VOL. LXX,] PHILOSOPHICAL TRANSACTIONS. 673 



algebra, for the most part, have not been so kind as to show us that they are so. 

 But still the question recurs, " How came Scipio Ferreus, of Bononia, who, as 

 Cardan tells us, was the first inventor of these rules, or the other person, who- 

 ever he was, that invented them, to think of making these lucky substitutions, 

 which thus transform the original cubic equations into equations of the 6th 

 power, which contain only the 6th and 3d powers of the unknown quantities 

 which are their roots, and consequently are of the form of quadratic equations?" 

 To answer this question as well as I can by conjecture (for I know of no 

 historical account of this matter in any book of algebra) and in a manner that 

 appears to me to be probable, is the design of the following pages. 



2. The most probable conjecture concerning the invention of these rules, 

 called Cardan's rules, by Scipio Ferreus, of Bononia, or whoever else was the 

 inventor of them, seems to be this: that the said inventor tried a great variety 

 of methods of reducing the 3 cubic equations of the 3d class, to wit, x^ -\- 

 qx = r and x^ — qx = r, and qx — x^ = r (to some one of which all other 

 cubic equations may, by proper substitutions, be reduced) to a lower degree, or 

 to a more simple form, by substituting various quantities in the stead of x, in 

 hopes that some of the terms arising by such substitutions might be equal to 

 others of them, and, having contrary signs prefixed to them, might destroy 

 them, and so render the new equation more simpleand manageablethanthe old one. 

 And, among other trials, it seems natural to imagine, that he would substitute 

 the sum or difference of 2 other quantities instead of x, as the most simple and 

 obsaous substitutions, that could be made. And by making these substitutions, 

 the abovementioned rules would of course come to be discovered, as well as the 

 aforesaid limitation of them in the resolution of the equation x^ — qx = r, 

 which restrains the rule to those cases only in which r is greater than 

 ii^, or — o-reater than ^, and their utter inutility in all the cases of the equa- 

 tion qx — x^ = r. This will appear by examining each of these equations sepa- 

 rately in the following manner. 



All.?,. In the 1st equation x^ -\- qx ■= i\, the investigator of these rules 

 would naturally be inclined to substitute the difference of 2 quantities, y and z, 

 instead of x, rather than their sum, or would suppose x to be equal to _y — z, 

 rather than to ?/ 4- z; because, if he was to suppose x to be equal to the sum 

 of the 2 quantities y and z, and was to substitute that sum, or the binomial 

 quantity y -\- z, instead of x in the equation x^ -\- qx = r, it is evident, that 

 (as the signs x^ and qx are both affirmative) the terms of tlie new equation, arising 

 from such substitution, would be all likewise affirmative; and consequently none 

 of them, though they should happen to be exactly equal to each other, could 

 exterminate each other, and thus render the new equation more simple than the 

 old one; which was the only view with which the substitution would have 



VOL. XIV. A R 



