**Guido Grandi**'s parents were Pietro Martire Grandi, an embroiderer in gold, and Caterina Legati. Guido was not given that name when he was baptised, rather he was given the name Francesco Lodovico but changed his name to Guido when he entered the Order of the Camaldolese. The family were of modest means and lived their lives in a religious manner. Guido's family had a number notable people in it, perhaps the most significant being Lorenzo, a maternal uncle, who was a physician and professor of Greek at the University of Bologna, and the 17th century writer Domenico Legati.

Grandi was educated first by the priest Pietro Canneti (1659-1730) and then at the Jesuit college in Cremona. At the College he studied rhetoric and Latin. An important influence on him was Giovanni Saccheri who he met at this time. Saccheri was teaching Latin but, Grandi wrote, that:-

... he had the goodness to instil in me a first love of philosophy and led me to the point where I could progress.

At this stage Saccheri was not interested in mathematics and there was no mathematics in the courses that Grandi studied. He became a member of the Order of the Camaldolese around Christmas 1687, perhaps being influenced to join this order by his first teacher Pietro Canneti. The Camaldolese Order, an offshoot of the Benedictine Order, was founded about 1012 at Camaldoli near Arezzo, Italy. A monastery and hermitage formed one unit in this Order and Grandi studied at the monastery of Sant' Apollinare in Classe, Ravenna. He completed his novitiate at Sant' Apollinare studying philosophy with Father Casimiro Galamini, but still had no contact with mathematics. He also studied literature, hagiography and history in a group led by Pietro Canneti. Students were excluded from the local Accademia dei Concordi (a cultural institution founded around 1580), the local branch being directed by Pietro Canneti. So, along with fellow students, he founded the Accademia dei Gareggianti. He studied poetry and wrote a work on the theory of music in 1691. Then, in 1692, he was sent to another monastery of the Camaldolese Order in Rome, namely San Gregorio al Celio, where he studied Father Galamini's theology course. He also studied canon law in Rome and in 1693 completed writing his commentary on Peter Damian's *Life of Blessed Romuald*, the founder of the Camaldolese Order. Peter Damian wrote the Life of Blessed Romuald about fifteen years after Romuald's death.

The following year, Grandi became a teacher of philosophy and theology at the Camaldolese monastery of Santa Maria degli Angeli in Florence. Up to this time he had shown little interest in mathematics but now his thoughts turned in that direction and he studied on his own the works of Euclid, Apollonius, Pappus and Archimedes. In Florence he met with Vincenzo Viviani and learned from him and his students the methods of classical geometry and also the infinitesimal methods of Bonaventura Cavalieri. Grandi published *Geometrica divinatio Vivianeorum problematum* in 1699. In this work he discussed Viviani's question:

A hemisphere has4equal windows of such a size that the remaining surface can be exactly squared - how is this possible?

Viviani had answered the question using geometry but had not given a proof. Grandi was able to solve the problem using a modification of Cavalieri's infinitesimal methods. The commentary in [19] explains that *Geometrica divinatio Vivianeorum problematum* is:-

... a work which contains more than its title would lead us to expect, and in which the author remarks many other curiosities in geometry of the same kind, and among others, a portion of the surface of a right cone which can be squared.

A review of Grandi's work in *Acta eruditorum* in 1701 brought him recognition in Italy and in other countries. Grand Duke Cosimo III learnt of Grandi's fame and arranged to meet with him around this time. Having learnt much geometry, between 1699 and 1700 Grandi began to look at applications to optics, mechanics, astronomy. Having now shown a deep understanding of mathematics, Grandi began to teach mathematics at the monastery of Santa Maria degli Angeli. Also around this time, Grandi also began to correspond with Tommaso Ceva who was at the Jesuit college of Brera in Milan. In fact Grandi was a vigorous correspondent, exchanging letters with many scientists and theologians. Many of the articles in the references discuss Grandi's correspondence, see for example [7], [8], [10], [11], [13], [16], [17], [18], and [29].

