ART. 5. The rectangle under the secant and cosine of an angle

is equal to the square of the radius, 46.

The sine of an angle is equal to the sine of its supple-

ART. 6.

ment, 47.

ART. 7. The sine of an angle is equal to half the chord of twice

the corresponding arch, ibid.

ART. 8. The rectangle under the radius and the sine of twice an

angle is equal to twice the rectangle under its sine and cosine, 48.

ART. 9. The rectangle under the radius and the cosine of twice

an angle is equal to the difference between the square of the radius

an twice the square of the sine of the angle, 50.

ART. 10. The tangent of half a right angle is equal to the

radius, 53.

ART. 11. The sine is equal to the cosine of half a right angle,

and the square of each is equal to half the square of the radius, 54.

ART. 12. The cosine of an angle equal to one third of two right

angles is equal to half the radius, ibid.

ART. 13. The square of the sine of an angle equal to one-third of

two right angles is equal to three-fourths of the square of the ra-

dius, 55.

ART. 14. The sine of an angle equal to one-third of a right angle

is equal to half the radius, ibid.

ART. 15. The square of the cosine of an angle equal to one-third

of a right angle is equal to three-fourths of the square of the radius,

ibid.

ART. 18. In a triangle the sides are to each other as the sines of

their opposite angles, 74.

ART. 82. The rectangle under the radius and sine of the sum of

two angles is equal to the sum of the two rectangles under the sine

of each and the cosine of the other, 48.

ART. 83. The rectangle under the radius and sine of the differ-

ence of two angles is equal to the difference of the two rectangles

under the sine of each and the cosine of the other, 49.

ART. 92. The rectangle under the radius and the cosine of the

sum of two angles is equal to the difference of the rectangles under

the cosines and the sines of the angles, 51.

ART. 93. The rectangle under the radius and the cosine of the

difference of two angles is equal to the sum of the rectangles under

the cosines and the sines of the angles, 52.

ART. 102. The square of the secant of half a right angle is equal

to twice the square of the radius, 53.

ART. 103. The cotangent of half a right angle is equal to the

radius, ibid.

ART. 10. The square of the cosecant of half a right angle is

equal to twice the square of the radius, 54.

ART. 19. In a triangle the tangent of half the sum of any two

angles is to the tangent of half their difference, as the sum of the

opposite sides is to their difference, 75.

ART. 20. In a triangle the cosine of any angle multiplied by

twice the product of the sides which contain it, is equal to the sum of

the squares of those sides diminished by the square of the third

side, 76.

ART. 21. In a right-angled triangle given the two sides contain-

ing the right angle, the triangle itself may be determined, 79.

ART. 22. In a right-angled triangle given the side opposite the

right angle and one of the remaining sides, the triangle itself may be

determined, 82.

ART. 23. In a right-angled triangle given the side opposite the

right angle, and one acute angle, the triangle itself may be deter-

mined, ibid.

ART. 24. In a right-angled triangle given either side about the

right angle, and either acute angle, the triangle itself may be deter-

mined, 83.

ART. 25. In any triangle given two sides and the included angle,

the triangle itself may be determined, ibid.

ART. 26. Given two angles of a triangle, and the side opposite

either of them, the triangle itself may be determined, 85.

ART. 27. Given two angles of a triangle, and the side between

them, the triangle itself may be determined, ibid.

ART. 28. Given the three sides of a triangle, the triangle itself

may be determined, ibid.

ART. 29. Given two sides of a triangle, and the angle opposite

either of them, the triangle itself may be in some cases determined,

86.

APPLICATION OF TRIGONOMETRY TO PRACTICE.

I. The perpendicular height of an object may be determined

though the foot be inapproachable, 88.

II. The distance of an elevated object may be determined though

its foot be inapproachable, 89.

III. The distance of an object on the ground may be determined,

though it be inapproachable, 90.

IV. The distance between two inapproachable objects may be

determined, 91.

ADDENDA TO TRIGONOMETRY.

ART. 30. In a very small arch the sine is equal to the tangent, 92.

ART. 31. The circumference of a circle is nearly equal to three

times and a seventh its diameter, ibid.

SPHERICAL TRIGONOMETRY.

DEF. I. A right line is said to be perpendicular to a plane when

it is perpendicular to all the right lines in that plane which pass

through the point where the line meets the plane, 97.

