Chapter III: Part 3

TUTOR. As there are different degrees of heat and cold, the earth has been divided into five zones, namely, one torrid, two temperate, and two frigid zones.

PUPIL. How are they distinguished?

TUTOR. The torrid zone is all that space surrounding the globe contained between the tropics, having the equator running through the middle of it. It is so called on account of its excessive heat, for, twice every year the sun is vertical to the inhabitants, that is, he shines directly on their heads, and casts no shadow, but under their feet, at noon.

PUPIL. We find it sometimes extremely hot here in our summer; surely, in the torrid zone it must be almost insupportable?

TUTOR. They are inured to it from their infancy.—But we are departing from our subject.—The temperate zones are comprehended between the tropics and polar circles, that between the tropic of Cancer and the arctic circle is called the north temperate zone, and that between the tropic of Capricorn and the antarctic circle the south temperate zone.

PUPIL. I suppose they are called temperate because the heat is not so intense as in the torrid zone?

TUTOR. True. Neither is the cold so severe as in the frigid zones, which are those regions comprized within the polar circles, and are denominated north and south, as they are contiguous to the north or south poles.

PUPIL. Why are they called frigid?

TUTOR. They are called frigid or frozen zones, because near the poles there are perpetual fields of ice, the heat of the sun, even in summer, being insufficient to dissolve it.—Now try if you can tell me the breadth of each zone in degrees.

PUPIL. The torrid zone being twenty-three degrees and a half on each side the equator must be forty-seven degrees, which must also be the breadth of the frigid zones, as the polar circles are distant twenty-three degrees and a half from the poles, which are their centers. And, as from the equator to either pole is ninety degrees, from the equator to the tropics twenty-three and a half, and from the polar circle to the pole twenty-three and a half, if the sum of these, that is, forty-seven, be taken from ninety, the remainder, forty-three, will be the breadth of each of the temperate zones.

TUTOR. Very well.

PUPIL. From what you have told me I have no doubt but that the earth is globular, but I have no proof of it: I must therefore beg your assistance.

TUTOR. That it cannot be an extended plane, as some have imagined, is very evident; for, if it were, the angle made with that plane and the north pole star would be always equal, for reasons I have before given you: neither can it be cylindrical, that is like a garden roller, as others have supposed.—If a person travel northward the pole star becomes more elevated, and if he could penetrate to the north pole of the earth the star would be in the zenith, or directly over his head: on the contrary, if he travel southward, it is more and more depressed till he arrives at the equator, where the star is in the horizon; as he proceeds it disappears, and other stars rise to his view, invisible to us. Here then you see it must be circular northward and southward.

PUPIL. I am convinced it must be so.

TUTOR. And it is as certain that it is so east and west: for, navigators have often sailed round it steering the same course: that is, if they sail an easterly or westerly course at setting off, by continuing the same course they will return to the port whence they departed. This you know they could not do if it were not round, any more than an insect could, by crossing a round table, arrive at the place it set out from; but, by going round the edge it would be still going forward and come again to the point it had left.

PUPIL. It is very evident.

TUTOR. Again. In every direction, if a ship be seen at a distance, the first things observed are the top-mast and rigging, whilst the hull or body of the ship is hidden behind the convexity, that is roundness of the water, just as you would see a man coming over a hill, you would first see his head, he would be rising more and more to your view till he arrived at the top, where he would be full in sight.

PUPIL. I am at a loss to account for the convexity of the water. How can its surface be round?

TUTOR. Have you never observed the drops of water falling from the eaves of a house?

PUPIL. Often, Sir.

TUTOR. Of what shape were they?

PUPIL. Globular.—But what is the cause of their being so?

TUTOR. Attraction.—For as every particle of water which composes the drop tends to the same center, every part of the surface must be equidistant from the center, it must therefore be spherical. In like manner if you separate quicksilver, each portion will form itself into a globe.

PUPIL. All this is very clear. And, for the same reason, the water in the ocean must be convex; for, I remember you told me that it gravitated towards the center of the earth.

TUTOR. Once more.—I think you must have seen an eclipse of the moon.

PUPIL. I have, Sir.

TUTOR. Of what figure was the darkened part?

PUPIL. Circular.

