82 AN ESSAY ON THE ROOTS OF INTEGERS, 



or first Rank ; the next upper division the Rank of the Square ; the next 

 the Rank of the Cube ; the next the Rank of the Biquadrate ; the next the 

 Rank of the Quadratus Cubi ; and the next, which constitutes the Pulpit 

 Diagram, call the Rank of the Number. This I have divided from the 

 other Ranks by a triple cross line. Again, in reverse order, call the Rank 

 of the Quadratus Cubi, also the last Rank or second Rank of the Num- 

 ber ; and call the Rank of the Biquadrate, also the second last Rank or 

 third Rank of the Number ; and call the Rank of the Cube, also the third 

 last Rank or fourth Rank of the Number ; and call the Rank of the 

 Square, also the fourth last Rank or fifth Rank of the Number ; 



Before proceeding farther, two phrases are yet to be explained. 



(39.) When, in this operation, one number is said to be written oppo- 

 site to another, it is hereby meant, that whatever be the distance between 

 these two numbers, yet with respect to the interstices formed by the 

 whole length of the longitudinal lines, the units of the one number are 

 written in the same interstice as the units of the other, the tens of the 

 one number in the same interstice as the tens of the other, the hundreds 

 of the one number in the same interstice as the hundreds of the other, and 

 so on ; all the higher corresponding degrees of the two numbers being 

 respectively in the same interstice. 



Thus in the Rank of the Biquadrate, at the letter of reference %', the 

 number 986,144,576 is written opposite the number 18,535,911,758,593, 

 which is at the letter i' in the Rank of the number or Pulpit Diagram, 

 for 6, the units of the first number is written in the same interstice of the 

 longitudinal lines as 3, the units of the second ; and the tens of both num- 

 bers, viz. 7 and 9, are in the same interstice to the left of the units, and 

 the hundreds of both numbers, viz. 5, and 5, are in the same interstice to 

 the left of the tens, and so of the thousands 4 and 8, &c. 



