In this volume Professor Mitchell gives a very clear, and, in the general plan pursued, a very good account of the methods and results of investigation in modern astronomy. He has explained with great fulness the laws of motion of the heavenly bodies, and has thus aimed at giving more than the collection of disconnected facts which frequently form the staple of elementary works on astronomy.
In doing this, however, he has fallen into errors so numerous, and occasionally so grave, that they are difficult to be accounted for, except on the supposition that some portions of the work were written in great haste. Passing over a few mere oversights, such as a statement from which it would follow that a transit of Venus occurred every eight years, mistakes of dates, etc., we cite the following.
On page 114, speaking of Kepler's third law, the author says, "And even those extraordinary objects, the revolving double stars, are subject to the same controlling law." Since Kepler's third law expresses a relationship between the motions of three bodies, two of which revolve around a third much larger than either, it is a logical impossibility that a system of only two bodies should conform to this law.
On page 182, it is stated, that Newton's proving, that, if a body revolved in an elliptical orbit with the sun as a focus, the force of gravitation toward the sun would always be in the inverse ratio of the square of its distance, "was equivalent to proving, that, if a body in space, free to move, received a single impulse, and at the same moment was attracted to a fixed centre by a force which diminished as the square of the distance at which it operated increased, such a body, thus deflected from its rectilinear path, would describe an ellipse," etc. Not only does this deduction, being made in the logical form,
If A is B, X is Y;
but X is Y;
therefore A is B,
not follow at all, but it is absolutely not true. The body under the circumstances might describe an hyperbola as welt as an ellipse, as Professor Mitchell himself subsequently remarks.
The author's explanation of the manner in which the attraction of the sun changes the position of the moon's orbit is entirely at fault. He supposes the line of nodes of the moon's orbit perpendicular to the line joining the centres of the earth and sun, and the moon to start from her ascending node toward the sun, and says that in this case the effect of the sun's attraction will be to diminish the inclination of the moon's orbit during the first half of the revolution, and thus cause the node to retrograde; and to increase it during the second half, and thus cause the nodes to retrograde. But the real effect of the sun's attraction, in the case supposed, would be to diminish the inclination during the first quarter of its revolution, to increase it during the second, to diminish it again during the third, and increase it again during the fourth, as shown by Newton a century and a half ago.
In Chapter XV. we find the greatest number of errors. Take, for example, the following computation of the diminution of gravity at the surface of the sun in consequence of the centrifugal force,--part of the data being, that a pound at the earth's surface will weigh twenty-eight pounds at the sun's surface, and that the centrifugal force at the earth's equator is 1/289 of gravity.
"Now, if the sun rotated in the same time as the earth, and their diameters were equal, the centrifugal force on the equators of the two orbs would be equal. But the sun's radius is about 111 times that of the earth, and if the period of rotation were the same, the centrifugal force at the sun's equator would be greater than that at the earth's in the ratio of 111 2 to 1, or, more exactly, in the ratio of 12,342.27 to 1. But the sun rotates on its axis much slower than the earth, requiring more than 25 days for one revolution. This will reduce the above in the ratio of 1 to 25 2, or 1 to 625; so that we shall have the earth's equatorial centrifugal force (1/289) × 12,342.27 ÷ 625 = 12,342.27/180,605 = 0.07 nearly for the sun's equatorial centrifugal force. Hence the weight before obtained, 28 pounds, must be reduced seven hundredths of its whole value, and we thus obtain 28 - 0.196 = 27.804 pounds as the true weight of one pound transported from the earth's equator to that of the sun."
In this calculation we have three errors, the effect of one of which would be to increase the true answer 111 times, of another 28 times, and of a third to diminish it 10 times; so that the final result is more than 300 times too great. If this result were correct, Leverrier would have no need of looking for intermercurial planets to account for the motion of the perihelion of Mercury; he would find a sufficient cause in the ellipticity of the sun.