178 Notices respecting New Books. 
tifying and serviceable to the advanced mathematician, is not 
in our opinion perfectly calculated for the initiation of a stu- 
dent, at least in the present state of other branches of ma- 
thematics in this. country. Besides this, there is a very im- 
- portant advantage in exhibiting the algebraic formula as Mr. 
Bonnycastle has done, together with the logarithmic rules ; 
because by properly attending to the signs (+ and —) of 
the various expressions for the sines, tangents, &c., of ares 
or of angles, and particularly by adopting those forniule 
which furnish results in cosies, or cotangents, or the tan- 
gents of half arcs, or tangents of 45°+4 half arcs, or angles, 
every ambiguity which would otherwise arise on the resoln- 
tion of spherical triangles may be kept clear of, except 
those which appertain to the two cases that are necessarily 
ambiguous: even in these two cases the student may pro- 
ceed without difficulty by attending properly to this author’s 
observations at pp. 77, 78, 79. Had he, indeed, inserted in 
his work, Bertrand’s table, given by Lacroix, to which he 
refers In a note at p. 79, a table which, though it com- 
prises all the possible varieties of these two cases, does not 
fill a page, there would have been little or nothing wanted to 
render this part of his treatise complete. 
After the doctrine of spherical trigonometry, Mr. Bonny- 
castle treats of the mutations of the signs of trigonometrical. 
quantities ; and then presents, in fifty-four pages, a most 
gopious and interesting collection of trigonometrical for- 
mula, relating to what is usually termed the arithmetic of 
sines and cosines, the values of sines, cosines, &c. in terms 
of circular ares, &c., and vice versi, exponential quantities, 
logarithmic series, the series for logarithmic sines, tangents, 
secants, &c, This part of the work alone is sufficient to 
stamp its value, did not every other part bear evident traces 
of the same hand. 
The demonstrations of the principal theorems in plane 
and spherical trigonometry, made use of in the earhier part 
of the volume, succeed the trigonometrical formule ; and 
these are followed by demonstrations of the leading theorems 
in the stereographic projection of the sphere,—some miscel- 
taneous problems relative to spherical areas,—solutions of 
alt 
