250 Proceedings of Royal Society of Edinburgh. [sess. 
pared, the rest of the work was carried on in the usual manner 
with verifications at frequent intervals. 
Logarithmic Sines. 
The logarithmic sines were deduced from the sines themselves 
by help of my fifteen-place table of logarithms of numbers from 
100 000 to 370 000, using the auxiliary table. Beginning at 
100° 00', the computations were made on scroll paper for each single 
minute down to 50° 00', each step being verified by first, second, 
and third differences. The third differences were then copied into 
their places on the actual manuscript, and the others were thence 
reconstructed. In this way all errors of transcription were avoided, 
and any mistake in the previous work detected. 
From this half of the canon, the other half, namely from 50° 00' to 
00° 00', was deduced according to the formula sin a = \ sin 2 a .sec a ; 
the method of proceeding being to compute each tenth log sine 
directly, and to fill in the single terms by differences easily got from 
the differences already written in the first part. This operation 
supplied a check on all the previous work. 
Logarithmic Tangents. 
The logarithmic tangents were computed in the same way, that is 
to say, each tenth term was found directly, and the single terms by 
means of the preceding differences, thus furnishing another verifica- 
tion of the whole. But, seeing that the log tangents of the one 
half are the arithmetical complements of those of the other half, it 
was enough to write out the log tangents of arcs from 50° 00' to 
00 ° 00 '. 
With the exception of the arrangement for computing by differ- 
ences, and for assuring exactitude, this is the very process used by 
JSepair in the construction of his Canon Mirificus ; and, indeed, 
this volume of Logarithmic Sines and Tangents might, with all 
propriety, have been entitled : — 
“ John Repair’s wonder-working Canon, changed by his express 
desire, to suit the Denary System of Arithmetical dotation.” 
After a labour which must have occupied his leisure time for 
more than the quarter of a century, Repair had published his 
