The Catholic World, Vol. 19, April 1874‐September 1874

by Various

When two planets, as is the case with the earth and Venus, both revolve in elliptical orbits around the sun, in virtue of the law of gravitation, then their respective times of orbital revolution are to each other as the cubes of their respective mean distances from the sun.

This is one of the laws of Kepler. It was announced by him as the wonderful result of seventeen long years of calculations. He took the data given by the observations of Tycho Brahe and of others, and those made by himself. He tried, by every imaginable form of arithmetical supposition, to combine them together somehow, and under the form of some mathematical law. This was his last result, perhaps the most surprising result of hard plodding, long-continued labor in the field of science. All honor to his memory. There are few discoveries in the mathematics of astronomy to be compared to this and the other laws of Kepler. He established them as experimental facts. The mathematical reason of them he did not learn.

Since his day, gravity has been discovered to be the bond which binds the solar system together, and its laws have been studied out. The differential and integral calculus, also discovered and perfected since his day, has enabled the scholar to grapple with intricate questions of higher mathematics, which, without its aid, would have remained insoluble. Availing themselves of the laws of gravity and of the aid of the calculus, astronomers have been able to give us a mathematical demonstration of Kepler's laws, which, from being mere isolated facts or numerical coincidences, have passed into the realm of scientific truths.

Now, we know the length of our own year—365.2422414 days; we know also the length of the year of Venus—224.70048625 days. If we divide the former by the latter, square the quotient, and then extract the cube root of this quotient, we shall obtain the number which indicates the proportion between the two mean distances. Applying this, we learn that if the distance of the earth from the sun be taken as 100,000,000 miles, the mean distance of Venus will be 72,333,240 miles. And consequently, when they are in the same direction from the sun, and supposing both to be at their mean distances from that luminary, the distance between them must be, according to the same proportion, 27,666,760 miles. It is obviously enough to know the actual value of either of those three distances to learn very easily the other two. The observations of the transit are intended to ascertain the last and smaller one. How this is done, and what [pg 150] difficulties are to be surmounted in doing it, we shall see further on. Just now we will remark that supposing the observer to have ascertained to the very furlong this distance, during the transit, between the planets, he must still do much before he can apply his proportion. That holds good only for the mean distances. There are only two points in the orbit or ellipse of each planet around the sun which are at the mean distance from that focus. Were those points for both planets to be found on the lines of the nodes, the matter would be easy. But it is not so. In June, the earth is approaching her greatest distance; in December, she is nearing her smallest distance from the sun. A similar embarrassment exists for the orbit of Venus. But the astronomer can bravely grapple with this double difficulty. He has learned the eccentricity and consequent shape of each ellipse, and he can calculate how far, proportionately, the actual distance of either planet, at any given point of its orbit, exceeds or falls short of the true mean distance. Such calculations have to be made for the earth and for Venus as they will stand on the 8th of next December. When this is done, the astronomer is at liberty to make use of the actual distance learned by observation, and to apply the Keplerian formula.

But perhaps the question suggests itself, why take all this trouble of a circuitous route? Why not measure the distance of the sun directly, if such things can be done at all? If it is possible to measure the distance of Venus by observations, surely the sun, which has an apparent diameter thirty times as great, and which can be seen every day, and from any accessible point of the earth's surface, gives a far ampler field for such observations. If we have instruments so delicate as to disclose to us the presence in the sun of iron, copper, zinc, aluminium, sodium, manganese, magnesium, calcium, hydrogen, and other substances, surely it will be possible to determine that comparatively gross fact—its distance from the earth. And, in truth, what becomes of the lesson we learned in our school-days, that the sun was just ninety-five millions of miles away from us?

And yet, strange as it may seem to those unacquainted with the subject, it has been found impossible to decide, by direct observations, the actual distance; and the distance usually accepted was not derived from such observations. As for our lately acquired knowledge of some of the constituent substances of the sun, that is derived from the spectroscope, which as yet throws no light on the question of distance.

