Gornell University Library Ithaca, Nem York BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 Cornell University Library TA 590.D31 projection with applicat WHAT 003 898 271 eos Serial No. 146 ‘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 Cartographer AND “OSCAR S. ADAMS Geodetic Computer Special Publication No. 68 PRICE, 50 CENTS Sold only by the Superintendent of Documents, Government Printing Office, u Washington, D. C. WASHINGTON GOVERNMENT PRINTING OFFICE 1921 Serial No. 146 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 Cartographer AND OSCAR S. ADAMS Geodetic Computer Special Publication No. 68 PRICE, 50 CENTS Sold only by the Superintendent of Documents, Government Printing Office, Washington, D. C, WASHINGTON GOVERNMENT PRINTING OFFICE 1921 w 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. 2 CONTENTS. ’ Page. General ‘statertien tii.s:55n0cisica wserriatene sca cmnmivacimamssiccemmminiee aes Bach uti bnanidearceancuseie a Analysis of the basic elements of map projection...........----.-0-0-+- Miocene Heeb eiarantianle 9 Problem to be solved........- Stacie ek aa tiranaame a ciaen be aieiclcha tiene tah agatedafaisleitiarra-a 9 Reference points on the sphere...... 2.22.0... 0 eee cee eee eee eee ee eee cece c ee eeeeeeeeenes Il Determination of latitude..........-..-..-.-- BS paresis ara as reise ate Se Wie ane ciara 12 Determination of longitude. .:...........- E acallted nan ES cite ye hecho wyatinicts ole SBR ae nie.c 13 Plotting points by latitude and iouglindle BEE acc onesies ecto OR Miaiesre 14 Plotting points by latitude and longitude on a plane map.....:......-..--: aon 14 How to:draw a'straicht line s:< oc. ccscsescecsuseeeseeiiees sci seeneeeenaeess 15 How to make a plane surface... ..2..-. concen ecwie cer ecese cee denl et eee ed ll cbecccunens 16 How to draw the circles representing meridians and parallels on a sphere.........- Lachinauanee 17 The terrestrial Pl OWS wwii ccsccessisieiccrermaes She ccteaeterare ae area wate aN Gah Beare ae maraveancteverscesicies 19 Representation of the sphere upon a plane... .-- 2-22 eee ete eee eee n eee 22 The problem ‘of map: projection a 66:50..ss + came sensing eeees seen eeeeas nes sa ereeweeeeeemees 22 Definition of map projection.........22-.20--22ce ence cece at bee belee bec ececcececsceeeeneees 22 ion. bh 5B es bevels » 22 ' Conditions fulfilled by a map projection............2. 2.22220. e eee eee eee ee eee eee Set slyagacas 25 - Classification of projections. .......22. 222.20 e eee eee ec ee eee ee cence eeeee Ei a Nateisese 25. MTOM Cea INS DP Srapiaws wieisreysicl oaion we eieatsicynianreneeieniel hq ahga ate hig tid At atone et og de : 27 ’ Projections considered without mathematics.............2.-2-2-- eae ede eee eee eee eee eens 28 Elementary discussion of various forms of projection.........- eee Pt cla tM Reaen ots » 30 Cylindrical equal-area projection.........-..-----------+-+ ; S ' 30 Lylindrical equal-spaced projection..............------ etcene Edad ueivn einen tema yee ' 30 Projection from the center upon a tangent cylinder 30 Mercator projections vic sctesteta rors eiviartofajrsborsieia inde alereieiareieshade vic o fete eteeteheeer sd Meleiale vie eiesmeitetaill 32 Geometrical azimuthal projections. ..........--.--- ibe Aw Beaten Oy oD arenes Aon 935 Stereographic polar projection. ......-.....2 2. eee eee ee eee eee eee Sevateiaitsthnrnrecinn ts » 85 Centraliorgniomionie! projection =..q, «sei ecciaie.cic.2:cjere.e.c.se:mieies cia ose eseveintoimenseieveis'e pia veleidieveisieuemiecenieimes Lan" 87 Lambert azimuthal equal-area projection. ........ afovasenes levee ch yepenevtaeyayaraaladiaanns mivaeaasr a 88 Orthopraphic polar ProjeCtion ys <.c. ie ccicngivinigig vistaterere vo eae wae eiroinmarnre ene ei ieleaiinerncng peg 38 Azimuthal equidistant projection.............22-.00200 eee e ee eee eee e eee ee eee eeeeneeee _, 40 Other projections in frequent use......---.20- 20-0. eee eee eee eee aust. A Construction of a stereographic meridional projection............-.--.. 2-02. cece eee eee eeee 2 Construction of a gnomonic projection with point of tangency on the equator Conical projet ON jeans te cacncmnnsaisascetanmanasdses Shee ce meno ees semcciaed fees Central projection upon a cone tangent at latitude 30° seose eet, Bonneprojectiont 0 - me d ys, | ? i PROB IE PLZ pe wo = —————— SS Fic. 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 retain their shape even at the cest 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 asphere 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 all points of the map. CONDITIONS FULFILLED BY A MAP PROJECTION. If, then, we endeavor to securé 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 mep 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 make 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 a is strictly pulled, 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 i 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. Hach 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 Fic, 18.—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 J = Z \ Fie. 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. Fie. 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 fedtures 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 10° intervals. The length of the map is evi- dently 7 (about 34) 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 70° of latitude. The poles could not be shown, since as the projecting line approaches them 30 31 ELEMENTARY DISCUSSION OF VARIOUS FORMS OF PROJECTION. suorjoefoid peoeds-penbe peoupuryAg—* ZT “Ol, c *uorofoid vare-penbe peotmpulyég—'9T “old 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. 0 Fig. 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 80° 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. 22864°—21-——-3 33 & Fic. 19.—Perspective projection upon a tangent cylinder 34 U. §. COAST AND GEODETIC SURVEY. 20 20 60 70 0 Fie. 20.—Mercator 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 60° 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 east 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 60° 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 80° of latitude the area is increased to 36 times its real size, and at 89° an area would be more than 3000 times as large as an equal-sized area at the Equator. This 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 reduced 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. 35 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 360° 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 : 5 Fic. 21.—Determination of radii for stereographic polar projection. the projection is made is generally taken as the equatorial plane. 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 ES W be the plane of a meridian with NV represent- ing the North Pole. Then NP will be the trace of the plane tangent at the North Pole. Divide the arc WN 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 n p, those at 20° on a circle with radius nq, 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 10° 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 86 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. Fic. 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 ascale 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 Je wee ane ee ee Fie, 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 10°, then Pg, 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 30° 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- 388 U. S COAST AND GEODETIC SURVEY. 180 ¢ Fie. 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 joming 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 10° 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. Fie, 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. Fie. 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, aswehaveseen. Wecan 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. 41 ' Fre, 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 Fie. 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. §. 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 YM YON Ue YM. Uf Ly YY) UM YN) Bu" Ph ij WY ify iY || Ly yy jj Yy ze a Yy . | -—__| Yj, z Paha . |_ fro Wye: ey Vy YY iy — Yy Yy Ue UM Y Yy /// ee oa YY Yy | ' Yy L, 20) | Ly | : i, LL su" ae mae Yy 7 120" no” 100° = 90" — 80” 70° 60° 50° 40° Fig. 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. Fic, 32.—Lambert equal-area projection of the Western Hemisphere. 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. Fie. 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. 70° 60° 40° 30° ‘60° 0” Fig. 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 ares of ellipses. 44 U. §. 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 eople. oo 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 Pp’ Fic. 35.—Determination of the elements of a stereographic projection on the plane of a meridain. projection. Take the arc PQ, equal to 30°; that is, Q will lic 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 arc PSP’; then this arc 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- i [fz ie oe lie Laon mo Lan eee | |_| | | ] oe > Ps yy * , s ‘ Lee I i Qo a ATLAS 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 asector of a circle. The magnitude of the angle at the center of this sector has to be determined by computation from the condition imposed c 5 Fia. 37.—Cone tangent to the sphere at latitude 30°. 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 pee and in the second the scale is maintained true along two parallels. CENTRAL PROJECTION UPON A CONE TANGENT AT LATITUDE 30°. 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 30°. (See fig. 37.) In this case the full circuit of 360° of longitude will be c ry Fie. 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 30° of latitude on the projection. OR, OS, CT, etc., are the radii for the parallels of 80°, 70°, 60°, etc., respectively. The map of the Northern Hemisphere on this projection is shows j 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. § 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, i 5 <7 Ly wy SO K > 70. gO 90 Fia. 39.—Central perspective projection on cone tangent at latitude 30°. 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 arc rep- resenting this parallel. A system of concentric circles is then drawn to represent the other parallels with the spacings along the central meridian on the same scale as that of the standard parallel. Along the ares of these circles the longitude dis- tances are laid off on the same scale in both directions from the central meridian, Yi) yy Y} ype Yj Yj Yy / YVLLYY Loe ZY Mh Ly Ly rer. LU ) Be yy Yy YY “iy Li a LL Yi 7 Yj a a ee ie Ly a Y yy Z - YY, Z, z YY Yy =% 7 LO ( fees i : Yj - mg : g Ue Yj MoT a gee nee We 105" 0" 95" 40" 85° 80° 7s" 20° Fic. 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 22864°—21——-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. 1 Fia. 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 can be mapped with an accuracy such that no departure from true scale can be de- tected. A quadrangle of 1° 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. Fie. 42.—Man’s head drawn on globular pro- Fie, 43.—Man’s head plotted on orthographic Br os jection. - jection. / é Fic. 44.—Man’s head plotted on stereographic Fie, 45.—Man’s head plotted on Mercator projec- projection. 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. U. S. COAST AND GEODETIC SURVEY. 52 epig wea *aqnd peqiosumor & uo ereyds 94} Jo uorjdeloid ormoMIOUNH—"gF “OTT “opis wines au nner as. oe Pit THON Pin wey woueg PART II. INTRODUCTION. It is the purpose in Part IT 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 ‘tioreased 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 be 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 H. 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). ConrorMaL Mapprinc 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. Equat-ArEa Maprina.—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........---+---+++-e+eeceee ete sc erect tet e ete ce teens eee eens die thine cht aad 7 Lambert conformal conic projection with standard parallels of latitude at 33° and ADS ois akeleratclsberna 24 (Between latitudes 303° and 473°, 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... .. Shalala shape ord See eatea ©GSeo0e00 Cooeo seoeSoo Soooo 19 A mathematical account of this projection is given in: Zéppritz, Prof. Dr. Karl,Leitfaden der Kartenentwurfslehre, 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 40°—Continued. Longitude 30° Longitude 35° Longitude 40° Longitude 45° Longitude 50° Longitude 55° Latitude y Meters , Meters 301 461 591 966 871 326 1139 309 1395 644] +4012 017 +3 527 345) 2 091 574 2 298 001 2 490 992 2670123 2 834 946)— 3 238 685] 3 341 257 3 426 851 3 494 820) 3 544 456 1 002 146 1311 367 +3 031 666]1 607 577 +2,526 155|1 890 367 +2011 987)2 159 301 +1 490 314/2 413 918 + 962 283:2 653 723/+1 108 095 + 429 035/2 878 225)+ 583 330) — 108 302)3 086 874/+ 53 007 648 604/3 279 120|— 481 739) 2 984 985|—1 190 758/3 454 376/—1 019 784 3119 741]—1 733 658)3 612 032/—1 580 010 —2 276 209)3 751 441/—2 101 311 —2 817 32113 871 917|—2 642 587| —3 355 917/3 972 724|/—3 182 747 —3 890 925/4 053 078|—3 720 706} —4 421 288)4 112 118] —4 255 393 4 148 912)—4 785 731 Meters +5 387 885) +4 966 260 +4 528 232 +4 075 098 +3 608 121 +3 128 536) +2 637 551 +2136 366) +1 626 160 Meters” +5 387 885) +4 992 087 +4578 013 +4 147 003) +3 700 352 +3 239 317/2 001 571 +2 766 548|2 357 656 +2279 018)2 697 424 +1 782 160)3 020 156) +1 275 7383 325 112 + 762 697/3 611 530 + 238 848/3 878 620 — 289 326/4 125 567 — 822 474)/4 351 526 4.091 331/—1 359 474/4 555 609) 4 251 505|—1 899 2114 736 897 4 390 271|—2 440 57914 894 407, 4 506 751)—2 982 475)5 027 097, 4 600 002)—3 523 806)5 133 855) 4 669 003] —4 063 478)5 213 471 4 712 638|—4 600 397|5 264 634 Meters 388 315] 763 928 1126 401 1 475 260) 1 809 998 2131174 2 434 962 2724 049 2 996 737 3 252 512} 3 490 384! 3.710 011 3 910 572| S 1 243 216 1 629 866) +5020 815 +4 633 520 +4 227 345 +3 803 605) +3 363 571 +2 908 476) — 598 827/475 104 1 131 980|5 002 765|— 877 44815 430 599 —1 669 779)5 205 559|—1 412 861 —2 211 115)5 382 331/—1 953 979) —2 754 883/5 531 855|—2 499 703 —3 299 97915 652 831|—3 048 927 —3 845 293 —4 389 713) Meters +5 387 885) +5 052 256) +4 694 410) Meters 0 463 906 914 747 1351 741 1 770 009) +4 315 689]1 451 160) +8 908 385)1 911 977 2 180 973/43 500 777/2 346 903) 2.571 559|+-3 067 068)/2 770 280 +2 439 543/2 944 994/+-2 617 467/3 175 970 +1 957 965,3 300 406)-+2 153 154,3 562 936 +1 464 921'3 636 906) +1 675 2943 930 123 + 961 57513 953 579] +1 185 033/4 276 453/+1 430 961 + 449.076)4 249 488/-+ 683 507/4 600 820/-+ 941 911 71 43414 523 664/+ 171 842/4 902 092)+ 440377 —' 348 847/5 179 093 72 554 Meters +5 387 885 +5 086 182 +4 760 300 +4411 541 +4 051 990 +3 650 347 +3 240321 +2 812 236 +2 367 231 +1 906 435 Meters 496 749 980 794 2 — 595 799 Longitude 60° Longitude 65° Longitude 70° Longitude 75° Longitude 80° Longitude 85° Latitude y Meters +5 387 885) +5 122 361 “Meters 0 525 944) 1 039 898 1 540 690 2 027 143) 2 498 081 2 952 313 3 388 643 3 805 858 Meters +4 830 776]1 091 579/-+-4 905 382 +4 514 344/1 619 593)/+-4 623 500 +4174 238]2 133 939]+-4 315 959) +3 811 608/2 633 253/+-3 983 794) +3 427 563/3 116 166) +3 628 015) +3 023 203)3 581 299/-+-3 249 622, +2 599 608/4 027 258) 4 202 726)+-2 157 841/4 452 631]+ 2 428 909 Meters +5 387 885) 551 259/+-5 160 540 S +2 849 595) Meters Meters 0 885) 572 489/+-5 200 446 1 135 398]+-4 983 620 1 687 179|+-4 738 343 2 226 296|-+-4 465 524)2 303 315 2 751 208/+4-4 166 05212 850 778 3 260 368/+-3 840 796/3 383 486| 3.752 226)+3 490 627/3 899 721 4 225 202/+3 116 386)/4 397 736 4 677 842|/+-2 719 004)4 875 760 589 457; 1170961 1742 813 0/+-5 387 885) +5 241 790) 602 013/+5 284 269) 610041 +5 064 964/1 197 928/+5 148 854/1 216 010) +4 858 149/1 785 923]-+4 982 152/1 816 003 +4 622 066/2 364 175|+4 784 658/2 408 109 +4 357 428/2 930 855/+-4 556 864)/2 990 401 +4 064 92013 484 119/-+-4 299 248/3 560 930 +3 745 227/4 022 100) +4 012 292/4 117 713 +3 399 040/4 542 899) -+3 696 46114 658 728 +3 3027 a 5 044 577)+-3 352 226/5 181 893 Meters 0)+-5 387 Meters 885 Meters Meters +5 387 885 +5 327 574 +5 234 696 +5 109 509 +4 952 257 +4 763 192 +4 553 004 +4 290 523 +4 007 376 +3 693 310 Meters So 4.577 995|-+ 1 698 9574 855 977|-+1 988 52515 108 094|-+2 299 076|5 331 972|-+2 4 930 376|-+1 223 99715 235 819|-+1 529404)... -..-|..2...-00-|-ccceee-[oceeeeeeeelocee 5269 B71 42: (794. O04 cn case cle ouoe cecclurciccoelsccccic cc dhe ace lc na teas clleaeacacdl- ccgeecallecde Seca Longitude 90° Longitude 95° Longitude 100° Latitude 2 y 2 y 2 y Meters Meters Meters Meters Meters Meters +5 387 885 +5 387 885 +5 387 885 613 457] 15371 383|........---.|-----.--e2-.[ecceee 1.225008) +5 321.870| 1 294 6731 5 409 7351 1832 631| +5 239340, 1835468| 5370715 2.434454} +5123 768| 2 442 638| +5 297 996 3.028 467] +4975 129, 3044202 +5191 278| 3 036 885| 4.5 410 124 3.612 636) +4 793373, 3638131] +5050207/ 3636304| +5311 321 4175 342| 14567928] 4 222340| +4874 439] 4228 414| +5 176 606 4743 288] +4330321| 4 794678| +4 663592 4811080 +5005 259 5285 429| +4048 873 THE LAMBERT ZENITHAL EQUAL-AREA PROJECTION. 75 TABLE FOR THE CONSTRUCTION OF THE LAMBERT ZENITHAL EQUAL-AREA MERIDIONAL PROJECTION. [Coordinates in units of the earth’s radius.] Longitude 0° Longitude 5° Longitude 10° Longitude 15° Longitude 20° Longitude 25° Lati- tude z y z y z y z y z y Zz y 0 0. 0. 087239 | 0.000000 | 0.174311 | 0. 0. 261052 | 0.000000 | 0.347296 | 0.000000 | 0. 432879 | 0.090000 0 0. 087239 | 0.086991 | 0. 087323 | 0.173812 | 0.087571 | 0.260302 } 0.087990 | 0.346294 | 0. 088582 | 0. 431623 | 0.089353 0 0.174311 | 0.086241 | 0.174476 | 0.172313 | 0.174972 | 0.258051 | 0.175804 | 0. 343285 | 0.176979 | 0.427851 | 0. 178510 0 0. 261052 | 0.084992 | 0, 261297 | 0, 169813 | 0.262032 | 0.254295 | 0. 263265 |. 0. 338266 | 0.265002 | 0. 421558 | 0. 267277 0 0. 347296 | 0.083240 | 0.347617 | 0.166306 | 0.348531 | 0.249026 | 0.350199 | 0.331226 | 0.352484 | 0.412733 | 0.355457 Oo 0, 432879 | 0.080981 | 0.433272 | 0.161785 | 0.434451 | 0. 242235 | 0, 43 0.322153 | 0.439222 | 0. 401363 | 0. 442855 0 0. 517638 | 0.078211 | 0.518096 | 0.156241 | 0.519473 | 0.233908 | 0. 521780 | 0.311030 | 0. 525038 | 0.387426 | 0. 529273 0 0. 601412 } 0.074923 | 0.601928 | 0. 149660 | 0.603479 | 0.224026 | 0.60607: 297835 | 0.609748 | 0.370897 | 0.614515 0 0. 684040 | 0.071109 | 0.684605 | 0. 142028 | 0.686305 | 0.212568 | 0.680152 | 0.282533 | 0. 693167 | 0.351743 | 0.698379 0 0. 765367 | 0.066759 | 0.765971 | 0.133325 | 0.767787 | 0.199504 | 0.770825 | 0.265103 | 0.775110 | 0.329244 | 0.779058 0 0. 845237 | 0.061860 | 0.845866 | 0.123525 | 0.847760 | 0.184800 | 0.850929 | 0, 245487 | 0.855389 , 0. 305387 | 0. 861169 0 0.923497 | 0.056398 | 0.924139 | 0.11 926064 | 0.168412 | 0.929286 |. 0. 223635 | 0.933818 | 0.278071 | 0. 939682 Q 1. 00001 0.050351 | 1.000635 | 0.100511 | 1.002542 | 0.149939 | 1.005727 | 0.199480 | 1.010205 | 0.247901 | 1.015991 0 1.074599 | 0.043698 | 1.075207 | 0.087211 | 1.077032 | 0.130054 | 1.080079 | 0.172940 | 1.084356 | 0.214781 | 1.089874 0 1. 147153 | 0.036408 | 1.147710 | 0.072644 | 1.149380 | 0. 108537 | 1.152166 | 0.143914 | 1.156072 | 0.178601 | 1.161099 0 1, 217523 1.218000 | 0.056739 | 1.219429 | 0.084733 | 1.221810 | 0.112277 | 1.225142 | 0. 139220 | 1. 229422 0 1.285575 | 0.019762 | 1.285937 | 0.039407 | 1.287022 | 0.058818 | 1.288828 | 0.077878 | 1.291350 | 0.096471 | 1. 294579 0 1.351180 | 0.010305 | 1.351387 | 0.020542 | 1.352150 | 0.030638 | 1.353030 } 0. 040529 | 1.354459 | 0.050147 | 1.356283 0 1, 414214 | 0. 1) | 1.414214 | 0.000000 | 1, 414214 |} 0.000000 } 1. 414214 | 0.000000 | 1.414214 | 0.050000 } 1.414214 i 7 Longitude 25° Longitude 30° Longitude 35° Longitude 40° Longitude 45° Longitude 50° Lati- tude z y z y z y z y z y £ y ° 0....] 0.432879 | 0.000000 | 0.517638 | 0.000000 | 0.601412 | 0.000000 | 0.684040 | 0.000000 | 0.765367 | 0.000000 | 0. 845237 | 0.000000 5....| 0.431623 | 0.089353 | 0.516124 | 0.090310 | 0.599638 | 0.091464 | 0.682000 | 0.092826 | 0. 763056 | 0.094411 | 0. 842647 | 0. 096237 10...| 0.427851 | 0.178510 | 0.511581 | 0.180411 | 0. 594311 | 0.182701 | 0.675879 | 0.185404 | 0. 756122 | 0.188550 | 0. 834881 | 0. 192172 15...| 0.421558 | 0.267277 | 0.504001 | 0.270093 | 0.585428 | 0. 273485 | 0.665670 | 0.277488 | 0.744560 | 0.282142 | 0. 821934 | 0. 287499 20...| 0, 412733 | 0.355457 | 0.493374 | 0.859147 | 0.572975 | 0.363589 | 0.651364 | 0.368827 | 0.728365 | 0.374912 | 0. 803803 | 0.381911 25...) 0.401363 | 0. 442855 | 0.479684 | 0.447361 | 0.556939 | 0. 452782 | 0.632946 | 0. 459168 | 0.706066 | 0. 465622 | 0. 780484 | 0.475097 30...| 0.387426 | 0.529273 | 0. 462910 | 0.534523 | 0.537297 | 0.540832 | 0.610397 | 0, 548258 | 0.682022 | 0. 556868 | 0.751972 | 0. 566744 35...) 0.370897 | 0.614515 | 0. 443023 | 0.620417 | 0.514021 | 0.627504 | 0.583694 |"0. 635835 | 0.651842 | 0.645482 | 0.718257 | 0.656527 40...| 0.351743 | 0.698379 | 0.419990 | 0. 704826 | 0.487078 | 0.712559 | 0.552805 | 0.721635 | 0.616961 | 0.732126 | 0.679328 | 0. 744114 45...| 0. 0.779058 | 0.393765 | 0.787531 | 0, 456425 | 0.795753 | 0.517691 | 0. 805385 | 0.577350 | 0.816497 | 0.635176 | 0. 829164 50...; 0.305387 | 0.861169 | 0.3! 0. 868302 | 0, 422007 | 0.876829 | 0.478307 | 0. 886800 | 0.532976 | 0. 898275 | 0. 585785 | 0. 911320 55...| 0.278071 | 0.939682 | 0.331516 | 0.946908 | 0.383762 | 0.955528 | 0. 434595 | 0.965586 | 0. 483798 | 0.977129 | 0. 531139 | 0.990210 60...| 0.247901 | 1.015991 | 0.295345 | 1.023106 | 0.341338 | 1.030750 | 0.386490 | 1.041432 | 0. 429767 | 1.052708 | 0.471219 | 1.065441 65...| 0.214781 | 1.089874 | 0.255687 | 1.096644 | 0.295462 | 1.104684 | 0.333910 | 1.114008 | 0.370826 | 1.124640 007 | 1.136597 70...| 0.178601 | 1.161099 | 0.212423 | 1.167253 | 0.245202 | 1.174540 | 0.276761 | 1.182962 | 0.306915 | 1. 192524 0. 334709 | 1.203229 75...| 0.139220 | 1.229422 | 0.165411 | 1.234646 | 0.190699 | 1.240809 | 0.214932 ] 1.247906 | 0.237959 | 1.255925 | 0.259626 | 1.264857 80...) 0.096471 | 1.294579 | 0.114481 | 1.298509 | 0.131794 | 1.303128 | 0, 148297 | 1.308420 | 0.163878 ; 1.314370 | 0.178427 | 1.320056 85...| 0.050147 | 1.356283 | 0.059427 | 1.358496 | 0.068301 | 1.361083 | 0.076708 | 1.364033 | 0.084588 ) 1.367329 | 0.091882 | 1.370953 90... 1. 414214 | 0.000000 | 1.414214 | 0.000000 | 1.414214 000000 } 1.414214 | 0.000000 | 1.414214 | 0.000000 | 1.414214 Longitude 50° Longitude 55° Longitude 60° Longitude 65° Longitude 70° Longitude 75° Lati- tude z y z y z "oy I y z y z y 0....| 0.845237 | 0.000000 | 0.923497 | 0.000000 | 1.000000 | 0.000000 | 1.074599 | 0.000000 | 1.147153 | 0.000000 | 1.217523 | 0.000000 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 **! 6, 821934 | 0.287499 | 0. 897621 | 0.293617 | 0.971458 | 0.300570 | 1.043276 | 0.308444 | 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 | 0.621083 | 1.069235 | 0.639100 35. -*! Q. 718257 | 0. 656527 | 0. 782723 | 0.669068 | 0. 844341 | 0.632676 | 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. 830435 45_,_| - 635176 | 0. 829164 0.690934 | 0. 843475 | 0.744377 | 0. ae 0. 795240 | 0.877451 | 0. 843242 | 0. 897359 | 0. 888073 | 0.919401 50. "| 0.585788 | 0.911320 | 0.636495.| 0.926012 | 0.684853 |_0. 942: 0. 730590 | 0.960693 } 0.773421 | 0.980881 | 0.813035 | 1.003117 Bs -+! 0, 531139 | 0. 990210 | 0.576381 | 1.004891 | 0.619275 | 1. : 81936 0, 659555 | 1.039318 | 0.696939 | 1.059210 | 0. 731128 | 1.080994 55... 0. 471219 | 1.065441 | 0.510618 | 1.079673 | 0.547723 | 1.095445 | 0.582282 | 1.112802 } 0.614031 | 1.131788 | 0.642692 | 1.152445 oo: --! 0. 406007 | 1.136597 | 0. 439234 | 1.149898 | 0.470291 | 1.164563 | 0. 498947 | 1.180610 | 0.524968 | 1.198048 } 0. 548109 | 1. 216887 ro --! 9, 334709 | 1.203229 | 0.362271 | 1.215076 | 0.387095 | 1.228063 | 0.409756 | 1.242180 | 0.430061 | 1.257414 | 0.447808 | 1. 273745 “"") 9, 259626 | 1.264857 | 0.279782 | 1.274684 | 0.298274 | 1.285385 | 0.314953 | 1.298935 | 0.329669 | 1.309303 | 0.342275 | 1.322449 75... 0. 178427 | 1.320956 | 0. 191837 | 1.328156 | 0. 204003 | 1.335940 | 0.214824 } 1.344276 | 0.224204 | 1.353126 | 0.232051 | 1.362449 80... 0. 091882 | 1.370953 | 0.098534 | 1.374885 | 0.104491 | 1.379104 | 0.109706 | 1.383581 | 0.114135 | 1.388292 | 0.117736 | 1.393206 aa --} 9.000000 | 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 90° Latitude z y z y z y “2 y 1.217523 | 0.000000] 1.285575 | -0.000000| 1.351180] 0.000000} 1.414214 0. 000000 1.213365 | 0,109901 | 1.281044} 0.113806} 1.346245 | ‘0, 118231 . 408832 | | 0. 123257 1.200903 | 0.219222} 1.267469 | 0.:226937 | 1.331607 | 0.235695 | 1.392729 0. 245576 1.180179 | 0.327383 | 1.244912 | 0.338721] 1.306926 | ‘0.351527 | ‘1.3660! 0. 366025 1, 151257 . 433805 | 1.213472 | 0.448481 | 1.272775} 0.465022 | 1.328926 0. 483690 1.114235 | 0.537905 | 1: 173287 |. 0.555553 | 1.229210} 0.575380 | 1.281713 0. 597672 1.069235 | 0.639100} 1.124542 | 0.659270 | 1.176491 | 0.681843 | | 1.224745 0. 707107 1.016411 | 0.736805 | 1.067459} 0.758974 | 1.114934] 0.783667 | 1. 158456 0. 811160 0.955952 | 0.830435 | 1.002808 | 0.854010 | 1.044910] 0.880132] 1.083351 0. 909039 0. 888073 | 0.919401 | 0.929400} 0.943738 | 0.966848 | 0.970541 | 1.000000 1.000000 0. 813085 | 1.003117} 0.849094 | 1.027521} 0,881231| 1.054223! 0.909039} 1. 083351 0.731128 | 1.080994 | 0.761799 | 1.104745) 0.788602 | 1.130542] 0.811160 | 1.158456 0.642692 | 1.152445] 0.667970 | 1.174806 | 0.689552 | 1.198901] 0.707107 224745 0.548109 | 1.216887} 0.568115 | 1.237122 | 0.584727] 1.258741] 0.597673 1, 281713 0.447808 | -1.273745 | 0.462796 | 1.291138 |’ 0.474823 | 1.309551 | 0.483690 1. 328926 342275 | 1.322449] 0.352628 | 1.336326] 0.360588 | 1.350874 | | 0.366025 1. 366025 0.232051 | 1.362449 | 0.238279 | 1.372193 | 0.242811 | 1.382308 | 0. 245576 1.392729 0.117736 | 1.393206 | © 0: 120476 | 1.398291 | 0.122324} 1.403512 | 0. 123257 1. 408832 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. proj Fastin ‘devised by Johann Heinrich Lambert, first came to notice in his Beitrige zum Gebrauche der Mathematik und deren Anwendung, volume 3, Berlin, 1772. | UMITS OF PROJECTION Fie. 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, ofa secant cone. The parallels thet are true to scale do not exactly coincide with those of the earth, " gine they’ ‘are ‘spaced i in such a way as to produce conformality. Although used for a map of Russia, the hata 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 ‘“ CONFORMALITY.”’ 