The Accademia Arcadia was a literary academy which was founded in Rome in 1690 to promote a more natural, simple poetic style and around 1700 Grandi attended this Academy. Poetry, especially Latin poetry, was a favourite with Grandi throughout his life. His other mathematical friends like Tommaso Ceva were also lovers of poetry. Grandi was offered a chair of mathematics in Rome but the Grand Duke Cosimo did not want to lose his star mathematician and in May 1700, the Grand Duke Cosimo offered Grandi a position of professor of philosophy at Pisa. Grandi chose to go to Pisa rather than to Rome. In Pisa, he published *Geometrica demonstratio theorematum Hugenianorum circa logisticam* in 1701. In this he studied the logarithmic curve proposed by Christiaan Huygens, using generalised algebraic methods, series expansions and infinitesimal methods. This is [19]:-

... an excellent specimen of the ancient geometrical method; in which piece also, as well as in his letter to the Jesuit Ceva, which follows it, are found several other curious and novel particulars.

He discussed the conical loxodrome, the curve that cuts the generators of a cone of revolution in a constant angle.

In fact Grandi taught methods of the infinitesimal calculus from 1702 in private lessons he was giving; he was the first to do so in Italy. In 1703, he published the book *Quadratura circoli et hyperbolae per infinitas hyperbolas et parabolas quadrabiles geometrice exhibita* which, although it does not contain anything particularly original, nevertheless was important in introducing infinitesimal methods into Italy. Grandi writes in the Preface:-

But I have here and there also inserted the dx, dy typical of differential calculus, and their methods of being differentiated and added. Thus had I been able to introduce them in my previous pamphlets too! But then, the secrets of that method had been inaccessible to me, while now, their usefulness and fecundity having been proven, why not insert them among the other methods I am familiar with? Also, the significance of the symbols is very clear, because it only signifies an infinitely small difference between the same x and y, and you will easily find the very rules of calculus if you observe and peruse this tract carefully - unless you want to recourse to the illustrious De L'Hôpital who explains them in a more complete way in his tract on the infinitely small.

Grandi had studied Newton's fluxions and Leibniz's differentials and used both approaches although he favoured the approach by Leibniz. He sent copies of his work to both these mathematicians and received thanks from Leibniz and copies of the *Opticks* and the *Principia* from Newton. One of Grandi's results in the *Quadratura* caused a lot of interest. He used the series expansion

^{1}/_{1+x}= 1 -x+x^{2}-x^{3}+x^{4}- ...

and putting *x* = 1, obtained 1 - 1 + 1 - 1 + 1 - 1 + ... = ^{1}/_{2}. However, he argued that (1 - 1) + (1 - 1) + (1 - 1) + ... = 0 + 0 + 0 + 0 +... . In the first draft of the work, Grandi claimed that since the sum of infinitely many 0s equaled ^{1}/_{2} he had proved that God could create the word out of nothing. The censor allowed the mathematics to be published but required the removal of the comment that it showed God could create the word out of nothing. Grandi reluctantly agreed to remove the comment but many mathematicians across Europe discussed Grandi's result that 0 + 0 + 0 + 0 +... = ^{1}/_{2}. The *Quadratura* also contains the first study of the curve which today is known as the Witch of Agnesi. Grandi gave the curve the name Scala, the scale curve, because it can serve as a measure of light intensity.

Now although Grandi continued to study the latest developments in mathematics, he also undertook other projects. One major project was the publication of the four-volume *Dissertationes Camaldulenses* (1707) in which he traced the origins of the Order of the Camaldolese. This work was the result of extensive research undertaken by Grandi. However, several scholars were unhappy with Grandi's treatment and he was forbidden to use the archives at Camaldoli. The Grand Duke Cosimo supported Grandi's publication of the *Dissertationes* and decided that he would move to protect him. The result was Grandi's first mathematical appointment which came in 1707 when he became mathematician to the Grand Duke of Tuscany, Cosimo III de' Medici. In May 1708 he sent a work on the theory of music to Isaac Newton who published it in the *Transactions* of the Royal Society of London in the following year. In 1709 he was proposed by Newton for a Fellowship of the Royal Society and duly elected as a fellow. In the same year, shortly after his election, he published *On the nature and properties of sound* in the *Philosophical Transactions *of the Royal Society of London.

