UC-NRLF 
 
LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORN 
 
 GrlK'T OK 
 
 f v ' n . v * <2 
 
 Received ^}j2^r.* .....> f 9 
 Accession No. Q / Q / <j . Class No. 
 


 IN 
 
 TOPOGRAPHER and ASSISTANT, 
 Showing mode of carrying instruments. 
 
PHOTOGRAPHIC SURVEYING 
 
 INCLUDING THK ELEMENTS OF 
 
 DESCRIPTIVE GEOMETRY 
 
 AND 
 
 . 
 
 PERSPECTIVE 
 
 E. DEVILLE 
 
 Surveyor General of Dominion Lands 
 
 OTTAWA 
 GOVERNMENT PRINTING BUREAU 
 
 1895 
 
PREFACE 
 
 The first edition of this book was issued in 1889. It was prepared 
 solely for the use of the surveyors employed by the Department of the 
 Interior on photographic surveys ; the edition of fifty copies was litho- 
 graphed in the surveys office. Some copies found their way outside, 
 and, owing to the favourable comments which they received, the whole 
 edition was exhausted in less than a year. Photographic surveys have 
 since been undertaken for many purposes which were not contemplated 
 at their inception, and a new edition of the book has become necessary 
 for the guidance of the surveyors engaged in the work. 
 
 The photographic method is knownj^many names : photogrammetry, 
 metrophotography, topophotographyJ^Bbnometry, etc. 
 
 It seems that iconometry is the most appropriate name, because 
 it expresses the principle of the method, which is to measure by means 
 of perspectives. Whether the perspectives are photographs, or whether 
 they are produced otherwise, is immaterial so far as the method is con- 
 cerned. Neither this term nor -any other has yet been generally 
 adopted. 
 
 The conception of this method is due to Laussedat : his first 
 experiments were made in 1849, the perspectives being drawn with a 
 camera lucida. His paper on the subject, written in 1850, was pub- 
 lished in 1854. Shortly after, he substituted photography for the 
 camera lucida. He gave a full exposition of the method in various 
 papers, and his work was so complete that little has been added to it 
 since. Wherever photographic surveys are now made, they are exe- 
 cuted by the application of the principles laid down by Laussedat. 
 
 In Germany, Meydenbauer was the first to give his attention to the 
 new method. His investigations, commenced in 1865 or thereabouts, 
 were continued by Jordan, Hauck, Koppe and many others. 
 
 In Italy, the celebrated engineer Porro, who was acquainted with 
 Laussedat and knew his work, proposed for surveying purposes, a 
 
iv PHOTOGRAPHIC SURVEYING, 
 
 camera built to receive spherical plates. The idea was not practical and 
 was never carried out. In 1875 and 1876, some experiments were made 
 by Lieutenant Manzi Michele, but they were not favourably received. 
 To Major General Annibale Ferrero, Director of the Geographical Mili- 
 tary Institute, is due the credit of initiating the ordnance photographic 
 surveys of the present day. Their execution was entrusted to engineer 
 L. Pio Paganini with a staff of able assistants. The work of the insti- 
 tute is very remarkable and deserves careful study. 
 
 In Austria, the first important surveys were made by Mr, Vincenz 
 Pollack, chief engineer of the state railways. Some beautiful moun- 
 tain maps have lately been printed by Giesecke and Devrient, of Leip- 
 zig, for the German and Austrian Alpine club. They are the work of 
 S. Simon. He made use of the original ordnance survey, filling in the 
 topographical details from photographs. These maps represent one of 
 the most successful applications of photographic surveying. Schiff- 
 ner's and Steiner's writings on the subject are among the best, and the 
 Austrian instrument makers, si^ch as R. Lechiier or Starke and Kam- 
 merer, have surveying cameras of many patterns to select from. 
 
 It is in Canada that the method has received its most extensive 
 application ; it was first employed when the surveys of Dominion 
 lands were extended to the Rocky Mountains. In the prairies, our 
 operations are limited to defining the boundaries of townships and 
 sections ; these lines form a network over the land by means of which 
 the topographical features, always scarce in the prairies, are sufficiently 
 well located for general purposes. 
 
 In passing to the mountains, the conditions are entirely different. 
 The topographical features are well marked and numerous ; the survey 
 of the section lines is always difficult, often impossible, and in most 
 cases useless. The proper administration of the country required a 
 tolerably accurate map, and means had to be found to execute it rapidly 
 and at a moderate cost. The ordinary methods of topographical sur- 
 veying were too slow and expensive for the purpose ; rapid surveys, 
 based on triangulations and sketches, were tried and proved in- 
 effectual ; then photography was resorted to. 
 
 Up to 1892, the photographic surveys were confined to the Rocky 
 Mountains, in the vicinity of the Canadian Pacific Railway ; at the 
 
PREFACE. r 
 
 end of that year, they covered about 2,000 square miles. In the same 
 year, an International Boundary Commission was appointed to examine 
 the country along the boundary between Canada and the United States 
 Territory of Alaska. The Canadian Commissioner, Mr. W. F. King, 
 decided to carry out his share of the work by photography. In 1893 
 and 1894, his parties surveyed about 14,000 square miles. 
 
 Irrigation surveys were commenced last year in the south-westerly 
 part of the North-west Territories, where the rainfall is not quite 
 sufficient for agricultural purposes. In addition to the gauging of 
 streams, the establishment of bench marks, etc., it is necessary to 
 ascertain the catchment areas and to define the sites best adapted for 
 reservoirs. For this purpose photography has again been resorted to 
 in the foot hills and on the eastern slope of the mountains. It has, 
 in this case, a peculiar advantage. Whether or not a site is a favour- 
 able one for a reservoir cannot be known until the plan has been partly 
 plotted. It must be possible to bring water to the proposed place and 
 to run it off; the capacity must also be adequate. If favourable, a 
 detailed survey of the site is required. With the ordinary surveying 
 instruments, a preliminary survey has to be made ; if, after plotting 
 it, the site is found favourable, the topographer has to go over the 
 ground a second time to make a detailed survey. Or, the whole of 
 the work may be executed at once, with the contingency that the 
 detailed survey may turn out useless. With the camera, the plan may 
 be plotted so far, and so far only, as required ; the photographs which 
 furnish a general plan, can be made to give all the detail wanted with- 
 out going again into the field. Whether the site is a good one or not, 
 there is no labour wasted. 
 
 Notwithstanding the many publications on photographic surveying, 
 the great advantages assigned to it and the numerous experimental 
 surveys executed, it has not yet come into general use ; in many quarters 
 there is still an adverse feeling against it. There is such a fascinating 
 simplicity about the method that it is at first difficult to understand 
 the reasons which prevent its adoption. Can anything more convenient 
 be conceived than a method which enables a topographer to gather 
 rapidly on the ground the material for his maps and to construct them 
 afterwards at leisure in his office 1 In the first edition of this 
 book, I endeavoured to explain this anomaly. The large scales of 
 
vi PHOTOGRAPHIC SUE VEYING. 
 
 European surveys were given as one of the reasons. Col. Laussedab 
 took very strong exception to this explanation, contending that photo- 
 graphy can be used to advantage whenever views can be had covering 
 a large space of ground. His contention is no doubt correct ; but the 
 advantages are not so great then as with small scales. This can be 
 illustrated by comparing a plane-table-stadia survey in a dry climate 
 with a photographic survey. When the scale is so large, and the points 
 fixed are so close together, thatthe topographer takes as much time to plot 
 one point as the rodman takes to move from this point to the next 
 one, then there is nob a very great difference in the cost of the two 
 surveys. The advantage of the photographic method is that the plot- 
 ting being done in the office, the field expenditure of the topographer 
 and the cost of his party during the construction of the map, are dis- 
 pensed with. Against this, there is the disadvantage that the plotting by 
 intersections is more laborious than the plane table plotting by direc- 
 tions and distances. It may be that the camera still has the advantage, 
 but not to such an extent as with a small scale survey. All this, of 
 course, rests on the assumption that the climate permits of outside 
 work every day with the plane table. 
 
 The experience of eight years has modified my views on the causes 
 which have prevented the adoption of the method. I am now inclined 
 to believe that they are simply to be found in the failure of those who 
 tried it. I soon discovered that the apparent simplicity of photographic 
 surveying is a delusion, and, that under no circumstances has a topo- 
 grapher to display more skill and ability than when using the camera. 
 He requires not only experience, but a combination of the faculties 
 which make an accomplished topographer. Unlike other methods, it 
 presents nothing' before his eyes to show the progress of the work or the 
 gaps that may exist in it. His undeveloped plates are his only records. 
 Every time he exposes one, he must have present in his mind what it 
 will give, what amount of information he can extract from it, what 
 constructions he will apply, what further views are necessary and 
 how they will combine. These acquirements are not the lot of 
 every topographer, and unless a man is well qualified, his attempts at 
 photographic surveying cannot be successful. All this, however, is 
 only the beginning of the surveyor's troubles ; it is in the purely 
 photographic part of the work that he finds the most frequent cause 
 of failure. The kind of views wanted, if they include distance, are the 
 
PREFACE. - vii 
 
 most difficult to obtain. They cannot be made on ordinary plates ; 
 orthochromatic plates, although suitable, present difficulties of their " 
 own. The prints made from the negatives are another source of 
 trouble ; if printed direct on thin albumen paper, they are distorted 
 in washing, and unless a cumbrous camera be employed, they are too 
 small for accurate plotting. Enlargements on bromide paper will, v 
 nine times out of ten, be of the soot and chalk kind, without any half- 
 tones. The production of these enlargements in large numbers, and of 
 uniform excellence, was almost impossible before we were taught by 
 Messrs. Hurter and Driffield the relation between photographic nega- 
 tives and their positives. 
 
 Taking all this into consideration, what chance is there that a 
 surveyor, at a first trial, and generally with an extemporized apparatus, 
 will be successful 1 A failure is almost inevitable, and it is quite 
 natural that he should discard the method as an unreliable one. 
 
 There is another difficulty which I would mention with diffidence if I 
 had not as good an authority on my side as Colonel Laussedat. He 
 was the first to elaborate this method, and still seems to be the one 
 who has the clearest conception of its essential principles. I refer to 
 the refined phototheodolites, which appear to be the favourite European 
 instruments. Their object is to secure great precision, the negatives 
 being measured, often under a microscope, and these measurements 
 submitted to calculation or to some other elaborate process. To apply 
 photography in this way is to misunderstand the function of the camera, 
 which is to replace the plane table, and to sacrifice its main advantages. 
 Any degree of precision may be attained, but it must be done, as with 
 the plane table, by the multiplication of stations and views and not 
 by the employment of logarithmic tables. 
 
 I hope to show in these pages how the difficulties assigned as causes 
 of failure can be overcome : if they are not, the method is not to be 
 blamed, but the men who apply it. Properly used, it gives results far 
 beyond what can be accomplished by any other process. 
 
 The plane table is accepted generally, although not universally, as 
 the instrument of the topographer. Let us compare the cost of plane 
 table and camera surveys, assuming that the plotting is made by in- 
 t ersections in both cases. 
 
viii PHO TOGRA PHIG 8 UR VEYINQ. 
 
 In the climate of the Rocky Mountains, where the work under my 
 direction has been executed, one half of the number of days during a 
 season is lost through smoke, fog, rain and snow storms, and our ex- 
 perience is that it takes three days in the office to plot the work 
 of one day in the field. Thus for every day's work in the field, there 
 is another day lost, on account of the weather and three days spent in 
 the office, or five days altogether. 
 
 Assuming that the plane table can be used in the field whenever the 
 weather is fair enough for the camera, which is not the case, also that 
 the topographer can plot and draw in the field as quickly as in the 
 office, where he has every convenience at hand, the same survey by 
 the plane table would require the same length of time at actual work 
 or four days. To this we must add four days lost on account of the 
 weather, or eight days altogether. 
 
 The cost of our parties in the field is $20.50 per diem : at office work 
 the only expense is the salary of the topographer, $5,00 per diem.' 
 Summing up we find the comparative cost as follows : 
 
 PLANE TABLE. 
 
 8 days in the field, at $20.50 per diem $164 
 
 CAMERA. 
 
 2 days in the field, at $20.50 per diem $41 
 
 3 " office, at 5.00 " 15 
 
 $56 
 
 This shows that the plane table survey would cost at least three 
 times as much as the camera survey. In reality the difference is greater, 
 because part of the work, as well on the ground as in the office, is 
 executed by the assistant, an arrangement which cannot very well be 
 made with the plane table. The figures above are derived from our 
 practice ; with more views or more detailed plotting, the difference in 
 cost would be still more in favour of the camera. 
 
 If we analyse the causes of the superiority of the camera, we find 
 that a very small portion of the topographer's time is spent in survey- 
 ing operations. Nearly the whole of it is devoted to travelling for 
 
PREFACE. ix 
 
 the purpose of seeing the country and he can map all that he, or rather 
 his camera, can see. His work consists of two distinct parts ; on th e 
 ground, he simply collects data, and, with the exception of a few 
 angles, does not waste any of his time in plotting or making measure- 
 ments. This is left for the office, where the only expenses are the 
 salaries of the surveyor and assistant. In the next place, the party 
 consisting of an assistant and two men is, if not smaller, at least as 
 small and inexpensive as for any other kind of survey. One man is 
 sufficient to carry our camera and tripod almost anywhere, while an 
 ordinary plane table, if it could be taken where our camera has 
 been, could not be carried there by any single man. 
 
 It is objected that plotting from photographs is more laborious than 
 plotting on the plane table. There is indeed a slight additional 
 labour ; against this may be set off the fact that no useless line is 
 ever drawn, as when, on the plane table, a point is sighted upon which 
 cannot be recognized from the next station. The greater convenience 
 of working in an office, instead of in the open air, turns the scale in 
 favour of the camera. But photography has an overwhelming advan- 
 tage in the numerous processes which the laws of perspective place 
 at the disposal of the topographer. The plane table cannot compete 
 with the perspectometer or the perspectograph. 
 
 Another objection is, that points cannot be so easily identified on 
 photographs, nor the forms of the surface so truly represented, as when 
 the topographer has the ground under his eyes. This is a mistaken 
 idea ; the're is no difficulty whatever in identifying any number of 
 points on moderately good photographs, and, moreover, the topographer 
 does not need, as with the plane table, to trust to his memory in order 
 to recognize them. The undulations of the ground are, it is true, less 
 distinct on the photographs, but this is more than compensated by the 
 advantage of having, side by side, views of the same place from several 
 stations. 
 
 The practice of photographic surveying requires a thorough knowledge 
 of descriptive geometry and perspective. These sciences are not in- 
 cluded in the programme of examination of Dominion Land Surveyors, 
 and, as it is for them that this book is prepared, I deem it proper to 
 explain the principles of both in a concise form, keeping in view the 
 special purpose of their application to Perspective Surveying. 
 
x PHOTOGRAPHIC SURVEYING. 
 
 I have reproduced, as nearly in their own words as possible, the 
 remarkable investigations of Messrs. Hurter and Driffield ; they 
 represent the most important advance in photography during recent 
 years, and may be given as a model of accurate scientific research. An 
 intimate understanding of their work is indispensable to the photo- 
 graphic surveyor. 
 
 A few explanations are given on the principles and mode of action 
 of photographic lenses ; the description of anastigmats and of their 
 peculiarities is from a lecture by their inventor, Dr. Paul Rudolf. 
 
 Considerable space is devoted to perspective instruments ; as now 
 constructed, they are almost useless for purposes of topographical 
 surveying, but, there is no reason why they should not be made 
 sufficiently precise, all that is required being more perfect workmanship. 
 For architectural surveys, they may prove most useful. 
 
 A short reference is made to secret and balloon surveying. Of course 
 the subject is of no practical interest to Canadian surveyors, but it is 
 well that those engaged on photographic surveys should have some 
 knowledge of everything pertaining to photographic surveying. 
 
 The Canadian Surveys are by far the most extensive that have ever 
 been attempted. For this reason, if for no other, it is hoped this 
 account of their mode of execution will prove acceptable to those 
 interested in the development of the science of Surveying. 
 
 E. DEVILLE. 
 
LIST OF THE MOST IMPORTANT PUBLICATIONS ON 
 
 * 
 
 PHOTOGRAPHIC SURVEYING. 
 
 1854. Laussedat. Memoire sur 1'emploi de la chambre claire dans les reconnais 
 sances topographiques. Memorial de 1'officier du genie, N 16, 1854. 
 
 1859. Laussedat. Analyse d'un memoire sur 1'emploi de la photographie dans le 
 leve des plans. Comptes rendus de 1'Academie des Sciences, 1859. 
 
 1860. Liwjier et Daussy. Rapport sur le memoire de M. Laussedat. 'Comptes 
 rendus de 1'Academie des Sciences, 1860. 
 
 1862. Pate. Application de la photographie a la topographic militaire. 
 
 1864. Lctussedat. Memoire sur 1'emploi de la photographie dans le leve des plans. 
 Memorial de 1'officier du genie. No. 17, 1864. 
 
 1865. Girard. Laussedat's Arbeiten in Bezug auf die Anwendung der Photo- 
 graphie zur Aufnahme von Planen. Photographisches Archiv.' 
 
 1866. Jouart. Application de la photographie aux leves militaires. 
 
 1867. Meydenbauer. Uber die Anwendung der Photographie zu Architektur und 
 Terrain aufnahmen. Zeitschrift fur Bauwesen. 
 
 1874. Javary. Memoire sur Tap plication de la photographie aux arts militaires. 
 Memorial de 1'officier du genie. No. 22, 1874. 
 
 1876. Jordan. Verwerthung der Photographie zu geometrischen Aufnahmen. 
 Zeitschrift fur Vermessungskunde. 
 
 1883. Hauck. Neue Construction der Perspective und Photogrammetrie. Theo- 
 rie der trilinearen Verwandtschaft ebener Systeme. Journal fur reine und ange- 
 wandte Mathematik. 
 
 1884. Hauck. Mein perspectivischer Apparat. Festschrift der K. techn. Hochs- 
 chule in Berlin. 
 
 1886. Bornecque. La photographie appliquee au lever des plans. 
 
 1887. Stolze. Photogrammetrie. Stein's " Das Licht im Dienste wissenschaf tlicher 
 Forschung. 
 
 1888. Reed. Photography applied to surveying. 
 
xii PHOTOGRAPHIC SURVEYING. 
 
 1889. Paganini. La fototopografia in Italia. 
 
 1889. Le Bon. Les levers photographiques et la photographie en voyage. 
 
 1889. Koppe. Die Photogrammetrie oder Bildmesskuhst. 
 
 1889. Moessard. Le cylindrographe, appareil panoramique. 
 
 1891-93. Steiner. Die Photographie im Dienste des Ingenieurs. 
 
 1892. Schiffner.DiQ Photographische Messkunst. 
 
 1892. Legros. Elements de photogrammetrie. 
 
 1892. Meyderibauer. Das photographische Aufnehmen zu wissenschaftlichen 
 Zwecken. 
 
 1893. Pollack. Die Beziehungen der Photogrammetrie zu den topographischen 
 Neuaufnahmen im bairisch-osterreichischen Grenzgebirge. Archiv fur die officiere des 
 deutschen Reichsheeres. 
 
 1893. Wang. Die Photogrammetrie oder Bildmesskunst im Dienste des Forst- 
 technikers. 
 
 1893-94. Laussedat. Les applications de la perspective au lever des plans 
 Annales du Conservatoire des Art et Metiers* 
 
CONTENTS 
 
 PAGE 
 
 Preface iii 
 
 List of the most important publications on photographic surveying xi 
 
 CHAPTER I. 
 
 DESCRIPTIVE GEOMETRY. 
 
 1. Definition, planes of projection. . . . ! . . . , . 1 
 
 2. Ground line 1 
 
 3. Representation of a point 1 
 
 4. Representation of a straight line 3 
 
 5. Through a given point, to draw a parallel to a given line 6 
 
 G. Representation of a plane 7 
 
 7. Line contained in a plane 8 
 
 8. Point in a plane 9 
 
 9. Through a point, to draw a plane parallel to another plane 9 
 
 10. Line perpendicular to a plane 10 
 
 11. Revolving a plane upon one of the planes of projection 10 
 
 12. Intersection of two planes 14 
 
 13. The intersecting planes are both parallel to the ground line 15 
 
 14. The intersecting planes cut the ground line at the same point 15 
 
 15. Intersection of two planes, one of which is horizontal or parallel to the ver- 
 
 tical plane 16 
 
 16. Planes perpendicular to one of the planes of projection 16 
 
 17. Intersection of a line and a plane * 16 
 
 18. Intersection of three planes 17 
 
 19. Through a point, to draw a straight line which will meet two given lines 17 
 
 20. Distance of two points 17 
 
 21. To lay off a given length on a line 17 
 
 22. Distance from a point to a line 18 
 
 23. Distance from a point to a plane 18 
 
 24. Distance of two parallel planes 19 
 
 25. Distance of two straight lines 19 
 
 26. Angles of a line with the planes of projection 19 
 
 27. Angle of two lines 20 
 
 28. Angles of a plane with the planes of projection 20 
 
 20. Angle of two planes .' 21 
 
 30. Through a given line in a plane to draw another plane making a certain 
 
 angle with the given plane. 22 
 
 81. Angle of a line with a plane 22 
 
 32. Method of rotations 22 
 
 33. Rotation of a point 23 
 
 84. Rotation of a line 23 
 
 35. Rotation of a plane 23 
 
 36. Distance of two points 24 
 
 87. Solution of spherical triangles 25 
 
 38. Given three sides, to find the angles 25 
 
 89. Given two sides and the included angle, to find the remaining side and angles. 26 
 
riv PHOTOGRAPHIC SURVEYING. 
 
 PAGE 
 
 40. Given two angles and the side opposite one of them, to find the remaining 
 
 sides and angle 26 
 
 41. Given two sides and the angle opposite one of them, to find the remaining 
 
 side and angles 27 
 
 42. Other cases. Supplementary triangles 27 
 
 43. Reduction of an angle to the horizon 28 
 
 CHAPTER II. 
 
 PERSPECTIVE. 
 
 44. General remarks 29 
 
 45. Definitions 29 
 
 46. Perspective of a point in the ground plane 31 
 
 47. Perspective of a line in the ground plane 32 
 
 48. Perspective of a point not in the ground plane 32 
 
 49. Perspective of a line not in the ground plane 33 
 
 50. Positions of the vanishing point 34 
 
 51. Vanishing line 34 
 
 52. Lines or figures in front planes 35 
 
 53. Measuring lines and measuring points 36 
 
 54. Reduction of a perspective to scale 37 
 
 55. To place in perspective a point of the ground plane 39 
 
 56. To place in perspective a line or figure of the ground plane 40 
 
 57. To place in perspective a point outside of the ground plane 41 
 
 58. To place in perspective a line outside of the ground plane 41 
 
 59. The distance line is an axis of symmetry of the perspective 41 
 
 60. Given the heights of two points and their perspectives, to find the vanishing 
 
 point and trace on picture plane of the line joining the given points 42 
 
 61. To find the intersections of a vertical line by a series of horizontal planes . . 42 
 
 62. To mark on the perspective of any line or curve contained in a vertical plane, 
 
 the intersections by a series of horizontal planes 44 
 
 63. To mark on the perspective the intersections of a plane, line, or curve, by a 
 
 series of horizontal planes 45 
 
 64. Intersections of a prism, pyramid, or conic surface, by a series of horizontal 
 
 planes 45 
 
 65. To place a point of the ground plane by means of its perspective. 46 
 
 66. To place a line on the ground plane by means of its perspective 46 
 
 67. To draw a figure on the ground plane by means of its perspective 47 
 
 68. Vanishing scale 48 
 
 69. Use of the measuring line 49 
 
 70. Precision of the method 49 
 
 71. To determine from the perspective, the projections of a point not in the 
 
 ground plane, but of which the height is known 50 
 
 72. To construct from its perspective a figure in any horizontal plane 50 
 
 73. To find the traces and vanishing point of a line given by its horizontal pro- 
 
 jection and perspective 51 
 
 74. Given the slope of a line and the horizontal projection of one of its points, to 
 
 find the horizontal projection and traces of the line 52 
 
 75. To find the traces of the plane containing three given points or two given 
 
 lines 53 
 
 76. Given the line of greatest slope, to find the traces of the plane 54 
 
 77. Change of ground plane 54 
 
 78. To find the horizontal projection of a figure from its perspective when the 
 
 figure is contained in a plane perpendicular to the principal plane 55 
 
 79. To find from its perspective the horizontal projection of a figure in a plane 
 
 perpendicular to the picture plane 57 
 
 80. Change of ground plane and distance line 59 
 
TJN1 
 
 CONTENTS. 
 
 PAGE 
 
 81. From the perspective of a figure in any given plane, to construct the horizon- 
 
 tal projection of the figure 61 
 
 82. Change of station, ground and picture planes 61 
 
 83. Reflected images 63 
 
 84. Shadows 64 
 
 85. Heights 66 
 
 80. Relations between two perspectives of the same object 67 
 
 CHAPTER III. 
 
 PERSPECTIVE INSTRUMENTS. 
 
 87. Simplest form of perspective instrument 70 
 
 88. Diagraph 70 
 
 89. Camera lucida 72 
 
 90. Camera obscura 75 
 
 91. Perspectograph 77 
 
 92. To draw the trace of the principal plane on the drawing board 82 
 
 93. To find the distance from the station to a front line of the ground plane 82 
 
 94. To find the distance between the two slides 83 
 
 95. To draw the graduation for the height of the station 84 
 
 96. To draw the horizon, ground and principal lines of the perspective 84 
 
 97. Centrolinead 85 
 
 98. Perspectometer 88 
 
 99. Drawing the ground plan with the camera lucida 89 
 
 100. Drawing the ground plan with the camera obscura 90 
 
 91 
 
 . 92 
 
 J.W. .L.'l.C* VV lli UilC *VUA*U |*M** V7AVU UUV \^CV11L^L G) UILFCW^UXG** 
 
 101. Drawing the ground plan with the perspectograph . 
 
 102. Change of scale 
 
 CHAPTER IV. 
 
 FIELD INSTRUMENTS. 
 
 103. Lenses 94 
 
 104. Aberrations 99 
 
 105. Diaphragms 103 
 
 106. Anastigmats 107 
 
 107. Surveying cameras 118 
 
 108. Canadian equipment 139 
 
 109. Use of instruments ... . 145 
 
 CHAPTER V. 
 
 HURTER AND DRIFFIELD's INVESTIGATIONS. 
 
 110. General remarks v 147 
 
 111. Density, opacity, transparency ". 147 
 
 112. Photometer 150 
 
 113. Development 154 
 
 114. Gradation 159 
 
 11 5. Action of light on the sensitive film 162 
 
 116. Speed of plates 167 
 
 117. Negatives and positives 171 
 
 118. Contact printing and enlarging 175 
 
x vi PHO TOGRA PHIG S UR VEYING. 
 
 CHAPTER VI. 
 
 PHOTOGRAPHIC OPERATIONS. 
 
 PAGE 
 
 119. Dry plates 181 
 
 120. Exposure 185 
 
 121. Development 191 
 
 122. Enlargement 193 
 
 CHAPTER VII. 
 
 FIELD WORK. 
 
 123. Triangulation 203 
 
 124. Camera stations 205 
 
 CHAPTER VIII. 
 
 PLOTTING THE SURVEY. 
 
 125. Scale of plan 207 
 
 126. Plotting the triangulation 208 
 
 127. Checking the photographs 209 
 
 128. Plotting the traces of the picture and principal planes 210 
 
 129. Identification of points 211 
 
 130. Plotting the intersections 212 
 
 131. Plotting with the perspectograph 213 
 
 132. Heights 214 
 
 133. Vertical intersections 216 
 
 134. Photograph board ! 216 
 
 135. Construction of the traces of a figure's plane 218 
 
 136. Contour lines 219 
 
 137. Photograph protractor 221 
 
 138. Precision of the method of photographic surveying 223 
 
 CHAPTER IX. 
 
 PHOTOGRAPHS ON INCLINED PLATES. 
 
 139. General remarks '. 224 
 
 140. Plotting the directions of points of the photographs 224 
 
 141. Determination of heights . . 225 
 
 142. Determination of the horizon line and vanishing point of verticals 226 
 
 143. Transferring the perspective to a vertical plane 227 
 
 144. Photographs on horizontal plates 228 
 
 SUPPLEMENTARY NOTE. 
 
 Laussedat's new photo-theodolite 280 
 
tTN 
 
 PHOTOGRAPHIC SURVEYING 
 
 CHAPTER I. 
 DESCRIPTIVE GEOMETRY. 
 
 1. DEFINITION, PLANES OF PROJECTION. The object of descriptive 
 geometry is to represent bodies and to solve problems on figures in 
 space by means of their projections on certain planes called "planes of 
 projection.'" 
 
 2. GROUND LINE. For this purpose, two planes intersecting each 
 other are employed ; they divide space into four solid angles. Usually, 
 one of the planes is vertical and the other one horizontal ; their line of 
 intersection is called "ground line " and is denoted by the letters XY. 
 
 3. REPRESENTATION OF A POINT. Let 
 XAY, Fig. 1, be the vertical plane, XBY 
 the horizontal or ground plane, and P a 
 point in space. From P, draw the perpen- 
 diculars Pp, Pp to the ground and vertical 
 planes ; p is the horizontal projection of the 
 point P and p its vertical projection. 
 
 Let the vertical plane be revolved round 
 the line XY as an axis, until it coincides 
 with the ground plane ; the point p will fall 
 at a point p" such that the line pp" will be 
 perpendicular to XY. 
 
 For let a plane be drawn through Pp and Pp' ; it is perpendicular 
 to the ground plane as containing Pp and perpendicular to the vertical 
 plane as containing Pp' ; but when a plane is perpendicular to two 
 other planes, it is perpendicular to their intersection, therefore, the 
 plane pPp is perpendicular to XY, and its traces op and op on the 
 ground and vertical planes, are also perpendicular to XY, since a line 
 perpendicular to a plane is perpendicular to all the lines passing through 
 its foot in the plane. 
 
 But op being perpendicular to XY, op" must also be perpendicular 
 to XY; it follows that pop" is a straight line perpendicular to XY. 
 The ground plane is now as shown in Fig. 2, op being the distance of 
 1 R 
 
DESCRIPTIVE GEOMETRY. 
 
 the point P from the ground plane and op its distance from the vertical 
 plane ; both points p and p are on the same perpendicular to the 
 ground line, as explained above. 
 
 ff It is usual to represent points in space by 
 
 capital letters, the horizontal projections by italic 
 letters and the vertical projections by the same 
 !r italic letters accented. 
 
 Fig-. 2 
 
 It has been shown that the two projections of 
 a point are on a perpendicular to the ground line. 
 Inversely, any two points on a perpendicular to 
 the ground line are the projections of a point of 
 space. 
 
 For let the part of Fig. 2 above the ground line be revolved round 
 the line XY until its plane be vertical as in Fig. 1. Through p draw a 
 parallel to op and through p draw a parallel to op ; they will meet in 
 a point P. 
 
 But op is perpendicular to XY by hypothesis and it is also perpen- 
 dicular to op, since pop is the angle of the vertical and ground planes ; 
 therefore op is perpendicular to the ground plane, because it is per- 
 pendicular to two lines in that plane. It follows that pP, parallel to 
 op' is perpendicular to the same plane. 
 
 In the same manner it may be shown that p'P is perpendicular to 
 the vertical plane : 
 
 Therefore p and p are the horizontal and vertical projections of 
 the point P. 
 
 (V (2) (3) (4) (5) (6J (7) (8) (9j 
 
 
 a 
 
 
 a 
 
 a 
 a 
 
 
 a 
 
 a 
 
 
 
 
 
 
 i 
 
 ', 
 
 d 
 
 a t 
 
 i ad 
 
 
 
 a 
 
 
 
 
 
 
 
 
 * 
 
 a 
 
 d 
 
 
 d 
 
 
 
 m 
 
 Fig-. 3 
 
 Fig. 3 is the representation of a point in various positions. 
 
STRAIGHT LINE. 3 
 
 (1.) Is a point in front of the vertical plane and above the ground 
 plane. 
 
 (2.) Is in front of the vertical plane and below the ground plane. 
 (3.) Is behind the vertical plane and below the ground plane. 
 (4.) Is behind the vertical plane and above the ground plane. 
 (5.) Is in the ground plane in front of the ground line. 
 (6.) Is also in the ground plane but behind the ground line. 
 (7.) Is in the vertical plane above the ground line. 
 (8.) Is also in the vertical plane but below the ground line. 
 (9.) Is on the ground line. 
 
 4. REPRESENTATION OF A STRAIGHT LINE. If perpendiculars be 
 drawn to a plane from every point of a straight line, the locus of the 
 feet of the perpendiculars is a straight line and is the orthogonal pro- 
 jection of the first one. 
 
 The projection of a straight line may also be defined as the inter- 
 section of one of the planes of projection by a second plane perpen- 
 dicular to the first one and containing the given line. This second 
 plane is called the projecting plane. 
 
 A straight line is perfectly defined by its projections, because it is 
 the intersection of the two projecting planes. There is, however, an 
 exception when the given line is contained in a plane perpendicular 
 to the ground line ; the two projecting planes coincide and the projec- 
 tions of the line are not sufficient to define it ; the traces must be 
 given. 
 
 Tiie " traces " of a line are the points where it intersects the planes 
 of projection. The vertical trace c (Fig. 4), being in the vertical 
 plane, its horizontal projection is on the 
 ground line, but it is also on the horizontal 
 projection ab of the given line, therefore it 
 must be at the intersection c of both lines. 
 So by erecting at c a perpendicular to the 
 ground line, the vertical trace will be found 
 at c. 
 
 Similarly, the horizontal trace is obtained 
 by erecting at d' a perpendicular d'd to the 
 ground line ; d is the horizontal trace. 
 
 p. . Inversely, the projections of a line may 
 
 be obtained from the traces. By drawing a 
 
 perpendicular from the vertical trace c' to the ground line, a point c of 
 the horizontal projection is obtained, which, joined to the horizontal 
 trace o?, gives the horizontal projection cd. The vertical projection is 
 obtained in a similar manner by finding the vertical projection d' of 
 the horizontal trace d and joining cd'. 
 
4 DESCRIPTIVE GEOMETRY. 
 
 A straight line is defined by the projections of two of its points. 
 Let aa'j bb', Fig. 4, be the points. The projections of the line are 
 a'b', ab. 
 
 A straight line may occupy various positions with reference to the 
 planes of projection ; these positions are illustrated below. 
 
 Fig. 5 shows a line intersecting the vertical plane 
 at b', above the ground line and the ground plane 
 at a, in front of the vertical plane. 
 
 Fig-. 6 
 
 Y Fig. 6. The vertical trace, b', is below the 
 ground line ; the horizontal trace, a, is in front of 
 it. The portion of the line between the traces 
 is in the lower front dihedral angle. 
 
 Fig. 7. The vertical trace, b', is below the 
 ground line ; the horizontal trace, a, is behind it. 
 
 Fig-. 7 
 
 Fig. 8. The vertical trace, b', is above the 
 ground line ; the horizontal trace, a, is behind 
 it. 
 
 Fig-. 8 
 
STRAIGHT LINE. 
 
 Fig. 9. Line parallel to the vertical plane, 
 with horizontal trace at a. In this case, the pro- 
 ~ Y jecting plane through ab is parallel to the verti- 
 cal plane; therefore its intersection with the 
 "* ground plane, ab, is parallel to the ground line. 
 
 Fig-. 9 
 
 Fig-. 10 
 
 Fig. 10. Horizontal line intersecting the vertical 
 Y plane at a'. The projecting plane through ab' is 
 parallel to the ground plane; therefore its inter- 
 section with the vertical plane, ab', is parallel to the 
 ground line. 
 
 Fig-. 11 
 
 Fig. 11. Any line in a plane perpendicular to the ground 
 line. The horizontal and vertical projections coincide, and 
 are on a perpendicular to the ground line. As explained 
 above, the line in this case is not defined by its projec- 
 tions, which do not change, whatever may be the direction 
 of the line in the projecting plane, but when the traces are 
 given the line is defined. 
 
 Fig-. 12 
 
 Fig. 12. Line perpendicular to thevertical plane at a'. 
 The vertical projection is a point, a, since any perpendicu- 
 lar to the vertical plane from a point of the given line will 
 intersect the plane at a. The horizontal projection, ab, is 
 a line perpendicular to the ground line, because the project-, 
 ing plane is perpendicular to the two planes of projection, 
 and therefore is perpendicular to their intersection XY. 
 There is no horizontal trace. 
 
DESCRIPTIVE GEOMETRY. 
 
 Fig-. 13 
 
 Fig. 13. Line perpendicular to the ground plane at a. 
 The perpendicular to the ground plane from any point of 
 the given line intersects the plane at a, which is the hori- 
 zontal projection of the line. The vertical projection, a'b', 
 is perpendicular to the ground line, because the projecting 
 plane, being perpendicular to both planes of projection, is 
 perpendicular to their intersection, the ground line. 
 
 Fig-. 14 
 
 Fig-. 15 
 
 Fig. 14. Line parallel to the ground line. In 
 this case each of the projecting planes is parallel 
 to the ground line ; therefore their intersections 
 with the corresponding planes of projection are 
 also parallel to the ground line. 
 
 Fig. 15. Line intersecting the ground line. 
 The point of intersection, a, is at the same time 
 the horizontal and the vertical trace of the line, 
 and both projections intersect there. 
 
 When a line is in the ground plane its hori- 
 zontal projection is the line itself and its vertical 
 projection is the ground line. 
 
 When a line is in the vertical plane its ver- 
 tical projection is the line itself and its horizontal projection is the 
 ground line. 
 
 The ground line is its own horizontal projection and its own verti- 
 cal projection. 
 
 5. THROUGH A GIVEN POINT, TO DRAW A PARALLEL TO A GIVEN LINE. 
 When two lines are parallel their projections of same denomination 
 are also parallel, because their projecting planes, being perpendicular 
 to the same plane of projection and passing through parallel lines, 
 are themselves parallel to each other, and therefore their intersections 
 with the plane of projection are parallel lines. 
 
 It follows that when a parallel to a 
 line ab, a'b' (Fig. 16) has to be drawn 
 Y through a point c, c', it is sufficient to 
 draw through c a parallel to ab and 
 through c a parallel to a'b' ; then cd, c'd' 
 is parallel to ab, a'b'. 
 
 When two lines ab, a'b' ; cd, c'd' (Fig. 
 17) intersect each other, the points of 
 
 Fig-. 16 
 
PLANES. 
 
 intersection p and p of the projections are on the same perpendicular 
 to the ground line. It has been shown in 3 that this is necessary in 
 order that p and p may represent a point in space. 
 
 It follows that when the points p and p 
 are not on the same perpendicular to the 
 ground line, the lines a&, a'b' '; cd, c'd', do 
 not intersect, that is to say they are not 
 contained in one plane. 
 
 6. REPRESENTATION OF A PLANE. A 
 plane is represented by its traces on the 
 planes of projection, that is to say by its 
 intersections with the said planes. These 
 traces meet in a point , Fig. 18, of the 
 Fig". 17 ground line, which is the point where the 
 
 plane cuts it. The vertical trace of the 
 plane is a/", the horizontal trace is aP. 
 
 When the plane is vertical, its trace P', Fig. 
 T 1 9, on the vertical plane, being the intersection of 
 two vertical planes, is a vertical line. 
 
 But a vertical line is perpendicular to the ground 
 plane and to all lines contained in this plane by which 
 it is intersected ; therefore it is perpendicular to the 
 ground line. 
 
 Fig-. 19 
 
 
 It may be shown in the same way that the horizontal 
 trace P, Fig. 20, of a plane perpendicular to the 
 vertical plane, is a line perpendicular to the ground 
 line. 
 
 Fig-,20 
 
DESCRIPTIVE GEOMETRY. 
 
 21 
 
 The plane may be parallel to the vertical plane, 
 in which case the vertical trace disappears. The 
 horizontal trace PQ, Fig. 21, is parallel to the 
 ground line, because the two lines are the inter- 
 sections of two parallel planes by a third one. 
 
 P' 
 
 Fig-. 22 
 
 Similarly, a horizontal plane is represented by 
 its vertical trace P'Q', Fig. 22, parallel to the 
 ground line ; the horizontal trace disappears. 
 
 Q' 
 
 When the plane is parallel to the ground line, the 
 two traces PQ, P'Q', Fig. 23, are also parallel to it. 
 
 Fig-. 23 
 
 Fig-. 24 
 
 A plane PaP', Fig. 24, perpendicular to the ground 
 line, has its traces perpendicular to it. The ground 
 line being perpendicular to the plane is perpendicular 
 to all the lines passing through in that plane and 
 therefore is perpendicular to the traces P, aP'. 
 
 Two parallel planes have their traces parallel, because the traces are 
 then the intersections of two parallel planes by a third one. 
 
 7. LINE CONTAINED IN A PLANE. A line contained in a plane has 
 its traces on the traces of the plane, since any point of the planes of 
 projection not on the traces is outside of the 
 given plane. Hence the following method 
 for finding a line contained in a plane PaP', 
 Fig. 25, when one of its projections ab is 
 7 given. The point a, where the horizontal 
 projection of the line intersects the horizontal 
 trace aP of the plane, is the horizontal 
 trace of the line. Its vertical projection a' 
 is a point of the vertical projection of the 
 Fig*. 25 line. But the point of intersection b of ab 
 
PLANES. 
 
 9 
 
 with the ground line is the projection of a point of the line AB con- 
 tained in the vertical plane, that is, the projection of the vertical trace 
 of AB ; then if at b a perpendicular bb' be erected to XY, its intersec- 
 tion b' with aP will be the vertical trace of AB and the vertical 
 projection will be obtained by joining a'b'. 
 
 8. POINT ix A PLANE. When a point M is contained in a plane 
 PaP', Fig. 26, one of the projections, m, of the point is sufficient to 
 define it. 
 
 To find the other projection, m, of M, let a 
 horizontal line be drawn through Min the plane 
 PaP' ; its horizontal projection is a line ma 
 Y parallel to aP, its vertical trace is at the in- 
 tersection a' of the vertical trace of the plane 
 PaP' with a perpendicular at a to the ground 
 line and its vertical projection is a line dm' 
 parallel to XY. The vertical projection of M 
 is then found by drawing through m a per- 
 r igf. _,o pendicular to XY and producing it to its inter- 
 
 section with am. The point mm being on the line am, am, is in the 
 plane PaP'. 
 
 It is not necessary that the line drawn through M be horizontal ; 
 any other line might be employed, but it is more convenient to use a 
 line parallel either to the vertical or to the ground plane. 
 
 9. THROUGH A POINT, TO DRAW A PLANE PARALLEL TO ANOTHER 
 PLANE. Let a point mm', Fig. 27, and a plane PaP' be given ; 
 through mm', it is required to draw a plane parallel to PaP' . Through 
 
 mm, draw a line parallel to Pa; its 
 horizontal projection is a parallel through 
 m to Pa and its vertical projection a 
 parallel through m to the ground line 
 ( 5). Find the vertical trace a by 
 erecting at a a perpendicular to the 
 ground line and producing it to its inter- 
 section with m'a. Then through a draw 
 PQ' parallel to aP' and through $ draw 
 BQ parallel to aP. The plane QPQ' is 
 parallel to the plane PaP', and it con- 
 tains the point mm, since it contains the 
 p^g, 27 line ma, m'a. 
 
10 DESCRIPTIVE GEOMETRY. 
 
 10. LINE PERPENDICULAR TO A PLANE. Let the line ma, m'b', Fig. 
 28, be perpendicular to the plane PaP f ; the projecting plane MA 
 ma is perpendicular to the ground plane ; it is also perpendicular to 
 the plane PaP\ since it contains a line MA 
 perpendicular to this plane ; therefore it is 
 perpendicular to the intersection aP of these 
 two planes and inversely the intersection aP 
 is perpendicular to the projecting plane. But 
 being perpendicular to the plane, aP is per- 
 pendicular to all lines passing through a and 
 Y contained in the plane, therefore it is perpen- 
 dicular to am. 
 
 In the same manner it may be shown that 
 b'm is perpendicular to aP', 
 
 Fig*. 28 g o when through a point mm, it is required 
 
 to draw a line perpendicular to a plane, perpendiculars ma and m'b' to 
 the traces of the plane are drawn through the projections m and m of 
 the point. 
 
 To draw through a given line a plane perpendicular to a given 
 plane, a line perpendicular to the plane is drawn, as explained above, 
 from any point of the given line, and then a plane is drawn through 
 the two lines by joining the traces of same denomination of the lines. 
 
 When it is required to draw through a given point a plane per- 
 pendicular to a given line, perpendiculars to the projections of the line 
 are drawn from any point of the ground line; they represent the traces 
 of a plane perpendicular to the given line and there remains only to 
 draw a plane parallel to the first one and passing through the given 
 point as explained in 9. 
 
 11. REVOLVING A PLANE UPON ONE OP THE PLANES OF PROJECTION. 
 For making constructions in a plane other than one of the projection 
 planes, it is often convenient to revolve the plane round one of its 
 traces upon the ground or vertical planes ; the construction is then 
 effected and, if necessary, the plane is revolved back to its original 
 position. 
 
 The problem can always be reduced to finding the position of a point 
 M of a plane PaP , Fig. 29, after this plane has been brought into coin- 
 cidence with one of the planes of projection, the ground plane for 
 instance, by a revolution round its horizontal traqe aP. 
 
REVOLVING PLANES. 
 
 11 
 
 Fig-. 29 
 
 From the point m, draw a perpendi- 
 cular mK to the trace uP of the plane 
 and join in space MK and Mm. The 
 plane MKm is perpendicular to the 
 Y ground plane as containing Mm ; hence 
 it contains the vertical line at K to 
 which A" is perpendicular. But aK 
 is also, by construction, perpendicular 
 to Km, therefore it is perpendicular 
 to the plane mKM and to KM which 
 is in this plane. Consequently when 
 the plane is revolved round its trace, 
 M will fall on a perpendicular KM 1 
 toaP. 
 
 Let us suppose now that the triangle MKm be revolved round Km 
 on the ground plane. The angle KmM being a right angle, the 
 side mM will fall in mm^ parallel to aP ; nim 19 which is the height of 
 M above the horizontal plane, is equal to hm. The triangle Kmm lt 
 can therefore be constructed and by taking KM^ equal to Km^ the 
 position of M 1 is obtained. 
 
 The construction lines on the figure are merely to show that rara x is 
 made equal to hm', KM^ equal to Km^ and that the point mm lies in 
 the plane PaP' ( 8). 
 
 A similar construction would be employed to revolve a plane upon 
 the vertical plane. 
 
 It may be observed that the angle mKm^ is the angle of the given 
 plane with the ground plane. 
 
 The position wf a line revolved upon the horizontal plane is deter- 
 mined by finding the positions of two of its points ; its traces for 
 
 instance. Let ab, a'b\ Fig. 30, be 
 the line and PaP' the plane. From 
 b draw a perpendicular bK to aP. 
 The same demonstration as in the 
 case of Fig. 29 shows that aP is per- 
 pendicular to the line Kb' in space, 
 therefore b' in its revolution round 
 aP will fall on Kb produced, and 
 KB l will be equal to Kb'. But Kb' 
 is the hypothenuse of the right angle 
 triangle Kbb', which can be con- 
 structed at Kbbi ; by making KB l 
 equal to Kb l the position of l is 
 obtained. The position of the hor- 
 izontal trace a of the given line has 
 not changed ; therefore this line 
 Fig-. 30 after its revolution will fall in aB^ 
 
12 DESCRIPTIVE GEOMETRY. 
 
 Here again the angle bKb-^ is the angle of the given plane with the 
 ground plane, and the construction indicated affords a simple method 
 of finding it. 
 
 The line aP^ is the position of the vertical trace after the revolution 
 of the plane ; the angle PaP l is the angle formed in space by the 
 traces of the plane, and aB^ is equal to ab'. Hence the following 
 construction to find the revolved line, when a is within the limits of 
 the drawing : 
 
 Draw bK perpendicular to aP and instead of constructing the tri- 
 angle Kbb^, describe a circle with a as a centre and ab' as radius. Join 
 to a the point of intersection B of the circle with bK produced ; B^a 
 is the revolved line, and aP l the revolved trace of the plane PaP'. 
 
 To revolve a plane back into its original position, inverse construc- 
 tions are employed. Let it be required, for instance, to find the pro- 
 jections of the point M l (Fig. 29) when the plane PaP^ is revolved 
 back to PaP'. The angle of PaP' with the ground plane is first deter- 
 mined by the construction given above ; then from M 1 a perpendicular 
 is drawn to aP, and at the point of intersection K an angle mKin^ is 
 constructed equal to the inclination of the given plane on the ground 
 plane. Km l is taken equal to KM^ and from m 1 a perpendicular 
 in^m is drawn to M^m; m is the horizontal projection of M, and mm^ 
 its height above the ground plane, from which the vertical projection 
 is easily found. 
 
 A line is revolved back into its original position by repeating in 
 inverse order the constructions given for revolving it upon the pro- 
 jection plane. Let aB^ (Fig. 30) be the line; from.5 1 and a draw the 
 perpendiculars B^b and aa to aP and XY respectively. At b erect a 
 perpendicular bb' to XY, produce it to its intersection b' with aP', and 
 join ab, ab', which are the projections required. 
 
 The constructions are simplified when the vertical trace has been 
 revolved on the ground plane. Let it be required to find the position 
 
 of the point mm (Fig. 31) on the plane 
 PaP' revolved in PaP l upon the ground 
 plane. From m draw a perpendicular 
 nin to aP ; it has been shown that the 
 point M of space in revolving round aP 
 will fall upon this line. Through mm 
 and in the plane PaP' draw a line ab, 
 ab', cutting the two traces of the plane 
 (7); on aP 1 take aA l , equal to aa, 
 and join A^b. As explained above, A t b 
 is, in the plane PaP i the line represented 
 in projection at ab, ab', and the point 
 Fig". 31 required, M l must be on this line. But 
 
 it has been shown that it is also on the line mn ; therefore it is at the 
 intersection of both lines, in M l . 
 
CALIFOH^J 
 
 REVOLVING PLANES. 
 
 To revolve back this point into its original position, a line A^b, 
 cutting the traces aP l and aP, is drawn through M l ; ad is taken 
 equal to uA lt and perpendiculars a a and bb' are drawn from a and b 
 to the ground line ; ab and a'b' are the projections of the line bA l 
 when revolved back to its original place. A perpendicular to aP is 
 next drawn from M^ ; its intersection with ab gives the horizontal 
 projection m of the point M ; the vertical projection is obtained by 
 drawing through m a perpendicular to the ground line and producing 
 to its intersection m with a'b'. * 
 
 Instead of the line ab, a'b', a parallel to the vertical plane may be 
 employed. Let mm, Fig. 32, be the point, PaP' the given plane and 
 
 PaP^ the same plane revolved upon the 
 ground plane. From m draw a perpen- 
 dicular mn to aP : the point M " x will 
 fall on this line. Then through mm, 
 draw ab, a'b', parallel to the vertical trace 
 _ Y aP' of the plane ( 5). When PaF is 
 revolved, this line still remains parallel 
 to aP' and as its trace a does not move, 
 the line falls in aJ$ l9 parallel to aP^. 
 The point Jf x is on aB 1 , but it is also 
 on mn, therefore it is at the intersection 
 MI of mn and aB^. 
 
 V 
 
 To find the projections of the point M 
 Fig*. 32 ^ 8 P ace when it is given revolved on the 
 
 ground plane in M lt draw through M^ 
 
 a parallel aB l to the trace aP ^ and a perpendicular M^m to aP. 
 Through a draw ab parallel to the ground line ; it is the horizontal 
 projection of a line parallel to the vertical plane and passing through 
 the point M. But the horizontal projection m of M is also on the line 
 M^m, therefore it is at the intersection of M^m and ab. 
 
 The vertical projection m of M is on the perpendicular mm' drawn 
 through m to the ground line ; it is also on the vertical projection a'b' 
 of the parallel to the vertical plane, which is obtained by drawing 
 from a a perpendicular aa to XY and through a a parallel to the 
 trace aP '. The intersection of a'b' and mm gives the vertical projec- 
 tion m of M. 
 
 The constructions are still further simplified when the given plane 
 is perpendicular to one of the planes of projection. 
 
14 DESCRIPTIVE GEOMETRY. 
 
 \P' m . Ms Let PP', Fig. 33, be a plane perpendicular to 
 
 the ground plane and mm a point of the plane. 
 The point M in space is on the vertical line passing 
 through m, which line is in the plane PaP and is 
 
 X ^ -T 4- Y perpendicular to the horizontal line aP. Therefore, 
 
 when PaP' is revolved round P, the line mM 
 \ ' / remains perpendicular to aP and the point M falls 
 
 ,/ ;X \D in M t at a distance mM^ from aP equal to the 
 height of M above the ground plane. But this 
 M/ height is km, therefore to determine the point M ^ 
 
 Fig". 33 draw at m a perpendicular to aP and take mM ^ 
 
 equal to hm. 
 
 Instead of revolving the plane round aP, it may be revolved round 
 aP' on the vertical plane. The point M then describes in space an arc 
 of circle of which the vertical projection is the line m'M 2 parallel to 
 XY, and the horizontal projection is an arc of circle mt, with a as a 
 centre and am as radius. When the plane PaP' coincides with the 
 vertical plane, the point M of the plane must be somewhere on the 
 line m'M 2 and its horizontal projection is t. Then if a perpendicular 
 to XY be erected at t and produced to its intersection M 2 with m'M 2 , 
 M 2 will be the required point. 
 
 To find the projections of the point J/", whose position M^ revolved 
 on the ground plane is given, draw from M^ a perpendicular M^m to 
 aP and from m a perpendicular mm to XY. Take hm equal to inM^, 
 height of the point M above the ground plane ; m, m are the projec- 
 tions of the point. 
 
 The projections of M 2 are found by drawing through M 2 a parallel 
 M 2 v to XY, which is the vertical projection of the arc of circle de- 
 scribed by M 2 when revolved back to its original position ; take am 
 equal to the distance M 2 v of M 2 from the trace aP and through m 
 draw the perpendicular mm' to XY. m, m' are the projections of the 
 point. 
 
 12. INTERSECTION OF TWO PLANES. Let PaP' 
 and QftQ', Fig. 34, be two planes : the points 
 M and N where the traces of the planes meet, 
 are the traces of the line of intersection of 
 the planes. The projections Mn, m'N of the 
 intersection are found as explained in 4 by 
 letting fall the perpendiculars Mm and Nn to 
 Fig". 34 the ground line and joining Mn, m'N. 
 
INTERSECTIONS. 
 
 15 
 
 13. THE INTERSECTING PLANES ARE BOTH PARALLEL TO THE GROUND 
 
 LINE. Let PQ, P'Q', RS, R'S', (Fig. 35), be the traces of two inter- 
 secting planes parallel .to the ground line ; the 
 
 r r |fe-_ s' construction given in 12 does not apply and 
 
 recourse must be had to an auxiliary plane. 
 P' rf' . Q' Draw a plane TOT' perpendicular to the 
 
 pQ\ 1 f ground line. The line of intersection of the 
 
 two given planes is parallel to the ground line 
 and so are its projections. If the projections 
 m and m of the point M where this line 
 intersects the plane TOT were known, the 
 projections of the line itself would be obtained 
 at once by drawing through m and m 
 parallels to the ground line. 
 Fig 1 . 35 To obtain M, let us revolve TOT around 
 
 OT upon the ground plane ; the intersection of TOT and PQ P'Q', 
 of which the traces are r, and d', will fall in cD^; OD 1 being equal to 
 Od'. Similarly the intersection of TOT and RS R'S will fall in aB } , 
 OB i being equal to Ob' and the point M will come in M 1 , at the 
 intersection of cZ> T and aB^. From M r draw a perpendicular to OT ; 
 the point of intersection m is the horizontal projection of M. The 
 vertical projection is obtained by making Om equal to mM I9 this 
 being the height of M above the ground plane. Then through m and 
 m' draw the parallels ef, e'f, to the ground line ; they are the pro- 
 jections of the line of intersection. 
 
 14. THE INTERSECTING PLANES CUT THE GROUND LINE AT THE SAME 
 
 POINT. Let PaP', QaQ', (Fig. 36), be two intersecting planes cutting 
 the ground line at a. Draw a plane TOT' perpendicular to this line ; 
 is a point of the line of intersection of the 
 planes and if the projections wand m of the 
 point M where this line cuts the plane TOT' 
 were known, the projections of the intersec- 
 tion would be obtained by joining am and 
 am'. 
 
 Let us revolve the plane TOT' around OT: 
 the intersection of TOT and PaP', of which 
 Y a and b' are the traces, will fall inaJB^ OB 
 being equal to Ob'. Similarly the intersec- 
 tion of TOT andlQaQ' will fall in cZ> 15 OD l 
 being equal to Od', and M will come in M lt at 
 the intersection of aB^ and cDj . From M^ 
 draw a perpendicular M^m to OT, the point 
 m is the horizontal projection of M. The 
 vertical projection m is obtained by making 
 Om equal to mM^. Then draw am and am! 
 which are the projections of the intersection. 
 
