2. Divide the greater number by the smaller, and reserve the quotient ; then make the division a new dividend, and the remainder a new division, to procure a second reserved quotient ; and continue the same process until seven, eight, or more quotients are obtained, according to the usual mode of obtaining a common measure in vulgar fractions.
3. Put - 0 0 for the first ratio, or fraction, and - for the se 1 cond, and then the succeeding fractions may be obtained from the reserved quotients, taken successively in the or der of their occurrence : thus, multiply the numerator of the last fraction by the first quotient, and add to the pro duct the numerator of the next preceding fraction, and the number so obtained will he the numerator of the next fol lowing new fraction ; treat the denominators in the same way, that is, multiply the last denominator by the same quotient, and add thereto the deno ninator of the next pre ceding denominator, and the amount will constitute the de nominator of the said new succeeding fraction. Let this process he continued till all the quotients are involved in so many successive fractions, which will approach gradu ally to the true fraction wanted ; and if the division is con tinued till there happens to be no remainder, the last quo tient will produce a fraction representing the truth itself; namely, the two numbers constituting such fraction will be to each other in precisely the same ratio as the two pe riods themselves, from which the succession of quotients was derived by the continual division.
As an exemplification of this method of determining a series of planetary numbers. for the wheelwork of a plane tarium, or orrery, either for single pairs, or for trains, we shall take the periods of Mercury and the Garth, as before proposed for the sliding rule, and from the accordance of the two methods, as to the results, it will appear that the arithmetical process has so far the advantage over the me chanical operation, that the fractions can be carried into terms of a higher denomination than can be read on a slid ing rule, however large its logarithmic scale; and, conse quently, the arithmetical series is preferable, as it affords a greater variety for the choice of practical trains, which must be derived from some single fraction composed of high numbers.
The same quotients would have been obtained, and con sequently the same series of fractions, if the two periods reduced into seconds had been used as the first divisor and 525948.800
dividend, viz. In the last column the same small fractions succeed one another, as are given by the logarithtnic lines, as far as to but beyond this fraction the numbers become too high. It is fortunate that the last fraction which is the most correct, is divisible into 737 90X34 the factors ---, which will constitute a train, of which 11 X67 the value differs almost imperceptibly from the truth ; for I I x67 =. 90X34 3060 of 365.24222 days, gives 81d. 23h• 14m•35.726'• for tropical period ; in which train the error is only 0.526'• in the whole period.
If the last fraction is found to contain a prime or incom posite number, too large to constitute a wheel, one of the preceding, but less accurate fractions, must be taken in stead ; or another quotient may be substi uted for the last quotient in the formula, provided that it be nearly of the same value. In this case, the quotient 2 or 4 might have 2213 been put for 3, and then --- = and 2X204+125 533 4Y847-1-519 = would have been the resulting 4X2041-125 941 3060 bers, of nearly equal value with 737 ; but, on consulting a table of prime numbers, it will be seen, that 2213 is not divisible into any two factors, and that both the numbers in the latter fraction are incomposite ; on which account, they must have been rejected as impracticable, and another quotient, 1 or 5, must have been substituted for the 3, in case practical numbers had not been otherwise obtained from one of the direct fractions of the series.
In like manner, a series of continuous fractions of a solar year may be obtained for all the planetary system ; and the trains, arising out of the last fractions of each series, or out of fractions of nearly the same value,(to be substituted by means of a change in the last quotient,) may be pro cured for the construction of a very perfect machine, so far as the mean motions are concerned. It may not be un acceptable to some of our readers to have a list of conti nuous fractions, which we have taken the trouble to com pute, according to the mode which we have just laid down and exemplified ; and we, therefore, subjoin the results in the form of a table.