only ; according to 365 20 40 59 which computation one day will be deficient in every four years ; but this is not the only source of error. The first wheel 365, though fixed in the form of an indented plate, gives motion to the rest of the train that is carried round it in a year by the annual bar, and the last pinion 10, which was intended to rotate in, a solar day, may be called the diurnal pinion. One condition with respect to this pillion is, that it shall turn from west to east, in order that the earth may rotate in the proper direction. An additional arbor is introduced therefore, and the numbers which 20 8 might have been omitted if — had been made , may be 40 80 considered as answering no other purpose than chang ing the direction of motion of the diurnal pinion ; or the two wheels of 59 teeth each might have been omit ted, so far as the computation of the train is concerned, one being a multiplier and the other a divisor, if the direc tion of motion had not been a condition. But while the earth rotates on its axis, during its progress in its annual orbit, another condition is, that its inclined axis should be kept parallel to its original situation during the whole year, which is done by giving one retrograde motion in a year to the small bar that supports the earth, which bar is called the bar of parallelism. On this bar of parallelism are placed two additional pinions, every way similar to the diurnal pinion of 10 teeth, to afford the means of giving inclination to the earth's axis, and of transmitting the mo tion of the diurnal pinion to the said axis unaltered ; but unfortunately the annual retrograde motion of the bar of parallelism round the diurnal pinion, produces a deduction of one revolution of the two additional pinions, and conse quently diminishes the number of the earth's rotations by one in each year ; so that instead of 3654- days, there are only 364 in each year produced by the train of wheelwork, and therefore an index to point out the hour, will always be at variance with the position of the earth in such a con struction. But there is yet another cause of error in this machine, independently of what is produced by the ope ration of the two additional pinions of the train. The same contrivance which preserves the parallelism of the earth's axis, makes her turn round from east to west in each year, while the train makes her turn, as we have ex plained, 364 times only from west to east, so that in fact another day is deducted in each year ; and an inhabitant of any given spot on the earth will transit the sun, or the Sun will appear to transit him, only 363 times in the year. In the more perfect machines these sources of error have been guarded against, as will be seen hereafter.
Another case, in which a planet's motion in its orbit is not altogether produced by its own train, is, when it has a part of its train in common with another planet, and the remainder carried by that other planet's revolving arm ; which is generally the case when the synodic revolution is represented by the mechanism. We have an instance of this sort in Ferguson's orrery, where the motions of Ve nus and Mercury are so connected, that the latter derives its motion from the former in a two fold manner. The 25 69 73 i train that actuates Venus x 20b. 47m. P. and Mercury's revolution is occasioned by only two addi tional wheels in 28 of Venus' period. Hence the revolu- tions of Mercury would be to those of Venus as 1 : 1.555, 18 if the wheels were all placed on a bar at rest ; but the 28 of Mercury are fixed on the moving arm of Venus, which carries :Mercury round once, without reference to the ration of the train, in every revolution of Venus. Hence the period of Mercury's revolution round the sun is 244.86606d• 23h. ; but as it 1+1.55555 2.55555 244.86606d• regards Venus, the synodic revolution will be -- 1.55555 = 157.414 days. Hence it is of the utmost importance in the examination of a planetary machine, not only to count the teeth of the wheels and pinions with care, hut to notice particularly, whether the whole train be so si tuated as to position, that the intended effect is correctly produced. The small orrery of Ferguson is more correct than any of its predecessors ; we shall therefore subjoin a table of its trains, and of their corresponding values in time, as an example that includes almost all the different modes of computation that are likely to occur in any one machine.
It may be proper to notice here, that when a retrograde motion, as compared with another motion, like that of the moon's node, is produced by mechanism, the effect is es timated by dividing the number of teeth in the driver by the difference of the two wheels, or products, if there be a train ; but if the motion is progressive, like that of the moon's apogee, then the difference of the two wheels, or products, contained in the teeth of the driven wheel, or wheels, will be the value, and in the same denomination of time as to years, months, or weeks, &c. as the period
is, in which the first wheel or driver revolves. In the ta ble before us, years is the period of the node's motion ; 3 and in another of Ferguson's machines the wheels for the 55 6 apogee are —, and the progressive period = years. In 62 examining whether the earth's axis has its parallelism pre served during the whole year, it is only necessary to notice whether the wheels employed, if only two, (with an inter mediate one to change the direction of motion,) have the same number of teeth each ; or, if a train be used, whether the products of the numerators and of the denominators are alike. In Martin's tellurian, the train for this purpose is 59 X 365 1.002192, which is therefore erroneous. In Ferguson's orrery, 25 69 which we have examined, the portion of the train which converts the year into a day, is common to the trains of the sun's rotation, and of Venus' revolution, which cir cumstance diminishes the number of wheels and pinions required by the computations.
Computation of Planetary Having explained how the value in time may be ascer tained for any given pair of wheels, or train of wheelwork, acting under different circumstances, we shall next pro ceed to show, how wheelwork may be computed to pro duce revolutions or rotations in any given time. The operations required for this purpose are just the reverse of those that have been explained for determining the time, when the wheels and pinions arc examined, as to their number of teeth and position. But before any computa tion can be effectually entered upon, the plan of the in tended machine, and disposition of all the parts employed to produce the respective motions, must be clearly com prehended and determined upon, otherwise the mechanism may not effect the calculated periods. Of this there is a remarkable instance in the small common orrery to be met with in the shops, said to have been contrived by one Ryley, who computed his wheelwork for the periodic re volutions of Mercury, Venus, and the Earth, as they re gard the sun, but placed them on planetary arms in such a way, that they actually produce synodic times in stead of periodic, i. e. they overtake one another in their orbits, in the same time that they ought to have employed in going round the sun; for instance, Mercury overtakes Ve nus in 864 instead of 115.877 days, and Venus overtakes the earth in 2194, instead of 585.923 days. In the latter case the velocity of the planet is more than double what it was computed to be, and would have been, if the wheels had been placed in a stationary position; hence the ma chine becomes nothing more titan an expensive toy, cal culated to mislead rather than to instruct the juvenile stu dent in asuonomy. It is a disgrace to the venders of such an imperfect piece of mechanism, that they have never examined and detected the errors arising out of its construction. For the sake of method, we will follow the same order in our computations, that we observed in our preceding examination of the different motions effected by wheelwork. The first and simplest case is that where a wheel and pinion, or two wheels only, are required to act to gether for the production of a motion to be effected in a given period, say in seven days ; here, if the driver, or first mover, be assumed to have a revolution in one day, and to have 15 teeth, or any convenient number, the wheel to be driven by it must e% idently have 15 x 7=105 teeth, 1u5 i in order to revolve in days. An instance of this sort may happen where a weekly index is introduced in con junction with a daily hand, to indicate the 24 hours of each of the seven successive days as they recur. But it will hardly ever happen, that any planetary motion can be cor rectly produced by a computation so simple. The perio dic or synodic times of any two planets to be compared together must first be reduced into the lowest denomina tion ; and if the large simple fraction, composed of such high numbers, could be brought, by continual division, into terms low enough to constitute the numbers of two wheels, without a remainder, the value would thus be re tained, and the reduced fraction would at once give the respective numbers for the teeth of the required wheels, or wheel and pinion. But two periods taken in this way will very rarely be found exactly commensurate, and there fore some method of approximating to the truth must be adopted in the practical computation of such planetary numbers as shall in all respects answer the best purpose.