Now Alessandro Marchetti (1633-1714) was a mathematician at Pisa who was a follower of Galileo. He did not like Grandi's international fame and, in an attempt to discredit him, criticised his *Quadratura*. As a result, Grandi published a second edition in 1710 and this time he was allowed to include the comment that he had proved that God could create the word out of nothing. Marchetti then published an attack on this second edition in 1711 to which Grandi responded with* Dialoghi ... circa la controversia eccitatagli contro dal sig. dottore Alessandro Marchetti *(1712). The controversy continued and, even after Marchetti's death in 1714, Grandi wanted to continue the argument since he argued that Marchetti's children had supported their father and so the argument should continue. Perhaps this episode justifies the comment that [19]:-

Grandi it seems was of a turbulent and quarrelsome disposition, being almost always engaged in disputes on various subjects, geometrical, theological, metaphysical or philological.

In fact, his quarrelsome nature meant that even his students, while recognising his talents and merits, nevertheless criticised him or left him. However, he lived a simple retiring life among a small circle of friends. His output of scholarly works continued to be exceptional. In 1710, Grandi published *De infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica* in which he thanked the Royal Society for electing him. In 1714 Grandi was appointed Professor of Mathematics at the University of Pisa, a position which became vacant on the death of Marchetti and, in the same year, Newton sent him a copy of the second edition of the *Principia*. He took on further duties in 1716 when he was appointed superintendent of water in Tuscany. In the following year, he was also appointed Pontifical Mathematician to deal with issues of hydraulics in the Romanga, and he helped to carry out an extensive survey of the Po system. In this capacity, he was involved with a number of projects such as ones to drain the Chiana Valley and the Pontine Marshes. Another major project which he undertook around the same time was collaborating with Tommaso Bonaventuri (?-1731) and Benedetto Bresciani (1658-1740) on publishing a 3-volume *Works of Galileo Galilei* (1718) [1]:-

Grandi contributed to it a "Note on the Treatise of Galileo Concerning Natural Motion," in which he gave the first definition of a curve he called the 'versiera'(from the latin 'sinus versus').

One of the results for which Grandi is best known today is his definition of the *rodonea curve*. Rodonea is the Latin for rose and Grandi first defined these curves in December 1713 in a letter he wrote to Leibniz. He did not publish his results on these curves until ten years after this when he published the paper *Handful or bouquet of geometrical roses* in the *Philosophical Transactions* of the Royal Society of London. He expanded the material on these curves in *Flores geometrici ex Rhodonearum, et Cloeliarum curvarum descriptione resultantes *(1728). In the same work he defines the *clelie curve* which, like the rodonea curve, he had first mentioned in a letter to Leibniz in 1713. He named the curve after Countess Clelia Borromeo and dedicated his book to her. If the longitude and colatitude of a point *P* on a sphere is denoted by *θ* and *φ* and if *P* moves so that *θ* = *m* *φ*, where *m* is a constant, then the locus of *P* is a clelie. Grandi also applied the term "clelies" to the curves determined by certain trigonometric equations involving the sine function

asinθ=bsinmφ

asinθ=a-bsinmφ

The *Flores geometrici* was, as can be seen from the title, a Latin text. However, Grandi wanted this work to be more widely read than just by scholars so, in 1729, he produced an Italian version of the work which contained additional explanations and proofs.

Grandi now travelled much, often undertaking practical work on hydraulics and was sometimes in Rome where he advised the Pope Clement XII on calendar reform. He also undertook projects writing biographies of religious figures. However, he did not neglect mathematics, publishing an Italian version of Euclid's *Elements* in 1731 and a series of works on mechanics *Instituzioni meccaniche* (1739), arithmetic *Instituzioni di aritmetica pratica* (1740) and geometry *Instituzioni geometriche *(1741). However, by this time he was suffering from severe health problems. From 1737 he began to suffer from memory loss, clearly a form of dementia, and realising that he did not have too long left, he hurried to produce his final publications. By late 1740 his mind had almost completely gone although he still managed to dictate a letter in the early part of 1741. In May 1742 his physical health also became a serious issue and on 26 June he collapsed in the church attached to the monastery in which he lived. He died about a week later and was buried in the monastery church in a tomb with a marble bust by the sculptor Giovanni Baratta (1670-1747) and an inscription by Father A Forzoni, one of Grandi's students.

**Article by:** *J J O'Connor* and *E F Robertson*