Why the sun and the moon appear larger at the horizon than at the meridian

Malebranche’s theory, refuted by experience

Wallis was the first to believe that the long interposition of land, and even of clouds, makes the sun and the moon appear larger at the horizon than at the meridian.

Malebranche reinforced this opinion which Régis disputed.

Régis attributed it to refractions occurring in the vapors of the earth.

He was mistaken, because refractions actually produce the opposite effect of what Régis attributed to them.

But Father Malebranche was no less mistaken in maintaining that the imagination, struck by the long stretch of land and clouds at our horizon, represents the same star as larger at the end of these lands and clouds than when, having reached its highest point, it is seen without any interposition.

The simplest experiments refute Malebranche’s theory.

A few years ago, I examined this phenomenon directly.

I had cardboard tubes made, 7-8 feet long and half a foot in diameter.

I had children look at the sun on the horizon through these tubes — children whose imagination was not at all accustomed to judge the size of the sun by the expanse appearing between the star and their eyes.

They did not even see either the ground or the clouds. The tube left them only the view of the sun, and all saw it — as I did — much larger than at noon.

This experiment and several others led me to imagine another cause.

I had already begun, unfortunately, to form a “system,” when I cane upon the mathematical solution by Mr. Smith and spared me the errors of a hypothesis.

First, we must establish that, according to the rules of optics, the sky must appear to us as a flattened vault.

Here is a simple proof.

Our sight extends distinctly only to the point where objects form in our eye an angle of at least one eight-thousandth of an inch, according to Hooke’s observations.

A man OP (figure 20) five feet tall looks at object AB, also five feet tall and 25,000 feet away: he sees it under angle AB; but since this angle AB is not within the one eight-thousandth of an inch threshold in his eye, he does not see it distinctly.

If he looks at object C, the angle is even smaller; he sees it as if this object were in AD; thus everything behind C becomes even less distinct.

The houses and clouds behind C must appear to skim the horizon near G; therefore, all clouds sink for us toward the horizon at the distance of 25,000 feet — that is, about a league and two-thirds of 3,000 paces — and they sink gradually.

Consequently, all clouds that rise to G (figure 21), about three-quarters of a league high, must appear to skim our horizon.

Thus, instead of seeing the clouds at G as high as the cloud N, we see the clouds at G touch the earth, and the cloud N rise about three-quarters of a league above our head.

We therefore should see the sky neither as a ceiling nor as a circular arch, but as a flattened dome, whose large diameter BB is about six times larger than its small one AD.

We therefore see the sky in this manner BAB; and when the sun or moon are at B on the horizon, they appear farther away (to us who are at D) by about one-third more than when these stars are at A.

Now, we must see them under the angles that come to our eyes from B and A; it remains to examine those angles (figure 22).

At first it might seem that these angles should be smaller when the object is farther away, and larger when it is closer; but here the opposite is true.

The real star, the tangible star, moves in BDRE; but the apparent star follows the curve BAGG.

Now, the angles are formed by the apparent object.

Draw, therefore, lines of sight from the eye at P to the real positions of the star D — these lines necessarily graze the apparent positions of the stars.

You see, for example, that the angle is considerably larger at the horizon at G, and becomes quite small at C; the difference is greater at the meridian.

The star at the meridian has its disk like 3, and at the horizon roughly like 9; because the diameters of the star are proportional to its apparent distances:

• the apparent distance of the star is about 9 at the horizon, and 3 at the meridian — so is its apparent size.

This truth is confirmed by another experiment of a similar kind: look at two stars actually one-tenth of a degree apart; they appear much farther apart at the horizon, and much closer toward the meridian.

These two stars, always equally distant, are seen under angle FCD toward the horizon (figure 23), which is much larger than angle FAB at the meridian: you see that this apparent difference comes precisely from the same reason I have just reported.

Thus, according to this rule and to the observations that confirm it, here are the proportions of the apparent sizes and distances of the sun and the moon:

• At the horizon, these stars are seen as size 100
• At 15 degrees above, size 68;
• At 30 degrees, size 50;
• At 90 degrees, size 30.

Similarly, two stars that always maintain between them their true distance appear at the horizon as separated by 100, and at the meridian as 30 — which, as you see, is always the proportion of about 9 to 3.

This theory is confirmed by another observation. The moon appears considerably larger at certain times of the year than at others.

The sun also appears larger in winter than in summer; and the differences in this apparent size, being more noticeable toward the horizon than at the meridian, are more easily perceived.

The reason for this increase in size is that when the diameters of the moon and sun appear larger, these stars are indeed closer to us: the sun is closer to the earth in winter than in summer by about 1.2 million leagues; thus in winter it appears larger — but the width of its disk is somewhat diminished by the refractions of thick winter air.

The moon, in summer, is in its perigee; thus it appears under a larger diameter, and the width of its disk at the horizon is less diminished in summer than in winter, because the air, in summer, is finer and thinner.

This phenomenon, therefore, belongs entirely to geometry and optics.

Dr. Smith has found the solution to a problem on which the greatest minds had built useless systems.