Maps, their uses and construction. A short popular treatise on the advantages and defects of maps on various projection ..

Front matter

ivl J\ r o

THEIR USES AND CONST!

G. JAMES MORRISON

Contents

CHAPTER

I. INTRODUCTORY

II. MAP PROJECTION CONSIDERED POPULARLY . . . 14

Chapter I

EVERYONE from time to time makes use of maps, but hardly anyone ever consults a globe, and it is not going too far to say that in the case of ninety-nine people out of a hundred all their ideas of the relative size, shape, and positions of the various countries in the world are derived entirely from maps, and generally from those on Mercator's projection ; yet of these people probably not one in ten knows that the ideas so derived areinaccurate to a degree which may fairly be called misleading.

On Mercator's maps, for instance, the size of all polar countries is enormously exaggerated. Greenland appears to be many times the size of India, whereas it has approximately the same area ; as regards position, the direct route from, say, London to Shanghai appears to pass through the Caspian Sea, while as a matter of fact it passes north of St. Petersburg. In these and various other ways not only maps on Mercator's

i;

projection but all maps of large areas are and must be faulty.

It is proposed in the following pages, in the first place to explain in a popular way why it is that no absolutely correct map can be made, and what are the good and bad points of maps constructed on various systems ; and to follow this up with a general explanation of the mathematical bases of the most useful projections, for the benefit of those interested in this branch of the subject.

It is hardly necessary to state that no difficulties of any magnitude are encountered in making maps of moderate areas. A flat map of England may be made, which, for the purpose of studying geography, may be looked upon as absolutely accurate ; but a map showing half the world or the whole world must be distorted to a considerable extent. While, however, this is true, it is also true that in such maps it is generally possible to preserve some one important quality by sacrificing others. Thus (i) a map may be made on which all small portions of the world retain very nearly their proper shapes, but countries near the edges of the map are on a larger scale than those near the centre. Or (2) a map may be made on which all countries retain their areas accurately, but the shape of each is somewhat distorted. Again (3) a map may be made on which all points retain the proper compass bearings of the routes between them ; that is to say, if on the map one port is north-east of another port, then a ship starting from the latter and sailing north-east will reach the former, but the course will not be the shortest possible : a statement which will be made clear hereafter. Or (4) a map may be made on which a straight line drawn from one point to another will indicate accurately the shortest route on the surface of the globe ; but in both these maps, (3) and (4), the distortion near the edges is very great.

The first-named map is very commonly used for giving people general ideas ; that is to say, for teaching geography. The second is useful for giving good ideas as to relative size, and as the distortion of shape is not very great it deserves to be in general use, while, as a matter of fact, it is hardly known. The third, or Mercator map, is useful as a sailing chart for moderate voyages, and has the advantage of showing the entire inhabited world on one sheet, a quality which has led to its being used for general purposes to an extent which is quite unjustified. The fourth is useful for laying down long ocean routes, but as it is not on the whole a convenient sailing map it is much less used than it ought to be.

It may seem strange to begin a treatise on maps by what may appear to be a general condemnation of the entire species. The fact is that maps of the world, or of large portions of the world, are at best miserable makeshifts ; but as they are absolutely necessary for many purposes, it has been thought it would be useful if the principles on which they are founded could be explained in a manner that would be clear to those who have not made a study of mathematics, so that they might see for themselves how far any particular map might be relied on, and how far people must be on their guard against using a map for a purpose for which it is not

B 2 suited, and so being misled, and badly misled, byappearances.

As everyone knows, the earth is a sphere, and though it is slightly flattened at the poles it will be assumed in this portion of the treatise, as is usual in general treatises on projections, that it is a perfect sphere. An accurate model of the earth may therefore be made in the form of a globe, but the inconvenience of using globes is so great that people are practically driven to flat maps. Now, if a correct representation of the world can be made on a globe, the simplest way in principle to make a flat map would be to lay a piece of tracing paper on the globe and trace the map through. A single attempt will show the futility of this method. Unless a very small piece of paper be used, it will not lie flat. Of course no piece, however small, will lie absolutely flat, but a comparatively small piece will lie so close to the surface of the globe that no appreciable crumpling takes place, and the portion of the globe covered can be traced so as to give a flat map without appreciable error. The larger the globe the larger the piece of paper that can be used, but the larger also will be the countries, so that the area to be thus shown cannot be increased. A map of the British Isles might perhaps with difficulty be accomplished, but if one tries to cover half the globe one finds the paper crumple up so as to make tracing impossible. Now this difficulty lies at the root of all map-making, and it cannot be abolished ; but there are certain methods by which, to a limited extent, it can be evaded.