 Fig-. 36 
 
16 
 
 DESCRIPTI VE GEOMETR Y. 
 
 Fig-. 37 
 
 15. INTERSECTION OP TWO PLANES, ONE OF 
 WHICH IS HORIZONTAL OR PARALLEL TO THE 
 VERTICAL PLANE. When one of the planes is 
 horizontal, the intersection is parallel to the 
 horizontal trace of the other plane : its ver- 
 tical projection is the trace Q R\ of the hor- 
 izontal plane (Fig. 37) and the horizontal pro- 
 jection a parallel qr to aP. 
 
 In thejcase of a plane parallel to the vertical 
 plane (Fig. 38), the horizontal projection of 
 7 the intersection is the trace QR of the vertical 
 plane. J | The vertical projection is a parallel 
 q's' tojthe vertical trace aP of the other 
 plane. 
 
 Fig-. 38 
 
 16. PLANES PERPENDICULAR TO ONE OF THE PLANES OF PROJEC- 
 TION. W T hen the two planes are both perpendicular to one of the 
 planes of projection, their intersection is also perpendicular to this 
 plane and its projection on it is the point where the traces of the planes 
 meet. The projection on the other plane is a perpendicular to the 
 ground line passing through the above point. 
 
 17. INTERSECTION OF A LINE AND A PLANE. To find the intersection 
 of a line and a plane, another line intersecting the first one is drawn 
 
 in the plane ; the point required is the inter- 
 V/ section of the two lines. 
 
 a' 
 
 Fig". 39 
 
 Let ab, a'b', (Fig. 39), be the line and PaP' 
 the plane. For auxiliary line the intersection 
 of PaP' by one of the projecting planes of 
 the given line, abb', for instance, may be 
 employed. 
 
 To obtain the projections of this intersec- 
 tion, draw the perpendiculars bb" and cc to the 
 ground line and join c'b" : cb, c'b" is the inter- 
 section. It meets the line ab ab' at mm' 
 which is the point where the line cuts the 
 plane PaP'. 
 R 
 
DISTANCES. 17 
 
 18. INTERSECTION OF THREE PLANES. The intersection of three 
 planes may be found either by constructing the line of intersection of 
 two of the planes, and then determining the point where this line cuts 
 the third plane, or by constructing the lines of intersection of one of 
 the planes with each of the others ; the point where the two lines meet 
 is the point of intersection of the three planes. 
 
 19. THROUGH A POINT, TO DRAW A STRAIGHT LINE WHICH WILL 
 MEET TWO GIVEN LINES. To draw through a point a straight line 
 which will meet two given lines not in the same plane, a plane is 
 passed through the point and one of the lines. The point where the 
 second line pierces the plane is ascertained ( 17), and by joining 
 this point of intersection to the given point, the required line is 
 obtained. 
 
 20. DISTANCE OF TWO POINTS. Let aa, bb', 
 (Fig. 40), be two points ; to obtain their dis- 
 tance, one of the projecting planes of the line 
 AB may be revolved about its trace upon the 
 corresponding projection plane. 
 
 Let us revolve, for instance AB ab around ab. 
 The point A will fall in A on a perpendicular 
 a A T to ab, the line aA x being the height of A 
 above the ground plane, that is the distance ra. 
 Similarly B will fall in B 1 , on a perpendicular 
 b l to ab, and at a distance from b equal to sb'. 
 Fig-. 40 The required distance of the points is A^B^. 
 
 The construction may be somewhat simplified by observing that if a 
 line be drawn through a parallel to A^B^, its length ac is equal to 
 A^B^t therefore, instead of constructing the trapezoid aA 1 B l b, it is 
 sufficient to erect a perpendicular to ab at b and to lay off on it a dis- 
 tance be equal to the difference between sb' and ra. 
 
 21. To LAY OFF A GIVEN LENGTH ON A LINE. The construction 
 
 given in 20 may be employed for laying off a given length on a line 
 AB (Fig. 40). Turn the projecting plane on the line ab as an axis 
 and lay off the required length A^D^ on A^B^. Then revolve the 
 projecting plane back to its natural position ; the horizontal projection 
 of D will be at d, foot of the perpendicular drawn from D^ to ab, and 
 its vertical projection will be at d", intersection of a'b' by a perpendi- 
 cular through d to the ground line. 
 2 R 
 
18 
 
 DESCRIPTIVE GEOMETRY. 
 
 22. DISTANCE FROM A POINT TO A LINE. The distance from a 
 
 point to a straight line is obtained 
 by passing a plane through the line 
 and the point, and revolving it upon 
 one of the planes of projection. Let 
 
 ab, ab', be the line and mm the 
 point (Fig. 41). Through mm draw 
 a parallel cd, c'd to ab, a'b' ; the line 
 ac is the horizontal trace of the plane 
 containing the two parallel lines. 
 Revolve this plane around its trace 
 
 ac, until it coincides with the ground 
 plane ( 11). 
 
 Let aB and M^ be the revolved 
 positions of ab' and M. From M l 
 let fall a perpendicular M^ K to a l ; 
 Fig*. 41 it is the distance required. 
 
 "f 23. DISTANCE FROM A POINT TO A PLANE. The distance from a point 
 to a plane may be obtained by dropping a perpendicular from the point 
 to the plane ( 10), finding the point where it pierces the plane ( 17) 
 and determining the distance of the two points. 
 
 ^ It is more convenient to pass through the point a plane perpendi- 
 cular to one of the traces of the given plane. This auxiliary plane, 
 being perpendicular to the other one, contains the perpendicular from 
 the point to the given plane ; after revolving it around its trace upon 
 one of the planes of projection, a simple construction gives the solution 
 -of the problem. 
 
 Let PaP' (Fig. 42) be the plane and mm the point. Through mm 
 
 pass the plane QPQ' perpendicular to 
 aP' and revolve it around pQ upon 
 the ground plane. The point A de- 
 scribes the arc of circle A A 1 , and 
 BA^ is the intersection of the two 
 planes revolved upon the ground 
 Y plane. 
 
 The point M is on a parallel to fiQ 
 passing through m. In revolving the 
 auxiliary plane m describes the arc 
 of circle me and the line m M falls in 
 cM 19 still parallel to 0Q. The point 
 M remaining during the revolution 
 of the plane at a constant distance 
 from the vertical plane falls on a 
 
 M, 
 
 Fig-. 42 
 
 parallel to the ground line passing through m ; therefore M^ comes at 
 the intersection of c M T and mM ., . There remains only to let fall a 
 perpendicular M ^ K l from M 1 to BA 1 ; it is the distance required. 
 
 R 
 
ANGLES. 
 
 19 
 
 24. DISTANCE OF TWO PARALLEL PLANES. The distance of two parallel 
 
 planes may be obtained by in- 
 tersecting them by a third 
 plane perpendicular to both 
 and revolving it upon one of 
 the planes of projection. 
 
 Let PaP', QfrQ' (Fig. 43) be 
 the parallel planes. Draw a 
 -Y plane ROR' perpendicular to 
 the vertical traces and revolve 
 it upon the ground plane around 
 OR as an axis. The points 
 R' and S' describe the arcs of 
 circles R'R-^, S'S l ; the lines 
 RR^ and SS r being the inter- 
 sections of the given planes by 
 the auxiliary one. These lines 
 are parallel and their distance is the distance of the planes . 
 
 25. DISTANCE OF TWO STRAIGHT LINES. Let AB and CD (Fig. 44) 
 be two straight lines not contained in one plane ; it is required to find 
 
 their shortest distance. This distance is the length of 
 the perpendicular to both lines. Through any point 
 of AB, A for instance, draw a parallel AF to CD, 
 and from a point G of CD let fall a perpendicular GH 
 on the plane BAF. Through the foot of Gil in the 
 plane BAF draw a parallel HK to AF and through K 
 {D another parallel KM to HG. The line KM is perpen- 
 dicular to both lines. 
 
 . 43 
 
 Figr- 44 
 
 Although presenting no difficulty, the construction 
 requires many lines and is omitted here. 
 
 26. ANGLES OF A LINE WITH THE PLANES OF PROJECTION. Let it 
 be required to find the angles formed by the line ab, a'b', (Fig. 45), 
 6 with the planes of projection. 
 
 The angle of the line with the ground 
 plane is the same as with the line ab, 
 since the plane b'ab is perpendicular to 
 the ground plane. This angle can be 
 obtained by revolving the triangle b'ba 
 around b'b as an axis upon the vertical 
 plane. The vertex a describes the arc 
 of circle ac and the triangle comes in b'bc, 
 the angle at c being the angle of the line 
 with the ground plane. 
 R 
 
 Fig-. 45 
 
20 DESCRIPTIVE GEOMETRY. 
 
 Similarly the angle with the vertical plane is obtained by revolving 
 the triangle aa'b' upon the ground plane around aa as an axis. The 
 vertex b' comes in d, the angle ada being the angle of the line with 
 the vertical plane. 
 
 When the line is contained in a plane perpendic- 
 ular to the ground line, such as ab', Fig. 46, the 
 angles are found by revolving this plane upon one 
 of the planes of projection, the ground plane for 
 instance : the vertical trace, b' describes the arc 
 of circle b'JB^ and the revolved position of the line 
 is aB^ ; a and E^ are the angles with the ground 
 and vertical planes respectively. 
 
 In the case of a line parallel to one of the planes 
 Fig". 46 o f projection, the angle of the line with the other 
 
 plane is the angle of its projection with the ground line. 
 
 27. ANGLE OF TWO LINES. To find the angle formed by two in- 
 tersecting lines, their plane is revolved about its trace upon one of the 
 planes of projection. Let ab, a'b' ; cd, c'd', (Fig. 47), be the lines. The 
 horizontal trace of their plane is the line ac passing through their 
 traces ; it forms with the two lines a triangle a Me in which M is the 
 angle to be found. 
 
 Revolve this triangle around ac upon the 
 ground plane ; the point M moves in the 
 plane perpendicular to ac whose horizontal 
 trace is the perpendicular mn to ac ; it will 
 therefore fall in Jf 19 somewhere on mn pro- 
 duced. The distance nM l is the same as 
 the distance from n to M and the latter is 
 the hypothenuse of the right angle triangle 
 Mnn. The side Mm of this triangle being 
 \ ie height of M above the ground line, the 
 triangle can be constructed by erecting at in 
 a perpendicular to mn and laying off mt 
 equal to m's ; M^ is then determined by 
 rig". 47 making nM l equal to nt. Joining M^a and 
 
 M^c, the angle required is aM^c. 
 
 It may happen that the traces of the lines are outside of the draw- 
 ing, and that the trace of their plane can not be obtained as explained 
 above. In that case, the lines are cut by an auxiliary horizontal 
 plane on which the construction of Fig. 47 is effected. 
 
 When the lines are parallel to one of the planes of projection, their 
 angle is the angle of their projections on that plane. 
 
 28. ANGLES OF A PLANE WITH THE PLANES OF PROJECTION. The angles 
 of a plane with the planes of projection are obtained by cutting it by 
 auxiliary planes perpendicular to the traces. Let PaP' t (Fig. 48), be 
 
ANGLES. 
 
 21 
 
 the plane. Draw a plane TOT' per- 
 pendicular to aP : its intersections 
 with the planes of projection and the 
 given plane form a right angled 
 triangle TOT' in which the angle 
 at T is the angle of PaP' with the 
 ground plane. Revolve this triangle 
 upon the vertical plane around OT 
 as an axis : T describes an arc of 
 circle TT 1 , of which is the centre, 
 and the triangle comes in T'OT l , 
 the angle T l being the angle of PaP' 
 with the ground plane. 
 
 Similarly, the angle with the ver- 
 .Fig". 48 tical plane is obtained by drawing 
 
 the plane SOS' perpendicular to aP' and revolving the triangle SOS' 
 
 upon the ground plane in SOSj^. The angle at S^ is the angle of PaP' 
 
 with the vertical plane. 
 
 The line T'T is the line of greatest declivity of the plane PaF : any 
 
 other line contained in the plane PaP' and not parallel to 'T'T forms 
 
 with the ground plane an angle smaller than T'TO. 
 
 29. ANGLE OF TWO PLANES. Let PaP*, QftQ', (Fig. 49), be two planes, 
 of which it is required to find the angle. Their intersection is pro- 
 jected horizontally in ab. Cut the planes by another one perpendicular 
 to both : it is perpendicular to their intersection and consequently the 
 horizontal trace cd is perpendicular to ab. The intersections of this 
 plane with the two given planes form with the trace cd a triangle in 
 
 which the angle opposite cd is 
 the angle of the two planes. 
 
 tfY 
 
 .^he intersection of the auxil- 
 
 iary plane with the projecting 
 plane abb' is the perpendicular 
 let fall from the vertex of the 
 triangle on cd because cd being 
 ~Y perpendicular to the projecting 
 plane is perpendicular to all 
 lines contained in it passing 
 through its foot K. The same 
 intersection is also perpendicu- 
 lar to the intersection ab' of the 
 two given planes, because ab' 
 being perpendicular to the aux- 
 iliary plane, is perpendicular to 
 all lines contained in that plane 
 by which it is intersected. 
 
 Fig-. 49 
 
22 DESCRIPTIVE GEOMETRY. 
 
 Now revolve the triangle abb' about its side ab upon the ground 
 plane. The angle at b being a right angle, the point b' will fall in B^ 
 on a perpendicular to ab at 6, bB l being equal to bb'. Join B^a and 
 let fall on it from K a perpendicular KH l : this is the height of the 
 triangle formed by cd and the intersections of the two given planes by 
 the auxiliary plane. Then revolve this triangle around cd upon the 
 ground plane ; its vertex will fall on the line ab at a distance Kh equal 
 to KH^ . Join he, hd ; did is the angle required. 
 
 When the planes are in such a position as to make the above con- 
 struction inconvenient, they may be replaced by parallel planes, whose 
 positions are selected at pleasure. This may be done, for instance, 
 when the planes cut the ground line at the same point or when their 
 traces do not meet within the limits of the drawing. 
 
 When the planes are both parallel to the ground line, the construc- 
 tion is the same as in Fig. 35 ; aM t c is the angle of the planes. 
 
 30. THROUGH A GIVEN LINE IN A PLANE TO DRAW ANOTHER PLANE 
 MAKING A CERTAIN ANGLE WITH THE GIVEN PLANE. The converse pro- 
 blem consists in drawing through a given line of a plane, another 
 plane making with the first one a given angle. The construction is 
 the same as in Fig. 49, but is inverted. The given line is the inter- 
 section of the two planes ; the triangle chd is constructed by means of 
 the line KH^ and the angle h ; it gives a point d of the horizontal 
 trace of the plane required. Another point of the trace is found at a, 
 then join ad, produce to ft and join to fib 1 : the required plane is aftb. 
 
 31. ANGLE OF A LINE WITH A PLANE. The angle of a line with a 
 plane is the complement of the angle of the line with a perpendicular 
 to the plane. So in order to find the first angle, a perpendicular may 
 be erected to the plane through a point of the given line ( 10) ; the 
 angle of the two lines is then determined ( 27). 
 
 32. METHOD OF ROTATIONS. The method of rotations is a process 
 employed in Descriptive Geometry for facilitating the solution of 
 problems. It consists in rotating the whole system of the projections, 
 or only part of it, around an axis perpendicular to one of the planes of 
 projection, until the system assumes a position favourable to the 
 solution of the problem. 
 
ROTATIONS. 
 
 23 
 
 Fig-. 50 
 
 Fig-. 51 
 
 33. ROTATION OP A POINT Let it be required 
 to rotate a point mm, (Fig. 50), through an angle 
 <y, around a vertical axis a, a'b'. The projection 
 ra, describes an arc of circle mm 15 with centre 
 at a and subtending an angle equal to tu. But 
 -y the point J/, during its motion, remains at the 
 same distance from the ground plane ; therefore 
 its vertical projection m, travels on a parallel 
 mm l to the ground line. So when the point m 
 has described the arc (o ) the point m' is in m' lt 
 at the intersection of the perpendicular to the 
 ground line through m l with the parallel to the 
 same line through m. 
 
 34. ROTATION OF A LINE. Let ab, a'b', 
 (Fig. 51), be a straight line to be rotated 
 around a vertical axis c, c'd' until paral- 
 lel to the vertical plane. From c let fall 
 the perpendicular cm on a b, and rotate 
 the projecting plane containing ab 
 around the axis. The point m describes 
 an arc of circle and stops at m T on the 
 perpendicular to the ground line drawn 
 through c. The projecting plane is then 
 parallel to the vertical plane and so are 
 the lines ab and aB. The new position 
 of ab is obtained by drawing through m l 
 a parallel a 1 b l to the ground line and 
 
 making a 
 
 bm. The 
 
 height above tho ground plane of the points A and B of the given line 
 does not change during the revolution around the axis ; the vertical 
 projection a of the trace a therefore moves on the ground line and the 
 projection b' on a parallel b'b' \ to the ground line. But a\, the new 
 vertical projection of a, must be on the perpendicular through a^ to 
 the ground line and since it is also on the ground line, it must be at 
 their intersection in a l . Similarly, b\ must be on the perpendicular 
 l^b\ to the ground line and also on the parallel b'b' \, therefore, it 
 must be at their intersection b\. The rotated vertical projection is 
 then a^b' \. 
 
 35. ROTATION OF A PLANE A plane may be rotated by turning three 
 of its points, not on a straight line (33), or a point and straight line, 
 both in the plane, or two of its lines (34). The following method is 
 a simple one : 
 
 PaP', (Fig. 52), is a plane to be rotated until perpendicular to the 
 vertical plane, about a vertical axis of which the horizontal trace is c. 
 
 R 
 
24 
 
 DESCRIPTIVE GEOMETRY, 
 
 From c let fall a perpendicular cd on aP and rotate aP until cd is 
 parallel to the ground line : a l P l is then perpendicular to XY. It is 
 the rotated horizontal trace of the plane. 
 
 Now draw any horizontal line gh, 
 g'h', in the plane PaP' produce cd to 
 its intersection f with gh and rotate 
 the line gh, g'h' through the same 
 angle, w, as the trace aP of the plane. 
 The pointy* of cd describes the arc of 
 circle ff^ and stops on cd^ produced. 
 The rotated horizontal projection is 
 the line g^h^ perpendicular to XY. 
 
 To obtain the vertical projection, 
 it must be observed that the height 
 of gh, g'h', above the ground plane is 
 
 . - gg' and that it does not change dur- 
 
 ing the rotation. The vertical trace 
 
 g moves on the parallel g'h' to XY, and stops at the intersection of 
 g'h' and g^h\ produced, since g 1 h l is perpendicular to XY. 
 
 The rotated line g 1 h 1 g\, is still parallel to the ground plane and is 
 now contained in a plane perpendicular .to the vertical plane ; there- 
 fore it is itself perpendicular to the vertical plane. Its vertical projec- 
 tion is the point g\, which is also its trace and consequently a point 
 of the vertical trace of the rotated plane. But 1 is another point 
 of the new vertical trace, therefore the rotated plane is P^a^P\. 
 
 The angle g^ a \g \ is the inclination of the rotated plane on the 
 ground plane : this inclination is the same before and after rotation. 
 
 The plane might now be brought parallel to the ground plane by a 
 second rotation about an axis perpendicular to the vertical plane. 
 
 36. DISTANCE OF TWO POINTS. As an application of this method, 
 the determination of the distance of two points may be given. 
 
 Let ad, bb', (Fig 53), be the points. Rotate 
 the vertical projecting plane containing a and 
 b around the vertical line through a until it 
 is parallel to the vertical plane. The point b 
 describes an arc of circle bb ^ and stops at b^ 
 on the parallel ab l to XY ; b' moves on a 
 -^parallel to XY and stops at b\ at the inter- 
 section of the parallel b'b\ and the perpendi- 
 cular b^ b' j to the ground line. The rotated 
 line ab^, ab\ is parallel to the vertical plane; 
 it is therefore equal to its vertical projection 
 a'b' l . The inclin ation of the line on the ground 
 plane is equal to the angle formed by the ver- 
 Fig". 53 tical projection a'b' 1 and the ground line. 
 
SPHERICAL TRIANGLES. 
 
 25 
 
 Another solution of this problem is given in 20. 
 
 37. SOLUTION OF SPHERICAL TRIANGLES. A spherical triangle may 
 be assimilated to a trihedral angle by supposing the vertex of the angle 
 to be at the centre of the sphere. The sides of the spherical triangle 
 are then subtended by the plane angles of the faces of the trihedral 
 angle and the angles of the triangle are the same as the dihedral angles 
 of the trihedral angle. 
 
 As usual, the sides of the spherical triangle are designated by a, b 
 and c, the opposite angles being A, B and C. 
 
 38. GIVEN THREE SIDES, TO FIND THE ANGLES. The three sides of 
 the triangle correspond to the three faces of the trihedral angle. 
 
 Develop them on the ground 
 plane, placing one of the 
 edges, OQ, (Fig. 54), per- 
 pendicular to the ground 
 line and revolve the faces 
 a and c about the edges OQ 
 and OR, upon the ground 
 plane. The intersection of 
 the trihedral angle by the 
 vertical plane is the base 
 / of a pyramid of which is 
 the vertex and OQR one 
 of the faces in its natural 
 position. Since OQ is per- 
 pendicular to the vertical 
 r ig\ 54 plane, the planes of the two 
 
 faces intersecting along OQ are also perpendicular to the vertical plane, 
 therefore OQP^ is one of the faces of the pyramid, revolved upon the 
 ground plane about OQ, and OP ^ is the third edge of the pyramid, 
 the vertical trace of which is on the arc of circle described from Q as 
 a centre with QP l as radius. 
 
 The third edge of the pyramid is also shown in OP 2 which must be 
 taken equal to OP l ; P 2 , like P lt is the vertical trace of the third 
 edge OP revolved upon the ground plane. Let now the face c be 
 revolved back to its natural position, by turning it about OR : the 
 horizontal projection of P 2 will move on the perpendicular P%m ^ 
 fall from P 2 on OR, and when P 2 comes to its original place in the 
 vertical plane, its horizontal projection will have moved along P z m 
 up to its intersection p with the ground line. The vertical trace P 
 will therefore be on the perpendicular pP to the ground line, but being 
 also on the arc of circle P t P, it is at their intersection. 
 
 Having now obtained the trace P of the edge OP on the vertical 
 plane, the dihedral angle C is found at once in PQR, since both faces 
 are perpendicular to the vertical plane. 
 
26 
 
 DESCRIPTIVE GEOMETRY. 
 
 Generally, only one angle is required : in making the construction, 
 the edge corresponding to this angle is placed perpendicular to the 
 ground line. 
 
 Should the other angles be wanted, A could be obtained from the 
 triangle pmP revolved around Pp on the vertical plane; Pm^p is 
 the angle A of the spherical triangle. B is constructed as explained 
 in 29 or by any other method. 
 
 39. GIVEN TWO SIDES AND THE INCLUDED ANGLE, TO FIND THE REMAIN- 
 ING SIDE AND ANGLES. Let a, I and (7, be given ; required c, A and B. 
 
 Place the intersection of a and b in OQ, (Fig. 54), perpendicular to 
 XY, and the face b on the ground plane, draw QP making the angle 
 PQY equal to C : QP is the vertical trace of the face a. Make the 
 angle QOP equal to ' : QOP l is the face a of the trihedral angle 
 revolved about OQ on the ground plane. Taking QP equal to QP l , 
 the point P is the vertical trace of the third edge of the trihedral angle. 
 
 To obtain A, let fall from P and p the perpendiculars Pp and pm to 
 XY and OR respectively. Revolve about Pp on the vertical plane the 
 triangle formed in space by Pp and pin : Pm^p is the angle A. 
 
 Then produce pm and take mP 2 equal to m l P : join OP 2 : ROP 2 
 is c. B is obtained as explained in 29. 
 
 40. GIVEN TWO ANGLES AND THE SIDE OPPOSITE ONE OF THEM, TO FIND 
 THE REMAINING SIDES AND ANGLE. Let a, A, and B be given : required 
 C, b and c. 
 
 Place the face c on the ground plane and the intersection of a and c 
 in OP, (Fig. 55), perpendicular to the ground line. Through P draw 
 
 PQ making with XYthe angle 
 B ; PQ is the vertical trace of 
 the face a, since a and c are 
 both perpendicular to the ver- 
 tical plane. Draw OQ- mak- 
 ing the angle a with PO ; 
 POQi is the face a of the tri- 
 hedral angle revolved upon the 
 ground plane about OP as an 
 axis, Qi is the revolved ver- 
 tical trace of the edge OQ. 
 Making then PQ equal to PQ l 
 gives the trace Q. Through Q 
 
 tig*, oo draw Qm^ making the angle A 
 
 with XY, and let fall the perpendicular Qq to XY. From q as a centre 
 and qm l as radius, describe the arc of circle rtun^ and through draw 
 the tangent OR to the circle. The angle of the plane ORQ with 
 the ground plane is equal to A, therefore ROQ is the face b of the 
 trihedral angle and POR is the face c. 
 
 b and the angle C are obtained as in former cases. 
 
UNIVERSITY 
 
 SPHERICAL TRIANGLES. 
 
 Fig:. 56 
 
 41. GIVEN TWO SIDES AND THE ANGLE OPPOSITE ONE OP THEM, TO 
 FIND THE REMAINING SIDE AND ANGLES. Let a, b and B be given : 
 required A, C and c. 
 
 Place the face a on the ground plane with the intersection of a and 
 b in OP, (Fig. 56), perpendicular to XY. Make POR and POQ l equal 
 
 to a and b respectively : POQ l 
 is the face b of the trihedral 
 angle revolved about OP on the 
 ground plane, therefore the ver- 
 tical trace Q of the edge oppo- 
 site to a is on a circle described 
 from P as a centre with PQ l as 
 radius. Through P pass a 
 plane perpendicular to OR : its 
 horizontal trace is a line Pm 
 perpendicular to OR and its 
 vertical trace the perpendicular 
 PS to XY. The intersections 
 of this plane with the two planes of projection and the plane of the 
 face c, form in space a triangle SPm in which P is a right angle and 
 m is the angle B. Revolving this triangle about SP upon the ver- 
 tical plane, in SPm lt the point S is obtained. But S is a point of 
 the vertical plane of projection and is also a point of the plane of the 
 face c, therefore it is a point- of the trace of the last plane. Joining 
 then fiS, the intersection of this line with the circle Q 1 Q is the ver- 
 tical trace of the edge of the trihedral angle opposite to a. 
 A, C and c are now constructed as in former cases. 
 
 42. OTHER CASES SUPPLEMENTARY TRIANGLES. The other cases of 
 spherical triangles are generally solved by the use of the supplementary 
 
 triangles or trihedral angles. The direct 
 solution, although possible, is not so con- 
 venient. The angles A lf S 1) C l) of the 
 supplementary triangle are the supple- 
 ments of the sides a, 6, c, of the other 
 triangle and the sides a ^ , b l , c l , of the sup- 
 plementary triangle are the supplements 
 of the angles A, B y (7, of the other one. 
 From any point 0, (Fig. 57), in the 
 interior of the trihedral angle, let fall 
 perpendiculars OS, OT, 0V, on the 
 faces. The angle of OT and V is the 
 supplement of the angle of the planes 
 to which they are perpendicular. But 
 the angle of the planes is the angle B 
 of the trihedral angle; therefore TOY 
 or b l is equal to 180 B. 
 
DESCRIPTIVE GEOMETRY. 
 
 Similarly : TOS=c l =lQO C 
 
 The plane TOY containing perpendiculars to a and c, is 
 perpendicular to both ; therefore, it is perpendicular to their inter- 
 section DQ and conversely DQ is perpendicular to TO V. For the 
 same reasons DR is perpendicular to TOS and DP to VOS. There- 
 fore the angle of DQ and DR or a, is the supplement of the angle 
 formed by VOT and TOS or A l ; 
 
 A 1 =ISO C a. 
 
 In the same manner, it may be shown that : 
 
 J? 1= 180 b, 
 and C 1 =180 c, 
 
 Hence the trihedral angle OTSY is the supplementary angle of 
 DPQR. 
 
 43. REDUCTION OF AN ANGLE TO THE HORIZON. The reduction of an 
 angle to the horizon is an application of the solution of spherical 
 triangles. "When an angle is observed between two points which are 
 not in the horizontal plane of the observer, the observed angle requires 
 a correction to reduce it to the angle formed by the projections of the 
 points on the ground plane. For that purpose the observer measures 
 the angular elevations or depressions of the points. 
 
 Take as vertical plane of projection the plane passing through the 
 observer and one of the points. Assume any point P, (Fig. 58), as the 
 place of observation and draw through it the lines PA and PB^ making 
 with the ground line angles equal to the elevations or depressions and 
 ft of the points. 
 
 ? ^ 
 
 The lines PA, PB and the vertical Pp 
 form a trihedral angle in which the faces 
 are 90 a, 90 ft and the observed angle. 
 A pyramid is cut off this trihedral angle by 
 the ground plane, the base of the pyramid 
 Y being the triangle p AB, in which p A and 
 p B are two sides and p the observed angle 
 reduced to the horizon. The third side 
 may be found by revolving the face APE 
 of the pyramid around AP, upon the 
 vertical plane. This face will come in 
 APB, the angle at P being the observed 
 Fig-. 58 angle'and 
 
 We have now the th'rd side of the triangle AB : hence describing 
 arcs of circles from A and p as centres with AB 2 and pB l as radii 
 respectively, their intersection is the point B and ApB is the required 
 angle. 
 
DEFINITIONS. 29 
 
 CHAPTER II. 
 
 PERSPECTIVE. 
 
 44. GENERAL REMARKS. Perspective is that branch of Geometry 
 which treats of the representation by figures drawn on a surface of 
 objects placed beyond it. Generally this surface is a vertical plane ; 
 it is called "picture plane" The figures drawn on it, according to the 
 rules of perspective, produce on the eye, as far as form is concerned, 
 the same impression as the objects themselves seen in their actual 
 places. 
 
 Suppose a transparent plane surface, such as glass, placed between 
 the eye and the objects to b-^ represented. If the outlines of the objects 
 seen through the glass could be traced on it, the image thus formed 
 would be an exact perspective. 
 
 Consider the visual ray from the eye to a point of space : this ray 
 pierces the picture plane in a second point, which is called the "per- 
 spective " of the first one. 
 
 The visual rays from the eye to all the points of a straight line form 
 a plane whose intersection with the picture plane is the perspective of 
 the line. Consequently, the perspective of a straight line is another 
 straight line. 
 
 When the line is a curve, the visual rays to its various points form 
 a conic surface whose vertex is at the eye and whose intersection with 
 the picture plane is the perspective of the curve. A surface of the 
 same nature is formed by the visual rays tangent to the visible outline 
 of an object ; the perspective of the object is the intersection of this 
 surface by the picture plane. 
 
 45. DEFINITIONS. The " ground plan" is the horizontal projection 
 of the objects to be represented ; thus for the perspective of a landscape, 
 the ground plan is the topographical plan of the ground ; for a building, 
 it is the horizontal or ground plan of the building (ABCD, Fig. 59). 
 
 The " ground plane " is the plane on which the ground plan is 
 placed (KXsY, Fig. 59). For a landscape, it may be, for instance, the 
 horizontal plane passing through the datum point of the topographical 
 plan and for a building, the basement or first floor plane. Any 
 horizontal plane may, however, be used as ground plane, provided its 
 altitude be taken into account ; the ground plan does not change, 
 whatever the altitude may be. 
 
30 
 
 PERSPECTIVE. 
 
 The " elevation " is the vertical projection of an object ; the elevations 
 of a building are those plans of the building which show the front, 
 rear, or sides. 
 
 The " picture plane ," as already explained, is the plane on which the 
 perspective is drawn (FFXY, Fig. 59). Generally, it is vertical and 
 placed between the eye and the object to be represented, but none of 
 these rules is absolute. Perspectives are sometimes drawn on planes 
 which are not vertical and objects are represented which are between 
 the picture plane and the eye. Such a position of objects is the rule 
 and not the exception In perspectives used for surveying, when they 
 are taken as representations not of the ground itself, but of a model 
 of it reduced to the scale of the map. This convention will be found 
 further on. Objects are even represented which are behind the 
 observer, the origin of light, for instance, in the construction of 
 shadows, but this is merely a geometrical conception to which the 
 usual definition of a perspective does not apply. 
 
 The " ground line " is the intersection of the ground and picture 
 planes (XT, Fig. 59). 
 
 The " station " is the 
 point supposed to be 
 occupied by the eye of 
 the observer. (S.l?ig.5$). 
 The "foot of the 
 station " is the point 
 where the vertical of 
 the station pierces the 
 ground plane, (s. Fig. 
 59). 
 
 The " principal point" 
 is the foot of the per- 
 pendicular drawn from 
 the station to the 
 picture plane ; it is 
 shown in P, Fig. 59. 
 
 Fig-. 59 
 
 The " distance line " is the line between the station and the principal 
 point (SP, Fig. 59). It length is the distance from the station or 
 from the foot of the station to the picture plane. 
 
 The " horizon plane " is the horizontal plane passing through the 
 station. It contains the distance line and cuts the picture plane on a 
 horizontal line passing through the principal point and called "horizon 
 line " (If If, Fig. 59). The distance between the horizon line or the 
 principal point and the ground line is equal to the altitude of the 
 station. 
 
POINT IN GROUND PLANE. 
 
 31 
 
 The " principal plane " is the vertical plane perpendicular to the 
 picture plane and passing through the station (SNQ, Fig. 59). It 
 contains the foot of the station, the principal point and the distance 
 line. 
 
 The "principal line " is the intersection of the principal and picture 
 planes (QN, Fig. 59). It is perpendicular to the ground and horizon 
 lines and intersects the latter at the principal point. 
 
 A "front plane " is a plane parallel to the picture plane. 
 
 A "front line " is any line contained in a front plane, therefore any 
 line parallel to the picture plane. 
 
 :N 
 
 In Fig. 60 these points, lines and planes are 
 represented by their orthogonal projections ; the 
 Y ground plane is taken for horizontal plane and the 
 picture plane for vertical plane ; ss is the station, 
 s the foot of the station, s or P the principal point, 
 HH' the horizon line, sp, s' the distance line and 
 spN the principal plane. 
 
 Fig-. 60 
 
 46. PERSPECTIVE OF A POINT IN THE GROUND PLANE. Let XYs, 
 (Fig. 61), be the ground plane, XYN the 
 picture plane. S the station and M a point 
 in the ground plane. The perspective of 
 M on the picture plane is the point where 
 the straight line SM pierces the picture 
 plane, that is the vertical trace of SM, M 
 being the horizontal trace. Thus we have 
 the first relation between a point of the 
 Fig". 61 ground plane and its perspective ; they are 
 
 the traces of the visual ray on the ground and picture planes respec- 
 tively. 
 
 v- L ' 
 
 A m' 
 
 Fig. 62 represents in orthogonal projection the 
 construction of Fig. 61 ; ss' is the station, M the 
 point of the ground plane sa, s'm the visual 
 ray and <>. the perspective of M. The points M 
 and /JL are the traces of- sa, s'm. 
 
 Fig-. 62 
 
32 
 
 PERSPECTIVE. 
 
 47. PERSPECTIVE OF A LINE IN THE GROUND PLANE. It has been 
 shown in 44 that the perspective of a straight line is the inter- 
 
 section with the picture plane of the 
 plane containing the station and the 
 given line. 
 
 Draw a plane through the straight 
 line AB and the station S (Fig. 63). 
 The intersection up of this plane with 
 the picture plane is the perspective of 
 AB. Thus we have this relation be- 
 tween a straight line in the ground 
 
 I..- 1 :-- plane and its perspective ; they are the 
 
 rig", bo traces on the ground and picture planes 
 
 of the plane containing the station and the line itself. 
 
 In orthogonal projection, the line being in the ground plane, the 
 
 horizontal projection is the line itself, 
 AB (Fig. 64) ; the vertical projection is 
 the ground line. To pass a plane through 
 the station sP and the line A, draw 
 through sP a parallel to AB ; the hori- 
 zontal projection is a parallel to AB 
 through s, and the vertical projection a 
 parallel through P to the ground line. 
 The vertical trace is at c', the intersec- 
 tion of c'P with the perpendicular c'c to 
 the ground line. The horizontal trace 
 f the plane containing sP and AB is 
 the line AB itself, since it is in the 
 
 Fig". 64 
 
 ground plane. The vertical trace passes through c', trace of the line 
 sc, PC', which is contained in the plane ; and as it must also pass 
 through A, therefore the vertical trace of the plane is the line Ac'. 
 Hence Ac' is the perspective of AB. 
 
 48. PERSPECTIVE OF A POINT NOT IN THE GROUND PLANE. The con- 
 
 struction given in $ 46 does not change when 
 
 ,,.-'; the point to be placed in perspective is not in 
 
 ^5-'' the ground plane. A line is still drawn through 
 
 the station sP, (Fig. 65), and the point mm. 
 
 x ^T~ ~~^ Y The vertical trace , is the perspective of mm. 
 
 \ ^' The horizontal trace a of the visual ray is 
 
 ; .,-' the perspective of the point mm on the ground 
 
 ' plane ; hence it may be stated as a general 
 
 rule, that the perspectives of a point on the 
 
 ground and picture planes are the traces of the 
 
 rig", bo }- ne j o i n i n g t ne station to the point. 
 
VANISHING POINT. 
 
 33 
 
 49. PERSPECTIVE OF A LINE NOT 
 
 IN THE GROUND PLANE. Let ab, 
 
 a'b', (Fig. 66), be a line not in 
 the ground plane : to obtain its 
 perspective, a plane must be 
 passed through the station sP 
 Y and the line ab, a'b'; the inter- 
 section of this plane with the 
 picture plane, that is the verti- 
 cal trace of the plane, is the per- 
 spective of the line. 
 
 Through sP, draw a parallel 
 
 Fig-. 66 sd, Pd' to ab, a'b'; both lines 
 
 are contained in the plane to be 
 
 drawn, therefore the traces of the plane are the lines ac, d'b' joining 
 the traces of same denomination of the parallels ; and d'b', the vertical 
 trace of the plane, is the perspective of ab, a'b'. 
 
 Let us now consider another line, ef, e'f parallel to ab, a'b'; the 
 plane passing through ef, e'f arid the station sP must again contain 
 the parallel sd, Pd, through the station ; therefore the vertical trace 
 of the plane, which is the perspective of ef, e'f, is the line/W joining 
 the vertical traces of the two parallels. Hence the perspective of any 
 line parallel to ab, a'b' will pass through the point d'. This result 
 could be foreseen, because when a system of parallels has to be placed 
 in perspective, all the planes serving to project them on the picture 
 plane have a common line of intersection, parallel to the general 
 direction of the system and passing through the station. Its trace 
 on the picture plane must therefore be the common point of inter- 
 section of the perspectives. This point is called the " Vanishing 
 point " of the parallel lines, because it represents the parts of the 
 lines which are at infinity ; their perspective ends or vanishes at that 
 point. 
 
 The horizontal traces of the planes are the perspectives of the par- 
 allel lines on the ground plane. Like the perspectives of the picture 
 plane, they all meet in a common point, which is the horizontal trace 
 of the parallel line through the station ; it is the vanishing point of 
 the perspectives of the ground plane. Therefore, it is seen that when 
 a plane is drawn through the station and a line in space, the traces of 
 the plane on the picture and ground planes are the perspectives of the 
 line on those planes. 
 
 3 R 
 
34 PERSPECTIVE. 
 
 50. POSITIONS OP THE VANISHING POINT. A horizontal line has its 
 vanishing point on the horizon line because the parallel drawn through 
 the station, being horizontal, is all contained in the horizon plane and 
 has its vertical trace on the horizon line. 
 
 Perpendiculars to the picture plane, being parallel to the distance 
 line, have for vanishing point the vertical trace of the distance line 
 which is the principal point of the perspective. 
 
 The vanishing points of horizontal lines making an angle of 45 with 
 the distance line are called " distance points," DD, (Fig. 67) ; their 
 distance from the principal rjbint is equal to the distance line, because 
 a horizontal line inclined at 45 to SP, forms an isosceles triangle SPD 
 in which SP = PD. 
 
 Lines in the principal plane have their van- 
 
 7^ ishing point on the principal line. Two of 
 
 /' these lines form angles of 45 with the distance 
 
 / line, one above and the other below the hori- 
 
 zon. Their vanishing points are known as 
 " upper and lower distance points "; they are 
 also at the same distance from the principal 
 7 point as the station. 
 
 Lines parallel to the picture plane have no vanishing point. It will 
 be shown later on that their perspectives are parallel to the lines them- 
 selves and do not meet. 
 
 51. VANISHING LINE. Through the station 
 sP (Fig. 68), pass a plane TVZ parallel to a 
 given plane QRM. The vertical trace VZ 
 contains the traces of all the lines drawn 
 through the station parallel to QRM, it is 
 therefore the locus of the vanishing points of 
 parallels to the plane QRM. This trace VZ 
 may be called the " vanishing line " of the 
 plane QRM or of any other plane parallel to 
 it (1). 
 
 The horizontal trace VT is in like manner 
 the vanishing line of the perspectives of the 
 ground plane. 
 
 Fig-. 68 
 
 (1) The term "vanishing line " is usually applied to the 
 lines : admitting that the expression " vanishing point 
 cannot be called otherwise than 
 acceptation only. 
 
 perspectives of parallel 
 
 anishing point " is a proper one, the line VZ 
 vanishing line." This term is used here with that 
 
FIGURES IN FRONT PLANES. 
 
 35 
 
 Fig-. 69 
 
 52. LINES OR FIGURES IN FRONT PLANES. The perspective of a 
 straight line contained in a front plane is another straight line parallel 
 to the first one. For the plane containing the station and the given 
 line being cut by two parallel planes, the picture and front planes, the 
 intersections are parallel lines. But these intersections are the line 
 itself and its perspective, therefore the perspective is parallel to the 
 given line. 
 
 Let &, (Fig. 69), be the station, 
 PP',FF', the picture and front planes 
 and A BCD a polygon in the front 
 plane, Join SA, SB, SC, SZ> ; these 
 lines intersect the picture plane at 
 8 a,b,c,d t the polygon abed being the 
 perspective of ABCD. The lines 
 drawn from S form a pyramid of 
 which the polygon of the front plane 
 is the base and the perspective a 
 section by a plane parallel to the 
 base. It is shown in geometry that 
 when a pyramid is cut by a plane 
 parallel to the base, the section is a 
 figure similar to the base. The front 'plane being parallel to the 
 picture plane, the perspective must be similar to the original figure. 
 
 It follows that a curve in the front plane is represented by a sim- 
 ilar curve in perspective, because such a line can be assimilated to a 
 polygon with a great number of sides. 
 
 When the front plane is beyond the picture plane, as in Fig. 69, 
 the perspective is smaller than the original figure ; it is larger when 
 the front plane is between the station and the picture plane, but in 
 either case it is an exact representation of the figure itself, on a differ- 
 ent scale. This scale, or the proportion between the perspective and 
 the original figure is called the "scale of the front plane." It is the 
 ratio between the distance line and the distance from the station to 
 the front plane. 
 
 A straight line parallel to the picture plane is contained in a front 
 plane and is represented in perspective by a line parallel to itself \ 
 therefore parallel lines, which are also parallel to the picture plane 
 have parallel lines for perspectives and have no vanishing point. The 
 parallel to the given lines passing through the station, being parallel 
 to the picture plane, has no trace on it. 
 
 Vertical lines are parallel to the picture plane and appear in per- 
 spective as parallels to the principal line. 
 
36 
 
 PERSPECTIVE. 
 
 Horizontal lines parallel to the picture plane are in perspective par- 
 allel to the horizon line. 
 
 53. MEASURING LINES AND MEASURING POINTS. Let PP', (Fig. 70), 
 be the picture plane, S the station and AB a straight line piercing the 
 
 picture plane at A. Through S, draw the 
 parallel SV to AB : V is the vanishing 
 point of AB whose perspective is VA, 
 since the vertical trace A is a point of the 
 s perspective and the vanishing point is 
 another one. 
 
 Through V, draw VM equal to VS and 
 through A the line AD parallel to VM. 
 
 Take a point O of AB and join CS, the 
 intersection f with VA is the perspective 
 of C. Join M Y and produce to its inter- 
 section Cj, with AD. 
 
 VS and AB being parallel the triangles 
 V Y S an d A Y C give the proportion : 
 
 Fig-. 70 
 
 VS 
 AC 
 
 YL 
 
 AY 
 
 (i) 
 
 The triangles VMf and AC ^ Y are a ^ so similar, VM being parallel 
 
 Vr 
 
 to AC V therefore : 
 
 Hence from (1) and (2) : 
 
 VM 
 
 AC 1 
 
 VS VM 
 
 (2) 
 
 But by construction 
 therefore 
 
 VM VS 
 
 The line AC, represented in perspective at Ay, is equal to the line 
 AC^ 
 
 Fig. 71 shows the picture plane with the same letters as in Fig. 70. 
 The part of the line seen in perspective at AY is equal to AC l . On 
 AC i take another point D, join to M, and call d the intersection with 
 VA. The line seen in Ad is equal to AD, therefore the part seen in 
 Yd is equal to C^D. 
 
SCALE. 37 
 
 M The line AD is called the "measuring 
 
 line" of A F, because it serves to measure 
 the length of the line in space correspond- 
 ing to any portion of its perspective A V ; 
 M is the "measuring point" 
 
 VM was not drawn in any particular 
 
 direction, therefore the direction of the 
 
 measuring line, parallel to VM, is indeter- 
 
 Fiff. 71 minate. It is usual to make it parallel to 
 
 the horizon line. 
 
 The position of the measuring point depends only on the vanishing 
 point ; therefore the same measuring point may serve for all lines 
 parallel to the same direction. 
 
 The same measuring line \vill serve for all lines having their vertical 
 traces on it. Should the line VM be drawn parallel to the vertical 
 trace of a plane, this trace would be a measuring line for all lines 
 contained in the plane. 
 
 If the measuring line is taken parallel to the horizon the measuring 
 point of any horizontal line is on the horizon line, since the vanishing 
 point is on that line. All lines in the same horizontal plane have 
 then for measuring line the vertical trace of the plane, and lines in 
 the ground plane have the ground line. 
 
 There is no measuring line or point for lines in a front plane, because 
 they have no vertical traces or vanishing points ; the scale of the front 
 plane has to be employed when the length of such a line is wanted. 
 
 The distance points are measuring points for lines parallel to the 
 distance line. 
 
 54. REDUCTION OF A PERSPECTIVE TO SCALE. Hitherto it has been 
 assumed that in the constructions, the real dimensions of the figures 
 were employed. It would be quite impracticable to do so in the 
 generality of cases. The dimensions must be reduced to a certain 
 scale in order not to exceed the limits of the paper. 
 
 Y M By changing the position of the measuring 
 
 line reduced distances can be used. Let F, M t 
 and AC (Fig. 72) be the vanishing and measur- 
 ing points and the measuring line of the per- 
 spective A V. The part of the line seen in ft? 
 is equal to BC. Through a point a of A V 
 / \ draw the parallel ac to AC and let us use it as 
 
 _y a measuring line ; the length corresponding to 
 
 PY is 6c, and we have the proportion : 
 
 be _Va 
 Fig". 72 ~BC = yA 
 
38 
 
 PERSPECTIVE. 
 
 Thus the lengths obtained are all reduced in the proportion of ^7" 
 
 Therefore, in order to obtain at once the length, on a certain scale, of 
 a line seen in perspective, it is sufficient to reduce the distance between 
 the measuring line and the vanishing point in the proportion of the 
 scale to be employed. Thus, if Va be made the one-thousandth part 
 of VA, the distances will be obtained on a scale of YWQTS- ^ * s ^ ne 
 measuring point, and ac the measuring line, of a line having V for 
 vanishing point and a for trace on the picture plane ; the new line is 
 therefore parallel to the line joining V to the station and to the 
 original line seen in perspective, but its distance from the station has 
 been reduced to the scale adopted. 
 
 Hence, to obtain the length reduced to scale of a line seen in per- 
 spective, reduce to scale the distance of the line from the station, 
 moving it parallel to itself in the plane containing the station. 
 
 The same conclusion is otherwise 
 arrived at in a more direct manner. 
 A figure ABCD (Fig. 73) forms, with 
 the visual rays joining it to the sta- 
 tion, a pyramid, the intersection of 
 which by the picture plane is the 
 perspective a 
 
 Fig-. 73 
 
 Let the pyramid be cut by a plane parallel to the base ABCD ; the 
 intersection A i B 1 C l D 1 is similar to ABCD, the proportion being 
 
 ~^~r. The lines A 1 lt B^C \ measured by means of their per- 
 spectives a/?, ft? are therefore the lines AB, EC reduced to 
 
 ^ A 
 the scale --r. The same demonstration applies to any system of 
 
 figures, whenever every point of the system has been moved in a 
 straight line towards the station, so as to reduce its distance from the 
 station in the proportion of the scale given. Hence we deduce the 
 following important rule : 
 
 To lay off dimensions reduced to scale or to measure them from a 
 perspective, assume that the system formed by the station and the 
 original figures or objects had been reduced to scale when the perspec- 
 tive was executed. 
 
CONSTRUCTION OF THE PERSPECTIVE. 39 
 
 55. TO PLACE IN PERSPECTIVE A POINT OP THE GROUND PLANE. 
 
 1st. By means of the principal point and a distance point. 
 
 Let M t (Fig. 74), be the point, XY the ground line, P and D the 
 principal and distance points, the picture plane 
 p D b?ing revolved upon the horizontal plane. 
 
 Draw MA at an angle of 45 J and MB perpen- 
 dicular to the ground line. The perspective 
 X,-'\ : of AM is the line AD joining the trace on the 
 
 X -f^- ^ K picture plane to the distance point. The per- 
 spective of MB, vanishing at the principal 
 point is PB ; therefore the perspective of M 
 Fig". 74 is at //. 
 
 2nd. By means of the distance of the point from the ground line. 
 
 Draw MB perpendicular to XY and take AB equal to MB; join 
 AD and PB. 
 
 3rd. By means of the station and principal point. 
 
 C 
 
 | /| Join the foot of the station s (Fig. 75) to the 
 
 j \ point M. The line sM is the horizontal trace of 
 
 !\ / the vertical plane containing M and the station, 
 
 j /\ which plane cuts the picture plane on a line AC 
 
 (/ \! perpendicular to XY. From M draw the perpen- 
 
 ' A B dicular MB to the ground line ; it is represented 
 
 in perspective by PB, therefore // (intersection of 
 
 AC and PB) is the perspective of M. 
 
 Fig-. 75 
 
 4th. By means of the projection on the principal plane. 
 
 Revolve the principal plane around its trace sp (Fig. 76) upon the 
 M ' ground plane ; the station will come in S^ on 
 
 a perpendicular to sp, s<S l being equal to the 
 altitude of the station. Draw Mm perpen- 
 dicular to sm and join Sm ; it is the projec- 
 tion on the principal plane of the visual ray 
 Ajf - Y from the station to the point M, and its 
 intersection // a with p Yis the projection of the 
 perspective of M revolved upon the ground 
 plane. Join sM and at m erect a perpen- 
 \ dicular m>j. to the ground line ; the perspec- 
 ....... '? tive of M is on that perpendicular at a dis- 
 
 tance m>j. equal to pi>. l . 
 Fig-. 76 
 
40 
 
 PERSPECTIVE. 
 
 When a great number of points have to be placed in perspective, this 
 last method is very convenient. In practice the perspective is not 
 constructed on the ground plan itself, as the operations would become 
 confused ; the plan and perspective are kept separate. 
 
 Let AJ3CD, (Fig. 77), be the ground plan, X 2 Y 2 the ground line, and 
 the trace of the principal plane. Join s 2 to A, , C and D. 
 
 On the paper which is to 
 serve for the perspective, 
 draw the ground line XY 
 and take a point p as inter- 
 section of the principal 
 Y plane. Take on the edge of a 
 piece of paper the distances 
 from p z to 3 , /S 3 , f 3 , f) 3 , and 
 carry them on XY in 2 , /? 2 , 
 T 2 , ^ 2 ; at the last mentioned 
 points erect perpendiculars to 
 the ground line. 
 
 At another place draw a line 
 s l 1 to represent the inter- 
 section of the ground and 
 principal planes ; place the 
 station S^ at its height h above 
 the ground plane, take s^p^ 
 equal to the distance line and 
 draw the trace of the picture 
 
 plane, 
 
 Fig-. 77 
 perpendicular to 
 
 On the edge of a piece of paper, take the distances of A, B, C, D, 
 from the ground line X 2 Y 2 and carry them onp l B 1 . Join S l to A lt 
 lt C\, D^. Again take on the edge of a piece of paper the distances 
 of ctj, &!, n, ^i f rom Pit an( i lav them on the perpendiculars 
 a 2 a, 2 /9, ^2^, ^ 2 ^. This gives a/5^<? as the perspective of ABCD. 
 
 56. TO PLACE IN PERSPECTIVE A LINE OR FIGURE OF THE GROUND 
 
 PLANE. A line of the ground plane may be placed in perspective by 
 determining the perspectives of two of its points. 
 
 When the vanishing point is known, only one additional point is 
 required to define the perspective. 
 
 With a figure composed of straight lines, the perspectives of the 
 points of intersection are fixed and joined together by straight 'lines. 
 
CONSTRUCTION OF THE PERSPECTIVE. 41 
 
 The perspective of a curve is found from the perspectives of a 
 sufficient number of points or by tangents to the curve. 
 
 57. TO PLACE IN PERSPECTIVE A POINT OUTSIDE OF THE GROUND 
 
 PLANE. When a point is not in the ground plane, the perspective of 
 its horizontal projection is first" found : the height of the point above 
 or below the ground plane is next reduced to the scale of the front 
 plane and laid on the vertical of the perspective previously found. 
 
 * Let m, (Fig. 78), be the projection of the point on 
 
 the ground plane and B the perspective of m, ob- 
 * X V --.y. tained as in 55. From m let fall the perpendicular 
 
 \ I "f 1 mA on XT and take Am equal to the height of the 
 \'jj point. Join m to the principal point, P ; Pm is the 
 
 \ I perspective of the perpendicular through M to the 
 
 X Y picture plane, therefore the perspective of M must be 
 on Pm'. But the given point M is on the vertical 
 line passing through m whose perspective is the per- 
 Fig*. 78 pendicular B;* to the ground line, therefore the per- 
 
 spective of the point M is at the intersection // of the two lines. 
 
 Comparing Fig. 78 with Fig. 65, 48, it will be seen that the con- 
 struction is precisely the same, although made on different principles. 
 
 58. TO PLACE IN PERSPECTIVE A LINE OUTSIDE OF THE GROUND 
 
 PLANE. When a line is in a horizontal plane, that plane may be taken 
 temporarily as ground plane and changed when the perspective has 
 been obtained. 
 
 If in any other plane, the perspective may be found by means of 
 the vanishing point and horizontal trace. The latter is placed in 
 perspective as explained in 55, and joined to the vanishing point. 
 
 For lines in front planes, one point of the line is placed in per- 
 spective and through it, a parallel to the line is drawn. 
 
 59. THE DISTANCE LINE is AN AXIS OF SYMMETRY OF THE PER- 
 SPECTIVE. A perspective is symmetrical with reference to the distance 
 line, all points of the picture plane at the same distance from the prin- 
 cipal point having the same geometrical properties. Therefore, any 
 plane perpendicular to the picture plane may be taken as ground plane, 
 or any line through the principal point as horizon line. So when 
 figures are contained in. a plane perpendicular to the picture plane, 
 their perspectives can be obtained by taking the plane of the figures 
 for ground plane and its vertical trace for ground line. 
 
42 PERSPECTIVE. 
 
 60. GIVEN THE HEIGHTS OF TWO POINTS AND THEIR PERSPECTIVES, TO 
 FIND THE VANISHING POINT AND TRACE ON PICTURE PLANE OF THE LINE 
 JOINING THE GIVEN POINTS. Let HH and P, (Fig. 79), be the horizon 
 line and principal point, a and ft two points of the perspective. Draw 
 EF parallel to the horizon line at a distance 
 equal to the height of a ; it is the trace of 
 /T the horizontal plane containing the point of 
 
 space seen in . 
 H 
 
 The perspective of the perpendicular to 
 the picture plane passing through this point 
 is Pa : its vertical trace is C. Draw CD 
 perpendicular to the horizon line and equal 
 Fig\ 79 to the height of ft above the plane of . D is 
 
 a point of the picture plane at the same 
 
 height as /5, and PD is the perspective of the perpendicular to the pic- 
 ture plane passing through D ; PD is in the same vertical plane as 
 PC and if produced will meet the vertical of a seen in perspective at 
 aB. The point of intersection A is at the same height as D and , 
 therefore Aft is a horizontal line and its vanishing point is on the 
 horizon line at G. But Aft and a. ft are in the same vertical plane having 
 for vertical trace the perpendicular G V to the horizon line, therefore 
 the vanishing point of aft is at its intersection V with GV. 
 
