in the foregoing figure, is the obliquity of the ecliptic, which is to the sun's greatest declination, which, as before observequal ed, is twenty-three degrees twenty-eight minutes either north or south. Now the curve line in the figure representing the ecliptic, call the hypothenuse; the equinoctial, or straight line, the base; and a line perpendicular to the equinoctial, on any part, is the sun's declination. To find the Declination of the Sun by Spherical Trigonometry. As radius is to The sine of the obliquity of the ecliptic, 23° 28' 10.00000 9.60012 117° 35' 9.94760 To the sine of the declination, 20° 40′ 5" 9.54772 To find the Sun's Right Ascension. As the co-tangent of the hypothenuse or 117° 35' 9.71802 To the co-tangent of the hypothenuse, 117° 35' 9.71802 As radius, To find the Obliquity of the Ecliptic. Is to the co-tangent of the hypothenuse, 117° 35' To the co-sine of the obliquity of the 10.00000 9.71802 119° 39′ 45′′ 10.24449 ecliptic, 23° 28' 9.96251 If either side exceed ninety degrees, take it from one hundred and eighty degrees; and if it exceed one hundred and eighty degrees, take it from three hundred and sixty degrees. Reformation of the Calendar. BEFORE the year 1800 many mistakes were made in the latitude, particularly near the beginning of that year, when the declination was advanced one day; but it was at that time corrected; the cause of which error I shall here explain. The year consists of only three hundred and sixty-five days, five hours, forty-eight minutes, and forty-five and a half seconds; and adding, therefore, to every fourth year a whole day, gives eleven minutes and fourteen seconds and a half too much. This quantity, though small in itself, in the course of many years amounts to a considerable sum. In the year 1582, during the pontificate of Gregory XIII. the equinoxes had advanced ten days, so that the vernal equinox, instead of falling on the 20th, fell on the 30th day of March. This irregularity caused much inconvenience with respect to the festivals of the church. The pope, therefore, with the advice of the ablest astronomers, reformed the calendar, having calculated that the surplus time, viz. eleven minutes and fourteen seconds, which had been added to the leap year, must amount to a whole day in one hundred and thirty-three years, and that, consequently, from the time of the Nicene council, A. D. 326, the equinox had fallen back eleven days, they struck off at once eleven days from the current year; so that the vernal equinox then fell on the 21st of March; and it was further agreed to omit three of the leap years in four hundred years. This arrangement has already been put in practice, in the years 1700 and 1800, which would have been leap years in the usual course, but were not observed as such. Even this plan is not quite perfect; for, since the eleven minutes and fourteen seconds compose an entire day at the end of one hundred and twentyeight years, instead of one hundred and thirty-three years, this will still cause an error of one day in three thousand two hundred years; so that, at the end of the year 4800, it will again be necessary to retrench another day. Navigators, in working for their latitude, should always allow the variation, or correction of the sun's declination, between Greenwich and the place of observation: for, by omitting this correction, there will be a considerable difference between the latitude found in March and the latitude found in Septem ber, though both observations are taken with the same instrumaet; which error is considerable, if the longitude should differ much from that of Greenwich: for in March the sun is altering its declination at the rate of twenty-three miles and a half every twenty-four hours to the northward, which is nearly a mile an hour; and in September the sun is altering his declination to the southward at the same rate that it did to the northward, and of course the same error will take place as before, but the contrary way. I am of opinion that the omission of this correction has been the cause of the errors in latitude in many places; and in working for the apparent time, it is also highly necessary to allow this correction; directions for which are given in the epitomes of navigation. To Ascertain the Apparent Time by the Rising or Setting of the Sun. As it may sometimes happen that, for want of a better opportunity, it may be important to find the apparent time by the rising or setting of the sun, I shall endeavour to explain the best method of doing it, as it is subject to errors arising from the variation of horizontal refraction, to which the sun is liable when in the horizon. In high latitudes, the time observed by this method wilk vary from one to four minutes, depending on the latitude and the variation of the refraction. Within the latitude of 35 I have found it, by a comparison with observations had the same afternoon to agree with a good watch, previously regulated by an altitude of the sun, within. from fifteen to thirty seconds. In an extraordinary refraction no dependence can be placed in the time obtained by the above method. This, however, will be perceived by the oval appearance of the sun, the lower edge being considerably more refracted than the upper. The declination, in the tables for finding the time of the sun's rising and setting are calculated only to degrees, and the time to hours and minutes, but may be proportioned to the inferior denominations. The declination is to be reduced to the meridian of the place. of observation in the same manner as in finding the apparent time by the sun's altitude. Allowing that the refraction of the sun, when in the horizon, to be thirty-three miles, and the dip about four miles, this will make thirty-seven miles which the sun's centre will appear above the horizon. The tables in the epitomes of navigation are calculated for the centre to be in the horizon; therefore, taking sixteen miles, which is equal to the sun's semi-diameter, from thirty-seven miles, there remains twenty-one, agreeing to about two-thirds of the sun's diameter; so that, when the sun's lower limb appears two-thirds of his diameter above the horizon, in reality his centre is in the horizon, or, in other words, it is then sun-set. Figure 17 represents the sun with his lower edge two-thirds of his diameter above the horizon; and when he appears here, he is in reality set, as in figure 18; and in rising, he will not be risen until his lower edge appears two-thirds of his diameter above the horizon, as in figure 17. A B represents the horizon. Of the Spheroidal Figure of the Earth. THE spheroidal figure of the earth, or its being flattened at the poles, and swelled at the equator, which is occasioned by its rotation on its axis, will be understood by a reference to the following figure. A B Equator. B CD Poles. From the above figure it is clear that a ship must sail a greater distance on a meridian in a high latitude, to obtain a degree, than near the equator. A degree of latitude was measured in England, between London and York, and it was found to contain sixty-nine and a half English statute miles, differing from the sea miles. The length of a mean degree of latitude is settled at sixty-nine and one-tenth of the same miles. Care should be taken not to confound this increase of the degrees of latitude (which is so small as to be of little or no consequence in navigation) with the lengthening of the degrees on Mercator's chart; that being another affair, in which the degrees are extended, to make them correspond with the degrees of longitude, which are all equal on the charts, and without which the globe could not be correctly represented on a chart. Remarks on Sixteen of the most remarkable fixed Stars. THE first nine are given in the Nautical Almanac, and are used for the lunar observations: they will serve also for obtaining the latitude and the apparent time. In this work they are respectively designated by the first nine letters of the alphabet. The other seven will serve for obtaining the latitude and apparent time, and are respectively marked with figures from 1 to 7. Their appearance, with respect to each other, is hereinafter shown, and their right ascension and declination are given in the different epitomes of navigation. The figures of these stars will appear contrary, when you pass their declination; that is, they will then appear upside down, and also when they have passed the meridian; but will always retain their bearings and distance with respect to each other. ARIETES bears west distance 23° from the Pleiades, or Seven Stars, and is of the second magnitude. ALDEBARAN bears east by south 35° from Arietes, and appears, as described by the figure, with six or seven stars near it, of the third magnitude, forming with Aldebaran the letter V. POLLUX bears east north-east, distance 45° from Aldebaran, and is nearly of the first magnitude: north-west distant from Pollux 5° is the star Castor, nearly of the same magnitude, and you will almost always sweep them at the same time. The southernmost is the one to be used for measuring the distance |