68 Expression of the sides of Right-angled Triangles, SfC. 



minitas above Pittsburgh to Shawaneetown in Illinois amounts to be- 

 tween two and three millions of busljels annually. The works at 

 Kenhawa alone furnish twelve hundred thousand bushels, and those 

 on the Muskingum between three and four hundred thousand. Large 

 quantities' are made at Kiskiminitas or Konemaugh, Pa. and at Yel- 

 low Creek in Columbiana County,-Ohio. Salt is also made in con- 

 siderable quantities on the Big Hockhocking and on Leading Creek. 

 When that beautiful, but simple process of manufacturing by atmos- 

 pheric evaporation, and that still more interesting one by steam, are 

 carried more extensively irito use, the people of the west will have 

 nothing farther to desire, either in the abundance, or in the excel- 

 lency of that great and indispensable article, marine or culinary salt. 



Art. VL — On the Expression of the sides of Right-angled Trian- 

 gles, by Rational and Integral numbers ; by Rev. Daniel Wilkie. 



In a late number of your excellent Journal,* there appear two in- 

 genious methods of finding the sides of right-angled triangles in in- 

 tegral numbers. 



The first of these methods is the following — Rule. — "Take any two 

 numbers whose difference is 2. Their sum will be the root of one 

 square ; their product that of the other. Add 2 to the product just 

 found, and you obtain the root of the sum of the squares, or the hy- 

 pothenuse." 



This method affords a number of curious results, and furnishes a 

 great number of answers,f yet it is evidently limited to those cases in 

 which the hypothenuse exceeds the next greater side by 2, or to 

 multiples of these. 



The second method is simpler and more general. It is as follows. 



" Assume m, n, p, any rational numbers, so that n be greater than 

 p, then, m{n^ -\-p^), m{n^—p^) and 2 mnp, will be the sides of 

 the required triangle." 



As the quantity m is a multiplier of all the terms, and does not 

 otherwise enter into the formation of the rule, it may be omitted. 

 The limitation that n be greater than p, is also unnecessary. 



The following method is proposed, as somewhat simpler, and 

 more general than those above stated. 



• V^jl. XX, No. 2. t And is accompanied by a demonstration. 