 To find the trace, draw through D the parallel DL to the horizon 
 line : it is the trace on the picture plane of the horizontal plane con- 
 taining AG, and the trace of AG is at its intersection L with DL. 
 But A G and a V being in the same vertical plane, the trace of a V is in 
 M, on the perpendicular LM to the horizon line. 
 
 aG is a horizontal line also in the same vertical plane as AG and a V : 
 consequently its trace is K, on LM produced. But aG is in the hori- 
 zontal plane whose trace is EF, therefore K is the intersection of aG 
 and EF. 
 
 61. To FIND THE INTERSECTIONS OF A VERTICAL LINE BY A SERIES 
 
 OF HORIZONTAL PLANES. Let HH' and FG, (Fig. 80), be the horizon 
 and principal lines of a perspective, /JL a point of the perspective yO of 
 a vertical line, of which the altitude above or below the station is 
 known. Take PM equal to this altitude : M is a point of the picture 
 plane at the same altitude as //. Join >J.M : it is the perspective of 
 
INTERSECTIONS BY CONTOUR PLANES. 
 
 43 
 
 Fig-. 80 
 
 a horizontal line having its trace in Mand 
 its vanishing point at V. Mark on FG 
 the intersections A, J3, C, D, E of the 
 horizontal planes, join to V and produce 
 VA, VJ3, VC t VD, F^to yy/ these lines 
 are the perspectives of horizontal lines 
 parallel to >j.M and contained in the hori- 
 f zoiital planes. Their intersections , /9, y, 
 ~ H ' d, s, with the perspective of the vertical 
 line tfl are the points required. 
 
 This construction is employed for de- 
 termining the intersections of a vertical 
 line by contour planes : the equidistance 
 is marked on the edge of a piece of paper 
 which is pinned along GF so that P cor- 
 
 responds to the altitude of the station. A straight edge is placed on //. 
 and the point of same height of the equi- 
 distance scale, then a pin is planted at F and 
 the straight edge moved through each of the 
 points A, B, . . . .E, always keeping it in con- 
 tact with the pin. 
 
 Another solution consists in projecting the 
 vertical line and its perspective on the prin- 
 --'.\^5 cipal plane. 
 
 ^--' Let SP, Fig. 81, be the distance line, r { the 
 
 principal line and FG the intersection of the 
 front plane containing the vertical line, by the 
 principal plane. Mark on FG the intersections 
 pi- .o-i of the horizontal planes, join to S and pro- 
 
 duce to r t O ; the intersections are the projections 
 on the principal plane of the points required. 
 
 In practice, the construction is made 
 on the perspective : r t O (Fig. 82), being the 
 perspective of the vertical line, NM is 
 taken on the horizon line equal to the dis- 
 tance line and NQ equal to the distance 
 of the vertical line from the picture plane. 
 At Q a perpendicular is erected to HH' 
 and the equidistance scale pinned along- 
 side, so that Q shall correspond to the 
 altitude of the station. The construction 
 is completed as in Fig. 81. 
 
 Fig-. 82 
 
PERSPECTIVE. 
 
 Contour planes being equidistant, the divisions , f$ of the 
 
 perspective are equal : it is therefore sufficient to find the length of 
 one division and to carry it on the perspective of the vertical line. 
 
 62. To MARK ON THE PERSPECTIVE OF ANY LINE OR CURVE CONTAINED 
 IN A VERTICAL PLANE, THE INTERSECTIONS BY A SERIES OF HORIZONTAL 
 
 PLANES. Let ;j.d (Fig. 83), be the perspective of a line contained in a 
 vertical plane : that plane contains the vertical seen in perspective at 
 <y/, perpendicular to the horizon line. Mark the points of division 
 A,J3,C, of dM by the horizontal planes, (61) and join // to the point 
 
 of the perspective SM of same altitude. 
 This being the perspective of a hori- 
 zontal line, its vanishing point is V. 
 Join V to A,B,C : these lines are the 
 perspectives of parallels to My., there- 
 fore they are in the plane fij/d and 
 . intersect the curve seen in perspective 
 at fj.8, but they are also contained in 
 the horizontal planes, hence , /9, 7% 
 are the points required. 
 
 Instead of first dividing the vertical 
 line 8M, the trace on the picture plane 
 and vanishing point of p-M may be 
 determined as in 60 and the points 
 of intersection marked at once on the 
 line 'J-d by placing the equidistance 
 scale on the perpendicular to the horizon line passing through the 
 vertical trace and joining the points of division to the vanishing point. 
 
 When the horizontal projection of the line is known, the vanishing 
 point and trace are obtained as follows : let a/9, (Fig. 84), be the per- 
 spective of the line, ab its horizontal projection, HH' the horizon and 
 
 XY the ground line. The intersection of 
 the ground plane by the projecting plane 
 containing the line seen at ap is ab ; the 
 trace on the picture plane of this intersec- 
 tion is at E, where ab produced meets XY. 
 Through the foot of the station s, draw 
 sv parallel to ab and v V perpendicular to 
 XY, meeting the horizon line in V. 
 V is the vanishing point of parallels to 
 ab. But the intersections of the hori- 
 zontal planes by the plane of /3, being 
 parallel to ab, V is their vanishing point; 
 and since they are all in the ^ame vertical plane, their traces are 
 on the vertical ED of the picture plane. Hence the equidistance 
 
 Fig-.'SS 
 
 H 
 
 Fig-. 84 
 
INTERSECTIONS BY CONTOUR PLANES. 45 
 
 scale is to be placed along ED, taking care that the point E of 
 the scale corresponds to the altitude of the ground plane ; the divi- 
 sions of the scale are joined to the vanishing point and produced to 
 their intersection with the perspective. 
 
 63. TO MARK ON THE PERSPECTIVE THE INTERSECTIONS OP A PLANE, 
 
 LINE, OR CURVE, BY A SERIES OF HORIZONTAL PLANES. The intersections 
 of a plane by a series of horizontal planes are horizontal lines parallel 
 to the trace, on the ground plane, of the plane intersected ; the vanish- 
 ing point of these lines is the point of intersection of the horizon line 
 by a parallel to this trace drawn through the station. 
 
 Let af (Fig. 85) be the perspective of a line or curve in the plane 
 POP, XY the ground line and HE' the 
 horizon line. Through the foot of the 
 station s draw sv parallel to OP, and 
 erect the perpendicular vV to the ground 
 r , line, meeting the horizon line in V : V is 
 the vanishing point of horizontal lines in 
 the plane POP', and, consequently, of the 
 intersections of that plane by the horizon- 
 tal planes. The traces of these lines on 
 the picture plane are on OP' and the ver- 
 tical distance between them is that of the 
 horizontal planes : therefore place at 0, on 
 a perpendicular to the ground line, the 
 Fiff 85 distances of the horizontal planes or the 
 
 scale of equidistance, draw parallels to the 
 
 ground line through the divisions A, B, C of the scale, and join A', B ', 
 C' to the vanishing point. These lines are the perspectives of the 
 intersections of the plane POP' by the horizontal plants. 
 
 64. INTERSECTIONS OP A PRISM, PYRAMID, OR CONIC SURFACE, BY A 
 SERIES OF HORIZONTAL PLANES. The intersections of a prism or 
 pyramid by a series of horizontal planes can be drawn on the per- 
 spective by determining the intersections of the edges of the prism or 
 pyramid by the planes and joining the corresponding points by straight 
 lines. 
 
 A similar process can be applied to a conic surface by using generat- 
 ing lines instead of edges, and also by employing tangents to the inter- 
 sections, parallel to the tangents drawn to the curve of the ground 
 plane forming the base of the cone. 
 
46 PERSPECTIVE. 
 
 65. To PLACE A POINT OF THE GROUND PLANE BY MEANS OF ITS 
 
 PERSPECTIVE. To restore a figure by means of its perspective is the 
 converse of perspective. Let us consider first the case of a point of 
 the ground plane ; its place can be found by inverting any of the 
 constructions given in 55. 
 
 For instance, in Fig. 74, the perspective ;j. of the point is joined to 
 the principal and distance points, P and D. At B a perpendicular 
 BM is erected to the ground line and from A a line AM is drawn at 
 an angle of 45 with the ground line. M is the point of the ground 
 plane. 
 
 In Fig. 75, join P/JL and draw /J.A and BM perpendicular to the 
 ground line. Join sA and produce to intersection with BM. 
 
 In Fig. 76, p^i is taken equal to the distance ,u.m of the perspective 
 /JL from the ground line and /JL I is joined to the station S 1 revolved on 
 the ground plane. The foot of the station s, is joined to the foot of 
 the perpendicular /tni to the ground line and the point M is at the 
 intersection of sm produced with the parallel to the ground line m'M. 
 
 When a number of points have to be placed, the constructions are 
 made as in Fig. 77, but in inverse order. The perspective u-pyfi is 
 given ; the distances pa 2t pfi 2 , .......... are carried on X 2 Y 
 
 2 
 
 
 
 Then s 2 is joined to a 3 j3 3 ........ and on 
 
 these lines produced the points A, J3, C, D, are so placed that their 
 distances from the ground line are equal to p-^A^, p l l , p 1 C l , p- l D l . 
 
 66. TO PLACE A LINE ON THE GROUND PLANE BY MEANS OF ITS PER- 
 SPECTIVE. The trace of the line on the picture plane is the point 
 where its perspective intersects the ground line : this point is common 
 to the perspective and to the line (/5, Fig. 86). 
 
 The vanishing point, F, is the intersection of 
 the perspective by the horizon line ; it gives the 
 direction of the line of the ground plane. 
 
 From V, let fall the perpendicular Vv to XY 
 and join vs. Through /? draw J3A parallel to sv ; 
 \ it is the required line. 
 
 > 
 
 . In the case of a front line, a point of the line 
 
 Fig 1 . 86 ig fixe( | k v one O f tne me thods of 65 and a 
 
 parallel to the ground line drawn through the point. 
 
INVERSE PROBLEM OF PERSPECTIVE. 
 
 47 
 
 67. To DRAW A FIGURE ON THE GROUND PLANE BY MEANS OF ITS PER- 
 SPECTIVE. A figure of the ground plane may be constructed by 
 means of its perspective as described in 65, each of the summits of 
 the figure being determined separately. 
 
 It may also be constructed by determining each of the lines form- 
 ing the figure, as in 66. 
 
 An irregular figure is inclosed between straight lines and drawn at 
 sight. 
 
 A convenient method is that known as the " method of squares." 
 The ground plane is divided into squares by lines parallel and 
 perpendicular to the ground line ; the network of squares is pro- 
 jected on the perspective and the figure drawn at sight in the corre- 
 sponding squares. 
 
 To construct the perspective of the squares, the distances of the 
 parallel lines are marked on the ground line in A, B, C, D, E, (Fig. 87), 
 the perspectives of the perpendiculars to the ground line are obtained 
 by joining these points to the principal point P. 
 
 The principal plane is next plotted sepa- 
 rately, sK^ being the trace of the ground 
 plane, S the station and PP' the trace of 
 the picture plane. Mark the intersections 
 
 F ly G 19 
 
 of sK by the lines par- 
 
 allel to the ground line, join to S and 
 carry to Pp the distances from p l to F^ 
 G 2 , H 2 , K% : through the points so 
 obtained, F, G, H, K, draw parallels to 
 s the ground line, which wil] complete the 
 perspective of the squares. 
 
 It is not necessary that the sides of 
 the squares be parallel or perpendicular to 
 the ground line. Any other direction may 
 be adopted, as for instance, north and 
 south, and east and west ,in the case of 
 topographical perspectives. 
 
48 
 
 PERSPECTIVE. 
 
 Grofind Plan 
 
 The vanishing points F and F a , 
 (Fig. 88), of these lines are found as 
 usual by drawing through the station 
 parallels to their directions until they 
 meet the horizon line. 
 
 The points of intersection with the 
 ground line of the north and south 
 lines, which will be supposed to van- 
 ish ^at F 1? are taken from the ground 
 plan, carried to the ground line of 
 the perspective, in A, B, C\ D, E, F, 
 and joined to V l : this gives the per- 
 spective of one set of parallel lines. 
 The other set is obtained by a similar 
 process, carrying the points G lt H^ 
 K! from the ground plan to the per- 
 spective and joining to the vanishing 
 F point F. 
 
 The squares must be made small 
 enough to guide the draughtsman 
 . , accurately in transferring the figure 
 
 from the perspective to the ground 
 .Fig-. 88. plan. 
 
 68. VANISHING SCALE. The direction of a point of the ground plane 
 is easy to find : it is sufficient to join the foot of the station to the 
 projection of the perspective on the ground line. Were the distance 
 of the point determined, it could be located at once. This is done by 
 means of the "vanishing scale". 
 
 Fig. 89 represents the principal plane : Pp and pA are the traces of 
 the picture and ground planes and S the station at a height h above the 
 ground plane. On pA and on each side of p, mark equal distances, 100, 
 200, etc. : they represent the intersections of pA by parallels to the 
 ground line. Join these points to S : the perspectives of the above 
 S parallels are parallels to the ground line 
 passing through the points of division of pP. 
 Suppose now that the distance of a point of 
 the perspective from the ground line be 
 found equal to pm : then the point of 
 the perspective is on a parallel to the ground 
 line passing through m. But this line is the 
 perspective of a parallel to the ground line 
 passing through M, therefore the point to be 
 found, being on that parallel, is at the dis- 
 tance pM from the ground line. M and m 
 Fig". 89 corresponding to the same divisions of the 
 
 scales pP and pA, the distance of the point is obtained at once by 
 reading the division of p A corresponding to m. 
 
 ,''',' 
 
 If*,',' I 
 300 , 1 
 
 eP i 
 
 h 
 
 ''' ''M/' /' 
 
 .' MS .' 
 
 loo , 
 
 
 ft ,100 
 
 100 
 
 
INVERSE PROBLEM' OF PERSPECTIVE. 
 
 49 
 
 The scale constructed as above on pP is called a "vanishing scale" 
 
 M Y 69. USE OF THE MEASURING LINE. Some- 
 
 times the greater part of an irregular figure 
 in a horizontal plane, may be inclosed be- 
 tween two parallel lines, as in Fig. 90. A 
 point Fis taken on the horizon line such that 
 two lines drawn from it inclose the figure 
 a ft Y $ s as well as possible. These lines are 
 the perspectives of two parallel lines in the 
 ground plane and their vanishing point is V. 
 Draw these parallels on the gr ound plan in 
 A j E^ and B l J) l and place on the perspective 
 the measuring point by taking VM equal to 
 the distance of V from the station. The 
 measuring line is the ground line XY. Find 
 the distances from F and G to the points of 
 the parallels corresponding to various points 
 FijET. 00 f * ne irregular figure and transfer them in 
 
 A l , B lt Cj, D lt E^, to the ground plan. 
 
 Draw the intermediate parts of the figure at sight. 
 
 Should two parallel lines prove insufficient, the number can be 
 
 increased. 
 
 The method of squares, the vanishing scale and the measuring line 
 
 can be employed for finding the perspective from the ground plan. 
 
 The operations are the converse of the preceding ones and require no 
 
 further explanation. 
 
 70. PRECISION OF THE METHOD. Let Ss and J/, (Fig. 91) represent the 
 vertical passing through the station and a point of T/he ground plane, 
 Sm and ,u.A t the traces of the horizon and picture planes and //. the 
 perspective of M. Draw Mm perpendicular to Sm: it is the height, h, 
 of the station above the ground plane. 
 
 The similar triangles SA/j. and SmM give : 
 
 Sm __SA 
 mM AV 
 
 or y I (1) 
 
 To find the effect on the distance y from the station to M, of an 
 error dx in the perspective, equation (1) must be differentiated, 
 considering x and y as variables ; this gives : 
 
 dy = y dx = 
 
 hi 
 
50 PERSPECTIVE. 
 
 Fig-. 91 
 
 So the error in the position of M caused by an 
 error in the perspective increases as the square of 
 the distance : therefore the method must not be em- 
 ployed for points or figures at too great a distance 
 from the station. 
 
 The error decreases as the height of the station increases : thus if 
 the height be doubled, the error will be reduced to one half. Hence, 
 perspectives intended for the reproduction of figures in the ground 
 plane should be taken from as great a height as possible. 
 
 The error decreases also as I increases, or as the size of the per- 
 spective increases. 
 
 71. To DETERMINE FROM THE PERSPECTIVE, THE PROJECTIONS OF A 
 POINT NOT IN THE GROUND PLANE, BUT OF WHICH THE HEIGHT IS KNOWN. 
 
 The perspective of a point is not sufficient to determine its position ; 
 other data must be furnished, such as the traces of a plane containing 
 it, its distance, or its height above the ground plane. 
 
 ^T>;. If the height be known, draw a parallel RT, Fig. 
 
 --"-r< T 92, to the ground line representing the trace on the 
 
 I/* picture plane of the horizontal plane containing the 
 
 x _ I j } r point. The projections of the visual ray joining the 
 
 j^'' ^ station to the point are s/^, P/J. ( 47) : it pierces 
 
 the horizontal plane RT in m, ra', and as the point to 
 
 be found is in that plane and on the line s//. 15 P,u, it 
 
 p,. . is the point of intersection, mm. 
 
 The construction is not always possible. For in- 
 stance RT may pass through P : this means that the point is in the 
 horizon plane, in which case it cannot be located by means of its per- 
 spective. 
 
 Pp. may coincide, or very nearly, with Ps, and the construction be- 
 come impossible or unceitain. The visual ray joining the station to 
 the point is then projected on the principal and ground planes instead 
 of the picture and ground planes : the different steps are precisely 
 the same in both methods. 
 
 72. TO CONSTRUCT FROM ITS PERSPECTIVE A FIGURE IN ANY HORIZON- 
 
 TAL PLANE. The methods given in 65, 66 and 67 apply to figures in 
 any horizontal plane, by using the planes of the figures as ground 
 planes ; all that is required being to shift the ground line on the 
 perspective to its proper position. 
 
TRACES. 
 
 51 
 
 73. To FIND THE TRACES AND VANISHING POINT OF A LINE GIVEN BY 
 
 ITS HORIZONTAL PROJECTION AND PERSPECTIVE. Before proceeding to 
 consider figures in various planes, it is necessary to show how the 
 plane of a figure and the traces of straight lines can be determined. 
 
 Let a/5 and ab (Fig. 93), represent the perspec- 
 tive and horizontal projection of a line. At b 
 draw a perpendicular to the ground line : the 
 trace on the picture plane must be on that per- 
 pendicular and also on a/S, therefore it is at 
 their intersection /5. 
 
 p 
 
 The vanishing point is the trace of a par- 
 allel to the line drawn through the station ; the 
 horizontal projection of this parallel is sv, 
 drawn through the foot of the station parallel 
 
 Fis* 93 to a ^' an< ^ ^ s trace * s on ^6 perpendicular v V 
 
 to the ground line. But this trace is the van- 
 ishing point of a/5 ; therefore it is at the intersection of v V and a/9 
 produced. 
 
 vertical projection passes through the trace V and the principal 
 point P ; producing it to the intersection with XT and drawing the 
 perpendicular v 1 V 1 to XT, the trace on the ground plane is found 
 at F r 
 
 The line joining F a to is the perspective on the ground plane of 
 the given line ( 49) whose trace is the intersection. of aV 1 and ab. 
 
 The trace on the ground plane may also be found by revolving the 
 projecting plane of ab around its vertical 
 trace bft (Fig. 94) upon the picture plane. 
 Draw the horizon line Pd ; the trace of the 
 given line on the horizon plane is seen in 
 Y on the perspective ; its horizontal projec- 
 tion is at the intersection c of ab by the 
 line joining the foot of the station to the 
 foot f of the perpendicular iff to XT. 
 When the projecting plane revolves, c de- 
 scribes the arc of circle cCj, with b as 
 centre : the point of the given line corre- 
 sponding to Y moves in the horizon plane ; 
 therefore it comes in y l on the horizon line, 
 at the intersection with the perpendicular 
 
 C I Y I to XT. The revolved line is fta^, and 
 
 Fi^.94 
 
 the revolved trace on the ground plane is a l . Revolving a 1 back to 
 ab, the trace is obtained in a. 
 
52 PERSPECTIVE. 
 
 The angle formed in a^ by the revolved line and XY is the angle of 
 the line with the ground plane. 
 
 A third method consists in determining from the perspective the 
 heights of two points of the given line, as will be explained later on. 
 
 Fig-. 95 
 
 The projecting plane of the line is 
 revolved on the ground plane around the 
 horizontal projection ab, Fig. 95. The 
 points A and B fall in A^ and B^, the 
 Y perpendiculars a A l and bB 1 to cd being 
 the heights of A and B above the ground 
 plane. A^B^ is the revolved line and 
 c its trace on the ground plane. The 
 revolved trace on the picture plane is at 
 the intersection of A l i produced with 
 the perpendicular dD to cd ; it is 
 revolved back to the picture plane by 
 
 drawing a perpendicular dd' to XY and describing an arc of circle 
 with d as centre dD l as radius. 
 
 74. GIVEN THE SLOPE OF A LINE AND THE HORIZONTAL PROJECTION OF 
 ONE OF ITS POINTS, TO FIND THE HORIZONTAL PROJECTION AND TRACES OF 
 THE LINE. Let a (Fig. 96), be the horizontal -projection of a point of 
 the line seen in perspective in /3, s the foot of the station and XY 
 
 the ground line. Join sa, pro- 
 duce to m and erect the perpen- 
 dicular ma to XY ; a and a are 
 the perspective and projection 
 of the same point, A, of the 
 given line. Draw the horizon 
 line HH' : d is the perspective 
 of the trace of the given line on 
 the horizon plane. Rotate the 
 projecting plane of the line 
 around the vertical of a until 
 parallel to the vertical plane ; 
 the point A of the given line, 
 being on the vertical of a, does 
 
 not move, and its perspective 
 
 remains in a. The perspective d of the trace on the horizon plane 
 moves on the horizon line : when the projecting plane is parallel to 
 the vertical plane, the perspective of the rotated line is parallel to the 
 line itself and may be drawn in ad 19 since the angle ad^H is given. 
 
TRACES. 
 
 53 
 
 The trace of the projecting plane on the ground plane has come in 
 ad l parallel to the ground line. The point d^ of the horizontal pro- 
 jection corresponding to ^ of the perspective is obtained by drawing 
 d^O perpendicular to XY and joining sO. Rotating back the projecting 
 plane to its original position, d^ comes in '?, the corresponding point 
 of the horizontal projection being on the line sn joining the foot of 
 the station to the foot of the perpendicular from d to the ground line. 
 But this corresponding point is the new position of d lt and d l moves 
 on an arc of circle with a as centre, therefore d l comes in d and da is 
 the horizontal projection of the given line. 
 
 The vertical trace is found at c' by the usual construction : the 
 vertical projection and horizontal trace may be determined as in 73 
 or the triangle formed by cc', cb and the given line may be revolved 
 around cc on the vertical plane. The axis cc does not move, cb falls 
 on the ground line and the hypothenuse c'6 15 becomes- parajlel to a^. 
 Revolving the triangle back to its original position, b^ comes in b, 
 which is the trace, on the ground plane, of the given line. Having 
 now the two traces, the vertical projection can be drawn by the usual 
 construction. 
 
 75. To FIND THE TRACES OF THE PLANE CONTAINING THREE GIVEN 
 
 POINTS OR TWO GIVEN LINES. 
 
 R Whether two lines or three 
 
 points be given, the problem 
 consisting in passing a plane 
 through them is the same, .and 
 
 i_ consists in finding the traces of 
 
 ; the given lines or of those join- 
 ing the given points. The traces 
 >' of same denomination are joined 
 
 ~A[ by straight lines, which are the 
 
 traces of the required plane. 
 
 Fig-. 97 
 
 The traces of the lines are 
 obtained by any of the processes 
 of 60, 73 or 74. 
 
 In Fig. 97, the heights of the three points A, B and C are supposed 
 to be known and the traces are determined by revolving the pro- 
 jecting planes on the ground plane around the horizontal projections 
 ab and be; ( 73). QOR is the required plane. 
 
 Sometimes the traces of the plane are required on the picture and 
 principal planes. Revolve the principal plane around its trace Rp t (Fig 
 
54 
 
 Figv98 
 
 PERSPECTIVE. 
 
 98) on the picture plane, the front part 
 of the principal plane turning to the 
 left. The station comes in S. 
 
 Let a, /?, Yt be the perspectives of 
 Y three points A, B, C, of which the pro- 
 jections on the ground plane are given, 
 a and c the traces on the picture and 
 principal planes of the horizontal pro- 
 jection of the line AB, d and b those 
 of the horizontal projection of AC. 
 
 Produce a/3 to the intersection/ with the principal line ; f is the per- 
 spective of the trace on the principal plane, of the line of space AB, 
 therefore the trace on the revolved principal plane is on Sf. But the 
 trace is on the vertical of c, therefore it is at c. The trace of the other 
 line is found in a similar manner at d' and the trace of the plane con- 
 taining the two lines- is c'd'. 
 
 The traces of the two lines A.B and AC on the picture plane are 
 obtained in a and b' as in 73, and being joined, give the trace of 
 their plane on the picture plane. The result is the plane QRT. 
 
 76. GIVEN THE LINE OF GREATEST SLOPE, TO FIND THE TRACES OF THE 
 PLANE. The line of greatest slope of a plane is perpendicular to the 
 
 trace on the ground plane. Hence, to draw the 
 traces of the plane, find those of the line and 
 through the ground plane trace a, Fig. 99, 
 draw aQ perpendicular to the horizontal pro- 
 jection, ab, of the line : it is the ground plane 
 trace of the required plane. The trace of the 
 plane on the picture plane is obtained by 
 joining Q to the vertical trace, /?, of the line. 
 
 In Fig. 99 the line of greatest slope is sup- 
 posed to be given by its horizontal projection, 
 Fiff 99 a ^' anc ^ * ts P ers P ec tive /3 : the traces are found 
 
 by the method of 73. Should the line be 
 
 known by the heights and perspectives of two of its points or by the 
 heights and horizontal projections, or by its slope, the traces could 
 be determined by the methods given in 60, 73 and 74. 
 
 77. CHANGE OF GROUND PLANE. A change in the ground plane 
 does not produce any change in the points or lines of the ground plan : 
 the traces of planes are displaced, but remain parallel to the original 
 trace. 
 
FIGURES IN INCLINED PLANES. 
 
 55 
 
 Fig. 100 shows the ground line moved from XY to X 1 Y l ; the left 
 
 .jt hand figure contains 
 
 R . m , ,\ the projections of a 
 
 point, of a line and 
 
 y j b'f \ \Q" v the traces of a plane 
 before the change of 
 ground plane. 
 
 i\ 
 
 Xr 
 
 Pig-. 100 
 
 ground line from XY to X. 
 
 In the first place, it 
 may be observed that 
 there is no change in 
 the vertical plane, be- 
 yond moving the 
 
 In the ground plane, the projections of the point m and of the line 
 ab remain the same, but the trace of the line is now in c instead of b. 
 The new trace is obtained by producing the vertical projection a'b' 
 across the old ground line XY to the new one, drawing the perpendi- 
 cular c'c and producing ab to meet c'c. 
 
 The trace of the plane has been moved from OT to O^T^. To find 
 the new one, produce the vertical trace O'R across the old ground line 
 XY to the new one X^ T lt and through the point of intersection 0^ 
 draw O^T^ parallel to OT. 
 
 78. TO FIND THE HORIZONTAL PROJECTION OP A FIGURE FROM ITS 
 PERSPECTIVE WHEN THE FIGURE IS CONTAINED IN A PLANE PERPENDI- 
 CULAR TO THE PRINCIPAL PLANE. Take for vertical plane of projection 
 the principal plane and let QZ, (Fig. 101), be the trace of the plane 
 
 containing the figu re. Take for ground plane 
 the horizontal plane passing through the 
 point of intersection of QZ with the trace 
 QR of the picture plane, XY being the ground 
 line. Let S be the station, s the foot of the 
 station, nri a point of the given figure and 
 mm its perspective. The given plane, being 
 perpendicular to the principal plane, the ver 
 tical projection of any point of the former 
 is on the trace QZ. The picture plane RQT 
 Fie 1 101 * s Perpendicular to both planes of projection, 
 
 therefore the projections of any point of the 
 picture plane are on its traces. 
 
 Produce QZ to meet the vertical of the station in A and take SS^ 
 equal to sA t S^ being above or below S according as A is below or 
 
56 PERSPECTIVE. 
 
 above s. Join S^m' and produce it to meet the ground line in b : join 
 Sri and rib. The line Sri passes through m, since in' is the per- 
 spective of ri. 
 
 The similar triangles rim'Q, n'SA give : 
 
 Qm n'Q 
 
 SA ri A 
 
 From the triangles bQiri, bsS^, we have : 
 
 Qm' _ bQ 
 
 s/S\ bs 
 
 But SA=--sS l 
 
 Therefore : n'Q bQ 
 
 n'A bs 
 
 or, n'Q bQ 
 
 Hence the triangles ribQ, QsA are similar, as having one angle equal 
 and the sides about it proportional, consequently bri is parallel to sA 
 or perpendicular to XY and the point n is the trace on the ground 
 plane of the visual ray sn, Sj^b. Were the eye placed in S 19 the point 
 of the ground plane which would be found to correspond to mm of 
 the perspective would be the horizontal projection n of the point of 
 the plane QZ. Should the new station S^ be used in connection with 
 the perspective of a figure in the plane QZ, the result obtained, when 
 constructing the corresponding figure of the ground plane, would be 
 the horizontal projection of the figure of the plane QZ. 
 
 Therefore to obtain the horizontal projection of a figure in a plane 
 
 p perpendicular to the principal plane, take for ground 
 
 X K line the trace XY, (Fig. 102), of the given plane on the 
 
 picture plane, find the height of the station above the 
 
 H _ \p n - point of intersection of its vertical by the given plane 
 
 x __ : _ Y (75) use it as height of the new station and draw 
 
 /l the horizon line H 1 H' l on the perspective at that 
 
 height above the ground line. The figure constructed 
 
 Fig". 102 from the perspective by any of the methods of 65, 
 
 66 or 67 will be the horizontal projection of the figure 
 
 in space. 
 
FIGURES IN INCLINED PLANES, 
 
 57 
 
 It has been shown that the perspective is the same as if the horizontal 
 projection had been seen from the station S 1 (Fig. 101) instead of 
 observing the original figure from S ; consequently, the precision of the 
 
 result ( 70) is increased in the proportion of *- by the inclination 
 
 S*j 
 
 of the plane of the figure. Were the plane falling instead of rising in 
 front of the observer, sS-^ would be smaller than sS and the precision 
 would be decreased. 
 
 Hence a perspective taken for the purpose of constructing a figure 
 in an inclined plane should always be taken in the direction of the 
 rising plane ; thus a river at the bottom of a sloping valley should be 
 taken looking up the valley. 
 
 79. TO FIND FROM ITS PERSPECTIVE THE HORIZONTAL PROJECTION OF A 
 FIGURE IN A PLANE PERPENDICULAR TO THE PICTURE PLANE. The 
 
 method of squares of 67, can be applied to a figure in any inclined 
 plane, by conceiving vertical planes containing the sides of the squares. 
 The intersections of these planes by the inclined plane form a series of 
 parallelograms corresponding to the squares of the ground plane. 
 
 Let QR (Fig. 103) be the 
 trace on the picture plane of a 
 plane perpendicular to it, XY 
 the ground line, P the principal 
 point, and abed one of the 
 squares of the ground plan. 
 The projecting planes of ab and 
 cd cut the trace QR in m and 
 n. Through the station, S, 
 draw a parallel to the inter- 
 section of the projecting planes 
 with the plane QR ; the hori- 
 zontal projection st is parallel to 
 ab and cd; the vertical pro- 
 ,.,. inq jection passes through P and 
 
 is parallel to QR, since all lines 
 
 in the plane QR are projected vertically on QR. At t draw the per- 
 pendicular t V to the ground line; V is the vanishing point of the 
 intersections of the projecting planes with the plane QR and the lines 
 Vm and Vn are the perspectives of these intersections. The distance 
 ran can be carried on QR and as many parallels placed in perspective 
 as necessary. 
 
 The same operation is repeated for ad and be, and the figure 
 obtained on the perspective corresponds to the square abed. 
 
t 5S PERSPECTIVE. 
 
 Another process consists in constructing the figure in the inclined 
 plane by one of the methods of 65, 66 or 67, using the plane of the 
 figure as ground plane ( 59). 
 
 Let QR, (Fig. 104) be the trace of the plane of the figure on the 
 picture plane, HIT the horizon line and P the principal point. 
 
 To construct the figure in the plane QR, that 
 line is taken as ground line : the new horizon line 
 H - is a parallel H^ H' l to QR through the principal 
 point. The height of the station is the distance 
 between these two lines, Pp ^ . The line which 
 will appear as the projection of the principal line 
 on the constructed figure is the perpendicular to 
 the picture plane at p t . On the true ground 
 plane, the distance between the two projections 
 of the principal line is equal to pp t . 
 
 .Fig-, 104 
 
 Having obtained the figure in the plane QR, let us now take for true 
 ground plane the horizontal plane of p lt the ground line being XT. 
 
 Let ABCD, (Fig. 105), be the figure in its plane, 
 s iPi the projection of the distance line, QR the 
 trace of the picture plane and s 1 the foot of the sta- 
 tion. The projection of A on the true ground plane 
 is at the same distance from the ground line as A is 
 from QR, but the distance of this projection from s^t 
 is equal to mA multiplied by the cosine of the incli- 
 nation co of the plane QR, for let a', (Fig. 104), be 
 Fig". 105 the vertical projection of A ; the right-angled triangle 
 p^a'n gives : 
 
 p^n = p^a cos. co. 
 
 Therefore, if Am (Fig. 105), be drawn parallel to QR, am taken equal 
 to Am cos. co and the same operation repeated for B, C, and D, the re- 
 sulting figure abed is the ground plan of ABCD. 
 
 The ground plan may be obtained in another way. For join s-^A : 
 the intersection a with QR is the projection on QR of the point of the 
 perspective corresponding to A. Take s-^p equal to s^p^ sec. co and 
 through p' draw Q^R^ parallel to QR : join s^a. The similar tri- 
 angles s^p^a, s^mA, give : 
 
 p 1 a 
 
 mA 
 
 (i) 
 
FIGURES IN INCLINED PLANES. 59 
 
 From the similar triangles s 1 pa l) s^^ma, we have : 
 
 f!L = i!! (2) 
 
 pa^ ma 
 
 Dividing (1) by (2), replacing sp and ma by s^^ sec. <o and mA cos. 
 10 respectively, we find : 
 
 P a i = Pi a - 
 
 This means that if the perspective be moved in Q 1 R l , the directions 
 obtained from the perspective for the different, points of the plane QR 
 will be the directions of the horizontal projections of these points. 
 
 Therefore to construct the horizontal projection of the figure seen 
 in perspective, find the distances of the various points of the figure 
 from the picture plane by means of a vanishing scale (68) made with 
 Pp lt Fig. 104, as height of the station and the real distance line. 
 Then find the directions of the projections, using QR as ground line 
 
 and a distance line increased in the proportion The figure con- 
 cos. 10 
 
 structed with the above distances and directions is the horizontal pro- 
 jection of the figure in the plane QR. 
 
 80. CHANGE OF GROUND PLANE AND DISTANCE LINE. Let A, (Fig* 
 106), be a point of a figure in a plane perpendicular to the picture 
 g plane and a its perspective. Take the plane of 
 '/: the figure as ground plane and let s^p^ be the trace 
 of the assumed principal plane. Revolve this 
 y principal plane around its trace on the ground 
 plane : the station comes in S, b and b' being the 
 ^ projections of and a the projection of A on the 
 assumed principal plane. Move the perspective to 
 
 b } b\ so thatSj// Sl ^ 1 , co being the angle of 
 cos. to 
 
 the assumed and true ground planes ; it has been 
 shown that s 1 b 1 is the direction of the projection 
 a of A on the true ground plane revolved around s^p-^ on the assumed 
 ground plane. The visual ray, however, does not pierce the ground 
 plane in a, its projection on the principal plane having been changed 
 from Sb' to Sb' 1 by the displacement of the perspective. But join Sp 
 and take as new ground plane the plane passing through c : s'a\ is 
 the trace of the assumed principal plane on this new ground plane 
 and a l the projection of the trace of the visual ray on the last plane. 
 Consequently this trace is at the intersection of s^a with the perpen- 
 dicular drawn from a l to s^'. 
 
 r 
 
60 PERSPECTIVE. 
 
 Similar triangles give the following proportions : 
 
 x 
 
 pa p'f cb' db\ da! \ 
 b'b' i fb\ b'm b'm b'b' \ 
 
 or pa da l 
 
 and p'a' = da' 1 
 
 pa! being equal to da \ the figure p'da\a is a parallelogram and a\a 
 is perpendicular to s^a, therefore the visual ray will pierce the new 
 ground plane in a. 
 
 Hence, if the perspective be moved from p l to p' and s'a \ taken as 
 ground plane, the perspective viewed from the station corresponds 
 on the new ground plane to the projection of the figure on the true 
 ground plane : this projection can consequently be constructed by 
 the methods of 65, 66, 67. 
 
 Fig. 106 gives the proportion : 
 
 Ss s'c 
 
 - = , - = COS. W. 
 
 or Ss'=Ss 1 cos. u>. 
 
 The heights Ss', Ss^, of the station above the various ground planes 
 being equal to the distances of the principal point from the correspond- 
 ing ground lines, the new ground and distance lines can be found as 
 follows : 
 
 It, Let QR, (Fig. 107), be the trace on the picture 
 plane of the plane containing the figure. From the 
 principal point P, let fall Pp^ perpendicular to QR 
 and draw Pp and p-^p perpendicular and parallel to 
 the true horizon line HH'. Take Pd equal to Pp 
 and draw Q 1 R 1 parallel to QR : it is the ground 
 line to be used in the construction, because 
 
 Pd = Pp^ cos. a). 
 
 Fiff 107 At the distance point, draw DD l perpendicular 
 
 to HH' and draw PD^ parallel to QR : PD is the 
 length to be used as distance line. 
 
 The height of the station Pd used for the construction is always 
 smaller than the real height Pp^ above the plane of the figure, there- 
 fore the precision of the construction is less than if the figure had 
 been in a horizontal plane. 
 
FIGURES IN INCLINED PLANES. 
 
 61 
 
 81. FROM THE PERSPECTIVE OF A FIGURE IN ANY GIVEN PLANE, TO 
 CONSTRUCT THE HORIZONTAL PROJECTION OF THE FIGURE. The method 
 
 of squares can be again employed 
 in this case. Let QOR, Fig. 108, 
 be the traces of the plane of the 
 figure on the ground and picture 
 planes, and abed one of the squares 
 of the ground plan. The project- 
 ing plane of ad intersects the 
 traces of QOR in Q and L ; the 
 vertical projection of the intersec- 
 tion of the two planes being Lq'. 
 Through the station draw a 
 parallel to ad, Lq : the horizontal 
 projection is sv parallel to ad, the 
 vertical projection is PV parallel 
 to Lq and the vertical trace, V, is 
 the vanishing point of the inter- 
 section of QOR with the projecting 
 plane of ad. The perspective of 
 this intersection is VL : the per- 
 spective of the intersection of the 
 
 projecting plane of cb is VK and all the lines required may be drawn 
 in perspective by carrying the distance LK on the trace OR and joining 
 the points of division to V. 
 
 The perspectives of the intersections with the plane QOR, of the pro- 
 jecting planes of ab and cd are obtained in a similar manner by draw- 
 ing through the station a parallel to ab, n'R, for instance, and joining 
 the vanishing point V to R and T. The resulting figure upyd cor- 
 responds, on the perspective, to the square abed of the ground plan. 
 
 It is also possible to construct a vanishing scale ( 68) for measuring 
 the distances of the various points from the picture plane. 
 
 Through the station, a plane is drawn perpendicular to the vertical 
 trace of the given plane : the intersections of the latter with the 
 picture and perpendicular planes and the station point are placed in 
 their actual positions and the vanishing scale is constructed by measur- 
 ing equal distances from the trace of the picture plane. 
 
 Fig-. 108 
 
 82. CHANGE OF STATION, GROUND AND PICTURE PLANES. The same 
 result is arrived at by changes in the relative positions of the station, 
 perspective and ground planes. 
 
PERSPECTIVE. 
 
 Let QR, (Fig. 109,), be the trace, on the principal plane, of the 
 plane containing the figure, which we will call A. 
 Take for ground plane the horizontal plane passing 
 through the intersection p of this trace with the 
 principal line and suppose the principal plane 
 revolved around its trace sp on the ground plane. 
 
 Let S be the station and /*// the perspective of 
 the point mm, in the plane A. Take SS 1 equal 
 to Qs and suppose that $ t be used as station in 
 connection with a new plane passing through sp 
 and the trace on the picture plane of the plane A. 
 Call this plane B. The visual ray from the new 
 Fig". 109 station to /V-A', is projected in S^', sp-. 
 
 Cut the planes A and B by a third one parallel to the principal 
 plane and passing through the point mm'. 
 
 The horizontal projection of both intersections is mn, parallel to sp. 
 The projection on the principal plane of the intersection with plane A 
 is m'ri parallel to QR and the intersection with plane B is projected 
 in n'a parallel to sp. 
 
 Join Sp and produce it to m ; produce S^fS to its intersection with 
 n'a, m'ri and n'a to their intersection with S^Q. Join ma. Similar 
 triangles give : 
 
 n p.' m p! 
 
 SD = 'm r S' 
 
 (1) 
 
 (2) 
 
 But SD = SE + ED 
 terms of (1) and (2) are identical and we have 
 
 , hence the first 
 
 m fj. 
 
 aS, 
 
 which is transformed into 
 
 ' n'S //#! 
 
 The triangles SS 1 /J.' and am'/jf having one angle equal and the two 
 sides about it proportional, are similar, and m'a is parallel to SS^ 
 Consequently a is on the perpendicular mm to sp. 
 
REFLECTED IMAGES. 
 
 63 
 
 The line sm, /S^ a, is the visual ray from the new station through the 
 point ,'j./j.' of the perspective : inn, an, is a line of the plane B. These 
 two lines intersect since the intersections m and a of their projections 
 are on the same perpendicular to the ground line, and the point of 
 intersection is the trace of the visual ray on the plane B since the 
 line mn, an is in that plane. The same point is also the trace on the 
 plane B of the vertical through mm. 
 
 Therefore, if verticals are drawn from all the points of the figure in 
 plane A, their traces on plane B form a new figure which corresponds 
 to the perspective viewed from S 1 . 
 
 The problem is thus reduced to construct from its perspective the 
 horizontal projection of a figure contained in a plane 
 perpendicular to the picture plane, which is done 
 by a change of ground and picture planes ( 79). 
 The process now involves changes of station, ground 
 plane, picture plane and trace of principal plane as 
 follows : 
 
 N 
 
 Fig". 110 
 
 to sT. Draw 
 
 principal point of the perspective. 
 
 Let Ql\, (Fig 110), be the principal line. Revolve 
 the principal plane on the picture plane around 
 $P 15 the front part of the principal plane being 
 turned to the left : the station comes in S, and 
 -##! is the vertical of the station. Let TQR be 
 the plane containing the figure seen in perspective. 
 Draw Qs perpendicular to QP l and take $oj equal 
 -^ parallel to sQ. The point P l is to be used as 
 
 Draw 
 equal to 
 assumed ground line. 
 
 j/?! perpendicular to QR, pp l parallel to sQ and take P^d 
 P p. Through d draw Q 1 R l parallel to QR ; it is the 
 
 Produce QR to N : QN is the length to be assumed as distance 
 line. 
 
 On the constructed figure, the perpendicular to the picture plane at 
 p 1 will appear as trace of the principal plane on the ground plane. 
 
 The traces of the plane containing the figure are found as in 75. 
 
 83. REFLECTED IMAGES. The case of horizontal reflecting surfaces 
 is the only one that will be considered. 
 
 When a perspective contains the direct and reflected images of the 
 same point, the point can be located in space, provided the altitude of 
 the station above the reflecting surface be known. 
 
64 PERSPECTIVE. 
 
 Take for ground plane the reflecting surface and revolve the 
 principal plane on it, around its trace. Let a, a, Fig. Ill, be the 
 point in space, , ' its perspective and a a l the perspective of its re- 
 flected image. The horizontal projection is the same for both images, 
 because the reflecting surface being horizontal, the direct and reflected 
 visual rays are in the same vertical plane having for trace sa. 
 
 Let sa, SOa, be the reflected visual ray : according to the laws of 
 reflection, the direction of SO is the same as if a were placed at a 
 distance equal to ca below the reflecting surface and on the same 
 vertical. 
 
 Produce a'O to S l : cb being equal to ca', sS is equal to sS^ Hence 
 to find the position in space of a, a', take sS-^ equal to sS : join Su, 
 Sa\ and b\0 : the point of intersection of So.' and S^O is the vertical 
 projection of the point of space. 
 
 Join sa and produce to the intersection with a a, perpendicular to 
 the ground line ; aa is the required point. 
 
 The construction gives not only the position a 
 of the point on the ground plane, but also its 
 height ca. 
 
 The middle of the vertical between the direct 
 and reflected images corresponds to a, the hori- 
 zontal projection of the point on the ground 
 plane. This shows that when the shore of a 
 $'' * lake, for instance, is indistinct on a perspective, 
 
 it would be incorrect to take for shore line the 
 Fig". Ill middle line between objects and their images in 
 
 the lake, because this would give for the distance 
 of the shore that of the objects themselves. 
 
 84. SHADOWS. The subject of shadows is an important branch of 
 perspective, but only those cast by the sun need be considered here. 
 
 Let and /? (Fig. 112) be the perspectives of two points A and , 
 m and n their shadows. The line joining A to its shadow is the 
 direction of the sun, and so is the line joining B to its own shadow ; 
 therefore these lines are parallels and their vanishing point is F, at 
 the intersection of ma and nft. 
 
SHADOWS. 
 
 65 
 
 A line drawn from the station to the sun is parallel to the first two 
 lines, because it is also the direction of 
 the sun ; .therefore V is its trace on the 
 picture plane or the perspective of the sun. 
 B' From V draw Vv perpendicular to XT; 
 
 m/ : \/l . i -i 
 
 X }r \ y sv is, on the ground plane, the direction 
 
 of the sun. On the horizon line take CF 
 I equal to sv and join FV : FVC repre- 
 
 sents, revolved on the picture plane around 
 VC, the triangle having its vertex at the 
 ,./ station and VC as opposite side. There- 
 
 fore VFC is the altitude of the sun. 
 Fie* 112 Having the sun's altitude, the azimuth 
 
 of the line sv of the ground plan can be 
 calculated, provided the latitude and approximate time are known. 
 
 Fig. 112 represents the sun in front of the observer. When it is 
 behind, the line between the station and the sun does not pierce the 
 picture plane ; it has to be produced to intersect it below the horizon 
 line. The trace of this line on the picture plane, V, Fig. 113, is still 
 considered as the perspective of the sun ; it is obtained in the same 
 manner as when the sun is in front and all demonstrations apply to 
 one case as well as to the other. 
 
 The calculation of the azimuth can be made by the method given in 
 
 37 for the solution of spherical triangles. 
 
 Tj 
 
 4 S i 
 
 \ /", 
 
 ft' 
 
 \ 
 
 "XV 
 
 v ty 
 
 
 
 \ \ 
 
 / / 
 
 
 
 *\Jli 
 
 
 
 Fig-. 113 
 
 Find the altitude CFV. of the sun 
 by the construction given above, 
 make EVC equal to the colatitude 
 of the place and FVM to the polar 
 distance of the sun. Take VM equal 
 to VE and from C and F as centres 
 with CE and FM respectively as 
 radii, describe arcs of circle. Join 
 their point of intersection, G, to (7, 
 and GCF is the azimuth of the 
 sun. 
 
 When the perspective has been 
 taken in the morning, plot the angle 
 Z on the left of sv in vsO, and the 
 line Os is the north and south line 
 of the ground plan. 
 
66 PERSPECTIVE. 
 
 In the afternoon, the angle Z should be plotted 011 the right of sv. 
 The rules are reversed when the perspective of the sun is above the 
 horizon line. 
 
 85. HEIGHTS. As a rule, one perspective is' not sufficient to deter- 
 mine the height of a point, although there are exceptions, as for instance, 
 points on the horizon line which are at the same height as the station. 
 
 The horizontal projection of the point being known, the height above 
 the ground plane is measured with a scale in the same manner as a 
 vertical is divided into equal parts ( 61). 
 
 For instance, a and a, Fig. 114, being the perspective and horizontal 
 projection of a point, and s the foot of the station, draw aF parallel 
 toXY. 
 
 From the trace p of the principal line, take pB 
 equal to the distance of a from XY. Join sl>, 
 and FE is the height of the point above a. 
 
 Fig-. 114 
 
 \ E This height being a fourth proportional to three 
 
 known lines, can be found with an ordinary 
 sector. Take with a pair of compasses the dis- 
 tance from a to XY t place one of the points on 
 the division p of the sector (Fig. 115) which 
 expresses the length of the distance line, and 
 open the sector until the second point of the 
 compasses coincides with the corresponding 
 division of the other branch, sj) and sB being 
 equal. Now take with the compasses the distance 
 from a to XY (Fig. 114) and place one of the points 
 in p (Fig. 115). The other point being placed on sp, 
 will coincide with a division of the scale, E for 
 instance ; then turn the compasses around and take 
 the distance from E to the same division F of the 
 other scale ;EF is the height of the point above the 
 ground plane. 
 
 Fig-. 115 
 
RELATIONS BETWEEN TWO PERSPECTIVES. 
 
 67 
 
 86. RELATIONS BETWEEN 
 
 TWO PERSPECTIVES OF THE SAME OBJECT. 
 
 Prof. G. Hauck* has shown some useful 
 properties of two perspectives of the 
 same object taken from different sta- 
 tions. In Fig. 116, S and S' are the 
 two stations, ff is the perspective of S' 
 on the picture plane of S, and ff' is the 
 perspective of S on the picture plane of 
 S'. Prof. Hauck calls <r and ff the kern 
 points. For facility of reference, we 
 will call any plane containing the two 
 kern points, a kern plane. 
 
 Let SAS' be such a plane passing 
 Fig-. 116 through the point A of the object. 
 
 The intersection EC of the two picture 
 
 planes pierces the kern plane in ; Off and Off' are the traces of the 
 last plane on the first ones, Off is therefore the perspective of Off' and 
 inversely Off' is the perspective of 0<r. The five lines SA, S'A, SS' t 
 Off and Off' are in the kern plane \ the intersection a of SA and Off is 
 the perspective of A from S and the intersection ' of S'A and Off' is 
 the perspective of A from S'. Hence the rule that the lines aa and 
 o!f>' joining the two perspectives of the same point to the correspond- 
 ing kern points meet the intersection BC of the picture planes at the 
 same point 0. 
 
 n 
 
 H' 
 
 'Fig-. 117 
 
 Revolve the two picture 
 planes around their inter- 
 section BC, until they coin- 
 cide (Fig 117); GG' is the 
 ground line, P, P' are the 
 principal points. If a scale 
 is placed on .6(7, the zero 
 being on the ground line 
 GG' t it is intersected at the 
 same division by T and 
 V, and the space inter- 
 cepted in HH' between the 
 two horizon lines is equal 
 to the difference of altitude 
 
 of the stations. So when the perspectives are separated, scales, placed 
 on the line BC on the two perspectives with their zeros on the ground 
 lines, are intersected at the same division by a<r and a' a'. The same 
 
 ' Theorie der trilinearen Verwandschaft ebener Systeme. Journal fur reine und 
 angewandte Mathematik. 95 Band, 1883. 
 
68 PERSPECTIVE. 
 
 relation holds good for the perspectives /9 and /?' of any other point ; 
 $a and ft' a meet the same division B of the scales. 
 
 The scales may be placed elsewhere, as at DE and FK parallel to 
 J3C, provided 
 
 Similar triangles show that 
 
 DX=FZ 
 
 The spaces intercepted on the scales are still the same, but the zeros 
 are no longer on the ground line unless the stations are at the same 
 altitude. 
 
 The scales can be placed on the opposite sides of the kern points at 
 LR and MQ. The spaces intercepted are equal provided 
 
 Instead of setting the scales in DX and FZ, the second one may 
 be placed at F'Z'. The spaces intercepted, DX and F'Z', are equal 
 when 
 
 But, it must be observed that in this case the graduations run in 
 opposite directions; DX being upright, F'Z' must be put upside 
 down. 
 
 As a general rule, the scales must be at distances from the kern 
 
 points proportional to the distances 
 
 . _ N^ _ . _ B / from the latter to the intersection 
 
 of the picture planes. The proper 
 positions are readily found from 
 the ground plan. 
 
 Let Aff, Bff', (Fig. 118), be the 
 traces of the picture planes, p, p' 
 the traces of the principal lines 
 Fig*. 118 an d s, s' the stations. Draw AB 
 
 , parallel to ss. A scale parallel to, 
 and at the distance Ap from the principal line of the first perspective 
 
RELATIONS BETWEEN TWO PERSPECTIVES. 69 
 
 will correspond to a scale parallel to, and at the distance Bp from the 
 principal line of the second perspective, because 
 
 ffO ff'O 
 
 which is the condition to be fulfilled. 
 
 It is convenient to draw AB so that both points shall lie outside of 
 the pictures, otherwise the scales would hide portions of the views. 
 
70 
 
 PERSPECTIVE INSTRUMENTS. 
 
 CHAPTER III. 
 
 PERSPECTIVE INSTRUMENTS. 
 
 87. SIMPLEST FORM OF PERSPECTIVE INSTRUMENT. Many instru- 
 ments have been devised for producing perspectives, either by 
 mechanical or optical means. 
 
 One of the simplest forms is probably the wire grating represented 
 
 in Fig. 119. Wires are 
 stretched on a frame so 
 as to divide it into small 
 squares. The frame is 
 placed in front of the 
 object or view to be 
 reproduced, and the 
 draughtsman looks 
 through an eye-hole in a 
 fixed position. Dividing 
 his paper into squares in 
 the same manner as the 
 frame, he is able to re- 
 produce the outlines of 
 the subject by drawing 
 his lines through the 
 squares of the paper cor- 
 responding to those of 
 the frame. The distance 
 from the frame to the eye- 
 
 rv ^Q . hole is the distance line 
 
 of the perspective when 
 the squares of the paper are equal to those of the frame. 
 
 88. DIAGRAPH. While for artistic purposes the grating is quite 
 sufficient, there is some uncertainty in drawing the figures of the 
 corresponding squares. To obviate this defect, it has been proposed 
 to follow the outlines of the subject with a pointer moved by the 
 hand, as in Fig. 120. 
 
D 1 AGRA PH. 
 
 71 
 
 A drawing board, on which is stretched a piece of paper, is placed 
 in an upright position in front of the subject of the perspective. It 
 is provided with a rod, supported at both ends by cords attached to a 
 counterpoise at the back of the board. The rod may be moved up or 
 down and to right or left, but, owing to the mode of suspension, it is 
 
 Fig-. 120 
 
 always parallel to the same direction. At the middle it carries a pencil 
 resting on the paper and at the end it has a pointer corresponding 
 to the eye-hole of the instrument. The draughtsman takes the pencil 
 with his hand and placing his eye at the eye-hole, he follows with the 
 pointer the outlines of the subject by moving the rod in the proper 
 direction. The pencil reproducing exactly the motion of the pointer, 
 describes the perspective on the drawing board. 
 
 The plane in which the pointer moves is the picture plane, the eye- 
 hole is the station and its shortest distance from the pointer is the 
 distance line. 
 
 The upright position of the drawing board is inconvenient ; in a 
 modification of the same instrument called the "Diagraph," the paper is* 
 placed horizontally and the motion of the pointer is transmitted to the 
 pencil by a cord and pulley. 
 
 None of these instruments have come into general use. 
 
72 
 
 PERSPECTIVE INSTRUMENTS. 
 
 89. CAMERA LUCIDA. The camera lucida 
 
 Fig-, 121 
 
 is the invention of 
 Wollaston. It con- 
 sists essentially of a 
 four sided prism 
 having a right angle, 
 two angles of 67 30' 
 and one angle of 
 135 (Fig. 121). The 
 eye is placed close 
 to and above the 
 edge of the prism, so 
 that the pupil re- 
 ceives at the same 
 time the rays of 
 light emitted by 
 objects placed in 
 front and those com- 
 ing from the surface 
 of the paper. The 
 outlines of the sub- 
 ject may be followed 
 on the paper with a 
 
 pencil, the point of which is seen directly while the subject appears 
 after a double reflection. 
 
 With this form of instrument the eye receives impressions simul- 
 taneously from objects at different distances. The pencil and paper 
 are quite close and the object is generally far away. The eye cannut 
 accommodate itself to both distances ; one of the images is always 
 more or less confused, and the work is very trying. 
 
