Popular Astronomy: A Series of Lectures Delivered at Ipswich/Lecture 4

by George Biddell Airy

LECTURE IV.

 

Recapitulation of Lecture III.—General notions on Parallax.—Method of finding the Moon's Parallax and Distance.—Methods of finding the Sun's Parallax by Transits of Venus across the Sun's Disc.—Causes of failure of other methods.

IN the course of the present lecture I shall depart completely from that part of astronomical observation with which we have been engaged for some time, relating to the apparent motion of the heavens; and in leaving this part of the subject, I shall remark, as I have done before, that there is nothing of so much importance as that you should know the stars and the apparent diurnal motion of the heavenly bodies yourselves; and for this purpose the thing that you should have is a celestial globe. Every person who wishes to know anything of Astronomy should become acquainted with the principal constellations, so as to be able to recognize them at sight in the heavens; to observe their diurnal motion, and the difference in the appearances of the stars at different seasons of the year. This is of the greatest importance, in order to give you an idea of the apparent motion of the sun and planets among the stars.

There are two particular subjects which I have omitted to mention so fully as I would have wished, but I will now allude to them finally. The first is this. In treating of the apparent motion of the sun among the stars. I pointed out to you that it appears to describe a path through the stars which is inclined to the axis round which the heavens appear to turn; but I said that it is a great circle; which amounts to the same as saying that the path is in a plane which passes through the centre of the globe; or, to express it otherwise, the path is in a plane which passes through the eye of the spectator. If we take a pair of compasses, and open them so that one leg is square to the other, or (as we usually express it) that their angle of inclination is 90 degrees; and if we hold one leg fixed in position, and make it serve as a spindle round which the compasses are to be turned; then the other leg will move in a plane: and if the eye of an observer be placed at the angle of the compasses, the plane which is described by that moving leg will be seen by him as a great circle of the celestial sphere; while by looking along the fixed leg he will see a point in the sphere which is the Pole of that great circle: (the word "Pole" being here used in a general sense, as related to any great circle). From this it appears that the angular distance of the Pole of a great circle, from any point of the great circle is 90 degrees. If we then suppose a circle to be traced through the heavens, of which every point is 90 degrees from the North Pole, that circle will be a great circle, or will be in a plane passing through the eye of the spectator. That great circle is the equator. Now, as I mentioned in the second lecture, if we trace (by the use of the transit instrument and mural circle) the annual path of the sun through the stars, (which is called the ecliptic), we find that the ecliptic is also a great circle; but it is not the same great circle as the equator. It is inclined to the equator, and crosses it at two points, which are called the first point of Aries and the first point of Libra. The first point of Aries is that crossing of the equator at which the sun is seen at the Spring Equinox. The first point of Libra is that point at which the sun is seen at the Autumnal Equinox. Neither of these points is exactly marked by any star or other mark in the heavens: the first point of Aries is not very far from the third star of Pegasus (Algenib), and the first point of Libra is not far from the bright star in the tail of Leo, towards the bright star of Virgo.

The importance of acquiring a knowledge of these points is this. I have spoken of the method of using the transit instrument in combination with the use of the clock, which I said was to determine the interval of time between the meridian passage of some known bright star, and the meridian passage of any other object which we see in the heavens. It was plain, therefore, that by using it in this way we can determine the interval between the passage of the object and that of any star or every star; and also that we can determine, with the same precision, the intervals of the passages of all the bright stars; and we must do this, if we wish to make our representation of the heavens and of the position of an object in the heavens at all complete. This, however, would be a most tedious way of doing it. There is no good way of doing what is equivalent to this, except by referring every interval of passage to some one imaginary point or Zero: and the imaginary point or Zero which all Astronomers have found it convenient to adopt, is the first point of Aries. It is a point as I have said, which we cannot see in the heavens, but which we can determine by tracing the motion of the sun among the stars. We must observe, by means of the mural circle, on what day or between what days the sun is 90 degrees from the Pole; then the sun is necessarily at the place where his path crosses the equator, (which, as I have said, is 90 degrees from the Pole,) and therefore is necessarily either at the first point of Aries or the first point of Libra. We must, at the same time, by means of the transit instrument, determine the intervals between the passages of the sun and several fixed stars on those days; and then we shall have the interval between the passages of the first point of Aries and those stars. We can then use the first point of Aries so determined, as a starting point for sidereal time; and then instead of measuring our sidereal time from the passage of any star, we shall measure it from the passage of that imaginary point. Now, suppose that our observations of transits on any evening are to be compared with the observation of the transit of the bright star of Aquila; even though we do take that bright star of Aquila as the practical starting point of our observation, yet we do not make our clock to point hours, minutes, and seconds, when the star comes to the meridian, but we put our clock to point 19 hours, 43 minutes, and some seconds; because if we put our clock to point hours, minutes, and seconds, when the first point of Aries is passing the meridian, it shows 19 hours, 43 minutes, and some seconds, when the bright star of Aquila passes the meridian. And that is the use of the first point of Aries, which cannot be seen in the heavens. It is better that an imaginary Zero be chosen for the starting point than any one star; and the first point of Aries is peculiarly convenient, on account of its relation to the sun's path.

The next point of which I omitted to speak is, the difference between a sidereal day and a solar day. I pointed out to you how, by very rough means, the passage of the sun through the stars might be observed. I said that it might be observed by any person, if he watched, at a given hour of the night, the appearance of the stars on successive days and months: when he would find, on going from one month to the next in the order of succession, taking always the same hour of the night, that the stars appear to go round towards the right, or towards the west, (our faces being turned towards the south); which, as I explained, proves that the sun appears to go through the stars towards the left or the east. From this it is plain that the stars set a little earlier every day in reference to sun-time: or, that they pass the meridian a little earlier every day in reference to sun-time; and therefore, if we define the sidereal day to be the time that elapses from the passage of a star over the meridian one day to the passage of the same star another day, that interval or the sidereal day will be less than a solar day. It is, in fact, about 23 hours, 56 minutes, and 4 seconds of ordinary clock-time: the mean solar day being 24 hours. To sum it up in a few words, the stars appear to be going every day in their diurnal motion from east to west, and they appear to be passing the meridian quicker than the sun does. The sun appears to be travelling from west to east among the stars; and therefore, though the apparent diurnal motion of the sun through the heavens from east to west is quick, yet, in consequence of this apparent motion through the stars in an opposite direction, it passes the meridian slower than the stars do.

In speaking of surveys, I explained that by the use of the transit instrument and the theodolite, we might ascertain the angle made by any one side of a triangle with the meridian. This can of course be done at as many stations as we please; it is commonly done at two or three. In the English language we have no term for expressing that peculiar act of determining the direction of a side of a triangle, or the direction of a chain of triangles, and therefore we have adopted a word from the French, "orientation"; it is, however, a bad word, used only for the want of a better word in the English language. Where the word "orientation" is used, it is understood to mean the ascertaining the general direction of a chain of triangles.

