NUMBERS. 
Procefs. Ratios, Values. 
Il X 23 — 253 snag 
26x 23—31§ 597 4237 
II xX 22 742 nag 
26 X 22 — sh = Fy = 4338 
Il X 21 _ 23t ag 
een Te 423°5 
HX X 207] 220 sig 
26 x 20 —1 = 519 47399 
Il xX 19 209 
= = «42 
26 X Ig —I1 493 ee 
rr x 18 198 
Zt = 2 = 142398 
26 x 18 — 1 467 4239 
mx yy 187 _ 
6 ei7oil 4e° °° 
_ ii x 16 _ 176 ue 
26 X 16 —1 =e 
II xX 15 __ 165 wae 
26x 15 —t$ 389 — 
2 bas 1424242 = —* as before. 
56x 14-15 uae 
The moft accurate ratio of this table is a but 29°33 
is incapable of becoming a practical train when reduced into 
afimple fraétion, one of the factors of the numerator being 
18341 
623 
which is not reducible into numbers low enough for a train ; 
and 29342 is alfo incapable of redudtion, by reafon of the 
the large prime 8783, the next in accuracy is 29294 = 
dencminator 571 being a prime number; but 29273 = 
19096 is reducible into the train ale x 4 the value of 
649 59 
which is 10746" h 54™ 53°; this train is of the fame 
value as 2924, sae from the ‘fliding-rule alone, and may 
otherwife be more conveniently expreffed thus, = x _. 
6 
ag 23% .. 16036 
545 545 
rd x ss the value of which is 10746* 20% 0” 
109 5 
will give us the lefs accurate 
175.26. 
Wheels eas the ae Revolution of the Geergian cae 
—We come, laftly, to the Georgian ry ae performs 
its circuit through ie clit in 305899 8" 27%, if we con- 
ande’s tables, as ene by 
e 83.750865 tropical years for the length 
of the fame ae which happens to be fo nearly 833, or 
oa ©=-, that an approximation feems unneceffary: the train 
oe to this fraGtion may be a x 5 = 305898 
A 52™ 0%, which time is fo near re period ae by the 
as that the error in {pace is only about 14” in the whole 
revolution 
Thefe examples will {uffice to fhew the application . this 
mode of approximation in any calculation of wheelwork 
that may occur in the pra€tical conftruction of an iets 
though, it muft be confeffed, the following method is more 
concife, as well as more ftrittly mathematical, and therefore 
will aoe be preferred by thofe who underftand it, and 
mal have occafion to make new calculations for wheel- 
wor 
It - was deemed ie a to exemplify both methods, i in 
order that, when one wi 
The Betaintion and an Orrery © w Cone 
ftru€tion, Martin. Coneeet i ey @Al ebre, 
roix 
ede IntroduSio in Analyfin Infinitum, 
Euler, Ovpvfeuia res ae Euler. Elements of 
gebra, vol. ii. Euler, La Refolution des Eaten Nu- 
meriques, La ange. Theorie des Nombres, Le Gendre. 
e l’Academie de Peterfbourg, 1779 
Chie du Journal de Ecole Polytechnique, ae HA *g 
‘on iot’s A ftron 
however, in olteine n of eit the 
prefent tables afford for the data of their calculations, and 
by giving only the theory and refult of their method, inftead 
of the arithmetical procefs at full length, have puzzled the 
artift, and left him at a lofs how to proceed in the praétical 
application of rules which have not been fufliciently exem- 
plified ; indeed, the fifteen theorems of Martin, which were 
intended to illuftrate, by algebraical expreflions, the theory 
of determining a converging feries of ratios, have deterred 
the initrument maker, who is unacquainted with algebra, 
from entering inci oe which carry with them the 
ira of intr 
, therefore, diveft the method of approximation, 
ter, of algebraical 
a 
paar neta by either vulgar nddeaw 
a line’ between them for the true ratio, or unr ed  frablon, 
2. Divide the greater by the {maller part of this ratio, 
and note the quotient ; then make the divifor a new dividend, 
and the remainder a new divifor, to procure a fecond qua- 
tient ; and continue the fame procefs until feven or eight 
quotients are obtained, according to the c ommon rule of 
obtaining a common meafure in valgar fra& 
ut © for the firft ratio, and for i eat and all 
e fucceffive 
m 
by th 
ratio laft obtained, and add to the produ the 
numerator ae the next os ratio, and this will be the 
numerator of a zew ratio to follow ; do the fame with the 
denominators of the = ae next preceding ratios for the 
denominator of the fam ratio; and continue this pro.’ 
cefs till ratios are eee of all the cures in ee 
Ee ue 