 In looking at a point of the perspective through the diagraph or 
 
 the camera lucida, it will be 
 noticed that the corresponding 
 position of the pencil is liable 
 to slight variations. This is 
 the effect of parallax, and will 
 "~A be understood better by con- 
 sidering a perspective drawn 
 on a pane of glass, B (Fig. 
 122), placed in front of the 
 subject. By looking through 
 the eye-hole .Z), the distant point 
 A will be seen in the directions 
 Ea or Fb, according to the 
 
 Ffg-. 122 
 
GAMES A LUCID A. 73 
 
 position of the eye. The pencil may thus be placed either at a or b and 
 still coincide with A. The parallax or displacement of the pencil is 
 equal to the diameter of the eye-hole, and might be eliminated by 
 making the hole very small, but practically it cannot be made smaller 
 than the pupil, because it would cut off part of the light entering the 
 eye and the images would be too dark. The parallax can be eliminated 
 otherwise. 
 
 Substitute for the eye-hole 
 a plano-concave lens B (Fig. 
 123); the parallel rays from 
 A will, after passing through 
 .Z?, diverge from 0, the focus 
 of the lens, and the appear- 
 
 Fie* 123 ance to the eye is as if A had 
 
 been brought to coincide with 
 
 0. By placing such a lens between the subject and the prism of the 
 camera lucida, close to the latter, and selecting a lens of a focal length 
 equal to the distance between the prism and the paper, the virtual 
 image of the subject is brought into the plane of the paper, the par- 
 allax disappears, and the eye, accommodating itself to this one dis- 
 tance, is relieved from strain. Wollast.on, to whom this improvement 
 is due, explained that the same effect was obtained by making the 
 upper face of the prism concave and dispensing with the plano-concave 
 lens. 
 
 It was with the camera lucida, in 1849 and 1850, that the first per- 
 spective surveys were made. The improved instrument which Col. 
 Laussedat devised for surveying is the hemi-periscopic camera lucida 
 (1). He takes 15 centimetres for the radius of the spherical concavity 
 in the upper face of the prism, and places the centre of the sphere on a 
 perpendicular to the upper face of the prism passing through its inner 
 edge (Fig. 124). The nodal point is thus located very nearly on the 
 
 ., Dinner edge of the prism, which is also its axis of rota- 
 
 ^/j /^\ tion. Unless these two conditions are fulfilled a motion 
 
 ^^ Jot the prism round the axis causes a displacement 
 
 V j /^ of the image on the paper. The focal length is about 
 
 ^J^^ 30 centimetres, which is the average distance of distinct 
 
 vision, and for which the parallax of tke instrument 
 
 pv is entirely eliminated. It may still be used within 
 
 ig*. 1J4 certain limits at shorter or greater distances from the 
 
 paper without any great amount of parallax, but it must be noted that 
 
 (1) Col. Laussedat's investigations were first published in the " Memorial de 
 1'officier du genie " for 1854. Very full instructions for the use of the camera lucida 
 in surveying are given in the " Annales du Conservatoire des Arts et Metiers for 1891. 
 
74 
 
 PERSPEC TI VE INS TR UMENTS. 
 
 it would not be available for copying or enlarging drawings, although 
 it might answer for reducing. In copying a drawing full size, the 
 parallax is the same as when looking at a distant point through a 
 prism with a plane upper face. For work of this kind, the mounting 
 of the prism should have a second eye hole to the right or left of the 
 concave portion. The parallax may p then be eliminated and the rays 
 of light brought to the same divergence by means of auxiliary lenses. 
 Laussedat's camera lucida is fixed to the drawing board by two 
 arms with clamps, which are set exactly opposite each other by means 
 
 of the scales drawn on the edges of the board (Fig. 126). The ad- 
 justments are made in the following order : 
 
 1. The drawing board is levelled in the same way as a plane table. 
 
 2. The upper face of the prism is levelled by means of the level 
 shown above (Fig. 125) and the slow motion screw at the side. 
 
 3. The principal point is found with a piumb line touching the 
 edge of the prism (Fig. 126). The distance from this edge to the 
 principal point is the distance line : it is measured with a scale. It 
 can be made longer or shorter by changing the length of the two side 
 arms of the instrument. 
 
 4. Horizon and principal lines Suspend a plumb line at some 
 distance in front : turn the drawing board around its vertical axis 
 till the image of the plumb line is seen passing through the principal 
 point. This image is the principal line : it is traced with the pencil. 
 A perpendicular through the principal point is the horizon line. 
 Other constructions for these lines will be found in Col. Laussedat's 
 memoir. 
 
CAMERA OBSCURA. 
 
 75 
 
 The angles of the prism must be very accurate : otherwise, their 
 errors must be calculated and taken into account. 
 
 Fig. 126 
 
 Neutral tint glasses placed between the eye and the paper, or the eye 
 and the subject, serve to equalize the brightness of the images. 
 This form of camera lucida is a perfect surveying instrument. 
 
 90. CAMERA OBSCURA. A camera obscura, in its simplest form, is a 
 box hermetically closed to extraneous light, except that coming through 
 a lens placed on one of the sides. The opposite side of the box being 
 in the focal plane of the lens, an image is formed on it of the distant 
 objects situated in front of the lens. 
 
 Disregarding the errors introduced by lenses, the image of the 
 camera obscura is a true perspective, for it is the same as would be 
 drawn on a picture piano placed in front of the lens at a distance 
 equal to the focal length. 
 
76 
 
 PERSPECTI VE INS TR UMENTS. 
 
 Let 
 
 R 
 
 0, (Fig. 127) be the optical centre of the lens and OA, OB, OC, 
 
 three rays of light coming from 
 three distant points of space. 
 The images of the points form 
 a triangle ABC in the focal 
 plane R of the camera. The 
 same rays form, by their inter- 
 section with a plane Q parallel 
 to R and at the same distance 
 from 0, another triangle abc 
 in which every side is equal and 
 parallel to the corresponding 
 side of ABC, therefore, abc= 
 ABC. 
 
 Fig-. 127 
 
 The same demonstration applies to any other triangle, and as a figure 
 can always be resolved into a number of triangles, any figure obtained 
 on the plane R is the same as on the plane Q reversed. But the figure 
 on Q is the perspective seen from the station on the picture plane Q, 
 therefore the image of the camera obscura is the perspective seen from 
 the optical centre of the lens on a picture pLine placed at the first focus. 
 The focal length of the lens is the distance line. 
 
 Before photography was known, the camera obscura was used for 
 drawing perspectives ; various forms were devised to adapt it to that 
 purpose. 
 
 One form consists of a rectangular prism with two spherical faces 
 and a plane hypothenuse reflecting face. The parallel rays of light 
 emitted by the subject are brought to a focus by the two spherical 
 faces while they are reflected at right angles by the hypothenuse face. 
 
 The prism is placed on the top of a tripod, which supports a drawing 
 board at the proper height to receive the image formed at the focus of 
 the object glass. The tripod is covered with a black cloth to shut off 
 extraneous light, so as to enable the draughtsman to see the image 
 projected on the paper and to follow it with a pencil. 
 
 The point of the pencil being between the lens and the paper, casts 
 a shadow just at the point where the image is wanted. The instrument 
 shown in Fig. 128 is not open to the same objection and requires 
 neither tripod, drawing board, nor even a black cloth. It is merely a 
 box with a lens in front and a mirror in PQ inclined at 45 to the axis 
 of the box. 
 
bb 
 
PERSPEQTOGRA PH. 77 
 
 The image formed by the lens is reflected by the mirror on a ground 
 glass placed in AB\ being inverted a second time 
 by the reflection, it now appears upright. The lid 
 which covers the ground glass, when not in use, is 
 open at an angle of about 45, and cuts off 
 sufficient light for the image to be visible. Under 
 these conditions, the image is not bright enough 
 to work on paper and has to be traced on the 
 ground glass, but with a black cloth covering the 
 box and the head of the draughtsman, it is 
 possible to work through thin paper. 
 
 Fig-. 128 91. PERSPECTOGRAPH. The perspectograph is 
 
 the invention of Hermann Ritter, a German 
 
 architect (1), its object being to draw a perspective from the plans of 
 the subject and not from the subject itself. The lines of the plans are 
 followed with a tracer, and the perspective is drawn by a pencil carried 
 by another part of the instrument. 
 
 ' As constructed by Chp. Schroder & Cie (Frankfort-on-the-Main), it 
 is a large instrument made partly of wood and partly of metal ; well 
 adapted for drawing perspectives of buildings from an architect's 
 plans, but useless for drawing a topographical plan from a perspec- 
 tive. It is represented in Fig. 129. 
 
 For surveying purposes, the instrument should be of small size, made 
 entirely of metal, and all the parts should be fitted with precision. It 
 should work easily and the amount of dead motion should be as small 
 as possible. 
 
 Its theory is, with slight modifications, applicable to any other per- 
 spective instrument and for that reason is given here at length, but 
 the use of the perspectograph, in its present form, cannot be recom- 
 mended for photographic surveys. 
 
 However perfect an instrument may be it always introduces in the 
 final result some errors of its own, due to dead motion, to imperfections 
 in the adjustments, and to the slight errors unavoidable in the deter- 
 mination of the constants. Whenever the precision of the survey 
 requires it, a geometrical construction should be employed, and the use 
 of perspective instruments should be restricted to reconnaissance or 
 rough surveys, in which rapidity is more important than perfect 
 accuracy. 
 
 (1) Perspectograph, von Hermann Hitter, Architekt, Frankfurt a. M. Druck 
 von J. Maubach & Co. 
 
78 PERSPECTIVE INSTRUMENTS. 
 
 The principle of the apparatus is as follows : 
 
 Let S (Fig. 130) be the station, M a point 
 of the ground plane and /JL its perspective. 
 Take siS\ parallel to the ground line and 
 equal to sti ; join S-^M. 
 
 The similar triangles sSM and 0/J.M give 
 
 sS Ms 
 
 r.130 
 
 From the triangles sS^M and 
 
 0^ " HO 
 , also similar, we have : 
 
 Hence, 
 
 But, by construction, sS is equal to sS^ therefore 
 
 This relation furnishes a new method for constructing a perspective. 
 Take on the ground plan (Fig. 131) sS^ parallel to the ground line 
 and equal to the height of the station. To find the perspective of a 
 
 point M of the plan, join sM and 
 S-^M, which intersect the ground 
 line in and //.j. On another 
 part of the paper, draw the ground 
 line of the perspective X'Y' and 
 take on it a point 0' to represent 
 the point of the ground plan. 
 At 0' erect 0','j. perpendicular to 
 
 P,. 
 
 X'Y and equal to Ol>-^ \ ;j. is the 
 perspective of M. Owing to the 
 
 position of the figure, the perspective appears upside down. 
 
 The perspective of another point N of the ground plan is obtained 
 in a similar manner by taking O'Q' equal to OQ and Q'v equal to Qv l . 
 
 This is done mechanically by the perspectograph : sM and S^ M 
 (Fig. 132) are two wooden arms joined in M and carrying the tracer. 
 They slide through four adjustable pieces s, S lt and /^ ; s and S^ 
 
PBRSPECTOQR4 PH. 
 
 79 
 
 can be adapted to any part of a rule RT, s is fixed at the point of 
 the ground plan representing the foot of the station, and the rule or 
 slide RT is firmly clamped to the drawing board parallel to the ground 
 line. The second piece S^ is placed at a distance from s equal to the 
 height of the station and fixed in that position. 
 
 The third piece is attached 
 to a rod which moves in the 
 groove of a slide XY, and car- 
 ries a pantograph system, with 
 axis at D fixed to the rod, so 
 that the distance from to D is 
 invariable while the instrument 
 is in use. When the arm s M is 
 moved, 6- being a fixed point, 
 follows the motion of the arm, 
 and carries with it, along the 
 groove XT, the movable rod 
 and the pantograph system. 
 
 Fig-. 132 
 
 The fourth piece //j is con- 
 nected with the joint A of the 
 pantograph system, so that the 
 
 distance i>.^A is invariable during the operation ; it is also bound to 
 
 slide on the movable rod. 
 
 The pantograph system is composed of four straight arms (AB, AC, 
 F<). and F/JL) and two arms (CDE and BDG) bent at right angles in 
 D. They are joined in A, B, C, Z>, E, F and G, the sides of the 
 parallelograms ABDC and DGFE being all equal. The arms F>>. and 
 FH' are double the length of one side of the parallelograms, and the 
 pencil which is to describe the perspective may be placed either in 
 //- or //. 
 
 The sum of the four angles at D is equal to four right angles ; two 
 of these angles, CDE and BDG, being right angles, the sum of the 
 two remaining ones must be equal to two right angles ; that is 
 
 But in a parallelogram the sum of two adjacent angles is equal to two 
 right angles ; that is, 
 
 hence, 
 
 CDB + DC A 180 C 
 EDO DC A. 
 
80 PERSPECTIVE INSTRUMENTS. 
 
 Therefore the two parallelograms are equiangular and their sides being 
 equal, the parallelograms are equal, but not placed in the same direc- 
 tion. The diagonal DA of one is equal to the diagonal GE of the other, 
 and BC is equal to DF. 
 
 The line //// is parallel to GE because Fp. is equal to F>J! ; it is there- 
 fore perpendicular to X Y since the diagonals of a rhombus intersect at 
 right angles, and it passes through D, because E;J. is equal and parallel 
 to GD. We have also 
 
 D<J. GE = DA 
 It is now easy to understand the working of the instrument. 
 
 The slide XY is placed on the ground line or rather on the line 
 representing the trace of the picture plane on the ground plan. When 
 the tracer in M is moved on a parallel to XY, the arm Ms carries 
 with it the movable rod and the pantograph system attached. The 
 distance from to ,u.^ does not vary, since the similar triangles MsS^ 
 and MOp-i always give the same proportion between O/JL I and the 
 constant length sS^. The distance from /^ to A and from to D 
 being invariable, and 0<J.^ being constant, AD and consequently Dp. 
 do not change, and the pencil in //. describes a parallel to XY : it is 
 the perspective of a line of the ground plane, parallel to the picture 
 plane. 
 
 When the tracer is moved away from XY, in the direction of Ms, 
 the points and D do not change, but 0;j-^ is lengthened and /^ 
 moves towards the right carrying with it the joint A and increasing 
 the diagonal DA to the same extent as 0/J.^ ; D;J. being equal to DA, 
 is also lengthened and JL moves down, precisely the same distance 
 as /jij moved to the right. 
 
 The construction thus effected mechanically is that of Fig. 
 1.31. The ground line of the perspective, X' Y', is the line which 
 would be described by the pencil in y., if the tracer M could be brought 
 to the centre of the groove and moved along XY : and ;j. r would then 
 coincide. 
 
 Drawing the tracer away from XY, but in the direction sM, /JL I sep- 
 arates from 0, and /JL moves down by the same quantity from its for- 
 mer position 0' on the ground line, O'p. being perpendicular to X Y' 
 
 and O'/J- = O/j-i 
 
 Now if M be placed on any other point of the ground plan, the per- 
 pendicular D;J. to X Y' will be carried away the same distance as the 
 point 0, and // will be at a distance from X Y ' equal to the new value 
 of O. 
 
PERSPECTOGRAPH. 81 
 
 The perspective is upside down, the draughtsman having to place 
 himself near J/to guide the tracer. 
 
 The end // of the arm F;i describes the symmetrical figure of the 
 perspective, or the image which would be seen in a mirror, but were 
 the fixed point S-^ placed on the left of s, // would describe the true 
 perspective, the direction of the motions of p. and // being reversed. 
 The ends ;i and //' of the arms of the pantograph system are both fitted 
 to receive the pencil, which can be changed from one to the other as 
 required. 
 
 The instrument, set as in Fig. 132, can only work on points or figures 
 beyond the picture plane : it is possible to place the slide XY on the 
 other side of RT, so as to work on points between the picture plane 
 and the station, but the obliquity of the arms prevents them from 
 sliding freely and the working of the instrument is unsatisfactory. 
 
 It may happen that with a high station and points at the extreme 
 right or left (it would be the extreme left on the figure), the obliquity 
 of the arm S^M becomes too great to work. S- must then be changed 
 from one side of s to the other, (from the right to the left of s on the 
 figure) and the perspective from one side of the ground line to the 
 other. The pencil is at the same time changed to the opposite arm of 
 the pantograph system. 
 
 The sliding rod XY may be reversed end for end in its groove, the 
 pantograph system coming on the opposite side of the movable arm 
 Ms. The pencil does not require to be changed, but the arm bearing 
 it, /jf for instance, instead of being between RT and XY, will now be 
 on the other side of XY. 
 
 A scale must be drawn on RT, the zero corresponding to the pointer 
 carried by the sliding piece s : the graduation extends to right and 
 left, and the pointer of the sliding piece S^ is set opposite the division 
 corresponding to the height of the station above the ground plane. 
 
 The distance between RT and XY is equal to the distance line of 
 the perspective. 
 
 In the case of figures in planes which are not perpendicular to the 
 principal plane, it has been shown that the solution of problems 
 involves changes in the distance line : the edges of the drawing board 
 must, therefore, carry graduations to permit of XY being moved any 
 given quantity, keeping it parallel to RT. 
 
 The different pieces of the instrument are adjustable, and must first 
 be placed in proper position for the work in hand. This done, the 
 6 R 
 
82 PERSPECTIVE. 
 
 slide RT is firmly clamped to the drawing board and XY placed parallel 
 to RT. All that is necessary now, is to draw the scales and determine 
 the position of the various lines and points on which rests the construc- 
 tion of the perspective. Hitherto, it has been assumed that the points 
 s, Si, 0, p., etc., were mathematical points and that their distances 
 could be measured directly, but, there is nothing on the instrument 
 to define their exact position and no such measure could be taken 
 with precision, consequently these quantities, which are the constants 
 of the instrument, have to be determined indirectly. 
 
 92. To DRAW THE TRACE OF THE PRINCIPAL PLANE ON THE DRAWING 
 
 BOARD. It is assumed that the grooves in the slides RT and XY are 
 parallel to their edges, and that the sliding motion in these grooves is 
 also parallel to the same direction. This assumption is practically 
 correct. 
 
 Place S above s (Fig. 133) and draw a 
 perpendicular M. M^ to RT, which, if pro- 
 T duced, would pass as nearly as possible 
 through s. Should this line pass exactly 
 through s, it would be the trace of the 
 principal plane on the ground plane, and 
 ' were the tracer M moved along the line 
 from M to M 15 the point of the slide XY 
 would not move. To ascertain whether 
 has moved or not, mark the position of p. 
 when the tracer is in M t then place the 
 - tracer in M^ and if p. has moved to the 
 
 right, in p lt this shows that MM^ is too 
 
 much to the left. Should MO and M^O be equal respectively to one- 
 half and twice the distance line, the error in the position of MM^ 
 would be equal to three times the displacement of //., but it is sufficient 
 for estimating the quantity by which MM 1 has to be shifted to repeat 
 the trial two or three times. The motion of is indicated by the 
 displacement of //. to the right or to the left : a motion of p- perpen- 
 dicularly to XY indicates merely that S^ is not precisely over s. 
 
 93. TO FIND THE DISTANCE FROM THE STATION TO A FRONT LINE OF 
 
 THE GROUND PLANE. The trace of the principal plane on the drawing 
 board being now determined, the ground plan could be placed in its 
 proper position on the board were it possible to measure exactly the 
 distance from the point s of the instrument to a front line AB, (Fig. 
 134), of the drawing board. This determination is made as follows : 
 Draw a second line CD parallel to AB ; place the two long arms one 
 above the other and bring the tracer to M ; mark the position of p.. 
 Then carry the tracer to CD, and follow the line until the pencil has 
 
 M 
 
DETERMINATION OF CONSTANTS. 
 
 83 
 
 returned to the same point n ; the line NM, if produced, would pass 
 
 through s. Repeat the same 
 operation in M^ N\ and fj.^. 
 Having the lengths of M. M l and 
 NN^ and the distance d be- 
 tween AB and CD, a simple pro- 
 portion gives the distance x from 
 
 Pie*. 134 
 
 two lines and the points 
 must be taken as far apart 
 as e instrument will allow. 
 
 Having the distance of the front line AB, other front lines at fixed 
 distances from the foot of the station are permanently marked on the 
 drawing board. 
 
 The ground plan can now be placed on the board by putting the 
 trace of the principal plane in coincidence with the line previously 
 drawn on the board, and placing a front line, of which the distance is 
 known, upon the corresponding one of the drawing board. 
 
 94. TO FIND THE DISTANCE BETWEEN THE TWO SLIDES. The distance 
 
 between the two slides is equal to the distance line of the perspective; 
 it must be indicated by a scale on the edge of the drawing board. In 
 order to locate the zero of the graduation the precise distance has to 
 be determined in one position of the instrument. 
 
 Draw a front line AB (Fig. 135). Put the tracer first in A and 
 
 then in B, marking in each case the 
 
 T positions /-/ and /Jt x occupied by the 
 
 pencil. Let d be the distance between 
 the slides, and m the distance from s 
 to AB ; we have : 
 
 Fiar.136 
 
 m AB 
 
 AB being a front line, its perspective 
 fLfjLi is parallel to XY and equal to 
 00 j ; consequently : 
 
 6J 
 
84 PERSPECTIVE. 
 
 The three lengths which form the second term can be measured and 
 the value of d calculated. 
 
 95. To DRAW THE GRADUATION FOR THE HEIGHT OF THE STATION. 
 
 The height of the station is represented on the instrument by sS^ 
 (Fig. 132). It is necessary to determine this distance for one position 
 of s and S^ in order to draw the graduation for setting S^ to any 
 required height. 
 
 Place the tracer M on the trace of the principal plane afc a distance 
 Ms equal to one and a half times sO, and note the place occupied by 
 the pencil //-. In this position of the instrument we have : 
 
 Then place the tracer M, still on the trace of the principal plane, but 
 at a distance Ms equal to three times sO : we have : 
 
 O/J.-L = ^sS 1 . 
 
 The change in the value of 0/J-^ is thus equal to one-third of the *" 
 height of the station ; but, this change is represented by the displace- 
 ment of the pencil ,u, which can be measured with a scale. Three 
 times this displacement is the height of the station. 
 
 The tracer, instead of being placed at the distances from the- foot of 
 the station given above, may be set at any distance which may be 
 convenient ; the fraction of the height of the station obtained will 
 be different, but the process will be the same. 
 
 96. TO DRAW THE HORIZON, GROUND AND PRINCIPAL LINES OF THE 
 
 PERSPECTIVE. The principal line is the perspective of the trace of the 
 principal plane on the ground plane. The latter has been marked on 
 the board ( 92) ; following it with the tracer, the pencil describes the 
 principal line. 
 
 The ground line cannot be drawn directly, because the tracer would 
 have to be carried along the front slide, 
 and the construction of the instrument 
 does not permit it. The difficulty is over- 
 come by drawing the perspective of a front 
 line between the ground and horizon lines. 
 Let Fig. 136 represent the principal 
 plane and M the trace of a front line of 
 the ground plane at a distance sM from 
 the foot of the station equal to twice the 
 distance line. The similar triangles 
 T. 136 fJ-p M give 
 
 pM 
 
CENTEOLINEAD. 
 
 85 
 
 Ss is equal to the height Pp of the principal point above the ground 
 line, therefore 
 
 up = \Pp 
 
 Following then with the tracer the front line drawn on the board 
 at a distance from the station equal to twice the distance line, the 
 pencil describes a horizontal line midway between the ground and 
 horizon lines. One-half the height of the station is now measured on 
 each side of the line so obtained and parallels drawn to it. The line 
 nearest to the front slide is the ground line, the other one is the horizon 
 line. 
 
 Processes similar to those given for the perspectograph can be 
 employed for any other perspective instrument. 
 
 97. CENTROLINEAD. In addition to the instruments already de- 
 scribed, others have been devised merely to facilitate the construction 
 of perspectives. They are not, properly speaking, perspective instru- 
 ments, since they do not enable the draughtsman to draw the per- 
 spective directly. 
 
 The vanishing point of a line nearly parallel to the picture plane, 
 being at a great distance from the principal point, may fall outside 
 the paper, in which case special constructions are necessary to draw a 
 line which, if produced, would pass through the vanishing point. 
 
 A line vanishing at any point may be drawn with the " centrolinead" 
 no matter how far it is from the principal point. The instrument con- 
 sists of a straight edge (Fig. 137) with two arms whose inclination to 
 
 Fig-. 137 
 
 the straight edge can be varied at pleasure. Two studs, six or eight 
 inches apart, are fixed to the edge of the drawing board. The arms of 
 
86 
 
 PERSPECTIVE. 
 
 the centrolinead being placed in contact with the studs, the various 
 directions of the straight edge intersect at a common point. 
 
 Let 00 (Fig. 138) be the straight edge, OA and OB the arms, A 
 and B the studs. Through A, and B pass a circle ; the arms being 
 clamped, the angle is constant and is bound to move on the circum- 
 ference of the circle whenever the position of the centrolinead is 
 changed, as from OC to O'G'. 
 
 Produce OC and O'G' to their second intersection with the circum- 
 ference : they must cut it at the same point V because the angle AO V 
 
 being invariable, must always 
 subtend the same arc AV, no 
 matter on what point of the 
 circumference the apex may 
 be. Consequently the straight 
 edge will draw all the lines 
 vanishing at V. 
 
 In plotting surveys, the cen- 
 trolinead is employed only for 
 horizontal lines, whose vanish- 
 ing point is on the horizon line. 
 The studs A and B are placed 
 on a perpendicular to the hori- 
 ; it follows that the horizon line 
 
 c r 
 
 Fig-. 138 
 
 zon line and at equal distances from it 
 HH' is a diameter of the circle, and, 
 
 VA= VB 
 
 The arms are equally inclined to the straight edge. The line OC, 
 bisecting the angle AOB, must pass through V which is the middle of 
 the subtended arc BVA. 
 
 The distance of the vanishing point, F, can be varied either by 
 changing the positions of the studs or the inclination of the arms. 
 Increasing the distance AB between the studs, the size of the circle is 
 increased in the same proportion and V moved to the left. 
 
 It is not usual to disturb the studs, the changes in the distance of 
 the vanishing point being obtained by altering the inclination of the 
 arms of the centrolinead. Were the arms perpendicular to the straight 
 edge, the vanishing point would be at infinitum, and the instrument 
 would describe parallels to the horizon line. 
 
 Closing the arms gradually, V comes near to AB ; when AOB be- 
 comes a righb angle, the intersection of AB and HH' being the centre 
 of the circle, the distance of V from AB is one half of the latter. 
 
 Closing the arms again, V continues to move towards AB without 
 ever reaching it. 
 
CENTROLINEA D. 
 
 87 
 
 In reality the studs are not mathematical points, but cylinders ; the 
 direction of the straight edge is, however, the same as if the arms 
 rested against the axes of the cylinders. 
 
 The direction of the vanishing point 
 may be given by a line of the ground 
 plan or by a line of the perspective. 
 In either case, the arms of the centro- 
 linead have to be set to correspond to 
 the vanishing point. 
 
 In the first case, revolve the picture 
 plane on the horizon plane around the 
 horizon line as an axis. The station 
 Fig". 139 comes i n s (Fi g< i39) } $p i s the dis- 
 
 tance line, A and B the two studs. Let SV be the direction of the 
 given line on the ground plan, V is its vanishing point. Through A, 
 B and V pass a circle ; the centrolinead should be set so that the 
 straight edge being on DH', the arms should be on DA and DB. 
 
 Join VB ; the angle VDB, the inclination of the arm on the straight 
 edge, is equal to the angle VBA, because they subtend equal arcs. 
 Join CS and US, and draw Me and cv parallel to AB and HH' ' ; join 
 bv. By reason of the similarity of triangles, vb is parallel to VB and 
 the angle 
 
 vbc VBC BDV. 
 
 Therefore, place the straight edge on Mb, the axis of rotation on b, 
 and adjust the left arm of the centrolinead to coincide with bv. The 
 other arm may be set by placing the straight edge on vb, the axis on b 
 and adjusting the arm to coincide with bM, or better by placing the 
 straight edge on the horizon line, the arm being already adjusted in 
 contact with the stud B and moving the other arm until it comes in 
 contact with the stud A. 
 
 The lines BS, CS, Me and cv are drawn once for all. bv need not be 
 drawn, so that the only line to be marked for setting the centrolinead 
 is Sv, the direction of the given line on the ground plan. 
 
 When the given line belongs to 
 the perspective, the construction 
 must be slightly modified. VE 
 (Fig. 140), being the given line, 
 take any point, F, on the horizon 
 line, join FE and FB, and draw 
 cM parallel to AB. Through e, 
 draw ev parallel to ^Fand join vb. 
 Owing to the similarity of tri- 
 angles, vb is parallel to VB and 
 the angle vbc is equal to VBA, the 
 inclination of the arm on the straight edge of the centrolinead. 
 
PERSPECTIVE. 
 
 FB and c M are drawn once for all, but FE and ve have to be marked 
 for every given line : that is, two lines instead of one by the former 
 construction. 
 
 Gentrolineads are usually sold in pairs, one to work on the right of 
 the principal point, and one on the left. 
 
 98. PERSPECTOMETER. In 6 7, a method has been given for transfer- 
 ring a figure from the perspective to the ground plan by means of 
 squares formed of lines parallel and perpendicular to the ground line. 
 The object of the "perspectometer" is to dispense with the construc- 
 tion of the squares' perspective. 
 
 On a piece of transparent material, glass, horn or celluloid, draw 
 two parallel lines AB, DD' (Fig. 141), and a common perpendicular Pp. 
 Take DP, PD', pA and pB equal to the distance line, and from p, lay 
 j) p j)' on AB equal distances pm, 
 
 mn, pm .... Join to P the 
 points of division m, n, m, 
 n . . . . Part of these lines in- 
 tersect AD and BD' at r, t, 
 r r , t', . . . . The corresponding 
 points are joined together by 
 lines which are parallel to 
 AB and DD. 
 
 The instrument is now 
 placed on a perspective, with 
 P on the principal point and DD' on the horizon line. The ground 
 line falls in XY, for instance : it is divided into equal parts by the 
 lines converging in P, and the figure of the perspectometer is the per- 
 spective of a network of squares in the ground plane having the equal 
 parts of XY as sides. By referring to 67, it will be seen that this 
 is the construction given for the " method of squares." 
 
 This instrument is useful for constructing from the perspective a 
 figure of the ground plane. By placing it on the perspective, the 
 squares covering the figure are at once apparent and only those 
 required are drawn on the ground plan. 
 
 The side of the squares is equal to the length intercepted on the 
 ground line between two of the converging lines : this distance is 
 laid on the ground plan from the trace of the principal plane, and 
 parallels drawn to the trace through the points of division. 
 
 The front line, nearest to the ground line, is laid on the ground 
 plan either by estimating its distance from the ground line or by 
 constructing it. The estimation is made by noting the fraction of a 
 square's side which represents the distance from the ground line. 
 
 
 
 
 I 
 
 ^/y/// 1 \ 
 
 s / / / / i \ 
 
 \ \\\VOs^ 
 \ \ \ \ \ \\ 
 
 
 -/--/// i\ 
 
 \--\--Y-\-\-X; 
 
 r 
 
 / / / 1 \ 
 
 \ \ \\\ 
 
 A 
 
 g o n m, ) 
 
 t m' n' o' q' 
 
 Fig-. 141 
 
DRAWING THE GROUND PLAN. 89 
 
 Figures in planes, inclined to the horizon but perpendicular to the 
 picture plane, are transferred to the ground plan by placing the centre 
 of the perspectometer on the principal point and the parallel lines in 
 the direction of the trace of the inclined plane on the picture plane. 
 The trapezoids of the instrument are the perspectives of squares in the 
 inclined plane, which squares are projected as rectangles on the ground 
 plane ( 79). The longest sides of the rectangles are perpendicular 
 to the picture plane, and equal to the length intercepted between two 
 converging lilies of the instrument on the trace of the inclined plane. 
 The shortest sides are the projections on the ground line of these 
 intercepted lengths. 
 
 The rectangles are constructed on the ground plan and the transfer 
 made from the perspective as in the preceding case. 
 
 When the plane containing the figure is inclined to the picture and 
 ground planes, the principal point must first be displaced on the 
 principal line, so as to project the figure on a plane perpendicular to 
 the picture plane and having the same trace as the given plane, ( 82) 
 the problem is then the same as the last one. 
 
 The perspectometer can only serve for perspectives having the same 
 distance line, such as photographs of distant objects taken with the 
 same lens ; every distance line requires a new instrument. The width 
 Pp should be equal to the height of the horizon line above the foot 
 of the picture ; the length DD' need not be longer than the picture, 
 the distance points being placed on the figure merely for the purpose 
 of demonstration. 
 
 The length of the equal parts of AB should be such that the divisions 
 of the lowest ground line employed are not too large for the degree of 
 accuracy required. These divisions are the sides of the squares or 
 rectangles of the ground plan and the larger their size, the less 
 accurate will be the transfer of the figure. 
 
 The instrument can be made by drawing it on a large scale on paper, 
 and taking a reduced negative from which a positive is made on a 
 transparency plate. The transparency is bleached in bichloride of 
 mercury and varnished ; the lines originally black, are now white on 
 clear glass. 
 
 99. DRAWING THE GROUND PLAN WITH THE CAMERA LUCIDA. The 
 distinction between the picture and the ground planes is purely con- 
 ventional ; the picture plane may be taken for ground plane, and the 
 ground plane for picture plane. If a be the perspective of the figure 
 A of the ground plane, inversely, A is the perspective of the figure a of 
 the picture plane. Consequently, any perspective instrument can be 
 employed to draw the ground plan from the perspective by a change 
 in the setting of the instrument, the distance line being now what was 
 
90 
 
 PERSPECTIVE. 
 
 formerly the height of the station, and inversely, the new height of the 
 
 station being the for- 
 mer distance line. 
 
 With the camera 
 lucida, the prism must 
 be fixed permanently 
 so that the height of 
 the virtual station 
 S'P (Fig. 142) above 
 the plane on which 
 the perspective is 
 placed, is equal to the 
 distance line of the 
 perspective (89). As 
 long as the latter does 
 not change, the prism 
 must remain in the 
 same position. The 
 perspective is placed 
 in HH'XY, in such a 
 
 Fig-. 142 
 
 manner that the line S'P joining the virtual station to the principal 
 point of the perspective is perpendicular to the plane of the table or 
 drawing board. 
 
 The ground plan is on a platform sX^ Y^ , parallel to the plane of 
 the table, and which can be moved up or down. It must be so placed 
 that the perpendicular Ss from the real station to the platform, is 
 equal to the height of the station ; s is the foot of the station, and the 
 ground line is somewhere in X^ Y r . 
 
 The eye looks by reflection at the perspective and sees directly the 
 pencil on the platform. 
 
 The determination of the constants is made by methods similar to 
 those of the perspectograph ( 92, 93, 94, 95, 96). 
 
 With the camera lucida, the ground plan may be drawn either within 
 or beyond the ground line ; it is a very great advantage. The par- 
 allax must be eliminated by concave or convex lenses of appropriate 
 focal length. 
 
 100. DRAWING THE GROUND PLAN WITH THE CAMERA OBSCURA. In 
 order to transfer a figure from the perspective to the ground plane, the 
 camera obscura would have to be set as in Fig. 143, the perspective 
 HH'X'Y' being placed vertical and the ground plan horizontal in sAD. 
 The distance from the optical centre of the lens, S t to the perspective 
 
DRAWING THE GROUND PLAN. 
 
 91 
 
 should be equal to the distance line, and the height of S above the 
 
 plane sAD equal to the height 
 of the station. The difficulty is 
 that only one line would be in 
 proper focus, the other portions 
 of the image being more or less 
 blurred. It might be possible, 
 by using a lens of proper focus 
 and a pin hole diaphragm, to 
 obtain fair definition within a 
 limited space ABCD, but it is 
 doubtful if the process would 
 Fig". 143 prove practical. 
 
 101. DRAWING THE GROUND PLAN WITH THE PERSPECTOGRAPH. In 
 order to draw the ground plan from the perspective with the perspec- 
 tograph, the distance between RT and XY (Fig. 132) must be equal 
 to the height of the station, and sS^ to the distance line. The princi- 
 pal point of the perspective must be placed in s, the horizon line 
 under RT and the ground line under XY. The instrument thus 
 arranged would not work with the perspectives used in surveying ; it 
 could not even be set, the obliquity of the arms being too great. It 
 may, however, be employed to transfer a figure of the perspective to 
 other planes than the ground plane, as, for instance, to obtain the 
 elevation of a building from the perspective of the facade ; the method 
 is fully described in Ritter's pamphlet (1). 
 
 In other cases, and particularly for topographical purposes, the 
 pencil must be placed in M and the tracer in //. The perspective is 
 placed under <j., with the ground and principal lines on the lines pre- 
 viously marked on the drawing board ( 96). Taking M with the 
 hand, the arms are moved so as to follow the lines of the perspective 
 with the tracer /;.. The operation being precisely the same as for 
 drawing the perspective from the ground plan ( 91), it is evident 
 that the pencil in M will now reproduce the ground plan. The use 
 of the instrument in this manner is at first a little difficult, owing to 
 the point M being guided by the hand while the perspective has fco be 
 traced with /*, whose motion is entirely different. Some practice is 
 necessary before being able to handle it successfully. 
 
 A certain amount of dead motion is inevitable in an instrument 
 of this kind, particularly when changing the direction of the motion 
 perpendicular to the slides XY and RT. In order to avoid the errors 
 which this would introduce, the pencil M should always be moved in 
 the same direction away from XY, for instance. When the draughts- 
 
 (1) See note, page 77. 
 
92 
 
 PERSPECTIVE. 
 
 man comes to a part of a line or curve which is directed towards XY, 
 he should lift the pencil, push M back to the other end of the curve, 
 and trace it in the opposite direction. 
 
 The position of the horizon line 
 HH' of the perspective (Fig. 144) 
 varies each time the distance from 
 s to S is changed, for it corresponds 
 to the tracer fj. when M is at infini- 
 turn and the two arms parallel ; rj, 1 
 would then be in N. Now, change 
 the height of the station from S l to 
 S 2 ; N comes in Q, carrying with it 
 the joint A of the pantograph sys- 
 tem to which it is rigidly fixed ; DA 
 is increased by NQ and ,u moves 
 down the same quantity. So the 
 horizon line is moved towards the 
 front of the drawing board the 
 is moved up. The ground line is 
 is provided with means of adjust- 
 these two points being connected 
 
 Fig-. 144 
 
 same distance as the station 
 not affected. The instrument 
 ment for the distance from A to / 
 
 by an iron rod sliding in a ring at A, to which it can be fixed by a 
 clamping screw. A graduation placed alongside the rod permits the 
 increase in the height of the station to be added to A/J.^. This adjust- 
 ment prevents the displacement of the horizon line. Both the perspec- 
 tive and the ground plan occupy invariable positions on the drawing 
 board, no matter what ground plane may be used. 
 
 When the distance line is changed, as for figures in inclined planes, 
 the simplest manner to place the perspective on the board is to put 
 the pencil in M on the trace of the principal plane (Fig. 144), MO 
 being equal to Os ; then slide the perspective under the tracer until 
 the latter is on a point midway between the ground and horizon lines. 
 The principal line of the perspective must, of course, coincide with the 
 line previously drawn on the drawing board. 
 
 102. CHANGE OF SCALE. The perspectograph, set as in Fig. 132, 
 will only work on points beyond the picture plane. It is possible, by 
 placing the slide XY on the other side of RT, to reach within the 
 picture plane, but the instrument does not work well. It is preferable 
 to use it as set in Fig. 132, and to resort to a change of scale when 
 figures within the picture plane are to be operated upon. 
 
 Let sXY and XYX^ 7 t (Fig. 145), be the ground and picture 
 planes. Take a new ground plane, s 1 X l Y 1} at a distance ss 1 below the 
 
DRAWING THE GROUND PLAN. 
 
 93 
 
 other plane equal to the height of the station Ss ; the figures obtained on 
 
 the new plane from the perspective, will 
 be on a scale double the former scale, 
 ( 54). The new ground line X I Y^ cor- 
 responds to the front line A B of the plane 
 sXY, midway between the foot of the 
 station, s, and the picture plane, so that 
 it is now possible to work with the per- 
 spectograph on the part of the ground 
 plane comprised between AB and XY, 
 but the result has to be reduced to half 
 
 145 
 
 sze. 
 
 By doubling the scale again, one half 
 of the space between AB arid s is covered ; the result must be reduced 
 four times. 
 
 In practice, the draughtsman commences by working on the figures 
 beyond the picture plane. After transferring them to the ground 
 plane, he moves S lt (Fig. 132) so as to double sS^ and draws a new 
 ground line on the perspective, by doubling the distance of the first 
 one from the principal point. He then places the perspective in proper 
 position for the new ground line and continues to operate the instru- 
 ment as before. 
 
 The ground plan is drawn on cross section paper, having squares of 
 four sizes, distinguished by lines of different intensities. The largest 
 squares are divided into four smaller ones which are also divided into 
 four. The sides of the squares are even divisions of the scale. When 
 the scale is increased two or four times, the reduction is made at sight 
 by transferring the figures from one set of squares to the other. Front 
 lines have been marked on the drawing board at even distances from 
 the foot of the station. The cross section paper is pinned to the board 
 with some of its lines upon the front lines of the board and a perpen- 
 dicular to the former on the trace of the principal plane. 
 
 The distance from the foot of the station is marked on one of the 
 front lines of the paper forming the sides of the squares : this distance, 
 with the trace of the principal plane permits of transferring to the 
 general ground plan the portion of it which has been obtained by the 
 perspectograph. 
 
 When the scale is changed, the distance of the front lines should be 
 changed accordingly. Thus, if the scale is doubled, the front line 
 marked 10 on the drawing board corresponds to a real distance of 5 
 and must be so marked on the paper. 
 
94 
 
 FIELD INSTRUMENTS. 
 
 CHAPTER IV. 
 
 FIELD INSTRUMENTS. 
 
 103. LENSES. The photographic lens is perhaps the most important 
 part of the instrumental outfit : upon its careful selection depends the 
 perfection and accuracy of the results. 
 
 There is no perfect lens ; all are subject to aberrations and deficien- 
 cies of various kinds. By combining glasses and lenses of different 
 shapes, the optician 'corrects some of these imperfections, and 
 it is for the photographer to select from the numerous combina- 
 tions available that particular one the deficiencies of which will have 
 the least detrimental effect on the work contemplated. 
 
 The curves ascribed to lenses are spherical. The line joining the 
 centres of the spheres forming the faces of a lens is the principal axis. 
 The lens is centered when the principal axis is perpendicular to the 
 planes of the circumferences limiting the faces. 
 
 Let #! and (7 9 , 
 Figs. 146 and 147, 
 be the centres of the 
 spheres ; ^ 1 and 
 R z their radii. 
 Draw the parallel 
 radii C x / and C 2 E 
 and join IE. The 
 tangent planes at / 
 and E being paral- 
 lel, the incident ray 
 PI, which is re- 
 fracted along IE, 
 emerges parallel to 
 its original direc- 
 tion. is the opti- 
 cal centre of the 
 lens ; it is fixed by 
 the relation : 
 
 Fig*. 146 
 
 c^o 
 
 ~c^o' 
 
LENSES. 
 
 95 
 
 and JV a , where the incident and emergent rays intersect the axis, 
 
 are the nodal 
 points. Their 
 position is con- 
 stant for rays 
 making a small 
 angle with the 
 optical axis. 
 Their distances 
 from the corre- 
 sponding sum- 
 mits of the len- 
 ses are given by 
 F ' e* 1 4.7 * k e equations 
 
 n 
 BO 
 
 in which n is the refractive index of the glass. 
 
 are nearly equal to one : if they are left out and AN- and 
 designated by a x and [a 2 , AO and BO by q^ and q 2 , the equations 
 become 
 
 a, -l 
 
 The distance between the nodal points is : 
 
 n 
 in which e is the thickness of the lens. 
 
 The planes Off, LJf, perpendicular to the principal axis at jy^ 
 and N<i are the principal planes. 
 
 A small pencil of rays parallel to the axis of a lens, converges to or 
 diverges from a point after refraction through the lens. In Fig. 148, 
 the pencil AG, BH, converges to the point F 2 on the axis. Similarly, 
 a pencil coming from the other direction converges to F 1 . These 
 
96 FIELD INSTRUMENTS. 
 
 two points are the principal foci. Their distances f^ f zi from the 
 corresponding nodal points J^ lt JV 2 , are equal : this distance is called 
 the jocal length. The planes F l K, F 2 S, perpendicular to the axis at 
 F l F 2 , are the focal planes. It is customary to number the planes 
 and points of a lens from the side from which the light comes. Thus in 
 
 Fig-. 148 
 
 Fig. 148, the light coming from the left, F l is the first principal 
 focus, NI the first nodal point, F l K the first focal plane and GX the 
 first principal plane ; F 2 and N 2 are the second principal focus and 
 second nodal point, 7^$ and LZ&re the second focal plane and second 
 principal plane. In formulas relating to lenses, lengths are considered 
 positive in the direction of the light's course and negative in the 
 opposite direction. In Fig. 148, the light coming from the left, the 
 focal length, F^N^ or f^ is negative and the focal length F 2 N 2 or 
 f 2 is positive. In Fig. 146, the radius Ji l is positive and R 2 is 
 negative. In Fig. 147 both R l and R 2 are negative. 
 The value of the focal length is 
 
 /--A- 
 
 (n-l) 
 
 i I 
 
 n 
 
 n being the refractive index of the glass and e the thickness of the 
 lens. A positive value of f 2 indicates that parallel rays converge to 
 the principal focus after refraction ; a negative value of f 2 indicates 
 that the rays diverge from the principal focus. 
 
 In thin lenses, where e can be neglected, the formula becomes : 
 
LSNSttS. 
 
 The average value of n is 1'5; it gives for the approximate value of 
 the focal lenth : 
 
 ./"##! 2Rz 
 
 An oblique pencil AF l} TK (Fig. 148) converges after refraction to 
 a point S in the focal plane ^2^- 
 
 A point T on the axis forms its image at V; V and T are called 
 conjugate foci. 
 
 The point A and its image P are called conjugate points. 
 Designating by f l and f 2 the distances TN l and VN%, we have : 
 
 
 
 7 = A~A 
 
 Also: 
 
 in which c^ and ^ 2 are the distances TF 1 and F/^ 2 j from the object 
 and its image to the principal foci. 
 
 The image P of any point A in a plane AT perpendicular to the 
 axis is in the conjugate plane VP. 
 
 The intersections X and Z of an incident ray TX. with the first 
 principal plane and of the refracted ray ZV with the second principal 
 plane are on a parallel to the principal axis. 
 
 The above definitions hold good only for lenses of small aperture 
 and for rays making small angles with the optical axis. When these 
 conditions are not fulfilled, discrepancies occur which are called aber- 
 rations. 
 
 A lens is fully defined by its principal planes and foci. When 
 these are known, the course after refraction of any ray or pencil of 
 rays is easily found. 
 
 To find the image of a point A, join AF l and produce to the inter- 
 section with the first principal plane at H ; this ray passing through 
 the first focus is, after refraction, parallel to the axis in HP. 
 
 Draw AL parallel to the axis until it intersects the second principal 
 plane in L. This ray being parallel to the axis, is refracted in LF 2 
 to the second focus. 
 
 Join AN^ : the refracted ray is parallel to its original direction and 
 passes through the second nodal point N z . 
 7 R 
 
FIELD INSTRUMENTS. 
 
 The image P of A is at the intersection P of the three refracted 
 rays. 
 
 The course of any ray, TX, after refraction, is found by considering 
 a parallel ray F^H through the first focus. Its course after refrac- 
 tion is in HS parallel to the axis. ' A pencil parallel to F^ H forms 
 its image at S in the second focal plane ; TX must therefore after 
 refraction converge to S. As the intersections of the incident and 
 refracted rays with the principal planes are on a parallel to the axis, 
 the course of the refracted ray is obtained by drawing XZ parallel to 
 F^F^ and joining ZS. 
 
 The conjugate foci T and F are fixed in the same manner. 
 The rays, of course, follow these directions only before entering the 
 lens and after emerging therefrom. 
 
 Their course inside 
 of the lens is shown 
 by solid lines in the 
 figure. 
 
 These construc- 
 tions and definitions 
 apply also to nega- 
 tive lenses, but in 
 the latter, a pencil 
 parallel to the axis 
 diverges from the 
 principal focus after 
 refraction instead of 
 converging to it. 
 
 In Fig. 149, the 
 same constructions 
 have been made and 
 the same letters used 
 as in Fig. 148. With 
 
 Fie 1 149 
 
 image P of the point 
 
 A is real : with the negative lens, it is virtual. A point between a 
 positive lens and its principal focal plane also gives a virtual image. 
 
 To find the principal and focal planes of a combination of any 
 number of lenses, a ray parallel to the axis is supposed to enter 
 the lenses on one side ; its course is constructed as already explained, 
 and the point where it intersects the axis after the last refraction is one 
 of the principal foci of the combination. The corresponding principal 
 plane is given by the intersection of the original and final directions 
 of the ray. The same process applied to a ray parallel to the axis and 
 coming from the opposite direction, gives the other focus and principal 
 plane. 
 
ABERRATIONS. 
 
 Fig. 150 illustrates the combination of a positive lens F^ N^ F 2 N 2 
 and a negative lens F\ N\ F' 2 N' 2 . The ray AB parallel to the axis 
 
 Fig, 150 
 
 intersects the second principal plane of the first lens at B and is refracted 
 in the direction of the second focus F 2 ; it then intersects the first 
 principal plane N\ of the second lens at C, and therefore must be 
 refracted in a direction passing through D in the second principal 
 plane N' z on the parallel CD to the axis. GF\ is drawn parallel to 
 F 2 and EG parallel to the axis ; ED is the direction of the refracted 
 ray and/ 2 the second focus of the combination. The intersection H 
 of the original and refracted rays is a point of the second principal 
 plane n 2 . 
 
 By the same process the ray KL gives the first focus f and the 
 first principal plane n 1 . 
 
 104. ABERRATIONS. The lenses hitherto considered had small aper- 
 tures and the pencils of light 
 were not much inclined on 
 
 the axis. When the condi- 
 
 = tions are different, the rules 
 given above are only approxi- 
 mate. A pencil parallel to 
 the axis does not, after refrac- 
 tion through a single convex 
 lens, converge exactly to one 
 Pig 1 . 151 point. The central rays meet 
 
 '2" 
 
100 
 
 FIELD INSTRUMENTS. 
 
 in /'(Fig. 151), and the marginal rays in F '. In trying with this lens 
 to focus a luminous point on the ground glass of the camera, it is 
 impossible to obtain good definition. The focus is somewhere between 
 F and F', and the image appears as a circular patch of light decreasing 
 in intensity from the centre to the edge. FFi* the loDgitudinal aber- 
 ration ; it increases as the square of the aperture of the lens and 
 inversely as its focal length. Off is the lateral aberration : it increases 
 as the cube of the aperture and inversely as the square of the focal 
 length. The aperture is reduced by diaphragms : their effect is to 
 cut off the marginal rnys. In focussing a camera, the general practice 
 is to do so with a large diaphragm, and to insert a smaller one for expo- 
 sing. When there is much uncorrected spherical aberration, the two 
 foci may be quite different. 
 
 greater refraction of 
 marginal rays towards 
 the cause of an- 
 
 Fig*. 152 
 
 The 
 the 
 
 the axis is 
 other imperfection, called dis- 
 tortion, in the images pro- 
 duced by single lenses with 
 diaphragms. Fig. 152 shows 
 the diaphragm C in front of 
 the lens : the square grating 
 A, perpendicular to the prin- 
 barrel-shape " distortion. The 
 
 cipal axis, gives an image like B with 
 
 axial rays are not displaced, but the other rays are refracted more and 
 more towards the axis as they approach the edge of the lens. The 
 single rays represented may be taken as the axes of pencils. The 
 A " pin-cushion " distortion oc- 
 
 curs when the diaphragm is 
 behind the lens (Fig. 153). 
 The marginal rays are still 
 more refracted than the cen- 
 tral rays, but after they have 
 crossed the axis at the dia- 
 phragm, they become diver- 
 Fig*. 153 g en t and the opposite dis- 
 tortion occurs. This explanation suggests a remedy : by placing the 
 diaphragm between two lenses, the " pin-cushion " distortion of the 
 front one is compensated by the "barrel-shape " distortion of the back 
 one. Although this is the simplest form of a rectangular lens, it is not 
 the only way to correct distortion. 
 
 In passing through a single lens, the different rays of which white 
 light is composed are unequally refracted, the deflection being greater 
 for the violet than for the red part of the spectrum. With a pencil of 
 
ABERRATIONS. 101 
 
 white light the violet rays come to a focus at V (Fig. 154), the green 
 
 rays at G and the red 
 rays at JH. At none of 
 these points is a sharp or 
 uncoloured image ob- 
 tained. This is called 
 chromatic aberration. 
 The brightest part of the 
 spectrum is near the line 
 FigT.154 j)^ an( j i n adjusting a 
 
 camera, the best image is seen on the ground glass when these rays 
 are in focus. On the other hand, the chemically active part of the 
 spectrum on ordinary plates is about the lines G and H, so, with an un- 
 corrected or undercorrected lens, the dry plate requires to be placed at 
 some distance in front of the ground glass in order to be in the focus 
 of the blue violet rays. The lens has a " chemical focus " different 
 from the visual focus. Chromatic aberration is corrected by combin- 
 ing lenses made of glass of different refractive and dispersive powers. 
 A positive lens of small dispersive power associated with a negative 
 lens of great dispersive power can be made achromatic although still 
 remaining convergent. With two kinds of glass, two lines of the spec- 
 trum may be brought to the same focus, but some slight discrepancies 
 remain for the other rays and form secondary spectra. If n kinds of 
 glass are employed, n lines can be brought to the same focus. Photo- 
 graphic lenses are corrected for the G or H lines, which are the most 
 active rays on ordinary dry plates. It will be shown later that some 
 orthochromatic plates exposed behind deep orange glass are sensitive 
 to yellow ra} 7 s only the chromatic correction of photographic lenses 
 is, therefore, not as good as it might be for these plates. 
 
 In focussing a figure AB in a plane perpendicular to the optical axis 
 
 EF, (Fig. 155), different positions of the 
 ground glass will generally be found 
 for the centre F, and the sides C and 
 D of the figure ; this indicates curvature 
 of the field. A diaphragm GH reduces 
 the curvature by cutting off the rays of 
 shorter foci. The farther the diaphragm 
 is from the lens, the flatter the field 
 becomes but the distortion is increased. 
 
 Fifi* 155 
 
 Another difficulty is frequently ex- 
 perienced in focussing extra axial points. In a position of the ground 
 glass, the image of a luminous point is a short line directed towards 
 the centre of the field ; in another position, the image is also a short 
 line but perpendicular to the first one. Between the two positions, 
 the images are patches of light -elongated in a radial or tangential 
 
102 FIELD INSTRUMENTS. 
 
 direction, according as the ground glass is nearest to the first or last 
 position. This is astigmatism; it increases from the centre to the 
 margin of the field. It is due to the convergence after refraction 
 being smaller in the plane containing the optical axis and radiating 
 
 point, or me- 
 ridian plane, 
 than in the 
 perpendicu- 
 lar or sagittal 
 plane. Let^C 
 C7i(Fig.l5G) 
 be a small 
 rectangular 
 element of 
 a lens on 
 which falls a 
 pencil of par- 
 allel rays ; 
 
 the lens by the meridian plane and DE its section by the sagittal 
 plane. The meridian plane containing the centres of the spherical 
 faces, its intersections BH by the latter are great circles of the spheres. 
 The intersections of the parallel planes AG and C K are very nearly 
 great circles, because they are at very short distances from the meridian 
 plane. The sagittal intersections AC, DE, GK are small circles of the 
 spheres ; the curvature of the faces is thus greatest in the sagittal 
 plane and smallest in the meridian plane. The convergence of the 
 rays Z>, and E is therefore greater than the convergence of , 
 and ff, the first ones meeting in F' while the latter intersect at F. 
 The convergence in the planes of parallel sections being approximately 
 the same, there are formed two focal lines, one, F', directed toward the 
 centre of the field or radial, and another one, F, farther off and 
 tangential. In the other small rectangular elements of the lens, the 
 sagittal sections are still small circles ; the sections parallel to the 
 meridian plane are not exactly great circles, but the same general 
 character is preserved throughout of greatest curvature in the sagittal 
 sections and smallest in the meridian sections, each small element of 
 the lens giving in the same way a radial and a tangential image. 
 
 There are thus two focal surfaces tangent on the optical axis, and 
 becoming more and more separated as the distance from the centre 
 increases. The length FF is the astigmatic difference. In focussing 
 on a radial line, the best definition is at F'; on a tangential line, it 
 is at F. The best mean focus is somewhere between the two, acgk being 
 the image of the radiating point and the figure of least confusion. 
 
