176 Memoir of John Napier, 



of powers, ami the extraction of roots ; and lastly, several 

 books upon different questions applicable to commercial 

 transactions. These questions extend from the rule of 

 three to the summation of very complicated numerical 

 series, such as the series of squares, of natural numbers, 

 and that of triangular numbers. 



At the third page of the book upon the extraction of 

 roots, (6 sect.) we find a table entitled, ' Table which gives 

 the method of finding the union {lien angles), a term used in 

 the work, to indicate the co-efficients of the developement 

 of any powpr of a binomial. This table is disposed exactly 

 like the arithmetical triangle of Pascal, and presents the 

 successive series of powers of 1 + ] from the power 1 up to 6. 



a Origin of successive numbers. 



If- \1 Root extracted. 



/V/\ 



j / ___\/____\l Equal figure or figure for one mul- 



/\3 t'\3 /\ tiplication. 

 ' \ ' * / \ 



■gi \C \l -vl Solid figure or figure for two mul- 



/ \ A / \ 6 ' \ 4- / \ tiplications. 

 ' x / \ i \ ' v 



t/i V x s' \t Figure for three multiplications. 



/\f/ V/VAVx 



//' « Y — — ) yi. \f. Figure for four multiplications. 



/ \6 / \lS;\20/\&/\6 /\ 



•*'"' ■'"' l * Figure for five multiplications. 



The exterior line to the left, says the text, contains 

 the numbers tsi, (the numbers which are to be added). The 

 exterior line to the right, contains the numbers yu, (coins). 

 It is the expression employed in the work to express the last 

 term of a binomial raised to a power. The numbers which are 

 placed in the middle of the others form the union (angles 

 or co-efficients). With this union, we have the true process 

 for extracting the roots of powers. 



Disposition of the table.— In proceeding from the top, we 

 observe 2 which forms the sign of the equal figure (phing- 

 fang). We observe then 3 and 3, which are the signs 

 of the right or solid figure (li-fang). Then we have 4, 6, 4, 

 which are the signs of the figure for three multiplications 



