obscura, but that he was the first to explain
by it the structure of the eye. An account of the Camera obscura
first occurs in CESARE CESARINI's Italian version of Vitruvius, pub.
1523, four years after Leonardo's death. Cesarini expressly names
Benedettino Don Papnutio as the inventor of the Camera obscura. In
his explanation of the function of the eye by a comparison with the
Camera obscura Leonardo was the precursor of G. CARDANO, Professor
of Medicine at Bologna (died 1576) and it appears highly probable
that this is, in fact, the very discovery which Leonardo ascribes to
himself in section 21 without giving any further details.]

71.

HOW THE IMAGES OF OBJECTS RECEIVED BY THE EYE INTERSECT WITHIN THE
CRYSTALLINE HUMOUR OF THE EYE.

An experiment, showing how objects transmit their images or
pictures, intersecting within the eye in the crystalline humour, is
seen when by some small round hole penetrate the images of
illuminated objects into a very dark chamber. Then, receive these
images on a white paper placed within this dark room and rather near
to the hole and you will see all the objects on the paper in their
proper forms and colours, but much smaller; and they will be upside
down by reason of that very intersection. These images being
transmitted from a place illuminated by the sun will seem actually
painted on this paper which must be extremely thin and looked at
from behind. And let the little perforation be made in a very thin
plate of iron. Let _a b e d e_ be the object illuminated by the sun
and _o r_ the front of the dark chamber in which is the said hole at
_n m_. Let _s t_ be the sheet of paper intercepting the rays of the
images of these objects upside down, because the rays being
straight, _a_ on the right hand becomes _k_ on the left, and _e_ on
the left becomes _f_ on the right; and the same takes place inside
the pupil.

[Footnote: This chapter is already known through a translation into
French by VENTURI. Compare his '_Essai sur les ouvrages
physico-mathematiques de L. da Vinci avec des fragments tires de ses
Manuscrits, apportes de l'Italie. Lu a la premiere classe de
l'Institut national des Sciences et Arts.' Paris, An V_ (1797).]

The practice of perspective (72. 73).

72.

In the practice of perspective the same rules apply to light and to
the eye.

73.

The object which is opposite to the pupil of the eye is seen by that
pupil and that which is opposite to the eye is seen by the pupil.

Refraction of the rays falling upon the eye (74. 75)

74.

The lines sent forth by the image of an object to the eye do not
reach the point within the eye in straight lines.

75.

If the judgment of the eye is situated within it, the straight lines
of the images are refracted on its surface because they pass through
the rarer to the denser medium. If, when you are under water, you
look at objects in the air you will see them out of their true
place; and the same with objects under water seen from the air.

The intersection of the rays (76-82).

76.

The inversion of the images.

All the images of objects which pass through a window [glass pane]
from the free outer air to the air confined within walls, are seen
on the opposite side; and an object which moves in the outer air
from east to west will seem in its shadow, on the wall which is
lighted by this confined air, to have an opposite motion.

77.

THE PRINCIPLE ON WHICH THE IMAGES OF BODIES PASS IN BETWEEN THE
MARGINS OF THE OPENINGS BY WHICH THEY ENTER.

What difference is there in the way in which images pass through
narrow openings and through large openings, or in those which pass
by the sides of shaded bodies? By moving the edges of the opening
through which the images are admitted, the images of immovable
objects are made to move. And this happens, as is shown in the 9th
which demonstrates: [Footnote 11: _per la 9a che dicie_. When
Leonardo refers thus to a number it serves to indicate marginal
diagrams; this can in some instances be distinctly proved. The ninth
sketch on the page W. L. 145 b corresponds to the middle sketch of
the three reproduced.] the images of any object are all everywhere,
and all in each part of the surrounding air. It follows that if one
of the edges of the hole by which the images are admitted to a dark
chamber is moved it cuts off those rays of the image that were in
contact with it and gets nearer to other rays which previously were
remote from it &c.

OF THE MOVEMENT OF THE EDGE AT THE RIGHT OR LEFT, OR THE UPPER, OR
LOWER EDGE.

If you move the right side of the opening the image on the left will
move [being that] of the object which entered on the right side of
the opening; and the same result will happen with all the other
sides of the opening. This can be proved by the 2nd of this which
shows: all the rays which convey the images of objects through the
air are straight lines. Hence, if the images of very large bodies
have to pass through very small holes, and beyond these holes
recover their large size, the lines must necessarily intersect.

[Footnote: 77. 2. In the first of the three diagrams Leonardo had
drawn only one of the two margins, et _m_.]

78.

Necessity has provided that all the images of objects in front of
the eye shall intersect in two places. One of these intersections is
in the pupil, the other in the crystalline lens; and if this were
not the case the eye could not see so great a number of objects as
it does. This can be proved, since all the lines which intersect do
so in a point. Because nothing is seen of objects excepting their
surface; and their edges are lines, in contradistinction to the
definition of a surface. And each minute part of a line is equal to
a point; for _smallest_ is said of that than which nothing can be
smaller, and this definition is equivalent to the definition of the
point. Hence it is possible for the whole circumference of a circle
to transmit its image to the point of intersection, as is shown in
the 4th of this which shows: all the smallest parts of the images
cross each other without interfering with each other. These
demonstrations are to illustrate the eye. No image, even of the
smallest object, enters the eye without being turned upside down;
but as it penetrates into the crystalline lens it is once more
reversed and thus the image is restored to the same position within
the eye as that of the object outside the eye.

79.

OF THE CENTRAL LINE OF THE EYE.

Only one line of the image, of all those that reach the visual
virtue, has no intersection; and this has no sensible dimensions
because it is a mathematical line which originates from a
mathematical point, which has no dimensions.

According to my adversary, necessity requires that the central line
of every image that enters by small and narrow openings into a dark
chamber shall be turned upside down, together with the images of the
bodies that surround it.

80.

AS TO WHETHER THE CENTRAL LINE OF THE IMAGE CAN BE INTERSECTED, OR
NOT, WITHIN THE OPENING.

It is impossible that the line should intersect itself; that is,
that its right should cross over to its left side, and so, its left
side become its right side. Because such an intersection demands two
lines, one from each side; for there can be no motion from right to
left or from left to right in itself without such extension and
thickness as admit of such motion. And if there is extension it is
no longer a line but a surface, and we are investigating the
properties of a line, and not of a surface. And as the line, having
no centre of thickness cannot be divided, we must conclude that the
line can have no sides to intersect each other. This is proved by
the movement of the line _a f_ to _a b_ and of the line _e b_ to _e
f_, which are the sides of the surface _a f e b_. But if you move
the line _a b_ and the line _e f_, with the frontends _a e_, to the
spot _c_, you will have moved the opposite ends _f b_ towards each
other at the point _d_. And from the two lines you will have drawn
the straight line _c d_ which cuts the middle of the intersection of
these two lines at the point _n_ without any intersection. For, you
imagine these two lines as having breadth, it is evident that by
this motion the first will entirely cover the other--being equal
with it--without any intersection, in the position _c d_. And this
is sufficient to prove our proposition.

81.

HOW THE INNUMERABLE RAYS FROM INNUMERABLE IMAGES CAN CONVERGE TO A
POINT.

Just as all lines can meet at a point without interfering with each
other--being without breadth or thickness--in the same way all the
images of surfaces can meet there; and as each given point faces the
object opposite to it and each object faces an opposite point, the
converging rays of the image can pass through the point and diverge
again beyond it to reproduce and re-magnify the real size of that
image. But their impressions will appear reversed--as is shown in
the first, above; where it is said that every image intersects as it
enters the narrow openings made in a very thin substance.

Read the marginal text on the other side.

In proportion as the opening is smaller than the shaded body, so
much less will the images transmitted through this opening intersect
each other. The sides of images which pass through openings into a
dark room intersect at a point which is nearer to the opening in
proportion as the opening is narrower. To prove this let _a b_ be an
object in light and shade which sends not its shadow but the image
of its darkened form through the opening _d e_ which is as wide as
this shaded body; and its sides _a b_, being straight lines (as has
been proved) must intersect between the shaded object and the
opening; but nearer to the opening in proportion as it is smaller
than the object in shade. As is shown, on your right hand and your
left hand, in the two diagrams _a_ _b_ _c_ _n_ _m_ _o_ where, the
right opening _d_ _e_, being equal in width to the shaded object _a_
_b_, the intersection of the sides of the said shaded object occurs
half way between the opening and the shaded object at the point _c_.
But this cannot happen in the left hand figure, the opening _o_
being much smaller than the shaded object _n_ _m_.

It is impossible that the images of objects should be seen between
the objects and the openings through which the images of these
bodies are admitted; and this is plain, because where the atmosphere
is illuminated these images are not formed visibly.

When the images are made double by mutually crossing each other they
are invariably doubly as dark in tone. To prove this let _d_ _e_ _h_
be such a doubling which although it is only seen within the space
between the bodies in _b_ and _i_ this will not hinder its being
seen from _f_ _g_ or from _f_ _m_; being composed of the images _a_
_b_ _i_ _k_ which run together in _d_ _e_ _h_.

[Footnote: 81. On the original diagram at the beginning of this
chapter Leonardo has written "_azurro_" (blue) where in the
facsimile I have marked _A_, and "_giallo_" (yellow) where _B_
stands.]

[Footnote: 15--23. These lines stand between the diagrams I and III.]

[Footnote: 24--53. These lines stand between the diagrams I and II.]

[Footnote: 54--97 are written along the left side of diagram I.]

82.

An experiment showing that though the pupil may not be moved from
its position the objects seen by it may appear to move from their
places.

If you look at an object at some distance from you and which is
below the eye, and fix both your eyes upon it and with one hand
firmly hold the upper lid open while with the other you push up the
under lid--still keeping your eyes fixed on the object gazed at--you
will see that object double; one [image] remaining steady, and the
other moving in a contrary direction to the pressure of your finger
on the lower eyelid. How false the opinion is of those who say that
this happens because the pupil of the eye is displaced from its
position.

How the above mentioned facts prove that the pupil acts upside down
in seeing.

[Footnote: 82. 14--17. The subject indicated by these two headings is
fully discussed in the two chapters that follow them in the
original; but it did not seem to me appropriate to include them
here.]

Demostration of perspective by means of a vertical glass plane
(83-85).

83.

OF THE PLANE OF GLASS.

Perspective is nothing else than seeing place [or objects] behind a
plane of glass, quite transparent, on the surface of which the
objects behind that glass are to be drawn. These can be traced in
pyramids to the point in the eye, and these pyramids are intersected
on the glass plane.

84.

Pictorial perspective can never make an object at the same distance,
look of the same size as it appears to the eye. You see that the
apex of the pyramid _f c d_ is as far from the object _c_ _d_ as the
same point _f_ is from the object _a_ _b_; and yet _c_ _d_, which is
the base made by the painter's point, is smaller than _a_ _b_ which
is the base of the lines from the objects converging in the eye and
refracted at _s_ _t_, the surface of the eye. This may be proved by
experiment, by the lines of vision and then by the lines of the
painter's plumbline by cutting the real lines of vision on one and
the same plane and measuring on it one and the same object.

85.

PERSPECTIVE.

The vertical plane is a perpendicular line, imagined as in front of
the central point where the apex of the pyramids converge. And this
plane bears the same relation to this point as a plane of glass
would, through which you might see the various objects and draw them
on it. And the objects thus drawn would be smaller than the
originals, in proportion as the distance between the glass and the
eye was smaller than that between the glass and the objects.

PERSPECTIVE.

The different converging pyramids produced by the objects, will
show, on the plane, the various sizes and remoteness of the objects
causing them.

PERSPECTIVE.

All those horizontal planes of which the extremes are met by
perpendicular lines forming right angles, if they are of equal width
the more they rise to the level of eye the less this is seen, and
the more the eye is above them the more will their real width be
seen.

PERSPECTIVE.

The farther a spherical body is from the eye the more you will see
of it.

The angle of sight varies with the distance (86-88)

86.

A simple and natural method; showing how objects appear to the eye
without any other medium.

The object that is nearest to the eye always seems larger than
another of the same size at greater distance. The eye _m_, seeing
the spaces _o v x_, hardly detects the difference between them, and
the. reason of this is that it is close to them [Footnote 6: It is
quite inconceivable to me why M. RAVAISSON, in a note to his French
translation of this simple passage should have remarked: _Il est
clair que c'est par erreur que Leonard a ecrit_ per esser visino _au
lieu de_ per non esser visino. (See his printed ed. of MS. A. p.
38.)]; but if these spaces are marked on the vertical plane _n o_
the space _o v_ will be seen at _o r_, and in the same way the space
_v x_ will appear at _r q_. And if you carry this out in any place
where you can walk round, it will look out of proportion by reason
of the great difference in the spaces _o r_ and _r q_. And this
proceeds from the eye being so much below [near] the plane that the
plane is foreshortened. Hence, if you wanted to carry it out, you
would have [to arrange] to see the perspective through a single hole
which must be at the point _m_, or else you must go to a distance of
at least 3 times the height of the object you see. The plane _o p_
being always equally remote from the eye will reproduce the objects
in a satisfactory way, so that they may be seen from place to place.

87.

How every large mass sends forth its images, which may diminish
through infinity.

The images of any large mass being infinitely divisible may be
infinitely diminished.

88.

Objects of equal size, situated in various places, will be seen by
different pyramids which will each be smaller in proportion as the
object is farther off.

89.

Perspective, in dealing with distances, makes use of two opposite
pyramids, one of which has its apex in the eye and the base as
distant as the horizon. The other has the base towards the eye and
the apex on the horizon. Now, the first includes the [visible]
universe, embracing all the mass of the objects that lie in front of
the eye; as it might be a vast landscape seen through a very small
opening; for the more remote the objects are from the eye, the
greater number can be seen through the opening, and thus the pyramid
is constructed with the base on the horizon and the apex in the eye,
as has been said. The second pyramid is extended to a spot which is
smaller in proportion as it is farther from the eye; and this second
perspective [= pyramid] results from the first.

90.

SIMPLE PERSPECTIVE.

Simple perspective is that which is constructed by art on a vertical
plane which is equally distant from the eye in every part. Complex
perspective is that which is constructed on a ground-plan in which
none of the parts are equally distant from the eye.

91.

PERSPECTIVE.

No surface can be seen exactly as it is, if the eye that sees it is
not equally remote from all its edges.

92.

WHY WHEN AN OBJECT IS PLACED CLOSE TO THE EYE ITS EDGES ARE
INDISTINCT.

When an object opposite the eye is brought too close to it, its
edges must become too confused to be distinguished; as it happens
with objects close to a light, which cast a large and indistinct
shadow, so is it with an eye which estimates objects opposite to it;
in all cases of linear perspective, the eye acts in the same way as
the light. And the reason is that the eye has one leading line (of
vision) which dilates with distance and embraces with true
discernment large objects at a distance as well as small ones that
are close. But since the eye sends out a multitude of lines which
surround this chief central one and since these which are farthest
from the centre in this cone of lines are less able to discern with
accuracy, it follows that an object brought close to the eye is not
at a due distance, but is too near for the central line to be able
to discern the outlines of the object. So the edges fall within the
lines of weaker discerning power, and these are to the function of
the eye like dogs in the chase which can put up the game but cannot
take it. Thus these cannot take in the objects, but induce the
central line of sight to turn upon them, when they have put them up.
Hence the objects which are seen with these lines of sight have
confused outlines.

The relative size of objects with regard to their distance from the
eye (93-98).

93.

PERSPECTIVE.

Small objects close at hand and large ones at a distance, being seen
within equal angles, will appear of the same size.

94.

PERSPECTIVE.

There is no object so large but that at a great distance from the
eye it does not appear smaller than a smaller object near.

95.

Among objects of equal size that which is most remote from the eye
will look the smallest. [Footnote: This axiom, sufficiently clear in
itself, is in the original illustrated by a very large diagram,
constructed like that here reproduced under No. 108.

The same idea is repeated in C. A. I a; I a, stated as follows:
_Infra le cose d'equal grandeza quella si dimostra di minor figura
che sara piu distante dall' ochio_.--]

96.

Why an object is less distinct when brought near to the eye, and why
with spectacles, or without the naked eye sees badly either close or
far off [as the case may be].

97.

PERSPECTIVE.

Among objects of equal size, that which is most remote from the eye
will look the smallest.

98.

PERSPECTIVE.

No second object can be so much lower than the first as that the eye
will not see it higher than the first, if the eye is above the
second.

PERSPECTIVE.

And this second object will never be so much higher than the first
as that the eye, being below them, will not see the second as lower
than the first.

PERSPECTIVE.

If the eye sees a second square through the centre of a smaller one,
that is nearer, the second, larger square will appear to be
surrounded by the smaller one.

PERSPECTIVE--PROPOSITION.

Objects that are farther off can never be so large but that those in
front, though smaller, will conceal or surround them.

DEFINITION.

This proposition can be proved by experiment. For if you look
through a small hole there is nothing so large that it cannot be
seen through it and the object so seen appears surrounded and
enclosed by the outline of the sides of the hole. And if you stop it
up, this small stopping will conceal the view of the largest object.

The apparent size of objects defined by calculation (99-105)

99.

OF LINEAR PERSPECTIVE.

Linear Perspective deals with the action of the lines of sight, in
proving by measurement how much smaller is a second object than the
first, and how much the third is smaller than the second; and so on
by degrees to the end of things visible. I find by experience that
if a second object is as far beyond the first as the first is from
the eye, although they are of the same size, the second will seem
half the size of the first and if the third object is of the same
size as the 2nd, and the 3rd is as far beyond the second as the 2nd
from the first, it will appear of half the size of the second; and
so on by degrees, at equal distances, the next farthest will be half
the size of the former object. So long as the space does not exceed
the length of 20 braccia. But, beyond 20 braccia figures of equal
size will lose 2/4 and at 40 braccia they will lose 9/10, and 19/20
at 60 braccia, and so on diminishing by degrees. This is if the
picture plane is distant from you twice your own height. If it is
only as far off as your own height, there will be a great difference
between the first braccia and the second.

[Footnote: This chapter is included in DUFRESNE'S and MANZI'S
editions of the Treatise on Painting. H. LUDWIG, in his commentary,
calls this chapter "_eines der wichtigsten im ganzen Tractat_", but
at the same time he asserts that its substance has been so
completely disfigured in the best MS. copies that we ought not to
regard Leonardo as responsible for it. However, in the case of this
chapter, the old MS. copies agree with the original as it is
reproduced above. From the chapters given later in this edition,
which were written at a subsequent date, it would appear that
Leonardo corrected himself on these points.]

100.

OF THE DIMINUTION OF OBJECTS AT VARIOUS DISTANCES.

A second object as far distant from the first as the first is from
the eye will appear half the size of the first, though they be of
the same size really.

OF THE DEGREES OF DIMINUTION.

If you place the vertical plane at one braccio from the eye, the
first object, being at a distance of 4 braccia from your eye will
diminish to 3/4 of its height at that plane; and if it is 8 braccia
from the eye, to 7/8; and if it is 16 braccia off, it will diminish
to 15/16 of its height and so on by degrees, as the space doubles
the diminution will double.

101.

Begin from the line _m f_ with the eye below; then go up and do the
same with the line _n f_, then with the eye above and close to the 2
gauges on the ground look at _m n_; then as _c m_ is to _m n_ so
will _n m_ be to _n s_.

If _a n_ goes 3 times into _f b, m p_ will do the same into _p g_.
Then go backwards so far as that _c d_ goes twice into _a n_ and _p
g_ will be equal to _g h_. And _m p_ will go into _h p_ as often as
_d c_ into _o p_.

[Footnote: The first three lines are unfortunately very obscure.]

102.

I GIVE THE DEGREES OF THE OBJECTS SEEN BY THE EYE AS THE MUSICIAN
DOES THE NOTES HEARD BY THE EAR.

Although the objects seen by the eye do, in fact, touch each other
as they recede, I will nevertheless found my rule on spaces of 20
braccia each; as a musician does with notes, which, though they can
be carried on one into the next, he divides into degrees from note
to note calling them 1st, 2nd, 3rd, 4th, 5th; and has affixed a name
to each degree in raising or lowering the voice.

103.

PERSPECTIVE.

Let _f_ be the level and distance of the eye; and _a_ the vertical
plane, as high as a man; let _e_ be a man, then I say that on the
plane this will be the distance from the plane to the 2nd man.

104.

The differences in the diminution of objects of equal size in
consequence of their various remoteness from the eye will bear among
themselves the same proportions as those of the spaces between the
eye and the different objects.

Find out how much a man diminishes at a certain distance and what
its length is; and then at twice that distance and at 3 times, and
so make your general rule.

105.

The eye cannot judge where an object high up ought to descend.

106.

PERSPECTIVE.

If two similar and equal objects are placed one beyond the other at
a given distance the difference in their size will appear greater in
proportion as they are nearer to the eye that sees them. And
conversely there will seem to be less difference in their size in
proportion as they are remote from the eve.

This is proved by the proportions of their distances among
themselves; for, if the first of these two objects were as far from
the eye, as the 2nd from the first this would be called the second
proportion: since, if the first is at 1 braccia from the eye and the
2nd at two braccia, two being twice as much as one, the first object
will look twice as large as the second. But if you place the first
at a hundred braccia from you and the second at a hundred and one,
you will find that the first is only so much larger than the second
as 100 is less than 101; and the converse is equally true. And
again, the same thing is proved by the 4th of this book which shows
that among objects that are equal, there is the same proportion in
the diminution of the size as in the increase in the distance from
the eye of the spectator.

On natural perspective (107--109).

107.

OF EQUAL OBJECTS THE MOST REMOTE LOOK THE SMALLEST.

The practice of perspective may be divided into ... parts [Footnote
4: _in_ ... _parte_. The space for the number is left blank in the
original.], of which the first treats of objects seen by the eye at
any distance; and it shows all these objects just as the eye sees
them diminished, without obliging a man to stand in one place rather
than another so long as the plane does not produce a second
foreshortening.

But the second practice is a combination of perspective derived
partly from art and partly from nature and the work done by its
rules is in every portion of it, influenced by natural perspective
and artificial perspective. By natural perspective I mean that the
plane on which this perspective is represented is a flat surface,
and this plane, although it is parallel both in length and height,
is forced to diminish in its remoter parts more than in its nearer
ones. And this is proved by the first of what has been said above,
and its diminution is natural. But artificial perspective, that is
that which is devised by art, does the contrary; for objects equal
in size increase on the plane where it is foreshortened in
proportion as the eye is more natural and nearer to the plane, and
as the part of the plane on which it is figured is farther from the
eye.

And let this plane be _d e_ on which are seen 3 equal circles which
are beyond this plane _d e_, that is the circles _a b c_. Now you
see that the eye _h_ sees on the vertical plane the sections of the
images, largest of those that are farthest and smallest of the
nearest.

108.

Here follows what is wanting in the margin at the foot on the other
side of this page.

Natural perspective acts in a contrary way; for, at greater
distances the object seen appears smaller, and at a smaller distance
the object appears larger. But this said invention requires the
spectator to stand with his eye at a small hole and then, at that
small hole, it will be very plain. But since many (men's) eyes
endeavour at the same time to see one and the same picture produced
by this artifice only one can see clearly the effect of this
perspective and all the others will see confusion. It is well
therefore to avoid such complex perspective and hold to simple
perspective which does not regard planes as foreshortened, but as
much as possible in their proper form. This simple perspective, in
which the plane intersects the pyramids by which the images are
conveyed to the eye at an equal distance from the eye is our
constant experience, from the curved form of the pupil of the eye on
which the pyramids are intersected at an equal distance from the
visual virtue.

[Footnote 24: _la prima di sopra_ i. e. the first of the three
diagrams which, in the original MS., are placed in the margin at the
beginning of this chapter.]

109.

OF A MIXTURE OF NATURAL AND ARTIFICIAL PERSPECTIVE.

This diagram distinguishes natural from artificial perspective. But
before proceeding any farther I will define what is natural and what
is artificial perspective. Natural perspective says that the more
remote of a series of objects of equal size will look the smaller,
and conversely, the nearer will look the larger and the apparent
size will diminish in proportion to the distance. But in artificial
perspective when objects of unequal size are placed at various
distances, the smallest is nearer to the eye than the largest and
the greatest distance looks as though it were the least of all; and
the cause of this is the plane on which the objects are represented;
and which is at unequal distances from the eye throughout its
length. And this diminution of the plane is natural, but the
perspective shown upon it is artificial since it nowhere agrees with
the true diminution of the said plane. Whence it follows, that when
the eye is somewhat removed from the [station point of the]
perspective that it has been gazing at, all the objects represented
look monstrous, and this does not occur in natural perspective,
which has been defined above. Let us say then, that the square _a b
c d_ figured above is foreshortened being seen by the eye situated
in the centre of the side which is in front. But a mixture of
artificial and natural perspective will be seen in this tetragon
called _el main_ [Footnote 20: _el main_ is quite legibly written in
the original; the meaning and derivation of the word are equally
doubtful.], that is to say _e f g h_ which must appear to the eye of
the spectator to be equal to _a b c d_ so long as the eye remains in
its first position between _c_ and _d_. And this will be seen to
have a good effect, because the natural perspective of the plane
will conceal the defects which would [otherwise] seem monstrous.

_III._

_Six books on Light and Shade._

_Linear Perspective cannot be immediately followed by either the_
"prospettiva de' perdimenti" _or the_ "prospettiva de' colori" _or
the aerial perspective; since these branches of the subject
presuppose a knowledge of the principles of Light and Shade. No
apology, therefore, is here needed for placing these immediately
after Linear Perspective._

_We have various plans suggested by Leonardo for the arrangement of
the mass of materials treating of this subject. Among these I have
given the preference to a scheme propounded in No._ III, _because,
in all probability, we have here a final and definite purpose
expressed. Several authors have expressed it as their opinion that
the Paris Manuscript_ C _is a complete and finished treatise on
Light and Shade. Certainly, the Principles of Light and Shade form
by far the larger portion of this MS. which consists of two separate
parts; still, the materials are far from being finally arranged. It
is also evident that he here investigates the subject from the point
of view of the Physicist rather than from that of the Painter._

_The plan of a scheme of arrangement suggested in No._ III _and
adopted by me has been strictly adhered to for the first four Books.
For the three last, however, few materials have come down to us; and
it must be admitted that these three Books would find a far more
appropriate place in a work on Physics than in a treatise on
Painting. For this reason I have collected in Book V all the
chapters on Reflections, and in Book VI I have put together and
arranged all the sections of MS._ C _that belong to the book on
Painting, so far as they relate to Light and Shade, while the
sections of the same MS. which treat of the_ "Prospettiva de'
perdimenti" _have, of course, been excluded from the series on Light
and Shade._

[Footnote III: This text has already been published with some slight
variations in Dozio's pamphlet _Degli scritti e disegni di Leonardo
da Vinci_, Milan 1871, pp. 30--31. Dozio did not transcribe it from
the original MS. which seems to have remained unknown to him, but
from an old copy (MS. H. 227 in the Ambrosian Library).]

GENERAL INTRODUCTION.