DIAPHRAGMS. 103 
 
 Some lenses give what is called " a flare spot " ; it is a circular patch of 
 light, more or less intense, in the middle of the image. J. H. Dall- 
 ineyer found by experiment that it was an image of the opening in the 
 diaphragm caused by the back lens of combinations, but he could not 
 understand how a real image was formed with such a short distance 
 between the lens and the diaphragm. The explanation was given by 
 Sir John Herschell. The rays entering the back lens are not all 
 refracted ; some are reflected by the back surface, and reflected again by 
 the front surface and it is by these rays that the image of the diaphragm's 
 opening is formed. In lenses of good construction, the curves of the 
 surfaces are calculated to throw the greater portion of the reflected 
 rays outside of the field and to spread evenly over it the remaining 
 ones. 
 
 105. DIAPHRAGMS. The aberrations increasing rapidly with the 
 aperture of a lens, the use of diaphragms naturally suggests itself. In 
 a single lens they are placed in front, because the barrel shape distortion 
 which they produce is less unsightly than the spindle shape. In 
 combinations, they are between the lenses. 
 
 The brightness of the image depends on the quantity of light ad- 
 mitted by the lens, and this is proportional to its aperture or to the 
 square of its diameter. The larger the aperture, the more light is 
 admitted. The brightness further varies inversely as the square of the 
 focal length ; for instance by doubling the focal length, the dimensions 
 of the image are doubled and the light admitted through the lens is 
 distributed over an area four times larger. The brightness of the 
 image is reduced in proportion. 
 
 Representing by d the diameter of the aperture, and by f the focal 
 length, the brightness is proportional to 
 
 d 2 
 
 This fraction is the measure of the rapidity of a lens, and the number 
 by which the camera exposure for different lenses or diaphragms is 
 computed. There are, however, several allowances to be made. A 
 porrion of the light is lost by reflection on the surface of the lenses, 
 and some is absorbed by the glass. If an exposure of 100 seconds be 
 required with a single lens and if each lens absorbs or reflects one- 
 tenth of the light, then the proper exposure with a double combination } 
 such as an aplanat, is 111 seconds ; with a triplet, it is 123 seconds 
 The colour of the glass has a great influence on the rapidity of lens* 
 
104 
 
 FIELD INSTRUMENTS. 
 
 The breadth of the pencils admitted by the diaphragm diminishes 
 with their inclination to the axis ; q being the area of the section of 
 the axial pencil, the section of a marginal one making an angle a with 
 
 the axis is : 
 
 q cos a 
 
 But an object forms an image larger 
 in A (Fig. 157) than in P, the pro- 
 portion being : 
 
 AO 
 OP 
 
 I 
 
 cos a 
 
 The proportion of the areas is 
 1 
 
 Fig-. 157 
 
 cos 2 a 
 
 and as the image at A is formed by pencils of reduced section, its 
 brightness, taking as unit the middle of the field, is 
 
 For several reasons -the decrease in illumination from the centre to 
 the margin of the field is in reality larger than this. It is well marked 
 in photographs taken with wide angle lenses ; the edges are always 
 much darker than the middle. 
 
 The exposure required by the photographic plate is inversely propor- 
 tional to the brightness of the image formed on its surface ; it increases 
 in the ratio : 
 
 (i) 
 
 It is important that each diaphragm should be numbered to indicate 
 its rapidity. 
 
 In the " uniform system," the exposure for a diaphragm of an aper- 
 ture of -^ is taken as unit, and each diaphragm is marked with the 
 corresponding exposure as follows : 
 
DIAPHRAGMS. 105 
 
 No. 2for^- 
 0-66 
 
 " " * 
 
 8 
 8 " 
 
 11-31 
 
 16 " ~J 
 16 
 
 32 
 
 (C QO 
 
 22-6 
 
 Zeiss takes as unit _^ : his numbers are : 
 100 
 
 No. ! for 
 
 " 
 
 " 4 " ^ 
 50 
 
 " 8 " 
 
 36 
 16 " J 
 
 25 
 
 " 32 " f 
 18-5 
 
 64 " ^ . 
 12-5 
 
 u 128 " * 
 
 u 
 
 256 " 
 
 6-3 
 
 Dallmeyer adopts for his unit 
 
106 FIELD INSTRUMENTS. 
 
 the International Congress of Photography has recommended 
 
 10 
 
 and several opticians have systems of their own. Many mark on their 
 diaphragms nothing but the value of 
 
 that is not, perhaps, the most convenient system, but it is the least 
 liable to cause confusion. 
 
 The section of the axial pencil passing through the lens is equal to 
 the opening of the diaphragm when it is in front of the lens : d is the 
 diameter of the aperture and may be measured with a scale. When a 
 lens or lenses are in front of the diaphragm, the effective and real 
 
 apertures are no longer 
 equal. The axial pencil con- 
 verges after refraction to the 
 focus F, Fig. 158, of the posi- 
 tive front combination, and its 
 section is reduced when it 
 reaches the diaphragm. The 
 diameter of the effective aper- 
 ture is : 
 
 Fig-.. 158 j_ A f 
 
 d is the diameter of the opening, / the focal length of the front lens or 
 combination, and I the distance from its second nodal point to the 
 plane of the diaphragm. 
 
 The effective aperture d' and the co-efficient 
 
 may be found by the following process due to Steinheil. After focus- 
 sing on a distant object, replace the ground glass of the camera by a 
 screen with a small hole in the middle. Place a bright light close to 
 and behind the hole. Cover the lens with a ground glass ; an illumi- 
 nated circle is seen on it which is the effective aperture. The diame- 
 ters of this circle and of the diaphragm are measured ; their ratio is 
 the co-efficient by which d must be multiplied. 
 
ANASTIGMAT8. 
 
 107 
 
 106. ANASTIGMATS. The lens of a surveying camera must be free 
 from distortion ; it must cover an angular field of about 60, and the 
 definition must be uniform all over the plate. Rapidity is not essential. 
 These conditions restrict the choice to what are 
 known as " wide angle lenses." They are doublets 
 consisting of two lenses placed close together and 
 between which is the diaphragm. The lens adopted 
 for the Canadian camera is Zeiss' anastigmat of f/l 8 
 aperture and 141 millimetres focus. Fig. 159 repre- 
 sents the lens full size. The radii, R, thicknesses, e, 
 and refractive indices for the line D of the spectrum, 
 n, are as follows : 
 
 FRONT LENS. 
 
 71 = 1-55247 
 
 ft?! = 4. 13*14 millimetres 
 K 2 =+6-61 
 
 e=l-25 " 
 
 Fig-. 159 
 
 ^ 3 =+6-61 millimetres 
 ^=+14-71 " 
 
 = 1-76 
 
 BACK LENS. 
 
 71=1-51674 
 
 R 5 = -28-33 millimetres 
 6 =+ 20-13 
 
 e=l-00 
 
 n = l-57360 
 
 7 = +20-13 millimetres 
 s =- 28-33 
 
 e = 2-89 
 
 The axial distance between the two lenses is 2*88 millimetres. The 
 diaphragm is midway between the lenses. The special features of the 
 anastigmats cannot be illustrated better than by giving the following 
 extracts from a lecture by Dr. Paul Rudolf at the Photographic Con- 
 vention of the United Kingdom ; * 
 
 The Zeiss anastigmatic lenses are dissymmetrical doublets consisting 
 of an achromatic anterior part, whose flint has the higher refractive in- 
 
 *British Journal of Photography, 28th July, 1893. 
 
108 FIELD INSTRUMENTS. 
 
 dex, and an achromatic posterior part in which the crown has the higher 
 refractive index. These two cemented parts of the doublet possess, 
 therefore, opposite differences of refractiveness in the crown and flint 
 glasses employed for achromatisation. "Crown" and "flint" are 
 here placed in opposition, not with respect to their chemical composi- 
 tion, but are considered with respect to their optical properties. The 
 same glass may therefore appear in two different achromatic com- 
 binations either as crown or flint. In the following remarks, crown 
 glass is always understood to refer to that glass of a binary lens 
 which is less in relative dispersion, while the term flint glass refers to 
 that glass which has the greater relative dispersive power. 
 
 The combination of the two cemented parts having opposite 
 differences of refractiveness in the crown and flint, embodies the 
 important, principle by which it became possible to effect anastigmatic 
 aplanatism of a system of lenses corrected spherically and chromatically 
 for large apertures. 
 
 The series of new glasses produced by the works of Messrs. Schott 
 & Co., of Jena, rendered it possible to realize this principle in the 
 construction of Zeiss anastigmatic lenses. 
 
 Refractive and dispersive powers were, with the older glasses, 
 dependent upon each other in a certain manner ; an increase in the 
 one corresponded to an increase in the other. In order that anastig- 
 mats may be constructed, it is, however, necessary that a range of 
 glasses be obtained in which any refractive inde^x may be coupled 
 with any desired dispersion. This is accomplished by the glasses made 
 by Messrs. Schott & Co. 
 
 All photographic lenses preceding the anastigmats had either a 
 very much curved field, or, if flat, an astigmatic field. In the latter 
 case the image was perfect in the centre, but deteriorated towards the 
 edge. 
 
 It may be useful here to define what is meant by a curved, flat and 
 astigmatic image. 
 
 Let the object be placed in a plane which is accurately normal to 
 the axis of the objective, or let the object be at so great a distance 
 from the objective that the difference in the distance of the different 
 points may be neglected. Then the field of an objective is considered 
 to be curved if different positions of the focussing screen are required 
 for sharply focussing a point in the axis (centre of the image) and a 
 point lying outside the centre of the image. 
 
ANASTIGMATS. 1C9 
 
 The whole of the older photographic lens types exhibit curvatures 
 of the image, such as to necessitate the dis- 
 tance of the focussing screen being shorter 
 for marginal points of the image than for 
 axial points. The points of distinct delinea- 
 tion are, therefore, situated upon a curved 
 surface whose concave side is presented to 
 the objective. Fig. 160 represents the 
 section of the image surface, and a plane 
 passing through the axis of the objective. 
 The dotted line a represents the ideal 
 ..,. image plane, which intersects the axis in 
 
 the axial image point, and is at right 
 
 angles to the axis of the objective, while curve b represents the actual 
 
 surface of the image. 
 
 .The field is flat if the position of the focussing screen is the same 
 for central and extra-axial points i.e., if the sharply focussed points 
 are all contained in a plane which is at right angles to the axis of the 
 objective, or if these points lie in the ideal plane of the image. 
 
 The image is astigmatic if sharp images of lateral, i.e., extra-axial, 
 points may be obtained by two different positions of the focussing 
 screen. The two images are not exactly similar to the object ; one of 
 them shows distortions in the directions radiating from the axis (radial 
 distortion), while the other exhibits distortions in the directions at 
 right angles to the radii (tangential distortion). This fact may easily 
 be demonstrated by means of an " aplanatic " lens, say, by using a 
 small circular disc having a diameter of only a few millimetres situ- 
 ated outside the axis and attempting to sharply focus it. With the 
 nearer position of the focussing screen, the image of the disc appears 
 as a radial line of a .breadth proportional to the diameter of the disc. 
 With the longer distance of the screen, the image is an arc of a 
 circle concentric to the axis of the objective (tangential distortion). 
 The lengths of the radial and tangential portions of the line are essenti- 
 ally dependent upon the difference of the two positions of the screen, 
 and increase continuously from centre to margin in the case of the 
 " aplanatic " lens. With objectives yielding astigmatic images, there 
 are thus two image surfaces conjugate to one and the same object plane. 
 These two image surfaces touch each other in the axial image point, 
 and the distance between them increases continuously from centre to 
 margin. " Mean curvature " may be defined as that surface which 
 represents the arithmetical mean of the deviations of the two image 
 surfaces from the ideal surface. The dissimilarity between the details 
 in the two image surfaces inter se and the original, increases from the 
 centre to the margin. The following is an interesting experiment : 
 
110 FIELD INSTRUMENTS. 
 
 Arrange in one plane (Fig. 161), along the radii of 
 
 concentric circles, bright discs. The angles between 
 * * * s * the radii should be chosen according to the astigmatic 
 
 9 9 9 aberrations and the focus of the objective. Direct 
 
 * the axis of the objective at right angles towards the 
 
 * * centre of the system of radially grouped discs, and 
 
 focus one of the extra-axial discs. The image obtained 
 at the shorter distance of the focussing screen from 
 
 the objective is, as Fig. 162 shows, a portion of a 
 
 radial line which, in proportion to the curvature of 
 
 Fig*. 161 th e image plane, becomes more and more indistinct 
 
 towards either side, and is more or less interrupted 
 
 radially in proportion to the degree of astigmatic 
 
 S S deviation. Fig. 1 6 2 is an image obtained by focussing 
 
 \ t a disc on a circle situated midway between the axis 
 
 ..._ mimm ^ and the outside circle. It will be seen that images 
 
 of all the discs grouped along the same circle are of 
 
 S I \ the same degree of distinctness or indistinctness, 
 
 * I ^ and also exhibit the same amount of distortion. 
 
 I 
 
 Focussing at the greater distance of the screen 
 
 shows the object (Fig. 163) tangentially distorted. 
 
 Pig*. 162 The image thus becomes composed of a series of 
 
 circles concentric to the centre of the image, which 
 
 are more or less interrupted, or perfectly uninter- 
 
 f * "\ rupted. The discs grouped along another circle, 
 
 *f """ ^^ which has not been sharply focussed, are similarly 
 
 J|| ill distorted, though in a less marked degree. 
 
 . \ y If, now, a screen having drawn upon it circles con- 
 
 X / centric to the axis of the objective and radii, be sub- 
 "*"** stituted for the system of discs, the astigmatic ob- 
 
 jective would reproduce the original with partial 
 similarity, but both systems of lines could not simul- 
 Fig*. 163 taneously be delineated with the same position of the 
 screen. The shorter distance would yield sharpness 
 of the radial elements, the circles at the same time being badly defined ; 
 the longer distance would show the circles sharply, and at the same 
 time the radial elements badly defined. 
 
 A sufficiently large screen bearing the two systems of lines, radial 
 and concentric circular lines, appears thus to form the most natural 
 test for astigmatism. Such a screen would, however, be too uniform 
 and too little adapted for exhaustive tests. 
 
 The screen 2x2 metres area, as it is used in the photographic lab- 
 oratory of Carl Zeiss, of Jena, is for this reason arranged somewhat 
 
ANASTIGMATS. 
 
 Ill 
 
 differently, and it may not be uninteresting to give here a short 
 description of it. Upon strips of paper of 18 x 21 cm. in area are two 
 systems of parallel lines of varying thickness crossing each other at 
 right angles and placed at varying distances ; the strips themselves 
 are fixed to the screen with one of their ends at the centre of the 
 screen, and their sides parallel to the sides of the screen in such a 
 manner that in each strip the radial and tangential lines alternate. 
 The middle of the screen contains a field consisting of rectangular 
 cross lines, which is intended for testing sharpness of definition. The 
 tangential parallels form substitutes for the system of concentric 
 circles, while the radial parallel lines take the place of the radii pro- 
 ceeding from the centre. The difference between the two positions of 
 the focussing screen for sharp delineation of the marginal portions of 
 the systems of straight lines represents the astigmatic difference. For 
 the purpose of demonstrating the incorrectness of the image caused by 
 astigmatism, the screen has square fields of more or less fine rectangu- 
 lar cross lines, diagonally attached to it in such a manner that, in one 
 the system of lines is parallel and at right angles respectively to the 
 diagonal, while in the other case they are inclined at 45 to the diag- 
 onal. The screen has also samples of writing and printing attached 
 to it. 
 
 When focussing square cross lines at the edge of the image, the 
 astigmatic objective produced, in the two characteristic cases furnished 
 by the test screen, the following deformations : 
 
 1. The straight lines composing the net at the edge of the image are 
 parallel and at right angles respectively to the direction of the radial 
 lines. 
 
 In this case represented by Fig. 164, in which A is the point of 
 
 intersection of the axis of the 
 objective and the plane of the 
 object, A' that of the axis and the 
 plane of the image sharp focus- 
 sing of the tangential lines causes 
 the lines which are at right angles 
 to the radius to appear nearly 
 sharp, while the lines which are 
 parallel to the radius are almost 
 entirely invisible (image a). Focus- 
 sing of the radial lines produces 
 the converse of the last test. The 
 lines parallel to the radius appear 
 sharp, the lines at right angles to 
 it disappear (image b). Mean 
 focussing results in a totally ill- 
 
 4' 
 
 Fig-. 164 
 
112 
 
 flELD INSTRUMENTS. 
 
 defined image, and eventually in more or less marked reversion of 
 the cross lines, such as a white net in a black field. 
 
 2. The straight lines of the net 
 
 "Hi 
 
 are inclined at 45 to the radial 
 direction. In Fig. 165, let A 
 and A' again be the points of 
 intersection of the objective axis 
 with the planes of the object and 
 image respectively. Tangential 
 focussing causes the rectangular 
 cross lines to be distorted so as to 
 present the appearance of taii- 
 gentially elongated hexagons, 
 and, in the case of great astig- 
 matic difference, it may result in 
 almost precise commutation of 
 the cross lines into a single system 
 of tangential lines (image a, Fig. 
 165.) 
 
 Radial focussing produces 
 
 Fig*. 165 radially elongated hexagons and 
 
 may, with great astigmatic 
 
 difference, result in changing the cross lines into a single system of 
 radial lines (image b, Fig. 165.) 
 
 If we focus between these two limits, the net may, similarly as above, 
 eventually be reversed so as to appear as black points in a white field ; 
 the same effect may also be produced in anastigmatic images by un- 
 sharp focussing. Similar results of a more or less marked character 
 may be obtained by replacing the quadratic net by one formed of oblongs, 
 rhombi, circles, &c. 
 
 In order that these relations might be illustrated, photographs of the 
 test screen were taken in the photographic laboratory of the optical 
 works of Carl Zeiss, and the photographs so obtained were reproduced 
 by photo-lithography. There are four plates, of which the two most 
 characteristic ones are Nos. I and IV. Here an " aplanat " and " an- 
 astigmat " are subjected to direct comparison. Plate I has been taken 
 with an " aplanat " made by a renowned firm. The objective had a 
 focal distance of 14 cm. and a relative aperture of F/6, and was stopped 
 down to F/12'5. Image and object are in the ratio of 75 to 1000, and 
 the angle subtended by the object is about 67. 
 
 The centre of the screen is sharply focussed. In this part the 
 delineation is extremely good, a sufficient proof that the objective, 
 per se, was a good specimen of its kind. As the margin is approached, 
 
A NASTIGMA TS. 113 
 
 the definition, however, loses more and more in distinctness, and 
 astigmatic distortion increases more and more. While the tangential 
 lines are fairly sharp up to the edge, the radial lines rapidly decrease 
 in definition past the third part of the field. In the diagonal squares, 
 the bounding lines of which are at right angles and parallel respec- 
 tively to the radius, it is noticed that the tangential lines are mark- 
 edly sharper than the radial lines, the latter being almost invisible, 
 and in the squares, whose sides are inclined at 45 to the radial direc- 
 tion, the distortion at the margin of the tangential lines may readily 
 be observed. The squares appear, in fact, as hexagons. 
 
 Plate IV was taken with a, Zeiss anastigmat, viz., anastigmat F/6*3 
 of 14 cm. focus, all other conditions being the same as those existing 
 in the former case. There, too, the centre was accurately focussed, 
 but barely any traces of those details which point to astigmatic 
 imperfections of the margin of the image will be noticed. 
 
 Plates II and III were taken with the same "aplanat" as that 
 used for Plate I. In the first case the tangential marginal lines 
 were focussed ; in the second case the marginal radial lines formed 
 the critical part of the object. While in the former case the centre 
 appeared to be fairly sharp, in the latter case it was totally worth- 
 less. The characteristics of astigmatism, as above explained, become 
 apparent in both plates. 
 
 The older types of lenses (aplanatic, antiplanats, portrait lenses, 
 single lenses, &c.) admitted of astigmatic correction, but they could 
 not at the same time be corrected for flatness of field. The Zeiss 
 " anastigmat " was the first lens in which, as we pointed out, anastig- 
 rnatic aplanatism was combined with the realization of other requisites 
 of a good photographic lens. 
 
 A lens having anastigmatic curvature yields sharply defined points 
 from centre to margin. These cannot, however, simultaneously be 
 fixed upon the plane negative plate of the photographic apparatus 
 when the points constituting the object are nearly in one plane at 
 right angles to the axis, or when they are at a relatively great dis- 
 tance from the objective. If it were desired to simultaneously fix 
 these sharp points upon the plate, it would be necessary to use a 
 curved sensitive surface corresponding to the curvature of the image. 
 Clearly the use of such curved sensitive strata is impossible, for it 
 must not be forgotten that for each lens type, each focal length, and 
 even each degree of magnification -or reduction, there is a distinct 
 corresponding curvature, to say nothing of the practical inconveni- 
 ence attached to curved photographic plates. At present we are, at 
 any rate, limited to flat negative plates. From an optical point of 
 view this is an undesirable limitation, which seriously affects definition 
 and depth of the curved image, 
 
 8 R 
 
114 
 
 FIELD INSTRUMENTS. 
 
 The flat plate cannot be covered uniformly and sharply from centre 
 to edge, unless the objects are grouped on a curved surface correspond- 
 ing to the curvature of the image. With portrait groups, photo- 
 graphers have a means of compensating the anomaly by arranging 
 the persons in a semicircle, in the centre of which the objective is 
 placed. With landscape and instantaneous photographs, however, 
 such an expedient is only rarely, if at all, applicable. In order to 
 obtain tolerable distinctness in the image from centre to edge it 
 would be necessary to work with narrow angles or to stop the lenses 
 down considerably. 
 
 In working in this manner it must be* borne in mind that both in 
 the centre and at the edge, near and far objects are to be depicted 
 simultaneously ; the objective, yielding a curved image, causes, how- 
 ever, distinct objects on the photographic plate to appear indistinct, 
 
 and sufficiently near objects 
 sharp at the edge when the 
 focussing is sharp for the 
 centre. On tin oblong flat 
 negative plate ABCD (Fig. 
 166) let near objects be 
 depicted at A, distant 
 objects at CD; then, if the 
 centre, M t be accurately 
 focussed, the points of sharp 
 delineation are situated 
 upon a curve, and are 
 represented by E, which intersects AB in two points, and is 
 symmetrical with respect to AB. By stopping the lens down 
 we obtain, as is well known, greater depth for distant objects 
 when focussing for near objects. In the present case the depth 
 necessarily diminishes from the centre to the margin. The limits 
 may be graphically represented by curves T l and T 2 , which, being 
 symmetrical with respect to E, have their greatest distance apart 
 at M. The depth of focus is represented by the area contained 
 between T l and T%. By this area the imperfections of the marginal 
 image may readily be ascertained. When it is important to improve 
 the distinctness at the edge it is necessary to sharply focus a point 
 situated at a distance from the centre and to sacrifice the distinctness 
 at the centre. 
 
 The deficient depth of focus of a lens yielding a curved image does 
 not, under certain conditions, become apparent in street scenes. In 
 such cases it may happen that the position of the camera is such that 
 the rows of houses are delineated simultaneously on both sides of the 
 street, the distant houses being thus shown in the centre, the near 
 ones at the edges of the plate. In such a case the curvature of the 
 
 Fig" 166 
 
ANASTIGMATS. 
 
 115 
 
 image may even become the very cause of greater marginal distinct- 
 ness than is obtainable with the flat field. But distant objects have 
 nearly always to be shown simultaneously at centre and edge, and in 
 such cases it is absolutely necessary to have a flat field. 
 
 It is possible to partly flatten the field of the aplanat. This is most 
 conveniently done with those points of the image which are due to 
 the meridional rays, i. e., for the tangential directions in the image. 
 Under these circumstances, one would, however, have to abandon the 
 anastigmatic correction of the image, and to rest content with partial 
 distinctness. Those points of the image, which are due to rays contained 
 in a sagittal section, yield another image surface (image points of radial 
 directions), which touches the former surface in the axis of the objec- 
 tive, and deviates from it with continuous curvature towards the edge, 
 as already explained. 
 
 With angles of 50, this deviation 
 amounts to one-fifteenth ; with 70, to 
 one-sixth; and with 90, nearly to one- 
 third of the focal length of the lens. The 
 section of these image surfaces by a plane 
 passing through the axis of the lens 
 would present the appearance shown in 
 Fig. 167. Curve a appertains to the 
 image points in the meridional section 
 (tangential distortion), b to those of the 
 sagittal section (radial distortion). H 
 
 . 
 
 represents the axis of the lens, H n a secondary axis. 
 
 This result may be obtained with aplanatic lenses if their halves be 
 sufficiently widely separated. The marginal distinctness is then very 
 defective, and the stopping down has to be carried very far if it is at 
 all desired to obtain sharp definition at the edge or fair definition ex- 
 tending over a considerable field, 
 n c 
 
 An objective having an anastig- 
 matically flat field, such as the 
 " anastigmat," produces, however, 
 a sharp image upon the flat plate, 
 which, as Fig. 168 shows, is 
 bounded by lines T l T 2 parallel 
 to the focussing line E. This 
 objective delineates near and 
 distant objects with the desired 
 uniform sharpness at centre and 
 
 Fig-. 168 
 
 The Zeiss anastigmats yield, 
 therefore, a uniform depth of focus 
 from centre to margin without 
 
116 FIELD INSTRUMENTS. 
 
 necessitating the same amount of stopping down that is imperative 
 with the " aplanats." The anastigmats have in proportion to their 
 covering power a considerable relative rapidity. 
 
 Owing to the better concentration of light in the anastigmatic flat 
 image, ns compared with the anastigmatic curved or astigmatic flat 
 images with an objective of the former type, the intensity necessarily 
 diminishes I s > from centre to edge than with a lens belonging to 
 either of the last named types. This advantage of the Zeiss anastig- 
 mats cannot be overrated, as the oblique incidence of rays at the 
 edge of the image is, in itself, productive of a continuous diminution 
 of intensity towards the edge. The anastigmats yield a negative 
 which is uniformly exposed from centre to margin. The advantages 
 resulting from the anastigmatic flatness of field greatly extend the 
 range of the applicability of these lenses. 
 
 The advantage of being able to use a large stop when a certain size 
 of plate is prescribed, and the advantage of the uniformly bright field 
 assist in the solution of the problem of using short focus lenses for 
 relatively large plates. With a given rapidity of the objective, essen- 
 tially shorter foci may be used in the case of anastigmatic lenses than is 
 admissible with other types. For instance, anastigmat F/6'3 (Series 
 II) of a focus of 105 to 120 mm. is quite sufficient for sharply cover- 
 ing a plate 9x12 cm. (3J x 5 inches) at F/9 ; with the older types the 
 focus would have to be 190 mm. (7J inches) at least. In order to 
 cover 13x18 cm. (5x7 inches) at F/9, it was necessary to employ a 
 lens of, say, 350 min. (14 inches) focus, whereas, with the anastig- 
 mats this result can be obtained with a focus of 210 mm. (8J inches), 
 and even with 170 mm. (7 inches). Short foci give, however, at 
 equal distances of the object, a better depth than long foci ; they yield 
 a sharper image of objects situated at different distances from the 
 apparatus. The anastigmatic lenses have, therefore, in another sense, 
 greater depth of focus than the older lenses. 
 
 These advantages become particularly apparent in instantaneous and 
 wide-angle lenses, and in the photography of architecture and interiors 
 and in copying. Detective cameras may be made of smaller dimensions, 
 as they may be fitted with short-focus lenses. Photographs of archi- 
 tecture and interiors, and reproductions of maps and paintings, may 
 be taken by means of rapid lenses, i.e., at short exposures. 
 
 In conclusion, the other advantages which the Zeiss anastigmats com- 
 bine with the anastigmatic flatness may be shortly enumerated. They 
 are as follows : 
 
 1. The reflection images have a most favourable position. 
 
 2. They admit of the most colourless glasses being used ; and 
 
 3. The two parts of the doublet are in close proximity. 
 
ANASTIGMATS. 
 
 117 
 
 The images formed by reflection at the boundary surfaces between 
 glass and air are all at a considerable distance from the plane of the 
 image. By this means the appearance of fogged images, which gener- 
 ally increases with the number of isolated lenses, is reduced to a mini- 
 mum, and thereby the. image rendered exceedingly brilliant. 
 
 The existence of this property is amply proved by photographs taken 
 with the an astigmatic lenses. 
 
 None of the anastigmatic lenses can be shown to have a flare-spot, 
 even when dazzling light enters the objective. 
 
 The use of colourless glasses is an advantage which cannot be over- 
 rated. Apart from sensitive plates, this is the only means of satisfy- 
 ing the universal postulate, depth of definition with short exposures. 
 
 With objectives of the same type, a certain desired amount of depth 
 can, with a given focal length, only be obtained by corresponding 
 stopping down of the lens. The further, however, this stopping down 
 is carried, the less becomes the light which can pass through the lens. 
 If, in addition to this, the scanty light thus admitted is further 
 impaired by detrimental colouring in the glasses, as war, the case with 
 the glasses formerly used in the construction of aplanats, it becomes 
 naturally impossible to work at short exposures. 
 
 The anastigmats, when applied to outdoor photography at F/18, give 
 fully exposed negatives, the usual commercial instantaneous dry plates 
 being used. Before the application of the Schott baryta glasses to 
 the construction of photographic lenses, this belonged to the province 
 of impossibilities. Even with stops F/25 and F/36 instantaneous 
 photographs are still obtainable. 
 
 The short structure of the anastigmats restrains the diminution of 
 the rapidity with which the intensity decreases from centre to edge. 
 It diminishes that part of the decrease of the intensity which is caused 
 by partial stopping of the pencils by the edges of the lenses. 
 
 Let L and L 2 (Fig. 169) be the lenses forming a doublet of a 
 
 . 169 
 
 diameter 2Z>, let B be the plane of the diaphragm, and let the dia- 
 
118 FIELD INSTRUMENTS. 
 
 phragm be situated midway between L r and L 2 . Let a be the dis" 
 tance of the diaphragm from the apex of the anterior surface, and let 
 2d be the diameter of the aperture of the diaphragm. 
 
 If, for the sake of simplification, the collective effect of the parts of 
 the doublet be neglected, i. e., if it be assumed that the diameter of 
 the pencil passing through the diaphragm 2d is also 2d previous to the 
 passage through the lens, and also if we disregard the curvature of the 
 external surfaces and the thickness of the lenses, then the oblique 
 pencil passing through 2d is stopped at that particular moment when 
 the principal ray, II, is of that degree of obliquity which is represented 
 by a straight line contained in a plane passing through the axis of the 
 objective, and connecting the edge of the lens with the edge of the dia- 
 phragm on the same side of the axis. Let the angle between this 
 principal ray, ZT, and the axis be a, then it will easily be seen that 
 
 D d 
 
 tan. a = 
 
 a 
 
 This limit is increased in a measure as the difference D d increases, 
 i. e., in a measure as the aperture of the diaphragm d becomes less. 
 
 When D d is constant, then a increases in a measure as a de- 
 creases. 
 
 From this we infer : the shorter the distance of the diaphragm 
 from the extreme apices of the lenses, the later is the moment of the 
 stoppage of oblique pencils by the edge of the lenses. 
 
 107. SURVEYING CAMERAS. The number of instruments devised or 
 proposed for photographic surveying is considerable. They are divided 
 into three classes : 
 
 1st. Ordinary cameras. 
 
 2nd. Surveying cameras or " photogrammeters." 
 
 3rd. Photo-theodolites. 
 
 The first and second ones require an auxiliary instrument for meas- 
 uring angles ; photo-theodolites are intended to be employed alone. 
 
 Ordinary cameras must be provided with a level ; the relative posi- 
 tions of the plate and lens must be invariable, and when adjusted, the 
 plate must be exactly vertical. 
 
 The horizon line is determined by two zenith distances of well-defined 
 objects as far apart as possible. The principal point is ascertained 
 from the azimuths of at least three points. It is expedient to make 
 these determinations for every photograph. 
 
 The employment of ordinary cameras for surveying cannot be re- 
 commended ; satisfactory results cannot be expected from imperfect 
 instruments. 
 
SURVEYING CAMERAS. 
 
 119 
 
 There are many patterns of surveying cameras or photogrammeters. 
 One of the earliest is Meydenbauer's. It is a camera with tapering 
 bellows set on a horizontal circle ; it moves on a vertical axis. A clamp 
 and tangent screw serve to bring the optical axis in any desired azimuth 
 Two metal rods connect rigidly the object glass and the frame bearing* 
 the plate. The levelling screws are part of the head of the tripod, the 
 horizontal circle resting on the top of the screws. This arrangement 
 dispenses with the usual triangular base of instruments and reduces 
 the height considerably. 
 
 Finsterwalder's photogrammeter is, in external appearance, some- 
 what like Meydenbauer's, but the horizontal circle is set on the ordinary 
 triangular base with the three levelling screws. 
 
 Meydenbauer has devised also a small camera which is made in two 
 sizes, 6x8 and 9x12 centimetres. It is a magazine camera ; the plates 
 are changed through a bag placed underneath. The camera is supported 
 in a peculiar manner ; it is placed on the top of a vertical rod, which 
 is jointed to the head of a tripod. The free ends of the tripod and the 
 upper end of the vertical rod are connected by wires to which tension 
 is given by ratchet wheels and pawls. 
 
 R. Lechner, of 
 Vienna, Austria, 
 makes the three 
 following patterns. 
 
 W e r n e r's (Fig. 
 170), is an ordinary 
 folding camera set 
 on a triangular base 
 with 1 e v e 1 1 i n'g 
 screws. 
 
 Fig. 170. 
 
120 
 
 FIELD INSTRUMENTS. 
 
 Pollack's photogrammeter (Fig. 171) is a rectangular metal camera 
 mounted on a graduated circle with levelling screws. The rising 
 
 front is moved by a rack 
 and pinion, the displace- 
 ment being read on a scale 
 by a microscope and ver- 
 nier. The object glass is 
 an anastigmat y/18. In 
 front of the ground glass 
 is a metal frame gradu- 
 ated in centimetres. 
 There is a special con- 
 trivance for pressing the 
 plate in the dark slide 
 against this frame, so 
 that the centimetre scale 
 is impressed on the photo- 
 graph. This serves, firstly, 
 to determine the vertical 
 and horizontal lines, 
 secondly, to eliminate 
 any error in the registra- 
 tion of the plate holder, 
 and, thirdly, to mark any 
 distortion of the paper. 
 
 In the middle of the 
 focussing screen is an eye 
 piece with cross threads 
 exactly opposite the 
 object glass, with which 
 it forms a telescope. It 
 is so adjusted that the 
 intersection of the threads 
 coincides with the princi- 
 pal point of the perspec- 
 tive. 
 
 Adjusting screws are 
 provided for tilting the 
 camera until the plate is 
 vertical, and for adjust- 
 ing the frame in front of 
 the photographic plate so 
 as to bring the horizon 
 and principal lines in 
 Fig. 171. correct position. 
 
SURVEYING CAMERAS. 
 
 121 
 
 Huebl's " Plane table photogrammeter," by the same maker (Fig. 
 172), is a compact and very well conceived instrument. It has a 
 camera base or carrier, on which are the levelling screws. It is put on 
 the tripod and the camera is connected with it by a central screw with 
 spiral spring. The tops of the le veiling-screws tit into slits at the bottom 
 of the camera. 
 
 K 
 
 Fig. 172. 
 
 The back frame is, like Pollack's, graduated in centimetres, and the 
 plate is brought in contact with it by a forward motion of the slide 
 carrying the plate holder. The peculiarity of the instrument is the 
 top, which is disposed for use as a plane table. A sheet of drawing 
 paper is fixed to it, and the directions of important points are drawn 
 by means of an alidade with telescope and vertical circle. When the 
 alidade is employed, no additional instrument is needed, the azimuths 
 being registered on the plane tableland the zenith distances read off 
 the vertical circle and noted. 
 
122 
 
 FIELD INSTRUMENTS. 
 
 Passing to photo-theodolites, we find among them the first instru- 
 ment specially constructed for surveying purposes, Col. Laussedat's 
 photo-theodolite, made by Brunner in 1858-59. 
 
 Fig. 173 shows this instrument as now constructed by E. Ducretet 
 and L. Lejeune. The base of the camera is the lower part of a 
 theodolite, with levelling screws, graduated hoi izontal circle and vernier. 
 The camera proper is a wooden rectangular box ; the sliding front 
 board carrying the object glass is moved up or down by a rack and 
 pinion. A telescope with striding level and vertical arc is fixed to one 
 side ; on the other is a sight for rapidly bringing the optical axis in 
 any given direction. The instrument is levelled by means of the strid- 
 ing level in the same way as any other theodolite ; vertical and hori- 
 zontal angles are measured also in the same manner. When the line of 
 
 Fig. 173. 
 
 collirnation of the telescope is horizontal, it must be directed on the 
 principal point of the perspective. The registering marks of the horizon 
 and principal lines are checked by bringing, with the striding level, 
 the telescope to this position. Tt may occasionally be desired to use 
 the instrument for measuring angles only without taking photographs, 
 as, for instance, in making a subsidiary triangulation. It would be in- 
 convenient to carry the camera when it is not needed. In this case, it 
 is detached and the telescope set up as in Fig. 174 ; reduced to this 
 form, it is an ordinary surveying transit. 
 
S UR VE YING CA MERA S. 
 
 123 
 
 Inversely, the telescope and vertical arc may be detached ; it then 
 becomes a surveying camera. 
 
 Fig. 174. 
 
 The metal parts of the instrument are made of aluminium ; it is com- 
 pact, light and well executed. It answers well the purpose intended, 
 which is to make detailed topographical surveys without having recourse 
 
 Fig. 175. 
 
 to auxiliary instruments, and it has the great merit of simplicity in its 
 construction. 
 
124 
 
 FIELD INSTRUMENTS. 
 
 R. Lechner makes Pollack's photo- theodolite (Figs. 175, 176 and 
 177). The camera is the same as in the photogrammeter, and there is 
 
 Fig. 176. 
 
 adied a vertical circle, telescope and striding level. Without the 
 camera, the instrument is an ordinary theodolite. 
 
 Fig. 177. 
 
SURVEYING CAMERAS. 
 
 125 
 
 The photo-theodolite of the Geographical Military Institute of Italy 
 is shown in Figs. 178 and 179. The camera is a pyramidal metal box 
 moving on a horizontal axis, so that it may be inclined to the horizon 
 when the angle of view in the vertical position is not sufficient to 
 take in all the points of the landscape. 
 
 This arrangement exists also in Koppe's theodolite, and may be useful 
 in a very broken country. Generally a vertical field of 45 is quit 
 
126 FIELD IN STB UMENTS. 
 
 sufficient, and most of the instruments are made for vertical views only, 
 
 Fig-. 179 
 
 The levelling screws are part of the tripod head, as in Meydenbauer's 
 photogrammeter, and they support the horizontal circle. The vertical 
 circle, telescope and level are on an upright at the side. The 
 pbject glass moves in its tube so that it can be adjusted to the focus 
 
128 
 
 FIELD INSTRUMENTS. 
 
 of near objects. The displacement is read on a scale with a vernier. 
 The camera can be detached and the instrument used for measuring 
 angles only, as an ordinary theodolite. It is packed in three boxes of 
 convenient shape for transport ; one contains the camera proper, 
 another one the two circles, tripod head and telescope, and the last 
 one, the plate holders. 
 
 Fig. 181. 
 
 The tripod is formed of the alpenstocks of the three men carrying 
 the cases ; they are fixed with thumb-screws to the tripod head. This 
 instrument was constructed for use in the Alps, where men must have 
 alpenstocks and where the tripod is the most awkward part of the 
 engineer's equipment. The alpenstocks being very strong make a firm 
 
SURVEYING CAMERAS. 
 
 129 
 
 stand- for the theodolite, and the transportation of an ordinary 
 tripod is dispensed with. 
 
 Koppe's photo-theodolite (Figs. 180, 181 and 182) is an ordinary 
 theodolite with telescope on one side and vertical circle on the other. 
 In the middle of the horizontal axis, and fixed to it, is the camera. The 
 
 Fig. 182 
 
 difference in the Italian instrument is that the telescope moves 
 with the camera, their axes remaining parallel. There are no dark 
 slides ; the plates are changed in the packing case, where they are 
 stored in two boxes K K z , Fig. 180. 
 
 In the lids are two circular holes A A, to which are attached sleeves 
 of flexible material impervious to light. For the purpose of changing 
 a plate, the case is closed after placing the camera in it ; the hands are 
 inserted through the sleeves, the exposed plate is taken out and pushed 
 into box K l ; a new plate is then taken from box K^ put in position, 
 and the camera closed. 
 
 It is evident from the descriptions given that some instruments 
 are intended only for surveys plotted by geometrical constructions, 
 while others are designed to furnish results of the utmost precision, 
 the measurements on the photographs being made with special 
 appliances and submitted to calculation. The photo-theodolite of 
 9 R 
 
130 
 
 FIELD INSTRUMENTS. 
 
 Starke & Kammerer, of Vienna, is an instrument representative of the 
 latter class, and will be described with a little more detail ; the remarks 
 apply to other instruments of the same kind.* The general principle 
 of these instruments is to have a telescope moving in a plane parallel 
 to the principal plane of the perspective ; in this one, the object glass 
 of the camera takes the place of the object glass of the telescope, and 
 for that purpose is combined with an eye piece in the middle of the 
 ground glass. With this telescope, not only can the horizontal angles 
 required by the survey be measured, but also vertical angles The 
 object, however, is not to make extensive series of angular measure- 
 ments, but only to take readings on the points of the triangulation. 
 The three levelling screws of the base rest in slits on the head of the 
 tripod to which the base is fastened by a central screw and spiral 
 spring. In Fig. 183, the levelling screws appear in A, A, A. The base 
 
 Fig. 183 
 
 carries the vertical axis and horizontal circle ; the latter is graduated 
 to 20' on its vertical face, and is read to 1' by means of a vernier N 
 and microscope L. The end of the vertical axis carries three arms, 
 .Z? t , -B 2 ^3' t which the vernier is screwed. The vertical axis is 
 adjusted by means of the three levelling screws and of the two levels 
 Zj, 1 2 . The plate, on which are the levels, is screwed to the arm 2 . 
 A clamp E and tangent screw M complete the base. 
 
 *This description is from a paper by G. Starke in the "Zeitschrift de Oester- 
 reischischen Ingenieur und Architekten Vereines, " No. 5 of 1894. 
 
SURVEYING CAMERAS. 
 
 131 
 
 In the slits r 1 r 2 r 3 of the arms B l B 2 . 3 rest the three levelling 
 screws F^ F. 2 F. } of the camera (Fig. 184) ; the camera is screwed from 
 
 Fig. 184. 
 
 the inside to the lower part. A handle, Q, is on top for convenience in 
 lifting it. The camera has, as well as the lower part, two cross levels 
 1 3 / 4 (Fig. 185), for adjustment with the foot screws F 1 F 2 F 3 . Fig. 
 184 shows the camera from the lens side, and Fig. 185 from above. 
 Fig. 186 is the frame securely bound to the camera which fixes the 
 position of the focal plane ; the ground glass and the dry plates are 
 brought in contact with it. It consists of four scales at right angles 
 with notches at every centimetre. Those in a ^a 2 and 6j6 2 are the 
 middle marks; when joined, the intersection is the middle of the 
 picture. The size of the picture is 17-8x22-8 centimetres. By 
 construction, the notches are so placed that a^a 2 is exactly per- 
 pendicular to 6,5 2 ; thus when a ^a 2 is horizontal, b^ 2 is in the 
 vertical plane perpendicular to a.a 2 . 
 
132 
 
 FIELD INSTRUMENTS. 
 
 The lens is a Zeiss anastigmat, f/l 8 of about 212 millimetres focus. 
 It is screwed on the plate S, Figs. 184 and 185, which has a sliding 
 vertical motion, and it is at such a distance that parallel rays falling 
 on it converge after refraction to points in the plane of the frame 
 RR, in which plane also are the face of the dry plate, the dull side of 
 the ground glass and the threads of the diaphragm in front of the 
 eye piece. The distance of objects photographed for surveying is 
 generally so great that a fixed focus has no disadvantage, and the same 
 
 Fig. 185. 
 
 focal length may be employed throughout. For distances of 500, 400, 
 300, 200 and 100 metres and with a lens of 212 millimetres focus, 
 the displacement of the focal plane should only be 0'09, 0*11, 0'15, 
 0-22 and 045 millimetres respectively. For shorter distances, the lens 
 has a motion in the direction of the axis to the extent of two 
 millimetres. This brings in focus objects situated at 23 metres from 
 the instrument. lii the helicoidal slit cut in the outer tube of the lens 
 is a small block t screwed to the inner tube. Both tubes are clamped 
 together by the screw shown at the end of the slit. Unscrewing it and 
 turning the inner tube from right to left brings the lens out. The 
 nclination of the slit to the axis is such, as to correspond to a motion 
 
SURVEYING CAMERAS. 
 
 133 
 
 of two millimetres in the direction of the axis when the block t travels 
 from one end of the slit to the other. These two extreme positions 
 are indicated by the graduations and 2 on the outer tube opposite 
 the index mark on t. The interval is divided into twenty parts, so 
 that the displacement of the lens is read directly to O'l millimetre. 
 This graduation is, however, seldom used ; in the generality of cases 
 the index is set at zero and the lens clamped in that position. 
 
 Fig. 186. 
 
 A vertical motion is given to the lens by the rack and pinions K l K% 
 Figs. 184 and 185 ; the latter, 7T 2 , is for slow motion, H is the clamp. 
 The displacement is measured on a scale m divided to millimetres with 
 the vernier n reading to O05 millimetre. This scale is 140 millimetres 
 long ; from to 70 the optical axis is directed below 
 the horizon, and from 70 to 140 is directed above. 
 At 70 it should be exactly on the horizon. 
 
 Two frames T and II (Figs. 187 and 188), serve to 
 make a light, tight connection between the single 
 plate holders or ground glass and the camera. They 
 are connected by the bellows W ; I is fixed to 
 the camera, but II can move as far as the bellows 
 extend. Each frame carries two hooks ; the frame 
 I has the upper right one, h lt and an exactly similar 
 
 Fig. 187. 
 
134 
 
 FIELD INSTRUMENTS. 
 
 one, lower left. The frame II has the upper left hook h 2 and a lower 
 right one. 'These hooks fasten the ground glass and plate-holders to 
 the camera. The ground glass is shown in Fig. 189 ; in Fig. 187, it is 
 represented in section attached to the camera. 
 The outer wooden frame carries two metal 
 diagonals connected to a central ring. In the 
 middle of the ring is the eye piece movable on 
 the axis x l x 2) but which may be fastened by 
 the two flaps p^p^ in its normal position with 
 the optical axis perpendicular to the ground 
 glass. Opposite the notches a^a* 6 t 6 2 of the 
 back f came are four circular openings in the 
 ground glass, through which the notches may 
 be examined for adjusting the instrument. 
 The ground glass is attached as follows : 
 
 . The movable frame II, which does not come 
 into operation in this case, is fastened to I by 
 the upper left hook k. 2 and the lower right one. 
 The ground glass is supported by the screws 
 Z^Z (Figs. 187 and 189), the p6ints of which 
 rest on the plates ~ -' of the fixed frame I 
 (Figs. 185 and 187). The face of the glass is 
 then brought in contact with the back frame 
 through the upper right and lower left hooks. 
 The position of the eye piece is adjusted by the 
 screws Z^Z until the line of collimation is 
 horizontal when the object glass is in its nor- 
 mal position at 70 of the vertical scale. In 
 this position, or not far from it, points can be 
 sighted through the eye piece of the ground 
 glass, but when the lens is further away from 
 the normal position, the use of the eye piece 
 with its axis perpendicular to the ground glass becomes inconvenient or 
 impossible. In such cases the flaps p j) 2 are unfastened, and the eye 
 piece is rotated on the axis x^x. 2 until its optical axis is in the direction 
 of the object glass. No great precision is required ; a few trials are 
 sufficient to find the inclination for which the best image is seen. 
 
 The plate holder A", shown in secflon, Fig. 188, contains the dry 
 plate G which is pressed by springs at the four corners. The hard 
 rubber shutter t is completely withdrawn during exposure. The 
 adjustment of the plate holder to the camera takes place as follows : 
 Frame II is set free from I, and the holder hung on frame II by 
 the bent plate I : the projecting edge of the holder engages in the 
 rebate of frame II and makes a light, tight joint. It is then fastened 
 to frame II by the pair of hooks upper left and lower right. Fig. 188 
 
 Fig. 188. 
 
S 67? VK YING CA HER A S. 
 
 135 
 
 shows this position of the holder. Having made sure that the cap is 
 on the lens, the shutter can now be withdrawn, after which the pair 
 of hooks upper right and lower left are brought into play and draw 
 the holder forward until the dry plate is brought in contact with the 
 back frame R R of the earner.:. The exposure is then given by un- 
 capping the lens and the holder withdrawn by repeating the same 
 operations in inverse order, first unfastening the pair of hooks upper 
 right and lower left, inserting the shutter and drawing back the two 
 last hooks, upper left and lower right. 
 
 Fisr. 189. 
 
 If the stations are points of the triangulation or have previously 
 been determined, the only angles to be measured are those required 
 for the orientation of the views : a vertical arc or circle is unnecessary. 
 But it may happen that the station has not been fixed or could not 
 be seen from the summits of the triangulation ; in such cases, vertica^ 
 angles must be measured to obtain the altitude of the photographic 
 
136 
 
 FIELD INSTRUMENTS. 
 
 station. Let 0, Fig. 190, be the second nodal point of the lens, C the 
 intersection of the diaphragm threads ; the instrument being adjusted, 
 CO is horizontal when the lens is at 70 of the vertical scale. Let M 
 be a known point from which the position of the photographic station 
 must be ascertained ; HM is the altitude to be found. The eye piece 
 is turned in the direction CM, and the object glass raised until the 
 image of M is seen at the intersection of the threads of the diaphragm ; 
 the object glass plate is clamped and its height read on the vertical 
 scale. Let V be the intersection of the vertical axis of the instrument 
 with the horizontal axis of the camera. In the triangles MHC, 
 
 Fig-. 190 
 
 O'OC, the distance D must have been previously ascertained, 
 00' = h has been measured on the vertical scale, OC =p is the focal 
 length, and d is a constant of the instrument which can be scaled on 
 it. These two similar triangles give the proportion 
 
 H 
 
 H being the difference of altitude HM. 
 Therefore 
 
 h 
 
 p p p 
 
 For this instrument d = 100, p = 212 mm. and the maximum value of 
 h is 70 mm. Then - d cannot exceed 0.33 mm., a quantity which can 
 always be neglected, the expression of H taking the simpler form of 
 
SURVEYING CAMERAS. 137 
 
 Let A H be the error of H due to an error of A h in the reading 
 of A: 
 
 7t is read on the scale to 0*05 mm., so 
 
 A#= 0-0001 18 D. 
 This gives 120 mm. for D = 1000 metres. 
 
 The instrument is adjusted by means of the four levels so as to fulfil 
 the following requirements : 
 
 1. The axis of the instrument must be vertical. 
 
 2. The line joining the notches a 1 a 2 of the back frame must be 
 horizontal : consequently, b-J) z is vertical when the next condition is 
 fulfilled. 
 
 3. The plane of the back frame must be vertical. 
 
 The object glass being at 70 of the vertical scale and the ground 
 glass in contact with the back frame, it is necessary : 
 
 4. That the horizontal and vertical threads of the diaphragm coincide 
 with the lines a^a^ 6 1 6 2) of the back frame. 
 
 5. That the line of collimation be horizontal. 
 
 6. That the threads of the diaphragm lie in the plane of the dull side 
 of the ground glass. 
 
 7. That the horizontal thread of the diaphragm coincides with the 
 axis of rotation x^x 2 of the eye piece. 
 
 8. That the vertical motion of the object glass shall be on a line 
 parallel to the line b^ 2 of the back frame. 
 
 Some of these adjustments must have been made by the instrument 
 maker, and if incorrect can only be rectified by him. Those which 
 follow are made by the engineer. 
 
 1. The cross levels of the base are adjusted as in any other 
 theodolite 
 
 2 and 3 Cross levels of the camera. Find a distant point of 
 which the image comes in the apex of the notch a 15 and verify the 
 coincidence with a magnifying glass through the circular hole in the 
 ground glass. Then revolve the instrument on its vertical axis until 
 the image comes into the notch 2 . Should it coincide exactly with 
 the apex, then the line a 1 2 is horizontal ; if not, it must be adjusted 
 by the levelling screw under a. A few trials will soon bring it to the 
 
138 FIELD INSTRUMENTS. 
 
 right position ; the level parallel to a 1 2 can now be adjusted and the 
 bubble brought between its marks. 
 
 The verticality of the plane of the back frame is checked by a light 
 plumb bob suspended from one of the hooks. It is adjusted by means 
 of the foot screw F 3 under the object glass. When exactly vertical, 
 the level Z 4 is adjusted until the bubble comes within the marks. It is 
 clear now that a a 2 will be horizontal and 6 1 6 2 vertical whenever the 
 camera is levelled, so long a? the adjustment of the cross levels does 
 not change. 
 
 4. Selecting a distant point very close to the horizon, its image is 
 brought into the apex of one of the notches a^ or a 2 and brought to exact 
 coincidence by a vertical motion of the object glass. The instrument 
 is then revolved on its vertical axis until the image is seen in the 
 eye piece. Should it coincide with the horizontal thread of the 
 diaphragm, then the latter is in proper adjustment ; if not, it must be 
 brought to coincide by raising or lowering the ground glass by means 
 of the screws Z^Z^ half of the error being corrected by one screw and 
 half by the other. They must both be screwed or unscrewed the same 
 number of turns. 
 
 The vertical thread of the diaphragm must coincide with the line 
 6 1 & 2 . To verify this, select a distant point and bring its image, by a 
 slow motion of the horizontal circle, into exact coincidence with the 
 edge of the back frame above or below the notch a 1 and read the angle 
 on the horizontal circle. Revolve the instrument, and bring in the 
 same way, the image of the point in contact with the edge of the frame 
 above or below the notch & 2 ; read again the angle on the horizontal 
 circle. Set the vernier in the middle of the measured angle and the 
 image must coincide with the vertical thread of the diaphragm. The 
 error, if any, is corrected by again using the screws Z^Z^. Should for 
 instance, the thread be on the left of the image, Z- is unscrewed till half 
 of 'the error is corrected and Z ' 2 screwed in to correct the other half. In- 
 versely, Z^ must be screwed in and Z^ unscrewed when the thread is on 
 the right of the image. It is clear that this correction will somewhat 
 disturb the preceding one and that both will have to be repeated 
 several times. 
 
 5. The line of collimation is adjusted to honzontality by the usual 
 methods of the level instrument. The object glass may then not be 
 exactly at 70 of the vertical scale. The index error may be employed 
 to correct all the readings, or it may be eliminated by loosening the 
 two screws of the vernier and moving it vertically until the reading is 
 exactly 70. 
 
 The vertical motion of the object glass is verified by setting it at 
 about 130, selecting a distant point and bringing its image into exact 
 coincidence with the apex of the notch b ly then lowering the object 
 
Fig. 191-TRANSIT THEODOLITE. 
 
I 
 
 Fig. 192- CAMERA 
 of the Canadian Surveys (horizontal position). 
 
IP 
 
 
 Fig. 193- GAMER A 
 of the Canadian Surveys (vertical position). 
 
SURVEYING CAMERAS. 139 
 
 glass till the image comes into the notch 6 2 , where it must coincide 
 with the apex. The image must also be seen on the vertical thread of 
 the diaphragm when the object glass is at the proper height. These 
 conditions being fulfilled, the displacement of the object glass is on a 
 line parallel to 6 1 6 2 and therefore vertical, and the vertical thread 
 coincides with the line b^b^. 
 
 108. CANADIAN EQUIPMENT. The equipment of a party on the 
 Canadian Surveys consists of a transit theodolite and two cameras. 
 The transit theodolite and its tripod are carried by the surveyor, and 
 a camera without the tripod by one of the men who always accom- 
 panies the surveyor. The assistant has his own camera with a tripod. 
 
 The transit is one of the ordinary patterns used by surveyors and is 
 shown in Fig. 191. It has three inch circles and reads to minutes. 
 The tripod is a short one, specially designed for mountain work. It is 
 three feet four inches long arid has sliding legs, the joint being 
 perfectly stiff. The surveyor observes either in a sitting or kneeling 
 position. For the purpose of packing, the head of the tripod is taken 
 off and put in the transit box ; when folded, the legs are twenty inches 
 long and are placed under the box of the transit as shown in frontis- 
 piece. The heavy parts of the instrument are made of aluminium ; 
 the whole, including tripod and case, and also the camera base, weighs 
 fourteen pounds and eight ounces. 
 
 The camera is shown in Figs. 192 and 193; Fig. 194 represents 
 sections of the instrument. The camera proper is a rectangular metal 
 box AJ3, open at one end. It carries the lens L and two sets of cross 
 levels (7(7, which are read through openings in the outer mahogany 
 box. The metal box is supported by wooden blocks and by a frame 
 FF held in position by two bolts DD. The plate holder is made 
 for single plates ; it is inserted in a carrier EE, which can be moved 
 forwards and backwards by means of the screw G. A folding shade 
 HH hooked in front of the camera, and diaphragms KK inside of the 
 metal box, intercept all light which does not contribute to the forma- 
 tion of the image on the photographic plate. The camera rests on a 
 triangular base with foot screws, identical with the base of the transit, 
 so that both may fit on the same tripod. It may be set up with the 
 longer side either horizontal or vertical. 
 
