Here’s a romantic tale, if ever there was one:

In 18^{th} century Italy, when there were serious impediments to the education of women, a young, attractive and extremely caring woman achieved academic success across a range of subjects. And, under the watchful eye of her father, who was a professor of mathematics, she devoted her late teens and twenties to mathematical study and produced an innovative textbook that was a model of clarity – and which almost served as a summary of 18^{th} century mathematical knowledge.

In her early thirties, exhausted by such labours, and affected by the death of her father, she gave up mathematics forever. And her religious convictions caused her to spend the last fifty years of her life tending to the sick and the poverty stricken. Years of self-sacrifice yielded spiritual rewards, but resulted in her last years being spent in relative penury and obscurity.

Sadly, despite a life governed by altruistic commitment, the fates conspired to permit an error in the English translation of her two-volume mathematical tract, meaning that Maria Agnesi’s name has since been inextricably linked with that of a witch.

For those seeking validation of this tale, there is no better place to start than with this biography by Antonella Cupillari. It is largely based upon the earliest known biography of Agnesi, written by a family friend, Antonio Frisi, in the year of her death (1799). This is about eighty pages long, and has been translated for the first time into English by Cupillari. But, as she says, Frisi’s encomiastic description of Agnesi’s life and achievements establishes the need for a more objective biographical appraisal.

Cupillari gives a good account of the social and cultural milieu of 18^{th} century Italy, and she explains why the education of women was such a contentious issue. However, in Agnesi’s case, there was the advantage of her father having been a professor of mathematics who provided her with a sound education. More than that, Cupillari describes Agnesi’s short mathematical career and considers how and why she embarked upon it.

Much of the Frisi biography portrays the last fifty years of Agnesi’s life spent in terms of helping the poor and the infirm, to which she was driven by her religious convictions. In fact, by making reference to two earlier biographies (both based upon Frisi) and by making use of other primary sources, it seems that Cupillari has greatly extended available knowledge of Maria Agnesi’s life and work

Agnesi did become famous in her lifetime as the author of the first textbook to provide a unified treatment of algebra, Cartesian geometry and calculus (*Instituzioni Analitiche ad uso della Gioventu Italiana,* 1748). Not only did that book improve upon l’Hôpital’s presentation of Leibnizian calculus, but it also introduced much of the work done by Euler and Descartes.

Written in Italian, it was obviously intended to disseminate the latest mathematical ideas amongst a readership other than the Latin literate scholars of the day; but how many copies were sold, and the extent to which the book achieved its intended goal, remain open questions. Nonetheless, the book received the prestigious approval of the French royal academy of sciences, whose report has been included by Cupillari.

That report gives a good description of the scope and nature of the contents of Agnesi’s two-volume work, and it recognises that, amongst a vast amount of other material, all the ideas of Descartes have been presented ‘with order, clarity and precision’. The report also recognises the difficulties faced by Agnesi in the compilation of this work and it says ‘A lot of art and sagacity were necessary to collect, as it has been done, with methods almost uniform, the discoveries spread in the Works of modern Geometers, and often explained with methods very different from one another’.

Although Cupillari says that the Frisi biography of 1799 is the core of her book, the last quarter of it can be considered no less important. This provides a range of translated excerpts from Agnesi’s textbook, ranging from the introduction to the rules of elementary algebra to her treatment of differential equations of the first degree. These have been reproduced with great clarity and one gets a very good idea of the style and substance of Agnesi’s approach.

Although in the Foreword to this book Cupillari says that Agnesi combined the Newtonian and Lebinizian ways of doing the calculus, this did not come across at all clearly. I checked some of Cupillari’s translated excerpts against John Colson’s English translation, published in 1801, and I found very good agreement between the two. But Colson represented the calculus using Newton’s fluxion notation, as opposed to Agnesi’s use of Leibniz’s notation. For example, where Colson would write:

Cupillari has:

25. Now that the fundamentals of this calculus are established, I shall pass to the manners, or rules for differentiating formulas. In the first place, let’s take the difference of several quantities added together, or subtracted one from the other, for example a + x + z + y + y – u. Since the difference of x is dx, of z is dz, etc, and of the constant a is nothing, and considering every quantity increased of its difference with that sign that belongs to it, the proposed formula will change into the following one a + x + dx + z + dz + y + dy – u – du. If we subtract from this the first one will produce the residue dx + dz + dy – du, which is exactly the increase of the proposed quantity, that is, its difference.

As for Colson, it was he who, in the process of translating Agnesi’s work from Italian into English, mistook the word ‘l’aversiera’ (wife of the devil) for the word ‘la versiera’ (free to move). Hence the current misnomer for the curve 1 – x(1 + y^{2}) = 0, which had been studied by Fermat and other mathematicians long before it came to the attention of Agnesi.

One of the concluding remarks made by Cupillari concerns the status of Agnesi as a mathematician. She says that, in the course of the last century, most mathematicians have shared the opinion that Agnesi was not an original discoverer, but an excellent writer of a mathematical textbook for her times. But it should be said that the task of collating, and synthesising, so much mathematics from inscrutable original sources, and with no prototypes with which to form comparisons, must count as a highly original achievement.

I enjoyed reading this biography by Antonella Cupillari, but it was not easy going. This was because the narrative is somewhat fragmented due to the inclusion of many quotations and the listing of many biographical sources. Also, I found that the Frisi biography too glowing to ring true — rather like one of those eulogistic epitaphs on Jacobean and Tudor monuments frequently found in English churches.

Finally, I think the book would have benefited by the inclusion of a clearer description of developments in European mathematics that preceded, and led to the writing of, Maria Agnesi’s ground-breaking textbook.

Peter Ruane now spends most of his time exploring the wonderful world of algebraic curves and recommends the following link for the meaning of ‘la versiera’: http://mathworld.wolfram.com/WitchofAgnesi.html.