PHYSICAL AND MATHEMATICAL SCIENCES. 75

Progress of Mathematics.

Diophantus flourished at Alexandria in an earlyage of the Christian era; but his precise date is matterof question. He has left a treatise, in which theprinciples of what is in fact algebra, though not givenin the form in which that science is now used, arelaid down with great ingenuity and ability, in thirteenhooks, entitled “ Arithmetical Questions.” Some ofthe problems are of considerable intricacy; and muchaddress is displayed in stating them, so as to bring outequations in a form involving only one power of theunknown quantity. The whole processes are expressedin common language, assisted by a very few symbols,which are mere abbreviations. Considering how littlepower and convenience the instrument had yet acquiredwith which he worked, it is remarkable how much hewas able to effect. He directed his attention particu-larly to the class of problems called indeterminate, orwhich admit of a number of solutions : they have beenhence known by the name of Diophantine problems.Though algebra was not at all yet reduced into a sym-bolical form, it is remarkable that Diophantus distinctlyexpresses the rule for the signs in multiplication, bysaying, that into gives iirapf 15 , &c. (minus

into minus gives plus).

Pappus. — Decline of ancient Science .

’ The declining science of the fourth century was ablyupheld by Pappus, one of the last distinguished orna-ments of the Alexandrian school, who flourished betweena. d. 350 and 400 : he cultivated, with success, almostall branches of geometry ; and, in most of them, hasleft some investigations of considerable value.

The subject of the loci was one to which his atten-tion was especially directed: in particular, he consideredthat case which was called the “ locus ad quatuorrectos ■ ” a problem which baffled the powers of the