Part 15

Thus we have here a case where two observers, without leaving their window, can tell the size of a distant object.

And it is quite clear that wherever the dove may pass between the window and the house, the observers will be equally able to determine the size of the cot, if only they know the relative distances of the dove and dovecot.

Thus, if D _a_ is twice as great as D A, as in fig. 43, then _a b_ is twice as great as A B, the length which the observers know; and if D _a_ is only equal to half D A, as in fig. 44, then _a b_ is only equal to half the known length A B. In every possible case the length of _a b_ is known. Take one other case in which the proportion is not quite so simple:--Suppose that D _a_ is greater than D A in the proportion of 18 to 7, as in fig. 45; then _b a_ is greater than A B in the same proportion; so that, for instance, if A B is a length of 7 inches, _b a_ is a length of 18 inches.

We see from these simple cases how the actual size of a distant object can be learned by two observers who do not leave their room, so long only as they know the relative distances of that object and of another which comes: between it and them. We need not specially concern ourselves by inquiring _how_ they could determine this last point: it is enough that it might become known to them in many ways. To mention only one. Suppose the sun was shining so as to throw the shadow of the dove on a uniformly paved court between the house and the dovecot, then it is easy to conceive how the position of the shadow on the uniform paving would enable the observers to determine (by counting rows) the relative distances of dove and dovecot.

Now, Venus comes between the earth and sun precisely as the dove in fig. 45 comes between the window A B and the dovecot _b a_. The relative distances are known exactly, and have been known for hundreds of years. They were first learned by direct observation; Venus going round and round the sun, within the path of the earth, is seen now on one side (the eastern side) of the sun as an evening star, and now on the other side (the western side) as a morning star, and when she seems farthest away from the sun in direction E V (fig. 46) in one case, or E _v_ in the other case, we know that the line E V or E _v_, as the case may be, must just touch her path; and perceiving how far her place in the heavens is from the sun's place at those times, we know, in fact, the size of either angle S E V or S E _v_, and, therefore, the shape of either triangle S E V or S E _v_. But this amounts to saying that we know what proportion S E bears to S V--that is, what proportion the distance of the earth bears to the distance of Venus.[17]

This proportion has been found to be very nearly that of 100 to 72; so that when Venus is on a line between the earth and sun, her distances from these two bodies are as 28 to 72, or as 7 to 18.

These distances are proportioned precisely then as D A to D _a_ in fig. 45; and the very same reasoning which was true in the case of dove and dovecot is true when for the dove and dovecot we substitute Venus and the sun respectively, while for the two observers looking out from a window we substitute two observers stationed at two different parts of the earth. It makes no difference in the essential principles of the problem that in one case we have to deal with inches, and in the other with thousands of miles; just as in speaking of fig. 45 we reasoned that if A B, the distance between the eye-level of the two observers, is 7 inches, then _b a_ is 18 inches, so we say that if two stations, A and B, fig. 47, on the earth E, are 7000 miles apart (measuring the distance in a straight line), and an observer at A sees Venus' centre on the sun's disc at _a_, while an observer at B sees her centre on the sun's disc at _b_, then _b a_ (measured in a straight line, and regarded as part of the upright diameter of the sun) is equal to 18,000 miles. So that if two observers, so placed, could observe Venus at the same instant, and note exactly where her centre seemed to fall, then since they would thus have learned what proportion _b a_ is of the whole diameter S S' of the sun, they would know how many miles there are in that diameter. Suppose, for instance, they found, on comparing notes, that _b a_ is about the 47th part of the whole diameter, they would know that the diameter of the sun is about 47 times 18,000 miles, or about 846,000 miles.

