TRIGONOMETRY. 



TRIGONOMETRY. 



angle*, but we might aa well attempt to confine geometry within 

 etymological limiu, aa the Kienoe of which we are going to gire some 

 account in this article : the measurement of the earth U now only an 

 tabled application of the former ; and the meaiurement of triangle*, 

 of the latter. 



In the modern division of the mathematical science*, trigonometry, 

 though "till defined in book* aa the art of measuring triangles, really 

 mean* the consideration of alternating or periodic magnitude ; in which 

 quantity u imagined to go through alteration* of increase and diminu- 

 tion without end ; that U <nr, a function of .r, is trigonometrical, u hen, 

 ai jc varies through all stages of magnitude (or. in technical language, 

 increases from to + ), it takes an infinite number of alternate 

 increases and diminutions. It u perfectly possible to contrive a common 

 algebraic function which shall go through any given number of uclj 

 change* a thousand, a million, or more ; but without recourse to infi- 

 nite series, it is impossible to find one in which the number of alterna- 

 tions U unlimited. If the properties of algebraical series were as visible 

 to the unassisted apprehension as those of figure in geometry, it would 

 be seen that the two following series (afterwards known aa those for 

 the sine and cosine of x), 



2.3 2.3.4.5 



X> 3* 



- 4c., and 1 - -- + 



4 



- *c , 



are periodic in value : and tliat, ir being a certain incommensurable 



number (3-141592 ), all the changes of magnitude that they can 



possibly take are only repetitions of what take place while x increases 

 from to 2*. We cannot form a more adequate idea of an intelligence 

 superior to that of the human race, than by imagining one to which 

 this truth ahould be, in consequence of sufficient rapidity of power of 

 computation, a purely elementary one. We are obliged to come by this 

 knowledge through our perceptions of space, and by the application of 

 algebra to geometry ; and the construction and use of our alphabet for 

 the expression of periodic magnitude is contained in whatrare called 

 the elements of trigonometry. 



The most simple notion of periodic magnitude lies in supposing that 

 the changes made are purely cyclical, or repetitions of the same for 

 ever ; aa for instance, those which occur in turning a handle in a verti- 

 cal plane. The number of revolutions traced out by the handle may 

 be as great as we please, and the quantity of length of the circular arc 

 described by its extremity may be as many times the circumference of 

 the circle as we please, that is, as long as we please ; but the distance 

 of the handle from the ground is periodic, exhibiting perpetual increase 

 and diminution as it rises and falls. Hence the circle naturally becomes 

 a sort of standard of reference, and circular motion the primary idea, 

 in all consideration of periodically changing magnitude. The arc, or 

 the angle which it subtends at the centre, is the magnitude which 

 increases without limit, all past revolutions being counted ; and the 

 lines which only depend on the position of the moving point in the 

 circle, anil nt on the number of revolutions by which it has attained 

 that position, are the periodic magnitudes in terms of which all others 

 are expressed. 



The periodic magnitudes connected with a varying angle, so far as 

 they have separate designation, are the sine, cosine, tangent, cotangent, 

 secant, cosecant, versed sine, Governed sine, and chord. A change has 

 taken place in the mode of conceiving these quantities, and one which 

 it is very desirable thoroughly to establish : though slight in appear- 

 ance, and producing no difference in results, it gives a great advantage 

 in the confederation of formula:. These elements were lines ; they now 

 often art, and in future always icill lit, the ratios of lines to lines. The 

 following figure exhibits the old definitions : 



fit. 1 



c ( the cotangent, o T the secant, o ( the cosecant, A M the versed sine, 

 c N the covened sine, of the angle AOB. If A B should make a com- 

 plete revolution, so as to come into the same position again, the angle 

 under consideration would now be 



four right angles + < A o B, 



but the sine, cosine, &c. would all be the same as before. 



As the line o B moves round, the signs of all these lines are to be 

 taken positively when they are in the same directions a* when A o B is 

 leas than a right angle. The following table will show them for the 

 angles A o B, A o B', A o B*, A o B'", all measured in the same direction 

 of revolution : 



In this system an angle has an infinite number of lines of each sort, 

 one to every radius which can be token. It is therefore necessary, 

 either to introduce into the formula; the value of the radius in 

 case, or to adhere to some one particular value of the radius, which is 

 always understood. The plan usually adopted is first to embarrass 

 the formula; with the general value of the radius, then gradually 

 to accustom the student to consider the radius as one unit, but to make 

 an exception when trigonometrical tables are used, by considering the 

 radius as ten thousand millions. These inconveniences are avoided in 

 the new system of definitions, which is as follows : 



Fig. 1. 



M 1 



From any jxiint in the line OB (or OB 1 , &c.) which is the variable 

 boundary of the angle, draw BM (or B'M', &c.) perpendicular to OA. 

 Let OM and MB be positive, OM' and M'II", &c., negative, as in the 

 usual method of reckoning co-ordinates. Call B M, B'M', &c., opposite 

 to the angles, o M, o M', 4c., adjacent ; and let o u, o B', Sic., be t-.illi'-l 

 bypothenuses (and always considered positive). Then the sine of B o M 

 is the fraction which B M is of o B, with its proper sign, in this case 

 positive : but the sine of AOB" is the fraction which M" B" is of o B", 

 taken negatively, because M"B" is negative. It is indifferent what 

 hyiKithenuse is taken, by the property of similar triangles, and the 

 following is the complete system of definitions, with the values v. 

 for the four angles. They give a slight degree more trouble at first, 

 which is amply compensated in the superior ease with which all 

 formula; may be deduced, to say nothing of the advantage of avoiding 

 the indefinite radius. 



Let o be the centre of a circle, and A o B an angle measured fr< mi .1 

 fixed radius o A, the direction of revolution in which angles are mea- 

 sured positively being denoted by the arrow. From B draw B M ]>er- 

 peodicular to o A, and at A and c draw tangents to the circle. Then, 

 in these old definitions, B M is the sine, o M the cosine, A T the tangent, The chord has long ceased to bo regarded as one of the trigono- 



