NUMBERS. 
give a produ& equal to the numerator, and fubftitute 
thofe numbers, with ign of multiplication between 
each, for the numerator “itfelt 5 do alfo the fame for 
denominator ; and thefe wi 
ame value as the original fin- 
ple fra¢tion. An inftance or two will render this cafe fa- 
a 62 ; : 
miliar: let us take —-2 as the ratio of a fuperior planet 
21 
compared with the earth, the numerator of which is too 
for the numerator are 37 and 17, their produé& being equal 
to 629, and the two compofite numbers for 21 are 7 and 3 ; 
37 X 17 
7% 7 
: 629 : 
hence, inftead of =. we have obtained > OF; 
21 
which is the fame thing in the for rm of a train, ox! x - or 
otherwife 22 x 47, ix being a matter of no confequence 
how the pinions are placed with refpeCt to the wheels, pro- 
vided they be both drivers: let now both parts of the ~ 
be doubled, fince 3 cannot be made into a good pinion, 
37, 34 
and the requifite train will be 7 x 
Again, let us take? for the ratio of an inferior 
21 
planet ; the compofite numbers will be ae 7 Se. 
and 27 x 23, conllituting a train of ~ a ort x << 
at ae aoa —se 
ama . 23° 54 46 
if they have all their lt doubled. “By this contrivance 
two {mall wheels with corref{ponding pinions anfwer the 
fame purpofe as one very large wheel and its pinion, and 
are not only much more eafily made, but rendered more 
portable = the conftruction of an inftrument. 
he aprens that there is a great difparity between 
the anne: and denominator of a fra€ti ] 
the large 
an example of this kind, let us take 2, the lunar num- 
bers of Fergufon’s ipeeals oes though more accu- 
rate than thofe of his o e has not attempted to re- 
duce into fmaller meet preaes the numerator is a 
prime number, for which we may put 19 x 1, and the de- 
nominator is equal to 47 x 5 3 hence i or 2 x Js 
47 * 5 47 5 
will give us a train of a x ~, equal in value to es, 
47 235 
In order to hagend the greateft poflible velocity with 
the ewe teeth in a train, the ratio of eac to its 
pinion a to be, fays Dr. T. Young, in his Syllabus, 
nearly as 36 to 10. 
@ : ~—lIn every inftance where x or 2 ftands in the 
t it muft be under t the fcrew be al- 
oie the driver, pean the fiGion. and indire a€tion 
of the teeth upon its thread will prevent the communica- 
tion of motion 
Cafe 4.—~This cafe is more intricate ie the preceding 
ones, | and feems not to have been un 
triver of an orr 
diurnal motion, which, 
muft be the numerator, and the one driven by it on the bar 
iste rae mult be added to, or fubtracted from, the 
e fi difference, as the cafe may be: if the motions of 
1e of the wheel borne by it be in the fame di- 
rection, the fraction muft be /ubtraged, but if they be in 
contrary directions, the fame mutt be added, before.the proper 
heels are calculated for the number thus obtained. a 
inftance of this f{pecies of aa let any central wheel 
P 
be fuppofed to have 15 teeth, and the wheel 
arm moveable abou 
arm is going once round: in the 
firft place, becaufe dea arm and the wheel 10 move both 
in the fame direCtion, 3, or 14, mutt be /ubtraded from 15, 
the number of revolutions required, which fubtra€tion leaves 
137; in the next place, becaufe the central wheel o 
goes only twice ms - the wheel of 10, which is driven 
by it three times, ~ ~ or — 2 of 1 33, which is equal 9, is the 
value to be expreffed by the fraction which is to reprefent 
12 
108 
ina fimilar ratio, will be the wheel and pinion required ; 
and the whole value, eftimated on Cafe 4 under our laft 
15 
Again, let two fuch wheels < ea as will produce 
the fame number of revolutions while the arm and the wheel 
borne by it move in contrary direGtions, the numbers of the 
two given wheels remaining the fame as before. Here 
the wheel and pinion required ; therefore, or any pair 
head, will be thus expreffed, 
~, or 13, muft be added to 15, which fum will be 164; 
then = or of 163 is rr, the value to be expreffed by the 
required wheels, which therefore may be ar producing 
two more revolutions of — central wheel than was produced 
in the other inftance ; the computation of the whole 
value will be 32 13\_ 35 
132 15) 
Henee it appears, as was intimated before, that when a 
negative 
