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HISTORY OF ALGEBRA.

TRACT 33.

V; as V3) for the cube root. He also introduced the cha-racters > and <, for greater and less ; and in the reductionof equations, he arranged the operations in separate stepsor lines, setting the explanations in the margin on the lefthand, for each line. By which, and other means, he mayhe considered as the introducer of the modern state of Alge-bra, which quite changed its form under his hands.

2d. He showed the universal generation of all the com-pound or affected equations, by the continual multiplicationof so many simple ones, or binomial roots ; thereby plainlyexhibiting to the eye the whole circumstances of the nature,mystery and number of the roots of equations ; with thecomposition and relations of the coefficients of the terms ;and from which many of the most important properties havesince been deduced.

3d. He greatly improved the numeral exegesis, or extrac-tion of the roots of all equations, by clear and explicit rulesand methods, drawn from the foregoing generation or com-position of affected equations of all degrees.

of oughtred’s clavis.

Outred was contemporary with Harriot, but lived a longtime after him, as he died only in 1660, at 87 years of age.His Clavis was first published in 1631, the same year inwhich Harriot’s Algebra was published by his friend Warner.In this work, Oughtred chiefly follows Vieta, in the nota-tion by the capitals a, b, c, d, 8lc, in the designation ofproducts, powers, and roots, though with some few varia-tions. His work may be comprehended under the followingparticulars.

1. Notation. This extends to both Algebra and Arith-metic, vulgar and decimal. The Algebra chiefly after themanner of Vieta, as abovesaid. And he separates the deci-mals from the integers thus, 21^56, which is the first time Ihave observed such a separation, and the decimals set downwithout their denominator.