116 Mr. T. S. Davies's Properties 



sufficient to communicate motion to the distant parts of the 

 fluid in the direction of the motion of the body. That such 

 increase of resistance as this formula indicates, takes place in 

 great velocities has been ascertained by experiments on the 

 motion of cannon balls. — I hope this paper contains, for those 

 who are acquainted with the subject in theory and experience, 

 sufficient evidence of the accuracy of my views; and if I have 

 any where trodden in the steps of others it is not knowingly, 

 and there is strong presumption that I have not done so, in 

 the fact that I have not the slightest recollection of any theory 

 founded on the same principles. 

 August 1826. Thomas Tredgold. 



XVI. Properties of the Trapezium. By T. S. Davies, Esq. 



of Bath*. 

 A T a very early period of my mathematical studies the tra- 

 "*"*- pezium seemed to offer a far more open field of inquiry, 

 and to yield far more interesting results, than any other figure, 

 the circle excepted, that I examined. The properties of the 

 triangle, numerous as they were, had been so often examined, 

 and their characters stated under almost every possible variety 

 of aspect, that little really novel appeared likely to result from 

 any inquiries, however well directed or vigorously pursued. 

 The trapezium on the contrary had been investigated very lit- 

 tle further than its elementary and more obvious properties; 

 and the majority even of these were such as either had some 

 analogy to the triangle, or resulted from dividing it into sub- 

 ordinate triangles. The ground has been occasionally trod- 

 den by others ; but I think it will be found that every one 

 who has done so has made his discoveries more by accident, 

 or to meet some mathematical exigency, than from any syste- 

 matic plan of inquiry : and the best proof of this is, that 

 amongst the great men who have lioticed any of the proper- 

 ties of the trapezium, not one of them has cared to pursue 

 those results into any of their remoter consequences. 



In the course of my researches I have been so fortunate as 

 to meet with several classes of properties which, whether viewed 

 as so many elegant theorems, or in reference to their applica- 

 tion to subsequent inquiries, cannot fail to interest the geome- 

 trical student ; and shall do myself the honour to present to this 

 Society a series of short papers containing a few of the pro- 

 perties that I have noticed, together with a general outline of 

 their demonstrations. 



* Read before the Society of Inquirers, Bristol, August 5, 1825, and 

 communicated by the Author. 



ECKD 



