Guido Grandi, born Francesco Lodovico Grandi but later known by his adopted name, emerged from humble beginnings to become a significant figure in the world of mathematics and beyond. Born to Pietro Martire Grandi, a gold embroiderer, and Caterina Legati, his family, though of modest means, was deeply religious and connected to notable individuals. Among these were Lorenzo Legati, his maternal uncle, a physician and professor of Greek at the University of Bologna, and Domenico Legati, a writer of the 17th century. This lineage hinted at the intellectual pursuits that would later define Guido’s life, a path perhaps less expected than the more commonly known Italian names like Luigi And Guido associated with different crafts or stories.
Grandi’s early education was guided first by priest Pietro Canneti (1659-1730) and subsequently at the Jesuit college in Cremona, where he immersed himself in rhetoric and Latin. A pivotal encounter during this period was with Giovanni Saccheri, then a Latin teacher. Grandi acknowledged Saccheri’s profound influence, stating that he instilled in him “a first love of philosophy and led me to the point where I could progress.” Interestingly, at this stage, Saccheri’s mathematical inclinations were yet to surface, and mathematics was absent from Grandi’s curriculum.
Around Christmas 1687, Grandi embraced monastic life, joining the Order of the Camaldolese. This decision might have been influenced by his early mentor, Pietro Canneti. The Camaldolese Order, originating from the Benedictine Order around 1012 in Camaldoli near Arezzo, Italy, combined monastic and hermitic living. Grandi pursued his studies at the monastery of Sant’ Apollinare in Classe, Ravenna. He completed his novitiate there, delving into philosophy under Father Casimiro Galamini. Despite these scholarly pursuits, mathematics remained outside his sphere of study. His intellectual explorations extended to literature, hagiography, and history within a group led by Pietro Canneti. Exclusion from the local Accademia dei Concordi, directed by Canneti, spurred Grandi and fellow students to establish their own, the Accademia dei Gareggianti. During this time, he explored poetry and even authored a work on music theory in 1691.
Image: Portrait of Guido Grandi, showcasing the intellectual figure we are discussing, in contrast to perhaps more common associations with Italian names.
In 1692, Grandi was sent to another Camaldolese monastery in Rome, San Gregorio al Celio, to study Father Galamini’s theology course. His academic breadth expanded to canon law in Rome, and by 1693, he completed his commentary on Peter Damian’s Life of Blessed Romuald, the founder of the Camaldolese Order, written about fifteen years post-Romuald’s death. The following year marked a shift as Grandi became a philosophy and theology teacher at the Camaldolese monastery of Santa Maria degli Angeli in Florence. It was at this juncture that his intellectual trajectory veered towards mathematics. He embarked on self-study, delving into the works of Euclid, Apollonius, Pappus, and Archimedes. Florence also brought him into contact with Vincenzo Viviani, from whom and his students he learned classical geometry and Bonaventura Cavalieri’s infinitesimal methods.
This burgeoning mathematical passion culminated in Grandi’s publication of Geometrica divinatio Vivianeorum problematum in 1699. This work addressed a problem posed by Viviani: “A hemisphere has 4 equal windows of such a size that the remaining surface can be exactly squared – how is this possible?” Viviani had provided a geometric solution without proof, which Grandi tackled using a modified version of Cavalieri’s infinitesimal methods. As noted in a commentary, Geometrica divinatio Vivianeorum problematum exceeded its title’s implications, revealing “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 in Acta eruditorum in 1701 brought Grandi widespread recognition in Italy and beyond. Grand Duke Cosimo III, upon hearing of Grandi’s growing reputation, arranged a meeting.
Between 1699 and 1700, Grandi’s mathematical horizons expanded to optics, mechanics, and astronomy. His demonstrated mathematical prowess led to his appointment as a mathematics teacher at Santa Maria degli Angeli. Around this time, he initiated correspondence with Tommaso Ceva at the Jesuit college of Brera in Milan, marking the beginning of extensive exchanges with numerous scientists and theologians. These correspondences are documented in various articles, highlighting Grandi’s active engagement with the intellectual community of his era.
