146 AN ESSAY ON THE ROOTS OF INTEGERS, 



and let us add the product to that which is in the Rank of the Cube, and 

 thus until we multiply it into the sum in the third Rank of the Number, 

 and transfer the sum of this product and the number in the second Rank of 

 the number, to the right hand in this Rank one place. Then let us add 

 the top number the second time to that which is in the Rank of the Latus 

 for the third Rank of the number, and let us multiply it into the sum, and 

 let us add the product to that which is in the Rank of the Square, and 

 let us multiply it into the sum, and let us add the product to that which 

 is in the Rank of the Cube, and so on till we have added its product 

 into the sum in the fourth Rank of the Number to that which is in 

 the third Rank, and let us transfer the sum to the right hand two 

 places, then let us add the top number to that which is in the Rank 

 of the Latus a third time, for the fourth Rank of the Number, and 

 let us operate with it as I have explained, and so on until we arrive 

 at the addition of the top number to that which is in the Rank of 

 the Latus for that same Rank, and its transference to the right hand, 

 so that its units should be opposite the second place of the preced- 

 ing period. And let it be known that we write the products in the Ranks, 

 so that their units should be under the single top figure, and we write 

 the result of the addition above the items after erasing them by a latitu- 

 dinal line, and this will be the Number which is above the lines in all the 

 Ranks, except the Rank of the Number, because the progress of the ope- 

 ration in all, except that Rank, is upwards. And that the product of the 

 multiplication of the top Number into that which is written in each Rank 

 is added to that which is in the Rank above it. Then let us seek the 

 greatest of the units, which, if we write it in the external Row opposite 

 to the first place of the preceding period, and below it in the lowest part 

 of the Rank of the Latus to the right of the Number written there and 

 multiply it into that which is in the Rank of the Latus, and add the pro- 

 duct to that which is in the Rank of the Square, then multiply it into 

 that which is in the Rank of the Square, and add the product to that 



