of Edinburgh, Session 1883-84. 
541 
logarithms of such ratios are given for each of the ten thousand 
numbers from 1 000 000 000 to 1 000 010 000. This list serves the 
purposes of both of Nepair’s Tabulce, iwima et secmida, and gives us, 
by help of this easy division, the fifteen-place logarithm of any 
number whatever. Kot only so; it also enables us to solve tlie 
converse problem, by help of a multiplication as easy. 
But we may approach to the required result, from the tabulated 
numbers immediately above. So, in order to supply the means of 
verification, the auxiliary table is carried, on the other side, to ten 
thousand numbers below the same 1 000 000 000. 
This addition to the canon, besides greatly lessening the labour in 
interpolating, lends itself readily to systematic computation. 
The fundamental canon for trigonometry is that of sines : these 
to 25 places for each two thousandth part of the quadrant, and 
to 15 places for each ten thousandth part, have been computed 
strictly by second differences, verified at short intervals. In the 
volumes placed before the Society the actual calculations are con- 
tained : they are recorded in such a form that each sine may be 
instantly examined. The manner of the calculation afforded a 
continuous and complete check, and the table is believed not to 
contain a single error. From these, the canon of logarithmic sines 
and the other usual trigonometrical tables may easily be compiled to 
an exactitude far beyond the requirements of practice. 
In a paper entitled ‘‘Nouveau Calcul des Mouvements elliptiques,'’ 
printed in the Memoirs of the Turin Academy for 1879, the mean 
anomaly of a planet is deduced from its position by taking the sum 
or the difference of two circular segments. In order to reap the 
advantage of this exceedingly simple solution of Kepler’s problem, 
we need first to compute the sines, measured, not in parts of the 
radius, but in parts of the circumference. The volume marked 
“Sines measured in Degrees” contains the whole calculation of this 
canon for each centesimal minute, and to ten decimal places of the 
quadrant. 
From this table, that of circular segments, measured in degrees of 
the surface of the circle, for each of the 40 000 minutes of the 
entire circumference, has been composed. This table, though 
designed expressly for astronomical purposes, has its uses in other 
branches of science. 
