of Edinburgh, Session 1883-84. 
543 
The lengths are given for each second of the ancient division up 
to one degree, in all 3600 values ; thereafter for each degree up to 
1 800°, or five complete revolutions ; the values for fractions of a 
second being got by the transposition of the numbers at the 
beginning of the table, this facility being due to the adoption of 
the decimal division for seconds. 
For the modern division 1000 terms suffice, because by mere 
transposition the table may be extended indefinitely both ways. 
In order to pass from the one system of subdivision of the quad- 
rant to the other, a table of equivalent modern and ancient degrees 
is given, first from 10^^ to 10® or 9° to 9°; then for centesimal 
minutes up to 10° (computed for the sake of verification); next for 
each tenth decimal second up to the same limit; and, lastly, for 
each hundredth part of a second up to ten seconds. By this table 
the conversion of ancient into modern or of modern into ancient 
degrees is easily effected. 
Similar tables for the conversion of ancient and decimal time are 
exhibited. 
In the reduction of astronomical observations we have very often 
to exchange solar and sidereal time. In 1868 the writer published 
Time Conversion Tables for each tenth second of the whole day. 
The counterpart to these is herewith presented ; it is continued from 
day to day up to 1000 days; and this suffices for minutes, seconds, 
and fractions by simple transposition. 
Lastly, there is appended a Traverse Table, for plain and mean 
latitude sailing, for each of the 400 degrees of azimuth and for dis- 
tances up to 100. 
These form at least a beginning in the collection of requisite 
decimal tables. That which is first wanted beyond them is the 
canon of logarithmic sines. The preparation of this canon would 
be greatly facilitated by the extension of the fifteen-place logarithms 
up to the whole million — that is, for all six-figure numbers. Those 
of them already prepared need the aid of the auxiliary multipliers 
2 and 3 ; had they been carried to the half million, the auxiliary 2 
would have sufficed. 
In conclusion, it may be remarked that five and seven place 
tables are exact enough for almost all business purposes ; but that, 
in order to have these true to the last figure, the original calculations 