 The lens is a Zeiss anastigmat, No. 3 of series V, 141 millimetres 
 focus, with a deep orange screen in front. 
 
 Having set up the camera on the tripod, the plate holder carrier E 
 is moved back as far as it will go by turning the screw G ; the plate 
 holder is inserted through the opening J/, the slide is withdrawn, and 
 the carrier moved forward by the screw G until the plate is in contact 
 
140 
 
 FIELD INSTRUMENTS. 
 
 with the back of the metal box. In order to secure perfect contact, 
 the carrier has a certain amount of free motion. The camera must 
 
 p 
 
 Fig-. 194 
 
 
 now be turned in the proper direction ; the field embraced by the plate 
 is indicated by lines drawn on the outside of the mahogany box. The 
 
SURVEYING CAMERAS. 141 
 
 camera is then carefully levelled, the exposure given, and the plate 
 holder withdrawn by repeating the same operations in inverse order. 
 
 The levels are rigidly fixed to the camera without any means of 
 adjustment. They are, however, very nearly adjusted by the maker. 
 For this purpose, he takes out the metal box and places it on a piece 
 of plate glass which has been levelled like an artificial horizon. By 
 filing one end or the other of the levels' outer case, he brings each 
 bubble very nearly in the middle of its tube. These tubes bear con- 
 tinuous graduations as illustrated in Fig. 195. 
 
 4 12 Accompanying the camera is a piece 
 
 I I 
 
 I I 
 
 | | i of plate glass, ^ in. thick and 11 in. 
 
 long, which can be inserted in the 
 carrier instead of the plate holder. 
 The end of the glass, which projects 
 
 Fig*. 195 outside of the camera, is coated on the 
 
 back with a varnish of gum guaiacum dissolved in alcohol to which 
 some lamp black has been added. This varnish has very nearly the 
 same refractive index as glass, and stops all reflections from the back 
 of the plate glass. 
 
 The first thing to be done when the camera is received from the 
 maker is to ascertain the exact readings of the levels when the back 
 of the metal box, on which the photographic plate is pressed, is 
 vertical. To do this, the bolts P (Fig. 194), next to the opening 
 M, are unscrewed and removed : Q may then slide backwards and be 
 taken out. The piece of plate glass is now inserted in the carrier U, 
 and pressed in contact with the metal box. The camera is placed on 
 its tripod and levelled. Immediately in front and at the same height, 
 a transit T, Fig. 196, is set up, and after carefully adjusting it, a 
 $ . distant point P is selected 
 
 on the same level as the 
 transit and camera. The 
 intersection of the threads 
 of the telescope is brought 
 to coincide with P, and the 
 telescope is clamped to the 
 
 Fig-. 196 vertical circle. Turning it 
 
 around the vertical axis, the image of P reflected by the plate glass 
 should appear upon the intersection of the telescope's threads. If it 
 does, the face of the plate glass is vertical and the position of the 
 bubble in the level tube is the correct one for adjusting the instru- 
 ment. If it does not, the camera must be inclined forward or 
 backward by means of the foot screws until coincidence is established. 
 The bubble of the level may or may not be now in the middle of the 
 
142 
 
 FIELD INSTRUMENTS. 
 
 tube, but its position, whatever it is, is the correct one for adjusting 
 the camera. The divisions of the graduation between which the 
 bubble is comprised are therefore noted, and whenever the camera is 
 to be levelled, it must be remembered that the bubble is to be brought 
 between these two same divisions. 
 
 This determination is made for the two positions of the camera, 
 horizontal and vertical. 
 
 The next step is to fix the place of the principal point on the photo- 
 graphic plate, and to measure the distance line or focal length. 
 
 Select a station so 
 that a number of distinct 
 and well defined distant 
 points may be found on 
 the horizon line. The view 
 may be, for instance, the 
 distant shore of a lake, or 
 a large building, or rows of 
 buildings. Set up the tri- 
 pod and adjust the transit. 
 Find two points E and F 
 on the horizon line, or with 
 a zenith distance of 90, 
 ~2) so that they both come 
 within the field of the 
 camera, when set hori- 
 zoiital, and near the edges 
 of the plate. Measure the angle w between them. Find two 
 other points G and If, also on the horizon line, and such a distance 
 apart that they both come within the field of the camera when set 
 vertical. Now, replace the transit by the camera in the horizontal 
 position, turn it so that it tal^es in E and F, level carefully and ex- 
 pose a plate. Set the camera in the vertical position, turn it so that 
 it takes in G and H y level carefully and expose another plate. The 
 .first plate after development, shows the two points E and F, on a 
 line very nearly parallel to the edges AB and CD, Fig. 197, 
 of the metal box. The principal point is of course on this line. Cut 
 the line through the film with a fine needle point. 
 
 The second plate, exposed in the vertical position, gives another 
 horizon line GH which may be transferred to the first plate by means 
 of the distances AK, CL to the corners of the metal box. This line is 
 also cut through the film with a fine needle : the principal point P is at 
 the intersection of both horizon lines. 
 
SURVEYING CAMERAS. 
 
 143 
 
 The distance line is calculated from the angle w between E and F 
 and from the distances EP and PF. 
 
 Let fl,'Fig. 198 > be tne second 
 nodal point of the lens ; a and b 
 the distances EP and P^ 7 , a and 
 ft the angles between E and P, 
 and between P and ^. 
 
 Fig-. 198 
 
 a and 6 are measured on the plate. 
 Designating by f the focal length 
 PS, we have 
 
 tan. a 
 
 / 
 
 tan. = -r 
 
 tan. a tan. p = -^ 
 
 Hence : 
 
 tan. (a + ) = tan. w = 
 
 1+1 
 
 ab 
 
 or 
 
 tan. 
 
 and 
 
 /=? 
 
 2 tan. 
 
 + ab 
 
 Having now found the focal length and principal point, reference 
 marks have to be made on the edges of the metal box to indicate the 
 horizon and principal lines and the focal length on the prints or 
 enlargements from the negatives. 
 
 Measure the distance m (Fig. 197), from P to AC. From the cor- 
 responding corners A and C (Fig. 199), of the metal box, lay out m 
 in AR and CT. With a very fine and sharp file, held in the direction 
 
144 
 
 FIELD INSTRUMENTS. 
 
 of the lens, cut in the edge of the metal a clean and sharp notch at T 
 
 and another one at R. 
 
 3*, r-~ 
 
 Repeat the same 
 operation at the cor- 
 ners A and B with the 
 distance n from P to 
 AB. 
 
 The lines OQ and 
 .S^Z 7 will be the horizon 
 and principal lines of 
 the photographs when 
 the camera is properly 
 
 levelled. 
 
 J /> 
 
 From R and T, mea- 
 Fig*.l99 sure * ne distances Rr, 
 
 Rr, Tt, Tt', equal to 
 
 one half o? the focal length. From and Q measure Oo, Oo, Qq, Qq, 
 equal to one quarter of the focal length, and at each one of these points 
 make another notch with the file held in the direction of the lens. 
 Every photograph will now show, like those which accompany the 
 specimen plan, twelve triangular projections into the black border of 
 the photograph. Four of these projections serve to fix the horizon and 
 principal lines ; the remaining eight give the focal length. 
 
 It is now necessary to find the correct readings of the transverse 
 levels, when the horizon and principal lines pass exactly through the 
 notches of the metal box. 
 
 Set up the camera 
 again in front of the same 
 distant view as before, 
 but in adjusting it, bring 
 the bubble of the trans- 
 verse level near one end 
 of the tube ; note the read- 
 ing of the graduation and 
 expose a plate. When 
 developed, it will give an 
 horizon line EF (Fig. 
 200), cutting the border 
 of the negative in A and 
 B, at some distance from 
 the notches and Q. 
 Fig-. 200 Now change the adjust- 
 
 B 
 
SURVEYING CAMERAS. 145 
 
 ment of the camera by bringing the bubble of the transverse level to 
 the otherendof tho tube, note the reading of the level and expose another 
 plate. This one gives another horizon line E'F', cutting the border of 
 the negative in C and D. 
 
 After measuring CO and OA or BQ and QD, a simple proportion 
 gives the reading of the level which shall bring the horizon line 
 through the two notches and Q. 
 
 The correct reading of the other transverse level is found by the 
 same method, with the camera in the vertical position. 
 
 All these operations must be executed with great care and precision, 
 and with the help of a microscope of moderate power. 
 
 It has been assumed that the levels were placed very nearly in 
 correct adjustment by the maker. If found too much out, they must, 
 of course, first be approximately adjusted by setting the metal box on 
 a well levelled plane. For this purpose, the plate glass supplied is set 
 on the camera base and levelled like an artificial horizon. 
 
 109. USE OF INSTRUMENTS. The instruments and tripod being very 
 light, steadiness is secured by a net between the tripod legs in which 
 a heavy stone is placed. With this device, better photographs and 
 observations are obtained, and there is no risk of the instrument being 
 blown away during one of the wind blasts so frequent in the 
 mountains. 
 
 After coming to a triangulation station, the surveyor adjusts his 
 transit, and measures the azimuths and zenith distances of the 
 triangulation points and of the camera stations. If accompanied by 
 his assistant, each reads one vernier and enters the reading in his 
 book. After completing the observations, they compare notes : any 
 error is corrected on the spot. 
 
 The camera is carried in a leather case containing also twelve plate 
 holders ; when more holders are wanted, they are carried separately. Tak- 
 ing the camera out of the case, the levelling base is screwed to it and put 
 on the tripod ; the shade is unfolded and attached to the front. A 
 plate holder is inserted in the carrier, and its number noted as well as 
 the approximate direction of the view. Having made sure that the 
 cap is on the lens, the surveyor draws the slide and screws the plate 
 in contact with the metal box. He now turns the camera around 
 until the lines on the upper face show that it is properly directed. He 
 looks along the lines of the side face to see whether the view reaches 
 high or low enough ; if it does not, he puts the longer dimension of 
 10 
 
146 FIELD INSTRUMENTS. 
 
 the camera upright, unless already in that position. He levels care- 
 fully and exposes the plate. When the sun shines inside of the front 
 hood, it should be shaded off by holding something above the hood. 
 On no account must the sun be allowed to shine upon the lens. 
 
 The most important camera stations are occupied by the surveyor ; 
 the other stations by the assistant with his own camera. 
 
 All views are taken with the same stop, F/36. 
 
DENSITY, OPACITY, TRANSPARENCY. 147 
 
 CHAPTER V. 
 
 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 110. GENERAL REMARKS. The laws which govern the behaviour of 
 photographic plates under the nction of light and developers were not 
 cleared up until the publication, a few years ago, of Messrs Hurter 
 & Dri {field's photo-chemical investigations. A knowledge of their 
 remarkable researches is so essential to a proper understanding of photo- 
 graphy that an abstract of their papers is given here.* 
 
 111. DENSITY, OPACITY, TRANSPARENCY. They commence by asking, 
 " What is a perfect negative 1 " Their answer is, that a negative is 
 theoretically perfect when the amount of light transmitted through its 
 various gradations is in inverse ratio to that which the corresponding 
 parts of the original subject sent out. The negative is mathematically 
 the true inverse of the original when the opacities of its gradations 
 are proportional to the light reflected by those parts of the original 
 which they represent. 
 
 In order that this definition may be understood, they explain the 
 laws of absorption of light by black substances and define clearly the 
 meaning which they attach to the terms opacity, transparency and 
 density of a negative. The whole of their investigations depends 
 upon these laws. 
 
 For substances, which do not reflect much light, such as black 
 opaque bodies, or transparent coloured bodies, the relation between 
 the light absorbed and the quantity of the substance present is very 
 simple. If, between the eye and a source of light, we place a thin 
 
 *Photo-Chemical Investigations and a new method of the determination of the 
 sensitiveness of photographic plates, by Ferdinand Hurter, Ph.D., and V.C. Driffield. 
 Journal of the Society of Chemical Industry, 31st May, 1890. 
 
 The Action of light on the Sensitive Film. Photography, 19th and 29th. Feb., 
 1891. 
 
 Relations between photographic negatives and their positives. Journal of the 
 Society of Chemical industry, 28th Feb., 1891. 
 
 Latitude in exposure and speed of plates. British Journal of Photography, 21st 
 July, 1893. 
 
 The principles involved in enlarging, by V. C. Driffield. British Journal of 
 Photography, 9th and 16th Nov., 1894. 
 
 )liy 
 
 Oi 
 
148 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 layer of dilute Indian ink, that layer absorbs light and thereby 
 reduces the intensity of the light transmitted. Assume that such a 
 layer absorbs one-half of the light, then one-half of the light will be 
 transmitted. Whatever may be the intensity of the original light, 
 the intensity after passing this layer of ink will be one-half of what it 
 was. The inter-position of two such layers will reduce the light to 
 one-quarter of the original intensity, three such layers will reduce it to 
 one-eighth and so on, each layer reducing the intensity to one-half of 
 what it receives. 
 
 Had the first layer allowed J- of the light to pass through, then two 
 such layers would reduce the intensity to ^, three layers to ^, etc. 
 In general, any number of layers would reduce the intensity of the 
 light to a fraction, which is equal to the fraction the first layer allows 
 to pass, but raised to a power the index of which is the number of 
 layers employed. If n equals layers employed and the first one 
 
 reduced the intensity of the light to a fraction the n layers would 
 
 m 
 
 reduce it to 
 
 If, instead of using so many layers, the first layer were made to contain 
 as much Indian ink as the n successive layers contain altogether, we 
 should find that the one layer now reduces the intensity of light by 
 exactly the same amount as the n layers did. The reduction of the 
 intensity is of course due to the black particles, and depends simply 
 upon the number of them which are interposed per unit area. We 
 can thus replace the number of layers by the number of particles and 
 the law takes this form : The intensity I x of light after passing A 
 molecules of a substance is a fraction of the original intensity /, such 
 that: 
 
 For mathematical reasons, they express as a negative power of the 
 
 G 
 base of the hyperbolic logarithms by making : 
 
 1 
 
 C ~ 
 
 and they write : 
 
 where k is the coefficient of absorption. The fraction 1 represents and 
 
DENSITY, OPACITY, TRANSPARENCY. 149 
 
 measures the transparency of the substance. The inverse of that frac- 
 tion, or = s** measures the opacity of the substance. It indicates 
 
 *a 
 
 what intensity of light must fall on one side of the substance in order 
 that unit intensity may be transmitted. 
 
 T being the transparency and the opacity : 
 
 OxT=l 
 
 Density is quite distinct from opacity. By density, they mean the 
 number of particles of a substance spread over unit area multiplied by 
 the coefficient of absorption ; kA is what they term density and mark 
 by the letter D. 
 
 In its application to negatives, the density is directly proportional 
 to the amount of silver deposited per unit area, and may be used as a 
 measure of that amount. 
 
 The relations between the three terms, transparency, opacity, . and 
 density, are the following : 
 
 T=e~ D 
 = D 
 
 D = logeO = -logeT 
 
 The density is the logarithm of the opacity or the negative logarithm 
 of the transparency. 
 
 These relations hold good for some substances with regard to ordinary 
 white light, for others only with regard to monochromatic light, and 
 for others they do not hold good at all. Messrs Hurter and Prim* eld 
 have satisfied themselves that they do hold good for the silver deposited 
 as a black substance in negatives, so long as the silver does not assume 
 a metallic lustre and reflects but a very small amount of light. 
 
 By means of these definitions we are now in a position to trace the 
 connection between the densities of a theoretically perfect negative and 
 the light intensities which produced them. 
 
 Since the- density is the logarithm of the opacity, and since in a 
 theoretically perfect negative the opacities are directly proportional to 
 the intensities of the light which produced them, it follows that each 
 density must be proportional to the logarithm of the light intensity 
 which produced it. 
 
 The result is this : in a theoretically perfect negative, the amounts of 
 silver deposited in the various parts are proportional to the logarithms 
 of the intensities of light proceeding from the corresponding parts of 
 the object. 
 
 The question arises, can such a negative be produced in practice I 
 
150 
 
 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 In order to answer this question, it was necessary first to find a 
 simple method of measuring the density of the silver deposited in 
 negatives and then to study the influence of the developers upon the 
 density of the deposits. The action of the light itself could then be 
 investigated. 
 
 112. PHOTOMETER. The instrument for measuring the density of 
 the deposit is based on the relation existing between density and 
 opacity. The opacity of the plate is measured, and in order to avoid 
 calculations and references to tables of logarithms, the scale of the 
 instrument is so arranged as to read the logarithm of the opacity, which 
 is the density. The reason it is preferable to have the results expressed 
 as density is because the density is a measure of the amount of silven 
 deposited or of the chemical work done by the light. 
 
 The instrument devised by Messrs Hurter and Driffield is repre- 
 sented iix Figs. 201, 202 and 203. It consists essentially of a small 
 
 Fig-. 201 
 
 Bunsen photometer, similar to those used for testing the illuminating 
 power of gas, etc. The screen shown in section in Fig. 204 and 
 
PHOTOMETER. 
 
 151 
 
 marked by a heavy black line, is a piece of paper with a grease spot 
 in the centre about one millimetre in diameter ; it is placed in a small 
 cubical box or chamber. The chamber carries an eye piece, through 
 which an image of each side of the disc can be viewed in two small 
 
 Fig-. 202 
 
 mirrors, and so compared. The chamber can be made to slide in a 
 straight line on a support by a rack and pinion. This arrangement is 
 placed within a larger box, the ends of which have apertures through 
 which light is admitted trom two powerful petroleum lamps. Corres- 
 ponding exactly with these apertures similar apertures are bored into 
 the sides of the small chamber, which admit the light to either side of 
 the Bunsen disc. The dimensions adopted for the larger box are 12 
 inches long, 6 in. high and 4 in. deep. The small chamber is a cube 
 measuring 2 inches inside. Except the scales, everything inside of 
 the box is blackened, and it is important to exclude all extraneous 
 light by means of a screen. The heat of the lamps very soon injures 
 the woodwork unless it is covered with asbestos cardboard and sheet 
 metal. 
 
152 
 
 HURTER <k DRIFFIELD'S INVESTIGATIONS. 
 
 The aperture in the left hand end of the large box is reduced to 
 about inch diameter by a diaphragm. At this end is placed the 
 
 Fig-. 203 
 
 plate to be measured, held in position by springs. The hole at the 
 right hand end of the box is reduced by a circular metal diaphragm, 
 revolving around its centre, and bearing seven circular apertures of 
 various sizes. 
 
 The instrument is provided with a fixed scale which indi- 
 cates the position of the disc chamber. It is constructed as 
 follows: Let 21 be the length of the box between the two 
 diaphragms, and x the distance of the disc from the middle 
 point between the diaphragms. The numbers to be inscribed 
 on the scale are given by the formula : 
 
 F' on/i 
 ig". zU4 
 
 ^ e cen ^ re 
 
 on 
 
 instrument is marked with zero ; the scale 
 
 sides is symmetrical. 
 
 The circular holes in the revolving diaphragm are used for the purpose 
 of reducing the light admitted through the aperture in the right hand 
 
PHOTOMETER. 153 
 
 end of the box. B being the diameter of the largest hole, the substitution 
 of another hole of diameter b reduces the area of the aperture, and, with 
 
 B* 
 
 it, the intensity of the light, in the proportion -p-, and the vulgar 
 
 .S 2 
 logarithm of the fraction y^ is the density which is to be added to 
 
 that read on the scale of the instrument when the hole B is replaced 
 by the hole b. 
 
 Two examples will show how the instrument is used. 
 
 1. When measuring a small density, the largest opening of the 
 circular diaphragm is turned opposite the aperture in the side of the 
 box, and the disc chamber is moved to such a position that the two 
 images of the Bunsen disc are alike. The number shown on the scale 
 by the index of the disc chamber is read, the plate to be measured is 
 inserted, and the disc chamber is moved towards the left until equality 
 of the images is restored. Again reading the scale, the density is the 
 difference between the two readings, those on the right hand of the 
 zero being considered as negative and those on the left hand as positive. 
 
 2. In the case of a high density, the largest hole of the circular 
 diaphragm is placed opposite the aperture of the box, and by inserting 
 a piece of opal glass between it and the lamp, the light on the right 
 hand side is reduced until the disc chamber requires to be moved 
 almost up to the right hand end of the box in order to secure equality 
 of the images. If necessary, the lamp is moved farther away. When 
 equality is thus secured, the scale is read. The plate to be measured 
 is then inserted and the disc chamber moved to the left until equality 
 is again restored. If that cannot be done by the movement of the disc 
 chamber alone, it can be obtained by using a smaller hole of the circular 
 diaphragm. 
 
 Suppose the index stood at 1-10 on the right and afterwards at T55 
 on the left of zero, then the density would be 1-10 + 1-55 = 2-65. 
 
 If the index stood at 1-10 to the right and afterwards at 1-7 to the 
 left, and equality could then only be restored by changing the largest 
 hole of the circular diaphragm to one corresponding to a density of 0*7, 
 then the density would be 1 -10 + 1 -7 + 0*7 = 3-50. Higher numbers than 
 3-55 do not occur in ordinary negatives. A plate, the density of which 
 
 is 3-55, only transmits OKAQ f ^ ne light which it receives. 
 
 The general rule for finding the density is : consider the numbers to 
 the right of zero as negative numbers, those to the left as positive. 
 Subtract the first reading from the second ; the result is the density. 
 
154 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 If the circular diaphragm be used as well, the amount it indicates must 
 be added. 
 
 It will hardly be necessary to say that a plate of density 1, permits 
 y 1 ^ of the light to pass and that a plate of density 2, permits T ^- of the 
 light to pass since 1 is the logarithm of 10 and 2 that of 100. 
 
 With this instrument, fairly accurate results are obtained. Messrs 
 Hurter and Driffield tried it on mixtures of Indian ink and water, 
 indigo solution and water, and many other substances. The greatest 
 error did not reach four per cent, an accuracy quite sufficient for photo- 
 graphic purposes, where, from other causes still greater errors are liable 
 to arise, as will presently be shown. 
 
 The lamps should be powerful petroleum ones with duplex burners. 
 The flames should be on the planes at right angles to the axis of the 
 instrument. Very erroneous results are obtained if Argand burners 
 are used. The lamps should be placed close to the diaphragms, and it 
 is advisable to provide a small stage outside of the diaphragm to hold 
 coloured glasses, when a substance requires investigation in the light 
 of a- particular colour. 
 
 The importance of this instrument justifies the lengthy description 
 given. It is for photographic experiments as indispensable, as the 
 balance is in analysis. 
 
 113. DEVELOPMENT. There is a generally accepted belief among 
 photographers, that a great amount of control can be exercised in de- 
 velopment over the density and the general gradations of a negative. 
 With the plates experimented upon by Messrs Hurter and Driffield, 
 they found that no such control existed.* 
 
 The plan adopted in carrying out these experiments was to subject 
 pieces of one and the same plate to the varying conditions, the influ- 
 ence of which, on the density or the gradation, was the subject of 
 investigation A precaution always taken is never to develop a piece 
 of a plate which has been exposed to the light without simultaneously 
 submitting to the same developer a piece of the same plate which has 
 not been exposed, and which is termed the fog strip. The object of 
 this precaution is to ascertain exactly how much of the resulting 
 density is due to the action of the light and how much is due to 
 
 * It is contended by some phot6graphers that since the date of these investigations, 
 such progress has been made in the manufacture of plates, that some, principally 
 rapid ones, can, to a certain extent, be controlled in development. In their first 
 paper, Messrs Hurter and Driffield pointed out the theoretical possibility of a plate 
 being slow with one developer and rapid with another. They have recently given an 
 instance of a plate which was three times faster with rodinal than with oxalate of 
 iron. E. D. 
 
DEVELOPMENT. 
 
 155 
 
 incidental fog, including therein fog inherent in the plate or caused by 
 injudicious development, and also the density due to glass and 
 gelatine. 
 
 In the series of experiments made to ascertain the influence of time 
 of development and composition of the developer on the density, one- 
 half of the plate was covered and the other half exposed to a standard 
 light as will be presently more fully explained. After exposure, the 
 plate was cut up in such a way that each piece included a portion of 
 the unexposed and a portion of the exposed plate. Each strip was 
 then developed, such modification in time of development or composi- 
 tion of developer being made, as formed the subject of investigation. 
 The resulting intensities were then measured after fixing, washing and 
 drying. 
 
 The first series of experiments was made with a normal developer 
 containing pyrogallol ammonia and bromide of ammonia for the pur- 
 pose of investigating the influence of the time of development. The 
 times were 2-5, 5, 7 '5, 10, 12*5 and 15 minutes. 
 
 The outcome of the experiments (Fig. 205) was that the total density 
 grows with the time of development, but that the density due to light 
 
 V. 
 
 k 
 
 1" 
 
 io 
 MINUTES 
 
 i i \ r . 
 
 16 80 
 
 i. 205 
 
 reaches a limit in about 15 minutes. The continued growth of the total 
 density is due to the action of the developer upon the bromide of silver, 
 which had not been affected by the light. 
 
156 
 
 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 In the next series of experiments (Fig. 206) the amount of pyrogallol 
 was varied ; they were repeated with the addition of sodium sulphite 
 
 r=E= i= E 
 
 Fig. 206 
 
 and citric acid. They show that an excess of pyrogallol beyond a cer- 
 tain limit tends to retard development and the production of density. 
 With sulphite of soda and citric acid, there is a falling off in density. 
 
 Then the amount of ammonia was varied (Fig. 207) and it was found 
 
 2-0 
 
 0-.E 
 
 v. 
 
 Fig. 207 
 
 that its addition, up to a certain extent, increases the density in a 
 given time, but that the amount of ammonia which can be added 
 
DEVELOPMENT. 
 
 157 
 
 without giving rise to fog, and without simultaneously adding bromide, 
 is very limited. The so-called accelerating action of ammonia being 
 due almost entirely to its solvent action on bromide of silver, which, 
 if the ammonia is increased sufficiently, results in greatly diminishing 
 the density. 
 
 If, to any solution of silver bromide in very dilute ammonia, 
 such as is used for development, bromide of ammonium be added, an 
 immediate precipitate of bromide of silver is the result. The so-called 
 accelerating action of ammonia, and the retarding action of ammonium 
 bromide, are probably due entirely to this solvent action of the one, 
 and the anti-solvent action of the other of these two reagents. The 
 rapid production of fog, when ammonia is increased, is due to the fact 
 that when pyrogallol solution is added to an ammoniacal solution of 
 bromide of silver, the silver in solution is precipitated immediately in 
 the metallic state. 
 
 The influence of the variation of ammonium bromide is shown in Fig. 
 208. Development in four minutes was entirely prevented when the 
 amount of bromide was about ten times that of ammonia present. 
 
 10 15 20 2 
 
 ffELAT/VE AMOUNTS Of BROMIDE 
 
 Fig-. 208 
 
 It is interesting to point to Figs. 205 and 207, just to show the 
 great amount of action which the alkaline developer may have on the 
 bromide of silver, although it has never been exposed. This disagree- 
 
158 
 
 HURTER & DRIFflELD'S INVESTIGATIONS. 
 
 able property is common to all alkajine developers, and it renders them 
 unsuitable for scientific investigations. In all important work, the 
 ferrous oxalate developer should be used for the reason, that it 
 attacks unexposed bromide of silver so slowly that in one hour, and 
 even more, no appreciable density can be developed upon a really 
 good plate. Nor does its action vary much with its composition. 
 The addition or omission of bromide from the constitution of this 
 developer does not seem to have any great influence, and a greater or 
 less concentration of the reagents within considerable limits, does not 
 affect its action ; indeed, no variation was found to arise from altera- 
 tions in its composition, excepting the length of time needed for 
 completion of development. Several experiments were made with 
 various times of development and different compositions of the 
 developer. The mean is shown in Fig. 209. In not one instance did 
 
 X "* 
 
 k 
 
 I 
 
 ^xfterirnental 
 
 10 
 
 30 40 
 
 M/NUTES 
 
 Fig-. 209 
 
 the density of the unexposed portions of the plate amount to more 
 than 0-098, which is the density due to clear glass and gelatine. That 
 ferrous oxalate does not, however used, attack silver bromide which 
 has not been exposed to light, is a most valuable and characteristic 
 property of this developer. 
 
 An important result of this series of experiments is, that the density 
 reached is dependent upon the time of development, as well as upon 
 the exposure of the plate. The time it takes to reach a given density 
 varies much with the gelatine employed in making the emulsion and 
 the age of the plate, but with each plate it obeys a certain law, which 
 
GRADATION. 159 
 
 is more or less clearly visible in every experiment. It may be sur- 
 mised that the number of particles of bromide of silver affected by the 
 light is greatest in the front layer of the film, and decreases in 
 geometrical progression as each succeeding layer of the film is reached, 
 an idea which will be better appreciated when the action of light 
 upon the film is explained. This idea expressed algebraically leads to 
 the formula : 
 
 where D t is the density after t minutes development, D the limit of 
 density reached by very prolonged development and t the time of de- 
 velopment ; a is a fraction depending upon the nature of the film, 
 concentration of developer, temperature, etc. The relation of the 
 calculated figures to the experimental data is seen in Fig. 209. 
 
 A very important conclusion can be shown to proceed from the 
 formula representing the course of development. 
 
 If, on any one plate, two exposures were given, one of which would 
 ultimately yield density D l and the other J9 2 , and if this plate were 
 developed for a time t, then two densities, d^ and c? 2 , would result 
 so that : 
 
 arid it will be seen that on dividing these equations : 
 
 A. = A. 
 d 2 ~ j} 2 - 
 
 The resulting ratio is independent of the time of development, and 
 is equal to the ratio of the ultimate densities which would be reached, 
 so that the gradation of negatives appears to be independent of the 
 time of development. 
 
 114. GRADATION. The above experiments have shown that with a 
 well balanced developer, there is a limit to density, which depends 
 upon the action of the light, and that, so far, the only control the 
 photographer has lies in deciding whether he will reach that limit 
 or not. 
 
 It has also become evident that if two different densities be developed 
 upon the same plate to their extreme limits, the ratio existing 
 between these limits must depend solely upon the action of the light. 
 The question now to be considered is, whether it is possible by any 
 modification of development to influence this ratio and whether this 
 same ratio exists at all stages of development. 
 
160 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 In making these experiments, the source of light adopted was a 
 standard candle placed at one meter distance from the plate. A 
 number of gradations were then produced upon the plate by exposing 
 different portions of it to the light for different periods of time, always 
 leaving one portion of the plate uiiexposed. It must be admitted thab 
 the candle is by no means an ideal standard, but there did not seem 
 to be at the time of these experiments any satisfactory substitute, and 
 if the suggestions for its use which are about to be made be adopted, 
 its accuracy is quite sufficient for photographic purposes. 
 
 Assuming that the candle has been used before, it is lighted and 
 the hardened tip of the wick snipped off' with scissors ; the flame of 
 the candle will now be found to grow steadily in height, and as soon 
 as the distance from the tip of the flame to the lowest point at which 
 the wick blackens has reached forty-five millimetres, the exposure 
 may commence. The candle flame may now be relied upon to remain 
 sufficiently constant for about ten minutes, and that is amply long 
 enough for our purpose. If, after this time, the light is required for 
 any other purpose, it will be well to again trim the wick and start 
 de novo. The height of the flame may be measured by a strip of card- 
 board upon which two marks are made at a distance of forty-five 
 millimetres. It is, of course, obvious that these experiments should 
 be made in a room free from draught, and it is often a wise precaution 
 to place the candle in a tall box, open on one side and well blackened 
 inside. The candle should be well in view during the entire exposure, 
 so that the operator may be aware of any fluctuation in the light. 
 If the candle be used in the open room, all white or bright surfaces 
 capable of reflecting light should be removed. The candle should be 
 extinguished by an extinguisher and kept covered while nob in use. 
 
 If a plate be examined by placing it between the eye and the red 
 lamp, it will be found that the opacity of the film falls off at the 
 edges. The edges should, therefore, be scrupulously avoided and strips 
 for experimentation should be cut from the centre of the plate, or, at 
 any rate, well away from the margin. The operation of cutting the 
 plate should be conducted as quickly as possible, and as far away as 
 possible from the red light, so as to avoid all fogging action of the 
 light upon the plate. The width of the strips may conveniently be 
 made about one inch. 
 
 In order to secure a constant ratio of illumination between the dif- 
 ferent exposures, and to be independent of any fluctuations that may 
 take place in the light, the exposures are made behind a rapidly re- 
 volving disc Z>, Fig. 210, in which are cut sectors proportional to the 
 exposures to be given. After inserting the plate in the dark slide, the 
 latter is placed in its position behind the disc. The distance from the 
 candle to the plate is carefully adjusted, and the candle is lighted and 
 
GRADATION. 
 
 161 
 
 trimmed. . When the flame has reached the requisite height, the ex- 
 posure may commence. The disc is caused to revolve, and, at a given 
 moment, the slide protecting the plate is drawn, and the exposure con- 
 
 Fig". 210 
 
 tinued for the requisite length of time. The candle may be brought 
 nearer or placed farther away from the plate so as to curtail or increase 
 the exposure ; at a distance of O707 metre, the light of the candle is 
 equal to 2 candle meters, and at a distance of two metres it is equal to 
 J candle metre. 
 
 The unit of exposure adopted is the candle metre second, which is 
 the exposure to the light of a standard candle at one metre distance 
 during one second. The time may be measured with a watch or metro- 
 nome. Within such limits as the experiments embrace, it has been 
 ascertained that it is immaterial whether an exposure be made with a 
 light of J candlemetre for 40 seconds, or a light of one candlemetre 
 for 10 seconds. It has also been proved by experiment, that as far 
 as the ratios of densities are concerned, 'they remain constant whether 
 the exposure be made with a candle, with a petroleum lamp or with 
 daylight, so long as the product of intensity of light and time of ex- 
 posure be the same. 
 
 11 
 
162 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 Several experiments were made to show that the length of time of 
 development does not affect the ratio of densities among themselves, 
 but increases every density by proportional amounts. The results 
 clearly showed that the ratio of densities was given by the light alone, 
 and was not affected by the time of development nor by modifications 
 in the developer. Experiments with ferrous oxalate, pyrogallol, hydro- 
 quinone and eikonogen did not give any material difference in the ratio 
 of the various densities obtained. 
 
 These experiments confirmed Messrs. Hurter and Driffield's belief 
 that the gradations of a negative were independent of the time of 
 development, and could not be affected by alterations in the composition 
 of the developer, and they concluded that the photographer had no 
 control over the gradations of the negative, the ratios of the amount 
 of silver deposited on the film being solely dependent upon the ex- 
 posure.* 
 
 115. ACTION OP LIGHT ON THE SENSITIVE FILM. These investiga- 
 tions have not only revealed the fact that one single density taken by 
 itself is not characteristic of the exposure which the sensitive film 
 received, since the density may be partially du^ to " fog,' : or may not 
 be developed to its extreme limit, but the experiments have also 
 clearly shown that with the usual developers, the ratio of two densities 
 exclusive of fog, is a function of the action of the light oil the plate. 
 In all the experiments, the exposures given varied between 10 and 
 80 c.m.s. In tabulacing the ratios found between the two exposures, 
 it was discovered that the ratio, though constant for one particular plate, 
 is very different for different plates. The ratio i, for the same ex- 
 posures, smaller for rapid than for slow plates, but even with the same 
 plate, the ratio between two densities varies for exposures which bear 
 the same ratio to each other, but are different in absolute value. It is 
 certain, therefore, that the ratio between two densities depends not only 
 on the ratio of the exposures, but also 011 the sensitiveness of the plate 
 and the absolute value of the exposures. 
 
 To discover the connection between expo ure, sensitiveness and 
 density produced, numerous experiments were made. The fir.>t investi- 
 gation was as to the general effect of prolonged exposure on the density. 
 A plate received exposures commencing with 0'625 c.m.s., then doubling 
 every time until 5120 c.m.s. The plate was developed with ferrous 
 oxalate and measured. The results are represented in Fig. 211 ; the 
 exposures being chosen as abscissae, the densities as ordinates. It will 
 be seen that every time the exposure is doubled, the density increases, 
 at first slowly, then considerably, and from 40 c.m.s. up to 1280 c.m.s., 
 
 *As already observed, it is contended by many photographers that this law no 
 longer holds good with some of the plates of recent manufacture E.D. 
 
ACTION OF LIGHT ON THE SENSITIVE FILM. 
 
 163 
 
 every time the exposure is doubled, nearly an equal addition to density 
 is the result, the addition to density being on an average 0*266, but 
 after an exposure of 1280 c.m.s. further doubling produces less and 
 less increase in density. The first few densities are too small to admit 
 of accurate measuring. From the figure it will be seen at once how 
 rapidly densities grow at first as exposure is increased, and how slowly 
 at last, densities tend towards a limit. 
 
 i | i i i i | i i i i i i i i i | 
 
 1 1 i i 1 1 1 1 17 
 
 100O 2000 300O -KJOO 
 
 EXPOSURE . CANDLE-METRE -SECONDS 
 
 Fig-. 211 
 
 If, in any part of the curve, the densities were proportional to the 
 logarithms of the exposures, that portion of the curve should be dis- 
 covered, if, instead of choosing the exposures a* abscissae, the logarithms 
 of the exposures were used. This is easily done when the exposures 
 progress, as they do in these experiments, in a geometric series. Each 
 new exposure has only to be marked equidistant from the previous one 
 as abscissa. In this manner the results of another experiment with 
 exposures from 1 to 524288 c.m.s. are plotted in Fig. 212. It will be 
 perceived that the curve now consists of four distinct branches. It 
 proceeds from exposure 1 in almost a horizontal direction, ascends slowly 
 to exposure 16, from thence it proceeds almost in a straight line to 
 exposure 2048, when the growth of densities becomes slow. The densities 
 reach their maximum at exposure 16384 and from thence the curve 
 returns, the densities diminishing slowly with increased exposures. 
 
 "i 
 
164 
 
 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 Four different periods are accordingly distinguished. The first period 
 is termed the period of "under exposure ; " it is comprised in the first 
 curved portion. The second period, that during which the curve is 
 
 3C 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ^ 
 
 ^ 
 
 
 x 
 
 \ 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 1-0 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 f'l 
 
 j 
 
 m 
 
 7* 
 
 e 
 n 
 
 M 
 /// 
 
 --// 
 
 <v/ 
 
 V/ 
 ^/y 
 
 fa 
 
 (^ ^ 
 
 -Vv 
 
 >Y 
 
 ^ 
 
 r//' 
 
 ftCC' 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~ w < 
 
 * 2 n i 
 
 si : 
 
 \ \ 
 
 \ I 
 
 . Q. 1-1 i 
 
 J ^ 5D C 
 
 ^S'g I ? 5 
 f ft } i 9 1 
 
 EXPOSURE. CANDLE-METRE -SECONDS 
 
 Fig-. 212 
 
 almost a straight line, is the period of "correct representation. " The 
 third period is that during which the curve is again strongly bent as 
 far as its maximum, this is the period of "over exposure "; and the last 
 portion of the curve is the period of " reversal. " 
 
 Period of Under Exposure. During this period, the ratio between 
 two densities is at first accurately equal to the ratio of the correspond- 
 ing exposures. Of course, there is no defining point which marks the 
 find of this period and the beginning of the next, but we learn from it 
 that, for short exposures, the amount of silver reduced is directly pro- 
 portional to the exposure. 
 
 Period of Correct Representation. The second period of exposures 
 has thus been named because, during this period, a plate is capable 
 of giving a negative differing as little as possible from that which, 
 at the beginning, was defined as theoretically perfect. That defini- 
 tion required that the densities of the negative should be pro- 
 
ACTION OF LIGHT ON THE SENSITIVE FILM. 165 
 
 portional to the logarithms of the exposures which produced them. It 
 is characteristic of this period, that the densities are proportional to 
 the logarithms of the exposures. This is shown in Fig. 212, where the 
 densities are the ordinates, the logarithms of exposures are abscissae, 
 and the period of correct representation a straight line. Dozens of 
 plates were measured, and the densities falling within this period were 
 found to conform to the very simple linear equation : 
 
 D = r [Log It C] 
 
 D being the density, f a constant depending on time of development, 
 It the product of intensity of light and time, i.e., the " exposure," and 
 C a constant depending upon the speed of the plate. 
 
 An answer can now be given to the question : " Can negatives be 
 produced such as were defined to be theoretically perfect 1 " And the 
 answer is, they can be produced, but only by so carefully adjusting 
 the time to the intensity of the light that the exposures may fall 
 within the period of correct representation. 
 
 Period of Over Exposure. Little need be said about this period. 
 As the curve tends to become parallel to the axis of abscissae, it is 
 clear that when exposures fall within this period, shadows and high 
 lights will all be represented by densities which are almost equal. 
 There will be no contrasts. In the first period, that of under exposure, 
 the contrasts are too great ; in the period of over exposure, they are 
 too small. 
 
 Period of Reversal. Within this period happens that peculiar 
 phenomenon, the transformation of the negative into the positive, the 
 " so arization," " reversal," etc. It is easy to understand how the 
 nega ive becomes a positive. While the deep shadows still act upon 
 the plate increasing the density, the high lights have passed their 
 maximum, and their densities grow less and less. The more the 
 exposure is prolonged, the less dense the high lights become, the 
 shadows exceeding them in density. 
 
 The period of reversal, although very interesting, requires such 
 enormous exposures that it need not be considered from a practical 
 point of view. The three first periods, that of under exposure, that 
 of correct representation and that of over exposure, are the only 
 practically interesting portions of the curve. 
 
 From a clever and well-reasoned mathematical investigation, based 
 on the idea that a certain definite amount of energy is needed to bring 
 a particle of silver bromide into the condition in which it can after- 
 wards be developed, and that it is only to the light absorbed by 
 unaltered silver bromide that increase of density consequent on 
 
166 HURTER <fc DRIF FIELD'S INVESTIGATIONS. 
 
 increased exposure is due, Messrs. Hurter & Driffield deduce the 
 following formula for the density of the plate after development : 
 
 > (0 1)^4- 
 
 in which is the opacity of the unexposed plate, / the intensity of 
 light ; t the time of exposure, /? a 1 ractioii the hyperbolic logarithm of 
 
 which is - -y, i a constant which is a measure of the slowness of 
 
 silver bromide, and which they call the inertia of the plate, Y a co ~ 
 efficient depending upon the time of development, which 'hey call the 
 " development factor" and log^ the hyperbolic logarithm of the 
 
 expression between brackets. To illustrate the close accordance 
 between their theory and experiments, the calculated curve is shown 
 by a black line in Pig. 212, while the measured densities are indicated 
 by dot-. For this purpose, the opacity of the unexposed plate was 
 measured for the rays of the spectrum from F to H and found to be 
 3.32. An inspection of Fig. 212 leaves little doubt that the action of 
 light on the sensitive film is fairly represented by the formula, and 
 consequently, it may be assumed as proved that the action of light at 
 any moment is proportional to the amount of light absorbed by 
 unaltered silver bromide. 
 
 To further elucidate this question, plates were prepared of different 
 opacities, by spreading on equal areas different amounts of silver 
 bromide. These plates were measured to ascertain their opacity to 
 blue light, and the curves calculated which are represented in Fig. 213. 
 The plates were then exposed and measured, the results being shown 
 by dots. 
 
 It will at once be perceived that the more thinly the plates are 
 coated, the shorter is that portion of the curve which is a straight line. 
 This means that the period of correct representation is very short and 
 great contrasts cannot be truly rendered by a thinly coated plate. It 
 will also be found on closer inspection that the centre of the straight 
 portion is in each curve in a different place, and that the thinner the 
 plate, the shorter is the exposure necessary to reach the centre portion. 
 This means that a thinly coated plate is somewhat faster than a thickly 
 coated me, though they are made of the same emulsion. A thinly 
 covered plate, however, appears very much faster than it is in reality. 
 It is incapable of rendering wide contrasts, hence the negative 
 always looks flat, and thereby gives to the eye the impression of over 
 exposure. Thickly coated plates give also very much greater latitude 
 in exposure. The plate illustrated by Fig. 212 would have given good 
 pictures of subjects with contrasts varying from 1 to 20, though the 
 
SPEED OF PLATES. 
 
 167 
 
 exposures had varied from 1 to 8, so that an exposure of 10 seconds, 
 or one of 80 seconds, would have resulted in but little difference in 
 the prints, but one of the negatives would have been much slower in 
 
 30 240 
 
 CANDLE-METRE - SECONDS 
 
 Fig-. 213 
 
 printing because generally denser. Thinly coated plates, on the other 
 hand, need very accurately timed exposures. 
 
 116. SPEED OF PLATES. In the formula : 
 
 -I may be replaced by when that represents a large numb* 11 ' that 
 is when the plate is richly coated and as log /5 is - -^-, the equation 
 can be transformed into another, viz. : 
 
168 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 which equation holds good only when the numerical value of -=-r- is 
 
 greater than 1 and less than the opacity 0. Tt is between these two 
 limits only that this equation gives tolerably correct results. 
 
 Suppose two richly coated plates, with different inertias i and i it 
 on which the same density is to be impressed by a given intensity of 
 light /; they would require different exposures, which would have to 
 be such, that : 
 
 It It 
 
 or the times would have to be chosen so that : 
 
 .*L 
 
 *i 
 
 This means that the values of i being known for different plates, the 
 exposures required to obtain the same results are also known for those 
 plates, if the exposure for any one of them has previously been ascer- 
 tained. 
 
 The determination of the numerical value of the symbol i is there- 
 fore an important problem. 
 
 Since the density of the image is an abstract number, it follows that 
 
 ^- is also an abstract number, and that i is therefore an exposure. It 
 
 ^ 
 
 was calle I the inertia, and it really measures that exposure which will 
 suffice to change a particle of silver bromide into the developable con- 
 dition. But for its practical application, it has another meaning. It 
 measures the least exposure which will just mark the beginning of the 
 period of correct representation. 
 
 The speed of the plate is the inverse value ; the longer the exposure 
 needed to bring the plate just to the beginning of the period of correct 
 representation, the slower is the plate. Therefore, the speed of the 
 
 plate is measured by the value 
 
 ^ 
 
 The method adopted for measuring the value of i is as follows. The 
 plate is given a series of exposures falling within the period of correct 
 representation and then developed and measured, fog being deducted. 
 The values of D and of the exposures It, permit of j and i in the formula 
 expressing the density being calculated. 
 
 It is preferable, however, to obtain the result by a graphic method, 
 by means of which all calculations and references to tables of logarithms 
 
8PEED OF PL A TES. 
 
 169 
 
 are avoided. Printed diagrams are used similar to Fig. 212. The 
 horizontal border is, like the scale x>f an ordinary slide rule, a logarith- 
 mic graduation for the exposures, but it is repeated four times instead 
 of twice, as in the case of the slide rule. Vertical lines are drawn at 
 the points 0-156, 0-312, 0-625, 1-25, 2-5, 5, etc., and they aie divided 
 into 25 equal parts, making the highest density 2-5 and the lowest 0. 
 Having measured the densities and deducted from each the density of 
 the "fog strip/' which is that due to the incipient fog of the plate and 
 to the glass and film, they are plotted on the vertical or exposure lines. 
 A piece of black thread is then stretched alon;4 that part of the curve 
 which practically forms a straight line, and which indicates the position 
 and extent of the correct period. In this way, the position of the 
 straight line may be ascertained before being actually drawn on the 
 diagram. After drawing it, it is continued till it intersects the hori- 
 zontal scale at the bottom of the diagram. The point at which the 
 intersection takes place gives the inertia. The remaining points may 
 now be connected by curves to the ends of the coriect period line. The 
 curve at the upper end represents the period of over exposure and that 
 at the lower end the period of under exposure, the whole representing 
 
 M 
 30 
 
 
 
 o 
 
 EXPOSURE 
 
 3O -625 1-25 ,' 2-S 5 1O 20 4O 8O 160 320 640 
 
 S S 4 S 
 
 DEVELOPMENT FACTO* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^- ' 
 
 
 
 
 
 
 
 
 
 
 / 
 
 X 
 
 
 
 
 
 
 
 
 
 
 / 
 
 /1 
 
 
 
 
 
 
 / 
 
 X 
 
 
 
 ^ 
 
 / 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 1 -2 -3 -5 -7 1 23571 
 
 INER 
 
 D 20 30 50 70 WO 200 300 50O 70O V. 
 TIA 
 
 Fig-. 214 
 
 the most characteristic features of the plate. The details just described 
 will be better understood by a reference to Fig. 214, which shows the 
 curve of an Edward's " isochromatic medium " plate.* The develop- 
 
 * This is not one of the diagrams of Messrs Hurter and Driffield's paper, but is 
 introduced here as an illustration of the plate used on the Canadian Surveys. 
 
170 HURTER cfc DRIFFIELD'S INVESTIGATIONS. 
 
 ment factor is obtained by drawing through the point 100 of the 
 exposure scale, a line parallel to the straight part of the characteristic 
 curve, noting its point of intersection with the density scale. This 
 gives the value of the tangent of the angle of inclination, which is the 
 development factor. In the example given, the inertia is 0*85 and 
 development factor - 98. 
 
 When the inertia of the plate is known, it is possible to time the 
 exposures in the camera so that the densities of the gradations are 
 almost exactly proportional to the logarithms of the light intensities 
 which produced them, and negatives can be produced which satisfy 
 very nearly the definition given ot a theoretically perfect negative. It 
 must be borne in mind, however, that such a negative is not neces- 
 sarily true to nature. If the negative is to be true to nature, a plate 
 must be used which is richly coated, the exposure must be carefully 
 timed, and the development must be carried only so far that the 
 value of the development factor is numerically equal to one. On the 
 other hand, such a negative would not itself generally give a print 
 true to nature; but that subject will be treated later on. 
 
 The exposure to be given in the camera can, when the inertia is 
 known, be found by means of the actinograph, an ingenious instru- 
 ment devised by Messrs. Hurter and Driffield. For isochromatic 
 plates, the inertia must be multiplied by a constant coefficient before 
 using it on the actinograph for daylight exposures. 
 
 The "actinograph speed" of a plate is obtained by dividing 34 by 
 the inertia. 
 
 Before proceeding further, it may be well to give Messrs. Hurter 
 and Driffield's recommendations on development, although they have 
 met with considerable opposition. They assert that, for all ordinary 
 photographic work, there is no developer superior to ferrous oxalate. 
 It- is preierable, because of the uniformity of the colour of the silver 
 deposited by it, a point of very great importance for printing and 
 enlarging by developing processes, in which the exposure is arrived at 
 by calculation ; it is preferable, because they have not yet found one 
 plate with which it disagreed, and this is more than can be said of 
 other developers. It will also develop an old plate which may have 
 been carelessly laid by for years ; while, with another developer, it 
 would be hopeless to obtain a passable result. It is preferable, be- 
 cause, of all developers, it is least liable to attack silver salts which 
 have not been acted upon by the light, and because it will not lend 
 itself to the production of foggy messes. It is not implied that other 
 developers may not have their special uses ; for instance, rodinal is 
 of the greatest value in the case of certain plates when dealing with 
 extremely short shutter exposures. 
 
NEGA 11 VES AND POSITIVES. 171 
 
 Proceeding with the operation of development, it is advisable that 
 it be conducted at a fixed temperature, 65 Fahr. for instance. The 
 developer itself should be brought to this temperature, and maintained 
 at it by placing the developing dish in a water bath of the same tem- 
 perature. The cons.ituents of the developer are intimately mixed by 
 stirring, and, at the moment of pouring on to the plate, the time is 
 noted. The dish should only be rocked for a few moments, in order 
 to expel any air bubbles from the surface of the plate, and should 
 then be covered, so as to expose the plate no more to the red 
 light than is absolutely necessary. Examination of the plate during 
 development should be avoided as far as possible, as no red light 
 whatever is safe in the case of even a fairly sensitive plate. It is 
 important to know beforehand what development factor corresponds 
 to a given time of development. This is found, once for all, by de- 
 veloping gradated exposures for different lengths of time and measur- 
 ing the development factor in each case. For other times the factor 
 is obtained by interpolation. Having decided on the density to b : 
 obtained and knowing the exposure, the density is plotted on the 
 diagram on the proper exposure line, and the point obtained joined to 
 the division of the horizontal scale representing the inertia of the 
 plate. A parallel to this line through the point 100 of the bottom 
 scale gives the development factor, from, which the time of develop- 
 ment is deduced. 
 
 After development, the plate is fixed in clean hyposulphite of soda 
 and washed in the ordinary way. After washing, it is well to wipe 
 the surface of the film gently with a plug of wet cotton wool. 
 When the plate is dry, the back of it should be thoroughly cleaned 
 and the film wipnd with a silk handkerchief. 
 
 117. NEGATIVES AND POSITIVES. Let a sensitive plate behind a 
 negative be illuminated by a light of known intensity, and for simpli- 
 city's sake, let the negative have only two or three different opaci- 
 ties. Let the op i cities of the negative be 40, 20 and 10; then we 
 should expect the plate to be illuminated behind the negative with 
 
 > and Yf\ f th e original intensity of the light. Experiment 
 
 reveals, however, that this is not so, but that the results of exposures 
 to the light behind the negative are greater than those which would be 
 
 produced by - 5 - and y^ of the original light intensity. The 
 
 reason f >r this is not far to seek. When the light shines on the plate 
 directly, s >y a '-out 70 to 80 per cent of the light is reflected by the 
 plate into space. When a negative is placed in front of the plate, the 
 light is similarly reflected by the sensitive surface, but a considerable 
 
172 HURTER cfc DRIFFIELD'S INVESTIGATIONS. 
 
 portion of it is at once reflected back again by the two reflecting sur- 
 faces of the negative, so that behind a negative, less of the light trans- 
 mitted by it is lost by reflection from the sensitive film, and conse- 
 quently, more work is done on the film than would be the case if the 
 same intensity of the light were to act upon a film free to reflect. The 
 amount of light reflected by the sensitive film and back again by the 
 negative depends upon the co-efficients of reflection of both the film and 
 the negative ; it may be foreseen that the same negative will give 
 different results upon different printing surfaces, according to the 
 amount of light which these surfaces reflect. 
 
 There is still another point to be considered. The opacity found 
 with the photometer, is measured chiefly to the yellow rays of the lamp, 
 whilst those rays are least active upon the plate. The opacity of the 
 negative to the blue rays is, in all cases tested, greater than the opacity 
 to the yellow rays. 
 
 These considerations explain why, when a negative is used for con- 
 tact printing, its opacity must be considered as less than that indicated 
 by the photometer ; and when it is used for enlargement, the opacity 
 must be considered as greater than that measured with the photometer, 
 because, in the one case, the sensitive surface cannot reflect freely, 
 whilst, in the other, it does so. 
 
 The exact amount by which the value of the opacity of the negative 
 is to be increased or decreased depends, therefore, upon the reflection 
 of the film, upon its sensitiveness to the different portions of the 
 spectrum and upon the colour of the negative. 
 
 If it were not for these corrections, the relation between a negative 
 and a positive could be at once deduced from the formula 
 
 In this formula, the intensity of the light, I, is reduced by the 
 negative and must be divided by the opacity ; which, in logarithmic 
 calculation, means deducting the logarithm of the opacity from the 
 logarithm of the intensity of the light. But, the logarithm of the 
 opacity is the density measured by the photometer, which density is 
 simply subtracted. In order to prevent confusion, the densities of 
 negatives, including fog, are denoted by JV 7 ", and the densities of posi- 
 tives by P. The equation which results, after introducing the correc- 
 tion a of the negative density for the reasons just explained, stands 
 thus : 
 
NEGATIVES AND POSITIVES. 
 
 173 
 
 and this equation represents the general relation between a negative 
 and its positive. P is the density of the positive produced behind the 
 negative of density N upon a plate of inertia i by means of the light 
 intensity / in the time t. 
 
 The coefficient , which converts the density as measured into the 
 printing density, is, for negatives developed by ferrous oxalate, usually 
 a fraction ; for pyro-developed negatives, it is generally nearly 1, if the 
 negative be used for contact printing; but, when the negative is used 
 for enlarging, this factor M, which changes the usual density of the 
 negative into the enlarging density, is always greater than 1, even for 
 negatives developed with ferrous oxalate. 
 
 The method for finding the printing factor a is illustrated by Fig. 
 215. Assuming that a speed determination has already been made of 
 
 
 31 
 
 2 -6'. 
 