In regard to the use of the Zenith Sector, of which I have spoken so frequently, I should wish you to charge your memory with this one notion: when that instrument is used for determining the measure of a degree of the earth, by being transported to two different stations, as A and B, Figure 18, and by being employed for observing the same chosen star at both places, the direction of the telescope is really the same at the two places, but the direction of the plumb-line is different at the two places. But, if we consider it only as a matter of observation at each of the places, then we fancy that the direction of the plumb-line is apparently the same in the two places, and that the direction of the telescope is apparently different. Thus the direction of the telescope, when pointed to the same star, is apparently different at Shanklin from what it is at Balta; but, in point of fact, the direction of the telescope is the same at both, and the direction of the plumb-line, or the direction in which a stone would fall, is different at the two places.

In yesterday's lecture I entered upon the subject of the motion of the planets; and I endeavoured to give you a notion of the complexity of the phenomena of the planets, and of the system first used to explain them. I remarked that for the inferior planets, this was made easier by referring their apparent motion to the sun. I pointed out that Venus moved (apparently) sometimes to the left of the sun, and sometimes to the right of the sun; never exceeding a certain angular limit to the right or left of the sun. I told you that her motion was sometimes backward and sometimes forward; and when referred to the sun, was pretty nearly similar, first on one side of the sun and then on the other side of the sun. All these conclusions are obtained by determining the places of Venus and the sun, with respect to the stars, by means of the transit instrument and mural circle, in the way described in my first lecture, and then laying them down on a globe, or treating them by other methods of calculation. There is no difficulty at all in figuring to ourselves, as the ancients supposed, that this apparent motion of Venus may be generally represented by supposing that she describes a circle round some imaginary centre, which centre is always in the line joining the earth and sun; and that the earth is at no very great distance outside that circle: as, for instance, in Figure 26, if we suppose Venus to revolve in tha circle V, whose centre v is in the line ES. They had similar notions in regard to Mercury, the centre of whose orbit might be supposed to be in some other place, as m.

With regard to Mars, Jupiter, and Saturn it was supposed by the ancients that they also moved in circles, and that the centres were somewhere in the same line ESmn, but that the circles were so large that they completely surrounded the earth. If we consider the apparent motion of Jupiter at the present time, when he is seen nearly opposite to the sun, rising nearly at sunset, and setting nearly at sunrise, and if we watch his course among the stars from day to day, or if we determine his place on different days by observations with the transit instrument and mural circle, we find that at this time Jupiter is moving among the stars towards the right hand.

It is convenient to have astronomical terms to describe this direction, without speaking of the right hand or left hand. Now, I have explained to you that the appearance of the stars, at different months of the year shows that the sun must be supposed to move through the stars towards the left; and the moon moves visibly towards the left. Astronomers therefore have agreed to describe this kind of motion by the term "direct," and the opposite motion by the term "retrograde." Now, the planets sometimes move in a retrograde direction: thus, when Mercury or Venus is in that part of its orbit which is nearest to the earth, its motion, as referred to the stars, is retrograde. And the apparent motion of Jupiter at the present time, from the description of it which I have just given, is retrograde; and so in all cases is that of Mars, Jupiter, Saturn, Uranus, Neptune, and the smaller planets, when they are seen on the side opposite to the sun. At other times their apparent motions are direct with respect to the stars.

All these motions are tolerably explained by the construction adopted by Ptolemy and the Greek astronomers: taking the assumption that the earth was fixed; that there was something like a bar, (Figure 26,) of which one end was fixed in the earth and which turned round in a year; that that bar carried the sun, and carried also the centres of the orbits of Mercury, Venus, Mars, Jupiter, and all the planets; and that all the planets revolved in their own orbits round their respective centres; the planets Mars, Jupiter, and Saturn, being supposed to have a retrograde motion with respect to the bar. I also said that it was found necessary, in order to account for the motion a little more accurately, to suppose that the planets did not revolve strictly in circles, but that the radial bar as nMa, Figure 26, carried another bar as MaN, jointed on it, and moving on the joint, and that this second bar carried the planet. Supposing a similar construction of each of the planets, we get a terrible complexity of motions, and all independent of the sun.

I then remarked, it was a strange thing that persons did not think of connecting these notions more closely with the sun. It would have answered their purpose quite as well to take one centre as another centre for the orbits of the planets, provided it were in the same direction, and provided the proper dimension were given to the orbit: that for instance, in Figure 27, the apparent motion of Venus, as seen from the earth E, would be the same, whether it moved in the small orbit V, whose centre is v, or in the large orbit V', whose centre is S. Having, then, the power of choosing the centre of the orbit as we please, we might as well take the sun for the place of the centre; and it is wonderful to me that such a simplification was not sooner adopted by the ancient astronomers.

The system Copernicus fixed upon in his successive steps was, first to bring all the centres of the orbits to the sun—still retaining the notion that this sun, together with the various orbits connected with it, were carried round the fixed earth—and then to suppose that the earth was in motion round the sun, (which would explain the appearances just as well,) or that the earth, as well as the other planets, moved round the sun. With this supposition, the motions of Mars, Jupiter, and Saturn are direct, but slower than that of the earth. And that was the state to which Copernicus reduced it; but still Copernicus was not freed from the notion that the small bars attached to the large ones carried the planets, as I have described. Kepler, a man who never spared his labour in working out any theory, after an infinity of trials, at last found out that he could represent everything perfectly well, by supposing that everyone of the planets moves in an ellipse, of which the sun is the focus; that the orbits of the different planets have different degrees of ellipticity; that the long axis of the ellipse is in different positions for each different orbit: of course it required an infinity of trouble to work that out. He established what is called Kepler's first law—that each of the planets revolves in an ellipse, of which the sun is one focus.

I will mention at once the third law which Kepler established, and which relates to the proportion of the periodic times of the different planets, and the proportion of their distances from the sun. Knowing the proportion of the distances of the planets from the sun, and knowing the periodic times of the planets round the sun, he was able to work out this rule. If we square the number of the days in the time of each of the planets going round the sun, we shall find that the squares of the times of revolution of the different planets are in the same proportion as the cubes of their mean distances from the sun. That was a most important thing to establish.

The second of Kepler's laws was this. In Figure 30, let S be the place of the sun in the focus of a planet's orbit; suppose that in one day the planet goes from K to L, and that in another part of the orbit it goes in one day from k to l; the law which Kepler made out was this: taking the areas which are included by the lines drawn from the extremities of these arcs straight to the sun; then the area KSL is equal to the area kSl. You will observe from this law, that it is quite evident that each planet moves quicker in that part of its orbit which is nearest to the sun than in that part of the orbit which is more distant from the sun; because the whole area described in one day or in a certain number of days is the same in the two cases. This was ascertained from observations, and without any notion of mechanical theory; Kepler did not possess any notion of that kind.

