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diagonals, be divided into triangles, it is easy to foresee that the subject, whose fundamental principles we are about to explain, though limited to the consideration of figures of three sides, may, nevertheless, comprehend implicitly many practical enquiries about plane figures in general. And such is, in reality, the case; it being found, in nearly all the problems which occur in surveying, and other surface measurements, to be most convenient, and, in some problems, to be absolutely indispensable to subdivide, in this manner, the surface into triangles, and then to compute the required lengths, &c. by Trigonometry.

But the immediate and most obvious purpose of practical Plane Trigonometry is to discover heights and distances, inaccessible to actual measurement, by connecting with each of these two other lines, so as to form a plane triangle, of which a sufficient number of parts to render the inaccessible part determinable by computation, may be actually measured. The angles, in such cases, are usually found by observation, with a suitable instrument—a Quadrant or a Theodolite; yet these, as well as the sides, are sometimes to be computed, as in the case of extensive surveys, for the purpose of mapping a district or country; or even for the purpose of ascertaining the figure of the earth itself. Surveys on so large a scale are, by way of pre-eminence, called TRIGONOMETRICAL SURVEYS, and require all the more refined resources of the science itself, combined with the utmost delicacy of observation, and the most finished accuracy in the instruments employed. But into any details of these important operations we cannot, of course, enter in this brief introduction, the object of which is merely to present the geometrical theory of the fundamental principles upon which all such operations proceed; and to exhibit the practical application of these principles in examples of common occurrence in the ordinary routine of Mensuration, Surveying, and Navigation.

Of the Measurement of Angles.

(2.) Since angles are quantities altogether differing in kind from the straight lines which include them, the computations of Trigonometry must involve two distinct kinds of measures. As we cannot speak of the length or breadth of an angle, there seems at first a difficulty in the way of an accurate numerical representation of this species of quantity, that does not exist with respect to lines, which may be measured by any convenient unit of length, as an inch, a foot, &c.; and the number of these lengths will properly stand for the measured line. But the difficulty soon disappears when we reflect that we have as much right to introduce an angular unit, for the purpose of measuring angles, as to introduce a linear unit, for the measurement of lines.

That the unit, applied to any measurement, shall be the same, in kind, as the thing measured, is obviously essential; that it shall be of any stated magnitude, is arbitrary; although, as matter of convenience, a unit of certain prescribed limits may be preferable to any other. The fixing upon the extent of the angular unit has been governed by these considerations of convenience; but, although not absolutely indispensable, yet it will facilitate the conception of this subject of angular measurement to call in the intervention of the circle. Let, then, BAC be any angle; and with centre A at any distance AB, let a circle be described. The intercepted arc BC varies precisely as the angle BAC; for if this should enlarge, and become the angle BAC,, the intercepted arc would, in like manner, enlarge, and become BC1, so that (prop. xxxiii. Book vi.),





As, therefore, the intercepted arcs always vary as the angles

at the centre, the former become competent representatives of the latter. These intercepted arcs are hence frequently said to measure the angles they subtend at the centre; the term measure being here used not in the strict geometrical sense that a part or submultiple is said to measure the whole, but as merely expressive of the constant proportionality of the arcs and the angles they subtend.

It is found convenient, for the purposes of computation, to consider the circumference of the circle, as divided into 360 equal parts or arcs, called degrees; and to regard the angle at the centre, subtended by one of these, as the angular unit. Each degree is moreover supposed to be subdivided into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds, and so on. The angle subtended by any number of these will, therefore, be some fractional part of the angular unit; and, to prevent a needless multiplicity of terms, the names degree, minute, second, &c., are applied as well to the angles themselves as to the subtending arcs. There is no risk of confusion in this twofold application of terms, since the expressions "an angle of so many degrees and minutes," and an arc of so many degrees and minutes," are sufficiently distinctive to prevent all ambiguity.