Prolegomena.

110.

You must first explain the theory and then the practice. First you
must describe the shadows and lights on opaque objects, and then on
transparent bodies.

Scheme of the books on Light and shade.

111.

INTRODUCTION.

[Having already treated of the nature of shadows and the way in
which they are cast [Footnote 2: _Avendo io tractato._--We may
suppose that he here refers to some particular MS., possibly Paris
C.], I will now consider the places on which they fall; and their
curvature, obliquity, flatness or, in short, any character I may be
able to detect in them.]

Shadow is the obstruction of light. Shadows appear to me to be of
supreme importance in perspective, because, without them opaque and
solid bodies will be ill defined; that which is contained within
their outlines and their boundaries themselves will be
ill-understood unless they are shown against a background of a
different tone from themselves. And therefore in my first
proposition concerning shadow I state that every opaque body is
surrounded and its whole surface enveloped in shadow and light. And
on this proposition I build up the first Book. Besides this, shadows
have in themselves various degrees of darkness, because they are
caused by the absence of a variable amount of the luminous rays; and
these I call Primary shadows because they are the first, and
inseparable from the object to which they belong. And on this I will
found my second Book. From these primary shadows there result
certain shaded rays which are diffused through the atmosphere and
these vary in character according to that of the primary shadows
whence they are derived. I shall therefore call these shadows
Derived shadows because they are produced by other shadows; and the
third Book will treat of these. Again these derived shadows, where
they are intercepted by various objects, produce effects as various
as the places where they are cast and of this I will treat in the
fourth Book. And since all round the derived shadows, where the
derived shadows are intercepted, there is always a space where the
light falls and by reflected dispersion is thrown back towards its
cause, it meets the original shadow and mingles with it and modifies
it somewhat in its nature; and on this I will compose my fifth Book.
Besides this, in the sixth Book I will investigate the many and
various diversities of reflections resulting from these rays which
will modify the original [shadow] by [imparting] some of the various
colours from the different objects whence these reflected rays are
derived. Again, the seventh Book will treat of the various distances
that may exist between the spot where the reflected rays fall and
that where they originate, and the various shades of colour which
they will acquire in falling on opaque bodies.

Different principles and plans of treatment (112--116).

112.

First I will treat of light falling through windows which I will
call Restricted [Light] and then I will treat of light in the open
country, to which I will give the name of diffused Light. Then I
will treat of the light of luminous bodies.

113.

OF PAINTING.

The conditions of shadow and light [as seen] by the eye are 3. Of
these the first is when the eye and the light are on the same side
of the object seen; the 2nd is when the eye is in front of the
object and the light is behind it. The 3rd is when the eye is in
front of the object and the light is on one side, in such a way as
that a line drawn from the object to the eye and one from the object
to the light should form a right angle where they meet.

114.

OF PAINTING.

This is another section: that is, of the nature of a reflection
(from) an object placed between the eye and the light under various
aspects.

115.

OF PAINTING.

As regards all visible objects 3 things must be considered. These
are the position of the eye which sees: that of the object seen
[with regard] to the light, and the position of the light which
illuminates the object, _b_ is the eye, _a_ the object seen, _c_ the
light, _a_ is the eye, _b_ the illuminating body, _c_ is the
illuminated object.

116.

Let _a_ be the light, _b_ the eye, _c_ the object seen by the eye
and in the light. These show, first, the eye between the light and
the body; the 2nd, the light between the eye and the body; the 3rd
the body between the eye and the light, _a_ is the eye, _b_ the
illuminated object, _c_ the light.

117.

OF PAINTING.

OF THE THREE KINDS OF LIGHT THAT ILLUMINATE OPAQUE BODIES.

The first kind of Light which may illuminate opaque bodies is called
Direct light--as that of the sun or any other light from a window or
flame. The second is Diffused [universal] light, such as we see in
cloudy weather or in mist and the like. The 3rd is Subdued light,
that is when the sun is entirely below the horizon, either in the
evening or morning.

118.

OF LIGHT.

The lights which may illuminate opaque bodies are of 4 kinds. These
are: diffused light as that of the atmosphere, within our horizon.
And Direct, as that of the sun, or of a window or door or other
opening. The third is Reflected light; and there is a 4th which is
that which passes through [semi] transparent bodies, as linen or
paper or the like, but not transparent like glass, or crystal, or
other diaphanous bodies, which produce the same effect as though
nothing intervened between the shaded object and the light that
falls upon it; and this we will discuss fully in our discourse.

Definition of the nature of shadows (119--122).

119.

WHAT LIGHT AND SHADOW ARE.

Shadow is the absence of light, merely the obstruction of the
luminous rays by an opaque body. Shadow is of the nature of
darkness. Light [on an object] is of the nature of a luminous body;
one conceals and the other reveals. They are always associated and
inseparable from all objects. But shadow is a more powerful agent
than light, for it can impede and entirely deprive bodies of their
light, while light can never entirely expel shadow from a body, that
is from an opaque body.

120.

Shadow is the diminution of light by the intervention of an opaque
body. Shadow is the counterpart of the luminous rays which are cut
off by an opaque body.

This is proved because the shadow cast is the same in shape and size
as the luminous rays were which are transformed into a shadow.

121.

Shadow is the diminution alike of light and of darkness, and stands
between darkness and light.

A shadow may be infinitely dark, and also of infinite degrees of
absence of darkness.

The beginnings and ends of shadow lie between the light and darkness
and may be infinitely diminished and infinitely increased. Shadow is
the means by which bodies display their form.

The forms of bodies could not be understood in detail but for
shadow.

122.

OF THE NATURE OF SHADOW.

Shadow partakes of the nature of universal matter. All such matters
are more powerful in their beginning and grow weaker towards the
end, I say at the beginning, whatever their form or condition may be
and whether visible or invisible. And it is not from small
beginnings that they grow to a great size in time; as it might be a
great oak which has a feeble beginning from a small acorn. Yet I may
say that the oak is most powerful at its beginning, that is where it
springs from the earth, which is where it is largest (To return:)
Darkness, then, is the strongest degree of shadow and light is its
least. Therefore, O Painter, make your shadow darkest close to the
object that casts it, and make the end of it fading into light,
seeming to have no end.

Of the various kinds of shadows. (123-125).

123.

Darkness is absence of light. Shadow is diminution of light.
Primitive shadow is that which is inseparable from a body not in the
light. Derived shadow is that which is disengaged from a body in
shadow and pervades the air. A cast transparent shadow is that which
is surrounded by an illuminated surface. A simple shadow is one
which receives no light from the luminous body which causes it. A
simple shadow begins within the line which starts from the edge of
the luminous body _a b_.

124.

A simple shadow is one where no light at all interferes with it.

A compound shadow is one which is somewhat illuminated by one or
more lights.

125.

WHAT IS THE DIFFERENCE BETWEEN A SHADOW THAT IS INSEPARABLE FROM A
BODY AND A CAST SHADOW?

An inseparable shadow is that which is never absent from the
illuminated body. As, for instance a ball, which so long as it is in
the light always has one side in shadow which never leaves it for
any movement or change of position in the ball. A separate shadow
may be and may not be produced by the body itself. Suppose the ball
to be one braccia distant from a wall with a light on the opposite
side of it; this light will throw upon the wall exactly as broad a
shadow as is to be seen on the side of the ball that is turned
towards the wall. That portion of the cast shadow will not be
visible when the light is below the ball and the shadow is thrown up
towards the sky and finding no obstruction on its way is lost.

126.

HOW THERE ARE 2 KINDS OF LIGHT, ONE SEPARABLE FROM, AND THE OTHER
INSEPARABLE FROM BODIES.

Of the various kinds of light (126, 127).

Separate light is that which falls upon the body. Inseparable light
is the side of the body that is illuminated by that light. One is
called primary, the other derived. And, in the same way there are
two kinds of shadow:--One primary and the other derived. The primary
is that which is inseparable from the body, the derived is that
which proceeds from the body conveying to the surface of the wall
the form of the body causing it.

127.

How there are 2 different kinds of light; one being called diffused,
the other restricted. The diffused is that which freely illuminates
objects. The restricted is that which being admitted through an
opening or window illuminates them on that side only.

[Footnote: At the spot marked _A_ in the first diagram Leonardo
wrote _lume costretto_ (restricted light). At the spot _B_ on the
second diagram he wrote _lume libero_ (diffused light).]

General remarks (128. 129).

128.

Light is the chaser away of darkness. Shade is the obstruction of
light. Primary light is that which falls on objects and causes light
and shade. And derived lights are those portions of a body which are
illuminated by the primary light. A primary shadow is that side of a
body on which the light cannot fall.

The general distribution of shadow and light is that sum total of
the rays thrown off by a shaded or illuminated body passing through
the air without any interference and the spot which intercepts and
cuts off the distribution of the dark and light rays.

And the eye can best distinguish the forms of objects when it is
placed between the shaded and the illuminated parts.

129.

MEMORANDUM OF THINGS I REQUIRE TO HAVE GRANTED [AS AXIOMS] IN MY
EXPLANATION OF PERSPECTIVE.

I ask to have this much granted me--to assert that every ray
passing through air of equal density throughout, travels in a
straight line from its cause to the object or place it falls upon.

FIRST BOOK ON LIGHT AND SHADE.

On the nature of light (130. 131).

130.

The reason by which we know that a light radiates from a single
centre is this: We plainly see that a large light is often much
broader than some small object which nevertheless--and although the
rays [of the large light] are much more than twice the extent [of
the small body]--always has its shadow cast on the nearest surface
very visibly. Let _c f_ be a broad light and _n_ be the object in
front of it, casting a shadow on the plane, and let _a b_ be the
plane. It is clear that it is not the broad light that will cast the
shadow _n_ on the plane, but that the light has within it a centre
is shown by this experiment. The shadow falls on the plane as is
shown at _m o t r_.

[Footnote 13: In the original MS. no explanatory text is placed
after this title-line; but a space is left for it and the text
beginning at line 15 comes next.] Why, to two [eyes] or in front of
two eyes do 3 objects appear as two?

Why, when you estimate the direction of an object with two sights
the nearer appears confused. I say that the eye projects an infinite
number of lines which mingle or join those reaching it which come to
it from the object looked at. And it is only the central and
sensible line that can discern and discriminate colours and objects;
all the others are false and illusory. And if you place 2 objects at
half an arm's length apart if the nearer of the two is close to the
eye its form will remain far more confused than that of the second;
the reason is that the first is overcome by a greater number of
false lines than the second and so is rendered vague.

Light acts in the same manner, for in the effects of its lines
(=rays), and particularly in perspective, it much resembles the eye;
and its central rays are what cast the true shadow. When the object
in front of it is too quickly overcome with dim rays it will cast a
broad and disproportionate shadow, ill defined; but when the object
which is to cast the shadow and cuts off the rays near to the place
where the shadow falls, then the shadow is distinct; and the more so
in proportion as the light is far off, because at a long distance
the central ray is less overcome by false rays; because the lines
from the eye and the solar and other luminous rays passing through
the atmosphere are obliged to travel in straight lines. Unless they
are deflected by a denser or rarer air, when they will be bent at
some point, but so long as the air is free from grossness or
moisture they will preserve their direct course, always carrying the
image of the object that intercepts them back to their point of
origin. And if this is the eye, the intercepting object will be seen
by its colour, as well as by form and size. But if the intercepting
plane has in it some small perforation opening into a darker
chamber--not darker in colour, but by absence of light--you will see
the rays enter through this hole and transmitting to the plane
beyond all the details of the object they proceed from both as to
colour and form; only every thing will be upside down. But the size
[of the image] where the lines are reconstructed will be in
proportion to the relative distance of the aperture from the plane
on which the lines fall [on one hand] and from their origin [on the
other]. There they intersect and form 2 pyramids with their point
meeting [a common apex] and their bases opposite. Let _a b_ be the
point of origin of the lines, _d e_ the first plane, and _c_ the
aperture with the intersection of the lines; _f g_ is the inner
plane. You will find that _a_ falls upon the inner plane below at
_g_, and _b_ which is below will go up to the spot _f_; it will be
quite evident to experimenters that every luminous body has in
itself a core or centre, from which and to which all the lines
radiate which are sent forth by the surface of the luminous body and
reflected back to it; or which, having been thrown out and not
intercepted, are dispersed in the air.

131.

THE RAYS WHETHER SHADED OR LUMINOUS HAVE GREATER STRENGTH AND EFFECT
AT THEIR POINTS THAN AT THEIR SIDES.

Although the points of luminous pyramids may extend into shaded
places and those of pyramids of shadow into illuminated places, and
though among the luminous pyramids one may start from a broader base
than another; nevertheless, if by reason of their various length
these luminous pyramids acquire angles of equal size their light
will be equal; and the case will be the same with the pyramids of
shadow; as may be seen in the intersected pyramids _a b c_ and _d e
f_, which though their bases differ in size are equal as to breadth
and light.

[Footnote: 51--55: This supplementary paragraph is indicated as being
a continuation of line 45, by two small crosses.]

The difference between light and lustre (132--135).

132.

Of the difference between light and lustre; and that lustre is not
included among colours, but is saturation of whiteness, and derived
from the surface of wet bodies; light partakes of the colour of the
object which reflects it (to the eye) as gold or silver or the like.

133.

OF THE HIGHEST LIGHTS WHICH TURN AND MOVE AS THE EYE MOVES WHICH
SEES THE OBJECT.

Suppose the body to be the round object figured here and let the
light be at the point _a_, and let the illuminated side of the
object be _b c_ and the eye at the point _d_: I say that, as lustre
is every where and complete in each part, if you stand at the point
_d_ the lustre will appear at _c_, and in proportion as the eye
moves from _d_ to _a_, the lustre will move from _c_ to _n_.

134.

OF PAINTING.

Heigh light or lustre on any object is not situated [necessarily] in
the middle of an illuminated object, but moves as and where the eye
moves in looking at it.

135.

OF LIGHT AND LUSTRE.

What is the difference between light and the lustre which is seen on
the polished surface of opaque bodies?

The lights which are produced from the polished surface of opaque
bodies will be stationary on stationary objects even if the eye on
which they strike moves. But reflected lights will, on those same
objects, appear in as many different places on the surface as
different positions are taken by the eye.

WHAT BODIES HAVE LIGHT UPON THEM WITHOUT LUSTRE?

Opaque bodies which have a hard and rough surface never display any
lustre in any portion of the side on which the light falls.

WHAT BODIES WILL DISPLAY LUSTRE BUT NOT LOOK ILLUMINATED?

Those bodies which are opaque and hard with a hard surface reflect
light [lustre] from every spot on the illuminated side which is in a
position to receive light at the same angle of incidence as they
occupy with regard to the eye; but, as the surface mirrors all the
surrounding objects, the illuminated [body] is not recognisable in
these portions of the illuminated body.

136.

The relations of luminous to illuminated bodies.

The middle of the light and shade on an object in light and shade is
opposite to the middle of the primary light. All light and shadow
expresses itself in pyramidal lines. The middle of the shadow on any
object must necessarily be opposite the middle of its light, with a
direct line passing through the centre of the body. The middle of
the light will be at _a_, that of the shadow at _b_. [Again, in
bodies shown in light and shade the middle of each must coincide
with the centre of the body, and a straight line will pass through
both and through that centre.]

[Footnote: In the original MS., at the spot marked _a_ of the first
diagram Leonardo wrote _primitiuo_, and at the spot marked
_c_--_primitiva_ (primary); at the spot marked _b_ he wrote
_dirivatiuo_ and at _d deriuatiua_ (derived).]

Experiments on the relation of light and shadow within a room
(137--140).

137.

SHOWS HOW LIGHT FROM ANY SIDE CONVERGES TO ONE POINT.

Although the balls _a b c_ are lighted from one window,
nevertheless, if you follow the lines of their shadows you will see
they intersect at a point forming the angle _n_.

[Footnote: The diagram belonging to this passage is slightly
sketched on Pl. XXXII; a square with three balls below it. The first
three lines of the text belonging to it are written above the sketch
and the six others below it.]

138.

Every shadow cast by a body has a central line directed to a single
point produced by the intersection of luminous lines in the middle
of the opening and thickness of the window. The proposition stated
above, is plainly seen by experiment. Thus if you draw a place with
a window looking northwards, and let this be _s f_, you will see a
line starting from the horizon to the east, which, touching the 2
angles of the window _o f_, reaches _d_; and from the horizon on the
west another line, touching the other 2 angles _r s_, and ending at
_c_; and their intersection falls exactly in the middle of the
opening and thickness of the window. Again, you can still better
confirm this proof by placing two sticks, as shown at _g h_; and you
will see the line drawn from the centre of the shadow directed to
the centre _m_ and prolonged to the horizon _n f_.

[Footnote: _B_ here stands for _cerchio del' orizonte tramontano_ on
the original diagram (the circle of the horizon towards the North);
_A_ for _levante_ (East) and _C_ for _ponete_ (West).]

139.

Every shadow with all its variations, which becomes larger as its
distance from the object is greater, has its external lines
intersecting in the middle, between the light and the object. This
proposition is very evident and is confirmed by experience. For, if
_a b_ is a window without any object interposed, the luminous
atmosphere to the right hand at _a_ is seen to the left at _d_. And
the atmosphere at the left illuminates on the right at _c_, and the
lines intersect at the point _m_.

[Footnote: _A_ here stands for _levante_ (East), _B_ for _ponente_
(West).]

140.

Every body in light and shade is situated between 2 pyramids one
dark and the other luminous, one is visible the other is not. But
this only happens when the light enters by a window. Supposing _a b_
to be the window and _r_ the body in light and shade, the light to
the right hand _z_ will pass the object to the left and go on to
_p_; the light to the left at _k_ will pass to the right of the
object at _i_ and go on to _m_ and the two lines will intersect at
_c_ and form a pyramid. Then again _a_ _b_ falls on the shaded body
at _i_ _g_ and forms a pyramid _f_ _i_ _g_. _f_ will be dark because
the light _a_ _b_ can never fall there; _i_ _g_ _c_ will be
illuminated because the light falls upon it.

Light and shadow with regard to the position of the eye (141--145).

141.

Every shaded body that is larger than the pupil and that interposes
between the luminous body and the eye will be seen dark.

When the eye is placed between the luminous body and the objects
illuminated by it, these objects will be seen without any shadow.

[Footnote: The diagram which in the original stands above line 1 is
given on Plate II, No 2. Then, after a blank space of about eight
lines, the diagram Plate II No 3 is placed in the original. There is
no explanation of it beyond the one line written under it.]

142.

Why the 2 lights one on each side of a body having two pyramidal
sides of an obtuse apex leave it devoid of shadow.

[Footnote: The sketch illustrating this is on Plate XLI No 1.]

143.

A body in shadow situated between the light and the eye can never
display its illuminated portion unless the eye can see the whole of
the primary light.

[Footnote: _A_ stands for _corpo_ (body), _B_ for _lume_ (light).]

144.

The eye which looks (at a spot) half way between the shadow and the
light which surrounds the body in shadow will see that the deepest
shadows on that body will meet the eye at equal angles, that is at
the same angle as that of sight.

[Footnote: In both these diagrams _A_ stands for _lume_ (light) _B_
for _ombra_ (shadow).]

145.

OF THE DIFFERENT LIGHT AND SHADE IN VARIOUS ASPECTS AND OF OBJECTS
PLACED IN THEM.

If the sun is in the East and you look towards the West you will see
every thing in full light and totally without shadow because you see
them from the same side as the sun: and if you look towards the
South or North you will see all objects in light and shade, because
you see both the side towards the sun and the side away from it; and
if you look towards the coming of the sun all objects will show you
their shaded side, because on that side the sun cannot fall upon
them.

The law of the incidence of light.

146.

The edges of a window which are illuminated by 2 lights of equal
degrees of brightness will not reflect light of equal brightness
into the chamber within.

If _b_ is a candle and _a c_ our hemisphere both will illuminate the
edges of the window _m_ _n_, but light _b_ will only illuminate _f
g_ and the hemisphere _a_ will light all of _d e_.

147.

OF PAINTING.

That part of a body which receives the luminous rays at equal angles
will be in a higher light than any other part of it.

And the part which the luminous rays strike between less equal
angles will be less strongly illuminated.

SECOND BOOK ON LIGHT AND SHADE.

Gradations of strength in the shadows (148. 149).

148.

THAT PORTION OF A BODY IN LIGHT AND SHADE WILL BE LEAST LUMINOUS
WHICH IS SEEN UNDER THE LEAST AMOUNT OF LIGHT.

That part of the object which is marked _m_ is in the highest light
because it faces the window _a d_ by the line _a f_; _n_ is in the
second grade because the light _b d_ strikes it by the line _b e_;
_o_ is in the third grade, as the light falls on it from _c d_ by
the line _c h_; _p_ is the lowest light but one as _c d_ falls on it
by the line _d v_; _q_ is the deepest shadow for no light falls on
it from any part of the window.

In proportion as _c d_ goes into _a d_ so will _n r s_ be darker
than _m_, and all the rest is space without shadow.

[Footnote: The diagram belonging to this chapter is No. 1 on Plate
III. The letters _a b e d_ and _r_ are not reproduced in facsimile
of the original, but have been replaced by ordinary type in the
margin. 5-12. The original text of these lines is reproduced within
the diagram.--Compare No 275.]

149.

The light which falls on a shaded body at the acutest angle receives
the highest light, and the darkest portion is that which receives it
at an obtuse angle and both the light and the shadow form pyramids.
The angle _c_ receives the highest grade of light because it is
directly in front of the window _a b_ and the whole horizon of the
sky _m x_. The angle _a_ differs but little from _c_ because the
angles which divide it are not so unequal as those below, and only
that portion of the horizon is intercepted which lies between _y_
and _x_. Although it gains as much on the other side its line is
nevertheless not very strong because one angle is smaller than its
fellow. The angles _e i_ will have less light because they do not
see much of the light _m s_ and the light _v x_ and their angles are
very unequal. Yhe angle _k_ and the angle _f_ are each placed
between very unequal angles and therefore have but little light,
because at _k_ it has only the light _p t_, and at _f_ only _t q_;
_o g_ is the lowest grade of light because this part has no light at
all from the sky; and thence come the lines which will reconstruct a
pyramid that is the counterpart of the pyramid _c_; and this pyramid
_l_ is in the first grade of shadow; for this too is placed between
equal angles directly opposite to each other on either side of a
straight line which passes through the centre of the body and goes
to the centre of the light. The several luminous images cast within
the frame of the window at the points _a_ and _b_ make a light which
surrounds the derived shadow cast by the solid body at the points 4
and 6. The shaded images increase from _o g_ and end at 7 and 8.

[Footnote: The diagram belonging to this chapter is No. 2 on Plate
III. In the original it is placed between lines 3 and 4, and in the
reproduction these are shown in part. The semi circle above is
marked _orizonte_ (horizon). The number 6 at the left hand side,
outside the facsimile, is in the place of a figure which has become
indistinct in the original.]

On the intensity of shadows as dependent on the distance from the
light (150-152).

150.

The smaller the light that falls upon an object the more shadow it
will display. And the light will illuminate a smaller portion of the
object in proportion as it is nearer to it; and conversely, a larger
extent of it in proportion as it is farther off.

A light which is smaller than the object on which it falls will
light up a smaller extent of it in proportion as it is nearer to it,
and the converse, as it is farther from it. But when the light is
larger than the object illuminated it will light a larger extent of
the object in proportion as it is nearer and the converse when they
are farther apart.

151.

That portion of an illuminated object which is nearest to the source
of light will be the most strongly illuminated.

152.

That portion of the primary shadow will be least dark which is
farthest from the edges.

The derived shadow will be darker than the primary shadow where it
is contiguous with it.

On the proportion of light and shade (153-157).

153.

That portion of an opaque body will be more in shade or more in
light, which is nearer to the dark body, by which it is shaded, or
to the light that illuminates it.

Objects seen in light and shade show in greater relief than those
which are wholly in light or in shadow.

154.

OF PERSPECTIVE.

The shaded and illuminated sides of opaque objects will display the
same proportion of light and darkness as their objects [Footnote 6:
The meaning of _obbietti_ (objects) is explained in no 153, lines
1-4.--Between the title-line and the next there is, in the
original, a small diagram representing a circle described round a
square.].

155.

OF PAINTING.

The outlines and form of any part of a body in light and shade are
indistinct in the shadows and in the high lights; but in the
portions between the light and the shadows they are highly
conspicuous.

156.

OF PAINTING.

Among objects in various degrees of shade, when the light proceeds
from a single source, there will be the same proportion in their
shadows as in the natural diminution of the light and the same must
be understood of the degrees of light.

157.

A single and distinct luminous body causes stronger relief in the
object than a diffused light; as may be seen by comparing one side
of a landscape illuminated by the sun, and one overshadowed by
clouds, and so illuminated only by the diffused light of the
atmosphere.

THIRD BOOK ON LIGHT AND SHADE.

Definition of derived shadow (158. 159).

158.

Derived shadow cannot exist without primary shadow. This is proved
by the first of this which says: Darkness is the total absence of
light, and shadow is an alleviation of darkness and of light, and it
is more or less dark or light in proportion as the darkness is
modified by the light.

159.

Shadow is diminution of light.

Darkness is absence of light.

Shadow is divided into two kinds, of which the first is called
primary shadow, the second is derived shadow. The primary shadow is
always the basis of the derived shadow.

The edges of the derived shadow are straight lines.

[Footnote: The theory of the _ombra_ dirivativa_--a technical
expression for which there is no precise English equivalent is
elaborately treated by Leonardo. But both text and diagrams (as Pl.
IV, 1-3 and Pl. V) must at once convince the student that the
distinction he makes between _ombra primitiva_ and _ombra
dirivativa_ is not merely justifiable but scientific. _Ombra
dirivativa_ is by no means a mere abstract idea. This is easily
proved by repeating the experiment made by Leonardo, and by filling
with smoke the room in which the existence of the _ombra dirivativa_
is investigated, when the shadow becomes visible. Nor is it
difficult to perceive how much of Leonardo's teaching depended on
this theory. The recognised, but extremely complicated science of
cast shadows--_percussione dell' ombre dirivative_ as Leonardo
calls them--is thus rendered more intelligible if not actually
simpler, and we must assume this theory as our chief guide through
the investigations which follow.]

The darkness of the derived shadow diminishes in proportion as it is
remote from the primary shadow.

Different sorts of derived shadows (160-162).

160.

SHADOW AND LIGHT.

The forms of shadows are three: inasmuch as if the solid body which
casts the shadow is equal (in size) to the light, the shadow
resembles a column without any termination (in length). If the body
is larger than the light the shadow resembles a truncated and
inverted pyramid, and its length has also no defined termination.
But if the body is smaller than the light, the shadow will resemble
a pyramid and come to an end, as is seen in eclipses of the moon.

161.

OF SIMPLE DERIVED SHADOWS.

The simple derived shadow is of two kinds: one kind which has its
length defined, and two kinds which are undefined; and the defined
shadow is pyramidal. Of the two undefined, one is a column and the
other spreads out; and all three have rectilinear outlines. But the
converging, that is the pyramidal, shadow proceeds from a body that
is smaller than the light, and the columnar from a body equal in
size to the light, and the spreading shadow from a body larger than
the light; &c.