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. Hinks, 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. Fic. 53.—Scale distortion of the Lambert conformal conic projection with the standard parallels at 29° and 45°. 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 beyorid 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. ad 80 U. S. COAST AND GEODETIC SURVEY. Fic. 54.—Scale distortion of the Lambert conformal conic projection with the standard parallels at 33° 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 24 or 14 per cent in the Lambert projection, depending upon what parallels are chosen as standard. The maximum scale error of 24 per cent, noted above, applies to a base map of the United States, scale 1:5 000 000, in which the parallels 33° 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 304° and 474° 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 24 per cent occurs in southernmost Florida. The. scale error for southernmost Texas is some- what less. With standard parallels at 29° and 45° (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 7° 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 iguld ‘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” 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, . _ » Inthe 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 i in 2000 and are practically negligible, while the angles measured on the map are practically equal to those on the earth. Itshould be remembered, however, that in the Lambert conformal conic, as well as in all other conic projections, the scale errors vary increas- ingly with the 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. _ Argas.—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 1 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 Tembart 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 i is preferable. epee _ 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 i 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 j in. the polyconic projection every parallel of latitude i is developed upon its own. cone, the multiplicity of cones so employed necessarily adds strength | to the proj jection; but this is not true. The ordi- nary polyconic projection has, in fact, only one line of strength; that i 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. _. Aline 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 seale 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. 20 In the Lambert projection, every point has a scale factor characteristic of that point, so that the area of any restricted locality ession isn acai by uieexin , meses areaon map | (scale factor)? Without a knowledge ot scale errors in Dtojections that are not equivalent, erroneous results in areas are often obtained. Inthe 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. 22864°—21—6 Area = 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 4° 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 ”! 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. 2 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 000000. 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 13 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 usc 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-4-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 i voresenrs field sheets? wey will sooner or later fall into disuse. On all recent French maps ie name of the cores 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),?2 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 1rom 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 36° and 54° north latitude. The coordinates for the construction of a proj jection 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 tlie 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 25° of arc 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 39° 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 25° 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 column, opposite 25°), and trom this peu strike an arc with the x value 2 5 581 184 meters, first column). ye at w Fig. 55.—Diagram for constructing a Lambert projection of small scale. The intersection with parallel 24° 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 giveri 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 1336305 © i]! : 50 6 605 970 1 223 308 # * * * xe OK | * KOK xe OF 23 9615 911 1 786 633 22 9 730 456 1 901 178 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 PROJECTION. 85 To determine the meridians, a chord distance (chord =2 rsin 5) may be com- ribs 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. Chord distances for this map are given in the table. 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 cs 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 36° AND 54°, [This table was used i in the construction of U.S. Coast and Geodetic Survey Chart No. 3070, Ni orth Atlantic Ocean, scale 1:10 000 000. i See Plate I for reduced copy. J. ee a [2=0.710105; log 7=9.8513225; log K=7.0685567.] t ao . Spaci Latitude Radii pacing of Meters “Meters ~ 2 787 926.3 3 495 899.8 ~3 430 293.7 2 853 532.4 4 035 253.3 2 248 572.8 . 4615 578.7 1 668 247.4 5 179 773.8 1 104 052.3 _ 5.734 157.3 549 668.8 * 6 283 826.1 000 000.0 * 6 833 182.5 549 356. 4 » 7 386 250.0 1 102 423.9 7 946 910.9 1 663 084.8 ' 8519 064.7 2 235 238.6 9 106 795.8 2 822 969.7 ' 9 714 515.9 3 430 689,.8 Coordinates of parallel 60° Coordinates of parallel 30° | Coordinates of parallel 40° Latitude 2 : z y _f y z y ° Meters Meters Meters Meters 285 837 8 859 492 142 * 15 253 |_. 570 576 35 403 982 394 | » 60 955 |: 853 125 79 529 1 468 876 136 930 1 132 400 141 069 1949 718 242 887 |... 1 407 327 219 785 2 423 076 378-417 |._.. 1 676 851 815 377 2 887 132 543 002 |.... 1 939 939 427 476 3 340 105 736 010 2195 579 555 652 3 780 256 956 699 2 442 790 699 415 4 205 804 1 204 222 2 680 625 858 210 4 615 387 1 477 630 2 908 169 1 031 480 5 007 163 1775 872 |.... 3124 549 1 218 408 5 379 716 2 097 804 |... 3 328 933 1 418 428 5 731 616 2 442190 |... 3 520 589 1 630 721 6 061 615 2 807 708 |.... 3 698 630 1 854 473 6 368 146 3 192 953 3 862 522 2 088 825 5 718 312 3 092 422 4 011 588 2 332 875 5 938 997 3 453 720 4145 251 2 585 689 6 136 881 3 828 010 86 U. S. COAST AND GEODETIC SURVEY, ScALE ALONG THE PARALLELS. Latitude—Degrees. Scale factor. Latitude—Degrees. Scale factor. QL vreisicsccxe ne a eidieiatere a aitwisinicatese'oxtsiannice: 1-079) D0. oui cas siidinulsieiisis tech aweweniteseweioesis O/99T DO. cieisjeiewauneaiactd djsie di toeeie Sane 8 L021). | D4 aces cnewescaa sas Gaasaeciness tin «------- 1.000 DO secrcaradsicreke daicirecine Aenea tiaea aia sneictar woe: Ly 000) CO se cscc)c eee ace Bar Be esis ios asia shes dx'022 40 oso eegeveioorcossteseceess ae een O92 710 cc. ciscinnicwiajenistesecitiaeeciinonnacyeesreumen dad lS AD wise ap nets siateratsin aleiGigisiae Guesalen hese tie ate 0. 988 (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 10° AND 48° 40’. [This table was used in the construction of a map of the Northern and Southern Hemispheres. See Plate VII.] [J=3; log K=7.1369624.] , r ve Scale Scale - Latitude Radius Difference | along the. Latitude Radius | Difference | along the parallel : parallel Meters Meters Degrees Meters Meters 13 707 631 1.0746 118 306 Rees | | 117398 : 116 378 13 355628 | 116378 13.240 149 | 15 479 13 125 526 13011719 | 113 807 113 026 12 898 693 12 786 406 112 287 111 587 12674819 | 1587 12 563 899 12 453 605 teen 12.343 906 | 109 699 12234766 | 108140 12 126 148 12 018 025 11 910 357 Le ies 11 803 114 ean 11 696 264 eR 11 589 778 11 483 614 11 377 751 105 863 11272 123 105 598 11166792 | 105361 11 061 628 105 164 10 956 642 10 851 795 104 847 9 380 898 0.9588 9274267 | 108629 | 9619 9167236) 307031) 99656 9059763 | 107273 | 0, 9696 8951802} 107981) 9.9740 8 843 311 0.9787 8734252 | 109058 | 0.9839 8.624 569| 100883 | 0.9806 8514220| 10849) 0.9056 403148] LOT? | 1002 8 291 302 1.0092 173630| 11267) 0167 065070] 113580) y'o048 950 560) 24510) yoga g35042| 115518) 19495 718 438 1.0525 117 759 679{ 117759) 10630 1.0743 120 308 378| Tor a1g | 1.0868 665 | 45yo11| 1.0092 454 1.1129 124 812 642| T2es05| 1.1278 1. 1433 128 355 762 330 316 1. 1601 132418 | 1.1782 134676 | 1.1975 Bees a g n = S885 = rs a > ° S oo 39 352 1.2184 10747059 | 104736 202249! 137208} 1" 2408 10 B47 aor | 104 609 Neoee| saapin| «| 2680 ay 019986 | M2545) 2012 10 433 197 7743841 — sygqp9| 1.3195 152 538 156 491 160 829 165 612 170 919 mAb Awan ROD BLB 10 014 334 104 852 9 909 332 DD CNM AT TAARA AAM@EW ANN aI00 5 g S 8 2 _ a SPLVS ORD Bk 199 652 4 068 075 209 656 3 858 419 3636997 | 21 422 8458 879 |. ec ewnen g96 | 106 265 on oOo P< & - 8 i &R SOPPr peree pee 3 S88 aSa&se Sea 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 Rene in France under the name “‘Quadrillage kilométrique systéme 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 360°, 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 8° 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 i 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 9° of longitude, and have overlaps of 1° of longitude with adiacent zones east and west. ~~ 87 U. S. COAST AND GEODETIC SURVEY. 88 ~ shh, 250 “891819 Po} Og} Jo sdeur Arey ror @ATSSOIS0Id oy B8IIOZ PLIN—"gg ‘DIT 62 fi Sh 601 sls $9U0Z 40 dej4aag +6 + BU0Z 40 UIPIAA s3LON Q ne o . qt oy oO 2601 all fll slat ser THE GRID SYSTEM OF MILITARY MAPPING. 93°30° 99° 84°30" 000,000 500,000 000,000 500,000 ° ° 2. S So o ,000000 1,400,000 Fic. 57.—Diagram of zone C, showing grid system. 89 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 Albers” 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 * 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 % Dr. H. C. Albers, the inventor of this projection, was a native of Lineburg, 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, Liineburg, 1821, may have been the work of the inventor of this rojection. = ” The standards chosen for a map of the United States on the Albers projection are parallels 293° and 453°, and this selection provides for 2 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. Hart] 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 mene 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 i 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 1} per cent, which amount is about the Ioinimum 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 294° and 45$° 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 i 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 proj jection 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. 98 structed on their own central meridians have a “rolling fit,”” because meridians are curved in opposite directions. wep 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 tseotional 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 Beférderung der Erd-und Himmels-Kunde, under the title. “Beschreibung einer neuen Kegelprojection von H. - aes pavened at Gotha, November, 1805, pages 450 to 459. A more recent development of the vote: is given i in Studien tiber flachentreue Kegelprojectionen by Heinrich Hartl, Mittheilungen des K. u. K. Militar-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 walls 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 30° 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, ¢ the eccentricity, and ¢ the latitude, the radius of curvature of the meridian * is given in the form _ a (l—é). | ea ain? go)? and the radius of curvature perpendicular to the meridian ** is equal to fa. paca wtp w a Pa~ (I —@ sin? yg)?" % See U. S, Coast and Geodetic Survey Special Publication No. 57, pp. 9-10. ne, Ae a! NDAD ah Yop ‘ ne . 94 U. S. COAST AND GEODETIC SURVEY. The differential element of length of the meridian, is therefore equal to the expression a (1—é) dy dm =q2 é sin? gs?’ and that of the parallel becomes a cos g dy p= G—esin? 9) in which ) is the longitude. The element of area upon the spheroid is thus expressed in the form a (1—ée) cospdgan | a a (1—é sin? ¢)? 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 @ 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 dé. p and @ must be expressed as functions of ¢ and \, and therefore Op Op dp= ap de +sy dn and 06 00 d=s, de+s dy. We will now introduce the condition that the parallels shall be represented by concentric circles; p will therefore be a function of ¢ alone, or —2P dp= 5) de. 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 0 should be independent of ¢, or 08 d6= nr dn. Furthermore, if 6 and ) 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, @ must be equal to some constant times ), or 6=nh, in which 7 is the required constant. This gives us dé =ndn. By substituting these values in the expression for dS’, we get dS’=pSendy dy. THE ALBERS CONICAL EQUAL-AREA PROJECTION. 95 Since the projection is to be equal-area, dS’ must equal —dS, or : Op 4. ai — 2) cos g de dy Poo” dg dh= — (1—é sin? )? The minus sign is explained by the fact that p decreases as ¢ increases. By omitting the dd, we find that p is determined by the integral e e Op, @#(l—e) cos g dg [e%a0- n af (1-—¢ sin? ¢)? liek represents the radius for ¢ =0, this becomes 2_ pa _28(1—e) : cos ¢ de e — N : ome sin? or If 8 is the latitude on a sphere of radius c, the igithend member would be represented by the integral We may define 8 by setting this quantity equal to the above right-hand member, or : ~=07(1 - ef” (cos y+ 2esin?¢ cos ¢ + 3e'sint pcosy +4esin®ycosp+:+++s: Waa Therefore, . eee ot sin’ et ke sin® ott sin? p+.+---- ): As yet c is an undetermined constant. We may determine it by introducing the condition that, when e=5 8 shall also equa = This gives 3e* onan (1428 re +24 imeee ): The latitude on the sphere is thus defined in the form 1476 sin? etfs sin* pee 7 22 3e Ae® lt pte ta + eaeaewn sin 6 =sin 9 96 U. S. COAST AND GEODETIC SURVEY. This latitude on the sphere has been called the authalic latitude, the term authalic meaning equivalent or equal-area. A table of these latitudes for every half degree of geodetic latitude is given in U. S. Coast and Geodetic Survey Special Publication No. 67. With this latitude the expression for p becomes p? = 26 sin B- The two constants n and F are as yet undetermined. Let us introduce the condition that the scale shall be exact along two given parallels. On the spheroid the length of the parallel for a given longitude differ- ence ) is equal to the expression ad COS ¢ Pra asin ot On the map this arc is represented by p0 =pnn. On the two parallels along which the scale is to be exact, if we denote them by subscripts, we have _ ad COS 9, pynh= —é sin’ ¢,)”’ or, on omitting \, we have aime @ COS, 1 n(l—é sin? ¢,)” and ie a COS ¢, 2 n(l—é sin? g,)* Substituting these values in turn in the general equation for p, we get 2c. a cos? 9 pac Se SOS RE R ao ae nv (1—é sin? 9,) and 2CF os @ cos? ~ 9b ee 2 & i OD B,= nm (1—é sin? 9.) - In U. S. Coast and Geodetic Survey Special Publication No. 8 a quantity called A’ is defined as (1 —é sin? 9’)}, asn 1” ” A’= and is there tabulated for every minute of latitude. Hence ae ea 1 (l1—é@ sin? g,) A,? sin? 1’’ (The prime on A is here omitted for convenience.) The equations for determining Ff and n, therefore, become 2¢ cos? ¢ gees es ee Pe 8 & n sin B,= A, sin? 1”” 2¢ cos? 9 R- — sin so ee Ey n B,= A,7m* sin? 1” THE ALBERS CONICAL EQUAL-AREA PROJECTION, 97 By subtracting these equations and reducing, we get cos? y, | cos? ¢, _ Asin? 1” A,? sin? 1” 2¢ (sin 6, —sin B,) COs? ¢, COS? G, | A; sin? 17” A, sin? 1” re—r,? aersind O20 B,) cos} (8, +8,) 4c? sin } (8,— B,) cos} @, 4B,” r, and r, being the radii of the respective parallels upon the spheroid. By substituting the value of n in the above equations, we could determine F, but we are only interested in canceling this quantity from the general equation for p. Since n is determined, we have for the determination of p, @ COS 9, cos, Ty Pin (1—é sin? eo)! nA, sin 1” ~ n But 2c. CL sim By. By subtracting this equation from the general equation for the determination of p, we get 22 ,. 2 p?—p, =—~ (sin 8, —sin 8) or 4c. p?=p,+—~ sin + (8,—8) cos + @, +8). In a similar manner we have @ COS $3 COS 9, = Pa n(1—é sin? ¢,)4 — “nA, sin 17” and paper sin $ (8,—8) eos $ (8,+8). The radius c is the radius of a sphere having a surface equivalent to that of the spheroid. For the Clarke spheroid of 1866 (c in meters) log c=6.80420742 To obviate the difficulty of taking out large numbers corresponding to logarithms, it is convenient to use the form ene 4 sin $ (8,—8) cos 4 (8,+8), until after the addition is performed in the right-hand member, and then p can be found without much difficulty. For the authalic latitudes use the table in U. 8. Coast and Geodetic Survey Special Publication No. 67. 22864°—21——_7 98 U. S. COAST AND GEODETIC SURVEY. Now, if \ is reckoned as longitude out from the central meridian, which becomes the Y axis, we get d=), x=p sin @, y= —p cos 0. In this case the origin is the center of the system of concentric circles, the central meridian is the Y axis, and a line perpendicular to this central meridian through the origin is the X axis. The y coordinate is negative because it is measured downward. If it is desired to refer the coordinates to the center of the map as a single system of coordinates, the values become x=p sin 9, Y =Py—p Cos 9, in which p, is the radius of the parallel passing through the center of the map. The coordinates of points on each parallel may be referred to a separate origin, the point in which the parallel intersects the central meridian. In this case the coordinates become r=p sin 86, Yy =p—p Cos = 2p sin? $8. If the map to be constructed is of such a scale that the parallels can be con- structed by the use of a beam compass, it is more expeditious to proceed in the following manner: If \’ is the of the meridian farthest out from the central meridian on the map, we get 6’ =nn’, We then determine the chord on the circle representing the lowest parallel of the map, from its intersection with the central meridian to its intersection with the meridian represented by 2’, chord = 2p sin 4 0’. With this value set off on the beam compass, and with the intersection of the parallel with the central meridian as center, strike an arc intersecting the parallel at the point where the meridian of \’ intersects it. The arc on the parallel represents »’ degrees of longitude, and it can be divided proportionately for the other intersections. Proceed in the same manner for the upper parallel of the map. Then straight lines drawn through corresponding points on these two parallels will determine all of the meridians. The scale along the parallels, kp, is given by the expression kp= NPs Ts? in which p, is the radius of the circle representing the parallel of ¢,, and r, is the radius of the same parallel on the spheroid; hence 7, = 098 ¥e * A’, sin 1” THE ALBERS CONICAL EQUAL-AREA PROJECTION. 99 The scale along the meridians is equal to the reciprocal of the expression for the scale along the parallels, or syne ka NPs CONSTRUCTION OF AN ALBERS PROJECTION. This projection affords a remarkable facility for graphical construction, requir- ing practically only the use of a scale, straightedge, and beam compass. In a map for the United States the central or ninety-sixth meridian can be extended far enough to include the center of the curves of latitude, and these curves can be drawn in with a, beam compass set to the respective values of the radii taken from the tables. To determine the meridians, a chord of 25° of longitude (as given in the tables) is laid off from and on each side ‘of the central meridian, on the lower or 25° parallel of latitude. By means of a straightedge the points of intersection of the chords with parallel 25° can be connected with the same center as that used in drawing the parallels of latitude. This, then, will determine the two meridians distant 25° from the center of the map. The lower parallel can then be subdivided into as many equal spaces as may be required, and the remaining meridians drawn in simi- larly to the outer ones. If a long straightedge is not available, the spacings of the meridians on parallel 45° can be obtained from chord distance and subdivision of the arc in a similar manner to that employed on parallel 25°. Lines drawn through corresponding points on parallels 25° and 45° will then determine the meridians of the map. This method of construction is far more satisfactory than the one involving rectangular coordinates, though the length of a beam compass required for the construction of a map of the United States on a scale larger than 1:5 000 000 is rather unusual. In equal-area projections it is a problem of some difficulty to make allowance for the ellipticity of the earth, a difficulty which is most readily obviated by an intermediate equal-area projection of the spheroid upon a sphere of equal surface. This amounts to the determination of a correction to be applied to the astronomic latitudes in order to obtain the corresponding latitudes upon the sphere. The sphere can then be projected equivalently upon the plane and the problem is solved. The name of authalic latitudes has been applied to the latitudes. of the sphere of equal surface. A table ** of these latitudes has been computed for every half degree and can be used in the computations of any equal area projection. This table was employed in the computations of the following coordinates for the con- struction of a map of the United States. 26 Developments Connected with Geodesy and Cartography, U.S. Coast and Geodetic Survey Special Publication No. 67. 100 U. S. COAST AND GEODETIC SURVEY. TABLE FOR THE CONSTRUCTION OF A MAP OF THE UNITED STATES ON ALBERS EQUAL- AREA PROJECTION WITH TWO STANDARD PARALLELS. Radius of | Spacings of pa parallel Parallels Additional data. ep 2 10 258.177 eee fo} 3502771 10 145 579 108 039 colog n= 0.2197522 0 ey rat 108 460 log c= 6.8042075 9 820 218 109 249 oe : po 9710 969 Longitude from central Chords on | Chords on 9 601 361 109 608 meridian. latitude 25°. | latitude 45°. 9 491 409 | 109 952 9381139} 355 383 9 270 576 eT. 13 9215188 | 110 838 : 303 652.00 eee) | araad 2 352 568 8 937 337 111 311 gs25827) ini 677 . 8714 150 111 822 Scale factor | Scale factor Latitude. along along : se oy 111 936 the parallel | the meridian serearz | Hol 8286312} 315 984 1.0126 0.9876 8 154 228 i : 112 065 1.00 io $0818! npon rom} 71853] | 111 921 1.0124 - 9878 7708 444 i 781 1.0310 9699 7594 823) 411 402 7 483 426 a a 111 138 7261 459 | 110 829 zene, | 100082 POEAO) a00i8e2 6 931 333 109 069 6 822 264 6713780 | 108 484 THE MERCATOR PROJECTION. DESCRIPTION. [See fig. 67, p. 146.] This projection takes its name from the Latin surname of Gerhard Kramer, the inventor, who was born in Flanders in 1512 and published his system on a map of the world in 1569. His results were only approximate, and it was not until 30 years later that the true principles or the method of computation and construction of this type of projection were made known by Edward Wright, of Cambridge, in a publication entitled “‘Certaine Errors in Navigation.” In view of the frequent misunderstanding of the properties of this projection, a few words as to its true merits may be appropriate. It is by no means an equal- area representation, and the mental adjustment to meet this idea in a map of the world has caused unnecessary abuse in ascribing to it properties that are peculiarly absent. But there is this distinction between it and others which give greater accuracy in the relative size or outline of countries—that, while the latter are often. merely intended to be looked at, the Mercator projection is meant seriously to be worked upon, and it alone has the invaluable property that any bearing from any point desired can be laid off with accuracy and ease. It is,therefore, the only one that meets the requirements of navigation and has a world-wide use, due to the fact that the ship’s track on the surface of the sea under a constant bearing is a straight line on the projection. GREAT CIRCLES AND RHUMB LINES. The shortest line between any two given points on the surface of a sphere is the arc of the great circle that joins them; but, as the earth is a spheroid, the shortest or minimum line that can be drawn on its.ellipsoidal surface between any two points is termed a geodetic line. In connection with the study of shortest distances, how- ever, it is customary to consider the earth as a sphere and for ordinary purposes this approximation is sufficiently accurate. A rhumb line, or loxodromic curve, is a line which crosses the successive merid- ians at a constant angle. A ship “sailing a rhumb” is therefore on one course ”’ continuously following the rhumb line. The only projection on which such a line is represented as a straight line is the Mercator; and the only projection on which the great circle is represented as a straight line is the gnomonic; but as any oblique great circle cuts the meridians of the latter at different angles, to follow such a line would necessitate constant alterations in the direction of the ship’s head, an opera- tion that would be impracticable. The choice is then between a rhumb line, which is longer than the arc of a great circle and at every point of which the direction is the same, or the are of a great circle which is shorter than the rhumb line, but at every point of which the direction is different. The solution of the problem thus resolves itself into the selection of points at convenient course-distances apart along the great-circle track, so that the ship may be steered from one to the other along the rhumb lines joining them; the closer the 27 A ship following always the same oblique course, would continuously approach nearer and nearer to the pole without ever theoretically arriving at it. 101 102 U. S. COAST AND GEODETIC SURVEY. points selected to one another,—that is, the shorter the sailing chords—the mote nearly will the track of the ship coincide with the great circle, or shortest sailing route. 5 . é 39° ° 150° 100 i) 120° 130 140' 59° 55° 50° 45° 40° 7 Fre. 58.—Part of a Mercator chart showing a rhumb line and a great circle. The dotted line shows the rhumb line which is a straight line on this projection. The curve shown by a full line is the great circle track which lies on the polar side of therhumb line. Any great circle or straight line drawn between two given points on the gnomonic projection may be plotted on the Mercator projection by noting the latitudes of the points where the track crosses the various meridians. 80° 90° 100° 0° 120° 130° 149° 150° 160° 5S : 7 ee : sor 45° 40° 35° 30° Fia. 59.