MAP GA 110 U6 1928 "{*a 7 $ t I< ' . '.. ,) f . Y .':Y ' 4 T t V , . r i, =V § 2l ti° , .. _ }. Y . ' -- 9 .) r i i '. f +t i' * vC' 'a EMENTSsOFkMAP PROJEC."'1 TIO N ' ° 1s ; "6x#f . 4y 4. f R1i : r {" t tWIT H" 3: 7 , APPLICy, 1iArtTIONS TO MAP AND t^YCHART $. Ci .c; ;CON-; STRUCT rIN .'b .. ;ra& gf3 S°". . . CHARLES H. DEETZ AND OSCAR S. ADAMS ( " Special Publication No. 68 (Revised May 1, 1928) 5_r T 1 i ZfO: l w . '.f. .y, - .. . ._ 1:.. I " a: J K ] t . I ' )}!. f .,4 - _ r t ,' rho _ .q. ,, __ ? .F ^ } 9 ., , ,. . " r . R ;, :.x 1 ' , . ,. ,i; , : : ? . . . , .; j . d + "1; d 'f w, ;^ tX Z y: a_ , r.. - : .' Y DEPARTMENT OF COMMERCE U. S. COAST AND GEODETIC SURVEY . 1 : .. f 1 i }} {" , ", : ' S r. 1 . . . ' _ r .R p.l ' . i L? I iI I,' ** j!b r11-, ( , ..I - 1 , 1 ff '*-. 1' 2 / J III:) ;' ( *I I- 'a' 1f I ' ' / 1 I/ NA I . p 2 ti'- , , r Sh r2 ?72 " ' - 2/ / I)~ r l 4 1 /z f -'I i - a /r/ , /,,/ U (1 ' al '1 1., . 1? . 1 " r , 'I . 12 ( ) }" F?, ; _ a #)I J / f .1 '/1' 1 - a / t r 2 j 'N > l °". _!t';. /Orc " Y / _ i .. r 4 h 1 t, t . _ + , ( !f . >i_.. , ._ E .... + - r ; r % C , r t i G r _ y ,: : . ' irrl _ i i ~- al 1 r' t 1 +r 2 AS it. ,. ' y ),, ''i /t 1; y r: 1.: ; i _ t "1 _ . ,. + r': f'A--. _,E.. : V t - N/I 5 - y Kr. -; ; ;-. -. i" _ , \1; ;;'; -- - : : ' : . = ; ,::, = - ,1 . 1- __ ,. f, i..11 _. .ti ' .. --.' _<<,+ is .'t - i _ !:I%; : t tf '; it S11 t.: r-:, s' _.. r. ' /, !;- ice '' { = : ' = - f, 1 w; t.i. l l .:. ..'".; , -'1 f r , . .,r,', THE GIFT OF -t.4 EI=EI3=czm=cz= i f_' ? ) }, ; 'I < - , "' tp h J t ) ~ f ' 1) r 2) 2)1t _ - -N' ' )' (, cyK _.ll }j( ; , (27 .1 f/1 } _ a 1 t 2 t4- ' r + y, ' k I 31r' r * I .t- E r ft' r ' y F 4 r 2 "')r #'Y * 2 \It y..*1 -. t .l ( /% ,:= ;' :.: :, ,: ;;= !. ; . ; -, .;, ' ' .; ,: i;'(;. . ;- ,, ,,! J ;J ;^     2 3 3 3 -7 r Serial No. 146 (2d ed.) DEPARTMENT OF COMMERCE U. S. COAST AND GEODETIC SURVEY E. LESTER JONES, Director ELEMENTS OF MAP PROJECTION WITH APPLICATIONS TO MAP AND CHART CONSTRUCTION BY CHARLES H. DEETZ AND OSCAR S. ADAMS Special Publication No. 68 (Revised May 1, 1928) PRICE 50 CENTS $old only by the superintendent of Documents, Government Printing Office Washington, D. C. UNITED STATES GOVERNMENT PRINTING OFFICE WASHINGTON 1928 PREFACE. In this publication it has been the aim of the authors to present in simple form some of the ideas that lie at the foundation of the subject of map projections. Many people, even people of education and culture, have rather hazy notions of what is meant by a map projection, to say nothing of the knowledge of the practical con- struction of such a projection. The two parts of the publication are intended to meet the needs of such people; the first part treats the theoretical side in a form that is as simple as the authors could make it; the second part attacks the subject of the practical construction of some of the most important projections, the aim of the authors being to give such detailed directions as are necessary to present the matter in a clear and simple manner. Some ideas and principles lying at the foundation of the subject, both the- oretical and practical, are from the very nature of the case somewhat complicated, and it is a difficult matter to state them in simple manner. The theory forms an important part of the differential geometry of surfaces, and it can only be fully appreciated by one familiar with the ideas of that branch of science. Fortunately, enough of the theory can be given in simple form to enable one to get a clear notion of what is meant by a map projection and enough directions for the construction can be given to aid one in the practical development of even the more complicated projections. It is hoped that this- publication may meet the needs of people along both of the lines indicated above and that it may be found of some interest to those who may already have a thorough grasp of the subject as a whole. PREFACE TO SECOND EDITION. In the preparation of a second edition no effort has been made to revise the general text, only a few necessary corrections having been incorporated. Some new material listed in the index to addenda on page 163 continues the discussion of world maps. General observations on the azimuthal equidistant projection and on some recent contributions to mathematical cartography are also included. 2 CONTENTS. PART I. Page. General statement. . . . . . . . . ...---------.--------- -------------------------------------- 7 Analysis of the basic elements of map projection.....-..................- ........-----.------- -9 Problem to be solved-------- . . . . ..------------------------.---------------------- 9 Reference points on the sphere. . . . .. .. ..------------------------------------------------ 11 Determination of latitude.. . . .. . . . ...----------------------------------------------------12 Determination of longitude.. . .. . . . ..--------------------------------------------------- 13 Plotting points by latitude and longitude on a globe......---------------------------------14 Plotting points by latitude and longitude on a plane map... ....-----------------------------14 How to draw a straight line...-............................................- .......--.------- 15 How to make a plane surface. . . . . . ....--------------------------------------------------16 How to draw the circles representing meridians and parallels on a sphere---------.-.-.------ 17 The terrestrial globe.............--------------------------------------------------------1-19 Representation of the sphere upon a plane. . . . . . ..------------------------------------------- 22 The problem of map projection. . . . . . . . ..------------------------------------------------22 Definition of map projection. . . . . . ... ..--------------------------------------------------22 Distortion.. ..---------------------------------------------------------------- 22 Conditions fulfilled by a map projection.........------------------------------------------25 Classification of projections..............------------------------------------------------25 The ideal map. . . . . . . . . . . ..------------------------------------------------------------27 Projections considered without mathematics. . . . . . ..--------------------------------------23 Elementary discussion of various forms of projection.......------------------------------------30 Cylindrical equal-area projection. . . . . .. . ..----------------------------------------------- 30 Cylindrical equal-spaced projection....................................................... 30 Projection from the center upon a tangent cylinder........-................................ 30 Mercator projection. . . . . .. ..---------------..------------------------------------------ 32 Geometrical azimuthal projections.-...................................................... 35 Stereographic polar projection.. . . . . . . .. . ...---------------------------------------------3y Central or gnomonic projection. . . . . . . . . ..------------------------------------------------37 Lambert azimuthal equal-area projection.. . . . . .. ...--------------------------------------39 Orthographic polar projection----------------------------------------------- 3 Azimuthal equidistant projection. . . . . . . . . ..----------------------------------------------40 Other projections in frequent use. . . . . . . . ..-----------------------------------------------42 Construction of a stereographic meridional projection--------.------------------------ 44 Construction of a gnomonic projection with point of tangency on the equator------------ 45 Conical projections...................................... -............................... 46 Central projection upon a cone tangent at latitude 30.-------------------------------------47 Bonne projection...............................................................4-------- 49 Polyconic projection. . . . . . . .. ..--------------------------------------------------------49 Illustrations of relative distortions. . . . . . . ..----------------------------------------------51 PART II. Introduction. . . . . . . . . . . . . . . . ...-------------------------------------------------------------53 Projections described in Part II...........------------------------------------------------53 The choice of projection. . . . . . . . .. . ..-----------------------------------------------------54 Comparison of errors of scale and errors of area in a map of the United States on four different projections........... ....- . .-.-.-- .-.- ...--. ..... ...... ......54 The polyconic projection. . . . . . . . . . ..--------------------------------------------------------58 Description. . . . . . . . . .. . . . ..--------------------------------------------------------------58 Construction of a polyconic projection..........................................60 Transverse polyconic projection.................................-------------------- 62 Polyconic projection with two standard meridians, as used for the international map of the world, scale 1:1 000 000................................................................. 62 3,. 4 CONTENTS. Page. The Bonne projection......................................................................... 67 Description.. . . . . . . . . . . . . ..--------------------------------------------------------------67 The Sanson-Flamsteed projection.. . . . . .. . ..---------------------------------------------- 68 Construction of a Bonne projection. . . . .. . . . ..---------------------------------------------68 The Lambert zenithal (or azimuthal) equal-area projection............... ---............-... ...-. 71 Description.. . . . . . . . . . . . . ..--------------------------------------------------------------71 The Lambert equal-area meridional projection.- .......-...-.............. -................. -73 Table for the construction of a Lambert zenithal equal-area projection with center on parallel 40' 73 Table for the construction of a Lambert zenithal equal-area meridional projection...-..- 75 The Lambert conformal conic projection with two standard parallels-..................-........-77 Description.. . . . . . . . . . . . . ..--------------------------------------------------------------77 Construction of a Lambert conformal conic projection....-................................... -83 Table for the construction of a Lambert conformal conic projection with standard parallels at 360 and 540...- ...- ...-.- ......--...............-...........--------------------------- 85 Table for the construction of a Lambert conformal conic projection with standard parallels at 100 and 480 40'. . . . . . . . . . . ...----------------------------------------------------------86 The Grid system of military mapping. . . . .. . . ..-----------------------------------------------87 Grid system for progressive maps in the United States-.-...-.....-....-...........-.........-87 The Albers conical equal-area projection with two standard parallels.........--....................-91 Description... . . . . . . . . . . . ..--------------------------------------------------------------91 Mathematical theory of the Albers projection....................... --..................-.... 93 Construction of an Albers projection.........................-....-....................... -99 Table for the construction of a map of the United States on Albers equal-area projection with two standard parallels. . . . . . . . . . ..----------------------------------------------------100 The Mercator projection...............--------------------------------------------------------101 Description ... . . . . . . . . . . . ..--------------------------------------------------------------101 Development of the formulas for the coordinates of the Mercator projection....-..............-105 Development of the formulas for the transverse Mercator projection-...........................108 Construction of a Mercator projection...........-------------------------------------------109 Construction of a transverse Mercator projection for the sphere with the cylinder tangent along a meridian.. . . . . . . . . . . . ..--------------------------------------------------------114 Mercator projection table..............----------------------------------------------------116 Fixing position by wireless directional bearings.................-.-.............. -... ............ 137 The gnomonic projection...........-------------------------------------------------------140 Description. .. . . . . . . . . . . ..--------------------------------------------------------------140 Mathematical theory of the gnomonic projection-........................................... 141 WORLD MAPS. The Mercator projection..............--------------------------------------------------------146 The stereographic projection.. . . . . . . . .. . ..----------------------------------------------------147 The Aitoff equal-area projection of the sphere............-.....-...... -... -......... -.......... 150 Table for the construction of an Aitoff equal-area projection of the sphere-----................. 152 The Mollweide homalographic projection------- -- --------------------------------153. Construction of the Mollweide homalographic projection of a hemisphere.........-..............-154 Homalographic projection of the sphere..........................--............-......... -155 Table for the construction of the Mollweide homalographic projection..........................-155 Goode's homalographic projection (interrupted) for the continents and oceans-....................-156 Lambert projection of the northern and southern hemispheres....................................-.-158 Conformal projection of the sphere within a two-cusped epicycloid........................-160 Guyou's doubly periodic projection of the sphere..........-------------------------------------160 Index........................-------------....-------------...------------------------.. .161 ILLUSTRATIONS. 5 ILLUSTRATIONS. FIGURES. Page. Frontispiece. Diagram showing lines of equal scale error or linear distortion in the polyconic, Lambert zenithal, Lambert conformal, and Albers projections...-...-.........(facing page) 7 1. Conical surface cut from base to apex. . . . . . ..---------------------------------------------9 2. Development of the conical surface3........................................................ 10 3. Cylindrical surface cut from base to base................................................... 10 4. Development of the cylindrical surface. . . . . . . ..-------------------------------------------11 5. Determination of the latitude of a place..........------------------------------------------32 6. Construction of a straight edge............-------------------------------------------------16 7. Constructing the circles of parallels and meridians on a globe---------- .--------------- - 18 8. Covering for a terrestrial globe...........-------------------------------------------------20 9. Pack of cards before "shearing". . . . . . . . ..------------------------------------------------23 10. Pack of cards after "shearing"..-.-..---......... .... .... .... .... .... .... ...23 11. Square "sheared" into an equivalent parallelogram. . . .. ..----------------------------------23 12. The Mollweide equal-area projection of the sphere......-.-.-------------------------------24 13. Earth considered as formed by plane quadrangles. . . . . . ..-----------------------------------28 14. Earth considered as formed by bases of cones. . . . . . ..--------------------------------------29 15. Development of the conical bases. . ... ...-------------------------------------------..---.29 16. Cylindrical equal-area projection. . . . . . . . . ..-----------------------------------------------31 17. Cylindrical equal-spaced projection. . . . . . . . . ..---------------------------------------------31 18. Modified cylindrical equal-spaced projection-------.--.------.----------------------- 32 19. Perspective projection upon a tangent cylinder. . . . . ..-------------------------------------33 20. Mercator projection. . . . . . . . .. . ..---------------------------------------------------------34 21. Determination of radii for stereographic polar projection.....-------------------------------35 22. Stereographic polar projection. . . .. . . . . ..-------------------------------------------------36 23. Determination of radii for gnomonic polar projection.........---------------------------------37 24. Gnomonic polar projection. . . . . . . . . ..----------------------------------------------------38 25. Determination of radii for Lambert equal-area polar projection--------- -----------------39 26. Lambert equal-area polar projection.........-................. -............................ -39 27. Determination of radii for orthographic polar projection......-------------------------------40 28. Orthographic polar projection. . . . . . . . . . ..--------------------------------------------------40 29. Azimuthal equidistant polar projection. . . . . . ..-------------------------------------------41 30. Stereographic projection of the Western Hemisphere-----.----------------------------41 31. Gnomonic projection of part of the Western Hemisphere.......------------------------------42 32. Lambert equal-area projection of the Western Hemisphere...........------------.-.----------------42 33. Orthographic projection of the Western Hemisphere--......---.----------------------------43 34. Globular projection of the Western Hemisphere...-......................................... 43 35. Determination of the elements of a stereographic projection on the plane of a meridian...-.... 44 36. Construction of a gnomonic projection with plane tangent at the Equator........-............ 45 37. Cone tangent to the sphere at latitude 30*. . . . ..-----------------------------------------46 38. Determination of radii for conical central perspective projection--------.--- --------------47 39. Central perspective projection on cone tangent at latitude 30------------------------------- 48 40. Bonne projection of the United States.. .....--------------------------------------------49 41. Polyconic projection of North America. . . . . . . ..-------------------------------------------50 42. Man's head drawn on globular projection. . . . . . ..-----------------------------------------51 43. Man's head plotted on orthographic projection.........--------------------------------------51 44. Man's head plotted on stereographic projection. . . . .. . ..-------------------------------------51 45. Man's head plotted on Mercator projection. . .. .. ..----------------------------------------51 46. Gnomonic projection of the sphere on a circumscribed cube..................................-52 47. Polyconic development of the sphere. . . .. . . . ..--------------------------------------------58 48. Polyconic development. . . . . . . . .. ..------------------------------------------------------59 49. Polyconic projection--construction plate. . . . . . ..------------------------------------------61 50. International map of the world-junction of sheets. . . . . ..----------------------------------65 51. Bonne projection of hemisphere. . . . . .. . . . ..------------------------------------------------67 52. Lambert conformal conic projection................................------.------.......... 77 53. Scale distortion of the Lambert conformal conic projection with the standard parallels at 290 and 45. ............................................................................... 79 6 ILLUSTRATIONS. Page. 54. Scale distortion of the Lambert conformal conic projection with the standard parallels at 330 and 45 .............................................................................. 80 55. Diagram for constructing a Lambert projection of small scale................................-84 56. Grid zones for progressive military maps of the United States............... ----------. ..... 88 57. Diagram of zone C, showing grid system .............. -............ ...-...................... 89 58. Part of a Mercator chart showing a rhumb line and a great circle..............................-102 59. Part of a gnomonic chart showing a great circle and a rhumb line..........-.................-102 60. Mercator projection-construction plate.. . . . . .. . ..------------------------------------------111 61. Transverse Mercator projection-cylinder tangent along a meridian-construction plate------- 115 62. Fixing positions by wireless directional bearings..........................-......-(facing page) 138 63. Diagram illustrating the theory of the gnomonic projection................-.....................-140 64. Gnomonic projection-determination of the radial distance...................................-142 65. Gnomonic projection-determination of the coordinates on the mapping plane..-............-142 66. Gnomonic projection-transformation triangle on the sphere.....-............................-143 67. Mercator projection, from latitude 600 south, to latitude 780 north... -.................... 146 68. Stereographic meridional projection.........................................................148 69. Stereographic horizon projection on the horizon of Paris..-................-...-.................-149 70. The Aitoff equal-area projection of the sphere with the Americas in center...................-151 71. The Mollweide homalographic projection of the sphere......................................-153 72. The Mollweide homalographic projection of a hemisphere....-..........-.............154 73. Homalographic projection (interrupted) for ocean units.....................-...................-157 74. Guyou's doubly periodic projection of the sphere.......-----------------------------------159 75. Sinusoidal or Mercator equal-area projection.................- ........(facing page) 164 76. Azimuthal equidistant projection of the world...--.....--....-...--...(facing page) 166 FOLDED PLATES. Following page. I. Lambert conformal conic projection of the North Atlantic Ocean-...................167 II. Transverse polyconic projection of the North Pacific Ocean, showing Alaska and its relation to the United States and the Orient, scale 1:40 000 000..-....-.-....-.-....-..-167 III. Albers projection of the United States.....-.-..-..-.-....---.-..-.-....-.-.. -.167 IV. Gnomonic projection of part of the North Atlantic Ocean..........................167 V. The Aitoff equal-area projection of the sphere..-.-..-.-..-..-.-..-.-..-..-.-.. -.167 VI. The world on the homalographic projection (interrupted for the continents)...-.-..-....167 VII. Lambert projection of the Northern and Southern Hemispheres.....................167 VIII. Conformal projection of the sphere within a two cusped epicycloid................-..167  f, Z Sca~ error of Lam bert Znta I °I Lines o scaleerr r o linirhdlstortion@ Poyoicpoecin 2,@%ad %s!onb eria oe ines La6rtCnfral2Iad4%sew y ote ins(es adwet Lamer Zeital 2%s2wnby ondngcicl Ales .................. and..._ show..ydotand.dahlines(east..d.west F~oNrssPucE--Digra shoingLines ofeq scale error or linear distortion i h oyo~c abr zenithal, :Lambert conformal, and Albers projections. (See statistics on pp. 54, 55.) 105877-28. (T1o faco p~age 7.) ELEMENTS OF MAP PROJECTIONS WITH APPLICATIONS TO MAP AND CHART CONSTRUCTION. By CHARLES H. DEET, Cartographer, and OSCAR S. ADAMS, GeOdetiC Computer. PART I. GENERAL STATEMENT.' Whatever may be the destiny of man in the ages to come, it is certain that for the present his sphere of activity is restricted to the outside shell of one of the smaller planets of the solar system-a system which after all is by no means the largest in the vast universe of space. By the use of the imagination and of the intellect with which he is endowed he may soar into space and investigate, with more or less certainty, domains far removed from his present habitat; but as regards his actual presence, he can not leave, except by insignificant distances, the outside crust of this small earth upon which he has been born, and which has formed in the past, and must still form, the theater upon which his activities are displayed. The connection between man and his immediate terrestrial surroundings is there- fore very intimate, and the configuration of the surface features of the earth would thus soon attract his attention. It is only reasonable to suppose that, even in the most remote ages of the history of the human race, attempts were made, however crude they may have been, to depict these in some rough manner. No doubt these first attempts at representation were scratched upon the sides of rocks and upon the walls of the cave dwellings of our primitive forefathers. It is well, then, in the light of present knowledge, to consider the structure of the framework upon which this representation is to be built. At best we can only partially succeed in any attempt at representation, but the recognition of the possibilities and the limitations will serve as valuable aids in the consideration of any specific problem. We may reasonably assume that the earliest cartographical representations con- sisted of maps and plans of comparatively small areas, constructed to meet some need of the times, and it would be later on that any attempt would be made to extend the representation to more extensive regions. In these early times map mak- ing, like every other science or art, was in its infancy, and probably the first attempts of the kind were not what we should now call plans or maps at all, but rough per- spective representations of districts or sketches with hills, forests, lakes, etc., all shown as they would appear to a person on the earth's surface. To represent these features in plan form, with the eye vertically over the various objects, although of very early origin, was most likely a later development; but we are now never likely to know who started the idea, since, as we have seen, it dates back far into antiquity. Geography is many-sided, and has numerous branches and divisions; and though it is true that map making is not the whole of geography, as it would be well for us 1 Paraphrased from "Maps and Map-making," by E. A. Reeves, London, 1910. 7 8 U. S. COAST AND GEODETIC SURVEY. to remind ourselves occasionally, yet it is, at any rate, a very important part of it, and it is, in fact, the foundation upon which all other branches must necessarily depend. If we wish to study the structure of any region we must have a good map of it upon which the various land forms can be shown. If we desire to represent the distribution of the races of mankind, or any other natural phenomenon, it is essen- tial, first of all, to construct a reliable map to show their location. For navigation, for military operations, charts, plans, and maps are indispensable, as they are also for the demarcation of boundaries, land taxation, and for many other purposes. It may, therefore, be clearly seen that some knowledge of the essential qualities inherent in the various map structures or frameworks is highly desirable,. and in any case the makers of maps should have a thorough grasp of the properties and limitations of the various systems of projection. ANALYSIS OF THE BASIC ELEMENTS OF MAP PROJECTION.1a PROBLEM TO BE SOLVED. A map is a small-scale, fliat-surface representation of some portion of the sur- face of the earth. Nearly every person from time to time makes use of maps, and our ideas with regard to the relative areas of the various portions of the earth's surface are in general derived from this source. The shape of the land masses and their positions with respect to one another are things about which our ideas are influenced by the way these features are shown on the maps with which we become familiar. It is fully established to-day that the shape of the earth is that of a slightly irregular spheroid, with the polar diameter about 26 miles shorter than the equatorial. The spheroid adopted for geodetic purposes is an ellipsoid of revolution formed by revolving an ellipse about its shorter axis. For the purpose of the present discussion the earth may be considered as a sphere, because the irregularities are very small compared with the great size of the earth. If the earth were represented by a spheroid with an equatorial diameter of 25 feet, the polar diameter would be approxi- mately 24 feet 11 inches. FIG. 1.-Conical surface cut from base to apex. The problem presented in map making is the question of representing the sur- face of the sphere upon a plane. It requires some thought to arrive at a proper appreciation of the difficulties that have to be overcome, or rather that have to be dealt with and among which there must always be a compromise; that is, a little of one desirable property must be sacrificed to attain a little more of some other special feature. In the first place, no portion of the surface of a sphere can be spread out in a plane without some stretching or tearing. This can be seen by attempting to flatten out a cap of orange peel or a portion of a hollow rubber ball; the outer part must be stretched or torn, or generally both, before the central part will come into the plane with the outer part. This is exactly the difficulty that has to be contended with in map making. There are some surfaces, however, that can be spread out in a plane without any stretching or tearing. Such surfaces are called developable surfaces and those like the sphere are called nondevelopable. The cone and the 'a Some of the elementary text in Part I and a number of the illustrations were adapted from a publication issued by G. Philip & Son (Ltd.), of London, England. The work is entitled "A Little Book on Map Projection," and its author is William Garnett writing under the pseudonym of Mary Adams. We wish to acknowledge our indebtedness to this work which sets forth in simple illustrations some of the fundamental facts in regard to map construction. 9 10 U. S. COAST AND GEODETIC SURVEY. cylinder are the two well-known surfaces that are developable. If a cone of revo- lution, or a right circular cone as it is called, is formed of thin material like paper Fir. 2.