There is only one more matter which I will mention before I proceed to the mechanical consideration of the subject. Without knowing the distance of the earth from the sun, and without knowing the distances of the planets from the sun, we do know the proportion of their distances. This is because, as a matter of observation, we know how much the planet (Mercury or Venus suppose) appears to go to the right or to the left of the sun. In Figure 27, it will do just as well to explain the phenomena of the planet Venus, whether we suppose that the sun is at S, or whether we suppose that the sun is at v, provided we suppose that the dimensions of the orbit V' are large, and those of the orbit V are small, in the same proportion as the proportion of the distances of S and v from E. For instance we may make one supposition that the earth is a hundred millions of miles from the sun; and that Venus is seventy-two millions of miles from the sun. Or. we may make another supposition—that the earth is only fifty millions of miles from the sun, and that Venus is only thirty-six millions of miles from the sun. With the latter supposition (in which the distances are in the same proportion as in the former) we should find that Venus will appear to go just as much to the right or to the left of the sun as with the former. And, therefore, when we find that the apparent motions, computed on the supposition that the distances of the earth and Venus from the sun are respectively one hundred and seventy-two millions of miles, do agree with those which are really observed, we cannot tell whether the real distances are one hundred and seventy-two millions of miles, or fifty and thirty-six millions, or any other number; all that we know is, that they must be in that proportion. It is important to observe that this was the foundation of the third of Kepler's laws, and that he knew, as well as we do at the present time, what is the proportion of the distance of Venus from the sun to the distance of the earth from the sun, although he had not the slightest knowledge of the absolute distance of the earth from the sun.

I shall now proceed with the mention of the mechanical laws of orbital motion. In the first place, I shall take into consideration the general effects of attraction, or force; in which expression by the word force I mean pressure producing an effect on the motion of bodies that are free. Suppose we drop a stone from our hand, or from a high building, everybody knows that it begins to fall with a very small velocity, and that it gains velocity as it falls. If I were to drop a stone from my hand at the height of a foot from the floor, it would fall lightly; but if I were to drop it from the height of a hundred feet, it would fall with a great shock. It is therefore evident that any falling body is accelerated in its motion. This shows that the effect of gravitation is not to create a sudden velocity, but to add velocity to velocity, and continually to increase velocity. Now, that is a thing which you must consider in regard to the motion both of bodies falling in a straight line and of bodies projected and allowed to fall in a curve.

It will be remembered that I exhibited an experiment to this effect—that if any body were projected horizontally at the same time that another body was allowed to fall freely, they would both reach the ground at the same time. The apparatus, Figure 32, by which the experiment was made, is so constructed that, of necessity, when one body is projected forward the other is allowed to drop at the same time. Now. let us consider what sort of a curve the projected body would describe. Suppose a shot is projected from a cannon, as A in Figure 35; as I said before, if a ball would fall from the cannon's mouth to the point C in a second of time, then the shot which was fired out of the cannon would have dropped to D in one second of time. What sort of course would it have described? It would have fallen from its original direction in exactly the same proportion, so far as regards the divisions of the time, as the ball which dropped from the cannon's mouth. The ball dropping from the cannon's mouth does not acquire all its velocity downwards at once, but by degrees. In like manner, if this other ball is supposed to be thrown out horizontally towards B, it does not begin to drop suddenly, but drops more and more rapidly; it follows, therefore, that the path of the cannon-shot begins to turn more and more downwards, and assumes the form of the curve which is shown in the figure. You see that the form differs very little from the circle, and it is what mathematicians call a parabola.

The same consideration is applied to the motion of a planet, in this manner. We must suppose that the planet has been put in motion; we cannot tell how the planets have been put in motion, but they are in motion; that is sufficient for our purpose. The planets, if there were nothing to pull them aside, would go on in a straight course, without altering their velocity. The supposition which Newton made, and on which is founded the theory of gravitation, and which is perfectly conformable with every result of observation, is, that all the planets are attracted towards the sun; that the force is different in different parts, but still always directed towards the sun; that the force is of such a character that it is greater the nearer they go to the sun. Thus, if a planet started from P, Figure 30, in the direction PQ, it would go on in a straight line if it were not pulled by the attraction of the sun; but, by the attraction of the sun, the orbit becomes bent, and the planet describes the curved orbit PklMKL. Now, though this reasoning shows most clearly that the planet will move in a curved orbit of some kind, it is entirely impossible for me in this oral lecture to tell you how the precise nature of this curved orbit is found out; it is, however, found out completely; and I must beg of non-mathematicians to take my word for the result. When the investigation is conducted thoroughly we obtain these results. First, taking for granted Kepler's second law "that each planet, considered without reference to other planets, does in equal times describe equal areas by the line connecting it with the sun," which law was ascertained purely from observation: it is found that this is explained by supposing that the planet is at all times drawn by some force toward the sun. Secondly, supposing a planet put in motion, and then continually acted upon by any force whatever, directed always to the sun, it is found that it will describe equal areas in equal times by the line connecting it with the sun. Thirdly, taking for granted Kepler's first law, which was ascertained from observation only, "that the planets in their revolutions describe ellipses," we can ascertain what is the force with which they are drawn towards the sun, and which causes them to describe ellipses: it is found to be an attraction towards the sun following the law of the inverse square of the distance; that is to say, when the planet is half-way distant from the sun, it will be drawn with four times the strength, or, when the planet is one-third distant from the sun, it will be drawn with nine times the strength. Fourthly, we may propose to ourselves this problem: suppose the planet to be once put in motion, and then continually attracted by the sun with a force inversely as the square of the distance from the sun, what curve will it describe? It is found by the investigation which I have spoken of, that the curve which it will describe will be one or other of the following a circle, an ellipse, a parabola, or a hyperbola; and that the sun will be in the centre of the circle, or in the focus of the ellipse, parabola, or hyperbola. In nature we do not know any instance of the hyperbola; comets, as we shall see hereafter, for the most part move in parabolas; some comets and all the planets move in ellipses; and some of these ellipses approach very nearly to circles. I am exceedingly sorry that it is impossible for me to give you an idea of the steps of these investigations; but I say, and I am sure you will agree with me, that half a man's life would be well spent in mastering them.

Kepler's third law is this: that if we compare the orbits of the different planets, the squares of the periodic times are in the same proportion as the cubes of the mean distances from the sun. This, also, is in conformity with the result of the theory of attraction following the law of the inverse square of the distance. I suppose there is no science in the world in which such important laws have been first discovered independently from observation only, and which have afterwards been shown to be the result of one grand principle of theory.

I next endeavoured to point out to you how it is that planets do not entirely depart from the sun. It has been wondered by some persons that, when the planets approach to the sun, they are not compelled by its attractive force to fall into the sun; and when they go away from the sun, it has been wondered that they do not go quite away from the sun's influence. I endeavoured to give you a notion of the mechanical causes which produced that alteration in the velocity of the planets, which is in fact embodied in that law of Kepler's, which states that a planet describes, by the lines connecting it with the sun, equal areas in equal times. In introducing that matter, I said, that the curvature of the orbit of a planet, in the same manner as a curvature of the path of a cannon ball, depends not only upon the force which pulls the planet or the ball so as to curve its orbit, but also upon the amount of velocity with which it is; moving. Therefore, in order to ascertain what is the curvature of an orbit, we must consider not only the amount of the sun's attraction at any part of it, but also what is the amount of velocity with which the planet is moving at that point. If, then, it can be shown that when the planet is nearest to the sun it moves with a greater velocity, it will follow that, though the attractive force is greater than when farthest from the sun, its orbit may not be more curved than when it is farthest from the sun.