OF COMPOUND DERIVED SHADOWS.

Compound derived shadows are of two kinds; that is columnar and
spreading.

162.

OF SHADOW.

Derived shadows are of three kinds of which one is spreading, the
second columnar, the third converging to the point where the two
sides meet and intersect, and beyond this intersection the sides are
infinitely prolonged or straight lines. And if you say, this shadow
must terminate at the angle where the sides meet and extend no
farther, I deny this, because above in the first on shadow I have
proved: that a thing is completely terminated when no portion of it
goes beyond its terminating lines. Now here, in this shadow, we see
the converse of this, in as much as where this derived shadow
originates we obviously have the figures of two pyramids of shadow
which meet at their angles. Hence, if, as [my] opponent says, the
first pyramid of shadow terminates the derivative shadow at the
angle whence it starts, then the second pyramid of shadow--so says
the adversary--must be caused by the angle and not from the body in
shadow; and this is disproved with the help of the 2nd of this which
says: Shadow is a condition produced by a body casting a shadow, and
interposed between this shadow and the luminous body. By this it is
made clear that the shadow is not produced by the angle of the
derived shadow but only by the body casting the shadow; &c. If a
spherical solid body is illuminated by a light of elongated form the
shadow produced by the longest portion of this light will have less
defined outlines than that which is produced by the breadth of the
same light. And this is proved by what was said before, which is:
That a shadow will have less defined outlines in proportion as the
light which causes it is larger, and conversely, the outlines are
clearer in proportion as it is smaller.

[Footnote: The two diagrams to this chapter are on Plate IV, No. 1.]

On the relation of derived and primary shadow (163-165).

163.

The derived shadow can never resemble the body from which it
proceeds unless the light is of the same form and size as the body
causing the shadow.

The derived shadow cannot be of the same form as the primary shadow
unless it is intercepted by a plane parallel to it.

164.

HOW A CAST SHADOW CAN NEVER BE OF THE SAME SIZE AS THE BODY THAT
CASTS IT.

If the rays of light proceed, as experience shows, from a single
point and are diffused in a sphere round this point, radiating and
dispersed through the air, the farther they spread the wider they
must spread; and an object placed between the light and a wall is
always imaged larger in its shadow, because the rays that strike it
[Footnote: 7. The following lines are wanting to complete the
logical connection.] would, by the time they have reached the wall,
have become larger.

165.

Any shadow cast by a body in light and shade is of the same nature
and character as that which is inseparable from the body. The centre
of the length of a shadow always corresponds to that of the luminous
body [Footnote 6: This second statement of the same idea as in the
former sentence, but in different words, does not, in the original,
come next to the foregoing; sections 172 and 127 are placed between
them.]. It is inevitable that every shadow must have its centre in a
line with the centre of the light.

On the shape of derived shadows (166-174).

166.

OF THE PYRAMIDAL SHADOW.

The pyramidal shadow produced by a columnar body will be narrower
than the body itself in proportion as the simple derived shadow is
intersected farther from the body which casts it.

[Footnote 166: Compare the first diagram to No. 161. If we here
conceive of the outlines of the pyramid of shadow on the ground as
prolonged beyond its apex this gives rise to a second pyramid; this
is what is spoken of at the beginning of No. 166.]

167.

The cast shadow will be longest when the light is lowest.

The cast shadow will be shortest when the light is highest.

168.

Both the primary and derived shadow will be larger when caused by
the light of a candle than by diffused light. The difference between
the larger and smaller shadows will be in inverse proportion to the
larger and smaller lights causing them.

[Footnote: In the diagrams _A_ stands for _celo_ (sky), _B_ for
_cadela_ (candle).]

169.

ALL BODIES, IN PROPORTION AS THEY ARE NEARER TO, OR FARTHER FROM THE
SOURCE OF LIGHT, WILL PRODUCE LONGER OR SHORTER DERIVED SHADOWS.

Among bodies of equal size, that one which is illuminated by the
largest light will have the shortest shadow. Experiment confirms
this proposition. Thus the body _m_ _n_ is surrounded by a larger
amount of light than the body _p q_, as is shown above. Let us say
that _v c a b d x_ is the sky, the source of light, and that _s t_
is a window by which the luminous rays enter, and so _m n_ and _p q_
are bodies in light and shade as exposed to this light; _m n_ will
have a small derived shadow, because its original shadow will be
small; and the derivative light will be large, again, because the
original light _c d_ will be large and _p q_ will have more derived
shadow because its original shadow will be larger, and its derived
light will be smaller than that of the body _m n_ because that
portion of the hemisphere _a b_ which illuminates it is smaller than
the hemisphere _c d_ which illuminates the body _m n_.

[Footnote: The diagram, given on Pl. IV, No. 2, stands in the
original between lines 2 and 7, while the text of lines 3 to 6 is
written on its left side. In the reproduction of this diagram the
letter _v_ at the outer right-hand end has been omitted.]

170.

The shadow _m_ bears the same proportion to the shadow _n_ as the
line _b c_ to the line _f c_.

171.

OF PAINTING.

Of different shadows of equal strength that which is nearest the eye
will seem the least strong.

Why is the shadow _e a b_ in the first grade of strength, _b c_ in
the second; _c d_ in the third? The reason is that as from _e a b_
the sky is nowhere visible, it gets no light whatever from the sky,
and so has no direct [primary] light. _b c_ faces the portion of the
sky _f g_ and is illuminated by it. _c d_ faces the sky at _h k_. _c
d_, being exposed to a larger extent of sky than _b c_, it is
reasonable that it should be more lighted. And thus, up to a certain
distance, the wall _a d_ will grow lighter for the reasons here
given, until the darkness of the room overpowers the light from the
window.

172.

When the light of the atmosphere is restricted [by an opening] and
illuminates bodies which cast shadows, these bodies being equally
distant from the centre of the window, that which is most obliquely
placed will cast the largest shadow beyond it.

173.

These bodies standing apart in a room lighted by a single window
will have derivative shadows more or less short according as they
are more or less opposite to the window. Among the shadows cast by
bodies of equal mass but at unequal distances from the opening by
which they are illuminated, that shadow will be the longest of the
body which is least in the light. And in proportion as one body is
better illuminated than another its shadow will be shorter than
another. The proportion _n m_ and _e v k_ bear to _r t_ and _v x_
corresponds with that of the shadow _x_ to 4 and _y_.

The reason why those bodies which are placed most in front of the
middle of the window throw shorter shadows than those obliquely
situated is:--That the window appears in its proper form and to the
obliquely placed ones it appears foreshortened; to those in the
middle, the window shows its full size, to the oblique ones it
appears smaller; the one in the middle faces the whole hemisphere
that is _e f_ and those on the side have only a strip; that is _q r_
faces _a b_; and _m n_ faces _c d_; the body in the middle having a
larger quantity of light than those at the sides is lighted from a
point much below its centre, and thus the shadow is shorter. And the
pyramid _g_ 4 goes into _l y_ exactly as often as _a b_ goes into _e
f_. The axis of every derivative shadow passes through 6 1/2
[Footnote 31: _passa per_ 6 1/2 (passes through 6 1/2). The meaning
of these words is probably this: Each of the three axes of the
derived shadow intersects the centre (_mezzo_) of the primary shadow
(_ombra originale_) and, by prolongation upwards crosses six lines.

This is self evident only in the middle diagram; but it is equally
true of the side figures if we conceive of the lines 4 _f_, _x n v
m_, _y l k v_, and 4 _e_, as prolonged beyond the semicircle of the
horizon.] and is in a straight line with the centre of the primary
shadow, with the centre of the body casting it and of the derivative
light and with the centre of the window and, finally, with the
centre of that portion of the source of light which is the celestial
hemisphere, _y h_ is the centre of the derived shade, _l h_ of the
primary shadow, _l_ of the body throwing it, _l k_ of the derived
light, _v_ is the centre of the window, _e_ is the final centre of
the original light afforded by that portion of the hemisphere of the
sky which illuminates the solid body.

[Footnote: Compare the diagram on Pl. IV, No. 3. In the original
this drawing is placed between lines 3 and 22; the rest, from line 4
to line 21, is written on the left hand margin.]

174.

THE FARTHER THE DERIVED SHADOW IS PROLONGED THE LIGHTER IT BECOMES.

You will find that the proportion of the diameter of the derived
shadow to that of the primary shadow will be the same as that
between the darkness of the primary shadow and that of the derived
shadow.

[Footnote 6: Compare No. 177.] Let _a b_ be the diameter of the
primary shadow and _c d_ that of the derived shadow, I say that _a
b_ going, as you see, three times into _d c_, the shadow _d c_ will
be three times as light as the shadow _a b_. [Footnote 8: Compare
No. 177.]

If the size of the illuminating body is larger than that of the
illuminated body an intersection of shadow will occur, beyond which
the shadows will run off in two opposite directions as if they were
caused by two separate lights.

On the relative intensity of derived shadows (175-179).

175.

ON PAINTING.

The derived shadow is stronger in proportion as it is nearer to its
place of origin.

176.

HOW SHADOWS FADE AWAY AT LONG DISTANCES.

Shadows fade and are lost at long distances because the larger
quantity of illuminated air which lies between the eye and the
object seen tints the shadow with its own colour.

177.

_a b_ will be darker than _c d_ in proportion as _c d_ is broader
than _a b_.

[Footnote: In the original MS. the word _lume_ (light) is written at
the apex of the pyramid.]

178.

It can be proved why the shadow _o p c h_ is darker in proportion as
it is nearer to the line _p h_ and is lighter in proportion as it is
nearer to the line _o c_. Let the light _a b_, be a window, and let
the dark wall in which this window is, be _b s_, that is, one of the
sides of the wall.

Then we may say that the line _p h_ is darker than any other part of
the space _o p c h_, because this line faces the whole surface in
shadow of [Footnote: In the original the diagram is placed between
lines 27 and 28.] the wall _b s_. The line _o c_ is lighter than the
other part of this space _o p c h_, because this line faces the
luminous space _a b_.

Where the shadow is larger, or smaller, or equal the body which
casts it.

[First of the character of divided lights. [Footnote 14: _lumi
divisi_. The text here breaks off abruptly.]

OF THE COMPOUND SHADOW _F, R, C, H_ CAUSED BY A SINGLE LIGHT.

The shadow _f r c h_ is under such conditions as that where it is
farthest from its inner side it loses depth in proportion. To prove
this:

Let _d a_, be the light and _f n_ the solid body, and let _a e_ be
one of the side walls of the window that is _d a_. Then I
say--according to the 2nd [proposition]: that the surface of any
body is affected by the tone of the objects surrounding it,--that
the side _r c_, which faces the dark wall _a e_ must participate of
its darkness and, in the same way that the outer surface which faces
the light _d a_ participates of the light; thus we get the outlines
of the extremes on each side of the centre included between them.]

This is divided into four parts. The first the extremes, which
include the compound shadow, secondly the compound shadow between
these extremes.

179.

THE ACTION OF THE LIGHT AS FROM ITS CENTRE.

If it were the whole of the light that caused the shadows beyond the
bodies placed in front of it, it would follow that any body much
smaller than the light would cast a pyramidal shadow; but experience
not showing this, it must be the centre of the light that produces
this effect.

[Footnote: The diagram belonging to this passage is between lines 4
and 5 in the original. Comp. the reproduction Pl. IV, No. 4. The
text and drawing of this chapter have already been published with
tolerable accuracy. See M. JORDAN: "_Das Malerbuch des Leonardo da
Vinci_". Leipzig 1873, P. 90.]

PROOF.

Let _a b_ be the width of the light from a window, which falls on a
stick set up at one foot from _a c_ [Footnote 6: _bastone_ (stick).
The diagram has a sphere in place of a stick.]. And let _a d_ be the
space where all the light from the window is visible. At _c e_ that
part of the window which is between _l b_ cannot be seen. In the
same way _a m_ cannot be seen from _d f_ and therefore in these two
portions the light begins to fail.

Shadow as produced by two lights of different size (180. 181).

180.

A body in light and shade placed between two equal lights side by
side will cast shadows in proportion to the [amount of] light. And
the shadows will be one darker than the other in proportion as one
light is nearer to the said body than the other on the opposite
side.

A body placed at an equal distance between two lights will cast two
shadows, one deeper than the other in proportion, as the light which
causes it is brighter than the other.

[Footnote: In the MS. the larger diagram is placed above the first
line; the smaller one between l. 4 & 5.]

181.

A light which is smaller than the body it illuminates produces
shadows of which the outlines end within [the surface of] the body,
and not much compound shadow; and falls on less than half of it. A
light which is larger than the body it illuminates, falls on more
than half of it, and produces much compound shadow.

The effect of light at different distances.

182.

OF THE SHADOW CAST BY A BODY PLACED BETWEEN 2 EQUAL LIGHTS.

A body placed between 2 equal lights will cast 2 shadows of itself
in the direction of the lines of the 2 lights; and if you move this
body placing it nearer to one of the lights the shadow cast towards
the nearer light will be less deep than that which falls towards the
more distant one.

Further complications in the derived shadows (183-187).

183.

The greatest depth of shadow is in the simple derived shadow because
it is not lighted by either of the two lights _a b, c d_.

The next less deep shadow is the derived shadow _e f n_; and in this
the shadow is less by half, because it is illuminated by a single
light, that is _c d_.

This is uniform in natural tone because it is lighted throughout by
one only of the two luminous bodies [10]. But it varies with the
conditions of shadow, inasmuch as the farther it is away from the
light the less it is illuminated by it [13].

The third degree of depth is the middle shadow [Footnote 15: We
gather from what follows that _q g r_ here means _ombra media_ (the
middle shadow).]. But this is not uniform in natural tone; because
the nearer it gets to the simple derived shadow the deeper it is
[Footnote 18: Compare lines 10-13], and it is the uniformly gradual
diminution by increase of distance which is what modifies it
[Footnote 20: See Footnote 18]: that is to say the depth of a shadow
increases in proportion to the distance from the two lights.

The fourth is the shadow _k r s_ and this is all the darker in
natural tone in proportion as it is nearer to _k s_, because it gets
less of the light _a o_, but by the accident [of distance] it is
rendered less deep, because it is nearer to the light _c d_, and
thus is always exposed to both lights.

The fifth is less deep in shadow than either of the others because
it is always entirely exposed to one of the lights and to the whole
or part of the other; and it is less deep in proportion as it is
nearer to the two lights, and in proportion as it is turned towards
the outer side _x t_; because it is more exposed to the second light
_a b_.

[Footnote: The diagram to this section is given on Pl. V. To the
left is the facsimile of the beginning of the text belonging to it.]

184.

OF SIMPLE SHADOWS.

Why, at the intersections _a_, _b_ of the two compound shadows _e f_
and _m e_, is a simple shadow pfoduced as at _e h_ and _m g_, while
no such simple shadow is produced at the other two intersections _c
d_ made by the very same compound shadows?

ANSWER.

Compound shadow are a mixture of light and shade and simple shadows
are simply darkness. Hence, of the two lights _n_ and _o_, one falls
on the compound shadow from one side, and the other on the compound
shadow from the other side, but where they intersect no light falls,
as at _a b_; therefore it is a simple shadow. Where there is a
compound shadow one light or the other falls; and here a difficulty
arises for my adversary since he says that, where the compound
shadows intersect, both the lights which produce the shadows must of
necessity fall and therefore these shadows ought to be neutralised;
inasmuch as the two lights do not fall there, we say that the shadow
is a simple one and where only one of the two lights falls, we say
the shadow is compound, and where both the lights fall the shadow is
neutralised; for where both lights fall, no shadow of any kind is
produced, but only a light background limiting the shadow. Here I
shall say that what my adversary said was true: but he only mentions
such truths as are in his favour; and if we go on to the rest he
must conclude that my proposition is true. And that is: That if both
lights fell on the point of intersection, the shadows would be
neutralised. This I confess to be true if [neither of] the two
shadows fell in the same spot; because, where a shadow and a light
fall, a compound shadow is produced, and wherever two shadows or two
equal lights fall, the shadow cannot vary in any part of it, the
shadows and the lights both being equal. And this is proved in the
eighth [proposition] on proportion where it is said that if a given
quantity has a single unit of force and resistance, a double
quantity will have double force and double resistance.

DEFINITION.

The intersection _n_ is produced by the shadows caused by the light
_b_, because this light _b_ produces the shadow _x b_, and the
shadow _s b_, but the intersection _m_ is produced by the light _a_
which causes the shadow _s a_, and the shadow _x a_.

But if you uncover both the lights _a b_, then you get the two
shadows _n m_ both at once, and besides these, two other, simple
shadows are produced at _r o_ where neither of the two lights falls
at all. The grades of depth in compound shadows are fewer in
proportion as the lights falling on, and crossing them are less
numerous.

186.

Why the intersections at _n_ being composed of two compound derived
shadows, forms a compound shadow and not a simple one, as happens
with other intersections of compound shadows. This occurs, according
to the 2nd [diagram] of this [prop.] which says:--The intersection
of derived shadows when produced by the intersection of columnar
shadows caused by a single light does not produce a simple shadow.
And this is the corollary of the 1st [prop.] which says:--The
intersection of simple derived shadows never results in a deeper
shadow, because the deepest shadows all added together cannot be
darker than one by itself. Since, if many deepest shadows increased
in depth by their duplication, they could not be called the
_deepest_ shadows, but only part-shadows. But if such intersections
are illuminated by a second light placed between the eye and the
intersecting bodies, then those shadows would become compound
shadows and be uniformly dark just as much at the intersection as
throughout the rest. In the 1st and 2nd above, the intersections _i
k_ will not be doubled in depth as it is doubled in quantity. But in
this 3rd, at the intersections _g n_ they will be double in depth
and in quantity.

187.

HOW AND WHEN THE SURROUNDINGS IN SHADOW MINGLE THEIR DERIVED SHADOW
WITH THE LIGHT DERIVED FROM THE LUMINOUS BODY.

The derived shadow of the dark walls on each side of the bright
light of the window are what mingle their various degrees of shade
with the light derived from the window; and these various depths of
shade modify every portion of the light, except where it is
strongest, at _c_. To prove this let _d a_ be the primary shadow
which is turned towards the point _e_, and darkens it by its derived
shadow; as may be seen by the triangle _a e d_, in which the
angle _e_ faces the darkened base _d a e_; the point _v_ faces the
dark shadow _a s_ which is part of _a d_, and as the whole is
greater than a part, _e_ which faces the whole base [of the
triangle], will be in deeper shadow than _v_ which only faces part
of it. In consequence of the conclusion [shown] in the above
diagram, _t_ will be less darkened than _v_, because the base of the
_t_ is part of the base of the _v_; and in the same way it follows
that _p_ is less in shadow than _t_, because the base of the _p_ is
part of the base of the _t_. And _c_ is the terminal point of the
derived shadow and the chief beginning of the highest light.

[Footnote: The diagram on Pl. IV, No. 5 belongs to this passage; but
it must be noted that the text explains only the figure on the
right-hand side.]

FOURTH BOOK ON LIGHT AND SHADE.

On the shape of the cast shadows (188-191).

188.

The form of the shadow cast by any body of uniform density can never
be the same as that of the body producing it. [Footnote: Comp. the
drawing on PI. XXVIII, No. 5.]

189.

No cast shadow can produce the true image of the body which casts it
on a vertical plane unless the centre of the light is equally
distant from all the edges of that body.

190.

If a window _a b_ admits the sunlight into a room, the sunlight will
magnify the size of the window and diminish the shadow of a man in
such a way as that when the man makes that dim shadow of himself,
approach to that which defines the real size of the window, he will
see the shadows where they come into contact, dim and confused from
the strength of the light, shutting off and not allowing the solar
rays to pass; the effect of the shadow of the man cast by this
contact will be exactly that figured above.

[Footnote: It is scarcely possible to render the meaning of this
sentence with strict accuracy; mainly because the grammatical
construction is defective in the most important part--line 4. In the
very slight original sketch the shadow touches the upper arch of the
window and the correction, here given is perhaps not justified.]

191.

A shadow is never seen as of uniform depth on the surface which
intercepts it unless every portion of that surface is equidistant
from the luminous body. This is proved by the 7th which says:--The
shadow will appear lighter or stronger as it is surrounded by a
darker or a lighter background. And by the 8th of this:--The
background will be in parts darker or lighter, in proportion as it
is farther from or nearer to the luminous body. And:--Of various
spots equally distant from the luminous body those will always be in
the highest light on which the rays fall at the smallest angles: The
outline of the shadow as it falls on inequalities in the surface
will be seen with all the contours similar to those of the body that
casts it, if the eye is placed just where the centre of the light
was.

The shadow will look darkest where it is farthest from the body that
casts it. The shadow _c d_, cast by the body in shadow _a b_ which
is equally distant in all parts, is not of equal depth because it is
seen on a back ground of varying brightness. [Footnote: Compare the
three diagrams on Pl. VI, no 1 which, in the original accompany this
section.]

On the outlines of cast shadows (192-195).

192.

The edges of a derived shadow will be most distinct where it is cast
nearest to the primary shadow.

193.

As the derived shadow gets more distant from the primary shadow, the
more the cast shadow differs from the primary shadow.

194.

OF SHADOWS WHICH NEVER COME TO AN END.

The greater the difference between a light and the body lighted by
it, the light being the larger, the more vague will be the outlines
of the shadow of that object.

The derived shadow will be most confused towards the edges of its
interception by a plane, where it is remotest from the body casting
it.

195.

What is the cause which makes the outlines of the shadow vague and
confused?

Whether it is possible to give clear and definite outlines to the
edges of shadows.

On the relative size of shadows (196. 197).

196.

THE BODY WHICH IS NEAREST TO THE LIGHT CASTS THE LARGEST SHADOW, AND
WHY?

If an object placed in front of a single light is very close to it
you will see that it casts a very large shadow on the opposite wall,
and the farther you remove the object from the light the smaller
will the image of the shadow become.

WHY A SHADOW LARGER THAN THE BODY THAT PRODUCES IT BECOMES OUT OF
PROPORTION.

The disproportion of a shadow which is larger than the body
producing it, results from the light being smaller than the body, so
that it cannot be at an equal distance from the edges of the body
[Footnote 11: H. LUDWIG in his edition of the old copies, in the
Vatican library--in which this chapter is included under Nos. 612,
613 and 614 alters this passage as follows: _quella parte ch'e piu
propinqua piu cresce che le distanti_, although the Vatican copy
agrees with the original MS. in having _distante_ in the former and
_propinque_ in the latter place. This supposed amendment seems to me
to invert the facts. Supposing for instance, that on Pl. XXXI No. 3.
_f_ is the spot where the light is that illuminates the figure there
represented, and that the line behind the figure represents a wall
on which the shadow of the figure is thrown. It is evident, that in
that case the nearest portion, in this case the under part of the
thigh, is very little magnified in the shadow, and the remoter
parts, for instance the head, are more magnified.]; and the portions
which are most remote are made larger than the nearer portions for
this reason [Footnote 12: See Footnote 11].

WHY A SHADOW WHICH IS LARGER THAN THE BODY CAUSING IT HAS
ILL-DEFINED OUTLINES.

The atmosphere which surrounds a light is almost like light itself
for brightness and colour; but the farther off it is the more it
loses this resemblance. An object which casts a large shadow and is
near to the light, is illuminated both by that light by the luminous
atmosphere; hence this diffused light gives the shadow ill-defined
edges.

197.

A luminous body which is long and narrow in shape gives more
confused outlines to the derived shadow than a spherical light, and
this contradicts the proposition next following: A shadow will have
its outlines more clearly defined in proportion as it is nearer to
the primary shadow or, I should say, the body casting the shadow;
[Footnote 14: The lettering refers to the lower diagram, Pl. XLI,
No. 5.] the cause of this is the elongated form of the luminous body
_a c_, &c. [Footnote 16: See Footnote 14].

Effects on cast shadows by the tone of the back ground.

198.

OF MODIFIED SHADOWS.

Modified shadows are those which are cast on light walls or other
illuminated objects.

A shadow looks darkest against a light background. The outlines of a
derived shadow will be clearer as they are nearer to the primary
shadow. A derived shadow will be most defined in shape where it is
intercepted, where the plane intercepts it at the most equal angle.

Those parts of a shadow will appear darkest which have darker
objects opposite to them. And they will appear less dark when they
face lighter objects. And the larger the light object opposite, the
more the shadow will be lightened.

And the larger the surface of the dark object the more it will
darken the derived shadow where it is intercepted.

A disputed proposition.

199.

OF THE OPINION OF SOME THAT A TRIANGLE CASTS NO SHADOW ON A PLANE
SURFACE.

Certain mathematicians have maintained that a triangle, of which the
base is turned to the light, casts no shadow on a plane; and this
they prove by saying [5] that no spherical body smaller than the
light can reach the middle with the shadow. The lines of radiant
light are straight lines [6]; therefore, suppose the light to be _g
h_ and the triangle _l m n_, and let the plane be _i k_; they say
the light _g_ falls on the side of the triangle _l n_, and the
portion of the plane _i q_. Thus again _h_ like _g_ falls on the
side _l m_, and then on _m n_ and the plane _p k_; and if the whole
plane thus faces the lights _g h_, it is evident that the triangle
has no shadow; and that which has no shadow can cast none. This, in
this case appears credible. But if the triangle _n p g_ were not
illuminated by the two lights _g_ and _h_, but by _i p_ and _g_ and
_k_ neither side is lighted by more than one single light: that is
_i p_ is invisible to _h g_ and _k_ will never be lighted by _g_;
hence _p q_ will be twice as light as the two visible portions that
are in shadow.

[Footnote: 5--6. This passage is so obscure that it would be rash to
offer an explanation. Several words seem to have been omitted.]

On the relative depth of cast shadows (200-202).

200.

A spot is most in the shade when a large number of darkened rays
fall upon it. The spot which receives the rays at the widest angle
and by darkened rays will be most in the dark; a will be twice as
dark as b, because it originates from twice as large a base at an
equal distance. A spot is most illuminated when a large number of
luminous rays fall upon it. d is the beginning of the shadow _d f_,
and tinges _c_ but _a_ little; _d e_ is half of the shadow _d f_ and
gives a deeper tone where it is cast at _b_ than at _f_. And the
whole shaded space _e_ gives its tone to the spot _a_. [Footnote:
The diagram here referred to is on Pl. XLI, No. 2.]

201.

_A n_ will be darker than _c r_ in proportion to the number of times
that _a b_ goes into _c d_.

202.

The shadow cast by an object on a plane will be smaller in
proportion as that object is lighted by feebler rays. Let _d e_ be
the object and _d c_ the plane surface; the number of times that _d
e_ will go into _f g_ gives the proportion of light at _f h_ to _d
c_. The ray of light will be weaker in proportion to its distance
from the hole through which it falls.

FIFTH BOOK ON LIGHT AND SHADE.

Principles of reflection (203. 204).

203.

