PHOTOGRAPHY THEORY AND PRACTICE PUBLISHED BY PITMAN STUDIO PORTRAIT LIGHTING By H. Lambert, F.R.P.S. A valuable work by a well-known portrait photographer. The illustrations are an extremely helpful feature of the work, and the variety of articles taken under various forms of lighting are of real assistance to the reader. In each case the author explains how the result has been obtained. In crown 4to, 92 pp., fully illustrated in photogravure and half-tone. 15s. net. Second Edition. COLOUR PHOTOGRAPHY By Robert M. Fanstone, A.R.P.S. Being a new edition of the book written by the late Capt. Owen Wheeler, F.R.P.S. An up-to-date manual, describing colour photography in simple language, and providing a practical guide to colour processes for the amateur photo- grapher with some experience of ordinary monochrome work, and for the young professional. In demy 8vo, cloth gilt, 171 pp., with 12 plates showing the whole process of colour in detail. 12s. Gd. net. Second Edition. PHOTOGRAPHIC CHEMICALS AND CHEMISTRY By J .Southworth and T. L. J. Bentley, D.I.C., A.R.C.Sc., B.Sc. (Hons. Lond.) The authors give a simple description of the principles of chemistry, and explain what takes place in the development, fixing, and after-treatment of photographic plates, films, and papers. A list is included giving a brief des- cription of all the chemicals likely to be met with. In crown 8vo, cloth gilt, 124 pp. Us. Gd. net. TELEPHOTOGRAPHY With Special Reference to the Choice and Use of the Telephoto Lenses in Connection with Photographic and Cinematograph Cameras Employing 8 mm., mm., Id mm., and Uo mm. Film. By Cyril F. Lan-Davis, F.R.P.S. Fourth Edition by H. A. Carter, F.R.P.S. A textbook for both professional and amateur photographers who wish to understand the many ways in which the telephoto lens will enable them to extend the range and scope of their photographic work. In crown 8vo, 1 17 pp., illustrated. Us. Gd. net. RETOUCHING AND FINISHING FOR PHOTOGRAPHERS By J. Spencer Adamson This book is written for professional photographers and keen amateurs who wish to know the latest methods and processes which are available for the retouching of photographic negatives and prints. In demy 8vo, 137 pp., with 8 full-page plates and 16 other illustrations. 4s. net. Third Edition. CAMERA LENSES By A. W. Lockett, Hons. Silver Medallist, City and Guilds Intended for all serious amateurs, and also for professional photographers. After a simple treatment of the principles involved, practically every type of camera lens now available is described. A study of this book will enable the photographer to select the most suitable lens or lenses for any class of work. In crown 8vo, 108 pp., with 100 illustrations. 2s. Gd. net. a ken on Ilford Panchromatic Pi.ati-: on Infra-red Plate, with Infra-red Filter VIEW FROM MARLFY, SUSSEX Originals by courtesy of Ilford Ltd. PHOTOGRAPHY THEORY AND PRACTICE BEING AN ENGLISH EDITION OF “LA TECHNIQUE PHOTOGRAPHIQUE 55 BY L. P. CLERC, Hon. F.R.P.S. EDITED BY THE LATE GEORGE E. BROWN, F.I.C., Hon.F.R.P.S. OFFICIER D'ACADEMIE EDITOR OF THE “ BRITISH JOURNAL OF PHOTOGRAPHY ” SECOND EDITION SIR ISAAC PITMAN & SONS, LTD. PARKER ST., KINGSWAY, W.C 2 BATH, MELBOURNE, TORONTO, NEW YORK HENRY GREENWOOD & CO., LTD. 24 WELLINGTON STREET LONDON, ENGLAND 037 First Edition, 1930 Second Edition, 1937 SIR ISAAC PITMAN & SONS, Ltd. PITMAN HOUSE, PA R HER STREET, KINGS WAY, LON BON W.C.2 THE PITMAN PRESS, BATH PITMAN HOUSE, LITTLE COLLINS STREET, MELBOURNE ASSOCIATED COMPANIES PITMAN PUBLISHING CORPORATION 2 WEST 45TH STREET, NEW YORK 205 WEST MONROE STREET, CHICAGO SIR ISAAC PITMAN & SONS (CANADA), Ltd. (INCORPORATING THE COMMERCIAL TEXT BOOK COMPANY) PITMAN HOUSE, 381-383 CHURCH STREET, TORONTO MADE IN GREAT BRITAIN AT THE PITMAN PRESS, BATH C7— (T.5630) PREFACE TO THE SECOND EDITION Opportunity has been taken in preparing the second English edition of this book to bring it into line with the second French edition (1934) of La Technique Photo- graphique , and to incorporate such further revision as has been necessitated by the considerable advances in photographic technique, which have been made since the appearance of that volume. At the same time certain ambiguities in translation which became apparent in the first English edition have been adjusted, and matter which had become out of date was either cut out altogether, or very considerably compressed PREFACE To inveigh against theory, even in the most elementary arts, proves only the ignorance of the inveigher. It is not the profoundness of the theory, but its imperfections, that should be blamed for the ill effects that so often follow its working out in practice. . . . Many data regarding the needs to be satisfied, the means of satisfying them, the time and expenditure involved, that are perforce ignored within the field of mere theory, come into the problem of working out a practical application. By bringing these factors into play with the skill that marks practical genius, it is possible both to extend the narrow limits within which prejudices against theorizing tend to confine the arts, and to guard against the mistakes to which an unskilful use of a particular theory may give rise. — Condorcet ( Tableau des Progres de V Esprit humain, 1793) This book does not represent an attempt to compile an encyclopaedia, a work of a kind which is always loaded with descriptions of obsolete methods and appliances and the details of numerous applications of interest only to a few technical people. The author’s aim has been to bring into one volume as complete a treatise as possible on modern working methods and apparatus in conjunction with the mini- mum of theoretical considerations which he considers necessary for their proper understanding. The beginner is recommended to study at first the parts of practical instruction which he requires, and to postpone a reading of the whole work until he has acquired some practical experience. In the interval he can make use of the book in the manner of a dictionary, by aid of the full alphabetical index at the end. Objection may be taken to the absence of bibliographical references. But such references would have been exceedingly numerous and would have added consider- ably to the already large number of pages without much advantage to most readers. Therefore the references have been confined to a mention of the names of the authors (and dates) of various discoveries, improvements, and experiments, a plan which at any rate will serve to narrow down the scope of any bibliographical search which some readers may wish to carry out. The author’s professional duties for thirty years past have rendered it necessary for him to read nearly everything which has been published on photography in the technical journals of the chief countries of the world, and to experiment or supervise experiments in regard to a large number of the data thus collected. In VI PREFACE a certain measure therefore this work may be considered the work of all ; and the author would excuse himself to experimenters and writers whose ideas and recom- mendations have been mentioned, possibly without acknowledgment. Since the publication in 1926 of the original French edition, the author has made a considerable number of additions and corrections which are embodied in the present English translation. The author desires to express to the translators and editor of this volume his sincere thanks for the evident care with which their work has been done. Their critical comments have enabled the author to correct several errors in the original edition. L. P. C. r CONTENTS PREFACE PART i INTRODUCTION : VISION AND PHOTOGRAPHY CHAPTER I LIGHT AND COLOUR ........... White light — Light-waves — Wavelength — Luminosity of different spectral colours — The ultra- violet and the infra-red — Natural and pigmentary colours — Absorption by coloured media — Sources of artificial light CHAPTER II QUANTITY OF LIGHT ........... Intensity and brightness — Illumination CHAPTER III LIMITS OF LUMINOSITY IN PHOTOGRAPHIC SUBJECTS .... Sensitivity of the eye — Perception of details of luminosity CHAPTER IV PHOTOGRAPHIC IMAGES : THE IDEAL SCIENTIFIC IMAGE : THE AESTHETIC IMAGE . Negative and positive — Range of extreme luminosities in a positive — The ideal scientific reproduction — The aesthetic image CHAPTER V perspective: monocular and binocular vision Geometrical perspective — Deformations due to displacement of the viewpoint — Normal distance of vision and angle of visual field — Anomalies of an exact perspective — Influence of choice of viewpoint — Binocular vision — Perspective on a non-vertical plane — Panoramic perspective — Depth of field PART 2 THE OPTICAL IMAGE BEFORE PHOTOGRAPHIC RECORDING CHAPTER VI THE CAMERA OBSCURA AND PINHOLE PHOTOGRAPHY . . . . . Pinhole photography — Making a pinhole CHAPTER VII GENERAL PROPERTIES OF OPTICAL SYSTEMS: ABERRATIONS . . . . Lenses — Images formed by convergent lenses — Real images — Virtual images — Optical centre — Nodal points — Chromatic and spherical aberration — Astigmatism — Tangential and radial images — Coma — Curvature of field — Distortion — Distribution of light in the field — Field illuminated — Field covered — Effect of internal reflection — Stereoscopic effects vii PACK V I 7 8 io 12 20 23 CONTENTS viii CHAPTER VIII FOCAL LENGTH OF LENSES: SCALE OF IMAGE : CONJUGATE POINTS . . .38 Relations between the size of object and image — Experimental determination of the focal length of a lens — Automatic adjustment of object and image CHAPTER IX DIAPHRAGMS AND RELATIVE APERTURE : EFFECT ON PERSPECTIVE AND INTENSITY . . . . . . . . . . 46 Different types of diaphragms — Pupils of an optical system — Photographic perspeclive — Depth of field — Hyperfocal distance — Influence of the corrections of the lens on the depth of field and hyperfocal distance — Fixed-focus cameras — Focussing scales — Different systems of numbering diaphragms — Measurement of the effective aperture of a diaphragm CHAPTER X CHOICE OF LENS : TESTING : CARE OF LENSES 6l Lens names — Single lenses — Petzval portrait lens — Rectilinear lenses — Wide-angle rectilinears — Anachromatic symmetrical lens — Antiplanats — The first anastigmats — Convertible and un- convertible anastigmats — The Hypergon wide-angle lens — Variable-power telephotos — Fixed- focus telephotos — Anachromatic telephotos — Sets of lenses — Different types of lens mounts — Choice of a focal length — Practical testing of lenses — Preservation and care of lenses CHAPTER XI LENS ACCESSORIES : SUPPLEMENTARY LENSES, LIGHT-FILTERS, PRISMS AND MIRRORS, LENS HOODS, SKY SHADES ....... 77 Supplementary lenses — Tele-attachments — Light-filters — Prisms and mirrors — Polarizers — Lens hoods — Skyshades — Soft focus attachments CHAPTER XII SHUTTERS 87 Different positions for the shutter — Efficiency of a shutter — Desirable characteristics of a shutter — Roller-blind shutters — Flap shutters — Diaphragm shutters — Focal plane shutters — Practical rules for the use of focal plane shutters — Shutter testing — Shutter releases CHAPTER XIII STAND CAMERAS : COMMERCIAL, PROFESSIONAL, AND SEMI-PROFESSIONAL . . 10 6 Names and functions of the parts of a camera — Cameras for commercial copying — Studio portrait cameras — Portable stand cameras — Tripod stands — Hand-stand cameras — Cameras for “ while-you-wait " photography — Testing of cameras CHAPTER XIV HAND CAMERAS 115 Miniature cameras — Rigid cameras — Folding cameras — Focussing of hand cameras — Ground- glass finders — The brilliant erecting finder — Direct-vision brilliant finders — Telemetric finders — Levels and plumbs — Safety devices — Reflex cameras — Sensitive material — Single metal slides — Changing boxes for plates — Roll-film cameras — Film packs — Tripods and pocket-supports for hand cameras — Tests of a hand camera PART 3 PRODUCTION OF NEGATIVES CHAPTER XV THE negative: general remarks on photographic negative processes . 130 The wet collodion process — The gelatino-bromide process CONTENTS IX CHAPTER XVI PREPARATION AND PROPERTIES OF NEGATIVES : GELATINO-BROMIDE EMULSIONS . 132 Preparation of emulsions — The sensitive emulsion — The latent image — Different actions on the photographic emulsion — The law of density — Measures of sensitivity — Reversal and solar- ization — The accuracy of photographic images CHAPTER XVII CHROMATIC SENSITIVITY, ORTHOCHROMATISM AND INFRA-RED PHOTOGRAPHY . . I44 Action of different spectral radiations on gelatino-bromide emulsions — Yellow filters — The factors of a colour filter — Experimental determination of the factors — Self-screened ortho- chromatic emulsions — Orthochromatism in practice — Photography of colour objects — Photography with infra-red rays — Colour-sensitizing CHAPTER XVIII PLATES, FILMS, AND NEGATIVE PAPERS ....... 157 Supports of sensitive coatings— Flexible supports — Halation — Prevention of halation — Photo- graphic plates — Photographic film — Relative merits of plates and films CHAPTER XIX NON-ACTINIC LIGHTING : DARK-ROOM LAMPS AND SAFELIGHTS .... l 66 Choice of dark-room illumination — Light-sources — Testing of safelights — Preparation of safelights CHAPTER XX EQUIPMENT OF THE DARK-ROOM ......... 172 The amateur’s dark-room — Public dark-rooms — Construction of the ideal professional dark- room — Passages — Ventilation — Heating — Sinks — White light in the dark-room CHAPTER XXI DARK-ROOM ACCESSORIES 177 Tanks and dishes — Material for tanks, hangers, and dishes — Glass-ware — Cleaning of utensils CHAPTER XXII chemicals: preparation of solutions 182 Choice of chemicals — Unstable substances — Storage of chemicals — Labelling of jars and bottles — Preparation of photographic baths — Filtration — Stock solutions — Solutions, concentration, solubility, saturation — Water used in the preparation of baths — Commercial preparations CHAPTER XXIII HANDLING OF SENSITIVE MATERIALS: LOADING AND UNLOADING OF DARK SLIDES: REPACKING .......... 188 Storage of sensitive materials — Dusting the sensitive surface — Identification of negatives — Unloading the dark slides CHAPTER XXIV LIGHTING OF THE SUBJECT I DAYLIGHT! ARTIFICIAL LIGHT .... I92 Daylight — Arc lamps — Incandescent lamps — Mercury-vapour lamps — Illumination of flat originals — Illumination by magnesium — Flash lamps — Flash powders CHAPTER XXV FOCUSSING OF THE IMAGE AND THE POSITION OF THE SUBJECT ON THE PLATE . 205 Focussing — Soft focus — Soft-focus negatives — Correction of focus with anachromatic lenses — Superimposed plates — Soft-focus printing — Choice of viewpoint X CONTENTS CHAPTER XXVI EXPOSURE 211 Time of exposure — Results of errors in exposure — Factors affecting time of exposure — Influence of relative displacements of the subject and of the camera — Instantaneous and time exposures — Actinometers — Photo-electric exposure-meters — Photometers — Exposure tables CHAPTER XXVII DESENSITIZING OF PHOTOGRAPHIC EMULSIONS ...... 222 Use of coloured developers — Loss in sensitivity of emulsions impregnated with developer — Desensitizers — Practical methods of desensitizing CHAPTER XXVIII THE DEVELOPMENT OF THE NEGATIVE ....... 226 Chemical development — Theory of development — Effect of dissolved bromide in the developing solution — Chemical fog — Effect of dilution of the developer — Effect of the temperature of the developing solution — The arithmetical coefficient of a developing solution — Mineral salts as developers — Ferrous oxalate developer — The developing function — Normal constituents of a developer — Mixture and oxidation products of developers — Role of sulphite — Role of the alkalis — Effect of soluble iodides — Use of various neutral salts — Organic developers — Poly- phenol developers — Amidophenol developers — Amidol — Addition products of developers — Par aphenylenedi amine — Sulphites, bisulphites, and metabisulphites — Alkalis and their substitutes — Practical equivalence of the usual alkalis — Alkali bromides — Wetting plates and films before development — Immersion in the developing bath — Local development — Factors influencing the duration of development — Judging the end of development — Critical examina- tion of finished negatives — Exhaustion and maintenance of the developer — Developer stains on hands and clothes — Formulae for developers — Rapid or slow development — Fine-grain developers — Rational automatic development — Development by visual inspection of the image — Development in several successive baths — Tentative development — Development in tropical climates — Combined developing and fixing — Method for the comparative testing of two developers — Physical development before fixing — Physical development after fixing CHAPTER XXIX FIXATION 270 The purpose of fixation — Solvents of the silver halides — Hyposulphite of soda — The chemistry of fixation — The mechanism of fixation — Additions to fixing baths — The chemistry of fixation in presence of alum — Fixing capacity of hyposulphite for silver halides — Speed of fixation — Fixation in two successive baths — Choice of the best concentration of fixer — Tests for exhausted fixer — Preparation of fixing baths — Fixation in practice — Recovery of silver from exhausted fixing solutions CHAPTER XXX WASHING 283 The function of washing — The mechanism of washing in several changes and in running water — Apparatus for washing — Control of washing — Washing in practice — Hypo eliminators CHAPTER XXXI drying 289 The purpose of drying — The physics of drying — Apparatus for drying — The operation of drying — Distortion of the image during drying — Rapid drying with a volatile liquid — Instantaneous drying by dehydration of the gelatine CHAPTER XXXII THE CHIEF FAILURES IN NEGATIVE-MAKING 294 Faults appearing during development and after fixing— Dichroic fog — Defects occurring during washing — Defects appearing during or after drying — Defects occurring in a negative after drying CONTENTS xi CHAPTER XXXIII REVERSAL PROCESS : METHODS FOR OBTAINING DIRECT POSITIVES . . . 3OI General considerations — Reversal by a second, determined exposure — Reversal by total black- ening of the residual silver bromide — Reversal by means of residual silver bromide of which the excess is dissolved to a determined degree — Other methods CHAPTER XXXIV METHODS OF AFTER-TREATMENT : INTENSIFICATION, REDUCTION, WORKING-UP, RETOUCHING ........... 306 Choice of method of intensification — Mercury intensification in two successive baths — Chromium intensification — Intensification with copper and silver — Choice of a reducer — Surface reducers — Farmer's reducer — Permanganate reducer — Proportional reducers — Ammonium persulphate super-proportional reducer — Indirect reduction of heavier densities — Local intensi- fication and reduction with a brush — Local tinting of the gelatine — Local abrasion of the negative — Work on the back of a negative — Reduction of contrasts by soft positive — Blocking- out — Spotting — The purpose of retouching — Retouching appliances — Technique of retouching — Added backgrounds CHAPTER XXXV VARNISHING, STRIPPING, NUMBERING, CLASSIFICATION, AND STORAGE OF NEGATIVES 323 Preparation of varnishes — Application of the varnish — De-varnishing of negatives — Oiling of paper negatives — Stripping glass negatives — Stripping film negatives — Transfer of the film to a new support — Removal of films from waste glass and celluloid negatives — Classification of negatives — Storage of negatives PART 4 PRINTING PROCESSES CHAPTER XXXVI PRINTING PAPERS AND PRINTING METHODS ....... 330 The principal printing processes — Supports for the photographic image — Photographic papers — Positive plates — Positive films — Colour of image and of base — The gradation of a sensitive paper — Printing frames and machines — Actinometers and light-integrators — Effect of illumin- ation on gradation — Effect of colour of the printing light on contrast — Calibration of negatives — Modes of printing — Printing with masks — Printing with a tinted border — Vignetting — Local control during printing — Reducing contrast and definition — Combination prints — Adding a sky to a landscape — General hints CHAPTER XXXVII SILVER PRINT-OUT PAPERS 34 ^ Deterioration of print-out papers — Salted paper — Albumenized papers — Sensitizing silk — Print-out silver emulsions — Use of collodion and gelatine P.O.P. — The printing room — Gold toning in alkaline baths — Sulphocyanide toning — Thio-urea toning — Platinum toning — Com- bined toning and fixing — Reactions in combined toning and fixing baths — Preparation and use of combined toning and fixing solutions — Development of part-exposed prints — Types of self-toning papers — Working methods XU CONTENTS CHAPTER XXXVIII PAPERS, PLATES, AND FILMS FOR POSITIVE PRINTS BY DEVELOPMENT . . 358 Different kinds of emulsion and their characteristic qualities — Abrasion — Manipulation of printing papers before exposure to light — Exposure — Soaking before development — Develop- ment — Developers for positives — Stopping development with an acid bath — Non-actinic light — Developers for bromide papers — Development of transparencies — Making of duplicates — Gaslight papers — Developers for gaslight papers — Method of development — Transparencies — Warm tones by development — Exposure — Warm tones on black-tone plates — Chloro-bromide papers for warm tones — Fixing developed images — Intensification — Reduction — Sulphide toning — Bleaching for sulphide toning — Direct sulphiding with hypo-alum — Single-solution sulphidinghvith polysulphides — Gold toning — Toning with selenium — Toning by the formation of coloured ferrocyanides — Uranium copper and iron toning — Dye-toning — Mordanting the images — Dyeing of mordanted transparencies — Tinting CHAPTER XXXIX WASHING, DRYING, AND GLAZING OF PAPER PRINTS ..... 39I Methods of washing papers — Apparatus for washing prints — Drying of paper prints — Rapid drying of prints — Deformation of the image during washing and drying — Glazing and enamel- ling of prints — Facing the prints with cellophane CHAPTER XL THE PRINCIPAL FAILURES IN MAKING PRINTS OF POSITIVES ON SILVER PAPERS . 397 Failures common to various printing processes — Failures with print-out papers — Failures with development papers — Failures in sulphide toning CHAPTER XLI PRINTING PROCESSES BASED ON THE SENSITIVITY OF IRON SALTS . . . 402 The ferro-prussiate process^Industrial papers for copying tracings — True-to-scale process — Printing on ferro-prussiate paper from ordinary negatives — Cyanotype paper — Ferro-gallic papers — Silver-iron printing papers — Sepia photo-copying papers — Kallitype and imitation platinum papers — Platinum-iron printing papers — Use of black-tone platinum papers — Sensi- tizing papers for platinum printing — Recovery of platinum residues CHAPTER XLII PIGMENT PROCESSES 413 The carbon process — Action of light on bichromated gelatine — Chromates and bichromates — Physiological action of bichromates — Transfer of the film — Preparation of carbon tissue — Sensitizing and exposing carbon tissue — Drying the sensitized tissue — Preparation of the negatives — Exposure to light — Development of tissue by single transfer — Development — Trans- parencies and positives — Intensification and toning of images on an impervious support — Development of tissue by double transfer — Preparation of the temporary support — Develop- ment — Preparation of the final support — Final transfer — Retouching and coloring — Carbon printing without transfer — The Artigue and Fresson processes — Papers exposed through the back — The gum-bichromate process and its variations — Coating the pigmented gum — Exposure and development — Multiple printing — Commercial gum-bichromate papers — Bichromated papers — Prints by dyeing films of bichromated gelatine — Imbibition without development — Dyeing of gelatine reliefs — Dusting-on processes — Direct reproductions by the powder process — Resinopigmentype — Colour process — Prints in greasy inks on bichromated gelatine — Materials and apparatus — Sensitizing and exposure to light — Washing, swelling, and drying — Inking the print — Special instructions for obtaining prints by transfer — Papers for transfer printing — — Transfer without press — Ozotype — Dye prints by photo-mordants — Dye prints with diazo compounds — Other processes CHAPTER XLIII PIGMENT PRINTS FROM SILVER PRINTS ....... 445 The Carbro or Ozobrome process — Theory of the Carbro process — The Bromoil process — The bromide print — Bleaching the image — Swelling of the image — Inking CONTENTS xiii CHAPTER XLIV FINISHING AND WORKING-UP PRINTS I TRIMMING, . MOUNTING, RETOUCHING, pack AND COLOURING 450 How much to trim — Choice of mounts — Placing the picture on the mount — Various methods of fixing the print on the mount — Mountants — Full mounting — Burnishing — Dry-mounting — Waxing and varnishing prints — Embossing prints — Outline photographs on wood — Combina- tion photographs — Spotting — Retouching of prints — Use of the air-brush — Retouching of commercial photographs for photo-mechanical reproduction — Coloured photographs — Colour- ing with dyes — Water-colouring — Pastel colours — Oil-painting and its variations — Framing photographs PART 5 SPECIAL TECHNIQUES CHAPTER XLV copying: restorations to the vertical: deformations .... 466 Contact printing — Use of reflected light — Copying with a camera — Factors affecting sharpness — Sensitive materials — Texture of the original print — Lighting for copying — Examination of documents — Cylindrical surface copied in strips — Anamorphoses— Caricature deformations CHAPTER XLVI enlargements 474 Condensed and diffused light — The negative — Condensers — Light-source — Vertical enlargers — Focussing the picture — Soft- focus effect — Exposure — Control — Enlarged paper negatives — Large-scale work CHAPTER XLVII LANTERN WORK ........... 490 Photographic lantern slides — Diagram and notice slides — Binding lantern slides — Storing and carrying — Projection lanterns — The light-box — Paraffin oil lamps with multiple wicks — Acety- lene burners — Oxy-acetylene limelight — Precautions to be taken in the manipulation of oxygen cylinders — Electric light — The condenser — Slide carriers — Projection lenses — Cooling devices — Opaque objects — Lantern screens — Opaque diffusing screens — Translucent screens — Daylight screens — The lantern lecture — Illumination of the hall — Centring the light source — Arrange- ment of slides CHAPTER XLVIII STEREOSCOPY ............ 506 The sensation of relief — Stereoscopic photographs — Stereoscopic transposition — Planoscopic pairs — Geometrical considerations — Parallax — Range of stereoscopic vision — Examination of stereograms — Stereoscopes with convergent eyepieces — Complementary stereoscope — Masking — Transparencies — Projection CHAPTER XLIX COLOUR PHOTOGRAPHY 533 Trichromatic selection — Additive synthesis — Three-colour filters — Triple projection — Chromo- scopes — Lenticular colour photography — Three-colour transparencies and prints — Kodachrome film — Colour-screen plates and films — Autochrome plates and films — The Dufaycolor process — Colour separation negatives from colour-screen transparencies CHAPTER L AN OUTLINE OF CINEMATOGRAPHY 552 Cine film — Cine cameras — Drying film — Titling — Printing positives — Projection — Stroboscopic illusions XIV CONTENTS CHAPTER LI PHOTO-MECHANICAL PROCESSES IN BRIEF 558 Line etching — Half-tone — Collotype — Photo-lithography — Photogravure — Colour prints CHAPTER LII GENERAL PRINCIPLES OF RADIOGRAPHY 563 Properties of X-rays — Protection — Radiographic images — Intensifying screens — Exposure — Printing positives APPENDIX: A CHRONOLOGY OF PHOTOGRAPHY 571 INDEX 575 INSET PLATE VIEW FROM MARLEY, SUSSEX — (A) TAKEN ON PANCHROMATIC PLATE (b) TAKEN ON INFRA-RED PLATE . . Frontispiece CONTRACTIONS A.U. . . Angstrom unit (§ 2) mm. . . millimetre B. . degree Baume hydrometer F. . degree Fahrenheit m. . metre grm. . . gramme C. . . degree Centigrade gr. • . grain c.c. . cubic centimetre* min. . . minim cm. . centimetre oz. . ounce drm. . dram * The term “cubic centimetre" is used in preference to “millilitre" throughout the text. In practice, the difference between the two measures is so small as to be negligible. xv PHOTOGRAPHY THEORY AND PRACTICE PART i INTRODUCTION: VISION AND PHOTOGRAPHY CHAPTER I LIGHT AND COLOUR i. White Light. When a beam of daylight, which we call white light, in spite of the fact that its quality is continually changing , 1 is decomposed, e.g. by passing it through a glass prism which separates it into the constituent elements of the incident light, the colours of the rainbow may be seen in their normal order. By using suitable apparatus, described in all books on optics, a pure spectrum can be obtained consisting of an infinite number of images of a fine slit, each image being formed by one of the elementary radiations which go to make up the light by which the slit is illuminated. Though there are an infinite number of colours in the spectrum, it is usual to divide them into groups, in each of which the eye experiences sensations which differ little from one another. For reasons as arbitrary as those which formerly caused the number of days in the week, or the wonders of the world to be fixed at seven, the number of spectral colours has also been fixed at seven, and a bad alexandrine of the Abbe Delisle — Violet, Indigo, Blue, Green, Yellow, Orange, and Red has contributed quite appreciably to keeping alive this unfortunate tradition, from which most exaggerated conclusions have often been drawn. Examination of the colours of the rainbow or of a slightly dispersed normal spectrum 2 shows that the spectrum may be divided into three chief regions of equal extent, the blue-violet, 1 Daylight nearly always shows some colour which is in excess of its average composition, and which varies with atmospheric conditions, time of day, and, to some extent also, with geographical latitude and with alti- tude. Light diffused by a lightly clouded sky is bluish ; while direct sunshine is yellowish, changing to red at sunset. 2 Spectrum as formed by passing the light through a grating, i.e. a system of parallel equidistant lines generally about 6oo per millimetre (15,240 per inch), instead of through a prism. 1 — (T. 5630) green, and vermilion-red, in each of which the variation of tint is almost imperceptible. Be- tween these large regions there are narrow tran- sition regions where the variation of colour is very rapid, one being a blue-greenish colour be- tween the blue-violet and the green ; the other, yellow between the green and the red. For most practical purposes it is sufficient to consider the three main regions into which the spectrum may, roughly speaking, be equally divided. Thus, if the spectrum is to be considered as divided into more than five regions (the three principal and the two transition regions), then one ought to speak of an infinite number of colours rather than only seven. 