For this purpose, I introduced to you the model, represented in Figure 36. I pointed out to you that the tension of the cord EF, acting in the direction EF, does produce the effect of keeping in equilibrium the tension of the two other cords, one acting in the direction of EA, and the other in the direction EB; and therefore, a force acting in the direction EF, does produce two forces acting in the directions AE, BE. This is what we mean by the resolution of forces. The way in which I desire to apply this consideration to the motion of the planets is this. The sun's attraction acting in the direction MS, Figure 30, can there be resolved into two forces, in conformity with the law just mentioned; one in the direction of the line OM, touching the orbit, or in the same direction as the motion of the planet, and the other force in the direction NM, perpendicular to the orbit. As regards this part of the force which is perpendicular to the orbit, its effect may be considered as similar to that of the force of gravity on the cannon ball; its action is square to the planet's path at that time, and its tendency is to curve the planet's path. But the other resolved part pushes the body along in its orbit, so that the planet, instead of being allowed to go on in one uniform speed, is by the sun's attraction accelerated in its course. This amounts to so great a quantity that, at the time the planet has arrived at the part L of its orbit, it is going at a very great speed. When the planet has arrived at the part of the orbit where the force is so great, it has been accelerated so much that its velocity also is very great. If the body had not been moving quicker, its path would have been much curved; but in consequence of its great velocity, its path may be very little curved; the power of the sun may be unable to bridle it any longer, and it may go on from that point increasing its distance from the sun. When it has thus reached a point as M', if we resolve the sun's force into two forces, one perpendicular to the orbit, and one in the direction of the orbit, then that which is perpendicular to the orbit bends it, but that which is in the direction of the orbit retards it; and when it has got to a certain distance, its velocity is small indeed; and though it is so far off that the sun's force is very small, nevertheless, in consequence of the planet's diminished speed, the attractive power of the sun may be able to pull it in and make it describe the same orbit again; and thus the planet need not either fall into the sun when nearest, or go quite away when farthest.

There is another thing which I think it very proper to mention, because many persons have a very erroneous notion upon it. Some persons have a notion that there is some remarkable adjustment, so that if anything however small was to disturb the motion of a planet, it would either fall to the sun or go quite away from the sun. That is not the case; the effect of the disturbance of a planet would be to change its orbit, but nothing else. I will endeavour to point out to you what I mean. Suppose that a planet has been going on describing the orbit LPRM, Figure 39 , for ages, continually describing the same curve in an ellipse round the sun. Now, I will suppose that when it is nearest to the sun at L, something comes in the way and retards its motion. Many persons suppose that, in consequence, the planet would fall into the sun.
Fig. 39. No such thing will happen; the only effect it would have is this: it would cause the planet to describe a different orbit, such as is shown by the dotted line Lprm. Its velocity would be diminished by the interruption at L, and it would consequently be more bridled in by the attraction of the sun there, and the planet would then describe a new orbit of such a nature as to have a greater curvature at L; but if nothing disturbed it again, it would then go on continually describing that new orbit over and over again. Whenever the series of disturbances ceases, whatever orbit the planet is then moving in, it will continue to go on moving in that orbit. The planet's orbit is changed by any sudden disturbance, but the orbit so changed will continue, and the planet will be no nearer destruction than if it had not been disturbed at all.

There is only one more point regarding the law of gravitation, on which I shall here speak; it is the velocity, or the change of motion, which an attracting body produces on another body. I have spoken of attraction as if it was directed towards the sun; but we shall find that experiments of various kinds lead us to this conclusion: that every particle of matter attracts every other particle of matter; and that every planet attracts every other planet, that every planet attracts the sun, that the sun attracts the planets, that the sun attracts the moon, and the moon attracts the sun, and that every body attracts every other body. Now, the thing I wish you to understand is this: suppose Venus and the sun are at equal distances from the earth, then the earth pulls the sun out of its way just as much as it pulls Venus out of the way. The enormous difference of magnitude of the attracted bodies makes no difference in the movement which the action of the attracting body produces on them. If there are two bodies a great one and a little one, and if something else attracts them, the great body is pulled through as many feet or miles in an hour as the little one.

I shall say nothing more about this subject at present, but proceed at once with the measure of the distance of the various heavenly bodies. I endeavoured to give you a notion of what we call parallax; I endeavoured to illustrate it by showing the combined effect of the two eyes in our head. I pointed out to you, by a familiar experiment, that it is not easy to obtain, with a single eye, and when the head is held unmoved, a correct notion of the distance, but that if we open both eyes, we then get an accurate knowledge of the distance. This, I remarked, is exactly similar to the effect of observing the same object at two observatories, which are planted at two parts of the earth at a considerable distance from each other.

Now, in Figure 40, let GC be the earth, M the moon. I wish to measure the distance of the moon from the earth. I have two observatories from which I view the moon. One of them, G, we will suppose to be at Greenwich or at Cambridge; the other, C, to be at the Cape of Good Hope. In remarking on the grounds of a person's judgment of the distance of an object D, Figure 38, as observed with the two eyes, I said that it depends on this: that the object is seen by the two eyes in two different directions. In like manner the measure of the distance of the moon, by means of observations made at the two observatories G and C, will be based upon this circumstance: that the moon is seen in two different directions.

Fig. 40.

How can that difference of direction be ascertained? It can be ascertained by observing, at each of these observatories, the Polar distance of the moon. It will be remembered (see page 33) that, by the use of the mural circle at Cambridge, or at Greenwich, we observe the elevation of the celestial North Pole; and by the use of the same mural circle, we observe the elevation of the moon on the other side of the Zenith; or to use a more convenient measure, which we more frequently employ, we measure the angular distances both of the Pole and of the moon from the Zenith. Thus, suppose that by observations made at G on one side the Pole is 38 degrees from the Zenith, on the other side the moon is 70 degrees from the Zenith; if we add together these two angles, we shall see that the moon is 108 degrees from the North Pole, as seen at Greenwich, G. This 108 degrees is the measure of the angle PGM, GP being supposed to be directed to the celestial North Pole. At the same time, observations are going on at the Observatory C, at the Cape of Good Hope, where they cannot see the North Pole, but they can see the South Pole, and therefore they must refer their observations there to the South Pole; suppose that there they find the angular distance of the moon from the South Pole to be 731/2 degrees, this is the measure of the angle P'CM, CP' being supposed to be directed to the celestial South Pole, and therefore parallel to GP.

Now, we have got these two measures from which we see, that by the combinations of the Zenith distances at Greenwich, the moon is seen at 108 degrees from the North Pole; and by a similar combination of the Zenith distances at the Cape of Good Hope, the moon is seen 731/2 degrees from the South Pole. If we were observing a star S at an immense distance, we should get this relation between the two angular measures; that the sum of the two angular measures, one from the South Pole and the other from the North Pole, must be 180 degrees; inasmuch as the two directions GP and CP' are exactly opposite, and the two directions GS and CS, on account of the immense distance of a star, are exactly parallel; and therefore, in turning a line first from the position GP to GS, and then from the position GS or CS to CP', we have turned it exactly half round. But the thing which we have found out with regard to the moon is this: that the sum of the two angular measures is more than 180 degrees, for the measure at Greenwich being 108 degrees, and that at the Cape of Good Hope being 731/2 degrees, the sum of these is 1811/2 degrees. How are these 731/2 degrees to be accounted for? This angle of 11/2 degrees is the angle made by the two lines GM, CM. I have arrived thus at the conclusion: that at the distance of the moon, the angle between the two lines, of which one is directed to the Greenwich Observatory, and the other to the Observatory at the Cape of Good Hope, is 11/2 degrees. Having the position of Greenwich and the Cape of Good Hope, knowing the angle is 11/2 degrees, and knowing the directions of the lines from the two Observatories, I can compute the distance of the moon at once.