-Development of the conical surface. and if it is cut from some point in the curve bounding the base to the apex, the conical surface can be spread out in a plane with no stretching or tearing. (See figs. 1 and 2.) Any curve drawn on the surface will have exactly the same length after development that it had before. In the same way, if a cylindrical surface is FiG. 3.-Cylindrical surface cut from base to base. cut from base to base the whole surface can be rolled out in the plane, if the surface consists of thin pliable material. (See figs. 3 and 4.) In this case also there is no stretching or tearing of any part of the surface. Attention is called to the develop- able property of these surfaces, because use will be made of them in the later dis- cussion of the subject of map making. ANALYSIS OF THE BASIC ELEMENTS OF MAP PROJECTION. 11 FIG. 4.-Development of the cylindrical surface. REFERENCE POINTS ON THE SPHERE. A sphere is such that any point of it is exactly like any other point; there is neither beginning nor ending as f ar as differentiation of points is concerned. On the earth it is necessary to have some points or lines of reference so that other points may be located with regard to them. Places on the earth are located by latitude and longitude, and it may be well to explain how these quantities are related to the terrestrial sphere. The earth sphere rotates on its axis once a day, and this axis is therefore a definite line that is different from every other diameter. The ends of this diameter are called the poles, one the North Pole and the other the South Pole. With these as starting points, the sphere is supposed to be divided into two equal parts or hemispheres by a plane perpendicular to the axis midway between the poles. The circle formed by the intersection of this plane with the surface of the earth is called the Equator. Since this line is defined with reference to the poles, it is a definite line upon the earth. All circles upon the earth which divide it into two equal parts are called great circles, and the Equator, therefore, is a great circle. It is customary to divide the circle into four quadrants and each of these into 90 equal parts called degrees. There is no reason why the quadrant should not be divided into 100 equal parts, and in fact this division is sometimes used, each part being then called a grade. In this country the division of the quadrant into 90° is almost universally used; and accordingly the Equator is divided into 3600. * After the Equator is thus divided into 3600, there is difficulty in that there is no point at which to begin the count; that is, there is no definite point to count as zero or as the origin or reckoning. This difficulty is met by the arbitrary choice of some point the significance of which will be indicated after some preliminary explanations. Any number of great circles can be drawn through the two poles and each one of them will cut the Equator into two equal parts. Each one of these great circles may be divided into 360°, and there will thus be 90° between the Equator and each pole on each side. These are usually numbered from 0* to 90° from the Equator to the pole, the Equator being 0° and the pole 90°. These great circles through the poles are called meridians. Let us suppose now that we take a point on one of these 300 north of the Equator. Through this point pass a plane perpendicular 12 U. S. COAST AND GEODETIC SURVEY. to the axis, and hence parallel to the plane of the Equator. This plane will intersect the surface of the earth in a small circle, which is called a parallel of latitude, this particular one being the parallel of 30 north latitude. Every point on this parallel will be in 300 north latitude. In the same way other small circles are determined to represent 200, 400, etc., both north and south of the Equator. It is evident that each of these small circles cuts the sphere, not into two equal parts, but into two unequal parts. These parallels are drawn for every 10°, or for any regular interval that may be selected, depending on the scale of the sphere that represents the earth. The point to bear in mind is that the Equator was drawn as the great circle midway between the poles; that the parallels were constructed with reference to the Equator; and that therefore they are definite small circles referred to the poles. Nothing is arbitrary except the way in which the parallels of latitude are numbered. DETERMINATION OF LATITUDE. The latitude of a place is determined simply in the following way: Very nearly L' U) 0- 0 O 0F B 0 4- A 0 y\ G Fia. 5.-Determination of the latitude of a place. in the prolongation of the earth's axis to the north there happens to be a star, to which the name polestar has been given. If one were at the North Pole, this star would appear to him to be directly overhead. Again, suppose a person to be at the Equator, then the star would appear to him to be on the horizon, level with his eye. It might be thought that it would be below his eye because it is in line with the earth's axis, 4,000 miles beneath his feet, but the distance of the star is so enormous that the radius of the earth is exceedingly small as compared with it. All lines to the star from different points on the earth appear to be parallel. ANALYSIS OF THE BASIC ELEMENTS OF MAP PROJECTION. 13 Suppose a person to be at A (see fig. 5), one-third of the distance between the Equator and the North Pole, the line BC will appear to him to be horizontal and he will see the star one-third of the way up from the horizon to the point in the heavens directly overhead. This point in the line of the vertical is called the zenith. It is now seen that the latitude of any place is the same as the height of the polestar above the horizon. Most people who have traveled have noticed that as they go south the polestar night by night appears lower in the heavens and gradually dis- appears, while the Southern Cross gradually comes into view. At sea the latitude is determined every day at noon by an observation of the sun, but this is because the sun is brighter and more easily observed. Its distance from the pole, which varies throughout the year, is tabulated for each day in a book called the Nautical Almanac. When, therefore, an observation of the sun is made, its polar distance is allowed for, and thus the latitude of the ship is determined by the height of the pole in the heavens. Even the star itself is directly observed upon from time to time. This shows that the latitude of a place is not arbitrary. If the star is one-third of the way up, measured from the horizon toward the zenith, then the point of observation is one-third of the way up from the Equator toward the pole, and nothing can alter this fact. By polestar, in the previous discussion, is really meant the true celestial pole; that is, the point at which the prolongation of the earth's axis pierces the celestial sphere. Corrections must be made to the observations on the star to reduce them to this true pole. In the Southern Hemis- phere latitudes are related in a similar way to the southern pole. Strictly speaking, this is what is called the astronomical latitude of a place. There are other latitudes which differ slightly from that described above, partly because the earth is not a sphere and partly on account of local attractions, but the above-described latitude is not only the one adopted in all general treatises but. it is also the one employed on all general maps and charts, and it is the latitude by which all navigation is conducted; and if we assume the earth to be a homogene- ous sphere, it is the only latitude. DETERMINATION OF LONGITUDE. This, however, fixes only the parallel of latitude on which a. place is situated. If it be found that the latitude of one place is 100 north and that of another 20 north, then the second place is 10° north of the first, but as yet we have no means of showing whether it is east or west of it. If at some point on the earth's surface a perpendicular pole is erected, its shadow in the morning will be on the west side of it and in the evening on the east side of it. At a certain moment during the day the shadow will lie due north and south. The moment at which this occurs is called noon, and it will be the same for all points exactly north or south of the given point. A great circle passing through the poles of the earth and through the given point is called a meridian (from merides, midday), and it is therefore noon at the same moment at all points on this meridian. Let us suppose that a chronometer keeping correct time is set at noon at a given place and then carried to some other place. If noon at this latter place is observed and the time indicated by the chronometer is noted at the same moment, the difference of time will be proportional to the part of the earth's circumference to the east or west that has been passed over. Suppose that the chronometer shows 3 o'clock when it is noon at the place of arrival, then the meridian through the new point is situated one-eighth of the way around the world to the westward from the first point. This difference is a definite quantity and has nothing arbitrary about it, but it would be 14 U. S. COAST AND GEODETIC SURVEY. exceedingly inconvenient to have to work simply with differences between the various places, and all would be chaos and confusion unless some place were agreed upon as the starting point. The need of an origin of reckoning was evident as soon as longitudes began to be thought of and long before they were accurately determined. A great many places have in turn been used; but when the English people began to make charts they adopted the meridian through their principal observatory of Greenwich as the origin for reckoning longitudes and this meridian has now been adopted by many other countries. In France the meridian of Paris is most generally used. , The adoption of any one meridian as a standard rather than another is purely arbitrary, but it is highly desirable that all should use the same standard. The division of the Equator is made to begin where the standard meridian crosses it and the degrees are counted 180 east and west. The standard meridian is sometimes called the prime meridian, or the first meridian, but this nomenclature is slightly misleading, since this meridian is really the zero meridian. This great circle, therefore, which passes through the poles and through Greenwich is called the merid- ian of Greenwich or the meridian of 00 on one side of the globe, and the 180th meridian on the other side, it being 180 east and also 180 west of the zero meridian. By setting a chronometer to Greenwich time and observing the hour of noon at various places their longitude can be determined, by allowing 150 of longitude to each hour of time, because the earth turns on its axis once in 24 hours, but there are 360° in the entire circumference. This description of the method of determining differences of longitude is, of course, only a rough outline of the way in which they can be determined. The exact determination of a difference of longitude between two places is a work of considerable difficulty and the longitudes of the principal observatories have not. even yet been determined with sufficient degree of accuracy for certain delicate observations. PLOTTING POINTS BY LATITUDE AND LONGITUDE ON A GLOBE. If a globe has the circles of latitude and longitude drawn upon it according to the principles described above and the latitude and longitude of certain places have been determined by observation, these points can be plotted upon the globe in their proper positions and the detail can be filled in by ordinary surveying, the detail being referred to the accurately determined points. In this way a globe can be formed that is in appearance a small-scale copy of the spherical earth. This copy will be more or less accurate, depending upon the number and distribution of the accurately located points. PLOTTING POINTS BY LATITUDE AND LONGITUDE ON A PLANE MAP. If, in the same way, lines to represent latitude and longitude be drawn on a plane sheet of paper, the places can be plotted with reference to these lines and the detail filled in by surveying as before. The art of making maps consists, in the first place, in constructing the lines to represent latitude and longitude, either as nearly like the lines on the globe as possible when transferred from a nondevelopable surface to a flat surface, or else in such a way that some one property of the lines will be retained at the expense of others. It would be practically impossible to transfer the irregular coast lines from a globe to a map; but it is comparatively easy to transfer the regular lines representing latitude and longitude. It is possible to lay down on a map the lines representing the parallels and meridians on a globe many feet in diameter. These lines of latitude and longitude may be laid down for every 100, for every degree, or for any other regular interval either greater or smaller. ANALYSIS OF THE BASIC ELEMENTS OF MAP PROJECTION. 15 In any case, the thing to be done is to lay down the lines, to plot the principal points, and then to fill in the detail by surveying. After one map is made it may be copied even on another kind of projection, care being taken that the latitude and longitude of every point is kept correct on the copy. It is evident that if the lines of latitude and longitude can not be laid down correctly upon a plane surface, still less can the detail be laid down on such a surface without distortions. Since the earth is such a large sphere it is clear that, if only a small portion of a country is taken, the surface included will differ but very little from a plane surface. Even two or three hundred square miles of surface could be represented upon a plane with an amount of distortion that would be negligible in practical mapping. The difficulty encountered in mapping large areas is gotten over by first making many maps of small area, generally such as to be bounded by lines of latitude and longi- tude. When a large number of these maps have been made it will be found that they can not be joined together so as to lie flat. If they are carefully joined along the edges it will be found that they naturally adapt themselves to the shape of the globe. To obviate this difficulty another sheet of paper is taken and on it are laid down the lines of latitude and longitude, and the various maps are copied so as to fill the space allotted to them on this larger sheet. Sometimes this can be done by a simple reduction which does not affect the accuracy, since the accuracy of a map is independent of the scale. In most cases, however, the reduction will have to be unequal in different directions and sometimes the map has to be twisted to fit into the space allotted to it. The work of making maps therefore consists of two separate processes. In the first place, correct maps of small areas must be made, which may be called survey- ing; and in the second place these small maps must be fitted into a system of lines representing the meridians and parallels. This graticule of the orderly arrange- ment of lines on the plane to represent the meridians and parallels of the earth is called a map projection. A discussion of the various ways in which this graticule of lines may be constructed so as to represent the meridians and parallels of the earth and at the same time so as to preserve some desired feature in the map is called a treatise on map projections. HOW TO DRAW A STRAIGHT LINE. Few people realize how difficult it is to draw a perfectly straight line when no straightedge is available. When a straightedge is used to draw a straight line, a copy is really made of a straight line that is already in existence. A straight line is such that if any part of it is laid upon any other part so that two points of the one part coincide with two points of the other the two parts will coincide through- out. The parts must coincide when put together in any way, for an are of a circle can be made to coincide with any other part of the same circumference if the arcs are brought together in a certain way. A carpenter solves the problem of joining two points by a straight line by stretching a chalk line between them. When the line is taut, he raises it slightly in the middle portion and suddenly releases it. Some of the powdered chalk flies off and leaves a faint mark on the line joining the points. This depends upon the principle that a stretched string tends to become as short as possible unless some other force is acting upon it than the tension in the direction of its length. This is not a very satisfactory solution, however, since the chalk makes a line of considerable width, and the line will not be perfectly straight unless extra precautions are taken. 16 U. S. COAST AND GEODETIC SURVEY. A straightedge can be made by clamping two thin boards together and by planing the common edge. As they are planed together, the edges of the two will be alike, either both straight, in which case the task is accomplished, or they will be both convex, or both concave. They must both be alike; that is, one can not be convex and the other concave at any given point. By unclamping them it can be seen whether the planed edges fit exactly when placed together, or whether they need some more planing, due to being convex or concave or due to being convex in places and concave in other places. (See fig. 6.) By repeated trials and with sufficient patience, a straightedge can be made in this way. In practice, of course, a straightedge in process of construction is tested by one that has already been made. Machines for drawing straight lines can be constructed by linkwork, but they are seldom used in practice. A B C FIG. 6.-Construction of a straight edge. It is in any case difficult to draw straight lines of very great length. A straight line only a few hundred meters in length is not easy to construct. For very long straight lines, as in gunnery and surveying, sight lines are taken; that is, use is made of the fact that when temperature and pressure conditions are uniform, light travels through space or in air, in straight lines. If three points, A, B, and C, are such that B appears to coincide with C when looked at from A, then A, B, and C are in a straight line. This principle is made use of in sighting a gun and in using the telescope for astronomical measurements. In surveying, directions which are straight lines are found by looking at the distant object, the direction of which from the point of view we want to determine, through a telescope. The telescope is moved until the image of the small object seen in it coincides with a mark fixed in the telescope in the center of the field of view. When this is the case, the mark, the center of the object glass of the telescope, and the distant object are in one straight line. A graduated scale on the mounting of the telescope enables us to determine the direction of the line joining the fixed mark in the telescope and the center of the object glass. This direction is the direction of the distant object as seen by the eye, and it will be determined in terms of another direction assumed as the initial direction. HOW TO MAKE A PLANE SURFACE. While a line has length only, a surface has length and breadth. Among sur- faces a plane surface is one on which a straight line can be drawn through any point in any direction. If a straightedge is applied to a plane surface, it can be turned around, and it will in every position coincide throughout its entire length with the ANALYSIS OF THE BASIC ELEMENTS OF MAP PROJECTION. 17 surface. Just as a straightedge can be used to test a plane, so, equally well, can a plane surface be used to test a straightedge, and in a machine shop a plate with plane surface is used to test accurate workmanship. The accurate construction of a plane surface is thus a problem that is of very great practical importance in engineering. A very much greater degree of accuracy is required than could be obtained by a straightedge applied to the surface in dif- ferent directions. No straightedges in existence are as accurate as it is required that the planes should be. The method employed is to make three planes and to test them against one another two and two. The surfaces, having been made as truly plane as ordinary tools could render them, are scraped by hand tools and rubbed together from time to time with a little very fine red lead between them. Where they touch, the red lead is rubbed off, and then the plates are scraped again to remove the little elevations thus revealed, and the process is continued until all the projecting points have been removed. If only two planes were worked together, one might be convex (rounded) and the other concave (hollow), and if they had the same curvature they might still touch at all points and yet not be plane; but if three surfaces, A, B, and C, are worked together, and if A fits both B and C and A is concave, then B and C must be both convex, and they will not fit one another. If B and C both fit A and also fit one another at all points, then all three must be truly plane. When an accurate plane-surface plate has once been made, others can be made one at a time and tested by trying them on the standard plate and moving them over the surface with a little red lead between them. When two surface plates made as truly plane as possible are placed gently on one another without any red lead between them, the upper plate will float almost without friction on a very thin layer of air, which takes a very long time to escape from between the plates, because they are everywhere so very near together. HOW TO DRAW THE CIRCLES REPRESENTING MERIDIANS AND PARALLELS ON A SPHERE. We have seen that it is difficult to draw a straight line and also difficult to construct a plane surface with any degree of accuracy. The problem of constructing circles upon a sphere is one that requires some ingenuity if the resulting circles are to be accurately drawn. If a hemispherical cup is constructed that just fits the sphere, two points on the rim exactly opposite to one another may be determined. (See fig. 7.) To do this is not so easy as it appears, if there is nothing to mark the center of the cup. The diameter of the cup can be measured and a circle can be drawn on cardboard with the same diameter by the use of a compass. The center of this circle will be marked on the cardboard by the fixed leg of the compass and with a straight edge a diameter can be drawn through this center. This circle can then be cut out and fitted just inside the rim of the cup. The ends of the diameter drawn on the card then mark the two points required on the edge of the cup. With some suitable tool a small notch can be made at each point on the edge of the cup. Marks should then be made on the edge of the cup for equal divisions of a semicircle. If it is desired to draw the parallels for every 100 of latitude, the semicircle must be divided into 18 equal parts. This can be done by dividing the cardboard circle by means of a protractor and then by marking the corresponding points on the edge of the cup. The sphere can now be put into the cup and points on it marked corre- sponding to the two notches in the edge of the cup. Pins can be driven into these points and allowed to rest in the notches. If the diameter of the cup is such that 105877*-28-2 18 U. S. COAST AND GEODETIC SURVEY. the sphere just fits into it, it can be found whether the pins are exactly in the ends of a diameter by turning the sphere on the pins as -an axis. If the pins are not correctly placed, the sphere will not rotate freely. The diameter determined by the pins may now be taken as the axis, one of the ends being taken as the North Pole FIG. 7.-Constructing the circles of parallels and meridians on a globe. and the other as the South Pole. With a sharp pencil or with an engraving tool circles can be drawn on the sphere at the points of division on the edge of the cup by turning the sphere on its axis while the pencil is held against the surface at the correct point. The circle midway between the poles is a great circle and will repre- sent the Equator. The Equator is then numbered 00 and the other eight circles on either side of the Equator are numbered 100, 20, etc. The poles themselves corre- spond to 90°. Now remove the sphere and, after removing one of the pins, insert the sphere again in such a way that the Equator lies along the edge of the cup. Marks can then be made on the Equator corresponding to the marks on the edge of the cup. In this way the divisions of the Equator corresponding to the meridians of 100 interval are determined. By replacing the sphere in its original position with the pins inserted, the meridians can be drawn along the edge of the cup through the various marks on the Equator. These will be great circles passing through the poles. One of these circles is numbered 00 and the others 100, 20, etc., both east and west of the zero meridian and extending to 1800 in both directions. The one hundred and eightieth meridian will be the prolongation of the zero meridian through the poles and will be the same meridian for either east or west. This sphere, with its two sets of circles, the meridians and the parallels, drawn upon it may now be taken as a model of the earth on which corresponding circles are supposed to be drawn. When it is a question only of supposing the circles to be drawn, and not actually drawing them, it will cost no extra effort to suppose them drawn and numbered for every degree, or for every minute, or even for every second of are, but no one would attempt actually to draw them on a model globe for inter- vals of less than 10. On the earth itself a second of latitude corresponds to a little more than 100 feet. For the purpose of studying the principles of map projection it is quite enough to suppose that the circles are drawn at intervals of 100. ANALYSIS OF THE BASIC ELEMENTS OF MAP PROJECTION. I9 It was convenient in drawing the meridians and parallels by the method just described to place the polar axis horizontal, so that the sphere might rest in the cup by its own weight. Hereafter, however, we shall suppose the sphere to be turned so that its polar axis is vertical with the North Pole upward. The Equator and all the parallels of latitude will be horizontal, and the direction of rotation corresponding to the actual rotation of the earth will carry the face of the sphere at which we are looking from left to right; that is, from west to east, according to the way in which the meridians were marked. As the earth turns from west to east a person on its surface, unconscious of its movement and looking at the heavenly bodies, naturally thinks that they are moving from east to west. Thus, we say that the sun rises in the east and sets in the west. THE TERRESTRIAL GLOBE. With the sphere thus constructed with the meridians and parallels upon it, we get a miniature representation of the earth with its imaginary meridians and parallels. On this globe the accurately determined points may be plotted and the shore line drawn in, together with the other physical features that it is desirable to show. This procedure, however, would require that each individual globe should be plotted by hand, since no reproductions could be printed. To meet this difficulty, ordinary terrestrial globes are made in the following way: It is well known that a piece of paper can not be made to fit on a globe but a narrow strip can be made to fit fairly well by some stretching. If the strip is fastened upon the globe when it is wet, the paper will stretch enough to allow almost a perfect fit. Accordingly 12 gores are made as shown in figure 8, such that when fastened upon the globe they will reach from the parallel of 700 north to 70° south. A circular cap is then made to extend from each of these parallels to the poles. Upon these gores the projection lines and the outlines of the continents are printed. They can then be pasted upon the globe and with careful stretching they can be made to adapt themselves to the spherical surface. It is obvious that the central meridian of each gore is shorter than the bounding meridians, whereas upon the globe all of the meridians are of the same length. Hence in adapting the gores to the globe the central meridian of each gore must be slightly stretched in comparison with the side meridians. The figure 8 shows on a very small scale the series of gores and the polar caps printed for covering a globe. These gores do not constitute a map. They are as nearly as may be on a plane surface, a facsimile of the surface of the globe, and only require bending with a little stretching in certain directions or contraction in others or both to adapt them- selves precisely to the spherical surface. If the reader examines the parts of the continent of Asia as shown on the separate gores which are almost a facsimile of the same portion of the globe, and tries to piece them together without bending them over the curved surface of the sphere, the problem of map projection will probably present itself to him in a new light. It is seen that although the only way in which the surface of the earth can be represented correctly consists in making the map upon the surface of a globe, yet this is a difficult task, and, at the best, expedients have to be resorted to unless the work of construction is to be prohibitive. It should be remembered, however, that the only source of true ideas regarding the mapping of large sections of the surface of the earth must of necessity be obtained from its representation on a globe. Much good would result from making the globe the basis of all elementary teaching in geography. The pupils should be warned that maps are very generally used because of their convenience. Within proper limitations they serve every purpose for 20 U. S. COAST AND GEODETIC SURVEY. 0 which they are intended. Errors are dependent upon the system of projection used and when map and globe 2 do not agree, the former is at fault. This would seem to be a criticism against maps in general and where large sections are involved and where unsuitable projections are used, it often is such. Despite defects which are inherent in the attempt to map a spherical surface upon a plane, maps of large areas, comprising continents, hemispheres or even the whole sphere, are employed because of their convenience both in construction and handling. liowever, before globes come into more general use it will be necessary for makers to omit the line of the ecliptic, which only leads to confusion for old and young when found upon a 2 Perfect globes are seldom seen on account of the expense involved in their manufacture. ANALYSIS OF THE BASIC ELEMENTS OF MAP PROJECTION. 