By Professor ELIAS SCHNEIDER.
APART of the theory of the tides presented in our textbooks has been pronounced absurd in my first article. It is also a matter of amazement that the effect of centrifugal force is entirely ignored in these textbooks. That the propelling force arising from this cause should be utterly disregarded in an explanation of the tides is very remarkable. And yet the existence of such a force is so easily demonstrated that nothing else seems necessary to prove it to be one of the causes of the tides, than what was presented in my first article. I will, however, give additional force to my reasoning by citing the results of actual experiment.
It may be shown that there is an actual difference in the amount of centrifugal force felt at any part of the earth's surface during different times of the twentyfour hours of one axial rotation; and also at different times of the earth's revolution around her centre of motion. Theory implies that when any portion of the earth's surface is moving toward that point in her orbit where such surface makes the most rapid sweep around the centre of motion, the greatest amount of centrifugal force must be felt at such surface; and that, when this part moves toward that point of the earth's orbit where it makes the slowest sweep around the centre of motion, the least amount of centrifugal force must be felt. Now, it is very evident that any portion of the earth's surface which is most remote from the centre of her motion, whether that centre be the sun or the centre of gravity between herself and the moon, makes the most rapid sweep, and that consequently her waters must feel the greatest amount of centrifugal force at that time.
Now, let us see what experiment tells us on this subject: A box has been made of proper dimensions, free within from all outside disturbance or motion of air. In this box is placed a steel frame, which moves like a gate on a very delicate hinge, so as to avoid all possible friction. A weight of nearly twenty pounds rests on this gate at about four feet from the hinge. The hinge, whose lower part is a mere point, or delicate pivot, and the weight, are in the same line, parallel with the meridian. The weight is free, as nearly as can be, to obey the power of its own inertia. In consequence of this it moves laterally once every twentyfour hours west and east, whenever the centrifugal force is increasing and decreasing.
From noon to midnight the earth's surface is moving toward that point where its motion is more rapid, and consequently it begins to feel an increasing amount of centrifugal force. This is indicated by the apparatus, for the weight, which rests on the gate, by virtue of its inertia, lags behind and makes an apparent motion westward. This motion is, of course, not real. The earth's surface moves eastward faster than the weight, and hence the weight appears to move westward. From midnight to noon the centrifugal force felt by the earth's surface diminishes, for it is then moving toward that point where its motion eastward is less rapid. This is also indicated by the apparatus, for the weight, having gradually acquired the same velocity eastward, remains stationary at midnight a very short time. But, soon after midnight, when the earth's surface begins to feel less centrifugal force, this weight, by virtue of its inertia, resists the change of motion, and therefore moves eastward as far as it moved westward before midnight.
This movement of the weight is greatest when newmoon occurs at midnight, for the earth then feels not only the centrifugal force produced by her revolution around the sun, but, in addition, that produced also by her revolution around the centre of gravity between herself and the moon.
The motion of the weight westward begins soon after midday, and reaches its highest acceleration at about 8 p. m.; the motion eastward begins soon after midnight, and reaches its highest acceleration at about 7 a. m.
I hope soon to make a new apparatus, which shall have a longer distance between the hinge and weight, and from it more marked results can be derived.
When a body moves in a curve around a centre, it feels the effect of two forces: the one, which I call centrifugal, is the impulse which puts the body in motion; the other, which I call centripetal, is the power which draws toward the centre and keeps the body from moving in a direct line. These are the only forces acting upon a body moving in a curve. The former is sometimes called tangential, but I prefer to call it centrifugal, for it is the only force which drives from the centre. There is no force acting directly from the centre. That which is often called centrifugal is really centripetal force, for the tension of the string in the following experiment is hot caused by any force acting on the body from the centre, but it is caused by a force drawing the body out of its rectilineal course, and toward the centre, compelling it to move in a curve.