2. Lightwaves — Wave-length. As the colours are infinite in number, obviously it is impossible to name them all accurately by words, and it is necessary to use numbers. Spectral radiations are universally specified by their wave-lengths . This numeration of colours, independent of any arbitrary convention, was made possible when the researches of Young and Fresnel proved that light is the propagation of a periodic vibration, which may be compared (provided the comparison is not pushed too far) to the propagation of sound (sound waves), or to the propagation of waves created on the surface of smooth water by the falling of a stone, or even to the course of a person walking in a straight line with a uniform velocity and a regular step. Without attempting here any justification of this wave theory, one may mention, however, amongst the numerous facts which may be quoted in support of it, the coloration of thin films (soap bubbles) and the direct process of colour photography due to G. Lippmann (inter- ference method). In all cases of the propagation of a periodic phenomenon there are three quantities involved : the velocity of propagation, the time of one period (or its reciprocal, the frequency ), and the “ step ” 1 2 PHOTOGRAPHY: THEORY AND PRACTICE or wave-length , which are connected by the relations — Wave-length = velocity x time of one period velocity — frequency’ The velocity of light in air is approximately 187,000 miles per second, and is the same for all radiations. While the pitch of a note is always character- ized by its frequency, radiations are always denoted by their wave-length (sometimes repre- sented by the Greek letter A, lambda ), which is who are colour-blind cannot perceive certain colours; they generally confuse red and green), the region which appears the most luminous in the normal spectrum of white light is the green. At high luminous intensities the maximum of luminosity is at 5,500 A. U. (yellow-green); at very weak illuminations the red, which normally is more luminous than the blue, begins to fade sooner than the blue, the sensation of colour disappears almost completely, and the maximum of luminosity approaches continually closer to 5,300 A. U. (blue-green). This effect, known as the Pnrkinje phenomenon, is, as we shall see Ultra-violet Violet Blue , Green Yellow 5RQP0 N M LKH h (y 1— !- Region | absorbed yyi b_E -sU Red Extreme Red . * . usually invisible l n ' ra_r ed 4J00 ' ‘shoo ' m 111 7 ( j 00 ' ' dioo Fig. 1. Distribution of Rays in the Solar Spectrum the distance from crest to crest of two successive waves, measured generally in ten-millionths of a millimetre, that is, in Angstrom units (A.U.). Expressed thus, the wave-lengths of visible radiations extend between 4,000 for the extreme violet and 7,000 for the extreme, easily visible red (with special precautions, trained observers have been able to see as far as 8,000). 1 Fig. 1 represents the distribution of wave-lengths between the different colours in a normal spectrum, the wave-length scale being equally graduated throughout. In the figure are also marked the positions of the black bands of the solar spectrum, called the Fraunhofer lines, denoted by the letters A to H, and constituting a series of reference marks which are sometimes used to denote the different spectral regions in cases where extreme precision is unnecessary. The frequency (number of waves per second) is constant for each radiation, irrespective of the medium, and is approximately expressed for yellow light by the figure 5, followed by 14 zeros. These numbers are not, as might be thought, simply theoretical speculations, but the expression of practical measurements; the wave-length is frequently used as a measure of length in precision work and even in certain industrial measurements. 3. Luminosity of Different Spectral Colours. For an observer who is not colour-blind (people 1 Note that the interval between the extreme visible radiations is less than a musical octave. presently, of special interest in regard to the choice of illuminants for the dark-room. About 95 per cent of the visual effect of the spectrum is confined between the lines C and F, whilst photographic action on ordinary emul- sions is limited only to those radiations of wavelength less than that of the F line. This peculiarity has important consequences in the photographic rendering of different colours. 4. The Ultra-violet and the Infra-red. The radiant energy of the sun and various artificial illuminants is not limited to the visible region of the spectrum, but covers a range, which is actually known, of at least 15 octaves. If, in the region beyond the spectrum violet, pieces of paper impregnated with fluorescein or rhodamine [fluorescent substances) are placed, it will be seen that they emit respectively green and red light. This effect they also show in the blue-violet but not to such an easily observable extent. The same phenomenon can be shown with crystals of uranium nitrate, with screens covered with barium platinocyanide (used in radioscopy), calcium tungstate (X-ray intensi- fying screens), or certain preparations of zinc sulphide. In order to demonstrate the existence of these invisible radiations a spectrum need not necessarily be used. By projecting on to a screen the image of an electric arc and then placing in the path of the beam a piece of special black glass (Wood’s glass) which absorbs all the visible light whilst transmitting the ultra- LIGHT AND COLOUR 3 violet, the image of the arc can be made to re-appear by placing at the point where the visible image existed one of the fluorescent screens mentioned above. The glasses generally employed in the con- struction of optical instruments transmit the ultra-violet down to about 3,500 A.U. The limit extends further to about 3,200 A.U. in the case of certain special glasses (Uviol). Thanks to the absorption of our atmosphere, the solar spectrum ends at about 3,000 A.U., which fact protects us from the very dangerous physiological effects of the shortest wave-length radiations such as are produced by arcs between metal electrodes and transmitted by quartz (rock crystal) down to about 2,000 A.U., which is also approximately the limit of transparency of gelatine and air. By means of suitable apparatus (reflection gratings in vacuum), and by using sensitive surfaces without gelatine, it has been possible to study photographically the ultra-violet down to about 100 A.U., where it joins the X-rays. Quartz lenses, used in conjunction with filters which transmit only ultra-violet, are employed in certain special applications of photography. A filter suitable for this purpose is a thin film of silver. In photographs obtained with these radiations alone, glass objects appear completely opaque ; certain white flowers and pigments are indis- tinguishable from pure blacks, and, further, no background and no shadows appear in a land- scape photographed in bright sunshine. The photograph looks as if it were taken in a dense fog (R. W. Wood, 1910). The infra-red, which extends the visible spec- trum beyond red, was of no photographic interest for a long time, its effects being chiefly thermal. The only known means for infra-red photography was an indirect method based on the fact that these radiations discharge almost instantaneously phosphorescent bodies. By uni- formly exciting a phosphorescent screen by ultra-violet, forming on it an infra-red image and then applying it against a sensitive layer, a positive image (§18) is impressed by the residual phosphorescence. Since the discovery of sensitizers (§223) for infra-red which are sufficiently easy to handle (E. Q. Adams and H. L. Haller, 1919; H. T. Clarke, 1925), it has been possible to place on the market plates and films for infra-red photo- graphy up to 10,000 or 13,000 A.U. These sensitive emulsions must be kept and manipu- lated with special care. The extreme trans- parency of the atmosphere for infra-red has enabled photographs to be taken, with these rays only, of landscapes up to 331 miles distance from a high viewpoint. If all radiations of wave- length below 7,000 A.U. are eliminated, green foliage is reproduced as white as snow, owing to the intense fluorescence of chlorophyll with a maximum near 7,400 A.U. (C. Dehre and A. Raff} 7 , 1935), and as the blue of the sky is ren- dered as black and there is no shadow detail the landscapephotographsthus taken in sunshine with a clear sky give an impression similar to that of a photograph taken by moonlight. 5. Natural and Pigmentary Colours. By the colour of an object is always meant the colour it appears when seen in white light, but this colour depends essentially on the nature of the light which illuminates it. If a paper covered with vermilion is placed successively in various regions of the spectrum, it appears red in the red region, yellow in the transition region, and black in all the others. Its surface, whilst readily reflecting the red and spectral yellow, is not able to reflect the green, blue-green, and blue-violet ; it absorbs and, as it were, destroys these radiations. In white light the surface appears red because of the pre- dominance of red in the light which it reflects. The vermilion layer will appear always red when illuminated by a light which contains red ; and black in all other cases. The colour is thus not actually in the vermilion, but in the light. Similarly, a yellow flower appears successively red, yellow, or green in the corresponding spectral regions, and black in all the others. No natural or manufactured object, no matter by what process its surface is covered, can even approximately be considered as monochromatic, that is to say, as reflecting only radiations of very nearly the same wave-lengths. Pure spectral colours do not exist in Nature (unless in the rain- bow) ; all coloured objects, natural or artificial, extinguish more or less completely some of the radiations of the light which shines on them, and it is the net result of the reflection of the other radiations which the eye observes. Colour is therefore the result of a subtraction process. White light becomes coloured because certain of the radiations in it are extinguished. It is therefore logical to define the colours of things by the spectral regions which they absorb. A bright yellow like that of the petals of a buttercup is due to the absorption of the blue-violet region and to the free reflection of the two other regions, green and red. If, on the other hand, 4 PHOTOGRAPHY: THEORY AND PRACTICE an object reflects to the eye only the yellow radiations, these are such a small portion of white light (less than i per cent) that the object appears almost black, or at least a very deep olive. By studying successively different colours, selected for the utmost purity and intensity, it can be shown, at least to a first approximation, that some of them absorb simultaneously two of the chief spectral regions that we have con- sidered, whilst others, absorbing only one of these regions, appear to be much more luminous. The table given below sets out some data on this point. Colours Spectral regions extinguished Spectral regions reflected Ultramarine blue. Green -f red. Blue-violet. Peacock blue. Red. Blue-violet + green. Emerald green. Red + blue-violet. Green. Cadmium yellow. Blue-violet. Green -f- red. Vermilion. Blue-violet + green. Red. Carmines and purples. Green. Red + blue-violet. This table shows us the existence of a colour not having any spectral equivalent. The purest type of this is the intense rose colour given by dyes such as rhodamine, rose Bengal, or erythrosine. 6. The colours which each reflect two of the chief regions of the spectrum are known as the primary colours by painters and printers, because by superposing them two at a time, thus adding together their respective absorptions (mixing of colours by subtraction of light), the inter- mediate colours can be obtained which only reflect one chief spectral region. For example, by superposing a peacock blue, which absorbs the red, and a yellow, which absorbs the blue- violet, we obtain a green, since this is the only spectral region transmitted simultaneously by the two superposed colours. The mixtures thus obtained are necessarily darker than each of the mixed colours. By placing close to one another little dots of colour so small that when looked at by the eye from a normal distance they appear blended into a single colour (the technique of the pointillist and of Autochrome), the mixing of colours is brought about by the addition of lights. The colour of the mixed light, for the reason that it is made up of the radiations reflected by each of the colours in the mixture, is more luminous than each of the colours separately. 7. The colour of a given substance depends to a very considerable extent on its state of division and on the medium in which it is. Many cases are known in which crushing a substance to a powder decreases its coloration. For example, blue copper sulphate appears white after being powdered. In addition to the light which has penetrated into the substance and which comes out from it coloured, due to the absorption of certain of the radiations in the incident white light, 1 there is always a certain proportion of white light reflected or diffused from the surface without alteration in colour, and which diminishes the actual coloration of the substance. In the case of a polished surface the white light is reflected almost entirely in the direction determined by the laws of reflection from mirrors. It is thus only in this one direction that the colour is diminished by reflection, and sometimes even almost completely lost in the considerable excess of white light reflected; in other directions the coloration appears with maximum intensity. In the case of a coloured powder or of a matt surface, the white light is scattered without change of colour approximately equally in all directions, so that it becomes impossible to see the true colour of the substance. This effect would be diminished by wetting the surface of the body with water (it is known that photo- graphic proofs printed on matt papers are always more beautiful and vigorous when wet than after drying), but this kind of colour intensification can be made more permanent by replacing the water, a volatile liquid, by a varnish, the effect of which will be all the more marked if the colour is contained in a more refracting medium. This is the reason why oil paintings or colour suspensions in gelatine give much more intense colorations than can be obtained by means of water colours in a medium of a very weak proportion of gum arabic, or by pastels, a medium-free coloured powder, of which the character of lightness disappears when it is treated with a fixative, i.e. covered with a varnish. 8. The colours given in the table in § 5 are pure saturated colours. On the other hand, the absorption band of a colour, i.e. the spectral region in which there is more or less complete extinction of the rays of the incident white light, may be of any extent, and its absorption 1 Only such metals as copper, gold, etc., appear to colour the light by reflection, without the light pene- trating inside the substance. The coloured light thus reflected has mixed with it, however, a greater propor- tion of the unchanged white light. LIGHT AND COLOUR 5 may be complete or only partial. If the absorp- tion band extends throughout the whole spec- trum, either a dark or broken hue will result (corresponding respectively with dilution of the pure saturated colour with black or grey) according as the absorption in certain regions is complete or not. If the absorption band is of limited extent, and if no rays are completely extinguished, a pale hue will result, equivalent to mixing white and the corresponding saturated colour. Thus, for example, pure saturated orange absorbs the blue-violet completely and an appreciable fraction of the green, whilst trans- mitting freely all the red. If the absorption is incomplete, with a maximum in the blue-green, the orange will give a flesh colour due to dilution with white. In like manner, the admixture of black or of grey with orange gives, in the former case, a dark terra-cotta, and in the other, a broken cream colour. All shades of colour can be accurately defined by means of the curve showing in the normal spectrum the percentage of diffused light for each of the radiations. 9. Nearly all objects, whether natural, dyed, or painted, even when they appear of very pure and intense hue, give actually only deep or broken hues, since no radiation is reflected completely. The purest yellows, oranges, and reds reflect generally about 70 per cent of the radiations which, theoretically, they ought to reflect (or scatter) completely. This proportion drops to about 20 per cent for the blues and violets and is less than 15 per cent for the greens. In this connection some measurements carried out by A. J. Bull (1911) on different leaves are given in the table below — Holly Ivy Pine Iris Birch (young sprouts) Beech ( sprouts) 12 % n% 8% 10 % 19% 32% This explains the difficulties that are always experienced in giving a satisfactory rendering of leaves. 10. Absorption by Coloured Transparent Media. These phenomena are much more definitely shown when the light is filtered through a coloured transparent medium, because the change which takes place in the light during its passage through the medium is not, as in the case of coloured surfaces, viewed by the light which they reflect, masked by the white light reflected from the surface. Let it be remembered, first of all, that, con- trary to a widespread belief, the light is not coloured by its passage through a coloured medium ; it only appears coloured because during its passage through the medium certain of the radiations which constitute white light are absorbed. A light-filter 1 always transmits less than it receives ; even the radiations which are transmitted most completely by the most perfect filter are slightly weakened, to an extent not less than 5 per cent. The use of light-filters is the simplest method which can be used to obtain a coloured light of any desired quality. They are constantly used in photographic practice for such purposes as lighting dark-rooms, correcting the colour ren- dering by photographic plates, etc. A light-filter, like a pigmentary colour, is defined by its absorption band. It is as easy to obtain light-filters in gelatine identical with one another by using suitable quantities of pure colouring matters, as it is difficult (it would be even more exact to say impossible) to obtain identical coloured glasses of different makes. When it is also stated that, as a rule, the coloured glasses available for photographic uses are taken from among those used for making stained glass windows or for railway signals, it will be seen how difficult it is to obtain results which are even roughly in agreement by using glasses specified only by their colour. The im- portance of the progress which has been made since the beginning of this century in replacing coloured glasses by scientifically-determined filters can easily be realized. A given light-filter absorbs a constant fraction of each of the radiations in the light which passes through it, whatever may be the intensity of the radiation. For example, if a certain orange filter absorbs 70 per cent of radiation 5,700 A.U. (yellowish-green) and 55 per cent of light of wave-length 5,900 A.U. (yellow), these propor- tions will be absorbed whatever may be the composition of the incident light (white light or a light already coloured), whatever may be its intensity, and irrespective of the position of the filter in the beam. Moreover, any object whatever will appear precisely the same to the eye, whether the filter is placed between the 1 In this volume the expression light-filter will be employed, thus conforming with the photographic nomenclature used in most languages. It is much more accurate than terms such as coloured screen, ray screen. 6 PHOTOGRAPHY: THEORY AND PRACTICE source of light and the object, or directly in front of the observer’s eyes, so that the object receives light directly from the source. If two light-filters are superposed, the trans- mission of the two together is, for each radiation, the product of the two transmissions separately. This holds in whichever order the two filters may be placed. For example, if a certain orange-yellow filter transmits 50 per cent of the radiation 5,600 A. U. (pure green), and if another blue-green filter transmits 10 per cent of this radiation, then the two filters together, whether placed in contact or not, and in what- ever order the light passes through them, will transmit 0-50 x o-io = 0-05, i.e. 5 per cent of the radiation under consideration. 11. Sources of Artificial Light. The light emitted by different common forms of illumin- ants differs enormously, as regards the relative proportions of the different radiations, from natural white light. The proportion of red radiations is always greater in artificial sources, and the proportion of violet considerably less. The composition of the light emitted by a solid source depends essentially on the temperature of the source. If the temperature of a body be gradually increased it emits first of all infra-red radiation, then red ; the other spectral radiations appear in their order as the temperature rises. Thus only very high-temperature sources emit an appreciable proportion of violet, which is, however, always less than that in solar radiation, since by no known means can a temperature be obtained comparable with that of the sun. A simple illustration of these facts can be seen in the case of an electric lamp when the voltage is varied, either accidentally or by means of a rheostat connected in the circuit. The light becomes more white and more active on the photographic plate as the voltage is raised. To a degree of precision quite sufficient for practical needs, one can determine the composi- tion of common artificial sources by considering three groups of radiations, blue-violet, green, and red, with limits respectively at 4,950 A.U. and at 5,800 A.U. instead of considering each individual radiation present. Source of light Red Green Blue Daylight .... % 33'3 % 33’3 % 333 Metal filament electric lamp 6 l 32 7 Half -watt electric lamp 50 30 20 Incandescent gas mantle . 54 38 8 Low voltage arc, ordinary car- bons ..... 50 32 18 Low voltage arc, impregnated white-flame carbons 40 35 25 A well-known effect of the particular composi- tion of the radiation from these light-sources is the changed appearance of certain colours when examined in artificial light; blues change to a deep grey; light greens to yellow; violets and pink colours to red. This drawback can be overcome by passing the light through a bluish glass, which, however, reduces considerably the luminous intensity. Special mention should be made of illuminants which consist of a tube containing a gas, made luminescent by the passage of an electric dis- charge through it. Two types of such lamps, mercury-vapour lamps and Neon tubes, are in everyday use, giving respectively a greenish- blue and an orange-red light. Neither of these sources gives a continuous spectrum, as do those enumerated above, but simply fine isolated lines. The mercury lamp gives, in addition to the ultra- violet, several lines in the violet, green and yellow, but nothing in the red. The Neon lamp, on the other hand, gives radiations in all the spectrum regions, but with marked predomin- ance of the red. It is plain that the light of the mercury-vapour lamp cannot be compensated by filtering the radiations present in excess, since it does not contain any red. Endeavours have been made to obtain from it white light by using it in con- junction with under-run incandescent electric lamps, or neon tubes, or by using reflectors coloured with rhodamine, which, by fluorescence, transform into red light a part of the ultra-violet by which they are illuminated (§4). CHAPTER II QUANTITY OF LIGHT 12. Intensity and Brightness. The luminous power of the source is determined by its intensity and indirectly by its brightness . The intensity is measured in decimal candles , or, for short, in candles, of perfectly definite intensity, which differs only slightly from that of the candles ordinarily used for lighting pur- poses. As the intensity varies according to the direction of the light, the mean value is generally taken, unless the direction is exactly specified. The apparent brightness of a source in a given direction is given by its luminous intensity, measured in candles, divided by the apparent area of the source in that direction. The idea of brightness is of special interest in the case of light-sources used for projection purposes, for in these cases the efficiency depends almost entirely on the brightness. For general lighting purposes it is usual to avoid using very bright sources of illumina- tion. For equal intensity a low-brightness source having a large surface gives a more diffused light which produces less sharply-defined shadows. When we come to the question of the lighting of subjects to be photographed, the considerable differences in the composition of the light given by different sources, and especially the very variable proportion of the radiations which affect ordinary sensitive emulsions, make practically worthless any comparisons which are based only on the values of visual intensity. 13. Illumination. The illumination of a sur- face is measured in lux or candle-metres , and is that produced on a screen, normal to the direc- tion of the rays and at one metre from a point source having an intensity of one candle. Using a given surface, it can be shown that the same illumination is obtained on that surface by placing a point source of one candle at one metre, or a source of four candles at two metres, nine candles at three metres, and so on. This fact is always expressed by the law which states that the illumination is inversely proportional to the square 1 of the distance. We shall have occasion to refer to this law 2 in dealing with 1 By the square of a number is meant the product of the number multiplied by itself ; thus, 64 = 8 X 8, or 64 is the square of 8. 2 The law of inverse squares is only valid for point sources, and cannot be applied to the " directed " beams from a lighthouse, a projector, or an enlarging lantern. times of exposure and printing. In practice, this law may, however, be applied with sufficient accuracy for practical purposes in all cases where the dimensions of the source are only a small fraction of the distance from the source to the illuminated surface . 1 When a large surface is illuminated by a source having small dimensions, the illumination of the surface falls off very rapidly on departing from the point of maximum brightness, at which the rays emitted by the source are incident normally. This decrease is still more rapid if, instead of having a radiation approximately symmetrical in all directions relative to the source, the source radiates mainly in one special direction, as is chiefly the case with an illumin- ated plane surface made incandescent. 14. Quantity of Light or Exposure. 2 The quantity of light received by a surface of unit area, or the exposure received by this surface, is the product of the illumination of the surface and the time of illumination. The unit of exposure is the lumen-second or candle-metre- secondy viz. the quantity of light received in one second by a surface of one square centimetre, exposed to a source of one candle at a distance of one metre. We shall see later that in the majority of photographic processes, equal ex- posures do not produce equal effects if the two factors of the exposure, the illumination and the time, are not the same. A diffusing surface (not polished) is in a way a source of light, so that a brightness can be assigned to it. For practical purposes this brightness can be considered as proportional to the illumination. 1 In the case of illumination by a linear source seen under a very wide angle (luminous tubes) the illumina- tion is inversely proportional to the distance. In illu- mination by a luminous surface (diffuser, ceiling, etc.) close to the surface to be lit and projecting considerably beyond it in all directions, the illunffe!?Uion is indepen- dent of distance. A window cannot be compared to a source of light. It constitutes a diaphragm which un- covers a variable extent of the source of diffused light (sky, or wall diffusing the light from the sky) ac- cording to the position of the point chosen in the room. 2 The word " exposure ” is unfortunately used in two senses, viz. that here defined, i.e. quantity of light, and that of the period of time during which a sensitive surface is exposed to light, e.g. an " exposure ” of so many seconds. It will be impossible in this work to avoid using the word in both senses, but it is hoped that no ambiguity will be involved. 7 CHAPTER III LIMITS OF LUMINOSITY IN PHOTOGRAPHIC SUBJECTS 15. Range between Extreme Luminosities in some Common Cases. There is a general ten- dency to exaggerate very considerably the ratio of the extreme luminosities of the various sub- jects which are photographed. Measurements which have been carried out, either indirectly by means of photographic plates (Hurter and Driffield, 1890) or by direct (visual) photo- metric tests of points in a subject (Mees, 1914; Goldberg, 1919), have allowed us to assign nu- merical values to the luminosities of various parts of photographic subjects, such as a land- scape, an interior scene, a portrait, etc. In a sunlit landscape, without any dense shadows in the foreground, the luminosity of the sky (comparable to that of a white paper receiving an illumination of about 16,000 lux (§ 13) ) is not more than about 25 to 30 times that of the deepest shadows. The ratio of the extreme luminosities for certain subjects is indicated roughly in the following table — Subject Ratio of extreme luminosities Landscape, with sun in the field of view . 2,000,000 : 1 Interior, with windows showing a sunlit landscape ..... 1,000 : 1 Portrait, artificial light, white clothes 100 : 1 Landscape with white sunlit areas and dense shadows in foreground 60 : 1 Lampblack on white paper . 20 : 1 Landscape in diffused light, with dark foreground . . . . . 15 : 1 Interior, no windows or reflections in field of view ..... 10 : 1 The earth, viewed from above : balloon, aeroplane (vertical view) . 4 : 1 Landscape in misty weather . 2 : 1 The relatively low values of these ratios are due to two facts : firstly, that absolute blacks do not exist in Nature, 1 and, secondly, that, with the exception of polished objects, even the whitest ones reflect only a part of the light which they receive. 1 The only way to get an absolute black is through a relatively small hole in a large box, the interior of which is entirely covered with black velvet, or, failing this, a coating of lamp black and dextrin (§ 236). A mass of magnesia, or a block of chalk, the whitest substances that are known, reflect only about 88 per cent of the light which falls on them, 1 even when the surface has been freshly scraped and made perfectly clean. For white paper this value falls to from 60 to 80 per cent according to the texture of the paper, its orienta- tion, and the degree of purity. It is only 78 per cent for freshly-fallen snow and 50 per cent for a white-washed wall. The blackest surface known, black silk velvet, reflects 0-4 per cent ; matt-blackened wood and matt-black cloth 2 to 3 per cent ; whilst black packing paper reflects up to 10 per cent of the light falling on it. In a landscape the ratio of the extreme lumin- osities becomes less as the sky gets more covered. In full sunlight and with a very clear sky, the shadows are only illuminated by the diffused light from nearby objects which receive the sun’s light directly. When the sky is clouded the whole of it acts roughly as a uniform illum- inating surface, and there are no longer any shadows. Between these two extremes, the more intense the diffused light from the sky relative to that coming directly from the sun, the more are the shadows illuminated. In a landscape, the ratio of the extreme luminosities is less for objects farther away. If the distant parts of a landscape are examined with a telescope (or even with a cardboard tube, so as to isolate part of the field of view), no heavy shadow can be observed ; diffused light from the atmosphere due to dust and water-vapour in suspension is superposed on the direct light from the object observed. At the farthest distance which can be seen in the direction of the horizon, no detail can be observed, all objects having the same luminosity as the sky, and becoming indistinguishable in a kind of bluish mist, called the atmospheric haze. Painters and draughtsmen make use of this fact (known to them as aerial perspective ) when they wish to convey the impression of extreme distance. 