TUTOR. Take this ball, and hold it before the candle between your finger and thumb, so that the shadow may be thrown on the wall, and in all positions you will find it circular.

PUPIL. It is so.

TUTOR. Apply this crown piece in the same manner, with the flat side to the candle.

PUPIL. It is a circle.

TUTOR. Turn it a little obliquely.

PUPIL. It is now an ellipsis.

TUTOR. Now turn the edge to the candle.

PUPIL. The shadow is a strait line.

TUTOR. You now see that no other body than that of a globe can in all positions cast a circular shadow.

PUPIL. I do, Sir.

TUTOR. The darkness on the disc of the moon at the time of an eclipse is the shadow of the earth, which in all situations is circular; the earth, therefore, which casts the shadow, must be a globe.

PUPIL. It must be so.—But——

TUTOR. The earth is mountainous.—It is so: but remember that the highest mountain bears no greater proportion to the bulk of the earth than the small irregularities on the peel of an orange bears to that fruit: that objection therefore is soon removed. And yet it is not a true sphere.

PUPIL. What then?

TUTOR. A spheroid, that is, it is a little flattened at the poles, and is in shape not unlike an orange or a turnip. This you will not be surprized at when I tell you that the equatorial parts are about four thousand miles from the center of motion.

PUPIL. I suppose then you infer that as the centrifugal force is greater the farther it is removed from the center, that the parts near the poles have a tendency to fly off towards the equator.

TUTOR. I do. And as we have finished this part of our subject, I shall take leave of you.

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DIALOGUE XI.

TUTOR.

I now propose giving you a description of the moon, and I doubt not it will afford you some degree of pleasure.

PUPIL. Indeed it will, as I know little more than that she is a secondary planet or satellite, revolving round the earth, and with it round the sun.

TUTOR. You know her mean distance from the earth.

PUPIL. I did not recollect that: 240 thousand miles.

TUTOR. Right. Her diameter is about 2161 miles, and her bulk about a fiftieth part of the earth’s. Her axis is almost perpendicular to the plane of the ecliptic, consequently she can have no diversity of seasons.

PUPIL. What is her period?

TUTOR. The time she takes to revolve from one point of the heavens to the same again is called her _siderial_ or _periodical revolution_, and is performed in 27 days, 7 hours, 43 minutes; but _synodical revolution_, or the time taken up to revolve from the sun to the same apparent situation with respect to the sun again, or from change to change, is 29 days, 12 hours, and 44 minutes.

PUPIL. I do not clearly comprehend it.

TUTOR. If the earth had no annual motion, the period of the moon would be uniformly 27 days, 7 hours, 43 minutes; but you are to consider that whilst the moon is revolving round the earth, the earth is advancing in its orbit, and of course she must be so much longer in completing her synodical revolution as the difference of time between that and her siderial revolution. This I will make clear to you in a few minutes.—What is the situation of the hour-hand and minute-hand of a watch at twelve o’clock?

PUPIL. They will be in conjunction.

TUTOR. And will they be in conjunction at one?

PUPIL. No, Sir.

TUTOR. Yet the minute-hand has made a complete revolution: but before they can be in conjunction again the minute-hand must move forward till it overtakes the hour-hand.

PUPIL. I now understand it, and must beg you to explain to me the different phases of the moon.

TUTOR. Take this ivory ball, and suspend it by the string with your hand between your eye and the candle. Let the candle represent the sun, the ball the moon, and your head the earth. In this situation, as the candle enlightens only one half of the ball, the part turned from you will be enlightened, and the part turned to you will be dark. This will be a representation of the moon at change, and as no part of her enlightened hemisphere is turned to the earth, she can reflect no light upon it, and consequently is invisible to us. She now rises and sets nearly with the sun.—Turn yourself a little to the left, and you will observe a streak of light like what is called the new moon.

PUPIL. I see it clearly.

TUTOR. Move round one quarter.

PUPIL. One half of the side next me is now enlightened.

TUTOR. You may conceive it to be the moon at first quarter.—Go on, and you will see the light increase till the ball is opposite to the candle, when the side next you will be wholly illumined, and will give you a just idea of the moon at full, which now rises about the time of sun-setting, being opposite to the sun: and, the farther she advances in her orbit the later she rises.

