8 PURE MATHEMATICS. 



to which I would now direct your attention: and as I 

 am unwilling to occupy too much of your time, I shall 

 endeavour to make appropriate selections. 



I commence with the subject of pure mathematics, 

 which we may truly characterise, in the language of 

 Dr. Barrow, as that which " effectually exercises, not 

 vainly deludes, nor vexatiously torments studious minds ; 

 which overcomes without opposition, triumphs without 

 pomp, compels without force, and rules absolutely 

 without the loss of liberty; which does not covertly 

 overreach a weak faith, but openly assaults an armed 

 reason, obtains a total victory, and puts on inevitable 

 chains ; which obtrudes no false science, but very science 

 itself, the mind, as soon as possessed of it, firmly adher- 

 ing to it, and never after deserting it of its own accord ; 

 nor can it be deprived of it by any force of others." 

 Among the earliest promoters of mathematics was the 

 celebrated PYTHAGORAS, author of the appellation 

 Philosopher, and " rendered immortal in the annals of 

 geometry," say the historians, by the invention of the 

 multiplication table, and by the discovery of three pro- 

 positions, viz. that only three regular plane figures, the 

 equilateral triangle, the square, and the hexagon, can 

 fill up the space about a point ; that the sum of the 

 three angles of every plane triangle is equal to two 

 right angles ; and that in any right angled plane triangle 

 the square on the longest side is equal to the sum of 

 the squares on the two other sides. The discovery of 

 this last proposition excited in the mind of Pythagoras 

 such ecstatic and devout feeling, that he is described as 

 offering a hecatomb to the gods on account of it. This 

 I am inclined to disbelieve, for the reason assigned by 

 Cicero, " that it was inconsistent with his principles, 



