FIG. 1.
Let B. D. be a small portion of the Earth’s circumference, whose centre of curvature is A. and consequently all the points of this arc will be on a level. But a tangent B. C. meeting the vertical line A. D. in C. will be the apparent level at the point B. and therefore D. C. is the difference between the apparent and the true level at the point B.
The distance C. D. must be deducted from the observed height to have the true difference of level; or the differences between the distances of two points from the surface of the Earth or from the centre of curvature A. But we shall afterwards see how this correction may be avoided altogether in certain cases. To find an expression for C. D. we have Euclid, third book, 36 prop. which proves that B. C² = C. D. (2 A D × C D); but since in all cases of levelling C. D. is exceedingly small compared with 2 A. D., we may safely neglect C. D² and then B C² = 2 A. D × C. D. or C. D = B. C² 2 A. D. Hence the depression of the true level is equal to the square of the distance divided by twice the radius of the curvature of the Earth.
For example, taking a distance of four miles, the square of 4 = 16, and putting down twice the radius of the Earth’s curvature as in round figures about 8000 miles, we make the depression on four miles = 168000 of a mile = 16 × 17608000 yards = 17650 yards = 52850feet, or rather better than 10¹⁄₂ feet.
Or, if we take the mean radius of the Earth as the mean radius of its curvature, and consequently 2 A. D = 7,912 miles, then 5,280 feet being 1 mile, we shall have C. D. the depression in inches 5280 × 12 × B C² 7912 = 8008 B. C² inches.
The preceding remarks suppose the visual ray C. B. to be a straight line, whereas on account of the unequal densities of the air at different distances from the Earth, the rays of light are incurvated by refraction. The effect of this is to lessen the difference between the true and apparent levels, but in such an extremely variable and uncertain manner that if any constant or fixed allowance is made for it in formulæ or tables, it will often lead to a greater error than what it was intended to obviate. For though the refraction may at a mean compensate for about a seventh of the curvature of the earth, it sometimes exceeds a fifth, and at other times does not amount to a fifteenth. We have, therefore, made no allowance for refraction in the foregone formulæ.”
If the Earth is a globe, there cannot be a question that, however irregular the land may be in form, the water must have a convex surface. And as the difference between the true and apparent level, or the degree of curvature would be 8 inches in one mile, and in every succeeding mile 8 inches multiplied by the square of the distance, there can be no difficulty in detecting either its actual existence or proportion. Experiments made upon the sea have been objected to on account of its constantly-changing altitude; and the existence of banks and channels which produce a “a crowding” of the waters, currents, and other irregularities. Standing water has therefore been selected, and many important experiments have been made, the most simple of which is the following:—In the county of Cambridge there is an artificial river or canal, called the “Old Bedford.” It is upwards of twenty miles long, and passes in a straight line through that part of the fens called the “Bedford level” The water is nearly stationery—often entirely so, and throughout its entire length has no interruption from locks or water-gates; so that it is in every respect well adapted for ascertaining whether any and what amount of convexity really exists. A boat with a flag standing three feet above the water, was directed to sail from a place called “Welney Bridge,” to another place called “Welche’s Dam.” These two points are six statute miles apart. The observer, with a good telescope, was seated in the water as a bather (it being the summer season), with the eye not exceeding eight inches above the surface. The flag and the boat down to the water’s edge were clearly visible throughout the whole distance! From this observation it was concluded that the water did not decline to any degree from the line of sight; whereas the water would be 6 feet higher in the centre of the arc of 6 miles extent than at the two places Welney Bridge and Welche’s Dam; but as the eye of the observer was only eight inches above the water, the highest point of the surface would be at one mile from the place of observation; below which point the surface of the water at the end of the remaining five miles would be 16 feet 8 inches (5² × 8 = 200 inches). This will be rendered clear by the following diagram:—
FIG. 2.