PHILOSOPHICAL SOCIETY OP WASHINGTON. 115 



Aimales, in 1810. It was not answered by his correspondents, 

 and accordingly he takes up the problem himself in the second 

 volume, and gives an analytic investigation; but no simple geo- 

 metric construction results from his analysis. 



Dr. A. L. Crelle, of Berlin, published a trigonometric solution 

 of the problem in a collection of mathematical problems published 

 in 1821, in Berlin, and in 1S2T, Prof. Lehmus, also of Berlin, 

 published a trigonometric solution of the problem in the second 

 volume of a course in pure and applied mathematics. 



Up to this time, then, viz., 1827, no geometrical proof of the 

 construction of the problem had appeared. 



The next writer upon Malfatti's problem was Prof Steiner, of 

 Berlin. In the first volume of Crelle's Journal he has a paper 

 entitled "Some Geometric Considerations," which is dated at 

 Berlin, in March, 1826. In this paper he says, that "about 

 three years since the author of this paper became interested, 

 accidentally, in the consideration of the problems: First, to de- 

 scribe a circle tangent to three given circles ; second, in Mal- 

 fatti's Problem; third, in Theorem, XY, Book IV of the Collect. 

 Mathemat. of Pappus; and fourth, in several porisms and the 

 purely geometric considerations of curves and areas of the second 

 degree. Pappus's theorem he was familiar with, but without 

 proof; so also Malfatti's Problem ; of the first, however, he was 

 acquainted with Vieta's geometrical solution." 



Further on he adds : " The effort of the author was, in the 

 solution of the different problems with reference to the tangencies 

 of circles, to find the underlying general principles by which they 

 were connected." 



Then three paragraphs follow in which the fundamental prin- 

 ciples of radical axes and centres of similitude are developed, 

 after which, "to show the fruit- 

 fulness of the preceding theory 

 by a suitable example," this 

 elegant solution of Malfatti's 

 Problem is given : Bisect the 

 angles of the triangle, and in 

 the three partial triangles so 

 formed inscribe circles. From 

 the point of tangency m of the 

 circle ilf (Fig. 1), with the side 