 55 1- 
 
 25 2- 
 
 5 s 
 
 >/ 
 
 , u 
 
 TSWff 
 
 ) 2 
 
 > 4< 
 
 > 8 
 
 It 
 
 3' 
 
 20 6 
 
 to 
 
 2-5 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 O.Q 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 R 
 
 i 
 i 
 
 
 
 
 
 
 / 
 
 ^ 
 
 
 
 
 
 
 
 
 O5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 1 
 
 0*5 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 I 
 
 i 
 
 i 
 
 
 
 
 1 
 
 1 f 
 
 
 1 
 
 
 I 
 
 2 -3 
 
 5 
 
 
 2 
 
 3 : 
 
 i i 
 
 t> 2 
 
 3O 
 
 SO 7O 
 
 100 
 
 200 30 
 
 5001 
 
 00 K 
 
 ioo 
 
 INERTIA 
 Fig-. 215 
 
 the plate of which the printing factor is required, and that the extent 
 and position of the line of correct representation are known, two more 
 exposures are made, which, so long as they fall within the correct 
 period, should lie as widely apart as possible. In the example, the 
 exposures given are 5 and 40 c.m.s. Tak'ing a negative of a uniform 
 density of, say, about 1 -0, the object now is to produce, through this 
 negative, upon the plate under examination, a density which shall lie 
 somewhere between the densities which will result from the two direct 
 exposures already given. The geometrical mean of 5 and 40 is, 
 roughly, 15 ; the intention is, therefore, to produce a density behind 
 
174 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 the negative (the total density of which is, in the example, 985) 
 equivalent to a direct exposure of 15 c.m.s. The necessary exposure 
 is calculated thus : 
 
 Log T=log 15 + 0-985. 
 
 2V 145. 
 
 This exposure of 145 c.m.s. is given through the negative and the 
 plate developed together with that produced by the direct exposures 
 of 5 and 40 c.m.s. Having measured the resulting densities and de- 
 ducted fog, those densities which result from the two direct exposures, 
 in the example 1-010 and 1-965, are marked upon the exposure lines 
 5 and 40 of a diagram. Through these two densities, a straight line is 
 drawn which coincides with the correct period of the plate. In the 
 example, the density produced through the negative is 1*730, exclusive 
 of fog ; this is marked on the density scale, and a horizontal line drawn 
 through it intersecting the straight line previously drawn. Through 
 this point of intersection a perpendicular is drawn to the inertia scale 
 and the point of intersection marked. This point of intersection 
 gives the direct exposure to which an exposure of 145 c.m.s. through 
 the negative is equivalent, in the example 24 c.m.s. The printing 
 factor is obtained by deducting log 24 from log 145, and dividing the 
 result by the density of the negative used : 
 
 Log 145 -log 24 
 ~985 
 
 To produce with certainty a good positive transparency on a given 
 plate, the exposure behind the highest density must be just sufficient 
 to slightly affect the plate, and, therefore, must be equal to the inertia. 
 The necessary exposure would be 
 
 log T=log i + aN 
 N being the highest negative density. 
 
 But, the calculation need not be made at all. The printing density 
 of the negative, measured on the density scale, is taken with a pair of 
 compasses and the same distance to the right on the exposure scale 
 measured from the inertia of the plate. The necessary exposure is 
 read off at once. 
 
 It is undoubtedly a difficulty that with one and the same negative, 
 its densities as measured photometrically have to be multiplied with 
 different factors according to the plates used with it. For contact 
 printing with negatives developed by ferrous oxalate, the factor varies 
 from about 0-6 to 1*0. It is, however, sufficient for practical purposes 
 
CONTACT PRINTING AND ENLARGING. 175 
 
 to use the factor O'S, but, of course, it is always better to ascertain the 
 correct factor by experiment. 
 
 It can be shown that one and the same negative is not equally suit- 
 able for all printing processes, and that a negative yielding a good 
 ordinary silver print is generally incapable of giving a first-class enlarge- 
 ment on bromide paper. By variations in the time of development, it 
 is possible to produce secondary negatives in which the scale of tones 
 is either contracted or extended, and this function of development is of 
 the utmost value in the production of special negatives for special 
 printing processes. 
 
 118. CONTACT PRINTING AND ENLARGING. The principles involved 
 in cont ict printing and enlarging on paper are practically the same. 
 In both cases, it is necessary that the negative shall have that range 
 of gradations which the paper is capable of registering. The subject 
 of contact printing has been treated separately in an able paper by 
 Mr. V. C. Driffield*. It will be sufficient for our purposes to give his 
 remarks on enlarging : with some slight and evident modifications, 
 they may be applied to contact printing. 
 
 There is a prejudice against enlargements on bromide paper, many 
 persons maintaining that satisfactory results cannot be produced on 
 this material. It must be admitted, that the best possible results are 
 probably obtained by a contact print in platinum or carbon from an 
 enlarged negative ; but at the same time, very much better bromide 
 enlargements can be produced than one very often sees, if only, the 
 conditions necessary for their production be observed. The failure, to 
 produce satisfactory enlargements on bromide paper, is probably chiefly 
 due to the fact that it is utterly impossible to produce a good enlarge- 
 ment from a negative which will yield a good contact print, and yet 
 one negative is generally expected to serve both purposes. 
 
 There is also a general impression that the production of a good 
 enlargement involves the use of a light of high intensity, preferably 
 daylight, or, at any rate, such an illuminant as lime light. This im- 
 pression is altogether a fallacy,; on the contrary, given a suitable 
 negative and a correct exposure, a perfectly satisfactory result can be 
 obtained with artificial light of extremely feeble intensity. The 
 mistake has arisen from attempting to produce enlargements from 
 negatives of ordinary density with light of low intensity. The pro- 
 longed exposure necessary in such cases has not been realized, and 
 under exposure has resulted ; while, with daylight, it has been easy 
 to secure a more adequate exposure and the better result has been 
 
 *The principles involved in the calculation of exposures for contact prints on 
 bromide paper British Journal of Photography of 22nd September, 1893. 
 
176 HURTER <k DRIF FIELD'S INVESTIGATIONS. 
 
 attributed to the higher illuminant. But, neither daylight nor any 
 other light will ever produce a satisfactory enlargement from an unsuit- 
 able negative. The only advantage, that appertains to the use of day- 
 light, is that it does not necessitate the possession of an optical lantern ; 
 but otherwise, the balance of advantage is infinitely in favour of the 
 lantern, Daylight is fickle, and more difficult to gauge accurately 
 than an oil lamp ; then again, it is only available in the day time, 
 while enlarging by oil light can be conducted at any hour, day or 
 night. 
 
 The essential conditions to consider in connection with enlarge- 
 ments are, firstly, the quality of the negative required ; and, secondly, 
 the estimation of correct exposure. 
 
 The consideration which determines the character of the negative 
 suited for enlargement is the range of gradation which bromide paper 
 is capable of yielding. The range of gradation of the negative must 
 coincide with the range of gradation of the paper, if a satisfactory 
 result is to be looked for. The range of the paper is the ratio exist- 
 ing between the two exposures, one of which just falls short of pro- 
 ducing any deposit, and the other which just suffices to produce the 
 deepest black which the paper is capable of recording when viewed by 
 reflected light. Say, for example, that an exposure of five seconds 
 just fails to produce a deposit and that one of 160 seconds just pro- 
 duces maximum blackness, the range of the paper would be as 5 is to 
 160, or as 1 is to 32. Now, obviously, for a negative to exactly cor- 
 respond with the paper range, its capacity for transmitting light must 
 have the same ratio that is to say, the ratio between the opacities 
 representing the deepest shadow and the highest light must also be as 
 1 is to 32. This means, that the difference between the maximum 
 and minimum densities of the negative must be equal to the logar- 
 ithm of 32, which is 1-505; but it must not be forgotten, that this is 
 the difference of the printing densities which are obtained by multi- 
 plying the densities measured in the photometer by the printing 
 factor a. 
 
 To investigate the range of bromide. paper, a series of exposures was 
 made and after developing, the gradations of the paper were measured 
 by reflected light, employing for that purpose a modification of Hurter 
 & Driffield's photometer. The readings do not, however, in this case 
 express density, but the fraction of light reflected by the gradations as 
 compared with light reflected by the normal white of the paper. A 
 
 reading of 1 '3 (log. 20) indicates that the gradation reflects of the 
 
 light reflected by the pure white of the paper. Having measured the 
 gradations of the entire strip in this way, they were plotted just as in 
 
CONTACT PRINTING AND ENLARGING. 
 
 177 
 
 the case of a speed determination, Fig. 216 shows the result. It is 
 at once seen that there is a strong resemblance between the paper scale 
 and the characteristic curve of a plate ; but, there is a peculiarity about 
 the upper part of the curve. In the case of a plate, this would merge 
 from the straight line of the correct period much more gradually : here, 
 when the ordinate 160 is reached, there is no further measurable 
 difference in the degree of blackness, so that, though exposures beyond 
 
 EXFOSUftf , C*HDIE-METRC-SECOMOS 
 
 OOOOO 
 
 <Q <O t- O 
 
 II 1 
 
 Fig-. 216 
 
 160 c.m.s. produce slight differences just discernible to the eye, 
 comparatively large density differences in the negative are repre- 
 sented by such trifling differences in the paper scale that they have 
 simply to be ignored. The same argument applies to the very faintest 
 indications of deposit at the other end of the scale, bo that the practi- 
 cally useful range of the paper is less than a mere inspection of the 
 gradations would lead one to infer. And, it must be remembered, that 
 the test strip was practically developed out, while in picture making 
 this is generally not the case. The figure, therefore, teaches that the 
 practically useful range of bromide paper is capable of recording light 
 intensities which are to each other as 5:160, or as 1:32 ; or, taking the 
 difference between the logarithms of these numbers, in order to com- 
 pare directly with the negative densities, it is at once learned that the 
 density range of a negative suitable for enlargement must not exceed 
 1 *5, and, as a matter of practical experience, a somewhat less range 
 gives even better results. The figure further teaches that, as the more 
 delicate intermediate tones of the negative correspond with the higher 
 part of the paper curve, where the gradations approach each other 
 
178 HURTER & DRIFFI ELD'S INVESTIGATIONS. 
 
 very closely; the small negative gradations should be kept as widely apart 
 as possible. This means that a negative for enlarging should be fully 
 exposed, so as to bring its lower densities into the period of correct ex- 
 posure. 
 
 The reason for preferring a negative with a rather too contracted, 
 than a too extended, range is that it is far better to make quite sure 
 of getting the delicate half tones, even at some loss of the deepest 
 black tones. 
 
 The range of the paper and its sensitiveness may, in practice, be 
 found quite easily. To ascertain the sensitiveness is to discover what 
 exposure just brings to the commencement of the paper range that 
 is to say, what is the maximum exposure which will only just pro- 
 duce evidence of deposit with full development. It is absolutely 
 necessary that this exposure be ascertained by the precise source of 
 illumination which is to be employed for the production of the 
 enlargement. A series of exposures is made on a strip of the paper 
 to the light emanating from the lantern, but the exposures progress 
 by 1'41 instead t of 2, thus providing an intermediate gradation, and 
 so rendering the decision as to the commencement of range more 
 exact. When making this determination the light of the lantern is 
 projected upon the screen, having, of course, secured an accurate 
 focus and a correct adjustment of the position of the light. The 
 amount of light falling upon the disc is then measured and the 
 exposures made. 
 
 A little difficulty may possibly be found in deciding which expo- 
 sure must be regarded as marking the commencement of the range. 
 It is always better, in case of doubt as to exactly where t 4 he range 
 commences, to take too great rather than too small an exposure. It 
 is advisable to be guided by the earliest unmistakable deposit, ignor- 
 ing a possible one or two gradations of an extremely faint and inde- 
 cisive character, and, while about to ascertain this information re- 
 garding the paper, it is well to mask a small portion so as to be able 
 to judge of the normal whiteness of the paper. If, after develop- 
 ment, there is evidence of fogging on the part to which light has not 
 had access, the paper is to be discarded altogether ; it will never 
 produce a satisfactory enlargement. Such fogging may arise in the 
 manufacture of the paper, or may be the result of deterioration from 
 long or careless keeping. 
 
 It was ascertained that the first evidence of deposit on the paper to 
 be used for enlarging was produced with an exposure to the light of 
 the lantern of a certain number of seconds ; it is transformed into c.m.s. 
 by measuring the intensity of the lantern's light in candle metres by 
 means of a simple shadow photometer. This, in the example given, 
 
CONTACT PRINTING AND ENLARGING. 179 
 
 corresponds to 5 c.ra. for the first evidence of deposit on the paper ; it 
 is one factor required before the exposure for the enlargement can be 
 calculated. The other factor involves a consideration of the negative 
 to be used, and is simply its maximum density multiplied by the enlarg- 
 ing factor a. From what has already been explained, it is clear that 
 there is no one number which correctly expresses the density of a given 
 deposit. It is perfectly obvious, that two deposits respectively developed 
 with pyrogallol and ferrous oxalate may have identical visual densities, 
 am| yet, owing to the wide difference in their colour, their actinic den- 
 sities will greatly differ ; but further than this, it is now seen that a 
 given deposit has three distinct density values, its visual value, its 
 contact printing value, and its enlarging value. The visual density of 
 a deposit, developed with ferrous oxalate, which measures I'O becomes 
 a contact printing density of 0-8 and an enlarging density of 1*4. 
 
 Let the negative to be used have visual maximum and minimum 
 densities of 1'18 and O12 respectively; the visual density range is 
 therefore 1 -18 - O12 = 1 -06. When, however, the densities come to be 
 applied in the operation of enlarging, they require to be multiplied by 
 1-4. This converts the visual densities 1-18 and O12 into the enlarg- 
 ing densities 1*652 and 0-168, thus making the enlarging range of the 
 negative 1 -484. From what has been seen in considering the question of 
 range generally, this negative, the enlarging range of which is so nearly 
 1-5, ought to produce a satisfactory enlargement. 
 
 The calculation of the exposure may now be proceeded with. All 
 that is to be kept in view is that the exposure must be so timed, that 
 the action of the light passing through the maximum density of the 
 negative shall just bring to the commencement of the paper range : or, 
 in other words, the exposure must be so timed that the light passing 
 through the maximum density of the negative shall produce the same 
 effect as an exposure, without the interposition of the negative, of 5 
 c.m.s. The maximum enlarging density of the negative is 1'652, and as 
 this density only transmits about one forty-fifth of the light it receives, 
 it is obvious that, in order to produce the same result upon the paper 
 as would be produced by an exposure of 5 c.m.s. to the naked light, it 
 is necessary to multiply 5 c.m.s. by forty-five, the opacity corresponding 
 to the maximum density. Five times forty-five is 225, and this is the 
 exposure required. 
 
 This cplculation is more conveniently made as follows : 
 
 Maximum enlarging density of negative ............. 1-652 
 
 Log 5, exposure marking commencement of range. . . 0"699 
 
 2-351 
 
180 HURTER & DRIFFIELD'S INVESTIGATIONS. 
 
 The number corresponding to this logarithm, _24 '4, say 225, is the 
 exposure required in c.m.s. 
 
 The factor 1 4 for converting the visual into the enlarging density, 
 applies only in the case of negatives developed with ferrous oxalate. 
 In the case of negatives very yellow in colour as from development 
 with unpreserved pyrogallol, this factor will be somewhat greater; 
 how much, will be easily decided by an experiment or two. 
 
DRY PLATES. 181 
 
 CHAPTER VI. 
 
 PHOTOGRAPHIC OPERATIONS. 
 
 119. DRY PLATES. For surveying purposes, photographs must be 
 clear and full of detail ; the points to be plotted must be well defined 
 and easily recognized. Just at the start, a serious difficulty is met 
 with ; distant points, which are those wanted by the surveyor, are 
 always more or less indistinct in a landscape, the distance merging into 
 a uniform tint by the effect of what is known as " aerial perspective." 
 This effect is due to the light diffused by the mass of air between the 
 observer and the distant point ; the greater the distance, and conse- 
 quently the greater the mass of intervening air, the more light is 
 diffused and the more indistinct the distance becomes. From the 
 artist's point of view, aerial perspective is necessary to give the 
 impression of distance ; but to the surveyor, it is most objectionable. 
 
 The light diffused by the atmosphere is that which it has previously 
 absorbed, and consists mainly of rays of shorter wave lengths, which, 
 although not very luminous, have the strongest actinic power. It 
 thus happens that aerial perspective is very much exaggerated when 
 translated by photography, the strong effect of the light emitted by 
 the blue haze, through which the distance is seen, completely blurring 
 on the plate the details of the image, so much so, that the photograph 
 becomes almost useless for surveying. Several causes contribute to 
 the same result, such as the presence of smoke in the air, dust raised 
 by wind, etc. 
 
 To get rid of this effect, it is necessary to have a plate which shall 
 not be acted upon by the rays of greater refrangibility. Ordinary 
 plates are, within the limits of exposure given in the camera, sensitive 
 to the blue and violet rays only, but orthochromatic plates are made 
 which are acted upon by the other end of the spectrum, although tLa 
 maximum of sensitiveness is still in the blue violet region. By using 
 a screen of a deep orange tint, it is possible to cut off nearly the whole 
 of the rays further than the green : such a screen in connection with 
 orthochromatic plates furnishes a partial solution of the difficulty. It 
 is, indeed, not entirely removed, because air absorbs, and therefore 
 diffuses, not only blue and violet rays, but also those of lesser refrangi- 
 bility ; the effect of aerial perspective still exists, but instead of being 
 exaggerated by photography, it is reduced. 
 
182 PHOTOGRAPHIC OPERATIONS. 
 
 All that is required in the plate is, that it shall be sensitive to other 
 rays than blue and violet, because these rays are cut off by the screen. 
 It is not necessary that it should be acted upon by the red rays, in 
 fact, it is preferable for convenience of manipulation, that it should 
 not. 
 
 Fig. 217 represents the action of the spectrum on " Edward's isochro- 
 matic medium" plates, exposed behind a deep orange screen. The 
 action of light commences at D and nearly ends at b. Between b and 
 
 F there is very little action ; just beyond F, is a band of slight sensi- 
 tiveness. It will be seen that the image on this plate is formed entirely 
 by the yellow and green rays ; consequently, the photograph may appear 
 quite unnatural. Yellow photographs as white ; an autumn landscape 
 when the leaves have turned yellow, looks as if the trees were covered 
 with snow. 
 
 The use of orthochromatic plates is not without its drawbacks. The 
 shadows, in a landscape, receive their light from the sky and it is this 
 light, reflected into the camera, which forms the image. But the light 
 of the sky is mostly blue and violet and does not act on the plate 
 behind the orange screen ; the result is that the shadows are much 
 more intense than on ordinary plates. 
 
 In order to find the proportion between the light coming from an 
 object in direct sunlight and that coming from the same object when 
 placed in the shade, Messrs. Hurter and .Driffield photographed a 
 folding screen, upon each of two folds of which had been placed a sheet 
 
DRY PLATES. 183 
 
 of white cardboard and a sheet of matt black paper. The screen was 
 so placed that one fold was illuminated by direct sunlight, and the 
 other by the light reflected from the sky. They cut a plate in two 
 and exposed one half in the camera to the screen ; on the other half 
 they made a f-eries of candle exposures. Both halves were developed 
 for the same time, and the densities measured and plotted. They 
 found that the densities of the black and white paper in the image of 
 the screen corresponded to the following candle exposures on the 
 characteristic curve plotted from the other half of the plate : 
 
 White in sun. , '. 22-50 
 
 shade 10-20 
 
 Black in sun . . 1 ' 62 
 
 " shade 0'77 
 
 The direct illumination from the sun was a little over twice that 
 from the sky. 
 
 The same experiment repeated with an orthochromatic plate behind 
 a deep orange screen gave the illumination as twelve times greater in 
 direct sunlight than in the shade. It is at once seen, what an efior- 
 mous difference there must be between a negative taken under these 
 conditions and one taken on an ordinary plate, and how much more 
 intense the shadows are in the first photograph. 
 
 The proportion between direct sunlight and skylight varies with the 
 altitude of the sun and with the absorption of the atmosphere : the 
 less light absorbed, the greater is the contrast. Shadows look more 
 intense when the sun is high than when it is low, but it is in the 
 mountains, at high elevations, that the contrast is greatest. In 
 general, the air is very pure and the coefficient of absorption small ; 
 only rays of very short wave lengths are absorbed and diffused, and 
 the sky assumes a strange deep blue tint. The thickness of the 
 overlying atmosphere being so much less than at sea level, the 
 absorption is still further reduced and the shadows of the landscape 
 become very intense. The effect is exaggerated by the orthochromatic 
 plates with deep yellow screen. 
 
 It is now easy to understand why good photographs of Alpine 
 scenery are so scarce : the wide contrasts present, ranging from snow 
 in sunlight to dark pines in shade, and the intensity of the shadows, 
 combine to make them the most difficult subjects to photograph. 
 Satisfactory results cannot be expected unless the very best plates 
 are used. 
 
 Fortunately, Messrs. Hurter and Driffield have shown how to recog- 
 nize a good plate. There must be silver enough on it to give a long 
 period of correct representation ; any plate, in which the straight part 
 
184 PHOTOGRAPHIC OPERATIONS. 
 
 * 
 
 of the characteristic curve is short, must be at once rejected. The 
 plates now used here are double coated "Edward's isochromatic me- 
 dium," specially made for the purpose, and backed with a non- 
 actinic coating for preventing halation. 
 
 When a subject presenting strong contrasts is photographed and a 
 long exposure given, it is observed that the action of the light appears 
 to spread upon the plate. The edge of the high lights, instead of 
 ending abruptly, merges into the shadow by a gradually decreasing 
 tint more or less extended, according to the intensity of the high light 
 and the length of exposure. This is called halation, and is caused by 
 the light which has passed through the film, struck the posterior sur- 
 face of the glass plate, and been reflected by it to the back of the 
 film. The fog which is seen on over-exposed plates is due, in part, to 
 halation. The remedy is to stop the light when it reaches the back 
 surface of the plate by coating it with some black or non-actinic 
 material that will absorb the light. Any kind of opaque material 
 will not cure halation. Two conditions are requisite : firstly, the 
 coating must be in optical contact with the glass ; a black cloth, for 
 instance, pressed on the plate, would not produce the least effect on 
 halation ; secondly, the refractive index of the coating must be the 
 same as the index of glass ; otherwise there will still be reflection at 
 the surface of the glass. 
 
 The backing is wiped out with a wet sponge before developing. 
 
 Dry plates must be preserved from heat and damp. It is advisable 
 to place the boxes, in which they are received from the manufacturers, 
 into a larger tin box securely locked. They are inserted in the plate 
 holders in weak ruby light, and are then dusted with a flat camel hair 
 brush. Two or three strokes of the brush are sufficient ; more would 
 develop electricity on the film which would attract the dust. The 
 plates are then numbered in pencil in one corner ; it is usual here, 
 to write on the plate the number of the dozen and the number 
 of the holder. After being exposed, the plates are removed from the 
 holders and packed face to face in the original boxes, without anything 
 between the plates. While unpacking and packing the plates, it is 
 convenient to have grooved metal boxes into which the whole contents 
 of a box are placed, and from which they can be handled more easily, 
 both in filling and emptying the holders. 
 
 Films were tried here before glass, but the results were not satisfac- 
 tory. Admitting that films could be made as good as glass plates, and it 
 does not seem that this point has yet been reached, their only advantages 
 over glass would be freedom from breakage and lessened weight ; there 
 is no material difference in the bulk of the holders to be carried by the 
 surveyor, whether he uses glass or films. 
 
EXPOSURE. 185 
 
 On these points, it may be said that small glass plates do not 
 break ; that is the experience on the surveys here and they are being 
 made in as rouglj a country as will be found anywhere. 
 
 The surveyor carries with him, at the utmost, eighteen plates ; if 
 they are half plates (4J x 6J in.) their weight is not a serious consider- 
 ation. The difference in weight between glass and films may become 
 an important matter when the supply for the whole survey is considered, 
 but the conditions must be very exceptional indeed when means can- 
 not be found for the transportation of the weight represented by the 
 plates. 
 
 120. EXPOSURE. The exposure to be given to a plate is inversely 
 proportional to the intensity of the light by which the subject is 
 illuminated. A subject requiring an exposure of ten seconds when the 
 intensity of the light is one, will, all other conditions unchanged, 
 require an exposure of five seconds with a light of intensity two. Be- 
 fore giving any rules for exposure, it is therefore necessary to investi- 
 gate the variations of daylight. 
 
 The light received by an object in direct sunshine consists of three 
 parts : firstly, the direct rays of the sun ; secondly, the light diffused by 
 the sky ; and thirdly, the light reflected or diffused by surrounding 
 objects. In a landscape and for our purposes, the third part need not 
 be considered. 
 
 The light of the sun, after passing through the atmosphere, has lost 
 a portion of its constituent rays, the loss being greatest for the 
 radiations near the violet end of the spectrum and smallest for those 
 near the red end. Taking as a unit, the intensity of a radiation in the 
 light of the sun just outside of the atmosphere, and denoting by a the 
 fraction of this radiation remaining after the light has passed through 
 a thickness of one atmosphere, the intensity, after passing through a 
 mass of air of M atmospheres, is : 
 
 M may be taken as proportional to the secant of the sun's zenith distance 
 up to 65 : for greater distances it becomes : 
 
 0174 x tabular refraction 
 
 cosine apparent altitude 
 
 The coefficient of transmission, a, is subject to great variations accord- 
 ing to the greater or less transparency of the atmosphere : hence, it 
 is quite impossible to predict accurately what the intensity of the 
 
186 PHOTOGRAPHIC OPERATIONS. 
 
 light will be when the sun is at a certain altitude, but when it is not 
 too low, the intensity may be found with sufficient accuracy for photo- 
 graphic purposes. 
 
 Edward's orthochroinatic plates behind a deep orange screen are, it 
 has been shown, sensitive to the rays from D to &, the greatest action 
 
 being about ^560 ; for exposing these plates, it is only the inten- 
 sity of these particular radiations which has to be considered and 
 investigated. An examination of the results obtained by different 
 
 observers shows that for 0' '"'560, the coefficient of transmission through 
 one atmosphere at sea level and under average conditions of the air, is 
 about 0'65 ; at an elevation of 10,000 feet, with dry and pure air, it 
 averages at least 80. These are the two coefficients upon which 
 subsequent calculations are based. 
 
 According to Clausius' theory of the diffusion of light, the intensity 
 of the light reflected and diffused by the sky is expressed by the 
 formula : 
 
 in which S is the light from the sky, Z the ratio of the lost portion of 
 the sun's light to that reflected and diffused by the sky, z the zenith 
 distance of the sun, a and M the coefficient of transmission and air 
 mass respectively. 
 
 The values given by this formula are too large and do not agree 
 with the results obtained on orthochromatic plates. To elucidate this 
 point, series of gradated exposures were made on white blotting 
 paper, in direct sunshine and in the shade, at various altitudes of the 
 sun. The characteristic curves were plotted and, for "each altitude, 
 two inertias were obtained, one for direct sunshine and one for sky- 
 light : the intensities were in the inverse ratio of the inertias. The 
 numbers so obtained were used for calculating the total light received 
 'by a surface exposed normally to the sun's rays, the light coming 
 directly from the sun being computed by the formula already given, 
 
 I=a* 
 
 with the coefficients of transmission 65 for sea level and 80 for 
 10,000 feet altitude. 
 
 The results are represented by Fig. 218 in which the abscissae are 
 the altitudes of the sun. The ordinates are, for convenience, made 
 equal to the reciprocals of the intensity of light, that is to the fraction 
 
 ~j to which the exposures must be proportional. The unit adopted is 
 
EXPOSURE. 
 
 187 
 
 the intensity at sea level when the sun is at the zenith. There are two 
 curves, one corresponding to sea level and the other one to an elevation 
 
 of 10,000 feet; it will be 
 noted, that there is very little 
 change in the exposures re- 
 quired at great elevations 
 until the sun approaches the 
 horizon : this agrees with the 
 experience of our surveyors. 
 The curves of Fig. 218 agree 
 fairly well with the results 
 obtained on orthochromatic 
 plates : there are occasionally 
 large discrepancies, but no 
 more than must be expected. 
 The transparency of the air 
 varies so much that no for- 
 
 15 20 
 
 40 50 60 70 
 
 AL T/TUDC or THE SUN 
 
 Fig-. 218 
 
 mula can express accurately 
 its absorption. 
 
 A number of actinometers 
 have been devised for the use 
 of photographers. The most 
 popular depend upon the 
 action of light on specially prepared bromide paper, the intensity of the 
 light being measured by the time of exposure required to produce a cer- 
 tain tint on the paper. It is hardly necessary to say, that these instru- 
 ments cannot be depended upon for ascertaining the exact exposure to 
 be given to orthochromatic plates behind a deep orange screen. The 
 prepared bromide paper is acted upon by the rays beyond F only, 
 while the orthochromatic plate answers only to the radiations between 
 D and b. For the indications of the actinometer to be accurate, the 
 proportion between the different radiations in daylight should not vary; 
 but that is not the case the proportion is changing constantly. If, the 
 exposure found by the actinometer were correct with the sun high in 
 the sky, it would be excessive with a low sun. The same remark 
 applies to all actinometers : every one indicates the intensity of some 
 particular radiations which are not those acting upon the ortho- 
 chromatic plate behind a deep orange screen, nevertheless, an actino- 
 meter may occasionally prove useful. 
 
 The next step to be taken is to find the inverse of the intensity of 
 light for every hour of the day, and for every day of the year, at the 
 place where the photographs have to be taken. The altitudes of the 
 sun are calculated for the latitude of the place and for one day in 
 
 every month, and the corresponding values of the fraction -= are taken 
 
188 
 
 PHOTOGRAPHIC OPERATIONS. 
 
 from the curves at Fig. 218. The results for 50 of north latitude 
 are plotted in Fig. 219; the abscissae are the months and days of the 
 year, and the ordinates the comparative exposures. The points cor- 
 responding to the same hour of the day are joined by curves ; the full 
 lines are for sea level and the broken lines for an elevation of 10,000 
 feet. 
 
 All this, however, applies only to light coming from a clear sky ; an 
 allowance has to be made when the sun is obscured by clouds. Messrs. 
 Hurter and Driffield, in the instructions for their actinograph, adopt 
 five degrees of brightness : very bright, bright, mean, dull, and very 
 
EXPOSURE. 189 
 
 dull. Very bright light is that coming from a pure sky, and mean is 
 when there is just sufficient sun to cast a very faint shadow. Very 
 dull is the dullest light in which it would be at all reasonable to take 
 a photograph, a definition which, it must be admitted, is somewhat 
 indefinite, but which a little practice will make clear. Bright is between 
 mean and very bright, dull is between mean and very dull. Taking as 
 a unit the exposure with a very bright light, the exposure for bright 
 light is 1^-, for mean light two, for dull light three, and for very dull 
 light four. 
 
 With orthochromatic plates and orange screen, the proportions are 
 as fojplows : 
 
 Light. Exposure. 
 
 Very bright , 1 
 
 Bright 1-5 
 
 Mean 2 
 
 Dull 4 
 
 Very dull 8 
 
 The increase of exposure in dull weather is due to the bluish 
 gray colour of the light coming from a cloudy sky ; the blue rays being 
 stopped by the orange screen, do not reach the plate. 
 
 We must now find the unit of exposure, or the exposure required at 
 sea level, when the sun is at the zenith ; with this unit, the diagram of 
 Fig. 219 gives at once the exposure at any time. Under ordinary 
 circumstances, there would be a different unit for each kind of subject; 
 thus, for views of the sea the unit would be* small, while for landscapes 
 with dark foliage in the foreground, it would be large. In surveying, 
 the character of views and the scale of tints to be reproduced are 
 tolerably uniform, and it will generally be found practicable to adopt 
 one unit for the whole of the work. 
 
 Let us consider Alpine scenery under a bright sun, snow being 
 present in quantities ; the scale of tints to be reproduced extends from 
 snow in sunshine to dark trees in the shade of the valleys. These 
 extreme contrasts being exaggerated by orthochromatic plates with 
 orange screen, it is easy to understand that unless the plate be given 
 all the exposure that it will stand, the shadows will be hopelessly 
 underexposed. It seems, therefore, that the exposure should be 
 proportional to the intensity of the high lights, reversing the old rule 
 of exposing for the shadows. The excessive contrast between snow 
 in sunshine and snow in the shade, may be decreased by making use of 
 the period of over-exposure in such a way, that snow in the shade and 
 the brightest rocks shall be rendered by densities corresponding about 
 to the end of the period of correct representation. An indirect 
 
190 PHOTOGRAPHIC OPERATIONS. 
 
 advantage of this mode of procedure is to increase considerably the 
 range of the plate and to secure better detail in the shadows. 
 
 Bearing these remarks in mind, the unit of exposure is ascertained 
 by trial plates on a landscape of the same kind as those of the survey. 
 The object is to find the greatest exposure which the plate will stand. 
 Let us suppose that on the 20th of September, at 2 p.m., and at sea 
 level, the proper exposure is found to be 34 seconds. The diagram of 
 Fig. 219 gives for this day and hour 1-4 as the coefficient by which the 
 unit of exposure must be multiplied : the unit is, therefore, 34 : 14 or 
 24 seconds. It is advisable to repeat the experiment on several days, 
 so as to obtain a good average unit, because the intensity of daylight 
 is subject to great fluctuations. 
 
 Suppose now, that it. is required to find the exposure at 4 p.m. on 
 the 20th of August. The diagram of Fig. 219 gives, for sea level, 1 65 
 times the unit, or 40 seconds. For an elevation of 10,000 feet, it 
 would be 92 times the unit, or 22 seconds. For any intermediate 
 elevation, the exposure can be obtained by interpolation. 
 
 The surveyor, however, does not make these calculations every time 
 he has to expose a plate : before starting for his day's work, he 
 examines his diagram and notes the exposures for that particular day 
 and for the elevation of the ground. These exposures may, during the 
 course of the day, be increased to suit the light, when the sky is not 
 clear ; a view that is generally dark also requires a longer exposure. 
 
 If desired, a new diagram like Fig. 219 may be plotted with the 
 unit of exposure found, giving directly the exposure instead of the 
 factor by which the unit must be multiplied. 
 
 If, it were attempted to give such prolonged exposures and to 
 handle the plate in accordance with ordinary practice, the negative 
 would be flat and foggy and present all the well known characteristics 
 of over-exposure. It is imperative to pay particular attention to the 
 following points : 
 
 1st. The plate must be backed, or it will be ruined by halation. 
 
 2nd. The inside of the camera must be dead black and provided 
 with one or two diaphragms. The sides of the camera are illuminated 
 very strongly by the light diffused by the plate, and also by some of 
 the rays coming through the lens : if the sides were not dead black, 
 this light would again be reflected on the plate which would be 
 fogged. 
 
 3rd. With the same object of preventing reflections inside of the 
 camera, the diameter of the back lens of the object glass must be 
 small. This condition is realized in Zeiss' anastigmat F/18. 
 
DEVELOPMENT. 
 
 191 
 
 4th. The light, which does not contribute to form the image on the 
 sensitive plate, must be prevented by a hood from entering the 
 camera ; otherwise, some of it would be reflected or diffused on the 
 plate, and cause more or less fog. 
 
 5th. Under no circumstances must the sun shine on the lens ; 
 when the line of sight is somewhere in the direction of the sun, it may 
 be advisable to cut off the sky, if very bright, by closing the upper 
 part of the hood's opening. 
 
 So far, our remarks have been confined to Alpine scenery, but, all 
 views taken for surveying purposes being distant landscapes, the rules 
 given are applicable to all cases. 
 
 It is useful to know the portions of the characteristic curve which 
 are brought into play by the exposures given. For this purpose, a plate 
 is given a series of gradated exposures to a steady light, like that of a 
 lamp which has been burning for some time. The plate is developed, 
 measured, and its densities plotted (Fig. 220). 
 
 2-5 
 
 312 -625 1-25 ' 2-5 
 
 EXPOSURE 
 10 20 
 
 
 -2 -3 -5 -7 
 
 57 10 
 INERTIA 
 
 Fig-. 220 
 
 i i ' r i i i.i - 
 
 20 30 50 70 100 tlOO 300 500 1W iCOO 
 
 In the same dish and at the same time, a landscape exposure is de- 
 veloped : its densities are measured and compared with the character- 
 istic curve. 
 
 121. DEVELOPMENT. The backing having been washed off, the 
 plates are put in grooved trays containing one dozen each, and are 
 
 B 
 
192 PHOTOGRAPHIC OPERATIONS. 
 
 developed with freshly prepared iron oxalate. The formula used 
 here is : 
 
 Oxalate of potash 1 oz. 
 
 Water ...*... \ ......'. 3 ozs. 
 
 Bromide of potassium 15 grs. 
 
 Acetic acid . 10 inins. 
 
 And: 
 
 Sulphate of iron 1 oz. 
 
 Water , 2 ozs. 
 
 Acetic acid . . 2 mins. 
 
 To each ounce of the oxalate solution are added two drams of the 
 iron solution. 
 
 The development is carried out for a definite leng.th of time, so 
 that the greatest densities of the negative be those adapted to the 
 enlarging process. Assuming, for instance, that bromide paper is 
 used, and that the highest negative density required is 1'05, the 
 development must be so timed as to produce this density. In Fig. 
 220, the density corresponding to' the exposure of 40 seconds is 2-35 
 and the development factor 1 -42. To reduce this density to 1 '05, the 
 development factor should be : 
 
 1-05 
 -2^x1-42 = 0-03 
 
 A few trials with plates having received series of exposures show 
 what the length of development must be. If the plates have been 
 properly exposed, the density due to snow is somewhere in the vicinity 
 of the exposures of 20 or 40 seconds in Fig. 220, and with a development 
 factor of ^0*63, the curve of the gradations should be as shown by the 
 dotted line. If snow were absent, the highest densities would probably 
 not exceed 0-90, but the negatives would still produce good enlarge- 
 ments. In a series of views containing no snow, the development 
 should be carried a little further, until, for instance, the 7 '5 seconds 
 exposure in Fig. 220, produces the requisite density of 1'05, the de- 
 velopment factor being 0*75. 
 
 With every batch of plates, the surveyor forwards a few duplicates 
 which are developed separately, for checking both the exposure and the 
 time of development. 
 
 After developing for the requisite time, the developer is emptied out 
 and the tray filled with water several times ; the plates are then taken 
 out and put in another grooved tray containing hyposulphite of soda. 
 When fixed, they are washed and before putting them to dry, the film 
 is wiped with a tuft of cotton. 
 
 R 
 
ENLARGEMENT. 
 
 193 
 
 122. ENLARGEMENT. The plates having been allowed to dry, the 
 highest density of each is measured and written in the margin. Messrs . 
 Hurter and Driffield's photometer is a very perfect and precise instru - 
 ment, but presents several disadvantages. It must be used in a dark 
 room, two powerful lamps have to be set up and attended to, and the 
 deposit of which the density is to be measured must have an area of at 
 least one-quarter of an inch in diameter, a condition seldom found in 
 landscape negatives. It has been found advisable to adopt here, 
 another form of photometer shown in Fig. 221 ; a section is given in 
 
 222 
 
 Fig. 222. Although not by any means an accurate instrument, the 
 densities measured by it are quite precise enough for the calculation of 
 exposures in enlarging. 
 
 It consists of a microscope A, of very low power, mounted over a plat- 
 form C on which rests the negative to be measured. The film being 
 in contact with the platform, its image is formed in the horizontal 
 plane passing through the axis of a large tube B closed at one end by 
 13 
 
194 PHOTOGRAPHIC OPERATIONS. 
 
 an iris diaphragm F. In p is a rectangular prism mounted on a dia- 
 phragm, which closes the other end of the tube B, and immediately in 
 front of the prism is a piece of opal glass E. The upper face of the 
 prism is in the focal plane of the microscope, and its edge divides the 
 field into two equal parts. 
 
 Under the platform C is a slide L carrying a photographic plate, on 
 which five strips of different densities have been impressed by a series 
 of exposures. These densities increase in arithmetical proportion, the 
 highest density being about equal to the density of the opal glass 
 E. Any of these strips may, by drawing the slide, be brought under 
 the hole H, over which the negative to be measured is placed ; by 
 pushing the slide as far as it will go, the negative strips are removed 
 from under the hole H ; there is then nothing under the negative 
 to intercept the light reflected by the mirror M in the direction of the 
 microscope's axis. 
 
 In front of the iris diaphragm, F, is a stage with springs for insert- 
 ing pieces of opal or coloured glass. 
 
 The object glass of the miscroscope is formed of two common opera 
 glass objectives of three-inch focal length. The eyepiece consists of 
 two piano convex lenses of two and three-quarter inch focus, placed 
 one and half inch apart. The magnifying power is about four times. 
 
 The platform C is lined with black velvet ; the whole instrument is 
 mounted on a simple wooden frame. 
 
 Coloured glasses of different tints may be adjusted to the end of the 
 eyepiece. 
 
 Turning the instrument towards a uniform source of illumination, 
 like the sky or a sheet of white paper, and looking through the eye- 
 piece, the field is seen divided into two parts differently illuminated \ 
 one of the halves receiving its light from the tube B through the opal 
 glass E and by reflection on the hypothenuse face of the prism p t the 
 other half receiving it by reflection on the mirror M through the 
 object glass of the microscope. The illumination of each of the two 
 halves of the field can be varied at will, the one belonging to the tube 
 B by opening or closing the iris diaphragm, and the one belonging to 
 the microscope by moving the slide L. A negative being placed over 
 the hole If, equality of illumination can be obtained by a judicious 
 use of the iris diaphragm and of the slide. 
 
 The principle of this instrument is, that the illumination of the opal 
 glass E is proportional to the area of the opening in the iris diaphragm. 
 Without the opal glass, the opening or closing of the diaphragm does 
 not cause any change in the illumination of the prism, so long as the 
 
ENLARGEMENT. 195 
 
 section of the bundle of rays coming out from the eyepiece is larger 
 than the pupil. When a diffusing screen, like opal glass, is inserted, 
 the light reaching the eye is composed of two parts : the light diffused 
 by the screen, which is proportional to its illumination or to the open- 
 ing of the diaphragm, and the light transmitted directly which is not 
 affected by the opening of the diaphragm. With opal glass of density 
 2, this last part of the light is not appreciable and the illumination 
 varies, within the limits of accuracy of the instrument, proportionally 
 to the aperture of the diaphragm. 
 
 Let us assume, that the density D of the darkest strip of the slide 
 L is such that, by placing it under the hole ff, the illumination / of 
 the two halves of the field is equal when the diaphragm is open to its 
 full diameter a. Now, place a negative over the hole ff, and close the 
 diaphragm until equality of illumination is restored : let a be its 
 diameter and /' the illumination; we have : 
 
 L ?LL 
 I' ~ a' 2 
 
 But ~jr is the opacity of the negative, and, as the density is the log- 
 arithm of the opacity, 
 
 A = 2(log a - log a') 
 
 A being the density of the negative. The graduation of the dia- 
 phragm is calculated by this formula, the full opening being marked 
 zero. The divisions of the scale are given hereunder : 
 
 Division of Diameter of dia- 
 
 the scale. phragm's aperture. 
 
 o-o i-oo 
 
 1 -89 
 
 2 79 
 
 3 -71 
 
 4 -63 
 
 5 -56 
 
 6 -50 
 
 7 -45 
 
 8 -40 
 
 9 -35 
 
 1-0 -32 
 
 We are now able to measure any density greater than D and less 
 than D + 1, D being the density which just secures equality of illumi- 
 nation when the diaphragm is fully open or when the index is at zero 
 13J 
 
196 PHOTOGRAPHIC OPERATIONS. 
 
 of the scale. The negative is placed over the ho'e H, the density 
 strips are removed from under it by pushing the slide, and the dia- 
 phragm is closed until the two halves of the field are equally bright. 
 Let a be the division of the scale opposite the index : the density is 
 
 When the density of the negative to be measured is less than Z>, 
 one of the strips of the slide L is drawn under the hole ff, so as to 
 form with the negative a density greater than D. The diaphragm is 
 again closed until the two halves of the field are equally illuminated. 
 Let a be the corresponding division of the diaphragm's scale, and d the 
 density of the slide's strip ; the total density measured is A + d, and 
 we have : 
 
 or: 
 
 We mark on the slide the value of D-d for each of the five 
 strips, and the instrument is ready for measuring any 'negative. The 
 diaphragm is first fully opened, and the negative placed over the hole H. 
 Looking through the eyepiece, the slide is drawn until the negative 
 appears darker than the prism, and the diaphragm is closed until 
 equality of illumination is obtained. The density is the sum of the 
 numbers read on the slide and on the diaphragm's scale. 
 
 Generally, there is a difference in the colour of the two halves of the 
 field ; to adjust accurately the illumination, it is necessary to place on 
 the eyepiece a glass of the complementary colour. 
 
 The values of D - d to be inscribed on the slide, may be ascertained 
 by using a negative of which the various densities have previously been 
 measured in a Hurter and Driffield's photometer. They may also be 
 found by measuring the densities of the slide, as follows : Take a 
 negative of density slightly greater than Z>, and place it over the hole 
 H ; push in the slide, and close the diaphragm until the illumination 
 of the field is uniform. Let a be the reading of the diaphragm's scale. 
 Then draw under the negative the first or lightest strip of the slide 
 and let B be the reading of the diaphragm's scale. The density of the 
 first strip is 
 
 d = .B-a 
 
 The difference between this strip and the next one is found in the same 
 way, and the operation is repeated until the density d' of the last strip 
 
ENLARGEMENT. 197 
 
 is obtained. Let y be the reading of the diaphragm's scale when 
 equality of illumination is produced with the last strip d', then : 
 
 from which D is readily obtained. 
 
 d' must be equal to or greater than D if greater, the number cor- 
 responding to it, to be marked on the slide, is negative. 
 
 For very great densities, a screen reducing the light is inserted in 
 front of the iris diaphragm ; this, however, is seldom necessary. 
 
 In measuring a landscape negative, it is advisable to cover it with a 
 sheet of black paper in which a small hole has been made. The nega- 
 tive is shifted on the platform until the spot to be measured is bisected 
 by the edge of the prism ; the black paper is then adjusted over the 
 negative so as to cover everything except this spot. 
 
 A little discrimination has to be exercised in selecting the spot of 
 highest density in a negative. We must bear in mind that all lesser 
 densities will be represented in the print by a deposit. It may occur 
 that one particular spot has a density much in excess of the other parts 
 of the negative, in which case it is not correct to regulate the exposure 
 by it. The density to be measured is the most transparent spot of 
 the negative which it is desired to have represented by pure white in 
 the print. 
 
 The measurement of densities, with the photometer just described, is 
 a very rapid operation. 
 
 Before developing the plates, it is necessary to ascertain the range 
 of gradation or the highest density required for enlargement. Messrs. 
 Hurter and Driffield have shown how this is done for enlargements on 
 transparency plates or bromide paper ; their instructions should be 
 carefully adhered to. But some simplifications may be introduced in the 
 details of the operations ; they are illustrated by a description of the 
 method followed here for enlarging on bromide paper. 
 
 The range of the paper is ascertained, as recommended by Mr. 
 Driffield, by a series of exposures progressing in geometrical ratio. If 
 the revolving disc is not available, the gradated exposures may be given 
 by covering successively different portions of the paper. The series 
 may be commenced, for instance, by exposing for three seconds the 
 whole of the paper strip, except a margin on which the white of the 
 paper is to be preserved. Then, a small portion of the strip is covered 
 and a further exposure of one second given ; a little more of the strip 
 is covered and another exposure of two seconds given, and so on. The 
 
198 PHO TOGRA PHIO OPERA TIONS. 
 
 first portion has received an exposure of three seconds, the second 
 portion four seconds and the third portion six seconds. The separate 
 exposure may be : 
 
 3*_1_.2 3 4 6 9 14 21 31 47 70, 
 the total corresponding exposures are : 
 
 3*_4_ 6 9 13 19 28 42 63 94 141 211. 
 
 That is very nearly a geometrical progression. The exposures need not 
 be made to the light of a candle ; indeed, it is preferable to use a lamp 
 which has been burning for some time, the light being more steady. 
 The distance of the lamp is adjusted so that the first exposures shall 
 not produce any appreciable deposit on development. 
 
 A strip of bromide paper was exposed as described above : 
 after full development, it was found that the first well-marked 
 change in the whiteness of the paper corresponded to the exposure of 
 4 seconds, while after the exposure of 141 seconds, there was no per- 
 ceptible increase in .the intensity of the black. The range of the 
 
 141 
 
 paper was, therefore, - or 35. If the enlarging factor of the 
 4 
 
 negatives were 1 4, their highest density, exclusive of fog, should be : 
 
 1-4 
 
 To find the exact value of this highest density, a plate is given 
 four or five exposures calculated to give densities, exclusive of fog, 
 little different from the highest density 1 10, already found. In the 
 example given, the densities may be 1 06, 1 08, 1 10, 1 12, 1 14. A 
 few trials may have to be made before the correct densities are 
 obtained. 
 
 This plate, which we will call the standard tint plate, is inserted in 
 the slide of the enlarging lantern, the image is projected on a piece of 
 bromide paper and a series of exposures is given by covering succes- 
 sively the paper across the images of the density strips. The mean 
 exposure must be timed to just produce a faint deposit on the 
 developed paper for the image of the middle density strip. Assuming 
 this mean exposure to be 45 seconds, the series should be : 
 
 39 s , 42 s , 45 s , 48 s , 52 s . 
 
ENLARGEMENT. 
 
 199 
 
 The resulting image on the paper consists of rectangular figures of 
 different tints, as shown by Fig. 223. The darkest tints are 
 DMS/T/ES corresponding to the 
 
 0-00 
 
 1-00 1-08 
 
 :;; ""I 1 
 
 HO 
 
 1-12 
 
 1-14 
 
 glass and 
 standard 
 
 those 
 clear 
 
 gelatine in the 
 plate ; if the ex- 
 posures have been properly 
 timed, one at least of the tints 
 in this strip must be the 
 darkest black of the paper. 
 In the figure, this tint has 
 been produced by the ex- 
 posures of 48 and 52 seconds; 
 48 seconds is, therefore, the 
 exposure for which the first 
 appreciable tint should be 
 produced through the highest 
 density of the negative. An 
 examination of the strip which 
 has received the exposure of 
 
 Fig". 223 4 s ec onds shows, that this 
 
 first tint was produced through the density 1 -08 ; we must, therefore, 
 endeavour to so develop our negatives that the highest density of each, 
 exclusive of fog, shall be as nearly as possible 1 '08, because they will 
 then correspond precisely to the range of the paper, an exposure of 
 48 seconds just producing black for the darkest shadows and the lightest 
 tint of the paper for the high lights. 
 
 To find the exposures for negatives 
 of different densities, take a logarith- 
 mic slide and glue a piece of paper 
 over the upper scale M (Fig. 224). 
 On the edge of the paper, measure a 
 . distance AB equal to the logarithm of 
 
 the range of the bromide paper, 35 in 
 
 the example : this distance AB represents the density found by the 
 experiment with the standard tint plate, 1 '08 in our case, and is to be 
 divided into equal parts, ttye point B being at 1 *08 of the scale. Slide 
 this scale until B coincides with number 48 of the logarithmic scale, 
 which number is the exposure found with the standard tint plate ; the 
 slide is now set to indicate the exposure for a negative of any density, 
 this exposure being the number of the logarithmic scale which coincides 
 with the density of the negative on the upper scale. For instance, a 
 negative of density 0'90 requires an exposure of 27 seconds, and one of 
 density 1-20 requires 71 seconds. 
 
 The explanation is as follows : Let E and D be the exposure and 
 density found with the standard tint plate, and r the range of the 
 
 All! 
 
 
 > 1 
 
 i i i I , 
 
 O 
 
 BJI . 
 
 "1 ! 1 
 1 J 
 
 i 
 
 II 1 1 I 1 I I ii 
 10 50 1< 
 
200 PHOTOGRAPHIC OPERATIONS. 
 
 bromide paper; since the exposure E has produced through the 
 density D the same effect on the bromide paper as an exposure r 
 times less through the clear glass and gelatine, the enlarging opacity 
 is r and the enlarging value of the density D is log r. Denoting by a 
 the enlarging factor, 
 
 aD = log r. 
 
 By construction, the density scale of the logarithmic slide is a scale 
 of the values of o.D. 
 
 The exposure e necessary to produce on the bromide paper the 
 
 lightest tint is , hence : 
 r 
 
 log e log E - log r 
 
 e is the number of the lower scale of the logarithmic slide which coin- 
 cides with zero of the density scale. 
 
 In order that another exposure E' through a density D' shall pro- 
 duce the same tint on the bromide paper, we must have : 
 
 log E' = log e + aD'. 
 
 This value of E' is, it will readily be seen, given by the lower scale of 
 the logarithmic slide opposite the division D' of the upper scale. 
 
 \ The standard tint plate is made on a plate of the same batch of 
 emulsion as those used on the survey, and is developed in the same 
 manner. 
 
 has been assumed that the fog densities of this plate and of all 
 the negatives were equal. With plates of one emulsion, exposed and 
 treated as they are here, this assumption is practically correct ; with 
 our plates the fog density never differs much from O15. As a matter 
 of fact, this fog is never measured, all densities employed including 
 fog : the only change which this necessitates, in the method outlined 
 above, is a shifting of the zero of the density scale of the logarithmic 
 slide. (Fig. 224.) The division formerly marked zero is now marked 
 0-15, the fog density, and the division 1*08 is marked 1*23 ; or, more 
 simply, the density scale is moved O15 to the left. 
 
 In deciding which is the lightest tint indicating the commence- 
 ment of the range of bromide paper, it is well to remember Mr. 
 Driffield's advice and to disregard the faintest deposits. The change 
 in the tints is so slow in this portion of the paper range that, if it 
 were used, the different shades of the high lights in the negative 
 would not bs adequately rendered. 
 
ENLARGEMENT. 201 
 
 Although these operations may appear complicated, they are in reality 
 very simple. Once the standard tint plate and the logarithmic slide have 
 been prepared, they serve for the whole of one season's work. The 
 only thing that remains to be done is to adjust the logarithmic slide 
 whenever enlargements are to be made. For that purpose, a series of 
 exposures is made with the standard tint plate, as already explained, 
 but this time, the only portion of the print examined is the strip cor- 
 responding to the exposure through the clear glass and gelatine. The 
 density scale M (Fig. 224) of the logarithmic slide is then shifted, until 
 the point B coincides with the number of the other scale expressing 
 the exposure which has first produced black on the print. The paper 
 does not require to be fixed, and the whole operation is complete in three 
 or four minutes. The enlargements can then be proceeded with. 
 
 In making this experiment, it will be observed that while the light 
 tints produced through the density strips merge imperceptibly into the 
 white of the paper, the gradations cease abruptly when black is attained; 
 this method of setting the logarithmic scale is therefore very precise. 
 From this peculiarity of bromide paper, two conclusions may be drawn : 
 firstly, that it may be accurately timed, and secondly, that it must be 
 accurately timed. Unlike dry plates, there is no latitude of exposure; 
 the negative and the exposure must both be adapted to the paper. If, 
 for some reason, the instructions given above cannot be followed, 
 enlargements on bromide paper should not be attempted ; they would 
 result in failure. In such a case, transparency plates should be 
 resorted to, because some use may be made of an imperfect trans- 
 parency, while an imperfect bromide print is useless.* 
 
 The prints are developed with iron oxalate, washed in acidulated 
 water and fixed. After a thorough washing, they are dried flat and in 
 such a way that the expansion or contraction of the paper is equal in 
 all directions. For this purpose, the prints may be soaked in alcohol 
 before drying, or the edges of the paper may be held by pins or weights, 
 where the contraction is too great. 
 
 Enlarging may be done either by daylight or by artificial light. 
 Considering that the enlargements are all of the same size, the day- 
 light apparatus may be extremely simple, but, it is essential that it 
 should be made with precision : otherwise, the prints will be distorted. 
 The plane of the negative and the plane of the paper must be parallel; 
 the lens must be rectilinear. The apparatus should be inclined at an 
 angle of 30 or 35, and point to the northern part of the sky. The 
 
 *We are now using a kind of bromide paper recently introduced, and called 
 " Platino Bromide." It has a much longer range than ordinary bromide paper (about 
 150), and requires a density of 1'80 in the negatives. The black tones are matt, and 
 giv<- to the prints a more artistic appearance. A special kind made on paper of the 
 thickness of a thin visiting card, is well adapted to survey photographs. 
 
202 PHO TOGRA PHIC OPERA TIONS. 
 
 disadvantage of daylight is its inconstancy ; when employed, its use 
 should be restricted to days when the sky is quite clear, and then 
 only to the middle of the day, when .there is little change in the 
 intensity of light. 
 
 Artificial illumination is more convenient and always available, but 
 requires more elaborate apparatus. For enlargements of moderate 
 amplitude, like those needed for surveying, the best source of light is 
 probably a spiral electric lamp,* provided the pressure on the mains 
 is uniform. While in use, other lights which may happen to be on 
 the same circuit should not be turned off or on. It is important to so 
 adjust the lamp, that the image of the spiral formed by the condenser 
 shall be exactly in the centre of the diaphragm of the enlarging lens. 
 The condenser not being achromatized, the image is strongly coloured 
 on the edges. If focussed for the actinic rays, the fringe of colour 
 must be red ; when it is blue, the lamp is too far from the condenser. 
 
