of Edinlurgli, Session 1883 - 84 . 
639 
way ; there are the prejudices of habit, the discomforts of transition, 
the existing mass of preparatory work suited to the old plan, and, 
above all, the mass of preparations needed for the new scheme, 
Here, indeed, the great obstacle lies. 
Aside from the proposal of a change of system, a new computa- 
tion of fundamental tables looms in the near future. The precision 
of modern measurements render it necessary, in astronomical specula- 
tions, to reckon to hundredths of a second of time, to tenths of a 
second of arc. How when we determine an arc by help of (say) 
the seven-place logarithm of its tangent true in the last figure, the 
uncertainty arising from the omitted parts may amount to the 
fortieth part of a second; so that, since the logarithm itself is 
subject to several similar uncertainties, we may, notwithstanding all 
care, err by the tenth part of a second. But it is a sound principle 
that the accuracy of the arithmetical work should be clearly beyond 
that of observation, in order that no perceptible new error may be 
introduced, and thus the time is not very far distant when eight- 
place tables may be indispensable. Hence, in designing the canons 
for the decimal system, we must also look forward to increased 
precision. 
Since by far the greater number of computations are done by help 
of logarithms, our first business is to see to the logarithmic canon. 
Beginning independently of all previous work, the logarithms of 
all primes up to 10,000 have been computed to 28 places, that 
they may be true to 25, each prime being put in relation to, at 
least, three others. The greatest discrepancy found amounted to 
unit in the 27th place, so that this fundamental table may be 
regarded as altogether free from error. The volumes I., II., III., 
placed on the Society’s table, contain all the articulate steps of the 
work, with indices to the primes and to the divisors used ; so that, 
if in any subsequent computation one of these divisors should 
recur, we are spared the labour of a new division. Thus for the 
logarithm of 6563, the three equalities were used 
32 130 000 001 11-599-743-6563 
627 600 001 = 7T9-719-6563 
36 930 001 = 17-331-6563 
and the agreement furnished presumptive evidence of the accuracy 
