Causes of the Earth! s rotary Motion. 181 



Hour which may attach to it ; and as this is often all that the 

 philosopher obtains, it should in justice be awarded to him to 

 whom, as being the first publisher, it is unquestionably due; 

 and that Mr. Herapath was so in the present instance you 

 will be convinced from the report of his lecture, which I have 

 taken the liberty to inclose, 



I am, sir, yours, &c. 

 Bristol, Feb. 24, 1825. James P. Bevan. 



XXVIII. Demonstrations of Trigonometrical Formulce. 



To the Editor of the Philosophical Magazine andjouriial. 

 Sir, 



FLATTER myself the demonstrations of the followino- 

 trigonometrical formulee are new: but if I am mistaken, per- 

 haps the circumstance of their not being in general use may 

 procure them insertion in your Magazine. 



I am, sir, yours, &c. 

 Lincoln's Inn, Feb. 14. 1823. C^ 



I 



Theorem 1. Let ABC be a 

 triangle having the 

 angles A, B, C to 

 which the sides op- 

 posite are a^b^c — . 

 Then by trigo- 

 nometry, 



a sm C = c sin A or a sin (A + B) = ^ sin A 

 Hence sin (A + B) = -^ sin A 



Again, if a perpendicular be drawn from C to the opposite 

 side Cy we have 



c — a cos B + i cos A 



.'. — = cos B H COS A = cos B H — r-,- cos A 



a a sin A 



Hence by substitution 



, . r.v / T-» sin \i A \ • . 



Sin ( A + B) = ( cos B + -:-— cos A sm A 



\ ' ' \ sin A 



and .-. sin (A + B) = sin A cos B 4- cos A sin B 



Theorem 