Now, finding the real size of an object like the sun, whose apparent size we can so easily measure, is the same thing as finding his distance. Any one can tell how many times its own diameter the sun is removed from us. Take a circular disc an inch in diameter,--a halfpenny, for instance--and see how far away it must be placed to exactly hide the sun. The distance will be found to be rather more than 107 inches, so that the sun, like the halfpenny which hides his face, must be rather more than 107 times his own diameter from us. But 107 times 846,000 miles amounts to 90,522,000 miles. This, therefore, if the imagined observations were correctly made, would be the sun's distance.

I shall next show how Halley and Delisle contrived two simple plans to avoid the manifest difficulty of carrying out in a direct manner the simultaneous observations just described, from stations thousands of miles apart.

We have seen that the determination of the sun's distance by observing Venus on the sun's face would be a matter of perfect simplicity if we could be quite sure that two observations were correctly made, and at exactly the same moment, by astronomers stationed one far to the north, the other far to the south.

The former would see Venus as at A, fig. 48, the other would see her as at B; and the distance between the two lines _a a´_ and _b b´_ along which her centre is travelling, as watched by these two observers, is known quite certainly to be 18,000 miles, if the observers' stations are 7,000 miles apart in a north-and-south direction (measured in a straight line). Thence the diameter S S´ of the sun is determined, because it is observed that the known distance _a b_ is such and such a part of it. And the real diameter in miles being known, the distance must be 107 times as great, because the sun looks as large as any globe would look which is removed to a distance exceeding its own diameter (great or small) 107 times.

But unfortunately it is no easy matter to get the distance _a b_, fig. 48, determined in this simple manner. The distance 18,000 miles is known; but the difficulty is to determine what proportion the distance bears to the diameter of the sun S S´. All that we have heard about Halley's method and Delisle's method relates only to the contrivances devised by astronomers to get over this difficulty. It is manifest that the difficulty is very great.

For, first, the observers would be several thousand miles apart. How then are they to ensure that their observations shall be made simultaneously? Again, the distance _a b_ is really a very minute quantity, and a very slight mistake in observation would cause a very great mistake in the measurement of the sun's distance. Accordingly, Halley devised a plan by which one observer in the north (or as at A, fig. 47) would watch Venus as she traversed the sun's face along a lower path, as _a a´_ fig. 49; while another in the south (or as at B, fig. 47) would watch her as she traversed a higher path, as _b b´_ fig. 49. By timing her they could tell how long these paths were, and therefore how placed on the sun's face, as in fig. 49; that is, how far apart, which is the same thing as determining _b a_, fig. 48. This was Halley's plan, and as it requires that the duration of the transit should be timed, it is called the method of durations. Delisle proposed another method--viz., that one observer should time the exact moment when Venus, seen from one station, _began_ to traverse the path _a a´_, while another should time the exact moment when she _began_ to traverse the path _b b´_; this would show how much _b_ is in advance of _a_, and thence the position of the two paths can be determined. _Or_ two observers might note the _end_ of the transit, thus finding how much _a´_ is in advance of _b´_ This is Delisle's method, and it has this advantage over Halley's--that an observer is only required to see _either_ the beginning or the end of the transit, not _both_.

I shall not here consider, except in a general way, the various astronomical conditions which affect the application of these two methods. Of course, all the time that a transit lasts, the earth is turning on her axis; and as a transit may last as long as eight hours, and generally lasts from four to six hours, it is clear that the face of the earth turned towards the sun must change considerably between the beginning and end of a transit. So that Halley's method, which requires that the whole duration of a transit should be seen, is hampered with the difficulty arising from the fact that a station exceedingly well placed for observing the beginning of the transit might be very ill placed for observing the end, and _vice versâ_.

