IH 



MATHEMATICS. TRIGONOMETRY. 



divide A D into the same numbnr 



DPI DlPi Again, ve nto te same numnr 



of equal parts, D N, DN, ---- and drw O, N, O, N 

 .... parallel to C D ; with centre O, and nuliu* C D 

 describe a circle, and with centre N.. and radius !)/, 

 describe an arc cutting the circle in the point P. This 

 is a point in the curve, similarly with the centre O, and 

 radius C D. Another circle may b described and inter- 

 sected by the arc of a circle described with centre N, 

 and radius Dp lt and thus another point be determined ; 



and so on for any number of points, which, being 

 joimil carefully, will give the curve in question. 



\\V thus conclude our treatise on Practical Geometry. 

 It has been our endeavour to oonGne it in extent to 

 tli.-it which it is absolutely necessary the scientific 

 draughtsman should be familiar with ; and in the 

 foregoing pages will be found all that is required for 

 practical purposes no really essential propositions 

 omitted. 



CHAPTER XL 

 PLANE TRIGONOMETRY. 



L Of* rtprettnting Line* and Angle* by Number*. 



Ir we have a line of any length, we can represent it 

 numerically by the number of times it contains a given 

 line, which we take to represent unity. Thus, if we 

 take a line a foot long to be the unit of length, a line 

 seven feet long can be represented by 7. Of course, the 

 same holds good of any other line. And so when we 

 speak of a line 8, 5, or whatever the number may be, we 

 mean that the line in question contains 8 or 5 of the 

 given unit, as 8 feet, or 6 feet. And of course, if we 

 can represent lines by numbers, we can generalise the 

 numbers by letters, and thus we can represent lines by 

 algebraical symbols; so that a be, xyz, c. , may bo 

 understood to represent lines. In the same manner as 

 before, if we speak of a line a, we mean a line contain- 

 ing as many units of length (e.g., feet) as a contains 

 units of number. 



On the same principle we may express angles by 



numbers or by letters. This is done by dividing the 



right angle into 90 equal parts, each of which is called 



degree, and dividing the degree into 60 equal parts, 



each colled a minute, and the minute into 60 equal 



parts, each called a second. An angle is then expressed 



, as being so many degrees, with odd minutes and seconds, 



j , jr., 36 degrees, 57 minutes, 31 seconds (which is usually 



; written 36" 57' 31"), in the same manner as a line is 



expressed by so many yards, with odd feet and inches. 



Of course, as we can thus represent angles by num- 

 bers, we may also represent them by letters, and may 

 have angles ABC; where the angle A (for instance) 

 means that the angle contains as many degrees ana 

 parts of a degree as A contains units and parts of a 

 unit. It is usual to denote angles either by Roman 

 capital letters, ABC; or eke by Greek small letters, 

 a, ft, -y . . . 0, <fr, ifr . . . while generally the small Roman 

 abe denote lines. This is, of course, only a conven- 

 tional arrangement. 



In the same manner as we may measure lines either 

 by feet or yards, or miles, so we might take, as the 

 unit of angular measure, any other part of the right 

 angle than the ^ th ; and, in fact, at the end of last 

 century, when the decimal notation was introduced into 

 France, it was proposed by certain French mathemati- 

 cians, to make the degree the ^th part of the right 

 angle. The proposition was at no time extensively 

 accepted, and is now quite abandoned. 



2. Definition of the Science of Trigonometry. 



We are thus enabled to express lines and angles by 

 numbers ; and this is the first step towards making 

 calculations in which lines and angles are the data. 

 However, before these calculations can be performed, it 

 U necessary that the relations which exist between 

 straight lines and angles should be investigated. It is 

 the object of the science of Trigonometry to make those 

 investigations. 



The object of the science will, perhaps, be more 

 clearly stated, if we limit the definition so as to make 

 it correspond more closely to its derivational meaning, 

 by saying that the tcience of Trigonometry hat for it* 

 object the intatiyation of the relation* which taut between 



the tide* and angles of triangle*, and the algebraical ex- 

 pression of those relation*. 



The immediate application of the science is to the 

 Fig. l. calculation of certain parts 



of a triangle from certain 

 given parts; e.g., having 

 given the sides B A, AC, 

 and the angle B A C of 

 the triangle ABC, we can 

 calculate the magnitude 

 * of the side B C. 

 The science has, however, very many other uses be- 

 sides the one from which its name is taken viz., the 

 measurement of triangles. 



3. The Circular Measure of an Angle. 



The measures above given, enable us to compare 

 arithmetically one straight line with another, and one 

 angle with another. But it is to be observed, that an 

 angle and a line are heterogeneous magnitudes ; and 

 therefore, if we would perform algebraical operations 

 into which lines and angles enter, we must devise some 

 plan of measuring angles that shall express them by 

 means of lines, or of tho ratios of lines. 



In fact, when we speak of an angle (of 57 suppose) it 

 tells us what the angle is, but does not at once give us the 

 moans of comparing that angle with given lines. 



The measure of the angle adopted for the purpose of 

 such calculations, is called the circular measure. 



It is founded on the two well-known geometrical pro- 

 positions. 



(a) That in circles of the same radius, the angle is 

 proportional to the arch 



which subtends it. 



(b) And that for the 

 same angle, in circles of 

 different radii, the arc 

 varies as the radius. 



If a = the arc B C, 



= the angle B A C, A 

 subtended by the arc B C, 



r = the radius A C. 

 we may express these propositions by the two variations 



a c/i when r is constant, 

 o t/i r when is constant. 

 .'. a v> rO when both vary. 



or the angle is measured by the ratio of the arc to the 

 radius. 



If we take the unit of angle to be the angle which is 

 subtended by an arc of the same length as tho nidius, 

 then 



In this case, the angle being measured by the ratio of 

 two lines, it can enter a calculation in which we are 

 dealing with lines. 



N.B. We can easily find the number of degrees in 

 the angle which is the unit of circular measure. 