When we use such an expression as "an arc of 24 degrees 42 minutes," we mean to denote a part of the circumference equivalent to 24 three hundred and sixtieths, or degrees, and 42 sixtieths of a degree; and when we speak of "an angle of 24 degrees 42 minutes," we mean the angle which that arc subtends at the centre.

It is worthy of notice that when we state the magnitude of an angle in this way, we convey an accurate idea of the amount of opening referred to, or furnish a datum sufficient for the actual construction of the angle. Thus an angle of 30 degrees is one which subtends the twelfth part of an entire circumference described about the vertex of the angle as centre, or a sixth part of the semi-circumference, or a third part of the quadrantal arc, or quadrant; and it is plain, from prop. xxxiii. of the sixth Book, that whatever be the length of the radius, the portion of the whole circumference subtended will always be the same for the same angle; because, by that proposition, any angle at the centre

is to four right angles as the subtended arc to the whole circumference, be the radius what it may. In order, therefore, actually to construct our angle, we have only to draw any straight line AB for one of its sides, and from A, the point intended for the vertex, to describe with any radius AB, a circle, and then to take BC equal to the twelfth part of the circumference, or equal to the third of the quadrant BD, and finally to draw AC.*

But when, instead of the angle, we speak of an arc of so many degrees, the case is very different: such an expression conveys no definite idea; and to be intelligible requires to be accompanied with information as to the length of the radius. A degree being only the technical name for the 360th part of a certain whole, it is obviously necessary that the length of this whole be stated before we can tell the extent of a degree.

Although, therefore, it is sufficient, in order to designate the magnitude of an angle, simply to state the number of degrees and minutes, &c. it contains, yet, to fix the magnitude of an arc, we must, in addition, fix the length of the radius, or of the circumference of the circle to which it belongs.

The notation employed to express degrees, minutes, &c. will be sufficiently explained by an example: thus, 36° is the notation for 36 degrees; 36° 23′ denotes 36 degrees 23 minutes; and 36° 23′ 17′′ stands for 36 degrees 23 minutes 17 seconds.

We have already remarked that the arc BD, equal to a fourth part of the entire circumference, is called a quadrant.

The part CD by which any proposed arc BC falls short of a quadrant is called the complement of BC; and the arc CDB, by which it falls short


There are instruments for the cutting off arcs of circles of any proposed number of degrees, or for laying down angles; the most common of these is the protractor. But the computations of Trigonometry have no need of such constructions; and we advert to them in the text only for the purpose of giving greater clearness to the conceptions of the student.

of a semi-circumference, is called the supplement of BC. An arc and its complement, therefore, make up 90°; an arc and its supplement make up 180o.

Thus CD is the complement of BC; C,D is the complement of BDC,; also CDB, is the supplement of BC, and CB, is the supplement of BDC,. Conformably to what has already been said, the terms applied to the arc equally belong to the angle measured by it; so that the complement of an angle is the difference between that angle and a right angle; the supplement of an angle the difference between it and two right angles. And thus, whether we refer to arc or angle, we say the complement of 36° 24' is 53° 56', and the supplement of it 143° 56′; also the complement of 126° 52′ 28′′ is 36° 52′ 28′′, and its supplement 53° 7′32′′, and so on.

Of the Trigonometrical Lines.

The Trigonometrical Lines are certain straight lines connected with the arcs or angular measures, and introduced in their stead for the purpose of restricting all the investigations of Trigonometry to linear quantities only. The principal of these lines are the sine and cosine, the tangent and cotangent, the secant and cosecant.

(3.) THE SINE. Let BDF be a circle whose centre is A, and radius AB, equal to the linear unit; or that length which we may agree to represent by the numeral unit 1, whether it be an inch, a foot, a yard, or any other measure.

Let BC be any arc, BAC the angle it subtends, and CS a perpendicular to the diameter BE.





line CS is called the sine of the arc BC, or of the angle BAC. Hence the sine of an arc, or of the angle which it measures, is the perpendicular from the termination of the

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