PHYSICAL AND MATHEMATICAL SCIENCES. 125

were first published about 1590, and afterwards col-lected by Schooten in one volume, in 16'46.

About the same period algebra became greatly in-debted to Albert Girard, a Flemish mathematician ;though his principal work, “ Invention Nouvelle enAlgebre,” was not printed till 1669- This ingeniousauthor perceived to a greater extent than Vieta haddone, but still not in all its generality, the principle ofthe successive formation of the co-efficients of an equa-tion from the sum of the roots; the sum of their pro-ducts taken two and two, the same taken three andthree, &c., whether the roots be positive or negative.He also conceived the notion of imaginary roots; andshowed that the number of the roots of an equationcannot exceed its dimensions. To these speculationshe added some others connected with the use of nega-tive roots in geometrical constructions, &c. He alsofirst introduced the phrase of “ a quantity less thannothing,” which has been so severely censured by somemathematicians, who cavil at terms without consideringthat they only become fair subjects of question whenthe wrong use of the term is likely to involve a mistakein the thing. This phrase, of course, never impliesany thing more than the very conception by which ne-gative quantities take their origin ; the supposedincrease of the quantity subtracted till it becomes equalto, and then greater than, that from which it is to betaken. The variable value of the whole is thus said topass through zero, and then become negative ; or, in thisfigurative sense, less than zero. That is, it becomes aquantity at variance with the first hypothesis; butwhich may still enter into our equation, if we changethe signs of all the terms ; or, which comes to the samething, if the signs remain, and we allow the fiction ofa quantity essentially negative. This is, in fact, one ofthose simple but refined artifices arising out of theSymbolical language of algebra, which has been so mys-tified by persons of obscure conceptions, and ignorantly