PUPIL. It is plain it must be so. She rises with the sun at change, being then in conjunction: and as she revolves in her orbit the same way as the earth does on its axis, the earth will have farther to revolve each day before it can see the moon. At the full she is in opposition, and of course rises when the sun sets: and so continues to rise later and later, till the change again.

TUTOR. You imagine that the moon rises exactly with the sun when she is at change; and when he sets, at full. I will presently convince you of your mistake; and would have you now proceed with your ball. Place it again opposite to the candle, and as you turn round you will find the light gradually decrease as it before increased, that the side that was before enlightened is now dark, and the dark side light. When you have gone three quarters round, one half of the side next you will be enlightened, and will resemble the moon at last quarter. As you go on the darkened part will increase, till you arrive at the place you set off from, where the light is quite obscured.

PUPIL. I have now completed the circuit, and am much delighted with it, as by this simple contrivance I can perceive the various changes of the moon, and that the western side is enlightened from the change to the full, and the eastern side from the full to the change.

TUTOR. I find then it has fully answered the purpose intended.

PUPIL. Indeed it has. But if you will give me leave I will use the ball again.

TUTOR. By all means.

PUPIL. I perceive, as I move round, that the same side of the ball is turned towards me whilst every part is turned to the candle. Is it so with the moon?

TUTOR. It is: and as every part of the moon is turned to the sun, she makes one revolution on her axis whilst she makes one in her orbit.

PUPIL. This is very singular. If the same side of the moon be always turned to the earth, the opposite side of course can never see it.

TUTOR. And they must likewise be deprived of the earth as a moon.

PUPIL. True. But how is it known that the same side of the moon is always opposed to the earth?

TUTOR. The moon, like our earth, consists of mountains and valleys, which, when seen through a good telescope, are very beautiful. The mountainous parts appear as lucid spots and bright streaks of light: and as the same spots, &c. are constantly turned to the earth, she must keep the same side to the earth.

PUPIL. It is very clear. Are there no seas?

TUTOR. It was formerly imagined that the dark parts were seas, but later observations prove that they are hollow places or caverns, which do not reflect the light of the sun. Besides, if there were seas there would consequently be exhalations, and if exhalations, clouds and vapours, and an atmosphere to support them. That there are no clouds is evident, because when our atmosphere is clear, and the moon above our horizon in the night-time, all her parts appear constantly with the same clear, serene, and calm aspect.

PUPIL. Has the moon then no atmosphere?

TUTOR. If she has it is imperceptible to us: for, when she approaches any star, we cannot discover with our best telescopes any change of colour or diminution of lustre in the star till the instant it is lost behind her: whence it is clear, that she can have no such gross medium as our atmosphere to surround her.

PUPIL. May we not then doubt whether she be inhabited or not, as without air we cannot breathe?

TUTOR. The same Almighty Being who created us and gave us air to breathe, may have provided a different way for their existence. It does not hold good that, because we could not live there, she is not inhabited. Fish will live a considerable time in water under an exhausted receiver: and, I have heard of a toad being found in a block of marble. Your doubt therefore, I think, ought not to be admitted.

PUPIL. I am satisfied. And must now beg to be informed how I may observe the moon’s motion.

TUTOR. Her real motion round the earth, may be easily known by remarking when she is near any particular star. Thus, suppose you see her west, that is to the right of it, she will be approaching, then in conjunction with, and afterwards pass it towards the east. Her apparent motion is that of rising and setting, which is occasioned by the rotation of the earth on its axis.

PUPIL. I remember not long since, when you shewed me Jupiter, that the moon was west of him: the next evening I saw her almost appear to touch him, and soon after at a great distance from him easterly. I now see that her real motion is from west by south to east, and her apparent motion from east by south to west.

TUTOR. If you have no objection, I will now explain the cause of eclipses.

PUPIL. So far from it, that it will give me the greatest pleasure.

TUTOR. Take your ivory ball, suspend it as before, in a right line between your eye and the candle.—Can you see the candle?

PUPIL. No, Sir.

TUTOR. For what reason.

PUPIL. Because the ball prevents the light coming to me.

