CHAPTER XMAP PROJECTIONSFig. 62To represent the curvedsurface of the earth, or anyportion of it, on the planesurface of a map, involves se-rious mathematical diﬃcul-ties. Indeed, it is impos-sible to do so with perfectaccuracy. The term pro-jection, as applied to therepresentation on a planeof points corresponding topoints on a globe, is notalways used in geographyin its strictly mathematicalsense, but denotes any rep-resentation on a plane ofparallels and meridians ofthe earth.190

191The Orthographic ProjectionThis is, perhaps, the most readily understood projection,and is one of the oldest known, having been used by the ancientGreeks for celestial representation. The globe truly representsthe relative positions of points on the earth’s surface. It mightseem that a photograph of a globe would correctly representon a ﬂat surface the curved surface of the earth. A glance atFigure 62, from a photograph of a globe, shows the parallelsnear the equator to be farther apart than those near the poles.This is not the way they are on the globe. The orthographicprojection is the representation of the globe as a photographwould show it from a great distance.Fig. 63. Equatorial orthographicprojectionParallels andMeridians FartherApart in Centerof Map. Viewinga globe from a dis-tance, we observethat parallels andmeridians appearsomewhat as repre-sented in Figure 63,being farther aparttoward the center andincreasingly nearertoward the outer por-tion. Now it is obviousfrom Figure 64 thatthe farther the eye isplaced from the globe, the less will be the distortion, althougha removal to an inﬁnite distance will not obviate all distortion.

192Thus the eye at x sees lines to E and F much nearer togetherthan lines to A and B, but the eye at the greater distance seesless diﬀerence.Fig. 64When the rays are perpendicular to the axis, as in Figure 65,the parallels at A, B, C, D, and E will be seen on the tangentplane XY at A0, B0, C0, D0, and E0. While the distance from Ato B on the globe is practically the same as the distance from Dto E, to the distant eye A0and B0will appear much nearertogether than D0and E0. Since A (or A0) represents a poleand E (or E0) the equator, line XY is equivalent to a centralmeridian and points A0, B0, etc., are where the parallels cross it.How to Lay oﬀ an Equatorial Orthographic Projec-tion. If parallels and meridians are desired for every 15◦, dividethe circle into twenty-four equal parts; any desired number ofparallels and meridians, of course, may be drawn. Now connectopposite points with straight lines for parallels (as in Fig. 65).The reason why parallels are straight lines in the equatorial or-thographic projection is apparent if one remembers that if the

193eye is in the plane of the equator and is at an inﬁnite distance,the parallels will lie in practically the same plane as the eye.Fig. 65To lay oﬀ themeridians, mark onthe equator pointsexactly as on thecentral meridian whereparallels intersect it.The meridians maynow be made as arcs ofcircles passing throughthe poles and thesepoints. With one footof the compasses in theequator, or equatorextended, place the other so that it will pass through the polesand one of these points. After a little trial it will be easy to layoﬀ each of the meridians in this way.Fig. 66. Western hemisphere, inequatorial orthographic projectionTo be strictly correctthe meridians should notbe arcs of circles as justsuggested but should besemi-ellipses with the cen-tral meridian as major axisas shown in Figure 66.While somewhat more dif-ﬁcult, the student shouldlearn how thus to lay themoﬀ. To construct the el-lipse, one must ﬁrst locatethe foci. This is done bytaking half the major axis

194(central meridian) as radius and with the point on the equa-tor through which the meridian is to be constructed as center,describe an arc cutting the center meridian on each side of theequator. These points of intersection on the central meridianare the foci of the ellipse, one half of which is a meridian. Byplacing a pin at each of the foci and also at the point in theequator where the meridian must cross and tying a string asa loop around these three pins, then withdrawing the one atthe equator, the ellipse may be made as described in the ﬁrstchapter.Fig. 67. Polar orthographicprojectionHow to Lay oﬀ a Po-lar Orthographic Pro-jection. This is laid oﬀmore easily than the for-mer projection. Here theeye is conceived to be di-rectly above a pole and theequator is the boundary ofthe hemisphere seen. It isapparent that from this po-sition the equator and par-allels will appear as circlesand, since the planes of themeridians pass through theeye, each meridian mustappear as a straight line.Lay oﬀ for the equator a circle the same size as the precedingone (Fig. 65), subdividing it into twenty-four parts, if meridiansare desired for every 15◦. Connect these points with the center,which represents the pole. On any diameter mark oﬀ distancesas on the center meridian of the equatorial orthographic pro-jection (Fig. 65). Through these points draw circles to repre-