Here we must remark, that the observations of which I have spoken would be of no use, if we had not got the measure of the earth. That measure, however, as I explained in a former lecture, has been obtained by the aid of our yard measure; and it is worth while to recall the principal steps of the process. By means of the yard measure we measured a base line—in a survey by means of the base line, with triangulation, we obtained the length of some very long lines upon the earth's surface; and by the use of the Zenith Sector at different parts of the earth, in combination with the measures of these long lines, we got the general dimensions of the earth as expressed in yards; then, knowing the position of the Observatory at Greenwich, and knowing the position of the Observatory at the Cape of Good Hope, we have the means of getting from these general dimensions of the earth, the length of the line GC in yards, and its position in regard to the other lines; we know also that the angle GMC is 11/2 degrees, and therefore we have the means of computing the length of GM or CM expressed in reference to our yard measure; that is, of ascertaining how far off the moon is from the earth as expressed by a yard measure.

In the middle of the last century, the celebrated French Astronomer, the Abbé de la Caille, was sent to the Cape of Good Hope to make the requisite observations; observations were also made at the same time at the Observatories at Paris and at Greenwich, to determine the angle spoken of. A few years ago, Mr. Henderson and Mr. Maclear, who were successively sent to the Cape of Good Hope, were partly employed in making observations for the same purpose; and from the observations also made at the same time at Greenwich and at Cambridge, the distance of the moon from the earth has been determined.

I will now mention the only failure likely to take place in consequence of pursuing the method which I have described. It is this: I pointed out to you in the first lecture that a correction for refraction is necessary for every observation made with the mural circle. In consequence of that refraction all objects, in every part of the heavens, appear higher than they really are; a correction is applied for that circumstance, but that correction may be liable to a small uncertainty. The angular distance of the line GM from the line GP, directed to the North Pole of the heavens, may therefore be slightly in error: we can come very near the truth indeed, and perhaps we should not be wrong a single second, or half a second—but still there is always some uncertainty; and I may say, that refraction is the very abomination of astronomers. In like manner there may be a small error in the angle MCP', and it may happen that the two errors are combined, so as to affect the determination of the angle GMC by a larger error.

Now there is another way in which this observation can be shaped, in which the effect will not be quite so bad. This is by referring the place of the moon, as seen at each Observatory, not to the North and South Pole as I have spoken, but to a star. Let S, Figure 40, be a star at a very great distance, and suppose we observe the moon when she is nearly in the direction of that star. Now, the thing which it is our object to ascertain, is the angle made by the two lines GM, CM. At G the moon is seen somewhat below the star; we have then only to measure the angle SGM, about which there can be very little uncertainty, either from refraction or from any other cause. At C the moon is seen still more nearly in the same direction as the star; the angle SCM can therefore be measured with great accuracy. The angle GMC is the difference between the two angles SGM, SCM. Suppose, for instance, that at G the moon is seen two degrees below the star, and at C is seen only half a degree below the star, then the difference or the angle GMC must be 11/2 degrees; and this angle is scarcely liable to any possible error. We have then got this angle GMC accurately, and we have got the directions which the two lines GM, CM, make in reference to the line GC; and the calculation is then much the same as with a triangle in a survey, where we have a base measured from which we can begin our computations. This is the way in which the distance of the moon is measured; and we may say, as a general result, that the distance of the moon from the earth is about thirty times the breadth of the earth.

It is necessary now to explain with a little more precision that we mean by the word parallax. It is convenient to make all our calculations of the moon's place with reference to the centre of the earth. Now, in Figure 40, it will be seen that the moon, if viewed from the centre of the earth, would be seen in the direction EM; but, as viewed from Greenwich, she is seen in the direction GM. The difference between these two directions is the angle GME, and this is called the moon's parallax at Greenwich. In like manner, the angle EMC is the moon's parallax at the Cape of Good Hope; and therefore the angle GMC, which has been found by observations in the way already described, is the sum of the parallaxes of the moon at Greenwich and the Cape of Good Hope.

Now, the method in which the calculation of the moon's distance is actually effected is this. From a knowledge of the earth's dimensions, the length of the line EG is known with considerable accuracy. And though (as I stated in the second lecture) the plumb-line at G is not directed actually to the earth's centre, but in a slightly different direction, H'GE', yet, from knowing the form of the earth, we can calculate accurately how much it is inclined to the line HGE, which is directed to the earth's centre. Thus, we know the angle H'GH, and we have observed the angle H'GM with the mural circle, and the difference is the angle HGM, which therefore is known. Then we assume, for trial, a value of the distance EM. With the length EM, the length EG, and the angle HGM, it is easy to calculate the angle GME. The same process is used to calculate the angle CME. We then add these two calculated angles together, and find whether their sum is equal to the angle GMC, which we have found from observation. If the sum is not equal to that quantity found from observation, we must try another assumption for the length of EM, and go through the calculation again. And this we must do over and over again, till the numbers agree.

It has been supposed that the observations are made at the same instant at Greenwich and the Cape of Good Hope. This is not strictly correct; but the difference of time is known, and the moon's motion is well enough known to enable us to compute how much the angle P'CM changes in that time; and thus we can find what would have been the direction of CM, if the observation had been made at exactly the same instant as the observation at G.

After having got this notion of the value of the moon's distance, and knowing the method of computing the parallax, it is necessary to apply that computed parallax to every observation made, in order to find the position of the moon as seen from the centre of the earth. Now, if we have made a considerable series of observations of the position of the moon as viewed from the Observatory, and then, by calculating every parallax, if we have got the corresponding places of the moon as viewed from the centre of the earth, we find this, that the same law holds in the motion of the moon round the earth as in the motion of the earth and planets round the sun; that is, that the moon moves in an ellipse which is in a plane passing through the earth's centre.