TUTOR. This then represents an eclipse of the sun, which can never happen but when the moon is between the sun and the earth, which must be at the change: for, as light passes in a right line, the sun is hidden to that part of the earth which is under the moon, and therefore he must be eclipsed. If the whole of the sun be obscured by the body of the moon, the eclipse is total: if only a part be darkened, it is a partial eclipse; and so many twelfth parts of the sun’s diameter, as the moon covers, so many digits are said to be eclipsed.

PUPIL. May not the word digit be applied to the moon as well as the sun?

TUTOR. It may: for it means a twelfth part of the diameter of either the sun, or the moon.

PUPIL. As you have now shewn me the cause of an eclipse of the sun, I am anxious to have that of the moon explained.

TUTOR. We must again have recourse to your little ball.—Turn yourself round till it is opposite to the candle in a line with your head, and you will see that no light can be thrown on it from the candle, because your head is between them. In like manner the rays of the sun are prevented falling on the moon, by the interposition of the earth: she must therefore be eclipsed.

PUPIL. I see it clearly. And as an eclipse of the sun happens when the moon is at change, that of the moon must be when she is at full; for, it is then only the earth’s shadow can fall on the moon, the earth being at no other time between the sun and her.

TUTOR. The diameter of the shadow is about three times that of the moon, and consequently the moon must be totally eclipsed whilst she continues in it. On the contrary, the shadow of the moon at an eclipse of the sun, covers so small a part of the earth’s surface, that the sun is totally or centrally eclipsed to but a small part of it; and its duration is very short. But a faint or partial shadow surrounds this darkened shade, in which the sun is more or less eclipsed, as the place is nearer to or farther from its center; this partial shadow is called the _penumbra_. I have prepared for you a little drawing, representing an eclipse both of the sun and moon, which I think will enable you better to understand what I have been explaining. (Plate IV. Fig. 1 and 2.) In the former, _p. p._ is the penumbra.

_T. Conder Sculp^t._]

PUPIL. In what does a central differ from a total eclipse?

TUTOR. An eclipse of the sun may be central, and not total; for, those who are under the point of the dark shadow, will see the edge of the sun like a fine luminous ring, all around the dark body of the moon when the sun is eclipsed at the moon’s greatest distance from the earth; but when she is nearest the earth at an eclipse of the sun, the eclipse is total. When the penumbra first touches the earth, the general eclipse begins; when it leaves the earth, the general eclipse ends. An eclipse of the moon always begins on the moon’s eastern side, and goes off on her western side; but an eclipse of the sun begins on the sun’s western side, and goes off on his eastern side. When the moon is eclipsed in either of her nodes, the eclipse is both central and total.

PUPIL. Pray, what is the reason we have not an eclipse at every full and change of the moon?

TUTOR. For the same reason that Mercury and Venus are not seen to pass over she sun’s disc at every inferior conjunction.

PUPIL. Is the orbit of the moon then inclined to the plane of the ecliptic?

TUTOR. It is: and no eclipse of the sun can happen but when the moon is within 17 degrees of either of her nodes: neither can there be one of the moon, unless she be within 12 degrees. At all other new moons she passeth either above or below the sun, as seen from the earth: and at all other full moons above or below the earth’s shadow, according as she is north or south of the ecliptic. You now see that the moon must sometimes rise before and sometimes after the sun at change, and before or after he sets at full.

PUPIL. I do, Sir, and am much obliged to you for this pleasing account of the moon, and of eclipses: and if you have any thing farther to observe, it will afford me additional pleasure.

TUTOR. You may, at some time or other, have an opportunity of seeing a total eclipse of the moon; it will therefore be necessary to prepare you for a phænomenon which otherwise you might be much surprized at, and that is, that after the moon is immersed in the earth’s shadow, she is still visible.

PUPIL. This is a phænomenon that I am not able to account for; for, the moon being an opaque body, she cannot shine by her own light[16], and the rays of the sun are prevented falling on her by the interposition of the earth, she cannot therefore shine by reflection.

[Footnote 16: Dr. Herschell supposes the moon and the rest of the
planets may have some inherent light: the side of the planet Venus,
turned from the sun, having been seen, as we see the moon soon after
the change.]

TUTOR. It is by reflection that we see her; for the rays of the sun which fall upon our atmosphere are refracted or bent into the earth’s shadow, and so falling upon the moon are reflected back to us. If we had no atmosphere, she would be totally dark, and of course invisible to us.