OF THE WAY IN WHICH THE SHADOWS CAST BY OBJECTS OUGHT TO BE DEFINED.

If the object is the mountain here figured, and the light is at the
point _a_, I say that from _b d_ and also from _c f_ there will be
no light but from reflected rays. And this results from the fact
that rays of light can only act in straight lines; and the same is
the case with the secondary or reflected rays.

204.

The edges of the derived shadow are defined by the hues of the
illuminated objects surrounding the luminous body which produces the
shadow.

On reverberation.

205.

OF REVERBERATION.

Reverberation is caused by bodies of a bright nature with a flat and
semi opaque surface which, when the light strikes upon them, throw
it back again, like the rebound of a ball, to the former object.

WHERE THERE CAN BE NO REFLECTED LIGHTS.

All dense bodies have their surfaces occupied by various degrees of
light and shade. The lights are of two kinds, one called original,
the other borrowed. Original light is that which is inherent in the
flame of fire or the light of the sun or of the atmosphere. Borrowed
light will be reflected light; but to return to the promised
definition: I say that this luminous reverberation is not produced
by those portions of a body which are turned towards darkened
objects, such as shaded spots, fields with grass of various height,
woods whether green or bare; in which, though that side of each
branch which is turned towards the original light has a share of
that light, nevertheless the shadows cast by each branch separately
are so numerous, as well as those cast by one branch on the others,
that finally so much shadow is the result that the light counts for
nothing. Hence objects of this kind cannot throw any reflected light
on opposite objects.

Reflection on water (206. 207).

206.

PERSPECTIVE.

The shadow or object mirrored in water in motion, that is to say in
small wavelets, will always be larger than the external object
producing it.

207.

It is impossible that an object mirrored on water should correspond
in form to the object mirrored, since the centre of the eye is above
the surface of the water.

This is made plain in the figure here given, which demonstrates that
the eye sees the surface _a b_, and cannot see it at _l f_, and at
_r t_; it sees the surface of the image at _r t_, and does not see
it in the real object _c d_. Hence it is impossible to see it, as
has been said above unless the eye itself is situated on the surface
of the water as is shown below [13].

[Footnote: _A_ stands for _ochio_ [eye], _B_ for _aria_ [air], _C_
for _acqua_ [water], _D_ for _cateto_ [cathetus].--In the original
MS. the second diagram is placed below line 13.]

Experiments with the mirror (208-210).

208.

THE MIRROR.

If the illuminated object is of the same size as the luminous body
and as that in which the light is reflected, the amount of the
reflected light will bear the same proportion to the intermediate
light as this second light will bear to the first, if both bodies
are smooth and white.

209.

Describe how it is that no object has its limitation in the mirror
but in the eye which sees it in the mirror. For if you look at your
face in the mirror, the part resembles the whole in as much as the
part is everywhere in the mirror, and the whole is in every part of
the same mirror; and the same is true of the whole image of any
object placed opposite to this mirror, &c.

210.

No man can see the image of another man in a mirror in its proper
place with regard to the objects; because every object falls on [the
surface of] the mirror at equal angles. And if the one man, who sees
the other in the mirror, is not in a direct line with the image he
will not see it in the place where it really falls; and if he gets
into the line, he covers the other man and puts himself in the place
occupied by his image. Let _n o_ be the mirror, _b_ the eye of your
friend and _d_ your own eye. Your friend's eye will appear to you at
_a_, and to him it will seem that yours is at _c_, and the
intersection of the visual rays will occur at _m_, so that either of
you touching _m_ will touch the eye of the other man which shall be
open. And if you touch the eye of the other man in the mirror it
will seem to him that you are touching your own.

Appendix:--On shadows in movement (211. 212).

211.

OF THE SHADOW AND ITS MOTION.

When two bodies casting shadows, and one in front of the other, are
between a window and the wall with some space between them, the
shadow of the body which is nearest to the plane of the wall will
move if the body nearest to the window is put in transverse motion
across the window. To prove this let _a_ and _b_ be two bodies
placed between the window _n m_ and the plane surface _o p_ with
sufficient space between them as shown by the space _a b_. I say
that if the body _a_ is moved towards _s_ the shadow of the body _b_
which is at _c_ will move towards _d_.

212.

OF THE MOTION OF SHADOWS.

The motion of a shadow is always more rapid than that of the body
which produces it if the light is stationary. To prove this let _a_
be the luminous body, and _b_ the body casting the shadow, and _d_
the shadow. Then I say that in the time while the solid body moves
from _b_ to _c_, the shadow _d_ will move to _e_; and this
proportion in the rapidity of the movements made in the same space
of time, is equal to that in the length of the space moved over.
Thus, given the proportion of the space moved over by the body _b_
to _c_, to that moved over by the shadow _d_ to _e_, the proportion
in the rapidity of their movements will be the same.

But if the luminous body is also in movement with a velocity equal
to that of the solid body, then the shadow and the body that casts
it will move with equal speed. And if the luminous body moves more
rapidly than the solid body, the motion of the shadow will be slower
than that of the body casting it.

But if the luminous body moves more slowly than the solid body, then
the shadow will move more rapidly than that body.

SIXTH BOOK ON LIGHT AND SHADE.

The effect of rays passing through holes (213. 214).

213.

PERSPECTIVE.

If you transmit the rays of the sun through a hole in the shape of a
star you will see a beautiful effect of perspective in the spot
where the sun's rays fall.

[Footnote: In this and the following chapters of MS. C the order of
the original paging has been adhered to, and is shown in
parenthesis. Leonardo himself has but rarely worked out the subject
of these propositions. The space left for the purpose has
occasionally been made use of for quite different matter. Even the
numerous diagrams, most of them very delicately sketched, lettered
and numbered, which occur on these pages, are hardly ever explained,
with the exception of those few which are here given.]

214.

No small hole can so modify the convergence of rays of light as to
prevent, at a long distance, the transmission of the true form of
the luminous body causing them. It is impossible that rays of light
passing through a parallel [slit], should not display the form of
the body causing them, since all the effects produced by a luminous
body are [in fact] the reflection of that body: The moon, shaped
like a boat, if transmitted through a hole is figured in the surface
[it falls on] as a boatshaped object. [Footnote 8: In the MS. a
blank space is left after this question.] Why the eye sees bodies at
a distance, larger than they measure on the vertical plane?.

[Footnote: This chapter, taken from another MS. may, as an
exception, be placed here, as it refers to the same subject as the
preceding section.]

On gradation of shadows (215. 216).

215.

Although the breadth and length of lights and shadow will be
narrower and shorter in foreshortening, the quality and quantity of
the light and shade is not increased nor diminished.

[3]The function of shade and light when diminished by
foreshortening, will be to give shadow and to illuminate an object
opposite, according to the quality and quantity in which they fall
on the body.

[5]In proportion as a derived shadow is nearer to its penultimate
extremities the deeper it will appear, _g z_ beyond the intersection
faces only the part of the shadow [marked] _y z_; this by
intersection takes the shadow from _m n_ but by direct line it takes
the shadow _a m_ hence it is twice as deep as _g z_. _Y x_, by
intersection takes the shadow _n o_, but by direct line the shadow
_n m a_, therefore _x y_ is three times as dark as _z g_; _x f_, by
intersection faces _o b_ and by direct line _o n m a_, therefore we
must say that the shadow between _f x_ will be four times as dark as
the shadow _z g_, because it faces four times as much shadow.

Let _a b_ be the side where the primary shadow is, and _b c_ the
primary light, _d_ will be the spot where it is intercepted,_f g_
the derived shadow and _f e_ the derived light.

And this must be at the beginning of the explanation.

[Footnote: In the original MS. the text of No. 252 precedes the one
given here. In the text of No. 215 there is a blank space of about
four lines between the lines 2 and 3. The diagram given on Pl. VI,
No. 2 is placed between lines 4 and 5. Between lines 5 and 6 there
is another space of about three lines and one line left blank
between lines 8 and 9. The reader will find the meaning of the whole
passage much clearer if he first reads the final lines 11--13.
Compare also line 4 of No. 270.]

On relative proportion of light and shadows (216--221).

216.

That part of the surface of a body on which the images [reflection]
from other bodies placed opposite fall at the largest angle will
assume their hue most strongly. In the diagram below, 8 is a larger
angle than 4, since its base _a n_ is larger than _e n_ the base of
4. This diagram below should end at _a n_ 4 8. [4]That portion of
the illuminated surface on which a shadow is cast will be brightest
which lies contiguous to the cast shadow. Just as an object which is
lighted up by a greater quantity of luminous rays becomes brighter,
so one on which a greater quantity of shadow falls, will be darker.

Let 4 be the side of an illuminated surface 4 8, surrounding the
cast shadow _g e_ 4. And this spot 4 will be lighter than 8, because
less shadow falls on it than on 8. Since 4 faces only the shadow _i
n_; and 8 faces and receives the shadow _a e_ as well as _i n_ which
makes it twice as dark. And the same thing happens when you put the
atmosphere and the sun in the place of shade and light.

[12] The distribution of shadow, originating in, and limited by,
plane surfaces placed near to each other, equal in tone and directly
opposite, will be darker at the ends than at the beginning, which
will be determined by the incidence of the luminous rays. You will
find the same proportion in the depth of the derived shadows _a n_
as in the nearness of the luminous bodies _m b_, which cause them;
and if the luminous bodies were of equal size you would still
farther find the same proportion in the light cast by the luminous
circles and their shadows as in the distance of the said luminous
bodies.

[Footnote: The diagram originally placed between lines 3 and 4 is on
Pl. VI, No. 3. In the diagram given above line 14 of the original,
and here printed in the text, the words _corpo luminoso_ [luminous
body] are written in the circle _m_, _luminoso_ in the circle _b_
and _ombroso_ [body in shadow] in the circle _o_.]

217.

THAT PART OF THE REFLECTION WILL BE BRIGHTEST WHERE THE REFLECTED
RAYS ARE SHORTEST.

[2] The darkness occasioned by the casting of combined shadows will
be in conformity with its cause, which will originate and terminate
between two plane surfaces near together, alike in tone and directly
opposite each other.

[4] In proportion as the source of light is larger, the luminous and
shadow rays will be more mixed together. This result is produced
because wherever there is a larger quantity of luminous rays, there
is most light, but where there are fewer there is least light,
consequently the shadow rays come in and mingle with them.

[Footnote: Diagrams are inserted before lines 2 and 4.]

218.

In all the proportions I lay down it must be understood that the
medium between the bodies is always the same. [2] The smaller the
luminous body the more distinct will the transmission of the shadows
be.

[3] When of two opposite shadows, produced by the same body, one is
twice as dark as the other though similar in form, one of the two
lights causing them must have twice the diameter that the other has
and be at twice the distance from the opaque body. If the object is
lowly moved across the luminous body, and the shadow is intercepted
at some distance from the object, there will be the same relative
proportion between the motion of the derived shadow and the motion
of the primary shadow, as between the distance from the object to
the light, and that from the object to the spot where the shadow is
intercepted; so that though the object is moved slowly the shadow
moves fast.

[Footnote: There are diagrams inserted before lines 2 and 3 but they
are not reproduced here. The diagram above line 6 is written upon as
follows: at _A lume_ (light), at _B obbietto_ (body), at _C ombra
d'obbietto_ (shadow of the object).]

219.

A luminous body will appear less brilliant when surrounded by a
bright background.

[2] I have found that the stars which are nearest to the horizon
look larger than the others because light falls upon them from a
larger proportion of the solar body than when they are above us; and
having more light from the sun they give more light, and the bodies
which are most luminous appear the largest. As may be seen by the
sun through a mist, and overhead; it appears larger where there is
no mist and diminished through mist. No portion of the luminous body
is ever visible from any spot within the pyramid of pure derived
shadow.

[Footnote: Between lines 1 and 2 there is in the original a large
diagram which does not refer to this text. ]

220.

A body on which the solar rays fall between the thin branches of
trees far apart will cast but a single shadow.

[2] If an opaque body and a luminous one are (both) spherical the
base of the pyramid of rays will bear the same proportion to the
luminous body as the base of the pyramid of shade to the opaque
body.

[4] When the transmitted shadow is intercepted by a plane surface
placed opposite to it and farther away from the luminous body than
from the object [which casts it] it will appear proportionately
darker and the edges more distinct.

[Footnote: The diagram which, in the original, is placed above line
2, is similar to the one, here given on page 73 (section 120).--The
diagram here given in the margin stands, in the original, between
lines 3 and 4.]

221.

A body illuminated by the solar rays passing between the thick
branches of trees will produce as many shadows as there are branches
between the sun and itself.

Where the shadow-rays from an opaque pyramidal body are intercepted
they will cast a shadow of bifurcate outline and various depth at
the points. A light which is broader than the apex but narrower than
the base of an opaque pyramidal body placed in front of it, will
cause that pyramid to cast a shadow of bifurcate form and various
degrees of depth.

If an opaque body, smaller than the light, casts two shadows and if
it is the same size or larger, casts but one, it follows that a
pyramidal body, of which part is smaller, part equal to, and part
larger than, the luminous body, will cast a bifurcate shadow.

[Footnote: Between lines 2 and 3 there are in the original two large
diagrams.]

_IV._

_Perspective of Disappearance._

_The theory of the_ "Prospettiva de' perdimenti" _would, in many
important details, be quite unintelligible if it had not been led up
by the principles of light and shade on which it is based. The word_
"Prospettiva" _in the language of the time included the principles
of optics; what Leonardo understood by_ "Perdimenti" _will be
clearly seen in the early chapters, Nos._ 222--224. _It is in the
very nature of the case that the farther explanations given in the
subsequent chapters must be limited to general rules. The sections
given as_ 227--231 _"On indistinctness at short distances" have, it
is true, only an indirect bearing on the subject; but on the other
hand, the following chapters,_ 232--234, _"On indistinctness at
great distances," go fully into the matter, and in chapters_
235--239, _which treat "Of the importance of light and shade in the
Perspective of Disappearance", the practical issues are distinctly
insisted on in their relation to the theory. This is naturally
followed by the statements as to "the effect of light or dark
backgrounds on the apparent size of bodies"_ (_Nos._ 240--250). _At
the end I have placed, in the order of the original, those sections
from the MS._ C _which treat of the "Perspective of Disappearance"
and serve to some extent to complete the treatment of the subject_
(251--262).

Definition (222. 223).

222.

OF THE DIMINISHED DISTINCTNESS OF THE OUTLINES OF OPAQUE BODIES.

If the real outlines of opaque bodies are indistinguishable at even
a very short distance, they will be more so at long distances; and,
since it is by its outlines that we are able to know the real form
of any opaque body, when by its remoteness we fail to discern it as
a whole, much more must we fail to discern its parts and outlines.

223.

OF THE DIMINUTION IN PERSPECTIVE OF OPAQUE OBJECTS.

Among opaque objects of equal size the apparent diminution of size
will be in proportion to their distance from the eye of the
spectator; but it is an inverse proportion, since, where the
distance is greater, the opaque body will appear smaller, and the
less the distance the larger will the object appear. And this is the
fundamental principle of linear perspective and it
follows:--[11]every object as it becomes more remote loses first
those parts which are smallest. Thus of a horse, we should lose the
legs before the head, because the legs are thinner than the head;
and the neck before the body for the same reason. Hence it follows
that the last part of the horse which would be discernible by the
eye would be the mass of the body in an oval form, or rather in a
cylindrical form and this would lose its apparent thickness before
its length--according to the 2nd rule given above, &c. [Footnote 23:
Compare line 11.].

If the eye remains stationary the perspective terminates in the
distance in a point. But if the eye moves in a straight [horizontal]
line the perspective terminates in a line and the reason is that
this line is generated by the motion of the point and our sight;
therefore it follows that as we move our sight [eye], the point
moves, and as we move the point, the line is generated, &c.

An illustration by experiment.

224.

Every visible body, in so far as it affects the eye, includes three
attributes; that is to say: mass, form and colour; and the mass is
recognisable at a greater distance from the place of its actual
existence than either colour or form. Again, colour is discernible
at a greater distance than form, but this law does not apply to
luminous bodies.

The above proposition is plainly shown and proved by experiment;
because: if you see a man close to you, you discern the exact
appearance of the mass and of the form and also of the colouring; if
he goes to some distance you will not recognise who he is, because
the character of the details will disappear, if he goes still
farther you will not be able to distinguish his colouring, but he
will appear as a dark object, and still farther he will appear as a
very small dark rounded object. It appears rounded because distance
so greatly diminishes the various details that nothing remains
visible but the larger mass. And the reason is this: We know very
well that all the images of objects reach the senses by a small
aperture in the eye; hence, if the whole horizon _a d_ is admitted
through such an aperture, the object _b c_ being but a very small
fraction of this horizon what space can it fill in that minute image
of so vast a hemisphere? And because luminous bodies have more power
in darkness than any others, it is evident that, as the chamber of
the eye is very dark, as is the nature of all colored cavities, the
images of distant objects are confused and lost in the great light
of the sky; and if they are visible at all, appear dark and black,
as every small body must when seen in the diffused light of the
atmosphere.

[Footnote: The diagram belonging to this passage is placed between
lines 5 and 6; it is No. 4 on Pl. VI. ]

A guiding rule.

225.

OF THE ATMOSPHERE THAT INTERPOSES BETWEEN THE EYE AND VISIBLE
OBJECTS.

An object will appear more or less distinct at the same distance, in
proportion as the atmosphere existing between the eye and that
object is more or less clear. Hence, as I know that the greater or
less quantity of the air that lies between the eye and the object
makes the outlines of that object more or less indistinct, you must
diminish the definiteness of outline of those objects in proportion
to their increasing distance from the eye of the spectator.

An experiment.

226.

When I was once in a place on the sea, at an equal distance from the
shore and the mountains, the distance from the shore looked much
greater than that from the mountains.

On indistinctness at short distances (227-231).

227.

If you place an opaque object in front of your eye at a distance of
four fingers' breadth, if it is smaller than the space between the
two eyes it will not interfere with your seeing any thing that may
be beyond it. No object situated beyond another object seen by the
eye can be concealed by this [nearer] object if it is smaller than
the space from eye to eye.

228.

The eye cannot take in a luminous angle which is too close to it.

229.

That part of a surface will be better lighted on which the light
falls at the greater angle. And that part, on which the shadow falls
at the greatest angle, will receive from those rays least of the
benefit of the light.

230.

OF THE EYE.

The edges of an object placed in front of the pupil of the eye will
be less distinct in proportion as they are closer to the eye. This
is shown by the edge of the object _n_ placed in front of the pupil
_d_; in looking at this edge the pupil also sees all the space _a c_
which is beyond the edge; and the images the eye receives from that
space are mingled with the images of the edge, so that one image
confuses the other, and this confusion hinders the pupil from
distinguishing the edge.

231.

The outlines of objects will be least clear when they are nearest to
the eye, and therefore remoter outlines will be clearer. Among
objects which are smaller than the pupil of the eye those will be
less distinct which are nearer to the eye.

On indistinctness at great distances (232-234).

232.

Objects near to the eye will appear larger than those at a distance.

Objects seen with two eyes will appear rounder than if they are seen
with only one.

Objects seen between light and shadow will show the most relief.

233.

OF PAINTING.

Our true perception of an object diminishes in proportion as its
size is diminished by distance.

234.

PERSPECTIVE.

Why objects seen at a distance appear large to the eye and in the
image on the vertical plane they appear small.

PERSPECTIVE.

I ask how far away the eye can discern a non-luminous body, as, for
instance, a mountain. It will be very plainly visible if the sun is
behind it; and could be seen at a greater or less distance according
to the sun's place in the sky.

[Footnote: The clue to the solution of this problem (lines 1-3) is
given in lines 4-6, No. 232. Objects seen with both eyes appear
solid since they are seen from two distinct points of sight
separated by the distance between the eyes, but this solidity cannot
be represented in a flat drawing. Compare No. 535.]

The importance of light and shade in the perspective of
disappearance (235-239).

235.

An opaque body seen in a line in which the light falls will reveal
no prominences to the eye. For instance, let _a_ be the solid body
and _c_ the light; _c m_ and _c n_ will be the lines of incidence of
the light, that is to say the lines which transmit the light to the
object _a_. The eye being at the point _b_, I say that since the
light _c_ falls on the whole part _m n_ the portions in relief on
that side will all be illuminated. Hence the eye placed at _c_
cannot see any light and shade and, not seeing it, every portion
will appear of the same tone, therefore the relief in the prominent
or rounded parts will not be visible.

236.

OF PAINTING.

When you represent in your work shadows which you can only discern
with difficulty, and of which you cannot distinguish the edges so
that you apprehend them confusedly, you must not make them sharp or
definite lest your work should have a wooden effect.

237.

OF PAINTING.

You will observe in drawing that among the shadows some are of
undistinguishable gradation and form, as is shown in the 3rd
[proposition] which says: Rounded surfaces display as many degrees
of light and shade as there are varieties of brightness and darkness
reflected from the surrounding objects.

238.

OF LIGHT AND SHADE.

You who draw from nature, look (carefully) at the extent, the
degree, and the form of the lights and shadows on each muscle; and
in their position lengthwise observe towards which muscle the axis
of the central line is directed.

239.

An object which is [so brilliantly illuminated as to be] almost as
bright as light will be visible at a greater distance, and of larger
apparent size than is natural to objects so remote.

The effect of light or dark backgrounds on the apparent size of
objects (240-250).

240.

A shadow will appear dark in proportion to the brilliancy of the
light surrounding it and conversely it will be less conspicuous
where it is seen against a darker background.

241.

OF ORDINARY PERSPECTIVE.

An object of equal breadth and colour throughout, seen against a
background of various colours will appear unequal in breadth.

And if an object of equal breadth throughout, but of various
colours, is seen against a background of uniform colour, that object
will appear of various breadth. And the more the colours of the
background or of the object seen against the ground vary, the
greater will the apparent variations in the breadth be though the
objects seen against the ground be of equal breadth [throughout].

242.

A dark object seen against a bright background will appear smaller
than it is.

A light object will look larger when it is seen against a background
darker than itself.

243.

OF LIGHT.

A luminous body when obscured by a dense atmosphere will appear
smaller; as may be seen by the moon or sun veiled by mists.

OF LIGHT.

Of several luminous bodies of equal size and brilliancy and at an
equal distance, that will look the largest which is surrounded by
the darkest background.

OF LIGHT.

I find that any luminous body when seen through a dense and thick
mist diminishes in proportion to its distance from the eye. Thus it
is with the sun by day, as well as the moon and the other eternal
lights by night. And when the air is clear, these luminaries appear
larger in proportion as they are farther from the eye.

244.

That portion of a body of uniform breadth which is against a lighter
background will look narrower [than the rest].

[4] _e_ is a given object, itself dark and of uniform breadth; _a b_
and _c d_ are two backgrounds one darker than the other; _b c_ is a
bright background, as it might be a spot lighted by the sun through
an aperture in a dark room. Then I say that the object _e g_ will
appear larger at _e f_ than at _g h_; because _e f_ has a darker
background than _g h_; and again at _f g_ it will look narrower from
being seen by the eye _o_, on the light background _b c_. [Footnote
12: The diagram to which the text, lines 1-11, refers, is placed in
the original between lines 3 and 4, and is given on Pl. XLI, No. 3.
Lines 12 to 14 are explained by the lower of the two diagrams on Pl.
XLI, No. 4. In the original these are placed after line 14.] That
part of a luminous body, of equal breadth and brilliancy throughout,
will look largest which is seen against the darkest background; and
the luminous body will seem on fire.

245.

WHY BODIES IN LIGHT AND SHADE HAVE THEIR OUTLINES ALTERED BY THE
COLOUR AND BRIGHTNESS OF THE OBJECTS SERVING AS A BACKGROUND TO
THEM.

If you look at a body of which the illuminated portion lies and ends
against a dark background, that part of the light which will look
brightest will be that which lies against the dark [background] at
_d_. But if this brighter part lies against a light background, the
edge of the object, which is itself light, will be less distinct
than before, and the highest light will appear to be between the
limit of the background _m f_ and the shadow. The same thing is seen
with regard to the dark [side], inasmuch as that edge of the shaded
portion of the object which lies against a light background, as at
_l_, it looks much darker than the rest. But if this shadow lies
against a dark background, the edge of the shaded part will appear
lighter than before, and the deepest shade will appear between the
edge and the light at the point _o_.

[Footnote: In the original diagram _o_ is inside the shaded surface
at the level of _d_.]

246.

An opaque body will appear smaller when it is surrounded by a highly
luminous background, and a light body will appear larger when it is
seen against a darker background. This may be seen in the height of
buildings at night, when lightning flashes behind them; it suddenly
seems, when it lightens, as though the height of the building were
diminished. For the same reason such buildings look larger in a
mist, or by night than when the atmosphere is clear and light.

247.

ON LIGHT BETWEEN SHADOWS

When you are drawing any object, remember, in comparing the grades
of light in the illuminated portions, that the eye is often deceived
by seeing things lighter than they are. And the reason lies in our
comparing those parts with the contiguous parts. Since if two
[separate] parts are in different grades of light and if the less
bright is conterminous with a dark portion and the brighter is
conterminous with a light background--as the sky or something
equally bright--, then that which is less light, or I should say
less radiant, will look the brighter and the brighter will seem the
darker.

248.

Of objects equally dark in themselves and situated at a considerable
and equal distance, that will look the darkest which is farthest
above the earth.

249.

TO PROVE HOW IT IS THAT LUMINOUS BODIES APPEAR LARGER, AT A
DISTANCE, THAN THEY ARE.

If you place two lighted candles side by side half a braccio apart,
and go from them to a distance 200 braccia you will see that by the
increased size of each they will appear as a single luminous body
with the light of the two flames, one braccio wide.

TO PROVE HOW YOU MAY SEE THE REAL SIZE OF LUMINOUS BODIES.

If you wish to see the real size of these luminous bodies, take a
very thin board and make in it a hole no bigger than the tag of a
lace and place it as close to your eye as possible, so that when you
look through this hole, at the said light, you can see a large space
of air round it. Then by rapidly moving this board backwards and
forwards before your eye you will see the light increase [and
diminish].

Propositions on perspective of disappearance from MS. C. (250-262).

250.

Of several bodies of equal size and equally distant from the eye,
those will look the smallest which are against the lightest
background.

Every visible object must be surrounded by light and shade. A
perfectly spherical body surrounded by light and shade will appear
to have one side larger than the other in proportion as one is more
highly lighted than the other.

251.

PERSPECTIVE.

No visible object can be well understood and comprehended by the
human eye excepting from the difference of the background against
which the edges of the object terminate and by which they are
bounded, and no object will appear [to stand out] separate from that
background so far as the outlines of its borders are concerned. The
moon, though it is at a great distance from the sun, when, in an
eclipse, it comes between our eyes and the sun, appears to the eyes
of men to be close to the sun and affixed to it, because the sun is
then the background to the moon.

252.

A luminous body will appear more brilliant in proportion as it is
surrounded by deeper shadow. [Footnote: The diagram which, in the
original, is placed after this text, has no connection with it.]

253.