21 terrestrial globe. It was probably copied upon a terrestrial globe from a celestial globe at some early date by an ignorant workman, and for some inexplicable reason it has been allowed to remain ever since. However, there are some globes on the market to-day that omit this anomalous line. Makers of globes would confer a benefit on future generations if they would make cheap globes on which is shown, not as much as possible, but essential geographic features only. If the oceans were shown by a light blue tint and the continents by darker tints of another color, and if the principal great rivers and mountain chains were shown, it would be sufficient. The names of oceans and countries, and a few great cities, noted capes, etc., are all that should appear. The globe then would serve as the index to the maps of continents, which again would serve as indexes to the maps of countries. Globes as made at present are so full of detail, and are so mounted, that they are puzzling to anyone who does not understand the subject well enough to do without them, and are in most cases hindrances as much as helpers to instruction. REPRESENTATION OF THE SPHERE UPON A PLANE. THE PROBLEM OF MAP PROJECTION. It seems, then, that if we have the meridians and parallels properly drawn on any system of map projection, the outline of a continent or island can be drawn in from information given by the surveyors respecting the latitude and longitude of the principal capes, inlets, or other features, and the character of the coast between them. Copies of maps are commonly made in schools upon blank forms on which the meridians and parallels have been drawn, and these, like squared paper, give assistance to the free-hand copyist. Since the meridians and parallels can be drawn as closely together as we please, we can get as many points as we require laid down with strict accuracy. The meridians and parallels being drawn on the globe, if we have a set of lines upon a plane sheet to represent them we can then transfer detail from the globe to the map. The problem of map projection, therefore, consists in finding some method of transferring the meridians and parallels from the globe to the map. DEFINITION OF MAP PROJECTION. The lines representing the meridians and parallels can be drawn in an arbitrary manner, but to avoid confusion we must have a one-to-one correspondence. In practice all sorts of liberties are taken with the methods of drawing the meridians and parallels in order to secure maps which best fulfill certain required conditions, provided always that the methods of drawing the meridians and parallels follow some law or system that will give the one-to-one correspondence. Hence a map projection may be defined as a systematic drawing of lines representing meridians and parallels on a plane surface, either for the whole earth or for some portion of it. DISTORTION. In order to decide on the system of projection to be employed, we must con- sider the purpose for which the map is to be used and the consequent conditions which it is most important for the map to fulfill. In geometry, size and shape are the two fundamental considerations. If we want to show without exaggeration the extent of the different countries on a world map, we do not care much about the shape of the country, so long as its area is properly represented to scale. For sta- tistical purposes, therefore, a map on which all areas are correctly represented to scale is valuable, and such a map is called an "equal-area projection." It is well known that parallelograms on the same base and between the same parallels, that is of the same height, have equal area, though one may be rectangular or upright and the other very oblique. The sloping sides of the oblique parallelograms must be very much longer than the upright sides of the other, but the areas of the figures will be the same though the shapes are so very different. The process by which the oblique parallelogram can be formed from the rectangular parallelogram is called by engineers "shearing." A pack of cards as usually placed together shows as profile a rectangular parallelogram. If a book be stood up against the ends of the cards as in figure 9 and then made to slope as in figure 10 each card will slide a little over the one below and the profile of the pack will be the oblique parallelogram shown in figure 10. The height of the parallelogram will be the same, for it is the 22 REPRESENTATION OF THE SPHERE UPON A PLANE. 23 thickness of the pack. The base will remain unchanged, for it is the long edge of the bottom card. The area will be unchanged, for it is the sum of the areas of the edges of the cards. The shape of the paralleogram is very different from its original shape. FIG. 10.-Pack of cards after "shearing." FIG. 9.-Pack of cards before"shearing." The sloping sides, it is true, are not straight lines, but are made up of 52 little steps, but if instead of cards several hundred very thin sheets of paper or metal had been used the steps would be invisible and the sloping edges would appear to be straight lines. This sliding of layer upon layer is a "simple shear." It alters the shape without altering the area of the figure. A 8 N Fio. 11.-Square 'sheared " into an equivalent parallelogram. This shearing action is worthy of a more careful consideration in order that we may understand one very imnportant point in map projection. Suppose the square A B CD (see fig. 11) to be sheared into the oblique parallelogram a b CD. Its base and height remain the same and its area is unchanged, but the parallelogram a b C D may be turned around so that C b is horizontal, and then C b is the base, and the line a N drawn from a perpendicular to b C is the height. Then the area is the product of b C and a N, and this is equal to the area of the original square and is constant whatever the angle of the parallelogram and the extent to which the side B C has been stretched. The perpendicular a N, therefore, varies inversely as the length of the side b C, and this is true however much B C is stretched. Therefore in an equal-area projection, if distances in one direction are increased, those measured in the direction at right angles are reduced in the corresponding ratio if the lines that they represent are perpendicular to one another upon the earth. If lines are drawn at a point on an equal-area projection nearly at right angles to each other, it will in general be found that if the scale in the one direction is increased that in the other is diminished. If one of the lines is turned about the 24 U. S. COAST AND GEODETIC SURVEY. point, there must be some direction between the original positions of the lines in which the scale is exact. Since the line can be turned in either of two directions, there must be two directions at the point in which the scale is unvarying. This is true at every point of such a map, and consequently curves could be drawn on such a projection that would represent directions in which there is no variation in scale (isoperimetric curves). In maps drawn on an equal-area projection, some tracts of country may be sheared so that their shape is changed past recognition, but they preserve their area unchanged. In maps covering a very large area, particularly in maps of the whole world, this generally happens to a very great extent in parts of the map which are distant from both the horizontal and the vertical lines drawn through the center of the map. (See fig. 12.) FnG. 12.-The Mollweide equal-area projection of the sphere. It will be noticed that in the shearing process that has been described every little portion of the rectangle is sheared just like the whole rectangle. It is stretched parallel to B C (see fig. 11) and contracted at right angles to this direction. Hence when in an equal-area projection the shape of a tract of country is changed, it follows that the shape of every square mile and indeed of every square inch of this country will be changed, and this may involve a considerable inconvenience in the use of the map. In the case of the pack of cards the shearing was the same at all points. In the case of equal-area projections the extent of shearing or distortion varies with the position of the map and is zero at the center. It usually increases along the diagonal lines of the map. It may, however, be important for the purpose for which the map is required, that small areas should -etain their shape even at the cost of the area being increased or diminished, so that different scales have to be used at different parts of the map. The projections on which this condition is secured are called "conformal" projections. If it were possible to secure equality of area and exactitude of shape at all points of the map, the whole map would be an exact counterpart of the corresponding area on the globe, and could be made to fit the globe at all points by simple bending without any stretching or contraction, which would imply alteration of scale. But a plane surface can not be made to fit a sphere in this way. It must be stretched in some direction or contracted in others (as in the process of "raising" a dome or cup by hammering sheet metal) to fit the sphere, and this means that the scale must be altered in one direction or in the REPRESENTATION OF THE SPHERE UPON A PLANE. 25 other or in both directions at once. It is therefore impossible for a map to preserve the same scale in all directions at all points; in other words a map can not accurately represent both size and shape of the geographical features at al points of the map. CONDITIONS FULFILLED BY A MAP PROJECTION. If, then, we endeavor to secure that the shape of a very small area, a square inch or a square mile, is preserved at all points of the map, which means that if the scale of the distance north and south is increased the scale of the distance east and west must be increased in exactly the same ratio, we must be content to have some parts of the map represented on a greater scale than others. The conformal pro- jection, therefore, necessitates a change of scale at different parts of the map, though the scale is the same in all directions at any one point. Now, it is clear that if in a map of North America the northern part of Canada is drawn on a much larger scale than the southern States of the United States, although the shape of every little bay or headland, lake or township is preserved, the shape of the whole continent on the map must be very different from its shape on the globe. In choosing our system of map projection, therefore, we must decide whether we want- (1) To keep the area directly comparable all over the map at the expense of correct shape (equal-area projection), or (2) To keep the shapes of the smaller geographical features, capes, bays, lakes, etc., correct at the expense of a changing scale all over the map (conformal pro- jection) and with the knowledge that large tracts of country will not preserve their shape, or (3) To mnake a compromise between these conditions so as to minimize the errors when both shape and area are taken into account. There is a fourth consideration which may be of great importance and which is very important to the navigator, while it will be of much greater importance to the aviator when aerial voyages of thousands of miles are undertaken, and that is that directions of places taken from the center of the map, and as far as possible when taken from other points of the map, shall be correct. The horizontal direction of an object measured from the south is known as its azimuth. Hence a map which preserves these directions correctly is called an "azimuthal projection." We may, therefore, add a fourth object, viz: (4) To preserve the correct directions of all lines drawn from the center of the map (azimuthal projection). Projections of this kind are sometimes called zenithal projections, because in maps of the celestial sphere the zenith point is projected into the central point of the map. This is a misnomer, however, when applied to a map of the terrestrial sphere. We have now considered the conditions which we should like a map to fulfill, and we have found that they are inconsistent with one another. For some particular purpose we may construct a map which fulfills one condition and rejects another, or vice versa; but we shall find that the maps most commonly used are the result of compromise, so that no one condition is strictly fulfilled, nor, in most cases, is it extravagantly violated. CLASSIFICATION OF PROJECTIONS. There is no way in which projections can be divided into classes that are mutually exclusive; that is, such that any given projection belongs in one class, and only in one. There are, however, certain class names that are made use of in practice principally as a matter of convenience, although a given projection may fall in two 26 U. S. COAST AND GEODETIC SURVEY. or more of the classes. We have already spoken of the equivalent or equal-area type and of the conformal, or, as it is sometimes called, the orthomorphic type. The equal-area projection preserves the ratio of areas constant; that is, any given part of the map bears the same relation to the area that it represents that the whole map bears to the whole area represented. This can be brought clearly before the mind by the statement that any quadrangular-shaped section of the map formed by meridians and parallels will be equal in area to any other quadrangular area of the same map that represents an equal area on the earth. This means that all sections between two given parallels on any equal-area map formed by meridians that are equally spaced are equal in area upon the map just as they are equal in area on the earth. In another way, if two silver dollars are placed upon the map one in one place and the other in any other part of the map the two areas upon the earth that are represented by the portions of the map covered by the silver dollars will be equal. Either of these tests forms a valid criterion provided that the areas selected may be situated on any portion of the map. There are other projections besides the equal-area ones in which the same results would be obtained on particu- lar portions of the map. A conformal projection is one in which the shape of any small section of the surface mapped is preserved on the map. The term orthomorphic, which is some- times used in place of conformal, means right shape; but this term is somewhat misleading, since, if the area mapped is large, the shape of any continent or large country will not be preserved. The true condition for a conformal map is that the scale be the same at any point in all directions; the scale will change from point to point, but it will be independent of the azimuth at all points. The scale will be the same in all directions at a point if two directions upon the earth at right angles to one another are mapped in two directions that are also at right angles and along which the scale is the same. If, then, we have a projection in which the meridians and parallels of the earth are represented by curves that are perpendicular each to each, we need only to determine that the scale along the meridian is equal to that along the parallel. The meridians and parallels of the earth intersect at right angles, and a conformal projection preserves the angle of intersection of any two curves on the earth; therefore, the meridians of the map must intersect the parallels of the map at right angles. The one set of lines are then said to be the orthogonal trajectories of the other set. If the meridians and parallels of any map do not intersect at right angles in all parts of the map, we may at once conclude that it is not a conformal map. Besides the equal-area and conformal projections we have already mentioned the azimuthal or, as they are sometimes called, the zenithal projections. In these the azimuth or direction of all points on the map as seen from some central point are the same as the corresponding azimuths or directions on the earth. This would be a very desirable feature of a map if it could be true for all points of the map as well as for the central point, but this could not be attained in any projection; hence the azimuthal feature is generally an incidental one unless the map is intended for some special purpose in which the directions from some one point are very important. Besides these classes of projections there is another class called perspective projections or, as they are sometimes called, geometric projections. The principle of these projections consists in the direct projection of the points of the earth by straight lines drawn through them from some given point. The projection is gen- erally made upon a plane tangent to the sphere at the end of the diameter joining the point of projection and the center of the earth. If the projecting point is the REPRESENTATION OF THE SPHERE UPON A PLANE. 27 center of the sphere, the point of tangency is chosen in the center of the area to be mapped. The plane upon which the map is made does not have to be tangent to the earth, but this position gives a simplification. Its position anywhere parallel to itself would only change the scale of the map and in any position not parallel to itself the same result would be obtained by changing the point of tangency with mere change of scale. Projections of this kind are generally simple, because they can in most cases be constructed by graphical methods without the aid of the analytical expressions that determine the elements of the projection. Instead of using a plane directly upon which to lay out the projection, in many cases use is made of one of the developable surfaces as an intermediate aid. The two surfaces used for this purpose are the right circular cone and the circular cylinder. The projection is made upon one or the other of these two surfaces, and then this surface is spread out or developed in the plane. As a matter of fact, the projection is not constructed upon the cylinder or cone, but the principles are derived from a consideration of these surfaces, and then the projection is drawn upon the plane just as it would be after development. The developable surfaces, therefore, serve only as guides to us in grasping the principles of the projection. After the elements of the projections are determined, either geometrically or analytically, no further attention is paid to the cone or cylinder. A projection is called conical or cylindrical, according to which of the two developable surfaces is used in the determination of its elements. Both kinds are generally included in the one class of conical projec- tions, for the cylinder is just a special case of the cone. In fact, even the azimuthal projections might have been included in the general class. If we have a cone tangent to the earth and then imagine the apex to recede more and more while the cone still remains tangent to the sphere, we shall have at the limit the tangent cylinder. On the other hand, if the apex approaches nearer and nearer to the earth the circle of tangency will get smaller and smaller, and in the end it will become a point and will coincide with the apex, and the cone will be flattened out into a tangent plane. Besides these general classes there are a number of projections that are called conventional projections, since they are projections that are merely arranged arbi- trarily. Of course, even these conform enough to law to permit their expression analytically, or sometimes more easily by geometric principles. THE IDEAL MAP. There are various properties that it would be desirable to have present in a a map that is to be constructed. (1) It should represent the countries with their true shape; (2) the countries represented should retain their relative size in the map; (3) the distance of every place from every other should bear a constant ratio to the true distances upon the earth; (4) great circles upon the sphere-that is, the shortest distances joining various points-should be represented by straight lines which are the shortest distances joining the points on the map; (5) the geographic latitudes and longitudes of the places should be easily found from their positions on the map, and, conversely, positions should be easily plotted on the map when we have their latitudes and longitudes. These properties could very easily be secured if the earth were a plane or one of the developable surfaces. Unfortunately for the cartographer, it is not such a surface, but is a spherical surface which can not be developed in a plane without distortion of some kind. It becomes, then, a matter of selection from among the various desirable properties enumerated above, and even some of these can not in general be attained. It is necessary, then, to decide what purpose the map to be constructed is to fulfill, and then we can select the projection that comes nearest to giving us what we want. 28 U. S. COAST AND GEODETIC SURVEY. PROJECTIONS CONSIDERED WITHOUT MATHEMATICS. If it is a question of making a map of a small section of the earth, it will so nearly conform to a plane surface that a projection can be made that will represent the true state to such a degree that any distortion present will be negligible. It is thus possible to consider the earth made up of a great number of plane sections of this kind, such that each of them could be mapped in this way. If the parallels and meridians are drawn each at 15° intervals and then planes are passed through the points of intersection, we should have a regular figure made up of plane quad- rangular figures as in figure 13. Each of these sections could be made into a self- consistent map, but if we attempt to fit them together in one plane map, we shall find that they will not join together properly, but the effect shown in figure 13 will F7 Fro. 13.-Earth considered as formed by plane quadrangles. be observed. A section 15° square would be too large to be mapped without error, but the same principle could be applied to each square degree or to even smaller sections. This projection is called the polyhedral projection and it is in substance very similar to the method used by the United States Geological Survey in their topographic maps of the various States. Instead of considering the earth as made up of small regular quadrangles, we might consider it made by narrow strips cut off from the bases of cones as in figure 14. The whole east-and-west extent of these strips could be mapped equally accu- rately as shown in figure 15. Each strip would be all right in itself, but they would not fit together, as is shown in figure 15. If we consider the strips to become very narrow while at the same time they increase in number, we get what is called the polyconic projections. These same difficulties or others of like nature are met with in every projection in which we attempt to hold the scale exact in some part. At REPRESENTATION OF THE SPHERE UPON A PLANE. 29 best we can only adjust the errors in the representation, but they can never all be avoided. Viewed from a strictly mathematical standpoint, no representation based on a system of map projection can be perfect. A map is a compromise between the FIG. 14.-Earth considered as formed by bases of cones. various conditions not all of which can be satisfied, and is the best solution of the problem that is possible without encountering other difficulties that surpass those due to a varying scale and distortion of other kinds. It is possible only on a globe to represent the countries with their true relations and our general ideas should be continually corrected by reference to this source of knowledge. FIG. 15.-Development of the conical bases. In order to point out the distortion that may be found in projections, it will be well to show some of those systems that admit of easy construction. The per- spective or geometrical projections can always be constructed graphically, but it is sometimes easier to make use of a computed table, even in projections of this class. ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. CYLINDRICAL EQUAL-AREA PROJECTION. This projection is one that is of very little use for the construction of a map of the world, although near the Equator it gives a fairly good representation. We shall use it mainly for the purpose of illustrating the modifications that can be. introduced into cylindrical projections to gain certain desirable features. In this projection a cylinder tangent to the sphere along the Equator is em- ployed. The meridians and parallels are straight lines forming two parallel systems mutually perpendicular. The lines representing the meridians are equally spaced. These features are in general characteristic of all cylindrical projections in which the cylinder is supposed to be tangent to the sphere along the Equator. The only feature as yet undetermined is the spacing of the parallels. If planes are passed through the various parallels they will intersect the cylinder in circles that become straight lines when the cylinder is developed or rolled out in the plane. With this condition it is evident that the construction given in figure 16 will give the net- work of meridians and parallels for 100 intervals. The length of the map is evi- dently 7r (about 3+) times the diameter of the circle that represents a great circle of the sphere. The semicircle is divided by means of a protractor into 18 equal arcs, and these points of division are projected by lines parallel to the line repre- senting the Equator or perpendicular to the bounding diameter of the semicircle. This gives an equivalent or equal-area map, because, as we recede from the Equator, the distances representing differences of latitude are decreased just as great a per cent as the distances representing differences of longitude are increased. The result in a world map is the appearance of contraction toward the Equator, or, in another sense, as an east-and-west stretching of the polar regions. CYLINDRICAL EQUAL-SPACED PROJECTION. If the equal-area property be disregarded, a better cylindrical projection can be secured by spacing the meridians and parallels equally. In this way we get rid of the very violent distortions in the polar regions, but even yet the result is very unsatisfactory. Great distortions are still present in the polar regions, but they are much less than before, as can be seen in figure 17. As a further attempt, we can throw part of the distortion into the equatorial regions by spacing the parallels equally and the meridians equally, but by making the spacings of the parallels greater than that of the meridians. In figure 18 is shown the whole world with the meridians and parallel spacings in the ratio of two to three. The result for a world map is still highly unsatisfactory even though it is slightly better than that obtained by either of the former methods. PROJECTION FROM THE CENTER UPON A TANGENT CYLINDER. As a fourth attempt we might project the points by lines drawn from the center of the sphere upon a cylinder tangent to the Equator. This would have a tendency to stretch the polar regions north and south as well as east and west. The result of this method is shown in figure 19, in which the polar regions are shown up to 700 of latitude. The poles could not be shown, since as the projecting line approaches them 30 ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. 3 31 o 0) 0 0 c 4 '_ 0 C CJ a_ r_ __ _ t I - -- -0 --4- 0__ _ __ - ~ -- ~ - - - - - - - - ,"-c fit f4m 0 C) C U) C) r-" c3 r I. 32 U. s. COAST AND GEODETIC SURVEY. indefinitely, the required intersection with the cylinder recedes indefinitely, or, in mathematical language, the pole is represented by a line at an infinite distance. - -- -- -- -- ---- - --- - --- -- - 30 Q d ~'- o 00 40 00 F1e. 18.-Modified cylindrical equal-spaced projection. MERCATOR PROJECTION. Instead of stretching the polar regions north and south to such an extent, it is customary to limit the stretching in latitude to an equality with the stretching in longitude. (See fig. 20.) In this way we get a conformal projection in which any small area is shown with practically its true shape, but in which large areas will be distorted by the change in scale from point to point. In this projection the pole is represented by a line at infinity, so that the map is seldom extended much beyond 800 of latitude. This projection can not be obtained directly by graphical construction, but the spacings of the parallels have to be taken from a computed table. This is the most important of the cylindrical projections and is widely used for the construction of sailing charts. Its common use for world maps is very misleading, since the polar regions are represented upon a very enlarged scale. ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. 33 a 0 0 N 0 0 ____ __ _N -7t __ _-7U ( t_ __ ___ _____ m b U C 0) G) IU O I% - 105877-28 3 34 U. S. COAST AND GEODETIC SURVEY. * 80 *0 FIG-. 2.