195sent parallels. You will then have the complete projection as inFigure 67.Projections may be made with any point on the globe ascenter, though the limits of this book will not permit the ratherdiﬃcult explanation as to how it is done for latitudes other than0◦or 90◦. With the parallels and meridians projected, the mapmay be drawn. The student should remember that all mapswhich make any claim to accuracy or correctness are made bylocating points of an area to be represented according to theirlatitudes and longitudes; that is, in reference to parallels andmeridians. It will be observed that the orthographic system ofprojection crowds together areas toward the outside of the mapand the scale of miles suitable for the central portion will not becorrect for the outer portions. For this reason a scale of milesnever appears upon a hemisphere made on this projection.SUMMARYIn the orthographic projection:1. The eye is conceived to be at an inﬁnite distance.2. Meridians and parallels are farther apart toward the center of the map.3. When a point in the equator is the center, parallels are straight lines.4. When a pole is at the center, meridians are straight lines. If the northernhemisphere is represented, north is not toward the top of the mapbut toward the center.Stereographic ProjectionIn the stereographic projection the eye is conceived to beupon the surface of the globe, viewing the opposite hemisphere.Points on the opposite hemisphere are projected upon a plane

196tangent to it. Thus in Figure 68 the eye is at E and sees A at A0,B at B0, C at C0, etc. If the earth were transparent, we shouldsee objects on the opposite half of the globe from the view pointof this projection.Fig. 68

197Fig. 69. Equatorial stereographicprojectionHow to Lay oﬀan Equatorial Stereo-graphic Projection. InFigure 68, E representsthe eye at the equator,A and N are the poles andA0N0is the correspond-ing meridian of the pro-jection with B0, C0, etc.,as the points where theparallels cross the merid-ian. Taking the line A0N0of Figure 68 as diameter,construct upon it a circle(see Fig. 69). Divide thecircumference into twenty-four equal parts and draw parallels as arcs of circles. Lay oﬀ theequator and subdivide it the same as the central meridian, thatis, the same as A0N0of Figure 68. Through the points in theequator, draw meridians as arcs of circles and the projection iscomplete.The Polar Stereographic Projection is made on thesame plan as the polar orthographic projection, excepting thatthe parallels have the distances from the pole that are repre-sented by the points in A0N0of Figure 68 (see Figs. 70, 71).Areas are crowded together toward the center of the mapwhen made on the stereographic projection and a scale of milessuitable for the central portion would be too small for the outerportion. This projection is often used, however, because it is soeasily laid oﬀ.

198Fig. 70. Polarstereographic projectionFig. 71. Northern hemisphere inpolar stereographic projectionFig. 72. Hemispheres in equatorial stereographic projectionSUMMARYIn the stereographic projection:1. The eye is conceived to be on the surface of the globe.

Meridians and parallels are nearer together toward the center of themap.3. When a point in the equator is the center of the map, parallels andmeridians are represented as arcs of circles.4. When a pole is the center, meridians are straight lines.Globular Projection.Fig. 73With the eye at an inﬁnite dis-tance (as in the orthographic pro-jection), parallels and meridians arenearer together toward the outsideof the map; with the eye on the sur-face (as in the stereographic pro-jection), they are nearer togethertoward the center of the map. Itwould seem reasonable to expectthat with the eye at some point in-termediate between an inﬁnite dis-tance from the surface and the sur-face itself that the parallels and meridians would be equidistantat diﬀerent portions of the map. That point is the sine of anangle of 45◦, or a little less than the length of a radius away fromthe surface. To ﬁnd this distance at which the eye is conceivedto be placed in the globular projection, make a circle of the samesize as the one which is the basis of the map to be made, drawtwo radii at an angle of 45◦(one eighth of the circle) and draw aline, AB, from the extremity of one radius perpendicular to theother radius. The length of this perpendicular is the distancesought (AB, Fig. 73).Thus with the eye at E (Fig. 74) the pole A is projected tothe tangent plane at A0, B at B0, etc., and the distances A0B0,

200B0C0, etc., are practically equal so that they are constructed asthough they were equal in projecting parallels and meridians.Fig. 74How to Lay oﬀan Equatorial Glob-ular Projection. Asin the orthographicor stereographic pro-jections, a circle isdivided into equalparts, according to thenumber of parallelsdesired, the centralmeridian and equatorbeing subdivided intohalf as many equalparts. Parallels andmeridians may bedrawn as arcs of cir-cles, being suﬃcientlyaccurate for ordinary purposes (see Fig. 75).Fig. 75. Hemispheres in equatorialglobular projectionThe polar globularprojection is laid oﬀ pre-cisely like the orthographicand the stereographic pro-jections having the poleas the center, exceptingthat the concentric circlesrepresenting parallels areequidistant (see Fig. 76).By means of starlikeadditions to the polar globular projection (see Fig. 77),the entire globe may be represented. If folded back,

