We have next to consider the works of Albert Girard, a Flemish mathematician. This writer, after having published an edition of Stevin's works in 1625, published in 1629 at Amsterdam a small tract on algebra which shows a considerable advance on the work of Vieta. Girard is inconsistent in his notation, sometimes following Vieta, sometimes Stevin; he introduced the new symbols ff. for greater than and sec. for less than; he follows Vieta in using the plus (+) for addition, he denotes subtraction by Recorde's symbol for equality (=), and he had no sign for equality but wrote the word out. He possessed clear ideas of indices and the generation of powers, of the negative roots of equations and their geometrical interpretation, and was the first to use the term imaginary roots. He also discovered how to sum the powers of the roots of an equation.
Passing over the invention of logarithms (q.v.) by John Napier, and their development by Henry Briggs and others, the next author of moment was an Englishman, Thomas Harriot, whose algebra (Artis analyticae praxis) was published posthumously by Walter Warner in 1631. Its great merit consists in the complete notation and symbolism, which avoided the cumbersome expressions of the earlier algebraists, and reduced the art to a form closely resembling that of to-day. He follows Vieta in assigning the vowels to the unknown quantities and the consonants to the knowns, but instead of using capitals, as with Vieta, he employed the small letters; equality he denoted by Recorde's symbol, and he introduced the signs > and < for greater than and less than. His principal discovery is concerned with equations, which he showed to be derived from the continued multiplication of as many simple factors as the highest power of the unknown, and he was thus enabled to deduce relations between the coefficients and various functions of the roots. Mention may also be made of his chapter on inequalities, in which he proves that the arithmetic mean is always greater than the geometric mean.
William Oughtred, a contemporary of Harriot, published an algebra, Clavis mathematicae, simultaneously with Harriot's treatise. His notation is based on that of Vieta, but he introduced the sign X for multiplication, @ for continued proportion, :: for proportion, and denoted ratio by one dot. This last character has since been entirely restricted to multiplication, and ratio is now denoted by two dots (:). His symbols for greater than and less than (@ and @) have been completely superseded by Harriot's signs.
So far the development of algebra and geometry had been mutually independent, except for a few isolated applications of geometrical constructions to the solution of algebraical problems. Certain minds had long suspected the advantages which would accrue from the unrestricted application of algebra to geometry, but it was not until the advent of the philosopher Rene Descartes that the co-ordination was effected. In his famous Geometria (1637), which is really a treatise on the algebraic representation of geometric theorems, he founded the modern theory of analytical geometry (see GEOMETRY), and at the same time he rendered signal service to algebra, more especially in the theory of equations. His notation is based primarily on that of Harriot; but he differs from that writer in retaining the first letters of the alphabet for the known quantities and the final letters for the unknowns.
Continued on page nine.
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