The positions of places on the earth are identified by latitude and longitude. Practically all educated people know what is meant by these words, and some readers may feel inclined to skip the next few pages, but it is advisable to explain how the latitude and longitude of

Fig. i north pole

SOUTH POI.E places are determined in order to make the succeeding pages of this treatise more easily understood. The surface of a sphere begins and ends nowhere. Any point on it bears the same relation to the whole that every other point bears, and there is no reason to choose one point more than another to start from. In the case of the earth, however (see fig. i), there is something. It revolves on its axis once a day, and this axis is therefore a definite line, the ends of which are called poles, one being called North and the other South. Here, then, is something definite to go upon. The earth is supposed to be divided into two equal parts by a line which goes round the world, half-way between the poles, and is called the Equator. The position of this, being defined with reference to the poles, is likewise definite. All circles which divide a sphere into two equal parts are called great circles, and the Equator is therefore a great circle. It has for a long time been the habit to divide a circle into 360 equal parts, called degrees. There is no reason why 360 should have been chosen rather than 400 or 100 or any other number, but it has been used so long that one has come to look upon it as a natural division. 90 degrees therefore means one fourth of a circle, 45 degrees means one eighth, 60 degrees means one sixth, and so on, a degree being indicated by a small circle after the figure, thus : 6o°.

Having decided to divide the Equator into 360 degrees, one is met by the difficulty of having no point at which to begin. It will be better to leave this question for a moment and return to it.

It is clear that any number of great circles can be drawn through the two poles, each of which will cut the Equator into two equal parts. Each of these may be divided into 3600, and in this case the Equator is taken as the starting-point, and there will be 900 between the

Equator and each pole on each side. It is usual to number these from the Equator towards the pole. Now suppose some point is chosen measuring io° north of the Equator on one of these great circles (which, as will be seen later on, are called meridians) and a circle is drawn through it parallel to the Equator, this circle will be the parallel of io° north latitude and will cut every meridian io° north of the Equator. In the same wayother circles may be drawn at 20°, 300, &c, to represent parallels of 200 and 300 of latitude, and further circles may be drawn at every io° or at every degree north and south, and all of these will be parallels of latitude. It is clear that each of these parallels of latitude cuts the sphere, not into two equal parts, but into two unequal parts. The parallels of latitude are therefore not great circles, but small circles. The point to be established, however, is that the Equator has been drawn with reference to the poles and the parallels have been drawn with reference to the Equator, and are therefore definite in position. There is absolutely nothing arbitrary except the numbering of the degrees. The parallel which lies half-way between the Equator and the pole is called the parallel of 450. It might be called by another name, but none the less it is a definite line on the surface of the globe, and not an arbitrary line.

The latitude of any place is determined on the following principles. There happens to be a star very nearly in the prolongation of the earth's axis to the north, to which the name pole-star is given. Suppose anyone to be at the North Pole, this star would appear to him to be directly overhead. Suppose again a person to be at the Equator, the star would appear to him to be on the horizon, level with his eye. It might be said that it would appear below his eye, because it is in line with the axis of the earth, 4,000 miles below his feet, but the distance of the star is so enormous that this is nothing. All lines to it from any point on the earth appear to be parallel. Suppose ,a person to be at F (see fig. 2), half-way between the Equator and the pole, the inclined line on the earth, D E, will appear to him to be horizontal, and he will see the star half-way up the heavens. Now this is equivalent to saying that the latitude of any place is indicated by the height of the pole-star. Most people who have travelled much have observed as they go south that the pole-star night by night appears lower in the heavens and gradually disappears while the Southern Cross gradually comes into view. It is true that at sea the latitude is fixed every day at noon by an observation of the sun, but this is because the sun is brighter and more easily observed. Its distance from the pole, which varies throughout the year, is tabulated for each day in a book called the Nautical Almanack, and when it is observed its polar distance is allowed for, and what is actually stated as the latitude of the ship is in fact the height of the pole in the heavens, and the star itself is as a matter of fact directly observed from time to time. This shows that latitude is not arbitrary. If the star appears one fourth of the way up, measured from the horizon towards the zenith, then the point of observation is one fourth of the way up from the Equator towards the pole, and nothing can alter this. It is of course to be understood that when the pole-star is spoken of, the pole (the prolongation of the earth's axis), to which the star is close, is really meant. When the star itself is observed the difference is allowed for. Southern latitudes are based on the same principles.