201the rays of the star would meet at the south pole.It should be noticed that “south” in this projection is in a linedirectly away from the center; that is, the top of the map issouth, the bottom south, and the sides are also south. Whileportions of the southern hemisphere are thus spread out, pro-portional areas are well represented, South America and Africabeing shown with little distortion of area and outline.Fig. 76. Polar globularprojectionThe globular projection is muchused to represent hemispheres, orwith the star map to represent theentire globe, because the parallels ona meridian or meridians on a par-allel are equidistant and show littleexaggeration of areas. For this rea-son it is sometimes called an equidis-tant projection, although there areother equidistant projections. It isalso called the De la Hire projectionfrom its discoverer (1640–1718).Fig. 77. World in polar globularprojectionSUMMARYIn the globular projection:1. The eye is conceived to be at a cer-tain distance from the globe (sine 45◦).2. Meridians are divided equidistantly byparallels, and parallels are dividedequidistantly by meridians.3. When a pole is the center of the map,meridians are straight lines.4. There is little distortion of areas.

202TheGnomonic ProjectionFig. 78When we look up atthe sky we see what ap-pears to be a great dome inwhich the sun, moon, plan-ets, and stars are located.We seem to be at the cen-ter of this celestial sphere,and were we to imaginestars and other heavenlybodies to be projected be-yond the dome to an imag-inary plane we should havea gnomonic projection. Be-cause of its obvious conve-nience in thus showing theposition of celestial bodies,this projection is a very oldone, having often been usedby the ancients for celestialmaps.Since the eye is at thecenter for mapping the ce-lestial sphere, it is con-ceived to be at the center of the earth in projecting parallelsand meridians of the earth. As will be seen from Figure 78, thedistortion is very great away from the center of the map and anentire hemisphere cannot be shown.All great circles on this projection are represented as straightlines. This will be apparent if one imagines himself at the center

203of a transparent globe having parallels and meridians tracedupon it. Since the plane of every great circle passes through thecenter of the globe, the eye at that point will see every portionof a great circle as in one plane and will project it as a straightline. As will be shown later, it is because of this fact that sailorsfrequently use maps made on this projection.Fig. 79. Polar gnomonic projectionTo Lay oﬀ a PolarGnomonic Projection.Owing to the fact that par-allels get so much fartherapart away from the centerof the map, the gnomonicprojection is almost nevermade with any other pointthan the pole for center,and then only for lati-tudes about forty degreesfrom the pole. The po-lar gnomonic projection ismade like the polar projec-tions previously described,excepting that parallels intersect the meridians at the distancesrepresented in Figure 78. The meridians, being great circles,are represented as straight lines and the parallels as concentriccircles.Great Circle Sailing. It would seem at ﬁrst thought thata ship sailing to a point due eastward, say from New York toOporto, would follow the course of a parallel, that is, wouldsail due eastward. This, however, would not be its shortestcourse. The solution of the following little catch problem inmathematical geography will make clear the reason for this. “Aman was forty rods due east of a bear, his gun would carry only

204thirty rods, yet with no change of position he shot and killedthe bear. Where on earth were they?” Solution: This couldoccur only near the pole where parallels are very small circles.The bear was westward from the man and westward is along thecourse of a parallel. The bear was thus distant forty rods in acurved line from the man but the bullet ﬂew in a straight line(see Fig. 80).Fig. 80The shortest distancebetween two points on theearth is along the arc of agreat circle. A great circlepassing through New Yorkand Oporto passes a littleto the north of the paral-lel on which both cities arelocated. Thus it is thatthe course of vessels plyingbetween the United Statesand Europe curves, some-what to the northward ofparallels. This following ofa great circle by navigatorsis called great circle sailing. The equator is a great circle andparallels near it are almost of the same length. In sailing withinthe tropics, therefore, there is little advantage in departing fromthe course of a parallel. Besides this, the trade winds and dol-drums control the choice of routes in that region and the Merca-tor projection is always used in sailing there. In higher latitudesthe gnomonic projection is commonly used.Although the gnomonic projection is rarely used excepting bysailors, it is important that students understand the principlesunderlying its construction since the most important projections

