352 On the Invention of the Circular Parts. 
data. He points out in the last chapter that the same formule 
will apply to all the cases of each triplicity, and his two for- 
mule resemble, of course, those of Napier in their structure. 
But Torporley has not accomplished the same amount either 
of symmetry or abbreviation which appears in the rules of 
Napier. The reduction of all the six cases to two, and the 
first exhibition of an organized mechanical mode of reducing 
each of the six cases to its primitive, belongs to him: Napier 
afterwards did the latter in a better manner, without the ne- 
cessity of mnemonical verses. 
Torporley has given two tables of double entry, which 
Delambre says are the most obscure and incommodious that 
ever were made. ‘The first is neither one nor the other; a 
and 6 being the arguments, and c the tabulated result, it 
amounts to tanc = tana x sin J, the double entry being con- 
trived like that of the common multiplication table. Of the 
second table, as the book is scarce, I subjoin half-a-dozen in- 
stances. 
30° 70° 
G GMM G GMM 
21 27 39 14 
° 0° 54° 
= § 33 : 30 46 
12 54 8 28 
90° 147° 
G GM M G GMM 
PE ate ae 59°| 88 3 
18 2 58 57 
53 56 29 6 
As far as the formule for right-angled triangles are con- 
cerned, this table applies as follows. ‘The sine of the angle 
on the left multiplied by the sine of the upper angle in the 
square compartment, gives the sine of the second angle in that 
compartment. Thus Torporley means to say that 
sin 24° x sin 21°27! = sin 8° 33). 
Those who like such questions may find out the meaning 
of the other parts of the table. 
The question whether Napier had seen Torporley’s work 
cannot easily be settled: he uses the-word ¢riplicity, which is 
one of frequent occurrence in Torporley, and the figure of 
his demonstration contains the three triangles put together in 
exactly the same way as Torporley’s mother and daughters 
come together on the mitre. These circumstances are not 
conclusive, but they are suspicious: ¢riplicitas was by no 
