NAVIGATION. 



"XOMETRT. 



on* circle U double the length of a degree of Another of 

 only half the radius of the former. Now, in trigono- 

 metry, angle* alone, independently of all conn. 

 wah their measuring arcs, are almost exclusively the 

 midjiwu of consideration ; and it U therefore necessary 

 that an invariable angle should have an invariable tine 

 M. i cosine, Ac, It U agreed, therefore, that the trigo- 

 nometrical sine, cosine, A-c., should be ettimated accord- 

 ing to the scale O A 1 : in other words, that the 

 abstract number 1, being the numerical representative of 

 the radius O A, the abstract numbers, which, con- 

 formably to this scale, are the numerical expressions for 

 Bn. OH, Ac., should be called the trigonometrical sine, 

 cosine, &c., of the angle A OB. The linos in th. juv- 

 eeding diagram, considered at lines, are the gtomttri>-al 

 sine, cosine, tangent, Ac. , of the arc A B, or of the angle 

 ..- O A ; but viewed as merely linear repre- 

 sentations of abstract numbers, O A denoting the abstract 

 unit, they are the trigonometrical sine, cosine, tangent, 



<>f the angle A OB. 



It is thus that the terms sine, cosine, tangent, A-c., as 

 employed in trigonometry, are regarded at ratios; for 

 from what has just been said, 



O A : Bn : : 1 : , v , ' the trigonometrical sine of O. 



O A : On : : 1 : 



OA 



OA 



AJ 



1 ! EC' 



Ac. 



Ac. 



And generally any geometrical sine, cosine, Ac. , divided 

 by the radius of the arc with which such sine, cosine, A-c. , 

 is connected, is the trigonometrical sine, cosine, A'c., of 

 the angle measured by that arc : but in trigonometry, 

 the terms sine of an angle, cosine of an angle, A-c. 

 without any prefix are sufficiently explicit, because 

 when geometrical lines are meant, the fact is always 

 ht 1. 



The learner will find it an assistance to keep the dia- 

 gram at page 1037 before his mind in investigating the 

 rotations among the trigonometrical quantities now con- 

 sidered, regarding the lines defined above as merely the 

 linear representatives of the abstract numbers, sine, 

 cosine, A-c., the radius being the representative of the 

 abstract number 1 ; for he will have the aid of geometry 

 to assist him in establishing the fundamental principles 

 of trigonometry. Thus, from Euclid, Prop. 47, Book I., 

 we know that 



Bn' + On' - OB, O A + A< - O, C* + Cu' _ On', 



so that we have from the first of these, 



sin.HJ-f cos.'O = 1 . . . (1) 



1 -j- tan.'O sec. 2 ... (2) 



an 1 from the third, 1 + cot,^ - cosec.^O .. .(3) 



Also from the property that the sides about il 

 angles of equiangular triangles are proportional (Km-, 

 1'rop. 4, B. Vl.), we find, upon comparing together the 

 .-ilar rijjlit-angled triangles B On, tO A, as also 

 the equiangular right-angled triangles B Otn, uO C, 

 that 



sin. O tan. O _ sin. O 



that i", tan. 0-cosT^ 



. cot. O - 



. sin. O- 



rnv. ( } 



sin. O 



1 



W 



(5) 

 (6) 

 (7) 



1 cosec. O' cosec. O 



oos O 1 1 



~~j - -aeTO " iScTo ' 



These fundamental relations will suffice for our present 

 purpose. We learn 



From (1). that sin. O - J (1 cos.O) ; and cos. O 

 - s '(l tin ' 



From (2), that teo. O - J (1 + tan.O) ; and tan. O 



From (3), that cosoc. O - V (1 + cot.O) ; and cot. O 



From (4) (5), that tan. O- 



-. ; and cot. O=. -.. 

 cot. O tan. O 



And from (C) (7), that cosec. O - - 



. ; and see. O 



cos. O' 



For the more general theory of the <n._'onometrical 

 ratios, the student is referred ; . <>n Trigo- 



nometry. 



HB TABLE* or SINKS. -, Ac. 



The numerical values of the trigonometrical lines con- 

 sidered above are computed according to the scale radius 

 1. and are arranged in tables called tables of natural 

 tine*, cosinet, A r c. As tangents, cotangents, secants, and 

 cosecants, may be deduced with but little trouble from 

 sines and cosines, as shown above, and as, moreover, tho 

 natural sines, cosines, <tc., are seldom employed in navi- 

 gation, the tables of these, which accompany books on 

 that subject, usually give only the sines and cosines. 

 But the tables which furnish the logarithms of these, 

 j indispensable in most of the calculations ] rformed 

 by the mariner, are always given in a complete form, 

 and contain the logarithmic sines, cosines, tangents, co- 

 tangents, and cosecants of all angles from up to 90. 

 In the former tables the numerical values of the sine and 

 cosine of every angle between and 90 is less than 1, 

 because the radius itself is only 1 : the logarithms of all 

 these sines and cosines would, therefore, have negative 

 indices (p. 1037), which is an inconvenience in pra< 

 To remedy this, the logarithmic tables are computed, not 

 to the scale rad. = 1, but to the scale rad. = 10', which 

 U so large as to preclude the possibility of any logarithm 

 in the tables requiring a negative index : in these tables, 

 the index, or integral part of the log., is always inserted 

 before the decimal part ; while in the logs, of number*, 

 as already stated, the index is omitted.* 



Now, in investigating the rules and formulas of trigo- 

 nometry, the argument always proceeds on tho supposi- 

 tion that the numerical value of the radius is 1 ; and 

 consequently, in the practical application of those rules 

 and formulae, it must be borne in remembrance, when 

 logs, are used, that the radius is taken 10 " times as 

 great that is, instead of the log. of a natural sine. 

 sine, A-c., we employ in reality the log. of 10' times that 

 nitural sine, cosine, <tc. For instance, if we call any 

 natural sine, cosine, itc., p, the logarithmic tallies will 

 give us, not log. p, but log. 10'"p that is, log. 10 " + 

 log. ; ; or since log. 10 is 10 log. 10 = 10 X 1, the 

 taUos will give us 10+ log. p, so that the log. of every 

 natural sine, cosine, Ac., is increased by 10. In order, 

 therefore, that our practical results may agree with our 

 theoretical formula-, these superfluous 10's introduced 

 merely to avoid negative logs. must be suppressed at 

 the close of our operation. If, in our work, as many of 

 these logarithmic trigonometrical quantities are used 

 subtractively as are used additivoly, then, of course, as 

 the superfluous 10's destroy one another, no correction 

 of the final result becomes necessary; but if more of 

 these logs, are additive than subtractive, as many 10's as 

 mark the excess of additive over subtractive logs, must 

 be suppressed in the result ; and if the excess be in favour 

 of the subtractive logs. , so many 10's must be introduced. 

 The suppression or the introduction of a 10, in any ; 

 amount, is so easy a matter, that the trouble attending 

 it is not worth consideration, in comparison with the 

 advantages gained by the avoidance of logs, with nega- 

 tive indices. 



"What has now been said being understood, let us sup- 

 pose that, by aid of tho algebraic series and formula! in 

 our Section on Mathematical Science, the two sets of 

 trigonometrical tables have been constructed namely, 

 tables of natural sines, cosines, &c.', and tables of loga- 

 rithmic sines, cosines, &c. ; and let us endeavour to see 

 a little into the use and advantages of what is thus sup- 

 plied, in the calculations of plane triangles. 



Pur ubln of Idfmrlthmie linet, oolloM, tc., *cr p. Odd, it uq. 



