THE LANOASHIBE GEOMETEBS AND THEIB WBITTNOS. 



133 



quite false ; viz., the Demonstration of a Theorem by Mr. Power, a 

 Problem by myself, and another by Mr. Merrit. I have corrected my 

 own, and Mr. Power will very easily do that for his own; but I think 

 Mr. Merrit's incurable, without beginning again de novo. » * « 

 But now to your Book. I have mostly corrected every thing as I 

 went on, with a pencil, in the Book itself, or in the Plate, and which 

 may easily be rubbed out if found necessary. * * * You may 

 send the next when convenient. 



" I am, yours, &c., 



"The Rev. John Lawson, "Jeremiah Ainswoeth. 



" Swanscombe, Kent." 



The principal portion of Mr. Ainsworth's writings appear 

 in Burrow's Diary. They relate to nearly every branch of 

 Mathematics, and their acknowledged superiority gained for 

 their author the "Prize of Twelve Diaries" for five successive 

 years. In the Diary for 1777, Mr. Thomas Moss proposed 

 for demonstration a property which in reality belongs to the 

 complete quadrilateral, although enunciated merely for the 

 case of the triangle. Its truth was very neatly proved by 

 Mr. Ainsworth, Mr. John Burrow, the proposer, and several 

 others, whose solutions are the more remarkable, since they 

 all but formally state the leading properties of the complete 

 quadrilateral, and add several to those already instanced under 

 another form by La Hire and Maclaurin. In reference to 

 Fig. 1, (see illustrative Diagram at end of Paper,) it is shown 

 that:— 



(1.) AB ! BC :: AM : CP :: AD : DC; 

 and SQ : QV :: SD : DV. 



(2.) Q m = Q m;, &c. 



(3.) AC, ac, &c., pass through the point D. 



(4.) The locus of the points N, n, &c., is the line T Q. 



(5.) If D be such that S D is conjugately divided, iri 

 the same ratio will all lines as A D, a D, &e«j 

 be divided. 