Strictly speaking, this is what is called the astronomical latitude of a place. There are, as will be shown later, other latitudes which slightly differ from the above, partly because the earth is not a true sphere and io MAPS: THEIR USES AND CONSTRUCTION

partly from local attractions, but the above described latitude is not only the one adopted in all general treatises, it is the one adopted in all general maps and charts, whether issued by the Admiralty or by others, and is the latitude by which all navigation is conducted ; and, assuming the earth to be a perfectly uniform sphere, it is the only latitude.

This, however, fixes only the parallel of latitude on which the point of observation is situated. If it be found that the latitude of one place is io° north and the latitude of another 200 north, then the second place is io° north of the former, but there is nothing to show whether it is east or west of it. Suppose at any point on the earth's surface a perpendicular pole be erected. The shadow of the pole in the morning will be on the west side, and in the evening on the east side, and there is a certain moment when it will lie due north and south. That moment is called noon, and it will be the same for all points exactly north or south of the point of observation. A great circle passing through the poles of the earth and through the point of observation is called a meridian (from meridies, mid-day), and it is therefore noon at the same moment at all points on that meridian. Suppose a chronometer keeping correct time to be set to noon and then carried to some other point of observation and noon observed at the new point and compared with the chronometer, and suppose it is found that at noon of the new point the chronometer marks two o'clock, then the meridian on which the new point is situated is one-twelfth of the way round the world to the westward of the first point. This difference is

INTRO D UCTOR Y i x

absolute and definite, there is nothing arbitrary about it, but it would be most inconvenient to have to work simply with differences of various places, and all would be chaos unless some one place was agreed upon as the starting-point. This question cropped up as soon as longitudes began to be thought of and long before they were accurately fixed, and a great many places in turn were used ; but as soon as English people began to make charts they adopted the meridian that passed through their principal observatory of Greenwich as the standard from which all longitudes should be measured, and this meridian has now been adopted by many other countries ; France, however, takes the meridian of Paris. The adoption of one meridian as standard rather than another is purely arbitrary. The divisions of the Equator are made to begin where the standard meridian crosses it, and degrees are counted 1800 each way. The great circle, therefore, which passes through Greenwich is called the meridian of Greenwich or meridian o° on one side of the globe, and the 180th meridian on the other side, it being 1800 east and also 1800 west of the zero meridian. By setting a chronometer to Greenwich time and observing the hour of noon at various places their longitude can be determined, allowing 150 to each hour of time, because the earth turns once on its axis in 24 hours and there are 3600 in the entire circumference. It is of course to be understood that this is only a rough outline of the principles on which longitude is determined. The exact determination of longitude is a work of considerable difficulty, and the longitudes of the principal observatories have not even at the present day been fixed with that degree of accuracy which is necessary for certain delicate observations.

