As an illustration of the relation of the scale of a map to the amount of detail, which can well be represented on it without confusion, assume for a moment that an observer is stationed in a balloon, which can be raised or lowered or placed at any desired height above the ground, and in addition that he is provided with a horizontal screen on which he is able to trace the details of the landscape below. The eye of the observer well represents the lens of a camera, and the screen the focussing plate. Therefore to produce a perfect image or map of the ground below it will be necessary to assume that all parts are stationary, balloon, plate and eye. For convenience assume that the eye remains over the centre of the screen at a distance of two feet. At a height of four miles above the ground the scale of the image on the screen would be exactly six inches to one mile, or a reproduction of the popular county map, on which every detail of importance such as houses, roads, paths, and fences is shown, and the smallest scale on which any attempt is made to preserve the relative proportions of such details.
On such a scale the 1/100th part of an inch represents a distance of very nearly nine feet on the ground and consequently however accurate the map might be in its projection, as an image showing the relative positions of all objects of importance on the ground, the scale is clearly too small for the measurement of areas for valuation purposes, and it is but a reproduction of the larger cadastral map.
Again assume that the balloon is stationed at a height of twenty-four miles above the ground, and that the observer places his eye at the same distance of two feet above the screen and attempts to construct a map from the image on the screen, which is now reproduced at a scale of one mile to one inch, or the exact scale of the general map. It needs but little imagination to foretell that houses would be mere specks, roads, faint lines, and forests, masses of color, in other words, that it would be more instructive to consult the general map, on which all details are magnified to be clearly visible and topographic features brought out with great distinctness than to attempt to trace with unaided eye, from the image of objects at a distance of twenty-four miles, the course of streams or roads through forest or moor, or to judge of the relative elevations or modeling of the ground from the values of light and shade. Without an intimate local knowledge of the county there would be nothing to indicate the name or boundaries of villages, or estates or the political and other subdivisions of the land, which are most clearly indicated on the map, in unmistakable styles of lettering.
Another and more serious problem which would be lessened as the balloon receded from the earth would be the distortion in perspective produced by the irregularities of the surface. The higher points being nearer the balloon would appear in the image on larger scale than the lower, and only in the case of a perfectly level country, would it be possible to produce a map without distortion by the method proposed, and then only for a limited area.
As the balloon receded, the relative differences of elevation would bear a smaller and smaller proportion or ratio to the distance, in other words, the distortion would grow less until at an infinite distance it might be neglected.
We might conceive that the observer was stationed at an infinitely great distance, and provided with a series of magnifying lenses of suitable powers to produce maps of any desired scale, yet, beyond a limited area, he would still be confronted with the problem of eliminating the distortion produced by the curvature of the earth.
Such is the conception of an accurate map which is an attempt to produce on a plain surface or sheet of paper, a horizontal projection of objects on the ground, which will show the relative positions of every detail on any desired scale with as little distortion as possible, and on which distances may be measured in any direction, and areas computed with a degree of accuracy only limited by the scale.
When a survey of a small area is made, such as an estate or parish, which bears but a small proportion in area to the surface of the earth, curvature is neglected, distortion due to this cause being imperceptible, but in the survey of a large country it is of primary importance.
Returning to the conception of an observer stationed at an infinite distance his position with reference to the new general one-inch map of England and Wales would be in the plane of a meridian passing through Delamere in Cheshire, and the published quarter sheets would be a series of rectangles each 18 miles by 12 miles, containing an area of 216 square miles whose edges were parallel to, and at right angles to the central meridian.
Those of Scotland and Ireland have for each country a central meridian and projection.