PHYSICAL AND MATHEMATICAL SCIENCES. 35

That notwithstanding these defects, and many objec-tions which have been brought against it, and attemptswhich have been made to improve upon it, Euclid’streatise should still sustain its high reputation, iscertainly no small praise. Its adoption as the text-book for elementary instruction is a different question.In England, it is still generally used; but on the Conti-nent , other works more specifically designed for the stu-dent’s initiation have superseded it. This, however, isindependent altogether of its merits as a philosophicalsystem. And in this point of view we are inclined tobelieve that its excellencies are hardly yet fully appre.ciated. The entire work has been usually edited asextending to fifteen books ; but the two last are admittednot to be Euclid’s: the Arabic translator expresslyascribes them to Hypsicles . It appears probable that thework we have is a combination of two or, perhaps, threetreatises, originally distinct. Besides geometry, thesebooks include the theory of arithmetic, and its applica-tion to geometry. The investigation of the philosophicalprinciples of reasoning adopted by Euclid in the differentparts of his writings is a subject of high interest, andone which deserves more attention than it has usuallyreceived, both with reference to a right apprehension ofhis excellencies, and to the questions which have arisenin modern times between the rival claims of geometryand algebra.

But we cannot here pursue questions of this kind.It will be more to our purpose to remark, that the bound-aries of what have been distinctively considered element-ary methods were reached by Euclid in his demonstrationof the proportionality of the areas of circles to thesquares, and of the contents of spheres to the cubes, oftheir diameters. The problem of squaring the circle, orassigning by geometrical methods the side of a squarewhose area is equal to that of a given circle, was onemuch celebrated among the ancients, and which liasindirectly proved of incalculable advantage to science ;since, in the unceasing and fruitless attempt to solve it,i) a