205yet to be discussed are based upon it.SUMMARYIn the gnomonic projection:1. The eye is conceived to be at the center of the earth.2. There is great distortion of distances away from the center of the map.3. A hemisphere cannot be shown.4. All great circles are shown as straight lines.a. Therefore it is used largely for great circle sailing.5. The pole is usually the center of the map.The Homolographic ProjectionThe projections thus far discussed will not permit the repre-sentation of the entire globe on one map, with the exception ofthe starlike extension of the polar globular projection. The ho-molographic projection is a most ingenious device which is usedquite extensively to represent the entire globe without distortionof areas. It is a modiﬁcation of the globular projection.How to Lay oﬀ a Homolographic Projection. Firstlay oﬀ an equatorial globular projection, omitting the parallels.The meridians are semi-ellipses, although those which are nomore than 90◦from the center meridian may be drawn as arcsof circles.Having laid oﬀ the meridians as in the equatorial globularprojection, double the length of the equator, extending it equallyin both directions, and subdivide these extensions as the equatorwas subdivided. Through these points of subdivision and thepoles, draw ellipses for meridians.

206Fig. 81. Homolographic projectionTo draw the outer elliptical meridians. Set the points ofthe compasses at the distance from the point through whichthe meridian is to be drawn to the central meridian. Place onepoint of the compasses thus set at a pole and mark oﬀ points onthe equator for foci of the ellipse. Drive pins in these foci andalso one in a pole. Around these three pins form a loop witha string. Withdraw the pin at the pole and draw the ellipse asdescribed on page 22. This process must be repeated for eachpair of meridians.The parallels are straight lines, as in the orthographic pro-jection, somewhat nearer together toward the poles. If nine par-allels are drawn on each side of the equator, they may be drawnin the following ratio of distances, beginning at the equator: 2,189, 179, 169, 159, 149, 139, 129, 119. This will give an approximatelycorrect representation.One of the recent books to make frequent use of this projec-tion is the “Commercial Geography” by Gannett, Garrison, and

207Fig. 82. World in homolographic projectionHouston (see Fig. 82).Equatorial Distances of Parallels. The following tablegives the exact relative distances of parallels from the equator.Thus if a map twenty inches wide is to be drawn, ten inches fromequator to pole, the ﬁrst parallel will be .69 of an inch from theequator, the second 1.37 inches, etc.φDis-tanceφDis-tanceφDis-tanceφDis-tanceφDis-tanceφDis-tance5◦.069 20◦.272 35◦.468 50◦.651 65◦.814 80◦.94510 .137 25 .339 40 .531 55 .708 70 .862 85 .97815 .205 30 .404 45 .592 60 .762 75 .906 90 1.000The homolographic projection is sometimes called the Moll-weide projection from its inventor (1805), and the Babinet, orBabinet-homolographic projection from a noted cartographerwho used it in an atlas (1857). From the fact that within anygiven section bounded by parallels and meridians, the area of thesurface of the map is equal to the area within similar meridians

208and parallels of the globe, is it sometimes called the equal-surfaceprojection.SUMMARYIn the homolographic projection:1. The meridians are semi-ellipses, drawn as in the globular projection,360◦of the equator being represented.2. The parallels are straight lines as in the orthographic projection.3. Areas of the map represent equal areas of the globe.4. There is no distortion of area and not a very serious distortion ofform of continents.5. The globe is represented as though its surface covered half of anexceedingly oblate spheroid.The Van der Grinten ProjectionThe homolographic projection was invented early in the nine-teenth century. At the close of the century Mr. Alphons Van derGrinten of Chicago invented another projection by which theentire surface of the earth may be represented. This ingenioussystem reduces greatly the angular distortion incident to the ho-molographic projection and for the inhabitable portions of theglobe there is very little exaggeration of areas.In the Van der Grinten projection the outer boundary is ameridian circle, the central meridian and equator are straightlines, and other parallels and meridians are arcs of circles. Thearea of the circle is equal to the surface of a globe of one halfthe diameter of this circle. The equator is divided into 360◦, butthe meridians are, of course, divided into 180◦.

209Fig. 83. World in Van der Grinten projectionFig. 84. World in Van der GrintenprojectionA modiﬁcation ofthis projection is shownin Figure 83. In thisthe central meridian isonly one half the lengthof the equator, andparallels are at uni-form distances alongthis meridian.

