With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra... - David Eugene Smith

Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates.

Vieta was the first algebraist after Ferrari to make any noteworthy advance in the solution of the biquadratic. He began with the type <math>x^4 + 2gx^2 + bx = c,</math> wrote it as <math>x^4 + 2gx^2 = c - bx,</math> added <math>gx^2 + \frac{1}{4}y^2 + yx^2 + gy</math> to both sides, and then made the right side a square after the manner of Ferrari. This method... requires the solution of a cubic resolvent.
Descartes (1637) next took up the question and succeeded in effecting a simple solution... a method considerably improved (1649) by his commentator Van Schooten. The method was brought to its final form by Simpson (1745).

It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B.C. In 's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position... and an arrangement of terms in a kind of notation. The rules given by Liu Hui form a kind of rhetorical algebra.
The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving s. ...Sun-tzï solved such problems by analysis and was content with a single result...
The Chinese certainly knew how to solve quadratics as early as the 1st century B.C., and rules given even as early as the K'iu-ch'ang Suan-shu... involve the solution of such equations.
Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should...
By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625).
The culmination of Chinese is found in the 13th century. ...numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner (1819).