138 Railway Curves. 



and half the angle at the centre ; that it is also the (_ of a right- 



C 



angled triangle, and that half the chord or _ is its opposite side 



2 



and that r, or the radius of the curve, is the hypothenuse, and 

 recollecting that in right-angled trigonometry, if we want a side, it 

 is convenient to make the given side radius ; that is, it is convenient 

 to put given and required sides into the form of a vulgar fraction, with 

 the given side for the denominator ; we should then have this fraction, 



hypothenuse r 



opposite side 7 



Now, according to trigonometrical definitions of sine, tangent, 

 secant, &c, 



hypothenuse r, 



1 = cosec. = ' c 



opposite side 7 



'.' r=_ cosec. 6. By this we find radius. 

 2 



c 

 And the same form of ecraation applies to the small half-chord - 



2 

 the radius r of the circle, and the little L &• 

 c 



viz. : r= - cosec. 6 (1) 



2 



2 r ° (u) 



and v cosec. 6= (n) ■.• Sin. 0- o v ' 



c x ' z r a 



c = 2 r sin. 6 (in) 



So that 6 may be found from its cosec. or sin. ; by the former gene- 

 rally, most easily by natural numbers and natural cosecant, because 

 c being arbitrary, may be assumed an easy divisor. 



Now if we have a very small table of natural cosecants to only a 

 few degrees extent, and for angles only that may be read on the limb 

 of the theodolite, we may then use the vernier to verify our readings 

 merely, which it will do by showing that the index and the last 

 division on the vernier are in complete coincidence with densions in 

 the limb, by ocular comparison with the intermediate divisions. We 

 have only to consult such a table to see instantly the nearest cosecant 



2 r 



to the value in decimals of and then assuming this, which is an 



c 

 angle which can be set off with great certainty and precision, as — use 

 the reverse formula, No. III. c = 2 r sin. 6. By this we find c, 

 which can be done in the field easily by means of table of natural sines, 

 of like extent, &c, as the above-mentioned table of cosecants. 



The angle 6 is thus made readable to very great accuracy, whilst 

 measuring of the sub-chord c to a nicety is a matter of no difficulty. 