—Part of a gnomonic chart showing a great circle and a rhumb line. The full line shows the great circle track. The curve shown by a dotted line is the rhumb line which lies on the equatorial side of the great circle track. THE MERCATOR PROJECTION, 1038 For this purpose the Mercator projection, except in high latitudes, has attained an importance beyond all others, in that the great circle can be plotted thereon from a gnomonic chart, or it may be determined by calculation, and these arcs can then be subdivided into convenient sailing. chords, so that, if the courses are carefully followed, the port bound for will in due time be reached by the shortest practicable route. It suffices for the mariner to measure by means of a protractor the angle which his course makes with any meridian. With this course corrected for magnetic variation and deviation his compass route will be established. It may here be stated that the Hydrographic Office, U. S. Navy, has prepared a series of charts on the gnomonic projection which are most useful in laying off great circle courses. As any straight line on these charts represents a great circle, by taking from them the latitudes and longitudes of a number of points along the line, the great-circle arcs may be transferred to the Mercator system, where bearings are obtainable. It should be borne in mind, moreover, that in practice the shortest course is not always necessarily the shortest passage that can be made. Alterations become necessary on account of the irregular distribution of land and water, the presence of rocks and shoals, the effect of set and drift of currents, and of the direction and strength of the wind. It, therefore, is necessary in determining a course to find out if the rhumb line (or lines) to deatinntion 4 is interrupted or impracticable, and, if so, to determine intermediate points between which the rhumb lines are uninterrupted. The resolution of the problem at the start, however, must set out with the great circle, or a number of great circles, drawn from one objective point to the next. In the interests of economy, a series of courses, or composite sailing, will I frequently be the solution. Another advantage of the Mercator projection is that meridians, or north and south lines, are always up and down, parallel with the east-and-west borders of the map, just where one expects them to be. The latitude and longitude of any place is readily found from its position on the map, and the convenience of plotting points or positions by straightedge across the map from the marginal divisions. prevents errors, especially in navigation. Furthermore, the projection is readily constructed. A true compass course may be carried by a parallel ruler from a compass rose to any part of the chart without error, and the side borders furnish a distance scale convenient to all parts of the chart, as described in the chapter. of ‘Construction of a Mercator projection”. In many other projections, when carried too far, spheri- cal relations are not conveniently accounted for. From the nature of the projection any narrow belt of intitude 4 in any part of the world, reduced or enlarged to any desired scale, represents approximately true form for the ready use of any locality. . _ All charts are similar and, when brought. to the same scale, will fit exactly. Adjacent charts of uniform longitude scale will join exactly and will remain oriented when joined. The projection provides for longitudinal repetition so that continuous sailing routes east or west around the world may be completely shown on one map. Finally, a: as stated before, for a nautical chart, if for no other purpose, the Mer- cator projection, except in high latitudes, has attained an importance which puts all others in the background. 2% The ie latitudescale will give the correct distaseel inthe see saatie latitude. If sufficiently important on thesmaller scale charts, a diagrammatic scale could be placed on the char-s, giving the Scale for various latitudes, as on a French Mercator chart of Africa, No, 2A, published by the Ministére de 1a Marine. 104 U. S. COAST AND GEODETIC SURVEY. MERCATOR PROJECTION IN HIGH LATITUDES, In latitudes above 60°, where the meridional parts of a Mercator projection increase rather rapidly, charts covering considerable area may be constructed advan- tageously on a Lambert conformal projection, if the locality has a predominating east-and-west extent; and on a polyconic projection, or a transverse Mercator, if the locality has predominating north-and-south dimensions. In regard to suitable projections for polar regions, see page 147. Difficulties in navigation in the higher latitudes, often ascribed to the use of the Mercator projection, have in some instances been traced to unreliable positions of landmarks due to inadequate surveys and in other instances to the application of corrections for variation and deviation in the wrong direction. For purposes of navigation in the great commercial area of the world the Mercator projection has the indorsement of all nautical textbooks and nautical schools, and its employment by maritime nations is universal. It is estimated that of the 15 000 or more different nautical charts published by the various countries not more than 1 per cent are constructed on a system of projection that is noticeably different from Mercator charts. The advantages of the Mercator system over other systems of projection are evident in nautical charts of small scale covering extensive areas,” but the larger the scale the less important these differences become. In harbor and coast charts of the United States of scales varying from 1:10 000 to 1:80 000 the difference of the various types of projection is almost inappreciable. This being the case, there is a great practical advantage to the mariner in having one uniform system of projection for all scales and in avoiding a sharp break that would require successive charts to be constructed or handled on different principles at a point where there is no definite distinction. The use of the Mercator projection by the U. S. Coast and Geodetic Survey is, therefore, not due to the habit of continuing an old system, but to the desirability of meeting the special requirements of the navigator. It was adopted by this Bureau within comparatively recent years, superseding the polyconic projection formerly employed. The middle latitudes employed by the U. S. Coast and Geodetic Survey in the construction of charts on the Mercator system, are as follows: Coast and harbor charts, scales 1:80 000 and larger, are constructed to the scale of the middle latitude of each chart. This series includes 86 coast charts of the Atlantic and Gulf coasts, each on the scale 1:80 000. The use of these charts in series is probably less important than their individual local use, and the slight break in scale between adjoining charts will probably cause less inconvenience than would the variation in the scale of the series from 1:69 000 to 1:88 000 if constructed to the scale of the middle latitude of the series. General charts and sailing charts of the Atlantic coast, scales 1:400 000 and 1:1 200 000 are constructed to the scale of latitude 40°. The scales of the different charts of the series are therefore variant, but the adjoining charts join exactly. This applies likewise to the following three groups: General charts of the Pacific coast, San Diego to Puget Sound, are constructed to the scale of 1:200 000 in latitude 41°. 29 On small scale chartsin the middle or higher latitudes, the difference between the Mercator and polyconic projections is obvi- ousto the eye and affects the method ofusingthecharts. Latitude must not be carried across perpendicular to the border of a poly- conic chart of small scale. THE MERCATOR PROJECTION, 105 General charts of the Alaska coast, Dixon Entrance to Dutch Harbor, are con- structed to the scale of 1:200 000 in latitude 60°. General sailing charts of the Pacific coast, San Diego to the western limit of the Aleutian Islands, are constructed to the scale of 1:1 200 000 in latitude 49°. Some of the older charts still issued on the polyconic projection will be changed to the Mercator system as soon as practicable. Information as to the construction of nautical charts in this Bureau is given in Rules and Practice, U. 8. Coast and Geodetic Survey, Special Publication No. 66. DEVELOPMENT OF THE FORMULAS FOR THE COORDINATES OF THE MERCATOR PROJECTION. The Mercator projection is a conformal projection upon a cylinder tangent to the spheroid at the Equator. The Equator is, therefore, represented by a straight line when the cylinder is developed or rolled out into the plane. The meridians are represented by straight lines perpendicular to this line which represents the Equator; they are equally spaced in proportion to their actual distances apart upon the Equator. The parallels are represented by a second system of parallel lines perpendicular to the family of lines representing the meridians; or, in other words, they are straight lines parallel to the line representing the Equator. The only thing not yet; deter- mined is the spacings between the lines representing the parallels; or, what amounts to the same thing, the distances of these lines from the Equator. Since the projection is conformal, the scale at any point must be the same in all directions. When the parallels and meridians are represented by lines or curves that are mutually perpendicular, the scale will be equal in all directions at a point, if the scale is the same along the parallel and meridian at that point. In the Mer- cator projection the lines representing the parallels are perpendicular. to the lines representing the meridians. In order, then, to determine the projection, we need only to introduce the condition that the scale along the meridians shall be equal to the scale along the parallels. An element of length along a parallel is equal to the expression - a cos ¢ dy “Ge sin’ g)*’ in which a is the equatorial radius, ¢ the latitude, \ 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 adh, or the scale along the parallel will be given in the form dp cosy ad\ (1—é sin? g)* An element of length along the meridian is given in the form a (1—é) dy dm= qe sin? o)™ 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 dm to ds equal to the scale along the parallel, if the projection is to be conformal. 106 U. S. COAST AND GEODETIC SURVEY. Accordingly, we must have dm___a (1—@) dy 2 cos ¢ ds ds (i—é sin’ yg)” (1—é’ sin? g)™’ or, a(l—é) de =e sin? g) cos @ The distance of the parallel of latitude ¢ from the Equator must be equal to the integral ip is a (1—é) dg ' (1—é sin? ¢) cos 9 ° _ E do. oa5 cos: ¢ dy de ¢ cos y de ee ee ae | See ey hi cos ¢ 5 l—esin ¢ 2 p ite sie (, e ws fuses cos ¢ ease fee g dy =a sin ( aGe” 1—esin ¢ 2 bites e " (i +3) ae _, (8 G5) cos —sin 4 2 dp _ 4 2) de vg [Fesgee- [ita =a — g l1—e sin 2 1+esin Bi sin (G+ sin (448) 7 “a(S 2 j 9g On integration this becomes s= a loge sin ($+ $)- .a loge.cos G+ S) +o log. (1— esin y)—$ log. (1+e sin g) = mT, g\, ae l—esing =a log, tan (G+ S)+5 log. oom ©) —esin ¢\*” =a log. [tan (i+ s). Gi +e sin oe) | The distance of the meridian \ from the central meridian is given by the integral x s’=a] dr oO =an. The coordinates of the projection referred to the intersection of the central meridian and the Equator as origin are, therefore, given in the form 1—esin ¢\ y= adie (G+ g) Git TFesing) |: THE MERCATOR PROJECTION. 107 In U. S. Coast and Geodetic Survey Special Publication No. 67, the isometric or conformal latitude is defined by the expression 9 1—e sin 9\ tan ($4 x) = tan (F+$) - (eee) , X=572 and g= 5~P) or, if He tan S=tan 2 . cts oe eee 2 2 1l—e cos p With this value we get . y=a log, cot 5? or, expressed in common logarithms, a Zz Y= log cot 3? in which ¥ is the modulus of common logarithms. M=0.4342944819, log M=9.6377843113. A table for the isometric colatitudes for every half degree of geodetic latitude is given in U.S. Coast and Geodetic Survey Special Publication No. 67. The radius a is usually expressed in units of minutes on the Equator, or | _ 10800 = log a =3.5362738828, a log (j= 3.8984805715. log y=3.8984895715 + log (log cot 5) , ‘or, y =7915’.704468 log cot 5- The value of z now becomes 10800 r, T ’ with \ expressed in radians; or, . r=iX, with \ expressed in minutes of arc. The table of isometric latitudes given in U. S. Coast and Geodetic Survey Special Publication No. 67 was: computed for the Clarke spheroid of 1866. If it is desired 108 U. S. COAST AND GEODETIC SURVEY. to compute values of y for any other spheroid, the expansion of y in series must be used. In this case y=7915’.704468 log tan (F+§) 4 6 8 —3437'.747 (e sin o+{ sin’ p+ { sin’ y +5 sin’ o+-- .); or, in more convenient form, y =7915’.704468 log tan (5+$)- 3437'.747 [(e+f+ 4 = +76 ei -) sin y (Bt iste sat +) sin 30+ (5+ gat -) sin 5e—(aa5+ + -) sin 7g-- -| If the given spheroid is defined by the ee é may be computed from the formula € 7 = Of i ) in which f is the flattening. The series for y in the sines of the multiple arcs can be written with coefficients in closed form, as follows: y=7915'. 704468 log tan (F+8)—s437’. var (of aw e— 2. sin 3¢ 8e DF +2 sin se 2G sin 7p+ +: > -), in which f denotes the flattening and e¢ the eccentricity of the spheroid. DEVELOPMENT OF THE FORMULAS FOR THE TRANSVERSE MERCATOR PROJECTION. The expressions for the coordinates of the transverse Mercator projection can be determined by a transformation performed upon the sphere. If p is the great- circle radial distance, and w is the azimuth reckoned from a given initial, the trans- verse Mercator projection in terms of these elements is expressed in the form L=4 w, y=a log. cot f. But, from the transformation triangle (Fig. 66 on page 143), we have cos p=sin a sin 9+ cos a cos ¢ cos), cos a sm y— sin a cos ¢ COS A sin \ cos ¢ tan w= in which @ is the latitude of the point that becomes the pole in the transverse pro- jection. By substituting these values in the equations above, we get ee tan=(°8 a sin g—sin @ cos ¢ cos X = sin \ COs @ and 7 i _@ 1+ cos p y=a log. cot 5= 5 loge iets p _@ Z Lin in e+ cos a cos ¢ cos A\ gq *08e 1—sin @ sin g—cos a cos ¢ COS A) THE MERCATOR PROJECTION, 109 If we wish the formulas to yield the usual values when a converges to 5 , we must replace \ by 5 or, in other words, we must change the meridian from which d is reckoned by 5 With this change the expressions for the coordinates become sin a cos ¢ sin A— in oma tan-( cos ~ cos a sin ¢ cos ¢ COS XA = Jog. (Ltsin a sin y +cos @ cos ¢ sin d =z W08e. 1—sin @ sin g—cos @ cos ¢ Sin With common logarithms the y coordinate becomes a8 4 1+sin a sin g+cos a cos ¢ sin X Yom °8 \i—sin a sin p—cosa cosy sin \)’ in which M is the modulus of common logarithms. A study of the transverse Mercator projections was made by A. Lindenkohl, U.S. Coast and Geodetic Survey, some years ago, but no charts in the modified form have ever been issued by this office. In a transverse position the projection loses the property of straight meridians and parallels, and the loxodrome or rhumb line is no longer a straight line. Since the projection is conformal, the representation of the rhumb line must intersect the meridians. on the map at a constant angle, but as the meridians become curved lines the rhumb line must also become a curved line. The transverse projection, therefore, loses this valuable property of the ordinary Mercator projection. The distortion, or change of scale, increases with the distance from the great circle which plays the part of the Equator in the ordinary Mercator projection, but, considering the shapes and geographic location of certain areas to be charted, a transverse position would in some instances give advantageous results in the prop- erty of conformal mapping. CONSTRUCTION OF A MERCATOR PROJECTION. On the Mercator projection, meridians are represented by parallel and equidistant straight lines, and the parallels of latitude are represented by a system of straight lines at right angles to the former, the spacings between them conforming to the condition that at every point the angle between any two curvilinear elements upon the sphere is represented upon the chart by an equal angle between the representa- tives of these elements. . In order to retain the correct shape and comparative size of objects as far as possible, it becomes necessary, therefore, in constructing a Mercator chart, to increase every degree of latitude toward the pole in precisely the same proportion as the degrees of longitude have been lengthened by projection. TABLES. The table at present employed by the U.S. Coast and Geodetic Survey is that appearing in Traité d’Hydrographie by A. Germain, 1882, Table XIII. This table is as good as any at present available and is included in this publication, beginning on page 117. The outer columns of minutes give the notation of minutes of latitude from the Equator to 80°. 110 U. S. COAST AND GEODETIC SURVEY. The column of meridional distances gives the total distance of any parallel of latitude from the Equator in terms of a minute or unit of longitude on the Equator. The column of differences gives the value of 1 minute of latitude in terms of a minute or unit of longitude on the Equator; thus, the length of any minute of latitude on the map is obtained by multiplying the length of a minute of longitude by the value given in the column of differences between adjacent minutes. The first important step in the use of Mercator tables is to note the fact that a minute of longitude on the Equator is the unit of measurement and is used as an expression for the ratio of any one minute of latitude to any other. The method of construction is simple, but, on account of different types of scales employed by different chart-producing establishments, it is desirable to present two methods: (1) The diagonal metric scale method; (2) the method. similar to that given in Bowditch’s American Practical Navigator. ! DIAGONAL METRIC SCALE METHOD AS USED IN THE U. 8. COAST AND GEODETIO SURVEY, Draw a straight line for a central meridian and a construction line perpendicular thereto, each to be as central to the sheet as the selected interval of latitude and longitude will permit. To insure greater accuracy on large sheets, the longer line of the two should be drawn first, and the shorter line erected perpendicular to it. Example: Required a Mercator projection, Portsmouth, N. H., to Biddeford, Me., extending from latitude 43° 00’ to 43° 30’; longitude 70° 00’ to 71° 00’; scale on middle parallel 1:400 000, projection interval 5 minutes. The middle latitude being 43° 15’, we take as the unit of measurement the true value of a minute of longitude as given in the Polyconic Projection Tables, U. S. Coast and Geodetic Survey Special Publication No. 5 (general spherical coordinates not being given in the Germain tables). Entering the proper column on page 96, we find the length of a minute of longitude to be 1353.5 meters. As metric diagonal scales of 1:400 000 are neither available nor convenient, we ordinarily use a scale 1:10 000; this latter scale, being 40 times the former, the length of a unit of measurement on it will be one-fortieth of 1353.5, or 33.84. Lines representing 5-minute intervals of longitude can now be drawn in on either side of the central meridian and parallel thereto at intervals of 533.84 or 169.2 apart on the 1:10 000 scale. (In practice it is advisable to determine the outer meri- dians first, 30 minutes of longitude being represented: by 6 x 169.2, or 1015.2; and the 5-minute intervals by 169.2, successively.) THE PARALLELS OF LATITUDE. The distance between the bottom parallel of the chart 43° 00’ and the next 5-minute parallel—that is, 43° 05’—will be ascertained from the Mercator tables by taking the difference between the values opposite these parallels and multiplying this difference by the unit of measurement. Thus: Meridional Latitude. distance. o + 43° 05 2853. 987 @& 00 2847.171 6.816 111 THE MERCATOR. PROJECTION, 100 o€9 Ae _ ‘ayeyd woryonsyst0d—uoroeloid 1o}eo1e—'09 ‘PLT LEO COI eC ODT or MoE meen eneeer”” “ Sl ok7 2 oy 6 269 Se : = er E fh i ~ OQ] perez S{f ; Q < : [affesed | fe4UaD >= oy} Me QI. \. ay} gle? § » f : : A] | pozez owen es A a 6569 teem ent” |e . . See eons, ‘ %, Rs, \ ; >€ee see erecta eT eaY NS ‘s. ‘ N, 08 069 L- : ,00 20L 106 00 100012 112 U. S, COAST AND GEODETIC SURVEY. 6.816 multiplied by 33.84 = 230.6, which is the spacing from the bottom parallel to 43° 05’. The spacings of the other 5-minute intervals obtained in the same way are as follows: Latitude. Spacings. 43 BS8E65 From the central parallel, or 43° 15’, the other parallels can now be stepped off and drawn in as straight lines and the projection completed. Draw then the outer neat lines of the chart at a convenient distance outside of the inner neat lines and extend to them the meridians and parallels already constructed. Between the inner and outer neat lines of the chart subdivide the degrees of latitude and longitude as minutely as the scale of the chart will permit, the subdivisions of the degrees of longitude being found by dividing the degrees into equal parts; and the subdivisions of the degrees of latitude being accurately found in the same manner as the full degrees of latitude already described, though it will generally be sufficiently exact on large-scale charts to make even subdivisions of the degrees of latitude, as in the case of the longitude. In northern latitudes, where the meridional increments are quite noticeable, care should be taken so as to have the latitude intervals or subdivisions computed with sufficient closeness, so that their distances apart will increase progressively. The subdivisions along the eastern, as well as those along the western neat line, will serve for measuring or estimating terrestrial distances. Distances between points bearing north and south of each other may be ascertained by referring them to the subdivisions between their latitudes. Distances represented by lines (rhumb or loxodromic) at an angle to the meridians may be measured by taking between the dividers a small number of the subdivisions near the middle latitude of the line to be measured, and stepping them off on that line. If, for instance, the terrestrial length of a line running at an angle to the meridians, between the parallels of latitude 24° 00’ and 29° 00’ be required, the distance shown on the neat space between 26° 15’ and 26° 45’ (=30 nautical miles)®° may be taken between the dividers and stepped off on that line. An oblique line of considerable length may well be divided into parts and each part referred to its middle latitude for a unit of measurement. TO CONSTRUCT A MERCATOR PROJECTION BY A METHOD SIMILAR TO THAT GIVEN IN BOWDITCH’S AMERICAN PRACTICAL NAVIGATOR. If the chart includes the Equator, the values found in the tables will serve directly as factors for any properly divided diagonal scale of yards, feet, meters, or miles, these factors to be reduced proportionally to the scale adopted for the chart. If the chart does not include the Equator then the parailels of latitude should be referred to a principal parallel, preferably the central or the lowest parallel to be %0 Strictly speaking, a minute of latitude is equal to a nautical mile in latitude 48° 15’ only. The length of a minute of lati- tude varies from 1842.8 meters at the Equator to 1861.7 meters at the pole. THE MERCATOR PROJECTION, 113 drawn upon the chart. The distance of any other parallel of latitude from the principal parallel is the difference of the values of the two taken from the tables and reduced to the scale of the chart. If, for example, it be required to construct a chart on a scale of one-fourth of an inch to 5 minutes of arc on the Equator, the minute or unit of measurement will be 4 of 4 inch, or 5 of an inch, and 10 minutes of longitude on the Equator (or 10 meridional parts) will be represented by 42 or 0.5 inch; likewise 10 minutes of latitude north or south of the Equator will be represented by yy X 9.932 or 0.4966 inch. The value 9.932 is the difference between the meridional distances as given opposite latitudes 0° 00’ and 0° 10’. If the chart does not include the Equator, and if the middle parallel i is latitude 40°, and the scale of this parallel is to be one-fourth of an inch to 5 minutes, then the measurement for 10 minutes on this parallel will be the same as before, but the measurement of the interval between 40° 00’ and 40° 10’ will be », x 13.018, or 0.6509 inch. The value 13.018 is the difference of the meridional distances as given opposite these latitudes, i. e., the difference between 2620.701 and 2607.683. (It may often be expedient to construct a diagonal scale of inches on the drawing to facilitate the construction of a projection on the required scale.) Sometimes it is desirable to adapt the scale of a chart to a certain allotment of paper. Example: Let a projection be required for a chart of 14° extent in longitude between the parallels of latitude 20° 30’ and 30° 25’, and let the space allowable on the paper between these parallels be 10 inches. Draw in the center of the sheet a straight line for the central meridian of the chart. Construct carefully two lines perpendicular to the central meridian and 10 inches apart, one near the lower border of the sheet for parallel of latitude 20° 30’ and an upper one for parallel of latitude 30° 25’. Entering the tables in the column meridional distance we find for latitude 20° 30’ the value 1248.945, and for latitude 30° 25’ the value 1905.488. The differ- ence, or 1905.488—1248.945 =656.543, is the value of the meridional arc between these latitudes, for which 1 minute of arc of the Equator is taken as a unit. On 10 in 656.543 ~ 00152 inch, which will be the unit of measurement. By this quantity all the values derived from the table must be multiplied before they can be used on a diagonal scale of inches for this chart. As the chart covers 14° of longitude, the 7° on either side of the central meridian will be represented by 0.0152 x 607, or 6.38 inches. These distances can be laid off from the central meridian east and west on the upper and lower parallel. Through the points thus obtained draw lines parallel to the central meridian, and these will be the eastern and western neat lines of the chart. In order to obtain the spacing, or interval, between the parallel of latitude 21° 00’ and the bottom parallel of 20° 30’, we find the difference between their meridional distances and multiply this difference by the unit of measurement, which is 0.0152. Thus: (1280.835—1248.945) x 0.0152 or 31.890 x 0.0152 = 0.485 inch. the projection, therefore, 1 minute of arc of longitude will measure 7-,-=73 22864°—21——-8 114 U. S. COAST AND GEODETIC SURVEY. On the three meridians already constructed lay off this distance from the bottom parallel, and through the points thus obtained draw a straight line which will be the parallel 21° 00’. Proceed in the same manner to lay down all the parallels answering to full degrees of latitude; the distances for 22°, 23°, and 24° from the bottom parallel will be, respectively: 0.0152 X (1344.945— 1248.945) =1.459 inches 0.0152 x (1409.513—1248.945) =2.441 inches 0.0152 x (1474.566— 1248.945) =3.429 inches, ete. Finally, lay down in the same way the parallel 30° 25’, which will be the northern inner neat line of the chart. A degree of longitude will measure on this chart 0.0152 x 60=0.912 inch. Lay, off, therefore, on the lowest parallel of latitude, on the middle one, and on the highest parallel, measuring from the central meridian toward either side, the dis- tances 0.912 inch, 1.824 inches, 2.736 inches, 3.648 inches, etc., in order to determine. the points where meridians answering to full degrees cross the parallels drawn on the chart. Through the points thus found draw the straight lines representing the. meridians. a. If it occurs that a Mercator projection is to be constructed on a piece of paper where the size is controlled by the limits of longitude, the case may be similarly treated. : CONSTRUCTION OF A TRANSVERSE MERCATOR PROJECTION FOR THE SPHERE WITH THE CYLINDER TANGENT ALONG A MERIDIAN. The Anti-Gudermannian table given on pages 309 to 318 in ‘‘Smithsonian Mathematical Tables—Hyperbolic Functions” is really a table of meridional dis- tances for the sphere. By use of this table an ordinary Mercator projection can be constructed for the sphere. Upon this graticule the transverse Mercator can be plotted by use of the table, ‘‘Transformation from geographical to azimuthal coordi- nates—Center on the Equator” given in U. S. Coast and Geodetic Survey. Special Publication No. 67, ‘Latitude Developments Connected with Geodesy and Cartography, with Tables, Including a Table for Lambert Equal-Area Meridional Projection.”’ oe Figure 61 shows such a transverse Mercator projection for a hemisphere; the pole is the origin and the horizontal meridian is the central meridian. The dotted. lines are the lines of the original Mercator projection. Since. the projection is turned 90° in azimuth, the original meridians are horizontal lines and the parallels are vertical lines, the vertical meridian of the transverse projection being the Equator of the original projection. The numbers of the meridians in the transverse projec-. tion are the complements of the numbers of the parallels in the original projection. The same thing is true in regard to the parallels in the transverse projection.and the meridians in the original projection. That is, where the number 20 is given for the transverse projection, we must read 70 in the original projection. The table in Special Publication No. 67 consists of two parts, the first part giving the values of the azimuths reckoned from the north and the second part giving the great-circle central distances. From this table we get for the intersection of latitude 10° with longitude 10°, ° 7 “ azimuth =44 33 41.2 radial distance=14 06 21.6 THE MERCATOR PROJECTION, , 115 To the nearest. minute these become vies as a=44° f=i4 06 The azimuth hammer longitude in the- original projection sat is laid off upward from the origin, or the point marked ‘‘pole” in the figure. The radial distance is the complement of the latitude on the original projection; hence the chosen. inter- section lies in longitude 44° 34’ and latitude 75° 54’ on the original projection. #0. 2030 40___§0 60 70 80 90 80 70 60 50 40 3020, ' Equator 1 ‘ ' ! Ne 1 ne ! ! ! ES 1 ! ; ' ' eae ee r\---------t4---- - =m 4j\-— 1\—-\-- - min — FE --H ~~ yp fete oe fp 4 1 ry yy ae ae ee | i be-\~-~---- eee ee eer aN. eel =fr—Pp—f-F= J ef cacuenPlcaweccnfad ! ae I , 1 ' —t 3|O 1 t i | : Se site bo K-40 AL ot bap fel Pat fo Jy fo ney fon ' a + i ‘ : Zl0 r ~~ . hs ' ' t t 1 ! ' JN - t o ase Daa ee Jf-- —- JR A-- KE ff -- - Hee i t 1 J 5 I ! ! tees : ! ! Ii \ VTS : | ets Lf OR Se RNS pe 1 VATeoT Ss | TANT i ' 1 ' \ | ! I Li I 1 2 t ee rn art ere ae = See qT ee 4 710} Y S71 =a t Pee ey See pe ee 1 { ! i I p j I ‘ th Sth TET ENON et aaa ee Monee Nee ee ees SSeS si | | 7 + alo ! T Fe Pole I wee ee eee ah----=4 ----4 Sm o. ok a) Pawnee bh He ee ane oe een oo een, t t 1 4 1 : ] , I J t ' 4 71 Ne ee ps at Me te ate 3. ' ry eon se ep ra NS ee tw S79 We ee 1 at ia ! 1 t y 1 1 ' A‘ 1 ! t I ' ! : pM anne an NOK ate AN OK SL PK Pd i 1 1 ! q ! ! slo \! ' | 1 1 ’ poor okn wan tene ~~ ee hye af--4-45 -k-- 4t--- -——-— SF aaieateeindents feteteten Catan dateatententen | ' ' y 1 1 1 } t ' b---74- PHP t-- Pst iF ~— --I-f- -—4{—- {==> 210 sap} -~ fee ht - - + ----4- r oo t { 1 i i : I I | ' t ' i | No ! > : - tl om wee = ipo e? Pe See foe ae ee ~ ia 10 — 20 30 ~#0~«0 «60-70 80 90 80 70 60 50 40 So 20 Equator 70 Fie. 61 lll ianevenee Merenter projection—cylindar tangent along a meridian—construction plate. It can be seen from the figure that there are three other points symmetrically situated with respect to this point, one in each of the other three quadrants. If the inter- sections in one quadrant are actually plotted, the other quadrants may be copied from this construction. Another hemisphere added either above or below will complete the sphere, with the GRP MAN of course, of the part that passes off to infinity. In practice the original projection need not be drawn, or, if it is drawn, the lines should be light pencil lines used for guidance only. If longitude 44° 34’ is laid off upward aiong a vertical line from an origin, and the meridional distance for 75° 54’ is laid off to the right, the intersection of the meridian of 10° with the parallel of 10° is located upon the map. In a similar manner, by the use of the table in Special Publication No. 67, the other intersections of the parallel of 10° can be located; then a smooth curve drawn through these points so determined will be the parallel of 105 Also the other intersections of the meridian of 10° can be located, and a smooth curve drawn through these points will represent the meridian of 10°. 116 U. S. COAST AND GEODETIC SURVEY. The table in Special Publication Nc. 67 gives the intersections for 5° intervals. in both latitude and longitude for one-fourth of a hemisphere. This is sufficient for the construction of one quadrant of the hemisphere on the map. As stated above, the remaining quadrants can either be copied from this construction, or the values may be plotted from the consideration of symmetry. In any case figure 61 will serve as a guide in the process of construction. In the various problems of conformal and equal-area mapping, any solution that will satisfy the shapes or extents of the areas involved in the former system has generally a counterpart or natural complement in the latter system. Thus, where we map a given locality on the Lambert conformal conic projection for purposes of conformality, we may on the other hand employ the Albers projection for equal- area representation of the same region; likewise, in mapping a hemisphere, the stereographic meridional projection may be contrasted with the Lambert meridional projection, the stereographic horizon projection with the Lambert zenithal; and so, with a fair degree of accuracy, the process above described will give us conformal . representation of the sphere suited to a zone of predominating meridional dimensions as a counterpart of the Bonne system of equal-area mapping of the same zone. Such a zone would, of course, for purposes of conformality, be more accurately mapped by the more rigid transverse method on the spheroid which has also been described and which may be adapted to any transverse relation. MERCATOR PROJECTION TABLE. [Reprinted from Traité d’Hydrographie, A. Germain, ar i de la Marine, Paris, MDCCCLXXXII, to latitude ° only. NOTE. It is observed in this table that the meridional differences are irregular and that second differences frequently vary from plus to minus. ‘The tables might well have been computed to one more place in decimals to insure the smooth construction of a projection. In the use of any meridional distance below latitude 50° 00’ the following process will eliminate irregularities in the construction of large scale maps and is within scaling accuracy: To any meridional distance add the one above and the one below and take the mean, thus: Meridional Latitude. | aistances, 28 35 1779. 745 28 36 1780. 877 28 37 1782. O11 5342. 633 The mean to be used for latitude 28° 36’ is 1780.8777 Meridi {Meridional distances for the spheroid. Compression m2" | THE MERCATOR PROJECTION, MERCATOR PROJECTION TABLE. 117 ates. | - = a° utes. eridional 3° distance, | Difference. Meridional | p Merid. - : distanae! | Difference.|| Meridional | pigerence.|| Motidional | piterence ates, 0 | 0.000 1 0. 998 59. 596 ‘ , , 1 | 9.993 | 93 | 60-990 | Sos 10.2 | ope | s2e- eee "986 993. || 120.204 | ™ 0.904 | 9 3 | 2.980 ppd | GF. B88 993. || 121-198 904 | tho get 995 | 2 3.973 993 . 122. 192 994 : 2 993. || 63-570 994 994 || 181. 845 04 5 | 4.966 993 || 128. 186 18 g95 | 3 "a 994 /82. 840 6 | 5.959 ae op eed 993 || 124.180 183. 834 eS 7 | 6.952 998 | 66. 550 gg4 || 125.174 oo4 || 184. 829 005 | 8 9 | 8.939 993 || 67-543 | So 127. 169 gos || 385-824 | Soa ; ; 993 68. 537 ; 993 || 186.818 94 10 9. 932 993 || 128. 155 ; 1 995 8 9. 994 87. 813 ut Xs 925 | 0- 998 an a 0.993 || 129-149 | 9 188. 808 pe : | wae ey ae 130.143 | 29% | igosoz | 99 | at Bos | om) ieee | Se | ell | ge) itt 388s . 993 73. B04 . 994 || 191. 792 995 ae |. sacnos 993 || 133.125 192. 78 g95 | 13 16 993 74, 497 994 ee | 14 17 | 16884 g93 || 75.491 94 | seria | 9% | tee ar7 25. | 2 47 | 16. 993 | 76. 484 993 eo 994. || 194.777 995 | 46 18 | 17.878 77,477 poe || Tee aee 195. 772 995 | 4 18. 871 gee | yBarh | oes 138, 088 goa || 196. 7e7 | $85 rH 20 | 19.864 993 "095 197, 995 | 38 : 994 7. 762 $ a 20. 857 0. Bes ee pie 0. 994 189. 089 198. 757 05 | a | 21.851 81.451 993 || 140.083 | ° 904 | jog ree | 0-995 | by | ee | a8) a | ge) arte |e | ae | ze ; 83. 438 3 : 9 201. 742 995 25 | 24.831 994 go, || 143. 066 94 | 309: 73 995 | 28 6 25. 993 84, 432 994 737 24 a | 36.817 998 | 5425 | 993 | iasose | 994 |) doe a | ys | 3 28 o7, 993 86. 419 994 - 054 9 204. 727 995 : 2 | 27.810 86.419 | go4 || 146. 048 @ || Sos voz | 995 | Be 28. 804 994 Sopra 993. || 147.042 994 | O06, 995 | 2 30 | 29.797 an8 oe tase | Il 20. eee ee 31 | 30.790 | 9-993 89. 400 149.0 a 995 | 79 lore ont 90.393 | 0-993 030 | % go4 || 208. 707 33 | 32.77 pee | less? | peat Tei. | S06: guy ay? 0.995 | 34 | 33.770 993 92. 380 ae 152. 013 gee tees 998 32 i ee 993 || 98.374 994 || 253.007 ee ae 995 | 33 : spre 994. || 222. 687 34 36 | 35.757 994 154. 001 995 37 36. 750 993 95. 361 993 154. 995 213. 682 35 el oe $98 96. 355 994 . 996 214. 677 995 3 | 37.743 9 oe a i 155.000 | 2a, ft aise | Se - =| sam 993 | Sasi | 99 | i5r 978 || aie.ees. | eee 33 0 } 9 : . 78 : ? a9 730 Kens 99. 336 . 94 995 217. 663 ee 39 o . 983 | 100.329 | % 993 158, 973 218. 658 41.716 ‘i eae || teas ae7: | “Gaz ' 0.996 | # 44 | 43, 993 || 102.316 G53: ger eee 220. 649 995 | 49 43. 703 993 | io3-310 | 9% 161. 956 af || aeig4, | | 288 z | ae Ba 994 || 105. 298 gm Ter ay g95 || 228. 685 45 Ge ae 3og || 106. 291 gy aan. 994 || 224- 681 996 | 4 107, 28 994 oe 995 | 45 49 | 48. 669 gag || 108 279 994 || 266.928 oe 226. 622 996 | 43 : 167. 922 9 : 48 oe | eeeee 1 Ohans || eee a || 6s. 917 995 | ey 396 | #9 be Pra 993 110. 266 0. 993 169. 0. 994 228. 613 : aio 994 911 . 0.995 | 5 53 52 643 994 60 170 994 229. 608 5 ' ta 38 994 . 905 9 go5 | 5t 54 | 53. 636 g08 | saa 5 ve 993. || 171. 900 995 ro oe gog | 5% ie 993 ar | PS | ae. gon | a See | gos |) 8 86 | 55.623 g94 || 114.241 173. 8 995 | een g96 | Of : 993 115, 235 994 . 889 994 233. 590 56. 616 994 || 174. 883 9 99 55 58 | 57.609 993 | 116229 | eq | 175.878 | og 235, 581 ove | 58 59,596 | a ae Boe ih ae bre ee : 862 | 238.568 | 9-996 a 118 U. S. COAST AND GEODETIC SURVEY. MERCATOR PROJECTION TABLE—Continued. [Meridional distances for the spheroid. Compression ral a 4° 5° 6° 7° < — Min- utes, Meridional Diff é Meridional Diff Meridional Diff Meridional ‘Din utes. distance. erence. || “distance. | ~terence-|/ “Gistance. terence. || “distance. derence: , , , ¥ * , , , 0 | 238.568 298.348 | - 358, 222 418. 206 0 0.996 0.997 || 3 0.998 ||. 1.001 ae ee ee 3 | 241.555 996 | 301.340 998 || 361.918 999 || 491. 209 ool | 3 996 a ee ees o00 | 3 : 242. 551 eee || 302.837 a, ae eee be ae 243. 547 303.334 | 363. 216 é 423. 210 5 . 996 997 | 999 001 7 | zus.sse | 995 | sooaze | 98% | Sonas | $8 | eau | gr) 8 | 246.534 | 385 |) 306.326 | 398 | gee o12 .| 4 289 | 426.213 | or | 8 9 | 247.530 aoe | 307.923 foe | OS | ee | em 8 40 |. 248,526 ; 308. 320 368.211 | 9 428.216 10 4 0.996 || 308-320 | 9 998 |} 0.999 1.001 | . H | Bose | ae) Sotaee | ose | gobo | me | aaa | Me | a 13 | 251.514 996 | 317.312 | 297 || 371.208 | 2999 |} 4gz.219 | 90 | 43 14 | 252.510 | $88 | gio'sio |: 828 | srei207 | , 999 || asz.220 | OOl | 44 ile g96 |) 312. 997 || 372. 0.999 002 253. 313. 307 - 373.206 433. 222 2 15 | 996 998 |) 3 1.000 | 4 001 Paes ae ee ae ee ee: “38 | 256.404 | $88 | 316.300 | $98 || g7e.204 ) 9:299 || 4s6.206 | “UO | as ; 256. 996 || 326. gos |) 376. i009 |aae 226 oo | 38 tan ; a 0 996 (317. 298 997 377. 204 0'999 437.2 002 258. 4 318.295 | \,- 378.203 | 438, 229 s: 20 id 8. 0.996 0.998 |} 378 1.000 |) © 1.001 a2 | deo.ars | 985 | So0cas1 | 888 | San' ano | 2-999 | Sioa52 | 902 | Bp 23 | 261.474 poe || S21 288 [> Gey || genoa | 2080 || aa1 gaa cn 24 | 262.470 gee |} 322. 286 jog || 362-201. |. Obes I aga.o35 por | 24 & ahs i 27 | 265.459 cog || 325.279 gon || 385.200 | 3-000 | 445. 241 oo ot 28 | 266.455 poe || 326.277 338 || 386.200 | 2-900 | 446. 240 et 88 29 | 267.451.| 398 || 327.275 oe8 |) 337.199 | 0-800 | 447. 244 | a ai | 2oa.4se | 0996 | S5:370 | 9-997 | Sep aos | 1-000 | Zas'oes | 1-002 | St | eede | gm | Boas | se | geae | a | doe | | ae 1264 | 998 || 399° 198 000 |) 459. 954 ae at 272. 433 wee | a2 ee peal He Be | 2rdazs | 998 | 35¢a60 | 998 | goxtos | 090 | Z5td58 | 902 | Se 33 | geaig | 96 | score | 998 | Socaes | 000 | Zexeee | one | & 997 998. || 326. 000 oo2 | 38 39 | 277.416 oat || 937,254 388 || “397.198 000 || 457. 264 ee | a 4 ; 253.) 2, ai | 270409 | 9-997 | Saeoet | 0-998 |) Soe a9s | 1.000 | 48-587 | a.002 | 48 42 | 280.406 oan || 340. 249 oes || 400. 198 Opp || 460.272 tee 43 | 281.402 gee |] 341. 247 998 || 401. 198 000 || 461. 274 eee sas 44 | 282.399 oes ||. 342.245 | 998 | 402’ 198 poo || 462.27 | ae 4g | zesaoz | 999 | ga dar | 998 | Gorton | 92 || dexdm | 008 | fg 47 | 285. 389 345, 240 405, 199 465, 284 47 48 | 286.386 oe! |} 346. 289 3ag || 406. 199 boy || 466. 287 toe | (48 49 | 287.383 oo? || 347.237 geq || 407.200 | 2 | 467, 289 ane se | asso | oon | 220 | oom | 220 | on | 4022 | ag | 52 | 290.373 oo 350. 233 jos || 410. 201 boy || 470.297 |e 53 | 291.370 og” |] 351.231 age || ALL, 202 ool || 471. 300 | 54 | 292.367 oo5 || 352. 230 aioe 000 || 472. 303 Oo8 | 5 se | 204300 | 987 | B54-aor | 989 | die ang | 99 | Grado | 008 | Se a ee eee 997 -224 | 999 001 003 eo | goraet |g aay || 29%229 | goss || 417-208 | 4 00! array | , OB | os 60 | 298.348 | ° 358.222 | 2: 418.206 | 1: 478.321 | 1 60 , ne HE MERCATOR PROJECTION JE I ontinued. _[Meridional di al distances for the spheroid, ) id. Compression 07 | 119 Mi a ae Meridi - 8 eri ona. cae Difference. Meridional m1 rT distance, | Difference.|| " cece Dift 4 4 £ ; ‘ance, | Difference.|| — ‘eridional | p; utes 1 ne 354 1.003 || 538-585 eal. : 480. 327 008: | eae | 1-006 | 0. 0 | . » 482. 333 003. || B42. 603 0p: || eRe O09 | Go co “2 | 5 | 483.337 oot || 542.609” 006. || - 802. 046 009 | oa. 6 % 6 484, 340 003 543. 615: | sae oe 009 ss 90 | 2 3 6 | 481340 093 Baanol £006 - 604. 063 - 9 . 690 012 4 $) ace | pas.ont | “Be |. 605. 072 009 || 864. 702 . 9 | 487.350 | 003 546. 633 006 || .806. 081 009 |) Gos. 737 01 10 |. 488, 354° 004 || 547-689 006 |): 607. 091 010 | Gor. 740 a 11 |. 489. | 007 608. 100 wong: |) Beereee |: OL 489.357. | 2-003 |) 548. 646 |" 1 om | 875 i a . 490. 361 004 || 549. 652 - 1.006 ||: 609- 109 09. . 013 9 : io. 004 || Bocese | oos | 620.118 | 1.009 || 669. 765 me Des | 3 491. 805 a 551.664 | 00g || 611. 128 010 670. 778 1.013 |, 20 ‘15 | 493.372 003 || 552.671 | 007 63 17 | oop sr ss | a i 16 |, 494.376 004 553. 677 xis ae oi a a | z a 47 | 495, 380 one | Be 600 | 5 | GiE ee a 2 | 3 : i oo B64. 684 Ree 615. 166 o1o ||. 874. 829 15 | 490.3 004 a eer 007 616. 175 009 675. 842 — 013 15 20 | 498.39 av |) Bona | 88 sits | 81° | grr sas | 21 392 | 007 |} 818.195 010 | 78.881 au | eece | ede eee 710 | 10 O08 | 04 8 | 8 2 209,39 04 559, 717 1.007 ||: 619. 204° . ' 913 19 2 i a ane | 38 mr a 620. 214 1.010 679. 894 - 95 | 503.4 004 || 562.737 ooe | 822. 234 cle | ao ee | oe | Boe. 12 007 || 623. 244 010 || “Gee gar a | 504. 416 004 563. 744 v ee oi z i | Soe he oo | See Tee oo7 ||. 624-254 | 9 . 947 o14 | 7 zn S042 004 oeP ee || 007. ||. 625. 264 010 ||: 684. 961 : B | ide cos | 578 a 626, 275 O11 || 885. 974 013 25 31 | 509. oe | 3.004 || 568. 780 a. a | a : : a | so | a | oe | cas. 205 010 015 a 29 z se ra 570. 795 is 631. 326 010 691. 043 1.014 30 35 | 513 004 || 572.809 007 oF eG ois | eae. on ui | oe oo || 888.347 010 | God, 085 au | ~ 514, 460 005 || Bre 354 mn | a a7 | 515.465 005) Bre Boo Org | 35 30 , ve | se 8 he Bra. 824 oe 635, 369 011 || 695. 099 s ses 0s 516, $3 poe 636. 379 010 696. 113 014 35 : sa i 576, 839 er 637. 390 011 697. 128 015 36 f na oF sie 008 638. 401 011 be 142 a 5 a | 548 005 Ge | 4.007 639. 412 011 . 156 os 39 4 sa ae 579, 802 Wi 640, 423 1.011 700. 171 i sua ms 580. 870 Hs 641. 434 oll 701. 185 1.014 40 45 | 523.50 005 || 582.886 - 00g || 842-445 OL | Fos. 215 ae | 46 . 504 5 008 G23.458 bil 704, 329 0 i 524, 509 005 || 588. 894 — ae a | 47 | 525.514 005 | Fes" 910 weg | a , |e ‘ san ie 584. 002 nog 645. 478 011 705. 244 f ae Ws 585, 910 08 646, 489 O11 706. 259 015 45 Bo | 528.5 005 || 587.926 00g -|| 847.500 OL | 703,280 a | Bl . 580 oog || 648.512 012 | fos, 304 a | 599,535 | 2-005 588. 934 a noms a8 | 8 52 | 530.540 005 | 259 951 Bn | it 58 : fe | seg | 008 | 50. 535 Log ||. 710.319 ae co 590. 951, one 651. 546 011 711.334 | 2-015 50 55 | (533. 006 || 992-968" ugg || eer oz | ne 30 a i 56 _ B57 5 gos | G88 620 m2 | nia.ar9 a | 534. 563 og || 293.976 Ot | ras 95 8 | 87 | 535.568 005 | 395" 993 tog | 55.38 2 20 | 8 | Bon 000 | Sor 002 009 || 858. 605. o12 || 716.410 015 | ee 3 om 288 597. 002 oo 657. 617 012 717. 425 015 6 solo | $8 657.617 O13 | 718.441 ois | 33 658.629 | 4 O19 719. 457 016 | 35 720.472 | 3-015 a 120 U. S. COAST AND GEODETIC SURVEY. MERCATOR PROJECTION TABLE—Continued. [Meridional distances for the spheroid. Compression sor] oe 12° 13° 14° 15° in- Min- utes, ve Ace Difference. pleco Difference. et dienel Difference. Meri bionet Difference. utes. 0 | 720.472 781. 532 842. 842 904, 422 0 1 | oiags | PETE | vende | PEN | ae ce | OY conan | eee | a 2 | 722.504 Ore | 783. 572 020 || 844. 890 pee || 906. 480 | 8 3 | 723.520 Ore | 784. 592 020 || 845. 915 bog || 907. 509 | 8 : a a ole || 785. 612 020 || 846. 939 O24 || 908. 538 ae 786. 632 847. 963 909. 567 5 6 | 726.567 | ip || 787.652 | 02 | geg'oss | 025 I gio.sa6 | 028 | 6 7 | 727.584 ole || 788. 672 020 | 850. 012 tos || 911. 626 eet 8 8 | 728.600 Ole |) 789. 692 020 || 851. 037 025 || 912. 655 | 8 9 | 729.616 ore | 790. 712 Oat || 852. 061 O2f || 913. 684 ree oy 10 | 730.632 791. 733 853. 086 914. 714 10 u | 731.649 | 1017 | 799,753 | 1 mn 954.211 | 1025 | ois zag | 1028 | a4 12 | 732. 665 OL || 793.773 p20 || 855. 136 025 || 916. 73 | of 13 | 733. 682 ore || 794. 794 O21 || 856. 161 02> || 917. 803 eae 14 | 734.698 oie |) 795. 814 020 || 857. 186 025 |} 918. 832 oe) la 15 | 735. 715 796. 835 858. 211 919. 862 15 16 | 736.732 O17 || 797. 856 021 | 359, 236 025 | 920. 892 030 | 46 17. | 737.749 Ore | 798.877 Ozh || 860. 262 O28 || 921. 922 Te 18 | 738.765 Ole || 799. 898 O21 || 861. 287 025 || 922: 958 ee | te . ze 782 Oly || 800.919 021 | 862. 312 025 || 923. 983 ae ae | tore | or | soya | on | eae | som | Bee | em | 2 22 | 742. 833 Oly || 803. 982 021 || 865. 389 O26 || 927. 074 | as 23 | 743. 850 O17 || 805. 003 O21 || 866. 415 028 || 928. 105 | os 24 | 744. 868 Ore || 806. 025 022 || 867. 440 O25 | 929. 135 ae a 25 | 745. 885 807. 046 868. 466 930. 166 25 26 | 746.902 Ory || 808. 068 a 869. 492 Ge A BL, 187 ot | ae a7 | 747.919 Org || 809. 089 O2l || 870.518 026 || 932. 228 ee | ee 28 | 748.937 oe || 810.111 022 || g71. 544 026 || 933, 259 er | oS 811. 133 872. 571 934. 290 29 018 022 026 031 30 | 750.972 | 4 gig || 812-185 | 4 goo | 873-597 | 4 one | 935822 | a9, | 88 Si | 7otg00 | Or) | sigavy | 1022 | srtees | 1023 | ose ane | SOR | a 32 | 753.007 big || 814.199 022 || 875. 649 p26 || 937. 384 or | 33 | 754. 025 Ore | 835.221 022 || 876. 676 027 || 938. 415 ae 34 | 755.043 oie. || 816. 243 022 || 877. 702 026 || 939. 447 032 «| A 35 | 756.061 817. 265 878. 729 940. 478 - 35 018 022 027 032 36 | 757.079 818. 287 879. 756 941. 510 36 37 | 758.097 Ore || 819. 309 bo || 380. 782 Ooo || 942. 542 feet ag 38 | 759.115 O18 || 820. 332 023 |] 881. 809 027 | 943. 573 Ss : a i i ae S 022 || 882, 836 027 || 944: 605 | M8 883. 863 945. 637 40 41 | 762.170 | 1018 | 823.399 | 1 fae | 8a 80L | bor || 946.669 | 2 one 42 | 763.189 oie | 824. 422 bee || 885. 918 boy | (947. 702 tee | ae 43 | 764.207 O18 | 825. 444 022 | 886. 946 poe | 948. 734 | 4a 44 | 765.226 ore | 826. 467 O23 || 887. 973 paz || 949. 766 ee | at 45 | 766.244 827. 490 889. 001 950. 799 45 46 | 767.263 ie 828, 513 oe 890. 028 nea 951. 832 ie 46 47 | 768.282 O19 || 829. 536 023 || 801. 056 Gos || 952. 864 | at 48 | 769.301 Org. || 880. 559 O23 || 892. 084 Soe || 953. 896 O32 | 48 s rm _ ae a O23 || 893. 112 O28 || 954. 929 tee 894. 140 955. 962 50 bi | 772.358 | O19 | 33.629 | 1 oes (|| 898.168 | > oe 956.995 | +033 | 54 52 | 773.377 Ory || 834 652 023 || 896. 196 028 || 958. 028 ea | ge 53 | 774.396 O19 || 885. 676 O28 | 897. 224 028 || 959. 061 | ie) ee, |) 434 961. 128 55 56 | 777. 454 oa 838. 747 fee 900. 308 oe 962. 161 ee 56 By | 778.473 ory || 888. 774 bog || 901. 337 028 |} 963. 195 oe | et 58 | 779.493 p20 || 840. 794 023 || 902: 365 038 || 964. 228 033 | 68 59 | 780.513 | 4 Ory || S41-si8 | 1 oot || 903.904 | , 028 | 965.262 | , O84 | a9 6o | 781.532 | 2 g42.842 | 1 904.492 | 2 966.296 | 2 60 THE MERCATOR PROJECTION. MERCA’ TOR PROJECTION TABLE—Continued C ‘Meridional distances for th Spher ‘oid. Compression re 5 ie 1d, | 594 121 iia 16° i utes. Meridiona! 3 distanea Difference. || Meridional i 19° : - distance, | Difference. Mortdicsia! ae a i 0 ne ; - stance. ifference. eridional | p in- 1 cee as #088 1098. 483 ? - ; distance. ifference. utes. 2 | 968.364 034 99.522 |, 1-039 1091. 007 : r 3 |: 969.398 034 30.561 039 92.052 | 2-045 1153. 893 ‘ 4 | 970.432 034 31. 600 039 93. 098 046 54.943 | 2-052 i 5 Oe ae | Bae tere | | Oe | ee | tor 971. 466 ozo || 95-188 045 || 57-046 052 | 2 6 | 972.500 034 33. 680 , 046 58. 097 051 3 7 | 973.534 034 34.719 039 96. 234 052 4 8 | 974.568 034 35. 759 040 97.279 045 59. 149 5 9 | 975.603 035 36.799 040 98. 325 046 60. 201 052 é 976. 638 0 1100. 4 046 62. 305 o52 | 7 He eee | ates eee say aie | Oi esaer yp ae 8 12 | 978.707 034 39.920 | 1-041 1101.462 | , ae 052 9 13 | 979.742 035 40. 960 040 02.508 | 1-046 1164. 411 i 44 | 980.777 035 42.000 040 03. 554 046 65.46) | 2-052 10 “af 035 43.041 041 04. 601 047 66.514 | 93 1 981. 812 041 05. 647 046 67. 566 o52 | & 16 | 982.847 035 44. 082 ; 046 68. 619 053 | 18 17 | 983. 882 035 45.122 040 06. 693 053 | 14 18 | 984.918 036 46, 163 041 07.740 047 69. 672 1 19 | 985.953 035 47. 204 041 08.787 047 70.724 o52 | 38 20 035 48. 245 041 09. 833 046 71.777 053 6 a1 988. ah 1.036 || 1049. 286 on 10: S20 ae 3 ee Oba 18 22 | 989.060 036 50.327 | 1-041 1111. 927 i ee 053 19 24 | 991.131 036 52. 409 041 14. 021 047 75.990 | 1-053 | 25 Oi agen | ee | ose | 088 | fags 054 | 3 992. 167 042 16. 116 047 78. 097 053 | 22 26 | 993. 203 036 54. 498 : 047 79.151 054 | 23 27 994. 239 036 55. 534 041 17. 163 24 054 28 | 995.276 037 56.576 042 18. 211 048 80. 205 | eee | ee. bee te ieso |] a 81.250 | 54 | d¢ 30 | 99 036 58. 660 042 20. 307 048 82. 313 054 | 36 rf ou 348 1.037 || 1059-702 042 21. 354 ee a a os a 32 | 999. 421 a eh ers 1122. 402 - eer 055 | 79 eae | || ee Ce || Ba. te 1185. 478 a | onde | 07 | os:870 oa || 24499 | Oge 86.030 | 1054 | gy oe | 037 63. 870 042 25.547 048 87. 585 055 | 3 Be . 582 043 26.595 048 88. 640 055 | 2 03.569 037 64. 913 049 89. 695 055 | 38 37 04. 606 037 65. 956 043 27. 644 055 of 38 05. 643 037 66. 998 042 28, 693 049 90. 750 39 06. 680 037 68. 041 043 29.741 048 91. 805 055 35 sa: N60 038 69.084 043 30. 790 049 92. 860 055 | 36 . 07. 718 oar | 2070-227 043 31. 839 49 of 915 O06 2 42 | 09.793 038 ea ee ad ae 888 o pr 055 39 is | Sotso | 82 | enone Wed SBopay | Rey 1196. 028 44 11.878 038 73. 257 044 34. 987 050 Fe ea | 4eObE | aa 45 038 74. 300 043 36. 036 049 98. 137 055 | 3 6 12. 906 043 37.086 050 ‘(|| 2299. 193 056 42 13. 943 037 75. 343 049 ‘|| 1200. 249 ose | 43 47 14, 981 038 76. 387 044 38.135 ong | 44 48 16.019 038 77,431 044 ' 39.185 050 01.305 | aose:| (C2) toate tae} eae ||P 02.361 | 956 | 4g 50 | 1018 oe || 7ee18 |. be oa | on | 7 | oe “7 51 ee 1. 038 1080. 562 044 42. 335 pk ig eR 088 48 52 20.172 038 81.607 | 1-045 1148. 385 sea a 057 | 49 54 ee 08h 83. 695 044 45. 485 050 07.643 | 1-056 50 55 039 84.739 - 044 46. 536 051 08.700 057 51 . 23. 288 045 || 47-586 050 || 09-757 057 | 5 66 | 24,327 039 85. 784 051 10. 814 057 | 53 57 25.366 039 86. 828 044 48. 637 : 057 | 54 58 | 26.405 039 87. 873 045 49. 688 051 ee 5 59 | 27.444 039 || 88-918 045 || 50-738 050 || 12-928 058 | 56 60 | 1028.483 | 2-039 89. 963 045 || 51-789 os1 | 22-986 057 | 3 i 1091007 | 2-044 || 52. 840 051 15. 044 058 7 , 1153. 891 1. 051 16.101 057 58 1217. 159 1. 058 . (122 U. S. COAST AND GEODETIC SURVEY. MERCATOR PROJECTION TABLE—Continuéd. [Meridional distances for the spheroid. Compression a’ | F 20° 21° 22° 23° : fits = in- utes. pie Difference. eripioal Difference. Meriacms Difference. eee Difference. utes. Be | aes | ee | cane ee aoa eS | aodeo | 4 2 | 19.275 os 82. 965 065 | 47. 089 ve 11. 673 080 | 9 19. 058 065 - 089. 073 || 4 081 3 | 20.333 oee || 84. 030 bee || 48.162 He Typ, | fee |e 4 | 21.392 eee 85. 095 pee || 49. 285 ae [sada fa | 4 5 | 22.450 86. 161 50. 307 14, 915 5 059 065 073 | ost (oe | S/o) By) ae | a] Bey ays 8 | - 25.626 059 | 89. 357 065 |. 53.596 |. 078 18. 158 081 | 059 066 f° 3526 | o7q |] 18-158.) ogy 9 | 26.685 ae 90. 