When we consider it sufficiently established that the moon does revolve in a plane passing through the earth's centre, we can take that assumption as the basis of calculation to be compared with observations; and we can find what the moon's distance is, from observations at a single Observatory. The effect of parallax is always to make the object appear lower. If the moon were viewed from the centre of the earth, its path (which is in a plane passing through the centre of the earth) would appear to he a great circle inclined to the equator; and if we compared its places when nearest to the celestial North Pole, and when furthest from it, one of these angular distances from the celestial North Pole would be as much less than 90 degrees as the other is greater than 90 degrees; and their sum would he 180 degrees. But, in consequence of parallax, each of these angular distances as viewed at Greenwich is increased and therefore their sum is greater than 180 degrees. By ascertaining, therefore, what each of these distances is, and what their sum is, and how much that sum exceeds 180 degrees, we have the sum of two parallaxes; and from this we can find the moon's distance by calculation, (assuming a distance for trial, and altering it as often as may he necessary, and for every alteration computing the two parallaxes, adding them together, and seeing whether their sum agrees with the observed sum), nearly in the same way as when the moon was observed at Greenwich and the Cape of Good Hope. It is remarkable that this principle was used as long ago as by Ptolemy, (about A.D. 130) and a respectable estimation of the proportion of the moon's distance to the earth's diameter was obtained by him; but, as he did not know the dimensions of the earth, he was unable to express the moon's distance by an absolute measure.

I shall now proceed to a subject of much greater difficulty, viz., the computation of the distance of the sun from the earth. This most difficult problem might not have been accurately solved but for a suggestion of Dr. Halley, who, in the year 1716, published a paper in the Philosophical Transactions of the Royal Society of London. He was an old man at the time, nearly sixty years old. He explained what, as he said, would be a satisfactory method of solution, by observations of the "transit"^[1] (or "passage") of Venus over the sun's disc or face; and he pointed out the possibility of seeing the transits of Venus across the sun's disc, which were to occur in 1761 and 1769; and he bequeathed, as a task to posterity, the problem of ascertaining the distance of the sun from the earth. For understanding the following statement, it is important to remark, that the method of finding the distance of the sun by observation of the transit of Venus, requires that observations be made (as in the first method for the moon) at two Observatories at widely different positions on the earth. In 1761 a transit of Venus occurred, which was visible in many parts of Europe; it was necessary to observe it in other parts of the earth, and expeditions were sent out for that purpose; amongst others, Dr. Maskelyne was chosen by the Government to go to St. Helena, (where, however, clouds prevented any part of it being seen), and a Mr. Mason to the Cape of Good Hope. In 1769; another transit of Venus occurred, which was visible in the North of Lapland, but in no other part of Europe; it was necessary, in order to procure a good station for observations to be compared with those in Lapland, to send out an expedition to the Pacific Ocean. Captain Cook was sent out by the British Government to the South Seas, in the year 1769, in order to observe the transit of Venus in the island of Otaheite. I wish to mention with regard to this expedition, that so far as I can understand, the expenses incurred in that part which related to the ascertaining the distance of the sun from the earth, were defrayed from the private purse of George the Third.

The next transit of Venus will occur in the year 1874. It will be followed by one in 1882: after which there will be none for more than a century.

From the transits of Venus in 1761 and 1769, but especially from the latter, the sun's distance from the earth was ascertained to be about 951/2 millions of miles. It was long believed that this determination was all but perfectly accurate; but recent investigations seem to point to the conclusion that the value of the sun's distance so found is too great by more than three millions of miles. The source of error appears to be traced to the untrustworthy nature of the observations made in Lapland; and astronomers now look to the coming transits in 1874 and 1882 for the precise settlement of this important question.

When the distance of the sun is obtained, the distances of the other planets are easily found by calculation from the proportion of distances, which, as I said, was known long before the real distance of any one was known.

Before entering on the explanation of the principles of this method, I will point out to you the nature of other attempts which had been made to ascertain the distance of the sun from the earth, some of which were totally unsuccessful. In the first place, we might use the same method to ascertain the distance of the sun from the earth as that used for ascertaining the distance of the moon from the earth. We might take the angular distance of the sun from the North Pole, as viewed from an Observatory on one part of the earth (as Greenwich); and the angular distance of the sun from the South Pole, as viewed from another Observatory at another part of the earth (as the Cape of Good Hope); and then, by adding these two angles together, if the sum exceeded 180 degrees, we should consider that excess as due to the parallax, and should calculate the sun's distance in the same manner as in the case of the moon. But then comes in refraction, which is here a serious matter. The sum of the parallaxes of the sun at the two stations that is to say, the angle GMC, Figure 40, (supposing M now to represent the sun), does not exceed eight or nine seconds; it is an exceedingly small angle. The uncertainty of refraction in the observation of the sun is always two or three seconds; the air being hot at the time the observations are made, and the sun almost always appearing, in a telescope, tremulous and ill-defined. It is plain, therefore, that where there is such an uncertainty, it is useless to attempt to determine an angle which is not greater than eight or nine seconds. You may say, perhaps, that we could do the same thing with the sun as with the moon, by referring it to the stars. But there is this difference: we can see the moon, and we can see small stars close to the moon, and at the time that we observe the moon and the stars the air is not disturbed by the heat of the sun; everything is steady and is seen well; but in the heat of the day, objects are unsteady and are never seen well: we cannot see a faint star at all; and we can only see bright stars at a distance from the sun; therefore we are cut off entirely from observing the parallax of the sun in that way. We cannot observe its angular distance from the North and South Poles from the uncertainty of refraction, and we cannot compare the sun with the stars, because we cannot see stars near to the sun when the sun is shining.

There is a second method, a very ingenious one, proposed by astronomers in the middle ages. You will observe, by the preceding operations which I have mentioned, that we have made several good steps for the purpose of measuring the distance of the sun from the earth. By a yard measure we have got the length of an arc on the meridian, and from that we have found the dimensions of the earth, and from these we have got the distance of the moon; and possibly, from the distance of the moon from the earth we shall be able to get the distance of the sun from the earth. In Figure 41, let S be the sun, E

Fig. 41.

the earth, and M the moon, and suppose the distance of the sun to be not particularly greater than that of the moon. The sun illuminates half the moon at once, and no more. In the position which is represented in Figure 41, when the angle SME is a right angle, persons on the earth will see the moon half illuminated, but not at an angle of 90 degrees from the sun. For when the angle SME is 90 degrees, the angle SEM must be less than 90 degrees, inasmuch as the sum of the two angles SEM and ESM is 90 degrees. From this it is plain, as a matter of theory, that when we see the moon half illuminated she is less than 90 degrees from the sun, and the angle at which she really is distant from the sun depends upon the proportion of her distance to the sun's distance. If the sun's distance from the earth is not excessively greater than the moon's, then that angle SEM will be sensibly less than 90 degrees; but if the sun's distance from the earth is very much greater than the moon's, then that angle will be very nearly equal to 90 degrees. This observation, therefore, gives us, theoretically, the way to determine the sun's distance from the earth, if we can determine exactly the time when the moon is half illuminated. Unfortunately, the roughness of the moon's surface, which resembles very much a volcanic surface, makes it impossible to observe, with any degree of exactness, at what time the moon is half illuminated, and this principle fails totally in practice, from that cause.

