FOUNDED ON ORDINARY GEOMETRY, &c. 171 



These Propositions, I believe, will suffice for treatment of the first thirteen Propositions 

 of Euclid's Sixth Book (Prop. I. excepted), and for all the Theorems and Problems appa- 

 rently involving proportions of straight lines (not of areas, &c.) which usually present them- 

 selves. As an instance of their application, I will take the theorem to which I alluded at 

 the beginning of this paper. 



Theorem. If pairs of tangents are drawn externally to each couple of three unequal 

 circles, the three intersections of the tangents of each pair will be in one straight line. 



I shall omit the demonstration that, for each couple of circles, the pair of tangents and 

 the line passing through the two centers all intersect at the same point ; and I shall use only 

 the intersection of one tangent with the line passing through the center. Also I shall omit 

 the construction and its demonstration, for inserting between the greatest and least of the 

 three circles a circle equal to the remaining circle, having its center upon the line joining 

 their centers, and being touched by their tangent. 



M 



Let A, B, C, be the centers of the given circles. Let N be the center of the circle 

 whose radius NO is equal to the radius BK, and which is touched at O by the tangent 

 DE. Join NB, MF, FI, MN, NI, FB. 



First we shall prove that MF is parallel to NB. 



The triangles NOF, CEF, have each one right angle, and they have another angle 

 common; hence they are equiangular; and by Proposition (C), the rectangle under CF, NO, 

 is equal to the rectangle under NF, EC; or, the rectangle under CF, BK, is equal to the 

 rectangle under NF, CL. Again, the triangles BMK, CML, are equiangular, for each has 

 one right angle, and they have another angle common; therefore the rectangle under CL, MB, 

 is equal to the rectangle under BK, MC. Consequently, by Proposition (B), the rectangle 

 under CF, MB, is equal to the rectangle under NF, MG. Therefore, by Proposition (F), 

 the parallelogram under CF, MB, which has one angle equal to MCF, is equal to the paral- 

 lelogram under NF, MC, which has one angle equal to MCF. But the former of these 



22—2 