16. Sensitivity of the Eye. Thanks to the reflex movements of the pupil, which, by ex- panding continuously in a dark place and 1 These coefficients of diffusion are sometimes called the albedo of the substance under consideration. 8 LIMITS OF LUMINOSITY 9 contracting almost instantaneously in bright light (varying from about 2 to 8 mm. in dia- meter), automatically regulates the quantity of light which falls on the retina, and owing to the adaptive power of the retina (§246), the human eye can see objects of which the illuminations lie between some millionths of a lux and several million lux. Such extreme differences of lumin- osity cannot, however, be perceived simul- taneously. In full daylight, the minimum per- ceptible is about 20,000 times more luminous than that perceptible during the night. The presence in the field of view of an object which is brighter than those surrounding it, and especially of an actual source of light, results in a kind of dazzling of the eye, which diminishes its sensitivity considerably and produces fatigue. The less the intensity of the surrounding illumination to which the eye is adapted, the less may be the illumination of the object in order to give the sensation of dazzle. The ease with which the eye adapts itself to very different illuminations usually results in feeble intensities being evaluated much too highly. At a very rough approximation the following table indicates the average relative values of the luminosity under different condi- tions — Open air. Interiors, Interiors, normal lighting Streets, fine weather day-time at night 1,000 10 0*1 0*001 17. Perception of Details of Luminosity. In monocular vision (one eye only), we distinguish different objects, or different parts of the same object, only by their differences of colouring, or by the variation of luminosity when we pass from one to another. If one examines a land- scape or any other object through a blue, green, or red filter of sufficient strength to destroy practically all differences of colour, then the details are perceived solely because of the variation of luminosity from point to point. In a good light, which is neither too strong nor too weak, the eye can generally perceive the contrast between two adjacent surfaces when their illuminations differ by from 1 to 2 per cent (Nutting, 1914; Goldberg, 1919). It should be noted that, as in acoustics, the smallest perceptible interval is determined by a ratio and not by a difference. The ease with which luminosity differences can be perceived becomes less as one of the surfaces becomes smaller, or when the surfaces compared have a coarser structure. Thus it is easy to perceive on a smooth wall a luminosity difference of 2 per cent, and yet it is difficult to see a difference of 5 per cent on a rough-cast wall, or on one of bricks. The sensitivity of the eye to luminosity differences becomes very much less both in a lighting strong enough to cause dazzle and in the case of poorly-lighted surfaces. In the shadows in a sunlit landscape, luminosity differ- ences cannot be seen unless they are as much as 20 per cent, or 30 per cent, or even 50 per cent in the case of leaves or other masses of a very pronounced structure. CHAPTER IV PHOTOGRAPHIC IMAGES I THE IDEAL SCIENTIFIC IMAGE; THE AESTHETIC IMAGE 1 8. Negative and Positive. The image obtained by the usual photographic processes is a nega- tive (Fig. 2), in which the lights in the subject are reproduced as opacities and the shadows as transparencies. The photographic reproduction of this negative by a further inversion of lumin- osities gives a normal image, or positive (Fig. 3). It cannot be too strongly impressed on a beginner not to judge the value of a negative by its appearance ; the negative is only a means to an end, and should be judged only by the prints which it is capable of giving. A pretty looking negative is not always the best. 19. Range of Extreme Luminosities in a Positive. The following table indicates the ratio of the extreme luminosities in images on paper, obtained by different processes and viewed under normal conditions — Typographical impression . . from 10 : 1 to 35 : 1 Black tone photographs, matt sur- face 1 ..... from 15 : 1 to 20 : 1 Intaglio print (photogravure) . less than 35 : 1 Carbon prints, black tone . . about 40 : 1 Black tone photographs, best qual- ity glossy surface 1 . P.O.P. prints, gold toned and glazed ..... about 100 50 : 1 These values should be considered as the maxima, corresponding with materials of the best quality and with perfect technique. They vary with the conditions under which prints are viewed; an image in which the whites are more glossy than the blacks appears more contrasty when it is viewed in the open air by diffused light than in the light from a source which is almost a point. It appears still more contrasty when it is illuminated under good conditions near a window (Nutting, 1914). 20. The Ideal Scientific Reproduction. In a photograph which reproduces a subject with absolute fidelity, there ought to be equality between each of the luminosities of the image and the luminosity of the subject at the corre- sponding point. Obviously this equality is only 1 A glossy surface reflects, according to the laws of specular reflexion, at most 10 per cent of the light falling on it (e.g. porcelain and glazed papers). A surface which has a specular reflexion factor of less than 2 per cent appears perfectly matt. possible for a certain value of the illumination of the image, and for all other values reduces to a proportionality. Even supposing that the photographic pro- cesses were able to reproduce the subject faith- fully over the limited range of luminosities which can be obtained with different papers, 1 it can be seen that reproduction under exact conditions is impossible with an image viewed by reflection, since the range of extreme lumin- osities of the subject would be limited to 20 : 1 in the case of matt prints, or 50 : 1 in the case of glossy prints. Note in passing the superiority, for purely record purposes, of papers with glossy surfaces, which not only allow any details to be read under considerable magnification (which cannot be done with a print the surface of which has a more or less coarse structure), but which also permit of a more correct representation of an extended range of luminosities. Thus one is often led deliberately to depart from the ideal proportionality between the luminosities of the image and those of the subject, and to “ compress ” the scale of luminosities of the image in such a way as to bring it between the limits which are available in practice. 21. The Aesthetic Image. It would obviously be correct to reproduce strictly the various tones which occur in a dark cave if the photo- graph obtained was going to be used to ornament the walls of this cave, or of any other place of the same illumination. Since, however, photo- graphs are usually intended to be looked at in a well-lit room, they ought therefore to ren- der the physiological relations of the different 1 On the contrary, there is no limit to this interval in the case of images viewed as transparencies (dia- positives) ; there is in this case a much greater liberty. Note that this advantage is to a great extent lost if such an image is examined by the reflection from its projection on to an opaque screen, instead of exam- ining it directly. It may be stated as a general rule that as regards the commonest subjects (photographs taken in good daylight or designed to give the impression of being thus taken), the image on paper should, in order to give natural sensations, differentiate, in the different regions, luminosities of which the ratios should be respectively — High lights, 5 per cent. Half lights, 10 per cent. Deep shadows, 25 per cent. 10 PHOTOGRAPHIC IMAGES IT luminosities of the object, and not their physical values. The apparent relative luminosities of any scene or object change to a more or less marked degree when the intensity of the illumination Fig. 2. The Negative Image in which it is examined is modified, just as if the intensity scale were transposed into a new key (F. F. Renwick, 1918). A lump of coal illuminated by direct sunshine can send back more light than a lump of chalk in the shade, and yet we see the coal as black and the chalk as white. This physiological interpretation does not occur when we look at a photograph in which we may take the image of a black object for that of a white one, or con- versely, according to their relative luminosities (H. Arens, 1932). In order to give to a painting the impression of dazzling light, the artist often has recourse to the suppression or the weakening of the details in the brightest parts of the subject, whilst he conveys the sensation of obscurity to the observer by suppressing the details in the Fig. 3. The Positive Image shadows. These methods are based on a correct observation of Nature, and just as the artist endeavours to reproduce Nature as he sees it, so in the same way the photographer ought, with the same aim in view, to make use of know- ledge derived from a study of the characteristics of the sensitive surfaces which he uses. Such effects as the foregoing may be supple- mented by others, e.g. by tinting very lightly with yellow an image representing a sunshine ellect ; and with blue one which is to give the ellect of night, but such general treatment must be done with extreme discretion. CHAPTER V PERSPECTIVE : MONOCULAR AND BINOCULAR VISION 22. Geometrical Perspective. The perspective of an object or of a group of objects (from the Latin : to see across) is the trace of all the points of intersection of all the straight lines from a fixed point (the viewpoint or centre of projection) to all the points of the objects to be represented, with a certain surface called the surface of projection. This surface is generally a vertical plane, but is sometimes a cylindrical surface (panoramas), or a segment of a sphere (cupolas), or, more rarely, some other surface. Practically, according to Leonardo da Vinci, the perspective may be defined as the trace which would be obtained on a transparent surface (glass, or gauze stretched on a frame in the case of a plane perspective), when one eye is kept in a fixed position determined by a sighthole, and the other is closed, in such a way that each of the points or outlines of this trace exactly masks the point or the corresponding outline in the subject to be represented. The perspective of anything of which all the parts, whether real or imaginary, have known dimensions and occupy known positions can be obtained by relatively simple geometrical constructions. Conversely, if a perspective con- tains the images of certain known objects, it is possible to deduce from it the dimensions and the relative positions of other unknown objects whose images figure in that perspective. Such a perspective regarded by one eye only from exactly the position of the viewpoint would appear to us, at least as far as the forms are concerned (without considering colours and luminosities), just like the object represented would appear when viewed from the correspond- ing point, the same outlines being seen in the same relative positions. In conformity with this definition, the surface of a projection plane only plays the role of an open window through which appears the land- scape or the scene which was represented. 23. If we consider an object (Fig. 4) which for clearness has been purposely chosen of simple form, a viewpoint 0 , and a vertical plane T, then the perpendicular OP dropped to the plane from the viewpoint meets the plane at a point P (called the principal point) , the distance OP being the principal distance of the perspective obtained. Any group of straight lines parallel to one another and to the plane of projection will be reproduced in the perspective by straight lines parallel to those considered. In particular, all vertical lines in the subject will be represented by vertical lines in the perspective. Any groups of parallel straight lines which are not parallel to the projection plane will be represented in the perspective by a group of straight lines converging to the same vanishing point , which is defined by the intersection of the projection plane with a straight line dropped from the viewpoint parallel to the direction in question in the subject. The vanishing points of all the horizontal lines are situated on the principal horizontal HHj the intersection of the projection plane with the horizontal plane through the viewpoint and also (in this case of a vertical projection plane) through the principal point P. In particular, all the horizontals contained in the facade of the shed (Fig. 4), or parallel to this facade, are represented by straight lines which converge to the vanishing point F, defined by theintersection of the plane of projec- tion with the straight line OF dropped from the viewpoint parallel to the straight lines being considered in the subject. All other groups of lines parallel to the facade of the shed will have their vanishing points on the vertical line FG. 24. Once the position of the viewpoint and the direction of the projection plane have been determined, the perspective obtained is to a close degree independent of the principal dis- tance. The perspectives obtained from a single viewpoint but on several parallel planes are geometrically similar; any one can be changed into any other merely by proportional ampli- fication or reduction ; for example, by means of a pantograph. The principal distance only affects the scale of the images, which all vary proportionally. 25. Deformations due to Displacement of the Viewpoint. When a perspective is looked at from a point other than its viewpoint, the different parts of the image are no longer seen at the same angles as the corresponding parts of the subject. The representation in this case is falsified, and one no longer appears to see the 12 MONOCULAR AND BINOCULAR VISION 13 object but only a more or less distorted form of it. If we suppose at first that the eye with which the perspective is observed remains at a distance Fig. 4. Elements in Geometrical Perspective from it equal to the principal distance, but without being placed at the viewpoint, the object undergoes a torsion. For example, if the eye is in a position higher than the viewpoint, all the horizontals of the sub- ject appear to slope down from the observer to the horizon ; their vanishing points become in fact lower than the eye, and the apparent slope of each horizontal will be that of a straight line joining the eye to the corresponding vanishing point. Next, suppose that the eye, while remaining at a distance from the pro- jection plane equal to the principal distance, is displaced laterally. To make this clear, suppose it is placed opposite the vanishing point F (Fig. 4). This point, being now substituted for the principal point, would be on a perspective examined under correct con- ditions, the vanishing point of the straight lines of the subject perpendicular to the projection plane. Under the actual conditions of examin- ation one is thus led to consider the facade of the shed as perpendicular to the plane of pro- jection, which is not the case. Every combination of the two displacements of the eye, the effects of which we have just considered separately, will result in a double torsion of the object. Notably, the straight lines, which in the object were perpendicular to the plane of projection, will appear always pointing towards the eye, whatever may be its position relative to the projection plane. 1 Now suppose that the eye, whilst being kept on the perpendicular from the projection plane to the principal point, is displaced along the length of this line. The object will appear drawn oat in depth or compressed , according as the distance of observation is greater or less than the principal distance, the deformation being in every case proportional to the ratio of these two distances. Imagine two objects at A and B in the horizontal plane (Fig. 5). In the perspective traced from the viewpoint 0 on the projection plane T, the images of these two points are at a and b. If, instead of observing this perspective from its viewpoint, the eye is moved to O', at double the dis- tance, obviously the objects cannot be considered as hanging freely in the air, but must be resting on the plane shown. One is, therefore, compelled to assign to these points the positions A' and B' } the object thus being drawn out in the ratio of 1 to 2. If 0 p b / / / 1 1 1 1 j 1 1 I y I I A BA' i Fig. 5. Distance and Perspective 1 Consider the case of a projection image containing a weapon which is being aimed straight in front. It is no more astonishing that whatever position an observer occupied relative to this image, the weapon would always appear to be pointing straight at him, wherever he might be, than to find that such an image viewed from the side does not show the profile of the weapon. H PHOTOGRAPHY: THEORY AND PRACTICE the distance A B is more or less fixed (the case of a man lying down to whom cannot reasonably be attributed double the normal stature), the details of the object situated at A' , which we intuitively consider as being at A", will be on an exaggerated scale for the position that we attribute to them in the object. The front planes are expanded relatively to the back planes. Obviously, these deformations may occur in addition to those due to the displacement of the observer upwards or across. 1 26. Normal Distance of Vision and Angle of Visual Field. A normal-sighted person generally chooses a distance of 10 or 12 in. as the distance from his eyes at which to examine such objects as printed matter, etc. This distance is usually known as the normal distance of vision. The smallest distance of distinct vision at which things can be seen without any abnormal effort is rarely less than 6 or 8 in. ; a normal eye can often see distinctly an object only from 4 to 6 in. away, but in such cases fatigue sets in so rapidly that this can only be done for a few seconds. In order that the eye may perceive simul- taneously all the objects represented in a picture, the latter must not be too extended. The eye places itself at distances from the picture ranging between the length of the diagonal of the picture and three times this length, the extreme angle between the rays used varying between 53 0 and 19 0 . 27. In order that a perspective may be examined with avoidance of the distortions described in § 25, it must be looked at from its viewpoint. The principal distance should then be at least equal to 10 in. (or, as an extreme, 8 in.), unless the picture is examined by means of a magnifying glass, which allows it to be brought nearer to the eye; 2 moreover, it is 1 Frequent examples of such deformations are met with in the use of painted backgrounds of certain design, seen or photographed from a point other than the viewpoint of the original projection. 2 When an image is observed with a magnifying glass under conditions such that the image is at infinity (this condition is instinctively fulfilled by an observer of normal vision, or by one who keeps on his correcting glasses), the image is seen as it would be if the centre of rotation of the eye coincided with the optical centre of the glass. In order to examine under perfect con- ditions a perspective with a principal distance less than the distance of distinct vision, a magnifying glass must be chosen of which the focal length is equal to this principal distance. Incidentally, note that the magnification of a magnifying glass is expressed by a quarter of the number expressing its converging power in diopters. Thus a glass of 8 diopters (focal length 125 mm.) has a magnification of 2. essential that the included angle (angle between extreme rays converging to the viewpoint) does not appreciably exceed 50°. If these conditions are not conformed with, the perspective can only be seen falsely. Accord- ing to what has already been said, it can easily be realized that the tolerances in the position of the eye during the examination become greater the greater the principal distance. In particular, if the principal distance is at least equal to 10 times the mean separation of the eyes, there will no longer be a very marked difference between the objects as received individually by each of the eyes, and the binocular view of the picture will no longer cause any inconvenience. 28. Anomalies of an Exact Perspective. A perspective, traced directly on glass or resulting from correctly carried out graphical construc- tion, is of necessity exact in the geometrical sense, but it may be either picturesque or defective , according to the value chosen for the principal distance and the included angle. If the eye can be placed at the viewpoint, it will obviously see an object identical with the object seen from the same viewpoint, but as soon as one moves from the normal position (and this will necessarily be the case if the principal distance is very short, or if the included angle exceeds the angle of the visual field) serious distortions will appear, especially towards the limits of the field. These distortions are due especially to the fact that the image projected on our retina is projected on to a sphere, a very different case from a plane perspective. From whatever angle we may look at a sphere its outline always appears exactly circular. On the contrary, the plane perspective of a sphere is an ellipse, except in the case where the centre of the sphere is on the perpendicular from the viewpoint to the projection plane. As the visual ray to the centre of the sphere makes an increasing angle with this perpendicular, so the distortion also becomes greater. Nevertheless, if one stands in front of a colonnade, all the columns appear the same diameter. If there is a difference, the columns farthest away appear somewhat smaller ; in the perspective of a colonnade seen from the front, the images of the columns become larger as one moves farther away from the principal point. Fig. 6 (from an old paper by Moessard), showing in elevation, in plan, and in perspective a series of identical vertical cylinders, each being surmounted by a sphere, is an excellent example MONOCULAR AND BINOCULAR VISION 15 of anamorphosis (i.e. a perspective which is displeasing, although correct), due to the fact that an excessive angle 1 has been included (by means of the angular graduations given, the obliquities corresponding with different deforma- tions may be seen). In fact, the artist, painter, engraver, or draughtsman always modifies the strict laws of geometrical perspective by means of certain tricks of which the greatest masters have given examples. He generally limits the included angle to between 15 0 and 20° by choosing a principal distance somewhere between twice and three times the greatest dimension of the image. Further, even if he respects the laws of perspective whilst tracing the principal lines, he departs from them for the details, each object being represented almost as if it were seen from the front. It can almost be said that the painter only adopts the plane perspective for the placing of the different elements, the tracing of these 1 Distortion similar to that of the spheres repre- sented in Fig. 6 is often noticed in the faces of people photographed in the foreground under a relatively large angle (photographs of crowds, banquets, etc.) resulting from the drawing on the plane of their spherical perspectives. Notice, however, that the observer who can only see with one eye and who cannot move, though provided for by the theorists of per- spective, is not found amongst Nature artists, who always judge their effects with both eyes open, and frequently move about so as to look at their picture from points far removed from the actual viewpoint ; by doing this they can correct the anomalies which would show to badly-placed spectators. This explains why pic- tures in museums can be examined from very different positions, and often even abnormal positions, without appearing displeasing. Un- fortunately, this wide tolerance is not found in the examination of a perspective, unless its principal distance is very great and the included angle very small. 29. Influence of Choice of Viewpoint. The choice of viewpoint affects the aspect of the Perspective image of each of the different objects and at the same time the ratio of the respective sizes of the images of objects situated at different distances. Consider the case of a sphere (Fig. 7), and let us determine the perspectives from the two viewpoints 0 and O'. It will be realized at once that, seen from very near, the sphere will show only a small fraction of the surface which can be seen from a farther distance away; all the shaded zone will be seen from 0 and not from O'. It can be seen that if we substitute for the sphere a human face seen from the front, then from the viewpoint O' the ears will be hidden, and the mouth (the opening of which represents about a quarter of the diameter) will occupy a third of the apparent diameter and seem to be enormous. Now consider the case of two objects of the same dimensions situated at different distances from the viewpoint, in the same direction. If the nearer of the two objects is at a distance from the viewpoint equal to n times the distance between the objects, the respective scales of i6 PHOTOGRAPHY: THEORY AND PRACTICE their images will bein the ratio n/(n + i). Thus the images will become less different as n gets greater, as is shown in the following table, where the values of nf(n + i) are given for different values of n. tt I 2 3 4 5 TO 20 TOO «/(« + 1) 0*50 o*66 o*75 o-8o 0*83 0-90- o-95 o-99 Thus it can be seen that if the distance of two equal objects is equal to the distance of the nearer of them from the viewpoint, one of the objects will be represented twice the size of the other. If we multiply by ten the distance to the first of the objects, and compensate for this increase of distance by extending the principal distance until an image of the nearer object is obtained which is the same as previ- ously, the more distant object will not differ from it more than io per cent. Returning now to the case of the front view of a portrait, and bearing in mind that the point of the nose is about 4! in. or 5J in. in front of the back outline of the ears, it can be calculated that in a portrait taken at about 4 ft. from the sitter, a rigorous application of the laws of perspective would result in the nose being represented on a scale greater by 10 per cent than the scale of the ears. A painter, when sketching a portrait, is always at least 10 or 15 ft. from his model. Let us take the case of a house, and consider its perspective at a distance of about 300 yd. At this distance the house is in correct relation with the distant landscape. If now we approach to within 20 yd. of it, whilst keeping the same principal distance, the image of the house will be magnified 15 times, but the distance will be practically the same size as before, and will thus be on a much smaller scale. Similarly, a painter, when prevented from going back far enough to see properly, would design the background on a magnified scale in order to correct this effect, which, though it would be scarcely noticeable in the examination of a landscape itself, because our brain corrects the sensations which our eyes transmit to it, might be displeasing in the case of a plane image. 30. Binocular Vision. Only a rough idea of the relative distances of objects can be obtained by monocular vision (using one eye only). One knows how difficult it is to place a finger in the neck of a bottle placed by someone else at the height of the observer's eyes, when one eye is shut. The factors to be appreciated are the varia- tion of the apparent dimensions of an object of known size, the changes in the relative position of the objects when the observer moves transversely, the aerial perspective (§ 15), and the variations in the effort necessary to accom- modate the eye (focussing the eye) according to the distance of the object. The causes which give rise to the sense of relief in binocular vision (using two eyes) are, on one hand, the dissimilarity of the two retinal images, each eye seeing a single near point projected on two different points of the back- ground, and on the other hand, the effort of convergence of the ocular axes towards the fixed point, this effort becoming greater as the point becomes nearer. These two circumstances only play a part for not very distant objects. Aviators and balloonists verify daily that at a few hundred yards above the earth all sensation of relief disappears, even for the highest buildings. Consider two perspectives of a single subject, each perspective having the same principal distance, on two parts of the same plane, from two viewpoints the separation between which is equal to the mean separation of the MONOCULAR AND BINOCULAR VISION 17 eyes (about 65 mm.). If the centres of rotation of one's eyes be placed at the viewpoints, each eye only seeing the perspective of its own viewpoint, the same sensation of relief will be experienced as in direct observation of the object with the two eyes (the variations of the accommodation no longer come in in this case). This relief may be so striking that an observer who did not already know would scarcely believe that the solid image which he could see was actually the result of two plane images. This fact forms the basis of stereoscopy. Stereoscopic vision implicitly assumes that the observer has two equal and symmetrical eyes. 31. Perspective on a Non-vertical Plane. If from the viewpoint 0 (Fig. 8) the perspective of a solid body 5 be drawn on the non-vertical plane T, the images of all the vertical lines of the solid will converge to the vanishing point V where V is the intersection of the plane T with the vertical dropped from the viewpoint. In presenting this plane under the same obliquity to an observer whose eye, placed at 0, would be forced to look in the direction of the principal point P, especially in the absence of external marks which would indicate to him the obliquity of the plane on which was the perspective, he might have the illusion of the object represented. But an observer who did not know, examining such a perspective under the same conditions as he would regard a normal one, would be led to conclude that the solid object represented was not a parallelepiped, but a truncated pyramid. He might not unnaturally conclude that the solid figure was represented as in the act of falling. 1 A vertical plane of projection is the essential condition for the reproduction of vertical lines as verticals in the perspective. Experience shows that once the perspective has been drawn under these conditions, the projection image can be then shown obliquely without its being displeasing (the case of pictures hung rather high in such a way that their view- point is at the height of the observer's eyes when standing), though from the moment when we realized this obliquity we should not consider as admissible the representation of an object on an inclined surface, even if this was viewed under its normal inclination. 32. Panoramic Perspective. In cylindrical per- spective, known as panorama, the viewpoint is 1 We are not envisaging here the case of views inten- tionally taken looking downwards or upwards for documentary purposes or in order to achieve some special effect. 