The straight edges of a body will appear broken when they are
conterminous with a dark space streaked with rays of light.
[Footnote: Here again the diagrams in the original have no
connection with the text.]

254.

Of several bodies, all equally large and equally distant, that which
is most brightly illuminated will appear to the eye nearest and
largest. [Footnote: Here again the diagrams in the original have no
connection with the text.]

255.

If several luminous bodies are seen from a great distance although
they are really separate they will appear united as one body.

256.

If several objects in shadow, standing very close together, are seen
against a bright background they will appear separated by wide
intervals.

257.

Of several bodies of equal size and tone, that which is farthest
will appear the lightest and smallest.

258.

Of several objects equal in size, brightness of background and
length that which has the flattest surface will look the largest. A
bar of iron equally thick throughout and of which half is red hot,
affords an example, for the red hot part looks thicker than the
rest.

259.

Of several bodies of equal size and length, and alike in form and in
depth of shade, that will appear smallest which is surrounded by the
most luminous background.

260.

DIFFERENT PORTIONS OF A WALL SURFACE WILL BE DARKER OR BRIGHTER IN
PROPORTION AS THE LIGHT OR SHADOW FALLS ON THEM AT A LARGER ANGLE.

The foregoing proposition can be clearly proved in this way. Let us
say that _m q_ is the luminous body, then _f g_ will be the opaque
body; and let _a e_ be the above-mentioned plane on which the said
angles fall, showing [plainly] the nature and character of their
bases. Then: _a_ will be more luminous than _b_; the base of the
angle _a_ is larger than that of _b_ and it therefore makes a
greater angle which will be _a m q_; and the pyramid _b p m_ will be
narrower and _m o c_ will be still finer, and so on by degrees, in
proportion as they are nearer to _e_, the pyramids will become
narrower and darker. That portion of the wall will be the darkest
where the breadth of the pyramid of shadow is greater than the
breadth of the pyramid of light.

At the point _a_ the pyramid of light is equal in strength to the
pyramid of shadow, because the base _f g_ is equal to the base _r
f_. At the point _d_ the pyramid of light is narrower than the
pyramid of shadow by so much as the base _s f_ is less than the base
_f g_.

Divide the foregoing proposition into two diagrams, one with the
pyramids of light and shadow, the other with the pyramids of light
[only].

261.

Among shadows of equal depth those which are nearest to the eye will
look least deep.

262.

The more brilliant the light given by a luminous body, the deeper
will the shadows be cast by the objects it illuminates.

_V._

_Theory of colours._

_Leonardo's theory of colours is even more intimately connected with
his principles of light and shade than his Perspective of
Disappearance and is in fact merely an appendix or supplement to
those principles, as we gather from the titles to sections_ 264,
267_, and _276_, while others again_ (_Nos._ 281, 282_) are headed_
Prospettiva.

_A very few of these chapters are to be found in the oldest copies
and editions of the Treatise on Painting, and although the material
they afford is but meager and the connection between them but
slight, we must still attribute to them a special theoretical value
as well as practical utility--all the more so because our knowledge
of the theory and use of colours at the time of the Renaissance is
still extremely limited._

The reciprocal effects of colours on objects placed opposite each
other (263-272).

263.

OF PAINTING.

The hue of an illuminated object is affected by that of the luminous
body.

264.

OF SHADOW.

The surface of any opaque body is affected by the colour of
surrounding objects.

265.

A shadow is always affected by the colour of the surface on which it
is cast.

266.

An image produced in a mirror is affected by the colour of the
mirror.

267.

OF LIGHT AND SHADE.

Every portion of the surface of a body is varied [in hue] by the
[reflected] colour of the object that may be opposite to it.

EXAMPLE.

If you place a spherical body between various objects that is to say
with [direct] sunlight on one side of it, and on the other a wall
illuminated by the sun, which wall may be green or of any other
colour, while the surface on which it is placed may be red, and the
two lateral sides are in shadow, you will see that the natural
colour of that body will assume something of the hue reflected from
those objects. The strongest will be [given by] the luminous body;
the second by the illuminated wall, the third by the shadows. There
will still be a portion which will take a tint from the colour of
the edges.

268.

The surface of every opaque body is affected by the colour of the
objects surrounding it. But this effect will be strong or weak in
proportion as those objects are more or less remote and more or less
strongly [coloured].

269.

OF PAINTING.

The surface of every opaque body assumes the hues reflected from
surrounding objects.

The surface of an opaque body assumes the hues of surrounding
objects more strongly in proportion as the rays that form the images
of those objects strike the surface at more equal angles.

And the surface of an opaque body assumes a stronger hue from the
surrounding objects in proportion as that surface is whiter and the
colour of the object brighter or more highly illuminated.

270.

OF THE RAYS WHICH CONVEY THROUGH THE AIR THE IMAGES OF OBJECTS.

All the minutest parts of the image intersect each other without
interfering with each other. To prove this let _r_ be one of the
sides of the hole, opposite to which let _s_ be the eye which sees
the lower end _o_ of the line _n o_. The other extremity cannot
transmit its image to the eye _s_ as it has to strike the end _r_
and it is the same with regard to _m_ at the middle of the line. The
case is the same with the upper extremity _n_ and the eye _u_. And
if the end _n_ is red the eye _u_ on that side of the holes will not
see the green colour of _o_, but only the red of _n_ according to
the 7th of this where it is said: Every form projects images from
itself by the shortest line, which necessarily is a straight line,
&c.

[Footnote: 13. This probably refers to the diagram given under No.
66.]

271.

OF PAINTING.

The surface of a body assumes in some degree the hue of those around
it. The colours of illuminated objects are reflected from the
surfaces of one to the other in various spots, according to the
various positions of those objects. Let _o_ be a blue object in full
light, facing all by itself the space _b c_ on the white sphere _a b
e d e f_, and it will give it a blue tinge, _m_ is a yellow body
reflected onto the space _a b_ at the same time as _o_ the blue
body, and they give it a green colour (by the 2nd [proposition] of
this which shows that blue and yellow make a beautiful green &c.)
And the rest will be set forth in the Book on Painting. In that Book
it will be shown, that, by transmitting the images of objects and
the colours of bodies illuminated by sunlight through a small round
perforation and into a dark chamber onto a plane surface, which
itself is quite white, &c.

But every thing will be upside down.

Combination of different colours in cast shadows.

272.

That which casts the shadow does not face it, because the shadows
are produced by the light which causes and surrounds the shadows.
The shadow caused by the light _e_, which is yellow, has a blue
tinge, because the shadow of the body _a_ is cast upon the pavement
at _b_, where the blue light falls; and the shadow produced by the
light _d_, which is blue, will be yellow at _c_, because the yellow
light falls there and the surrounding background to these shadows _b
c_ will, besides its natural colour, assume a hue compounded of
yellow and blue, because it is lighted by the yellow light and by
the blue light both at once.

Shadows of various colours, as affected by the lights falling on
them. That light which causes the shadow does not face it.

[Footnote: In the original diagram we find in the circle _e_
"_giallo_" (yellow) and the cirle _d_ "_azurro"_ (blue) and also
under the circle of shadow to the left "_giallo_" is written and
under that to the right "_azurro_".

In the second diagram where four circles are placed in a row we find
written, beginning at the left hand, "_giallo_" (yellow), "_azurro_"
(blue), "_verde_" (green), "_rosso_" (red).]

The effect of colours in the camera obscura (273-274).

273.

The edges of a colour(ed object) transmitted through a small hole
are more conspicuous than the central portions.

The edges of the images, of whatever colour, which are transmitted
through a small aperture into a dark chamber will always be stronger
than the middle portions.

274.

OF THE INTERSECTIONS OF THE IMAGES IN THE PUPIL OF THE EYE.

The intersections of the images as they enter the pupil do not
mingle in confusion in the space where that intersection unites
them; as is evident, since, if the rays of the sun pass through two
panes of glass in close contact, of which one is blue and the other
yellow, the rays, in penetrating them, do not become blue or yellow
but a beautiful green. And the same thing would happen in the eye,
if the images which were yellow or green should mingle where they
[meet and] intersect as they enter the pupil. As this does not
happen such a mingling does not exist.

OF THE NATURE OF THE RAYS COMPOSED OF THE IMAGES OF OBJECTS, AND OF
THEIR INTERSECTIONS.

The directness of the rays which transmit the forms and colours of
the bodies whence they proceed does not tinge the air nor can they
affect each other by contact where they intersect. They affect only
the spot where they vanish and cease to exist, because that spot
faces and is faced by the original source of these rays, and no
other object, which surrounds that original source can be seen by
the eye where these rays are cut off and destroyed, leaving there
the spoil they have conveyed to it. And this is proved by the 4th
[proposition], on the colour of bodies, which says: The surface of
every opaque body is affected by the colour of surrounding objects;
hence we may conclude that the spot which, by means of the rays
which convey the image, faces--and is faced by the cause of the
image, assumes the colour of that object.

On the colours of derived shadows (275. 276).

275.

ANY SHADOW CAST BY AN OPAQUE BODY SMALLER THAN THE LIGHT CAUSING THE
SHADOW WILL THROW A DERIVED SHADOW WHICH IS TINGED BY THE COLOUR OF
THE LIGHT.

Let _n_ be the source of the shadow _e f_; it will assume its hue.
Let _o_ be the source of _h e_ which will in the same way be tinged
by its hue and so also the colour of _v h_ will be affected by _p_
which causes it; and the shadow of the triangle _z k y_ will be
affected by the colour of _q_, because it is produced by it. [7] In
proportion as _c d_ goes into _a d_, will _n r s_ be darker than
_m_; and the rest of the space will be shadowless [11]. _f g_ is
the highest light, because here the whole light of the window _a d_
falls; and thus on the opaque body _m e_ is in equally high light;
_z k y_ is a triangle which includes the deepest shadow, because the
light _a d_ cannot reach any part of it. _x h_ is the 2nd grade of
shadow, because it receives only 1/3 of the light from the window,
that is _c d_. The third grade of shadow is _h e_, where two thirds
of the light from the window is visible. The last grade of shadow is
_b d e f_, because the highest grade of light from the window falls
at _f_.

[Footnote: The diagram Pl. III, No. 1 belongs to this chapter as
well as the text given in No. 148. Lines 7-11 (compare lines 8-12 of
No. 148) which are written within the diagram, evidently apply to
both sections and have therefore been inserted in both.]

276.

OF THE COLOURS OF SIMPLE DERIVED SHADOWS.

The colour of derived shadows is always affected by that of the body
towards which they are cast. To prove this: let an opaque body be
placed between the plane _s c t d_ and the blue light _d e_ and the
red light _a b_, then I say that _d e_, the blue light, will fall on
the whole surface _s c t d_ excepting at _o p_ which is covered by
the shadow of the body _q r_, as is shown by the straight lines _d q
o e r p_. And the same occurs with the light _a b_ which falls on
the whole surface _s c t d_ excepting at the spot obscured by the
shadow _q r_; as is shown by the lines _d q o_, and _e r p_. Hence
we may conclude that the shadow _n m_ is exposed to the blue light
_d e_; but, as the red light _a b_ cannot fall there, _n m_ will
appear as a blue shadow on a red background tinted with blue,
because on the surface _s c t d_ both lights can fall. But in the
shadows only one single light falls; for this reason these shadows
are of medium depth, since, if no light whatever mingled with the
shadow, it would be of the first degree of darkness &c. But in the
shadow at _o p_ the blue light does not fall, because the body _q r_
interposes and intercepts it there. Only the red light _a b_ falls
there and tinges the shadow of a red hue and so a ruddy shadow
appears on the background of mingled red and blue.

The shadow of _q r_ at _o p_ is red, being caused by the blue light
_d e_; and the shadow of _q r_ at _o' p'_ is blue being caused by
the red light _a b_. Hence we say that the blue light in this
instance causes a red derived shadow from the opaque body _q' r'_,
while the red light causes the same body to cast a blue derived
shadow; but the primary shadow [on the dark side of the body itself]
is not of either of those hues, but a mixture of red and blue.

The derived shadows will be equal in depth if they are produced by
lights of equal strength and at an equal distance; this is proved.
[Footnote 53: The text is unfinished in the original.]

[Footnote: In the original diagram Leonardo has written within the
circle _q r corpo obroso_ (body in shadow); at the spot marked _A,
luminoso azzurro_ (blue luminous body); at _B, luminoso rosso_ (red
luminous body). At _E_ we read _ombra azzurra_ (blue tinted shadow)
and at _D ombra rossa_ (red tinted shadow).]

On the nature of colours (277. 278).

277.

No white or black is transparent.

278.

OF PAINTING.

[Footnote 2: See Footnote 3] Since white is not a colour but the
neutral recipient of every colour [Footnote 3: _il bianco non e
colore ma e inpotentia ricettiva d'ogni colore_ (white is not a
colour, but the neutral recipient of every colour). LEON BATT.
ALBERTI "_Della pittura_" libro I, asserts on the contrary: "_Il
bianco e'l nero non sono veri colori, ma sono alteratione delli
altri colori_" (ed. JANITSCHEK, p. 67; Vienna 1877).], when it is
seen in the open air and high up, all its shadows are bluish; and
this is caused, according to the 4th [prop.], which says: the
surface of every opaque body assumes the hue of the surrounding
objects. Now this white [body] being deprived of the light of the
sun by the interposition of some body between the sun and itself,
all that portion of it which is exposed to the sun and atmosphere
assumes the colour of the sun and atmosphere; the side on which the
sun does not fall remains in shadow and assumes the hue of the
atmosphere. And if this white object did not reflect the green of
the fields all the way to the horizon nor get the brightness of the
horizon itself, it would certainly appear simply of the same hue as
the atmosphere.

On gradations in the depth of colours (279. 280).

279.

Since black, when painted next to white, looks no blacker than when
next to black; and white when next to black looks no whiter than
white, as is seen by the images transmitted through a small hole or
by the edges of any opaque screen ...

280.

OF COLOURS.

Of several colours, all equally white, that will look whitest which
is against the darkest background. And black will look intensest
against the whitest background.

And red will look most vivid against the yellowest background; and
the same is the case with all colours when surrounded by their
strongest contrasts.

On the reflection of colours (281-283).

281.

PERSPECTIVE.

Every object devoid of colour in itself is more or less tinged by
the colour [of the object] placed opposite. This may be seen by
experience, inasmuch as any object which mirrors another assumes the
colour of the object mirrored in it. And if the surface thus
partially coloured is white the portion which has a red reflection
will appear red, or any other colour, whether bright or dark.

PERSPECTIVE.

Every opaque and colourless body assumes the hue of the colour
reflected on it; as happens with a white wall.

282.

PERSPECTIVE.

That side of an object in light and shade which is towards the light
transmits the images of its details more distinctly and immediately
to the eye than the side which is in shadow.

PERSPECTIVE.

The solar rays reflected on a square mirror will be thrown back to
distant objects in a circular form.

PERSPECTIVE.

Any white and opaque surface will be partially coloured by
reflections from surrounding objects.

[Footnote 281. 282: The title line of these chapters is in the
original simply _"pro"_, which may be an abbreviation for either
_Propositione_ or _Prospettiva_--taking Prospettiva of course in its
widest sense, as we often find it used in Leonardo's writings. The
title _"pro"_ has here been understood to mean _Prospettiva_, in
accordance with the suggestion afforded by page 10b of this same
MS., where the first section is headed _Prospettiva_ in full (see
No. 94), while the four following sections are headed merely _"pro"_
(see No. 85).]

283.

WHAT PORTION OF A COLOURED SURFACE OUGHT IN REASON TO BE THE MOST
INTENSE.

If _a_ is the light, and _b_ illuminated by it in a direct line,
_c_, on which the light cannot fall, is lighted only by reflection
from _b_ which, let us say, is red. Hence the light reflected from
it, will be affected by the hue of the surface causing it and will
tinge the surface _c_ with red. And if _c_ is also red you will see
it much more intense than _b_; and if it were yellow you would see
there a colour between yellow and red.

On the use of dark and light colours in painting (284--286).

284.

WHY BEAUTIFUL COLOURS MUST BE IN THE [HIGHEST] LIGHT.

Since we see that the quality of colour is known [only] by means of
light, it is to be supposed that where there is most light the true
character of a colour in light will be best seen; and where there is
most shadow the colour will be affected by the tone of that. Hence,
O Painter! remember to show the true quality of colours in bright
lights.

285.

An object represented in white and black will display stronger
relief than in any other way; hence I would remind you O Painter! to
dress your figures in the lightest colours you can, since, if you
put them in dark colours, they will be in too slight relief and
inconspicuous from a distance. And the reason is that the shadows of
all objects are dark. And if you make a dress dark there is little
variety in the lights and shadows, while in light colours there are
many grades.

286.

OF PAINTING.

Colours seen in shadow will display more or less of their natural
brilliancy in proportion as they are in fainter or deeper shadow.

But if these same colours are situated in a well-lighted place, they
will appear brighter in proportion as the light is more brilliant.

THE ADVERSARY.

The variety of colours in shadow must be as great as that of the
colours in the objects in that shadow.

THE ANSWER.

Colours seen in shadow will display less variety in proportion as
the shadows in which they lie are deeper. And evidence of this is to
be had by looking from an open space into the doorways of dark and
shadowy churches, where the pictures which are painted in various
colours all look of uniform darkness.

Hence at a considerable distance all the shadows of different
colours will appear of the same darkness.

It is the light side of an object in light and shade which shows the
true colour.

On the colours of the rainbow (287. 288).

287.

Treat of the rainbow in the last book on Painting, but first write
the book on colours produced by the mixture of other colours, so as
to be able to prove by those painters' colours how the colours of
the rainbow are produced.

288.

WHETHER THE COLOURS OF THE RAINBOW ARE PRODUCED BY THE SUN.

The colours of the rainbow are not produced by the sun, for they
occur in many ways without the sunshine; as may be seen by holding a
glass of water up to the eye; when, in the glass--where there are
those minute bubbles always seen in coarse glass--each bubble, even
though the sun does not fall on it, will produce on one side all the
colours of the rainbow; as you may see by placing the glass between
the day light and your eye in such a way as that it is close to the
eye, while on one side the glass admits the [diffused] light of the
atmosphere, and on the other side the shadow of the wall on one side
of the window; either left or right, it matters not which. Then, by
turning the glass round you will see these colours all round the
bubbles in the glass &c. And the rest shall be said in its place.

THAT THE EYE HAS NO PART IN PRODUCING THE COLOURS OF THE RAINBOW.

In the experiment just described, the eye would seem to have some
share in the colours of the rainbow, since these bubbles in the
glass do not display the colours except through the medium of the
eye. But, if you place the glass full of water on the window sill,
in such a position as that the outer side is exposed to the sun's
rays, you will see the same colours produced in the spot of light
thrown through the glass and upon the floor, in a dark place, below
the window; and as the eye is not here concerned in it, we may
evidently, and with certainty pronounce that the eye has no share in
producing them.

OF THE COLOURS IN THE FEATHERS OF CERTAIN BIRDS.

There are many birds in various regions of the world on whose
feathers we see the most splendid colours produced as they move, as
we see in our own country in the feathers of peacocks or on the
necks of ducks or pigeons, &c.

Again, on the surface of antique glass found underground and on the
roots of turnips kept for some time at the bottom of wells or other
stagnant waters [we see] that each root displays colours similar to
those of the real rainbow. They may also be seen when oil has been
placed on the top of water and in the solar rays reflected from the
surface of a diamond or beryl; again, through the angular facet of a
beryl every dark object against a background of the atmosphere or
any thing else equally pale-coloured is surrounded by these rainbow
colours between the atmosphere and the dark body; and in many other
circumstances which I will not mention, as these suffice for my
purpose.

_VI._

_'Prospettiva de' colri' (Perspective of Colour)_

_and_

_'Prospettiva aerea' (Aerial Perspective)._

_Leonardo distinctly separates these branches of his subject, as may
be seen in the beginning of No._ 295. _Attempts have been made to
cast doubts on the results which Leonardo arrived at by experiment
on the perspective of colour, but not with justice, as may be seen
from the original text of section_ 294.

_The question as to the composition of the atmosphere, which is
inseparable from a discussion on Aerial Perspective, forms a
separate theory which is treated at considerable length. Indeed the
author enters into it so fully that we cannot escape the conviction
that he must have dwelt with particular pleasure on this part of his
subject, and that he attached great importance to giving it a
character of general applicability._

General rules (289--291).

289.

The variety of colour in objects cannot be discerned at a great
distance, excepting in those parts which are directly lighted up by
the solar rays.

290.

As to the colours of objects: at long distances no difference is
perceptible in the parts in shadow.

291.

OF THE VISIBILITY OF COLOURS.

Which colour strikes most? An object at a distance is most
conspicuous, when it is lightest, and the darkest is least visible.

An exceptional case.

292.

Of the edges [outlines] of shadows. Some have misty and ill defined
edges, others distinct ones.

No opaque body can be devoid of light and shade, except it is in a
mist, on ground covered with snow, or when snow is falling on the
open country which has no light on it and is surrounded with
darkness.

And this occurs [only] in spherical bodies, because in other bodies
which have limbs and parts, those sides of limbs which face each
other reflect on each other the accidental [hue and tone] of their
surface.

An experiment.

293.

ALL COLOURS ARE AT A DISTANCE UNDISTINGUISHABLE AND UNDISCERNIBLE.

All colours at a distance are undistinguishable in shadow, because
an object which is not in the highest light is incapable of
transmitting its image to the eye through an atmosphere more
luminous than itself; since the lesser brightness must be absorbed
by the greater. For instance: We, in a house, can see that all the
colours on the surface of the walls are clearly and instantly
visible when the windows of the house are open; but if we were to go
out of the house and look in at the windows from a little distance
to see the paintings on those walls, instead of the paintings we
should see an uniform deep and colourless shadow.

The practice of the prospettiva de colori.

294.

HOW A PAINTER SHOULD CARRY OUT THE PERSPECTIVE OF COLOUR IN
PRACTICE.

In order to put into practice this perspective of the variation and
loss or diminution of the essential character of colours, observe at
every hundred braccia some objects standing in the landscape, such
as trees, houses, men and particular places. Then in front of the
first tree have a very steady plate of glass and keep your eye very
steady, and then, on this plate of glass, draw a tree, tracing it
over the form of that tree. Then move it on one side so far as that
the real tree is close by the side of the tree you have drawn; then
colour your drawing in such a way as that in colour and form the two
may be alike, and that both, if you close one eye, seem to be
painted on the glass and at the same distance. Then, by the same
method, represent a second tree, and a third, with a distance of a
hundred braccia between each. And these will serve as a standard and
guide whenever you work on your own pictures, wherever they may
apply, and will enable you to give due distance in those works. [14]
But I have found that as a rule the second is 4/5 of the first when
it is 20 braccia beyond it.

[Footnote: This chapter is one of those copied in the Manuscript of
the Vatican library Urbinas 1270, and the original text is rendered
here with no other alterations, but in the orthography. H. LUDWIG,
in his edition of this copy translates lines 14 and 15 thus: "_Ich
finde aber als Regel, dass der zweite um vier Funftel des ersten
abnimmt, wenn er namlich zwanzig Ellen vom ersten entfernt ist
(?)"_. He adds in his commentary: "_Das Ende der Nummer ist wohl
jedenfalls verstummelt_". However the translation given above shows
that it admits of a different rendering.]

The rules of aerial perspective (295--297).

295.

OF AERIAL PERSPECTIVE.

There is another kind of perspective which I call Aerial
Perspective, because by the atmosphere we are able to distinguish
the variations in distance of different buildings, which appear
placed on a single line; as, for instance, when we see several
buildings beyond a wall, all of which, as they appear above the top
of the wall, look of the same size, while you wish to represent them
in a picture as more remote one than another and to give the effect
of a somewhat dense atmosphere. You know that in an atmosphere of
equal density the remotest objects seen through it, as mountains, in
consequence of the great quantity of atmosphere between your eye and
them--appear blue and almost of the same hue as the atmosphere
itself [Footnote 10: _quado il sole e per leuante_ (when the sun is
in the East). Apparently the author refers here to morning light in
general. H. LUDWIG however translates this passage from the Vatican
copy "_wenn namlich die Sonne (dahinter) im Osten steht_".] when the
sun is in the East [Footnote 11: See Footnote 10]. Hence you must
make the nearest building above the wall of its real colour, but the
more distant ones make less defined and bluer. Those you wish should
look farthest away you must make proportionately bluer; thus, if one
is to be five times as distant, make it five times bluer. And by
this rule the buildings which above a [given] line appear of the
same size, will plainly be distinguished as to which are the more
remote and which larger than the others.

296.

The medium lying between the eye and the object seen, tinges that
object with its colour, as the blueness of the atmosphere makes the
distant mountains appear blue and red glass makes objects seen
beyond it, look red. The light shed round them by the stars is
obscured by the darkness of the night which lies between the eye and
the radiant light of the stars.

297.

Take care that the perspective of colour does not disagree with the
size of your objects, hat is to say: that the colours diminish from
their natural [vividness] in proportion as the objects at various
distances dimmish from their natural size.

On the relative density of the atmosphere (298--290).

298.

WHY THE ATMOSPHERE MUST BE REPRESENTED AS PALER TOWARDS THE LOWER
PORTION.

Because the atmosphere is dense near the earth, and the higher it is
the rarer it becomes. When the sun is in the East if you look
towards the West and a little way to the South and North, you will
see that this dense atmosphere receives more light from the sun than
the rarer; because the rays meet with greater resistance. And if the
sky, as you see it, ends on a low plain, that lowest portion of the
sky will be seen through a denser and whiter atmosphere, which will
weaken its true colour as seen through that medium, and there the
sky will look whiter than it is above you, where the line of sight
travels through a smaller space of air charged with heavy vapour.
And if you turn to the East, the atmosphere will appear darker as
you look lower down because the luminous rays pass less freely
through the lower atmosphere.

299.

OF THE MODE OF TREATING REMOTE OBJECTS IN PAINTING.

It is easy to perceive that the atmosphere which lies closest to the
level ground is denser than the rest, and that where it is higher
up, it is rarer and more transparent. The lower portions of large
and lofty objects which are at a distance are not much seen, because
you see them along a line which passes through a denser and thicker
section of the atmosphere. The summits of such heights are seen
along a line which, though it starts from your eye in a dense
atmosphere, still, as it ends at the top of those lofty objects,
ceases in a much rarer atmosphere than exists at their base; for
this reason the farther this line extends from your eye, from point
to point the atmosphere becomes more and more rare. Hence, O
Painter! when you represent mountains, see that from hill to hill
the bases are paler than the summits, and in proportion as they
recede beyond each other make the bases paler than the summits;
while, the higher they are the more you must show of their true form
and colour.

On the colour of the atmosphere (300-307).

300.

OF THE COLOUR OF THE ATMOSPHERE.