PUPIL. What is her appearance?

TUTOR. It is that of a dusky colour, somewhat like tarnished copper.—I have one thing more to remark before we quit this subject, which is, that the moon’s nodes have a retrograde or backward motion, in a direction contrary to the earth’s annual motion, and go through all the signs and degrees of the ecliptic in little less than nineteen years, when there will be a regular period of eclipses, or return of the same eclipses for many ages.

PUPIL. Pray, Sir, what do you propose for our next subject?

TUTOR. The ebbing and flowing of the sea, or cause of the tides.

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DIALOGUE XII.

TUTOR.

In order to explain the cause of the tides, I have since I saw you last prepared a little drawing for you, (Plate IV. fig. 3.) where S represents the sun, M the moon at change, E the center of the earth, and A B C D its surface, covered with water. It is obvious, from the principles of gravitation, that if the earth were at rest the water in the ocean would be truly spherical, if its figure were not altered by the action of some other power. But, daily experience proves that it is continually agitated.

PUPIL. What is the cause of this agitation?

TUTOR. The attraction of the sun and moon, particularly the latter: for, as she is so much nearer the earth than the sun, she attracts with a much greater force than he does, and consequently raises the water much higher, which, being a fluid, loses as it were its gravitating power, and yields to their superior force.

PUPIL. What proportion does the attractive power of the sun bear to that of the moon?

TUTOR. As three to ten. So when the moon is at change, the sun and moon being in conjunction, or on the same side of the earth, the action of both bodies is on the surface of the water, the moon raising it ten parts,[17] and the sun three, the sum of which is thirteen parts, represented by B _b_. Now it is evident, that if thirteen parts be added by the attractive power of those bodies, the same number of parts must be drawn off from some other part, as A _a_, C _c_. It will now be high-water under the moon at _b_, and its opposite side _d_, and low-water at _a_ and _c_.

[Footnote 17: By part here I do not mean any specific measure.]

PUPIL. That the attraction of the sun and moon must occasion a swelling of the waters on the side next them, I can readily conceive, and that this swell must cause a falling off at the sides: but that the tide should rise as high on the side opposite to the sun and moon, in a direction contrary to their attraction, is what I am not able to account for.

TUTOR. This difficulty will be removed when you consider that all bodies moving in circles have a constant tendency to fly off from their centers. Now, as the earth and moon move round their center of gravity, that part of the earth which is at any time opposite to the moon will have a greater centrifugal force than the side next her, and at the earth’s center the centrifugal force exactly balances the attractive force: therefore, as much water is thrown off by the centrifugal force on the side opposite to the moon, as is raised on the side next her by her attraction. Hence, it is plain, that at D, fig. 3, the centrifugal force must be greater than at the center E, and at E than B, because the part D is farther from the center of motion than the part B. On the contrary, the part B being nearer the moon than the center E, the attracting power must there be strongest, and weakest at D. And, as the two opposing powers balance each other at the earth’s center, the tides will rise as high on that side from the moon, by the excess of the centrifugal force, as they rise on the side next her by the excess of her attraction.

PUPIL. In this explanation you have mentioned nothing of the sun.

TUTOR. From what I have already said it must be plain to you that if there were no moon the sun by his attraction would raise a small tide on the side next him; and, it is as evident that the tides opposite would be raised as high by the centrifugal force: for the sun and earth, as well as the earth and moon, move round their center of gravity. This may be exemplified by an easy experiment. Take a flexible hoop, suppose of thin brass, tie a string to it and whirl it round your head, and it will assume an elliptical shape; the tightness of the string drawing out the side next to your hand, and the centrifugal force throwing off the other.

PUPIL. This I clearly comprehend.

TUTOR. I shall now refer you to the next figure, (fig. 4.) where F represents the moon at full: the sun and moon are in opposition, and yet the tide is as high on each side as in the former case. I wish you to shew me the cause.

PUPIL. I will use my endeavour to do it, Sir.

TUTOR. Then I doubt not you will accomplish it.

PUPIL. When the moon is at full, ten parts of water are raised from that side of the earth next her, by her attraction; and, as the side which is next her is opposite to the sun, three parts must be thrown off by his centrifugal force, the sum of which will be thirteen parts next the moon.—From the side opposite to the moon, and under the sun, ten parts are thrown off by her centrifugal force, and three raised by his attraction, making thirteen, the same as before.

