349 
of Edinburgh, Session 1876-77. 
was copied on thin transfer paper, on which no alteration can be 
made without being obvious. 
The method of computing might have been extended to differ- 
ences of the fourth order, in which case the common multiplier, 
4 vera 2 , would have been much smaller, and the terms of the 
products fewer. But, on the other hand, the entries in the record 
would have been more, and the effects of the residual errors would 
have been much greater and more troublesome in correction. For 
the interpolation of each quarter degree, the saving obtained by the 
use of fourth differences would have been unappreciable ; for those 
of each fifth and of each single minute, it would have been hardly 
such as to compensate for the inconveniences just mentioned; but 
for the future interpolations of each tenth second and of each single 
second, the fourth differences may be advantageously used. 
When the canon of sines shall have been completed, the com- 
putation of the working table, that of logarithmic sines, will be 
easy, particularly by help of my fifteen-place table of logarithms. 
Beginning, as John Nepair did, at the sine of the whole quad- 
rant and proceeding downwards, the computed logarithmic sines 
may be verified by their differences, which are small. When the 
work has been brought down to the log sin of the half-quadrant, 
the farther progress is easy and rapid, for the formula 
2 sin a . cos a — sin 2 a 
gives log sin a = log \ + log sin 2 a - log cos a, so that the dif- 
ferences of any order may be got at once from the previously 
tabulated differences of that order, and, what is most worthy of 
remark, may be used without the fear of an accumulation of 
residual error. 
The table of logarithmic tangents follows as a matter of course. 
3. On the Precautions to be taken in recording and using the 
Records of Original Computations. Edward Sang, Esq. 
The real utility of tables of numerical results is only secured by 
making them accessible to those computers who may require them ; 