423 cee || -eeseoo" | Oe I Toreegh) ee |g 10 | 1227.744 |, ; 1291. 489 1355.673 | yey |] 1420.321 | ' 7 9. 10 - 1.059 1. 066 1.074 |) doer f Bl ees | Se eae | Se |B | ie Be | ae | 13 | 30.921 oe 94. 688 067 || (58, 894 O74 || 93 566]. 082 | ‘43 . 80. 059 o66 || 58 074 081 14 | 31.980 ae 95. 754 re 59. 968 Aa | Baar ORS be a 15 | 33.040 96. 821 61.042 | | 25.729 |‘, 15 059 066 “074 |. 083 #) ee) BS) ge | Se) ee | ee) ee -159 | 059 067 || 8 074 || 27-894 | ogo | 18 | 36.218 ban || 1300. 021 bey || 64. 265 i 28. 976 ae alk ” es cs oe LORE Oey || 65.340 we 30.059 | O83 | 49 1302. 155 1366. 415 1431.142 | 1 noo | 20 1.061 1. 068 | 1.074 | 1.083 BRS |S [ees |e eae | ea | ae | 2 . 060 068 076 -308 || 083 23 | 41.519 ae 05. 358 hea 69.640 | 078 34. 391 ee ae 24 | 42.580 Deo || (06.425 ee eee. | ee 35.474 | O83 °| 24 25 | 43,640 07. 493 71. 790 36.557: | 25 061 068 | o76 084 ay | agree | 08 | omesg | 68 | Teece | 076 |) 32-88 | ogg | 36 061 o6s ||: 73 075 725 | gga | | 28 | 46. 823 he 10. 697 ce 75.017 ove || 39. 809 et as op inte ee cian No Nae cee Nee et ee a 1377. 169 1441. 977 30 1.061 1. 068 1.076 1. 084 gem | ae [eae | as |e | ogg |e | oa | tae 061 069 || 2% o7e || 44 084 52. 129 pe 16. 040 O6° || 80. 398 ue 45. 230 | Bs 34 | 53.191 ae 17. 109 ea 81. 475 tre | 46. 815 nee | es 35 | 54,252 18.178 82. 551 47. 400 35 062 069 077 085, He) BB) B) Ee) e) ee] sls oe | ee 062 070 or7 || 4% 085 38 | 57.488 21. 386 a 85. 782 ve 50. 655 | 085 | a8 : Ss a oe 2 if oy || 86.860 oe BL. 741 coe | 39 41 | 60.626 | 2-063 ee | L nn Mee |e Hes aa |e ees 1 e/ ge | S| Be) ie) ee) me) eel es 8 : 063 070 : 078 56. 084 ose | 43 44 | 63.814 bes || 27.805 O70 || 92. 248 a 57.170 Oe | ae ac | 65.940 | 8 | aoous | 07 || Sedoe | 978 | Exgas | O87 | 47 | 67.003 31.016 95,482 | 60. 429 AY 48 | 68.067 hee 32. 086 rie 96. 561 61. 516 ts, | 48 ee ve Fs i i 97. 639 oe 62. 603 ba, | 49 oh | iby | Be 35, a9 | oe 1390. 797 ee oe | oe it ee ee ee 064 071 . 079 66. 951 os7 | 53 54 | 74. 449 Oeg | 38.513 a 03. 034 co 68. 038 oer | a 55 se | rear | 8% | Seer | 072 | Geddes | ovo | $2228 | ogg | 58 57 | 77. 642 065 41, 728 071 06. 273 080 71. 302 088 | sy 58 | 78. 706 064 42. 800 072 07. 353 080 72. 390 083 | 5 a ae oe ere 072 . 080 088 1. 064 LOG Voge | i080 lage | ices. | 28 60 | 1280.835 | + 1344945 | 1 1409.513 |b 1474.566 | 1 60 THE MERCATOR PROJECTION. TIC Continued. MERCATOR PROJECTION TABLE—Continue 1 an. € i ion sax" [Meridional distances for the spheroid. Compress 562 ] 123 24° ‘250 26° 27° ‘Min- - utes. : > Meridional | niserence. ; Meridional | piserence. ae Difference.|! “distance. etal Mec eanal | Difference. "aeteeae, . pablo. - 7 ; oh 0 1S! a ‘ a ‘ - Fi oy et 1672. 92: 1.117 ; gi 0.134 | = y7 || 1606. 243 1.106 74,040 116: i (0 | 1474.566 | 4 og9 ee = i : Ve ine a 75. 156 117 3 a ee 088 | 42/308 |: OB" ||. ee eee dor: 4] < ee ur | 4 2 76.7 4 089 43. 426 098 10. 670 107 ae 5 3 77. 88 089 | 44504 098 > f 178.507 | aq7 6 4 78.921 089 622 VL.777 | ay 79, 624. 117 7 0.010 80 | 48 7o0 | 098 fo ah fe | eh ees 118 8 | sro | 090 cae P an Te | lO "81. 859 |S “ayy ; ay | | 47, [. 15. 0 “108 | 976 |' “'F78 7 | 82.189 | 99 48.916 |’ g99 207 | * rere en a 8 | 83.278 090 50.015 "|" ggg. | 16. 2¢ cake ee 1684.094 | 4 118 7 9 |, . 84.368 090 3 | 1617. 315° 1,108 |} 85.212 am | at AD aS | cog. | Web O99 19532 08 ere 11 | ~ 86.548 090 “3 37] 099 “aan to 9 |] ~87-449°) 338 ‘44 “BR Nee 87, = } 090 1 54° 410 ‘| 099° IP ; a 74g °° ee 88.567 119 |. 16 13 | 88.72 (O91 | 55.509 100 |)" 7 89. 686 119 | 46 14 | 89.819 090 || 5H «22.858 1 09 | “90. 805. 119 | 49 90.909 | — 991 |}. 56-809 | ggg aie oe a pamee A ear ‘119 | 4g te | 28 000 oat “Begs | 0g 36.185" | 209 || g3° 043 120 |. 38 i‘ > . 1 eral . uv 6.1 - 4 W 93. 091 per || 89.908 | 100 a 295 oe 94.163 119 - 18 | . 94.18 091 1: “61,008 100 | ‘| 1695. 282 | 4 199 a1 19 | 95.273 091 08 "| 4 401. |} 1628-404 |} 4 449 96.402] ““io9 29 96. 364 yz |] 1562.1 1.101. ||” ,29.514 || 110 -97. 522 120 20 | 14 ps | 1-09 63.209 | "499. 30.624 | 349 ee eae io0 | 23 a1 97. ele 02 64.309 | 401 |} - 31.734 | 749 1699. 762 11 | 24 “a | _ Boar | age “ee |, 10L | arg ee a8 25 23 | 1499. 639 091 ||. 66.511 lol |< || 170.883 | 499 36 24 | 1500. 730 092 612 |° 83.955 | 110 02. 003 “121 | oy 1.822 |" ggp |} 67.612 |° yo Be a: AN |. gees iat | ee a eat & a ee HL ge, me || tll) ee Ir | ik a oe aes Obs 37.2! ‘Ly. 95) 366° 121 27 |, 04.007 | Qgo _ 70. 915 102. | 39° 398 i fj 95: So) Oe] ase |: ee 102 | 38 | 0g ag? Lise | 3 29 | ...06.192 - 092 1673.119 |. 4 160 1630. a 1. _ 07. 609 121 | 95 f . on 40. ‘ 0 3 peed it 15. 102 | 49) gaa 12 197 32 09.470 _ 093 76. 425 102 |). 43. 956 ie 10. 974 122 = 33 10. 563 093 77. 527 102 - 12.096 123 | 36 34 11. 656 094 29 45.068 — 113 13. 219 122 37 12.750 gy) 2e8 ios eas) ane 14. 341 18 | Se or 4 80. J) . 48. : 112 16. 586 193 37 |, 14.937 O94 || 81.938 103 518 88. |. 16.031 094 83.041 | 493 7 ee 1717.709 | 4 494 u 39 |. 17.125 094 4A 1650. 631 | 4 443 18. 833 123 | 45 8.219 4 || 1584.1 . 1.104 51.744 || "493 19. 956 124 3 aon ee ee 103 ||. 52.857 114 || 57" 080 is: || ae du. . ‘I 53.971 113 22. 203 124 42 20. 408 094 87.455 104 55. 084° : | 2507 | 0% | goss | 104 | BS bug dL esse) toe it 44 22. 597 095 663 ' 56.198 114 94.451 124 a7 3. 692 oe ioe, eeesie | ag | 25.575 105 |“ae 46 | 24.787 095 1 ari tog. Bye dt ee eee 124 | $8 eee nen, 2 91. 59. 54 114 || 97’ g04 125 re ee a an peers 105 || 60. 654 : 0 48 26. 978 095 94. 081 105 oa 1728. 949 1.125 2 49 28. 073 096 1595.186 | 4 ios 186% es 1. te 30. a 125 | Bo an : : ; 11 31.19 Be 1 2620 0 | aoa: || Ae a 105 | sa.s98 | ie +32 and t22 | 88 52 | sige, | 09 sso | 205 | gas | 28 33.450 | 8 | 5a 52 . s 1 229 | Be 53 | 32.457 096 || 1599. 607 105 sa te 34.575 126 2 54 | 33.553 097 12 67. 344 115 35.701 126 | ga 650 g || 2600.7 106 68. 459 116 36. 827 126 58 se | 5.746 | 096 2924 | 208 || Gp, ga. | iS | gr953 ] 128 | 59 56 vo» 02, 70. 69 116 . 080 BT 36. i 097 04. 030 106 LOR | atte on one 1.126 | ¢% 58 37.940 | — gg7 05.136 | 4 407 1672. 923 59 39.037 | 4 997 1606. 243 . 60 | 1540. 134 124 U. S. COAST AND GEODETIC SURVEY, MERCATOR PROJECTION TABLE—Continued. [Meridional distances for the spheroid. Compression yor | = 28° 29° 30° 31° ee utes. Meeiionel Difference. ag Difference. eee Difference. ve Difference.| U*e8- ? ’ , , , , t , 0 | 1740. 206 1808. 122 1876. 706 1945, 992 0 11) atag Poh ae | opaeo | EE Nene | eee | aaa | Ee | 2 | 42.460 127 || 10.398 jay | 79.004 io || 48.314 aa 8 3) 3. 587 gr |) 1.585 138 |) 80-258 19 || 49.476 |S ; ao a6 1, 303 150 | 50.687 -| fen | 4 5 | 45.841 13. 812 82. 453 51. 799 5 6 | 46969 | 128 ) is'950 | 138 | 3.603 | 250 | s2'961 | G5 | 5 7 | 48.096 t2g || 16.089 igg || 84.753 150 | 54.128 ae iin ct 8 | 49.224 | 1.288 tag || 85.903 150 | 55. 285 aa 9 | 50.352 ize || 18.367 MD | '87.058 150 | 56.448 i | 8 10 | 1751. 481 1819. 506 1888. 204 1957. 611 10 1 | 52.609 | 2228 | “90.645 | 2289 | 'g9,355 | 1252 | sei77a | 12688 | a1 12 | 53.738 aL res 135 || 90.506 151 | 59.937 ae aa | Be) B| ee) gb) oe) Be] ae | ie |e : "064 92. 62. 263 14 129 140 151 164 15 | 57.124 25, 204 93. 960 63. 427 15 16 | 58.254 | 389 | 26345 | Mi | ose | 12 | essen | 18% | 46 17 | 59.383 28 || 27.485 in || 98.264 152 || 65.756 1 | Al 18 | 60.513 13) || 28. 626 ial || 97.416 152 | 66. 920 is | 48 19 | 61.643 130 || 29.767 14t || 98.569 153 | 68.085 165 | 19 20 | 1762.773 1830. 908 1899. 721 1969. 249 20 aa | 63.903 | 1330 | “32.049 | 7341 | y900.874 | 1253 | “70.414 | 1268 | a4 22 | 65.033 130 || 33.190 14h |i 02.027 153 | 71,580 a | at 23 | 66.164 it 94.338 12 | 03.181 15¢ || 72.745 7 24 | 67.295 13t || 35.474 12 || 04.334 153 | 78.911 ie | ee 25 | 68.426 36. 616 05. 488 75.077 25 26 | 69.557 et | Br 708 142 || 06. 642 134 || 76.243 166 | 26 27 | 70.688 131 || 38,900 142 || 07.796 154 | 77.409 ace | 82 28 | 71.820 132 || 40.043 08. 950 78. 575 28 | 72.051 | 12) | aise | 343 | 10.105 | 385 | zarez | 287 | ag 30 | 174.083 1842. 329 1911. 259 1980. 909 30 a abi | RE aa | iaaia | 1135 ao.ore | Aer | af 32 | 76.347 132 || 44.615 143 || 13.569 159 | 83.244 oe | 3 33 | 77.479 a 45.759 i 14. 724 ae 84. 411 |e 34 | 78.612 133 || 46. 902 13 || 15.880 158 || 95.579 yee. |) #8 35 | 79.745 48. 046 17.035 86. 747 36 | 80.877 132 || 49.190 uP || 18.191 ae 87. 915 168 | $e 37 | 82.011 ne 50. 335 Ce 19. 347 Hs 89. 084 | a 38 | 83.144 a 51. 479 ifs || 20.503 ae 90. 252 | 8s 39 | 84.277 133 || 52.624 Me || 21,660 1 || OL.aal 1 | 39 40 | 1785. 411 1853. 769 1922. 816 1992. 590 41 | 36.545 | 1134 adou | 128 bs.ors | dP oo pry i aaa ay 42 | 87.679 i 56. 059 ae 25. 130 ioe 94. 929 170 | 2 43 | 88.813 13f || 57.204 it 28.287 157 || 96.098 a | 8 44 | 89.948 138 || 58.350 18 || 27.445 158 || 97.268 wo | ad 45 | 91.082 59. 496 28. 603 98. 438 46 | 92.217 ie 60. 642 14g || 29.760 | 38% | 199.609 | 272 re 43 | 94.487 ise 62. 934 ue 3 OTF 5? a i | 48 49 | 95.622 | 438 || e408 | 377 | 33.235 | 388 | osia. | 272 | 49 50 | 1796. 758 1865. 228 1934. 394 2004. 292 bi | 97.893 | 2-135 nears | Set 35.553 | 2-259 Ooaes | Lit | Bf 62 | 179.029 | 336 || 67.522 | 347 | 36.712 | 369 | 06.635 | 172 | 52 53 | 1800165 | 436 || 68.669 | 14% || 37-a71 | 188 | o7.go7 | 272 | 53 54 | 01.301 136 || 69.817 ie | 39.031 160 || 08.979 Ate | oe 55 | 02.438 70. 964 40.191 a6 | oss | jr | wma | 4 | ane | 280 11323 Ae 56 ss | uses | 187 | Gaaop | 49 | Serr | 160 12488 | ang | 1187 || ef 05? | a4ag | ..44-882 | 4.48) | tae | 28 | so 60 | 1808. 122 1876. 706 1945.992 | 2. 2016.015 | 60 THE MERCATOR PROJECTION. RO Cc T 0 ontin} ed . [ Meridional dis or 5 | istances for the spheroid. pression phi id. Com 594 125 =F, 32° 33 utes, Meridiona! - dis a Difference.|| Meridional Tt “7 distance, | Difference. pice Dift ‘i 0 | 20 7 ; — sera AL aoa s a ee ee i lass | eee ; et i |S 174 86.814 | 1 197 || 2158.428 ; 3 | 19.537 fos) Sede | hes | ab ae 4 | 20.711 174 90. 376 188 ee 201 Sas | 3 5 21 174 91. 563 187 oe 201 35 i 6 - 885 188 |) 93-282 | 300 mie) as | cya | AUB Beate ae | 7 | 24.235 175 || 3s tor ise a | ar | nae i 93. 989 ne 65. 636 202 36.977 . :| a is 85.127 188 ager 202 38. 194 a7 | 5 | 8 i 96.315 A 68. 041 203 39.411 217 6 it .761 189 69. 243 A 41. 845 BY : ate | Aedes. | Geee eee 2 anus og | 12 | 30.112 76 || 2099-882 | 89 1 69 3 us | i ae 176 o1-O71 ae 72, 853 204 44,281 1.218 10 15 33 177 03. 450 190 Sot 204 ae7m | 8 16 641 190 75. 260 ol 47.036 a | Seen. “HR Ih Ge cag See : 17 | 35.995 47 one igo | ae ae | ae iH 05 820 190 EP eas 204 49.155 B| es if 07024 191 a 205 50. 374 og | 15 20 | 2039 178 09. 402 191 80. 077 ee 52.818 220 i 21 527 2 191 81. 282 a 54. 083 220 40.705 | 2-278 110. 593 = sat im | 3 gt eee | Se aoa 1.392 (Ss 6as : ar a) £3 i 11785 192 aa | 1208 2255. 253 i ian i 12. 976 a 84. 899 206 56. 473 1. 220 20 a | 45 179 || 15-360 192 || Byam ae | se ous m | 8 26 41g | 192 | 87.311 208 | 60.135 mi |B 46.597 a7g || . 16.552 a nest a | 27 | 47.776 179 18 aay 192 S72 | a i 17745 193 be Bon 206 61.357 g| oe i 18.287 192 ay asl 207 62.578 921 | 2% 30 . 180 21.323 Ee 8.336 208 0 | § 32 | 58.675 180 23.71 | iba gp. 782 23 : . 33 | 54.856 181 24. 904 id a 702 | os “Em a ae is 24 204 193 cea 208 ce fey. || eee toe 35 181 27.293 195 98.178 a0 71137 mi | om | Be300 189 || 28-487 1 | 595 soo | A 37 | 59.580 181 29. 682 195 | Or: sod 20 : : 38 60. 762 182 30. 877 Le 03.014 20 73 39 | 61.944 182 32. 072 195 03.014 508 76.033 Bi | 40 | 2063 182 33. 268 196 ae 210 7735 | 2 41 64. ie 1.182 || 2134-464 | ’ ae oe 210 i 9 7 : a i as a5e | op | 208 — 210 483 eee | ea 43 | 66.674 183 36. 856 He ce | a “ee 8 Sdn is 36. 856 we 09.065 211 80. 934 1. 226 40 45 | 69 183 39. 249 197 10.276 a 53.385 | 2 aie tr 3 oo ier 10.276 a 83. 385 one | 3 41 . 223 3 : uv ot 58 = 71.407 184 41. 643 iss “210 ae | é Hor is 41. 648 ie 13.910 212 85. 838 g| ae i 42.94 198 eRe 212 87.065 927 | 45 50 | 2074 184 45. 236 198 16. 334 rH 89.519 2 4 ot | | ee | en 1 | te ie a alia | 63 | 78.513 185 || 48-831 199 wae | Mas [aoe i ie 18 48. 831 Ha 21.185 213 93. 203 1. 228 50 55 80 186 51. 229 199 22. 398 a 95. 660 229 s 5 . 884 199 23. 611 aul 96,889 | § 6 | 82.069 185 || E3697 me | ogg 28 =|. 87 | 83.255 136 | 98-827 | 1.200 26 : ar 58 | 84.441 186 paeer. | oe 31.258 ai 8 59 _, 85. 628 187 56. 027 200 27. 253 ae 3800. 577 0 f 3 | oa Bt 56.027 ae 98 468 215 || 2300. 577 230 6 2158. 423 | 1-201 nee 215 080 =| § 2230.898 | 1-215 bf 307 20 | 5s osor 267 | 1-280 | Bp 126 U. S. COAST AND GEODETIC SURVEY. MERCATOR PROJECTION TABLE—Continued. [Meridional distances for the spheroid. Compression sor | Min 36° 37° 38° 39° _ Min- utes. mriioual Difference. weet Difference. ae Difference. oe Difference.| U8: , if , , , , , ’ 0 | 2304. 267 2378. 581 2453, 888 ay. || 2530.288 | 9° 95 0 rij 05.408. | 28 | hess | REE a se | eee ano | ee | 2 | 06.729 a 81. 075 ay || 86.416 264 || 32.801 ae, (2 3 | 07.960 ay) B88 ane 57. 680 26% || 34. 083 iis 4 | 09.192 zo |), £8,870 2A7 | 58. 945 265 || 35. 366 a= | sf 5 | 10.423 84. 818 60. 210 36. 649 5 B| em | 2 || Boose |. SB | tras | 2 | Snae | 3a] 8 ¢ |) aoeee || “Be |) grate ee ge eee Ml Bae) eg 233 249 741 | 966 284 8 | 14.120 233 || 88,564 ate | 64.007 266 || 40. 499 a | 3S 9 | 15.353 233 || 99.813 ey | 86273 268 || 41.783 ae | 8 10 | 2316. 586 2391. 062 2466.539 | 2543. 068 | 10 1} aneie | Pee ee | DEEP | geeog | ee | “eacees [a SSeS | ae 12 | 19.053 jaa || 93.562 25) || 69.073 267 || 45.637 a | a 13 | 20.287 et || 94-812 250 || 70. 340 267 || 46.922 zeb | 1s ‘a1 | 2 6. 71, 608 48, 208 : 14 234 251 268 286 15 | 22.755 97.313 72. 876 49. 494 15 16 | 23.990 235 98. 564 251 74. 144 268 50. 781 287 | 46 17 | 25.295 | 233 | oapo sig | 282 || zs.aia | 268 || sacey |. 388 | ay 18 | 26.460 ze? || 2401. 067 25) || 76. 681 5e0 53. 354 pee ae 20 | 2398. . a ne a 22 ooo - Boa oa | 103, 2479. 220 2555. 929 20 at | Sater | Ee foe ze | ee pose | EB obs | Doe | at a2 | 31.404 337 || 06.076 382 || 81. 759 ay 58. 504 | 2 23 | 32.640 236 || 07.329 253 || 83.030 zi || 59.793 a | a 24 | 33.877 337 | 08. 582 23 || 84.300 a 61. 081 | ae 2 | 35.14 | | 09. 836 85.571 62. 370 25 26 | 36.351 ae | 11.090 254 || 86. 942 27l || 63.660 | 300 | a6 27 | 37.589 338 || 12.344 | sae Bie 64.949 | 28° | ay as | 38.827 338 || 13.598 254 || 30, 385 he 66. 239 coe | a 30 | 2341. 303 oe ae i: aaa Piet BE | as atp 201 | ; 2491. 930 ono || 2568. 820 30 ai | aba | ee) | eae | ee | gag | Se | vem | eet | at 32 | 43.781 338 || (18.618 255 | 94. 475 oe 71, 402 ae ieee 33 | 45.020 ae || teers | | Be | op mas 273 || 72.694 ae Bs 34 | 46.260 ory. | ah da 256 || 97.022 amt 73.986 fen. | ae 35 | 47.500 22. 386 98. 296 75. 278 35 36 | 48.740 2D ||. 23. 643 257 || 2499. 570 ot 76. 570 ain | 38 37 | 49.980 ay | 24. 900 257 || 2500. 844 ae 77. 863 93 | 37 38 | 51.221 pat || 26 1b? ge | oaaig | ge 79. 156 oe || a8 = {eee (ae 27.415 258 | 03.394 am | 80. 449 ae | 38 40 | 2958.703 | 1 541 | 2428-672 | 4 95g |) 2504669 | 4 o76 | 2581-743 | 1 ag4 | $0 "Bd. BATT 29.-930 258 | 05. 945 ae 83. 037 a | 42 | 56.185 a || 3h. 189 gee || 7.221 ae 84.331 | 294 | 49 43 | 57.427 is 32. 448 ea 08. 497 ate 85, 626 ae 43 44 | 58.669 ate || 33.707 oF | e778 Be | 66.081 |. eee | ee 45 | 59.912 34. 966 11. 050 88. 216 46 | 61,154 22 | 36. 225 ee 12. 327 ae Bosh |. 295 | 36 47) 62.397 oar | BLASS 360 || 13-604 53 || 90.807 oe | ay 48 63.681 | 945 | 98.745 | oe) | 14.882 | 57g || 92.108 |. 3o7 | 48 S| ee zit | i te Bel | 16. 160 378 ||. 93.400 | 266 2517. 438 2594. 697 | . 50 at | e7.372 |. 2a ag.be | oot ae 95.994 | 2-297 | 54 52 | 68.616 oie 43.788 a 19. 996 oC 97.292 | 298 | 59 63 | 69.861 | 942 || 45.050 | 3h? | ‘a1a7s | 278 | og'500 | 288 | 53 54 | 71.106 | 542 || 46311 | 585 || 22.554 | 279 | asco ssa | 288 | 54 55 | 72.351 47.573 23. 834 2601. 186 55 oe | 73.507 | Sar || 48.836 | 263 | op iia | 280 || “op'4a5 |... 299 | 56 By | 74342 | 572 || 50.098 | 282 | - 26. 395 03.784 |. 129° | gy 58 | 76.088 | 24° | 51.361 27.675 | 289 || o5:084 | 2-300 | 58 mo | 7.288. a oak pao | 5 oo 28. 956 281 06.383 | 1:299 | 59 60 | 2378581 | 1 2453. 888 | 1: 2530.238 | 1-282 || o607.683 | 1-300 | 6 THE MERCATOR PROJECTION, [Meridional d istances for th 8 Sphienoid, Com pression il aad 127 T Min- 40° Zz utes.. | Meridiona: 1° tance. 1 Difference. Meridional |. 42° 43° ; distance. Difference. aeaialnel site es 0 | 260 i Ie midiemal | Digterence || Meridtenal | py Min- A 08, be 1.301 || 2686. 280 : 7 ; distance, | Piference. ares 2 | 10.284 sap | e8%eo0 |* 0 2766.089 | ’ 8 11. 585 301 || 88-920 320 67.430 | 21-341 2847. 171 4 12. 886 301 || ‘90-241. 321 68.771 341 48.533 | 2-362 : 5 14.18 302 || 91. 562 821 70. 112 _ 841 49, 896: 863 y 6 - 188 322 71. 454 342 foe | Bek | a 16. 792 302 94, 206 322 72. 796 2s | Ga | ee 8 18. 095 303 95. 528 322 74, 138 342 53. 987 : 9 | 19.398 303 || 26. 850 322 75. 481 343 || 55.352 365 | 5 10 | 2620. 701 303 98. 173 323 76. 824 aa, || | 14: 388 oY | | as 14 | 79.594 we) 88. 761 a an o on 16. 339 He af 15 | 81.390 90. 605 02. 836 18, 291 1s | 83.187 92. 451 oe 04. 734 a 20. 244 a3 | 16 17 | 84.984 a 94. 297 g8 | 06. 632 898 || 22.197 are. | a 18 | 86.783 i“ 96. 144 oar | 08. 531 899 | 24.152 gee i 8 19 | 88.582 709 || 97.992 848 | 10. 431 soy || 26. 108 oe} ae 20 | 4090. 382 4199. 840 4312. 332 4428, 064 20 a1 | 92192 | 1 a 4201.690 | 1 as 14.233 | 1 ie 30.022 | + aoe 2 22 | 93.983 SOL || 03. 540 850 | 16. 136 aoe || 31.981 el ee 23 | 95. 785 a 05. 391 oes 18, 040 oot || 33: 940 a | 38 24 | 97.588 803 | 07. 248 oe m3 au 20s || 85. 901 atl a6 25 | 4099. 392 09. 095 1. 84 37. 862 26 | 4101. 197 oe 10. 949 a 23. 755 oe 39. 825 ae | 3 27 | 03.002 g05 || 12.804 S54 || 25. 663 sug || 41. 788 oe ea 2s | 04.808 ae 14. 659 ned 27. 571 oo8 | 43. 753 a | 38 29 | 06.615 ov 16. 515 856 || 29. 480 909 || 45.718 oo aa 30 | 4108. 423 4218. 372 4331. 389 4447, 684 30 gi | toes | *o 20.230 | oe 33.300 | 13h) | 49.652 | 2 aa | st 32 | 12.040 808 | 22. 089 859 || 35. 212 aia || 51. 620 | ae 33 | 13.850 ay 23, 949 soy a7. 185 a 53. 589 oe || a 34 | -15. 661 gly || 25. 809 oy = a on ts oe a a 35 | 17.473 27. 671 95 7. 36 | 19.285 oo 29. 533 Bea 42, 868 ne 59, 503 a2 | 36 37 | 21.098 §13 || 31.396 S63 | 44.784 gi6 || 61. 476 ae | 3 3s | 22.912 git || 33. 260 S64 || 46. 701 gt7 | 63. 451 oe | 38 39 | 24.727 815 | 35. 125 S6> | 48.619 a 65. 426 oo | 38 40 | 4126. 543 4236. 991 4350. 538 4467. 402 40 41 | 28.360 | 1-817 ag.a57 | 1866 ngabe | ee 69.379 | h O77 | aa 42 | 30.177 Bi || 40.724 ead 54. 379 uel | TL Bar ae) aa 43 | 31.995 Sig | 42. 592 868 || 56. 301 oon || (73.336 te 44 | 33.814 | 44. 461 869 || 58, 224 o73 | 75.317 ce | a 45 | 35.634 46. 331 60. 148 77. 298 45 46 | 37.454 820 | 48, 202 ne 62. 072 oot || 79. 280 082 | 46 a7 | 39.275 $21 || 50. 074 are 63. 997 325 || 81. 263 a 4s | 41.097 a 51. 946 Hes 65, 924 oo7 | 83. 247 a ae 49 | 42.920 Eon 53. 819 a73 | 67. 851 pon ta. 232 385 | 49 Bo | 4144, 744 4255. 694 4369. 779 4487, 218 50 bi | 46.569 | 182 p7.bep | 1875 71.709 | »930 | “so\205 | 1987 | 54 52 | 48.394 825 || 59. 445 are || 73. 639 a 91. 193 ay | 88 53 | 50. 220 aoe 61. 322 a7? | 75. 870 oon || 93. 182 oon | 8 54 | 52.047 $27 || 68.200 an 77. 502 gan || 95.172 eee se 55 53. 875 65. 079 79. 434 97. 163 55 56 | 55.704 oon 66. 958 Ea 81. 368 ook || 4499. 155 bee | oe 87 | .57.534 $30 || 68. 839 ay 83.303 | 935 || 4501. 148 ma | 8 68 | 59.364 a 70. 720 a 85, 239 236 || 03. 142 coe | 8 59 | 61.195 | , 335 M0. | a eee |) 8h We | a bee Oster |g eae | ae co | 4163.027 | 1 4274. 485 | 1 4389.113 | 1- 4507.133 | ) 60 132 U. S. COAST AND GEODETIC SURVEY. MERCATOR PROJECTION TABLE—Continued. [Meridional distances for the spheroid. Compression aor | Min. e “ 62° 63° utes. Meridional : sae aay ae Min- distance’ | Difference. i ag Difference. ee Difference. a Difference.| 2*°S- 0 | 4507.133 4628.789 | 4754.350 | » 1 | ooo 4) Te" a6 ety be | See ee ea | eae) a @ | 11128 | 445. || » 82.910 Ora || 58.607 129 | 38.518 201 | 3 3) 18-127 | soo | 34972 | ogg | 90-730 |) Obl | oe | 8 he ool || 37-038 Oe | (62.867 By | 92.925 oe | % 39.099 65.000 6 | iii ope) Sen? oss || 65-000 133. || 93-180 seth 8 y 21. 134 003 066 ; 135 97.336 6 7 | 21.134 oe || 43-231 bes ||. 69.268 139 || 4899. 544 ae |g Bae 003 || 45-299 Oe || 71-403 132 || 4901. 753 eae as 10 | 4527 rt ae goa Ae cape Ce, os a an ee | 2.007 108. a 2.071 || 4775-878 | 9439 || 4906-175 | 9913 | 10 a |) aay 009] 1-508 | “orp |) 77-817 we | gees | e219 | at 13 | 33.175 009 |) « Be ora || de P28 | ~ gay || 10 GOS 2 1 010 || 55-853 |' g74 || 82-099 wey | oe | 35. 185 Oe | Be T27 Ore || 84.242 143 |) 45" 035 217 | 44 15 | 37.197 o12 || 89-802 x || 86.386 me az.258 7 16 | 39.209 ote || 41.879 O77 || 88.531 M5 | 19.472 a9 | 4g 17 | 41.222 63.956 | 077 146 : 901 | 16 18 | 43.237 | 925 | ‘g6. ay || 20877 | jaa | ghe6e8 1 13 | 43. Or || 66.085 | O78 || 92. 825 23.015 | 222 | a8 45. 252 oe | 68.114 bay || 94.973 148 | 96, 138 223 | 49 20 | 4547.269 | 4670.195 4797. 123 a : pes a | 49.286 a | weer | aes | domore | 48 170.388 | 2226 | at 1.305 | | 74. 360 ; 153 ; 927 019 4801. 427 : 32. 81 ee ee ee ee || oes8o | abe 8b. oss | 228 | 33 el ee qacay | * OS | ongse {2 Sore | AE Tae 80. 615 07.891 26 | 59.389 022 | 39703 oss |) 97-0 157 || 39-504 ozo | 25 a7 | 61143 | oes | a4791 | Oop || 12.208 | 388 eon | 22 Ss B/ Se) @) oe) a) ee) ig] Be] a |e ‘i 27 16. 48. 441 30 | 4567. 491 4691. 064 ve 162 a | 3 ai | 69.519 | 2-028 |“ o3:i57 | 2-093 at8-058 | gives | S088 | gona | 2 s | bay | Seo) Oe) Bae 165 || 22927 | "gay | SE 33 73.577 030 97.346 095 - 016 165 55. 157 94 32 a4 | 75.609 | 032 | ge99.443 | 097 7,348 ler | 58, 398 | a = 35 | 77.641 | 4g || 4701. 540 seme | oo a 36 | 79.674 03. 639 ogg |) 29-51 169 || 61-885 ee 37 | 81.708 Oot | 05.739 aye lh) Stk a | age | ae 247 | 36 B) Be] | cae) | wor) | eae] ae 8 40 | 4587. 817 ana.os | 5. a | Mee ey | 41 89.856 | 2039 ; 2.104 || 4840-374 | 9 476 |] 4973. 125 | 40 a ee 039 || 14-149 Toe || 22vebo | ae 7377 | 2252 | aa less. | male | eee |) ape) aad 176 | 77630 | 253 | ag 43 | 93.936 | gap | 18-861 | og | 46-904 | 379 || 79.885 | 3 nee o42 | 20-469 | 359 || .49.083 | 229 || gpaai |. 258 it 45 | 4598. 020 28 180 pee | 46 | 4600. 064 044 ae 110 51.263 84. 398 45 46 | 4000.064 | 4s | 24.688 | ayy |) 53.445 | agg | 86.657 | BHD | 46 48 | 04.155 a6 | gei, | as || pereee 134 || 88.917 260 | 47 49 | 06.202 047 || 37 095 113 || 57-812 185 91.178 261 | 43 50 | 4608. 250 4733. 140 4862. 18 a si | 10.299 | 2-049 || “°35°956 | 2-116 -183 | 9 jg || 4995.704 | . 50 52 | 12.349 050 : 117 64.371 | * 4997.970 | 2266 | 54 . 051 37. 373 66. 560 189 | 50 266 | 53 14. 400 39. 491 118 190 00. 236 52 3 052 ; 118 || 68.750 02.504 268 | : bea || 41-610 | 329 | zo.oa | 192 | grag | 270 | Ba 55 | 18.506 43.731 73 192 |) - 271 | 56 | 20.560 054 |} 45" 859 121 134 194 || 07.045 55 58 | 24.672 056 | 50.099 124 pee 196 || 1-590 | 373 | 5 go | 20700 | 5.08 | sear | 125 | Sram | tos | HRS | ae | 8 60 | 4628. 789 : . ; -141 “ 4754. 350 4s84.ii7 | 2-199 |l soig'aig | 2-278 | 62 T HE MERCATOR PROJECTION: MERCAT OR PROJECTION TABLE—Continued inued. [Meridional distances for the sphero id. 50n° | 1 Compression 133 Min- 64° age utes. | Meridiona aaa Difference.||. Meridional 66° 67° A P distance. Difference. Merlaional wie i 7 5018. 41 ’ ; - stance. erence. eridional | 1, Min- : 20. 698 2.279 || 5157. 629 ; distance, | Piference,| WSS : 22. 978 280 59. 993 2. 364 5302. 164 ae j ; 25. 259 281 62. 359 366 04.62] | 2-457 5452. 493 27.542 283 “64.726 367 07.079 ° 458 \ || 99: Ool 2.558 0 5 | -a08 285 67.094 ar | 09.038 f+ aa Br.610 | 229. 1 ‘ 827 Fs a ee 60.171 561 | 2 ; 32.113 286 69. 464 464 62.734 563 3 : 34. 400 287 71. 835 371 ||. 14.463 © 564 4 36.688 | 288 74.208 | 373 | 16. 928 465 ||: 85-298 - |) \ '5@7. 9 | 38.978 | 290 76. 583 375 9.394 .| 466 “67,865 ||) 67 5 10 | 5041. 269 291 78. 959 376 21. 862 468 .70. 433 568 6 at | ae cen | 2OBe. | PAB BRE 378 || 24-381 469 an 3 - 45.356 | » 294 83.716 | 2-379 ° 5326, 802 a et By4 9 1 48.151 995 || . 86.096 380 29.975 | 2-473: 5478. 148 4 | 50.447 996 || 88-478 382 31. 750 475 80.724 | 2576 10 15 52.7 298 90. 861 383 34.026 |.. 478 83. 301 577 | 12 16 . 745 ll 385 36. 704 478 85. 880 579 | 2 i 55. 045 300 93.246 | ; 479 88.461 | 8l 13 e 57.346 - 301 95. 632 386 39.183 |: NaN, seo | 14 i 59. 648 - 302 5198. 020 388 41. 664 4g7.. || 91-048 | 9 61.952 |... 304 5200. 410 390 || 44-147 433 |} 93-627 |. 534. | 28 20 | 5064.287 | 5 305 02. 801 391 46.631'| 484. | 5 96.213 | ., 286 7 V2 a2 | 2806 eqaae. | 393 49.117 » 486 5498. 801 a a6 ra 68.871 | 308 07.588 | 2304 5351. 605 | 1 ae 501: 19 of 71. 180 309 09.983 | » 3% "54.094 | 2 489° 5503. 981 73,491 311 72.380 | 397 56. 585 491: || 06-573 | 2. 592 ” 5 56. BO 312 ||. 14.779 399 ||. 59-078 |» 493 || 09-166 | 06; |e Jf -803 | 3, : Goo | 61-572 |. soa: || , 12-761 '|.. 595 | 2 : ae | eee | 179. 496 ||. . 24-358 597 | 33 az | go.432.| 315 AW 6BT 4) abe 64.068 | pee OR wt) Bog | Se 28 | 82.748 |. 316 21. 984 403 || : 66-565 "|. 497 16.957 © 29 | 35.066 | 318 II" 24. 389 405 69.064 | ~ 499 Ts9«| Gee 25 30 | 5087 BIS |e 26.795. | + 288 rests |< w0Ol! ae gee 603 | 38 31 89. ae 2.320 +5229, 203. 08 ae “O08 ie 37 a 808 28 82 | 92.028 ee) si. aia: 409 . ||:5376.572 ||: dog eee Seige 88 | 94.351 | 323 ||. 34-023 ‘al ||. 79-078" 2. 506 5529.985 | » gy | - 84 | 96.676 325 36. 435 412 || 81-586 508 32.597 | 2-612 30 gn |e 326 38.349.| 414 | 84.095 | . 009. I] 35. 212° | _ 615 31 8 one |. Zs 216 ||<; 86-607 | - 519: ||. , 37-829 e17_ |: 32 Ssasciit 101. 320 398 (|) «. 41-265 Re ae 512, |. 40-447. {gig [#83 87 | 03. 659. 399 || 43-682 Sane Wage 119 | 620 |: 34 B83 |. 05.989 - 339 |. 46-101 | > aig ||. 2- 634 BIB Iv 43.067 | p94) 39 | 08.321 | 382: | 48.522 4: 1420: 84.150 7 Big! | 45. 688 | 2k, = 40 Been ago.oaee | ceed IF er a es ee oe 24: | 36 5110. 655 “5399, 15 50.9 625 | 37 a | 13; 990.) 2-888 15258. 366 ‘oa || 8899-187") S99. || 83. a eae 42 |: '15.326:1. 386 BB. 791 | 2 dog || 5401-709 |} 628. | 39 43 47. 664 © (338 . 58. 217 «| 426. “04. 231 . 2. 522 “BBB6. 192. |: 44 | 20.003 339 "60. 645 428 06.756 | 222 58. 822 |. 2.630, |. 40 cag |e: ‘Baz: ||,/ 88-074 1. 429 || 09-282: |. B26, || oteeee 682 an 46 \- -24. a6 342 65.506 : ae 1.810}. B28 “eg 088). ae : 43 ay | °'97.029 |.» 343 at 93g |.. ..438° | 14. 340 - ace ia | a7 44 48 | "29. 974. | p45 | 70.373) | “ggg |) 16.871 531. ||-.. 69-360 .| 45 49 “31.721 ga7 dl. 72. 809. | - 436 (19.404 |: 533: ot 000 | : 640 | 46 50 | 513 - “ggg | 75-246 | - 487 21.989. |: 585: wa. 64. |, S41 | g el 4.069 5 440. || 24-476: Baz |} 77.284. . 643 | 43 36,419 | 2-390 277. 686 538). ||. 79-929 645 | 48 52 | 38.770 | Sot | 80.126: | 2-440 5427.014'|- o'ean 647 | 49 53 | 41.122 “352 |]. 82. 568 442 29. 554... 2.540 se 576 | 5 54 43.476. “g5q |. 95-012 | 444 ||, 32-096:|. . 542. ||, 85-225 2. 649 ne 55 355 || 87-457 an’ | se eho"). Bee |e eee |e «00, | By 45. 831 aay Pay. es! | 34D 90. 528 gg | 22 56 | 48.188 357 || 89-905 "547 93. 182 goa | 58 64 50. 