TUTOR. I could not have done it better. These are called _Spring Tides_. But when the moon is in her quarters, the action of the sun and moon are in opposition to each other; that is, they act in contrary directions (see fig. 5.) The moon of herself would raise the water ten parts under her, and throw off ten parts by her centrifugal force on the opposite side; but, the sun being then in a line with the low-water, his action keeps the tides from falling so low there, and consequently from rising so high under and opposite to her. His power, therefore, on the low-water being three parts, leaves only seven parts for the high water, under and oppose the moon. These are called _Neap Tides_.

PUPIL. This is very plain.

TUTOR. You would naturally suppose that the tides ought to be highest directly under and opposite to the moon: that is, when the moon is due north and south. But we find, that in open seas, where the water flows freely, the moon is generally past the north and south meridian when it is high-water. For, if the moon’s attraction were to cease when she was past the meridian, the motion of ascent communicated to the water before that time would make it continue to rise for some time after: as the heat of the day is greater at three o’clock in the afternoon than it is at twelve; and it is hotter in July and August than in June, when the sun is highest and the days are longest.

PUPIL. These are convincing reasons. And, pray what time after the moon has passed the meridian, is it high-water?

TUTOR. If the earth were entirely covered with water, so that the tides might regularly follow the moon, she would always be three hours past the meridian of any given place when the tide was at the highest at that place. But, as the earth is not covered with water, the tides do not always answer to the same distance of the moon from the meridian at the same places, because the regular course of the tides is much interrupted by the different capes and corners of the land running out into the oceans and seas in different directions, and also by their running through shoals and channels. But, at whatever distance the moon is from the meridian on any given day, at any place, when the tide is at its height there, it will be so again the next day, much about the time when the moon is at the like distance from the meridian again.

PUPIL. Are not the tides later every day than they were the preceding day?

TUTOR. Yes; and the reason is obvious: for, whilst the earth is revolving on its axis in twenty-four hours, the moon will be advancing in her orbit; therefore the earth must turn as much more than round its axis before the same place which was under her can come to the same place again with respect to her, as she has advanced in her orbit during that interval of time, which is 50 minutes. This being divided by 4, gives 12-1/2 minutes; so that it will be 6 hours 12-1/2 minutes from high to low-water, and the same time from low to high-water: or 12 hours 25 minutes from high-water to high-water again.

PUPIL. This I understand perfectly well.

TUTOR. I have now finished my description of the tides, and having a little time to spare, if you wish to know how to find the proportionate magnitude of the planets with that of the earth, and to calculate their distances from the sun, I will employ it that way.

PUPIL. At our first conference I remember you shewed me the proportion that the other planets bear to the earth, with their periods and distances from the sun; but to have it in my power to make the calculations myself, will certainly give me great pleasure.

TUTOR. To find what proportion any planet bears to the earth; or, that one globe bears to another, you must observe that, _all spheres or globes are in proportion to one another as the cubes of their diameters_. So that you have nothing more to do than to cube the diameter of each, and divide the greatest by the least number, and the quotient will shew you the proportion that one bears to the other.

PUPIL. The operation appears very simple; but, as I do not know what a cube number is, I cannot perform it.

TUTOR. You cannot forget what a square number is.

PUPIL. The product of any number multiplied into itself is a square number, as 4 is the square of 2.

TUTOR. Any square number multiplied by its root, or first power, will be a cube number. Thus 4 multiplied by 2 will be 8, which is the cube of 2; 9 is the square or second power, and 27 the cube or third power of 3, &c. This you will perhaps better understand by

A TABLE OF

Roots. 1. 2. 3. 4. 5. 6. 7. 8. 9.
Squares. 1. 4. 9. 16. 25. 36. 49. 64. 81.
Cubes. 1. 8. 27. 64. 125. 216. 343. 512. 729.

PUPIL. I do, Sir; and am now prepared for an example.

TUTOR. The diameter of the sun is 893552 miles, of the earth 7920 miles; how much does the sun exceed the earth in magnitude?