The next method pointed out for ascertaining the distance of the sun from the earth, and which has been used with considerable success, is founded upon the observation of the planet Mars. Although the distances of the planets from the sun were not known to the ancient astronomers, yet the proportions of their distances were known to them. I have already pointed out to you that Venus is the planet which exemplifies this best. By observing how far Venus goes to the right and to the left of the sun, we can ascertain the proportion of the distance of Venus from the sun, to the distance of the earth from the sun, which is a proportion of 72 to 100 nearly; although we do not know the absolute distances at all The same remark applies to other planets; although we do not know their absolute distances, yet we can see or ascertain the proportions of the diameters of their orbits, which explain their different apparent movements. With regard to Mars, we are able to assert (knowing the proportion of the distances of Mars and of the earth from the sun, and the form of their orbits) that at certain times, when Mars is in opposition to the sun, the distance of Mars from the earth is only one-third of the distance of the sun from the earth; therefore, if we can ascertain the distance of Mars from the earth at that time, and multiply it by three, we get the distance of the sun from the earth. The distance of Mars can be got at with considerable accuracy, by observations made at Greenwich and the Cape of Good Hope, in exactly the same method as that which I have explained for getting the distance of the moon. Our observations as to the angular distance of Mars from the celestial North Pole must be referred to the same fixed star at the two places. In the Nautical Almanack, for several years past (a work which is published three or four years in advance,) there is prepared a list of stars, which it is recommended should be observed and compared with Mars, not only at the European Observatories, but also at the Cape of Good Hope and elsewhere in the Southern Hemisphere. Accordingly, at the opposition in 1862, observations were made in pursuance of these recommendations: and from them it has been deduced, that the distance of the sun from the earth is rather more than 911/2 millions of miles, a smaller value it will be observed than that found from the observations of the last transit. This method is a pretty good one; but it is not the most accurate method, which is that founded upon the transit of Venus, and to the explanation of it I now proceed.

There is one point with regard to the motion of the planets, to which I should wish to allude in the first place, as I have hitherto made no mention of it. It is rather important in reference to the transit of Venus. The orbit of each planet is in a plane passing through the sun, but the orbits of all the planets are not in the same plane; the orbit of Venus is not in the same plane as the orbit of the earth. In Figure 42 let S be the sun, E, the earth in its orbit,

Fig. 42.

and V, Venus in her orbit. It is plain that Venus in her motion crosses the plane of the earth's orbit in two positions, and no more. Suppose, now, that E and V represent the relative positions of the earth and Venus, at the time when they are, generally speaking, in the same direction from the sun. If a person on the earth looks at Venus now, he will see her above the sun. If it happen that the two are, generally speaking, in the same direction from the sun at E' and V', then when the conjunction takes place she will be seen below the sun. But there is one state of things to be considered, viz.: when the conjunction occurs at V" and E", Venus being very near that part of her orbit where it crosses the plane of the earth's orbit. At this time, if an observer on the earth were to look at Venus, he would see her upon the sun's face. The same would happen if the conjunction took place on the side of the orbit opposite to V" and E". It is a matter of great importance to ascertain at what times this will occur; at what time the conjunction takes place when Venus is near that point (called the node) of her orbit. Now, whenever the conjunction occurs at V", or on the side exactly opposite, then there is an opportunity of seeing Venus on the sun's face. The most celebrated transits of Venus, as I have said, occurred in 1761 and 1769; and the next will occur in the years 1874 and 1882.

I will now proceed to show you the method by which these transits are made available for measuring the distance of the sun from the earth. I must first point out to you what we know and what we do not know. From observing the distance which Venus goes to the right and left of the sun, we do know the proportion of the distance of Venus from the sun to the distance of the earth from the sun. This must be remembered carefully. We do not know the absolute distance of the earth from the sun, nor the absolute distance of Venus from the sun, but we know their proportion. Now, with respect to the diameter of the sun: suppose that I have an instrument like a pair of compasses, and that I use this so as to be able to observe the apparent angular breadth of the sun. I apply the joint of the compasses to my eye; I direct one of the legs to the lower part of the sun, and the other to the upper part of the sun; I then know the apparent angular breadth of the sun; and from this I determine the proportion of the absolute diameter of the sun to the distance of the sun, and whatever distance I assume for the sun I must take the diameter proportionately to that distance. If the distance of the sun is one hundred millions of miles, the breadth of the sun (roughly speaking) must be one million; if the distance of the sun is fifty millions of miles, the diameter of the sun is half-a-million of miles. Let Figure 43 be a perspective view of the state of things in which the sun's distance is supposed to be one hundred millions of miles; and the breadth of the sun is one million of miles. Let Figure 44 be a perspective view of the state of things in which the sun's distance is supposed

Fig. 43.

to be fifty millions of miles, and its breadth to be half-a-million of miles. In Figure 43 the distance of Venus from the sun must be seventy-two millions

Fig. 44.

of miles; but in Figure 44, the distance of Venus from the sun is only thirty-six millions of miles. Of all these measures we do not know anything absolute, we only know their proportions.

But there is one measure which we do know accurately. The diameter of the earth is nearly eight thousand miles, and from the knowledge of that we shall be able to determine all the others. It must then be carefully borne in mind that the diameter of the earth AB, in Figure 43, is the same as the diameter of the earth A'B', in Figure 44.

Suppose, then, that there is to occur a transit of Venus: that is to say a conjunction of the sun and Venus is about to take place, when Venus is near the node of her orbit, in which case Venus is seen to pass across the sun's face. We will suppose that the observation is to be made at two Observatories, one of which is near the North Pole of the earth, and the other near the South Pole of the earth. From the relation of the motions of different planets expressed in Kepler's third law (that the squares of the periodic times are proportional to the cubes of the distances), it follows that the periodic times increase in a greater proportion than the distances from the sun, and that the actual motions of the more distant planets in their orbits are slower. Thus, suppose one planet is 4 times as far from the sun as another; then its periodic time is 8 times as great (since the square of 8 is equal to the cube of 4); but the orbit which it describes is only 4 times as large; and thus it moves with only half the speed in that orbit. In like manner the motion of the earth in its orbit is slower than the motion of Venus in her orbit. Therefore, as the earth is moving in the direction from AB towards a, and Venus is moving with greater speed in the direction from V towards v, the inhabitants of the earth will see Venus moving apparently across the sun in the direction from E towards F, or from C towards D, (or in a retrograde direction). Now, first, let us examine in Figure 43 what will be the apparent path of Venus over the sun's face, as seen at the point A, near the North Pole of the earth. Let a straight line be drawn from A through V till it meet the sun's face, and the end of that line will describe the path CD on the sun's face, (considering the sun's face as a flat disc, which will suffice for this purpose). Thus from the northern station we see Venus travelling along the line CD. In a similar way we find that, from the point B on the south side of the earth, Venus is seen to travel along the line EF.

Suppose the distance between the points A and B to be 7000 miles, and let us calculate the distance between the two lines CD, EF. The supposition is, that the distance of the earth from the sun is one hundred millions of miles, and the distance of Venus from the sun seventy-two millions of miles, and consequently the distance of Venus from the earth, twenty-eight millions of miles. It follows that the interval between the lines CD, EF (which must have the same proportion to AB that 72 bears to 28), is 18,000 miles.

