and printing Mathematical Tables. S7T 
I Would, however, premise, that if any one shall be of opinion, 
notwithstanding all the precautions I have taken and means I 
have employed to guard against the occurrence of error, that it 
may still be possible for it to arise, the method of differences en- 
ables me to determine its existence. Thus, if proper numbers 
are placed at the outset in the engine, and if it has composed a 
page of any kind of table, then by comparing the last number it 
has set up with that number previously caiculated, if they are 
found to agree, the whole page must be correct : should gny dis- 
agreement occur, it would scarcely be worth the trouble of look- 
ing for. its origin, as the shortest plan would be to make the en- 
gine recalculate the whole page, and nothing v/ould be lost but 
a few hours’ labour of the moving power. 
Of the variety of tables which such an engine could calculate, 
I shall mention but a few. The tables of powers and products 
published at the expence of the Board of Longitude, and calcu- 
lated by Dr Hutton, were solely executed by the method of 
differences ; and other tables of the roots of numbers have been 
caiculated by the same gentleman on similar principles. 
As it is not my intention in the present instance to enter into 
the theory of differences, a field far too wide for the limits of 
this letter, and which will probably be yet further extended in 
consequence of the machinery I have contrived, I shall content 
myself with describing the course pursued in one of the most 
stupendous monuments of arithmetical calculation which the 
world has yet produced, and shall point out the mode in which 
it was conducted, and what share of mental labour would have 
been saved by the employment of such an engine as I have con- 
trived. 
The tables to which I allude are those caiculated under the di- 
rection of M. Prony by order of the French Government ; — a 
work which will ever reflect the highest credit on the nation 
which patronized, and on the scientific men w^ho executed it. 
The tables computed were the following. 
1. The natural sines of each 10,000 of the quadrant calculat- 
ed to twenty-five figures with seven or eight orders of differen- 
ces. 
2. The logarithmic sines of each 100,000 of the quadrant cal- 
culated to fourteen decimals with five orders of differences. 
VOL. VII. NO. 14. OCT. 1822. 
T 
