To the Editor of the Popular Science Monthly.
I MAKE the following comments on Prof. Schneider's second article about "The "Tides." All the objections to the statements in the first article remain in full force. The chief points of this second installment are two: 1. The disturbing action of the sun on the moon's motion; 2. The fall of the earth and the moon below their respective tangents, whereby it is sought to be proved that the moon approaches the earth at the time of opposition. If these two statements are shown to be wholly in error, the second article goes the way of the first.
On page 231, December number of this Journal, we find this: "Thus our moon moves faster, and, by a radius drawn to the earth, describes an area greater for the time, and has its orbit less curved, and therefore approaches nearer to the earth in the syzygies than in the quadratures. . . . The moon's distance from the earth in the syzygies is to its distance in the quadratures, in round numbers, as 69 to 70."
This extract, which from its form would seem to be from a single paragraph, is in reality from two widely-separated parts of Newton's works, and is besides inaccurate. The phrase "in round numbers" is neither in the original nor in Motte's translation, but instead of it there is cœteris paribus, and it is hardly necessary to say that the phrases are not exactly equivalent. But let these slips pass. I give two other extracts from the "Principia," book iii., prop, xiv., cor. i.: "The fixed stars are immovable, seeing they keep the same positions to the aphelions and the nodes of the planets."
Herein is a double error:
Again, book iii., prop, xxxvii., cor. 3: "The density of the moon is to the density of the earth. . . . as 11 to 9."
This is very far from the truth, and scores of other mistakes in the "Principia" are known to those who are familiar with that work. So "the best of authority" is sometimes at fault, and his conclusions are not always to be accepted blindly and with-out investigation. But if they are to be so accepted, as Mr. Schneider's way of parading his authority seems to imply, would it not be better to accept Newton's theory of the tides, which is the true theory, and so make an end of it. But that theory excludes Prof. Schneider's.
If Newton's statement concerning the distance of the earth and moon in the syzygies and quadratures is to be taken without qualification, then it is plainly wrong; and the mathematical proof that it is wrong can be found more or less fully developed in any of the following works on astronomy, viz., those of Woodhouse, Herschel, Lardner, Gummere, Loomis, Norton, Olmsted, Robinson, and others.
Further, there is the practical proof of the correctness of the other view in the calculated and the observed positions of the moon for every hour of every day in every year, these positions being carefully noted by a score of observers every day. There is no more possibility of universal error in these observations and calculations than there is that a person who says that two and two are four should be in error on that point.
Prof. Schneider's first statement, then, is all wrong; and, this failing, the second goes with it. But it is also easily shown by his own figures that his conclusion should be exactly the opposite to what he makes it. He says, "The distance the earth falls, in one second of time, toward the sun is about .12144+ of an inch," the moon toward the sun .12084 of an inch, and the moon toward the earth .05386 of an inch. The expression "falling toward the sun" evidently means "falling from the tangent;" any other meaning is false and absurd. With the correctness of these numbers I have nothing to do. Consider them correct; then at the end of a second the earth and moon will be farther apart than they were at the beginning. Here is the proof:
In the accompanying figure let M be the place of the moon at opposition, E, that of the earth, and S the sun; A, the place of the moon at the end of a second; B, that of the earth at the same instant. Then, since the moon at opposition moves in a second about two-thirds of a mile farther than the earth, the curve M A is longer than E B, and A is farther to the right than B, and the moon at A is below the point C by .0538 of an inch: the quantity being ascertained by supposing the earth to stand fast, while the moon moves forward with the difference of their motions. On this there can be no disagreement. The distance C A is about two-thirds of a mile. But if A is below the tangent .12084 of an inch, and B .12144 of an inch below its tangent, then B is farther from A then when the bodies were at M and E respectively. When the earth is at B it is at the same distance from the sun as it was at E—i.e., E S and B S are