PUPIL. The cube of 893522, the sun’s diameter, is 713371492260872648; and of 7920, the earth’s, 496793088000. And 713371492260872648 divided by 496793088000 is equal to 1435952, and so many times is the bulk of the sun greater than that of the earth.

TUTOR. This one example may suffice, as I intend by and by to give you a table of diameters, &c.; you may then calculate the rest at your leisure.

PUPIL. I shall now, Sir, be glad to have the other explained.

TUTOR. The periods of the planets, or the times they take to complete their revolutions in their orbits, are exactly known; and the mean distance of the earth from the sun has been also ascertained. Here, then, we have the periods of all, and the mean distance of one, to find the distances of the rest; which may be found by attending to the following proportion:

As the square of the period of any one planet,
Is to the cube of its mean distance from the sun;
So is the square of the period of any other planet,
To the cube of its mean distance.

The cube root of this quotient will be the distance sought.

PUPIL. Here again I find myself at a loss, as I have not learnt to extract the cube root.

TUTOR. I will give you [18]Doctor Turner’s rule, which I think will answer your purpose.

[Footnote 18: Young Geometrician’s Companion.]

“First, having set down the given number, or resolvend, make a dot over the unit figure, and so on over every third figure (towards the left hand in whole numbers, but towards the right hand in decimals); and so many dots as there are, so many figures will be in the root.

Next, seek the nearest cube to the first period; place its root in the quotient, and its cube set under the first period. Subtract it therefrom; and to the remainder bring down one figure only of the next period, which will be a dividend.

Then, square the figure put in the quotient, and multiply it by 3, for a divisor. Seek how often this divisor may be had in the dividend, and set the figure in the quotient, which will be the second place in the root.

Now, cube the figures in the root, and subtract it from the two first periods of the resolvend; and to the remainder bring down the first figure of the next period, for a new dividend. Square the figures in the quotient, and multiply it by 3, for a new divisor; then proceed in all respects as before, till the whole is finished.”

The following example will, I trust, make it clear to you.

EXAMPLE.

It is required to find the cube root of 15625.

. .
15625 (25
8
─────
12) 76
15625
─────
.....
═════

Point every third figure, and the first period will be 15; the nearest cube to which, in the table I gave you just now, you will find to be 8, and its root 2; the 8 you must place under the 15, and the 2 in the quotient: take 8 from 15 and 7 will remain, to which bring down 6, the first figure of the next period, and you have 76 for a dividend. The figure put in the quotient is 2, the square of which is 4, which multiplied by 3 is 12, for a divisor. Now 12 in 76 will be 5 times; cube 25, and you will have 15625, which, subtract from the resolvend, and nothing will remain; which shews that the resolvend is a cube number, and 25 its root.

PUPIL. You say 12 in 76 is 5 times; I should have said 6 times.

TUTOR. In common division it would be so; but as the cube of 26 would be greater than the resolvend from which you are to subtract it, it can go but 5 times.

PUPIL. Now, Sir, I think I have a sufficient knowledge of the rule to solve a problem.

TUTOR. The earth’s period is 365 days, and its mean distance from the sun 95 millions of miles; the period of Mercury is 88 days—what is his mean distance?

PUPIL. As the distance of the earth is given, I must make the square of 365 the first term, the cube of 95 the second, and the square of 88 the third term of the proportion.

TUTOR. Certainly.—Take your slate, or a piece of paper, prepare your numbers, and make your proportion.

PUPIL. I find the square of 365 = 133225; of 88 = 7744; and the cube of 95 = 857375.

Then 133225 : 857375 :: 7744 to a fourth term.

I now multiply the second and third terms together, and divide the product by the first, the quotient 49836 is the cube of the mean distance of Mercury from the sun in millions of miles, and the fourth term sought.

TUTOR. So far you are right. Now extract the root.

. .
49836 (36 3 36
27 3 36
─── ── ─────
27) 228 Sq. of 3 = 9 216
46656 Mul. by 3 108
───── ── ─────
3180 Divisor 27 1296
═════ ══ 36
─────
7776
3888
─────
Cube of 36 = 46656
═════

PUPIL. The root I find to be 36, which is the mean distance of Mercury from the sun, in millions of miles.

TUTOR. You now see, that although 27 in 228 will go 8 times, yet here it will go but 6 times; and, as there is a remainder, it shews you that the resolvend is not a cube number.