I say that the blueness we see in the atmosphere is not intrinsic
colour, but is caused by warm vapour evaporated in minute and
insensible atoms on which the solar rays fall, rendering them
luminous against the infinite darkness of the fiery sphere which
lies beyond and includes it. And this may be seen, as I saw it by
any one going up [Footnote 5: With regard to the place spoken of as
_M'oboso_ (compare No. 301 line 20) its identity will be discussed
under Leonardo's Topographical notes in Vol. II.] Monboso, a peak of
the Alps which divide France from Italy. The base of this mountain
gives birth to the four rivers which flow in four different
directions through the whole of Europe. And no mountain has its base
at so great a height as this, which lifts itself almost above the
clouds; and snow seldom falls there, but only hail in the summer,
when the clouds are highest. And this hail lies [unmelted] there, so
that if it were not for the absorption of the rising and falling
clouds, which does not happen twice in an age, an enormous mass of
ice would be piled up there by the hail, and in the middle of July I
found it very considerable. There I saw above me the dark sky, and
the sun as it fell on the mountain was far brighter here than in the
plains below, because a smaller extent of atmosphere lay between the
summit of the mountain and the sun. Again as an illustration of the
colour of the atmosphere I will mention the smoke of old and dry
wood, which, as it comes out of a chimney, appears to turn very
blue, when seen between the eye and the dark distance. But as it
rises, and comes between the eye and the bright atmosphere, it at
once shows of an ashy grey colour; and this happens because it no
longer has darkness beyond it, but this bright and luminous space.
If the smoke is from young, green wood, it will not appear blue,
because, not being transparent and being full of superabundant
moisture, it has the effect of condensed clouds which take distinct
lights and shadows like a solid body. The same occurs with the
atmosphere, which, when overcharged with moisture appears white, and
the small amount of heated moisture makes it dark, of a dark blue
colour; and this will suffice us so far as concerns the colour of
the atmosphere; though it might be added that, if this transparent
blue were the natural colour of the atmosphere, it would follow that
wherever a larger mass air intervened between the eye and the
element of fire, the azure colour would be more intense; as we see
in blue glass and in sapphires, which are darker in proportion as
they are larger. But the atmosphere in such circumstances behaves in
an opposite manner, inasmuch as where a greater quantity of it lies
between the eye and the sphere of fire, it is seen much whiter. This
occurs towards the horizon. And the less the extent of atmosphere
between the eye and the sphere of fire, the deeper is the blue
colour, as may be seen even on low plains. Hence it follows, as I
say, that the atmosphere assumes this azure hue by reason of the
particles of moisture which catch the rays of the sun. Again, we may
note the difference in particles of dust, or particles of smoke, in
the sun beams admitted through holes into a dark chamber, when the
former will look ash grey and the thin smoke will appear of a most
beautiful blue; and it may be seen again in in the dark shadows of
distant mountains when the air between the eye and those shadows
will look very blue, though the brightest parts of those mountains
will not differ much from their true colour. But if any one wishes
for a final proof let him paint a board with various colours, among
them an intense black; and over all let him lay a very thin and
transparent [coating of] white. He will then see that this
transparent white will nowhere show a more beautiful blue than over
the black--but it must be very thin and finely ground.

[Footnote 7: _reta_ here has the sense of _malanno_.]

301.

Experience shows us that the air must have darkness beyond it and
yet it appears blue. If you produce a small quantity of smoke from
dry wood and the rays of the sun fall on this smoke, and if you then
place behind the smoke a piece of black velvet on which the sun does
not shine, you will see that all the smoke which is between the eye
and the black stuff will appear of a beautiful blue colour. And if
instead of the velvet you place a white cloth smoke, that is too
thick smoke, hinders, and too thin smoke does not produce, the
perfection of this blue colour. Hence a moderate amount of smoke
produces the finest blue. Water violently ejected in a fine spray
and in a dark chamber where the sun beams are admitted produces
these blue rays and the more vividly if it is distilled water, and
thin smoke looks blue. This I mention in order to show that the
blueness of the atmosphere is caused by the darkness beyond it, and
these instances are given for those who cannot confirm my experience
on Monboso.

302.

When the smoke from dry wood is seen between the eye of the
spectator and some dark space [or object], it will look blue. Thus
the sky looks blue by reason of the darkness beyond it. And if you
look towards the horizon of the sky, you will see the atmosphere is
not blue, and this is caused by its density. And thus at each
degree, as you raise your eyes above the horizon up to the sky over
your head, you will see the atmosphere look darker [blue] and this
is because a smaller density of air lies between your eye and the
[outer] darkness. And if you go to the top of a high mountain the
sky will look proportionately darker above you as the atmosphere
becomes rarer between you and the [outer] darkness; and this will be
more visible at each degree of increasing height till at last we
should find darkness.

That smoke will look bluest which rises from the driest wood and
which is nearest to the fire and is seen against the darkest
background, and with the sunlight upon it.

303.

A dark object will appear bluest in proportion as it has a greater
mass of luminous atmosphere between it and the eye. As may be seen
in the colour of the sky.

304.

The atmosphere is blue by reason of the darkness above it because
black and white make blue.

305.

In the morning the mist is denser above than below, because the sun
draws it upwards; hence tall buildings, even if the summit is at the
same distance as the base have the summit invisible. Therefore,
also, the sky looks darkest [in colour] overhead, and towards the
horizon it is not blue but rather between smoke and dust colour.

The atmosphere, when full of mist, is quite devoid of blueness, and
only appears of the colour of clouds, which shine white when the
weather is fine. And the more you turn to the west the darker it
will be, and the brighter as you look to the east. And the verdure
of the fields is bluish in a thin mist, but grows grey in a dense
one.

The buildings in the west will only show their illuminated side,
where the sun shines, and the mist hides the rest. When the sun
rises and chases away the haze, the hills on the side where it lifts
begin to grow clearer, and look blue, and seem to smoke with the
vanishing mists; and the buildings reveal their lights and shadows;
through the thinner vapour they show only their lights and through
the thicker air nothing at all. This is when the movement of the
mist makes it part horizontally, and then the edges of the mist will
be indistinct against the blue of the sky, and towards the earth it
will look almost like dust blown up. In proportion as the atmosphere
is dense the buildings of a city and the trees in a landscape will
look fewer, because only the tallest and largest will be seen.

Darkness affects every thing with its hue, and the more an object
differs from darkness, the more we see its real and natural colour.
The mountains will look few, because only those will be seen which
are farthest apart; since, at such a distance, the density increases
to such a degree that it causes a brightness by which the darkness
of the hills becomes divided and vanishes indeed towards the top.
There is less [mist] between lower and nearer hills and yet little
is to be distinguished, and least towards the bottom.

306.

The surface of an object partakes of the colour of the light which
illuminates it; and of the colour of the atmosphere which lies
between the eye and that object, that is of the colour of the
transparent medium lying between the object and the eye; and among
colours of a similar character the second will be of the same tone
as the first, and this is caused by the increased thickness of the
colour of the medium lying between the object and the eye.

307. OF PAINTING.

Of various colours which are none of them blue that which at a great
distance will look bluest is the nearest to black; and so,
conversely, the colour which is least like black will at a great
distance best preserve its own colour.

Hence the green of fields will assume a bluer hue than yellow or
white will, and conversely yellow or white will change less than
green, and red still less.

_VII._

_On the Proportions and on the Movements of the Human Figure._

_Leonardo's researches on the proportions and movements of the human
figure must have been for the most part completed and written before
the year_ 1498; _for LUCA PACIOLO writes, in the dedication to
Ludovico il Moro, of his book_ Divina Proportione, _which was
published in that year:_ "Leonardo da venci ... hauedo gia co tutta
diligetia al degno libro de pictura e movimenti humani posto fine".

_The selection of Leonardo's axioms contained in the Vatican copy
attributes these words to the author:_ "e il resto si dira nella
universale misura del huomo". (_MANZI, p. 147; LUDWIG, No. 264_).
_LOMAZZO, again, in his_ Idea del Tempio della Pittura Milano 1590,
cap. IV, _says:_ "Lionardo Vinci ... dimostro anco in figura tutte
le proporzioni dei membri del corpo umano".

_The Vatican copy includes but very few sections of the_ "Universale
misura del huomo" _and until now nothing has been made known of the
original MSS. on the subject which have supplied the very extensive
materials for this portion of the work. The collection at Windsor,
belonging to her Majesty the Queen, includes by far the most
important part of Leonardo's investigations on this subject,
constituting about half of the whole of the materials here
published; and the large number of original drawings adds greatly to
the interest which the subject itself must command. Luca Paciolo
would seem to have had these MSS. (which I have distinguished by the
initials W. P.) in his mind when he wrote the passage quoted above.
Still, certain notes of a later date--such as Nos. 360, 362 and 363,
from MS. E, written in 1513--14, sufficiently prove that Leonardo did
not consider his earlier studies on the Proportions and Movements of
the Human Figure final and complete, as we might suppose from Luca
Paciolo's statement. Or else he took the subject up again at a
subsequent period, since his former researches had been carried on
at Milan between 1490 and 1500. Indeed it is highly probable that
the anatomical studies which he was pursuing zvith so much zeal
between 1510--16 should have led him to reconsider the subject of
Proportion.

Preliminary observations (308. 309).

308.

Every man, at three years old is half the full height he will grow
to at last.

309.

If a man 2 braccia high is too small, one of four is too tall, the
medium being what is admirable. Between 2 and 4 comes 3; therefore
take a man of 3 braccia in height and measure him by the rule I will
give you. If you tell me that I may be mistaken, and judge a man to
be well proportioned who does not conform to this division, I answer
that you must look at many men of 3 braccia, and out of the larger
number who are alike in their limbs choose one of those who are most
graceful and take your measurements. The length of the hand is 1/3
of a braccio [8 inches] and this is found 9 times in man. And the
face [Footnote 7: The account here given of the _braccio_ is of
importance in understanding some of the succeeding chapters. _Testa_
must here be understood to mean the face. The statements in this
section are illustrated in part on Pl. XI.] is the same, and from
the pit of the throat to the shoulder, and from the shoulder to the
nipple, and from one nipple to the other, and from each nipple to
the pit of the throat.

Proportions of the head and face (310-318).

310.

The space between the parting of the lips [the mouth] and the base
of the nose is one-seventh of the face.

The space from the mouth to the bottom of the chin _c d_ is the
fourth part of the face and equal to the width of the mouth.

The space from the chin to the base of the nose _e f_ is the third
part of the face and equal to the length of the nose and to the
forehead.

The distance from the middle of the nose to the bottom of the chin
_g h_, is half the length of the face.

The distance from the top of the nose, where the eyebrows begin, to
the bottom of the chin, _i k_, is two thirds of the face.

The space from the parting of the lips to the top of the chin _l m_,
that is where the chin ends and passes into the lower lip of the
mouth, is the third of the distance from the parting of the lips to
the bottom of the chin and is the twelfth part of the face. From the
top to the bottom of the chin _m n_ is the sixth part of the face
and is the fifty fourth part of a man's height.

From the farthest projection of the chin to the throat _o p_ is
equal to the space between the mouth and the bottom of the chin, and
a fourth of the face.

The distance from the top of the throat to the pit of the throat
below _q r_ is half the length of the face and the eighteenth part
of a man's height.

From the chin to the back of the neck _s t_, is the same distance as
between the mouth and the roots of the hair, that is three quarters
of the head.

From the chin to the jaw bone _v x_ is half the head and equal to
the thickness of the neck in profile.

The thickness of the head from the brow to the nape is once and 3/4
that of the neck.

[Footnote: The drawings to this text, lines 1-10 are on Pl. VII, No.
I. The two upper sketches of heads, Pl. VII, No. 2, belong to lines
11-14, and in the original are placed immediately below the sketches
reproduced on Pl. VII, No. 1.]

311.

The distance from the attachment of one ear to the other is equal to
that from the meeting of the eyebrows to the chin, and in a fine
face the width of the mouth is equal to the length from the parting
of the lips to the bottom of the chin.

312.

The cut or depression below the lower lip of the mouth is half way
between the bottom of the nose and the bottom of the chin.

The face forms a square in itself; that is its width is from the
outer corner of one eye to the other, and its height is from the
very top of the nose to the bottom of the lower lip of the mouth;
then what remains above and below this square amounts to the height
of such another square, _a_ _b_ is equal to the space between _c_
_d_; _d_ _n_ in the same way to _n_ _c_, and likewise _s_ _r_, _q_
_p_, _h_ _k_ are equal to each other.

It is as far between _m_ and _s_ as from the bottom of the nose to
the chin. The ear is exactly as long as the nose. It is as far from
_x_ to _j_ as from the nose to the chin. The parting of the mouth
seen in profile slopes to the angle of the jaw. The ear should be as
high as from the bottom of the nose to the top of the eye-lid. The
space between the eyes is equal to the width of an eye. The ear is
over the middle of the neck, when seen in profile. The distance from
4 to 5 is equal to that from s_ to _r_.

[Footnote: See Pl. VIII, No. I, where the text of lines 3-13 is also
given in facsimile.]

313.

(_a_ _b_) is equal to (_c_ _d_).

[Footnote: See Pl. VII, No. 3. Reference may also be made here to
two pen and ink drawings of heads in profile with figured
measurements, of which there is no description in the MS. These are
given on Pl. XVII, No. 2.--A head, to the left, with part of the
torso [W. P. 5a], No. 1 on the same plate is from MS. A 2b and in
the original occurs on a page with wholly irrelevant text on matters
of natural history. M. RAVAISSON in his edition of the Paris MS. A
has reproduced this head and discussed it fully [note on page 12];
he has however somewhat altered the original measurements. The
complicated calculations which M. RAVAISSON has given appear to me
in no way justified. The sketch, as we see it, can hardly have been
intended for any thing more than an experimental attempt to
ascertain relative proportions. We do not find that Leonardo made
use of circular lines in any other study of the proportions of the
human head. At the same time we see that the proportions of this
sketch are not in accordance with the rules which he usually
observed (see for instance No. 310).]

The head _a_ _f_ 1/6 larger than _n_ _f_.

315.

From the eyebrow to the junction of the lip with the chin, and the
angle of the jaw and the upper angle where the ear joins the temple
will be a perfect square. And each side by itself is half the head.

The hollow of the cheek bone occurs half way between the tip of the
nose and the top of the jaw bone, which is the lower angle of the
setting on of the ear, in the frame here represented.

From the angle of the eye-socket to the ear is as far as the length
of the ear, or the third of the face.

[Footnote: See Pl. IX. The text, in the original is written behind
the head. The handwriting would seem to indicate a date earlier than
1480. On the same leaf there is a drawing in red chalk of two
horsemen of which only a portion of the upper figure is here
visible. The whole leaf measures 22 1/2 centimetres wide by 29 long,
and is numbered 127 in the top right-hand corner.]

316.

From _a_ to _b_--that is to say from the roots of the hair in front
to the top of the head--ought to be equal to _c_ _d_;--that is from
the bottom of the nose to the meeting of the lips in the middle of
the mouth. From the inner corner of the eye _m_ to the top of the
head _a_ is as far as from _m_ down to the chin _s_. _s_ _c_ _f_ _b_
are all at equal distances from each other.

[Footnote: The drawing in silver-point on bluish tinted paper--Pl.
X--which belongs to this chapter has been partly drawn over in ink
by Leonardo himself.]

317.

From the top of the head to the bottom of the chin is 1/9, and from
the roots of the hair to the chin is 1/9 of the distance from the
roots of the hair to the ground. The greatest width of the face is
equal to the space between the mouth and the roots of the hair and
is 1/12 of the whole height. From the top of the ear to the top of
the head is equal to the distance from the bottom of the chin to the
lachrymatory duct of the eye; and also equal to the distance from
the angle of the chin to that of the jaw; that is the 1/16 of the
whole. The small cartilage which projects over the opening of the
ear towards the nose is half-way between the nape and the eyebrow;
the thickness of the neck in profile is equal to the space between
the chin and the eyes, and to the space between the chin and the
jaw, and it is 1/18 of the height of the man.

318.

_a b_, _c d_, _e f_, _g h_, _i k_ are equal to each other in size
excepting that _d f_ is accidental.

[Footnote: See Pl. XI.]

Proportions of the head seen in front (319-321).

319.

_a n o f_ are equal to the mouth.

_a c_ and _a f_ are equal to the space between one eye and the
other.

_n m o f q r_ are equal to half the width of the eye lids, that is
from the inner [lachrymatory] corner of the eye to its outer corner;
and in like manner the division between the chin and the mouth; and
in the same way the narrowest part of the nose between the eyes. And
these spaces, each in itself, is the 19th part of the head, _n o_ is
equal to the length of the eye or of the space between the eyes.

_m c_ is 1/3 of _n m_ measuring from the outer corner of the eyelids
to the letter _c_. _b s_ will be equal to the width of the nostril.

[Footnote: See Pl. XII.]

320.

The distance between the centres of the pupils of the eyes is 1/3 of
the face. The space between the outer corners of the eyes, that is
where the eye ends in the eye socket which contains it, thus the
outer corners, is half the face.

The greatest width of the face at the line of the eyes is equal to
the distance from the roots of the hair in front to the parting of
the lips.

[Footnote: There are, with this section, two sketches of eyes, not
reproduced here.]

321.

The nose will make a double square; that is the width of the nose at
the nostrils goes twice into the length from the tip of the nose to
the eyebrows. And, in the same way, in profile the distance from the
extreme side of the nostril where it joins the cheek to the tip of
the nose is equal to the width of the nose in front from one nostril
to the other. If you divide the whole length of the nose--that is
from the tip to the insertion of the eyebrows, into 4 equal parts,
you will find that one of these parts extends from the tip of the
nostrils to the base of the nose, and the upper division lies
between the inner corner of the eye and the insertion of the
eyebrows; and the two middle parts [together] are equal to the
length of the eye from the inner to the outer corner.

[Footnote: The two bottom sketches on Pl. VII, No. 4 face the six
lines of this section,--With regard to the proportions of the head
in profile see No. 312.]

322.

The great toe is the sixth part of the foot, taking the measure in
profile, on the inside of the foot, from where this toe springs from
the ball of the sole of the foot to its tip _a b_; and it is equal
to the distance from the mouth to the bottom of the chin. If you
draw the foot in profile from the outside, make the little toe begin
at three quarters of the length of the foot, and you will find the
same distance from the insertion of this toe as to the farthest
prominence of the great toe.

323.

For each man respectively the distance between _a b_ is equal to _c
d_.

324.

Relative proportion of the hand and foot.

The foot is as much longer than the hand as the thickness of the arm
at the wrist where it is thinnest seen facing.

Again, you will find that the foot is as much longer than the hand
as the space between the inner angle of the little toe to the last
projection of the big toe, if you measure along the length of the
foot.

The palm of the hand without the fingers goes twice into the length
of the foot without the toes.

If you hold your hand with the fingers straight out and close
together you will find it to be of the same width as the widest part
of the foot, that is where it is joined onto the toes.

And if you measure from the prominence of the inner ancle to the end
of the great toe you will find this measure to be as long as the
whole hand.

From the top angle of the foot to the insertion of the toes is equal
to the hand from wrist joint to the tip of the thumb.

The smallest width of the hand is equal to the smallest width of the
foot between its joint into the leg and the insertion of the toes.

The width of the heel at the lower part is equal to that of the arm
where it joins the hand; and also to the leg where it is thinnest
when viewed in front.

The length of the longest toe, from its first division from the
great toe to its tip is the fourth of the foot from the centre of
the ancle bone to the tip, and it is equal to the width of the
mouth. The distance between the mouth and the chin is equal to that
of the knuckles and of the three middle fingers and to the length of
their first joints if the hand is spread, and equal to the distance
from the joint of the thumb to the outset of the nails, that is the
fourth part of the hand and of the face.

The space between the extreme poles inside and outside the foot
called the ancle or ancle bone _a b_ is equal to the space between
the mouth and the inner corner of the eye.

325.

The foot, from where it is attached to the leg, to the tip of the
great toe is as long as the space between the upper part of the chin
and the roots of the hair _a b_; and equal to five sixths of the
face.

326.

_a d_ is a head's length, _c b_ is a head's length. The four smaller
toes are all equally thick from the nail at the top to the bottom,
and are 1/13 of the foot.

[Footnote: See Pl. XIV, No. 1, a drawing of a foot with the text in
three lines below it.]

327.

The whole length of the foot will lie between the elbow and the
wrist and between the elbow and the inner angle of the arm towards
the breast when the arm is folded. The foot is as long as the whole
head of a man, that is from under the chin to the topmost part of
the head[Footnote 2: _nel modo che qui i figurato_. See Pl. VII, No.
4, the upper figure. The text breaks off at the end of line 2 and
the text given under No. 321 follows below. It may be here remarked
that the second sketch on W. P. 311 has in the original no
explanatory text.] in the way here figured.

Proportions of the leg (328-331).

328.

The greatest thickness of the calf of the leg is at a third of its
height _a b_, and is a twentieth part thicker than the greatest
thickness of the foot.

_a c_ is half of the head, and equal to _d b_ and to the insertion
of the five toes _e f_. _d k_ diminishes one sixth in the leg _g h_.
_g h_ is 1/3 of the head; _m n_ increases one sixth from _a e_ and
is 7/12 of the head, _o p_ is 1/10 less than _d k_ and is 6/17 of
the head. _a_ is at half the distance between _b q_, and is 1/4 of
the man. _r_ is half way between _s_ and _b_[Footnote 11: _b_ is
here and later on measured on the right side of the foot as seen by
the spectator.]. The concavity of the knee outside _r_ is higher
than that inside _a_. The half of the whole height of the leg from
the foot _r_, is half way between the prominence _s_ and the ground
_b_. _v_ is half way between _t_ and _b_. The thickness of the thigh
seen in front is equal to the greatest width of the face, that is
2/3 of the length from the chin to the top of the head; _z r_ is 5/6
of 7 to _v_; _m n_ is equal to 7 _v_ and is 1/4 of _r b_, _x y_ goes
3 times into _r b_, and into _r s_.

[Footnote 22-35: The sketch illustrating these lines is on Pl. XIII,
No. 2.]

[Footnote 22: a b _entra in_ c f 6 _e_ 6 _in_ c n. Accurate
measurement however obliges us to read 7 for 6.] _a b_ goes six
times into _c f_ and six times into _c n_ and is equal to _g h_; _i
k l m_ goes 4 times into _d f_, and 4 times into _d n_ and is 3/7 of
the foot; _p q r s_ goes 3 times into _d f, and 3 times into _b n_;
[Footnote: 25. _y_ is not to be found on the diagram and _x_ occurs
twice; this makes the passage very obscure.] _x y_ is 1/8 of _x f_
and is equal to _n q_. 3 7 is 1/9 of _n f_; 4 5 is 1/10 of _n f_
[Footnote: 22-27. Compare with this lines 18-24 of No. 331, and the
sketch of a leg in profile Pl. XV.].

I want to know how much a man increases in height by standing on
tip-toe and how much _p g_ diminishes by stooping; and how much it
increases at _n q_ likewise in bending the foot.

[Footnote 34: _e f_ 4 _dal cazo_. By reading _i_ for _e_ the sense
of this passage is made clear.] _e f_ is four times in the distance
between the genitals and the sole of the foot; [Footnote 35: 2 is
not to be found in the sketch which renders the passage obscure. The
two last lines are plainly legible in the facsimile.] 3 7 is six
times from 3 to 2 and is equal to _g h_ and _i k_.

[Footnote: The drawing of a leg seen in front Pl. XIII, No. 1
belongs to the text from lines 3-21. The measurements in this
section should be compared with the text No. 331, lines 1-13, and
the sketch of a leg seen in front on Pl. XV.]

329.

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 [fibula] joins the thigh bone [femur].

330.

_a n b_ are equal; _c n d_ are equal; _n c_ makes two feet; _n d_
makes 2 feet.

[Footnote: See the lower sketch, Pl. XIV, No. 1.]

331.

_m n o_ are equal. The narrowest width of the leg seen in front goes
8 times from the sole of the foot to the joint of the knee, and is
the same width as the arm, seen in front at the wrist, and as the
longest measure of the ear, and as the three chief divisions into
which we divide the face; and this measurement goes 4 times from the
wrist joint of the hand to the point of the elbow. [14] The foot is
as long as the space from the knee between _a_ and _b_; and the
patella of the knee is as long as the leg between _r_ and _s_.

[18] The least thickness of the leg in profile goes 6 times from the
sole of the foot to the knee joint and is the same width as the
space between the outer corner of the eye and the opening of the
ear, and as the thickest part of the arm seen in profile and between
the inner corner of the eye and the insertion of the hair.

_a b c_ [_d_] are all relatively of equal length, _c d_ goes twice
from the sole of the foot to the centre of the knee and the same
from the knee to the hip.

[28]_a b c_ are equal; _a_ to _b_ is 2 feet--that is to say
measuring from the heel to the tip of the great toe.

[Footnote: See Pl. XV. The text of lines 2-17 is to the left of the
front view of the leg, to which it refers. Lines 18-27 are in the
middle column and refer to the leg seen in profile and turned to the
left, on the right hand side of the writing. Lines 20-30 are above,
to the left and apply to the sketch below them.

Some farther remarks on the proportion of the leg will be found in
No. 336, lines 6, 7.]

On the central point of the whole body.

332.

In kneeling down a man will lose the fourth part of his height.

When a man kneels down with his hands folded on his breast the navel
will mark half his height and likewise the points of the elbows.

Half the height of a man who sits--that is from the seat to the top
of the head--will be where the arms fold below the breast, and
below the shoulders. The seated portion--that is from the seat to
the top of the head--will be more than half the man's [whole height]
by the length of the scrotum.

[Footnote: See Pl. VIII, No. 2.]

The relative proportions of the torso and of the whole figure.

333.

The cubit is one fourth of the height of a man and is equal to the
greatest width of the shoulders. From the joint of one shoulder to
the other is two faces and is equal to the distance from the top of
the breast to the navel. [Footnote 9: _dalla detta somita_. It would
seem more accurate to read here _dal detto ombilico_.] From this
point to the genitals is a face's length.

[Footnote: Compare with this the sketches on the other page of the
same leaf. Pl. VIII, No. 2.]

The relative proportions of the head and of the torso.

334.

From the roots of the hair to the top of the breast _a b_ is the
sixth part of the height of a man and this measure is equal.

From the outside part of one shoulder to the other is the same
distance as from the top of the breast to the navel and this measure
goes four times from the sole of the foot to the lower end of the
nose.

The [thickness of] the arm where it springs from the shoulder in
front goes 6 times into the space between the two outside edges of
the shoulders and 3 times into the face, and four times into the
length of the foot and three into the hand, inside or outside.

[Footnote: The three sketches Pl. XIV, No. 2 belong to this text.]

The relative proportions of the torso and of the leg (335. 336).

335.

_a b c_ are equal to each other and to the space from the armpit of
the shoulder to the genitals and to the distance from the tip of the
fingers of the hand to the joint of the arm, and to the half of the
breast; and you must know that _c b_ is the third part of the height
of a man from the shoulders to the ground; _d e f_ are equal to each
other and equal to the greatest width of the shoulders.

[Footnote: See Pl. XVI, No. 1.]

336.

