NUMBERS. 
wheels for the moon’s nodes are oe and py = 18}, gives 18 
years and 243.49481 days for the period of their retrograde 
revolution. Alfo, in the mechanical paradox of Mr. Fergu- 
fon, the wheels Pe produce the fame kind of revolution in uf 
or 18% half years. In this cafe the retrogradation of the 
driven wheel is quicker than that of the ecliptic plate, on 
which account the apparent motion of the nodes is retrograde, 
viz. the reverfe of ea ft cafe ; 3 for there the /o/s of velocity 
ence of two otions ion 
movement of wheels and pinions may here alfo be reduced 
a fimple Eamon and then eftimated by the prefent 
cafe. 
Cafe 7.—The paralleliim of a wheel’s arbor, which is 
elles o a moveable arm round a central aaa is preferved 
by means of one revolution of that wheel, in a direction coz- 
in the fame time that "the arm is 
: Ww 
is alae any individual point of the 
its ee it is only nece er Pit re- 
volves israel once in each eecide 
ports it a direCtion n contrary to “that of the arm; it is 
of no one uence, in point of accuracy, me -— — 
chanifm this effect is produced. It is 
eflect this purpofe, that the ae axis and clipe ie pate 
in all the tellurians, lunaria, and orreries a 
be, preferved parallel to a required origi Teo : but 
fhall ap. cafion hereafter to point out a defe& in this 
refpe& in f our modern a anes 
The third head that we propofed to treat of, is to deter- 
mine by calculation planetary numbers proper for wheel- 
work, which hall produce a given e effect. The moft fyftema- 
ing down the various rules for doing this, wi 
ich we have feen the e ap- 
e fundamental directions for determining 
laft head, for computing the value of wheelwork already con- 
ftru@ed, that analyfis does to fynthefis; the reader, there- 
fore, muft bear in og that in thofe cafes where multipli- 
cation or additio e ufed there, divifion or fubtraction 
menfurable fractions. 
afe 1.—When a pinion is required to revolve any given 
number of times for its wheel once, as fuppofe 10, it is only 
neceflary to fix upon a fuitable number for the pinion, and 
multiply the given number of revolutions thereby to afcer- 
5 
a the number of teeth for the required wheel; thus, ir 
. pinion chofen be g, the wheel muft be g x 10 = go, 
fo that 2 will be the pair required to produce an increafe 
of velocity in the ratio of 10 to 13 but if a decreafe be re- 
quired, then 9° will be the proper pair, 
n the number of revolutions confifts of an integer 
and a fractional part, or what is ufually called “a mixed 
eafe, it m 
number,” as is generally the 
to a fi mple fraétion, which will Faia the ratio of 
wheel and its proper pinion : for inftan 2 able aia 
were required to be effected in a ~~ time the fimple frac 
tion or ratio would be al orrather a as ; there is to be an in 
creafe of velocity, and as the denominator is to be made the 
driver ; here, however, the 4 is number to con 
ftitute a good pinion 5 let, ped both os of the fee 
tion be augmented i in the fame proportion, till they are both 
of a convenient practical fize; in‘this inflance, for moft 
common purpofes, the numerator and denominator may be 
both doubled, by which means = will be the wheel and pi- 
nion required. a 
This rule is not only eafy to be _underftood by any per- 
fon who underftands the nature of vulgar fraGions, but its 
application is general in all kinds of fimple wheelwork 
which are placed in ftationary pofitions. Had the tropical 
seca of the hae been given in books on aftro- 
xact fraétions of a folar year, or of any other 
igs pened, thefe ec would have been fufficient 
for determining the wheelwork which falls under this cafe ; 
but the periods are generally given in wee hours, minutes, 
and feconds, on which account it is neceffary to oun 
each period into its loweit den nomination, and alfo the pe- 
i i othe fame, af then 
tinual alternate ace till there is no remainder 
e€ted in the common fchool books of arithmetic ; 
the fraétion in its iawn terms ses ee a ratio for which 
the ati and pinion are to be det 
For the fake of exemplification, a ee us fuppofe that ae 
eel period of any heavenly body re 
fun, be 243° 115 52™ 308, 
quired to produce this revolution, whilft the driver revolves 
in 365% 5° 48™ 48°: thefe periods, reduced into a fimple 
. : 210 gi! ea ee . 2 
fraGtion, will be — bee” which, in its lowefl terms, is —; 
therefore, increafing both parts eight, nine, or ten times, we 
16 I 
fhall have re 
24 2 
20 : 
, OF = for the wheels required. Inevery 
inftance where a folar year is the denominator, the ratio 
an inferior planet will be a proper, and of a fuperior one 
an improper asa 
Ca ver two periods which are compared to- 
afe 2.— 
gether eeu a fra&tion, which, in its loweft terms, is 
peau ee of numbers large for fingle olea, a train 
- the fimple wheels, which may be done thus; find, by 
a repetition of — — or from a table for this 
purpofe, what two o mbers, ufually called compo- 
fite numbers or Reo, multiplied into each other, will 
give 
