Algebra

‘ALGEBRA is a Science by which the most abstruse or difficult Problems, in Arithmetick or Geometry, are Resolved and Demonstrated; that is, it equally interferes with them both; and therefore it is promiscuously named, being sometimes called Specious Arithmetick, as by Harriot, Vieta, and Dr Wallis, &c. And sometimes it is called Modern Geometry, particularly by the late ingenious and great Mathematician Dr Edmund Halley, Savilian Professor of Geometry in the University of Oxford, and Royal Astronomer at Greenwich; who, in giving the following Instance of the Excellence of our Modern Algebra, writes thus: “The Excellence of the Modern Geometry (saith he) is in nothing more evident, than in those full and Adequate Solutions it gives to Problems; representing all the possible Cases at one View, and in one general Theorem many Times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the Manner of the Ancients, might well become the subjects of large Treatises: For whatever Theorem solves the most complicated Problem of the Kind, does with a due Reduction reach all the subordinate Cases.” Of which he gives a notable Instance in the Doctrine of Dioptricks for finding the Foci of Optic Glasses universally. (Vide Philosophical Transactions, Numb. 205).’

John Ward, The Young Mathematician’s Guide, p. 2.

Works by the key players in the development of algebra in Early Modern Europe are well represented in the Edward Worth Library, including those by Cardano (1501-1576), Viète (or Vieta, 1540-1603), Harriot (1560-1621), Descartes (1596-1650) and Wallis (1616-1703). Progress in the sixteenth and seventeenth centuries was given an impetus by Cardano’s solution of the cubic; this was followed by the important insights of Viète and Harriot into the structure of the solutions of polynomial equations, Descartes’ application of algebra in geometry and Wallis’ overview of advances in ‘Specious Arithmetick’ (to use Kersey’s term) in his Treatise on Algebra (1685). Edmund Halley’s (1656-1742) works in the Worth Library are represented by the inclusion of his equations in the 1707 and 1732 editions of Sir Isaac Newton’s Arithmetica universalis and his own 1706 edition of the works of Apollonius of Perga (c.262 BC-c.190 BC).

The interaction between algebra and geometry was not confined to applications in astronomy. Analytic geometry (including coordinate geometry familiar to Irish students taking Junior Cycle maths), introduced algebra as a tool that changed forever how we think about geometry. Here, French mathematicians (Viète, Fermat, Descartes) were to the fore as Stedall (2008) describes in some detail. Their contributions are considered in the section on notation of this exhibition.

To conclude the algebra exhibit, let’s look at how someone in the library (perhaps Edward Worth himself?) worked at understanding an algebraic text. Attached (inside the front cover) to Harriot’s Artis analyticae praxis, we see the ‘rough work’ in manuscript dealing with three types of equations based on Harriot’s work:

x² – bx + cx = bc
x² – bx – cx = -bc
x³ + (- b + c + d) x² – (bc + bd – cd)x = bcd

Harriot notes detail

Thomas Harriot, Artis analyticae praxis (London, 1631), manuscript annotations detail.

For today’s reader, Harriot’s variable, a, has been replaced here by x; squares and cubes are denoted by x² and x³ (rather than by xx and xxx); and brackets are used to gather the coefficients of different powers of x. Focusing on the first equation, it is noted that x = b is a solution (‘posito b = a’) and that the equivalent canonical form (‘ab originalibus suis æquationes canonicæ deriuentur’ to quote from p. 4 of Harriot) of this equation is (x-b)(x+c) = 0. In this manuscript (following Harriot), the binomial factors, (x-b) and (x+c), are expressed as a-b above a+c, with a horizontal line beneath, and a vertical line to the right of the pair of factors. The other two equations are treated in a similar manner.

Sources

J J O’Connor and E F Robertson, Thomas Harriot, MacTutor History of Mathematics, (University of St Andrews, Scotland, February 2000).

J J O’Connor and E F Robertson, John Wallis, MacTutor History of Mathematics, (University of St Andrews, Scotland, February 2002).

Stedhall, Jacqueline, Mathematics Emerging. A Sourcebook 1540-1900 (Oxford University Press, 2008).

Text: Maurice OReilly.