Suppose the body E (Fig. 1) moves with a certain velocity in the Fig. 1. curve E C D, and that the string E S feels a known tension, just equal to its strength. Now, double the velocity, and the strength of the string must be increased fourfold to keep it from breaking, for the force drawing the body toward the centre must then be four times as great to keep it moving in the curve. Or, suppose the body moves from A toward B with a known velocity, and that on reaching E is acted upon by the string. The body is then made to take a curvilinear motion, and the string feels a tension drawing the body not directly from but toward the centre, and equal to a force necessary to keep the body from moving in a straight line. It may be remarked that, as action and reaction are equal, the tension is felt both ways. But the reader can easily see what I mean.
This law of motion can be still better illustrated by a reference to one of the satellites of the planet Neptune. The mean distance of this satellite is nearly equal to the distance of our moon from the earth. We may assume these distances to be exactly equal. Then, as at the same distance the centripetal force must increase as the square of the velocity, to keep the body moving in the curve, and as the velocity of this moon of Neptune is about four and a half times greater than that of our moon, the centripetal force, or the force of gravity produced by Neptune on this moon, must be (4.5)^{2} about twenty times as great as is the centripetal force or the gravitating power our earth produces on its moon. In other words, the planet Neptune is about twenty times as heavy as our earth, for weight is nothing else than the measure of gravity.
The preceding statements are sufficient to show what is meant by centrifugal and centripetal forces. Let us now see how these act on bodies moving in large and small curves, and how the waters on the earth's surface are driven by centrifugal force toward a line tangent to her orbit. Since the length of the orbital curve of the earth is very great, and therefore not much deflected from a straight line, the waters are driven very little above the usual surface, no matter how rapidly the earth herself may move in this curve. The centrifugal force or original impulse felt by the whole earth is very great, but that felt by her waters is hardly visible or sensible in midocean. For the tidewaves cannot get above the line tangent to the curve of the earth's orbit. The following illustration will show this:
Let A B C (Fig. 2) represent a part of the curve of the earth's orbit, in its motion around the central sun, and B D a line tangent
to the curve at the point B. Now it is very evident that no tidewave produced by centrifugal force can get higher above the curve of the orbit than this tangent line, and the distance between the curve and the tangent, as at E, is very small. The part of the earth's surface most remote from the sun has indeed a greater tendency to continue moving on in the straight line of the original impulse than any other part. The particles of water have a small degree of cohesion, and they will therefore continue to move a short distance along this tangent, but only a little above the usual surface of the earth.
The curve in which the surface of the earth moves around the centre of gravity between herself and the moon is much more deflected from a straight line. Here also the tidewave can rise no higher than to the line tangent to this curve. The distance of the point G (Fig. 3) from the curve is, however, much greater than the point E in Fig. 2 from its curve. The motion of the surface of the earth at H around the point C, the centre of gravity between herself and the moon, is only about sixtyfive miles an hour; while the surface at B (Fig. 2) Fig. 3 moves with a velocity of 68,000 miles an hour around the sun. Nevertheless, as the waters are driven toward these respective tangents by the effect of centrifugal force, the tidewave must be greatest where the distance between tangent and curve is the greater.

Fig. 4. 
Let us now proceed to prove by mathematical demonstration the falsity of the theory of the tides found in our textbooks. Herschel, in his "Outlines of Astronomy," uses the following language: "That the sun, or moon, should by its attractions heap up the waters of the ocean under it seems to them (objectors) very natural. That it should at the same time heap them up on the opposite side seems, on the contrary, palpably absurd. The error of this class of objectors. . . . consists in disregarding the attraction of the disturbing body on the mass of the earth, and looking on it as wholly effective on the superficial water. Were the earth, indeed, absolutely fixed, held in its place by an external force, and the water left free to move, no doubt the effect of the disturbing power would be to produce a single accumulation vertically under the disturbing body. But it is not by its whole attration, but by the difference of its attractions on the superficial water at both sides, and on the central mass, that the waters are raised; just as in the theory of the moon the difference of the sun's attractions on the moon and on the earth (regarded as movable, and as obeying that amount of attraction which is due to its situation) gives rise to a relative tendency in the moon to recede from the earth in conjunction and opposition, and to approach it in quadratures."