2— (T.5630) situated on the axis of the cylinder of revolu- tion (vertical cylinder), which constitutes the projection surface. In this system of per- spective, verticals are represented by verticals, the horizon line by a meridian circle, and all other straight lines by ellipses. When the pro- jection surface is maintained in its cylindrical form and looked at from the viewpoint, what is seen is identical with the subject, but if the projection surface is now unrolled and becomes a plane, all the straight lines of the subject, with the exception of the verticals and the horizon line, are represented by curves. Thus Fig. 9. Panoramic Perspective it is, for example, that the straight line ABC (Fig. 9) is represented in the panoramic per- spective, after this has been flattened out to a plane, by the curve abc, with vanishing points at F and F', which are common to the perspec- tives of all other straight lines parallel to that considered. Such deformations are obviously a drawback in cases where it is desired to represent subjects containing numerous straight lines other than the verticals, such as architectural works or views of towns having straight streets. But the suppression of all deformations due to excessive obliquity of the visual rays relatively to the projection of plane perspectives (§ 28, Fig. 6) gives to panoramic photographs, often limited to a fraction of the complete horizon, a special interest in such cases as the representation of a very extensive landscape, such as high moun- tainous country, or of a large number of people. Due to its being unfolded to give a plane surface, such an image no longer permits of i8 PHOTOGRAPHY: THEORY AND PRACTICE one viewpoint only, but of an infinite number of them, arranged on a straight line parallel to the horizon line, at a distance from the image equal to the principal distance. Such a projec- tion should be considered as the combination of a great number of projections each formed from a straight vertical band, each to be exam- ined from its particular viewpoint. The observer moving in front of the projection should thus Fig. io. ( A ) Curve for Placing Figures to Appear as in ( D ) on a Panoram Negative only look at the details of the image which he sees exactly opposite to him. 33. In practice, the question has arisen as to how to place a series of people in such a way that, after the panoramic photograph has been flattened out to form a plane surface, the figures appear exactly in line. To do this the people should be placed along an arc of a hyperbolic spiral (C. J. Stokes, 1919) having its origin at the viewpoint, that is on the axis of the cylinder. To trace the correct curve to give the desired result, the procedure is as follows — With a principal distance of, say, 10 Jin., we may use a projection, which, after being un- folded to form a plane, measures 24 in. long, corresponding to an arc of 132 0 in a circle of 10J in. radius. Considering only a part of this, we will include the people within an angle of 120 0 . We will allow a height of 4! in. for the image of the nearest person and 3 in. for the ones farthest away, assuming the people to have an average height of 66 in. Equal lengths measured on the horizon-line of the image on the plane correspond with equal angles on the ground. The diminution of the height of the images being continuous, and the ratio of the dimensions of a person to his image being equal to the ratio of their distances from the view- point, the distances of people in the different directions, defined by the angle they make with the direction chosen as origin, may be calculated as follows — Angle Height of image Distance from subject to viewpoint o° 12 cm. (168 X 2 6) / 1 2 = 3 m. 64 cm. 30 ° 11 ,, (168 x 26 )/ii = 3 m. 98 ,, 6o° 10 ,, (168 x 26 )/io = 4 m. 37 ,, 90° 9 (168 X 26)/9 = 4 m. 85 ,, 120° 8 ,, (168 X 26)/8 = 5 m. 46 Fig. ioa represents the curve along which people should be placed in order to obtain the effect shown in Fig. iob. The extensions of the spiral in parts other than those coming into the case considered have been traced. It will be noticed that in one direction it tends to approach more and more nearly to a circle, and in the other, to approach indefinitely to a straight line. 34. Sharpness of Vision. Sharpness of vision, variable with the illumination and the observer, is measured by the distance from centre to centre of black parallel lines of equal width, separated by white spaces of the same width, this distance being expressed as a fraction of Fig. ii. Test Object for Sharpness of Vision the greatest distance at which the lines can still be separately seen when viewed closely, i.e. when their images are formed on the most sensitive part of the retina. A good eye can distinguish two lines the distance apart of which measured from centre to centre corresponds to an angle of 1 minute. This would be given by a distance of 1/250 in. at 12 in. A kind of fan- shaped object such as that shown in Fig. 11 is used frequently, with equal sectors alternately MONOCULAR AND BINOCULAR VISION r 9 black and white. In this case the smallest distance from the thin end is determined at which the sectors are still distinguishable from one another. Practically, the sharpness of an eye is considered to be about the average when it can separate lines Y h) ^ n - a P ar t at a distance of 8 in., corresponding with an angle of -^uVo radian. 35. Depth of Field. When the eye is accom- modated for regarding an object at a certain distance with maximum sharpness, objects nearer and farther away do not give sharp images on the retina. There exists, however, a certain zone within which all objects appear to the eye with the same sharpness. The depth of this zone is known as the depth of field , which becomes greater as the object viewed is farther away. In fact, in looking at a scene of which the different elements are at very different distances from the observer, the accommodation varies constantly as the eye concentrates on the various points. Thus, in the average sensation which results, the most important points are seen more sharply than those of only secondary interest, which are, as it were, only seen accidentally. PART 2 THE OPTICAL IMAGE BEFORE PHOTOGRAPHIC RECORDING CHAPTER VI THE CAMERA OBSCURA AND PINHOLE PHOTOGRAPHY 36. The Camera Obscura. The camera obscura (Fig. 12) appears to have been known at a very early date. 1 In one of his undated manuscripts, the celebrated painter, engineer, and philo- sopher, Leonardo da Vinci, who died in 1519, describes this phenomenon in the following way, without giving any indication that it was either a recent or personal discovery: “ When the images of illuminated objects enter a very dark room through a very small hole and fall on a piece of white paper at some distance from the hole, one sees on the paper all the objects in their own forms and colours. They will be smaller in size and will appear upside down because of the intersection of the rays. . . . A suitable hole can be made in a very thin plate of iron/’ Outside the room each illuminated point, scattering the light in all directions, sends through the aperture a beam of light in the form of a very narrow cone. This cone has its apex at the object point in question, and its base is that of the aperture. Thus it illuminates the scattering or translucent screen on which it is received by a small spot, which is thus the image of the point object. Within certain limits, the spot formed by the projection of the aperture on the screen will become smaller, and consequently the whole image sharper, as the aperture itself becomes smaller and as the material in which the aperture is made is thinner. The image will also become sharper as the aperture is moved farther away from the screen. In fact, under these conditions, the sizes of the individual spots increase much less quickly than the dimensions of the image. 2 The camera obscura was much improved in the second part of the sixteenth century by 1 It was apparently mentioned by Ibn al Haitam in 1038 (Eder’s Jahrbuch fiir Photographie, 1910, p. 12). 2 Such images sometimes occur unintentionally on photographic plates as parasite images or " doubles/’ when there happens to be a small hole in the outside wall of the camera, such as a screw-hole which has not been stopped up. fitting a biconvex lens at the aperture. In the early part of the eighteenth century it was developed into a portable instrument similar to our present-day cameras, and was frequently used by artists as a means of making sketches from Nature. 37. Identity of the Camera Obscura Image with an Exact Perspective. In 1568 D. Barbaro recommended the use of the camera obscura for automatically making perspective drawings. Suppose that a sheet of glass is put in front of the camera at the same distance from the aperture as the screen on which the image is projected is behind, and parallel to, this screen. The perspective formed on this surface with the aperture as viewpoint will be accurately formed by the intersection with the plane of this glass of all the rays which, after passing through the aperture, go to make up the images of the external objects. The exact identity of this image and of the perspective obtained can be easily shown (Fig. 13). It is due to the fact that the traces of the points of intersection of the lines in a beam of concurrent straight lines, with two parallel planes symmetrically placed relative to the meeting point of the lines, are superposable one on the other. 38. Pinhole Photography. Although not much practised in recent years, pinhole photography can give very useful results in the case of inanimate objects; it even yields images under conditions in which it would be impossible to get comparable results with the objectives now available (Meheux, 1886). In order to obtain an image of sufficient sharpness it seems to be an advantage to use an aperture of the smallest possible diameter in a very thin plate. 1 A simple experiment, such as forming the image of a luminous filament of an electric lamp, shows that with each distance of the object from the camera there corresponds, 1 A thick plate would restrict the field included, and would decrease the contrasts in the picture, owing to light reflections from the cylindrical surface of the aperture. 20 CAMERA OBSCURA AND PINHOLE PHOTOGRAPHY 21 for a given diameter of the aperture, a distance from the aperture to the receiving screen (e.g. a matt glass) at which the greatest possible sharpness of the image is obtained. Actually, the phenomenon of the diffraction of light modifies, sometimes in one way and sometimes in another, according to the circumstances, the effective diameter of the spot, as ascertained in accordance with the laws of the rectilinear propagation of light. 1 There is quite an appreciable latitude in the Fig. 12. Formation of the Image in the Camera Obscura Fig. 13. Perspective Rendering by the Camera Obscura conditions for the best results, which explains why the rules formulated by different experi- menters show a certain amount of inconsistency. For photographing distant objects, the op- timum distance F from the aperture to the screen, or to the photographic plate which may be substituted for the screen, should lie between two limits calculated respectively by multiplying the square of the diameter d of the aperture by 625 (Abney; Dallmeyer), or by 1,250 (Colson; 1 The fact that the progressive decrease of the open- ing of a pinhole causes the sharpness of the image to pass through a maximum affords an example of the fact that geometrical optics (or, more exactly, collinear geometry), while able to indicate sufficiently the posi- tion and size of images, is quite unable to indicate the degree of their sharpness. Combes), 1 all distances being expressed in millimetres. For example, with an aperture of 0-4 mm. diameter, the distance F should be between the two limits calculated as follows — 625 X 0*4 X 0-4 = 100 mm. ; 1,250 X 0*4 X 0*4 = 200 mm. The sharpness of images photographed in this way, when the conditions are properly adjusted, is quite comparable to that of images given by soft-focus lenses, and particularly by anachrom- atic lenses. A considerable angle of field can be covered by a pinhole, which can thus be advantageously used for photographing monuments in cases when it is not possible to go a sufficient distance away. 2 The one disadvantage of this process is the relatively long exposures required. This, how- ever, is of considerably less account now that ,i r I i t — .1 — 1 Fig. 14. Making the Aperture for Pinhole Photography we have at our disposal plates of such extreme sensitivity. 3 39. Making a Pinhole. It is not very easy to obtain commercially metal plates having calibrated holes with clean edges suitable for 1 This rule, which is partly experimental and partly theoretical, may be expressed thus 625 d 2 < F < 1 250 d 2 In the case where it is desired to photograph very near objects the optimum extension is calculated in just the same way as when using an objective of focal length F under the same conditions (§ 60). 2 A minute camera with four pairs of pinhole lenses, fitted in an oesophagic probe, has been used for photographing the internal walls of the stomach (Heilpern, Back and Veitschberger, 1930). 3 The time of exposure on very rapid modern plates, for an open landscape photographed in summer in fine weather at mid-day, with an aperture of o-6 mm. diameter and a principal distance (or extension) of 200 mm. (8 in.), is about 5 seconds. 22 PHOTOGRAPHY: THEORY AND PRACTICE pinhole photography. It may, therefore, be worth while briefly to describe how such an aperture may be made. Experience has shown that there is no appreci- able difference between the results given by pinholes of circular apertures and those of square apertures. One can, therefore, proceed to construct one as follows (Malvezin ; Gabriely). On a piece of card draw two lines in the form of a cross (Fig. 14). At their point of intersection cut out a circle of about | in. diameter. Outside this circle stick four needles into the card (shown by the black circles), the diameter of the needles corresponding to that of the desired aperture, making sure to push them in far enough for their uniform cylindrical parts to be actually in the card. Next, fix with glue four bands of about | in. width cut in metal foil (extra thin copper) or in aluminium leaf, as shown in the figure. There will now remain a square aperture with true edges, the length of each side being equal to the diameter of the needles. Remove the needles and protect the whole thing by another card with a piece cut out of the centre, glued on to the first. Blacken the exposed surfaces of the cards. In order to make a circular aperture in a piece of metal foil resting on a strip of soft wood or on a piece of lead, a needle may be used as a punch. The needle should be stuck through the centre of a cork along the axis of the latter, cut off flush with one end, and the other end of the needle then cut about 1/25 in. from the other surface of the cork. This projecting end must be in the parallel cylindrical part of the needle. Now rub the protruding end on an oilstone until a plane and polished end with sharp edges is obtained. The hole is then made in the metal foil by giving the top of the cork a sharp blow. The edges of the hole thus made should be examined with a strong magnifying glass, and, if necessary, made perfectly smooth by means of the finest emery paper. To render it permanent, the metal foil may now be mounted, as described above, between two cards. The aperture made in this way may be fixed to the front of a camera or any light-tight box which can be loaded with a plate or film. A card running in grooves makes a sufficiently good shutter, since the necessary exposures are long. 1 1 The image given by a pinhole is generally so faint that it cannot be easily examined on a ground glass screen. In order to find out the width of field which the pinhole gives, an aperture of about £th in. diameter may be temporarily substituted for it, or, failing this, a spectacle lens of focal length equal to the distance previously ascertained as optimum distance between aperture and screen for photographing distant objects. CHAPTER VII GENERAL PROPERTIES OF OPTICAL SYSTEMS; ABERRATIONS 40. Lenses. Lenses are masses of glass, bounded, by successive moulding, grinding and polishing operations, by two spherical 1 surfaces, or a spherical and a plane surface. According as the beam of light emerging from a lens held up to the sun has a diminishing or increasing cross- section. the lens is said to be convergent or diver- gent ; convergent (or positive) lenses are thicker at the centre than at the edge (Fig. 15, I to III) ; on the other hand, the edges of divergent (or negative) lenses are thicker than the centre (Fig. 15, IV to VI). The optical (or principal) axis of a lens is the straight line joining the centres of the two spherical surfaces, or, in the case of lenses having one surface plane, the perpendicular on to that surface from the centre of curvature of the other. In every combination of lenses the optical axes must coincide ; this is known as a centred system . 41. Images Formed by Convergent Lenses. The elementary teaching of optics assumes an ideal simplicity in the instruments studied which is quite artificial (lenses of zero or negligible thickness ; rays at small inclination to the axis passing through the lenses close to the axis, etc . ) . These mathematical fictions can only with difficulty be applied to the complex system of the photographic lens, often working at a very large aperture over a very extended field; it is all the more necessary to call attention to this point, as the application of the rules thus simplified may lead, by mathematical deductions which are strictly logical but ill-founded, to grossly erroneous conclusions. When a convergent lens is placed at a suitable distance from a luminous object (or more generally any well-lighted object, stray light being excluded) it forms an inverted image which can be received sharply on a screen placed at a determined distance from the lens, this 1 Lenses have already been made with one or both surfaces non-spherical (toric, ellipsoidal, paraboloidal, etc.) ; some opticians consider that photographic lenses will not be further improved without recourse to such surfaces. It may be added that lenses in everyday use (magni- fiers, condensers, etc.) are generally not ground but moulded; moulding is sometimes used, with suitable precautions, to minimize the labour of roughing lenses of large size or very deep curves. screen being, for example, a piece of white paper viewed by reflected light, or ground glass viewed by transmitted light. 42. A simple lens (reading-glass of large diameter or condenser lens), when used to pro- ject the image of a window on white paper pinned to the opposite wall but placed not exactly opposite to the window, provides us an excellent lesson in optics. The image is rather poor, being spoilt by a number of defects or aberrations (the only optical instrument that can give perfect images is the plane mirror). The images of the bars will show rainbow colours [chromatic aberration), and even if this aberra- tion is removed by viewing through suitable coloured filters, the image is not sharp [spherical aberration due to the spherical form of the lens surfaces). The image can be improved by cover- ing the lens with a piece of paper pierced with a circular hole smaller than the lens [diaphragm or stop), but is then not so bright. Further, it is seen that the images of the bars are more or less curved [distortion), the curvature varying with the position of the diaphragm. The lens requires to be moved towards or away from the paper in order to bring the centre and edges of the image successively into focus [curvature oj the field). Finally, the image of the vertical bars is not sharp at the same time as that of the horizontal bars, especially at the edge of the image [astigmatism) . 43. Real Images — Virtual Images. An optical image (such as we have considered in the pre- vious paragraph), capable of being received on a matt screen, is called a real image. When a convergent lens is placed at too short a distance from an object, it is impossible to form a real image of the object at any position of the screen, but on looking through the lens an upright, magnified image of the object is seen. Such an image, visible only through the lens by an observer looking in the direction of the object, is called a virtual image. All the observational instruments (telescopes, micro- scopes, etc.), adjusted for an observer with normal sight, give virtual images. A divergent lens can give only a virtual, upright, diminished image of a real object, at a position closer to the observer than the object. 2 3 24 PHOTOGRAPHY* THEORY AND PRACTICE This property is utilized in the construction of " brilliant finders. ” 44. Optical Centre — Nodal Points. In the optical axis of a lens is always a point, called the optical centre , 1 such that every ray the path of which (or its prolongation) within the lens passes through this point has its paths outside the lens parallel. The interrupted path of this ray of light forms what is called a secondary axis . The centre can be inside the lens (Fig. 16a) or outside (Fig. 16b). 2 In the second case, it is not the effective path of the ray passing through the lens without angular deviation which meets this point ; the optical centre is the intersection 1 11 m iv v vi Fig. 15. Types of Lens Elements I. Biconvex lens 1 II. Plano-convex lens V Positive lenses III. Convergent meniscus ) I. Divergent meniscus 1 II. Plano-concave lens > Negative lenses III. Biconcave lens ) with the optical axis of the continuation of the internal part of the ray. The intersections of the optical axis with the external parts of a secondary axis (or their continuation) define two points N and N' (Figs. i6a and i6b). In a perfectly corrected system the positions of these nodal points 3 is invariable, whatever the direction of the rays considered. If the optical centre could be realized, it would be found that each of the nodal 1 To determine the position of the optical centre it is sufficient to draw two parallel radii in the plane under consideration, one for each surface ; the straight line joining the intersection of these radii with their respec- tive surfaces cuts the optical axis at the point C, the optical centre. (Figs. i6a and i6b.) If tangents are drawn at the points of intersection of the two parallel radii with the surface a parallel glass plate is formed, which coincides with the lens itself at the points of contact. Now a parallel plate produces no external deviation, so that the exterior parts of a single ray- through the centre are parallel. 2 Note that the lenses drawn in Figs. i6a and i6b have the same radii of curvature and the same distance between centres. 3 Where the exterior surfaces of an optical instru- ment are bounded by the same medium (air in the case of a photographic lens) the nodal points are identical with the principal or Gauss points. This is not the case with an immersion microscope objective where the outside surface touches a liquid in contact with the preparation. points is its image formed by one of the surfaces of the lens, but on the assumption that this sur- face is bounded by an infinitely thick lens on one hand and by air on the other. Each of the nodal points is the image of the other formed by the lens or lens system considered. In order to distinguish between the nodal points, the one towards which the secondary axes from different points of the object converge is called the nodal point of incidence ; that from which the secondary axes diverge to the different points of the image is the nodal point of emergence. The intersections PP' of the surfaces with the axis are sometimes called the poles. 45. Foci — Focal Length. The image of an infinitely distant point (e.g. a star) towards which the optical axis of a lens is pointed, is the focus of that lens. From considerations of symmetry this is necessarily situated on the optical axis. As the lens can be turned with either face to the point-object, it possesses two foci F and F' (Fig. 17). In the case of a con- vergent lens the foci are the nearest points to the lens at which a real image can be formed of a real object. The word “ focus ” recalls the use of “ burning glasses/' the concentration of rays being a maximum in the neighbourhood of the focus so that tinder or other inflammable material can be ignited there when the lens is directed towards the sun. When the two surfaces of the lens are in free contact with the air, the distance of each of the foci from the corresponding nodal point is the same ; this distance (NF = N'F') is called the focal distance or focal length . x For a rough approximation and where a thin lens is con- sidered, the nodal points can be ignored and the focal length reckoned from the optical centre C. In many lenses the optical centre is close to the diaphragm. Telephoto lenses are the chief exception. The focal length of an optical instrument is one of its essential characteristics. 46. Chromatic Aberration. The refraction of a ray of light passing from one medium to another (from air to glass, or vice versa) at non-normal incidence has not the same value for different colours. Therefore, when a beam of white light (§ 1) traverses a lens, the sharp images formed by light of different colours do not coincide. The rays which are refracted most, ultra-violet and violet, form their images nearer to the lens than those which are refracted 1 The focal length is sometimes wrongly called the focus. . GENERAL PROPERTIES OF OPTICAL SYSTEMS 25 less, green and red. 1 (Fig. 18.) There is thus an infinite number of images each corresponding with one of the component radiations. In par- ticular the position of the foci (images of infi- nitely distant points on the axis) and of the nodal points (images of the optical centre) vary with rays of different colours, as does also the focal length. The practical consequence of this is that whatever be the position of the viewing screen This inconvenience can be minimized by displacing the photographic plate by the correct amount after visual focussing, or by using, both for focussing and photographing, a coloured filter which transmits only a small portion of the spectrum. Generally it is preferable to correct the chromatic aberration more or less completely by the use of at least two glasses of different characteristics, usually a crown and a flint, the use of different material allowing or the photographic plate on which the image is to be recorded, the sharp image corresponding with the apex of one of the cones is surrounded by bright rings corresponding with sections of all the other cones. If the position of the screen has been determined visually, and if the image is photographed with it in the same position, the phenomenon is increased by the fact that the focus chosen is the best for the yellow-green images, which are the brightest visually, and consequently will not all agree with that for the ultra-violet and violet images, usually the most active on a photographic plate. 2 1 Calling n the mean refractive index of glass and n' and n" the values of the index for the two rays con- sidered, the difference of f ocallength(/-/") , expressed in n' — n" terms of the mean focal length /, is/'-/" = . /. (longitudinal chromatic aberration). n - 1 Considering the spectral lines G and E (corresponding with the maximum photographic activity on ordinary plates and the maximum physiological activity respec- tively), this expression represents about 0*14 per cent of the focal length for crowns and 0-16 per cent for flints, the general designations of two classes of optical glasses. 2 This defect is usually referred to by saying that the images formed by two different colours to be united. Lenses for photography are generally corrected for D (yellow) and G (blue-violet) rays of the solar spectrum, and are then called achro- matic (from Greek, meaning colourless). For some work, in particular colour photography, such a correction is insufficient, and coincidence of the nodal points and foci for three different colours is aimed at, generally by the employment of at least three glasses. A lens so corrected is such a lens possesses a chemical focus as distinct from the visual focus. 26 PHOTOGRAPHY: THEORY AND PRACTICE called apochromatic 1 (or with reduced secondary spectrum). Fig. 19 is drawn for a photographic lens of 16 ft. focal length (in order that the aberration can be easily read), and indicates approximately thef displacements (distances from the mean focal plane PP') of the images formed by different spectral regions. Achromatic and apochromatic lenses are not usually corrected for infra-red. When photo- graphing with infra-red emulsions it is therefore necessary to rectify the focussing, this correc- tion being made once and for all by methodical trial and error for each lens. As a rule the exten- sion of the camera must be increased, after visual focussing, by 0-3 to 0*4 per cent of its value. For the study of other aberrations we shall suppose that chromatic aberration is eliminated by means of a colour filter. 47. Spherical Aberra- tion. Among the aberra- tions due to the spherical curvature of the lens surface, the name spher- ical aberration is usually confined to that shown by light-rays at small inclinations to the axis. If we suppose a lens divided into zones con- centric to the optical axis, which can easily be realized in practice by means of diaphragms with annular apertures (Fig. 20) centred on the axis, the focal length of a convergent lens is found to diminish progressively from the central zone to the edge. For any posi- tion of the screen or photographic plate between the extreme foci F and F'" (Fig. 21), the image of a luminous point will be a circle, the bright- ness of which diminishes from the centre to the edge. 2 1 In the meaning given to this word by Abbe, an apochromatic objective ought also to be aplanatic (§ 47) for two colours. 2 The points of intersection of successive pairs of rays determine a surface, the form of which resembles the bell of a trumpet, along which there is a concentra- tion of light. This can easily be seen by blowing a smoke cloud into the beam or placing a screen in the beam close to the focus and nearly parallel with the axis. This surface, corresponding with a beam of light having aberration, is generally called the caustic of the beam. This aberration can be diminished by limiting the surface of the lens to a single narrow zone (in practice, the central zone) by using a diaphragm, but it is obvious that the position of the sharpest focus will depend on the aperture of this diaphragm. The spherical aberration can also be diminished by suitable choice of curva- tures of the lens. 1 . As a rule, spherical aberration is corrected by making the images produced by two zones of the lens coincide, generally the central zone and the extreme (marginal) zone, or one close to it. This correction, obtained by the employment of a more or less complex system in place of the single lens, is never absolute. In order to show the importance of residual aberrations a curve is drawn in the principal section, of which each point is defined (Fig. 22) by the intersection of the incident ray with a line drawn perpendicular to the axis through the corresponding focus (Fig. 22 indicates the aberration considerably exaggerated. It is usual to magnify the aberra- fig. 20. Diaphragm tions, which in this case to show Spherical are those of a 4 in. lens, Aberration by 20). An optical system rigorously corrected for this aberration is said to be aplanatic 2 (Greek Fig. 21. Spherical Aberration = free from error). As a matter of fact, no photographic lens is rigorously aplanatic. It is 1 Spherical aberration is at a minimum fora biconvex lens of which the surface on which the light is incident has a radius of curvature £ that of the surface of emergence (for glass of mean refractive index 1-5). This minimum aberration is only 64 per cent of that of an equiconvex lens of the same focal length. Spherical aberration is at a maximum in the case of a meniscus. 