It is clear that if a globe have the lines of latitude and longitude drawn on it on the principles above stated, and the latitude and longitude of a certain number of places be determined by observation, these places can be plotted in their proper positions on the globe and the detail can be filled in by ordinary surveying. If in the same way lines to represent latitude and longitude be drawn on a piece of paper, the places can be plotted with reference to these lines and the detail filled in by surveying as before ; and the art of making maps consists in the first place in laying down the lines to represent latitude and longitude, either as nearly like the lines on the globe as is possible in transferring lines from a curved to a flat surface, or else in such a way that some one property of the lines will be retained at the expense of others. To transfer the irregular coastlines from a globe to a map would be a work of superhuman difficulty ; but to transfer regular lines is comparatively easy, and it is of course possible to lay down on a map lines corresponding to those which would be drawn on an imaginary globe many feet or even many miles in diameter. The lines of latitude and longitude may be laid down for every io°, or every degree, or every quarter degree, or less if necessary; but the principle is — lay down the lines, plot the principal points, and fill in by surveying. After one map has been made it may of course be copied even on to another projection, care being taken that the latitude and longitude of every point are kept correct throughout. It is clear that if the lines of latitude and longitude cannot be accurately laid down on a fiat surface, still less can the detail be laid down. This difficulty is got over as follows : — For the first maps such a moderate area is taken that there is no practical difference between the flat surface and the curved surface, and it will be assumed that the maps so made are bounded by lines of latitude and longitude.1 When a large number of these maps have been made it will be found that they cannot be joined together to lie fiat.

If they are carefully joined along the edges it will be found that they naturally adapt themselves to the shape of a globe ; therefore another piece of paper is taken and on it are laid down lines of latitude and longitude, and each map is copied so as to fill the place prepared for it.

Sometimes this can be done by simple reduction which does not affect accuracy, as the accuracy of a map is independent of the scale, but sometimes the correct map has to be enlarged in one direction and reduced in another, and sometimes it has to be twisted, and in almost every case it has to be manipulated to get it into its place. The work of making maps of large areas, therefore, consists of two parts : first, making correct maps of small areas, which may be called surveying ; and secondly, laying down lines of latitude and longitude into which these maps must be fitted, and this second part is called map projection.

1 The arrangement by which maps may be made of rectangular shape to join up flat over a limited area such as an English county will be dealt with in the second part of this treatise.

H MAPS: THEIR USES AND CONSTRUCTION

Chapter II

THE art of map projection, as stated above, consists in laying down lines of latitude and longitude on a flat piece of paper, either in such a way that correct maps of the various spaces can be filled in with as little distortion as possible, or else so that some one good property may be retained at the expense of others which are sacrificed. The map projector's work is therefore finished when these lines are laid down, but in the diagrams in this part of the treatise an outline of the continents will generally be added, so as to make the diagrams clearer to the ordinary reader.

The first system of projection which naturally suggests itself is to make a picture of the globe. When one looks at a globe one sees a little less than half, but if the globe is at a great distance one may assume without appreciable error that one sees half. Figs. 3 and 4 show two views of the earth on this projection, which is called ORTHOGRAPHIC. It will be seen in each case that while the centre of the map is good the edges are very much crowded, and the result is so unsatisfactory that one at once asks if it is not possible to compress the central portion into a smaller area, and so leave a little more space for the outer portions. The projection which is now to be explained meets this difficulty to some extent.

The principle on which it is founded is shown in

Fig. t, fig- 5- Suppose a globe to be cut in two at A B, and a piece of paper placed between the two halves. Then suppose straight rods like knitting-needles to be run through from any points, F and H, to the point E, passing through the paper at the points G and K. By using a sufficient number of points the lines of latitude and longitude can be marked out on the paper.

Three specimens of this projection, which is called STEREOGRAPHIC, are shown in figs. 6, 7, 8. The first of

Fig. 4 these is the Northern Hemisphere, the second what is usually called the Eastern Hemisphere, and the third the hemisphere the central point of which is the intersection of the parallel of 50 with the meridian of

Greenwich (a point, that is to say, close to London). On the diagrams lines are drawn for each io° or 200 of latitude and longitude, but it is clear that on a larger map lines could be drawn for every degree. Now a practically correct map can be made of a portion of the

Fig. 5 world measuring i° each way, because curvature in such a size is of no consequence. Suppose, then, that correct maps were made separately of all the little foursided portions, it would be found that by simply reduchu; each of them to the requisite scale it could be fitted

C almost exactly into the space to which it belonged. The words ' almost exactly ' are used because the edge nearest the centre of the map would have to be compressed a little, but so little that only very careful measurement would detect the difference.