This language gives about the clearest presentation we have of the pullingaway doctrine. But there is no "tendency in the moon to recede from the earth in conjunction and opposition, and to approach it in quadratures." On the contrary, the tendency of the moon's motion is just the reverse—namely, to approach in conjunction and opposition, and to recede in quadratures. And if so in regard to the moon and earth, it must be still more so in regard to the earth and her waters under this influence alone, as can be demonstrated.
I am sustained in my position by the best of authority. "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." The authority I quote is Newton's "Principia."
Let us make a calculation, and apply it to the earth and her waters. The moon performs its revolution in 27^{d} 7^{h} 4349^{m}, which is equal to 2,360,60623 seconds. The seconds of time in which the moon makes one revolution around the earth is to one second of time as 1,296,000 seconds in a whole circle is to a fractional part of one second of a circle, which we will call x. Hence x $=$ 12960002360606_{2/3} $=$ .54901141 $+$, which is the fractional part of one second of the circle of the heavens the moon describes in one second of time. The semicircumference of a circle whose radius is one equals 3.141592653589 +. Hence one second of this semicircumference equals 3.141592653589648000 $=$ .0000048481368110 +, and the fractional part .54901141 $+$ of one second of this semicircumference is equal to .00000266168242648 +.
Let E M and E M’ represent the moon's distance from the earth, M M' the arc which the moon describes in one second of time, and A M’ the sine of this arc. Let E M’ equal 240,000 miles, the moon's distance, in round numbers, from the earth, and E C equal one mile. The arc B C, being very small, may be regarded as equal to its sine. The length of this arc we have already found. From similarity of triangles we have the following proportion: A M': B C:: E M': E C, or, by substituting the figures, A M': .00000266168242648:: 240000: 1. Therefore A M' $=$ .6388037823552 +, which is the sine of the arc passed over by the moon in one second of time. The cosine E A is equal to $\scriptstyle {\sqrt {{\frac {}{EM^{2}}}{\frac {}{AM^{2}}}}}={\sqrt {(240000)^{2}(.6388037823552)^{2}}}$ $+=$ 239999.9999991498535 +, which, subtracted E M, gives A M $=$ .0000008501464 +, and this fractional part of a mile, reduced to inches, gives .053865275+, the fractional part of an inch as the distance the moon falls from a tangent to its orbit in one second of time. Multiply this by the square of 60, and we get, when reduced, 16.159+ feet, the distance the moon descends in one minute, which is equal to 15.1+ Paris feet, the result obtained by Newton in his "Principia."
The distance the earth falls, in one second of time, toward the sun is about .12144+ of an inch, and the distance the moon falls toward the sun in one second, when in opposition, is about .12084 of an inch. This, added to the distance the moon falls toward the earth in one second, makes .17470 +. Now, .17470 $$ .12144 $=$ .05326. Hence the moon, when in opposition, moving faster toward the earth than the earth does toward the sun, by .05326 fractional part of an inch in a second, these two bodies have a tendency to get nearer to each other in this position. The same can be proved when the moon is in conjunction.
Now let us see how this same law affects the waters of the ocean. The earth moves toward the sun .12144 part of an inch in a second. The waters of the earth, on the side turned away from the sun, are only 4,000 miles farther from the sun than the centre of the earth. Gravity toward any body diminishes as the square of the distance increases. Hence these waters, influenced by the gravitating power of the sun alone, and not hindered by any intervening object, would fall toward the sun .12143 part of an inch in one second. Hence the earth has a tendency to move away from the waters with a velocity of .00001 part of an inch in one second—that is, if these waters were not influenced by the gravitating power of the earth, and only by that of the sun, the earth would be "pulled away" from its waters at the rate of only the 100,000th part of an inch in one second. But it must be remembered that the waters gravitate, in addition to this, toward the earth at the rate of 16.15+ feet in one second, and therefore these waters are depressed by gravity, and not elevated. The same may be proved in regard to lunar tides.
I close by saying that I am an earnest seeker of truth, and nothing but a sincere desire for truth has impelled me to write these two articles. Any person attempting to prove me in error, with the same good motive, will be kindly welcomed.