The most direct mechanical method of determining a series of approximate fractions, is by means of two loga rithmic lines of a long sliding rule, if one of the high num bers taken on the slider (A) be placed in coincidence with the other, taken on the stock (B), then every pair of coin cident strokes, read along the rule, will be so many frac tions of nearly the same value with each other, and with the high numbers from which they are derived ; and if they are written down in succession as they occur, their comparative values may be ascertained by considering them as so many fractions of the assumed period of the first mover, and by examining such fractional parts of the .
said period succession. Thus let the tropical revolu tion of Mercury be given, as compared with the revolu tion of the earth, viz. 525948.8: 126674.585 minutes, then their velocities will be to each other as 1 : 4.151967, or as 1: 4.152 very nearly ; if therefore 1 on A be put into coincidence with 4.152 on B, as before directed, the pairs of coincident numbers, read on the rule in succession, will be , with several others nearly coin 25 29 54 137 191 cident. The values of these fractions may be determined and arranged thus ; 46 Hence it appears that the numbers — would produce le 191 tropical revolution of mercury within 6' 25" of the truth itself, and that the errors of the other numbers increase as the numbers themselves diminish. If the driving wheel of 191 teeth were placed on an annual arbor, and 4.6 on M ercury's tube, in a planetarium, the planetary arm car ried thereby would perform a revolution in the period above specified. If, however, the numbers in any of the pairs are too small, they may both be increased in the same pro portion, so as to give wheels of a proper diameter for con 13 struction ; for instance, may have its numbers doubled, 2 and will make a pair of wheels of convenient practical 108 dimensions ; but in the last pair, 191 being a prime num ber, cannot be diminished ; and the computation that is best in theory, is frequently inconvenient to be introduced in practice, as the constituent portion of a simple machine, where the common distance between the arbors is limited. If we were to compare the tropical period of Venus, in like manner, with a solar year, the only small fraction procured by the sliding rule would be 8 —, which may be made 16, 13 26 24 32 40 48 , 52 65 78 , or accordingly as the numbers are both multiplied by 2, 3, 4, 5, or 6, till the wheels become of a suitable diameter to answer their proposed purpose. The
8 value of — of a solar year, is 224d. and the 13 true tropical period of Venus 224d. 41m• 30s•, so that the error in each period would be-1-0d• 38m• In a similar manner, the approximate fractions of a year may be obtained for all the other planets with very little trou ble ; and the wheels derived from them, when duly pro portioned, and cut with suitable cutters for equalizing the tooth and space, will constitute a planetarium of Roemer's construction.
\Vhen, however, the periods are intended to be exhibit ed with great accuracy, a train must necessarily be substi tuted for a single pair of wheels. In this case, when the high fraction, consisting of the periods of the two planets to be compared together, is reduced to a common deno rnination, and diminished into its lowest terms, each part of such fraction must be considered as a product, arising out of two, three, or more factors, according to the num ber of wheels intended to be employed in such train; and if each part of the fraction, reduced into its lowest terms, is found to be divisible into factors, having numbers suit able for constituting wheels, then such factors, put into the form of a compound fraction, in which the numerators and denominators must keep their respective places, will be the train required. But as it will very seldom be found, that a large portion is commensurable, when days, hours, minutes, and seconds, are all taken into both parts of the portion ; and as the occurrence of a large prime number, as a factor, renders the train impracticable, the direct me thod of approximation devised, as is generally understood by COTES, must be substituted for the two logarithmic lines, for determining a series of approximate fractions, out of which to choose a practicable pair of numbers for the com position of a train. The method of procuring a continu ous series of fractions, or ratios, by common arithmetic, may be easily practised by attending to the following rules: I. Reduce the periods of the two heavenly bodies, to be connected by a train into one common denomination, either in seconds, or in decimal parts of their comparative periods, where unity will represent the shorter period.