545 358 ||, 92-354 449: || 39-782 ; 657 | 54 58 |. 52.905 3e9_ || 94-803. 44g |f..; 42-281 54g. ||, 95-839 : 59 | -55. 266 381. 927 255° | 5d ge cee B51 |) Soe any 658. | 56 60_| 5157. 629 2.363. || 5299-709 454. || |47-384 552 (|) D601. 157 - 660 | 36 " © _ |] 5302. 164 2. 455. 49. 938 554: 03. 819 662 57 = 155: || 452.493. | 2588 06. 483 soa, | ~88 > | 5609.49. | 2 866 ai 134 U. S. COAST AND GEODETIC SURVEY. MERCATOR PROJECTION TABLE—Continued. [Meridional distances for the spheroid. Compression zor’ | i. 68° 69° 70° 71° = im- = utes Meine Difference. oe Difference. Menional Difference. Meridional Difference.| 2% 0 | 5609. 149 5772. 739 5943. 955 6123. 602 0 1 | asiz | *Gro | 75598 | "78? | ae ars | 2828 |” oe ers | 3072 | 4 2 | 14.487 672 || 78.319 793 || 49.803 927 || 29.746 on | 2 3 | 17.159 673 || 81-112 795 || 52.730 929 || 32.822 we los 5 | 22.508 86. 705 58. 591 38. 981 ae | 6 | 25.186 | G75 || 89.505 | 800 | Grisag | 985 | azogs | 8 |g 7 | 27.865 632 || 92.306 B04 || 64.463 939 || 45.151 ne s] mer] as [am | oe | cae | Re | Seo | oe | 10 | 5635.915 | 9 6g7 |) 5800.725 | » gi9 || 5973.287 | » gay |) 6154. 426 10 ; : , 3.097 11 | 38.602 690 || 03.535 813 || 76.234 945 || 57.523 Gea. | at 12 | 41.292 691 || 06.348 ait || 79.182 ory || (60.622 ioe | 22 13 | 43.983 693 || 09.162 aig |) Balas os || 63.724 ios | 28 cai | OP aie foe) Vee || ae : : 69. 937 15 16 | 52.068 | G99 || 17-620 | $22 | oro | 958 | 7s.047 | M9 | 4g 17 | 54.767 vo || 20.443 foe || 93.961 goa || 76.160 We | a mas | i) gee | ge | ees | se) mae | i | 705 830 ; beg || 52804 120 | 18 20 | 5662.876 |. 528.926 | , 6002, 858 6185, 514 >, | 20 ai | 65.583 | 7705 || 31.758 | 2832 | op. 823 | 2970 | ss 63g | 3-124 | ay az | 68.202 | fy || 34593 | $35 | ogsor | 973 | on: 764 Be | ee 23 | 71.003 m3 || 37.429 og | 11.776 ore |) 94. 893 ie P33 24 | 73.716 715 | 40.267 ess | 14.758 ono {| 6198. 025 i aa 25 | 76.431 43. 108 17.738 6201. 159 2g | ross | 712 | 45951 | 843 | ao.716 | 983 || os 206 | 187 |. 36 ay | 81.867 | fo) || 48.797 | BAe | aa7o1 | 985 | 7.436 | 140 | ay 28 | 84.588 | fo; || 51.644 | 887 | 66s | 987 | jo.579 | 143 | 9g 29 | 87.311 ree | 54. 494 go) || 29.678 ee Me | 8 30 | 5690. 036 5857. 346 6032. 670 6216. 872 ai | 92.763 | 7727 1 “60,200 | 2 for 35.665 | 2995 || “29.023 | 3.151 31 32 | 95.492 551 63. 057 pee 38.662 | » doy 23.176 153 | 39 33 | 5698. 223 733 | 65.915 ger | 41-661 | 2999 |) 26.332 156 | 33 34 | 5700. 956 aes 68. 776 ae 44.664 | 9-008 29, 491 ne 34 35 | 03.691 71. 639 47, 668 32, 36 | 06.429 | 735 | 74.505 | 866 | so.675 | 007 | 35° sis | 165 | $8 a7 | 09.168 | 74; | 77.372 | 88) 53.685 | O10 | 38 985 | 167 | gy 38 | 11.909 743 | 80.242 a7 | 56. 697 Ole || 42.155 aa 39 | 14.652 74g | 83-114 are 59.712 1 Oia 45, 328 ae | as 46 | 5717.398 5885. 989 6062. 729 : ai | 20.145 | 2747 W gg ge5 | 2.876 || gs'74g | 3.019 |) 6788-008 | 5 iq | 40 42 | 22. 894 749 1 91.744 879 | 68.770 022 || Bi bea 181 | 4 43 | 25. 646 752 Ht 94: 625 881 | or 794 024 || Be O47 ist | 45 44 | 28.399 | ee ft 5807.508 | 883 | 74’go1 | 027 | gy'934 | 187 | gg 45 | 31.155 5900. 394 an rr — ae 46 | 33.913 vee 03, 282 888 || 30. 983 om | Fea 12 | $ 47 36. 672 762 06. 172 an 83. 918 035 70, 811 195 | 4g 48 | 39.434 red. | 09.065 gee | 86. 955 Oy || 74.010 199 | 48 49 | 42.198 reg 11.960 ao? || 89. 995 SN 77.211 ne 50 | 5744. 964 5914. 857 6093. 038 si | 47.732 | 2798 | “iz. 756 | 2-899 |” 9603 | 3.045 || S80 4l4 | 3 og, | 50 770 902 83. 621 61 52 | 50.502 772 || 20.658 sor || 6099. 130 nee 86. 831 210 | 52 : ip : $06 | 05.282 poz || 93. 258 54 55 | 58,825 veg | Bea? ou || 08-287 eae aaa ee 56 61. 604 780 32. 288 ng 11. 345 058 || g299, 697 221 | 56 57 64. 384 733 35. 201 14. 406 061 6302. 921 224 57 58 | 67.167 38.117 916 17. 469 063 06. 148 227 59 | 69.952 | , 785 |} 41.035 918 90,534 065 |) 99.378 230 | 38 60 | 572.739 | * 5943, 955_| 2-920 ff 6193. 602 | 3.068 || 6312 610 | 3-232 | go THE MERCATOR PROJECTION. MERCATOR PROJECTION TABLE—Continued. {Meridional distances for the spheroid. Compression xr | 135 ix 712° 73° 74° 75° oe utes. | Meridional | 1, i idi i distance. Difference. ae Difference. eet Difference. peed Difference. utes. o | 6312. 610 . |esiz.on | » 6723.275 | > i | 15.865 | 225 | vpan1 | 222 | oe.o03 | 3-628 OP thes | ees] 4 2 | 19.083 38 | 18. 914 423 631 : 868 | 3 2 | 19.08 a 428 | 30.534 B31 || 55.493 2 pee ee ee aa ae a eee | 2 247 433 643 || 63-242 | gg 4 5 | 28.816 29. 203 41.4 : 55D 451 67.123 5 6 | 22.055 | 353 | 82.640 a |) Seer + ee os | ee 85.3 aes 36.079 me 48.747 pa 74, 898 890 | 8) 3 575 ae 89. 522 ae 52. 401 Re 78. 792 oa) 8 ee =e s Be 486 | 56. 059 658 || 82.690 ae | 8 ; és oe ; 6759. 721 6986. 592 10 48. 360 49,871 | 738 63.386 | > 668 90.498 | 3906 | ay 12 | 5. 627 pall 53. 327 ae 67.055 - 94. 409 ae 12 13 | 54.898 73 || 58.786 49 || 70.728 673 |) 6998. 324 | ae a ae ae "249 oo 74.404 676 || 7002. 243 ae ; 63.715 | + 78. 084 16 | 64.727 279 67.185 470 81. 768 684 16: Ose 928 | 46 tee ate 283 473 688 10. 095 16 “0 a 70. 658 ae 85.456 fee 14. 028 933 | 4y 18 71. 296 Sat 74, 134 aa 89. 148 ae 17.965 an | a8 20 | 6377.876 e ae — 4 i cas 0 el : Sian ; 6796. 543 7025. 852 a1 | 81.171 eee 84.583 | 3-486 | gg00.246 | 2-708 poger | 3-949 | St 22 | 384.468 2e7 || 88.073 ee 03. 953 ry | 88.988 954 | 99 23 87. 768 oe 91. 566 a 07.663 ae 37. 714 959 | 93 : ay 95. 063 aan 11.377 ee 41.677 fee | oad a5 | 94.379 310 || 6598. 563 15.096 45. 645 25 26 | 6397.689 6602. 067 504 18. 812 722 ; 972 313 507 76 || “ae ee S 27 | 6401. 002 a 05.574 ae 22. 545 53, 594 977 | oy 28 04.317 ate 09. 084 a 26.275 a 57.575 oot | 28 les oe a . 51¢ || 30.009 re || 61.561 | ae ; ; 6833. 747 7065. 551 si | 14.203 | 3826 19.636 | 3-520 || 37,489 See pa eap [eet | at 32 | 17.611 rae 23. 160 Fea 41, 236 oe 73.544 | 3999 | go a i pee 8.740 Toe || 81.555 ro tae 35 | 27.613 83.754 52. 498 oe : 85. 568 a6 | 30.954 a 37. 292 a 56. 260 ane 89. 585 our | 36 37 | 34.298 om 40, 833 oy 60. 027 766 || 93.607 022 | 37 38 | 37.645 oar || 44.378 63. 797 7097. 633 a6 S| tags 549 774 031 | 38 i 350 || 47.927 5m9 || 67.571 ary || 7101. 664 oak | ae 40 644.348 | 5 o5q | S051. 479 | 5,556 | 6871-342 | 5 rg | 705-699 | 4 o39 | 40 : es 035 | 8:858" || 95, 191 re? || 09.739 | * 41 42 | 51.063 359 || 58. 594 559 || 78.916 7e5 || 38.784 045 | 49 43 | 54.425 362 || 62.157 563 |) 82.706 17. 833 049 | 3B 44 | 57.790 365 | 65.723 566 | 86.500 oe | Beger OOF | aa 45 | 61.159 69.293 90. 298 25.946 46 | 64.531 | 372 |. 72.866. aC 94. 100 802 || - 30.009 063 | 4 47 | 67.906 375 | 76.443 B77 || 6897. 906 806 || 34.077 068 | 47 as | 724 | Sor | 80-02 | ges | 6ol-7is | gig | 38-149 He 381 || 83.609 585 || 05.531 815 || 42,296 ba | 49 50 | 6478.050 | 5 ag7 || 6687.197 | 5 59, || 6909.350 | 3 go || 7146. 308 50 51 | 81.437 387 | 90.788 | 358t | “ag.a72 | 3-822 | "50.304 4.085 | 54 52 | 84.828 au 94. 383 py 16. 998 ae 54. 485 091 | 5 53 | 88.222 30% || 6697. 982 pee ||, 20.829 58.581 096 | 5g 64 | 91.619 gor || 6701. 584 oe 24. 664 a 62. 682 AOL | ge 55 | 95.020 404 || 05-190 gio || 28-508 gag || 68-787 oi 56 56 6498. 424 oF 08. 800 te 32. 346 ce 70. 897 10 | 56 58 | 05.241 a0 | 2230 | 6i7 | ab ous | 852 || Fe S39 120 | 5% Bae 413 aa ae 621 . 856 79. 182 125 | 38 3.417 “651 | 3 695 || 43-901 | 3. g60 || _.83-257 | 4.339 | 59 60 | 6512.071 6723. 275 6947.761 | > 7187.387 | * 60 136 U. S. COAST AND GEODETIC SURVEY. MERCATOR PROJECTION TABLE—Continued. [ Meridional distances for the spheriod. Compression 27" ae 76° 77° 78° 79° om utes. ons Difference. ee Difference. oe Difference. aeons Difference.| ¥*€5: wf , a a , , té © | 7187.387 | 4 454 || 7444428 | 4 gay | 7721-700 | 4 91, || 8022758 | Sig | 0 1 | 91.521 1S 48. 875 Te 36. 511 ae 28. 001 oe tad 2 | 95.660 Tae 53. 327 ye 31. 329 ae 33. 252 ae les 3 | 7199. 804 i 57. 785 oe 36. 154 38. 511 ele 8 4 | 7203. 953 is, | 62.248 ae 40. 985 a 43. 778 a ie & | 08.107 66. 717 45, 823 49. 053 5 6 | 12.266 ie, | 71.192 ayo || 50. 668 oe | 84. 336 a 1s 7 | 16.499 a 75. 673 SS 55. 520 aoe 59. 628 an 8 | 20.598 in, | 80. 160 7 60. 378 Se 64. 927 ae 7489. 150 7770. 115 8075. 550 - i | aaaeg | 2 | ok a | 74.993 | *878 | go.873 | © 323 | aa 12 | 37.321 igg || 7498. 163 pe 79. 878 Be 86. 203 oe | 8 i | 6 4.6u tog || 7502. 678 84.770 ie 91.542 .| 389 | ag 4 | 45,712 as 07. 199 Pe 89. 669 or? |) 8096. 890 a de 15 | 49.915 11. 726 94. 575 8102. 246 15 16 | 54.193 208 | 16.258 232 | 7799. 487 ole || 07. 610 365 | 16 17 | 58.336 ee 20. 797 Paq || 7804. 407 foe 12, 983 ae | at 18 | 62.555 ce 25, 341 mes | (09. 334 ae 18. 364 =. | is as eon | ae | Nec oe ae ee . : 7819. 208 8129. 150 0 21 | 75.240 | +238 g9.008 | # Eel 2aiss | “8M 1 sass | 5: oa ae 22 | 79.478 238 || 43.575 567 || 29. 109 ne 39. 969 Poa 3 | 83.721 on 48. 149 ey 34.070 | Sel If 45, 301 2 | 28 _ ee dh a 2 = ae 39. 038 ane 50. 821 mae. fla 44, 013 56. 260. 25 26 | 7296. 482 aM 61. 905 ae 48. 996 ae 61. 708 co |: 28 27 | 7300. 747 a 66. 502 a 53.986 | 4 bor 67. 165 alee re 05. ~ 974 71. 106 610 58.983 | 5 oq ||. 72.630} Jaq |; 28 Sw lesa coe ene | eel eee lee ; ; 998 nee | cs, 30 BL. | “agesa.|) See Sider | a 7a.o1g "|, 2018 I go.o7e | 400 | oa 32 | 22.144 oe 89. 581 ae 70.041.) Ges |eioanrs | 98 | ge 33 | 26.439 a 94, 215 aR 84. 073 Oia. || 8200. 082 507 | 33 34 | 30.739 7598, 855 9.113 |. | “05, 598 516 | 34 35 | . 35.045 08 | 608, 502 = aL oe 35. ; 94, 160 11. 128 35 36 | 39.356 ae 08. 154 65° || 7899. 214 ae 16. 657 534 | 36 37 | 43.672 a | wets Gay || 780R 276 | apes e900. BES | a as | 47.904 | 357 | 12478 | rt |) 09-845 | og 27752 |B? I, 38 so | 7356.653 | 4 997 en = sis an 506 Se Niece HOT oe t le Ny. le 1 * “41 | 60.990 | 4337 alma | * Ge 24.596 |, 8-091 |) S208 $83 | 5.578 it 42 | 65.332 a 36. 199 one 29. 694 He 50. 047 586 | 49 45 | 69.680 | 353 | 40-895 | top |) 34799 | 113 |) 55.642 | 585 | ag : ts o aoe | (45.597 i 39, 912 i 61. 247 eno | ae : 50. 305 45. 033 66. 861 46 | 82.756 364 | 55.020 Tus || 50. 161 i 72.434 | 623 | 46 47 | 87.126 ao | 59. 741 = 55.297 |. 286 | 78° q77 633 | ay 48 | 91.501 na 64, 469 we | 60. 441 i 83. 759 ee | 48 49 | 7395. = oe 69. 203 rao || 65-592 izg || 89.409 G0 | 49 50 | 7400. 268 7673. 943 7970. 751 8295. 069 oi | 04659 | * 20) 78.689 | * 726 75.917 | %&166 || 359737 | 5.668 | 3e 52 | 09.055 ee 83, 442 Se 81. 090 173 06. 414 677 | 52 53 | 13.457 aE 88. 201 tie 86.271 qeh 12.101 ae 4 | 17.865 a 92. 967 _ 91. 460 te 17.798, | $7 | 54 55 | 22.278 7697. 740 7996. 656 56 | 26.697 faq || 7702. 519 772 | 8001. 861 205 a ol m5 | 3 57 | 31.121 430 || 07-304 ae 07. 074 213 34, 944 725 | By 68 | 35.551 | 43g | 12096 | 795 || 12.204 | 220 | ao gzg | 784 | 3g 59 SU087 | 4 aay a re 17.522 | . oe 46. 422 744 | 59 60 | 7444498 | * m721. 700 | 8022. 758 | > 3352.176 | > 754 | 69 FIXING POSITION BY WIRELESS DIRECTIONAL BEARINGS.” A very close approximation for plotting on a Mercator chart the position of a ship receiving wireless bearings is given in Admiralty Notice to Mariners, No. 952, June 19, 1920, as follows: I.— GENERAL. Fixing position by directional wireless is very similar to fixing by cross bearings from visible objects, the principal difference being that, when using a chart on the Mercator projection allowance has to be made for the curvature of the earth, the wireless stations being generally at very much greater distances than the objects used in an ordinary cross bearing fix. Although fixing position by wireless directional bearings is dependent for its accuracy upon the degree of precision with which it is at present possible to determine the direction of wireless waves, con- firmation of the course and distance made good by the receipt of additional bearings, would afford con- fidence to those responsible in the vessel as the land is approached under weather conditions that preclude the employment of other methods. At the present time, from shore stations with practiced operators and instruments in good adjustment, the maximum error in direction should not exceed 2° for day working,but it is to be noted that errors at night may be larger, although sufficient data on this point is not at present available. II.— TRACK OF WIRELESS WAVE. The track of a wireless wave being a great circle is represented on a chart on the Mercator projection by a flat curve, concave toward the Equator; this flat curve is most curved when it runs in an east and west direction and flattens out as the bearing changes toward north and south. When exactly north and south it is quite flat and is then a straight line, the meridian. The true bearing of a ship from a wireless telegraph station, or vice versa, is the angle contained by the great circle passing through either position and its respective meridian. ou ae: III. — CONVERGENCY. Meridians on the earth’s surface not being parallel but converging at the poles, it follows that a great circle will intersect meridians as it crosses them at a varying angle unless the great circle itself passes through the poles and becomes a meridian. The difference in the angles formed by the intersection of a great circle with two meridians (that is, convergency) depends on the angle the great circle makes with the meridian, the middle latitude between the meridians, and the difference of longitude between the meridians. Bs : This difference is known as the convergency and can be approximately calculated from the formula— Convergency in minutes=diff. long. in minutes X sin mid. lat. Convergency may be readily found from the convergency scale (see fig. 62), or it may be found by traverse table entering the diff. long. as distance and mid. lat. as course; the resulting departure being the convergency in minutes. Spee i IV.—TRUE AND. -MERCATORIAL BEARINGS. Meridians on a Mercator chart being represented by parallel lines, it follows that the true bearing of the ship from the station, or vice versa, can not be represented by a straight line joining the two positions, the straight line joining them being the mean mercatorial bearing, which differs from the true bearing a A valuable contribution to.this subject by G. W. Littlehales, appeared in the Journal of the American Society of Naval Engineers, February, 1920, under the title: “The Prospective Utilization of Vessel-to-Shore Radiocompass Bearings in Aerial and Transoceanic Navigation.” ; ; Since going to press our attention has been called to a diagram on Pilot Chart No. 1400, February, 1921, entitled ‘‘ Position Plotting by Radio Bearings” by Elmer B. Collins, nautical expert, U. S. Hydrographic Office. On this diagram there is given a method of fixing the position of a vessel on a Mercator chart both by plotting and by computation. © The Admiralty uses dead-reckoning position for preliminary fix whereas by the Hydrographic Office method the preliminary fix is obtained by laying the radiocompass bearings on the Mercator chart. The Hydrographic Office also gives a method of computation wherein the radiocompass bearings are used in a manner very similar to Sumner lines. See also the paragraph wireless directional bearings under the chapter Gnomonic Projection, p. 141. ae 138 U. §. COAST AND GEODETIC SURVEY. by +4 the convergency. As it is this mean mercatorial bearing which we require, all that is necessary when the true bearing is obtained from a W/T station is to add to or subtract from it 4 the convergency and lay off this bearing from the station. Nore.—Charts on the gnomonic projection which facilitate the plotting of true bearings are now in course of preparation by the Admiralty and the U. 8. Hydrographic Office. V.—SIGN OF THE 4 CONVERGENCY. Provided the bearings are always measured in degrees north 0° to 360° (clockwise) the sign of this } convergency can be simply determined as follows: Na laticccoicceenacktnetacsia 3 convergency is ++ to the bearing given by the W/T station when ship is E. of station. Nolatsccie-. os aisiteeigameinncavess 3 convergency is — to the bearing given by the W/T station when ship is W. of station. S Jatsccwek aks hecsiseuuiass The opposite. When the W/T station and the ship are on opposite sides of the Equator, the factor sin mid. lat. is necessarily very small and the convergency is then negligible. All great circles in the neighborhood of the Equator appear on the chart as straight lines and the convergency correction as described above is immaterial and unnecessary. VI.—EXAMPLE. A ship is by D. R.*? in lat. 48° 45’ N., long. 25° 30’ W., and obtains wireless bearings from Sea View 2443° and from Ushant 2774° What is her position? Sea View.....-..... Lat. 55° 22’ N. Long. 7° 193’ W. D: Rin cess osesasie Lat. 48° 45’ N. Long. 25° 30’ W. Mid. lat........... 52° 03/N. _— Diff. long. 1090.5 Convergency=1090.5 Xsin 52°=859’, or $ convergency=7° 09 The true bearing signaled by Sea View was 244}°; as ship is west of the station (north lat., see Par. V) the’ 4 convergency will be ‘‘minus”’ to the true bearing signaled. Therefore the mercatorial bearing will be 237}° nearly. Similarly with Ushant. Lat.Di Biuc.ccneee 48° 45’ N. Long. 25° 30’ W. Lat. Ushant........ 48° 263’ N. Long. 5° 053’ W. Mid latiresiecceseaies 48° 36’ N. Diff. long. 1224.5 W. Convergency=1224.5 Xsin 48° 36’=919’, or 4 convergency=7° 40’ The true bearing signaled by Ushant was 277}°; as ship is west of the station (north lat., see Par. V) the } convergency will be “‘minus”’ to the true bearing signaled. Therefore the mercatorial bearing will be 270° nearly. Laying off 2374° and 270° on the chart from Sea View and Ushant, respectively, the intersection will be in: Lat. 48° 273’ N., long. 25° 05’ W., which is the ship’s position. Notr.—In plotting the positions the largest scale chart available that embraces the area should be used. A station pointer will be found convenient for laying off the bearings where the distances are great. The accompanying chartlet (see Fig. 62), drawn on the Mercator projection, shows the above posi- tions and the error involved by laying off the true bearings as signaled from Sea View W/T station and Ushant W/T station. ; The black curved lines are the great circles passing through Sea View and ship’s position, and Ushant and ship’s position. The red broken lines are the true bearings laid off as signaled, their intersection ‘‘B’’ being in lati- tude 50° 14’ N., longitude 25° 46’ W., or approximately 110’ from the correct position. The red firm lines are the mean mercatorial bearings laid off from Sea View and Ushant and their intersection ‘‘C’’ gives the ship’s position very nearly; that is, latitude 48° 274’ N., longitude 25° 05’ W. 82 Dead reckoning. gel ss° } r 1 L H : Sue Incorrect Position’B” _L-~ H Hy pore Se : Long. 2 “46W - | 50% &. DE ae - H 50° H DR PositionA” ~~ _277° H Lat4esgSN. Long. 25°30 'W. oO 4 Correct Position’"C" 270° T Lat. 48° 274°. Long.25°05 W. perp CECE PEE FEE PEEEEET EE Scale of Mid.Latitude 9° 5° “10° 15° 20° 25° 30° Scale of Convergency 0 : a 2 i 3 : 4 bata tai 30° 35° 40° 45° 50° 55° 60° 65° _70°_78° 0°90" Scale of Mid. Latitude 6 6 7 8 3 to Scale of Convergency Example:- Mid. Lat. 50°30; diff. long. 282; To find the Convergency. Under 50°30; on Mid. Lat. scale read 7.70n scale of Convergency which multiplied by 28.2 gives 2/7* the Convergency. Fig 62 C.& G.S. Print. FIXING POSITION BY WIRELESS DIRECTIONAL BEARINGS. 139 Position ‘‘A’’ is the ship’s D. R. position, latitude 48° 45’ N., longitude 25° 30’ W., which was used for calculating the 4 convergency. Nore.—As the true position of the ship should have been used to obtain the 4 convergency, the quantity found is not correct, but it could be recalculated using lat. and long. ‘‘C’’ and a more correct value found. This, however, is only necessary if the error in the ship’s assumed position is very great, VII.—ACCURACY OF THIS METHOD OF PLOTTING. Although this method i is not rigidly accurate, it can be used for all practical purposes up to 1,000 mailes range, and a very close approximation found to the lines of position on which the ship i is, at the moment the stations receive her signals. VIll.—USE OF w/t BEARINGS WITH OBSERVATIONS OF HEAVENLY BODIES. It follows that W/T bearings may be used in conjunction with position lines obtained from observa- tions of heavenly bodies, the position lines from the latter being laid off as straight lines (although in this case also they are not strictly so), due consideration being given to the possible error of the W/T bearings. Moreover, W/T bearings can be made use of at short distances as ‘‘position lines,’’ in.a similar manner to the so-called ‘Sumner line’’ when approaching port, making the land, avoiding dangers, etc. IX.—CONVERSE METHOD. When ships are fitted with apparatus by which they record the wireless bearings of shore stations whose positions are known, the same procedure for laying off bearings from the shore stations can be adopted, but it is to be remembered that in applying the 4 convergency to these bearings it must be applied in the converse way, in both hemispheres, to that laid down in paragraph V. THE GNOMONIC PROJECTION. DESCRIPTION. [See Plate IV.] The gnomonic projection of the sphere is a perspective projection upon a tan- gent plane, with the point from which the projecting limes are drawn situated at the center of the sphere. This may also be stated as follows: The eye of the spectator is supposed to be situated at the center of the terrestrial sphere, from whence, being at once in the plane of every great circle, it will see these circles projected as straight lines where the visual rays passing through them inter- sect the plane of projection. A straight line drawn between any two points or places on this chart represents an arc of the great circle passing through them, and is, therefore, the shortest possible track line between them and shows at once all the geographical localities through which the most direct route passes. Fic. 63.—Diagram illustrating the theory of the gnomonic projection. The four-sided figure QRST is the imaginary paper forming a ‘‘tangent plane,” which touches the surface of the globe on the central meridian of the chart. The N.-S. axis of the globe is con- ceived as produced to a point P on which all meridians converge. Where imaginary lines drawn from the center of the earth through points on its surface fall on the tangent plane, these points can be plotted. The tangent paper being viewed in the figure from underneath, the outline of the island is reversed as in a looking glass; if the paper were transparent, the outline, when seen from the further side (the chart side) would be in its natural relation.—From charts: Their Use and Meaning, by G. Herbert Fowler, Ph. D., University College, London. Obviously a complete hemisphere can not be constructed on this plan, since, for points 90° distant from the center of the map, the projecting lines are parallel 140 THE GNOMONIC PROJECTION. 141 to the plane of projection. As the distance of the projected point from the center - of the map approaches 90° the projecting line approaches a position of parallelism to the plane of projection and the intersection of line and plane recedes indefinitely from the center of the map. The chief fault of the projection and ‘the one which is incident to its nature is that while those positions of the sphere opposite to the eye are projected in approxi- mately their true relations, those near the boundaries of the map are very much distorted and the projection is useless for distances, areas, and shapes. The one special property, however, that any great circle on the sphere is repre- sented by a straight line upon the map, has brought the gnomonic projection into considerable prominence. For the purpose of facilitating great-circle sailing the Hydrographic Office, U. S. Navy, and the British Admiralty have issued gnomonic charts covering in single sheets the North Atlantic, South Atlantic, Pacific, North Pacific, South Pacific, and Indian Oceans. - This system of mapping is now frequently employed by the Admiralty on plans of harbors, polar charts, etc. Generally, however, the area is so small that the difference in projections is hardly apparent and the charts might as well be treated as if they were on the Mercator projection. =. ~~ The use and application of gnomonic charts as supplementary in laying out ocean sailing routes on the Mercator charts have already been noted in the chapter on the Mercator projection. In the absence of charts on the gnomonic projection, great- circle courses may be placed upon Mercator charts either by computation or by the use of tables, such as Lecky’s General Utility Tables. It is far easier and quicker, however, to derive these from the-gnomonic chart, because the route marked out on it will show at a glance if any obstruction, as an island or danger, necessitates a modified or composite course. WIRELESS DIRECTIONAL BEARINGS. The gnomonic projection is by its special properties especially adapted to the plotting of positions from wireless directional bearings. Observed directions may be plotted by means of a protractor, or compass rose, constructed at each radiocompass station. The center of the rose is at the radio station, and the true azimuths indicated by it are the traces on the plane of the projection of the planes of corresponding true directions at the radio station. MATHEMATICAL THEORY OF THE GNOMONIC PROJECTION. A simple development of the mathematical theory of the projection will be given with sufficient completeness to enable one to compute the necessary elements. In figure 64, let PQP’Q’ represent the meridian on which the point of tangency lies; let ACB be the trace of the tangent plane with the point of tangency at C; and let the radius of the sphere be represented by #; let the angle COD be denoted by p; then, CD =OC tan COD =F tan p. All points of the sphere at arc distance p from C will be represented on the projec- tion by a circle with radius equal. to CD, or p= tan p. To reduce this expression to rectexigular cosrcinates: let us suppose the circle drawn on the plane of the projection. In figure 65, let YY’ represent the projection of the central meridian and XX’ that of the great circle through ( (see fig. 64) perpendicular to the central meridian. 142 U. S. COAST AND GEODETIC SURVEY, 2 D \ ® {| ‘ \ Pp Fie. 64,—Gnomonic projection—determination of the radial distance. Y f \ Y¥’ Fig. 65.—Gnomonic projection—determination of the coordinates on the mapping plane. If the angle XOF is denoted by w, we have z=p cosw=F tan p cos w y=psinw =F tan p sin w; or, _ sin p cos w COs p _sinpsine, cos p THE GNOMONIC PROJECTION. 1438 Now, suppose the plane is tangent to the sphere at latitude a. The expression just given for x and y must be expressed in terms of latitude and longitude, or ¢ and \, \ representing, as usual, the longitude reckoned from the central meridian. In figure 66, let T be the pole, Q the center of the projection, and let P be the point whose coordinates are to be determined. r Fic. 66.