instruction have proved, as hundreds of individuals would bear testimony, that the theorem here given will save, at the lowest estimate, two-thirds of the labor ordinarily incurred by the rectangular method. A further advantage is that, dispensing with a large and faulty table altogether, it is far more accurate—the computations being executed by aid of the common logarithmic numbers, calculated with greater care and usually extending to six or seven deci. mal places, and the operation being so ordered that, without any additional labor beyond what is absolutely essential to an honest confidence in the result, all gross errors, if any exist, whether of the field or the tables, are detected, and if these have no existence, the smaller and unavoidable ones very much reduced.

It is hoped that this book, often requested by my pupils, will prove acceptable to the schools generally.

Of the third Part, embracing the mensuration of solids, spherical trigonometry, and navigation, time will permit us to say little more than that, by the method pursued, we have been enabled, within moderate limits, to give a fuller development of these subjects than is usually found in our elementary books.

The modification and extension of Napier's Rules demands, how ever, a brief historic notice. I demonstrated and extended these rules by showing : I. When A = 90°, 2. = 90°-Q, B. = 90°- B, C. = 90° - C,

sinb = cosac cos B. = tan C. tanc,

sinc = cosac cos Cc tanB. tanb,

sinac = cosb cosc tanB. tang.

sinB = cos C. cosb tanac tanc,

sin C.= cos B, cosc

tana, tanb:

II. When a = 90°, B, = 180° — B, A, = A - 90°, &c.,

sinB = cos A, cosb, = tanc, tan C.,

sin C= cos A, cosc, = tanb, tanB.,

sin A,=cos B, cosC, = tanb, tanc,,

sinb, = cosc, cos B. = tan A, tan C.,

since = cosb, cos C. = tan A, tanB.: III. When c= a, or the triangle is isosceles,

sinac = tan Atan(1B). sin A= tana, tan(+b), sin(+b) = cosa, cos(B).. sin (1 B) = cos A, cos(16).