The crocodile said: "My friend and brother, give me your heart, so that my wife may eat it and give up starving herself."

Then the monkey laughed, and scolded him, and said: "You fool! You traitor! How can anyone get a second heart? Go home, and never come back under the rose-apple tree. You know the proverb:

"Whoever trusts a faithless friend
And twice in him believes,
Lays hold on death as willingly
As when a mule conceives."'

him my plan? I will do it." your heart.

Now the crocodile was embarrassed when he heard this, and he thought: "Oh, why was I such a fool as to tell If I can possibly win his confidence again, So he said: "My friend, she has no use for What I said was just a joke to test your sentiments. Please come to our house as a guest. Your brother's wife is most eager for you.'

The monkey said: "Rascal! Go away this moment. I will not come. For

The hungry man at nothing sticks;
The poor man has his heartless tricks.
Tell Handsome, madam, if he should
Inquire, I've left the well for good.

"How was that?" asked the crocodile. And the monkey

told this story.

ON THE UNIVERSALITY OF THE LAW OF

GRAVITATION*

A. O. LEUSCHNER

Newton deduced his law of gravitation from the previously known laws of falling bodies and from astronomical observations. According to his law, every particle of matter in the universe attracts every other particle of matter with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. It is not my purpose to review the fascinating history of science leading to the discovery of this fundamental law, nor is it my object to examine the law in the light of the remarkable new theories advanced in recent years in physics and physical chemistry. Newton did not prove the universality of the law of gravitation, but by a happy stroke of genius generalized a fact which he had found to be true in the case of the mutual attraction of the moon and the earth. The characterization of the law as universal implies that it should hold not only here on our earth, but also in our solar system, including the major and minor planets, comets, meteors, et cetera; and furthermore that it should hold beyond our solar system, among the stars surrounding us and forming our stellar system, and in fact in other stellar systems beyond our own, if there be such.

* Read February 7, 1916, before the Cosmos Club of the University of California, and March 25, before the Astronomical Society of the Pacific.

The remarkable confirmation which the law of gravitation has received since it was first pronounced by Newton in 1682, through the explanation of many previously unaccounted-for phenomena and the accurate prediction of the motion of the planets and other objects in our solar system, leaves no doubt of its general applicability in the sense in which it was pronounced by Newton; and yet, if we should investigate how far the astronomer can go in establishing the universality of the law of gravitation on the basis of observational evidence, we should find that, while we can go much farther than Newton did himself, the evidence available from astronomical science has its tremendous limitations. Furthermore, it is a remarkable fact that, although for over two centuries the law has been applied successfully to the explanation of the most intricate problems of motion, in and beyond our solar system, yet much confusion exists in the published works of some competent astronomers as to its complete demonstrability in regard to double stars, as evidenced in their attempts at a proof of the universality of the law. It therefore seems profitable to correct this confusion by setting forth, step by step, the manner in which astronomical science may serve to establish the law of gravitation and to point out the limitations which are imposed upon the astronomer in proving its universality. Fortunately, we can dispense with the complicated geometrical processes employed by Newton, mathematical methods of great elegance having become available since his time for the establishment of the law. Actual mathematical proofs, however, capable of verification by any mathematician, will be omitted here, in order to simplify the sequence of our conclusions.

The first result easy of demonstration is that if a particle is subject to a central force, the areas which are swept over by the line joining it with the center are proportional to the intervals of time in which they are described. The line joining the particle with the center is called the radius vector. Conversely, it can be shown that if the areas swept

over by the radius vector are proportional to the intervals of time, the attracting force acting on the particle is constantly directed towards the center. These results depend upon three laws, or axioms, of Newton, stated by him for the first time in 1686 in his Principia, though partly known to Galileo and Huyghens. These axioms are as follows:

(1) Every body continues in a fixed state of rest or of uniform motion in a straight line unless it is compelled to change that state by a force impressed upon it.

(2) The rate of change of motion is proportional to the force impressed and takes place in the direction of the straight line in which the force acts.

(3) To every action there is an equal and opposite reaction, or the mutual actions of two bodies are always equally and oppositely directed.

At the beginning of the seventeenth century the famous Kepler had deduced three laws of motion from the observations of Tycho Brahe. These laws are:

(1) The radius vector of each planet with respect to the sun as the origin sweeps over equal areas in equal times. (2) The orbit of each planet is an ellipse with the sun at one of its foci.

(3) The squares of the periods of the planets are to each other as the cubes of the semi-major axes of their respective orbits.

An ellipse is a curve which may readily be drawn by fastening two pins firmly on a piece of paper, attaching the ends of a string to the pins, then drawing the string taut with a pencil and carrying the pencil completely around the two pins, always keeping the string taut. The two points marked by the pins on the paper are called the foci of the ellipse. The straight line drawn through the two foci from one end of the ellipse to the other is called the major axis of the ellipse, its length being designated by 2a. The point midway between the two foci is the center of the ellipse. The distance of either focus from the center, in terms of the semi-major axis as unity, is called the eccen

It signifies the amount
Ellipses of different size

tricity and is denoted by e. that the focus is out of center. or shape may be drawn by varying the length of the string, or the eccentricity. The period of revolution of a planet is the time it takes the planet to move completely around the ellipse from a given point back to the same point. Kepler's laws, then, signify that each planet of the solar system moves in an ellipse of its own, around the sun, the sun being the common focus of all ellipses; that the radius vector, the line drawn from the sun to the position of a planet on its ellipse, describes equal areas in equal intervals of time; finally, that the square of the period of a planet divided by the cube of the semi major-axis gives the same quotient for every planet, or, expressed mathematically, is a constant. This is written:

p1
= c.

Kepler's laws are the result of a lifetime of empirical investigations or guesses, based on the observations of Tycho Brahe, particularly on his observations of Mars. These laws are by no means rigidly correct, but they represented the observations of Tycho Brahe within the accuracy with which he was able to make them. If Kepler had had at his disposal observations of the precision obtainable with present-day instruments, it is very doubtful whether he would have been satisfied to announce them. Their inexactness arises from the fact that they take no account of the interaction of the planets, but only of the action of the sun on each planet separately, so that the mass or weight of each planet is disregarded.

In any ellipse the radius vector drawn from the sun at the focus S to the planet at the point M shall be designated by r, and the angle formed by the radius vector r with the major axis by 0. From the converse of the law of areas, referred to above, it is evident that the planets of the solar system, according to Kepler's laws, are subject to a central force, but so far we have no indication as to