128 GEOMETRICAL PORISMS, 



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. About 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, fo that CM may be to MG as 

 fip 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 Q^join NQ^ meet- 

 ing the other fines in O and P, and NQ^fhall 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 through E and B, and mest DB, GB 

 in N and Q^ the line joining NQ^lliall be divided at O, io 

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

 as n 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 fine NQ_ being drawn, fliall be di- 

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

 confi:ru<5lion as n p tx^ p q. Hence, it is obvious, that NQ^ is 

 divided fimilarly to n q. 



It may be remarked, that the preceding cotiflrudlion 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 indeterminate. Now, this it is 

 evident will be the cafe, if the points A, E fliall both fall at M 

 the interfedlion of the circles. This cafe forms Prop. VJ. of this 

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



which 