210Cylindrical ProjectionsGnomonic Cylindrical Projection. In this projectionthe sheet on which the map is to be made is conceived to bewrapped as a cylinder around the globe, touching the equator.The eye is conceived to be at the center of the globe, projectingthe parallels and meridians upon the tangent cylinder. Figure 85shows the cylinder partly unwrapped with meridians as parallelstraight lines and parallels also as parallel straight lines. As inthe gnomonic projection, the parallels are increasingly fartherapart away from the equator.Fig. 85An examination ofFigure 86 will showthe necessity for theincreasing distancesof parallels in higherlatitudes. The eye atthe center (E) seesA at A0, B at B0, etc.Beyond 45◦from theequator the distancebetween parallelsbecomes very great.A0B0represents thesame distance (15◦oflatitude) as G0H0, butis over twice as longon the map. At A0(60◦north latitude)the meridians of the globe are only half as far apart as they areat the equator, but they are represented on the map as thoughthey were just as far apart there as at the equator. Because

211of the rapidly increasing distances of parallels, to representhigher latitude than 60◦would require a very large sheet, sothe projection is usually modiﬁed for a map of the earth as awhole, sometimes arbitrarily.G0H0is the distance from the equator to the ﬁrst parallel, andsince a degree of latitude is about equal to a degree of longitudethere, this distance may be taken between meridians.Stereographic Cylindrical Projection. For reasons justgiven, the gnomonic or central cylindrical projection needsFig. 86reduction to show thepoles at all or anyhigh latitudes with-out great distortion.Such a reduction iswell shown in thestereographic projec-tion. In this theeye is conceived to beon the equator, pro-jecting each meridianfrom the view pointof the meridian oppo-site to it. Figure 87shows the plan onwhich it is laid oﬀ,meridians being paral-lel straight lines andequidistant and paral-lels being parallel straight lines at increasing distances away fromthe equator.The Mercator (Cylindrical) Projection. In the ortho-graphic, stereographic, globular, gnomonic, homolographic, and

212Van der Grinten projections, parallels or meridians, or both,are represented as curved lines. It should be borne in mind thatdirections on the earth are determined from parallels and merid-ians. North and south are along a meridian and when a meridianFig. 87is represented as acurved line, north andsouth are along thatcurved line. Thus thetwo arrows shown atthe top of Figure 81,are pointing in almostexactly opposite di-rections and yet eachis pointing due north.The arrows at thebottom point oppositeeach other, yet bothpoint due south. Thearrows pointing to theright point the sameway, yet one pointsnorth and the otherpoints south. A linepointing toward thetop of a map may or may not point north. Similarly, paral-lels lie in a due east-west direction and to the right on a mapmay or may not be to the east.It should be obvious by this time that the map projectionsstudied thus far represent directions in a most unsatisfactorymanner, however well they may represent areas. Now to thesailor the principal value of a chart is to show directions to steerhis course by and if the direction is represented by a curved

213line it is a slow and diﬃcult process for him to determine hiscourse. We have seen that the gnomonic projection employsstraight lines to represent arcs of great circles, and, consequently,this projection is used in great circle sailing. The Mercatorprojection shows all parallels and meridians as straight lines atproportional distances, hence directions as straight lines, andis another, and the only other, kind of map used by sailors inplotting their courses.Fig. 89Maps in AncientTimes. Before themiddle of the ﬁfteenthcentury, sailors did notcover very great por-tions of the earth’ssurface in continuousjourneys out of sightof land where theyhad to be guided al-most wholly by thestars. Mathematicalaccuracy in maps wasnot of very great im-portance in navigationuntil long journeys hadto be made with no opportunity for veriﬁcation of calculations.Various roughly accurate map projections were made. The mapsent to Columbus about the year 1474 by the Italian astronomerToscanelli, with which he suggested sailing directions across the“Sea of Darkness,” is an interesting illustration of a commontype of his day.The long journeys of the Portuguese along the coast of Africaand around to Asia and the many voyages across the Atlantic

214Fig. 88

215early in the sixteenth century, made accurate map projectionnecessary. About the middle of that century, Emperor Charles Vof Spain employed a Flemish mathematician named GerhardKramer to make maps for the use of his sailors. The wordKramer means, in German, “retail merchant,” and this trans-lated into Latin, then the universal language of science, becomesMercator, and his invention of a very valuable and now widelyused map projection acquired his Latinized name.Plan of Mercator Chart. The Mercator projection is madeon the same plan as the other cylindrical projections, exceptingas to the distances between parallels. The meridians are rep-resented as parallel lines, whereas on the globe they converge.There is thus a distortion of longitudes, greater and greater,away from the equator. Now the Mercator projection makes theparallels farther apart away from the equator, exactly propor-tional to the meridional error. Thus at latitude 60◦the merid-ians on the earth are almost exactly half as far apart as at theequator, but being equidistant on the map, they are representedas twice as far apart as they should be. The parallels in thatportion of the Mercator map are accordingly made twice as farapart as they are near the equator. Since the distortion in lat-itude exactly equals the distortion in longitude and parallelsand meridians are straight lines, all directions are representedas straight lines. A navigator has simply to draw upon the mapa line from the point where he is to the point to which he wishesto sail in a direct course, measure the angle which this line formswith a parallel or meridian, and steer his ship according to thebearings thus obtained.To Lay oﬀ a Mercator Projection. Figure 89 shows the sim-plest method of laying oﬀ this projection. From the extremityof each radius drop a line to the nearest radius, parallel to thetangent A0L. The lengths of these lines, respectively, represent