PUPIL. I see it clearly.

TUTOR. You now seem perfect in the rule; I shall therefore not trouble you with any more examples, but shall give you the table I promised you.

┌─────────────────────────────────────────────────────────────────────┐ │ TABLE. │ ├──────────┬──────────┬───────────────┬───────────────┬───────────────┤ │ Names │Diameters,│ Magnitude, │ Periods, │ Mean Distance │ │ of the │in English│ compared │ in │ from the Sun, │ │ PLANETS. │ Miles. │with the Earth.│Years and Days.│ in Mil. of │ │ │ │ │ │ Miles. │ ├──────────┼──────────┼───────────────┼───────────────┼───────────────┤ │Sun │ [A]893522│ 1435952 │ —— │ —— │ │ │ │ │ │ │ │Mercury │ 3261│ 1/14 │ 0 —— 88 │ 36 │ │ │ │ │ │ │ │Venus │ 7699│ 5/49 │ 0 —— 224 │ 68 │ │ │ │ │ │ │ │Earth │ 7920│ 1 │ 1 or 365 │ 95 │ │ │ │ │ │ │ │Moon │ 2161│ 1/49 │ —— │ —— │ │ │ │ │ │ │ │Mars │ 5312│ 1/3 │ 1 and 322 │ 145 │ │ │ │ │ │ │ │Jupiter │ 90255│ 1479 │ 11 —— 314 │ 494 │ │ │ │ │ │ │ │Saturn │ 80012│ 1031 │ 29 —— 167 │ 906 │ │ │ │ │ │ │ │Georgian │ 34217│ 82 │ 83 —— 121 │ 1812 │ └──────────┴──────────┴───────────────┴───────────────┴───────────────┘

[Footnote A: The Diameters were taken from Adams’s Lectures, Vol. IV.
p. 39.]

PUPIL. I shall take the first opportunity of calculating the rest, in which I am certain I shall have great satisfaction.

* * * * *

TUTOR. I have now conducted you through the elementary parts of astronomy, have given you a general view of the system of the world, and prepared you to pursue the study with profit and pleasure.—In your future researches, the more accurate you are, the more you will discover of regularity, symmetry, and order in the constitution of the frame of nature.

“Hail, Sov’reign Goodness! all-productive Mind!
“On all thy works thyself inscrib’d we find;
“How various all, how variously endow’d,
“How great their number, and each part how good!
“How perfect then must the Great Parent shine, ⎫
“Who, with one act of energy divine, ⎬
“Laid the vast plan, and finish’d the design!” ⎭

THE END.

------------------------------------------------------------------------

Directions to the Bookbinder.

Plate I. _to face the_ Title.
———— II. —— _page_ 40.
———— III. —— —— 88.
———— IV. —— —— 131.

------------------------------------------------------------------------

Transcriber’s note:

All instances of ‘disk’ changed to ‘disc’

Errata, instance of ‘disk’ on page 79 added, “—— 79. — 5. ⎭”

Page 11, ‘Years’ changed to ‘years,’ “130 years after Christ”

Page 20, ‘h e’ changed to ‘the,’ “would have as much the appearance”

Page 24, ‘cannon ball’ changed to ‘cannon-ball,’ “the time a cannon-ball would”

Page 63, comma changed to full stop after ‘TUTOR,’ “TUTOR. Why?”

Page 65, ‘a’ changed to ‘_a_,’ “carry a planet from A to _a_”

Page 74, ‘itaxis’ changed to ‘its axis,’ “if the earth revolve on its axis every”

Page 78, ‘Mercury’ struck after ‘Sun,’ “Sun, Venus, Mars, and Jupiter are known to revolve on their axes”

Page 93, ‘cancer’ changed to ‘Cancer,’ “is the _tropic of cancer_; that”

Page 93, ‘capricorn’ changed to ‘Capricorn,’ “the _tropic of capricorn_”

Page 115, ‘othes’ changed to ‘other,’ “and other stars rise to his”

Page 115, ‘bnt’ changed to ‘but,’ “out from; but, by going round”

Page 116, ‘it’s’ changed to ‘its,’ “How can its surface be round”

Page 128, full stop inserted after ‘eclipses,’ “explain the cause of eclipses.”