BOOK XIV.

PLANE TRIGONOMETRY.

PLANE TRIGONOMETRY is the art of measuring and calculating the sides and angles of triangles; also it comprehends whatever relates to the properties and relations of certain right lines drawn in and about a circle, called sines, tangents, secants, &c.

The sides, of plane triangles are measured in feet, yards, chains, &c.; but the angles are measured in degrees, minutes, and seconds, by the arc of a circle contained between the two legs and the angular point for its centre, &c.

A chord is a right line which divides the circle into two unequal parts, as K-L is the chord of the arc KM L.

A versed sine is that part of the diameter contained between the right sine and the arc, as D-B is the versed sine of the arc B-H.

A tangent of an arc is a right line drawn perpendicular to the end of the diameter, just touching the arc, as B-C is the tangent of the arc B-H.

A secant of an arc is a right line drawn from the centre of the circle through the circumference, and produced until it cuts the tangent, as C-A; thus C-A is the secant of the arc B-H.

A sine of an arc is a right line drawn from one extremity of that arc, perpendicular to the radius, through the other extremity of the arc, as D-H, is the sine of the arc B-H.

A co-siǹe of an arc is the sine of the complement of that arc, thus G-H is the co-sine of the arc B-H.

A co-tangent of an arc is the tangent of the complement of that arc, as E-F is the co-tangent of the arc B—H.

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The sine, tangent, and secant of an arc, are said to be the measure of so many degrees as that arc contains parts o 360 degrees; so that the radius, being the sine of a quadrant, or a fourth part of a circle, contains 90°: thus the radius is always equal to the sine of 90°, as is the chord of 60°, and the tangent of 45°, all these three being equal to the radius; and the sine, tangent, and secant of an arc are equal to the sine, tangent, and secant of an arc, as much above 90 degrees, as the former was deficient of 90: thus the sine, tangent, or secant of 80° is the sine, tangent, or secant of 100°; of 70° is 110°; of 60° is = 120°; so that in taking out the logarithms of sines, tangents, or secants, for any number of degrees above 90, the given angle must be subtracted from 180 degrees, and the logarithm of the remainder be taken.

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If in a triangle one angle be right or obtuse, the rest are acute; and if one angle in a triangle be right, the other two taken together make one right angle; wherefore, if one of the

acute angles in a right-angled triangle be known, the other is found by subtracting the known angle from 90 degrees.

In every oblique plane triangle, if one of the angles be given, the sum of the other two is found by subtracting the given angle from 180 degrees; and if two of the angles be known, the third is found by subtracting their sum from 180 degrees.

The complement of an angle is what it wants of 90 degrees. The supplement of an angle is what it wants of 180 degrees.

LOGARITHMS.

Logarithms are a series of numbers, invented by Lord Napier, Baron of Marchiston, in Scotland, by which the work of multiplication may be performed by addition, and the operation of division may be done by subtraction, so that great time and trouble are saved thereby in the performance of all arithmetical computations; for if the logarithm of any two numbers be added together, the sum will be the logarithm of the product; and if from the logarithm of the dividend you subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient. Again, if the logarithm of any number be divided by 2, the quotient will be the logarithm of the square root of that number; or, if the logarithm of any number be divided by 3, the quotient will be the logarithm of the cube root of that number.

Logarithms are a series of numbers in an arithmetical progression, corresponding to as many others in geometrical proportion, in such manner that 0 in the arithmetical series, corresponds to, or is the exponent of 1, in the geometrical series: thus,

THE USE OF THE TABLE OF LOGARITHMS.

TO FIND THE LOGARITHM OF ANY NATURAL NUMBER CONTAINING LESS THAN FIVE FIGURES.

Examples.

1. Required the logarithm of 7.

Look in the table for the number 7 in the side column, and against it is 0.845098. This number having but one figure, the index is therefore 0.

2. Required the logarithm of 79.

Look in the table for the number 79 in the side column, and against it is 1.897627, to which 1 is the index, because the number contains two figures.

Note. When a natural number consists of one figure or units, the index to be prefixed to the logarithmic decimal is 0; but when it hath two figures or tens, the index is 1; when it hath three figures or hundreds, the index is 2; when it hath four figures or thousands, the index is 3; and so on uniformly as in the foregoing series, the index being always one less than the number of figures in the natural number. The decimal part of a logarithm is the same whether the natural number to which it belongs be a whole number, decimal, or partly a whole number and decimal, as below. For a more particular account of logarithms see Dr. Hutton's tables.

MULTIPLICATION OF LOGARITHMS.

Rule. Add the logarithms of the two numbers together, and the natural number answering to the sum will be the product required.

1. Required the product of 84 and 26.

Logarithm of 84 = 1.92428

Logarithm of 26 1.41498

Product 2184 = 3.33926

2. How many feet will a plank contain, whose length and breadth are 48 and 22 feet?

Answer, 1056 feet.

DIVISION OF LOGARITHMS.

Rule. From the logarithm of the dividend subtract the logarithm of the divisor, and the remainder will be the logarithm of the quotient.

If any of the indices be negative, or if the divisor be greater than the dividend, change the index of the divisor; then if the indices have unlike signs take their difference, and prefix the sign of the greater; if they have like signs take their sum and prefix the common sign. When there is a unit to carry from the decimal part of the logarithm of the divisor, subtract it from the index of that logarithm if it be negative, otherwise add it, before the signs are changed.

Divide 240 by 12.

Logarithm of 240 = 2.38021

Ditto of 12 1.07918

Quotient 20

1.30103

RULE OF THREE BY LOGARITHMS.

Rule. Add the logarithms of the second and third terms together, and from the sum subtract the logarithm of the first term.

1. What is the fourth proportional to 6; 3; 44?

Logarithm of 44 = 1.64345