216the distances∗between parallels. Thus N0M equals CP , K0N0equals BN, A0K0equals AK. The meridians are equidistantand are the same distance apart as the ﬁrst parallel is from theequator.Fig. 90. World in mercator projectionThe table of meridional parts on page 217 gives the relativedistances of parallels from the equator. By means of this table amore exact projection may be laid oﬀ than by the method justsuggested. To illustrate: Suppose we wish a map about twentyinches wide to include the 70th parallels. We ﬁnd in the tablethat 5944.3 is the distance to the equator. Then, since the mapis to extend 10 inches on each side of the equator,105944.3isthe scale to be used in making the map; that is 1 inch on themap will be represented by 10 inches ÷ 5944.3. Suppose wewish to lay oﬀ parallels ten degrees apart. The ﬁrst parallelto be drawn north of the equator has, according to the table,599.1 for its meridional distance. This multiplied by105944.3∗Technically speaking, the distance is the tangent of the angle of lati-tude and any table of natural tangents will answer nearly as well as the tableof meridional parts, although the latter is more accurate, being correctedfor the oblateness of the meridian.

217equals slightly more than 1. Hence the parallel 10◦should belaid oﬀ 1 inch from the equator. The 20th parallel has for itsmeridional distance 1217.3. This multiplied by the scale105944.3gives 2.03 inches from the equator. The 30th parallel has ameridional distance 1876.9, this multiplied by the scale gives3.15 inches. In like manner the other parallels are laid oﬀ. Themeridians will be105944.3× 60 or 600 inches ÷ 5944.3 for everydegree, or for ten degrees 6000 inches ÷ 5944.3, which equals1.01 inches. This makes the map 36.36 inches long (1.01 inches×36 = 36.36 inches).Table of Meridional Parts∗1◦59.6 18◦1091.1 35◦2231.1 52◦3647.1 69◦5773.12◦119.2 19◦1154.0 36◦2304.5 53◦3745.4 70◦5944.33◦178.9 20◦1217.3 37◦2378.8 54◦3846.1 71◦6124.04◦238.6 21◦1281.0 38◦2454.1 55◦3949.1 72◦6313.05◦298.4 22◦1345.1 39◦2530.5 56◦4054.9 73◦6512.46◦358.3 23◦1409.7 40◦2607.9 57◦4163.4 74◦6723.67◦418.3 24◦1474.7 41◦2686.5 58◦4274.8 75◦6948.18◦478.4 25◦1540.3 42◦2766.3 59◦4389.4 76◦7187.89◦538.6 26◦1606.4 43◦2847.4 60◦4507.5 77◦7444.810◦599.1 27◦1673.1 44◦2929.9 61◦4628.1 78◦7722.111◦659.7 28◦1740.4 45◦3013.7 62◦4754.7 79◦8023.112◦720.6 29◦1808.3 46◦3099.0 63◦4884.5 80◦8352.613◦781.6 30◦1876.9 47◦3185.9 64◦5018.8 81◦8716.414◦842.9 31◦1946.2 48◦3274.5 65◦5158.0 82◦9122.715◦904.5 32◦2016.2 49◦3364.7 66◦5302.5 83◦9583.016◦966.4 33◦2087.0 50◦3456.9 67◦5452.8 84◦10114.017◦1028.6 34◦2158.6 51◦3551.0 68◦5609.5 85◦10741.7We see, then, that the same scale of miles cannot be usedfor diﬀerent parts of the map, though within 30◦of the equator∗From Bowditch’s Practical Navigator.

