I. the second cause, we observe that the sun, being placed in BOOK one of the foci of the elliptic orbit of the earth, appears to move more slowly in the six northern signs than in the six southern; and this difference of velocity is sufficient to render unequal the diurnal arcs of the equator. It arises from the union of these two causes, that the length of the solar day, compared with the time of the earth's rotation, is sometimes less and sometimes greater than twenty-four hours; and this inequality will always be greatest when the two causes which we have just explained concur in accumulating the differences in the same direction. The series of these differences form what is called the equation of time, or the quantity which must, at certain seasons, be added to, and at other seasons subtracted from, the hour indicated by clocks, which are regulated by the sun, and mark the true time, if we wish to get the mean or astronomical time. Mean or Now, it is for mean time that the astronomical tables are cal time. constructed, by help of which we calculate the motions of the stars, and from these motions deduce the geographical positions of places. astronomi. the moon. We have now considered the earth in relation to the Motions of sun; but it is also very closely connected with the moon. This satellite of our planet performs its revolution round A month, the earth in 27 days 7 hours, 43' 11": this time is usually called a periodical month. It is observed, that the moon employs a little more than this time to return to the sun after each conjunction. The true cause of this difference is, that the earth, and consequently the moon its satellite, advances in the ecliptic, while the moon describes her orbit; so that before the moon comes into the same position relatively to the sun, 2 days and about 5 hours elapse, beyond the time required for completing a revolution. The whole time occupied in returning to the sun is 29 days 12 hours, 44' 3" 10"". This space of time is called a synodical month, or lunar month. It commences from the moment when the moon is directly between the sun and the earth, in which position the moon is said to be in conjunction. BOOK This aspect is represented in figure 10, where S represents I. Eclipses of the sun and moon. the sun, T the earth, and L the moon. In describing its orbit, the moon takes, with regard to the sun, many situations, from which arise the aspects, or phases, which it assumes. The moon being an opaque body, can be seen only in as far as it reflects the light that it receives from the sun; it can be visible to us, therefore, only when it begins, after having passed the point L, to turn towards the earth a portion of its enlightened disk. This portion increases according as the moon recedes from the sun, until it arrives at L', the opposite point of its orbits, when the earth being between it and the sun, we see the whole enlightened hemisphere; the moon then appears full, and is said to be in opposition with the sun. The conjunction and opposition of the moon with regard to the sun, or the new and full moon, are what are called the syzygies. When the moon is distant from the sun a fourth part of the circumference, as at i or i', it is in quadrature, and shews only one-half of its enlightened hemisphere. It is the first or last quarter, according as the round edge of the enlightened part is towards the west or cast. One would be led to suppose that the moon, every time it comes into conjunction with the sun, ought to conceal from us the whole, or, at least, a part of the disk of the sun; and that every time it is in opposition, it ought to pass through the shadow which the earth projects behind it; so that there would be, in the former case, an eclipse of the sun, and in the latter, an eclipse of the moon. These phenomena do not, however, occur at every new and full moon; and the reason is, that the plane of the moon's orbit is inclined to that of the ecliptic, and that these two planes meet one another only in their line of common section, which passes through the centre of the earth. It is evident that the moon is not in the plane of the ecliptic, except when it passes through one or other of the extremities of this line, that is to say, when it is in the nodes of its orbit. When the conjunctions and oppositions coincide with the 11, which I. nodes, there are eclipses; in the opposite case no eclipse BOOK happens. We shall comprehend better these particulars by comparing figure 10, which represents in a geometrical plane the orbits of the earth and of the moon, with fig. shows their section or profile, along the line ST. ST represents the plane of the ecliptic, and L lunar orbit. We proceed now to point out in what manner the observation of these phenomena enable us to determine the longitude of a place upon the earth. This line that of the We know, that in order to find the difference of longitude between two places, it is only required to ascertain precisely the hour which is reckoned at the same instant at each of these places, by the observation of some instantaneous phenomenon which can be seen at both. The eclipses of the moon appear at first view the most fa- Longitude vourable phenomena; for the entrance of the moon into the by the eclipses of shadow of the carth takes place at the same instant for all the moon, the points of the hemisphere which