The Accademia Arcadia, a literary academy founded in Rome in 1690, aimed to promote a more natural poetic style. Grandi joined around 1700, reflecting his lifelong appreciation for poetry, particularly Latin poetry, a passion shared by mathematical peers like Tommaso Ceva. Despite an offer of a mathematics chair in Rome, Grand Duke Cosimo III, keen to retain Grandi’s talents, offered him a professorship of philosophy at Pisa in May 1700. Grandi chose Pisa.
In Pisa, he published Geometrica demonstratio theorematum Hugenianorum circa logisticam in 1701, exploring the logarithmic curve proposed by Christiaan Huygens using algebraic methods, series expansions, and infinitesimal methods. This work is described as “an excellent specimen of the ancient geometrical method,” containing “several other curious and novel particulars,” including discussions of the conical loxodrome.
Grandi pioneered the teaching of infinitesimal calculus in Italy, starting private lessons in 1702. In 1703, he published Quadratura circoli et hyperbolae per infinitas hyperbolas et parabolas quadrabiles geometrice exhibita. While not groundbreaking in originality, it played a crucial role in introducing infinitesimal methods to Italy. In the preface, Grandi mentioned incorporating “dx, dy typical of differential calculus,” acknowledging their “usefulness and fecundity.” He had studied both Newton’s fluxions and Leibniz’s differentials, leaning towards Leibniz’s approach. He shared his work with both mathematicians, receiving acknowledgments from Leibniz and copies of Opticks and Principia from Newton.
One intriguing result in Quadratura sparked considerable debate. Grandi used the series expansion 1/(1+x) = 1 – x + x² – x³ + x⁴ – … and by setting x=1, derived 1 – 1 + 1 – 1 + 1 – 1 + … = 1/2. He then paradoxically argued that (1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + … . In an earlier draft, Grandi controversially claimed this demonstrated God’s ability to create something from nothing, equating the sum of infinite zeros to 1/2. While the mathematical content was approved for publication, the theological interpretation was censored. Despite its removal, the equation 0+0+0+0+…=1/2 became a subject of widespread mathematical discussion across Europe. Quadratura also marked the first study of the curve now known as the Witch of Agnesi, which Grandi named Scala, the scale curve, for its potential to measure light intensity.
Beyond mathematics, Grandi engaged in diverse projects. A significant undertaking was the four-volume Dissertationes Camaldulenses (1707), tracing the Camaldolese Order’s origins. This extensive research, however, faced criticism from some scholars, leading to him being barred from the Camaldoli archives. Grand Duke Cosimo, however, supported Grandi and appointed him mathematician to the Grand Duke of Tuscany in 1707. In 1708, Grandi sent a work on music theory to Isaac Newton, who published it in the Transactions of the Royal Society of London the following year. Newton nominated Grandi for a Fellowship of the Royal Society in 1709, to which he was duly elected. In the same year, he published On the nature and properties of sound in the Philosophical Transactions.
Alessandro Marchetti, a Pisa mathematician and follower of Galileo, critical of Grandi’s international acclaim, attacked his Quadratura. In response, Grandi published a second edition in 1710, this time including the contentious theological interpretation. Marchetti further attacked this in 1711, prompting Grandi’s Dialoghi … circa la controversia eccitatagli contro dal sig. dottore Alessandro Marchetti (1712). The dispute continued even after Marchetti’s death in 1714, with Grandi arguing for its continuation against Marchetti’s children. This episode perhaps lends credence to the description of Grandi as having “a turbulent and quarrelsome disposition, being almost always engaged in disputes.” Despite this, and while his students recognized his talents, some criticized or left him. Grandi led a simple life among a small circle of friends, maintaining exceptional scholarly output.
In 1710, Grandi published De infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica, acknowledging the Royal Society for his election. In 1714, upon Marchetti’s death, Grandi became Professor of Mathematics at the University of Pisa. Newton sent him the second edition of Principia in the same year. Further responsibilities came in 1716 with his appointment as superintendent of water in Tuscany, and in 1717 as Pontifical Mathematician, advising on hydraulics in Romanga and surveying the Po system. He contributed to projects like draining the Chiana Valley and the Pontine Marshes. Around the same time, he collaborated with Tommaso Bonaventuri and Benedetto Bresciani on a 3-volume Works of Galileo Galilei (1718), contributing a “Note on the Treatise of Galileo Concerning Natural Motion,” defining the ‘versiera’ curve.