--MeTCator projection. Since a degree is one three-hundred-and-sixtieth part of a circle, the degrees of latitude are everywhere equal on a sphere, as the meridians are all equal circles. The degrees of longitude, however, vary in the same proportion as the size of the parallels vary at the different latitudes. The parallel of 600 latitude is just one-half of the length of the Equator. A square-degree quadrangle at 60~ of latitude has. the same length north and south as has such a quadrangle at the Equator, but the extent cast and west is just one-half as great. Its area, then, is approximately one- half the area of the one at the Equator. Now, on the Mercator projection the longitude at 6O0 is stretched to double its length, and hence the scale along the meridian has to be increased an equal amount. The area is therefore increased fourfold. At 800 of latitude the area is increased to 36 times its real size, and at. 890 an area would be more than 3000 times as large as an equal-sized area at the Equator. Tfhis excessive exaggeration of area is a most serious matter if the map be used for general purposes, and this fact ought to be emphasized because it is undoubtedly true that in the majority of cases peoples' general ideas of geography are based on Mercator maps. On the map Greenland shows larger than South America, but. in reality South America is nine times as large as Greenland. As will be shown later, this projection has many good qualities for special purposes, and for some general purposes it may be used for areas not very distant from the Equator. No suggestiOn is therefore made that it should be abolished, or even rediuced from its position among the first-class projections, but it is most strongly urged that no one should use it without recognizing its defects, and thereby guarding against being misled by false appearances. This projection is often used because on it the whole inhabited world can be shown on one sheet, and, furthermore, it can be prolonged ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. in either an east or west direction; in other words, it can be repeated so as to show part of the map twice. By this means the relative positions of two places that would be on opposite sides of the projection when confined to 3600 can be indicated more definitely. GEOMETRICAL AZIMUTHAL PROJECTIONS. Many of the projections of this class can be constructed graphically with very little trouble. This is especially true of those that have the pole at the center. The merid- ians are then represented by straight lines radiating from the pole and the parallels are in turn represented by concentric circles with the pole as center. The angles between the meridians are equal to the corresponding longitudes, so that they are represented by radii that are equally spaced. STEREOGRAPHIC POLAR PROJECTION. This is a perspective conformal projection with the point of projection at the South Pole when the northern regions are to be projected. The plane upon which N p q r P P Q R W 0 - E S Fic. 21.-Determination of radii for stereographic polar projection. the projection is made is generally taken as the equatorial plaine. A plane tangent at the North Pole could be used equally well, the only difference being in the scale of the projection. In figure 21 let N E S W be the plane of a meridian with N represent- ing the North Pole. Then NP will be the trace of the plane tangent at the North Pole. Divide the are N E into equal parts, each in the figure being for 10 of latitude. Then all points at a distance of 10° from the North Pole will lie on a circle with radius nm1, those at 20° on a circle with radius n q, etc. With these radii we can construct the map as in figure 22. On the map in this figure the lines are drawn for each 100 both in latitude and longitude; but it is clear that a larger map could be constructed on which lines could be drawn for every degree. We have seen that a practically correct map can be made for a region measuring 1 each way, because curvature in such a size is too slight to be taken into account. Suppose, then, that correct maps were made separately of all the little quadrangular portions. It would be found that by simply reducing each of them to the requisite scale it could. be fitted almost exactly into the space to which it belonged. We say almost exactly, because the edge 36 U. S. COAST AND GEODETIC SURVEY. nearest the center of the map would have to be a little smaller in scale, and hence would have to be compressed a little if the outer edges were reduced the exact amount, but the compression would be so slight that it would require very careful measurement to detect it. 90 180U 90 Fco. 22.-Stereographic polar projection. It would seem, then, at first sight that this projection is an ideal one, and, as a matter of fact, it is considered by most authorities as the best projection of a hemi- sphere for general purposes, but, of course, it has a serious defect. It has been stated that each plan has to be compressed at its inner edge, and for the same reason each plan in succession has to be reduced to a smaller average scale than the one outside of it. In other words, the shape of each space into which a plan has to be fitted is prac- tically correct, but the size is less in proportion at the center than at the edges; so that if a correct plan of an area at the edge of the map has to be reduced, let us say to a scale of 500 miles to an inch to fit its allotted space, then a plan of an area at the center has to be reduced to a scale of more than 500 miles to an inch. Thus a moderate area has its true shape, and even an area as large as one of the States is not distorted to such an extent as to be visible to the ordinary observer, but to obtain this advantage ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. 37 relative size has to be sacrificed; that is, the property of equivalence of area has to be entirely disregarded. CENTRAL OR GNOMONIC PROJECTION. In this projection the center of the sphere is the point from which the projecting lines are drawn and the map is made upon a tangent plane. When the plane is tangent at the pole, the parallels are circles with the pole as common center and the meridians 5 R Q 0 q - - - - - - - - - FIG. 23.-Determination of radii for gnomonic polar projection. are equally spaced radii of these circles. In figure 23 it can be seen that the length of the various radii of the parallels are found by drawing lines from the center of a circle representing a meridian of the sphere and by prolonging them to intersect a tangent line. In the figure let P be the pole and let PQ, QR, etc., be arcs of 100, then Pq, Pr, etc., will be the radii of the corresponding parallels. It is at once evident that a complete hemisphere can not be represented upon a plane, for the radius of 90° from the center would become infinite. The North Pole regions extending to latitude 300 is shown in figure 24. The important property of this projection is the fact that all great circles are represented by straight lines. This is evident from the fact that the projecting lines would all lie in the plane of the circle and the circle would be represented by the intersection of this plane with the mapping plane. Since the shortest distance be- 38 U. S. COAST AND GEODETIC SURVEY. 180 0 9 Fin. 24.-Gnomonic polar projection. tween two given points on the sphere is an arc of a great circle, the shortest distance between the points on the sphere is represented on the map by the straight line joining the projection of the two points which, in turn, is the shortest distance joining the projections; in other words, shortest distances upon the sphere are represented by shortest distances upon the map. The change of scale in the projection is so rapid that very violent distortions are present if the map is extended any distance. A map of this kind finds its principal use in connection with the Mercator charts, as will be shown in the second part of this publication. LAMBERT AZIMUTHAL EQUAL-AREA PROJECTION. This projection does not belong in the perspective class, but when the pole is the center it can be easily constructed graphically. The radius for the circle representing a parallel is taken as the chord distance of the parallel from the pole. In figure 25 the chords are drawn for every 100 of arc, and figure 26 shows the map of the Northern Hemisphere constructed with these radii. ORTHOGRAPHIC POLAR PROJECTION. When the pole is the center, an orthographic projection may be constructed graphically by projecting the parallels by parallel lines. It is a perspective projection in which the point of projection has receded indefinitely, or, speaking mathematically, ELEMENTARY DISCUSSION OF, VARIOUS FORMS OF PROJECTION. FIG. 25.-Determination of radii for Lambert equal-area polar projection. 90 G80 . -0 09 F1G, 26.--Lambert equal-area polar projection. 39 40 U. S. COAST AND GEODETIC SURVEY. FIG. 27.-Determination of radii for orthographic polar projection. the point of projection is at infinity. Each parallel is really constructed with a radius proportional to its radius on the sphere. It is clear, then, that the scale along the parallels is unvarying, or, as it is called, the parallels are held true to scale. The 90 14 10 g0 FIG. 28.-Orthographic polar projection. method of construction is indicated clearly in figure 27, and figure 28 shows the North- ern Hemisphere on this projection. Maps of the surface of the moon are usually constructed on this projection, since we really see the moon projected upon the celestial sphere practically as the map appears. AZIMUTHAL EQUIDISTANT PROJECTION. In the orthographic polar projection the scale along the parallels is held constant, as we have seen. We can also have a projection in which the scale along the meridians is held unvarying. If the parallels are represented by concentric circles equally spaced, we shall obtain such a projection. The projection is very easily constructed, ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. 90 480n 41 "v" " 0 " 9i -r U .. o. Fio. 29.-Azimuthal equidistant polar projection. since we need only to draw the system of concentric, equally spaced circles with the meridians represented, as in all polar azimuthal projections, by the equally spaced tbo u i o f ico 0 .r r i r . y: 0 FIG. 30.-Stereographic projection of the Western Hemisphere. radii of the system of circles. Such a map of the Northern Hemisphere is shown in figure 29. This projection has the advantage that it is somewhat a mean between the stereographic and the equal area. On the whole, it gives a fairly good repre- 42 U. S. COAST AND GEODETIC SURVEY. sentation, since it stands as a compromise between the projections that cause dis- tortions of opposite kind in the outer regions of the maps. OTHER PROJECTIONS IN FREQUENT USE. In figure 30 the Western Hemisphere is shown on the stereographic projection. A projection of this nature is called a meridional projection or a projection on the W r W ):> oo- uo0 8 4 70 60> W . 40' Fic. 31.-Gnomonic projection of part of the Western Hemisphere. plane of a meridian, because the bounding circle represents a meridian and the North and South Poles are shown at the top and the bottom of the map, respectively. r on 70' O'' Fra. 2.-Labert qual-rea pojecton ofthe WseoH. ipee ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. 43 The central meridian is a straight line and the Equator is represented by another straight line perpendicular to the central meridian; that is, the central meridian and the Equator are two perpendicular diameters of the circle that represents the outer meridian and that forms the boundary of the map. ai FIG. 33.-Orthographic projection of the Western Hemisphere. In figure 31 a part of the Western Hemisphere is represented on a gnomonic projection with a point on the Equator as the center. 60 40 s0- 00 PIG. 34.--Globular projection of the Western Hemisphere. A meridian equal-area projection of the Western Hemisphere is shown in figure 32. An orthographic projection of the same hemisphere is given in figure 33. In this the parallels become straight lines and the meridians are arcs of ellipses. 44 U. S. COAST AND GEODETIC SURVEY. A projection that is often used in the mapping of a hemisphere is shown in figure 34. It is called the globular projection. The outer meridian and the central meridian are divided each into equal parts by the parallels which are arcs of circles. The Equator is also divided into equal parts by the meridians, which in turn are arcs of circles. Since all of the meridians pass through each of the poles, these conditions are sufficient to determine the projection. By comparing it with the stereographic it will be seen that the various parts are not violently sheared out of shape, and a comparison with the equal-area will show that the areas are not badly represented. Certainly such a representation is much less misleading than the Mercator which is too often employed in the school geographies for the use of young people. CONSTRUCTION OF A STEREOGRAPHIC MERIDIONAL PROJECTION. Two of the projections mentioned under the preceding heading-the stereo- graphic and the gnomonic-lend themselves readily to graphic construction. In figure 35 let the circle PQP' represent the outer meridian in the stereographic P S W .p K E FiG. 35.-Determination of the elements of a stereographic projection on the plane of a meridain. projection. Take the arc PQ, equal to 300; that is, Q will lio in latitude 60. At Q construct the tangent RQ; with R as a center, and with a radius RQ construct the are QSQ'. This arc represents the parallel of latitude 60. Lay off OK equal to RQ; with K as a center, and with a radius KP construct the are PSP'; then this are represents the meridian of longitude 60 reckoned from the central meridian POP'. In the same way all the meridians and parallels can be constructed so that the construction is very simple. Hemispheres constructed on this projection are very frequently used in atlases and geographies. ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. 45 CONSTRUCTION OF A GNOMONIC PROJECTION WITH POINT OF TANGENCY ON THE EQUATOR. In figure 36 let PQP'Q' represent a great circle of the sphere. Draw the radii OA, OB, etc., for every 10.of arc. When these are prolonged to intersect the tan- gent at P, we get the points on the equator of the map where the meridians inter- G N F P HR A 1 ILIIII'____Li_____r__ ' -- - i I I D E K w" I;t 0 T P' FIG. 36.-Construction of a gnomonic projection with plane tangent at the Equator. sect it. Since the meridians of the sphere are represented by parallel straight lines perpendicular to the straight-line equator, we can draw the meridians when we know their points of intersection with the equator. The central meridian is spaced in latitude just as the meridians are spaced on the equator. In this way we determine the points of intersection of the parallels with 46 U. S. COAST AND GEODETIC SURVEY. the central meridian. The projection is symmetrical with respect to the central meridian and also with respect to the equator. To determine the points of inter- section of the parallels with any meridian, we proceed as indicated in figure 36, where the determination is made for the meridian 30 out from the central meridian. Draw CK perpendicular to OC; then CD', which equals CD, determines D', the intersection of the parallel of 10° north with the meridian of 30 in longitude east of the central meridian. In like manner CE'= CE, and so on. These same values can be transferred to the meridian of 30 in longitude west of the central meridian. Since the projection is symmetrical to the equator, the spacings downward on any meridian are the same as those upward on the same meridian. After the points of intersection of the parallels with the various meridians are determined, we can draw a smooth curve through those that lie on any given parallel, and this curve will represent the parallel in question. In this way the complete projection can be constructed. The distortions in this projection are very great, and the representa- tion must always be less than a hemisphere, because the projection extends to in- finity in all directions. As has already been stated, the projection is used in con- nection with Mercator sailing charts to aid in plotting great-circle courses. CONICAL PROJECTIONS. In the conical projections, when the cone is spread out in the plane, the 360 degrees of longitude are mapped upon a sector of a circle. The magnitude of the angle at the center of this shctor has to be determined by computation from the condition imposed C S30* FIG. 37.-Cone tangent to the sphere at latitude 300. upon the projection. Most of the conical projections are determined analytically; that is, the elements of the projection are expressed by mathematical formulas ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. 47 instead of being determined projectively. There are two classes of conical projec- tions-one called a projection upon a tangent cone and another called a projection upon a secant cone. In the first the scale is held true along one parallel and in the second the scale is maintained true along two parallels. CENTRAL PROJECTION UPON A CONE TANGENT AT LATITUDE 300. As an illustration of conical projections we shall indicate the construction of one which is determined by projection from the center upon a cone tangent at lati- tude 300. (See fig. 37.) In this case the full circuit of 360 of longitude will be C R S P T t B O'P' FIG. 38.-Determination of radii for conical central perspective projection. mapped upon a semicircle. In figure 38 let P Q P' Q' represent a meridian circle; draw CB tangent to the circle at latitude 30, then CB is the radius for the parallel of 300 of latitude on the projection. CR, CS, CT, etc., are the radii for the parallels of 800, 70°, 60°, etc., respectively. The map of the Northern Hemisphere on this projection is shown in figure 39; this is, on the whole, not a very satisfactory pro- jection, but it serves to illustrate some of the principles of conical projection. We might determine the radii for the parallels by extending the planes of the same until they intersect the cone. This would vary the spacings of the parallels, but would not change the sector on which the projection is formed. A cone could be made to intersect the sphere and to pass through any two chosen parallels. Upon this we could project the sphere either from the center or from any other point that we might choose. The general appearance of the projection would be similar to that of any conical projection, but some computation would 48 U. S. COAST AND GEODETIC SURVEY. be required for its construction. As has been stated, almost all conical projections in use have their elements determined analytically in the form of mathematical formulas. Of these the one with two standard parallels is not, in general, an intersecting cone, strictly speaking. Two separate parallels are held true to scale, 00 600 150 80 00 OC but if they were held equal in length to their length on the sphere the cone could not, in general, be made to intersect the sphere so as to have the two parallels coin- cide with the circles that represent them. This could only be done in case the distance between the two circles on the cone was equal to the chord distance between the parallels on the sphere. This would be true in a perspective projection, but it would ordinarily not be true in any projection determined analytically. Probably the two most important conical projections are the Lambert conformal conical pro- ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. 49 jection with two standard parallels and the Albers equal-area conical projection. The latter projection has also two standard parallels. BONNE PROJECTION. There is a modified conical equal-area projection that has been much used in map making called the Bonne projection. In general a cone tangent along the parallel in the central portion of the latitude to be mapped gives the radius for the are rep- resenting this parallel. A system of concentric circles is then drawn to represent the other parallels with the spacihgs along the central meridian on the same scale as that of the standard parallel. Along the arcs of these circles the longitude dis- tances are laid off on the same scale in both directions from the central meridian, FIG. 40.-Bonne projection of the United States. which is a straight line. All of the meridians except the central one are curved lines concave toward the straight-line central meridian. This projection has been much used in atlases partly because it is equal-area and partly because it is compara- tively easy to construct. A map of the United States is shown in figure 40 on this projection. POLYCONIC PROJECTION. In the polyconic projection the central meridian is represented by a straight line and the parallels are represented by arcs of circles that are not concentric, but the cen- ters of which all lie in the extension of the central meridian. The distances between the parallels along the central meridian are made proportional to the true distances between the parallels on the earth. The radius for each parallel is determined by an element of the cone tangent along the given parallel. When the parallels are con- structed in this way, the arcs along the circles representing the parallels are laid off proportional to the true lengths along the respective parallels. Smooth curves drawn through the points so determined give the respective meridians. In figure 15 it may be seen in what manner the exaggeration of scale is introduced by this method of projection. A map of North America on this projection is shown in 105877-28-4 50 U. S. COAST AND GEODETIC SURVEY. figure 41. The great advantage of this projection consists in the fact that a general table can be computed for use in any part of the earth. In most other projections there are certain elements that have to be determined for the region to be mapped. F/. 41.-Polyconic projection of North America When this is the case a separate table has to be computed for each region that is under consideration. With this projection, regions of narrow extent of longitude cani be mapped with an accuracy such that no departur e Prom true scale can be de- tected. A quadrangle of 10 on each side can be represented in such a manner, and in cases where the greatest accuracy is either not required or in which the error in scale may be taken into account, regions of much greater extent can be successfully mapped. The general table is very convenient for making topographic maps of limited extent in which it is desired to represent the region in detail. Of course, maps of neighboring regions on such a projection could not be fitted together exactly to form an extended map. This same restriction would apply to any projection on which the various regions were represented on an unvarying scale with minimum distortions. ELEMENTARY DISCUSSION OF VARIOUS FORMS'OF PROJECTION. 51 ILLUSTRATIONS OF RELATIVE DISTORTIONS. A striking illustration of the distortion and exaggerations inherent in various systems of projection is given in figures 42-45. In figure 42 we have shown a man's head drawn with some degree of care on a globular projection of a hemisphere. The other three figures have the outline of the head plotted, maintaining the latitude and longitude the same as they are found in the globular projection. The distortions and exaggerations are due solely to those that are found in the projection in question. Fic. 42.-Man's head drawn on globular pro- jection. Fic. 43.-Man's head plotted on orthographic pro- jection. Fi. 44.-Man's head plotted on stereographic projection. FIG. 45.-Man's head plotted on Mercator projec- tion. This does not mean that the globular projection is the best of the four, because the symmetrical figure might be drawn on any one of them and then plotted on the others. By this method we see shown in a striking way the relative differences in distortion of the various systems. The principle could be extended to any number of projections that might be desired, but the four figures given serve to illustrate the method. 52 52 U. S: COAST AND GEODETIC SURVEY. fir. . ;" S t . ' r" i; ,. I z os c +'- - - - a ,arV p ;, , . s i _ z - - - -5 i; ll, ,> ",.,.;f , " ' . a 1n _ 1 ' ter' .. vxi ' - US N) .a) v) o v) y U a) cd b O) N) C C) O b O. .C.C V 0 c d- , ', . r' /, / ,; ' ' y, 9 ,r ' , . ,; l ,, t . . s . _ " ,. ° e r PART II. INTRODUCTION. It is the purpose in Part II of this review to give a comprehensive description of the nature, properties, and construction of the better systems of map projection in use at the present day. Many projections have been devised for map construction which are nothing more than geometric trifles, while others have attained prominence at the expense of better and ofttimes simpler types. It is largely since the outbreak of the World War that an increased demand for better maps has created considerable activity in mathematical cartography, and, as a consequence, a marked progress in the general theory of map projections has been in evidence. Through military necessities and educational requirements, the science and art of cartography have demanded better draftsmanship and greater accuracy, to the extent that many of the older studies in geography are not now considered as worthy of inclusion in the present-day class. The whole field of cartography, with its component parts of history and surveys, map projection, compilation, nomenclature and reproduction is so important to the advancement of scientific geography that the higher standard of to-day is due to a general development in every branch of the subject. The selection of suitable projections is receiving far more attention than was formerly accorded to it. The exigencies of the problem at hand can generally b met by special study, and, as a rule, that system of projection can be adopted which will give the best results for the area under consideration, whether the desirable con- ditions be a matter of correct angles between meridians and parallels, scaling prop- erties, equivalence of areas, rhumb lines, etc. The favorable showing required to meet any particular mapping problem may oftentimes be retained at the expense of other less desirable properties, or a compro- mise may be effected. A method of projection which will answer for a country of small extent in latitude will not at all answer for another country of great length in a north-and-south direction; a projection which serves for the representation of the polar regions may not be at all applicable to countries near the Equator; a projection which is the most convenient for the purposes of the navigator is of little value to the Bureau of the Census; and so throughout the entire range of the subject, particular conditions have constantly to be satisfied and special rather than general problems to be solved. The use of a projection for a purpose to which it is not best suited is, therefore, generally unnecessary and can be avoided. PROJECTIONS DESCRIBED IN PART II. In the description of the different projections and their properties in the follow- ing pages the mathemetical theory and development of formulas are not generally included where ready reference can be given to other manuals containing these features. In several instances, however, the mathematical development is given in somewhat closer detail than heretofore. In the selection of projections to be presented in this discussion, the authors have, with two exceptions, confined themselves to two classes, viz, conformal projections and 53 54 U. S. COAST AND GEODETIC SURVEY. equivalent or equal-area projections. The exceptions are the polyconic and gnomonic projections--the former covering a field entirely its own in its general employment for field sheets in any part of the world and in maps of narrow longitudinal extent, the latter in its application and use to navigation. It is within comparatively recent years that the demand for equal-area projec- tions has been rather persistent, and there are frequent examples where the mathe- matical property of conformality is not of sufficient practical advantage to outweigh the useful property of equal area. The critical needs of conformal mapping, however, were demonstrated at the commencement of the war, when the French adopted the Lambert conformal conic projection as a basis for their new battle maps, in place of the Bonne projection here- tofore in use. By the new system, a combination of minimum of angular and scale distortion was obtained, and a precision which is unique in answering every require- ment for knowledge of orientation, distances, and quadrillage (system of kilometric squares). CONFORMAL MAPPING is not new since it is a property of the stereographic and Mercator projections. It is, however, somewhat surprising that the comprehensive study and practical application of the subject as developed by Lambert in 1772 and, from a slightly different point of view, by Lagrange in 1779, remained more or less in obscurity for many years. It is a problem in an important division of cartography which has been solved in a manner so perfect that it is impossible to add a word. This rigid analysis is due to Gauss, by whose name the Lambert conformal conic projection is sometimes known. In the representation of any surface upon any other by similarity of infinitely small areas, the credit for the advancement of the subject is due to him. EQUAL-AREA MAPPING.-The problem of an equal-area or equivalent projection of a spheroid has been simplified by the introduction of an intermediate equal-area projection upon a sphere of equal surface, the link between the two being the authalic$ latitude. A table of authalic latitudes for every half degree has recently been com- puted (see U. S. Coast and Geodetic Survey, Special Publication No. 67), and this can be used in the computations of any equal-area projection. The coordinates for the Albers equal-area projection of the United States were computed by use of this table. THE CHOICE OF PROJECTION. Although the uses and limitations of the different systems of projections are given under their subject headings, a few additional observations may be of interest. (See frontispiece.) COMPARISON OF ERRORS OF SCALE AND ERRORS OF AREA IN A MAP OF THE UNITED STATES ON FOUR DIFFERENT PROJECTIONS. MAXIMUM ScALE ERROR. Per cent. Polyconic projection......----------------------------------------------------------------7 Lambert conformal conic projection with standard parallels of latitude at 330 and 450 .............. 2 (Between latitudes 30* and 47, only one-half per cent. Strictly speaking, in the Lambert conformal conic projection these percentages are not scale error but change of scale.) Lambert zenithal equal-area projection-----------------------------------............................ .. Albers projection with standard parallels at 290 3O and 45 30'---------------. . . . SThe term authalic was first employed by Tissot, in 1881, signifying equal arcs. INTRODUCTION. 55 MAXIMUM ERROR OF AREA. Per cent. Polyconic...........---------------------------------------------------------------------------7 Lambert conformal conic. . . . .. ..---------------------------------------------------------------5 Lambert zenithal. . . . . . . . ..---------------------------------------------------------------------0 Albers...........