2 A regrettable confusion due to the syllable -plan, has often arisen owing to the erroneous employment of this term to designate a lens free from curvature of field, i.e. giving a plane image of a plane object. P Fig. 19. Chromatic Error of Various Lenses (16 ft. focal length) Apochromatic Achromatic Non -achromatic GENERAL PROPERTIES OF OPTICAL SYSTEMS 27 possible to correct spherical aberration only for certain object distances, which are selected as being those at which the lens will most frequently be used, according to the purpose for which it is designed. In practice, the correction is sufficient for most requirements at intermediate distances. We shall see, however, that the residuals of this the rays of the meridian section converge, is then farther from the lens than the point C, to which the rays in the sagittal section converge. If a screen (white paper, ground glass, etc.) is held perpendicular to the optical axis and gradually moved away from the lens, the beam emerging from the lens and originating in a aberration determine the distortion of the image. 48. Astigmatism. Astigmatism (Greek = absence of point) is an aberration which is seen in oblique pencils, and arises from the assym- metry of the refraction in different sections of the beam; the most obvious effect is the con- centration of light into two distinct foci. To explain this effect, at least diagrammatic- ally, consider the section of the lens made by a plane containing the secondary axis AA'B'B of an oblique pencil (Fig. 23A), and also the sequence of sections of the lens and of the pencil by planes perpendicular to the first plane and Aberration of Oblique (a) and Axial (b) Rays containing successive elements of the secondary axis, the different sections being projected on to the middle plane containing the interior portion A'B' of the secondary axis (Fig. 23B). In the first case ( meridian section), the curvatures of the lens are less pronounced than in the second case [sagittal section ). The point B, to which single point source will describe on the screen successively the following shapes : a circle (when the screen is in contact with the lens) ; ellipses becoming flatter and flatter with their long axes in the meridian plane, which degenerate to a short straight line in that plane ; ellipses orient- ated as before but becoming more and more circular; a circle; ellipses getting flatter and flatter, with their long axes in the sagittal plane ; a short straight line in the sagittal plane ; ellipses again. 1 Fig. 24 represents this succession of “ images ” of a point source of light, consider- ably exaggerated. 49. Tangential and Radial Images. The fact that a stigmatic pencil gives a double image, viz. two straight lines (focal lines) in different planes when refracted by an astigmatic system, gives the following easily-observed phenomenon. If an object consists of circles, concentric with the axis, and radii thereto, the elements R of each point of a radius will merge into one another and give a sharp image (at least for a certain length), while the elements T will give a blurred image (Fig. 25). Conversely, the elements T will merge into one another and give a sharp image of the circles (or short tangents to them). For this reason the images R and T are often called radial (or sagittal) and tangential images respectively. If the two focal lines are not widely separated from one another, or if the angular aperture of the beam is sufficiently small, a more or less homogeneous image will be produced if the screen is placed at an inter- mediate position where the pencil gives a circular patch (circle of least confusion, C , Fig. 24). 1 This experiment can be best carried out by using an ordinary chemical flask filled with water as the “ lens,” as it will be slightly assymetric; or one of the elements of a condenser may be used. 28 PHOTOGRAPHY: THEORY AND PRACTICE The locus of the radial images of infinitely distant points (e.g. stars) given by a lens is a surface S r (Fig. 26), which (at least in the central region) is generally concave to the lens ; the tangential images lie on another surface S t , generally less curved than S r . These two sur- faces [focal surfaces ) have a point of contact at the focus F. The radial and tangential images of points in any plane perpendicular to the optical axis form analogous surfaces. 1 In order to represent the astigmatism of a Correction of astigmatism is only possible by the employment of at least three separated lenses, or, if the lenses are to be cemented in groups, at least four lenses of different material. Two at least of the glasses must form what is Fig. 24. Astigmatic Images of a Point lens, a graphic method is used similar to that already employed for spherical aberration (Fig. 22). The displacements of the two focal surfaces (multiplied by four to facilitate reading of the curves) for a lens of 4 in. focal length are plotted called an abnormal pair in which the refractive index varies in the opposite direction to the dispersion, thus behaving in a contrary manner to the old glasses. 1 A lens corrected for astigmatism is said to be stigmatic, or, more usually (in spite of the pleonasm), anastigmatic or an anastigmat. 50. Coma. Coma is due to the difference of refraction in oblique rays between the central and marginal zones of a lens; it may thus be said to be spherical aberration of pencils travers- ing the lens obliquely. On account of dissym- metry between the path of the rays in the Fig. 25. Astigmatism of Concentric Circles Fig. 26. Astigmatic Focal Planes on the horizontal scale, while the angle made by the secondary axis with the principal axis is plotted vertically on the scale of o-i in. to the degree. Figs. 27A and 27B, from von Rohr, show respectively the astigmatism curves for a lens partly corrected for astigmatism (Ortho- stigmat type II) and for one well-corrected (Planar). 1 In the case of simple lenses, the forms which reduce spherical aberration to a minimum are those which give a maximum of astigmatism, and vice versa. meridian and sagittal sections (already referred to in the explanation of astigmatism), an un- symmetrical patch is formed instead of a point 1 The first glasses to be produced which allowed of correction of astigmatism (glasses with small dispersion and high refractive index) were made experimentally in France by Feil in 1880; their manufacture was commenced in Germany towards 1890 and for a number of years gave a pronounced superiority to German optics, the legend of which still persists although English and French opticians have for a long while caught up and surpassed their German rivals. GENERAL PROPERTIES OF OPTICAL SYSTEMS 29 image, the appearance somewhat resembling the image of a comet (whence the name), the tail of which is generally directed away from the optical axis ( outward coma). Coma is often associated with astigmatism, but whilst in the case of a lens incompletely corrected, astigmatism attains a maximum and then decreases as the inclination of the rays to the axis increases, coma steadily increases. Also, being of zonal origin, coma is much more A- 20 V\-* Fig. 27B rapidly reduced by the use of a small diaphragm than is astigmatism. Coma is seen in its characteristic form chiefly when long exposures are made on an object having a number of brightly illuminated points off the axis, the tails sometimes stretching a great distance. Fig. 28, taken from S. P. Thompson, shows the cross-section of the beam of light by a plane perpendicular to the axis in the neighbourhood of the normal position of the image of a point formed by a plano-convex lens, having a dia- phragm like that of Fig. 20, but containing several annular apertures. 1 51. Curvature of the Field. For reasons of symmetry, it is easy to see that the images of 1 If a biconvex lens giving minimum spherical aberra- tion (§ 47, note) is compared with a meniscus of the same focal length with its convex surface towards the incident light, it is found that the meniscus, while giving very pronounced spherical aberration, has much less coma at an angle of incidence of 20°. infinitely distant points given by a sphere of glass would lie on a spherical surface concentric with that of the spherical lens, and of radius equal to the focal length. In these circumstances the image of a near plane would be a surface of still greater curvature. The focal surface of a lens of old type (achrom- atics, rectilinears, symmetricals) always has a very marked concavity towards the lens, the mean radius of curvature being between 1*5 times and twice the focal length. 1 In an astigmatic objective the surface that is to be con- sidered as the locus of the image is neither the radial nor the tangential surface, but an intermediate surface containing the circles of least confusion (C, Fig. 24). The practical consequence of curvature of the field is that, if a plane held perpen- dicular to the axis is dis- placed relatively to the lens, the position corresponding with maximum sharpness of the central region of the image is more or less distant Fig. 28. Coma from that corresponding (s. p. Thompson) with maximum sharpness of the marginal regions of the image. In spite of Fig. 29. Curvature of Field the fact that (as we shall see) there is a latitude in the position of the focussing screen or photo- graphic plate (depth of focus), in focussing the 1 The condition that must be fulfilled to flatten the field is incompatible with the condition for achro- matism, unless an abnormal combination of glasses (§ 48) is used. 30 PHOTOGRAPHY: THEORY AND PRACTICE image sharp, curvature of the field sets a limit in every case to the useful angle of field of the lens. Fig. 29, which is essentially only diagram- matic , shows the impossibility of having on a plane P a sharp image formed on the surface 5 . Fig. 30. Types of Distortion By adjusting the position of P to give a sharp focus for the intermediate zone cd, the central and marginal ( ab ) parts of the image can be considered nearly sharp. The useful field of the lens is then limited to the angle A OB. the image of a square AAA A centred on the axis by a meniscus lens will be either a pin- cushion shape BBBB (Fig. 30 or) barrel shaped CCCC y according to the position of the stop relatively to the lens. The deformation is greater the greater the angle the square subtends at the lens. Fig. 31 explains in a simple manner the mechanism of this phenomenon. According to the position of the stop, different portions of an oblique beam possessing aberration are used for forming the image, so that the concentration of light occurs at different distances from the axis, whilst for a pencil parallel to the axis the position of the image is independent of the position of the diaphragm. The more or less blurred images BBBB and CCCC result from the selection by the diaphragm of certain rays which, in its absence, would give an extremely blurred image combining these two partial images. 1 The distortion is reversed if the stop is placed behind instead of in front, from which the simple conclusion was arrived at that by placing the stop in the plane of symmetry of an objective formed The curvature of the field of anastigmats is always very much less than that of ordinary objectives. In the least favourable cases the radius of curvature of the field is at least four times the focal length. 1 52. Distortion. It has long been known that of symmetrical elements the distortion would be zero. Although, in fact, distortion is reduced under these conditions, it will only be zero if such an objective (said to be rectilinear) is used symmetrically, i.e. when producing an image of a plane surface the same size. In fact, a 1 For astronomical work of great precision (Harvard Observatory) the plates are bent into a spherical form, of curvature equal to that of the focal surface (which is very small). The glass plate, which is thin, is bent by suction against a concave support of cast iron. An earlier method of compensation was to place a plano- concave lens against the plate to lengthen the focus of the marginal parts of the image (Piazzi Smyth's cor- rector) . 1 In a lens incompletely corrected for spherical aberration of oblique pencils, a displacement of the diaphragm in its own plane will produce similar deform- ation of the image, which will not be symmetrical if the diaphragm is not correctly centered. The same effects may arise with any aperture limiting the beam of light, e.g. the shutter, when this occupies a position other than the normal plane of the stop, or the plane of the focussed image. GENERAL PROPERTIES OF OPTICAL SYSTEMS 3i symmetrical lens, when used with an angular field of 90° in the photography of distant objects, gives quite distinct pincushion distortion. Actually, distortion is a very general pheno- menon, being present (although to only a small extent) in lenses corrected for astigmatism and curvature of the field, and arises from spherical aberration of the nodal points, i.e. from a slight variation in the position of these points accord- ing to the obliquity of the axes considered. Fig. 32, where the nodal point aberration is considerably exaggerated, shows that in these circumstances the images abed of equidistant points ABCD cannot themselves be equidistant, the scale of the image (ratio of the object to the image) varying progressively from the centre to the edge. With pincushion distortion the scale increases from centre to edges, and the distortion is said to be positive ; with barrel dis- tortion [negative) the scale decreases from the centre to the edge. Aberration of the nodal points, like all manifestations of spherical aberration, is reduced by using smaller stops, which at the same time reduce the distortion. With an unsymmetr ical objective, the optician can remove distortion completely by a proper choice of the constructive elements at his dis- posal, 1 for a given scale of image, chosen at will, or, what amounts to the same thing, for a definite object distance (e.g. for an infinitely distant object in lenses designed for aerial photography or landscape work ; object distance of several yards for portrait lenses; scale approximately unity for process lenses). For every other distance or scale, distortion will be present (the more so with large-aperture lenses), although it may remain so small as only to be detected by laboratory methods. 2 1 It must not be inferred generally from this remark that because a lens is unsymmetrical it is necessarily freer from distortion than a symmetrical lens. A badly designed unsymmetrical lens has, on the contrary, more pronounced distortion than the worst of the symmetrical lenses. Distortion also varies from lens to lens in the same series. 2 The name orthoscopic has sometimes been given to images free from distortion, but, strictly speaking, an Distortion, like the other aberrations, can be represented graphically. In Figs. 33A and 33B, drawn respectively for a symmetrical and an unsymmetrical lens respectively (both by the same maker, of equal excellence, and of the same aperture), the divisions of the vertical scale correspond to the angles made by the secondary axes with the principal axis, while the horizontal scale indicates percentage variation of scale, positive ( + ) or negative (-). Two curves are shown for each lens, one for objects at infinity 53. Influence of Diaphragm Aperture on the Different Aberrations. The employment of dia- phragms with smaller apertures always (at any rate up to a certain limit) improves the definition given by a lens which is incompletely corrected, but the degree of improvement is different for the different aberrations. Chromatic aberration varies almost directly as the diameter of the stop ; spherical aberration on the axis varies as a rule almost as the “ cube ” of the diameter (the product of the diameter multiplied by itself twice). Astigmatism • and curvature of the field are approximately pro- portional to the diameter and the square (pro- duct when multiplied by itself) of the slope of the secondary axis to the optical axis ; coma is pro- portional to this slope and to the square of the diameter, approximately. image is free from distortion only if the sharp images of objects at infinity, or lying on a plane, themselves lie entirely on a plane. 32 PHOTOGRAPHY: THEORY AND PRACTICE As a general rule, it can be stated that no lens is completely corrected, but fortunately the blur produced instead of the ideal point image is not of uniform intensity of illumination over the area derived from geometrical considerations. There is generally a maximum intensity over a small fraction of the area of the blur, so that the photographic image is always better defined when the exposure is a minimum than when a prolonged exposure is given, which allows the poorly illuminated parts of the image patch to be recorded on the plate. These variations of sharpness with exposure are much more apparent with a badly corrected lens. As has been indi- cated, there is a limit beyond which the diaphragm aperture cannot be diminished with the hope of reduc- ing aberrations. Till now we have con- sidered the rays of light obeying the laws of geometrical optics, which is a perfectly . | _ legitimate convention, < 2- 5% -5% -i% +i°/o since the results agree Fig. 33A Fig. 33B with experiment when Distortion over Field pencils of light of of Image sufficient angular aper- ture are considered, but when the aperture is reduced to less than 0*04 in., or less than about one-seventieth of the distance from the image, geometrical optics fails. By reason of the propagation of light in concentric waves, the image of a point formed by an optical instrument is always a patch, even if the instrument is perfectly stigmatic and aplanatic. The distribution of light in such an area is shown diagrammatically on an enlarged scale in Fig. 34 for the image of a point on the optical axis (bright disc surrounded by concen- tric rings alternately black and faintly bright). The diameter of this diffraction disc is greater the narrower the pencil of light, and the photo- graphic definition would be spoilt if the limits mentioned above were passed. An exceedingly minute diaphragm would produce an image hardly better than that given by a pinhole. Astronomers, and especially microscopists, know that to get an image as sharp and detailed as possible it is necessary to use lenses of large aperture, the limit of resolution (minimum dis- tance between two parallel lines that can be reproduced separately) is smaller the larger the aperture used. 1 54. Distribution of Light in the Field. No objective can give a uniformlybright image of a uniformly illu- minated surface, even if this is of small extent. This can be explained by comparing the effects of a beam directed along the axis with one in an oblique direction, forming the images P and P l respectively (Fig. 35). Viewed from the point P the lens has the appearance of a uniformly illuminated circle, whilst viewed from P' the appearance is an ellipse, the area of which is smaller than that of the circle by an amount Fig. 35. Marginal Intensity of Image which increases with the obliquity. Further- more, P' is further from 0 than P, and, as is well-known, the illumination diminishes when the source of light is farther away. Finally, the oblique beam illuminates the screen or sensitive 1 The smallest angle subtended by two points that can be separated by a photographic lens is given by aA/D, where a is a coefficient the value of which is nearly 1, A is the wave-length of light used, and D the diameter of the effective aperture of the lens (P. Nutting, 1909). The earlier experiments of Foucault and Dawes showed that at the centre of the field, the limit of resolu- tion of a perfectly corrected lens, in seconds, is equal to the quotient of 13 (the mean of the values 14 and 12 given by these authors) b y the diameter o f the aperture in centimetres ; in practice, the limit is, in favourable circumstances, slightly less than the calculated value. Fig. 34. Diffraction Image GENERAL PROPERTIES OF OPTICAL SYSTEMS 33 surface in the plane PP', perpendicular to the optical axis, less than it would a screen placed at pp , perpendicular to its mean direction. Combining the effects of these different causes it is possible to calculate the maximum illumina- tion at different angles. 1 The numerical values of Definition A — Illumination of image (100 at centre) B => Angle of incidence of principal ray given below and plotted graphically (Fig. 36) are percentages of that at the centre. 2 Angle . . o° io° 20° 30° 40° 50° 6o° 70° Illumination 100 94-1 78-0 56-2 34-4 17-1 6*2 1-4 The latitude is fortunately so large that a variation of about 20 per cent (which allows of an angle of 18 0 in the extreme pencils, or a field of 36°) is negligible, and even a variation of 40 per cent (field of 56°) is not very harmful, but for wider angles the effects of this variation become excessive. 3 The construction of the lens almost always produces a still more rapid falling-off in illum- 1 If we suppose that the law of inverse squares applies to this case, the illumination would be proportional to cos -, n), where co is the angle the secondary axis makes with the optical axis. As a matter of fact the variation is not so great, but is certainly more rapid than would be expressed by cos 3 gl>. The numerical values given below and used for Fig. 36 are calculated for cos 4 o). 2 It is by no means uncommon for the illumination actually measured to be only 70 per cent of that calcu- lated at 20 0 from the axis and only 50 per cent of that at 25°, this being due to the partial stopping of the oblique rays by the lens mount. 3 Amongst the devices used for compensating for this variation, even approximately, may be men- tioned : (1) an opaque stop or a truncated cone placed at some distance in front of the lens to cut off some of 3 (T.5630) ination at the edges of the field than is accounted for above, because the oblique pencils are partly intercepted by the mount. To explain this without excessive complication let us see what would happen in the case of a lens mount without glasses. For a certain aperture of the diaphragm DD (Fig. 37) all those light-beams more oblique than AA would be partially inter- cepted by the lens cell and the tube. If the diaphragm is replaced by a smaller one D’D ' the limit o f obliquity for which there i s n o cutting off is increased, since in these circumstances the beam BB passes freely. From this it is seen that reducing the aperture of the stop reduces the inequality in the bright- ness of the field, but without entirely correcting Fig. 37. Cut-off of Marginal Rays it, since the causes previously mentioned still operate. A beam of obliquity equal to CC is completely stopped, and this denotes the limit of the field illuminated by the lens. These phenomena can easily be verified by moving the eye in the plane of a sharp image formed by a lens. It is then Fig. 38. Lens Aperture for Oblique Pencils seen that the aperture of the stop DD is more and more covered by the lens rims LL and L'L' , as the eye moves away from the optical axis (Fig. 38). the central rays; (2) a star-shaped diaphragm placed in front of the lens and rotated by blowing, during exposure ; (3) a graduated neutral filter placed in front of the photographic plate : this may consist of a nega- tive of a uniformly illuminated surface taken with the same lens, or a plano-convex lens of neutral glass cemented to a plano-concave lens of clear glass to form a plane-parallel plate. 34 PHOTOGRAPHY: THEORY AND PRACTICE 55. Field Illuminated ; Field Covered. We have just seen that the image plane receives no light at an angle greater than a certain value. Rotation of the secondary axis corresponding to this angle round the optical axis generates a cone of which the vertex angle (twice this limiting inclination to the axis) is the angle of the field illuminated. This cone cuts the image plane in a circle. The image, which is sharp at the centre of the circle, becomes as a rule useless at the edge as much from want of sharpness as from insufficient illumination. If we agree to a certain tolerance amounts to saying that all plate sizes of which the diagonal is less than this diameter will be sharply covered in the specified circumstances. 56. Loss of Light in Passing through a Lens. A beam of light passing through transparent matter undergoes loss, partly by absorption and partly by reflection at the entrance and emerg- ence surfaces. Loss by absorption within the glass of a modern lens is generally very small, often negligible, for visible rays. The mean values of transmission (not reckoning loss by reflection, to be examined later) are indicated below for of the diaphragm the images of distant objects will be useful within a circle, concentric with the circle of illumination, which is the circle of good definition under the given conditions. The vertex angle of the cone formed by the secondary rays passing through this circle is the angle of field covered sharply. If the plane of the focussed image moves away from the lens (as when the object approaches it) the angle of field sharply covered remains the same, but the circle of good definition, being the intersection of the cone with a plane farther from the vertex, increases. The employment of a small stop to improve the definition of the oblique images and to equalize the illumination over the field often has the effect of increasing the field of view as well, but this must not be taken as a general rule. Lens catalogues indicate (or should) the angle of field covered sharply for each lens at different apertures, or the diameters of the circles covered, for an object at infinity. Any rectangular shape that can be inscribed in that circle will then receive a sharp image. This Thickness in cm. .1 2 3 4 5 6 Transmission % . 97-6 95-3 93 907 88*5 86-4 This loss is much greater for ultra-violet radiation, which is, however, useless and indeed often harmful in current photographic practice. The loss of light by absorption may be consider- able in old lenses, certain glasses of which have a pronounced yellow colouration. Loss by reflection at the surfaces of the lens is generally more considerable than loss by absorption. In objectives containing one or more cemented lenses the loss is negligible at the cemented surfaces (about i per cent) ; we need therefore to consider only losses at glass-air surfaces. The mean values of transmission (not reckoning loss by absorption, examined above) are given below for one, two, three, or four lenses in air, supposing that the polish is perfect. Number of glass-air surfaces . 2 4 6 8 Transmission % ... . 897 80 4 72-1 64*6 To obtain the total transmission approxi- mately, reckoning both causes of loss, it would be sufficient to multiply one factor by the other, e.g. a lens containing six glass-air surfaces in GENERAL PROPERTIES OF OPTICAL SYSTEMS 35 which the total of the thicknesses of the com- ponents is 3 cm. transmits probably 72-1 X 0-93 = 67 per cent. 57. Effect of Internal Reflection. The light reflected at each free surface is, unfortunately, not lost ; a part of the beam which has suffered several internal reflections is sent back to the object, but another part passes on to the plate. In Fig. 39 it is seen that the beam which forms Fig. 40. False Images (R. Schiittauf) the image P also gives an image P x on the side of the object, after one internal reflection, and an image P 2 on the side of the image P after two internal reflections. If the incident beam is sufficiently intense, and if the exposure is sufficiently long, these images will be registered on the photographic plate as circular or elliptical areas of relatively large dimensions. The num- ber of parasite (ghost) images reflected to the plate will be greater the greater the number of glass-air surfaces. The intensity of the images diminishes according as the number of reflections the beam has undergone is greater. Number of glass-air surfaces . . 2 4 6 8 Number of ghost images . 1 6 15 28 These ghost images appear frequently in photographs taken at night which have had a long exposure and where the view contains light- sources of great intensity towards the edge of the field. Owing to the symmetry of the lens round its axis, the secondary axis of the different beams arising from the same original beam are contained in a meridian plane. The centres of the areas corresponding with a single point source are thus all situated on the straight line which joins the image of the point and the point where the optical axis cuts the sensitive surface. Fig. 40 (from R. Schiittauf) shows the limits of the six ghost images given by a rectilinear lens (symmetrical lens of two groups each consisting of two cemented lenses, so that there are in all four glass-air surfaces) where the object is a bright point on a black background. 1 In some of the old lenses one of the internally reflected beams gave almost a sharp image of the stop in the image plane, somewhat enlarged, centred on the optical axis, and superposing a bright patch (called the central flare spot ) on the image. In regular photographic work these ghost images are not seen individually, but the light directed towards the plate after internal reflec- tion forms a slight fog over the whole image, reducing contrast. Fig. 41 (from measurements made by E. Goldberg) shows the effect of these internal reflections for an objective containing four independent lenses (eight glass-air surfaces) photographing a landscape of which HH is the horizon. Successive reflections of the light from the sky produce on that part of the plate on which the landscape is recorded an amount of light which decreases as the distance from the horizon line becomes greater. The circles in dotted lines correspond with different obliquities of the beam; the curves in full line join points of the image in which the parasite light has an intensity equal to 6 per cent, 5 per cent, . . . 2 per cent of that in the image of the sky. The Fig. 41. Veil from Internal Reflection (E. Goldberg) intensity of this parasitical light is reduced appreciably when the lens is stopped down. 2 1 To observe this phenomenon easily, stick a small piece of opaquecard on the back surface of the focussing screen midway between the centre and a corner, and direct the camera towards the sun, so that the direct image of the sun is masked by the card. On the focuss- ing screen a number of bright circles will be seen, which would not have been the case if the image of the sun, incomparably brighter, had remained visible in the field. 2 The curves in Fig. 41 give results obtained with a lens at F/6-8 (§ 71). 36 PHOTOGRAPHY: THEORY AND PRACTICE The same author has been able to establish the fact that from these reflections and from the diffusion that is unavoidable at surfaces, even if perfectly polished and kept perfectly clean, the extreme contrast in the image yielded by a lens is always less than in the subject itself examined from the same viewpoint. A subject having a range of contrasts infinitely great is reduced to a contrast about 200 : 1 with a single lens and to about 60 : 1 with an anastig- mat giving a sharp image over a relatively large field. These measurements confirm the experience of the old photographers, who used for landscape work a single objective consisting of a number of Fig. 42. Small Cube as Fig. 43. Duplex Seen by a Large Lens Diaphragm lenses cemented together, considering that this type gave more brilliant images. 