PRE FACE.

The Reader is now presented with the Third Volume of our SCIENTIFICAL LIBRARY, containing four brief Treatises on LOGARITHMS, APPLICATION of ALGEBRA to GEOMETRY, PLANE and SPHERICAL TRIGONOMETRY. A slight knowledge of these sciences is indispensable to the student, but a very slight knowledge is sufficient. Their elementary principles, and enough of practice to confirm those principles in his mind, are all that is necessary to be given : it would be totally inconsistent with the plan of our Library, which professes to give all that is essential, but nothing more, to swell out our volume with formulæ which are only to be met with in profound investigations far beyond the aim of the general reader. In Spherical Trigonometry, especially, we have limited our Treatise to the exhibition of those formulæ alone which frequently occur in Astronomy and the Elementary Sciences: by which means we have been able to reduce this intricate branch of mathematical knowledge to a very simple form and very trifling dimensions. On the same principle, we have excluded from our Plane Trigonometry everything which was not absolutely necessary to the student, retaining, we hope, everything which will be found at all useful, In our second Treatise the Cartesian method of Algebraic Geometry has been omitted, as being much more remarkable for its elegance than utility. Of Logarithmic and Trigonometric Tables we have given just what may serve as examples of their use and construction: a complete set would occupy many such volumes as ours, and after all be of little use except to one class of readers.

We may add, that the same plan is followed in this as in our preceding Treatises, namely: that of subjoining, under DOUBLE RULES, such principles of each science, as, though not indispensable, might be found of advantage. These may or may not be read, according to the Student's fancy.

Of the above-mentioned four Sciences the various uses and applications will be evident to him who proceeds but a little way in the road to mathematical and physical knowledge. Even in the commonest affairs of life-in accompts, measurements, &c. their utility is scarcely imaginable. They enter, one and all, into every numerical investigation of any magnitude, and are latent in almost every mechanical invention. LAPLACE could not have written his MECANIQUE CELESTE without their help; nor the simplest artificer pursue his trade without, at least, a practical acquaintance with their principles. No more, we apprehend, need be said to gain for these Sciences the notice of every Reader who is desirous to improve his mind or worldly condition.

CONTENTS.

LOGARITHMS.

DEFINITION. The base of a system of logarithms is that number

whose different powers are made equal to the natural numbers, 6.

ARTICLE 1. In every system of logarithms the logarithm of

unity is cipher, 6.

ART. 2. In every system of logarithms, the logarithm of the base

is unity, ibid.

ART. 3. The logarithm of the product of any two numbers is

equal to the sum of their logarithms, ibid.

Art. 4. The logarithm of the quotient of any two numbers is

equal to the difference of their logarithms, 7.

ART. 5. The logarithm of the power of any number is equal to

the logarithm of the number multiplied by the index of the power, 8.

ART. 6. The logarithm of the root of any number is equal to the

logarithm of the number divided by the index of the root, 9..

THE APPLICATION OF ALGEBRA TO GEOMETRY.

DEF. The area of a - figure is the portion of space contained

within its boundaries, 21.

Art. 62. The logarithm of a proper fraction is a subtractive

number, 10.

ART. 63. The logarithm of a fraction is equal to the logarithm of

its reciprocal with a different sign, ibid.

Art. 1. The area of a parallelogram is equal to the product of

any side taken as base, and the corresponding altitude, 21.

ART. 2. The area of a triangle is equal to half the product of

any side taken as base, and the corresponding altitude, 22.

ART. 3. The side of a square is to its diagonal as unity to the

square-root of two, ibid.

"ART. 4. The altitude of an equilateral triangle is to the side as

the square-root of three to two, 23.

ART.5. Given any two sides of a right-angled triangle, the

third may be found, ibid.

ART. 6. In a right-angled triangle, given one of the sides about

the right angle, and the sum of the other two, the triangle itself may

be found, 24.

ART. 7. Given the base and altitude of a triangle, the inscribed

square may be found, 25.

"Art. 8. To divide a right line into two such parts that the

square of one may be equal to the rectangle under the other and the

whole line, 27.

Art. 9. Given the greater of two geometric extremes, and the

difference between the lesser and mean, the mean may be found, 29.

Art. 10. Given the three sides of a triangle, we may find its

area, 31.

PLANE TRIGONOMETRY.

DEF. I. The tangent of an arch, or angle, is that portion of the

geometrical tangent at one extremity of the arch, which is inter-

cepted between the sides of the corresponding angle, 43.

DEF. II. · The secant of an arch, or angle, is that portion of a geo-

metrical secant through one extremity of the arch, which is inter-

cepted between a tangent at the other and the centre, ibid.

DEF. III. The complement of an arch, or angle, is the difference

between it and a quadrant, or right angle, 44.

DEF. IV. The sine of an arch, or angle, is the perpendicular

drawn from one extremity of the arch to the opposite side of the

angle, ibid.

Der V. The versed sine of an arch, or angle, is the intercept of

the diameter between the sine and tangent, 46.

DEF. VI. The supplement of an arch, or angle, is the difference

between it and a semicircumference, or two right angles, 47.

ART. 1. The square of the secant of an angle is equal to the sum

of the squares of the tangent and of the radius, 43.

Art. 2. The rectangle under the tangent and cotangent of an

arch, or angle, is equal to the square of the radius, 44.

Art. 3. The square of the radius is equal to the sum of the

squares of the sine and cosine of an arch or angle, 45.

Art. 4. The ratio of the sine to the cosine of an angle is the

same as that of the tangent to the radius, ibid.