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 B. C² C. D = ------. 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 16 16 × 1760 176 528 = ---- of a mile = --------- yards = --- yards = --- feet, 8000 8000 50 50 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 = ---------------- = 8008 B. C² inches. 7912 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.”