218representations of areas will be in very nearly true proportions.The parallels in a map not wider than this, say for Africa, may bedrawn equidistant and the same distance apart as the meridians,the inaccuracy not being very great.SUMMARYIn the cylindrical projection:1. A cylinder is conceived to be wrapped around the globe, tangent to theequator.2. All parallels and meridians are represented as straight lines, the formerintersecting the latter at right angles.3. The parallels are made at increasing distances away from the equator:a. In the gnomonic projection, as though projected from the center ofthe earth to the tangent cylinder.b. In the stereographic projection, as projected from the equator upon anopposite meridian, the projection point varying for each meridian.c. In the Mercator projection, at distances proportional to the merid-ional excess.Directions are better represented in this projection than in any other.Here northward is directly toward the top of the map, eastwarddirectly toward the right, etc. For this reason it is the projectionmost commonly employed for navigators’ charts.4. There is great distortion of areas and outlines of continents in highlatitudes; Greenland appears larger than South America.5. The entire globe may be represented in one continuous map.6. The same scale of miles cannot be used for high latitudes that is usednear the equator.Conic ProjectionThe portion of a sphere between the planes of two parallelswhich are near together is very similar to the zone of a cone (seeFig. 91). Hence, if we imagine a paper in the form of a cone

219placed upon the globe and parallels and meridians projectedupon this cone from the center of the globe, then this conicalmap unrolled, we can understand this system.Fig. 91Along the parallel tangent to the cone,points on the map will correspond exactlyto points upon the globe. Parallels whichare near the line of tangency will be repre-sented very much in the relative positionsthey occupy on the globe. In a narrowzone, therefore, near the tangent parallel,there will be very little distortion in lati-tudes and longitudes and an area mappedwithin the zone will be very similar inform and area to the form and area as itappears upon the globe itself. For thisreason the conic projection, or some mod-iﬁcation of it, is almost always employedin representing small areas of the earth’ssurface.Fig. 92To Lay oﬀ a Conic Pro-jection. If the forty-ﬁfth paral-lel is the center of the area to bemapped, draw two straight linestangent to the forty-ﬁfth parallelof a circle (see Fig. 93). Projectupon these lines points for par-allels as in the gnomonic pro-jection. With the apex as cen-ter, draw arcs of circles throughthese points for parallels. Merid-ians are straight lines meetingat the apex and are equidistant

220along any parallel.It will be observed that parallels are farther apart away fromthe tangent parallel (45◦, in this case) as in the Mercator pro-jection they are farther apart away from the equator, which istangent to the globe in that projection. There is also an exag-geration of longitudes away from the tangent parallel. Becauseof this lengthening of parallels, meridians are sometimes curvedinwardly to prevent too much distortion of areas. The need forthis will be apparent if one draws parallels beyond the equator,for he will ﬁnd they are longer than the equator itself unlessmeridians curve inwardly there.Fig. 93By taking thetangent parallel tendegrees north of theequator and reducingdistances of parallels,a fan-shaped mapof the world maybe shown. In thismap of the world onthe conic projection,there is even greaterdistortion of parallelssouth of the equator,but since meridiansconverge somewhat north of the equator there is less distortionin northern latitudes. Since most of the land area of theglobe is in the northern hemisphere, this projection is muchbetter suited to represent the entire world than the Mercatorprojection.

221Fig. 94. North America in conicprojectionBonne’s (Conic) Pro-jection. This is a mod-iﬁcation of the conic pro-jection as previously de-scribed to prevent exaggera-tion of areas away from theparallel which is conceivedto be touching the globe.The central meridian is astraight line and parallelsare concentric equidistantcircles. The distance be-tween parallels is the lengthof the arc of the circle whichis used as a basis for theprojection. For ordinarypurposes, the distance AB(Fig. 93) may be taken foreach of the distances be-tween parallels.Fig. 96. Europe in conic projectionHaving laid oﬀ the cen-tral meridian and markedoﬀ the arcs for parallels,the true distance of themeridian on each parallelis laid oﬀ and the merid-ian is drawn through thesepoints. This gives a gen-tle inward curve for merid-ians toward the outside ofthe map of continents. In-stead of following Bonne’s

222Fig. 95. The world in conic projectionsystem with strict accuracy, the map maker sometimes makesthe curve a little less in lower latitudes, allowing a slight exag-geration of areas to permit the putting in of more details wherethey are needed.Fig. 97Intersecting Conic Projec-tion. Where a considerable extentin latitude is to be represented, thecone is sometimes conceived to cutinto the sphere. In this case, eachmeridian intersects the sphere attwo parallels (see Fig. 97) and sincealong and near the tangent paral-lels (A and B) there is little distor-tion, this plan is better adapted fora map showing greater width northand south than is the conic projec-tion.The map of Europe well illus-

