The Notebooks of Leonardo Da Vinci

The Notebooks of Leonardo Da Vinci

Author: Leonardo Da Vinci

Edition: 10

Language: English

The Notebooks of Leonardo Da Vinci

Volume 1

Translated by Jean Paul Richter

888

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should have remained so long unpublished, and indeed forgotten.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. forpublication (2)

--Admonition to readers (3).

--The disorder in theMSS. (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 byone 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.

97).--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.

.

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.]

0.

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. MÜLLER STRÜBING

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 dì prospettiva cioè

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.

1.

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.

2.

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.

3.

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 _partirò

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).

4.

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.].

5.

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.]

6.

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.

7.

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.

8.

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).

9.

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.

st. 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 entrerà_. _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 è quella parie della

quale he nulla--Linia he luncheza senza àpieza; &c.]

45.

, 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, biàcho--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 già 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, affinchè sappiamo qual fosse l'opinione

di cosi chiaro e famoso pittore intorno alla prospettiva . . Scrive

Bramantino che la prospettiva è una cosa che contrafà il naturale, e

che ciò 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 riferirò 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 più appresso. Adunque la prima

si dimanda prospettiva, cioè 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 può

crescere e sminuire, ma procede dagli effetti degli occhi, i quali

sono piccioli, e perciò volendo vedere tanto gran cosa_, bisogna che

mandino fuora la virtù visiva, _la quale si dilata in tanta

larghezza, che piglia tutto quello che vuoi vedere, ed_ arrivando a

quella cosa la vede dove è: _e da lei agli occhi per quello circuito

fino all' occhio, e tutto quello termine è 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 mathématiques 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.

523, 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-mathématiques de L. da Vinci avec des fragments tirés de ses

Manuscrits, apportés 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 ècrit_ 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 più 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.]

00.

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.

01.

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.]

02.

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.

03.

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.

04.

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.

05.

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

06.

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).

07.

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.

08.

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.]

09.

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.

10.

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.

11.

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).

12.

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.

13.

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.

14.

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.

15.

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.

16.

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.

17.

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.

18.

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).

19.

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.

20.

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.

21.

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.

22.

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).

23.

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_.

24.

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.

25.

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.

26.

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.

27.

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).

28.

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.

29.

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).

30.

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.

31.

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).

32.

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.

33.

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_.

34.

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.

35.

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.

36.

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).

37.

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.]

38.

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).]

39.

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).]

40.

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).

41.

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.]

42.

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.]

43.

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).]

44.

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).]

45.

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.

46.

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_.

47.

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).

48.

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.]

49.

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).

50.

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.

51.

That portion of an illuminated object which is nearest to the source

of light will be the most strongly illuminated.

52.

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).

53.

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.

54.

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

-4.--Between the title-line and the next there is, in the

original, a small diagram representing a circle described round a

square.].

55.

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.

56.

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.

57.

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).

58.

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.

59.

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).

60.

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.

61.

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.

62.

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).

63.

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.

64.

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.

65.

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).

66.

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.]

67.

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

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

68.

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).]

69.

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.]

70.

The shadow _m_ bears the same proportion to the shadow _n_ as the

line _b c_ to the line _f c_.

71.

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.

72.

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.

73.

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.]

74.

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).

75.

ON PAINTING.

The derived shadow is stronger in proportion as it is nearer to its

place of origin.

76.

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.

77.

_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.]

78.

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.

79.

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).

80.

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.]

81.

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.

82.

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).

83.

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.]

84.

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.

86.

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.

87.

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).

88.

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.]

89.

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.

90.

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.]

91.

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).

92.

The edges of a derived shadow will be most distinct where it is cast

nearest to the primary shadow.

93.

As the derived shadow gets more distant from the primary shadow, the

more the cast shadow differs from the primary shadow.

94.

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.

95.

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).

96.

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.

97.

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.

98.

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.

99.

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

2: 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

1-14, 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

480. 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