128 GEOMETRICAL FORI SMS. 



three points A, E, B in a circle, pafling through N and Q^ there- 

 fore NQjs given by pofition. 



Construction. Let DB, BG be the lines upon which the 

 extremities of NQ^are to be placed. Abovit the triangles BDF, 

 BCG, defcribe circles, draw BH parallel to FD, meeting the 

 circle DBF in H, and draw BK parallel to CO, meeting the 

 circle CBG in K. In DF find L, fo that DL may be to LF as 

 « to q, and in CG find M, fb that CM may be to MG as 

 }i p to p q, join HL meeting the circle DBF in E, join alfo 

 KM meeting the circle CBG in A. Through the points A, E, B 

 defcribe a circle meeting DB, BG in N and Qj^join NQ__ meet- 

 ing the other lines in O and P, and NQ^llaall be divided limi- 

 larly to n q. 



It has been proved in Prop. 5. that the point E being found 

 as above, if any circle pafs throvigh E and B, and mejt UB, GB 

 in N and Q^ the line joining NQ^lhall be divided at O, fo 

 that NO will be to OQ^as DL to LF, that is by conftrudion 

 as no to q. Likewife, that the point A being found as 

 above, if any circle be defcribed through A and B, to meet 

 DB, BG in N and Q^ the Hne NQ_ being drawn, fhall be di- 

 vided at P, fo that NP will be to PQjjs CM to MG, that is by 

 conflru(^ion as np to p q. Hence, it is obvious, that NQ^ is 

 divided fimilarly to n q. 



It may be remarked, that the preceding cohftrudlion points 

 out very clearly, a circumftance which appears to have efcaped 

 the notice of fome Mathematicians that have given folutions of 

 the problem, with a view to its application to Aftronomy. It 

 is that the given ratios of NO, OP, PQ^ to one another may be 

 fuch as to render the problem indeterininate. Now, this it is 

 evident will be the cafe, if the points A, E fhall both fall at jE 

 the interfed:ion of the circles. This cafe forms Prop. VI. of this 

 paper, fo that it may be fvifficient to add here, that the ratios 



which 



