52 AN ESSAY ON THE ROOTS OF INTEGERS, 



the above series, it is proposed to express Z, by a similar series, in the 

 following seven equations. 



I. A + B = Z. IV. A ~ B = Z. 



II. A — B = Z. V. A B = Z. 



III. A x B = Z. VI. A = Z. 



VII. A Z = B 



All other operations on numbers, belong either to the synthetical or 

 analytical part of Algebra. 



(4.) These operations, in the above order, successively become more 

 and more complicated, and hence to form an estimate of the state of arith- 

 metic among any people, it is sufficient to enquire into the method by 

 which they perform the most complicated of these operations with which 

 they are acquainted. 



(5.) Conformably to this, I here propose to enquire into the method 

 by which the Arabians, supposing A andB to be integers, express Z in the 



B 



sixth equation or A = Z, or in other words, the Arabian method of 

 extracting the Roots of integer powers. This method is contained in 

 the Ayoun-ul-Hisab, a book, respecting which the reader will see all 

 that I know in vol. XIII. of the Researches, p. 461. I believe the Arabs 

 never attempted any general method for the seventh equation, which is 

 the foundation of the Theory of Logarithms, except mere tentation. 

 The extent of their knowledge on the subject of negative exponents, 

 may be seen either in Mr. Strachey's History of Algebra, published 

 in the Asiatic Researches, vol. XII. p. 177, or in the Calcutta edition of 

 the Arabico- Persic Kholasut-ul-Hisab, p. 313, et seq. I do not find any 

 trace of their acquaintance with fractional exponents. 



