Writing the quotients in the reverse order and proceed

ing as before, we obtain the following set.

Now every one of the fractions in the last set consist

m

of the value of that belongs to one of the fractions

k

of the first set, as shown by the corresponding letters of reference; the fractions of the first set being supposed to be represented by the formula

Any other integrals substituted for m and k will give new approximate fractions; as for example,

the decimals serve to show the closeness of the approxi

252. If we apply this method to the example (Art. 247) of an annual movement, the approximate fraction becomes

in which k and m may have any values; for example,

corresponding to a period of 365d. 5h. 48'. 58".6944. (error 10".69). This is the annual train which has been calculated by a different method by P. Allexandre, in 1734, and afterwards by Camus and Ferguson.

which corresponds to a period of 365d. 5h. 48'. 55".38, is quite as convenient, and rather more accurate.

In a train of this kind one or more endless screws may be introduced, by way of saving teeth; for example, in the fraction last cited the numerator does not admit of being divided into less than three wheels; but the denominator may be distributed between two pinions and an endless screw, (remembering that the latter is equivalent to a

If

pinion of one leaf) thus, 1 × 20 × 13, or 1 × 10 × 26. the endless screw be not convenient, then the terms of the fraction must be multiplied by 4, to make the numbers of the denominator large enough for three pinions, and the train will stand thus,

44 x 89 x 97

8 x 10 x 13

253. Ex. To find a Lunar train that shall derive its motion from the twelve-hour arbor of a clock.

The mean synodic period of the Moon is 29a. 12h. 44′. 2". '.8032, which is exactly equal to 29.530588, or nearly 29.5306, and since twelve hours is equal to 0.5, the ratio will 147653 2500

be

295306 5000

, or, dividing each term by 2,

; from which the following quotients and fractions may be obtained.

Now as the whole number nearest to the original fraction is 59, which is less than 82, it is clear that two pair of wheels should suffice. The whole of the secondary fractions which would not admit of reduction, are omitted. The principal fractions are refractory, with the exception of (A), 32.5.7 42

945 16 which has been employed by Ferguson and by Mr. Pearson; it corresponds to a period of 29d. 12h. 45′ exactly,

and has an error in excess of 57".2; as it is a multiple of seven, it may be introduced into a clock which has a weekly arbor.

This fraction has been already obtained by a coarser method in (Art. 245.).

Of these a is a train given by Francœur, b and c by Allexandre, d by Camus, e by Mr Pearson; each of these writers having arrived at his result by a method of his own".

Vide Francœur, Mécanique Elementaire, p. 146. Allexandre, Traité Général des Horloges, p. 188. Camus on the Teeth of Wheels. Rees' Cyclopædia, art. Planetary Numbers.

254. The early mechanists were content with much more humble approximations, and employed a great number of unnecessary wheels. In the annual movement of the planetary clock, by Orontius Finæus (about 1700), the following annual train is employed, from a wheel which revolves in three days*.

12- -48

A train of half the number of wheels would do as well,

Again Oughtred†, in 1677, is satisfied to represent the synodic period of the Moon by 29 days, and employs the

train

40 × 59

10 × 4

Huyghens employed for the first time con

tinued fractions in the calculation of this kind of wheel

work.

255. Let it be required to connect an arbor with the hour arbor of an ordinary clock, in such a manner that it may revolve in a sidereal day; so as to indicate sidereal time upon a dial, while the ordinary hands of the clock shew mean time upon their own dial.

Twenty-four hours of sidereal time are equivalent to 23h. 56. 4".0906 of mean solar. Neglecting the decimals and reducing to seconds, we obtain 86400" of sidereal time equivalent to 86164" of mean time, and therefore one wheel must make 86400 turns while the other makes 86164, or dividing by the common factor 4, we get

+ Oughtred, Opuscula. Hugenii Op. posth. 1703.