--Top of the chin--hip--the insertion of the middle finger. The end
of the calf of the leg on the inside of the thigh.--The end of the
swelling of the shin bone of the leg. [6] The smallest thickness of
the leg goes 3 times into the thigh seen in front.

[Footnote: See Pl. XVII, No. 2, middle sketch.]

The relative proportions of the torso and of the foot.

337.

The torso _a b_ in its thinnest part measures a foot; and from _a_
to _b_ is 2 feet, which makes two squares to the seat--its thinnest
part goes 3 times into the length, thus making 3 squares.

[Footnote: See Pl, VII, No. 2, the lower sketch.]

The proportions of the whole figure (338-341).

338.

A man when he lies down is reduced to 1/9 of his height.

339.

The opening of the ear, the joint of the shoulder, that of the hip
and the ancle are in perpendicular lines; _a n_ is equal to _m o_.

[Footnote: See Pl. XVI, No. 2, the upper sketch.]

340.

From the chin to the roots of the hair is 1/10 of the whole figure.
From the joint of the palm of the hand to the tip of the longest
finger is 1/10. From the chin to the top of the head 1/8; and from
the pit of the stomach to the top of the breast is 1/6, and from the
pit below the breast bone to the top of the head 1/4. From the chin
to the nostrils 1/3 Part of the face, the same from the nostrils to
the brow and from the brow to the roots of the hair, and the foot is
1/6, the elbow 1/4, the width of the shoulders 1/4.

341.

The width of the shoulders is 1/4 of the whole. From the joint of
the shoulder to the hand is 1/3, from the parting of the lips to
below the shoulder-blade is one foot.

The greatest thickness of a man from the breast to the spine is one
8th of his height and is equal to the space between the bottom of
the chin and the top of the head.

The greatest width is at the shoulders and goes 4.

The torso from the front and back.

342.

The width of a man under the arms is the same as at the hips.

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 a man 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 arm pits and the bottom of the buttock.

[Footnote: The lower sketch Pl. XVI, No. 2, is drawn by the side of
line 1.]

Vitruvius' scheme of proportions.

343.

Vitruvius, the architect, says in his work on architecture that the
measurements of the human body are distributed by Nature as follows:
that is that 4 fingers make 1 palm, and 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 his
buildings. If you open your legs so much as to decrease your height
1/14 and spread and raise your arms till your middle fingers touch
the level of the top of your head you must know that the centre of
the outspread limbs will be in the navel and the space between the
legs will be an equilateral triangle.

The length 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 of
a man's height; from the bottom of the chin to the top of his head
is one eighth of his height; from the top of the breast to the top
of his head will be one sixth of a man. From the top of the breast
to the roots of the hair will be the seventh part of the whole man.
From the nipples to the top of the head will be the fourth part of a
man. The greatest width of the shoulders contains in itself the
fourth part of the man. From the elbow to the tip of the hand will
be the fifth part of a man; and from the elbow to the angle of the
armpit will be the eighth part of the man. The whole hand will be
the tenth part of the man; the beginning of the genitals marks the
middle of the man. The foot is the seventh part of the man. From the
sole of the foot to below the knee will be the fourth part of the
man. From below the knee to the beginning of the genitals will be
the fourth part of the man. The distance from the bottom of the chin
to the nose and from the roots of the hair to the eyebrows is, in
each case the same, and like the ear, a third of the face.

[Footnote: See Pl. XVIII. The original leaf is 21 centimetres wide
and 33 1/2 long. At the ends of the scale below the figure are
written the words _diti_ (fingers) and _palmi_ (palms). The passage
quoted from Vitruvius is Book III, Cap. 1, and Leonardo's drawing is
given in the editions of Vitruvius by FRA GIOCONDO (Venezia 1511,
fol., Firenze 1513, 8vo.) and by CESARIANO (Como 1521).]

The arm and head.

344.

From _b_ to _a_ is one head, as well as from _c_ to _a_ and this
happens when the elbow forms a right angle.

[Footnote: See Pl. XLI, No. 1.]

Proportions of the arm (345-349).

345.

From the tip of the longest finger of the hand to the shoulder joint
is four hands or, if you will, four faces.

_a b c_ are equal and each interval is 2 heads.

[Footnote: Lines 1-3 are given on Pl. XV below the front view of the
leg; lines 4 and 5 are below again, on the left side. The lettering
refers to the bent arm near the text.]

346.

The hand from the longest finger to the wrist joint goes 4 times
from the tip of the longest finger to the shoulder joint.

347.

_a b c_ are equal to each other and to the foot and to the space
between the nipple and the navel _d e_ will be the third part of the
whole man.

_f g_ is the fourth part of a man and is equal to _g h_ and measures
a cubit.

[Footnote: See Pl. XIX, No. 1. 1. _mamolino_ (=_bambino_, little
child) may mean here the navel.]

348.

_a b_ goes 4 times into _a c_ and 9 into _a m_. The greatest
thickness of the arm between the elbow and the hand goes 6 times
into _a m_ and is equal to _r f_. The greatest thickness of the arm
between the shoulder and the elbow goes 4 times into _c m_, and is
equal to _h n g_. The smallest thickness of the arm above the elbow
_x y_ is not the base of a square, but is equal to half the space
_h_ 3 which is found between the inner joint of the arm and the
wrist joint.

[11]The width of the wrist goes 12 times into the whole arm; that is
from the tip of the fingers to the shoulder joint; that is 3 times
into the hand and 9 into the arm.

The arm when bent is 4 heads.

The arm from the shoulder to the elbow in bending increases in
length, that is in the length from the shoulder to the elbow, and
this increase is equal to the thickness of the arm at the wrist when
seen in profile. And the space between the bottom of the chin and
the parting of the lips, is equal to the thickness of the 2 middle
fingers, and to the width of the mouth and to the space between the
roots of the hair on the forehead and the top of the head [Footnote:
_Queste cose_. This passage seems to have been written on purpose to
rectify the foregoing lines. The error is explained by the
accompanying sketch of the bones of the arm.]. All these distances
are equal to each other, but they are not equal to the
above-mentioned increase in the arm.

The arm between the elbow and wrist never increases by being bent or
extended.

The arm, from the shoulder to the inner joint when extended.

When the arm is extended, _p n_ is equal to _n a_. And when it is
bent _n a_ diminishes 1/6 of its length and _p n_ does the same. The
outer elbow joint increases 1/7 when bent; and thus by being bent it
increases to the length of 2 heads. And on the inner side, by
bending, it is found that whereas the arm from where it joins the
side to the wrist, was 2 heads and a half, in bending it loses the
half head and measures only two: one from the [shoulder] joint to
the end [by the elbow], and the other to the hand.

The arm when folded will measure 2 faces up to the shoulder from the
elbow and 2 from the elbow to the insertion of the four fingers on
the palm of the hand. The length from the base of the fingers to the
elbow never alters in any position of the arm.

If the arm is extended it decreases by 1/3 of the length between _b_
and _h_; and if--being extended--it is bent, it will increase the
half of _o e_. [Footnote 59-61: The figure sketched in the margin is
however drawn to different proportions.] The length from the
shoulder to the elbow is the same as from the base of the thumb,
inside, to the elbow _a b c_.

[Footnote 62-64: The arm sketch on the margin of the MS. is
identically the same as that given below on Pl. XX which may
therefore be referred to in this place. In line 62 we read therefore
_z c_ for _m n_.] The smallest thickness of the arm in profile _z c_
goes 6 times between the knuckles of the hand and the dimple of the
elbow when extended and 14 times in the whole arm and 42 in the
whole man [64]. The greatest thickness of the arm in profile is
equal to the greatest thickness of the arm in front; but the first
is placed at a third of the arm from the shoulder joint to the elbow
and the other at a third from the elbow towards the hand.

[Footnote: Compare Pl. XVII. Lines 1-10 and 11-15 are written in two
columns below the extended arm, and at the tips of the fingers we
find the words: _fine d'unghie_ (ends of the nails). Part of the
text--lines 22 to 25--is visible by the side of the sketches on Pl.
XXXV, No. 1.]

349.

From the top of the shoulder to the point of the elbow is as far as
from that point to the joints of the four fingers with the palm of
the hand, and each is 2 faces.

[5]_a e_ is equal to the palm of the hand, _r f_ and _o g_ are equal
to half a head and each goes 4 times into _a b_ and _b c_. From _c_
to _m_ is 1/2 a head; _m n_ is 1/3 of a head and goes 6 times into
_c b_ and into _b a_; _a b_ loses 1/7 of its length when the arm is
extended; _c b_ never alters; _o_ will always be the middle point
between _a_ and _s_.

_y l_ is the fleshy part of the arm and measures one head; and when
the arm is bent this shrinks 2/5 of its length; _o a_ in bending
loses 1/6 and so does _o r_.

_a b_ is 1/7 of _r c_. _f s_ will be 1/8 of _r c_, and each of those
2 measurements is the largest of the arm; _k h_ is the thinnest part
between the shoulder and the elbow and it is 1/8 of the whole arm _r
c_; _o p_ is 1/5 of _r l_; _c z_ goes 13 times into _r c_.

[Footnote: See Pl. XX where the text is also seen from lines 5-23.]

The movement of the arm (350-354).

350.

In the innermost bend of the joints of every limb the reliefs are
converted into a hollow, and likewise every hollow of the innermost
bends becomes a convexity when the limb is straightened to the
utmost. And in this very great mistakes are often made by those who
have insufficient knowledge and trust to their own invention and do
not have recourse to the imitation of nature; and these variations
occur more in the middle of the sides than in front, and more at the
back than at the sides.

351.

When the arm is bent at an angle at the elbow, it will produce some
angle; the more acute the angle is, the more will the muscles within
the bend be shortened; while the muscles outside will become of
greater length than before. As is shown in the example; _d c e_ will
shrink considerably; and _b n_ will be much extended.

[Footnote: See Pl. XIX, No. 2.]

352.

OF PAINTING.

The arm, as it turns, thrusts back its shoulder towards the middle
of the back.

353.

The principal movements of the hand are 10; that is forwards,
backwards, to right and to left, in a circular motion, up or down,
to close and to open, and to spread the fingers or to press them
together.

354.

OF THE MOTIONS OF THE FINGERS.

The movements of the fingers principally consist in extending and
bending them. This extension and bending vary in manner; that is,
sometimes they bend altogether at the first joint; sometimes they
bend, or extend, half way, at the 2nd joint; and sometimes they bend
in their whole length and in all the three joints at once. If the 2
first joints are hindered from bending, then the 3rd joint can be
bent with greater ease than before; it can never bend of itself, if
the other joints are free, unless all three joints are bent. Besides
all these movements there are 4 other principal motions of which 2
are up and down, the two others from side to side; and each of these
is effected by a single tendon. From these there follow an infinite
number of other movements always effected by two tendons; one tendon
ceasing to act, the other takes up the movement. The tendons are
made thick inside the fingers and thin outside; and the tendons
inside are attached to every joint but outside they are not.

[Footnote 26: This head line has, in the original, no text to
follow.] Of the strength [and effect] of the 3 tendons inside the
fingers at the 3 joints.

The movement of the torso (355-361).

355.

Observe the altered position of the shoulder in all the movements of
the arm, going up and down, inwards and outwards, to the back and to
the front, and also in circular movements and any others.

And do the same with reference to the neck, hands and feet and the
breast above the lips &c.

356.

Three are the principal muscles of the shoulder, that is _b c d_,
and two are the lateral muscles which move it forward and backward,
that is _a o_; _a_ moves it forward, and _o_ pulls it back; and bed
raises it; _a b c_ moves it upwards and forwards, and _c d o_
upwards and backwards. Its own weight almost suffices to move it
downwards.

The muscle _d_ acts with the muscle _c_ when the arm moves forward;
and in moving backward the muscle _b_ acts with the muscle _c_.

[Footnote: See Pl. XXI. In the original the lettering has been
written in ink upon the red chalk drawing and the outlines of the
figures have in most places been inked over.]

357.

OF THE LOINS, WHEN BENT.

The loins or backbone being bent. The breasts are are always lower
than the shoulderblades of the back.

If the breast bone is arched the breasts are higher than the
shoulderblades.

If the loins are upright the breast will always be found at the same
level as the shoulderblades.

[Footnote: See Pl. XXII, No. 1.]

358.

_a b_ the tendon and ankle in raising the heel approach each other
by a finger's breadth; in lowering it they separate by a finger's
breadth.

[Footnote: See Pl. XXII, No. 2. Compare this facsimile and text with
Pl. III, No. 2, and p. 152 of MANZI'S edition. Also with No. 274 of
LUDWIG'S edition of the Vatican Copy.]

359.

Just so much as the part _d a_ of the nude figure decreases in this
position so much does the opposite part increase; that is: in
proportion as the length of the part _d a_ diminishes the normal
size so does the opposite upper part increase beyond its [normal]
size. The navel does not change its position to the male organ; and
this shrinking arises because when a figure stands on one foot, that
foot becomes the centre [of gravity] of the superimposed weight.
This being so, the middle between the shoulders is thrust above it
out of it perpendicular line, and this line, which forms the central
line of the external parts of the body, becomes bent at its upper
extremity [so as to be] above the foot which supports the body; and
the transverse lines are forced into such angles that their ends are
lower on the side which is supported. As is shown at _a b c_.

[Footnote: See Pl. XXII, No. 3.]

360.

OF PAINTING.

Note in the motions and attitudes of figures how the limbs vary, and
their feeling, for the shoulderblades in the motions of the arms and
shoulders vary the [line of the] back bone very much. And you will
find all the causes of this in my book of Anatomy.

361.

OF [CHANGE OF] ATTITUDE.

The pit of the throat is over the feet, and by throwing one arm
forward the pit of the throat is thrown off that foot. And if the
leg is thrown forward the pit of the throat is thrown forward; and.
so it varies in every attitude.

362.

OF PAINTING.

Indicate which are the muscles, and which the tendons, which become
prominent or retreat in the different movements of each limb; or
which do neither [but are passive]. And remember that these
indications of action are of the first importance and necessity in
any painter or sculptor who professes to be a master &c.

And indicate the same in a child, and from birth to decrepitude at
every stage of its life; as infancy, childhood, boyhood, youth &c.

And in each express the alterations in the limbs and joints, which
swell and which grow thinner.

363.

O Anatomical Painter! beware lest the too strong indication of the
bones, sinews and muscles, be the cause of your becoming wooden in
your painting by your wish to make your nude figures display all
their feeling. Therefore, in endeavouring to remedy this, look in
what manner the muscles clothe or cover their bones in old or lean
persons; and besides this, observe the rule as to how these same
muscles fill up the spaces of the surface that extend between them,
which are the muscles which never lose their prominence in any
amount of fatness; and which too are the muscles of which the
attachments are lost to sight in the very least plumpness. And in
many cases several muscles look like one single muscle in the
increase of fat; and in many cases, in growing lean or old, one
single muscle divides into several muscles. And in this treatise,
each in its place, all their peculiarities will be explained--and
particularly as to the spaces between the joints of each limb &c.
Again, do not fail [to observe] the variations in the forms of the
above mentioned muscles, round and about the joints of the limbs of
any animal, as caused by the diversity of the motions of each limb;
for on some side of those joints the prominence of these muscles is
wholly lost in the increase or diminution of the flesh of which
these muscles are composed, &c.

[Footnote: DE ROSSI remarks on this chapter, in the Roman edition of
the Trattato, p. 504: "_Non in questo luogo solo, ma in altri ancora
osservera il lettore, che Lionardo va fungendo quelli che fanno
abuso della loro dottrina anatomica, e sicuramente con cio ha in
mira il suo rivale Bonarroti, che di anatomia facea tanta pompa_."
Note, that Leonardo wrote this passage in Rome, probably under the
immediate impression of MICHAELANGELO'S paintings in the Sistine
Chapel and of RAPHAEL'S Isaiah in Sant' Agostino.]

364.

OF THE DIFFERENT MEASUREMENTS OF BOYS AND MEN.

There is a great difference in the length between the joints in men
and boys for, in man, from the top of the shoulder [by the neck] 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 but one because Nature constructs in us the mass which is the
home of the intellect, before forming that which contains the vital
elements.

365.

OF PAINTING.

Which are the muscles which subdivide in old age or in youth, when
becoming lean? Which are the parts of the limbs of the human frame
where no amount of fat makes the flesh thicker, nor any degree of
leanness ever diminishes it?

The thing sought for in this question will be found in all the
external joints of the bones, as the shoulder, elbow, wrists,
finger-joints, hips, knees, ankle-bone and toes and the like; all of
which shall be told in its place. The greatest thickness acquired by
any limb is at the part of the muscles which is farthest from its
attachments.

Flesh never increases on those portions of the limb where the bones
are near to the surface.

At _b r d a c e f_ the increase or diminution of the flesh never
makes any considerable difference. Nature has placed in front of man
all those parts which feel most pain under a blow; and these are the
shin of the leg, the forehead, and the nose. And this was done for
the preservation of man, since, if such pain were not felt in these
parts, the number of blows to which they would be exposed must be
the cause of their destruction.

Describe why the bones of the arm and leg are double near the hand
and foot [respectively].

And where the flesh is thicker or thinner in the bending of the
limbs.

366.

OF PAINTING.

Every part of the whole must be in proportion to the whole. Thus, if
a man is of a stout short figure he will be the same in all his
parts: that is with short and thick arms, wide thick hands, with
short fingers with their joints of the same character, and so on
with the rest. I would have the same thing understood as applying to
all animals and plants; in diminishing, [the various parts] do so in
due proportion to the size, as also in enlarging.

367.

OF THE AGREEMENT OF THE PROPORTION OF THE LIMBS.

And again, remember to be very careful in giving your figures limbs,
that they must appear to agree with the size of the body and
likewise to the age. Thus a youth has limbs that are not very
muscular not 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
sinews very prominent.

HOW YOUNG BOYS HAVE THEIR JOINTS JUST THE REVERSE OF THOSE OF MEN,
AS TO SIZE.

Little children have all the joints slender and the portions between
them are thick; and this happens because nothing but the skin covers
the joints without any other flesh and has the character of sinew,
connecting the bones like a ligature. And the fat fleshiness is laid
on between one joint and the next, and between the skin and the
bones. But, since the bones are thicker at the joints than between
them, as a mass grows up the flesh ceases to have that superfluity
which it had, between the skin and the bones; whence the skin clings
more closely to the bone and the limbs grow more slender. But since
there is nothing over the joints but the cartilaginous and sinewy
skin this cannot dry up, and, not drying up, cannot shrink. Thus,
and for this reason, children are slender at the joints and fat
between the joints; as may be seen in the joints of the fingers,
arms, and shoulders, which are slender and dimpled, while in man on
the contrary all the joints of the fingers, arms, and legs are
thick; and wherever children have hollows men have prominences.

The movement of the human figure (368-375).

368.

Of the manner of representing the 18 actions of man. Repose,
movement, running, standing, supported, sitting, leaning, kneeling,
lying down, suspended. Carrying or being carried, thrusting,
pulling, striking, being struck, pressing down and lifting up.

[As to how a figure should stand with a weight in its hand [Footnote
8: The original text ends here.] Remember].

369.

A sitting man cannot raise himself if that part of his body which is
front of his axis [centre of gravity] does not weigh more than that
which is behind that axis [or centre] without using his arms.

A man who is mounting any slope finds that he must involuntarily
throw the most weight forward, on the higher foot, rather than
behind--that is in front of the axis and not behind it. Hence a man
will always, involuntarily, throw the greater weight towards the
point whither he desires to move than in any other direction.

The faster a man runs, the more he leans forward towards the point
he runs to and throws more weight in front of his axis than behind.
A man who runs down hill throws the axis onto his heels, and one who
runs up hill throws it into the points of his feet; and a man
running on level ground throws it first on his heels and then on the
points of his feet.

This man cannot carry his own weight unless, by drawing his body
back he balances the weight in front, in such a way as that the foot
on which he stands is the centre of gravity.

[Footnote: See Pl. XXII, No. 4.]

370.

How a man proceeds to raise himself to his feet, when he is sitting
on level ground.

371.

A man when walking has his head in advance of his feet.

A man when walking across a long level plain first leans [rather]
backwards and then as much forwards.

[Footnote 3-6: He strides forward with the air of a man going down
hill; when weary, on the contrary he walks like a man going up
hill.]

372.

A man when running throws less weight on his legs than when standing
still. And in the same way a horse which is running feels less the
weight of the man he carries. Hence many persons think it wonderful
that, in running, the horse can rest on one single foot. From this
it may be stated that when a weight is in progressive motion the
more rapid it is the less is the perpendicular weight towards the
centre.

373.

If a man, in taking a jump from firm ground, can leap 3 braccia, and
when he was taking his leap it were to recede 1/3 of a braccio, that
would be taken off his former leap; and so if it were thrust forward
1/3 of a braccio, by how much would his leap be increased?

374.

OF DRAWING.

When a man who is running wants to neutralise the impetus that
carries him on he prepares a contrary impetus which is generated by
his hanging backwards. This can be proved, since, if the impetus
carries a moving body with a momentum equal to 4 and the moving body
wants to turn and fall back with a momentum of 4, then one momentum
neutralises the other contrary one, and the impetus is neutralised.

Of walking up and down (375-379)

375.

When a man wants to stop running and check the impetus he is forced
to hang back and take short quick steps. [Footnote: Lines 5-31 refer
to the two upper figures, and the lower figure to the right is
explained by the last part of the chapter.] The centre of gravity of
a man who lifts one of his feet from the ground always rests on the
centre of the sole of the foot [he stands on].

A man, in going up stairs involuntarily throws so much weight
forward and on the side of the upper foot as to be a counterpoise to
the lower leg, so that the labour of this lower leg is limited to
moving itself.

The first thing a man does in mounting steps is to relieve the leg
he is about to lift of the weight of the body which was resting on
that leg; and besides this, he gives to the opposite leg all the
rest of the bulk of the whole man, including [the weight of] the
other leg; he then raises the other leg and sets the foot upon the
step to which he wishes to raise himself. Having done this he
restores to the upper foot all the weight of the body and of the leg
itself, and places his hand on his thigh and throws his head forward
and repeats the movement towards the point of the upper foot,
quickly lifting the heel of the lower one; and with this impetus he
lifts himself up and at the same time extends the arm which rested
on his knee; and this extension of the arm carries up the body and
the head, and so straightens the spine which was curved.

[32] The higher the step is which a man has to mount, the farther
forward will he place his head in advance of his upper foot, so as
to weigh more on _a_ than on _b_; this man will not be on the step
_m_. As is shown by the line _g f_.

[Footnote: See Pl. XXIII, No. 1. The lower sketch to the left
belongs to the four first lines.]

376.

I ask the weight [pressure] of this man at every degree of motion on
these steps, what weight he gives to _b_ and to _c_.

[Footnote 8: These lines are, in the original, written in ink]
Observe the perpendicular line below the centre of gravity of the
man.

[Footnote: See Pl. XXIII, No. 2.]

377.

In going up stairs if you place your hands on your knees all the
labour taken by the arms is removed from the sinews at the back of
the knees.

[Footnote: See Pl. XXIII, No. 3.]

378.

The sinew which guides the leg, and which is connected with the
patella of the knee, feels it a greater labour to carry the man
upwards, in proportion as the leg is more bent; and the muscle which
acts upon the angle made by the thigh where it joins the body has
less difficulty and has a less weight to lift, because it has not
the [additional] weight of the thigh itself. And besides this it has
stronger muscles, being those which form the buttock.

379.

A man coming down hill takes little steps, because the weight rests
upon the hinder foot, while a man mounting takes wide steps, because
his weight rests on the foremost foot.

[Footnote: See Pl. XXIII, No. 4.]

On the human body in action (380-388).

380.

OF THE HUMAN BODY IN ACTION.

When you want to represent a man as moving some weight consider what
the movements are that are to be represented by different lines;
that is to say either from below upwards, with a simple movement, as
a man does who stoops forward to take up a weight which he will lift
as he straightens himself. Or as a man does who wants to squash
something backwards, or to force it forwards or to pull it downwards
with ropes passed through pullies [Footnote 10: Compare the sketch
on page 198 and on 201 (S. K. M. II.1 86b).]. And here remember that
the weight of a man pulls in proportion as his centre of gravity is
distant from his fulcrum, and to this is added the force given by
his legs and bent back as he raises himself.

381.

Again, a man has even a greater store of strength in his legs than
he needs for his own weight; and to see if this is true, make a man
stand on the shore-sand and then put another man on his back, and
you will see how much he will sink in. Then take the man from off
his back and make him jump straight up as high as he can, and you
will find that the print of his feet will be made deeper by the jump
than from having the man on his back. Hence, here, by 2 methods it
is proved that a man has double the strength he requires to support
his own body.

382.

OF PAINTING.

If you have to draw a man who is in motion, or lifting or pulling,
or carrying a weight equal to his own, in what way must you set on
his legs below his body?

[Footnote: In the MS. this question remains unanswered.]

383.

OF THE STRENGTH OF MAN.

A man pulling a [dead] weight balanced against himself cannot pull
more than his own weight. And if he has to raise it he will [be able
to] raise as much more than his weight as his strength may be more
than that of other men. [Footnote 7: The stroke at the end of this
line finishes in the original in a sort of loop or flourish, and a
similar flourish occurs at the end of the previous passage written
on the same page. M. RAVAISSON regards these as numbers (compare the
photograph of page 30b in his edition of MS. A). He remarks: "_Ce
chiffre_ 8 _et, a la fin de l'alinea precedent, le chiffre_ 7 _sont,
dans le manuscrit, des renvois_."] The greatest force a man can
apply, with equal velocity and impetus, will be when he sets his
feet on one end of the balance [or lever] and then presses his
shoulders against some stable body. This will raise a weight at the
other end of the balance [lever], equal to his own weight and [added
to that] as much weight as he can carry on his shoulders.

384.

No animal can simply move [by its dead weight] a greater weight than
the sum of its own weight outside the centre of his fulcrum.

385.

A man who wants to send an arrow very far from the bow must be
standing entirely on one foot and raising the other so far from the
foot he stands on as to afford the requisite counterpoise to his
body which is thrown on the front foot. And he must not hold his arm
fully extended, and in order that he may be more able to bear the
strain he must hold a piece of wood which there is in all crossbows,
extending from the hand to the breast, and when he wishes to shoot
he suddenly leaps forward at the same instant and extends his arm
with the bow and releases the string. And if he dexterously does
every thing at once it will go a very long way.

386.

When two men are at the opposite ends of a plank that is balanced,
and if they are of equal weight, and if one of them wants to make a
leap into the air, then his leap will be made down from his end of
the plank and the man will never go up again but must remain in his
place till the man at the other end dashes up the board.

[Footnote: See Pl. XXIV, No. 3.]

387.

Of delivering a blow to the right or left.

[Footnote: Four sketches on Pl. XXIV, No. 1 belong to this passage.
The rest of the sketches and notes on that page are of a
miscellaneous nature.]

388.

