564 PHILOSOPHICAL TRANSACTIONS. [aNNO I67I. 



termined. Join DC, and produce the same; then a circle passing through the 

 points ABD intersects DC, produced at S, the place of station. 



Calculation. In the triangle ABD, all the angles and the side AB are known, 

 whence may be found the side AD. Then in the triangle CAD, the two sides 

 C A and AD are known, and their contained angle CAD, whence may be found 

 the angles CD A and ACD, the complement whereof to a semicircle is the 

 angle SCA ; in which triangle the angles are now all known, and the side AC; 

 whence may be found either of the distances SC or S A. 



Case IV. — If the station be without the triangle, as fig. 13, made by the ob- 

 jects, the sum of the angles observed is less than four right angles. The con • 

 struction is the same as in the last case, and the calculation likewise; saving 

 that you must make one operation more, having the three sides AC, CB, AB, 

 thereby find the angle CAB, which add to the angle EAD, then you have the 

 two sides, viz. AC, being one of the distances, and AD, found as in the former 

 case, with their contained angle CAD given, to find the angles CD A and ACD, 

 the complement whereof to a semicircle is the angle SCA; now in the triangle 

 SCA, the angle at C being found, and at S observed, the other at A is likewise 

 known, being the complement of the two former to a semicircle, and the side 

 AC given; hence the distances CS or AS may be found. 



Case V. — If the place of station be at some point within the plain of the tri- 

 angle, as in fig. 14, made by the three objects, the construction and calculation 

 is the same as in the last, saving only that instead of the observed angle ASC, 

 the angle ABD is equal to the complement thereof to a semicircle, to wit, it is 

 equal to the angle ASD; both of them insisting on the same arch AD; and in 

 like manner the angle BAD is equal to the angle DSB, which is the comple- 

 ment of the observed CSB, and in this case the sum of the three angles ob- 

 served is equal to four right angles. 



In these three latter cases no use is made of the angle observed between the 

 two objects A and B, that are made the base line of the construction; yet the 

 same is of ready use for finding the third distance or last side sought, as in the 

 4th case, in the triangle SAB, there is given the distance AB, its opposite angle 

 equal to the sum of the two observed angles, and the angle SAB attained, as in 

 the 4th case: hence the third side or last distance S B may be found. 



And here it may be noted, that the three angles CAS, A SB, SBC, are to- 

 gether equal to the angle ACB; for, the two angles CSB and CBS are equal 

 to ECB, as being the complement of SCB to two right angles; and the like in 

 the triangle on the other side. Ergo, &c. 



Case VI. — If the three objects be A, B, C, and the station at S, as before, it 

 may happen, according to the former constructions, that the points C and D 



