OF ARTS AND SCIENCES. 129 



Two hundred and ninety-fourth Meeting. 



April 6, 1847. — Monthly Meeting. 



The President in the chair. 



Professor Strong, of New Brunswick, New Jersey, commu- 

 nicated the following papers, viz. : — 



I. '■'An attempt to prove that the sum of the three angles of any 

 rectilineal triangle is equal to two right angles. 



^' Def Two quantities are said to be of the same kind, when the 

 less can be multiplied by some positive integer, so as to exceed the 

 greater. Thus, if A and B are quantities of the same kind, and if A 

 is greater than B^ then some positive integer, m, may be found, such 

 that the inequality m B^A shall exist. For if m is taken greater than 

 the quotient arising from the division of A by B, then evidently there 

 results the inequality m B'^A, as required. 



" Dem. If m denotes any positive integer, then shall the inequality 

 2 '" ^ m obtain. For the first member of the inequality denotes the 

 product arising from taking 2 as a factor as often as these units in m, 

 whereas the second member is the sum of the units represented by m, 

 and the inequality is evident. 



" Cor. If we take A and jB, as above, and mB^A, there results 

 jB^^; much more, then, shall the inequality B^^ have place. 

 This follows at once since it has been shown that 2 "^ is greater than 

 m ; and it is evident that if m is an indefinitely great number, - is in- 

 definitely greater than 2™* 



" Ax. No angle of a rectilineal triangle can exceed two right angles. 



" Prop. 1 . To find a triangle that shall have the sum of its angles 

 equal to the sum of the angles of any given triangle. 



" Let ABC denote the given triangle ; and suppose one of its sides, 



c 



B C, is bisected at D, and that D and the opposite angle A are con- 

 nected by the right line A D, which is produced in the direction A D 



17 



