Method of making the divisions on the circular pieces of the instrument. The circle of the largest piece is divided in such a manner, that 368 degrees 2 minutes 42 seconds comprehend 354 days and somewhat less than 9 hours; hence it follows, that this circle ought to contain 346 days, and 15 hours, which may be assumed without any sensible error as two thirds of a day. But to divide a circle into 346 parts and two thirds, reduce the whole into thirds, which in this case make 1040 thirds, and then find the greatest multiple of 3 that can be easily halved, and is contained in 1040. This number will be found in the double geometrical progression, the first or least term of which is 3; as for example 3, 6, 12, 24, 48, 96, 192, 384, 768. The ninth number of this progression, viz 768, is the one required: subtract this number from 1040; and by. the rule of three, find the number of degrees minutes and seconds contained in the remainder 272, by saying, as 1040 thirds: 360 degrees :: 272 thirds: 94 degrees 9 minutes 23 seconds. Then cut off from this circle an angle of 94° 9′ 23′′, and divide the rest of the circle always into halves. When eight sub-divisions have been made, you will come to the number 3, which will be the arc of one day; and if the arc of 94° 9′ 23′′ be divided by this also, the whole circle will be divided into 346 days and two thirds; for there will be 256 days in the larger arc, and 90 days two thirds in the other. Each of these spaces will correspond to 1° 2′ 18′′, as may be seen by dividing 360 by 3463, and 10 days cor respond to 10° 23'. By these means a table for dividing this circular piece might be formed. These days must afterwards be distributed to each of the months of the year, according to the number which belongs to them, beginning with March, and continuing to the fifteenth hour of the tenth of February, which corresponds to the commencement of March; and the remainder of the month of February passes beyond and above. The circle of the second plate must be divided into 179 equal parts. For this purpose, find the greatest number that can always be halved to unity, and which is contained in 179. This number is 128, which taken from 179, leaves for remainder 51. Then find, by the rule of three, what part of the circumference is equal to this remainder, by saying, as 179: 360 degrees:: 51 parts: 102 degrees 34 minutes 11 seconds. Having cut off from the circle an arc of 102° 34′ 11′′, divide the remainder always into halves, and after seven sub-divisions you will come to unity. This part of the circle therefore will be divided into 128 equal parts: then with the same aperture of the compasses, divide the remaining arc into 51 parts; and the whole circle will be divided into 179 equal parts, each corresponding to 2 degrees and 40 seconds, as may be easily seen by dividing 360 by 179. In the last place, to divide the circle of the upper plate, take the fourth of its circumference, and add to it one of the 179 parts or divisions of the edge of the middle plate: if the compasses, with an aperture equal to the fourth thus increased, be then made to turn four times, it will divide the circle in the manner in which it ought to be; since by subdividing each of these quarters into three equal parts, we shall have 12 spaces for the 12 lunar months; so that the end of the twelfth month, which forms the commencement of the twelfth lunar year, surpasses the first new moon by 4 of the 179 divisions marked on the middle plate. The method of using this machine is contained in the following problem. PROBLEM XVII. A lunar year being given; to find, by means of the preceding machine, the days of the solar year corresponding to it; and on which there will be new or full moon, or an eclipse of the sun or moon. Let the proposed lunar year be the 101st in the table of epochs, which corresponds to the division of the middle circular plate marked 101. Bring the edge of the index of the upper plate to the division marked 101 of the middle plate, where the commencement of the 101st lunar year falls; and as this commencement took place, according to the table of epochs, on the 26th February 1778, at 6 hours 34 minutes, turn both the upper plates together in that state, till the edge of the index, attached to the upper plate, corresponds to the 6th hour, or a little more than the fourth of the 26th of February marked on the lower plate, at which time the first new moon of the proposed lunar year happened. Then, without altering the situation of the three plates, extend a thread from the centre of the instrument, or turn the moveable index, till it pass through the middle of the aperture of the first full moon: the edge of the index will then correspond to the middle of the 13th of March, which ought to be the time of full moon, within a few hours; and as the aperture of this full moon does not present a red colour, there was no eclipse of the moon. To find what took place at the following full moon, add to the new moon of the epoch 29 days 12 hours 44 minutes, and you will have the time of new moon on the 27th of March, at 19 hours 16 minutes; and by performing the same operation, it will be found that there was no eclipse either at that new moon, or at the full moon following. But, proceeding in this progressive manner, you will come to the new moon of November, which took place on the 19th of that month, at 1 hour, 8 minutes; then performing the same operation, you will find the full moon following on the 3d of November, at about 8 in the evening, and it will be seen that there was a partial eclipse, the aperture of the full moon being in that part filled with the red colour. The eclipses of the sun will be found in like manner: they will be indicated by the black colour which will present itself at the aperture of the new moons. On the 24th of June 1778, for example, new moon took place at 19 hours 8 minutes, or 8 minutes past 7 in the evening; and as the aperture of this new moon will be in part occupied by the black colour which is below, we may conclude that there was a partial eclipse of the sun on the 24th June 1778 in the evening: which was indeed the case. By such a machine however, it is not possible, as may be readily conceived, to determine the exact hour and minute of an eclipse or of a lunation. It is enough, if it indicates whether a conjunction or opposition takes place in the ecliptic; the rest must be determined by calculation, for which precepts may be found in all works that treat expressly on this subject. To gratify the curiosity of the reader, we shall here give a table of the eclipses, both of the sun and moon, which will take place in the course of the present century; with the different circumstances attending them, such as the time of the middle of the eclipse, and its extent; and, in regard to eclipses of the moon, how many digits will be eclipsed, &c. We must however observe, that as this table is extracted from an immense labour*, undertaken for another purpose, This labour is a table of the solar and lunar eclipses since the commencement of the Christian æra, to the year 1900, inserted in l'Art de véri |