494 TRANSACTIONS OF THE 



use of it in his calculating machine. As it may be of service in expedit- 

 ing numerical calculations, he would explain it at the blackboard and give 

 the formula for its use. 



The law is, that the sum of a regular series of odd numbers, commenc- 

 ing with one, is always equal to the square of the number of terms used. 

 So the odd numbers may be written with the arithmetical series underneath 



them thus : 



1 3 5 7 9 11 13 15 17 19 

 12345 6 7 8 9 10 

 and the square of any number in the lower series is equal to the odd num- 

 ber above it added to all those preceding, thus : l-|-3-]-5+74- 9=5X5. 

 By the following formula, in which n is the last number, we may find the 



sum of a series of odd numbers : ( .^ y ; for example, if the sum of 13 

 and the odd numbers below it is required, we have (13-|-l-i-2)-, or 49, 

 In this case the process of adding seven terms is expedited by simply mul- 

 tiplying 7 by 7. I have also discovered that the sum of the even numbers, 

 below any odd numbers, is always equal to the square of the number of 

 terms minus the same number. So the formula for the sum of all the 

 numbers below a given odd number is 



/n4-l\2 , /n+l\2 ?i-f-l n+1 , / n+1 , n—\\ 



For example, the number 17 added to all the numbers below it is equal 

 to 9X9-|-9X8, or 153. By this rule, adding any great number of terms 

 is reduced to the simple process of two luultiplications and a single addi- 

 tion. 



Mr. Pierce remarked that the waters of the rivers of Hazleton, Luzerne 

 Co., Penn., possessed a peculiar property of corroding iron, so much so, 

 that when a shoal had been left on Saturday night under a dropping of it, 

 a hole would be eaten through it by Monday morning. 



Mr. Griffith exhibited a vial of it and iron destroyed by it. Ho described 

 the effect produced by the surface water also as destructive of iron, causing 

 boilers of some iron to produce very red color in the water. He produced 

 a white material, one of its effects. 



Mr. Meigs asked if it was not copperas. 



Mr. Seeley. — Yes. 



Conversation followed between Messrs. Griffith, Pierce, Seeley, Stetson, 

 Tillman, Cohen, Butler, Garbanati and Pell, relative to preventives by 

 tallow, potato, bran, &c., and the want of homogenity in iron, especially 

 boiler and other plates. 



Alanson Nash said that copper was found comby sometimes, owing to 

 impurities in it. A law suit in England grew out of it, against the maker 

 of the copper, and heavy damages awarded. He said that an assay 

 of iron was required as much or more than of gold and silver. The 

 British government assays copper sheets. Old iron v^ primitive rock, 

 after ages of exposure, made into steel, form excellent razors ; it had not 