------------------------------------------------------------------------------0 MAXIMUM ERROR O AZIMUTH. Polyconic...........------------------------------------------------------------------------10 56' Lambert conformal conic. . . . . . . ..-----------------------------------------------------------0 00' Lambert zenithal..........------------------------------------------------------------------10 04' Albers--..........-------------------------------.-----------------------------------------0 43' An improper use of the polyconic projection for a map of the North Pacific Ocean during the period of the Spanish-American War resulted in distances being distorted along the Asiatic coast to double their true amount, and brought forth the query whether the distance from Shanghai to Singapore by straight line was longer than the combined distances from Shanghai to Manila and thence to Singapore. The polyconic projection is not adapted to mapping areas of predominating longitudinal extent and should not generally be used for distances east or west of its central meridian exceeding 500 statute miles. Within these limits it is sufficiently close to other projections that are in some respects better, as not to cause any incon- venience. The extent to which the projection may be carried in latitude is not limited. On account of its tabular superiority and facility for constructing field sheets and topographical maps, it occupies a place beyond all others.4 Straight lines on the polyconic projection (excepting its central meridian and the Equator) are neither great circles nor rhumb lines, and hence the projection is not suited to navigation beyond certain limits. This field belongs to the Mercator and gnomonic projections, about which more will be given later. The polyconic projection has no advantages in scale; neither is it conformal or equal-area, but rather a compromise of various conditions which determine its choice within certain limits. The modified polyconic projection with two standard meridians may be carried to a greater extent of longitude than the former, but for narrow zones of longitude the Bonne projection is in some respects preferable to either, as it is an equal-area representation. For a map of the United States in a single sheet the choice rests between the Lambert conformal conic projection with two standard parallels and the Albers eqaal- area projection with two standard parallels. The selection of a polyconic projec- tion for this purpose is indefensible. The longitudinal extent of the United States is too great for this system of projection and its errors are not readily accounted for. The Lambert conformal and Albers are peculiarly suited to mapping in the Northern Hemisphere, where the lines of commercial importance are generally east and west. In Plate I about one-third of the Northern Hemisphere is mapped in an easterly and westerly extent. With similar maps on both sides of the one referred to, and with suitably selected standard parallels, we would have an interesting series of the Northern Hemisphere. The transverse polyconic is adapted to the mapping of comparatively narrow areas of considerable extent along any great circle. (See Plate II.) A Mercator projection can be turned into a transverse position in a similar manner and will give us conformal mapping. A The polyconic projection has always been employed by the Coast and GCoodetic Survey for field sheets, and general tables for the construction of this projection are published by this Bureau. A projection for any small part of the world can readily be on- structed by the use of these tables and the accuracy of this system within the limits specified are good reasons for its general use. 56 U. S. COAST AND GEODETIC SURVEY. The Lambert conformal and Albers projections are desirable for areas of pre- dominating east-and-west extent, and the choice is between confjormality, on the one hand, or equal area, on the other, depending on which of the two properties may be preferred. The authors would prefer Albers projection for mapping the United States. A comparison of the two indicates that their difference is very small, but the certainty of definite equal-area representation is, for general purposes, the more desirable property. When latitudinal extent increases, conformality with its pres- ervation of shapes becomes generally more desirable than equivalence with its resultant distortion, until a limit is reached where a large extent of area has equal dimensions in both or all directions. Under the latter condition-viz, the mapping of large areas of approximately equal magnitudes in all directions approaching the dimensions of a hemisphere, combined with the condition of preserving azimuths from a central point-the Lambert zenithal equal-area projection and the stereo- graphic projection are preferable, the former being the equal-area representation and the latter the conformal representation. A study in the distortion of scale and area of four different projections is given in frontispiece. Deformation tables giving errors in scale, area, and angular dis- tortion in various projections are published in Tissot's Mmoire sur la Repr6senta- tion des Surfaces. These elements of the Polyconic projection are given on pages 166-167, U. S. Coast and Geodetic Survey Special Publication No. 57. The mapping of an entire hemisphere on a secant conic projection, whether con- formal or equivalent, introduces inadmissible errors of scale or serious errors of area, either in the center of the map or in the regions beyond the standard parallels. It is better to reserve the outer areas for title space as in Plate I rather than to extend the mapping into, them. The polar regions should in any event be mapped separately on a suitable polar projection. For an equatorial belt a cylin- drical conformal or a cylindrical equal-area projection intersecting two parallels equidistant from the Equator may be employed. The lack of mention of a large number of excellent map projections in Part II of this treatise should not cause one to infer that the authors deem them unworthy. It was not intended to cover the subject in toto at this time, but rather to caution against the misuse of certain types of projections, and bring to notice a few of the interesting features in the progress of mathematical cartography, in which the theory of functions of a complex variable plays no small part to-day. Without the elements of this subject a proper treatment of conformal mapping is impossible. On account of its specialized nature, the mathematical element of cartography has not appealed to the amateur geographer, and the number of those who have received an adequate mathematical training in this field of research are few. A broad gulf has heretofore existed between the geodesist, on the one hand, and the cartographer, on the other. The interest of the former too frequently ceases at the point of presenting with sufficient clearness the value of his labors to the latter, with the result that many chart-producing agencies resort to such systems of map pro- jection as are readily available rather than to those that are ideal. It is because of this utilitarian tendency or negligence, together with the mani- fest aversion of the cartographer to cross the threshold of higher mathematics, that those who care more for the theory than the application of projections have not received the recognition due them, and the employment of autogonal5 (conformal) Page 75, Tissot's Mmoire sur la Repr6sentation des Surfaces, Paris, 1881-"Nous appellerons autogonales los projections qui conservent les angles, ot authaliques celles qui conservent les aires." INTRODUCTION. 57 projections has not been extensive. The labors of Lambert, Lagrange, and Gauss are now receiving full appreciation. In this connection, the following quotation from volume IV, page 408, of the collected mathematical works of George William Hill is of interest: Maps being used for a great variety of purposes, many different methods of projecting them may be admitted; but when the chief end is to present to the eye a picture of what appears on the surface of the earth, we should limit ourselves to projections which are conformal. And, as the construction of the rseau of meridians and parallels is, except in maps of small regions, an important part of the labor in- volved, it should be composed of the most easily drawn curves. Accordingly, in a well-known memoir, Lagrange recommended circles for this purpose, in which the straight line is included as being a circle whose center is at infinity. An attractive field for future research will be in the line in which Prof. Goode, of the University of Chicago, has contributed so substantially. Possibilities of other combinations or interruptions in the same or different systems of map projection may solve some of the other problems of world mapping. Several interesting studies given in illustration at the end of the book will, we hope, suggest ideas to the student in this particular branch. On all recent French maps the name of the projection appears in the margin. This is excellent practice and should be followed at all times. As different projec- tions have different distinctive properties, this feature is of no small value and may serve as a guide to an intelligible appreciation of the map. THE POLYCONIC PROJECTION. DESCRIPTION. (See fig. 47.] The polyconic projection, devised by Ferdinand Hassler, the first Superintend- ent of the Coast and Geodetic Survey, possesses great popularity on account of mechanical ease of construction and the f act that a general table11 for its use has been calculated for the whole spheroid. It may be interesting to quote Prof. Hassler7 in connection with two projec- tions, viz, the intersecting conic projection and the polyconic projection: 1. Projection on an intersecting cone.-The projection which I intended to use was the development of a part of the earth's surface upon a cone, either a tangent to a certain latitude, or cutting two given parallels and two meridians, equidistant from the middle meridian, and extended on both sides of the FIG. 47.-Polyconic development of the sphere. meridian, and in latitude, only so far as to admit no deviation from the real magnitudes, sensible in the detail surveys. 2. The polyconic projetion.-* * * This distribution of the projection, in an assemblage of sec- tions of surfaces of successive cones, tangents to or cutting a regular succession of parallels, and upon 6 Tables for the polyconic projection of maps, Coast and Geodetic Survey, Special Publication No. 5.. 7Papers on varloos subjects connected with the survey of the coast of the United States, by F. R. Hlassler; communicated Mar. 3, 1820 (in Trans. Am. Phil. Soc., new series, vol. 2, pp. 406-408, Philadelphia, 1825). 58 THE POLYCONIC PROJECTION. 59 regularly changing central meridians, appeared to me the only one applicable to the coast of the United States. Its direction, nearly diagonal through meridian and parallel, would not admit any other mode founded upon a single meridian and parallel withotit great deviations from the actual magnitudes and shape, which would have considerable disadvantages in use. K / / / // K, 3 --scale distortion F. 48.-Polyconic development. Figure on left above shows the centers (K, KI, K2, K3) of circles on the projec- tion that represent the corresponding parallels on the earth. Figure on right above shows the distortion at the outer meridian due to the varying radii of the circles in the polyconic development. A central meridian is assumed upon which the intersections of the parallels are truly spaced. Each parallel is then separately developed by means of a tangent cone, the centers of the developed arcs of parallels lying in the extension of the cen- tral meridian. The arcs of the developed parallels are subdivided to true scale and the meridians drawn through the corresponding subdivisions. Since the radii for the parallels decrease as the cotangent of the latitude, the circles are not concen- tric, and the lengths of the arcs of latitude gradually increase as we recede from the meridian. The central meridian is a right line; all others are curves, the curvature in- creasing with the longitudinal distance from the central meridian. The intersections between meridians and parallels also depart from right angles as the distance increases. From the construction of the projection it is seen that errors in meridional distances, areas, shapes, and intersections increase with the longitudinal limits. It therefore should be restricted in its use to maps of wide latitude and narrow longitude. The polyconic projection may be considered as in a measure only compromising various conditions impossible to be represented on any one map or chart, such as relate to First. Rectangular intersections8 of parallels and meridians. 8 The errors in meridional scale and area are expressed in percentage very closely by the formula in which l=distance of point from central meridian expressed in degrees of longitude, and =latitude. EXAMPLE-For latitude 39' the error for 10* 25',22" (560 statute miles) departure in longitude is 1 per cent for scale along the meridian and the-same amount for area. The angular distortion is a variable quantity not easily expressed by an equation. In latitude 30' this distortion is 1* 27' on the meridian 15* distant from the central meridian; at 30' distant it increases to 5' 36'. The greatest angular distortion in this projection is at the Equator, decreasing to zero as we approach the pole. The distortion of azimuth is one-half of the above amounts. 60 U. S. COAST AND GEODETIC SURVEY. Second. Equal scale0 over the whole extent (the error in scale not exceeding 1 per cent for distances within 560 statute miles of the great circle used as its central meridian). Third. Facilities for using great circles and azimuths within distances just mentioned. Fourth. Proportionality of areas9 with those on the sphere, etc. The polyconic projection is by construction not conformal, neither do the parallels and meridians intersect at right angles, as is the case with all conical or single-cone projections, whether these latter are conformal or not. It is sufficiently close to other types possessing in some respects better proper- ties that its great tabular advantages should generally determine its choice within certain limits. As stated in Hinks' Map Projections, it is a link between those projections which have some definite scientific value and those generally called conventional, but possess properties of convenience' and use. The three projections, polyconic, Bonne, and Lambert zenithal, may be con- sidered as practically identical within areas not distant more than 30 from a common central point, the errors from construction and distortion of the paper exceeding those due to the system of projection used. The general theory of polyconic projections is given in Special Publication No. 57, U. S. Coast and Geodetic Survey. CONSTRUCTION OF A POLYCONIC PROJECTION. Having the area to be covered by a projection, determine the scale and the interval of the projection lines which will be most suitable for the work in hand. SMALL-SCALE PROJECTIONS (1-500 000 AND SMALLER). Draw a straight line for a central meridian and a construction line (a b in the figure) perpendicular thereto, each to be as central to the sheet as the selected interval of latitude and longitude will permit. On this central meridian and from its intersection with the construction line lay off the extreme intervals of latitude, north and south (mm2 and mm4) and sub- divide the intervals for each parallel (mi and m) to be represented, all distances10 being taken from the table (p. 7, Spec. Pub. No. 5, "Lengths of degrees of the meridian "). Through each of the points (mi1, m2, m, m4) on the central meridian draw addi- tional construction lines (cd, ef, gh, if) perpendicular to the central meridian, and mark off the ordinates (x, x,, x2, x3, m4, x5) from the central meridian corresponding to the values10 of "X" taken from the table under "Coordinates of curvature" (pp. 11 to 189 Spec. Pub. No. 5), for every meridian to be represented. At the points (x, x, x2, x3, x4, x) lay off from each of the construction lines the corresponding values10 of "Y""from the table under "Coordinates of curvature' 0 Footnote on preceding page. 10 The lengths of the arcs of the meridians and parallels change when the latitude changes and all distances must be taken from the table opposite the latitude of the point in use. it Approximate method of deriving the' values of y intermediate between those shown in the table. The ratio of any two successive ordinates of curvature equals the ratio of the squares of the corresponding arcs. Examples.-Latitude 60 to 61. Given the value of y for longitude 50', 292.m8 (see table), to obtain the value of y for longitude 55'. (50)2292.8' hence y=354.m3 (see table). Similarily, y for 3*=3795m 2 ;hence y for 4*=6747m, which differs 2 from the tabular value, a negligible quantity for the intermediate values of y under most conditions. THE POLYCONIC PROJECTION. 61 y y X53 544 X3 5 , x C _- 5---- --x- - - - - - 73-- - -x - - - X - X5 x xi x 54, x5 6 "'~~~ - "L x --54------3-----4--- -J 7C4 k3X~ 4 FIG. 49.-Polyconic projection-construction plate. (pp. 11 to 189, Spec. Pub. No. 5), in a direction parallel to the central meridian, above the construction lines if north of the Equator, to determine points on the meridians and parallels. Draw curved lines through the points thus determined for the meridians and parallels of the projection. LARGE-SCALE PROJECTIONS (1-10 000 AND LARGER). The above method can be much simplified in constructing a projection on a large scale. Drav the central meridian and the construction line ab, as directed above. On the central meridian lay off the distances12 mm2 and mm4 taken from the table under "Continuous sums of minutes "for the intervals in minutes between the middle parallel and the extreme parallels to be represented, and through the points m2 and m4 draw straight lines ed and ef parallel to the line ab. On the lines ab, cd, and ef lay off the distances'2 mx5, m2A5, and m4x5 on both sides of the central meridian, taking the values from the table under "Arcs of the parallel in meters" corresponding to the latitude of the points m, m2, and m4, respectively. Draw straight lines through the points thus determined, x5, for the extreme meridians. 12 The lengths of the arcs of the meridians and parallels change when the latitude changes and all distances must be taken from the table opposite the latitude of the point in use. 62 U. S. COAST AND GEODETIC SURVEY. At the points x6 on the line ab lay off the value'3 of y corresponding to the inter- val in minutes between the central and the extreme meridians, as given in the table under "Coordinates of curvature," in a direction parallel with the central meridian and above the line, if north of the Equator, to determine points in the central parallel. Draw straight lines from these points to the point m for the middle parallel, and from the points of intersection with the extreme meridians lay off distances'1 on the extreme meridians, above and below, equal to the distances mm2 and mm4 to locate points in the extreme parallels. Subdivide the three meridians and three parallels into parts corresponding to the projection interval and join the corresponding points of subdivision by straight lines to complete the projection. To construct a projection on an intermediate scale, follow the method given for small-scale projections to the extent required to give the desired accuracy. Coordinates for the projection of maps on various scales with the inch as unit, are published by the U. S. Geological Survey in Bulletin 650, Geographic Tables and Formulas, pages 34 to 107. TRANSVERSE POLYCONIC PROJECTION. (See Plate II.) If the map should have a predominating east-and-west dimension, the poly- conic properties may still be retained, by applying the developing cones in a trans- verse position. A great circle at right angles to a central meridian at the middle part of the map can be made to play the part of the central meridian, the poles being transferred (in construction only) to the Equator. By transformation of coordinates a projection may be completed which will give all polyconic properties in a traverse relation. This process is, however, laborious and has seldom been resorted to. Since the distance across the United States from north to south is less than three-fifths of that from east to west, it follows, then, by the above manipulation that the maximum distortion can be reduced from 7 to 24 per cent. A projection of this type (plate II) is peculiarly suited to a map covering an important section of the North Pacific Ocean. If a great circle 1 passing through San Francisco and Manila is treated in con- struction as a central meridian in the ordinary polyconic projection, we can cross the Pacific in a narrow belt so as to include the American and Asiatic coasts with a very small scale distortion. By transforma- tion of coordinates the meridians and parallels can be constructed so that the projection will present the usual appearance and may be utilized for ordinary purposes. The configuration of the two continents is such that all the prominent features of America and eastern Asia are conveniently close to this selected axis, viz, Panama, Brito, San Francisco, Straits of Fuca, Unalaska, Kiska, Yokohama, Manila, Hongkong, and Singapore. It is a typical case of a projection being adapted to the configuration of the locality treated. A map on a transverse polyconic projection as here suggested, while of no special navigational value, is of interest from a geographic standpoint as. exhibiting in their true relations a group of important localities covering a wide expanse. For method of constructing this modified form of polyconic projection, see Coast and Geodetic Survey, Special Publication No. 57, pages 167 to 171. POLYCONIC PROJECTION WITH TWO STANDARD MERIDIANS, AS USED FOR THE INTER- NATIONAL MAP OF THE WORLD, ON THE SCALE 1:1 000 000. The projection adopted for this map is a modified polyconic projection devised by Lallemand, and for this purpose has advantages over the ordinary polyconic projection in that the meridians are straight lines and meridional errors are lessened and distributed somewhat the same (except in an opposite direction) as in a conic } The lengths of the arcs of the meridians and parallels change when the latitude changes and all distances must be taken from the table opposite the latitude of the point in use. 1 A great circle tangent to parallel 45 north latitude at 100* west longitude was chosen as the axis of the projection in this plate. THE POLYCONIC PROJECTION. 63 projection with two standard parallels; in other words, it provides for a distribution of scale error by having two standard meridians instead of the one central meridian of the ordinary polyconic projection. The scale is slightly reduced along the central meridian, thus bringing the parallels closer together in such a way that the meridians 20 on each side of the center are made true to scale. Up to 60° of latitude the separate sheets are to include 6° of longitude and 40 of latitude. From latitude 60° to the pole the sheets are to include 120 of longitude; that is, two sheets are to be united into one. The top and bottom parallel of each sheet are constructed in the usual way; that is, they are circles constructed from centers lying on the central meridian, but not concentric. These two parallels are then truly divided. The meridians are straight lines joining the corresponding points of the top and bottom parallels. Any sheet will then join exactly along its margins with its four neighboring sheets. The correction to the length of the central meridian is very slight, amounting to only 0.01 inch at the most, and the change is almost too slight to be measured on the map. In the resolutions of the International Map Committee, London, 1909, it is not stated how the meridians are to be divided; but, no doubt, an equal division of the central meridian was intended. Through these points, circles could be constructed with centers on the central meridian and with radii equal to pn cot p, in which pn is the radius of curvature perpendicular to the meridian. In practice, however, an equal division of the straight-line meridians between the top and bottom parallels could scarcely be distinguished from the points of parallels actually constructed by means of radii or by coordinates of their intersections with the meridians. The provisions also fail to state whether, in the sheets covering 12° of longitude instead of 6°, the meridians of true length shall be 4° instead of 2° on each side of the central meridian; but such was, no doubt, the intention. In any case, the sheets would not exactly join together along the parallel of 60° of latitude. The appended tables give the corrected lengths of the central meridian from 0° to 60° of latitude and the coordinates for the construction of the 40 parallels within the same limits. Each parallel has its own origin; i. e., where the parallel in question intersects the central meridian. The central meridian is the Y dxis and a perpendic- ular to it at the origin is the X axis; the first table, of course, gives the distance between the origins. The y values are small in every instance. In terms of the para- meters these values are given by the expressions , x=pn cot p sin (X sin o) y = p, cot [1-cos (X sin p)]= 2p, cot sin2esin ). The tables as given below are all that are required for the construction of all maps up to 60° of latitude. This fact in itself shows very clearly the advantages of the use of this projection for the purpose in hand. A discussion of the numerical properties of this map system is given by Lallemand in the Comptes Rendus, 1911, tome 153, page 559. 64 U. S. COAST AND GEODETIC SURVEY. TABLES FOR THE PROJECTION OF THE SHEETS OF THE INTERNATIONAL MAP OF THE WORLD. [Scale: 1:1 000 000. Assumed figure of the earth: a=6378.24 km.; b=6356.56 km.] TABLE 1.-Corrected lengths onc the central meridian, in millimeters. LaiueNatural Correc- Corrected Laiuelength tion length Frm0 t ---------------------------------4207 -.2 4.0 4rm0to 8------------------------------------------------------------------------ 442.31 -.2 442.05 48to 8------------------------------------------------------------------------ 442.40 .26 442.14 8to 12------------------------------------------------------------------------ 442.53 .26 442.27 12to21------------------------------------------------------------------------- 442.69 .24 442.45 20 to 24------------------------------------------------------------------------ 442.690 .24 442.66 24 to 28----------------------------------------------------------------------------- 44.13 .22 442.91 28 to2----------------------------------------------------------------------------- 443.139 .20 44.919 2to 32----------------------------------------------------------------------------- 443.68 .19 443.9 326to40------------------------------------------------------------------------------ 443.8 .17 443.61 460to 44----------------------------------------------------------------------------- 44.9 .15 44.814 44to48----------------------------------------------------------------------------- 44.0 .3 444 46to52 ------------------------------------------------------------------------------ 444.92 .11 444.61 482to 52----------------------------------------------------------------------------- 44.2 .09 44.813 56 to 560----------------------------------------------------------------------------- 445.22 -. 08 445.144 TABLE 2.-Coordinates of the intersections of the parallels and the meridians, in millimeters. Longitude from central Lati- Coordi- meridian tude nates ___ ___ ____ 10 2 30 0 x 111.32 222.64 333.96 y 0.00 0.00 0.00 4 x 111.05 222.0 333.16 yi 0.07 0.27 0.61 8 r 10.24 220.49 330.73 y 0.13 0. 54 1.21 12 r 108.90 27.81 326.71 yi 0.20 0.79 1.78 16 r 107.04 214.07 32.10 V 0.26 1.03 2.32 20 x 104. 65 209. 29 313. 93 y 0.31 1.25 2.81 24 z 103. 75 203.50 305. 24 "y 0.36 1.45 3.25 28 x 98.36 196.72 295.06 y 0.40 1.61 3.63 32 x 94. 50 188. 98 283. 45 y 0.44 1.75 3.93 36 x 90.16 180.32 270.46 y 0.46 1.85 4.16 40 r 85. 40 170. 78 26. 14 y 0.48 1.92 4.31 44 x 80. 21 160. 40 240. 58 y 0.49 1.95 4.38 48 x 74. 63 149. 24 223. 83 V 0.48 1.94 4. 36 52 x 68. 68 137. 34 205. 98 V 0.47 1.89 4.25 56 x 62. 39 124. 77 187. 13 y 0.45 1.81 4.06 60 x 55.80 111.59 167.35 y I 0.42 .69 3.80 In the debates on the international map, the ordinary polyconic projection was opposed on the ground that a number of sheets could not be fitted together on account of the curvature of both meridians and parallels. This is true from the nature of things, since it is impossible to make a map of the world in a series of flat sheets which shall fit together and at the same time be impartially representative of all meridians and parallels. Every sheet edge in the international map has an exact fit with the corresponding edges of its four adjacent sheets. (See fig. 50.) The corner sheets to complete a block of nine will not make a perfect fit along their two adjacent edges simultaneously; they will fit one or the other, but the THE POLYCONIC PROJECTION. 65 angles of the corners are not exactly the same as the angles in which they are required to fit; and there will be in theory a slight wedge-shaped gap unfilled, as shown in the figure. It is, however, easy to calculate that the discontinuity at the points a or b in a block of nine sheets, will be no more than a tenth of an inch if the paper Q, a Q Q b b b b FIG. 50.