1 58. Stereoscopic Effects. A lens of very large diameter, such as some at one time used as portrait lenses, gives an image of a near object in which appear certain parts of the subject which an eye (placed as close as possible to the lens) would see only if moved from right to left (Brewster, i860) and up and down. The image of a small cube isolated in space, e.g. a die suspended by a thread in the optical axis of such a lens, would show five faces (Fig. 42), presenting thus the appearance of a truncated pyramid seen from the direction of the small end. It has long been recognized that it is possible to obtain with such a lens, fixed with respect to the object photographed, two stereoscopic images by using an eccentric stop, rotated through 180 0 between the exposures, the aper- ture being in. in a horizontal direction from the centre, so that two successive positions of the aperture are at a distance apart equal to the 1 E. Goldberg defines brilliance of an objective by the logarithm of the ratio J 0 fl R where 7 0 = the illumina- tion of the image of a uniformly illuminated hemisphere centred at the optical centre of the lens, and I R = the illumination of a point of the image of an absolutely black object (§15) placed at the intersection of the hemisphere and the optical axis. mean separation of the eyes. The diaphragm merely extracts from the complete image certain details by isolating certain light-rays. It has also been proposed (Lehmann, 1878 ; Boissonas, 1900, etc.) to use with large lenses a diaphragm with two apertures (Fig. 43) to obtain a single image in which the doubling of certain outlines would suggest some idea of relief. The painter or draughtsman, observing his model with two eyes, synthesizes the two views. It thus seems logical for the portrait photo- grapher to use a lens of which the useful diameter is at least equal to, preferably greater than, the mean separation of the eyes (G. Cromer, 1921), without, however, falling into an exaggeration which, viewed at a short distance, would spoil the image. A lens of small diameter gives a view as seen by a one-eyed person. In scientific photography, in which a mathe- matically correct perspective is required, the use of lenses of large diameter should, on the contrary, be avoided. 59. Defects of Workmanship and of Material. Photographic lenses made by opticians of repute are always carefully examined before leaving the workshops, and run hardly any risk of showing any faulty material or bad workmanship, but these faults are sometimes met with in lenses which carry no maker’s name or bear a more or less fancy name. Defects of material comprise non-homogeneity of the glass and imperfect annealing. Want of homogeneity is not usually evident except in lenses of large diameter; it can be recognized by forming on a ground glass screen the image of a point source of light (e.g. the image of the sun in a well-polished metal ball or small silvered bulb) close to the axis of the lens. If now the screen is moved out of focus until a circle of light of about i in. diameter is obtained, any defect will be visible as strue or dark zones. Bad annealing, which gives rise to double refraction of the rays of light (leading to a doubling of the image) can only be seen by examination in polarized light in an optical laboratory provided with the proper equipment. Excessive pressure on the glass in its mount may also lead to double refraction. 1 Beginners have a tendency to consider the bubbles seen in every anastigmat as a defect. 1 It would be very desirable if the precaution, taken by some opticians, of engraving on the lens mount marks to indicate when the different parts are correctly screwed together, were more general. GENERAL PROPERTIES OF OPTICAL SYSTEMS 37 These bubbles, enclosed in the glass in the course of the second melting (after the first melting has been broken up and faulty pieces rejected) cannot be removed except by com- pletely liquefying the glass, which would have the effect of separating the constituents in the order of their density and thus cause a defect much more serious than the bubbles, of which the sole effect is to diffuse about one-thousandth part of the light — an absolutely negligible amount. The form of faulty workmanship most fre- quently met with in lenses of poor quality is bad centring (§ 40). This can easily be tested by looking at the images of a point or small source of light reflected at the various air-glass surfaces. These ought to lie exactly on a straight line. Another defect which is likely to arise when the lens has been remounted in a different mount from that supplied by the maker is incorrect separation of the different components, or incorrect placing of the diaphragm, an error of less than 0-004 m - having a fatal effect on the sharpness of the image, especially in lenses of short focal length. As a general rule, any modi- fication of the mount (particularly those with a between-lens shutter) should be made, wherever possible, by the lens maker, who, more than any- one, is interested for the sake of the reputation of his name or trade-mark in preserving the original perfection of the instrument. If it is impossible to do this, it would be as well to make a thorough test of the lens before and after any modification (§ 116). CHAPTER VIII FOCAL LENGTH OF LENSES; SCALE OF IMAGE; CONJUGATE POINTS 6o. Conjugate Points. When a point R' is the image of a point R (Fig. 44) formed by an optical system, the point R is also the image of R' (principle of reversibility of light rays 1 ); the points of such a pair are two conjugate points of the optical system considered. Various formulae and simple graphical constructions enable us, when the focal length of a lens and the positions of the nodal points or foci (§§ 44 and 45) are known, to determine the position of the image of a point the position of which is known. In order to avoid the complication of curva- ture of the held, we shall consider, in what follows, only points on the optical axis, the images of which are therefore also on the optical axis. The power of a lens [convergent power or con- vergence) is the reciprocal i/F of the focal length ; when the focal length is measured in metres the power of the system is expressed in diopters. Thus, for example, a lens of 0-20 m. (8 in.) focal length has a power of 1/0*20, or 5 diopters. 2 The effect of an optical system is to add its con- vergence (or subtract, in the case of a divergent system, i.e. of negative vergence) to that of the 1 The principle of reversibility is frequently misin- terpreted. For instance, if a number of points at different distances from the lens have been photo- graphed on a fiat plate it cannot be expected that by a reversal process the images of the photographed points will coincide with the system of points photo- graphed. Again, an image defaced by aberrations (distortion excepted) will not, by a system of reversal, form a sharp image. Even a sharp photograph cannot, by reversal, yield an image as sharp as the original, since the aberrations in each process add up. 2 The following table gives the focal lengths of lenses of various powers as expressed in diopters — Focal Power | Focal Length Focal Power | Focal Length Diopters In. j Cms. Diopters In. 1 Cms. •25 157-48 400 4-5 8*74 22*2 •5 78-74 200 5 7-87 20 •75 52-49 133*3 5-5 715 18*18 1 39*37 100 6 6*56 16*6 1-25 31-49 80 7 5-62 14*28 i ’5 26-24 66-6 8 4*92 12*5 i -75 22*49 57-14 9 4-37 1 1*1 2 19*68 50 io 3'93 10 2-25 17*49 44’4 11 357 9*09 2-5 15*74 40 12 3*28 8*3 2-75 I 4 - 3 1 36*36 13 3*02 7-69 3 13-12 33*3 14 2*8i 7-14 3*5 ir*24 28-57 15 2*62 6*6 4 9-84 25 20 1*96 5 light beams passing through it. Now a point at a distance p from the nodal point of incidence sends to the system a divergent beam of which the negative vergence is ijp (this is also some- times known as the proximity , [optical) of the point considered), which can also be expressed in diopters. If the convergence (positive vergence) i/F of the system is greater than the negative vergence of the beam 1 Ip, the emergent beam will have a convergence (1 /F -1 /p) diopters. The proximity 1 Ip' of the image [p' being measured from the nodal point of emergence) is then given by i/p’ = i/F - 1 Ip or 1 Ip + i/p’ = i/F which translates literally the mechanism of the alteration of the waves of light made by the optical system in question If, instead of considering, as above, the ultra- nodal distances (reckoned from the nodal points), we consider the ultra-focal distances d and d’ (reckoned from the foci), we shall obtain the law of conjugate points in the form given by Newton, which is often very advantageous, d x d' = F 2 or, in ordinary language, the product of the ultra-focal distances of a point and of its image is equal to the product of the focallength by itself. 1 61. Among the different methods of graphic- ally representing the law of conjugate points, the following (Lissajous, 1870) enables us to account for all the practical consequences of this rela- tionship at first sight. Construct a square (Fig. 45) NFMF', of which the sides are equal to the focal length of the lens considered, and produce the sides NF and NF' to X and Y respectively. From the origin N mark off NR on NX equal to the ultra-nodal distance [p) of the point object [FR is thus the extra focal distance d), join RM and produce it to meet NY in R'. The length NR' is equal to the ultra- nodal distance p' of the point-image and F'R' is the ultra-focal distance d'. If now RR' is rotated about M, its intersections (produced if necessary) with NX, NY correspond to two 1 To obtain the second of these expressions from the first it is only necessary to replace p and p' by ( d -f- F) and (d' 4- F) respectively, get rid of the denominators in the fractions, and eliminate identical terms appearing on both sides of the equation. 38 FOCAL LENGTH OF LENSES 39 conjugate points. 1 It is seen that as R moves already mentioned (§44) that the nodal points away from the lens, R' moves nearer to it, and are conjugate. vice versa. When R moves to infinity, the 62. Relations between the Size of Object and straight line MR becomes parallel to AX, and Image. If we consider a lens without appreciable R' coincides with F'. Inversely, if R approaches distortion or curvature of the field (which would F, the straight line RM becomes parallel to AY obviously not be the case with the meniscus and consequently R' recedes to infinity. represented in Fig. 46), we know that the images *- ' P * * -p’ -> Fig. 44. Image Formation If the point R approaches closer to the lens of all points on a plane perpendicular to the than the focal length, e.g. to the position marked optical axis lie on another plane also perpen- T, the straight line MT no longer meets AY, dicular to the optical axis. Knowing also that but its prolongation AY', in T\ corresponding the exterior parts of a secondary axis are to a virtual image (§43). parallel straight lines (lines joining object and A particularly interesting case is that in which image points to the corresponding nodal points), the object 5 has an ultra-nodal distance twice we can determine the scale of the image (relation the focal length. In this case FS = MF, and between corresponding dimensions of object and the straight line MS is inclined at 45 0 , and S' is such that NS' = NS. The two points 5 and S' at equal distances from their respective nodal points are called the symmetrical points of the lens ; their separation is the shortest distance that can exist between a point object and its real image. Another particular case, but of no practical interest, is that where the points R and R' coincide with A ; this leads to the fact 1 This property can be verified by consideration of the similarity of the triangles MFR, R'F'M or by equating the area of the triangle RNR' to the sum of the areas of the square and the two triangles MFR and R'F'M. image) of an object of which the position is known relatively to a lens of known focal length. Let us suppose that the length of the image of an arrow RQ (Fig. 46) of length l perpen- dicular to the axis, at an ultra-nodal distance p } is to be determined. Join by a straight line the point Q to the nodal point of incidence N, and draw through the nodal point of emergence N' a line parallel to QN to meet in Q' the straight line R'Q' through R' (the image of R), parallel to RQ. The point Q' is the image of Q, and the element of line R'Q' is the image of RQ. The length I' of this image can be ascertained from l , since the triangles RQN and R'Q'N' are similar. 40 PHOTOGRAPHY: THEORY AND PRACTICE The ratio of similitude or scale (n = i/N) of reproduction 1 is then equal to the ratio of the ultra-nodal distances of image and object n = i/N = I'/l = p'/p From this relation can be deduced simply 2 p = (i + N)F = (i + i fn)F p' = ( i + n)F = (i + i/N)F If we replace the ultra-nodal distances by the ultra-focal distances, we obtain the simpler and more easily expressed relations d = NF = F/n d' = F/N = nF To obtain an image of a plane object placed perpendicular to the axis at a reduction of i/N, it should be placed at an ultra-focal distance equal to N times the focal length ; the focussed farther from the lens than its image, 1 and magnification whenever the object is closer to the lens than to its image. When the object photographed has a certain depth, it is no longer possible to speak of the scale of the image, since this will vary from point to point. It should be mentioned here that in the most general case, where such an image is photographed on a plane perpendicular to the axis, the relative dimensions of the different parts of the image are inversely proportional to the ultra-nodal distances of the corresponding point objects, and not to their ultra-focal dis- tances, the point images being all on the same plane and no longer the conjugates of the point objects. image is formed at an ultra-focal distance i/iVth the focal length. For enlargement n times, the object should be placed at an ultra-focal distance equal to i/nth the focal length, and the image will be at an ultra-focal distance equal to n times the focal length. For example, if it is desired to reduce a square of 12 in. sides to an image of 4 in. sides, with a lens of 8 in. focal length, N = 3. The original must be placed at 3 X 8 = 24 in. from the anterior focus, and the image will be formed on a plane at 8/3 = 2§ in. from the posterior focus. Reversing the two positions, the image would be enlarged three times (n = 3). For a reproduction same size (N = n = 1) the planes of copy and image cut the axis at the symmetrical points 3 S and S'. There is reduction whenever the object is 1 When the scale of reproduction is less than unity, i.e. in the case of reduction , it is often expressed as a fraction i/N, N being the reciprocal of n. 2 By simply replacing p by its value Np' in the expression 1 /p 1 !p' = 1 /F, reducing the three fractions to a common denominator and simplifying the result. 3 The nodal planes, planes through the nodal points perpendicular to the axis, are also object and image of the same size but without reversal of the image ; this property is without practical application. 63. Graphical Construction of the Image Formed by an Optical System. Knowing the position of a point object Q relatively to the foci F and F', and the nodal points N and N' of an optical system, the position of the image Q' can be determined as follows : Draw the optical axis FF' (Fig. 47), and at N and N' draw per- pendiculars to it to indicate the nodal planes. From Q draw a straight line parallel to the axis, meeting the nodal plane of emergence in the point a ' ; a' is the image of a, the intersection of the ray with the nodal plane of incidence. The emergent ray will then pass through the focus F' } since all incident rays parallel to the axis, after refraction, meet the optical axis at the posterior focus. Draw another line QF and produce it to meet the nodal plane of incidence in b ; the image of b is b' on a line through b parallel to the axis, which is also the emergent 1 The scale is zero when the object is at an infinitely great distance (e.g. the stars). The distance between the images of two distant objects cannot be determined from considerations of scale. It is determined from the angle seen to subtend the two point objects, e.g. the solar disc is viewed from the earth under an angle of 32 minutes of arc ( apparent diameter ) i.e. about i/iooth of his distance. The image of the sun will thus be equal to i/iooth of the focal length of the lens used. The mean diameter of the lunar image is about the same size as that of the sun. FOCAL LENGTH OF LENSES 4i ray. Q' , the intersection of a'F' and bb', is the image required. The accuracy of the construc- tion can be tested by seeing whether QN, Q'N' of the secondary axis are parallel to one another. 64. Image of a Plane Inclined to the Axis. Consider a lens of which the foci and nodal points are FF' and NN' respectively (Fig. 48), and let R and R' be two conjugate points. If a plane perpendicular to the plane of the paper meets the optical axis obliquely at R , all the points in this plane (at least all those not far from the axis) form their images on another Fig. 47. Geometry of Image Formation plane, also meeting the optical axis obliquely at R r and perpendicular to the plane of the paper. This image plane is defined by the condition that its intersection M' with the nodal plane of emergence should be contained in the plane MM' parallel to the axis through M } the intersection of the object plane with the nodal plane of emergence. It is easily seen that the image is deformed relatively to the object. In particular the points on the line X, the intersection of the object plane with the anterior focal plane, will be imaged at infinity, the secondary axes NX, N'X' being parallel. In the same way infinitely distant points of the object plane in the direction NY will be imaged on the straight line Y', the intersection of the image plane with the posterior focal plane. On this straight line all the vanish- ing points of parallel straight lines in the object plane will be imaged, while all lines meeting in X will be parallels in the image plane. This manner of distorting is made use of for correcting the perspective of photographs acci- dentally taken on an inclined plane (by making all the vanishing points of vertical lines in such a perspective meet in X, they will be corrected in the image), or for making lantern slides which are to be projected obliquely. We shall return to this subject in more detail in Chapter XLV. 65. Experimental Determination of the Focal Length of a Lens. The method of measuring the focal length 1 usually described consists in focussing a distant object (distant at least 1,000 times the focal length to be measured) in the camera, then focussing an equal size image of an easily measurable geometrical figure (e.g. a circle or equilateral triangle). The amount the camera has to be extended between the two positions is exactly the focal length. 2 Unfortunately, it is not always possible to have a distant view at hand, and, on the other hand, to get an image exactly full size often requires a large number of trials when the use of a special camera such as a process camera is not avail- able. 3 Instead of measuring the focal length directly, as above (distance of a focus from the corre- sponding symmetrical point), it is easy to cal- culate it by focussing a test object on two different known scales, and measuring the dis- placement of the focussing screen between the two positions. As far as possible, the precautions recom- mended in process work (Chapter XLV) should be taken to ensure parallelism between the object plane and the focussing screen (this being supposed perpendicular to the axis), and accu- rate focussing should be done in the manner Fig. 48. Image Formation on Inclined Surface suggested in § 308. Let n and n' be the two scales, which may be chosen arbitrarily but very 1 The focal length indicated in catalogues, or en- graved on the mount of a lens, is the mean of the prob- able focal lengths of a number of specimens, and this may differ by 2 per cent to 5 per cent from the true value. 2 For these operations it is advantageous to reverse the focussing screen, as it is easier to measure, with the accuracy that i desirable, the image directly on the ground glass side. 3 If a distant object is available for focussing on, it would be sufficient to obtain the second position at a scale of reduction n. It is known that, in these condi- tions, the ultra-focal distance of the image is nF, equal to the change in camera extension measured ( e ). The focal length is then F = e/n. It would be necessary to avoid using a small value of n, for then e would be small, and owing to probable errors in the measurement of e and n, the value of F obtained would be inaccurate. 42 PHOTOGRAPHY: THEORY AND PRACTICE accurately measured, and e the change in camera extension between the two positions. We know that the extra-focal distances at scales n and n' are Fn, Fn' respectively; their difference is the length e, which is known. Thus, e e = (n - n r )F , whence F = , v ’ n - n In other words, the focal length is the differ- ence in extension measured, divided by the difference between the two scales of reproduc- tion. 1 To get sufficient precision, the two scales chosen should be as widely different as possible. If, for example, the test object is an equilateral triangle 2 of which the sides are 4-8 in. long, reduced in the two positions to 3-6 and i*6 in. (scales of reduction 0*75 and 0*33 respectively) and that the increase in camera extension is 2*8 in., the focal length will be given by 2-8/0*42 = 6*67 in. The position of the posterior focus could be easily found by measuring the ultra-focal dis- tance nF from one of the positions of the image towards the lens. The position of the nodal point of emergence would be given by measuring a further distance F towards the lens. 66. Where the only camera available is capable of only a small range of focus (which is the case of a large number of hand cameras) the above method is not possible, and the following method can be used (Debenham, 1879). Having focussed the image of a geometrical figure and determined the scale n , the total distance l between object and image is measured. This distance is the sum of the two ultra-nodal distances p and p' increased or diminished by the nodal interval i (separation of the nodal points) according as the nodal points are in the normal position or crossed (the nodal points are said to be crossed when the nodal point of emergence is the nearer to the anterior focus, which is opposite to the case of the systems previously considered). Replacing^ and p' by their values in terms of n and F, we obtain, after simplification 1 F = l ± i 2 + n + i/n If n is very small (which will always be the case in small cameras) i/n will be very large, and consequently the error arising from neglect- ing the internodal distance will be divided by a number generally greater than 10 and will therefore nearly always be negligible, except for telephotos and single lenses of convertible sets. If, for example, a lens of 6 in. focal length has an internodal distance of 0*12 in. if the scale of reduction is 1/10, and the total separation between object and image is 71-5 in., the for- mula gives a focal length of 5-9 in., which is sufficient approximation for all practical pur- poses. 67. A variation of this method will avoid the error arising from neglecting the internodal distance, whatever its value, and at the same time determine it. Having measured the total distances l and /' for two scales n and n ' , the focal length F and internodal distance i are given by F = i-r (* + £)-("' + i = l-(2 + n + £jF respectively. If, for example, it has been found that for n = 1/5 l = 43*6 in. n' = 1/3 r = 32-4 in. it follows that 77 ir2 a F = F866 = 6 -° in - and i = 43-6 - 6(2 + 5 + 1/5) = 43-6-43-2 = 0-4 in. 1 If at the same time the displacement of the camera between the two positions had been measured (£), the following formula would also give the focal length, which would furnish a useful check — F = E (i/n' - i/n) 2 Care should be taken to verify the equality of the sides in the image to be sure that there is no deforma- tion of the image, and that there is a scale of reduction. In general these scales will be expressed by less simple numbers than those given in the example, but this will not affect the method of calculation. 68. Direct Determination of the Position of the Nodal Points and the Focal Length. If the lens is rotated about an axis perpendicular to the optical axis, and containing the nodal point of emergence, the images of very distant points remain fixed during the rotation, at least if it 1 In fact Z = F(i + m) + F(i + i/n) ± i so that F ( 2 + n -f- i /«) = / ± i. FOCAL LENGTH OF LENSES 43 is not large and if the angle between the second- ary axis to the point object and the optical axis is never very great (Moessard, 1889). To explain this, consider a lens (Fig. 49) which, for the sake of simplicity, we suppose to be reduced t o the nodal planes N { N e . The image of an infinitely distant object in the direction NjM on the axis is formed at the focus F. After rotation about an axis perpendicular to the optical axis through the nodal point of emergence N e , the nodal point of incidence will move to N/. The point object being infinitely distant, the secondary axis N/M' to this point is parallel to N^M. By virtue of the definition of the nodal points (§ 44) the two exterior parts of the secondary axis are parallel to one another ; The displacements of the image noticed when the angle of rotation is large enable us to deter- mine the form of the focal surface point by point, and to study the various aberrations of the image. 69. Automatic Adjustment of Object and Image. The relations between the ultra-focal distances of two conjugate points, or of planes perpendicular to the axis passing through them, can be translated geometrically so that auto- matic linkages between these planes can be made, so dispensing with all focussing in en- larging or reproduction. The only adjustment to be made is that for the scale of reduction, obtained by the displacement of one of the conjugate planes, the image remaining sharp the secondary axis thus emerges from the lens in the direction N e F. This property, which is made use of in the greater number of panoramic cameras, can also be utilized to determine the position of the nodal points and the focal length directly, when the nodal point of emergence is within the mount (a condition which excludes telephoto and analogous lenses). The lens being mounted so that it can be moved to and fro on a platform which can rotate about a vertical pivot, the image is formed on a fixed screen and observed while the lens is moved on the platform until a position is found such that the images of distant points remain stationary while the lens is rotated. The nodal point is then on the axis of rotation and the distance of this axis from the screen is the focal length required (Moessard Tourniquet, 1893) . Turning the lens end for end, the other nodal point can be found similarly, and a second measurement made of the focal length, which gives a useful check on the first measurement. throughout. Numerous solutions of this prob- lem have been given ; we shall indicate only some of them, selected from the most charac- teristic. We shall suppose, in what follows, that the nodal points coincide with the optical centre. If this is not accurately true, it will be necessary to assign to one of the nodal points the position indicated by the centre, and move the conjugate point corresponding to the other nodal point in an appropriate direction by an amount equal to the nodal interval. (1) Consider (Fig. 50) two points 0 and /, free to move in a slot parallel to the optical axis. At C, the intersection of the slot with the plane drawn through the optical centre at right angles to the optical axis, erect a perpendicular CD, of length equal to the focal length F. By making a bent lever pivoted at D, having slots of which the axes meet at right angles in D ( J. Carpentier, 1898), rotate, it is possible to constrain two studs P and P' in the axial slot to move so that their distances d and d' from the point C will always 44 PHOTOGRAPHY: THEORY AND PRACTICE satisfy the relationship between the ultra-focal distances of two points, viz. 1 dd' = F 2 It will then only be necessary to join P and P' to 0 and I respectively by two connecting rods, of length equal to F, to make certain that 0 and I are conjugate points, and consequently also the two planes perpendicular to the axis through them. nifying the movements transmitted from P" to P by means of a pantograph CaP"a'Pb f P"bC ) the size will be considerably reduced. If the two coupled lozenges of the pantograph have sides of length l and L respectively, it will only be necessary to satisfy the relationship Fig. 50. Self-focussing Linkage (Carpentier) (2) Another linkage (G. Koenigs, 1900) is (3) Let (Fig. 51A) the points 0 , / on the optical formed of an articulated lozenge P'AP"B (Fig. axis of the lens C be the intersections of the 51) and two equal rods ^ 4 C, CB, pivoted at their copyholder and plateholder. From these points joint. If we represent by m the common length of the four sides of the lozenge and by n that of the connecting rods, the lengths d = CP' and d' = CP" (which, by reason of symmetry, are obviously in a straight line) will always 2 be such that dd' = m- - n 2 We then only need to give to the constant value of this product the square of the focal length in order to get the required linkage, but in these circumstances the jointed lozenge would generally be of very great dimensions ; by mag- 1 It is shown in the manuals of plane geometry that the product of two segments determined on the hypo- tenuse of a right-angle triangle by the foot of the perpendicular let fall from the apex is equal to the square of the perpendicular. 2 A circle with A as centre and radius m will pass through the points P f and P". Now the product of the distances of any point C from the two intersections of the circle by a chord through C, i.e. the product CP ' X CP", is equal to the difference between the squares of the radius in and the distance n of the point C from the centre of the circle. draw towards the lens, distances equal to the focal length/. The points^ and B thus obtained must be at distances from C such that CA . CB = f 2 . Using AB as a diameter draw a circle with a centre M and through C draw the chord DD' perpendicular to AB. Elementary geo- metry teaches that CA . CB = CD . CD' , hence CD = /. The rays MA, MB, MD are evidently Fig. 5 i a. Principle of Focussing Linkage. equal and, conversely, by this last equation the respective distances of the conjugate points OC, IC can be definitely determined (P. R. Burchall, 1933). Among the methods of linking based on FOCAL LENGTH OF LENSES 45 this principle may be mentioned (A. Bonnetain, 1934) a system of three racks engaging in M on one and the same toothed wheel. (4) Finally, there are numerous arrangements based more or less directly on the hyperbolic cam 1 (G. Pizzighelli, 1889). For instance, a table T on which the optical centre C is fixed can slide on two rails RR (Fig. 52) perpendicular to the object plane P. Movement of the table T is communicated by means of racks and pinions to a table T', but in a direction at right angles. A slot in T' in the form of a rectangular hyperbola acts on a stud P ' which is constrained to move in an axial slot in T. Any plane per- pendicular to the optical axis and containing P' will be conjugate to P. An obstacle to the employment of these devices in practice is the difficulty of obtaining delivery of a series of lenses of exactly equal focal length, so that it is impossible to make these apparatus in quantity. A number of linkage arrangements have been brought out in the last few years which have an adjustment for com- pensating for slight variations in the focal length. 