—Gnomonic projection—transformation triangle on the sphere. The angles between great circles at the point of tangency are preserved in the projection so that w is the angle between QP and the great circle perpendicular to TQ at Q; or, LTQP=5-«. Also, TQ=5- TP=5-¢, QP =P) and, ae ZQTP = From the trigonometry of the spherical triangle we have cos p=sin @ sin ¢+ cos @ cos A COs 9, sin p_ sind COS p COS w , or sin p cos w=sin \ cos 9, and . f . sin p sin w=CcOs @ SIN y—SsiN a@ COS A COS ¢. On the substitution of these values in the expressions for z and y, we obtain as definitions of the coordinates of the projection— ta F sin d cos ¢ sin a sin g+COs a COs \ cos ¢’ _ & (cos asin g—sin @ cos ) cos ¢) sin a sin 9+ cos @ COs A COS ¢ 144 U. S. COAST AND GEODETIC SURVEY. The Y axis is the projection of the central meridian and the X axis is the pro- jection of the great circle through the point of tangency and perpendicular to the central meridian. These expressions are very unsatisfactory for computation purposes. To put them in more convenient form, we may transform them in the following manner: = FR sin-d cos ¢ ~ sin @ (sin g+cos ¢ cot a cos d) _f cos a (sin g—cos ¢ tan @ cos A) | sin a (sin y+cos ¢ cot a cos A) Let cot B=cot @ cos A, tan y= tan a cos dX, then zB sin \ cos ¢ = in 0S ¢ COS nos gcsin g sin B+ cos ¢ cos B) Fes (sin cos y—cos ¢ sin ¥) 008 7 g cos ¥ g sin a co 0s (sin g sin B+ s ¢ cos B) But cos (p—B) =sin ¢ sin B+ cos ¢ cos 8, and ; sin (p—7) =sin g cos y—Ccos ¢ sin 7. Hence _# sin sin d cos 9 sin a cos (y— 8B) _# cot a sin B sin (y— Y). cos y cos (¢— 8B) These expressions are in very convenient form for logarithmic computation, or for computation with a calculating ‘machine. For any given meridian 6 and 7 are constants; hence the coordinates of intersection along a meridian are very easily computed. It is known, a@ priori, that the meridians are represented by straight lines; hence to draw a meridian we need to know the coordinates of only two points. These should be computed as far apart as possible, one near the top and the other near the bottom of the map. After the meridian is drawn on the projection it is sufficient to compute only the y coordinate of the other intersections. If the map extends far enough to include the pole, the determination of this point will give one point on all of the meridians. Since for this point \=0 and o=5 » we get B=a, Y=, z=0, y=R cota. If this point is plotted upon the projection and another point on each meridian is determined near the bottom of the map, the meridians can be drawn on the projec- tion. THE GNOMONIC PROJECTION, 145 If the map is entensive enough to include the Equator, the intersections of the straight le which represents it, with the meridians can be easily computed. When g=0, the expressions for the coordinates become | oR tah ood! aid ye Ritme x line s publi to the X axis at’ the distance /="—R tan’ a represents the: Anquator: ‘i e intersection of the meridian \ with this line is given’ by ae “o=R tan \ seca. - When: the Equator and the’ pole aré both on the map, the meridians may thus be determined in a very simple manner. The parallels may then be determined by computing the y coordinate of the various intersections with ‘these straight-line meridians. - ; 2th, If the poate of venom is it, vali > poe a=5 and the expressions for the: nai nates become | ta aR cot ¢ sin oe - yY=—-R cote cos, » In these expressions. X is reckoned from the central meridian from south to east. As usually given, dis reckoned from the east point to northward. ‘Letting A= ath and dropping the | prime, we e obtain: the usual forms: aR cot ¢ cos i, += R cot'g sin d. The parallels are reprecented by concentric circles. each with the radius a ie p= cot ¢. The masedians. are represented by the equally spaced radii of this system of circles. If the point of tangency is on the Equator, a=0, and the expressions become o=FR tan X, y= tan sec d. The meridians in this case are represented by straight lines perpendicular to the X axis and parallel to the Y axis. The distance of the meridian \ from the origin is given by. a= tan 2. , Any, gnomonic projection is symmetrical. with respect to the central meridian or ‘to the Y axis, so that the computation of the projection on one side of this axis is sufficient for the complete construction. When the point of tangency is at the pole, or on the Equator, the projection is symmetrical both with respect to.the Y axis and to the X axis. It is sufficient in either of these cases to compute the inter- sections for a single quadrant. | _ Another method for the construction of a gnomonic chart is given in the Admiralty Manual of Navigation, 1915, pages 31 to 38. 20864°—21—10 WORLD MAPS. As stated concisely by Prof. Hinks, ‘‘the problem of showing the sphere on a single sheet is intractable,’’ and it is not the purpose of the authors to enter this field to any greater extent than to present a few of the systems of projection that have at least some measure of merit. The ones herein presented are either conformal or equal- area projections. THE MERCATOR PROJECTION. The projection was primarily designed for the construction of nautical charts, and in this field has attained an importance beyond all others. Its use for world maps has brought forth continual criticism in that the projection is responsible for many false impressions of the relative size of countries differing in latitude. These details have been fully described under the subject title, ‘Mercator projection,” page 101. The two errors to one or both of which all map projections are liable, are changes. of area and distortion as applying to portions of the earth’s surface. The former Fic. 67.,—Mercator projection, from latitude 60° south, to latitude 78° north. error is well illustrated in a world map on this projection where a unit of area at the Equator is represented by an area approximating infinity as we approach the pole. Errors of distortion imply deviation from right shape in the graticules or network of meridians and parallels of the map, involving deformation of angles, curvature of meridians, changes of scale, and errors of distance, bearing, or area. In the Mercator projection, however, as well as in the Lambert conformal conic projection, the changes in scale and area can not truly be considered as distortion or as errors. A mere alteration of size in the same ratio in all directions is not considered distortion or error. These projections being conformal, both scale and area are correct in any restricted locality when referred to the scale of that locality, but as the scale varies with the latitude large areas are not correctly represented. Userut FEATURES OF THE MERCATOR PROJECTION IN WorLp Mars.—Granting that on the Mercator projection, distances and areas appear to be distorted relatively 146 : WORLD MAPS. 147 when sections of the map differing in latitude are compared, an intelligent use of the marginal scale will determine these quantities with sufficient exactness for any given section. In many other projections the scale is not the same in all directions, the scale of a point depending upon the azimuth of a line. As proof of the impossibilities of a Mercator projection in world maps, the critics invariably cite the exaggeration of Greenland and the polar regions. In the considera- tion of the various evils of world maps, the polar regions are, after all, the best places to put the maximum distortion. Generally, our interests are centered between 65° north and 55° south latitude, and it is in this belt that other projections present difficulties in spherical relations which in many instances are not readily expressed in analytic terms. Beyond these limits a circumpolar chart like the one issued by the Hydrographic Office, U. S. Navy, No. 2560, may be employed. Polar charts can be drawn on the gnomonic projection, the point of contact between plane and sphere being at the pole. In practice, however, they are generally drawn, not as true gnomonic projections, but as polar equidistant projections, the meridians radiating as straight lines from the pole, the parallels struck as concentric circles from the pole, with all degrees of latitude of equal length at all parts of the chart. However, for the general purposes of a circumpolar chart from latitude 60° to the pole, the polar stereographic projection or the Lambert conformal with two standard parallels would be preferable, In the latter projectton the 360 degrees of longitude would not be mapped within a circle, but on a sector greater than a semicircle. Nore.—The Mercator projection has been employed in the construction of a hydrographic map of the world in 24 sheets, published under the direction of the Prince of Monaco under the title “Carte Bathymétrique des Océans.” Under the provisions of the Seventh International Geographic Congress held at Berlin in 1899, and by recommendation of the committee in charge of the charting of suboceanic relief, assembled at Wiesbaden in 1903, the project of Prof. Thoulet was adopted. Thanks to the generous initiative of Prince Albert, the charts have obtained considerable success, and some of the sheets of a second edition have been issued with the addition of continental relief. The sheets measure 1 meter in length and 60 centimetersin height. The series is constructed on 1:10 000 000 equatorial scale, embracing 16 sheets up to latitude 72°. Beyond this latitude, the gnomonic projection is employed for mapping the polar regions in four quadrants each. The Mercator projection embodies all the properties of conformality, which implies true shape in restricted localities, and the crossing of all meridians and parallels at right angles, the same as on the globe. The cardinal directions, north and south, east and west, always point the same way and remain parallel to the borders of the chart. For many purposes, meteorological charts, for instance, this property is of great importance. Charts having correct areas with cardinal directions running every possible way are undesirable. While other projections.may contribute their portion in special properties from an educational standpoint, they cannot entirely displace the Mercator projection which has stood the test for over three and a half centuries. It is the opinion of the authors that the Mercator projection, not only i is a fixture for nautical charts, but that it plays a definite part in giving us a continuous conformal mapping of the world. re, THE STEREOGRAPHIC PROJECTION. The most widely known of all map projections are the Mecator projection already described, and the stereographic projection, which dates back to ancient Greece, having been used by Hipparchus (160-125 B. C.). The stereographic projection is one in which the eye is supposed to be placed at the surface of the sphere and in the hemisphere opposite to that which it is desired to project. The exact position of the eye is at the extremity of the diameter passing through the point assumed as the center of the map. 148 U. S. COAST AND GEODETIC SURVEY It is the only azimuthal projection which has no angular distortion and in which every circle is projected as a circle, It is a conformal projection and the most familiar form in which we see it, is in the stereographic meridional as-employed. to represent the Eastern and Western Hemispheres. In the stereographic . meridi- onal projection the center is located: on the Equator; in the stereographic: horizon projection the center is located on any selected parallel, © © 1 _Another method of projection more frequently employed by geographers for representing hemispheres is the globular projection, in which the Equator and central meridian are straight lines divided into equal ‘party’ and the other meridians are Pile fot ge wo ae Fig. 68.—Stereographic meridional proj ection. circular arcs uniting the equal divisions of the Equator with the wiflaay the parallels, except the Equator, are likewise circular arcs, dividing the extreme and central meridians into equal parts. In the globular representation, nothing is correct except the graduation of the outer circle, and the direction and graduation of the two diameters; distances and directions can neither be measured nor plotted. It is not a projection defined for the preservation of special properties, for it does not correspond with the surface of the sphere according to any law of cartographic interest, but is simply an arbitrary distribution of curves conveniently constructed. WORLD MAPS, 149 The. two projections, stereographic and globular, are noticeably different when seen side by side, .In the stereographic. projection ‘the meridians intersect. the parallels at right angles, as on the globe, and the projection is better adapted to the plotting and measurement of all kinds of relations * pertaining to the sphere than any other projection. Its use in the conformal representation of a hemisphere is not fully appreciated. ” In the ‘stereographic projection of a hemisphere we have the principle of Tchebicheff, namely, that a map constructed on a conformal projection is the best possible when the scale is constant along the whole boundary.. This, or an approxi- 272. #60 v0 she, so alo, abo, ao, F ( se, \. 7P ' Fele Word 4] 90, a 00 Bio @” a PARIS go jo lo ) + Le 50 a ‘ pe q : e tly a) a le eye iF ecm é ye & (Go a) so ‘Jo ; Slo ON $ Pal, o 2 & $e 45 #e: 70 Fie. 69.—Stereographic horizon projection on the horizon of Paris, mation thereto, seems to be the most, satisfactory solution that has been suggested in the problem of conformal mapping of a hemisphere. The solution of various problems, including the measurement of angles,’ direc- tions, and distances on this projection, is given in U. S. Coast and Geodetic Survey Special Publication. No. 57. _The mathematical theory of the projection, the con- 34 Aninteresting paper on this a spa in the earlapeats Journal of Science, Vol. XI, February, 1901, _ The ade graphic Projection and its Possibilities from a Geographic Standpoint, by 8. L. Penfield. The application of this projection to the solution of spherical problems is given in Notes on Stereographic Projection, by Prof. W. W. Henderickson, U. 8. N., Annapolis, U. 8. Naval Institute, 1905. ’ A practical use of the stereopraphic projection isillustrated in the Star Finder recently devised by G. T. Rude,. hydrographic and geodetic engineer, U. S. Coast and Geodetic Survey. 150 U. S. COAST AND GEODETIC SURVEY. struction of the stereographic meridional and stereographic horizon projection, and tables for the construction of a meridional projection are also given in the same publication. THE AITOFF EQUAL-AREA PROJECTION OF THE SPHERE. (See Plate V and fig. 70.) The projection consists of a Lambert azimuthal hemisphere converted into a full sphere by a manipulation suggested by Aitoff.% It is similar to Mollweide’s equal-area projection in that the sphere is repre- sented within an ellipse with the major axis twice the minor axis; but, since the parallels are curved lines, the distortion in the polar regions is less in evidence. The representation of the shapes of countries far east and west of the central meridian is not so distorted, because meridians and parallels are not so oblique to one another. The network of meridians and parallels is obtained by the orthogonal or perpen- dicular projection of a Lambert meridional equal-area hemisphere upon a plane making an angle of 60° to the plane of the original. The fact that it is an equivalent, or equal-area, projection, combined with the fact that it shows the world in one connected whole, makes it useful in atlases on physical geography or for statistical and distribution purposes. It is also employed for the plotting of the stars in astronomical work where the celestial sphere may be represented in one continuous map which will show at a glance the relative dis- tribution of the stars in the different regions of the expanse of the heavens. OBSERVATIONS ON ELLIPSOIDAL PROJECTIONS.—Some criticism is made of ellip- soidal projections, as indeed, of all maps showing the entire world in one connected whole. It is said that erroneous impressions are created in the popular mind either in obtaining accuracy of area at the loss of form, or the loss of form for the purpose of preserving some other property; that while these are not errors in intent, they are errors in effect. It is true that shapes become badly distorted in the far-off quadrants of an Aitoff projection, but the continental masses of special interest can frequently be mapped in the center where the projection is at its best. It is true that the artistic and mathe- matically trained eye will not tolerate ‘the world pictured from a comic mirror,” as stated in an interesting criticism; but, under certain conditions where certain properties are desired, these projections, after all, play an important part. The mathematical and theoretically elegant property of conformality is not of sufficient advantage to outweigh the useful property of equal area if the latter prop- erty is sought, and, if we remove the restriction for shape of small areas as applying to conformal projections, the general shape is often better preserved in projections that are not conformal. The need of critical consideration of the system of projection to be employed in any given mapping problem applies particulary to the equal-area mapping of the entire sphere, which subject is again considered in the following chapters. A base map without shoreline, size 11 by 224 inches, on the Aitoff equal-area projection of the sphere, is published by the U. S. Coast and Geodetic Survey, the radius of the projected sphere being 1 decimeter. Tables for the construction of this projection directly from z and y coordinates follow. These coordinates were obtained from the Lambert meridional projection by doubling the 2’s of half the longitudes, the y’s of half the longitudes remaining unchanged. # Also written, D. Aitow. A detailed account of this projection is given in Petermanns Mitteilungen, 1892, vol. 38, pp. 85-87. ze SCO x AIG: an el Se “3 0 Tae et i / ISIC. LL, / a aaa ‘naw Et at Ws Af [fff LP Ta GMLE. Gea RRP Xk ine st x 7 PELL. [Pe RE SEIS ESE EAS 009 SpnyyeT n v'EBOL | SPSOL | G°LZ0T | TSOOT | 6°086 | 2°006 | F'2r6 | 0°96 | EITG | £°868 | 8'988 | 8 ‘928 | 8°898 | ‘198 | F'Sc8 | 6'OS8 | 8'24b8 | B-GHs | Z'CFs 4 = TSI8T | S°VOLT | S691 | LOZOT | 8°OPST | S*TOFT | 2 °69ET | O'ELZT | OTATL | O'990L | 9°9S6 | O'FFB 9 “8tL | 8'OI9 | O'T6h | 9°69E | O'LH | L°EZT 1 0'0 z 009 OpNyyeyT o 0°606 T0838 | O'PS8 | F088 | Z°608 | T°06L | O'GLL | L°LGL | LOL | T°ZEL | OTOL | 9°OTL 8'FOL | F “869 | 2°69 | 2°689 | £°989 | 9°F89 | 0°F89 a A L°991Z@ | 8°680G | 9°FO0S | GTIGL | F°SISL | S°9OLT | O°SECT | FEEL | L°SGET | G'EEZT | 9 “SOIT S46 | 0°08 | GEOL | T°G9S | T°Seh | Lee | Zehr | 0'0 Eb (| SOR Ea Sara oP OPN448T, * T°LOL B°189 | €°6c9 | 1°69 | T'IZ9 | 0'SO9 | 2°069 | O'8zg | 2°99¢ | G'9Se | & "8h 8 ‘Ovo | G'PES | E°6cS | O'SeG | BIZ | G6Ig | E"sTs | 9-2I¢ Ah p 26h. | O°EGES | FEF G'SETS | L4°1Z0S | F'O6ST | L'LZT | LOFT | O'FOST | O'FOET | 8OVZT | O'FLOT | 8°S26 | B'PLL | T'229 1 8°49F | G'ZTe | F'9ST | 0'0- ah PE a paige ops naan o0€ OpNyyeyT L*€8h 0'Sop |G “SbF 8'eeh | 80ch | Z°60r | O'E6E | B'6se | 618s | 6'PLe | 8 "89 9 "E90 feces | Goce | GaSe | Z°OSe | 9°8FE | O°ZPE | E-LFE fA 6°LS9% | 9°SHGS | B'9ZFS | G'ZOES | L°SLTS | 6°LE0S | 9°SBST | O'SSLT | 9°LOST | LOT | L°COET | O'OPIT | L°986 | Ges | G°299 | T’s6h | O'zee | G°99T | 0'0 >) NBER RISES Meee cE a 002 OpnjyyeyT 9°SK L°S% | 6°9Z | Z°6IZ | F'ZIs | ¥'90Z | O'1OZ | S°96T | Z'26L | 9°S8T | F ‘a8 4 ‘I F'O8T | S'BLT | O°LLT | 8°SZT | OGLE | SPAT | O°PAT A S'G81Z | B°E99% | 6'HESS | 8 LOPS | 8°E9zZ | E'EZIST | 9'PLOL | O'FST | S°GOOT | S'SIST | 8 TSEL | O'SSIT | Z EZOl | 2°gc8 | O'989 | TOTS | 9°FFE | S'ZZT | 0'O Z [sseeree SSIpSiseitincismas Ol Opnyiney 0°0 0°0 00 0°0 0°0 0°0 0°0 00 0°0 00 0-0 0°0 0°0 0'0 0'0 00 00 0°0 0-0 4 F8E8Z | H'ZOLS | SILGS | O'GEFS | EFS | ZEFIZS | 0'000Z | O'LFST | F°OGOT | 2 °OSIT | T89EL | 8°ZOCT | ESeOL | L°s98 | 9°F69 | Leo | O'BPE | SPLAT 00 D> PRESEN rir eapae cise ~sogenbay o08T oOLT o09T 00ST oOhT 00ST 00ST oOIT o00T 006 008 004 J 009 00S o0P - 008 00 o0L eOt “ier PESTER “"*"° 9pngisue'y a (‘sxjourqTTONDep Uy seyeuTp10d0 Ie~MBuejooy ‘*ioyoutpSp T sperBe orsyds pogoofoad jo snqpeay) : = "aUHHdS AHL JO NOLLOGLOUd VAUV-1IVO0R AAOLIV NV JO NOILONUYLSNOO GAHL YOd ATAVL WORLD MAPS, 153 Thus, in the Lambert ‘meridional: Peer the coordinates at latitude 20°, longrude 20°, are z=0,33123 dectih ted! or' 331.23 doethillinlsters: ' y=0.35248 decimeter, or 352.48 decimillimeters. For die Aitoff projection, the coordinates at latitude 20°, longitude 40°, will be 2=2X 331.23 = 662.5 decimillimeters. y= 352. 5 decimillimeters. The coordinates for a Lambert equal-area meridional projection are given on page 75. THE MOLLWEIDE HOMALOGRAPHIC PROJECTION, This ranean is also known as Babinet’ s equal-surface projection. and its distinctive character is, as its name implies, a proportionality of areas on the sphere with the corresponding areas of the projection. The Equator is developed into a straight line and graduated equally from 0° to 180° either way from the central meridian, which is perpendicular to it-and of half: the length of the representative line of the Equator. The parallels of latitude are all straight lines, on each of which the degrees of longitude are equally spaced, but do not bear their true proportion in length to those on the sphere. Their distances from the Equator are determined by the law ‘of equal surfaces, and their values in the table have been tabulated between the limits 0 at the Equator and 1 for the pole. ganas 71.—The arene ee Projection, of ‘She sphere. . The meridian of 90° on sities side of the central meridian appears in the projection as a circle, and by intersection determines the length of 90° from the central meridian on all the parallels; ‘the other meridians are parts of elliptical arcs. Extending the projection to embrace the whole surface of the sphere, the bounding line of the projection becomes an ellipse; the area of the circle included by the meridians of 90° equals that of the hemisphere, and the crescent-shaped areas lying outside of this circle between longitudes + 90° and + 180° are together equal to that of the circle; also the area of the projection between parallels +30° is equal to the same. 154 U. S. COAST AND GEODETIC SURVEY, In the ellipse outside of the circle, the meridional lengths become exaggerated and infinitely small surfaces on the sphere and the projection are dissimilar in form. The distortion in shape or lack of conformality in the equatorial belt and polar regions is the chief defect of this projection. The length which represents 10 de- grees of latitude from the Equator exceeds by about 25 per cent the length along the Equator. In the polar regions it does not matter so much if distortions become excessive in the bounding circle beyond 80 degrees of latitude. The chief use of the Mollweide homalographic projection is for geographical illustrations relating to area, such as the distribution and density of population or the extent of forests, and the like. It thus serves somewhat the same purpose as the Aitoff projection already described. ; The mathematical description and theory of the projection are given in Lehrbuch der Landkartenprojectionen by Dr. Norbert Herz, 1885, pages 161 to 165; and Craig (Thomas), Treatise on Projections, U. S. Coast and Geodetic Survey, 1882, pages 227 to 228. CONSTRUCTION OF THE MOLLWEIDE HOMALOGRAPHIC PROJECTION OF A HEMISPHERE. Having drawn two construction lines perpendicular to each other, lay off north and south from the central point on the central meridian the lengths, sin #, which are given in the third column of the tables * and which may be considered as y coor- Fic. 72.—The Mollweide homalographic projection of a hemisphere. dinates, these lengths being in terms of the radius as unity. The points so obtained will be the points of intersection of each parallel of latitude with the central meridian. With a compass set to the length of the radius and passing through the upper and lower divisions on the central meridian, construct a circle, and this will represent the outer meridian of a hemisphere. Through the points of intersection on the central meridian previously obtained, draw lines parallel to the Equator and they will represent the other parallels of latitude. % These tables were computed by Jules Bourdin. WORLD MAPS. 155 For the construction of the meridians, it is only necessary to divide the Equator and parallels into the necessary number of equal parts which correspond to the unit of subdivision adopted for the chart. . HOMALOGRAPHIC PROJECTION OF THE SPHERE. _ In tho construction of a projection including the entire sphere (fig. 71), we proceed as before, excepting that the parallels are extended to the limiting ellipse, and their lengths may be obtained by doubling the lengths of the parallels of the hemisphere, or by the use of the second column of the tables under the values for cos 8, in which cos @ represents the total distance out along a given parallel from the central to the outer meridian of the hemisphere, or 90 degrees of longitude. In the projection of a sphere these distances will be doubled on each side of the central meridian, and the Equator becomes the major axis of an ellipse. Equal divisions of the parallels corresponding to the unit of subdivision adopted for the chart will determine points of intersection of the ellipses representing the meridians. TABLE FOR THE CONSTRUCTION OF THE MOLLWEIDE HOMALOGRAPHIC PROJECTION. [x sin e=2 6+sin 20.] Latitude : . Difference Latitude - Difference 2 cos @ sin @ sin @ © cos 6 sin @ sin @ ° , ° , 0 00 10000000 | 0.000000 22 30 0, 9522324 | 0,30537300 0 30 0,9999767 | 0. 00685431 peer 23 00 0.9500758 | 0.31201940 Saeed 1 00 09999060 | | 0,01370813 et 23 30 0.9478704 | 0.31865560 Sante) 1 30 0.999788 | 0.02056114 orn 24 00 09456170 | 0.32528210 sree 2 00 0.9996240 | 0.02741423 te 24 30 0,9433152 | 0,33189860 cree 2 30 09904127 | 0, 03428622 aennde 25 00 0,9409646 | 0. 33850520 saiey 3 00 0,9991542 | 0.04111710 ease 25 30 0, 9385654 | 0.34510150 pee 3 30 09988489 | 0. 04796660 eens 28 00 0.936174 | 0.35168730 a 4 00 0,.9984987 | 0: 05481465 ake 26 30 0.936210 | 0, 35826250 ey 4 30 0,9980970 | 0, 06168115 27 00 0,9310754 | 0. 36482680 non 5 00 0.997 : 27 30 0.9284809 | 0.371380 5 30 0.971572 | 0.07534880 ee 28 0,9258374 | 0,37792200 Her 6 00 0.9966169 | 0.08218950 883830 28 30 0.9231446 |. 