-International map of the world-junction of sheets. preserves its shape absolutely unaltered. What it will be in practice depends entirely on the paper, and a map mounter will have no difficulty in squeezing his sheets to make the junction practically perfect. If more than nine sheets are put together, the error will, of course, increase somewhat rapidly; but at the same time the sheets will become so inconveniently large that the experiment is not likely to be made very often. If the difficulty does occur, it must be considered an instruc- tive example at once of the proposition that a spheroidal surface can not be developed on a plane without deformation, and of the more satisfying proposition that this modified projection gives a remarkably successful approximation to an unattain- able ideal. Concerning the modified polyconic projection for the international map, Dr. Frischauf has little to say that might be considered as favorable, partly on account of errors that appeared in the first publication of the coordinates. The claim that the projection is not mathematically quite free from criticism and does not meet the strictest demands in the matching of sheets has some basis. The system is to some extent conventional and does not set out with any of the better scientific properties of map projections, but, within the limits of the separate sheets or of several sheets joined together, should meet all ordinary demands. The contention that the Albers projection is better suited to the same purpose raises the problem of special scientific properties of the latter with its limitations to separate countries or countries of narrow latitudinal extent, as compared with the modified polyconic projection, which has no scientific interest, but rather a value of expediency. In the modified polyconic projection the separate sheets are sufficiently good and can be joined any one to its four neighbors, and fairly well in groups of nine throughout the world; in the Albers projection a greater number of sheets may be joined exactly if the latitudinal limits are not too great to necessitate new series to 105877--28-5 66 U. S. COAST AND GEODETIC SURVEY. the north or south, as in the case of continents. The latter projection is further discussed in another chapter. The modified polyconic projection loses the advantages of the ordinary polyconic in that the latter has the property of indefinite extension north or south, while its gain longitudinally is offset by loss of scale on the middle parallels. The system does not, therefore, permit of much extension in other maps than those for which it was designed, and a few of the observations of Prof. Rosen, of Sweden,. on the limi- tations 1" of this projection are of interest: The junction of four sheets around a common point is more important than junctions in Greek-cross arrangement, as provided for in this system. The system does not allow a simple calculation of the degree scale, projection errors, or angular differences, the various errors of this projection being both lengthy to compute and remarkably irregular. The length differences are unequal in similar directions from the same point, and the calculation of surface differences is specially complicated. For simplicity in mathematical respects, Prof. Rosen favors a conformal conic projection along central parallels. By the latter system the sheets can be joined along a common meridian without a seam, but with a slight encroachment along the parallels when a northern sheet is joined to its southern neighbor. The conformal projection angles, however, being right angles, the sheets will join fully around a. corner. Such a system would also serve as a better pattern in permitting wider employment-in other maps. On the other hand, the modified polyconic projection is sufficiently close, and its adaptability to small groups of sheets in any part of the world is its chief advan- tage. The maximum meridional error in an equatorial sheet, according to Lallemand " is only 1, or about one-third of a millimeter in the height of a sheet; and in the direction of the parallels -I, or one-fifth of a millimeter, in the width of a sheet. The error in azimuth does not exceed six minutes. Within the limits of one or several sheets these errors are negligible and inferior to those arising from drawing,, printing, and hygrometric conditions. 15 See Atti del X Congrosso Internazionale di Geografia, Roma, 1913, pp. 37-42. 18 Ibid., p. 681. THE BONNE PROJECTION. DESCRIPTION. [See fig. 51.] In this projection a central meridian and a standard parallel are assumed with a cone tangent along the standard parallel. The central meridian is developed along that element of the cone which is tangent to it and the cone developed on a plane. BONNE PROJECTION OF HEMISPHERE Development of cone tangent along parallel 45* N. 60 60660 180 180 16o so q 80 160 qOrthpoe 140 8014 2020 100 40 00 8 60 60 40Equ tor 40 FIG. 51. The standard parallel falls into an are of a circle with its center at the apex of the developing cone, and the central meridian becomes a right line which is divided to true scale. The parallels are drawn as concentric circles at their true distances apart, and all parallels are divided truly and drawn to scale. Through the points of division of the parallels the meridians are drawn. The central meridian is a straight line; all others are curves, the curvature increasing with the difference in longitude. The scale along all meridians, excepting the central, is too great, increasing with the distance from the center, and the meridians become more inclined to the parallels, 07 68 U. S. COAST AND GEODETIC SURVEY. thereby increasing the distortion. The developed areas preserve a strict equality, in which respect this projection is preferable to the polyconic. USES.-The Bonne 1 system of projection, still used to some extent in France, will be discontinued there and superseded by the Lambert system in military mapping. It is also used in Belgium, Netherlands, Switzerland, and the ordnance surveys of Scotland and Ireland. In Stieler's Atlas we find a number of maps with this projection; less extensively so, perhaps, in Stanford's Atlas. This projection is strictly equal-area, and this has given it its popularity. In maps of France having the Bonne projection, the center of projection is found at the intersection of the meridian of Paris and the parallel of latitude 50G (= 450). The border divisions and subdivisions appear in grades, minutes (centesimal), sec- onds, or tenths of seconds. LnIITATIONS.-Its distortion, as the difference in longitude increases, is its chief defect. On the map of France the distortion at the edges reaches a value of 18' for angles, and if extended into Alsace, or western Germany, it would have errors in dis- tances which are inadmissible in calculations. In the rigorous tests of the military operations these errors became too serious for the purposes which the map was intended to serve. THE SANSON-FLAMSTEED PROJECTION. In the particular case of the Bonne projection, where the Equator is chosen for the standard parallel, the projection is generally known under the name of Sanson- Flamsteed, or as the sinusoidal equal-area projection. All the parallels become straight lines parallel to the Equator and preserve the same distances as on the spheroid. The latter projection is employed in atlases to a considerable extent in the mapping of Africa and South America, on account of its property of equal area and the comparative ease of construction. In the mapping of Africa, however, on account of its considerable longitudinal extent, the Lambert zenithal projection is preferable in that it presents less angular distortion and has decidedly less scale error. Diercke's Atlas employs the Lambert zenithal projection in the mapping of North America, Europe, Asia, Africa, and Oceania. In an equal-area mapping of South America, a Bonne projection, with center on parallel of latitide 100 or 15 south, would give somewhat better results than the Sanson-Flamsteed projection. CONSTRUCTION OF A BONNE PROJECTION. Due to the nature of the projection, no general tables can be computed, so that for any locality special computations become necessary. The following method involves no difficult mathematical calculations: Draw a straight line to represent the central meridian and erect a perpendicular to it at the center of the sheet. With the central meridian as Y axis, and this per- pendicular as X axis, plot the points of the middle or standard parallel. The coordi- nates for this parallel can be taken from the polyconic tables, Special Publication No. 5. A smooth curve drawn through these plotted points will establish the stand- ard parallel. The radius of the circle representing the parallel can be determined as follows: The coordinates in the polyconic table are given for 30 from the central meridian. 17 Tables for this projection for the map of France were computed by Plessis. THE BONNE PROJECTION. 69 With the x and y for 30, we get tan2=x; andr,=sin (0 being the angle at the center subtended by the arc that represents 300 of longitude). By using the largest values of x and y given in the table, the value of r, is better determined than it would be by using any other coordinates. This value of r can be derived rigidly in the following manner: r=N cot 0 (N being the length of the normal to its intersection with the Y axis); but S1 N=A' sin 1" (A' being the factor tabulated in Special Publication No. 8, U. S. Coast and Geodetic Survey). Hence, r=cot $ r'A' sin 1" From the radius of this central parallel the radii for the other parallels can now be calculated by the addition or subtraction of the proper values taken from the table of "Lengths of degrees," U. S. Coast and Geodetic Survey Special Publica- tion No. 5, page 7, as these values give the spacings of the parallels along the central meridian. Let r represent the radius of a parallel determined from r, by the addition or subtraction of the proper value as stated above. If 0 denotes the angle between the central meridian and the radius to any longitude out from the central meridian, and if P represents the are of the parallel for 10 (see p. 6, Spec. Pub. No. 5), we obtain P. 6 in seconds for 1 of longitude= P . r sn 1 chord for 1° of longitude= 2r sine. Arcs for any longitude out from the central meridian can be laid off by repeating this are for 10. 6 can be determined more accurately in the following way by the use of Special Publication No. 8: A''= the longitude in seconds out from the central meridian; then . N'' cos 0 6 in seconds = Xco r A' sin 1" This computation can be made for the greatest X, and this 0 cAn be divided proportional to the required A. If coordinates are desired, we get x=r sin 6. 6 y = 2r sin -" The X axis for the parallel will be perpendicular to the central meridian at the point where the parallel intersects it. 70 U. S. COAST AND GEODETIC SURVEY. If the parallel has been drawn by the use of the beam compass, the chord for the X farthest out can be computed from the formula 6 chord=2r sin 2- The arc thus determined can be subdivided for the other required intersections with the meridians. The meridians can be drawn as smooth curves through the proper intersections with the parallels. In this way all of the elements of the projection may be deter- mined with minimum labor of computation. THE LAMBERT ZENITHAL (OR AZIMUTHAL) EQUAL-AREA PROJECTION. DESCRIPTION. [See Frontispiece.] This is probably the most important of the azimuthal projections and was em- ployed by Lambert in 1772. The important property being the preservation of azimuths from a central point, the term zenithal is not so clear in meaning, being obviously derived from the fact that in making a projection of the celestial sphere the zenith is the center of the map. In this projection the zenith of the central point of the surface to be represented appears as pole in the center of the map; the azimuth of any point within the sur- face, as seen from the central point, is the same as that for the corresponding points of the map; and from the same central point, in all directions, equal great-circle distances to points on the earth are represented by equal linear distances on the map. It has the additional property that areas on the projection are proportional to the corresponding areas on the sphere; that is, any portion of the map bears the same ratio to the region represented by it that any other portion does to its corre- sponding region, or the ratio of area of any part is equal to the ratio of area of the whole representation. This type of projection is well suited to the mapping of areas of considerable extent in all directions; that is, areas of approximately circular or square outline. In the frontispiece, the base of which is a Lambert zenithal projection, the line of 2 per cent scale error is represented by the bounding circle and makes a very favorable showing for a distance of 220 44' of arc-measure from the center of the map. Lines of other given errors of scale would therefore be shown by concentric circles (or almucantars), each one representing a small circle of the sphere parallel to the horizon. Scale error in this projection may be determined from the scale factor of the 1. almucantar as represented by the expression cos in which 0=actual distance in arc measure on osculating sphere from center of map to any point. Thus we have the following percentages of scale error: Distance in arc from Scale error center of map Degrees Per cent 5 0.1 10 0.4 20 1.2 30 3.5 40 6.4 50 10.3 60 15.5 In this projection azimuths from the center are true, as in all zenithal pro- jections. The scale along the parallel circles (almiucantars) is too large by the amounts indicated in the above table; the scale along their radii is too small in inverse proportion, for the projection is equal-area. The scale is increasingly errone- ous as the distance from the center increases. 71 72 U. S. COAST AND GEODETIC SURVEY. The Lambert zenithal projection is valuable for maps of considerable world areas, such as North America, Asia, and Africa, or the North Atlantic Ocean with its somewhat circular configuration. It has been employed by the Survey Depart- ment, Ministry of Finance, Egypt, for a wall map of Asia, as well as in atlases for the delineation of continents. The projection has also been employed by the Coast and Geodetic Survey in an outline base map of the United States, scale 1 : 7 500 000. On account of the inclusion of the greater part of Mexico in this particular outline map, and on account of the extent of area covered and the general shape of the whole, the selection of this system of projection offered the best solution by reason of the advantages of equal- area representation combined with practically a minimum error of scale. Had the limits of the map been confined to the borders of the United States, the advantages of minimum area and scale errors would have been in favor of Albers projection, described in another chapter. The maximum error of scale at the eastern and western limits of the United States is but 1J per cent (the polyconic projection has 7 per cent), while the maxi- mum error in azimuths is 10 04'. Between a Lambert Zenithal projection and a Lambert conformal conic projection, which is also employed for base-map purposes by the Coast and Geodetic Survey, on a scale 1 : 5 000 000, the choice rests largely upon the property of equal areas repre- sented by the zenithal, and conformality as represented by the conformal conic pro- jection. The former property is of considerable value in the practical use of the map, while the latter property is one of mathematical refinement and symmetry, the projection having two parallels of latitude of true scale, with definite scale factors available, and the advantages of straight meridians as an additional element of prime importance. For the purposes and general requirements of a base map of the United States, disregarding scale and direction errors which are conveniently small in both pro- jections, either of the above publications of the U. S. Coast and Geodetic Survey offers advantages over other base maps heretofore in use. However, under the subject heading of Albers projection, there is discussed another system of map pro- jection which has advantages deserving consideration in this connection and which bids fair to supplant either of the above. (See frontispiece and table on pp. 54, 55.) Among the disadvantages of the Lambert zenithal projection should be men- tioned the inconvenience of computing the coordinates and the plotting of the double system of complex curves (quartics) of the meridians and parallels; the intersection of these systems at oblique angles; and the consequent (though slight) inconvenience of plotting positions. The employment of degenerating conical projections, or rather their extension to large areas, leads to difficulties in their smooth construction and use. For this reason the Lambert zenithal projection has not been used so exten- sively, and other projections with greater scale and angular distortion are more frequently seen because they are more readily produced. The center used in the frontispiece is latitude 400 and longitude 960, correspond- ing closely to the geographic center'8 of the United States, which has been determined by means of this projection to be approximately in latitude 390 50', and longitude 980 35'. Directions from this central point to any other point being true, and the law of radial distortion in all azimuthal directions from the central point being the same, this type of projection is admirably suited for the determination of the geo- graphic center of the United States. 18 '1cographic center of the United States" is here considered as a point analogous to the center of gravity of a spherical surface equally weighted (per unit area), and hence may be found by means similar to those employed to find the center of gravity. THE LAMBERT ZENITHAL EQUAL-AREA PROJECTION. 73 The coordinates for the following tables of the Lambert zenithal projection" were computed with the center on parallel of latitude 400, on a sphere with radius equal to the geometric mean between the radius of curvature in the meridian and that perpendicular to the meridian at center. The logarithm of this mean radius in meters is 6.8044400. THE LAMBERT EQUAL-AREA MERIDIONAL PROJECTION. This projection is also known as the Lambert central equivalent projection upon the plane of a meridian. In this case we have the projection of the parallels and meridians of the terrestrial sphere upon the plane of any meridian; the center will be upon the Equator, and the given meridional plane will cut the Equator in two points distant each 90 from the center. It is the Lambert zenithal projection already described, but with the center on the Equator. While in the first case the bounding circle is a horizon circle, in the meridional projection the bounding circle is a meridian. Tables for the Lambert meridional projection are given on page 75 of this publica- tion, and also, in connection with the requisite transformation tables, in Latitude Developments Connected with Geodesy and Cartography, U. S. Coast and Geodetic Survey Special Publication No. 67. The useful property of equivalence of area, combined with very small error of scale, makes the Lambert zenithal projection admirably suited for extensive areas having approximately equal magnitudes in all directions. TABLE FOR THE CONSTRUCTION OF THE LAMBERT ZENITHAL EQUAL-AREA PROJECTION WITH CENTER ON PARALLEL 404. Longitude 0* Longitude 5* Longitude 10* Longitude 15* Longitude 20 Longitude 25* Latitude - x Y x y x y x y x y x y Meters Meters Meters Meters Meters Meters Meters Meters Meters Meters ilfeters Meters 90 . 0+5387885 0+5387885 0+5387885 0+5387885 0+5387885 0+5387885 85* 0+4 878 763 52 414 +4 880 599 104 453 +4 886 085 155 742 +4895196 205 914 +4 907 863 254 604 +4 924 009 80*.. .-.-.-.-. -0 +4360 354 102 679 +4 363 859 204 665 +4 374 361 305 266 +4 391 792 403 799 +4 416 058 499 587 +4 447 015 75- ...-... 0 +3833 644 150 800 +36838 672 300 777 +3 855 490 448 560 +3 878 743 593 609 +3 913 587 734 842 +3 958 086 70* ....-.. 0 +3299 637 196 770 +3 306 041 392 357 +3 325 225 585 579 +3 357113 775 258 +3 401 565 960 222 +3 458 391 65.. . ...-. 0 +2 759 350 240 571 +2 766 994 479 775+2 789 898 716 248 +2 827 981 948 624+2 881 110 1 175 542 +2 949 088 60. 0 +2 213 809 282175 +2 222 561 562 835+2 248789 840 467 +2 292 419 1113 555 +2 353 321 1 380 581 +2 431 312 55* 0 +1664 056 321 546 +1673 787 641 463 +1702 962 958118 +1 751 509 1 269 876+1819 313 1 575 0?5 +1906 212 50*....... 0+1111133 358 645 +1121723 715 572+1153 474 1 069 062+1206 328 1 417 387 +1 280 187 1 758 808 +1374 910 45* 0+ 556 096 393 422 + 567 424 785 065+ 601 395 1 173 145 + 657 9611555 870 + 737 0461 931430 + 838 536 40. 0 0 425 827 + 11 951 8498355+ 47 792 1 270 200 + 107 490 168508S+ 190 989 2 092 644 + 298 207 35 0- 556 096 455 800 - 543 637 909 762 - 506 266 1 360 044 - 444 0051 804 787 - 356 887 2 242115 - 244 963 30*. .-.-..-.-. -0-1111133 483 280-1 098 277 964 722 -1 059 712 1 442 480 - 995 4431 914 696 - 905 490 2 379 489 - 789 868 25*.. ...-.- 0 -1 664 056 508 200 -1 650 911 1 014 578 -1 611480 1 517 303 -1 545 757 2 014 529 -1 453 735 2 504 389 -1 335 405 20. 0-2 213 809 530 490 -2 200 485 1059 186 -2 160 506 1 584 288 -2 093 872 2 103 978 -2 000 539 2 616 420 -1 880 485 15....... 0 -2 759 350 550 072 -2 745 953 1 098 391 -2 705 752 1 643 198 -2 638 727 2 182 718 -2 544 835 2 715 156 -2 424 020 10...... 0 -3 299 637 566 863 -3 286 269 1 132 024 -3 246157 1 693 776-3179 267 2 250 398 -3 085 552 2 800 148 -2 964 935 5'........ 0 -3 833 644 580 775 -3 820 4081 159 907 -3 780 690 1 735 750 -3 714 453 2 306 644 -3 621 639 2 870 912 -3 502166 0*..-...... 0 -4 360 354 591 710 -4 347 349 1 181 844 -4 308 330 1 768 820 -4 243 252 2 351 051 -4 152 00 2 926 92 -4 034 658 -5......- 0 -4 878 763 599 562 -4 866 0901 197 621 -4 828 0681 792 661 -4764 647 2 383 170 -4 675 776 2 967 618 -4 561 366 -10.----- -0-5387885 ........"............................................................... -............ 19 A mathematical account of this projection is given in: Z5ppritz, Prof. Dr. Kar,Leitfaden der Karteneutwurfslehre, Erster Theil, Leipzig, 1899, pp. 38-44. 74 U. S. COAST AND GEODETIC SURVEY. TABLE FOR THE CONSTRUCTION OF THE LAMBERT ZENITHAL EQUAL-AREA PROJECTION WITH CENTER ON PARALLEL 400-Continued. Longitude 30° Longitude 35° Longitude 40° Longitude 45° ILongitude 50° Latitude I I I 9o° .. 85°..... 80° .. 75°..... 70°.. 65° ... 55° .-. 50° ._ 45°--.- 40° .. 35° .. 30° .. 25° .. 20°--.- 10°--.- - 10°. Mel ters 0 301 461 591 966 871 326 1 139 309 1 395 644 1 640 025 1 872 122 2 091 574 2 298 001 2 490 992 2 670 12:3 2 834 946 2 984 985 3 119 741 3238685 3 341 257 3 426 851 3 494 828 3 544 456 Meters +5347 895 +4 943 517 +44S4441 +4 012 017 +3l 527 345 + 3 031 666 +2.526 155 +2 011 987 +1 490 314 + 962 2831 + 429 035 - 108 302 - 648 604 -1 190 758 -1 733 658, -2 276 209 -2 817 321 -3 355 917 - 3 890 925 -4 421 288 11lrcrs 346 141 690 290 1 002 148 1 311 367 1 607 577 1 890 367 2 159 301 2 413 918 2 653 72g 2 878 225 3 086 874 3 279 121 3 454 376 3 612 032 3 751 441 3 871 917 3 972 724 4 053 078 4 112 11 + 4148 912 31 ,Meters + 5387 885 +4 966 260 +4 528 232 +4 075 098 +3 608 121 + 3128 538 + 2 637 551 +2 136 368 +1626160 +1 108 095 +1 58.3 338 + 53 007 - 481 739 -1 019 784 -1 560 018 -2 101 311 - 2 642 587 -3 182 747 - 3720 708 -4 255 393 -4 785731 x Meters 0 388 315 763 928 1126401 1 475 260 1 809 998 2 131 174 2 434 962 2 724 049 2 996 737 3 252 512 3 49034 3710011 3 910 572 '1091 331 4 251 505 4 390 271 4 506 751 4 600 002 4 669 003 4 712 638 Meters +5 387 885 +4 992 087 +4 578 013 +4 147 003 +3 700 352 +3 239 317 +2 766 548 +2 279 018 + 1 782 160 + 1275 738 + 762 697 + 238 848 -289 326 -822 474 -1 359 474 -1 899 211 - 2440 579 - 2982 475 -3 523 806 -4 063 478 -4 600 397 Afters 427 669 842 271 1 243 218 1 629 868 2 001 571 2 357 658 2 697 424 3 020 158 3325112 3 611 538 3 878 620 6 125 567 6 351 528 6 555 605 6 736 893 6 894 403 5 027 097 5133 855 5 213 471 5 264 634 3/ x Meters Meters +45387885 0 +5 020 815 463 906 +4 633 520 914 747 +4 227 345 1 351 741 +3 803 605 1 770 009 +3 363 571 2 180 973 +2 908 476 2571 559 +2 439 543 2 944 994 + 1957 965 3300 406 +1 464 921:3 636 906 31 Meters +5 387 885 + 5052 256 +4 694 410 +4 315 689 +3 908 385 +3 500 777 + 3067 068 +2 617 467 +2 153 154 + 1 675 294 + 1 185 033 + 683 507 + 171 842 - 148 847 - 877 448 -1 412 861 -1 953 979 - 2499 703 - 3 048 927 Longitude 55° x y1 Meters Meters 0 +5 387 885 496 749 +5 086 182 980 794 +4 760 300 1 451 160 +4 411 541 1911977+4 051 990 2 346 901 2 770 281 3175 970 3 562 936 3 930 121 4 276 453 4 600 821 1 902 092 5179 091 5 430 598 +3 650 347 +3 240 321 +2 812 236 +2 367 231 +1 906 435 + 1 430 961 + 941 911 + 440 377 - 72 554 - 595 799 + 961 57,5 + 449 076 -- 71434 - 598 827 -1 131 980 -1669779 -2211115 - 2 754 8831 -3 299 979 -3 845 29)3 -4 389 713 3 953 579 4 249 488 4 523 664 4775104 5 002 765 5 205 559 5 382 331 5 531 855 5 652 831 Longitude 600 Longitude 65° Longitude 70° 1 Longitude 75° Latitude 90°--. - 85° -.- 80° .. 75°--. - 65° -. - 50°.- 45° -.- 40° -.- 35° -. - x Meters 0 525 944 1 039 898 1 540 696 2027143 31 Meters +5 387 885 +5 122 363 +4 830 776 +4 514 344 +4 174 238 .feters 551 251 1 091 571 1 619 591 2 133 939 31 Mfeters +5 387 881 +5 160 548 +4 905 382 +4 623 508 +4 315 959 x J1 J1 Longitude 80' x 31 Longitude 85° 3feters 0 572 489 1 135 398 1 687 179 2 226 296 Meters +5 387 883 +5 200 446 +4 983 620 +4 738 343 +4 465 524 2 498 081 +3 811 608 2 633 253 +3 983 794 2 751 208 +4 166 051 2 952 313 +3 427 563 3 116 166 +3 628 015 3 260 368 +3 840 791 3 388 043 +3 023 203 3 581 299 +3 249 622 3 752 226 +3 490 627 3 805 858 +2 599 608 4 027 2 58 +2 849 595 6 225 202 +3 116 381 Meters 0 589 457 1 170 961 1 742 813 2 303 315 2 850 778 3 383 488 3 899 721 41397 738 4 875 760 5 331 972 Alfters +5 387 885 +5 241 790 +5 064 964 +4 858 149 +4 622 066 +4 357 428 +4 064 920 + 3 745 227 +3 399 040 +3 027 033 + 2 629 879 Meters 602 013 1 197 928 1 785 923 236417,9 12 930 851 3484119 '4022101 4 542 891 5 044 571 Meters +5 387 881 + 5 284 269 +5 148 854 +4 982 151 +4 784 658 +4 556 864 +4 299 248 +4 012 291 + 3 696 461 + 3 352 228 Meters 0 610 041 1 216 016 1 816 003 2 408 109 2 990 401 3 560 930 4117 713 4 658 728 5 181 893 Mfeters +5 387 885 +5 327 574 + 5234 696 +5 109 509 +4 952,257 +4 763 192 +4 553 004 +4 290 523 +4 007 376 + 3 693 310 4 202 726 4 577 995 4 930 378 5 258 557 +x217 17841 + 1698 957 +1 223 9W7 + 734 004 48452 631 4 855 977 5 235 818 +2 428 903 + 1 988' 526) + 1529 40 4 4 677 842 5108 094 +2 719 004 +2 299 071 Longitude 90° Longitude 95° Longitude 100° Latitude -_____- _--_______ ____ ___ x 31 x 31 x 3 90 °.....................................---........ 8°................................................ 70° ............................................... 55° .............................................-- 450 .............................................-- 40°.............................."--- .........-"- Meters 0 613 457 1 225 008 1 832 631 2 434 454 3 028 467, 3 612 656 4 175 342 4 743 288 5 285 429' Meters +5 387 881 +5 371 383 +5 321 871 + 5 239 341 +5 123 768 +4 975 129 +4 793 371 +4 567 928 +4 330 321 +4 048 871 Afeters 0 .............................. 1 835 468 2 442 638 3 044 202 3 638 131 4 222 340 4 794 678 .fcters -I5 387 885 +5 370 715 d-5 297 996 +5 191 278 + 5 050 207 +4 874 439 +4 663 592 Afeters 3 636 304 4 228 414 4 811 080 Meters +5 387 885 +5 311 321 +5 176 606 + 5 005 259 THE LAMBERT ZENITHAL EQUAL-AREA PROJECTION. TABLE FOR THE CONSTRUCTION OF THE LAMBERT ZENITHAL EQUAL-AREA MERIDIONAL PROJECTION. [Coordinates in units of the earth's radius.] 75 Longitude 0° Longitude 5* Longitude 10* Longitude 15*° T d1i- Longitude 20* x y 1 u 1 I - I I ! tude 0.... 5.... 10... 15... 20... 25... 30... 35... 40... 45... 50... 55... 60... 65... 70... 75... 80... 85... 90... x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y 1 x 0.000000 0.087239 0.174311 0.261052 0.347296 0.432879 0.517638 0. 601412 0.684040 0.765367 0.845237 0.923497 1.000000 1. 074599 1. 147153 0.087239 0.086991 0.086241 0.084992 0.083240 0.080981 0.078211 0.074923 0.071109 0. 066759 0.061860 0.056398 0.050351 0. 043698 0.036408 0.000000 0. 087323 0.174476 0. 261297 0. 347617 0. 433272 0. 518096 0.601928 0. 684605 0. 765971 0.845866 0. 924139 1. 000635 1.075207 1. 147710 1. 218000 1.285937 1. 351387 1. 414214 0.174311 0.173812 0.172313 0. 169813 0. 166306 0.161785 0.156241 0. 149660 0.142028 0. 133325 0. 123525 0. 112600 0.100511 0.087211 0.072644 0.056739 0.039407 0. 020542 0. 000000 y 0.000000 0. 087571 0.174972 0. 262032 0. 348581 0. 434451 0. 519473 0. 603479 0. 686305 0. 767787 0. 847760 0. 926064 1. 002542 1.077032 1. 149380 1. 219429 1. 287022 1.352150 1. 414214 x 0.261052 0.260302 0.258051 0. 254295 0. 249026 0. 242235 0. 233908 0. 224026 0. 212568 0.199504 0. 184800 0. 168412 0. 149939 0. 130054 0. 108537 0. 084733 0.058818 0.030638 0. 000000 y/ 0.000000 0. 087990 0.175804 0. 263265 0.350199 0.436429 0. 521780 0.606079 0.689152 0. 770825 0.850929 0.929286 1.005727 1.080079 1. 152166 1. 221810 1.288828 1.353030 1. 414214 0.347296 0. 346291 0.343285 0. 338266 0. 331226 0.32215.3 0. 31130 0. 297835 0. 282538 0.265103 0.245487 0.223635 0. 199480 0. 172940 0. 143914 0. 112277 0. 077878 0. 040529 0. 000000 0.000000 0. 088582 0.176979 0. 265002 0. 352484 0. 439222 0. 525038 0. 609748 0. 693167 0. 775110 0. 855389 0. 933818 1. 010205 1. 084356 1. 156072 1. 225142 1. 291350 1. 354459 1. 414214 Longitude 25* S y 0.432879 0.010000 0. 431623 0. 089353 0.427851 0.178510 0. 421558 0. 267277 0.412733 0.355457 0.401363 0.442855 0. 387426 0. 529273 0.370897 0.614515 0.351743 0. 698379 0. 329244 0. 779058 0.305387 0.861169 0. 278071 0. 939682 0.247901 1.015991 0.214781 1.089874 0. 178601 1. 161099 0. 139220 1. 229422 0.096471 1.294579 0.050147 1.356283 0.000000 1.414214 1.217523 0.028444 1.285575 0.019762 1.351180 0.010305 1.414214 0.000000 I t Lati- tude 0.... 10... 15... 20... 25... 30... 35... 40... 45... 50... 55... 60... 65... 70... 75... 80... 85... 90... Longitude 25* x y Longitude 30* Longitude 3-*0 Longitude 40* Longitude 45* yJ 0. 432879 0. 431623 0. 427851 0. 421558 0. 412733 0. 401363 0. 387426 0. 370897 0. 351743 0.329244 0.305387 0. 278071 0. 247901 0. 214781 0. 178601 0. 139220 0. 096471 0. 050147 0. 000000 0.000000 0.089353 0. 178510 0. 267277 0. 355457 0. 442855 0. 529273 0.614515 0. 698379 0. 779058 0.861169 0.939682 1. 015991 1. 089874 1. 161099 1. 229422 1. 294579 1. 356283 1. 414214 0. 517638 0. 516124 0. 511581 0.504001 0.493374 0.479684 0.462910 0.443023 0. 419990 0.393765 0.364296 0. 331516 0. 295345 0. 255687 0. 212423 0. 165411 0. 114481 0. 059427 0. 000000 0. 000000 0.090310 0. 180411 0.270093 0.359147 0.447361 0. 534523 0. 620417 0. 704826 0. 787531 0.868302 0. 946908 1. 