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bservations has to be corrected for a wholly different series of conditions; that each observer's station is shifted continuously by two distinct forms of motion, which must both be taken into account (involving a careful reference to the question of time) before any satisfactory use can be made of the observed results.

Such, in a general sense, is the nature of the problem astronomers have to deal with. The conditions of the problem are not of their fixing; all they can do is to face as resolutely and skilfully as they can the difficulties which the problem presents to them. They have done this so well that the history of the problem has become to the thoughtful student of science as interesting as a romance. But they do not pretend to have secured a greater amount of accuracy than the nature of the problem and the means at their disposal render possible. Let this be distinctly understood beforehand, —Absolute accuracy in the solution of this problem is simply out of the question.

And now let us consider how the problem is to be attacked. In reality no less than six methods have been successfully applied; but there is only one method which is strictly geometrical in its nature, and this method must be the first to engage our attention.

In determining the distance of an inaccessible object, the geometer first measures a base-line, and then observes the bearing of the object from either end of that line. He thus has the means of determining the distance of the object with an exactness proportional to the accuracy of his observations.

Suppose, for example, the object is at o (fig. 2), and that the observer measures the base-line A B and observes the bearings B O and A o. Then, if he has done this accurately, he can draw a picture, as in the figure, accurately representing his observations, and he can measure either A o or b o in this picture, and, by comparing these measurements with the length of A B in his picture, he can tell what relation the actual distances A o or B o bear to the actual base-line A B.

Fio. 2.

Thus, if his base-line A B is 800 feet long, so that each of the divisions in the figure represents a length of 100 feet, then if B o be found to contain fifteen of these divisions, he will know that o is 1,500 feet from B. Usually, however, instead of merely drawing a figure (a process in which errors may creep in through the imperfection of rulers, compasses, and so on), the surveyor would apply trigonometrical calculation to determine B o or A o as accurately as his observations permitted.

But now, if either his base-line or his bearings be wrongly determined, it is clear the distances A o or b o will be wrongly estimated from them. The effect of a wrong measurement of the base-line is too obvious to need special discussion: clearly the error of b o or A o will be precisely proportional to the error of A B. But the error resulting from wrong estimates of the bearings requires to be attentively considered.

Suppose the bearing A o wrongly observed, and placed as A 1 or A 2. Then if the bearing b o be correctly observed, the resulting error will be o 1 or 0 2 respectively. On the other hand, if the bearing A o be correctly observed but B O misplaced as b 3 or B 4, the error will be o 3 or o 4 respectively. If the bearings A O and B O be both misplaced outside, or both inside, the true direction of o, the place of the point o will be calculated as if at 5 or 6 respectively. And, finally, if the bearings are misplaced in different wavs—that is, one inside o and the other outside—the point o will be calculated as if at 7 or 8, respectively.

Now, under favourable conditions, a skilful observer, though he must needs make some error in estimating his bearings (for no instruments can be absolutely perfect), would yet bring the estimated point relatively very near to o; in other words, though he might set it at a point out of place in the same way as any of the points 1, 2, 3, ... 8, the area of error corresponding to the area 5 7 6 8 would be small compared with the area A O B.

But now suppose that instead of such a triangle as A o B, our observer has to deal with a triangle shaped like a 0 b in the next figure — an ill-conditioned triangle, to use an expression of Sir John Herschel's. It is at once seen that a very small error in either of his bearings will set the observer far wrong in his estimate of the distance of o. Suppose he has rightly determined the position of a o, but has the bearing bo' or b o" in place of the true bearing b o. He

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has the large error o o' or o o", instead of the relatively small error o 4 or o 3 in the case pictured in fig. 2.

Now, the first important astronomical problem in distance-measuring—a problem infinitely less difficult than that of determining the Sun's distance—involves this very difficulty to a degree far greater than is indicated in fig. 3. I refer to the measurement of the Moon's distance.

If E (fig. 4) represent the Earth, the Moon would be placed somewhat as at M, and if it were possible to make use of two observatories situated as at a and b at opposite extremities of a diameter of the Earth, the

Fio. i.

actual difference of bearing of the Moon would be represented by the small angle a M b. As a matter of fact, however, even this small angle has to be reduced considerably, because from a or b, the Moon would be on the horizon, and the estimate of her position rendered unsatisfactory by atmospheric refraction. The angle a M b is about a degree and a quarter, and it affords a very satisfactory idea of the skill with which ancient astronomers employed their relatively ineffectiv e instrumental means, that their estimate of the Moon's distance differed from the truth by only a fiftieth part. Modern astronomy has so completely mastered the problem of determining the Moon's distance, that the estimate now adopted can scarcely exceed or fall short of the truth by so much as twenty miles, or less than a ten-thousandth part of the whole.

But when the method thus shown to be available in the case of the Moon is applied to the Sun, it is found to be absolutely ineffective. The nicest observation fails to show any measurable difference in the Sun's position according as he is viewed from one or another part of the Earth's surface. It is true that there is a difference, and indeed a difference which is large compared with some quantities which astronomers are in the habit of dealing with; but as a means of estimating the Sun's distance, this direct reference to what is called parallactic displacement may be regarded • as wholly ineffective.

Other methods, then, must be adopted. I proceed to consider two methods which suggested themselves to ancient astronomers. It is interesting to consider even those attempts which have failed; for they show the real difficulty of the problem we are engaged upon.

It occurred to Aristarchus of Samos (who flourished

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