It is somewhat remarkable, that the errors in this con struction are, in three out of the four periods, the same as in Roemer's ; but this machine had the advantage of a dial of 24 hours, as well as a spiral face to indicate each day of the year, by the simple addition of one wheel of 73 teeth, and an endless single screw. We shall have occa sion to show hereafter, that a more correct satellite ma chine was afterwards made by the same contriver.
As a third example, we will examine the periods produc ed by the wheel-work of Roemer's Planetarium, in which a pair of wheels for each planet at that time known may be taken as the fraction of a solar year. The numbers of the wheels employed, and the revolutions produced there by, when the denominators are fixed on one common arbor, and the numerators attached to each separate concentric tube, will appear from the following arrangement.
The true tropical periods,which are contained in the fourth column, are derived from the tables of La Lande; and if we were to compare the numbers that compose the wheels of the common planetarium, exhibited for sale by the different mathematical instrument makers, we should find the great est part of them bearing to the computations of the original contriver. In the instances of Mercury, Mars, and Saturn, the instrument maker is at liberty to choose either of the two pairs offered him, according to the size of the wheels he means to employ. It is hardly necessary to remark, that the diameters of the two wheels that act together must be to each other in the same proportion, that their respective numbers of teeth are to each other, exclusively of the acting portion of the teeth, which will be more or less, according to the fineness of the teeth em ployed ; and also that the teeth and spaces should fit each other well, that no loss of time may be occasioned by im proper shake ; but it may be proper to observe that the sum of the acting radii of all the pairs thus employed should be separately ihe same constant quality ; hence the sum of the teeth in each pair should not differ considerably from the sum of the numbers of the mean pair taken for the assumed motion of the earth ; which requisite Roemer has not sufficiently attended to, though he might have in creased or diminished any of his numerators, provided that its own denominators were also increased or diminish ed in the same proportion. It is only necessary that the
value of the fraction should be retained.
When the revolution of a planet is occasioned by a train consisting of several wheels and pinions, which is usually the case in orreries, where the rotations are effected as well as the revolutions of certain planets, the periods may be ascertained without difficulty, by considering the whole train as a compound fraction of the assumed period of the first mover, and by reducing it into a simple fraction in the following manner : Select all the driving wheels and pinions, which generally are found in the alternate places, viz. in the first, third, fifth, &c. each actuating its fellow, and having counted the teeth of each in succession, multi ply them into each other, and let the product be taken as the denominator of a large reduced fraction ; then do the same with the driven wheels and pinions, which will be found to occupy the second, fourth, sixth, &c. places, and make their product the numerator of the same large frac tion; and the reduced fraction thus obtained must be con sidered as a pair of large wheels, the period produced by which may be determined in the same manner as the periods of Roemer's satellite instrument and the planeta rium have been ascertained. For instance, in the automa tion of Huygens, a clock movement with a balance im pels all the planets, by first giving motion to a common annual arbor, from which all the planets derive their pe riods, thus : A pinion of four leaves, revolving in as many days by means of the clock-work, drives a wheel of 45 teeth, on the arbor of which another pinion of 9 teeth is made fast, which drives another wheel of 73 teeth round 45 73 3285in— 4 of— 36 9 = — of 4 days, =365. Here the two pinions are the drivers, and the two wheels are driven, which con sequently are made the numerators in the compound frac tion, and the velocity is thereby diminished ; but if it had been required to increase the velocity in the same ratio, the wheels must have been the drivers or denominators of the fraction. Again, the annual arbor, which carries all the driving wheels of the planetary pairs, and of the trains for Mercury and the Moon in this machine, must be con sidered as the first mover, from which the periods of all the last wheels are derived by their respective fractions of 365 days, according to the subjoined explanation.