345 
of Edinburgh, Session 1876 - 77 . 
In the second place, Henry Brigg’s original table had been col- 
lated by Prony’s assistants, and errors in the tenth and higher 
places had been found ; yet no notice had been taken of the very 
many errors in the fourteenth and even in the thirteenth place. 
Thirdly, the system of computation adopted was so imperfect 
that, although differences of the sixth order were extended to the 
thirty-sixth place, the results were liable to errors up to the 
twentieth figure. 
Lastly ; the Cadastre calculations were used by M. Lefort for 
correcting Adrian Vlacq’s ten-place table (that table of which all 
the subsequently published seven-place tables are abridgments); 
with the result of sometimes putting Vlacq in the wrong when he 
is right. That is to say, the Cadastre calculations cannot be trusted 
for the compilation of a ten-place logarithmic table. 
Such being the case with that part of Prony’s great work which 
was comparable with previously published tables, we are unable to 
place confidence in the trigonometrical portion, which necessarily 
is almost entirely new ; and we are forced, when contemplating the 
compilation of the Canon of Sines, to hold the Cadastre Tables as 
non-existent, or at best, as useful for controlling palpable errors of 
the press. 
In actual trigonometrical calculations we very seldom use the 
sines and tangents themselves, but employ their logarithms instead; 
wherefore, both the Canon of Sines and the Logarithmic Canon are 
needed as the joint foundation of our working tables. Having car- 
ried the table of logarithms as far as to 370 000, and being satisfied 
of the insufficiency of the work done in the Bureau du Cadastre, I 
resumed the computation of the sines, and have now proceeded to 
such a length as to be able to submit the methods employed to 
the Society, and through it to the mathematical public. 
The decimal division of the quadrant is effected by bisections 
and quinquisections, and the first thing to be determined is the order 
in which these operations should be taken. Now, we obtain the 
sine of the half arc, not from the sine of the whole arc, but from 
its cosine, or rather from its co-versed-sine ; when the arc is small 
the co-versed-sine, or defect of the cosine from the radius, is repre- 
sented by a very small decimal fraction, the number of whose 
