Note Book V. Chap. 2. Note (e) — BOOK V.
Reading edition
[e] If any should be so curious, to desire to know how far a Man’s Prospect reacheth, by Means of the Height of his Eye, supposing the Earth was an uninterrupted Globe; the Method is a common Case of right-angled plain Triangles, where two Sides, and an opposite Angle are given: Thus in Fig. 4. A H B is the Surface, or a great Circle of the terraqueous Globe; C the Center, H C its Semidiameter, E the Height of the Eye; and foreasmuch as H E is a Tangent, therefore the Angle at H is a right Angle: So that there are given H C 398,386 Miles, or 21034781 English Feet, (according to Book II. Chap. 2. Note (a);) C E the same Length with the Height of the Eye, on the Mast of a Ship, or at only a Man’s Height, &c. added to it; and E H C the opposite right Angle. By which three Parts given, it is easy to find all the other Parts of the Triangle. And first, the Angle at C, in order to find the Side H E, the Proportion is, As the Side C E, to the Angle at H; so the Side H C, to the Angle at E, which being substracted out of 90 gr. the Remainder is the Angle at C. And then, As the Angle at E, is to its opposite Side H C, or else as the Angle at H is to its opposite Side C E; so the Angle at C, to its opposite Side E H, the visible Horizon. Or the Labour may be shortned, by adding together the Logarithm of the Sum of the two given Sides, and the Logarithm of their Difference; the half of which two Logarithms, is the Logarithm of the Side requir’d, nearly. For an Example, We will take the two Sides in Yards, by Reason scarce any Table of Logarithms will serve us farther. The Semidiameter of the Earth is 7011594 Yards; the Height of the Eye is two Yards more, the Sum of both Sides, is 14023190.
Logar. of which Sum is, 7,1468468
Logar. of two Yards (the Differ.) is, 0,3010300
Sum of both Logar. 7,4478768
The half Sum, 3,7239384
is the Logar. of 5296 Yards = three Miles, which is the Length of the Line E H, or Distance the Eye can reach at six Feet Height.
This would be the Distance, on a perfect Globe, did the visual Rays come to the Eye in a strait Line; but by Means of the Refractions of the Atmosphere, distant Objects on the Horizon, appear higher than really they are, and may be seen at a greater Distance, especially on the Sea; which is a Matter of great Use, especially to discover at Sea the Land, Rocks, &c. and it is a great Act of the divine Providence, in the Contrivance and Convenience of the Atmosphere, which by this Means enlargeth the visible Horizon, and is all one, as if the terraqueous Globe was much larger than really it is. As to the Height of the Apparent above the true Level, or how much distant Objects are rais’d by the Refractions, the ingenious and accurate Gentlemen of the French Academy Royal, have given us a Table in their Measure of the Earth, Art. 12.