296 
Proceedings of the Royal Soeiety 
In applying such, a table to the business in hand, we should have 
to compute the circumference from the diameter, and afterwards 
the diameter from the circumference. This double arithmetical 
operation may be avoided by the simple contrivance of counting all 
in diameters ; for the length of the hand, we substitute the diameter 
of the circle which it would gird. Writing them 
W = diameter of wheel, 
P = diameter of pulley, 
B = diameter for band, we have 
W - P = 2 sin ^ , 
B _W=^-('l-~')sin^. 
hir \ Itt/ 
Prom a table constructed according to these formulge, the dimen- 
sions of the lathe may be got by little more than inspection. 
The actual table and the mode of using it, exhaust the mechani- 
cian’s interest in the subject, but, to the speculative mathematician, 
some connected points may appear to be deserving of notice. 
Our ordinary tables, arranged according to the ancient graduation, 
are exceedingly inconvenient, because the ratio— expressed in 
¥ 
degrees and minutes would involve, in each case, a troublesome 
division. The decimal subdivision of the gradient was therefore 
preferred, and the computations made for each thousandth part, 
that is for intervals of 10' centesimal. A manuscript canon of sines 
to 15 places (using however only 10 of these) rendered easy the 
computation of the terms 
1 + r~ ) ^ f 1 - ^ sin i . 
\ 
The canon of “ sines measured in degrees ” had been prepared, in 
order to compile the table of circular segments, used for the compu- 
tation of the anomalies of the planets ; the term was thus 
ready to hand. In this way the construction of the table for 
lathe-bands was greatly facilitated. 
Ill all such calculations, the residual last-place errors may happen 