0,38445240 oy 8 30 0,9960289 j 0, 08902780 crane 29 00 0, 9204030 | 0.39097120 es 7 00 0.9953042 | 0, 09586340 oo 29 30 0.917619 | 0. 39747840 rides! 7 30 0.9947127 | 0, 10269610 622070 30 00 0.914708 | 0, 40307380 sn 8 00 0,9930839 | 0, 10952580 ane 30 30 0. 9118800 | 0, 41045670 en 8 30 0. 0,11635235 teases 31 00 09080400 | 0, 41692680 oy 9 00 0,9923847 | 0.12317565 aes 31 30 0..9059504 | 0. 42338400 pan 9 30 0.9915144 | 0,12099545 peeee 32 00 0.9020108 | 0, 42982800 en 10 00 0.905070 | 0, 13681155 aieiie 32 30 0. 8998216 | 0, 43625840 nee 10 30 0.989632 | 0, 14362350 eae 33 00 0, 8966820 | 0, 44267510 aeier Ho 0.9886204 | 0. 15043095 caus 33 30 0, 8934924 | 0, 44907810 ee 30 0.9875614 | 0.15723380 pert 34 00 0, 8902524 | 0, 45548720 orn 12 00 0,.9884550 | 0. 16403190 Genny 34 30 0. 8869620 | 0, 46184240 pees 12 30 09853012 | 0,17082520 ried 35 00 0. 8836206 | 0, 48820350 she 13 00 0,9841004 | 0.1761365 Heer 35 30 0, 8802282 | 0, 47455020 en 13 30 g.9sz8517 | 0. 18499710 ereeoe 38 00 0. 8787850 | 0. 43088240 ee 14 00 09815556 | 0.1911 ee 8732008 | 0, 48719920 on 14 30 0,.9802124 | 0.19794810 See 37 00 0..8897454 | 0, 49350080 agoleg 15 00 09788217 | 0.20471500 676000 37 30 |. 0.8661484 | 0. 49978670 how 15 30 | 0.977380 | 0.21147500 oreseo 38 00 0. 8825002 | 0. 50806870 con 18 30 Q.origns7 | 0. 22407845 are 39 00 o.sssose2 | 051864860 e260 18 30 0, 9743837 . Gralis : 518548 oan 17 00 0.9727827 | 0.23171960 (esse 39 30 0,8512442 | 0.52476980 | 622130 17 30 0.971537 | 0,23845390 40. 00 0, 8473879 | 0, 53097420 18- 00 0,9694770 | 0.24518120 ane 40 30 0. 8434792 | 0.53716160 oan 18 30 0,9677529 | 0. 25190120 noe 41 00 | 08395179 | 0.54333170 ee 19 00 0, 9859809 | 0. 25861370 ee 41 30 0. 8355020 | 0, 54948450 OH 19 30 0.9641609 | 0.26531840 Ae 42 00 | . 0.8314364 | 0.55561960 euacig 20 00 0.9622929 | 0.27201520 sie 42 30 0.8273120 | 0, 56173680 hse 20 30 0.960370 | 0.27870400 43 00 0, 8231420 | 0, 58783530 21 00 | 0.9584130 | 0. 28538430 ee 43 30 08189142 | 0. 57391550 pie 21 30 09564009 | 0, 29205610 ae 44 00 0,8146326 | 0.579971 ae 22 00 0,.9543409 | 0, 29871950 ant 44 30 0, 8102968 | 0, 58602010 oe 22 30 0.9522324 | 0.30537390 45 00 0. 8059068 | 0.59204370 156 U. S. COAST AND GEODETIC SURVEY, TABLE. FOR THE CONSTRUCTION OF THE MOLLWEIDE HOMALOGRAPHIO PROJECTION— continued. [+ sin p=26+sin 26.] Latitude ae Difference Latitude ‘ Difference 2 cos 0 sin 6 sing ¢ cos 8 sin @ sin 6 o of o oF 4 00 0.9059058 | 0.59204370 67 30 0.5451794 | 0,83831940 8 30 0.8014604 | 0.59804760 ro 68 00 0.537379 | 0.84311240 arson 46 00 0. 7969604 0. 60403170 596360 68 30 0. 5302071 0. 84786820 ibe 48 30 0.792409 | 0.60999530 ae 69 00 0.5225861 | 0.85258660 a71840 47 00 07877940 | 0.61593870 ae 69 30 05148715 | 0.85726740 468080 47 30 07331270 | 0.62186190 7@ 00 0.5070603 | 0.86191080 48 00 0.784035 | 0.62776410 ae 70 30 0.4991511 | 0.86651480 uns 48 30 0,7736235 | 0.63364540 er 71 00 0.491423 | 0,87107920 oe 49 00 0.7687865 | 0.63950560 ee 71 30 0.4830314 | 0.87560300 ae 49 30 0.768025 | 0.64534360 an 72 00 0.4748167 | 0,88008460 ee 50 00 0.7580409 | 0.65115960 _. 72 30 04664042 | 0.88452400 50 30 0.7530317 | 0.65695270 pee 33 00 «| —0.4580813 | 0. 88892040 Be 51 00 07488643 | 0.66272350 neue | 3 30 0.4495146 | 0.89327300 ora 51 30 0.7437375 | 0.66847200 en 74 00 0.440851} 0.89758020 oe 52.00 07385513 | 0.67419710 an 74 30 0. 4320659 "90184180 aes 52 30 0.733054 | 0.67989910 75 00 0.421614 | 090605620 53 00 | 07279995 | 0.68557740 a 75 30 0.414156, | 0.910240 | 416800 53. 30 07226332 | 0.69123180 pene 76 00 0.4049354 | 0.91434520 | | 412100 54 00 0.7172058 | 0.69686130 pore 76 30 0, 3956158 | 0.91841600 407080 54 30 0.717175 | 0. 70246580 eee 77 00 03861534 | 0.92243460 401880 55 00 07081678 | 0. 70804460 77 30 0.3765409 | 0.92640010 55 30 0.700550 |. 0. 7135830 ery 7800 0.366705 | 0.93031150 zero 56 00 06948790 | 0.71912650 aes 78 30 0, 3568322 | 0.93416860 Sea 56 30 0.681300 | 0.72462920 oe 79 00 0,3467148 | 0.98797000 ate 57 00 0, 6833342 | 0. 73010570 pee 79 30 03364137 | 0.941 71410 ae 57 30 06774641 | 0.73555570 80 00 0.3259234 | 0.94539600 58 00 0.6715285 | 0.74097870 Bote 80 30 0.3152285 | 0.94901590 ae 58 30 06855270 | 0.74637350 pee 81 00 0.3043189 | 0.95257020 oe 59 00 06594590 | 0.75174020 ee 81 30 0.2921755 | 0.95605840 io 59 30 0.653232 | 0.75707900 ree 82 00 0.281763 | 0,95948020 a 60 00 0.647191 | 0. 76238870 82 30 0,2701079 | 0-96288000 60 30 0.6408456 | 0. 76766950 ey 83 00 0,2581516.. | 0.96610470 | 327470 61 00 0.634509 | 0.77292120 ete 83 30 0.245837 - | 0.96929940 ee 61 30 06280869 | 0.77814310 Pai 84 00 02332737, | 0.97241090 ae 62 00 0.626001 | 0.78333450 21607 84 30 0.202700 | 0.97543890 read 62 30 06150407 | 0.788495 85 00 0,2068365 | 0.97837520 ; 63° 00 0.608076 | 0. 79362470 Piece 83° 30 0.1929149, | 0.98121520 Pe, 63 30 0.6016988 | 0.79872290 ienie 86 00 0.1784407 | 0.98395070 arise 64 00 0,5049143 | 0.80378900 50400 88 30 0.1633412 | 0.98656970 | 61900 64 30 0.5880519 | 0,80882300 Hearts 87 00 0.1474833° | 0.98906470 ae 65 00 0.581107 | 0,81382420 408880 87 30 0.1306860 | 0.99142650 — 65 30 0.5740894 | 0.81879250 ein 88 00 0.1126372 | 0.99363620 ee 66 00 0.566987 | 0.82372660 489940 88 30 0.0929962 | 0.99566640 on 68 30 0.5598024 | 0,82862600 486440 89 00 0.0710530 | 0.99747270 ee 67 00 0.525330 | 0.88349040 cnn 89 30 0.0447615 | 0.99809770 | — jeaeay 87 30 05451704 | 0,83831940 90 00 00000000 | 1.000000 GOODE’S HOMALOGRAPHIC PROJECTION (INTERRUPTED) FOR THE CONTINENTS AND OCEANS. [See Plate VI and fig. 73.) : Through the kind permission of Prof. J. Paul Goode, Ph. D., we are able to include in this paper a projection of the world devised by him and copyrighted by the University of Chicago. It is an adaptation of the homalographic projection and is illustrated by Plate VI and by figure 73, the former study showing the world on the homalographic projection (interrupted) for the continents, the latter being the same projection interrupted for ocean units. The homalographic projection (see fig. 71) which provides the base for the new modification was invented by Prof. Mollweide, of Halle, in 1805, and'is an equal- area representation of the entire surface of the earth within an ellipse of which the ratio of major axis to minor axis is 2:1. The first consideration is the construction of .an equal-area hemisphere (see fig. 72) within the limits of a circle, and in this pro- WORLD MAPS.) L57 jection the radius of the circle is taken as the square root of 2, the radius of the sphere being unity. The Equator and mid-meridian are straight lines at: right angles to each ‘other, and are diameters of the:map, the parallels being projected in right lines parallel to the Equator, and the meridians in ae all of whieh pass through two fixed points, the poles. “Yat ache gb geve Gag beg hy aod & BG te oe ed Fie. 73.—Homalographic projection (interrupted) for ocean units. 158 U. S. COAST AND GEODETIC SURVEY. In view of the above-mentioned properties, the Mollweide projection of the hemisphere offers advantages for studies in comparative latitudes, but shapes become badly distorted when the projection is extended to the whole sphere and becomes ellipsoidal. (See fig. 71.) In Prof. Goode’s adaptation each continent is placed in the middle of a quad- rillage centered on a mid-meridian in order to secure for it the best form. Thus North America is best presented in the ‘meridian 100° west, while Eurasia is well taken care of in the choice of 60° east; the other continents are balanced as follows: South America, 60° west; Africa, 20° east; and Australia, 150° east. Besides the advantage of equal area, each continent and ocean is thus balanced on its own axis of strength, and world relations are, in a way, better shown than one may see them on a globe, since they are all seen at one glance on a flat surface. In the ocean units a middle longitude of each ocean is chosen for the mid-meridian of the lobe. Thus the North Atlantic is balanced on 30° west, and the South Atlantic on 20° west; the North Pacific on 170° west, and the South Pacific on 140° west; the Indian Ocean, northern lobe on 60° east, and southern lobe 90° east. We have, then, in one setting the continents in true relative size, while in another setting the oceans occupy the center of interest. i The various uses to which this map may be put for statistical data, distribution diagrams, etc., are quite evident. Section -3 (the eastern section) of figure 73, if extended slightly in longitude and published separately, suggests possibilities for graphical illustration of long-distance sailing routes, such as New York to Buenos Aires with such intermediate points as may be desired. While these could not serve for nautical charts—a province that belongs to the Mercator projection—they would be better in form to be looked at and would be interesting from an educational standpoint. Asa study i in world maps on an equal-area representation, this projection is a noteworthy contribution to economic geography and modern: cartography. LAMBERT PROJECTION OF THE NORTHERN AND SOUTHERN HEMISPHERES. [See Plate VII.] : This projection was suggested by Commander A. B. Clements of the U. S. Shipping Board and first constructed by the U. S. Coast and Geodetic Survey. It is a conformal conic projection with two standard parallels and provides for a repetition of each hemisphere, of which the bounding circle is the Equator. The condition that the parallel of latitude 10° be held as one of the standards combined with the condition that the hemispheres be repeated, fixes the other standard parallel at 48° 40’. The point of tangency of the two hemispheres can be placed at will, and the repetition of the. ‘hemispheres provides ample room for continuous sailing routes between any two continents in either hemisphere. A map of the world has been prepared for the U. S. Shipping Board on this system, scale 1:20000 000, the diameter of a hemisphere being 54 inches. By a gearing device the hemispheres may be revolved so that a sailing route or line of commercial interest will pass through the point of contact and will appear as a con- tinuous line on the projection. Tables for the construction of this projection are given on page 86. The scale factor is given in the last column of the tables and may be used if greater accu- racy in distances is desired. In order to correct distances measured by the graphic 159 sereqds eq} jo woryeloid otpotied ATqnop s,noAny—p/, ‘oLy, OA @ |@ $4 oF oc o WORLD MAPS. ol oF 0% a4 160 U. S. COAST. AND GEODETIC SURVEY. scale of the map, divide them by the scale factor, Corrections to area may be applied in accordance with the footnote on page 81. | With two of the parallels true to scale, and with scale variant in other parts of the map, care should be exercised in applying corrections. Tn spite of the great extent peed = this system: of projection, the property of form, with a comparatively small change of scale, i is retained, and a scale factor for the measurement of certain spherical relations is available. CONFORMAL PROJECTION OF THE SPHERE WITHIN A TWO-CUSPED EPICYCLOID. [See Plate VIII J = The shape of the sphere when developed. on a polyconic projection (see fig. ‘A7) suggested the development of a conformal projection within the area inclosed by a two-cusped epicycloid. The distortions i in this case appear in the distant quadrants, or regions, of lesser importance. Notwithstanding the appearance of similarity in the bounding meridians of the polyconic and the conformal development, the two projections are strikingly different and present an interesting study, the polyconic projection, however, serving no purpose in the mapping of the entire sphere. For the above system of conformal representation we are indebted to Dr. F. August and Dr. G. Bellermann. The mathematical development appears in Zeit- schrift der Gesellschaft fiir Erdkunde zu Berlin, 1874, volume 9, part 1, No. 49, pages 1 to 22. GUYOU’S DOUBLY PERIODIC PROJECTION OF THE SPHERE. [See fig. 74.] In Annales Hydrographiques, second series, volume 9, pages 16-35, Paris, 1887, we have a description of an interesting projection of the entire sphere by Lieut. E. Guyou. It is a conformal projection which provides for the repetition of the world in both directions—east or west, north or south, whence the name doubly periodic. The necessary deformations are, in this projection, placed in the oceans in a more successful manner than in some other representations. The accompanying illustration shows the Eastern and Western Hemispheres without the duplicature noted above. The above projection is the last one in this brief review of world-map projec- tions. In the representation of moderate areas no great difficulties are encountered, but any attempt to map the world in one continuous sheet presents difficulties that are insurmountable. Two interesting projections for conformal mapping of the world are not included in this review as they have already been discussed in United States Coast and Geodetic Survey Special Publication No. 57, pages 111 to 114. Both of these are by Lagrange, one being a double circular projection in which Paris is selected as center of least alteration with variation as slow as possible from that point; the other shows the earth’s surface within a circle with the center on the Equator, the variations being most con- spicuous in the polar regions. For conformal mapping of the world the Mercator projection, for many pur-- poses, is as good as any, in that it gives a definite measure of its faults in the border scale; for equal-area mapping, Prof. Goode’s interrupted homalographic projection accomplishes a great deal toward the solution of a most difficult problem. INDEX. Page. Aitoff equal-area projection. ............. 150-153 ADIGE.) ij dkso0 oc aa anesiieatns aenuennee 152 Albers oH. Ge seccncce aacaeaqkeeve'v ddaegass 91 Albers projection.....................0202- 55, 56 Comparison with Lambert conformal.... 92- 93, 116 Construction...........--.....2.22008- 99 Description.....-.-.....2-......220500- 91-93 Mathematical theory.......... sional 93-99 Table for United States................ 100 Anti-Gudermannian......................-- 114 August; Dr Poo cs sosacdeanss wavaaeuisuce 160 Authalic latitude......................- 54, 56, 99 | Azimuthal projection: Gnomonic meridional..............-.-- 48, 45 Gunomonic (or central)..........-......- 37, 43, 45, 52, 101-108, 140-145 Lambert meridional equal-area.. 43, 73,74, 116 Lambert polar equal-area.............-- 38 Lambert zenithal equal-area.... 56, 71-74, 116 Orthographic meridional........- acme 43, 51 Orthographic polar.......-....--- eieies 38 Polar equidistant... nnn es Polar gnomonic............-..----6-- 37, 147 Stereographic horizon... ......-. 116, 147-149 Stereographic meridional...........-..- 42, 44, 51, 116, 147-149 Stereographic polar......-......------. 35, 147 Behrmann, Dr. W.............2-------- 2 93 Bonne projection.......--..---2-+-++ 49, 60, 81, 116 Construction..... donk! aCe e es ees eas 68-70 Description...........222222222-2eeeeee 67-68 Bourdin, J., tables. ......-----.---2-20220- 154 Bowditch, American Practical Navigator... 112 British Admiralty: Fixing positions by wireless............. 137 Gnomonic charts...-....--2--2-40---;-. 141 Manual of navigation. .......-........- 145 Central projection (see also Gnomonic).....- 37 Gnomonic meridional.....-............ 48, 45 Gnomonic polar....--..---------------- 37 Tangent cone..-...-------------------- 47 Tangent cylinder................------ 30 Choice of projection........-.--.--------.-- 54 Clements, A. B........---.--0----- eee eee ee 158 Collins: Bi Beaisssicicieis oe ee stsie se -iscteieis’s oiniste 137 Conformal mapping. ....-...-.+--.---- 26, 54, 116 Conformal projection: Lambert. See Lambert. Mercator. See Mercator. Sphere. .......2.----- cece cee c cece ee eee 160 Stereographic horizon......-.-..- 116, 147-149 22864°—21——11 Page. Conformal projection—Continued. Stereographic meridional............ . 42,44, 51, 116, 147-149 Stereographic polar......./...---.---- 35, 147 Conformallity....cinsc.ccse eee sos Oey sacle 78 Conical projections (see also Albers and Lam- Wet coscondeiecunparedacuasnanieniemeake - 6 Craig, Thomas.........-..-.20.---20--eeee- 78, 154 Cylindrical conformal projection (see also Mercator) sc.uisi. ss aneusceseweewercen 32, 51, 56 Cylindrical equal-area projection......... 30, 56, 93 Cylindrical equal-spaced projection........-. 30 Deformation tables..................0--00-- 54, 56 Earth: 2 Polyhedron.......--.-.-.- ereeee eer < 28 Reference lines..........22..2.- dacattess FELT + SWAPS seg socicte dieu widielie Sasa oaaeciaeeeesiee 9 Truncated cone. ..-..-.--------------- 28 Ellipsoidal projections.............-....-..- 150 Equal-area mapping..-.-.......-.-...---.- 54, 116 Equal-area projections...-.................. 24, 26 ME OH otans cp staciaigheannbe ins Posegac: SOIOS Ail iris wisciieaticte» axe canes 55, 66, 91-100, 116 Be eet tee detsh Ate tas 49, 60, 67-70, 81, 116 Cylindrical.........-.. 2 racine ome 30, 56, 93 Goode’s homalographic........ 57, 156-158, 160 Lambert meridional azimuthal...... 43,73, 116 Lambert polar azimuthal..............-. 38 Lambert zenithal (or azimuthal) projec- HOD sds waves RDS ands Wa aia 56, 71-74, 116 Moll weide. j2..2cccce% osc censenss 24, 153-156 Sanson-Flamsteed........-.--..------- 68 Equal-spaced projection, cylindrical........ 30 Equidistant polar projection........-.-..-. 40, 147 Errors of projections. .-...- . 54-56, 59, 80-82, 91, 146 Compensation....... Cali usueneeawien ... 81-82 Fowler; G: Hojesseosevasseecs sstcceseeecene 140 Frischaul, Dre. isc: sees cemnnaeendccis 65 GaMeSscyacseotcceesattneneceseesadoasteeece 54,57 Geological Survey, tables................... 62 Geometrical azimuthal projections.......... 35 Germain, A., tables (Mercator)............ 116-136 Globe: Plotting points ................2------- 14 MOR OSTA § ce iwicrewmcec ise pesaw eeu 19 Globular projection.................- 43, 44, 51, 148 Gnomonic (or central) projection .......-- 101-103 Description santaec0's 3 se's eetoseacees 140-141 Mathematical theory..........---.---- 140-145 Meridional............-..-.-.--+----+--- 48, 45 POP i ccvtesmmceewees nest neaome ae wnt 37 Tangent cube.....-.--------------205- 52 162 INDEX. Page. Page. Goode; Prof. Ji Picescssevswenvscewss - 57,160 | Map projection—Continued. Homalographic projection... sgncesesemess 156-158 Geometrical azimuthal...............-. 35 Great circles (see also Plate 1)......... 83, 101-103 Perspective......22..2.cccceccccceceece 26 Grid system of military mapping........... 87 Polar equidistant...............2..0004 40, 147 Rrane. eecieariedoeet Waesinchinas gees 87 Problem: aecwsesenituecenscacouaemeiy 22 United States............-2-.2--2----- 87-90 | Mercator projection................2...02.. 32, 51, Guyou’s doubly periodic projection . ..... 159-160 101-137, 141, 145-147, 160 a asaoceseecnee coe neces Gi) ee ree as Hassler, Ferdinand...............-----+-++- 58 Tedae iat cose ee 101-105 Hendrickson, W. W........--2-2-2e2e0e28+ 149 ee % Herz, Dr. Norbert. DY Ed a fe ta eh at be 93, 154 Di e opmen of tormulas.......-.-..- 105-109 : Ree istances measured.............0.+25- 1038, 112 Po ACRE nen cisimetebilinnanatny a Bf Employed by Coast and Geodetic Sur , Bis ll lig Pee pic asia cecemnncaceras 60, 79, 146 ie ee ae Hydrographic Office, U. 8.......- 103, 137, 188, 147 High Jatitudeé vc. cccssaccsesesnuenene 82, 104 International map. See Polyconic projec- TADS oss sisecsinasnceacicjes 109, 110, 116, 117-136 tion with two standard meridians. ‘Transverse....-......-----05 104, 108, 109, 116 Isometric latitude.............-.-...20---- 107 Transverse construction for sphere.... 114-116 Isoperimetric curves.........- jest beeen: 24, 92 Useful features..................000. 146, 147 Ministére de la Marine..................... 108 Lagrange......-.------+---+22--+ eee reese 54, 160 | Mollweide homalographic projection. . . 24, 153-156 Lallemand..........-----------++++++-- 62, 63, 66 Construction........---....e2eeecee eee 154 Lambert azimuthal projection, polar_..-....- 38 MAG: ccaceecseiicwanuietinniagierene 66, 156 Lambert conformal conic projection......... 55, 56, | Monaco, Prince Albert of.........----+--++- 147 77-86, 116 Comparison with Lambert zenithal . .-. . 72 | North Atlantic Ocean map........-...--..-+ 82-84 Compensation of scale error..........--- 81 TTS LG craton deseteslesarehersce dete enddseees ek ese ene 85 Construction.....-..-..---2-----2-2--++- 83 General observations................---- 80 | Orthographic projection: Table for hemispheres................- 86 Meridional..: c.csessscocenecsen encenes 43, 51 Table for North Atlantic........... 85 Polar: csi saeurassiesins np eneeunmuncees 38 . Tables for United States, reference... a 83, 84 Lambert equal-area meridional projection... .. 43, 56, | Penfield, 8. Ly .eeeeeeseeee eee eee eeceeeeeee 149 71-73, 116 Perspective projection.............--.---.-- 26 Table for construction............----- 73, 74 Central on tangent cone............---- 47 Lambert, Johann Heinrich..............--- 78 Gnomonic meridional..............---.- 48, 45 Lambert projection of the northern and Gnomonic (or central) ....... 101-103, 140-145 southern hemispheres........--.---.--- 158-160 Gnomonic polar.....-...--.---+-.-- .. 37,147 Maples rst cd hades Gian 86 Orthographic meridional................ 43, 51 Lambert zenithal (or azimuthal) projection... 56, Orthographic polar................-.---. 38 71-73, 116 Stereographic horizon............ 116, 148, 149 Comparison with Lambert conformal... 72 Stereographic meridional.............. 42, 44, Table for construction ..............--. 73, 74 51, 116, 147-149 Tattende determination... ....scc0-r0-noc. ig | — Stereographic polar.................+. 35, 147 Lecky, reference to tables..............---- 141 Tangent cylinder..............-.++++-+- 30 Littlehales, G. W......222222020000ceeee eee 137 | Petermanns Mitteilungen................... 150 Longitude determination.................+ — 13 | Pillsbury, Lieut. J. B................--.-- 83 POlar CHArts sec wetacteseneiaviariniteniselsietats 35-40, 147 Map: Polyconic projection..............++. 49, 55, 58-60 Definition....-....-.-.-----+---+222+0+- 9 Compromise..............02222eeeeeeees 59-60 Ideal, the.......-.-----.-2+2+-2+--+-5- 27 Construction.............-2-0---eeeeee 61 Plotting points............----.----+.-. 14 International map..............2-.00-5 62-66 Problem.........--------------++-++++- 9 ADIOS: Sic sichet bd icravctceatctd te selected densi es 64 Process of making................-.-+.- 15 TranSVC©8C.-- ooo eee ccc cececccccccccee 62 Map projection: With two standard meridians............ 62-66 Azimuthal..........0-2-2-2222e-eeeeeee 26 Tables......-----cccecccccccccccece 64 Condition fulfilled.............--.-.-.-. 25 Conformal sic.cicercdcenwinaoimosenaren 26 | Reference lines Conventional.............----+--+-----+ 27 Hatt tac :cwiscistunieedsuies ecuiicseesceawiass 11 Definitloncen seoavaeriasisteneseecuriciie-te 22 GOD Ge ape t cee cence ieretoretwiarg eee eswietels 14,17 Distortion: . 2... jcccecssieemacdcsskencex 22, 51 Plane Map eecwedecseswesmswiiceiiccesaee 14 INDEX. 163 Page. Page. Rhumb lines. ..... 2.22.22. 0.0.0.22202002. 101-103 | Surface: Rosén; BIOL. 5 ctecccasaou ld Awe eme es vives 66 Developable and nondevelopable. ...... 9 Rude, Gi Vircew see bet cena cee ae 149 Developable, use.........------------- 27 Plane, construction..........--.---+-++-- 16 Sanson-Flamsteed projection. .............. 68 | Tchebicheff..............22eeeeceeeeeeeeees 149 Smithsonian tables..............2.2.2..--- BU, L14 | tPiss0t cok c coc ceecs oss ve wide eese oeebe eee 54, 56 Sphere: Construction of meridians and parallels. W7 United States, map...... 54, 55, 72, 79-85, 91-93, 99 INR ahaa nical aA ahaedd ll ne it Ste hi sey OTS eats eGithad aureclaewaeaeics a ¢ projection.........-..... 18 - 56, 147 Lambert conformal ..............--- 85, 86 Dae Tit eneie ornselt sie at citigie Biase A 6, 147-149 Lambert zenithal ...............-. 73, 74 Meridional............- 42, 44, 51, 116, 147-149 Construction.........-...0.222.--4 44 | Wangerin, A..............22-+-222--+--++- 78 Polar tec 35, 147 Wireless directional bearings.........- 137-189, 141 Siiieie ee ji | Moeller cranes -pocceee-neps 57, 146-160 Straight line, how to draw............-...... 15 ! Zéppritz, Prof. Dr. Karl................----. 73 $ 50° “0 30° w 10" T NORTH ATLANTIC O WITH ‘< EASTERN NORTH AMERICA A 60° 7 LAMBERT CONFORMAL CONIC PROJECTION § eS AZORES = GREAT C/RGLES ARE SHOWN Plate | 40° oy 10° 20° —— a IC OCEAN ICA AND EUROPE JECTION por CELAND Zoe NEA e es: x 4é>, Seale along the Parallels \ North Latitude Scale Factor 719 Statute Milos 100 o 100 200 300 400 500 600 700 ‘00 300 1000 g Nautical Miles beceremecet _ —- ——+. ——_ aS ° Sf oO 100 ° 100 200 300 ‘400 500 600 700 800 300 1000 Kilometers bec =—— = a 1000 500 =—T000 7500 2000 \ =i \ 20" 10° o 0° a = (North Atlantic Ocean, Lambert Projection ) USCEGS. U.S.COAST AND GEODETIC SURVEY e. - \e 7 e 0° Ge 130° \%, 60° 180-160 40"_—120” a \ ALASKA™~ \— Ne AND ITS RELATION TO THE © UNITED STATES AND THE ORIENT ee J Scale 40 600,000 (Transverse Polyconic Projection) e PEKIN d Yokohama | % Z Aleutian Island | eA ef 4132 Nautical Miles i Steame, 0 track between San Fr, . \ ST ULENCISCO and | Yokoh \ ome #536 Nautica Plate || Statute Miles Nautical Miles C.&G.S. Print. ALBERS EQUAL AREA PROJECTION STANDARD PARALLELS 29°30'anD 45°30’ Plate Ill 125° 120° 115° 110° 105° 100° 95° fi 65 setSscre soso le oce led een eee octoeRlee ss 39° eC, 2 ay 30 2 . { % \ e by : % \ 2 S C.& G.S. Print. noes = SS I t | . 100 50 0 500 1000 Statute Miles From Hytragraphi« Office Chart 1280 GNOMONIC PROJECTION OF THE NORTH ee OCEAN ; POINT OF TANGENCY Ttew¥ork) +, Corres! Piéree Al Z: 0: R: E: § eee OJECTION OF THE NORTH ATLANTIC OCEAN ; POINT OF TANGENCY 30°N. AND 30°W. i eS at eo faerers os ea) eS | Se Se at Al oe ae ee ee ea, ‘_ (C. Farewell Pa oo a : ao. 8 Plate IV C.& G.S.Print. FE PP ARES SS EE BEC ge RR Se LEE ETERS SS PPLLLIRGL TTT le a aS : RSS [EEE AAAS SD KEL A may | NEES RE ¥ DY. SR BEZ SSE ——— a THE WORLD ON THE HOMALOGRAPHIC PROJECTION (INTERRUPTED) FOR THE CONTINENTS (¢ in + ee : is Op SH SVs JOUER = vr c\ Seth CL He Ll % PROJECTION OF THE SPHERE A TWO CUSPED EPICYGLOID Plate VIII Devised by Dr. F August 150 SZ _—ne +33 SEK \Me . oa $ os © lf Lik [Ti 4 Ped ore \\ es IDV RES MOOS REALL TRS ae FN RRO TET OR NOS PHT Re A FRY Y Lo Testy CEL ae fee pce ee Poh (pee FRE ESCH ERP HS ee seat aT ECE Ark RE Se Kt 1 gH [pa “af | x / R\ \X WA x HECTOR F EE Fa ie . eae vd IT MENS 12 A C.& G.S.Print.