Delisle's method is free from this objection, because an observer has only to note the beginning _or_ the end, not both. But it is hampered by another. Two observers who employ Halley's method have each of them only to consider how long the passage of Venus over the sun's face _lasts_; and they are so free from all occasion to know the exact time _at_ which the transit begins and ends, that theoretically each observer might use such an instrument as a stop-watch, setting it going (right or wrong as to the time it showed) when the transit began, and stopping it when the transit was over. But for Delisle's method this rough-and-ready method would not serve. The two observers have to compare the two moments at which they severally saw the transit begin,--and to do this, being many thousand miles apart, they must know the exact time. Suppose they each had a chronometer which had originally been set to Greenwich time, and which, being excellently constructed and carefully watched, might be trusted to show exact Greenwich time, even though several months had elapsed since it was set. Then all the requirements of the method would be quite as well satisfied as those of the other method would be if the stop-watches just spoken of went at a perfectly true rate during the hours that the transit lasted. But it is one thing to construct a time-measure which will not lose or gain a few seconds in a few hours, and quite another to construct one which will not lose or gain a few seconds in a journey of many thousand miles, followed perhaps by two or three months' stay at the selected station. An error of five seconds would be perfectly fatal in applying Delisle's method, and no chronometer could be trusted under the conditions described to show true time within ten or twelve seconds. Hence astronomers had to provide for other methods of getting true time (say Greenwich time) than the use of chronometers; and on the accuracy of these astronomical methods of getting true time depended the successful use of Delisle's method.

Then another difficulty had to be considered, which affected both methods. It was agreed by both Halley and Delisle that the proper moment to time the beginning or end of transit was the instant when Venus was just within the sun's disc, as in fig. 50, either having just completed her entry, or being just about to begin to pass off the sun's face. If at this moment Venus presented a neatly defined round disc, exactly touching the edge of the sun, also neatly defined, this plan would be perfect. At the very instant when the contact ceased at the entry of Venus, the sun's light would break through between the edges of the two discs, and the observer would only have to note that instant; while, when Venus was leaving the sun, he would only have to notice the instant when the fine thread of light was suddenly divided by a dark point. But unfortunately Venus does not behave in this way, at least not always. With a very powerful and very excellent telescope, in perfectly calm, clear weather, and with the sun high above the horizon, she probably behaves much as Halley and Delisle expected. But under less favourable conditions, she presents at the moment of entry or exit some such appearance as is shown in figures 51, 52, and 53, while with a very low sun she assumes all sorts of shapes, continually changing, being for one moment, perhaps, as in one or other of figs. 51, 52, and 53, and in the next distorted into some such pleasing shape as is pictured in fig. 54.

Accordingly, many astronomers are disposed to regard both Halley's method and Delisle's as obsolete, and to place reliance on the simple method of direct observation first described. They would, however, of course bring to their aid all the ingenious devices of modern astronomical observation in order to overcome the difficulties inherent in that method. One of the contrivances naturally suggested to meet such difficulties is to photograph the sun with Venus upon his face. The American astronomers, in particular, consider that the photographic results obtained during the transit of 1874 will outweigh those obtained by all the other methods. The German and Russian astronomers, as well as those of Lord Lindsay's expedition, while placing great reliance on photography, employed also a method of measuring the position of Venus on the sun's disc, by means of a kind of telescope specially constructed for such work, the peculiarities of which need not be here considered.

The observations made in 1769 were so imperfect that astronomers deduced a distance fully 3,000,000 miles too great. Of late, other methods of observation had set them much nearer the true distance, which has been judged to lie certainly between 91,800,000 miles and 92,600,000 miles--a tolerably wide range.

But it may perhaps occur to some that the distance of the sun may be changing. The earth might be drawing steadily in towards the sun, and so all our measurements might be deceptive. Nay, the painful thought might present itself that when the observations of 1769 were made, the sun really was farther away than at present by more than 3,000,000 of miles. If this were so, the earth would, in the course of a century, have reduced her distance by fully one-thirtieth part, so that, supposing the approach to continue, she would in 3,000 years fall into the sun, while, long before that period had elapsed, the increased heat to which she would be exposed would render life impossible.