How do we ascertain the distance of bodies from us? Practice enables us to judge, and judge correctly, of the distance and size of things immediately around us almost without any consciousness of how we do it. But if we analyze the process, it will be found that we do it chiefly by using both eyes at the same time. They are separated by an interval of two and a half to three inches. As we look at an object near to us, the rays from each visible point of it must separate, in order to enter both eyes. The images thus formed on the retina of each eye differ sensibly, and we instinctively take cognizance of that difference. Speaking mathematically, the interval is a base line, at each end of which a delicate organism takes the angle [pg 151] of the object viewed, and our conclusion is based on our perception of the difference between them. Ordinarily, we estimate distances by the cross-sight thus obtained. When, however, the body is so far off that the lines of light from it to the eyes become so nearly parallel that the eyes fail to perceive the minute difference between the representations formed on the retina, then we must recur to the results of past experience, and judge, as best we may, of the distance from other data than that given us at the moment by our eyesight. Thus a sailor at sea judges of the distance of a vessel on the horizon from the faintness with which he sees her; for he knows that the intervening atmosphere absorbs some of the light, so that distant objects are dim. He judges from the fact that a vessel of the form and rig of the one he is looking at is usually of a given size, and a certain distance is required to cause the entire vessel to look so small, and certain portions, the size of which he is familiar with, to become indistinguishable. He is guided, also, by the amount to which, on account of the earth's curvatures, the vessel seems to be sunk below the horizon. These are data from experience. It is wonderful with what accuracy they enable him to judge. A landsman by the seaman's side, and without such aid, could give only the most random guesses as to the distance of the vessel.

That we really do make this use of both eyes in judging of the distance of bodies near us will be evident if we bandage one eye and try to determine their distances, only using the other. It will require caution to avoid mistakes. We knew an aged painter, who had lost the sight of one eye, but still continued to play, at least, with his brush. He had to use the finger of his left hand to ascertain by touch whether the tip of his brush, loaded with the proper color, was sufficiently near the canvas or not. If he relied on his eye alone, it often happened that when he thought it near, not the eighth of an inch away, it failed in reality by an inch and a half to reach the canvas. He would ply the brush, and, noticing that the color was not delivered, would smile sadly at what he called his effort to paint the air. So long as he had retained the use of both eyes, this mishap, of course, had never occurred to him.

When a surveyor desires to ascertain the distance of a visible object which he cannot approach, he must avail himself of the same principle of nature. He measures off on the ground where he is a suitable baseline, and takes the angle of the object from each end of it, not vaguely by his unaided eyesight alone, but with a well-graduated instrument. It is, as it were, putting his eyes that far apart, and taking the angles accurately. From the length of the measured base-line and the size of the two angles he can easily calculate the distance of the object. In taking such measurements, the surveyor must make his base sufficiently large in proportion to the distance sought. If the base be disproportionately small, the angles at the extremities will not serve. Their sum will be so near 180° that the possible errors which are ever present in observations will more than swallow up the difference left for the third angle, and the distance is not obtained. In our excellent Coast Survey, which, in exactness of working, is not surpassed anywhere in the world, the bases [pg 152] carefully measured may be five or seven miles long, and angles under 30° are avoided when possible.

From such measuring of distant objects on the surface of the earth, the passage was easy to an attempt to measure the distance of heavenly bodies. How far is the moon from us? It was soon found that a base of ten miles or of a hundred miles was entirely too short to give satisfactory angles. The moon was too distant. A far larger base was required. Suppose two places to be selected on the same meridian of longitude, and therefore agreeing in time, and situated sixty degrees of latitude apart. The distance between them will be equal to a radius of the earth. At each station, and at the same hours, the angles are taken which the moon makes with the zenith, or, better still, with some star near it, coming to the meridian at the same time. In such a case, the angles are, satisfactory. The base is large enough. The result of such observations, and of others which we need not dwell on, is that, when nearest to us, the centre of the moon is distant from the centre of the earth 222,430 miles; when at her greatest distance, 252,390 miles. These numbers are based on the fact that the equatorial radius or semi-diameter of the earth is 3962.57 miles. This value, however, may in reality be a quarter of a mile too short. The mean distance of the moon is roughly stated at 60 semi-diameters of the earth.