Goniometric Prollems. 135 



c, BDC s, ACD c, ADC - c, BCD s, BDC c, ACD s, ADC 

 = 5, DOB s, DCA c, ADB - s, BDC 5, ADC c, ACB, 

 which gives the relation betwixt the six angles at C and D. 



Snppose now BC to be perpendicular to CD, then the 

 s, BCD = 1 and c, BCD = O; cons; quenlly, by making 

 these alterations the equation becomes c, I'DC s, ACD 

 c, ADC = s, DCA c, ADB - s, BDC s, ADC c, ACB. 



In the same manner an equation may be found when AC 

 is perpendicular to CD, as follows : s, liCD c, BDC c, ADC 

 = s, DCS c, ADB - s, BDC s, ADC c, ACB. 



If both AC and BC be conceived perpendicular to CD, 

 then s, ACD = 5, BCD = 1, and c, ACD = c, BCD = ; 

 which alteration being made, we have c, BDC c, ADC = 

 c, ADB — 5, BDC s, ADC c, ACB ; agreeing exactly with 

 the equation before found for this case, Problem VIII. 



When the angle BDC = ADC, then s, BCD c', BDC 

 s, ACD - c, BCD 5% BDC c, ACD = s, DCB s, DCA 

 c, ADB - 5% BDC c,ACB. 



If at the same time AC and BC be both conceived per- 

 pendicular to CD, we have c\ BDC = c, ADB — 5% BDC 

 c, ACB. 



And if the angles BDA, BCA, be made equal to nothing 

 by the side AC moving up to HC and coinciding with it, 

 then c% BDC= rad.^ — s'-, BDC; which is the well known 

 property of a right-angled triangle. 



If AC be supposed perpendicular to BC, at the same time 

 that AC and BC are both perpendicular to CDj then c, ACB 

 being equal to 0, we get r, BDC c, ADC = c, ADB ; which 

 is the equation given for this case in Problem IV, p. 34. 



From the general equation we get c, ACB = s, DCB 

 s, DCA c, ADB <r, BDC <r, ADC + c, PCD c, ACD - 

 s, BCD s, ACD r, BDC r, ADC, and c, BDA = c, ACB 

 s, BDC s, ADC <r, DCB (r, DCA + c, BDC c, ADC - 

 T, BCD r, ACD s, BDC s, ADC; which equations irive 

 the angles ACB or ADB when the five other angles are 

 known. 



By a little attention to the solutions of lliesc problems we 



shall perceive that they proceed upon the same elementary 



principles as spheric trigonometry : lor in that science one 



1 4. convenient 