Fortunately, we know quite certainly that no such approach is taking place. It is known that the distance of the earth from the sun cannot change without a corresponding change in her period of revolution--that is, in the length of the year. The law connecting these two (indicated in the note, page 279) is such that, on the reduction of the distance by any moderate portion the period would be reduced by a portion half as great again. For instance: if the distance of the earth from the sun were reduced by a thirtieth part (or about 3,000,000 miles) the length of the year would be reduced by a thirtieth and half a thirtieth--that, is, by a twentieth part, or by more than eighteen days. We know that no such change has taken place during the last century, or since the beginning of history. Nay, from the Chaldean estimate of the length of the year, which only exceeded ours by about two minutes, it is easily shown that the distance of the earth from the sun has not diminished 200 miles within the last 2,500 years. So that, assuming even that the earth is approaching the sun at this rate, or eight miles in a century, it would be 1,250,000 years before the distance would be diminished by 100,000 miles, which is the probable limit of error in the determination of the sun's distance.

If, finally, it be asked, What, after all, is the use of determining the sun's distance? the answer we shall give must depend on the answer given to the question, What, after all, is the use of knowing any facts in astronomy other than those useful in navigation, surveying, and so on? And I think that this question would introduce another and a wider one--viz., What is the use of that quality in man's nature which makes him seek after knowledge for its own sake? I certainly do not propose to consider this question, nor do I think that the reader will find any difficulty in understanding _why_ I do not. But accepting the facts: (1) that we _are_ so constituted as to seek after knowledge; and (2) that knowledge about the celestial orbs is interesting to us, quite apart from the use of such knowledge in navigation and surveying, it is easy to show that the determination of the sun's distance is a matter full of interest. For on our estimate of the sun's distance depend our ideas as to the scale, not only of the solar system, but of the whole of the visible universe. The size of the sun, his mass, and therefore his might, the scale of those wonderful operations which we know to be taking place upon, and within, and around the sun; all these relations, as well as our estimate of the size and mass of every planet, and therefore our estimate of the earth's relative importance in the solar system, depend absolutely and directly on the estimate we form of the sun's distance. Such being the case (this being in point of fact the cardinal problem of dimensional astronomy) it cannot but be thought that, great as were the trouble and expense of the expeditions sent out to observe the transit of 1874, they were devoted to an altogether worthy cause.

FOOTNOTES:

[17] There is, however, a much more perfect way of determining this proportion, by applying the law which Kepler found to connect the distances of the planets from the sun with the times in which they complete the circuits of their orbits. The law is that, if we take any two planets, and write down the numbers expressing their periods of circuit (say in days), and the numbers expressing their distances from the sun (say in miles) in the same order; then if we multiply each number of the first pair into itself, and each number of the second pair twice into itself, the four numbers thus obtained will be proportional; that is to say, as the first is to the second, so will the third be to the fourth. Now, as every one knows who has worked sums in the rule of three, when any three are given out of four proportionals, the fourth can always be found; but we know the periods of circuit both of the earth and Venus (365·2564 days and 224·7008 days respectively) very exactly indeed, because they have traversed their orbits so many times since they began to be observed by astronomers. We can call the earth's distance 100, and then applying the rule just stated, we get Venus' distance relatively to the earth's. The reader who cares to work out this little sum will find no difficulty whatever--if at least he is able to extract the cube roots of any number. The proportion runs thus:--

365·2564 × 365·2564 : 224·7008 × 224·7008 :: 100 × 100 × 100 : (Venus' distance cubed.)

Work out this sum and we get for Venus' distance 72·333. The ratio of Venus' distance to the earth's is almost exactly expressed by the numbers 217 and 300.

Hazell, Watson, and Viney, Printers, London and Aylesbury.

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Transcriber's Notes:

Italic text is denoted by _underscores_.

Obvious printer's errors corrected.

Every effort has been made to replicate this text as faithfully as possible, including non-standard punctuation, inconsistently hyphenated words, and other inconsistencies.

End of Project Gutenberg's Flowers of the Sky, by Richard A. Proctor