Lowering of the visible horizon, [1] is amount by which the visible horizon is lower than the horizontal plane that touches the Earth. In order to make clear what is meant by this, let C be the center of the Earth represented ( Figure I, Géographie [2] ) by the circle or globe BEM. Having drawn from any point A raised above the surface of the globe the tangents AB, AE, and the line AOC, it is evident that a spectator whose eye would be situated at point A would see the entire portion BOE of the Earth ending with the points touching B and E; so that the plane BE is, in fact, the horizon of the spectator placed at A. See Horizon.

This plane is lower than the horizontal plane FOD by the distance OG, which touches the Earth at O; and if the distance AO is rather small with respect to the radius of the Earth, the line OG is almost equal to the line AO. Thus if one has the distance AO, or the elevation of the eye of the viewer, evaluated in feet, one can easily find the versine [3] OG of the arc OE. For example, let AO = 5 feet, the versine OG of the arc OE will therefore be 5 feet, the total sine [4] or radius [5] of the Earth being 19,000,000 feet in round numbers [6]: thus we will find that the arc OE [7] is about 2 and one half minutes; consequently the arc BOE [8] will be 5 minutes: and as a degree of the Earth is 25 leagues, [9] it follows that if the Earth were perfectly round and smooth without any eminences, a man of ordinary height would be able to view a space of about two leagues across, or one league in any direction: at a height of 20 feet, the eye would be able to see two leagues in any direction; at a height of 45 feet, 3 leagues, etc.

On account of mountains, one can sometimes see more or less far than the distances just given. For example, because of the mountain NL ( Figure 1, number 2, Geography [10] ) placed between A and the point E, viewer A would not be able to see the part NE; whereas in the case of the mountain PQ, situated beyond B, the same viewer would to be able to see terrestrial objects that are beyond B but situated on the mountain above the visual ray AB.

The lowering of a star below the horizon is measured by the arc of the vertical circle [11] that is found below the horizon, between the star and the horizon. See Star, Vertical.

1. Today a distinction is drawn between the visible horizon, which refers to the meeting of sky and whatever is on the surface of the earth that an observer can actually see, and the true horizon, which refers to the apparent meeting of the theoretically smooth surface of the Earth with the sky. The discussion by D’Alembert in this article refers, evidently, to the true horizon.

2. As reproduced from Recueil de planches, sur les sciences, les arts libéraux, et les arts méchaniques, avec leur explication. Quatrième livraison. Géographie. Planche I.

3. The versine ( sinus verse in French) is a trigonometric function of an angle that is now rarely used, though it appears in some of the earliest trigonometric tables made in India in the fourth and fifth centuries CE. Historically this function was used in calculations for the purposes of navigation. D’Alembert uses it here because the versine is equal to the ratio OG/OC. In the figure to which d’Alembert refers, OG = 5 feet and OC (the distance to the center of the Earth, i.e. its radius) = 19,000,000 feet. If OC were made the unit distance, it can be calculated that OG would be 0.00000026316 or 5/19,000,000. This angle can be calculated to be 0.41567 degrees, which is very close, as d’Alembert points out, to an angle of 2.5 minutes, which is equal to 0.41667 degrees.

4. This expression is synonymous with radius. In Ozanam’s Cours de mathématique, tome 2 (1697) we read : « Le Sinus Total, qu’on appelle aussi Rayon, est le Sinus du Quart de Cercle, ou de 90 degrez, qui est toûjours égal au Demi-diametre du Cercle, & c’est à cause de cela qu’on l’appelle Rayon, & encore Sinus Total, parce qu’il est le plus grand de tous les Sinus qui peuvent être décrits dans un même Cercle » (Définition XXI).

5. D’Alembert uses the term rayon or ‘ray’.

6. D’Alembert is not using feet of the same length as ours. The pied de roi in mid-18th century France was about 6.6% larger than an English foot today, and 19,000,000 of d’Alembert’s feet equal 20,249,200 of our “English” feet —much closer to the distance of 20,900,000 feet that is often used today in calculations involving the radius of the Earth.

7. That is, the arc intercepted by the angle OCE. In relation to OC and OE, the radius of the earth, the distance OG is the versine of the angle that intercepts that arc.

8. That is, the angle BCE intercepting the arc BOE.

9. Or about sixty miles. In Jaucourt’s article Lieue we find, unfortunately, no statement of what value is standard for the Encyclopédie. Jaucourt writes, unhelpfully: “[T]he common league of France is two thousand five hundred geometrical paces, the short [league] is two thousand, the great [league] is three thousand five hundred, or even more.”

10. As reproduced from Recueil de planches, sur les sciences, les arts libéraux, et les arts méchaniques, avec leur explication. Quatrième livraison. Géographie. Planche I.

11. A vertical circle or cercle vertical is a great circle perpendicular to the horizon that lies on the imaginary sphere of arbitrarily large radius concentric with Earth upon which all objects in the observer’s sky can be thought of as projected; this imaginary sphere is known as the celestial sphere.