We will now go on the supposition represented in Figure 44, that the distance of the earth from the sun is only fifty millions of miles, and therefore that the distance of Venus from the sun is thirty-six millions of miles, because, as I have said, the proportion of the distance of Venus from the sun to the distance of the earth from the sun is known beforehand, and the distance of Venus from the earth fourteen millions of miles; and let C'D' be the path of Venus as viewed from A', and E'F' the path of Venus as viewed from B'; and, still supposing the distance between A' and B' to be 7000 miles, let us calculate what is the interval between C'D' and E'F'. This interval is in the same proportion to A'B' as 36 is to 14, and therefore it will be 18,000 miles, exactly the same as in Figure 43. Thus the interval between the two lines of the apparent path of Venus is the same, whether we suppose the earth's distance from the sun to be one hundred millions of miles, or fifty millions of miles.

But there is this cause of difference in their effects, that on one of these suppositions the whole diameter of the sun is one million of miles, but on the other it is only half-a-million of miles. Thus, as taken in absolute measures of miles, the parallel lines CD, EF, are just as far apart as the parallel lines C'D', E'F'; but they are drawn across a circle whose diameter in one case is double what it is in the other. But, as viewed from the earth, the apparent diameter of the sun is the same on the two suppositions, but the lines CD, EF, (if we could see them painted on the sun), would appear nearer together, on the supposition of Figure 43, than on that of Figure 44. Now, inasmuch as in the two Figures we make no difference of supposition as to the position of Venus as viewed from the earth's centre, the lines will cross the sun's disc in the same general position in both figures; and, therefore, they will meet the edge of the sun's disc at nearly the same angle^[2]; and, therefore, as the interval between them is the same, the difference of length in miles between CD and EF is the same as the difference of length between C'D' and E'F'.^[3] But as the lines CD and EF are about double the length of C'D' and E'F', the proportion of the difference of lengths to the whole length in Figure 43 is only half what it is in Figure 44. Thus, suppose the position of the lines to be such that CD is longer than EF by one-thirtieth part: then C'D' is longer than E'F' by one-fifteenth part. And if the earth's distance from the sun is one hundred-millions of miles, and if Venus be seen to cross the sun's disc in the direction which we have supposed, the difference of the times occupied by the passage of Venus over the sun, as viewed from A and from B, will be only one-thirtieth part of the whole time; but if the distance of the earth from the sun is only fifty millions of miles, the difference of times will be one-fifteenth part of the whole time.

Now we have come to something that can be compared immediately with observation. We can observe at the two stations A and B, the whole time that is occupied in the passage of Venus across the sun's face. And having ascertained the whole time of transit at each of these stations, we can take the difference between the two times, and find what proportion it bears to the whole time. If, (supposing the lines to cross the sun's disc in the direction supposed above,) the difference is found to be one-thirtieth of the whole, then we conclude that the sun's distance is one hundred millions of miles; if it is one-fifteenth of the whole, we conclude that the distance is fifty millions of miles; and so from any result of observation as to the difference of times occupied in the passage, we draw an inference as to the sun's distance.

It is now proper to remark that the observations cannot be made strictly as we have supposed, at two stations, A and B, which preserve the same relative position during the whole transit, because the earth is during all this time revolving on its axis. And it will be worth while to remark how advantage may be taken of this circumstance to increase the difference of times of passage of Venus over the sun's disc as seen at the two places, and thus to render the result more accurate. (For if, on the supposition that the sun's distance is one hundred millions of miles, we can by proper choice of stations increase the difference of times from 16 minutes to 24 minutes, and if the uncertainty in observation from chance errors is five seconds, then that uncertainty is 1/192 of the whole in the former case, and 1/288 of the whole in the latter case; and the proportionate uncertainty in the sun's distance will be the same.) The transit of Venus in 1769, occurred on June 3, a day very near to the summer solstice. The North Pole of the earth was turned partly towards the sun. Venus appeared to pass across the sun's disc from left to right, or in the retrograde direction. The earth was revolving from right to left, or in the way which we call direct. Now let Figure 45 represent the view which would be had by a person (for instance, an inhabitant of the planet Mars,
Fig. 45. if it had happened to be in the proper position) looking over the earth, and seeing the sun beyond it. To avoid confusion, Venus is omitted from this figure. The South Pole, P', of the earth would be seen by him, but the North Pole, being nearer the sun, would be invisible. A station near the South Pole would, in the course of its diurnal revolution, describe a circle, of which the greater part (represented by the dark arc of a circle) would be towards his eye, and the smaller part would be on the side next the sun. A station near the North Pole would describe a circle, of which the smaller part would be towards his eye, and the greater part towards the sun. The whole passage of Venus occupied about six hours. Now, it was possible to choose a station A (Wardhoe, in Lapland) such that, at the beginning of the transit Wardhoe was at A, that it then passed the dark part of its revolution, and arrived on the sunny side at a at the end of the transit; in other words, that the transit began in the evening and ended the next morning, And another station B (Otaheite) might be chosen, such that the transit began soon after Otaheite entered into the light at B, (or in the morning) and ended shortly before Otaheite came to the darkness at b. Now, we have already seen that Venus as seen from A described the longer path CD, merely because A is higher in the figure than B; but now we may see that, in consequence of A being to the left, Venus will be seen to enter at C sooner than if A had been in the central line; and, in consequence of a being to the right, Venus will be seen to leave the sun at D later than if a had been in the central line; and for both reasons, the time in which Venus appears to describe the path CD will be lengthened. In like manner, it will be found that as seen from B and b, the time in which Venus appears to describe the path EF will be shortened. And, therefore, the difference of the whole durations of the transit as seen at the two stations will be considerably increased.

I have mentioned specially Wardhoe and Otaheite. Many other stations were used, but these were the best of all. The whole duration of the transit at Wardhoe was about 5 hours 54 minutes. The whole duration at Otaheite was about 5 hours 32 minutes. The difference of durations was about 22 minutes. We may perhaps say, that 7 minutes of this was caused by Wardhoe being north of the earth's centre, 5 minutes by Otaheite being south of the earth's centre, and 10 minutes by the circumstance that at Wardhoe the transit began near sunset, while at Otaheite it began near sunrise.

On computing, from the known dimensions of the earth, the distances of the points A,a,B,b, in miles, and finding (by the proportion which I have mentioned before) how distant in miles were the lines CD, EF, how much they differed in length, and how much they appeared to differ when respect was had to the different positions of the two stations at the beginning and end of the transit; and finding what must be the actual diameter of the sun in miles, in order that the difference between these lines CD, EF, thus altered for the changes of positions of the two stations, might bear to their whole length the same proportion which the difference of observed durations of transit bore to the whole duration of transit, and computing from this diameter of the sun in miles what his distance in miles must be to make his apparent angular diameter what we know it to be; it was found that the mean distance of the sun is 95,300,000 miles. If the distance of the sun had been only 47,650,000 miles, the difference of durations of transit at Wardhoe and Otaheite would have been 44 minutes.

The method of determining the sun's distance, which I have thus attempted to illustrate, is one of the most difficult subjects for a public lecture that I know; and if I have given you a few notions of it I shall be perfectly satisfied. I shall make some further remarks on this subject in the next lecture.

1. The word "transit" signifies nothing more than "passage across."
2. This will be true if the difference between the radii of the circles which are compared with one another be small compared with either radius.
3. It is to be borne in mind that the interval between CD and EF is small in comparison with the diameter of the sun.