Thomas Harriot (1560-1621) is recognized as a foundational and important figure in the mathematical world, though his work was not adequately appreciated until after he died. During his lifetime he did not publish any mathematical work; his *Artis Analyticae Praxis* was published posthumously in 1631.

Thomas Harriot, *Artis analyticae praxis* (London, 1631), title page.

Harriot’s patron, Henry Percy, 9th Earl of Northumberland (1564-1632), known as the ‘Wizard Earl’, is named on the title page. Percy had been implicated in the Gunpowder Plot of 1605 and, as a result, was imprisoned in the Tower of London until 1621. Harriot’s other patron, also involved in significant political intrigue at the time, was Sir Walter Raleigh (1554-1618). Harriot travelled to America (Virginia) in Raleigh’s service in the 1580s, and spent a short period at the end of that decade in Munster (Youghal), where he supported Raleigh’s colonizing efforts on behalf of Queen Elizabeth I.

In his online article relating to mathematics in England during the sixteenth century, Stephen Johnston (2013) notes that the Renaissance era was a time where academia began to recognize mathematics as more than arithmetic or essential knowledge that children must learn. Mathematics became an area of expertise in itself, a topic to be studied and analyzed well past the educational standards of that time. Ian Bruce (2006) emphasizes that the importance of *Artis Analyticae Praxis* lies in its attempt to move forward from ancient Greek mathematical forms. This was executed through Harriot’s use of symbols in order to shorten the long, complicated paragraphs and confusing wording associated with previous algebraic problems. Through careful analysis and consideration of Viète’s work on algebra, Harriot was inspired to implement a simplified notation through his use of symbols. In this way, Harriot took Viète’s earlier work in the algebraic field one step further. The page below shows Harriot’s own work in the first section of *Artis Analyticae Praxis*. This is highly unusual for seventeenth-century mathematics, because to the modern eye it requires almost no translation, due to the fact that many of Harriot’s symbols are used today including the signs ‘>’, ‘<’, and ‘=’ denoting ‘greater than’, ‘less than’, and ‘equivalent to’.

Thomas Harriot, *Artis analyticae praxis* (London, 1631), p. 10.

In addition to supplying a condensed, simplified version of the mathematics, Harriot provides small descriptions of his work suggesting a deeper analysis beyond simply solving a problem or problem set. In fact, he sought to show clear, concise solution pathways that could be understood and followed easily. For example, Harriot includes a problem that involves reducing a binomial equation to one with a single term.

Thomas Harriot,* Artis analyticae praxis* (London, 1631), p. 29.

More specifically, he starts with the binomial equation a² – ba + ca = bc and shows the expression reduces to the monomial a² = b² (where ‘a’ denotes the variable quantity). Using both his own notation and shortened word phrases, Harriot shows that, putting b = c in the given equation, it follows that a² – ba + ba = b² and hence a² = b². The reason that this kind of work gave rise to such a meaningful advancement during Harriot’s time was because a simple problem like this would have taken up much space and many words that would have left an average mathematician confused. By mapping out a way to display the algebra concisely, he made the mathematics accessible to a larger audience. When *Artis Analyticae Praxis* was published, Harriot became known as the ‘father of modern notation’ for his work.

Finally, look at Harriot’s proof of the arithmetic-geometric inequality, namely that for any two distinct positive numbers, p and q, their arithmetic mean, (p+q)/2, is always greater than their geometric mean, √(pq). To follow his proof, check out first the discussion on the algebra page.

Thomas Harriot, *Artis analyticae praxis* (London, 1631), p. 78.

Unfortunately the printed proof begins with two misprints; it should read: ‘*quarum pp. maxima est, qq. vero minima est*’ which presumes that p > q. Multiplying each side by the positive number p – q gives p(p-q) > q(p-q) or p²-pq > pq-q². Thus p²+q² > 2pq, and adding 2pq to each side gives: p²+2pq+q² > 4pq. However, the left-hand side is just (p+q)², so ¼(p+q)² > pq. The proof concludes at the top of the next page: ‘Ergo {(p+q)/2}² > pq. Quod erat probandum.’ (‘What was to be proved.’)

Harriot’s* Artis Analyticae Praxis* has an interesting history with the combination of its late publication and the fact that Harriot is considered to have been well ahead of his time when he developed the work covered in the book. Though he did not receive proper recognition for his ingenuity during his lifetime, his work on the development of mathematical notation proves him to be a foundational figure in the history of mathematics.

**Sources**

Bruce, Ian, Harriot’s Artis Analyticae Praxis, (17 *Century Maths*. August 2006).

Johnston, Stephen, The identity of the mathematical practitioner in 16th century England, (*Museum of the History of Science*, Oxford, February 2013).

Lee, Sidney, Henry Percy (1564-1632), entry in *Dictionary of National Biography*, (Oxford University Press, 2004).

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

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

**Text:** Danielle Lanting and Maurice OReilly.