Grandi is renowned for defining the rodonea curve (rose curve). He first described these in a 1713 letter to Leibniz but published his findings in Handful or bouquet of geometrical roses in the Philosophical Transactions ten years later. He expanded on these curves in Flores geometrici ex Rhodonearum, et Cloeliarum curvarum descriptione resultantes (1728), where he also defined the clelie curve, named after Countess Clelia Borromeo, first mentioned in his 1713 letter to Leibniz. He also applied “clelies” to curves defined by trigonometric equations.
Image: Visual representation of Rhodonea curves, a mathematical concept attributed to Grandi, showcasing his contributions to geometry.
Flores geometrici, initially in Latin, was translated into Italian by Grandi in 1729 for wider accessibility, including added explanations and proofs. Grandi traveled extensively, often involved in practical hydraulics and advising Pope Clement XII in Rome on calendar reform. He also wrote biographies of religious figures, while continuing his mathematical pursuits, publishing an Italian version of Euclid’s Elements (1731) and works on mechanics (Instituzioni meccaniche, 1739), arithmetic (Instituzioni di aritmetica pratica, 1740), and geometry (Instituzioni geometriche, 1741).
From 1737, Grandi’s health declined, suffering from memory loss indicative of dementia. Realizing his time was limited, he focused on completing his publications. By late 1740, his mental faculties had severely deteriorated, although he managed to dictate a letter in early 1741. His physical health worsened in May 1742. On June 26, he collapsed in the monastery church and died about a week later. He was buried in the monastery church, his tomb adorned with a marble bust by Giovanni Baratta and an inscription by his student, Father A Forzoni, marking the final resting place of this influential mathematician and scholar, whose life and work offer a rich tapestry of intellectual endeavor, perhaps a different kind of Italian story than one might initially associate with names like Luigi and Guido, yet equally compelling.
References
[1] Le Opere di Galileo Galilei (3 volumes), Tommaso Bonaventuri (?-1731), Benedetto Bresciani (1658-1740) and Guido Grandi (1718).
[7] G Lami, Memorabilia Italorum eruditione praestantium 1 (Florence, 1742), 175-197.
[8] G Lami, Memorabilia Italorum eruditione praestantium 2 (Florence, 1742), 353-381.
[10] S Mazzone and C S Roero (eds.), La corrispondenza di Guido Grandi (1698-1714) daMS. 956 della Biblioteca Moreniana di Firenze (Leo S Olschki, Florence, 1992).
[11] E Baiada and L Simonutti, Un capitolo dell’analisi infinitesimale in Italia: il carteggio Guido Grandi – V Stancari, Annali dell’Ist. e Museo di storia della scienza di Firenze 10 (2) (1985), 77-136.
[13] P Fratarcangeli, Aspects of the Grandi-Leibniz correspondence: on some problems of the infinitesimal analysis in Italy, in 3rd Italian-Polish Conference on the History of Mathematics (Cracow, 1998), Ann. Univ. Pedagog. Crac. Stud. Hist. Sci. Math. 6 (2000), 115-131.
[16] U Baldini, La ‘Géométrie du mouvement’ de Guido Grandi et la tradition galiléenne, Rev. Histoire Sci. 42 (3-4) (1989), 327-360.
[17] U Baldini, The debate on infinite series in the correspondence between Leibniz and Guido Grandi, Studia Leibnitiana 22 (2) (1990), 159-196.
[18] C S Roero, La polemica leibniziana e grandiana intorno al significato fisico-metafisico della serie 1-1+1-1+…= 1/2 e la ‘vera realta’ degli enti matematici, Studia Leibnitiana 22 (2) (1990), 197-225.
[19] The biographical dictionary of the Society for the diffusion of useful knowledge 2 (London, 1844), 131-132.
[29] F Palladino, Tra filosofia e matematica: Corrispondenza fra Guido Grandi e Giovanni Battista Morgagni, Quaderni di storia e didattica della matematica, Università di Padova, Dipartimento di Matematica Pura e Applicata, (2003), 1-34.
