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and others again from the differences in the length of the pendulum vibrating seconds in different latitudes. Still however we want some theory of the internal density of the earth; which, it is feared, will ever remain hid from human investigation. But if, to the want of this essential knowledge, we add the irregularity of the figure of the earth, as shown by the measurements of the degrees on its surface; if, to the doubts which some persons may still entertain of the truth of the compression of the poles*, we add the discordant opinions as to the quantity of that compression; we may surely assert that it will be no small advantage to point out a method (independent of all hypothesis) for determining, with facility and with the greatest precision, the differences between the terrestrial radii at an indefinite number of points on the earth's surface. indeed is the object of the present memoir.
2. It has always been said, when speaking of the compression or flattening of the poles of the earth, that the Parallax of the Moon would afford the best means of ascertaining it; provided the variations, arising therefrom, were of sufficient magnitude to be observed with perfect accuracy. But, since, by supposing this compression not to exceed of the semidiameter (which is the most received opinion at the present day) there would be
* See Lorgna "Principj di Geografia &c." § 31.
a difference of not more than 9" between the moon's parallax at the equator, and her parallax at a high latitude, for instance 60°; so indeed it is but too true that a difference so small might be easily concealed under the possible errors of observation. It was, on this account, that Manfredi and Maupertuis in vain suggested the determination of the compression by direct observations of the moon's parallax: nor has any advantage been hitherto obtained by this method.
§ 3. But this is not the only instance in which a quantity, extremely minute on one side, has been found to produce effects that are quite discernible on the other. The attention of the observer should be directed to those points where they are most sensible. This has been the object of my research; and I hope not entirely without success. For, there are circumstances in which scarcely a second in the parallax may cause a difference of 15, 20, or 30 seconds of time, or even more, in the duration of the occultation of a star by the moon. But, occultations, more particularly when the immersions take place behind the dark limb of the moon, can easily be observed without committing an error of a single second of time*: so that opportunities
* The duration of an occultation cannot probably be so well observed when the immersions take place behind the dark limb of the moon, as when they take place behind the illuminated side; since the instant of emersion cannot be so well ascertained. B.
exist not only of removing all doubt as to the reality of the compression of the poles, but also of such frequent occurrence that the gradation of the compression, (or variation of the curvature) at different latitudes, may be accurately distinguished. The importance of the subject induces me to treat it more fully in detail.
§ 4. I shall not take up the time of the reader in demonstrating that no hypothesis is required to deduce the variation of curvature from the observation of the parallax. For, since the variation of curvature is nothing more than the inequality of the terrestrial radii (and that is nothing else than the difference of the parallax), it is evident that finding by observation the variation of parallax at different latitudes is in fact finding the variation of curvature by immediate observation. One involves the other without any intermediate help. The only difficulty consists in freeing from uncertainty those observations which show the unequal parallax: and this is precisely the special character of the particular kind of occultations which I have in view. It is true that astronomers, in calculating occultations, have made the parallax vary in conformity with that quantity of compression which they have adopted as the true theory. Moreover, these variations do not in general produce differences which are discernible in the relative duration of the occultations.
observed in different latitudes. Those differences, which are discernible, arise only in certain circumstances, to which hitherto no particular attention has been paid. It is on this account that occultations have not, as yet, served at all to determine the amount or quantity of the compression of the polar axis.
§ 5. Before I proceed to describe what these favourable circumstances are, it will be proper, in order to appreciate their utility, to define the precise degree of accuracy of which the computation and observation of occultations are capable. With respect to the computation, it is evident that the apparent distance between the star and the moon cannot be accurately determined unless we know exactly not only the apparent place of the star, but also that of the moon and its diameter. These elements are obtained by calculating the phænomena from observations made in a place (the longitude of which is known) under circumstances where the alteration of the parallax, on account of the variation in the curvature of the earth, does not produce a perceptible difference in the duration of the occultation: conditions not uncommon, nor difficult to be obtained.
§ 6. Let us then suppose that the moon's diameter and the place of the star are well determined (in which elements the least uncertainty exists); and
that any small errors, errors, which may occur, be thrown on the place of the moon, Let us, moreover, endeavour to discover (by comparing the calculation with the observation) the errors of the tables in the longitude and latitude of the moon, mixed (as we have already stated) with the preceding errors*. In the present state of astronomy I do not see any great danger to be apprehended from this union of the two errors, notwithstanding the delicacy of the investigation in question. At any rate we may arrive at a sufficient degree of accuracy by verifying the place of the star by a sufficient number of observations, and also the diameter of the moon by means of the telescope which is used for the occultations.
7. Having corrected the errors of the lunar tables, we shall have accurate elements wherewith to represent the state of the heavens: and there can then be no doubt that we may arrive at great exactness in our calculation of the duration of an oc cultation (or of the moment of immersion and emersion) for the place in which we wish to ascer→ tain the curvature of the earth; since the geographical longitude of the place has been determined by
See the Memoir which obtained the prize from the Academy of Copenhagen: "Méthode pour calculer les longitudes géographiques." Vérone; chez Ramanzini.