70. Combination of Lenses or Optical Systems. It sometimes happens that another system, convergent or divergent, has to be added to a lens, and it is desirable to be able to determine the focal length of the combination, the optical axes of the different components being assumed to coincide (centred system). For thin lenses in contact the law that the power (§ 60) of the system is the sum of the powers of the components may be considered exact, it being understood that negative powers (corre- sponding with divergent lenses) are to be subtracted. Calling / and /' the focal lengths of the components and F that of the resultant system, then I/F =!//+!//' In general, however, this rule is not applicable, and account must be taken of the spacing of the combination, i.e. the separation between the nodal point of emergence of the first system and the nodal point of incidence of the second. 2 1 The relation ddf = F 2 is the equation of a rect- angular hyperbola with the asymptotes as axes. The vertex is at a distance F from the two asymptotes. 2 The optical interval is also sometimes used ; this is the separation between the posterior focus of the Referring the reader to a treatise on optics for the proof, we shall limit ourselves to formulating the rule. Calling e the separation as defined above, the resultant focal length is given by I/F = I// +1//'-*///' The resultant focus is at a distance D from the posterior focus of the second system, equal to /' 2 n ± “*-(/-/') These rudiments will have an application to the case of lenses in which focussing is effected by varying the separation of their components (§ 108), sets of lenses (§ 112), supplementary Fig. 52. Cam Self-focussing (Pizzighelli) lenses (convergent or divergent 1 ) and telephoto lenses (§ 109). first system and the anterior focus of the second. The optical interval (5 is connected with the separation e by the equation 5 = e - (/-/') Using this relationship, the resultant focal length is F =//'/< 5 It may be added that the resulting system is afocal (i.e. the focal length is infinite) if the optical interval is zero (normal adjustment of astronomical telescopes). 1 To determine the focal length of a divergent lens the procedure is the same as for a convergent lens but a virtual object must be used, which can be the image of an object formed by a convergent system of relatively great focal length, the divergent lens being placed between the convergent system and the real image. For an approximate value of the focal length, the lens may be directed towards the sun, and the distance from the lens to a screen measured when the diameter of the circle of light on it is double that of a circular aperture placed against the lens; or a thin divergent lens may be neutralized by placing it in contact with a thin convergent lens of the same focal length (the method used by oculists). CHAPTER IX DIAPHRAGMS AND RELATIVE APERTURE! EFFECT ON PERSPECTIVE AND INTENSITY 71. Relative Aperture of a Diaphragm. The diameter of the beam of rays incident parallel to the axis which, after refraction through the lens components in front of the diaphragm, completely fills the latter is called the effective diameter of the diaphragm. Thus, D, D' and D" (Fig. 53), although of different diameters, all have the same effective diameter d. If, without altering the position of the stop, the real diameter is altered, its effective aperture varies proportionally. The constant ratio be- tween the effective and the real aperture is 1 — / D D» ! A n Fig. 53. Effective Aperture sometimes called the coefficient of the effective aperture , and is equal to 1 only if the beam of light reaches the stop before meeting the lens (the case with single lenses). In the general case, in which the stop has in front of it one or more lenses forming a convergent system, the coefficient is greater than 1. As the value depends on the construction of a given lens, obviously no rule can be given, but it may be stated that with symmetrical anastigmats it generally lies between i-i and 1-15, whilst with anastigmats consisting of three separated lenses it often amounts to 1*3. If the diameter of the effective aperture is i/nth the focal length F, the aperture is said to be F/n, which is also called the relative aperture of the diaphragm considered. If, for example, the real diameter is o*8 in. and the effective aperture is 0-92 in. of a lens of 4-6 in. focal length, the relative aperture is i 7 / 5. The relative aperture of the largest stop a lens can use is called the maximum relative aperture , or, more simply, the maximum aper- ture of the lens. We shall see later (§ 90) that the maximum relative aperture of a lens is the principal factor governing its speed. 1 1 Most very large aperture lenses have the advertised relative aperture only for the bundle of rays parallel to the optical axis (§ 54). 72. Different Types of Diaphragms. In order to be able to get all possible apertures with a lens, modern objectives are usually fitted with an iris diaphragm (Fig. 54) having an aperture which can be varied by means of a rotating ring or external lever on the mount. 1 The thin blades of the iris are of ordinary steel or ebonite. Though ebonite has the advantage of not rust- ing like steel, in damp climates, care must be taken not to subject it to great heat. Hence an ebonite iris should not be used in the enlarging or projecting lantern using a condenser, or there will be danger of the blades melting or burning. With lenses of which the component glasses are too closely spaced to accommodate an iris, a rotating diaphragm is employed (Fig. 55). Here an eccentric disc has a number of different apertures, which, by rotation of the disc, are brought into position concentric with the axis of the lens. The size of the aperture in position is indicated by a number engraved on the part of the projecting disc opposite the aperture. In many old lenses and in modern lenses for process work, 2 Waterhouse stops (Fig. 56) are inserted through a slot in the side of the lens tube. 73. Pupils of an Optical System. The beams of light passing through an optical system are limited by the aperture of the diaphragm. Now the components of the system in front of the stop (lens L lt Fig. 57) form a virtual image (called the entrance pupil , P { ) of the stop D. The entrance pupil is such that the prolongation of rays through L v which afterwards are just bounded by the diaphragm D, reach the outline of the entrance pupil. The diameter of the entrance pupil is the effective aperture of the diaphragm just mentioned. In like manner the components behind the diaphragm ( L 2 in Fig. 57) form a virtual image of its aperture, called the exit pupil P e) the 1 In Fig. 54 the guiding slots of the movable ring are shown radial, which is the usual form. By sloping them it is possible to make the usual markings of apertures almost equidistant (Lan Davis, 1911). 2 The making of negatives through screens, as used in preparing half-tone blocks or in lithography, re- quires stops with non-circular openings (generally square), and capable of being variously orientated in the lens tube. 46 DIAPHRAGMS AND RELATIVE APERTURE 47 outline of which is reached by the prolongations of those rays which (before passing through L 2 ) just reached the outline of the diaphragm D (E. Abbe, 1890). The two pupils 1 are thus con- jugate with respect to the complete lens. If we suppose the diaphragm aperture gradu- ally reduced to a small opening 0 on the axis, admitting but a single ray of light, this single to the position and dimensions of this image, from the positions of the nodal points and the foci. 74. Photographic Perspective. The photo- graphic image is an exact perspective rendering of the objects represented, the viewpoint of which, relatively to the objects, is the centre of the entrance pupil; relatively to the image, the Fig. 54. Iris Diaphragm Fig. 55. Rotating Fig. 56. Waterhouse Diaphragms Diaphragms ray would be the one originally directed to the point /, the centre of the entrance pupil, and would appear to emerge, after passing through the lens, from the point E, the centre of the exit pupil. The assembly of the different parts of the ray forms what is sometimes called a principal ray. In the particular case where the stop is placed with its centre at the optical centre of the instrument (which frequently happens with symmetrical lenses), the centres of the pupils coincide with the nodal points, but this coinci- dence does not occur with single lenses, conver- tible sets, nor telephotos, and in other types of lenses is not always aimed at. Just as by consideration of nodal and focal points it is possible to determine the dimensions of the image without considering the construc- tion of the optical system forming it, so by consideration of the pupils it is possible to determine the perspective and the centre of projection of images without having to be con- cerned with the optical system. The pupils in fact determine which of the rays are used in the formation of the image, without upsetting in any way the conclusions that have been drawn, as 1 The pupils must not be confused with the windows (German, Luke), consideration of which is less frequent in treating of photographic lenses. The windows are the images formed by the two systems L x and L 2 of the aperture limiting the field of view (the mount of the lens or the aperture of some attachment to the lens), and thus correspond with the field stops in observational instruments. viewpoint is usually identical with the nodal point of emergence. Consider (Fig. 58) the two conjugate planes QQ' of an optical system represented by its nodal points, foci, and pupils, and let us find how the points R and 5 outside the plane Q will Pi Pe be reproduced on the plane Q', R' and S' being the respective images of R and 5 (the graphical construction is indicated by dotted lines) and being themselves outside Q’ . The bundle of rays used in the formation of the image of the point R is limited by the cone with apex R and base P t - (the entrance pupil). After passing through the lens it forms another cone, having the exit pupil P e as base and R' as apex. These two cones form circular patches ( circles of confusion ) on the focussed plane Q 4 8 PHOTOGRAPHY: THEORY AND PRACTICE and on its conjugate Q' , where the focussing screen or photographic plate is placed. The circles of confusion are conjugate, with their centres r and r' at the intersections of the principal ray with the respective planes, and the ratio of their sizes is equal to the ratio of the ultra-nodal distances of Q and Q' (say n ). If the patch at r' is sufficiently small and the photograph is viewed at a sufficiently large distance, the patch is indistinguishable from the geometrical point image of r (the secondary axis corresponding to r has been drawn in Fig. 58). 75. The fact that the construction of the lens (except in the case of very pronounced distortion) and its focal length have no influence on the perspective of the image can be proved by photographing an architectural subject from the same viewpoint successively with a pinhole (§ 38) and with lenses of very different focal lengths. The different images thus obtained will be identical except for size. It is thus incorrect to attribute to the use of a short-focus lens the unpleasant, almost dis- torted, views which are easily obtained with The photographic image thus coincides with a photograph, made on a scale of reproduction n , of the perspective of the objects projected on the plane Q from a point coinciding with the centre of the entrance pupil. It is thus itself a perspective view if the lens is free from distor- tion and the diameter of the entrance pupil is a small fraction of the distance of the objects represented, so that stereoscopic effects are avoided (§ 58). Now it is known that different perspectives obtained by proportional enlargement or reduc- tion have their principal distances proportional to their respective scales (§ 24). The principal distance of the perspective at Q' (the photo- graphic image) is thus the product of the distance of Q from the entrance pupil, multiplied by n. In every case where the object photographed is at a distance from the lens compared with which the distance of the entrance pupil from the nodal point of incidence is negligible, the prin- cipal distance of the photograph can be taken as the ultra-nodal distance, and the viewpoint as the nodal point of emergence (L. P. Clerc, 1923)- these lenses. These perspectives, exact but unpleasant, are due solely to the choice of too close a viewpoint. When a photographer pos- sesses only one lens, and that of short focus, he unfortunately tries to get as large a picture as possible, and so approaches closer than he would if he used a long-focus lens. Unless he is specially trained, he does not notice, when examining the view, the exaggerated perspective arising from too close a viewpoint, the brain making the objects appear at their correct relative size, whilst binocular vision places them in their correct relative positions. On the plane images these compensations do not exist, and the so-called distortion becomes actually offen- sive, especially if the image is viewed from a position other than the correct viewpoint (§§25 and 28). These anomalies disappear in stereoscopic vision if the images are viewed at the principal distance. When it is stated that a lens of short focus gives “ faulty perspective/' which should be translated as “ geometrically correct but un- pleasant perspective," it is understood, then, that the photographer has clumsily tried to DIAPHRAGMS AND RELATIVE APERTURE 49 compensate for the smallness of the scale of his image by approaching too close to his subject. The position of the camera should be chosen without any consideration of scale. If, when the viewpoint is chosen, it is found that the lens is not of sufficient focal length to give directly as large an image as desired, the small image should be subsequently enlarged. 76. Depth of Field. Knowing that the image of a point outside the plane focussed on is a circular patch on the image plane, the limits within which the objects should lie in order that these patches (circles of confusion) should be practically indistinguishable from points, can be determined. 1 First of all, a standard of latitude must be agreed upon ; usually a maximum value is assigned to the diameter of the circle of con- fusion, e.g. i/25oth in. (it is then said that a sharpness of 1/250 thin, is required), a blur which, viewed at 12 in., is indistinguishable from a point by a person having very good sight (§ 34). This convention is purely arbitrary, and is too severe for pictures which are to be viewed at a greater distance, as when placed on a wall, and is not sufficiently severe for a small image which has to be subsequently enlarged by projection or examination under a magnifier (the case of stereoscopic pictures). The image being normally examined from its viewpoint (§ 25), it is logical, at any rate in pictorial photo- graphy, to fix the diameter of the circle of confusion which can be tolerated as 1/2, oooth of the ultra-nodal distance of the image (J. Thovert, 1904). 77. Relative Depth of Field. Let q and q' (Fig. 58) be the ultra-nodal distances of the object plane Q and its conjugate Q' on which the photo- graphic image is recorded, respectively, F/n the relative aperture of the lens, r and 5 the ultra- nodal distances of point-objects respectively in front of and behind the plane Q, r' and s' the ultra-nodal distances of their focussed images. The dimensions of the circles of confusion in the plane of the image ( r ) and (s) are ex- pressed by _W = £jz£ M = £z£ F/n f- ’ F/n s' In order that the blurs (r) and (s) should have the maximum permissible diameter aq' (the 1 The calculations given here, in conformity with tradition, are based on geometrical optics, and should not, therefore (§ 38), be considered to give rigorously correct results. 4 — (T.5630) coefficient a being, for example, 1/2,000), the distances / and s' must be such that naq' r' — q' q' — s' = = “i 7- which may be written i/q' - i/r' = 1 /s' - i/q' = na/F and as (§ 60) 1 /q’ = i/F - i/q, i/r' = i/F - i/r 1 /s' — i/F - i/s it follows that i/r - i/q = i/q - i/s = najF The difference between the extreme conver- gences (§60) of R and 5 and the convergence of the plane Q focussed on is then represented by n/2,oooths of the power of the lens, all measure- ments being expressed in diopters ; and the total depth of field (distances between R and 5 measured parallel to the optical axis) corre- sponds with a difference in convergence equal to w/i,oooths of the power. For a lens of 4-4 in. focal length, i.e. o*n m., or a power of i/o-ii = 9-09 diopters, with an aperture of F/ 6, the total tolerance of conver- gence will be (9-09 X 6)/i,ooo, or 0*05454 diop- ters, which has to be divided between the near and far points. If the object focussed on was at 197 in. (5 m.) with a convergence of 1/5 = 0*2 diopters, the convergences of the two limits of depth of field will be 0*20 0*02727, correspond- ing with object distances of 1/0*22727 and 1/0*17273, or 4*40 and 5*80 m. (173 and 229 in.) respectively. A table of reciprocals of numbers from 1 to 1,000 will be useful for making calculations rapidly and of sufficient accuracy in problems relating to depth of field. It may be said that the depth of field is that portion of space from which the lens aperture is viewed under an angle approximately con- stant within the limiting angle of confusion (A. Jonon, 1925). It should be noted that the sharp field extends less in front of the plane focussed on than behind it (24 and 36 in. in the example above). Calculations of a similar degree of simplicity enable us to work out the aperture at which the lens must be used in order to give a sharp image of objects at different distances from the lens, and on what plane the lens ought to be focussed. The convergences of the extreme points at 50 PHOTOGRAPHY: THEORY AND PRACTICE distances of 80 and 320 in. (2 and 8 m.) respec- tively, are £ = 0-5 and £ = 0-125 diopters. The difference is thus 0-375 diopters. In order that the tolerance in convergence may be equal to this, which represents 375/9-09 thousandths of the power of the lens (say 41/1,000) the lens must be stopped down to E/41. In practice, the nearest marked aperture of the iris, F/45, is used, and this will give ample guarantee of the sharpness and depth required. The distance to focus on will be given by the mean of the extreme convergences (0-500 + o-i25)/2 = 0-312 corresponding with a distance of 1/0-312 m. = 3-2 m. (126 in.). 78. Absolute Depth of Field. To conform to tradition we shall deduce the formulae for depth of field in terms of an absolute diameter 1 of the circle of confusion e (e.g. e = i/25oth in.), and not, as above, a constant fraction of the ultra- nodal distance of the plane of the photographic plate. Keeping the same notation as in the preceding section, and calling ( R ) and (S) the image patches projected on the plane Q by beams having their apices at R and S, and bounded by the diaphragm, these are given (§ 58) by the relations R q - r S s - q F/n ~ r F/n ~ s but if the image Q' of the plane Q is reduced on a scale i/m, which implies that q = [m + 1 )F, the diameters (r) and (s) of the images are equal to (R)/m and (S)/m respectively. If these are to be equal to the maximum diameter e, then (R) and (5) are equal to me , and r and s (the ultra-nodal distances of the near and far planes which will be rendered sharply) will be calcu- lated from em q-r s - q F/n ~ r ~ s q/r = 1 nme/F, q/s = 1 - nme/F whence, after simplification [m + i)E 2 [m + i)E 2 F + nme s F - nme Using the same numerical values as in the previous example, all distances being reduced to metres and taking i/25oth in. (o-oi cm.) as the maximum diameter of the circle of con- 1 In scientific photography the limit of tolerance would be the limit of resolution of the sensitive surface used. fusion, we shall find, for the case of an object at 197 in. (5 m.), 500 m + 1 = -y = 45*45 »» = 44*45 = 45*45 X 11 X 11 5i5 QQ ” 11 + (6 x 44*45 X o-oi) — 13*667 = 402 cm. (160 in.) 45-45 x 11 X 11 = 54)00 s ~ 11 - (6 x 44*45 X o-oi) 8-333 = 660 cm. (260 in.) giving a more extensive field of sharp definition than previously calculated, because the toler- ance of definition is much greater in this case. Knowing that the near and far planes of an object are at the ultra-nodal distances r and s respectively, the plane to be focussed on is given by 2 rs V ~ r + s hi other words , the distance on which to focus is obtained by dividing twice the product of the distances of the near and far planes from the lens , by the sum of these two distances . 1 The relative aperture to give the desired sharpness once the focus is fixed can be calcu- lated: we omit the calculation but give the result — 1 2 rs - F(r + s) n ~ e F 2 (s - r) Unless the near plane is exceedingly close, the second term of the numerator is negligible compared with the first. The practical formula which, moreover, leads to a smaller aperture than is necessary, and thus gives complete assurance that the sharpness will be sufficient, reduces to — (s - r)F 2 n = — 2 ers In other words, to obtain the denominator n of the fraction F/n, expressing the relative aperture, multiply the focal length by itself, then by the distance between the near and far planes (depth of field to be photographed), and divide the product by the number obtained by multiplying twice the diameter of the permis- sible circle of confusion by the product of the two distances mentioned. 1 The distance thus calculated is absolutely inde- pendent of the tolerance fixed for the definition. The rigorous calculations of physical optics give the same result in this case as the approximations of geometrical optics. DIAPHRAGMS AND RELATIVE APERTURE 5i If, for example, objects at 120 and 280 in. from the lens are to be photographed with a 6 in. lens, the plane to focus on is at a distance 2 X 120 X 280 = 168 in. 400 and for a circle of confusion of i/25oth in., the relative aperture will be E/22. (280-120) X 6 x 6 576 576 = 672 x 25 = 26-88 X — X 120 X 280 250 = 21-5 79. If the camera carries a scale by which the change in camera extension can be measured, the above calculations can be avoided. Having focussed successively the near and far planes and noticed each time the position of the movable part on the scale, the camera is set to the mid-way position and will then be correct. To obtain a degree of sharpness equal to i/25oth in., the E/ No. is taken as one-eighth the number of thousandths of an inch the camera extension has been altered (G. Cromer, 1911). If, for example, the movement is o-i6in. (= i6o/i,oooths), the aperture will be F/ 20. For a sharpness of i/i25th in. or i/5ooth in., half or double the above number must be taken, i.e. F/ 10 or E/40 respectively in the above case. 80. Factors Affecting Depth of Field. The depth of field (distance between near and far planes in focus) can be expressed by one or other of the following formulae, according as the permissible circle of confusion is a constant fraction a of the ultra-nodal distance of the image, or a fixed amount e. The formulae give the difference (s - r) of the distances previously calculated — 2 naq 2 F t x 2 neq{q - E)E 2 W E 2 - n*a*q* W E 4 - n 2 e 2 (q - Ff in each of which the second term of the denom- inator is generally negligible unless q is very great, so that we can replace these by the simpler formulae below, which lead to a slightly smaller value — (i*) ^ (**) In this form it is seen at once that, all condi- tions remaining the same, with the exception of the one factor considered 1 — 1 The application of the laws of physical optics leads to the result that relative depth of field is independent of the focal length F, and is inversely proportional to the square of the diameter of the effective aperture (T. Smith, 1928). (1) Depth of field is less with a lens of greater focal length ; it is inversely proportional to F if the tolerance is defined as an angular constant, and inversely to the square of F (i.e. F X F) when the tolerance is fixed by an absolute value. (2) Depth of field is proportional to n and is thus greater the smaller the stop. (3) Depth of field is greater for greater object distances, being proportional to the square of q (i.e. q X q). (4) Depth of field, based on the value of a or e, is greater the less exacting the require- ments of definition are. 81. It is interesting to know, at least from a practical point of view, if, having to photograph an object of a certain depth, from a given point of view, a lens of focus to give a required scale of image can be used, or whether it is preferable to use a lens of very short focus, giving a small image which is afterwards enlarged. The method of calculation used in § 77 shows clearly that (neglecting loss of sharpness on enlargement) the small lens is decidedly more advantageous, for, at an equal relative aperture, it will give, after enlargement, an image having much greater depth of field than the image obtained directly on the same scale. We have, in fact, seen that i/r -i/q = i/q- i/s = na/F It is seen that, when photographing (the lenses having the same E/No.), an object from the same viewpoint (r and 5 constant), the angu- lar tolerance a varies proportionally to E instead of being constant. But it would require to be constant to obtain the same definition (in abso- lute value) after magnification of the small image (in the ratio of the ultra-nodal distances of the image planes) to bring it to the same dimensions as the large image photographed directly. Taking the same numerical data as in § 77, but supposing this time that the focal length is 13-2 in. (0*33 m.), or a power of 3*03 diopters, we will find the depth of field for the same circle of confusion after equalization of the sizes of the two images. The distance of the plane focussed on being the same in the two cases, the sizes of the two images are proportional to the respective ultra-nodal distances, and it will be sufficient to give a the same value, 1/2, oooth. The total tolerance of convergence is thus found to be (3*03 X 6)/i,ooo = 0-0182 diopters, and consequently the convergences of the limits of 52 PHOTOGRAPHY: THEORY AND PRACTICE the field are (o-2 ± 0-0091) diopters, correspond- ing with ultra-nodal distances of 1/0-2091 = 478 m. (188 in.) and 1/0-1909 = 5-24 m. (206 in.) with a total depth of field of 18 in. only, instead of 48 in. in the case of the lens of 4-4 in. focal length. It remains to examine whether, to obtain a reproduction at the same size by direct photo- graphy, supposing that aesthetic considerations allow an alteration of viewpoint, there is any advantage as regards depth of field, in using a lens of shorter focus, or if, on the contrary, it is preferable to get farther back from the view and use a long-focus lens. Contrary to the general opinion, it is still the shorter focus lens which possesses the greater depth of field. 1 82. Hyperfocal Distances. Particularly inter- esting problems in the application of the depth of focus formulae are the determination of the distance at which the lens must be focussed in order that the far plane in focus may be at in- finity, and the finding of the distance of the near plane in these circumstances. The distance of the plane focussed on which satisfies this condition for a lens of given focal length and aperture, is usually called the hyper- focal distance for that aperture. Let it be said at once that the hyperfocal distance will have different values according to the degree of unsharpness which can be tolerated. 2 If we agree to adopt as the tolerance a con- stant fraction a of the ultra-nodal distance of 1 From the formulae of § 77, remembering that q = (m 4- 1 )F, it is possible to deduce the expression for the total depth of field (s — r) _ 2 a F (m + 1 ) 2 n r 1 - (m 4- i) 2 a 2 n 2 The scale of reduction m being constant, the camera extensions are proportional to the focal lengths, and in order to have the same limits for the diameter of the circle of confusion in the photographs taken with lenses of different focal lengths, the product aF must be constant = k (say), whence a = h/F t and the above expression reduces to s _ _ 2 k (m + 1) 2 n r 1 - (m 4 - 1) 2 n 2 (h/F) 2 The F/ No. n being supposed constant, it is seen that, if F increases the denominator increases and (s - r) decreases. 2 In addition to these two factors (aperture and tolerance) affecting the hyperfocal distance of a given lens, an additional source of uncertainty arises from the fact that, although the greater number of writers have adopted the above definition, some (e.g. Moessard) have defined it as the distance of the near plane when the far plane is at infinity, i.e. half what we call the hyperfocal distance. the image (still supposing that the pupils coin- cide with the nodal points), the hyperfocal dis- tance H of a lens of focal length F and aperture number n , is easily calculated if we consider that the convergence of the far plane is zero, this being at infinity. Calling l the distance to the near plane, the preceding formulae for the depth of field become 1 1 /I - 1 /H = i/H = na/F whence H = F/na and l = Fl2na = H/ 2 Let us calculate, for example, on what dis- tance a lens of 4-4 in. focal length at F/6 ought to be focussed in order to give sharp focus right up to the horizon, the image of a point being considered sharp when its diameter does not exceed 1/1, oooth of its distance from the lens. All we need do is to divide the focal length 4-4 in. by the F/ No. 6 and multiply by 1,000, which gives 4-4 X 1,000 = 733 in. (61 ft.) The camera, once set for this distance, would give, with the same criterion of definition, a sharp image of all objects from 30^ ft. It may be remarked that when the limit of sharpness is defined by an angular value the hyperfocal distance is proportional to the focal length and inversely proportional to the F /No. n. It is also greater the more severe the standard of good definition is. This is easily explained by noticing that the hyperfocal distance thus defined is the distance from which the effective aperture subtends the angle of tolerance. If, for example, the angle is 1/2, oooth (circle of confusion = 1/2, oooth of the principal distance), the hyperfocal distance is 2,000 times the useful aperture of the stop. Similarly, if the angle were 1/1, 500th or 1/1, oooth, the hyperfocal distance would be 1,500 or 1,000 times the useful aperture. 83. If we agree to adopt, as limit of definition, a diameter e for the circle of confusion, we calcu- late H by the condition that 5 is infinitely great, which requires the denominator of the fraction for s (§ 78) to be zero, so F = nme, whence m = — ne 1 This value could also be found by finding the value of q which in expression (1), § 80, makes the denom- inator zero, and thus gives an infinite value to the depth of field. DIAPHRAGMS AND RELATIVE APERTURE 53 As the plane focussed on, reduced in the ratio i /m, is at an ultra-nodal distance (w + i)F, we get 1 H = F(F/ne + i), and for the correspond- ing length /, by replacing m by the value above in the expression for r, l = H/ 2 Let us calculate, for example, the hyperfocal distance in the same case as above, but with a circle of confusion of i/25oth in. All measurements being expressed in inches (any other unit could be adopted as long as it was used throughout the calculation) the recip- rocal of the tolerance is 250. The focal length, 4-4 in., is multiplied by 250 and divided by the F/ No. 6, giving 1,100/6 = 183-5. Increase this by 1 and multiply by the focal length, and we get for the hyperfocal distance 184-5 X 4-4 = 812 in. = 6 ft. The camera being focussed on this distance, all points at least half this distance from the camera, viz., 33 ft. 9 in., will give sharp images, which is usually expressed by saying that the camera thus set gives sharp images from 33 ft. 9 in. to infinity. It may be remarked that, when the limit of sharpness is thus defined as an absolute value, the hyperfocal distance is, for equal relative apertures, proportional to the focal length multiplied by itself, 2 i.e. for lenses of focal length half, double or triple that of the lens in the example, the hyperfocal distances would be respectively one-quarter, four times, and nine times, the value calculated above. 