Why an impetus is not spent at once [but diminishes] gradually in
some one direction? [Footnote 1: The paper has been damaged at the
end of line 1.] The impetus acquired in the line _a b c d_ is spent
in the line _d e_ but not so completely but that some of its force
remains in it and to this force is added the momentum in the line _d
e_ with the force of the motive power, and it must follow than the
impetus multiplied by the blow is greater that the simple impetus
produced by the momentum _d e_.

[Footnote 8: The sketch No. 2 on Pl. XXIV stands, in the original,
between lines 7 and 8. Compare also the sketches on Pl. LIV.] A man
who has to deal a great blow with his weapon prepares himself with
all his force on the opposite side to that where the spot is which
he is to hit; and this is because a body as it gains in velocity
gains in force against the object which impedes its motion.

On hair falling down in curls.

389.

Observe the motion of the surface of the water which resembles that
of hair, and has two motions, of which one goes on with the flow of
the surface, the other forms the lines of the eddies; thus the water
forms eddying whirlpools one part of which are due to the impetus of
the principal current and the other to the incidental motion and
return flow.

[Footnote: See Pl. XXV. Where also the text of this passage is given
in facsimile.]

On draperies (390--392).

390.

OF THE NATURE OF THE FOLDS IN DRAPERY.

That part of a fold which is farthest from the ends where it is
confined will fall most nearly in its natural form.

Every thing by nature tends to remain at rest. Drapery, being of
equal density and thickness on its wrong side and on its right, has
a tendency to lie flat; therefore when you give it a fold or plait
forcing it out of its flatness note well the result of the
constraint in the part where it is most confined; and the part which
is farthest from this constraint you will see relapses most into the
natural state; that is to say lies free and flowing.

EXAMPLE.

[Footnote 13: _a c sia_. In the original text _b_ is written instead
of _c_--an evident slip of the pen.] Let _a b c_ be the fold of the
drapery spoken of above, _a c_ will be the places where this folded
drapery is held fast. I maintain that the part of the drapery which
is farthest from the plaited ends will revert most to its natural
form.

Therefore, _b_ being farthest from _a_ and _c_ in the fold _a b c_
it will be wider there than anywhere else.

[Footnote: See Pl. XXVIII, No. 6, and compare the drawing from
Windsor Pl. XXX for farther illustration of what is here stated.]

391.

OF SMALL FOLDS IN DRAPERIES.

How figures dressed in a cloak should not show the shape so much as
that the cloak looks as if it were next the flesh; since you surely
cannot wish the cloak to be next the flesh, for you must suppose
that between the flesh and the cloak there are other garments which
prevent the forms of the limbs appearing distinctly through the
cloak. And those limbs which you allow to be seen you must make
thicker so that the other garments may appear to be under the cloak.
But only give something of the true thickness of the limbs to a
nymph [Footnote 9: _Una nifa_. Compare the beautiful drawing of a
Nymph, in black chalk from the Windsor collection, Pl. XXVI.] or an
angel, which are represented in thin draperies, pressed and clinging
to the limbs of the figures by the action of the wind.

392.

You ought not to give to drapery a great confusion of many folds,
but rather only introduce them where they are held by the hands or
the arms; the rest you may let fall simply where it is its nature to
flow; and do not let the nude forms be broken by too many details
and interrupted folds. How draperies should be drawn from nature:
that is to say if youwant to represent woollen cloth draw the folds
from that; and if it is to be silk, or fine cloth or coarse, or of
linen or of crape, vary the folds in each and do not represent
dresses, as many do, from models covered with paper or thin leather
which will deceive you greatly.

[Footnote: The little pen and ink drawing from Windsor (W. 102),
given on Pl. XXVIII, No. 7, clearly illustrates the statement made
at the beginning of this passage; the writing of the cipher 19 on
the same page is in Leonardo's hand; the cipher 21 is certainly
not.]

_VIII._

_Botany for Painters and Elements of Landscape Painting._

_The chapters composing this portion of the work consist of
observations on Form, Light and Shade in Plants, and particularly in
Trees summed up in certain general rules by which the author intends
to guide the artist in the pictorial representation of landscape._

_With these the first principles of a_ Theory of Landscape painting
_are laid down--a theory as profoundly thought out in its main
lines as it is lucidly worked out in its details. In reading these
chapters the conviction is irresistible that such a_ Botany for
painters _is or ought to be of similar importance in the practice of
painting as the principles of the Proportions and Movements of the
human figure_ i. e. Anatomy for painters.

_There can be no doubt that Leonardo, in laying down these rules,
did not intend to write on Botany in the proper scientific
sense--his own researches on that subject have no place here; it
need only be observed that they are easily distinguished by their
character and contents from those which are here collected and
arranged under the title 'Botany for painters'. In some cases where
this division might appear doubtful,--as for instance in No._
402--_the Painter is directly addressed and enjoined to take the
rule to heart as of special importance in his art._

_The original materials are principally derived from MS._ G, _in
which we often find this subject treated on several pages in
succession without any of that intermixture of other matters, which
is so frequent in Leonardo's writings. This MS., too, is one of the
latest; when it was written, the great painter was already more than
sixty years of age, so we can scarcely doubt that he regarded all he
wrote as his final views on the subject. And the same remark applies
to the chapters from MSS._ E _and_ M _which were also written
between_ 1513--15.

_For the sake of clearness, however, it has been desirable to
sacrifice--with few exceptions--the original order of the passages
as written, though it was with much reluctance and only after long
hesitation that I resigned myself to this necessity. Nor do I mean
to impugn the logical connection of the author's ideas in his MS.;
but it will be easily understood that the sequence of disconnected
notes, as they occurred to Leonardo and were written down from time
to time, might be hardly satisfactory as a systematic arrangement of
his principles. The reader will find in the Appendix an exact
account of the order of the chapters in the original MS. and from
the data there given can restore them at will. As the materials are
here arranged, the structure of the tree as regards the growth of
the branches comes first_ (394-411) _and then the insertion of the
leaves on the stems_ (412-419). _Then follow the laws of Light and
Shade as applied, first, to the leaves (420-434), and, secondly, to
the whole tree and to groups of trees_ (435-457). _After the remarks
on the Light and Shade in landscapes generally_ (458-464), _we find
special observations on that of views of towns and buildings_
(465-469). _To the theory of Landscape Painting belong also the
passages on the effect of Wind on Trees_ (470-473) _and on the Light
and Shade of Clouds_ (474-477), _since we find in these certain
comparisons with the effect of Light and Shade on Trees_ (e. g.: _in
No._ 476, 4. 5; _and No._ 477, 9. 12). _The chapters given in the
Appendix Nos._ 478 _and_ 481 _have hardly any connection with the
subjects previously treated._

Classification of trees.

393.

TREES.

Small, lofty, straggling, thick, that is as to foliage, dark, light,
russet, branched at the top; some directed towards the eye, some
downwards; with white stems; this transparent in the air, that not;
some standing close together, some scattered.

The relative thickness of the branches to the trunk (393--396).

394.

All the branches of a tree at every stage of its height when put
together are equal in thickness to the trunk [below them].

All the branches of a water [course] at every stage of its course,
if they are of equal rapidity, are equal to the body of the main
stream.

395.

Every year when the boughs of a plant [or tree] have made an end of
maturing their growth, they will have made, when put together, a
thickness equal to that of the main stem; and at every stage of its
ramification you will find the thickness of the said main stem; as:
_i k_, _g h_, _e f_, _c d_, _a b_, will always be equal to each
other; unless the tree is pollard--if so the rule does not hold
good.

All the branches have a direction which tends to the centre of the
tree _m_.

[Footnote: The two sketches of leafless trees one above another on
the left hand side of Pl. XXVII, No. 1, belong to this passage.]

396.

If the plant n grows to the thickness shown at m, its branches will
correspond [in thickness] to the junction a b in consequence of the
growth inside as well as outside.

The branches of trees or plants have a twist wherever a minor branch
is given off; and this giving off the branch forms a fork; this said
fork occurs between two angles of which the largest will be that
which is on the side of the larger branch, and in proportion, unless
accident has spoilt it.

[Footnote: The sketches illustrating this are on the right hand side
of PI. XXVII, No. I, and the text is also given there in facsimile.]

397.

There is no boss on branches which has not been produced by some
branch which has failed.

The lower shoots on the branches of trees grow more than the upper
ones and this occurs only because the sap that nourishes them, being
heavy, tends downwards more than upwards; and again, because those
[branches] which grow downwards turn away from the shade which
exists towards the centre of the plant. The older the branches are,
the greater is the difference between their upper and their lower
shoots and in those dating from the same year or epoch.

[Footnote: The sketch accompanying this in the MS. is so effaced
that an exact reproduction was impossible.]

398.

OF THE SCARS ON TREES.

The scars on trees grow to a greater thickness than is required by
the sap of the limb which nourishes them.

399.

The plant which gives out the smallest ramifications will preserve
the straightest line in the course of its growth.

[Footnote: This passage is illustrated by two partly effaced
sketches. One of these closely resembles the lower one given under
No. 408, the other also represents short closely set boughs on an
upright trunk.]

400.

OF THE RAMIFICATION.

The beginning of the ramification [the shoot] always has the central
line [axis] of its thickness directed to the central line [axis] of
the plant itself.

401.

In starting from the main stem the branches always form a base with
a prominence as is shown at _a b c d_.

402.

WHY, VERY FREQUENTLY, TIMBER HAS VEINS THAT ARE NOT STRAIGHT.

When the branches which grow the second year above the branch of the
preceding year, are not of equal thickness above the antecedent
branches, but are on one side, then the vigour of the lower branch
is diverted to nourish the one above it, although it may be somewhat
on one side.

But if the ramifications are equal in their growth, the veins of the
main stem will be straight [parallel] and equidistant at every
degree of the height of the plant.

Wherefore, O Painter! you, who do not know these laws! in order to
escape the blame of those who understand them, it will be well that
you should represent every thing from nature, and not despise such
study as those do who work [only] for money.

The direction of growth (403-407).

403.

OF THE RAMIFICATIONS OF PLANTS.

The plants which spread very much have the angles of the spaces
which divide their branches more obtuse in proportion as their point
of origin is lower down; that is nearer to the thickest and oldest
portion of the tree. Therefore in the youngest portions of the tree
the angles of ramification are more acute. [Footnote: Compare the
sketches on the lower portion of Pl. XXVII, No. 2.]

404.

The tips of the boughs of plants [and trees], unless they are borne
down by the weight of their fruits, turn towards the sky as much as
possible.

The upper side of their leaves is turned towards the sky that it may
receive the nourishment of the dew which falls at night.

The sun gives spirit and life to plants and the earth nourishes them
with moisture. [9] With regard to this I made the experiment of
leaving only one small root on a gourd and this I kept nourished
with water, and the gourd brought to perfection all the fruits it
could produce, which were about 60 gourds of the long kind, andi set
my mind diligently [to consider] this vitality and perceived that
the dews of night were what supplied it abundantly with moisture
through the insertion of its large leaves and gave nourishment to
the plant and its offspring--or the seeds which its offspring had
to produce--[21].

The rule of the leaves produced on the last shoot of the year will
be that they will grow in a contrary direction on the twin branches;
that is, that the insertion of the leaves turns round each branch in
such a way, as that the sixth leaf above is produced over the sixth
leaf below, and the way they turn is that if one turns towards its
companion to the right, the other turns to the left, the leaf
serving as the nourishing breast for the shoot or fruit which grows
the following year.

[Footnote: A French translation of lines 9-12 was given by M.
RAVAISSON in the _Gazette des Beaux Arts_, Oct. 1877; his paper also
contains some valuable information as to botanical science in the
ancient classical writers and at the time of the Renaissance.]

405.

The lowest branches of those trees which have large leaves and heavy
fruits, such as nut-trees, fig-trees and the like, always droop
towards the ground.

The branches always originate above [in the axis of] the leaves.

406.

The upper shoots of the lateral branches of plants lie closer to the
parent branch than the lower ones.

407.

The lowest branches, after they have formed the angle of their
separation from the parent stem, always bend downwards so as not to
crowd against the other branches which follow them on the same stem
and to be better able to take the air which nourishes them. As is
shown by the angle _b a c_; the branch _a c_ after it has made the
corner of the angle _a c_ bends downwards to _c d_ and the lesser
shoot _c_ dries up, being too thin.

The main branch always goes below, as is shown by the branch _f n
m_, which does not go to _f n o_.

The forms of trees (408--411).

408.

The elm always gives a greater length to the last branches of the
year's growth than to the lower ones; and Nature does this because
the highest branches are those which have to add to the size of the
tree; and those at the bottom must get dry because they grow in the
shade and their growth would be an impediment to the entrance of the
solar rays and the air among the main branches of the tree.

The main branches of the lower part bend down more than those above,
so as to be more oblique than those upper ones, and also because
they are larger and older.

409.

In general almost all the upright portions of trees curve somewhat
turning the convexity towards the South; and their branches are
longer and thicker and more abundant towards the South than towards
the North. And this occurs because the sun draws the sap towards
that surface of the tree which is nearest to it.

And this may be observed if the sun is not screened off by other
plants.

410.

The cherry-tree is of the character of the fir tree as regards its
ramification placed in stages round its main stem; and its branches
spring, 4 or five or 6 [together] opposite each other; and the tips
of the topmost shoots form a pyramid from the middle upwards; and
the walnut and oak form a hemisphere from the middle upwards.

411.

The bough of the walnut which is only hit and beaten when it has
brought to perfection...

[Footnote: The end of the text and the sketch in red chalk belonging
to it, are entirely effaced.]

The insertion of the leaves (412--419).

412.

OF THE INSERTION OF THE BRANCHES ON PLANTS.

Such as the growth of the ramification of plants is on their
principal branches, so is that of the leaves on the shoots of the
same plant. These leaves have [Footnote 6: _Quattro modi_ (four
modes). Only three are described in the text, the fourth is only
suggested by a sketch.

This passage occurs in MANZI'S edition of the Trattato, p. 399, but
without the sketches and the text is mutilated in an important part.
The whole passage has been commented on, from MANZI'S version, in
Part I of the _Nuovo Giornale Botanico Italiano_, by Prof. G.
UZIELLI (Florence 1869, Vol. I). He remarks as to the 'four modes':
"_Leonardo, come si vede nelle linie sententi da solo tre esempli.
Questa ed altre inessattezze fanno desiderare, sia esaminato di
nuovo il manoscritto Vaticano_". This has since been done by D.
KNAPP of Tubingen, and his accurate copy has been published by H.
LUDWIG, the painter. The passage in question occurs in his edition
as No. 833; and there also the drawings are wanting. The space for
them has been left vacant, but in the Vatican copy '_niente_' has
been written on the margin; and in it, as well as in LUDWIG'S and
MANZI'S edition, the text is mutilated.] four modes of growing one
above another. The first, which is the most general, is that the
sixth always originates over the sixth below [Footnote 8: _la sesta
di sotto. "Disposizione 2/5 o 1/5. Leonardo osservo probabilmente
soltanto la prima"_ (UZIELLl).]; the second is that two third ones
above are over the two third ones below [Footnote 10: _terze di
sotto: "Intende qui senza dubbio parlare di foglie decussate, in cui
il terzo verticello e nel piano del primo"_ (UZIELLI).]; and the
third way is that the third above is over the third below [Footnote
11: 3a _di sotto: "Disposizione 1/2"_ (UZIELLI).].

[Footnote: See the four sketches on the upper portion of the page
reproduced as fig. 2 on P1. XXVII.]

413.

A DESCRIPTION OF THE ELM.

The ramification of the elm has the largest branch at the top. The
first and the last but one are smaller, when the main trunk is
straight.

The space between the insertion of one leaf to the rest is half the
extreme length of the leaf or somewhat less, for the leaves are at
an interval which is about the 3rd of the width of the leaf.

The elm has more leaves near the top of the boughs than at the base;
and the broad [surface] of the leaves varies little as to [angle
and] aspect.

[Footnote: See Pl. XXVII, No. 3. Above the sketch and close under
the number of the page is the word '_olmo_' (elm).]

414.

In the walnut tree the leaves which are distributed on the shoots of
this year are further apart from each other and more numerous in
proportion as the branch from which this shoot springs is a young
one. And they are inserted more closely and less in number when the
shoot that bears them springs from an old branch. Its fruits are
borne at the ends of the shoots. And its largest boughs are the
lowest on the boughs they spring from. And this arises from the
weight of its sap which is more apt to descend than to rise, and
consequently the branches which spring from them and rise towards
the sky are small and slender [20]; and when the shoot turns towards
the sky its leaves spread out from it [at an angle] with an equal
distribution of their tips; and if the shoot turns to the horizon
the leaves lie flat; and this arises from the fact that leaves
without exception, turn their underside to the earth [29].

The shoots are smaller in proportion as they spring nearer to the
base of the bough they spring from.

[Footnote: See the two sketches on Pl XXVII, No. 4. The second
refers to the passage lines 20-30.]

415.

OF THE INSERTION OF THE LEAVES ON THE BRANCHES.

The thickness of a branch never diminishes within the space between
one leaf and the next excepting by so much as the thickness of the
bud which is above the leaf and this thickness is taken off from the
branch above [the node] as far as the next leaf.

Nature has so placed the leaves of the latest shoots of many plants
that the sixth leaf is always above the first, and so on in
succession, if the rule is not [accidentally] interfered with; and
this occurs for two useful ends in the plant: First that as the
shoot and the fruit of the following year spring from the bud or eye
which lies above and in close contact with the insertion of the leaf
[in the axil], the water which falls upon the shoot can run down to
nourish the bud, by the drop being caught in the hollow [axil] at
the insertion of the leaf. And the second advantage is, that as
these shoots develop in the following year one will not cover the
next below, since the 5 come forth on five different sides; and the
sixth which is above the first is at some distance.

416.

OF THE RAMIFICATIONS OF TREES AND THEIR FOLIAGE.

The ramifications of any tree, such as the elm, are wide and slender
after the manner of a hand with spread fingers, foreshortened. And
these are seen in the distribution [thus]: the lower portions are
seen from above; and those that are above are seen from below; and
those in the middle, some from below and some from above. The upper
part is the extreme [top] of this ramification and the middle
portion is more foreshortened than any other of those which are
turned with their tips towards you. And of those parts of the middle
of the height of the tree, the longest will be towards the top of
the tree and will produce a ramification like the foliage of the
common willow, which grows on the banks of rivers.

Other ramifications are spherical, as those of such trees as put
forth their shoots and leaves in the order of the sixth being placed
above the first. Others are thin and light like the willow and
others.

417.

You will see in the lower branches of the elder, which puts forth
leaves two and two placed crosswise [at right angles] one above
another, that if the stem rises straight up towards the sky this
order never fails; and its largest leaves are on the thickest part
of the stem and the smallest on the slenderest part, that is towards
the top. But, to return to the lower branches, I say that the leaves
on these are placed on them crosswise like [those on] the upper
branches; and as, by the law of all leaves, they are compelled to
turn their upper surface towards the sky to catch the dew at night,
it is necessary that those so placed should twist round and no
longer form a cross.

[Footnote: See Pl. XXVII, No. 5.]

418.

A leaf always turns its upper side towards the sky so that it may
the better receive, on all its surface, the dew which drops gently
from the atmosphere. And these leaves are so distributed on the
plant as that one shall cover the other as little as possible, but
shall lie alternately one above another as may be seen in the ivy
which covers the walls. And this alternation serves two ends; that
is, to leave intervals by which the air and sun may penetrate
between them. The 2nd reason is that the drops which fall from the
first leaf may fall onto the fourth or--in other trees--onto the
sixth.

419.

Every shoot and every fruit is produced above the insertion [in the
axil] of its leaf which serves it as a mother, giving it water from
the rain and moisture from the dew which falls at night from above,
and often it protects them against the too great heat of the rays of
the sun.

LIGHT ON BRANCHES AND LEAVES (420--422).

420.

That part of the body will be most illuminated which is hit by the
luminous ray coming between right angles.

[Footnote: See Pl. XXVIII, No. 1.]

421.

Young plants have more transparent leaves and a more lustrous bark
than old ones; and particularly the walnut is lighter coloured in
May than in September.

422.

OF THE ACCIDENTS OF COLOURING IN TREES.

The accidents of colour in the foliage of trees are 4. That is:
shadow, light, lustre [reflected light] and transparency.

OF THE VISIBILITY OF THESE ACCIDENTS.

These accidents of colour in the foliage of trees become confused at
a great distance and that which has most breadth [whether light or
shade, &c.] will be most conspicuous.

The proportions of light and shade in a leaf (423-426).

423.

OF THE SHADOWS OF A LEAF.

Sometimes a leaf has three accidents [of light] that is: shade,
lustre [reflected light] and transparency [transmitted light]. Thus,
if the light were at _n_ as regards the leaf _s_, and the eye at
_m_, it would see _a_ in full light, _b_ in shadow and _c_
transparent.

424.

A leaf with a concave surface seen from the under side and
up-side-down will sometimes show itself as half in shade, and half
transparent. Thus, if _o p_ is the leaf and the light _m_ and the
eye _n_, this will see _o_ in shadow because the light does not fall
upon it between equal angles, neither on the upper nor the under
side, and _p_ is lighted on the upper side and the light is
transmitted to its under side. [Footnote: See Pl. XXVIII, No. 2, the
upper sketch on the page. In the original they are drawn in red
chalk.]

425.

Although those leaves which have a polished surface are to a great
extent of the same colour on the right side and on the reverse, it
may happen that the side which is turned towards the atmosphere will
have something of the colour of the atmosphere; and it will seem to
have more of this colour of the atmosphere in proportion as the eye
is nearer to it and sees it more foreshortened. And, without
exception the shadows show as darker on the upper side than on the
lower, from the contrast offered by the high lights which limit the
shadows.

The under side of the leaf, although its colour may be in itself the
same as that of the upper side, shows a still finer colour--a colour
that is green verging on yellow--and this happens when the leaf is
placed between

426.

the eye and the light which falls upon it from the opposite side.

And its shadows are in the same positions as those were of the
opposite side. Therefore, O Painter! when you do trees close at
hand, remember that if the eye is almost under the tree you will see
its leaves [some] on the upper and [some] on the under side, and the
upper side will be bluer in proportion as they are seen more
foreshortened, and the same leaf sometimes shows part of the right
side and part of the under side, whence you must make it of two
colours.

Of the transparency of leaves (427-429).

427.

The shadows in transparent leaves seen from the under side are the
same shadows as there are on the right side of this leaf, they will
show through to the underside together with lights, but the lustre
[reflected light] can never show through.

428.

When one green has another [green] behind it, the lustre on the
leaves and their transparent [lights] show more strongly than in
those which are [seen] against the brightness of the atmosphere.

And if the sun illuminates the leaves without their coming between
it and the eye and without the eye facing the sun, then the
reflected lights and the transparent lights are very strong.

It is very effective to show some branches which are low down and
dark and so set off the illuminated greens which are at some
distance from the dark greens seen below. That part is darkest which
is nearest to the eye or which is farthest from the luminous
atmosphere.

429.

Never paint leaves transparent to the sun, because they are
confused; and this is because on the transparency of one leaf will
be seen the shadow of another leaf which is above it. This shadow
has a distinct outline and a certain depth of shade and sometimes is
[as much as] half or a third of the leaf which is shaded; and
consequently such an arrangement is very confused and the imitation
of it should be avoided.

The light shines least through a leaf when it falls upon it at an
acute angle.

The gradations of shade and colour in leaves (430-434).

430.

The shadows of plants are never black, for where the atmosphere
penetrates there can never be utter darkness.

431.

If the light comes from _m_ and the eye is at _n_ the eye will see
the colour of the leaves _a b_ all affected by the colour of _m_
--that is of the atmosphere; and _b c_ will be seen from the under
side as transparent, with a beautiful green colour verging on
yellow.

If _m_ is the luminous body lighting up the leaf _s_ all the eyes
that see the under side of this leaf will see it of a beautiful
light green, being transparent.

In very many cases the positions of the leaves will be without
shadow [or in full light], and their under side will be transparent
and the right side lustrous [reflecting light].

432.

The willow and other similar trees, which have their boughs lopped
every 3 or 4 years, put forth very straight branches, and their
shadow is about the middle where these boughs spring; and towards
the extreme ends they cast but little shade from having small leaves
and few and slender branches. Hence the boughs which rise towards
the sky will have but little shade and little relief; and the
branches which are at an angle from the horizon, downwards, spring
from the dark part of the shadow and grow thinner by degrees up to
their ends, and these will be in strong relief, being in gradations
of light against a background of shadow.

That tree will have the least shadow which has the fewest branches
and few leaves.

433.

OF DARK LEAVES IN FRONT OF TRANSPARENT ONES.

When the leaves are interposed between the light and the eye, then
that which is nearest to the eye will be the darkest, and the most
distant will be the lightest, not being seen against the atmosphere;
and this is seen in the leaves which are away from the centre of the
tree, that is towards the light.

[Footnote: See Pl. XXVIII, No. 2, the lower sketch.]

434.

OF THE LIGHTS ON DARK LEAVES.

The lights on such leaves which are darkest, will be most near to
the colour of the atmosphere that is reflected in them. And the
cause of this is that the light on the illuminated portion mingles
with the dark hue to compose a blue colour; and this light is
produced by the blueness of the atmosphere which is reflected in the
smooth surface of these leaves and adds to the blue hue which this
light usually produces when it falls on dark objects.

OF THE LIGHTS ON LEAVES OF A YELLOWISH GREEN.

But leaves of a green verging on yellow when they reflect the
atmosphere do not produce a reflection verging on blue, inasmuch as
every thing which appears in a mirror takes some colour from that
mirror, hence the blue of the atmosphere being reflected in the
yellow of the leaf appears green, because blue and yellow mixed
together make a very fine green colour, therefore the lustre of
light leaves verging on yellow will be greenish yellow.

A classification of trees according to their colours.

435.

The trees in a landscape are of various kinds of green, inasmuch as
some verge towards blackness, as firs, pines, cypresses, laurels,
box and the like. Some tend to yellow such as walnuts, and pears,
vines and verdure. Some are both yellowish and dark as chesnuts,
holm-oak. Some turn red in autumn as the service-tree, pomegranate,
vine, and cherry; and some are whitish as the willow, olive, reeds
and the like. Trees are of various forms ...

The proportions of light and shade in trees (436-440).

436.

OF A GENERALLY DISTRIBUTED LIGHT AS LIGHTING UP TREES.

That part of the trees will be seen to lie in the least dark shadow
which is farthest from the earth.

To prove it let _a p_ be the tree, _n b c_ the illuminated
hemisphere [the sky], the under portion of the tree faces the earth
_p c_, that is on the side _o_, and it faces a small part of the
hemisphere at _c d_. But the highest part of the convexity a faces
the greatest part of the hemisphere, that is _b c_. For this
reason--and because it does not face the darkness of the earth--it
is in fuller light. But if the tree has dense foliage, as the
laurel, arbutus, box or holm oak, it will be different; because,
although _a_ does not face the earth, it faces the dark [green] of
the leaves cut up by many shadows, and this darkness is reflected
onto the under sides of the leaves immediately above. Thus these
trees have their darkest shadows nearest to the middle of the tree.

437.

OF THE SHADOWS OF VERDURE.