223trates this diﬀerence. Europe lies between 35◦and 75◦north lat-itude. On a conic projection the tangent parallel would be 55◦.Near this parallel there would be no exaggeration of areas but atthe extreme north and south, 20◦away from this parallel, therewould be considerable distortion. If, instead, we make an inter-secting conic projection, we should have the cone pass throughparallels 45◦and 65◦and along these parallels there would beno distortion and no part of the map being more than 10◦awayfrom these lines, there would be very little exaggeration any-where.Fig. 98. Africa andEurope in polyconicprojectionIt should be noticed that theregion between the intersections ofthe meridians must be projectedback toward the center of the sphereand thus be made smaller in themap than it appears on the globe.The central parallel would be tooshort in proportion to the rest.Since this area of Europe (between45◦and 65◦) is the most impor-tant portion and should show mostdetails, it would be a serious de-fect, from the practical map maker’spoint of view, to minify it.Polyconic Projection. Thisdiﬀers from the conic projection inthat it is readjusted at each parallelwhich is drawn, so that each one istangent to the sphere. This makes the circumscribing cone bentat each parallel, a series of conic sections. The word polyconicmeans “many cones.” The map constructed on this projectionis thus accurate along each parallel, instead of along but one

224as in the conic projection or along two as in the intersectingconic projection. For representing small areas this is decidedlythe most accurate projection known. Since the zone along eachparallel is projected on an independent cone, the point whichis the apex for one cone will not be the same for any other(unless both north and south latitudes are shown in the samemap). In the conic projection the parallels are all made fromthe apex of the cone as the center. In the polyconic projectioneach parallel has its own conical apex and hence its own center.This may easily be observed by a comparison of the parallels inFigure 94 (conic projection, all made from one center) and thosein Figure 98 (polyconic projection, each made from a diﬀerentcenter).SUMMARYIn the conic projection:1. A cone is conceived to be ﬁtted about a portion of the globe, tangentto some parallel.2. The tangent parallel shows no distortion and portions near it have butlittle. This projection is therefore used extensively for mappingsmall areas.a. In the conic projection on the gnomonic or central plan, the eye isconceived to be at the center of the globe, parallels are crowdedcloser together toward the central parallel, and distant areas areexaggerated.The cone may be conceived to intersect the globe at two parallels,between which there is a diminution of areas and beyond whichthere is an exaggeration of areas.b. In the Bonne projection parallels are drawn at equidistant intervalsfrom a common center and meridians are slightly curved to preventdistortion in longitudes.c. In the polyconic projection many short conic sections are conceivedto be placed about the globe, one for each parallel represented.Parallels are drawn from the apexes of the cones.

225The ScaleThe area of any map bears some proportion to the actualarea represented. If the map is so drawn that each mile shallbe represented by one inch on the map, since one mile equals63, 360 inches, the scale is said to be 1 : 63, 360. This is oftenwritten fractionally,163, 360. A scale of two inches to the mileis 1 : 31, 680. These, of course, can be used only when smallareas are mapped. The following scales with their equivalentsare most commonly used in the United States Geological Survey,the ﬁrst being the scale employed in the valuable geological folioscovering a large portion of the United States.Scale 1 : 125, 000, 1 mile = 0.50688 inches.Scale 1 : 90, 000, 1 mile = 0.70400 inches.Scale 1 : 62, 500, 1 mile = 1.01376 inches.Scale 1 : 45, 000, 1 mile = 1.40800 inches.Some ConclusionsThe following generalizations from the discussion of map pro-jections seem appropriate.1. In all maps north and south lie along meridians and eastand west along parallels. The top of the map may or may notbe north; indeed, the cylindrical projection is the only one thatrepresents meridians by perpendicular lines.2. Maps of the same country on diﬀerent projections mayshow diﬀerent shapes and yet each may be correct. To makemaps based upon some arbitrary system of triangles or lines isnot scientiﬁc and often is not even helpful.3. Owing to necessary distortions in projecting the parallelsand meridians, a scale of miles can rarely be used with accuracyon a map showing a large area.

Straight lines on maps are not always the shortest dis-tances between two points. This will be clear if we rememberthat the shortest distance between two points on the globe isalong the arc of a great circle. Now great circles, such as merid-ians and the equator, are very often represented as curved lineson a map, yet along such a curved line is the shortest distancebetween any two places in the line on the globe which the maprepresents.5. To ascertain the scale of miles per inch used on any map,or verify the scale if given, measure the space along a meridianfor one inch and ascertain as correctly as possible the numberof degrees of latitude contained in the inch. Multiply this bythe number of miles in one degree of latitude, 69, and you havethe number of miles on the earth represented by one inch on themap.