1 We should get the same value by putting the denominator in expression (2), § So, equal to zero, and get q ~ F = F 2 /ne, whence q = F 2 /ne 4- F 2 This would only be exactly true if, instead of expressing the hyperfocal distance in the form of an ultra-nodal distance, it was expressed as an ultra-focal distance, H' = F 2 /na. If the diameter of the aperture is reduced to half or one-quarter its value, the hyperfocal distance will be reduced to half or quarter the value calculated above. Depth of held (circle of confusion = 1/1,000 the camera extension). The unit of length is the diameter of the effective aperture Fig. 60. Chart of Depth of Field (Scheffer) If, instead of fixing the limit at i/25oth in., it was taken as i/i25th tr i/5ooth in., the hyper- focal distance would be half or twice respectively the former value. In every case the distance of the near plane is half the hyperfocal distance. 84. The formula for hyperfocal distance can be obtained directly without using the formulae for depth of field. 54 PHOTOGRAPHY: THEORY AND PRACTICE Let P be the entrance pupil of the lens (Fig. 59). Every parallel pencil of light (i.e. coming from an infinitely distant point) will cut planes perpendicular to the axis (and Q in particular) in a circle of the same diameter as that of the pupil. The beam, limited by the pupil and having as apex any point R half way between Q and the pupil, cuts Q in a circle of the same size (which will coincide with the former if R is on the principal ray of the beam). If we focus the lens on Q , the condition that all point- objects from R to infinity shall be sharp is that the images of the circles on Q reduced i/mth, must not exceed the limit e assigned to the diameter of the circle of confusion. Now, the diameter of the pupil is F/n ; the condition is therefore F/n X m . = e, and gives m = F/n X e, whence, the distance H of the plane reduced i/mth being (m + 1 )F, H = (. F/ne + 1 )F 85. Tables and Abaci for Finding Depths of Field and Hyperfocal Distances. To avoid cal- culations of depth of field and hyperfocal dis- tances a number of numerical tables, abaci, and calculating rules or discs have been made, the forms of which are infinite. As examples we reproduce two abaci corre- sponding with the two methods of expressing tolerance in definition. In Fig. 60 (W. Scheffer, 1909), constructed for a circle of confusion equal to 1/1, oooth the camera extension, the unit of length adopted is the diameter of the effective aperture of the lens, say F/n. Thus, for a lens of 6 in. focal length at Fj 5 the diameter is 1-2 in. The lens being focussed at 180 in., i.e. 150 times the effective diameter, the vertical line through the mark 150 on the horizontal line is followed till the curves are reached corresponding to the depth of field in front (on the lower curve) and behind (on the upper curve) ; in this case giving the'divisions 20 and 30, i.e. 20 X 1-2 = 24 in., and 30 X i*2 = 36 in. Thus the distances of the near and far planes are 180 - 24 = 156 in., and 180 + 36 = 216 in. This graph shows plainly that, with the con- vention adopted, the depth of field depends only on the diameter of the useful aperture. It also shows that the depth of field in front of the plane focussed on is always less than the depth behind. Graphs made on the assumption of a fixed diameter for the circle of confusion (usually i/25oth in.) are necessarily much more compli- cated. The abacus given below (L. P. Clerc, 1923) is used with the table reproduced here. In the upper part of the table, follow the line corresponding to the degree of sharpness re- quired (from i/ioth to i/4oth mm., i/25oth to 1/1,000 in.) until the column giving the focal length is reached. Follow this downwards until the horizontal line indicating the aperture is reached, and note the letter. On the abacus (Fig. 61) follow the oblique line (ascending from left to right) designated by the letter obtained from the table, until its intersection with the oblique line (descending from left to right) corresponding with the dis- tance on which the lens is focussed is reached. Where they cross, follow the horizontal and vertical lines which indicate on their scales the DIAPHRAGMS AND RELATIVE APERTURE 55 distance of near and far planes in focus, respec- tively. Values intermediate between those given in the table or on the abacus can be estimated (between the nearest values marked). If using, for example, a lens of 135 mm. (5-4 in.) focal length, we wish to know the limits of the field when focussing on a plane at 5 m. (197 in.) taking o-i mm. (i/25othin.) as stan- dard of definition of a point, the aperture being F/16, follow the line F (indicated on the table for these conditions) to the oblique line corre- sponding with 5 m. ; from their intersection follow the vertical and horizontal lines, esti- mating their intermediate values ; the readings F/ 16, the smaller of the two ensuring fulfilment of the conditions. Finally, to determine the hyperfocal distance with the same standard of definition (1/10 mm. or i/25oth in.) for a lens of focal length 135 mm. (5-4 in.), stopped down to F/11 (letter G), follow the inclined line from right to left to the inter- section of the line G with the lower scale of distances. This inclined line meets the scale of distances for the focussed plane at the division 16 metres, which is the required hyperfocal distance. 86. Influence of the Corrections of the Lens on the Depth of Field and Hyperfocal Distance. It cannot be too strongly emphasized that the Distance focussed on are 3*45 m - U36 in.) on the lower scale and 9-2 m. (363 in.) on the vertical scale on the right. The depth of field is thus 9-20 - 3*45 m. = 5-75 m. (226 in.), i.e. 1-55 m. (61 in.) in front and 4-20 m. (165 in.) behind the plane focussed on. To know at what distance to focus and at what aperture to set the diaphragm to obtain images with a standard of definition of 1/10 mm. (i/25oth in.), between 2*50 m. (99 in.) and 6 m. (237 in.), look for the intersection of the vertical 2-5 and the horizontal 6, which is midway between 3-40 and 3-60, so that the focus should be set for 3-50 m. (138 in.). The point of inter- section occurs between the obliques E and F, so that the lens of 135 mm. (5-4 in.) focal length should be stopped down between F/ 22-5 and depth of field and hyperfocal distances, calcu- lated from the formulae given in the foregoing paragraphs or read on tables and abaci from them, useful as they are to guide the photo- grapher, by indicating the kind of variation which will arise from an alteration in working conditions, have only relative accuracy. On the one hand we have supposed, in deduc- ing these formulae, that the nodal points are coincident with the centres of the corresponding pupils, which is not always the case (the separa- tion may be considerable for telephoto and similar lenses). On the other hand, we have implied that the lens is an ideal (never realized in practice) having a flat field free from all aberrations. Finally it is supposed that the circles of confusion are uniformly illuminated, 56 PHOTOGRAPHY: THEORY AND PRACTICE while, in fact, quite a large patch may give a sharp image if the light is concentrated in a small area. It has emphatically been stated that depth of field is necessarily the same for different lenses of the same focal length and aperture, a con- clusion to which an examination of the formulae does, in fact, lead. These two factors are cer- tainly those of the most importance, but account must also be taken of the fact that, in practice, a lens perfectly corrected has nearly always much less depth of field than a lens having residual aberrations, chromatic (D. Brewster, 1867), or spherical aberration, these lenses giving more homogeneous images better suited for pictorial photography. 1 Finally, a lens having considerable curvature of field would appear to possess more depth than is given by the classic formulae when the near planes come close to the edge of the field of the lens. 87. Fixed-focus Cameras. In many very cheap cameras for beginners, no adjustment for focus is provided. The camera is adjusted by the maker to focus on the hyperfocal distance for the largest aperture of the lens, the limit of definition being fixed at i/i25th in. The lenses used on these cameras being nearly always simple or achromatic, very poorly corrected, with maximum aperture rarely exceeding F/16, the hyperfocal distance is very short, and conse- quently the minimum distance of sharp objects (half the hyperfocal distance) is small enough to allow almost all subjects (except portraiture, properly so-called) to be attempted. 1 Compare, for example, two lenses of the same focal length and aperture, one, A , free from spherical aberra- tion, the other, B, having positive spherical aberration at apertures greater than F/16. The aberration of B being positive, the caustics of each beam will be formed between the sharpest image and the lens, approaching the latter as the aperture increases, and thus affecting only the lower limit to the depth of field. At F/16 the lenses give equal depth ; at an aperture slightly larger the caustics due to spherical aberration make their appearance with B. A beam limited by a caustic would be restricted more rapidly than one limited by a cone, so that the depth of field would diminish less quickly with the uncorrected lens. If objects situated in front of or behind the focussed plane are considered, this difference between the two lenses increases as the aperture is increased. At a certain aperture the caustic, meeting the sensitive surface, gives practically a con- stant circle of confusion, and thus sets an almost invariable lower limit to the depth of field, whilst with the perfect lens every increase of aperture increases the diameter of the circles of confusion, and thus limits the depth of field. A negative aberration would affect the posterior limit in the same way (C. Welborne Piper, 1903)- For example, with a lens of 2-4 in. focal length at F/16, the hyperfocal distance, corresponding to a sharpness of i/i25thin., is (from the for- mula of § 83) 47 in., and all objects will be sharp from 24 in. from the camera, which will amply suffice for all needs. Similarly, for a 6 in. lens the hyperfocal distance is 283 in., so that every- thing will be sharp from 142 in. This would rule out many classes of work, but the depth of field could be increased by stopping down still further. 1 This interesting property of lenses of short focus and small aperture has frequently given rise to the unfortunate expression fixed-focus lens , which has led to much misunderstanding. It must therefore be insisted that this action is not the result of any special form of lens, but of the general application of the laws governing depth of field. 88. Focussing Scales. Portable cameras for use generally in the hand are not suitable for focussing the image on a focussing screen, and are therefore provided with a scale graduated in object distances so that the focus can be set for any distance within the limits of the scale. The mark on this scale corresponding with objects at a great distance (generally indicated by 00, the algebraic symbol for an infinitely great number) in some cases 2 indicates the focal plane for objects on the horizon, and, in others, the focus for objects at the hyperfocal distance of the lens at the maximum aperture (usually adopting i/25oth in. as the circle of confusion). It must be remembered that when the focus is set for infinity the nearest plane that is sharp is at the hyperfocal distance, while when focus is set for the hyperfocal distance the nearest plane that is sharp is at half this distance The distances on the scale are often chosen quite arbitrarily. It would be better to divide the hyperfocal distance (allowing 1/2, oooth the camera extension as the circle of confusion) by 1 Closing the diaphragm on a fixed focus camera does not bring into full use the depth of field corresponding with the aperture, since focussing which ought to be made for the corresponding hyperfocal distances is fixed for the hyperfocal distance of the maximum aperture. It would, however, be possible, without too much complication, to couple the diaphragm with the focussing movement automatically. To do this (G. Cromer, 1912) the stops are made in a sliding plate which increases in thickness by the required difference in focus for each aperture. The plate, acting as a cam, displaces the lens, which is pressed against it by a spring. 2 Some makers indicate the focus both for infinity and the hyperfocal distance, the latter being indicated by a mark of different colour or the letter H. DIAPHRAGMS AND RELATIVE APERTURE the consecutive numbers o, i, 2, 3, . . . , i.e. infinity, the hyperfocal distance, half, one-third, one-quarter, . . . the hyperfocal distance. Such a scale possesses an interesting property when used with the appropriate diaphragm and when the tolerance of sharpness used in its construc- tion is accepted. When the focus is adjusted for one of the distances on the scale, the depth of field extends to the contiguous distances. If, for example, the lens is of 5 in. focal length and E/8 aperture (hyperfocal distance 788 in. (Piper) for a circle of confusion of i/25oth in.), and the * scale is marked 00, 66', 33', 22', 17', 13', n', 9' 5", 8' 3", 7 ' 4", 6' 7", 6', 5 ' 6", 5' 1", 4' 8', 4' 4", 4', etc., when set for 13' all objects between 11' and 17' will be sharp. A still better arrangement, giving the limits of the field for all apertures, whatever the dis- tances scaled, has been suggested by C. Wel- borne Piper and used by some makers. Fig. 62 shows its application to a stereoscopic camera where the two lenses are adjusted simultaneously by means of a connecting rod carrying the scale of distances and moving over a scale of apertures marked off on each side of the zero indicating the sharp focus. Two marks are used for any aperture number, and indicate the limits of the field on the focussing scale. For example, in the position illustrated the lens focussed at 4 m. (158 in.) gives at E/14 a sharp image from 2-50 m. (100 in.) to 10 m. (400 in.). If it were required to photograph a subject extending from 3 to 5 m. (118 to 197 in.) from the lens, the zero of the aperture scale will be placed midway between 3 and 5 on the scale of dis- tances, and the best general focus is assured. The 3 and 5 marks then come within the marks for E/7, so that there will be sufficient depth at that aperture. With slight variations in construction, this arrangement is applicable to all focussing scales, 57 particularly when they are magnified relatively to the actual displacement of the lens. 89. Depth of Focus. 1 If the plane on which the image is formed is slightly separated from the position of the sharp image of a point, the image will be a disc which can be considered as a geometrical point if its diameter does not exceed a certain fraction a of the ultra-nodal distance of the sharp image, or does not exceed a constant limit e (§ 76). Knowing (Fig. 63) the diameter of the aper- ture Fin and the ultra-nodal distance q' of the focussed image (supposing again that the pupils coincide with the nodal planes) it is easy to calculate the distance v, the error in the position of the plate which can be tolerated. Calling the diameter of the disc of confusion at Q 2 (Q'), consideration of the similar triangles having their apices at Q' gives F/n q' Now, according to the convention adopted in fixing the tolerance in sharpness, we have ((?') = or ( the difference between the two focal lengths. If the image given by the complete system is magnified m times (ratio of lengths) relatively to that which the convergent system alone would have given, the distance E of the divergent lens from the magnified image is E = f 2 (m - 1) If, in the case of a telephoto, where fjf 2 = s, the distances 0 and I of the object and image respectively from the telephoto and the distances 0 and i for an ordinary lens of the same focal length giving an image of the same size, are calculated, it is found that 0 = o + [F(s - 1) + /J 1 = i- [F(i-i/s)+/J In one or other of these forms the telephoto requires careful handling on account of the very faint image, making focussing difficult when large magnifications are attempted. The use of an optical system with variable separation requires in fact that each of the components should be separately corrected, which is only possible with small apertures. This type of lens must thus be considered as suitable only for special purposes. There is no doubt, how- ever, that it can render very appreciable service. The use, in place of it, of enlargement of part of a negative taken with an ordinary lens is not to be thought of, for at the same time the granular structure of the image is enlarged, which limits the degree of enlargement to not more than ten times as a rule. 1 no. Fixed-focus Telephotos. Very great Fig. 84. Variable Power Telephoto Fig. 85. Fixed-focus Telephoto, F/4 8 (Lee) Fig. 8b. Puyo and de Pulligny’s Anachro- matic Telephoto The ratio s being always greater than (or at least equal to) unity, it is seen that, for a photo- graph on the same scale, the distance of the object from the telephoto is always greater than it would be with an ordinary lens of the same focal length, and the more so the smaller the focal length of the amplifying negative lens compared with that of the convergent system. Conversely, the distance from the image to the telephoto is always less than it would be with a lens of the usual construction. The first telephoto lenses used any ordinary photographic lens as the convergent system, and as amplifier a more or less complex divergent system. The two components were mounted so that the separation could be altered by a rack and pinion, the optical interval d of the above formulae being marked on a scale on the outer tube ; also, usually, the resulting magnification and the corresponding distance of the image. The first simplification took place in 1896 by the construction of a complete system shown in Fig. 84, the divergent system being shown in full lines in the position for a focal length of 4 in., whilst in the position shown by the dotted lines the focal length is about 12 in. improvement, at least in the construction of tele- photo lenses for general use, was made when K. Martin (1905) abandoned the variable mag- nification and made a lens which, in view of its employment on hand cameras, would be more correctly described as a long-focus lens for short-extension cameras. The possibility of compensating the aberrations of each of the components by aberrations of opposite sign in the other, and the removal of the difficulty of centring which arises when two sliding tubes are employed (which must necessarily have a little play) enabled him to obtain an aperture of jF/ 9 covering about 35 0 and giving an image of a quality comparable with that of a good rectilinear. The lens had the advantage that 1 It may be mentioned that convergent lenses have also been used as magnifiers, the arrangement being then similar to an astronomical telescope with increased separation between the components so as to obtain a real image. Also afocal magnifiers have been used such as a Galilean or terrestrial telescope (e.g. a field glass) adjusted for normal sight and placed in front of the lens, which thus takes the place of the eye of an observer. The magnification is then equal to the ratio of the focal lengths of the two components of the amplifier. The A dons of Dallmeyer (1899) are afocal magnifiers con- structed on this principle. 70 PHOTOGRAPHY: THEORY AND PRACTICE for a focal length of 9 \ in. (covering almost the whole of a 4I X 3J in. plate), the distance from the vertex of the back lens to the plate (prac- tically the same as the distance from the camera front to the plate) was only in., that is, only 58 per cent of the focal length. The aperture of these lenses was extended afterwards to F/y or Fjy^y (by the same designer), with a slightly smaller field, the focal length for the 4! X 3J in. plate being increased to 10 \ in., without increase of camera extension. This very important reaction against the use, on amateur cameras, of lenses of small focal length which, though useful for record work, are opposed to all aesthetic rules, has been supported by the greater number of English and German opticians, who have all kept to the same ratio, which is a very suitable one, between the focal length and the camera “ extension/' while striving to increase the aperture and the quality of the image. In England, where this type of lens seems to have reached very great perfection, the aperture has been increased, first to F/ 6-8 and then to E/3* 5, with a field of almost 35 0 . Fig. 85 shows a lens of F/ 4-8 aperture by H. W. Lee (1922). These different lenses have been made up to now only for use on hand cameras of size not exceeding 7 x 5 in. The most recent types give an image comparable with that given by the best modern lenses of normal construction. 1 hi. Anachromatic Telephotos. The use of a telephoto lens of low but variable magnification is of importance to the landscape photographer, who, having chosen his viewpoint, must make his image cover his plate. On the other hand, the distribution of sharpness between the suc- cessive planes, without exaggerated softness in the distance, leads him to the use of a smaller relative aperture as the focal length is increased. In fact, it may be said (C. Puyo) that the abso- lute diameter of the greatest effective aperture of service in artistic landscape work should be in the neighbourhood of o-8 in. For 18 X 24 cm. (9J X 7 in.) plates, the focal lengths mostly used lie between 16 in. and 25 in. The anachromatic telephoto for which C. Puyo and L. de Pulligny gave data in 1906, which they called adjustable landscape (Fig. 86), is very suitable for this purpose. It is constructed of two lenses of 1 Lenses of this type, but reversed, have been used in cameras for* simultaneous three-colour selection, the increase in camera extension then providing the neces- sary space for the system of reflectors (beam splitters) used to divide the beams of light among the three images. o-8 to i-2 in. diameter, one convergent and the other divergent, of the same focal length, e.g. about 4 in. for 9 1 X 7 in. plates, or 3 in. for 7 X 5 in. plates. These are mounted so that the separation can be varied without exceeding a quarter of their common focal length. A diaphragm of o-8 in. aperture (a little less for lenses of very short focal length) is placed in front of the convergent lens, and a variable diaphragm can usefully be placed behind the negative amplifying lens for varying the degree of softness. Such a combination, used over a small field, can be sufficiently corrected for astigmatism, and have an almost flat field without too much spherical aberration. The telephoto is of importance not solely for record and pictorial landscape photography, but is also extremely useful for portraiture, giving large heads with the camera placed at a very great distance from the sitter, thus avoiding the distortion, amounting almost to caricature, which results from too close a viewpoint. For the professional having at his disposal a triple-body camera, the telephoto lens of vari- able separation can be very simply rigged up without special mounting. The convergent sys- tem (which may be a Petzval or other type of portrait lens, or anachromatic symmetrical lens, according as a semi- or completely anachromatic lens is desired; the first-named gives a better defined image, more agreeable to the taste of the average customer) is mounted on the front body ; the divergent lens will be attached to the middle body, so that its centre is on the optical axis of the convergent lens. In this case a plano-concave lens should be chosen, placed with its concave surface towards the front lens, the focal length of which should be almost half that of the convergent system and the diameter about one-third its own focal length. With a portrait lens of 18 in. focal length at 33 to 50 ft. from the sitter, portraits of 9J X y\ in. size are obtained of very agreeable perspective, where the arrangement of the hands is easy and good instead of having rectilinear move- ment, are guided by horizontal grooves a 1 b 1 and a 2 b 2 , and joined by a long member to the end of the arm B X B 2> movable around the pivot arranged radially, thus giving the advantage of perfect symmetry in all directions. 1 The following description and the figures relating to it are taken from a general survey of shutters pub- lished in 1906 and 1907 by E. Wallon in La Revue de Photographic. g6 PHOTOGRAPHY: THEORY AND PRACTICE M and brought into its position of rest by the in this direction it engages the arm and spring r. The driving spring R is coiled round consequently moves the plates of the shutter, the piston rod in the body of the pump P. This the aperture of which is fully opened when the pump is full of air, which the piston, under cam has turned through about 90°, the pin G pressure of the spring R, compresses to the end being pressed against the cam. Before this pin of the cylinder, where it escapes through very is set free, thus allowing the shutter to close, the whole of the sector 5 shown by the arc xy must pass under it. The time this takes to pass is the time of full aperture, after which the plates close under the influence of the spring r. The air brake does not act appreciably during the actual opening or closing of the shutter, the air not being sufficiently compressed to have any effect on the driving spring R. In fact, the action of the brake only affects the duration of the exposure at full aperture. For very long “ time ” exposures, the holes in p through which the air escapes are almost completely closed. The move- ment of the cam is then very slow Fig. 109. Double-plate Shutter — Set when the pin G approaches the end X of the sector S, and if there is no longer small holes, the exact size of which may be any pressure on the rod K, the pin c is stopped regulated at will by rotation of the button p, by the notch /before the plates close; a second which has marked on it a series of numbers pressure on K releases c and the shutter corresponding to different exposures. closes. An extension of the piston rod carries a rack The times of opening and shutting are about which engages in the toothed wheel Q, which is 1 /400th second ; the shortest time at full aperture directly connected with the setting lever, and which is fixed to the cam 5 which forms the vital part in the mechanism. When the key for setting the shutter is pressed, the rack moves towards the right, thus compressing the driving spring and turning the sector 5 in the opposite way to the hands of a clock. This presses by the tip * on the pin G of the lever L, which is joined at M on the arm B 1 B 2 , and which is slightly displaced from its position of rest sufficiently for the sector to pass by, the lever returning quickly to its normal position, the pin G being thenceforth pressed against the sector. The triangular pin C of the cam is then caught in the notch 6 of the bolt Fig. ito. Double-plate Shutter — Open D, which pivots round N and is acted on by the spring t, the shutter thus being set with- is also about the same, so that the shortest total out any movement of the plates V. exposure is about i/i2oth second, with an If the bolt D is lowered by pressing on the rod efficiency greater than 60 per cent. This rises K, the pin c is set free and on being released the to about 80 per cent for an exposure of i/5oth spring impresses on the cam a rotation in the second, and increases continually with the opposite direction to that previously described, exposure, as is the case with the majority of The lever L is pushed towards the right, but shutters other than focal-plane. SHUTTERS 97 138. Diaphragm Shutters. The first shutter with several pivoted plates opening like the leaves of an iris diaphragm and operated simul- taneously by an internal ring concentric with the diaphragm appears to have been made in 1887 by Beauchamp and Dallmeyer. Some of these shutters, with a large number of plates (e.g. 10 in th eVolnte and the X -Excello), are actually iris diaphragms, the leaves opening just sufficiently to form the boundary of the desired aperture as indicated on a scale, during a given time, as shown by a second scale. The efficiency of these shutters is not particularly good, viz. a maximum of 50 per cent for a numbers of laminae, the diameter of the aper- ture being taken as unity. Figs, hi to 116 show the theoretical forms of leaves corresponding with shutters fitted with 2, 3, or 4 leaves, according as to whether these pivot around points (marked by small circles) situated on the edge of the aperture or on the edge of the casing. This advantage of shutters having several leaves is, however, to a great extent counter- balanced by the fact that, for a given method of construction and the same law of movement, the efficiency diminishes as the number of leaves increases. 1 The table given below gives for Fig. hi Fig. 113 Fig. 115 Fig. 112 Fig. 114 Fig. 116 Theoretical Forms of Two-, Three-, and Four-leaf Diaphragm Shutters uniform movement of the leaves, but this can be slightly increased by choosing a suitable movement. Also, the construction of these shutters is somewhat complicated, which usually results in a higher price and a more fragile construction than in shutters having an iris which is quite separate from the shutter leaves. Generally the number of leaves is three or four, but sometimes may be two or five. For a suitable form of leaf the external diameter of the casing of the shutter should, for a given aperture, be smaller as the number of leaves is made greater (Lan Davis, 1911), although manufacturers have not always made the best use of this fact. The table below gives the theoretical diameter of the casing for different Numberof leaves 2 or 3 4 5 6 8 10 20 30 Diameter of casing . 2 i -93 | 1-83 1-73 i -59 1-49 1*26 1*17 different numbers of leaves (J. Demar^ay, 1905) the efficiencies calculated for the case in which, instead of turning round pivots, each one slides in a direction parallel to a radius, the assumption being made, in agreement with general practice for the shortest exposures, that no brake is used. Number of leaves 2 3 4 6 cc Efficiency .... 0-424 0-367 0-351 0-341 0-333 Almost all shutters of this type are fitted with air brakes for regulating the exposure, the brake only being used during the period of full aperture 1 This law would not hold for leaves in the form of sectors dividing the diaphragm into equal parts and opening by a translatory movement of each of the sectors in a direction along the length of its centre line, or by rotation around a pivot at some distance away. Such arrangements, however, present consider- able difficulties in practice. 7 — (T .*>630)