12 DISSERTATION SECOND. [f ABT i. 



Maurolycus of Messina flourished in the middle of the 

 sixteenth century, and is justly regarded as the first geo- 

 meter of that age. Beside furnishing many valuable trans- 

 lations and commentaries, he wrote a treatise on the conick 

 sections, which is highly esteemed. He endeavoured also 

 to restore the fifth book of the conicks of Apolionius, in 

 which that geometer treated of the maxima and minima 

 of the conick sections. His writings all indicate a man of 

 clear conceptions, and of a strong understanding ; though 

 he is taxed with having dealt in astrological prediction. 



In the early part of the seventeenth century, Cavalleri 

 was particularly distinguished, and made an advance in the 

 higher geometry, which occupies the middle place be- 

 tween the discoveries of Archimedes and those of New- 

 ton. 



For the purpose of determining the lengths and areas of 

 curves, and the contents of solids contained within curve 

 superficies, the ancients had invented a method, to which 

 the name of Exhaustions has been given ; and in nothing, 

 perhaps, have they more displayed their powers of mathe- 

 matical invention. 



Whenever it is required to measure the space bounded 

 by curve Mines, the lenglh of a curve, or the solid contain- 

 ed wilhin a curve superficies, the investigation does not 

 fall within the range of elementary geometry. Rectilineal 

 figures are compared, on the principle of superposition, by 

 help of the notion of equality which is derived from the 

 coincidence of magnitudes both similar and equal. Two 

 rectangles of equal bases and equal altitudes are held to be 

 equal, because they can perfectly coincide. A rectangle 

 and an oblique angled parallelogram, having equal bases 

 and altitudes, are shown to be equal, because the same 

 triangle, taken from the rectangle on one side, and added 

 to it on the other, converts it into the parallelogram ; and 



