PROGRESS OF MATHEMATICS AND MECHANICS 233 



Some of Pacioli's commercial problems are exceedingly compli- 

 cated. He solves numerical equations of the first and second 

 degree, but admits only positive roots and considers the solution 

 of cubic equations, as well as the squaring of the circle, impossible. 

 Addition is denoted by p or p, equality sometimes by ae, a begin- 

 ning of syncopated algebra. The introduction of the radical sign 

 with indices V2> V3 and of the iigns -f and date from about 

 this time. 



In geometry Pacioli, like Regiomontanus, employs algebraic 

 methods. Among other problems he determines a triangle from 

 the radius of the inscribed circle and the segments into which 

 it divides one of the sides. His solution, though highly esteemed 

 at the time, is much less simple than he might have obtained by 

 the formulas at his command. 



In the spirit of the Renaissance he brings the feeble mathematics 

 of the universities into fruitful relations with the practical mathe- 

 matics of the artist and the architect. The inscribed hexagon and 

 the equilateral triangle play their part as gild secrets in the develop- 

 ment of Gothic architecture. The question is not " How to prove," 

 but "How to do." 



On the other hand, the current tendency to drift into mysti- 

 cal interpretation is exemplified by the following extract from 

 Pacioli : 



There are three principal sins, avarice, luxury, and pride; three 

 sorts of satisfaction for sin, fasting, almsgiving, and prayer ; three 

 persons offended by sin, God, the sinner himself, and his neighbour ; 

 three witnesses in heaven, Pater, verbum, and spiritus sanctus ; three 

 degrees of penitence, contrition, confession, and satisfaction, which 

 Dante has represented as the three steps of the ladder that leads to 

 purgatory, the first marble, the second black and rugged stone, and the 

 third red porphyry. There are three sacred orders in the church 

 militant, subdiaconati, diaconati, and presbyter ati ; there are three parts 

 not without mystery, of the most sacred body made by the priest 

 in the mass ; and three times he says Agnus Dei, and three times, 

 Sanctus ; and if we well consider all the devout acts of Christian wor- 

 ship, they are found in a ternary combination ; if we wish rightly to 