is then turned towards the moon; that is, for all the places where the eclipse can. be observed; besides, the spots visible upon the lunar disk, afford the means of making several observations upon the same eclipse, by marking with precision the time of the disappearing of each spot at its entrance into the shadow, or the immersion, and that of its reappearing at its coming out of the shadow, or the emersion supposing that we have determined at each place the true time of this observation, the difference of these times, converted into degrees of the equator, will give immediately the difference of the longitudes. If all the results obtained do not exactly agree, the mean of all the observations is commonly taken; but it is much better to examine in detail the circumstances which have accompanied each observation, appreciating accordingly the relative accuracy of each, and to compare only those which are free from all suspicion of inaccuracy.* *Burg, in Zach, Astronomical correspondence, iv. 629. Oltmanns, Researches upon the Geography of the New Continent; passim. BOOK I. Astronomi There is no absolute need of corresponding observations. The astronomical almanacks, such as the Connaissance des Tems of the French, the Nautical Almanack of the English, cal alma- or the Calendrier du Navigateur of the Danes, give the results of the calculations of eclipses made before-hand for a known point. nacks. Longitudes tellites of It is in this manner that M. Lalande has determined the longitude of Casbin, a city in the north of Persia, from the eclipse of the moon which happened on the 30th June, 1787, and was observed at that place by the astronomer Beauchamp. The end of the eclipse, or the total emersion of the lunar disk from the shadow of the earth, having taken place at Casbin at 7 hours 45' 30", true time, and the calculation giving for Paris, 4 hours 36′ 38′′, the difference, which is 3 hours 8' 52", is equal to the difference of the longitudes of Paris and Casbin. If we convert it into degrees, allowing 15 for an hour, which gives 15 minutes of a degree for a minute of time, and 15 seconds of a degree for a second of time, we shall have for 3 hours, 8′52′′ in time, an arc of 45° 15'. This is the longitude of Casbin relatively to the meridian of Paris, as deduced from the above observation. But eclipses of the moon present this great inconvenience, that the instant when the lunar disk enters into the true shadow of the earth, the instant which marks the commencement of the eclipse, can never be assigned with precision; we cannot therefore be certain of not erring a few seconds of time in the determination of the phases of an eclipse of the moon; for this reason, the use of lunar eclipses for determining longitudes is now generally abandoned. Cassini was the first who, in 1668, proposed to make by the sa- use of the eclipses of the satellites of Jupiter, for the purJupiter. pose of finding longitudes. The theory of these eclipses is the same with that of the eclipses of the moon; for the satellites of Jupiter, when placed in circumstances similar to those which produce the eclipses of the moon, fall, in like manner, into the shadow of their primary planet: if we observe at the same time, at several places, their immersions and I. emersions, we may make the same use of these, for the de- BOOK termination of the longitudes, as of the eclipses of the moon. There are two other planets, Saturn and Uranus, which are also accompanied by satellites, but their smallness, and their distance, permitting them to be seen only by means of powerful telescopes, render the observation of their eclipses almost impracticable, or, at least, of little use. Even the satellites of Jupiter do not all equally well answer the purposes of the observer; for here, as in the eclipses of the moon, the precise moment of immersion and of emersion is always a little uncertain; particularly in the second and third satellites.* The use which may, nevertheless, be made of these satellites of Jupiter, has induced astronomers to frame tables for predicting their immersions, in order that corresponding observations may not be necessary.t eclipses of The eclipses of the sun are no less proper than those of Longitudes the moon, for determining longitudes: it is sufficient for this purpose that we observe at each of the places of which the sun. we wish to know the difference of longitude, the commencement or the termination of one and the same eclipse; but the calculation is not so simple as in the case of a lunar eclipse. M. de Lalande, by great care and attention, has, by means of solar eclipses, corrected the positions of a great number of places. The calculation becomes more difficult, only because the relative situation of the sun and woon is not the same for the different points of the earth's surface at which these two bodies are visible at the same time. The case of the moon is the same as that of the clouds, which, seen from a particular point, appear situated under the sun, and project their shadow over a limited space, beyond which the sun shines in all its splendour. This spectacle varies continually, according as, from one instant to another, the sun, the cloud, and the spectator, change their * Rossel, Voyage d'Entrecasteaux, ii. 245. Zach, Astron. Correspond. i, 421. + Table of the Satellites of Jupiter, by M. Delambre. |