PROGRESS OF MATHEMATICS AND MECHANICS 235 



even enunciates the fundamental principle that for simple machines 

 forces in equilibrium are inversely as the virtual velocities. 



' Whoever,' he says, ' appeals to authority applies not his intellect 

 but his memory.' ' While Nature begins with the cause and ends with 

 the experiment, we must nevertheless pursue the opposite plan, 

 beginning with the experiment and by means of it investigating the 

 cause.' 'No human investigation can call itself true science, unless 

 it comes through mathematical demonstration.' 'He who scorns 

 the certainty of mathematics will not be able to silence sophistical 

 theories which end only in a war of words.' 



Unfortunately his work in this field remained unpublished, and 

 therefore relatively unfruitful. 



Leonardo and other great artists of his time - - notably Albrecht 

 Diirer of Nuremberg (1471-1528) developed the geometrical 

 theory of perspective. For the purpose of accurately representing 

 the human head Diirer made both plans and elevation. " Intelli- 

 gent painters and accurate artists," he says, " at the sight of works 

 painted without regard to true perspective must laugh at the blind- 

 ness of these people, because to a right understanding nothing im- 

 presses more disagreeably than falsehood in a painting, regardless 

 of the diligence with which it has been made. That such painters, 

 however, are pleased with their own mistakes is due to the fact 

 that they have not learned the art of measurement, without which 

 no one can become a true workman." All this had importance 

 both for modern art and modern geometry. 



Characteristic of this period is the so-called Margarita Philo- 

 sophica published in many editions from 1503 to 1600. It was the 

 first modern encyclopaedia printed, and gives in its twelve books " a 

 compendium of the trivium, the quadrivium, and the natural and 

 moral sciences." 



A younger contemporary of Pacioli, Michael Stifel (1487-1567), 

 a German monk converted to Lutheranism, developed a fantastic 

 arithmetical interpretation of the Bible, identifying Pope Leo X 

 with the beast in Revelation and predicting the immediate end 

 of the world, with results disastrous to his person as well as his 

 reputation. 



