‘*Mathematical Resolution, or the Analytical Art, commonly called Algebra, is that way of reasoning which assumes or takes the Quantity sought as if it were known or granted; and then with the help of one or more Quantities given or known, proceeds by Consequences, until at length the Quantity first only assumed of feigned to be known, is found equal to some Quantity or Quantities certainly known, and is therefore likewise known.*’

John Kersey, *The elements of that mathematical art commonly called algebra,
expounded in four books* (London, 1673), p.1.

John Kersey,* The elements of that mathematical art commonly called algebra, expounded in four books* (London, 1673), frontispiece portrait.

John Kersey was baptised four hundred years ago on the 23^{rd} November 1616; he died due to a stone in his bladder in 1677. He became an established mathematician and teacher of the subject in the 1650s in London. Kersey tutored Sir Alexander Denton’s grandchildren, Edmund and Alexander, and, judging from the dedication in this book, it appears he had a very strong bond with his pupils and their family:

*In testimony of his Gratitude, for signal Favours* *conferr’d on him by that truly Noble Family; Which also gave both Birth and Nourishment to his Mathematical Studies (*from the dedication to Kersey’s Algebra*).*

Although Kersey’s book was printed in 1673-74, it had been ready since 1667. He was having financial problems getting the book published, so he successfully sought support from subscribers, including Sir Isaac Newton (1642-1627), for the enterprise. His principal supporter was his influential friend, John Collins (1625-1683), Librarian of the recently founded Royal Society, to whom Kersey pays tribute:

*I cannot but declare to the World, that my old and much respected Friend, Mr. John Collins, a person well known to be both singularly skilfull in, and an industrious Promoter of the Mathematicks in general, hath been a principal Instrument Work to light, as well by animating me to Compile it, as by endeavouring to procure it to be well Printed (*from the preface of Kersey’s Algebra*).*

The work was printed by William Godbid, in London, as four books in two volumes, however, the copy in the Edward Worth Library has all four books rebound together. The binding is thought to be Irish and includes a title flap which is a piece of paper on the side of a page with the name of the book and the author on it.

John Kersey, *The elements of that mathematical art commonly called algebra, expounded in four books* (London, 1673), title flap, with algebra applied to a problem in geometry (book IV, p. 211), in the background.

Kersey’s *Algebra* was written in English, rather than Latin, and included several tables and diagrams to help visualise the concepts and understand them. These features indicate the expectation that the book would sell well, as they incurred an extra printing cost. The tables were used to summarize basic algebra such as the binomial theorem. However the mathematical notation was slightly different in its seventeenth-century form from what is in use today. For example, where we would write b², Kersey wrote ‘bb’.

John Kersey, *The elements of that mathematical art commonly called algebra, expounded in four books* (London, 1673), binomial expansion, book II, p. 139.

An example of Kersey’s use of a table to explain a simple algebraic idea is given on page 6 (Book I) where he indicates powers of 3 up to the eighth power:

INDICES. |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
&c. |

POWERS. |
3 |
9 |
27 |
81 |
243 |
729 |
2187 |
6561 |
&c. |

Much of Book III is devoted to the solution of problems from the third-century* Arithmetica* of Diophantus (of which there is an 1621 edition by Claude Gaspar Bachet (1581-1638) in the Edward Worth Library). A typical problem concerns finding three numbers, x, y and z, so that when each is added, in turn, to their product, xyz, the result, in each case, is a perfect square.

John Kersey,* The elements of that mathematical art commonly called algebra, expounded in four books* (London, 1673), book III, p. 79.

In his exposition of the solution, Kersey chooses ‘the Solid’, xyz = a² —2a, where a has yet to be specified. To achieve the required squares, x is taken to be 1, so that xyz + x = (a—1)², while y is taken to be 2a, so that xyz + y = a². He then takes xyz + z = (a—3)². This leads to the determination of a as 20/9, so that y = 40/9 and z = 1/9. It can then be verified that the three squares are (11/9)², (20/9)² and (7/9)². Note how this ‘modern’ explanation compares with Kersey’s wordier seventeenth-century one. Yet, Kersey’s is not at all as wordy as Cardano’s explanations of algebra from the middle of the previous century. It must certainly have been evident to Kersey (and to Diophantus, although his notation is known to have been very cumbersome indeed), that this solution was not unique, for, if we put xyz + z = (a—p)², replacing 3 by p, then it can be shown that a = 2(1+p)²/(4p—3). We have just seen the result of taking p = 3; if we were, instead, to take p = 7, then a = 4, giving rise to the integer solution: x = z = 1, y = 8, the three squares being 9, 16 and 9, respectively.

Kersey’s *Algebra* became the standard authority on the subject in the late seventeenth-century, appearing in ten editions, as well as influencing authors in the century that followed.

**Sources**

Ruth Wallis, ‘Kersey, John, the elder’, entry in Dictionary of National Biography, (Oxford University Press, 2004).

**Text:** Ciara Cowman and Maurice OReilly.