DEF. II. When two planes cut each other, the angle between

them, or their mutual inclination, is measured by the angle between

two right lines drawn in those planes from the same point of their

line of intersection, and perpendicular to it, 99.

DEF. III. Two planes are perpendicular to each other when the

angle between them is measured by a right angle, 100.

DEF. IV. Any figure made where a solid is cut by a surface, is

called a Section of that solid, ibid.

DEF. V. A sphere is a solid figure bounded by one surface, such,

that all right lines drawn from it to one and the same point within

the figure are equal to one another, 101.

DEF. VI. In a sphere, the point from which the surface is every-

where equally distant is called the centre of the sphere, ibid.

DEF. VII. A right line drawn from the centre of a sphere, and

terminated in the surface, is called a Radius of the sphere, ibid.

DEF. VIII. A right line drawn through the centre of a sphere,

and terminated both ways in the surface, is called a Diameter of the

sphere, ibid.

DEF. IX. A great circle of a sphere is that made by a plane

passing through the centre of the sphere, 102.

DEF. X. A lesser circle of a sphere is that made by a plane which

does not pass through the centre of the sphere, ibid.

DEF. XI. A spherical triangle is a spherical surface bounded by

three arcs of great circles, 103.

ART. 1. One portion of a right line cannot be in a plane, and

another portion out of it, 95.

ART. 2. Three right lines which meet one another are in the

same plane, 96.

ART. 3. If two planes cut one another, their line of intersection

will be a right one, ibid.

ART. 4. If at the point where two right lines intersect a right

line stand perpendicular to both, it shall also be perpendicular to the

plane in which they lie, 97.

ART. 5. Every section of a sphere, made by a plane, is a circle,

101.

ART. 6. All great circles of the same sphere are equal to each

other, 102.

ART. 7. Two great circles on the same sphere divide each other

into two equal parts, ibid.

ART. 8. All small circles equally distant from the centre of the

sphere are equal to each other, ibid.

ART. 9. In any right-angled spherical triangle, the product of

the sine of the side opposite the right angle and the sine of either

remaining angle is equal to the sine of the side opposite the latter

angle, 103.

ART. 10. In any right-angled spherical triangle, the tangent of

either side adjacent to the right angle is equal to the product of the

tangent of the side opposite the right angle and the cosine of the

angle between these sides, 105.

ART. 11. In any right-angled spherical triangle, the cosine of

the side opposite the right angle is equal to the product of the cosines

of the remaining sides, ibid.

ᎪᎡᎢ. ] 2.

In any right-angled spherical triangle the sine of

the middle is equal to the product of the tangents of

the adjacent extremes, 111.

In any right-angled spherical triangle the sine of

the middle is equal to the product of the cosines of the

opposite extremes, 112.

LOGARITHMS.

THE hardest part of this Science is its name. After what has been said on Powers and Progressions, in our ALGEBRA, the reader will find little difficulty in understanding the nature and use of LOGARITHMS. So much as is necessary and sufficient, we propose to explain in the present Treatise.

It is manifest from inspection that in a series of terms in geometric progression, the indices of the multiplying ratio themselves are a series in arithmetic progression; thus, if the former series be

a, ar1, ar2, ar3, art, ar5, ar6, &c. inasmuch as a is equal to a × 1, or a × ro (by Obs. Art. 37, ALG.), the series of indices will be

0, 1, 2, 3, 4, 5, 6, &c.

which is an arithmetic series. This corresponds exactly with the other, or more properly speaking, with r, the multiplying ratio of the other; the powers of which in the successive terms are indicated, or numbered, by the corresponding terms of the second series. Hence originates the word Logarithms; scil. from two Greek words (arithmos and logos), signifying the numeration of a ratio. The terms of the arithmetic series are called logarithms of the multiplying ratio in the geometric series; thus 0 is the logarithm of 7o, 1 the log. of r1, 2 the log. of r2, &c.

It is also apparent that the same arithmetic series will correspond to the geometric series

ro, r1, r2, r3, pt, r5, rổ, &c.

or to any geometric series whatever, the indices of whose multiplying ratio increase in like manner. Moreover, if there be geometric terms interposed between the above, thus,

ro, rì, rì, r3, r2, rž, r3, r‡, rt, &c.

B