Fig. 6

Here, then, at first sight, it appears that a perfect projection has been obtained, and as a matter of fact it is considered by most authorities to be the best projection of a hemisphere for general purposes, but it has one serious defect. It has been said 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

Fig. 7 other words, the shape of each space into which a plan has to be fitted is practically correct, but the size is less in proportion at the centre than at the edges, so that if a correct plan of an area at the edge of the map has to be reduced to a scale of 500 miles to an inch to

c 2 fit its allotted space, then a plan of an area at the centre has to be reduced to a scale of iooo miles to an inch. Thus a moderate area has its true shape, and even an area as large as India is not distorted to an extent to be

Fig. 8 visible to the ordinary observer, but to obtain this advantage relative size is sacrificed. It is somewhat as if a ruler was represented by a billiard cue. Any short length of one would look like a short length of the other, but in a long length there is great dissimilarity.

The projection which next naturally presents itself for consideration is the PROJECTION OF EQUAL AREA-, a projection very little used, and for which, apparently, there is no special name. In the projection just above

Fig. 9 described, area is sacrificed to shape. In this projection (see figs. 9 and 9a) shape is sacrificed to area, but there is this difference, that while the shape in stereographic projection is only approximately correct, the area in this projection is mathematically exact. Everyone knows that a piece of paper measuring 3 in. by 3 in. has the same area as a piece measuring 4 in. by 2\ in., and it is by altering shapes somewhat in this manner that true areas are obtained. The amount of distortion, though

Fig. 9A not very serious, is still sensible, as will be seen by comparing India on fig. 9 with India on Mercator's projection, where the shape of India is practically exact. But, before condemning this projection, it would be well to compare the relative sizes of India and Greenland on this map with their relative sizes on the Mercator (fig. 13). It may well be thought that the stupendous inaccuracy of Mercator as regards size is more misleading than the inaccuracy of this projection as regards shape. In any case this map is useful, and should be more widely used, in order to give people correct ideas

Fig. 10 as to relative sizes, particularly seeing that, as has been stated above, the ideas of areas so obtained will be absolutely and not only approximately correct.

The method of constructing this projection is shown on fig. 10 Suppose on this figure D is the North Pole and A B the Equator, and H F and K G the parallels of latitude of 6o° and 300 north respectively. With a pair of compasses take the length D B as radius and describe a circle. This will be the boundary of the map and will represent the Equator. With radii D F, D G, describe other circles. These will be the parallels of 6o° and 300. Describe as many more parallels in the same way as may be necessary. Draw the meridians radiating from the centre and the projection is made.

It will of course be seen that in filling in the details a difficulty arises. Suppose lines be drawn for every degree and the correct plans of the spaces be used which were used for stereographic projection. It will no longer be possible to make them fit by simply reducing the scale. What has to be done is to fit a square into a rectangfle thus :

It is clear that no alteration of scale will effect this object. What has to be done is to divide the spaces into squares and rectangles thus :

These divisions have to be made so small that a draughtsman can by eye copy into each rectangle the work shown in the corresponding square. The extent to which this dividing up is carried depends solely on the scale and the purpose for which the map is to be used. It is not suggested that it is carried to a very fine point in filling in the details in these diagrams, but the principle is the one universally adopted in copying from one projection to another.

Anyone who has followed this explanation so far will have noticed that on the stereographic polar projection the circles showing parallels of latitude are farther apart at the edge of the map than in the middle, while in the equal area projection the reverse is the case. This is not so easily seen on other hemispheres, but the principle there is the same, and the non-mathematical reader must accept the statement that the principles hold good throughout.

In the stereographic projection relative areas would be improved if the outer divisions were reduced in size and the inner divisions enlarged, and in the equal area projection shape would be improved if the outer divisions were enlarged and the inner ones reduced, and both these results would be obtained if the divisions between parallels of latitude were made equal throughout. It is possible to construct a projection on mathematical principles in which this will be nearly the case, but it seems much simpler to adopt equal divisions at once and call the projection the Arbitrary Projection. This projection is shown on figs. 11 and 12. A comparison between this and the stereographic will show that shapes are not greatly distorted, while a comparison with the equal area projection will show that areas have not been unduly sacrificed, and it seems difficult to decide definitely against this projection. It is very easily constructed, and if children were taught the geography of the world from projections of this sort their general ideas would probably be more correct than at present.