023106 1. 096644 1. 167253 1. 234646 1. 298509 1. 358496 1. 414214 x y 0.601412 0.000000 0.599638 0.091464 0.594311 0.182701 0. 585428 0. 273485 0.572975 0.363589 0.556939 0.452782 0. 537297 0.540832 0.514021 0.627504 0.487078 0.712559 0. 456425 0. 795753 0.422007 0.876829 0.383762 0. 955528 0.341338 1.030750 0. 295462 1.101684 0. 245202 1. 174540 0. 190699 1. 240809 0. 131794 1. 303128 0. 065301 1. 361083 0.000000 1.414214 x 0.684040 0.682000 0. 675879 0. 665670 0. 651364 0. 632946 0.610397 0. 583694 0. 552805 0. 517691 0.478307 0. 434595 0. 386490 0. 333910 0. 276761 0. 214932 0. 148297 0. 076708 0. 000000 0. 000000 0. 092826 0. 185404 0.277488 0. 368827 0. 459168 0. 548258 0.635535 0.721635 0. 805385 0.886800 0. 965586 1. 041432 1.114008 1. 182962 1. 247906 1. 308420 1. 364033 1. 414214 y y Longitude 50* x y 0. 765367 0. 763056 0. 756122 0.744560 0. 728365 0. 706066 0. 682022 0. 651842 0. 616961 0. 577350 0.532976 0. 483798 0. 429767 0. 370826 0. 306915 0. 237959 0. 163878 0. 084588 0.000000 0. 000000 0.094411 0. 188550 0.282142 0. 374912 0. 465622 0. 556868 0. 645482 0. 732126 0. 816497 0.898275 0. 977129 1. 052708 1. 124640 1. 192524 1. 255925 1. 314370 1. 367329 1. 414214 0. 845237 0. 842647 0. 834881 0. 821934 0. 803803 0. 780484 0. 751972 0. 718257 0. 679328 0. 635176 0.585785 0. 531139 0. 471219 0. 406007 0. 334709 0. 259626 0. 178427 0.091882 0. 000000 0. 000000 0. 096237 0. 192172 0. 287499 0. 381911 0. 475097 0. 566744 0.656527 0. 744114 0. 829164 0.911320 0. 990210 1. 065441 1. 136597 1. 203229 1. 264857 1. 320956 1. 370953 1. 414214 Longitude 50* Longitude 55* Longitude 60* Longitude 6a* Longitude 70* Longitude 75* tude x y x y x y x . Z y O 0.... 0.845237 0.000000 0.923497 0.000000 1.000000 0.000000 1.074599 0.000000 1.147153 0.000000 1.217523 O.000C00 5 0. 842647 0. 096237 0. 920622 0. 098326 0. 996827 0. 100703 1. 071115 0. 103398 1. 143342 0. 106449 1. 213365 0. 109901 10 ...0. 834881 0. 192172 0. 911995 0. 196312 0. 987311 0. 201021 1. 060670 0. 206359 1. 131919 0. 212397 1. 200903 0. 219222 15_.. 0. 821934 0. 287499 0. 897621 0.293617 0. 971458 0. 300570 1. 043276 0. 308144 1. 112907 0. 317341 1. 180179 0. 327383 20... 0.803803 0.381911 0.877502 0.389897 0.949282 0.398961 1.018962 0.409211 1.086352 0.420776 1.151257 0.433805 25... 0. 780484 0. 475097 0. 851641 0. 484802 0. 920800 0. 495801 0. 987761 0. 508217 1. 052313 0. 522193 1.114235 0. 537905 30... 0. 751972 0. 566744 0. 820046 0. 577981 0. 886036 0. 590691 0. 949722 0. 605007 1. 010871 0. 621053 1. 069235 0. 639100 35.._ 0.718257 0.656527 0.782723 0.669068 0.844341 0.682676 0.904904 0.699123 0.962126 0.716924 1.016411 0.736805 40 0. 679328 0. 744114 0. 739682 0. 757694 0. 797784 0. 772979 0. 853380 0. 790097 0. 906201 0. 809194 0. 955952 0. 830135 45 0. 635176 0. 829165 0. 690934 0. 843475 0. 744377 0. 859533 0. 795240 0. 877451 0. 843242 0. 897359 0. 888073 0. 919401 50 0. 585785 0. 911320 0. 636495 0. 926012 0. 684853 0. 942438 0. 730590 0. 960693 0. 773421 0. 980881 0.813035 1. 003117 ss.. 0.531139 0.990210 0.576381 1.004891 0.619275 1.021236 0.659555 1.039318 0.696939 1.059210 0.731128 1.080994 60... 0. 471219 1. 065441 0. 510618 1.079673 0. 547723 1. 095145 0. 582282 1. 112802 0. 614031 1. 131788 0. 642692 1. 152445 65... 0.406007 1.136597 0.439234 1.149898 0.470291 1.164563 0.498947 1.180610 0.524968 1.198048 0.548109 1.216887 70.. 0.334709 1.203229 0.362271 1.215076 0.387095 1.228063 0.409756 1.242180 ' 0.430061 1.257414 0.447808 1.273745 75 0. 259626 1. 261857 0. 279782 1. 274684 0. 298274 1. 285385 0. 314953 1. 296935 0. 329669 1. 309303 0. 342275 1. 322449 0. 178427 1. 320956 0. 191837 1. 328156 0. 204003 1. 335940 0. 214824 1. 344276 0. 224204 1. 353126 0. 232051 1. 362449 85. 0. 091882 1. 370953 0. 098534 1. 371885 0. 104491 1. 379104 0. 109706 1. 33581 0. 114135 1. 388292 0. 117736 1. 393206 90. 0.000000 1.414214 0.000000 1.414214 0.000000 1.414214 0.000000 1.414214 0.000000 1.414214 0.000000 1.414214 76 U. S. COAST AND GEODETIC SURVEY. TABLE FOR THE CONSTRUCTION OF THE LAMBERT ZENITHAL EQUAL-AREA MERIDIONAL PROJECTION--Continued. [Coordinates in units of the earth's radius.] Longitude 75° Longitude 80° Longitude 85° Longitude 000 Latitude -_____ ____ ____ __________ x y x y x y x y/ 0----------------------------.. 1. 217523 0.000000 1.285575 0.000000 1. 351180 0.000000 1. 414214 0.000000 5------------------------.... 1. 213365 0. 109901 1.281041 0. 113800 1.316245 0. 118231 1.408832 0. 123257 10............................ 1.200003 0. 219222 1. 267469 0. 226937 1.331607 0. 235695 1. 392729 0. 245576 15 ........................ 1. 180179 0. 327383 1.2.14912 0. 338721 1. 306926 0. 351527 1. 366025 0. 366025 20............................ 1. 151257 0. 433805 1. 213172 0. 418481 1. 272775 0. 465022 1. 328926 0. 483690 25............................ 1. 114235 0. 537605 1. 173287 0.555553 1.229210 0. 575380 1.281713 0.597672 30......................... 1. 069235 0. 639100 1. 124542 0. 659270 1. 176491 0. 681843 1. 224745 0. 707107 351........................... 1. 016411 0.736805 1. 067459 0. 758974 1.114934 0. 783667 1. 158456 0. 811160 40............................ 0. 955952 0.8.30435 1. 002308 0. 854010 1. 044910 0. 880132 1. 083351 0. 909039 45............................ 0. 888073 0. 919401 0. 929400 0. 943738 0. 966848 0. 970541 1. 000000 1. 000000 50............................ 0. 813035 1.003117 0.849094 1. 027521 0.881231 1. 054223 0. 909039 1.083351 55............................ 0. 731128 1. 080994 0. 761799 1.101745 0. 788602 1.130542 0.811160 1. 158456 60---------------------------.. 0.642692 1. 152445 0.667970 1.1374806 0.689552 1. 198901 0.707107 1.224745 65......................... 0. 548109 1. 216887 0.568115, 1.237122 0.584727 1. 258741 0. 597673 1.281713 70......................... 0. 447808 1. 273745 0. 462796 1. 291138 0. 474823 1. 309551 0. 483690 1. 328926 75......................... 0. 342275 1. 322449 0. 352628 1. 336326 0. 360588 1. 350874 0. 366025 1. 366025 80......................... 0. 232051 1. 362449 0. 238279 1. 372193 0. 242811 1. 382308 0. 245576 1. 392729 851........................ 0. 117736 1. 303206 0. 120476 1. 398291 0. 122324 1. 403512 0. 123257 1. 408832 90............................ 0. 000000 1. 414214 0. 000000 1. 414214 0. 000000 1. 414214 0. 000000 1. 414214 THE LAMBERT CONFORMAL CONIC PROJECTION WITH TWO STANDARD PARALLELS. DESCRIPTION. [See Plate I.] This projection, devised by Johann Heinrich Lambert, first came to notice in his Beitrige zum Gebrauche ~der Mathematik und deren Anwendung, volume 3, Berlin, 1772. / LIMITS OF PROJECTION - STANDARD PARALLEL dower) - " FiG. 52.-Lambert conformal conic projection. Diagram illustrating the intersection of a cone and sphere along two standard parallels. The elements of the projection are calculated for the tangent cone and afterwards reduced in scale so as to produce the effect of a secant cone. The parallels that are true to scale do not exactly coincide with those of the earth, since they are spaced in such a way as to produce conformalty. Although used for a map of Russia, the basin of the Mediterranean, as well as for maps of Europe and Australia in Debes' Atlas, it was not until the beginning of the World War that its merits were fully appreciated. 77 78 U. S. COAST AND GEODETIC SURVEY. The French armies, in order to meet the need of a system of mapping in which a combination of minimum angular and scale distortion might be obtained, adopted this system of projection for the battle maps which were used by the allied forces in their military operations. HISTORICAL OUTLINE. Lambert, Johann Heinrich (1728-1777), physicist, mathematician, and astrono- mer, was born at Milhausen, Alsace. He was of humble origin, and it was entirely due to his own efforts that he obtained his education. In 1764, after some years in travel, he removed to Berlin, where he received many favors at the hand of Frederick the Great, and was elected a member of the Royal Academy of Sciences of Berlin, and in 1774 edited the Ephemeris. He had the facility for applying mathematics to practical questions. The intro- duction of hyperbolic functions to trigonometry was due to him, and his discoveries in geometry are of great value, as well as his investigations in physics and astronomy. He was also the author of several remarkable theorems on conics, which bear his name. We are indebted to A. Wangerin, in Ostwald's Klassiker, 1894, for the following tribute to Lambert's contribution to cartography: The importance of Lambert's work consists mainly in the fact that he was the first to make general investigations upon the subject of map projection. His predecessors limited themselves to the investi- gations of a single method of projection, especially the perspective, but Lambert considered the problem of the representation of a sphere upon a plane from a higher standpoint and stated certain general condi- tions that the representation was to fulfill, the most important of these being the preservation of angles or conformality, and equal surface or equivalence. These two properties, of course, can not be attained in the same projection. Although Lambert has not fully developed the theory of these two methods of representation, yet he was the first to express clearly the ideas regarding them. The former-conformality-has become of the greatest importance to pure mathematics as well as the natural sciences, but both of them are of great significance to the cartographer. It is no more than just, therefore, to date the beginning of a new epoch in the science of map projection from the appearance of Lambert's work. Not only is his work of im- portance for the generality of his ideas but he has also succeeded remarkably well in the results that he has attained. The name Lambert occurs most frequently in this branch of geography, and, as stated by Craig, it is an unquestionable fact that he has done more for the advance- ment of the subject in the way of inventing ingenious and useful methods than all of those who have either preceded or followed him. The manner in which Lambert analyzes and solves his problems is very instructive. He has developed several methods of projection that are not only interesting, but are to-day in use among cartographers, the most important of these being the one discussed in this chapter. Among the projections of unusual merit, devised by Lambert, in addition to the conformal conic, is his zenithal (or azimuthal) equivalent projection already described in this paper. DEFINITION OF THE TERM "cCONFORMALITY." A conformal projection or development takes its name from the property that all small or elementary figures found or drawn upon the surface of the earth retain their original forms upon the projection. This implies that- All angles between intersecting lines or curves are preserved; THE LAMBERT CONFORMAL CONIC PROJECTION. 79 For any given point (or restricted locality) the ratio of the length of a linear element on the earth's surface to the length of the corresponding map element is constant for all azimuths or directions in which the element may be taken. Arthur R. Ilinks, M. A., in his treatise on "Map projections," defines ortho- morphic, which is another term for conformal, as follows: If at any point the scale along the meridian and the parallel is the same (not correct, but the same in the two directions) and the parallels and meridians of the map are at right angles to one another, then the shape of any very small area on the map is the same as the shape of the corresponding small area upon the earth. The projection is then called orthomorphic (right shape). The Lambert Conformal Conic projection is of the simple conical type in which all meridians are straight lines that meet in a common point beyond the limits of the map, and the parallels are concentric circles whose center is at the point of inter- section of the meridians. Meridians and parallels intersect at right angles and the angles formed by any two lines on the earth's surface are correctly represented on this projection. It.employs a cone intersecting the spheroid at two parallels known as the stand- ard parallels for the area to be represented. In general, for equal distribution of scale error, the standard parallels are chosen at one-sixth and five-sixths of the total length of that portion of the central meridian to be represented. It may be advisable in some localities, or for special reasons, to bring them closer together in order to have greater accuracy in the center of the map at the expense of the upper and lower border areas. NO) G \OO ol c 'ii) Ga FIG. 53.--Scale distortion of the Lambert conformal conic projection with the standard parallels at 29° and 450. On the two selected parallels, arcs of longitude are represented in their true lengths, or to exact scale. Between these parallels the scale will be too small and beyond them too large. The projection is specially suited for maps having a pre- dominating east-and-west dimension. For the construction of a map of the United States on this projection, see tables in U. S. Coast and Geodetic Survey Special Publication No. 52. 80 U. S. COAST AND GEODETIC SURVEY. Nvo 33 0\,9 FIG. 54.-Scale distortion of the Lambert conformal conic projection with the standard parallels at 330 and 45°. The chief advantage of this projection over the polyconic, as used by several Government bureaus for maps of the United States, consists in reducing the scale error from 7 per cent in the polyconic projection to 2J or 11 per cent in the Lambert. projection, depending upon what parallels are chosen as standard. The maximum scale error of 21 per cent, noted above, applies to a base map of the United States, scale 1: 5 000 000, in which the parallels 330 and 45* north latitude (see fig. 54) were selected as standards in order that the scale error along the central parallel of latitude might be small. As a result of this choice of standards, the maximum scale error between latitudes 30 ° and 47z° is but one-half of 1 per cent, thus allowing that extensive and most important part of the United States to be favored with unusual scaling properties. The maximum scale error of 2J per cent, occurs in southernmost Florida. The scale error for southernmost Texas is some- what less. With standard parallels at 29° and 450 (see fig. 53), the maximum scale error for the United States does not exceed 1 per cent, but the accuracy at the northern and southern borders is acquired at the expense of accuracy in the center of the map. GENERAL OBSERVATIONS ON THE LAMBERT PROJECTION. In the construction of a map of France, which was extended to 70 of longitude from the middle meridian for purposes of comparison with the polyconic projection of the same area, the following results were noted: Maximum scale error, Lambert =0.05 per cent. Maximum scale error, polyconic=0.32 per cent. Azimuthal and right line tests for orthodrome (great circle) also indicated a preference for the Lambert projection in these two vital properties, these tests indicating accuracies for the Lambert projection well within the errors of map con- struction and paper distortion. In respect to areas, in a map of the United States, it should be noted that while in the polyconic projection they are misrepresented along the western margin in one THE LAMBERT CONFORMAL CONIC PROJECTION. 81 dimension (that is, by meridional distortion of 7 per cent), on the Lambert projec- tion 2 they are distorted along both the parallel and meridian as we depart from the standard parallels, with a resulting maximum error of 5 per cent. In the Lambert projection for the map of France, employed by the allied forces in their military operations, the maximum scale errors do not exceed 1 part in 2000 and are practically negligible, while the angles measured on the map are practically equal to those on the earth. It should be remembered, however, that in the Lambert conformal conic, as well as in all other conic projections, the scale errors vary increas- ingly with th~e range of latitude north or south of the standard parallels. It follows, then, that this type of projections is not suited for maps having extensive latitudes. AREAs.-For areas, as stated before, the Lambert projection is somewhat better than the polyconic for maps like the one of France or for the United States, where we have wide longitude and comparatively narrow latitude. On the other hand, areas are not represented as well in the Lambert projection or in the polyconic projection as they are in the Bonne or in other conical projections. For the purpose of equivalent areas of large extent the Lambert zenithal (or azimuthal) equal-area projection offers advantages desirable for census or statistical purposes superior to other projections, excepting in areas of wide longitudes com- bined with narrow latitudes, where the Albers conical equal-area projection with two standard parallels is preferable. In measuring areas on a map by the use of a planimeter, the distortion of the paper, due to the method of printing and to changes in the humidity of the air, must also be taken into consideration. It is better to disregard the scale of the map and to use the quadrilaterals formed by the latitude and longitude lines as units. The areas of quadrilaterals of the earth's surface are given for different extents of latitude and longitude in the Smithsonian Geographical Tables, 1897, Tables 25 to 29. It follows, therefore, that for the various purposes a map may be put to, if the property of areas is slightly sacrificed and the several other properties more desirable are retained, we can still by judicious use of the planimeter or Geographical Tables overcome this one weaker property. The idea seems to prevail among many that, while in the polyconic projection every parallel of latitude is developed upon its own cone, the multiplicity of cones so employed necessarily adds strength to the projection; but this is not true. The ordi- nary polyconic projection has, in fact, only one line of strength; that is, the central meridian. In this respect, then, it is no better than the Bonne. The Lambert projection, on the other hand, employs two lines of strength which are parallels of latitude suitably selected for the region to be mapped. A line of strength is here used to denote a singular line characterized by the fact that the elements along it are truly represented in shape and scale. COMPENSATION OF SCALE ERROR. In the Lambert conformal conic projection we may supply a border scale for each parallel of latitude (see figs. 53 and 54), and in this way the scale variations may be accounted for when extreme accuracy becomes necessary. 25 In the Lambert projection, every point has a scale factor characteristic of that point, so that the area of any restricted locality is represented by the expression measured area on map Area =- (scale factor)2 Without a knowledge of scale errors in projections that are not equivalent, erroneous results in areas are often obtained. In the table on p. 55, "Maximum error of area," only the Lambert zenithal and the Albers projections are equivalent, the polyconic and and Lambert conformal being projections that have errors in area. 105877-28 6 82 U. S. COAST AND GEODETIC SURVEY. With a knowledge of the scale factor for every parallel of latitude on a map of the United States, any sectional sheet that is a true part of the whole may have its own graphic scale applied to it. In that case the small scale error existing in the map as a whole becomes practically negligible in its sectional parts, and, although these parts have graphic scales that are slightly variant, they fit one another exactly. The system is thus truly progressive in its layout, and with its straight meridians and properties of conformality gives a precision that is unique and, within sections of 2° to 40 in extent, answers every requirement for knowledge of orientation and distances. Caution should be exercised, however, in the use of the Lambert projection, or any conic projection, in large areas of wide latitudes, the system of projection not being suited to this purpose. The extent to which the projection may be carried in longitude 21 is not limited, a property belonging to this general class of single-cone projections, but not found in the polyconic, where adjacent sheets have a "rolling fit" because the meridians are curved in opposite directions. The question of choice between the Lambert and the polyconic system of pro- jection resolves itself largely into a study of the shapes of the areas involved. The merits and defects of the Lambert and the polyconic projections may briefly be stated as being, in a general way, in opposite directions. The Lambert conformal conic projection has unquestionably superior merits for maps of extended longitudes when the property of conformality outweighs the prop- erty of equivalence of areas. All elements retain their true forms and meridians and parallels cut at right angles, the projection belonging to the same general formula as the Mercator and stereographic, which have stood the test of time, both being likewise conformal projections. It is an obvious advantage to the general accuracy of the scale of a map to have two standard parallels of true lengths; that is to say, two axes of strength instead of one. As an additional asset all meridians are straight lines, as they should be. Conformal projections, except in special cases, are generally of not much use in map making unless the meridians are straight lines, this property being an almost indis- pensable requirement where orientation becomes a prime element. Furthermore, the projection is readily constructed, free of complex curves and deformations, and simple in use. It would be a better projection than the Mercator in the higher latitudes when charts have extended longitudes, and when the latter (Mercator) becomes objection- able. It can not, however, displace the latter for general sailing purposes, nor can it displace the gnomonic (or central) projection in its application and use to navi- gation. Thanks to the French, it has again, after a century and a quarter, been brought to prominent notice at the expense, perhaps, of other projections that are not con- formal-projections that misrepresent forms when carried beyond certain limits. 21 A map (chart No. 3070, see Plate I) on the Lambert conformal conic projection of the North Atlantic Ocean, including the eastern part of the United States and the greater part of Europe, has been prepared by the coast and Geodetic Survey. The western limits are Duluth to New Orleans; the eastern limits, Bagdad to cairo; extending from Greenland in the north to the West Indies in the south; scale 1:10 000 000. The selected standard parallels are 36* and 54* north latitude, both these parallels being, therefore, true scale. The scale on parallel 45* (middle parallel) is but 11 per cent too small; beyond the standard parallels the scale is increasingly large. This map, on certain other well-known projections covering the same area, would have distor- tions and scale errors so great as to render their use inadmissible. It is not intended for navigational purposes, but was con- structed for the use of another department of the Government, and is designed to bring the two continents vis-A-vis in an approxi- mately true relation and scale. The projection is based on the rigid formula of Lambert and covers a range of longitude of 165 degrees on the middle parallel. Plate I is a reduction of chart No. 3070 to approximate scale 1: 25 500 000. THE LAMBERT CONFORMAL CONIC PROJECTION. 83 Unless these latter types possess other special advantages for a subject at hand, such as the polyconic projection which, besides its special properties, has certain tabular superiority and facilities for constructing field sheets, they will sooner or later fall into disuse. On all recent French maps the name of the projection appears in the margin. This is excellent practice and should be followed at all times. As different projec- tions have different distinctive properties, this feature is of no small value and may serve as a guide to an intelligible appreciation of the map. In the accompanying plate (No. 1),22 North Atlantic Ocean on a Lambert con- formal conic projection, a number of great circles are plotted in red in order that their departure from a straight line on this projection may be shown. GREAT-CIRCLE COURSES.-A great-circle course from Cape Hatteras to the English Channel, which falls within the limits of the two standard parallels, indicates a de- parture of only 15.6 nautical miles from a straight line on the map, in a total distance of about 3,200 nautical miles. The departure of this line on a polyconic projection is given as 40 miles in Lieut. Pillsbury's Charts and Chart Making. DISTANCEs.-The computed distance rom Pittsburgh to Constantinople is 5,277 statute miles. The distance between these points as measured by the graphic scale on the map without applying the scale factor is 5,258 statute miles, a resulting error of less than four-tenths of 1 per cent in this long distance. By applying the scale factor true results may be obtained, though it is hardly worth while to work for closer results when errors of printing and paper distortion frequently exceed the above percentage. The parallels selected as standards for the map are 360 and 540 north latitude. The coordinates for the construction of a projection with these parallels as standards are given on page 85. CONSTRUCTION OF A LAMBERT CONFORMAL CONIC PROJECTION FOR A MAP OF THE UNITED STATES. The mathematical development and the general theory of this projection are given in U. S. Coast and Geodetic Survey Special Publications Nos. 52 and 53. The method of construction is given on pages 20-21, and the necessary tables on pages 68 to 87 of the former publication. Another simple method of construction is the following one, which involves the use of a long beam compass and is hardly applicable to scales larger than 1:2 500 000. Draw a line for a central meridian sufficiently long to include the center of the curves of latitude and on this line lay off the spacings of the parallels, as taken from Table 1, Special Publication No. 52. With a beam compass set to the values of the radii, the parallels of latitude can be described from a common center. (By computing chord distances for 250 of are on the upper and lower parallels of latitude, the method of construction and subdivision of the meridians is the same as that described under the heading, For small scale maps, p. 84.) However, instead of establishing the outer meridians by chord distances on the upper and lower parallels we can determine these meridians by the following simple process: Assume 390 of latitude as the central parallel of the map (see fig. 55), with an upper and lower parallel located at 24 and 49°. To find on parallel 24 the 22 See footnote on p. 82. 84 U. S. COAST AND GEODETIC SURVEY. intersection of the meridian 250 distant from the central meridian, lay off on the central meridian the value of the y coordinate (south from the thirty-ninth parallel 1 315 273 meters, as taken from the tables, page 69, second colum, opposite 250), and from this point strike an are with the x value (2 581 184 meters, first column). /2/ Y 1 9E I 6° a° 3 X a '. a .\ A N LO T n '2. >E---------- 2,581,184------- --- FIG. 55.-Diagram for constructing a Lambert projection, of small scale. The intersection with parallel 240 establishes the point of intersection of the parallel and outer meridian. In the same manner establish the intersection of the upper parallel with the same outer meridian. The projection can then be completed by subdivision for intermediate meridians or by extension for additional ones. The following values for radii and spacings in addition to those given in Table 1, Special Publication No. 52, may be of use for extension of the map north and south of the United States: Latitude Radius Spacings from 39* 51 6 492 973 1 336305 50 6 605 970 1 223 308 * * * * * * * 23 9615911 1 786 633 22 9 730456 1901 178 FOR SMALL SCALE MAPS. In the construction of a map of the North Atlantic Ocean (see reduced copy on Plate I), scale 1:10 000 000, the process of construction is very simple. Draw a line for a central meridian sufficiently long to include the center of the curves of latitude so that these curves may be drawn in with a beam compass set to the respective values of the radii as taken from the tables given on page 85. THE LAMBERT CONFORMAL CONIC PROTECTION. 85 To determine the meridians, a chord distance (chord= 2 r sin may be com- puted and described from and on each side of the central meridian on a lower parallel of latitude; preferably this chord should reach an outer meridian. The angle 6 equals l X, in which X is the longitude out from the central meridian and r is the radius of the parallel in question. On parallel 300, the chord of 65 degrees = 6 230 277 meters. By means of a straightedge the points of intersection of the chords at the outer ends of a lower parallel can be connected with the same center as that used in de- scribing the parallels of latitude. This, then, will determine the outer meridians of the map. The lower parallel can then be subdivided into as many equal spaces as the meridional interval of the map may require, and the meridians can then be drawn in as straight lines to the same center as the outer ones. If a long straightedge is not available, the spacings of the meridians on the upper parallel can be obtained from chord distance and subdivision in a similar manner to that employed on the lower parallel. Lines drawn through corresponding points on the upper and lower parallels will then determine the meridians of the map. This method of construction for small-scale maps is far more satisfactory than the one involving rectangular coordinates. Another method for determining the meridians without the computation of chord distances has already been described. TABLE FOR THE CONSTRUCTION OF A LAMBERT CONFORMAL CONIC PROJECTION WITH STANDARD PARALLELS AT 360 AND 54°. [This table was used in the construction of U. S. Coast and Geodetic Survey Chart No. 3070, North Atlantic Ocean, scale 1:10000 000. See Plate I for reduced copy.] (1= 0.710105; log 1=9.8513225; log K= 7.0685567.1 Latitude Radii Spacings of parallels Meters Afeters 75............................ ................................................ 2 787 926.3 3 495 899.8 70............................................................................3430 293.7 2853532.4 65............................................................................4 035 253.3 2 248 572.8 60............................................................................4615578.7 1668247.4 55 ............................................................................ 5 179 773.8 1 104 052.3 50..................................................................... ........... 5734157.3 549668.8 45 ............................................................................ 6 2835826.1 000 000.0 40 ............................................................................6 833 182.5 549 356.4 35............................................................................................. 7 386 250.0 1 102 423.9 30................................................................................................... 7946910.9 1663084.8 25.............................................. .................................................... 8 519 064.7 2 235 238.6 20....................................................................................... 9 106 795.8 2 822 969.7 15................................................................................................ 9 714 515.9 3 430 689.8 Coordinates of parallel 60* Coordinates of parallel 30* Coordinates of parallel 40* Longitude x y y xy *Meters Meters Meters Meters Meters Meters 5......................................... 285837 8859 492142 15253.......................... 10......................................... 570 576 35 403 982 394 60 955... ................... 15......................................... 853 125 79 529 1 468 876 136 930.......................... 20.................................. 1 132 400 141 069 1 949 718 242 887.......................... 25-......................................... 1 407 327 219 785 2 423 076 378 417"........ 30......................................... 1676851 315377 2887132 543002. ................... 35......................................... 1939 939 427 476 3 340 105 736 010....... .............. 40......................................... 2 195 579 555 652 3 780 256 956 699... .............. 45......................................... 2442 790 699 415 4 205 894 1 204 222.......................... 50......................................... 2 680 625 858 210 4 615 387 1 477 630........................ 55......................................... 2908169 1031430 5007163 1775872. 60......................................... 3 124 549 1 218 408 5 379 716 2 097 804.......................... 65......................................... 3 328 933 1 418 428 5 731 616 2 442 190 ........................ 70.................................. 3 520 539 1 630 721 6 061 515 2807 708.......................... 75......................................... 3698630 1854473 6368146 3192953 . 80.................................. 3 862 522 2 088 825 .. .. ..............5718312 3 092 422 85..................................4011588 2332875 ....... .........5938997 3 453729 90........................................4145251 2585689..............................6136881 3828010 86 U. S. COAST AND GEODETIC SURVEY. SCALE ALONG THE PARALLELS. Latitude-Degrees Scale factor. Latitude-Degrees. Scala factor. 20................................--.. 1.079 50...................-................-0.991 30------------- ------------- ------------1.021 54-----------------------------------1.000 36------------------------------------- 1.000 60------------------------------------- 1.022 40------------------------------------- 0.992 70-------------------------------...1.113 45------------------------------------- 0.98 (To correct distances measured with graphic scale, divide by scale factor.) TABLE FOR THE CONSTRUCTION OF A LAMBERT CONFORMAL CONIC PROJECTION WITH STANDARD PARALLELS AT 100 AND 480 40'. [This table was used in the construction of a map of the Northern and Southern Hemispheres. See Plate VII.] [1=[; log K=7.1369624.1 Latitude D egrees 0--------------------.. 1---------------------- 2...................... 3..................... 4..................... 5 ..................... 6...................... 7-----------------.--.- 8...................... 9..................... 10...................... 11..................... 12...................... 13...................... 14..................... 15..................... 16..................... 17...................... 18---------------------- 19...................... 20...................... 21...................... 22.................... 23...................... 24....................... 25...................... 26...................... 27...................... 28...................... 29...................... 30...................... 31..................... 12...................... 33...................... 34...................... 35...................... 36....................-- 37....................- 38...................... 39--"-".-...... 40. .............. Radius M1leters 13 707 631 13 589 325 13 472 006 13 355 628 13 240 149 13 125 526 13 011 719 12 895 693 12 786 406 12 674 819 12 563 899 12 453 605 12 343 906 12 234 766 12 126 148 12 018 025 11 910 357 11 803 114 11 696 264 11 589 778 11 483 614 11377 751 11 272 153 11 166 792 11 061 628 10 956 642 10 851 795 10 747 059 10 642 400 10 537 791 10 433 197 10 328 587 10 223 929 10 119 186 10 014 334 9 909 332 9 804 151 9 698 751 9 593 100 9 487 161 9 380 896 iffterence Scale along the parallel Latitude Radius I Difference Scale along the parallel 1 1 11 1 1- 1 Meiters 118 306 117 319 1136378 115 479 114 623 113 807 113 026 112 287 111 587 110 920 110 294 109 699 109 140 108 618 108 123 107 668 107 243 106 850 106 486 106 164 105 863 105 598 105 361 105 164 104 986 104 847 104 736 104 659 104 609 104 594 104 610 104 658 104 743 104 852 105 002 105 181 105 400 105 651 105 939 106 265 1. 0746 1.0655 1. 0567 1. 0484 1.0404 1. 0328 1.0256 1. 0187 1. 0121 1. 0059 1. 0000 0. 9944 0. 9891 0. 9842 0. 9795 0. 9751 0. 9711 0. 9673 0. 9638 0. 9606 0. 9576 0. 9550 0. 9526 0. 9505 I 0. 9487 0.9471 0. 9459 0. 9440 0. 9442 0. 9437 0. 9436 0. 9437 0. 9442 0. 9449 0. 9459 0. 9473 0. 9489 0. 9508 0. 9531 0. 9557 0. 9586 Degrees 40...................- 41...................-- 42...................- 43..................... 44..................... 45..................... 46..................... 47..................... 48................... - 49..................... 50..................... 51 .................... 52..................... 53................... 54..................... 53...................-- 56...................-- 57...................-- 58...................-- 59...................-- 60..................."- 61...................-- 62...................- 63..................... 64.................... 65.................... 66.................... 67..................... 68.................... 69..................... 70.................... 71.................... 72.................... 73..................... 74-1................... 75---..................... 76................... 77 .................... 78.................... 79.................... 80..................... 81.................... 82..................... 48° 30................-- Mleters 9 380 896 9 274 267 9 167 236 9 059 763 8 951 802 8 843 311 8 734 252 8 624 569 8 514 220 8 403 148 8 291 302 8 178 630 8 065 070 7 950 560 7 835 042 7 718 438 7 600 679 7 481 686 7 361 378 7 239 665 7 116 454 6 991 642 6 865 117 6 736 762 6 606 446 6 474 028 6 339 352 6 202 249 6 062 531 5 919 986 5 774 384 5 625 462 5 472 924 5 316 433 5 155 604 4 989 992 4 839 073 4 642 237 4 458 752 4 267 727 4 068 075 3 858 419 3 6:36 997 8 4 58 879 2feters 106 629 107 031 107 473 107 961 108 491 109 059 109 6&.3 110 349 111 072 111 846 112 672 113 560 114 510 115 518 116 604 117 759 118 993 120 308 121 713 123 211 124 812 126 525 128 355 130 316 132 418 134 676 137 103 139 718 142 545 145 602 148 922 152 538 156 491 160 829 165 612 170 919 176 836 183 485 191 025 199 652 209 656 221 422 0. 9586 0. 9619 0. 9656 0. 9696 0. 9740 0. 9787 0. 9839 0. 9896 0. 9956 1. 0021 1.0092 1. 0167 1. 0248 1. 0334 1.0426 1.0525 1. 0630 1.0743 1.0863 1.0992 1. 1129 1. 1276 1. 1433 1. 1601 1. 1782 1. 1975 1. 2184 1.2408 1. 2650 1. 2912 1. 3195 1.35 1.38 1.42 1.46 1.51 1.56 1.61 1. 67 1.75 1.83 1.93 2.04 0. 9988 THE GRID SYSTEM OF MILITARY MAPPING. A grid system (or quadrillage) is a system of squares determined by the rectangular coordinates of the projection. This system is referred to one origin and is extended over the whole area of the original projection so that every point on the map is coordinated both with respect to its position in a given square as well as to its posi- tion in latitude and longitude. The orientation of all sectional sheets or parts of the general map, wherever located, and on any scale, conforms to the initial meridian of the origin of coordinates. This system adapts itself to the quick computation of distances between points whose grid coordinates- are given, as well as the determination of the azimuth of a line joining any two points within artillery range and, hence, is of great value to military operations. The system was introduced by the First Army in France under the name "Quadrillage kilometrique systeme Lambert," and manuals (Special Publications Nos. 47 and 49, now out of print) containing method and tables for constructing the quadrillage, were prepared by the Coast and Geodetic Survey. As the French divide the circumference of the circle into 400 grades instead of 3600, certain essential tables were included for the conversion of degrees, minutes, and seconds into grades, as well as for miles, yards, and feet into their metric equiva- lents, and vice versa. The advantage of the decimal system is obvious, and its extension to practical cartography merits consideration. The quadrant has 100 grades, and instead of 80 39' 56", we can write decimally 9.6284 grades. GRID SYSTEM FOR PROGRESSIVE MAPS IN THE UNITED STATES. The French system (Lambert) of military mapping presented a number of features that were not only rather new to cartography but were specially adapted to the quick computation of distances and azimuths in military operations. Among these features may be mentioned: (1) A conformal system of map projection which formed the basis. Although dating back to 1772, the Lambert projection remained practically in obscurity until the outbreak of the World War; (2) the advantage of one reference datum; (3) the grid system, or system of rectangular coordinates, already described; (4) the use of the centesimal system for graduation of the cir- cumference of the circle, and for the expression of latitudes and longitudes in place of the sexagesimal system of usual practice. While these departures from conventional mapping offered many advantages to an area like the French war zone, with its possible eastern extension, military mapping in the United States presented problems of its own. Officers of the Corps of Engineers, U. S. Army, and the Coast and Geodetic Survey, foreseeing the needs of as small allowable error as possible in a system of map projection, adopted a succession of zones on the polyconic projection as the best solution of the problem. These zones, seven in number, extend north and south across the United States, covering each a range of 90 of longitude, and have overlaps of 10 of longitude with adiacent zones east and west. 87 88 U. S. COAST AND GEODETIC SURVEY. 0J N z'a aa * I 1 z ." J I F- Cl cC -~N ZL--- r rJ / 4 1J. el 0 0 - U Q L N I IN _/ ¢- I Y 0 N I r I ¢ y Z a ta-z THE GRID SYSTEM OF MILITARY MAPPING. o0 " 0co 89 4' __ 3,000,000 I i -i 4- 7! 44e 1 .-j i 2 40 . 2,500,000 2;00.0,000 1,500,000 1,000,000 500,000 32° i 28' 0 0 0 0 0 O_ O O~ 0 0 0 FIG. 57.-Diagram of zone C, showing grid system. 90 U. S. COAST AND GEODETIC SURVEY. A grid system similar to the French, as already described, is projected over the whole area of each zone. The table of coordinates for one zone can be used for any other zone, as each has its own central meridian. The overlapping area can be shown on two sets of maps, one on each grid system, thus making it possible to have progressive maps for each of the zones; or the two grid systems can be placed in different colors on the same overlap. The maximum scale error within any zone will be about one-fifth of 1 per cent and can, therefore, be considered negligible. The system is styled progressive military mapping, but it is, in fact, an inter- rupted system, the overlap being the stepping-stone to a new system of coordinates. The grid system instead of being kilometric, as in the French system, is based on units of 1000 yards. For description and coordinates, see U. S. Coast and Geodetic Survey Special Publication No. 59. That publication gives the grid coordinates in yards of the intersection of every fifth minute of latitude and longitude. Besides the grid system, a number of formulas and tables essential to military mapping appear in the publi- cation. Tables have also been about 75 per cent completed, but not published, giving the coordinates of the minute intersections of latitude and longitude. THE ALBERS CONICAL EQUAL-AREA PROJECTION WITH TWO STANDARD PARALLELS. DESCRIPTION. [See Plate III.] This projection, devised by Albers23 in 1805, possesses advantages over others now in use, which for many purposes give it a place of special importance in carto- graphic work. In mapping a country like the United States with a predominating east-and-west extent, the Albers system is peculiarly applicable on account of its many desirable properties as well as the reduction to a minimum of certain unavoidable errors. The projection is of the conical type, in which the meridians are straight lines that meet in a common point beyond the limits of the map, and the parallels are concentric circles whose center is at the point of intersection of the meridians. Merid- ians and parallels intersect at right angles and the arcs of longitude along any given parallel are of equal length. It employs a cone intersecting the spheroid at two parallels known as the standard parallels for the area to be represented. In general, for equal distribution of scale error, the standard parallels are placed within the area represented at distances from its northern and southern limits each equal to one-sixth of the total meridional distance of the map. It may be advisable in some localities, or for special reasons, to bring them closer together in order to have greater accuracy in the center of the map at the expense of the upper and lower border areas. On the two selected parallels, arcs of longitude are represented in their true lengths. Between the selected parallels the scale along the meridians will be a trifle too large and beyond them too small. The projection is specially suited for maps having a predominating east-and- west dimension. Its chief advantage over certain other projections used for a map of the United States consists in the valuable property of equal-area representation combined with a scale error 24 that is practically the minimum attainable in any system covering this area in a single sheet. In most' conical projections, if the map is continued to the pole the latter is represented by the apex of the cone. In the Albers projection, however, owing to the fact that conditions are imposed to hold the scale exact along two parallels instead of one, as well as the property of equivalence of area, it becomes necessary to give up the requirement that the pole should be represented by the apex of the cone; this 23 Dr. H. C. Albers, the inventor of this projection, was a native of Lneburg, Germany. Several articles by him on the subject of map projections appeared in Zach's Monatliche Correspondenz during the year 1805. Very little is known about him, not even his full name, the title "doctor" being used with his name by Germain about 1865. A book of 40 pages, entitled Unterricht im Schachsspiel (Instruction in Chess Playing) by H. C. Albers, Lneburg, 1821, may have been the work of the inventor of this projection. 24 The standards chosen for a map of the United States on the Albers projection are parallels 29° and 45*, and this selection provides for a scale error slightly less than 1 per cent in the center of the map, with a maximum of 1 per cent along the northern and southern borders. This arrangement of the standards also places them at an even 30-minute interval. The standards in this system of projection, as in the Lambert conformal conic projection, can be placed at will, and by not favoring the central or more important part of the United States a maximum scale error of somewhat less than 1 per cent might be obtained. Prof. Hartl suggests the placing of the standards so that the total length of the central meridian remain true, and this arrangement would be ideal for a country more rectangular in shape with predominating east-and-west dimensions. 91 92 U. S. COAST AND GEODETIC SURVEY. means that if the map should be continued to the pole the latter would be repre- sented by a circle, and the series of triangular graticules surrounding the pole would be represented by quadrangular figures. This can also be interpreted by the state- ment that the map is projected on a truncated cone, because the part of the cone above the circle representing the pole is not used in the map. The desirable properties obtained in mapping the United States by this system may be briefly stated as follows: 1. As stated before, it is an equal-area, or equivalent, projection. This means that any portion of the map bears the same ratio to the region represented by it that any other portion does to its corresponding region, or the ratio of any part is equal to the ratio of area of the whole representation. 2. The maximum scale error is but 12 per cent, which amount is about the minimum attainable in any system of projection covering the whole of the United States in a single sheet. Other projections now in use have scale errors of as much as 7 per cent. The scale along the selected standard parallels of latitude 291 and 452° is true. Between these selected parallels, the meridional scale will be too great and beyond them too small. The scale along the other parallels, on account of the compensation for area, will always have an error of the opposite sign to the error in the meridional scale. It follows, then, that in addition to the two standard parallels, there are at any point two diagonal directions or curves of true-length scale approximately at right angles to each other. Curves possessing this property are termed isoperimetric curves. With a knowledge of the scale factors for the different parallels of latitude it would be possible to apply corrections to certain measured distances, but when we remember that the maximum scale error is practically the smallest attainable, any greater refinement in scale is seldom worth while, especially as errors due to distortion of paper, the method of printing, and to changes in the humidity of the air must also be taken into account and are frequently as much as the maximum scale error. It therefore follows that for scaling purposes, the projection under consideration is superior to others with the exception of the Lambert conformal conic, but the latter is not equal-area. It is an obvious advantage to the general accuracy of the scale of a map to have two standard parallels of latitude of true lengths; that is to say, two axes of strength instead of one. Caution should be exercised in the selection of standards for the use of this projection in large areas of wide latitudes, as scale errors vary increasingly with the range of latitude north or south of the standard parallels. 3. The meridians are straight lines, crossing the parallels of concentric circles at right angles, thus preserving the angle of the meridians and parallels and facili- tating construction. The intervals of the parallels depend upon the condition of equal-area. The time required in the construction of this projection is but a fraction of that employed in other well-known systems that have far greater errors of scale or lack the property of equal-area. 4. The projection, besides the many other advantages, does not deteriorate as we depart from the central meridian, and by reason of straight meridians it is easy at any point to measure a direction with the protractor. In other words it is adapted to indefinite east-and-west extension, a property belonging to this general class of single-cone projections, but not found in the polyconic, where adjacent sheets con- THE ALBERS CONICAL EQUAL-AREA PROJECTION. 93 structed on their own central meridians have a "rolling fit," because meridians are curved in opposite directions. Sectional maps on the Albers projection would have an exact fit on all sides, and the system is, therefore, suited to any project involving progressive equal-area mapping. The term "sectional maps" is here used in the sense of separate sheets which, as parts of the whole, are not computed independently, but with respect to the one chosen prime meridian and fixed standards. Hence the sheets of the map fit accurately together into one whole map, if desired. The first notice of this projection appeared in Zach's Monatliche Correspondenz zur Befdrderung der Erd-und Himmels-Kunde, under the title "Beschreibung einer neuen Kegelprojection von H. C. Albers," published at Gotha, November, 1805, pages 450 to 459. A more recent development of the formulas is given in Studien iber fli chentreue Kegelproj ectionen by Heinrich Hartl, Mittheilungen des K. u. K. Militr-Geograph- ischen Institutes, volume 15, pages 203 to 249, Vienna, 1895-96; and in Lehrbuch der Landkartenprojectionen by Dr. Norbert Herz, page 181, Leipzig, 1885. It was employed in a general map of Europe by Reichard at Nuremberg in 1817 and has since been adopted in the Austrian general-staff map of Central Europe; also, by reason of being peculiarly suited to a country like Russia, with its large extent of longitude, it was used in a wall map published by the Russian Geographical Society. An interesting equal-area projection of the world by Dr. W. Behrmann appeared in Petermanns Mitteilungen, September, 1910, plate 27. In this projection equidistant standard parallels are chosen 300 north and south of the Equator, the projection being in fact a limiting form of the Albers. In view of the various requirements a map is to fulfill and a careful study of the shapes of the areas involved, the incontestible advantages of the Albers projection for a map of the United States have been sufficiently set forth in the above description. By comparison with the Lambert conformal conic projection, we gain the practical property of equivalence of area and lose but little in conformality, the two projections being otherwise closely identical; by comparison with the Lambert zenithal we gain simplicity of construction and use, as well as the advantages of less scale error; a comparison with other familiar projections offers nothing of advantage to these latter except where their restricted special properties become a controlling factor. MATHEMATICAL THEORY OF THE ALBERS PROJECTION. If a is the equatorial radius of the spheroid, E the eccentricity, and the latitude, the radius of curvature of the meridian 25 is given in the form a (1-E2) pm - (1 - E sin 10)32' and the radius of curvature perpendicular to the meridian 25 is equal to a pn - (1 -E2 sinl'- 1/2' 2 See U. S Coast and Geodetic Survey Special Publication No. 57, pp. 9-10. 94 U. S. COAST AND GEODETIC SURVEY. The differential element of length of the meridian is therefore equal to the expression a (1- E2) dy dm = (IE2sin2 0)32' and that of the parallel becomes a cos dX dp = (1 -E2 sin2 )12' in which X is the longitude. The element of area upon the spheroid is thus expressed in the form a2 (1-_E2) cos d dX dS=dmdp= (1-E2sin2 p)2 We wish now to determine an equal-area projection of the spheroid in the plane. If p is the radius vector in the plane, and 6 is the angle which this radius vector makes with some initial line, the element of area in the plane is given by the form dS' = p dp do. p and 6 must be expressed as functions of p and X, and therefore dp = j Pd ± -PdX and d6= - dy±+ d. We will now introduce the condition that the parallels shall be represented by concentric circles; p will therefore be a function of o alone, or dp- dv'. As a second condition, we require that the meridians be represented by straight lines, the radii of the system of concentric circles. This requires that ' should be independent of p, or 6 Furthermore, if 6 and X are to vanish at the same time and if equal differences of' longitude are to be represented at all points by equal arcs on the parallels, 6 must be equal to some constant times A, or 8 =nA, in which n is the required constant. This gives us d6 =ndX. By substituting these values in the expression for dS', we get dS'=p ands dN. THE ALBERS CONICAL EQUAL-AREA PROJECTION. 95 Since the projection is to be equal-area, dS' must equal -dS, or bp _ a2(1-E2) cosy dp dX p -ndpdX-- (1-E2 sin' )'2 The minus sign is explained by the fact that p decreases as (p increases. By omitting the dX, we find that p is determined by the integral -epd =a2(1 -E2) coscp dyp 0 a o n 0(1 - E2sin2 g,)2- If R represents the radius for in which a is the equatorial radius, the latitude, X the longitude, and E the eccen- tricity. For the purpose before us we may consider that the meridians are spaced equal to their actual distances apart upon the earth at the Equator. In that case the element of length dp along the parallel will be represented upon the map by a dx, or the scale along the parallel will be given in the form dp cosy a dX (1-e2 sin' 'p)*1 An element of length along the meridian is given in the form d=a (1-E2) dp -(1-2 sin2 P)",2 Now, if ds is the element of length upon the projection that is to represent this element of length along the meridian, we must have the ratio of dim to ds equal to the scale along the parallel, if the projection is to be conformal. 106 106 U. S. COAST AND GEODETIC SURVEYO Accordingly, we must have dm a (l-E2) dip cos p or, a(1-E2) d~o ds(1-=(E2 Sjn2 c0) Ccs The distance of the parallel of latitude

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Lambert Poection) UISC 4US  USCOASTAND GEODETIC SURVEY Plate 11 900 o o1100o0130°0 °61600 1800 1600 140 ANDALASKA ADITS RELATION TO THE o UNITED STATES AND THE ORIENT , -- Scale 40,000,000Oo j C, (Transverse Polyconic Projection) p HAW irN~ 40 15 1601700 0 17fir 120° 90° i.0 70° $ 50° b 11 e Q .ti 's ca Qo s distancef between Washington and Tokyo ortest rcle (s6 - QO 1gnchora$e , atio yam. P 1 Urea ° "°J\ eward -- --- ---- Sea \e of Q n Fuca \a d " e Poi 'autica/ Miles and Strait of " ' r o v x t- a 4.536 Nautical SC'o O Mies N a Fca nama o , a o ana o II Sao sec w t sec yea S o ON ISLANDS Statute Miles 0 So0 1000 2000 3000 Nautical Miles 0 500 1000 2000 3000 150° 440° 130° 120° 110° 100° net C. & G. S. Print - 99  ALBERS EQUAL AREA PROJECTION 0 T U~A XTTl A nU % Y .T. r TT"T C0 6bOoOn ._- P 0 ~1 AiNOAIAK)Dt'ALALLJ 25° 120° 115° 1100 05° 100° 91 ------------------------------------------- Standard Pars o>~ EL:LO ZU 3UAND 4° 30'Plate Ill 50 90° 85° 80° 75° 7065° v' pp~r0 45 0 C.& G.S.Pr-99 a0 50 0 500 IUU0 ttueMie 1 00 50 0 500 100 )o Statute M i les I.  GNOONI~ ~ROJECTI ON OF from Hydro~graphiC Office Chart 280 G O O .G of ~if. . . .' " Mota P . " .'C.5 tb ," ,l: ' ; " " , ;. THE NORTH ATLANTIC OCEAN; POIN T OF TANGENCY 3OdN. AND 30°w. C.Farewell _ I" 600": ." ........ .... . ......" ,, ,", Rao ........ ,Joh s --. "" " "" . , ' ; 50° . ''"" " ," " " ''. . : n . . . . . . . : . . . . Z: o3%: R g Plate IV YNORTH SEJA , tpoo. . Ile {1n. so epoA7 C t~l~cen I.~ ~.~~**~~*~~...~~ .....70° 60°00 4*0° 30° 20° 10 CA&G.S. Print-99  AITOFF'S E( UAL AREA PROJECTION OF THE SPHERE Plate V it , Q O O G S S O DOS 0 e 0 n a a i I a 0 0 a a a ° ° 0 ovv 0 0 l- .a o, 00 0 0 0 4 o O i o a e e g 11 = a e g a l A- : \' C.& G.SP-int - 99  I THE WORLD 00 THE HOMALOGRAPHIC PROJECTION (INTERRUPTED) FOR THE CONTINENTS (GOODE) Plate VI li MA / 80 80.. AV\ a 70 7 ______ __ ____ _ _ ______ 60 i _ _ _ _ _ _ _ _70 70 ~ " -T- _ 70l ILL -.---Tv FZ - - - - - - - - - - - - - - - --- it-Ili anW ,r ------- --- ----- ------ - 60 .sr - -- -r-r r--rtes --a - 1 O1 t " so :, T 50 Q 1 f " 10 i - A f n\ . 50, i 'wk 1~A ), Ip--,' inp- v, kAmC & -2 -- I' 'r I . - - r f A N. N\ '\X \ N N c. 50 in 0 b _ \40 - . , t " c . . .j S J rr. { _ t _ ' -'' j..... .... , _- ' ' "-'O' \\\ 40 i t X 41 i 7K i T rc i F 4i _7 k , k k-rv I -~ -- -I-- - .. X 40 30 h l 3A r" r x .. i. .. - :[ ' r .. _ '' .. 4 z-. 1 N -- \ 3n fvi f i . . 4Acr M- I - , i - - -w i i --- r , ice' t t - or - IF %t 1 1 lk :k , i V-N i ;z--- --" i -,- - I v .- k - 110,14 - , k -!. - -, IF, . -v 3 1, 1, \ 30 /-_ / / / I -'-I7 #1 : / /,d" / .... h \ ! r j i... ^ mil, "._ __ _ f= ----f°----r----a'-- - ---- - --------------------------- ------ ------------------z S+I - --- ----- -- wi, - 'h3iz--t--------A--- ---- ------ ---\ E---#=--- --f- ----- ---r -I l - - T- =---1 ---j- -- 1--I ------------ kw + - ---------- Xoowr-I------ ----- ----- ----- .4 - ---- ,1- ---_ _ _. J p, I I I I \0 / / 1.4k I I.....T_. 'N V \)..1 1' 'Iltw- 14 20 A i i . I- - - i zu; i f rc i N i A i _I 7 i / I / K: ...J " . / \1 1I s. f ,.. r r J j 1 ."' N n zo V f S ..._ 1 9 r M. _____10 140_0_0 -_10_9_SOY 0 )A w 13o 040O O I i\ 7 ' w \7 I \ 0 pf 0-O ?6 J10I0, I~ 7 w6 ..0 k50!40 ___ ___ __ _ .___ __ ___ 0 '"--.. f , ....- ;" t <" ,, . 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Prinf - 99 I  Plate VII LAMBERT- PROJECTION OF THE NORTHERN AND SOUTHERN HEMISPHERES 0 0 180 0180° o° ° ° 0 e 0 0 0 s t i ro 0 180° 1800 o 0 P d 0 0 O 0 i O)0 90. 0 ~~C. & G. S. Pit-9  CONFORMAL PROJECTION OF THE SPHERE WITHIN A TWO CUSPED EPICYCLOID Plate VIII 70 Devised by Dr. F.Augusf 70 0 \ E. .o _ 0 10 _ ------- ------ ----- ------ 0 d 0 g ° o q -- ----- ----- ----- ------ 10 y o --- ----- --- -------- ---- --- -- N 30 20 3040 510 70 8^° l so a 0 so 110 70 60 " 0 0 0 m O b 10 0 q. O 40 60 d v.. 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