 The insertion of diaphragms does not reduce the illumination, so 
 long as the aperture is larger than the image of the source of light ; a 
 lens of moderate aperture may therefore be employed with a spiral 
 electric lamp. With other sources of light, the useful portion is that 
 of which the image is inside of the aperture of the diaphragm ; the 
 other parts might as well be covered and, in fact, should be covered 
 on the general principle of photography that any light which is not use- 
 ful is prejudicial. 
 
 Slight distortions are caused by the play of the negative carrier in 
 the lantern and by the bromide paper not lying quite flat on the copy- 
 ing board. To minimize this cause of error, it is essential to employ 
 a lens of long focus. 
 
 There does not appear to be any special advantage to enlarge to 
 one size rather than another. The proportion adopted here (about 
 2-1) was simply calculated to fill in the width of the bromide paper. 
 Once fixed, however, the proportion should not be changed. 
 
 *This lamp is made by the Edison and Swan United Electric Light Co., Ltd. 
 They call it "Focus Lamp." 
 
TRIANGULATION. 
 
 CHAPTER VII. 
 
 FIELD WORK. 
 
 123. TRIANGULATION. The triangulation may be executed at the 
 same time as the topographical survey, but it is preferable to have 
 some of the principal points located in advance by a primary 
 triangulation. 
 
 The subject is fully treated in the standard works on surveying ; 
 very little requires to be added here. There exists, however, some 
 misconception as to the order to be followed in the operations : a few 
 words of explanation may prove useful. 
 
 A survey must be considered as consisting of two distinct opera- 
 tions ; one has for its object the representation of the shape or form of 
 the ground, the other the determination of its absolute dimensions. 
 A perfect plan or triangulation can be made without the measure of 
 any base or length ; the plan will exhibit the various features of the 
 ground in their exact proportions, but no absolute dimension can be 
 measured on it until the scale of the plan has been determined. This 
 is done by measuring on the ground one of the dimensions represented 
 on the plan : so, the object of the measure of a base is to fix the scale 
 of the survey and nothing more. 
 
 To execute a triangulation, the surveyor is recommended to com- 
 mence by measuring a base and making it the side of a triangle, on 
 which he is to build other triangles of increasing dimensions. There is a 
 certain logical sequence in the order followed, but in strict theory, the 
 order is immaterial, the triangulation may be executed first and when 
 completed, connected with a base by triangles decreasing in size as 
 they come near the base. 
 
 In practice, the ca^e is different : there are several advantages in 
 executing the triangulation first. 
 
 The selection of a base is governed by various considerations : the 
 ground must be tolerably level and free of obstacles, and the direction, 
 length, and position of the base must be such as to permit a good con- 
 nection, by triangles of proper shape, with the main triangulation. 
 The surveyor can make a better choice after he has been over the 
 
204 FIELD WORK. 
 
 whole ground, than on his arrival when he has seen little of it. 
 Having established the main triangles, he also knows best how to 
 connect them with a base. In a mountainous country, the principal 
 summits of the triangulation are fixed by nature and cannot be 
 changed, while the position or direction of the base may generally be 
 modified to some extent. Were the base measured first, it might be 
 found not to connect properly with the main triangles. 
 
 The secondary triangulation is the work of the topographer, and the 
 construction of signals on the secondary points should be his first act 
 upon arriving on the ground. 
 
 Should the time at his disposal allow, he will not commence the 
 survey proper until all signals have been established, otherwise he may 
 have to measure angles between points not very well defined. When 
 he does so, the closing error of a triangle is assumed to be due to the 
 want of definition of the points. 
 
 Let A, B and C represent the angles of a triangle whose summits have 
 been occupied in the order given. At A, the surveyor observes the 
 angle between B and (7, where there are no signals. t He puts up a 
 signal at A and moves to B. In measuring the angle between A and 
 (7, he has a signal at A and none at (7. Placing a signal at J3, he 
 measures the third angle C between two signals. 
 
 Call a the closing error of the triangle and e the probable error of a 
 sight on a point without signal. The probable errors of the angles 
 are: 
 
 For A, ej/2~ 
 
 " C, 0. 
 
 The corrections to the angles must be proportional to the probable 
 error of each ; they are : 
 
 For 
 
 B, 
 
 1 + 1/2 
 " <7, 0. 
 
 The closing error must not exceed a certain limit fixed by the degree 
 of precision of the survey : when the limit is exceeded, the stations 
 must be re-occupied, commencing at the most doubtful one. 
 
CAMERA STATIONS. 205 
 
 The stations of the primary triangulation are the last ones to be 
 occupied when they have been established by a previous survey. 
 
 To have a correct idea of the work he is doing, the surveyor must 
 make in the field a rough plot of his triangulation, on which he marks 
 all the stations occupied. It shows him the weak points of the survey 
 and enables him to plan his operations with more assurance. 
 
 The object of the secondary triangulation is to fix the camera sta- 
 tions : its summits are selected for that purpose only. All the topo- 
 graphical details of the plan are drawn from the camera stations. 
 
 124. CAMERA STATIONS. A camera station is fixed either by angles 
 taken from the station on the triangulation points, or by angles taken 
 from the latter, or by both. It is easier and more accurate to plot 
 a station by means of angles taken from the triangulation points than 
 by the angles measured at the station ; therefore, the camera stations 
 should, if possible, be occupied before the triangulation summits. 
 There are, however, other considerations which may interfere. 
 
 Camera stations must be chosen in view of the construction of the 
 plan by the method of intersections : other methods are to be employed 
 only, when this one fails or when the data collected on the ground are 
 not sufficient to furnish enough intersections. 
 
 A conspicuous mark or signal of some kind should be left at each 
 station ; it does not require to be very elaborate, a pole or a few stones 
 would be sufficient. Angles on this signal are measured from the tri- 
 angulation points, in order to place the station on the plan. 
 
 It seldom occurs that the camera is set up precisely at a triangula- 
 tion point. Generally, it is advisable to move a few feet in one direc- 
 tion or another, to include in the view a certain part of the land- 
 scape. Whenever there is any advantage in displacing the camera, the 
 surveyor should not hesitate to do so. The distance from the triangu- 
 lation point measured with a light tape and an angle read on the 
 instrument, locate the camera station. 
 
 For the same reason, it is not necessary that several views be taken 
 from each station : every view should be taken from the point where 
 it is best for the construction of the plan. The greater number of 
 stations gives very little extra work, either in taking the angles for 
 fixing their positions, or in plotting them. 
 
 In general, views taken from a great altitude and overlooking the 
 country are desirable, but there are numerous exceptions. Those taken 
 from low altitudes are of great assistance in drawing the contour lines 
 of the valleys. 
 
206 FIELD WORK. 
 
 Sometimes difficulties may exist in obtaining two views which will 
 furnish intersections over a certain part of the ground. In such a case, 
 the method of vertical intersections may be employed, views being 
 taken from different altitudes. Provided the difference of altitude is 
 large enough and the points to be determined not too far, the precision 
 is the same as with horizontal intersections. 
 
 It would be desirable to have the views of the same part of the 
 ground taken at the same time of day ; the shadows cast being identical, 
 it is more easy to recognize the different points. It would be well, 
 also, to avoid views taken looking towards the sun ; they are flat and 
 lack detail. But, the surveyor has other considerations to take into 
 account ; he is seldom able to choose his own time for occupying a sta- 
 tion or taking a photograph. He has often to take views against the 
 sun or dispense with them altogether. With care in cutting off the 
 sky, he may still obtain results remarkably good under the circum- 
 stances. 
 
 The identification of points, even under different lighting, does not 
 offer any serious difficulties. The number of photographs must be 
 sufficiently large to cover the ground completely : an additional view 
 causes very little extra work, either in printing or plotting, and may 
 save much trouble. The surveyor should not hesitate to take one 
 whenever he finds a place where it may be useful. 
 
 Two or three points in each view are observed with the altazimuth, 
 the altitudes and horizontal angles between them being noted. The 
 altitudes serve to check the horizon line on the photograph, in case the 
 camera should be slightly out of level, and the horizontal angles are 
 for the orientation of the view. 
 
 The notes of observations are kept in the usual manner for such 
 work, the points observed upon being shown on sketches made on the 
 spot. The sketches serve to identify points with more certainty than 
 a mere designation by a letter or figure. 
 
SCALE. 207 
 
 CHAPTER VIII. 
 
 PLOTTING THE SURVEY. 
 
 125. SCALE OF PLAN. The minutes of the Canadian Surveys are 
 plotted on a scale of ^-^J^y > they are afterwards reduced for publication 
 The equidistance is ] 00 feet. 
 
 The convention already adopted in perspective ( 54) must be recalled 
 here ; the angles measured and the photographs taken are conceived to 
 have been measured and taken on a model of the ground already 
 reduced to scale. 
 
 That the perspectives obtained from any point of such a model are 
 identical with those taken from the similar point of the ground has 
 already been shown ( 54) ; the same rule applies to photographs, in 
 theory at least. 
 
 The angles measured are also equal to those of the ground, for any 
 triangle ABC (Fig. 225), of the ground is represented on the model by 
 a similar triangle abc. The altazimuth set in a 
 gives between b and c the same angle as it would 
 between B and (7, if set at A. 
 
 Thus, if the plan be required on a scale of 2rnj(nr> tne 
 model is assumed to have been reduced to that scale, 
 and the problem consists in making a plan full size 
 by means of angles and photographs obtained on the 
 model. 
 
 No change being made to the camera, the focal 
 length preserves the same value ; if one foot, it covers 
 Fig".) 225 on the model a distance corresponding to twenty 
 
 thousand feet on the ground. 
 
 The plan and the model being both re Juced to the scale of 20 ^ 00 , it 
 is clear that, if this scale be uued to measure an actual dimension on 
 either, the result is the number expressing the corresponding actual 
 dimension on the ground. If a division of the scale be called a " scale 
 foot," a dimension of the ground is expressed in real feet by the same 
 number which expresses in " scale feet " the corresponding dimension 
 
208 PLOTTING THE SURVEY. 
 
 of the model or plan. A distance of a mile contains 5280 real feet on 
 the ground, and is represented on the model by 5280 " scale feei." 
 
 The focal length of one foot mentioned above would be a focal length 
 of 20,000 " scale feet." 
 
 It follows that, although the problem consists in representing a model 
 full size, the scale may be employed to measure the actual dimensions, 
 the value of one division being considered as an arbitrary unit. 
 
 In other words,aLiliputian surveyor must be imagined to be operating 
 in a Liliputian country of which he wants to make a plan full size. His 
 camera is of enormous dimensions, bearing to him the same proportion 
 as a camera several miles long would to an ordinary man. 
 
 Of course, all the constructions used in plotting the plan can be 
 demonstrated without such an hypothesis, but the explanations would 
 not be so simple, and it would not be so easy to grasp the whole 
 subject. 
 
 126. PLOTTING THE TRIANGULATION. The primary triangulation is 
 assumed to have been previously calculated ; the primary stations can 
 therefore be plotted at once by their co-ordinates. 
 
 The angles of the secondary triangles are now calculated, and the 
 corrections indicated by the closing errors applied. Some of these 
 triangles have common sides with the primary triangulation : they 
 are calculated first. With the values found for their sides, the adjoin- 
 ing triangles are calculated, and so on, until the lengths of all sides 
 have been obtained. 
 
 With these values, the differences of latitude and departure from 
 every summit of the secondary triangulation to the nearest primary 
 station are calculated. Unless the primary triangles be very large, 
 the secondary stations can be plotted on the plan by their latitudes 
 and departures without any appreciable error. 
 
 The camera stations are next placed by the angles observed upon 
 them from the triangulation points. These angles are plotted with a 
 vernier protractor, or by means of a table of chords ; either method is 
 accurate enough for the purpose. 
 
 As long as a sufficient number of readings have been taken on a 
 camera station from triangulation points, no difficulty is experienced 
 in placing the station ; it is not so when only a limited number of 
 readings, or none at all, are available. There are two cases to consider. 
 
CHECKING THE PHOTOGRAPHS. 209 
 
 Case I. The camera station has been observed from one or more 
 triangulation points. The camera station M (Fig. 
 226) having been observed from the triangulation 
 point A t triangles may be formed with M, A and 
 >j/ other triangulation points observed both from A 
 and M, such as B. In the triangle MAB the 
 angles at M and A have been observed, and 
 
 Similar calculations being made for other triangu- 
 lation points give the directions of the station as seen from these 
 points ; the plotting is done as if the station had been observed from 
 every such point. 
 
 Case II. The camera station has not been observed from any tri- 
 angulation point. In this case the station must be placed by the 
 angles which have been observed from it. This can be done either 
 by describing, through the points observed, circles containing the 
 angles between them, or by the use of a station pointer. The first 
 method requires complicated constructions and is not very accurate, 
 and the station pointer can serve only for three points at a time. 
 The following process will be found rapid and accurate when many 
 points have been observed from the station : 
 
 On a piece of tracing paper, take a point to represent the camera 
 station, and draw the directions of all the points observed. Put the 
 tracing paper upon the plan and try to bring every one of the direc- 
 tions drawn to pass through the corresponding point of the plan. 
 The camera station is then in its place. 
 
 From the foregoing it is evident that the surveyor should endeavour 
 to obtain at least one direction from a triangulation point on every 
 camera station: the plotting is less laborious and the result more 
 accurate. 
 
 The use of photographs for placing camera stations must be avoided; 
 the precision is not sufficient. 
 
 127. CHECKING THE PHOTOGRAPHS. Before making any use of the 
 photographs, it must be ascertained that they have not been distorted 
 in the operation of enlarging. Join the middle notches HH\ PP', 
 Fig. 227, and with a set square test these two lines for perpendicularity. 
 Take with a pair of compasses the distance of the two notches A and 
 , which is one-half of the enlarged focal length, and see whether it is 
 equal to the distance of the two notches C and D. Then apply one 
 of the points of the compasses in P ; the other point must come in 
 14 
 
210 
 
 PLOTTING THE SURVEY. 
 
 E and F. Transfer the point to F and check FG and FJ. If the 
 photograph stands all these tests, it may be depended upon as accurate ; 
 
 D< 
 
 fj ^ 
 
 A A 
 
 \ A 
 
 
 
 C< 
 
 
 
 V V, V 
 6 P J 
 
 Fig-. 227 
 
 if it does not, it is returned to the photographer with a request for 
 a better one. 
 
 128. PLOTTING THE TRACES OF THE PICTURE AND PRINCIPAL PLANES. 
 The traces of the picture and principal planes are now drawn on the 
 plan. Every photograph contains at least one and generally several 
 of the points marked on the plan. Find the distance Sa, Fig. 228, 
 
 from the station to the pro- 
 jection of such a point a of 
 the photograph on the horizon 
 line ; PS is taken on the 
 principal line equal to the 
 focal length and Pa equal to 
 ad. The whole of this con- 
 struction is made on the 
 " photograph board " which 
 will be mentioned further on. 
 
 On the line S 1 A of the 
 Fig-. 228 . plan representing the direc- 
 
 tion of a, take from the 
 
 station S l the distance S 1 a 1 equal to Sa ; from a x as centre with aa' 
 as radius, describe an arc of circle and draw S-^p tangent to it : it is 
 the trace of the principal plane. The trace of the picture plane is the 
 perpendicular to S^ passing through a^. 
 
IDENTIFICATION OF POINTS. 211 
 
 Instead of making the construction on the photograph board, it can 
 be made on the plan. On S^A take S^B (Fig. 229), equal to the 
 focal length, erect EG perpendicular to S^A and 
 equal to aa (Fig. 228). Join S 1 C and take S^ equal 
 to the focal length : at p erect a perpendicular to 
 $! C ; it is the trace of the picture plane, and S l C is 
 the trace of the principal plane. 
 
 The first method is preferable, because it does not 
 require so many construction lines on the plan. 
 
 O Q The trace of the principal plane is marked only 
 r ig". Z-M where it intersects the picture trace so as not to con- 
 fuse the plan. 
 
 129. IDENTIFICATION OF POINTS. The survey being plotted mainly 
 by intersections, it is necessary, after selecting the points on one pho- 
 tograph, to identify these same points on another photograph covering 
 the same ground. The points are chosen on those lines which best 
 define the surface, such as ridges, ravines, streams, crests, changes of 
 slope, etc. Each point is marked on the photographs by a dot and a 
 number in red ink. 
 
 Ability to identify points is acquired by practice ; it is surprising 
 to see with what rapidity and certainty a surveyor familiar with the 
 work can not only pick out the points on several photographs, but 
 also find as many as he wants. Beginners may, in case of difficulty, 
 resort to Prof. Hauck's construction ( 86). The two photographs are 
 pinned, side by side, on a drawing board. The images of the stations, 
 if they appear on the photographs, are the kern points ; if outside of 
 the pictures, they are plotted from the plan on the drawing board. 
 The parallels to the principal lines, on which the scales are to be 
 placed, are drawn as explained in 86, and the scales fixed in position. 
 A fine needle is fixed at each of tho_kern points ; to it is tied a fine 
 silk thread, the other end of which is/ fastened to a light paper weight 
 by a fine rubber band. A well defined point is identified on the 
 two photographs ; its image must be far enough from the kern 
 points. Taking one of the paper weights in one hand, the silk 
 thread is given enough tension to keep it straight ; it is displaced to 
 pass through the point which has just been identified and the weight 
 is deposited on the drawing board, still keeping the tension. This 
 operation is repeated with the other silk thread and the other photo- 
 graph. The two threads should intersect the scales at the same 
 division ; if they do not, one of the scales is moved until the divis- 
 ions intersected are identical. The identification of points may now 
 be proceeded with. Having selected a point on one of the photo- 
 graphs, the silk thread is displaced to pass through the point and the 
 14 
 
212 
 
 PLOTTING THE SURVEY. 
 
 intersection of the scale by the thread is noted. The other thread is 
 now moved to intersect the same division on the other scale; the 
 point looked for is under the thread. 
 
 130. PLOTTING THE INTERSECTIONS. The surveyor marks on theedga 
 of a band of paper the distance from each point of one of the photo- 
 graphs to the principal line, and adju>ts the paper on the trace of the 
 picture plane previously drawn on the plan, holding it by paper weights : 
 he repeats the same operation for the other photograph. Inserting a 
 fine needle at each station, he fastens to it a black silk thread con- 
 nected at the other end by a fine rubber band to a small paper weight. 
 Holding the weight in one hand, he moves the thread until it coincides 
 with one of the marks on the edge of the band of paper and he deposits 
 the weight on the plan, giving sufficient tension to the rubber to keep 
 the thread taut. Doing the same thing at the other station, the inter- 
 section of the two threads indicates the position on the plan of the 
 point of the photographs. 
 
 When the bands of paper overlap, as in Fig. 230, 
 the portion CD of the picture trace PQ is marked 
 on the band MN which is under ; the band PQ is 
 placed in proper position and the marks on its edge 
 transferred to the line CD. The band PQ is now 
 placed under MN, the marks on the latter along 
 CD serving the same purpose as those of PQ. 
 
 The station may be too close to the edge of the 
 plan to plot the trace of the picture plane, as 
 in A, Fig. 231, the picture trace falling in QR, out- 
 side of the plan. 
 
 M In this case the trace AC of the principal 
 
 plane is produced to , a distance equal to the 
 \ focal length, and MN is drawn perpendicular 
 \_p to EC or parallel to QR. The line MN occu- 
 \ pies with reference to QjR, the same position 
 "^ as the focal plane of the camera does to the 
 \ picture plane of the perspective. The direc- 
 tion of a point of the photograph projected in 
 Q on the picture trace, is found by joining 
 NA and producing to the opposite side of A. 
 
 Pig-. 231 
 
 The first two intersections should be checked 
 
 either by a third one or otherwise. They may, for instance, be checked 
 by determining the height of the point from the two photographs : 
 unless correctly plotted, the two heights obtained will not agree. This 
 check, however, does not indicate slight errors. 
 
 . 230 
 
 Q\~ 
 
 <* 
 
PLOTTING WITH THE PERSPECTOGRAPH. 
 
 213 
 
 The check may also be a line drawn by means of the perspectograph 
 or perspectometer and on which the point is situated, such as the shore 
 of a lake or of a river, but the best check is a third intersection. 
 
 The number of every point is inscribed in pencil on the plan. 
 
 131. PLOTTING WITH THE PERSPECTOGRAPH. To draw with the per- 
 spectograph the plan of a figure which 
 appears on a photograph, the figure must 
 be beyond the picture plane (102) or 
 above the ground line on the photograph. 
 Thus the lake AB (Fig. 232), being 
 f ff' below the ground line XY of the photo- 
 Y graph, cannot be drawn without such a 
 change of ground plane, as will bring the 
 new ground line X'Y' below the lake AB. 
 It has been explained that this is done 
 by doubling the height of the station 
 until the ground line is brought into cor- 
 Fig*. 232 rect position (102). 
 
 The slide XY of the perspectograph, Fig. 233, is, by means of the 
 scales drawn in X and Y on the drawing board, adjusted to a distance 
 from RT equal to the focal length. 
 
 After adjusting S, the pencil 
 is brought over a point M of 
 the trace GJ of the principal 
 plane at a distance sM from 
 s equal to twice the focal 
 length. 
 
 The photograph is pinned 
 under the tracer, the horizon 
 line HH' over the correspond- 
 ing line AB of the board and 
 the principal line over EF ': 
 the iron rod connecting V and 
 Z is then adjusted so as to 
 bring the tracer p. midway 
 between the horizon and 
 ground lines. 
 
 Fig". 233 The cross section paper is 
 
 pinned to the board, one of its 
 
 lines coinciding with the trace of the principal plane GJ, and other 
 lines with the front lines AB and CD, drawn at known distances from 
 the foot of the station s. 
 
 c*-- 
 
 Tt 
 
 A9J) 
 
 B \ 
 
214 PLOTTING THE SURVEY. 
 
 There will be no difficulty in tracing with the point // the part of 
 the photograph which, on the figure, is on the right of the principal 
 line, but it may happen that in moving >j. to the left, the obliquity 
 of the arm MS would prevent the free play of the instrument. It 
 should then be reversed, the slide XY being changed end for end, the 
 photograph transferred from EF to KL, the cross section paper moved 
 so as to bring, on the trace NQ of the principal plane, the line of the 
 paper which was formerly over GJ, and the point S placed to the left 
 of s ; p. being now between the two slides RT and JCF, the tracer has 
 to be changed to the opposite arm. 
 
 The perspectograph can be so adjusted that the trace of the principal 
 plane is the same in both positions of the instrument, it being 
 sufficient not to move s, when inverting the arms and slide XY : the 
 cross section paper then does not require to be displaced. 
 
 Having obtained the plan of the figure shown on the photograph, 
 the reduction to the proper scale is made at sight on the cross section 
 paper, and transferred to the general plan. The transfer should be 
 checked by points previously established by intersections. 
 
 The use of the instrument is possible every time the plane of the 
 figure can be determined, as for instance a lake, a river, a contour 
 line, or the foot of a mountain. Slight differences of level do not 
 affect the rosult when the height of the station h great. 
 
 The instrument can also be used for figures in inclined planes, such 
 as a river with a rapid slope, the outline of a stratification plane 
 which has not been distorted, a road, or a railway. 
 
 132. HEIGHTS. The heights of the points fixed by intersections are 
 found as explained in 85. The distance from the point to the 
 horizon line is taken with a pair of compasses on the photograph ; one 
 of the points of the compasses is placed on the 
 division A of the sector (Fig. 234), OA being 
 equal to the focal length. The sector is then 
 opened until the other point of the compasses 
 coincides with the corresponding division B of the 
 other arm. The distance on the plan from the point 
 to the picture line is taken with the compasses, 
 then one leg being placed in A on the sector, the 
 other will come somewhere in (7, the compasses are 
 then turned around on C and brought on the 
 division D of the other arm corresponding to C. 
 The line CD is the height of the point above or 
 below the horizon plane, which means the height 
 Fig". 234 above or below the station. 
 
HEIGHTS. 
 
 215 
 
 Another method consists in making use of the angular scale shown in 
 Fig. 235. Take SP equal to the focal length ; erect the perpendicular 
 
 PA to SP and divide both into 
 equal parts. Join to S the points 
 of division of PA and through 
 those of SP draw parallels to PA. 
 
 Now, with a pair of compasses, 
 take on the photograph the 
 distance from the point of the 
 perspective to the horizon line : 
 transfer it to P/J. and suppose 
 that it is found to correspond to 
 the line S/J. passing through the 
 
 Pig-. 235 
 
 point 9 of the graduation of PA. 
 
 Take with the compasses the distance on the plan from the 
 horizontal projection of the point to the picture line, and transfer it to 
 the right or left of P according as the point of the plan is beyond or 
 within the picture line. Then take with the compasses the distance 
 on a parallel inB to PA, between m and, the point M where the line 
 mB is intersected by />-, corresponding to 9 of the graduation. This 
 distance mM is the height of the point above or below the station. 
 
 A scale is now pinned somewhere, perpendicularly to a line AB 
 (Fig. 236), the division C of the scale corresponding to 
 AB being the height of the station. The compasses 
 are taken off the sector, and one of the legs being set 
 
 5 in (7, the other leg coincides with a division D of the 
 scale, above or jDelow (7, which is the height of the 
 point above the datum plane. This height is entered 
 in pencil on the plan, enclosed in a circle, to distin- 
 guish it from the number of the station. It is checked 
 
 by a second photograph, and, when the discrepancy 
 between the two heights is within the limits of error 
 admissible, the mean is entered in red ink on the plan 
 and the pencil figures erased. 
 
 Fig-. 230 
 
 A difference in the heights obtained from the two 
 photographs indicates that the two points identified do 
 not represent the same point of the ground, or that an 
 error has been made either in plotting it, or in finding 
 its height. 
 
 A third intersection disposes of the first two alternatives, and a 
 new measurement of the height shows whether any error has been 
 made. 
 
216 
 
 PLOTTING THE S UR VE Y. 
 
 133. VERTICAL INTERSECTIONS. In the method of horizontal inter- 
 sections the base line is projected on the horizontal plane; in this 
 method it is projected on a vertical plane. The difference of altitude 
 of the two stations must therefore be considerable. 
 
 The principal plane of one of the photographs is taken as vertical 
 plane of projection ; the ground plane is the horizontal plane contain- 
 ing one of the stations. In 
 Fig. 237, the ground line is the 
 trace of the principal plane of 
 the photograph taken from the 
 station A ; the ground plane is 
 the horizontal plane of the sta- 
 -Y tion B. On the ground plan, a 
 and B are the two stations, CD 
 and EF their picture traces. 
 The station A on the vertical 
 plane is on the perpendicular 
 a A to XY equal to the height 
 of A above B. A point such 
 as p plotted by the method of 
 
 Mg*. 267. horizontal intersections, would 
 
 not be accurately fixed because the angle of the directions aD and BF 
 is too small. 
 
 Project the visual rays from A and B on the vertical plane : the 
 visual ray from A is a line AQ passing through the projection Q of the 
 point's image on the principal line. It is drawn by taking CQ equal 
 to the height of the point on the photograph above the ground line, 
 and joining AQ. 
 
 The vertical projection of the visual ray from B is a line b'R pass- 
 ing through the vertical projections of the station b' and of the point's 
 image R, on the second photograph. To find R, let fall FG perpen- 
 dicular to XY and produce to R, GR being equal to the height, on the 
 photograph, of the point's image above the horizon line. 
 
 The intersection of AQ and b'R is the vertical projection^' of the 
 point. Letting fall the perpendicular p'o to XY and producing, deter- 
 mines the position p of the point on the ground plan. 
 
 The construction gives not only the point on the ground plan but 
 also its height op. This process is the best one for plotting a narrow 
 valley between two high walls : it has, however, the disadvantage of 
 requiring a complicated construction. 
 
 134. PHOTOGRAPH BOARD. So many construction lines are employed 
 on the photographs that it is advisable to have a photograph board on 
 which part of the lines are drawn beforehand, once for all. 
 
PHOTOGRAPH BOARD. 
 
 217 
 
 It consists of an ordinary drawing board, covered with strong draw- 
 ing paper. Two lines at right angles, DD' and&S', Fig. 238, represent 
 the horizon and principal lines ; PD, PD' PS and PS' are each equal to 
 the focal length, so that D, D,' S and S' are the left, right, lower and 
 upper distance points respectively. 
 
 Fig-. 238 
 
 The photograph is pinned in the centre of the board, the principal 
 line coinciding with SS' and the horizon line with DD'. Four scales, 
 forming the sides of a square OTYZ, are drawn in the centre, the side 
 of the square being a little larger than the length of a photograph. 
 
 They answer various purposes as, for instance, drawing parallels to 
 the horizon or principal lines by laying a straight edge on the corre- 
 sponding divisions of the scales or marking the ground line by joining 
 the divisions of the vertical scales representing the height of the 
 station. 
 
 At a suitable distance from the distance point D' a perpendicular 
 QR is drawn on which are marked, by means of a table of tangents, the 
 angles formed with DQ by lines drawn from D. This scale is employed 
 for measuring the altitudes or azimuthal angles of points of the photo- 
 graph, as will be explained later on (137). From S as a centre with 
 SP as radius, an arc of circle PL is described and divided into equal 
 parts. Through the points of division, and between PL and PD', 
 lines are drawn converging to S. Parallels MNto the principal line are 
 also drawn sufficiently close together. All these lines are used in con- 
 nection with the scale of degrees and minutes QR. 
 
 R 
 
218 
 
 PLOTTING THE SURVEY. 
 
 The studs of the centrolineads are fixed in A, , C and E ; the lines 
 AB and CE, joining their centres, and those required for adjusting the 
 centrolineads, are drawn and used as explained in 97. 
 
 A square FGKH is constructed on the four distance points. 
 
 135. CONSTRUCTION OF THE TRACES OF A FIGURE'S PLANE. When a 
 figure is in an inclined plane, it is necessary to have the traces of the 
 plane on the principal and picture planes for using a perspective instru- 
 ment on the photograph. 
 
 Two cases are met with in practice : the plane is given by the line 
 of greatest slope, or by three points. 
 
 Case /. The line of greatest slope may be an inclined road, or the 
 middle of a straight valley in which a river flows with a rapid current. 
 On the plan, this line is represented by a lino ab, Fig. 239, the altitude 
 of a being known. 
 
 Pin the photograph to the board 
 and take for ground plane the plane 
 of a : draw the ground line XY. 
 
 On the plan draw aO perpendicular 
 to ab and produce it until it inter- 
 sects the principal line S 1 p 1 and 
 picture trace JT i Y^ . 
 
 On the photograph take pE equal 
 to p^b ; at E erect a perpendicular 
 to XY and produce it to the inter- 
 section /? with the perspective of the 
 line of greatest slope. Take pN equal 
 to p^O and join N$ : it is the trace 
 of the required plane on the picture 
 plane. 
 
 Take pQ equal to p^L and join 
 MQ ; it is the trace of the required 
 plane on the principal plane, sup- 
 posed to be revolved around SS' on 
 the picture plane, the station falling 
 in D. Produce MQ to R : DR is 
 the vertical distance of the station 
 above the plane RMft. The new 
 horizon and ground lines are now 
 drawn as in 82' 
 
 Fig-. 239 
 
CONTOUR LINES. 
 
 219 
 
 o - 
 
 YK 
 
 Case II. Take for ground plane the plane containing one of the 
 points, a (Fig. 240), and draw the ground line XY on the photograph. 
 
 Join a on the plan to the two remain- 
 ing points and produce to the inter- 
 sections E 'and F with the picture 
 trace. 
 
 Take on the photograph p^K equal 
 to pE and draw KL perpendicular 
 to XY ; join the perspectives and 
 ft of the points shown in a and b 
 on the plan and produce to the inter- 
 section with KL. Take p T T equal 
 to pF, draw TN perpendicular to 
 XY and produce to the intersection 
 N with the line joining the perspec- 
 tives and f. Join NL : it is the 
 trace of the required plane on the 
 picture plane. 
 
 Produce LN to and take pG 
 equal to p^O ; join aG and take p^Q 
 equal to pH. The line MQ is the 
 trace of the required plane on the 
 principal plane supposed to be re- 
 volved around SS' 011 the picture 
 plane, the station being in D. Here 
 also, DR is the vertical height of the 
 station above the plane of the three 
 given points. The new horizon and 
 ground lines are constructed as previously explained. 
 
 136. CONTOUR LINES. A sufficient number of heights having been 
 determined, the contour lines are drawn by estimation between the 
 points established. In a rolling country, a limi' ed number of points 
 is sufficient to draw the contour lines with precision, but in a rocky 
 region the inflections of the surface are so abrupt and frequent that it 
 is utterly Impossible to plot enough points to represent the surface 
 accurately. The photographs are of great assistance to the draughts- 
 man ; having them under his eye, he is able to modify his curves so as 
 to represent the least inequalities of the ground. 
 
 Instead of drawing the contour lines at once on the plan, the draughts- 
 man may commence by sketching them on the photograph in the s.nne 
 way as he would on the plan. Every point plotted has been marked 
 on the photograph, and the altitudes may be taken from the plan. By 
 adopting this course, he is able to follow very closely the inequalities 
 
 240 
 
PLOTTING THE SURVEY. 
 
 of the surface. The curves serve to guide the draughtsman in drawing 
 those of the plan, or they may be transferred by the perspectograph or 
 the perspectometer. 
 
 As long as a sufficient number of points is obtained by intersections, 
 there is no difficulty in drawing the contour lines, but it may happen in a 
 rapid survey, that the points are too few and too far apart for denning 
 the surface. It is then necessary to resort to less accurate methods. 
 
 / A mountain ridge, which appears in a p on a photograph (Fig. 241), 
 can be divided by the contour planes, by assuming that it is contained 
 
 in a vertical plane. The construc- 
 tion, which has been explaned in 
 62, is carried out as follows: 
 
 On the plan produce the pro- 
 jection ab of the ridge, to the in- 
 tersection F with the picture trace 
 and draw through the station S^ C 
 parallel to ab. 
 
 Having pinned the photograph 
 to the photograph board, take from 
 the principal point on the horizon 
 line P V equal to p 1 C and PG equal 
 to p^F. At G, place the scale of 
 equidistances perpendicular to the 
 horizon line, the division G cor- 
 responding to the -height of the 
 station, and join the marks of the 
 scale to the vanishing point V. 
 
 Having now the points of inter- 
 section of the ridge by the contour 
 planes, their distances from the 
 principal line are marked on the 
 edge of a band of paper and their 
 These directions produced to ab 
 
 Fig-. 24 1 
 
 directions plotted in the usual way. 
 
 give the intersections of the contour lines. 
 
 When the mountain has rounded forms and no well-defined ridge, 
 the visible outline must be assumed to be contained in a vertical 
 plane perpendicular to the direction of the middle of the ridge. 
 The construction is made by drawing, on the photograph board, SV 
 perpendicular to the direction SM of the middle of the outline. (Fig. 
 242.) On the plan, p^M^ is taken equal to PM, and from the projec- 
 tion a of the summit of the mountain a perpendicular ab is let fall 
 on S 1 M 1 which represents the projection of the visible outline; it is 
 
PHOTOGRAPH PROTRACTOR. 
 
 221 
 
 produced to the intersection N with the picture trace, PQ is taken equal 
 
 to piN, and the scale of equidis- 
 tance placed at Q perpendicular to 
 the horizon line. The points of 
 division are joined to F, produced 
 to /3, and the plotting done as in 
 the preceding case; or the direc- 
 tions of the intersections of /3 by 
 the contour planes may simply be 
 plotted and the contour lines drawn 
 tangent to these directions. 
 
 The horizon line contains the pers- 
 pectives of all the points at the 
 height of the station ; it is the pers- 
 pective of a contour line when the 
 height is that of a contour plane. 
 
 Full details on the plotting of 
 contour lines being given in the text 
 books on surveying, it is not necess- 
 ary to repeat them here. The main 
 
 Fig". 242 point is to understand thoroughly 
 
 the mode of formation of the sur- 
 face and its variations under different circumstances ; the surveyor 
 should pay particular attention to the subject, making a special study 
 of it. Without this knowledge, the proper representation of the 
 ground would require the plotting of a very large number of points. 
 
 137. PHOTOGRAPH PROTRACTOR. The angle between the point of a 
 photograph and the principal and horizon lines, that is, the altitude 
 r or azimuthal angle, is sometimes wanted. 
 
 The azimuthal angle is obtained at 
 once on the photograph board by join- 
 ing the station S (Fig. 243) to the pro- 
 jection a of the point on the horizon 
 line. If required in degrees and minu- 
 tes, the distance Pti is transferred to 
 the principal line in PG ; D is joined 
 to G and produced to the scale of de- 
 grees and minutes J3C, where the gra- 
 duation K indicates the value of the 
 azimuthal angle. 
 
 Were many such angles to be meas- 
 ured, the horizontal scales TY and OZ 
 
 *" * .* 
 
 
 7. 
 
 
 
 *---*. 
 
 
 
 
 i : 
 
 P ~~~- : 
 
 r=*- _ 
 
 1 
 
 <*\ F 
 
 
 
 Fig-. 243 
 
PLOTTING THE SURVEY. 
 
 (Fig. 238) might be divided into degrees and minutes by means of a 
 table of tangents, using as radius the focal length SM. A straight 
 edge being placed on a point of the photograph, and directed to pass 
 through identical divisions of TY and OZ, would at once give the 
 azimuthal angle of the point. 
 
 The altitude is the angle S (Fig. 243) of the right angle triangle 
 having for sides Sa and aa. To construct it, take DF equal to >Sa, 
 draw FE parallel and equal to a, join DE and produce to the scale 
 of degrees and minutes jBC. This construction is facilitated by the 
 lines previously drawn on the board. With a pair of compasses, take 
 the distance from a to the principal line, carry it from P (Fig. 238) 
 in the direction PD', and from the point so obtained take the distance 
 to the arc ML, measuring it in the direction of the radii marked on 
 the board : this is the distance PF (Fig. 243). Then with the com- 
 passes, carry aa to FE, which is done by means of the parallel lines 
 MN of Fig. 238. The construction is now completed as already 
 explained. 
 
 A protractor may be constructed to measure these angles. It con- 
 sists of a plate of transparent material on which are lines parallel to 
 the principal line, containing the points of same azimuth, and curves 
 of the points of same altitude. 
 
 The azimuthal lines are constructed by plotting the angles in S and 
 drawing parallels to the principal line through the points of intersec- 
 tion with the horizon line. 
 
 Denoting by h the altitude of a point a and taking the horizon and 
 principal lines as axes of co-ordinates, the equation of the curve of 
 altitude h is : 
 
 2, = 
 
 tan. 2 A. 
 
 This is an hyperbola of which the principal and horizon lines are the 
 transverse and conjugate axes and the centre 
 is the principal point. One of the branches 
 contains the points above the horizon and the 
 other branch the points of same altitude below 
 the horizon. The asymptotes are lines inter- 
 secting at the principal point and making 
 angles equal to h with the horizon line. 
 
 This hyperbola is the intersection by the 
 picture plane of the cone of visual rays form- 
 ing the angle h with the horizon. 
 
 The curves of equal altitude may be calcul- 
 ated by the formula of the hyperbola or they 
 
 Fig-. 244 
 
PRECISION OF THE SURVEY. 223 
 
 may be plotted by points, reversing the construction given above for 
 finding the altitude of a (Fig. 243). The complete protractor is shown 
 in Fig. 244 : the angular distance between the lines depends on the 
 degree of precision required. 
 
 The instrument may be made, like the perspectometer, by drawing 
 it on paper on a large scale, photographing and making a transparency 
 which is bleached in bichloride of mercury. 
 
 138. PRECISION OF THE METHOD OP PHOTOGRAPHIC SURVEYING. 
 The precision of a survey executed by the methods exposed, when all 
 the points are established by intersections, is the same as that of a 
 plan plotted with a very good protractor or made with the plane 
 table. There is, however, this difference : the number of points 
 plotted by photography is greater than by the other methods. 
 
 Points plotted by means of their altitude below the station are far 
 less accurate, their positions being given by the intersection of the 
 visual ray with the ground plane, the angle of intersection being equal 
 to the angle with the horizon plane or to the angle of depression of the 
 point. With the camera employed, embracing 60, this angle is always 
 less than 30 ; even that is seldom obtained in practice, a declivity 
 of 30 being almost a precipice. Therefore, the intersection is always 
 a poor one and the uncertainty becomes considerable with points near 
 the horizon. 
 
 With perspective instruments, doing mechanically the same con- 
 struction, the results are still less precise, being affected by the 
 instrumental errors. 
 
 On the other hand, it must not be forgotten that when these 
 methods are employed, the ordinary topographer would fall back on 
 sketching ; the results furnished by photography are therefore 
 infinitely more precise. 
 
 The plan given at the end of this book and the photographs which 
 accompany it, are specimens of actual work on the Topographical 
 Survey. 
 
224 PHOTOGRAPHS ON INCLINED PLATES. 
 
 CHAPTER IX. 
 
 PHOTOGRAPHS ON INCLINED PLATES. 
 
 139. GENERAL REMARKS. Hitherto it has been assumed that the 
 photographs used for the survey were taken on plates perfectly vertical. 
 There are several cases in which this condition cannot be fulfilled : the 
 camera may be an ordinary one, without any means of adjusting the 
 plate, or the photographs may have been taken merely as illustrations, 
 their employment for the construction of the plan being decided upon 
 afterwards. 
 
 There are two classes of surveys in which the plates are always 
 inclined. The first are secret surveys, the views being taken with a 
 camera concealed about the person or otherwise. The scope of these 
 surveys is very limited ; the photographs, being instantaneous, lack 
 detail in the distance, and, unless objects present great contrasts of 
 light and shade, their images are somewhat indistinct as soon as the 
 distance attains a few hundred yards. Another cause of trouble is 
 the small size of the camera and plates : the views, being instantane- 
 ous, stand very little enlargement and the measurements are in 
 consequence not very accurate. 
 
 The other class of surveys comprises those made from balloons. It 
 is very doubtful whether the method will ever be found practical and 
 prove of more than theoretical interest. It requires the consideration 
 of an entirely new system of survey by means of photographs taken on 
 plates placed horizontally or nearly so. 
 
 140. PLOTTING THE DIRECTIONS OF POINTS OF THE PHOTOGRAPHS. 
 When the photographic plate is not vertical, the picture plane of the 
 perspective, which is parallel to the plate, is pierced by the vertical of 
 the station. This trace is the vanishing point of all the vertical lines, 
 which, having ceased to be front lines, are no longer represented by 
 parallels to themselves. 
 
 Let ABCD (Fig. 245), be a photograph on an inclined plate, P being 
 the principal point and HH' the horizon line. The perpendicular V~ 
 drawn through the principal point to the horizon line, is the principal 
 line. 
 
DETERMINATION OF HEIGHTS. 
 
 225 
 
 Revolve the principal plane on the picture plane around the prin- 
 cipal line as an axis : the station falls in S^ on a perpendicular PS 
 
 to VP t PS being equal to the focal 
 length. 
 
 Join xS'jTT and S 1 V \ the first line 
 is the revolved horizontal line from 
 the station to the picture plane; 
 /$! F is the revolved vertical of the 
 station, and V the vanishing point 
 of vertical lines. 
 
 Now revolve the horizon plane on 
 the picture plane around the horizon 
 line. The station comes in S t on 
 the principal line produced, at a 
 distance xS equal to nS^ . 
 
 To find the horizontal direction of 
 a point /JL of the photograph, draw 
 the perspective of its vertical line 
 \ ,' /' by joining /JL to V. The intersection 
 
 \ / y' n with the horizon line is the per- 
 
 ,*''' spective of the trace in the horizon 
 
 plane of the vertical of the point 
 and Sn is its direction. 
 
 Fig". 245 Comparing this construction with 
 
 the one for vertical plates, we see 
 
 that the same methods may be employed provided - be used as principal 
 point, 7tS L as focal length, and that every point of the photograph be 
 first projected on the horizon line by joining it to V, before measuring 
 its distance from the principal line. The points such as n can be 
 marked on a band of paper and used as wa,s done for vertical plates 
 
 With a plate nearly vertical, V is at a great distance from P, and 
 the perspectives of the vertical lines have to be drawn with the centro- 
 linead. 
 
 141. DETERMINATION OF HEIGHTS. Let m, Fig. 245 be, on the 
 ground plan, the point seen at /JL on the photograph. Project on the 
 principal plane the triangle formed by the visual ray, its projection on 
 the horizon and the line n/JL. On the revolved principal plane, the pro- 
 jection of the visual ray is S^ri, p.ri being perpendicular to F. The 
 projection of m is F; it is revolved to G and the perpendicular GK to 
 Si* is the projection of the vertical of the point or its height above 
 the horizon plane. 
 15 
 
226 PHOTOGRAPHS ON INCLINED PLATES. 
 
 Various devices may be imagined for constructing expeditiously the 
 heights of a number of points. 
 
 142. DETERMINATION OF THE HORIZON LINE AND VANISHING POINT OF 
 VERTICALS. In order to make use of a photograph for plotting the plan, 
 the horizon and principal lines and the vanishing point of verticals 
 must be marked on the photograph. 
 
 It is assumed that the camera is available, either before or after the sur- 
 vey, for experimenting upon and that the focal 
 V> length and principal point may be determined 
 by the usual methods, with the plate vertical. 
 If the zenith distances of several points of the 
 / photograph hctve been observed with a survey- 
 
 ing instrument, the determination of the hori- 
 zon line presents no difficulty. Assume a 
 vanishing point of vertical lines V, Fig. 246, 
 ^ ,p and join it to a point p. of the photograph of 
 which the zenith distance is known. Through 
 
 ___^ L , the principal point P, draw PE perpendicular 
 
 ~~ a ' to V.'t, and PS perpendicular to PE and equal 
 \\; / to the focal length. Draw EG perpendicular 
 
 1' to ES t take EF equal to Ey., join SF and 
 
 make the angle FSG equal to the altitude of 
 ft ; FG is the distance measured on Vp- from 
 Fig". 246 P- to the horizon line. 
 
 Making p.n equal to FG fixes one point n of the horizon line. A similar 
 construction repeated on another point of the photograph furnishes a 
 second point of the horizon line. 
 
 The first result will generally be inaccurate, because the position of 
 the vanishing point V is only approximate. A. new vanishing point 
 must, therefore, be fixed by means of the horizon line just obtained, and 
 the construction explained above is repeated. The second horizon line 
 
 found will likely be sufficiently precise ; if 
 not, the construction must be made a third 
 time. 
 
 In secret surveys, measured angles are 
 seldom available, but it is easy to devise 
 an attachment like a hand level, to mark 
 the horizon line on the plate when the 
 . g photograph is taken. 
 
 Failing this, the horizon line must be 
 
 furnished by the subject. "When the view 
 
 Fig*. 247 includes buildings, the vanishing point of 
 
TRANSFER TO A VERTICAL PLANE. 227 
 
 verticals is given by producing to their intersection the vertical lines 
 of the buildings. This point F, Fig. 247, -is joined to the principal 
 point, P, and PS is made perpendicular to VP and equal to the focal 
 length. Drawing S~ perpendicular to SV, the perpendicular HH to 
 V~ is the horizon line. 
 
 Horizontal lines vanish on the horizon line ; therefore, if the hori- 
 zontal lines of two faces of a building be produced to their intersection, 
 the line joining the two vanishing points is the horizon line. 
 
 If two intersecting horizontal lines appear on the photograph as a 
 straight line, the latter is the horizon line. 
 
 Although angles cannot be measured, it may be possible to ascertain 
 the points of the view which are at the same altitude as the observer; 
 these joined together give the horizon line. 
 
 143. TRANSFERRING THE PERSPECTIVE TO A VERTICAL PLANE. In- 
 stead of using exact copies of the negatives for plotting the plan, the 
 copies or enlargements can be made in such a way that the perspective 
 is restored to a vertical plane. 
 
 Have a copying, or enlarging camera, OCD (Fig. 248J, movable on a 
 horizontal axis passing through the first nodal point and parallel to 
 the negative. 
 
 Make an experimental negative with the field camera, the plate be- 
 ing vertical ; draw on it the horizon and principal lines, place it in the 
 
 holder of the copying camera, and mark 
 the points of the holder corresponding 
 r to the horizon and principal lines. 
 \ n After inserting the holder, the camera 
 is moved until the plate CD is vertical, 
 and fixed in that position. The screen 
 '", AB is now adjusted at the proper dis- 
 tance, parallel to the plate, and the 
 projected images of the horizon and 
 principal lines are marked 011 it, in such 
 a manner, that the marks will appear 
 on the prints. 
 
 Fig-. 248 
 
 To copy a negative taken in an 
 
 inclined position, the horizon and principal lines are drawn on it, also 
 a parallel to the horizon through the principal point. The negative 
 is placed in the holder with the principal line on the proper marks, and 
 the horizontal line of the principal point on the marks corresponding 
 to the horizon line of the experimental plate. The camera is moved, up 
 or down, until the image of the negative's horizon line TT coincides with 
 the horizon line Q previously marked on the screen ; in this position 
 
PHOTOGRAPHS ON INCLINED PLATES. 
 
 the image on the screen is the perspective restored to a vertical picture 
 plane, because the inclination of the camera being the same as when the 
 negative was taken, any point N' of the latter would have photographed 
 in N on a vertical plate and given the same image M on the screen. 
 
 With a lens of sufficiently long focus and photographs taken nearly 
 vertical, as is generally the case, the displacement of the camera is too 
 small to affect the definition on the screen. 
 
 The holder must be provided with means for adjusting the negative; 
 the principal point must always occupy the same position, the plate 
 turning around it as a pivot. 
 
 The horizon and principal lines are indicated on the print by the 
 marks fixed to the screen : the principal point has been displaced in 
 copying and is now on the horizon line. 
 
 The change of picture plane can also be effected with the perspec- 
 tograph, but the use of the instrument is not to be recommended when 
 the change can be made so simply by photographic process. 
 
 144. PHOTOGRAPHS ON HORIZONTAL PLATES. Photographs on hori- 
 zontal plates might be obtained by an ar- 
 rangement similar to the one described in 
 100, with a pin-hole stop in the lens; 
 they may be taken from a balloon with an 
 ordinary camera, but the plates are only 
 approximately horizontal. 
 
 The picture and ground planes being 
 parallel, the figures of one are similar to 
 those of the other ; thus the photograph 
 a /? (Fig. 249) of a lake AB is also its plan, 
 and only requires to be reduced to the 
 proper scale. The reduction is given by 
 the proportion between the distances Ss 
 and SF from the station to the ground 
 
 Fig-. 249 
 
 Fig-. 250 
 
 and picture planes. When the height of the station and 
 the focal length are equal, the photograph is a full size 
 plan. 
 
 To plot the directions of the various points, the prin- 
 cipal point P of the photograph is placed on the foot of 
 the station , and a line of known direction, such as Pa, 
 on the corresponding line of the plan sA. To find the 
 direction of any other point B, its perspective ft is joined 
 to the principal point P : this line coincides with sB on 
 the plan. 
 
 The height of a point is found by taking SP t Fig. 250, 
 equal to the focal length and Ss equal to the height of 
 
HORIZONTAL PLATES. 
 
 229 
 
 the station, drawing Pa and sa perpendicular to SP, Pa being equal to 
 the distance of the point's perspective from the principal point, and sa 
 equal to the distance on the plan from the station to the point. Join 
 Sa the parallel a A to SP is the height of the point above the ground 
 plane. 
 
 A photograph taken from a balloon cannot be perfectly horizontal ; 
 to make use of it for plotting the plan, the trace *, Fig. 251, of the 
 vertical of the station on the picture plane must be known. 
 
 The directions of the principal line sP and of the perpendicular to it, 
 
 AJB, are the same on the plan and on the 
 photograph; they are different for all 
 other lines. 
 
 To find the direction on the ground 
 plan of a point p. of the perspective, draw 
 
 B PS perpendicular to the principal line and 
 equal to the focal length, join Ss and take 
 SC equal to the distance p. A from p. to 
 AB. Draw p. A and CD parallel to Ps 
 
 . - and take Ap! equal to SD ; sp! is the direc- 
 
 tion of the point on the ground plan. For 
 
 Ap. forms with its horizontal projection a right angle triangle in which 
 the angle at A is the inclination of the plate to the horizon ; this 
 triangle is constructed in SCD. the angle JS being the inclination of 
 the plate. Therefore A /if, which is made equal to SD, is the horizontal 
 projection of Ap., p! is the trace, on the ground plane, of the vertical 
 of p., hence the vertical plane passing through s and the point .// of the 
 photograph cuts the ground plane along sp' . 
 
 A much better way to employ these photographs would be to restore 
 them to a horizontal picture plane in printing, by the process of 1 43, 
 using Ps and A B in the same manner as the principal and horizon 
 lines of the vertical photograph. 
 
 The great difficulty in balloon surveying is the determination of the 
 trace of the vertical of the station on the picture plane, or of the foot of 
 the station on the ground plan. The oscillations of the balloon prevent 
 the use of any kind of level inside of the camera, and instrumental 
 measurements of angles are open to the same objection. The angles 
 might, however, be measured by two observers located on the ground. 
 
 In a view containing vertical lines, their vanishing point gives the 
 trace of the vertical of the station ; for a photograph taken from a 
 short distance above buildings, this mode of determining the trace 
 would be very convenient. 
 
 Balloon surveying appears adapted to military purposes only, 
 although the advocates of the process are confident that it will 
 eventually take the place of all other surveying methods. 
 
SUPPLEMENTARY NOTE. 
 
 LAUSSEDAT'S NEW PHOTO-THEODOLITE. Col. Laussedat has recently 
 devised a light photo-theodolite shown in Figs. 252, 253 and 254 : the 
 following description is given by the makers, Messrs. E. Ducretet & L. 
 Lejeune, of Paris. 
 
 It consists of an ordinary sur- 
 veying transit theodolite with a 
 small camera on top. 
 
 Fig. 252 is the complete instru- 
 ment on its stand, to which it is 
 fixed by a central screw acting 
 on the base S with foot screws. 
 C is the camera, of very small 
 size, with changing box for 15 
 plates 6Jx9 centimetres. With 
 this box, plates are changed with- 
 out removing the camera C from 
 the instrument. 
 
 The changing box can be re- 
 placed by another one in full day 
 light ; the surveyor can carry 
 with him several loaded boxes. 
 Plates are subsequently enlarged 
 to 18x24 centimetres or under. 
 
 The wide angle rectilinear lens 
 has a focal length of 75 milli- 
 metres ; the angle between the 
 horizontal points marked on the 
 photographs is set at 60. The 
 lens is provided with an iris 
 diaphragm and can be used with 
 coloured screens, for obtaining 
 Fig. 252. better photographs of distances 
 
LAUSSEDATS NEW PHOTO-THEODOLITE. 
 
 231 
 
 OC 
 
 and clouds. Focussing is rapidly effected by a special arrangement, 
 with a graduation in metres on the plate H : the camera can thus be 
 used for infinity, as in surveying, or for short distances as in photo- 
 graphing groups or other subjects. 
 
 The plate H carrying the objective, has a vertical motion : the 
 camera must always be perfectly horizontal. 
 
 The finder V, with adjustment for focussing, shows the extent of 
 
 the view covered by the sensitive 
 plate. It is very bright and has a 
 large aperture. A shutter E covers 
 the objecb glass of this finder when 
 OMl'u- i L & tafe'i not required. 
 
 The telescope L, Oc, of the transit 
 theodolite is provided with stadia 
 wires. It is fixed to a vertical circle 
 Ce, with vernier, clamp and slow 
 motion screw, divided into half 
 degrees : this graduation is made in / 
 grades when so ordered. The telescope 
 L makes a complete revolution. It is 
 placed midway between the uprights 
 M in the vertical plane of the photo- 
 graphic objective 0. This disposition 
 secures the stability of the instru- 
 ment, which is not realized when the 
 telescope is at the side. N is the 
 adjustable level. A is the horizontal 
 circle with vernier divided into half 
 degrees or grades. It is fixed by the 
 clamp and slow motion screw P'. 
 
 D is a long compass with clamp 
 ^ and slow motion screw P, for read- 
 ing, on circle A, the direction of the 
 magnetic meridian. S is the base 
 with foot screws fitting in the slits 
 Fig. 253. of the stand Pi. 
 
 Fig. 253 represents the same combination, the changing box C being 
 removed and replaced by the ground glass. 
 
 Fig. 254 shows the camera C removed from the geodetic apparatus 
 and set alone on the stand Pi by means of the additional spindle S'. 
 
S UPPLEMENTA RY NO TE. 
 
 In this case, the whole camera and the geodetic apparatus are discon- 
 nected and can be employed separ- 
 ately. 
 
 A carrying case, with shoulder 
 straps, contains the whole instru- 
 ment, including two changing boxes, 
 each with 15 plates 6Jx9 centi- 
 metres. The weight of the case, 
 instrument complete, and one 
 changing box is 8 kil. 100. The 
 stand Pi is carried separately. 
 There is room in the case for three 
 changing boxes, one artificial 
 horizon, note books and reading 
 glasses. The external dimensions 
 are : length, 39 centimetres ; width, 
 17 centimetres ; height, 28 centi- 
 metres. 
 
 Fig. 254. 
 
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