Guido Grandi, born Francesco Lodovico Grandi but later known as Guido upon entering the Camaldolese Order, emerged as a significant figure in the world of 18th-century mathematics. His journey from a religiously inclined family of modest means to becoming a celebrated mathematician, theologian, and philosopher is a testament to his intellectual prowess and diverse interests.
Grandi’s parents, Pietro Martire Grandi, a gold embroiderer, and Caterina Legati, instilled in him a religious upbringing. His family boasted notable figures, including his maternal uncle Lorenzo, a physician and professor of Greek at the University of Bologna, and the 17th-century writer Domenico Legati. His early education began under the tutelage of priest Pietro Canneti (1659-1730) and continued at the Jesuit college in Cremona, where he delved into rhetoric and Latin. A pivotal encounter during this period was with Giovanni Saccheri, then a Latin teacher, who, in Grandi’s words, “…had the goodness to instil in me a first love of philosophy and led me to the point where I could progress.” Interestingly, Saccheri at this time had not yet ventured into mathematics, and Grandi’s curriculum was devoid of mathematical studies.
Around Christmas 1687, Grandi embraced monastic life, joining the Order of the Camaldolese, possibly influenced by his initial mentor, Pietro Canneti. The Camaldolese Order, originating from the Benedictine Order around 1012 in Camaldoli near Arezzo, Italy, combined monastic and hermitical living. Grandi pursued his studies at the Sant’ Apollinare monastery in Classe, Ravenna. He completed his novitiate at Sant’ Apollinare, studying philosophy under Father Casimiro Galamini, yet mathematics remained absent from his academic pursuits. His literary interests flourished as he studied literature, hagiography, and history within a group led by Pietro Canneti. Excluded from the local Accademia dei Concordi, Grandi and fellow students established the Accademia dei Gareggianti, where he explored poetry and even authored a work on music theory in 1691.
In 1692, Grandi relocated to the San Gregorio al Celio monastery in Rome, another Camaldolese institution, to study Father Galamini’s theology course. He further broadened his academic scope to include canon law and in 1693 completed his commentary on Peter Damian’s Life of Blessed Romuald, the revered founder of the Camaldolese Order, written about fifteen years after Romuald’s passing.
The year 1694 marked Grandi’s foray into teaching, as he became an instructor of philosophy and theology at the Camaldolese monastery of Santa Maria degli Angeli in Florence. It was during this Florentine period that Grandi’s intellectual trajectory took a significant turn towards mathematics. He embarked on self-study, immersing himself in the works of Euclid, Apollonius, Pappus, and Archimedes, laying the foundation for his future mathematical endeavors. Florence also facilitated a crucial meeting with Vincenzo Viviani, from whom, along with his students, Grandi learned the intricacies of classical geometry and Bonaventura Cavalieri’s infinitesimal methods.
This newfound mathematical passion culminated in Grandi’s first publication in 1699, Geometrica divinatio Vivianeorum problematum (“Geometric solutions of Viviani’s problems”). This work addressed a geometrical challenge 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?
While Viviani had provided a geometric solution, he lacked a formal proof. Grandi successfully tackled this problem by adapting Cavalieri’s infinitesimal methods. As noted in a commentary [19], Geometrica divinatio Vivianeorum problematum exceeded expectations, encompassing “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 propelled Grandi to international recognition. Grand Duke Cosimo III, upon hearing of Grandi’s rising fame, arranged a meeting. Between 1699 and 1700, Grandi expanded his mathematical explorations into optics, mechanics, and astronomy. His demonstrated mathematical acumen led to his appointment as a mathematics teacher at the Santa Maria degli Angeli monastery. Around this time, Grandi initiated correspondence with Tommaso Ceva at the Jesuit college of Brera in Milan, marking the beginning of his extensive network of scientific and theological exchanges, documented in various scholarly articles [7, 8, 10, 11, 13, 16, 17, 18, 29].
Grandi’s intellectual pursuits extended beyond mathematics. He engaged with the Accademia Arcadia, a literary academy founded in Rome in 1690, advocating for a more natural poetic style. Poetry, particularly Latin poetry, remained a lifelong passion for Grandi, shared by his mathematical peers like Tommaso Ceva. Despite receiving an offer for 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 over Rome.
In Pisa, Grandi published Geometrica demonstratio theorematum Hugenianorum circa logisticam in 1701, focusing on the logarithmic curve proposed by Christiaan Huygens. He employed generalized algebraic methods, series expansions, and infinitesimal techniques in his analysis. This work was hailed as an “excellent specimen of the ancient geometrical method” [19], containing “several other curious and novel particulars,” including discussions on the conical loxodrome. From 1702, Grandi privately taught infinitesimal calculus, becoming the first to introduce these methods in Italy. His 1703 publication, Quadratura circoli et hyperbolae per infinitas hyperbolas et parabolas quadrabiles geometrice exhibita, while not entirely original, played a crucial role in disseminating infinitesimal methods within Italy. In the preface, Grandi noted his inclusion of “dx, dy typical of differential calculus, and their methods of being differentiated and added,” acknowledging their “usefulness and fecundity.” He referenced De L’Hôpital for a more comprehensive explanation of these methods.
Grandi studied both Newton’s fluxions and Leibniz’s differentials, favoring Leibniz’s approach. He shared his work with both mathematicians, receiving acknowledgments from Leibniz and copies of Opticks and Principia from Newton. A particular result in Quadratura generated significant debate. Grandi used the series expansion:
$Largefrac{1}{1+x}normalsize = 1 – x + x^{2} – x^{3} + x^{4} – …$
Substituting x=1, he arrived at:
$1 – 1 + 1 – 1 + 1 – 1 + … = largefrac{1}{2}normalsize$
However, he also observed that (1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + 0 +… . In an initial draft, Grandi controversially claimed that the sum of infinite zeros equaling 1/2 demonstrated God’s ability to create the world from nothing. Censors permitted the mathematical content but demanded the removal of this theological interpretation. Grandi reluctantly complied, but the equation $0+0+0+0+… = largefrac{1}{2}normalsize$ sparked widespread discussion among European mathematicians. Quadratura also featured the first study of the curve known today as the Witch of Agnesi, which Grandi named Scala, or scale curve, due to its potential application in measuring light intensity.
While Grandi remained engaged with the latest mathematical advancements, he also pursued other scholarly projects. A significant undertaking was the four-volume Dissertationes Camaldulenses (1707), tracing the origins of the Camaldolese Order. Despite extensive research, his treatment faced criticism, and he was denied access to the Camaldoli archives. Grand Duke Cosimo, however, supported the publication and appointed Grandi as mathematician to the Grand Duke of Tuscany, Cosimo III de’ Medici, in 1707, marking his first official mathematical position. In May 1708, he 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, and he was duly elected. 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.
Alessandro Marchetti (1633-1714), a Pisan mathematician and follower of Galileo, publicly criticized Grandi’s Quadratura, attempting to undermine his growing international reputation. In response, Grandi published a second edition in 1710, this time including his controversial theological interpretation. Marchetti launched a further attack 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 keen to prolong the argument, claiming Marchetti’s children were perpetuating their father’s stance. This episode perhaps contributed to the assessment 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.” His contentious nature sometimes alienated even his students, despite their recognition of his talents. Yet, Grandi maintained a simple life within a small circle of friends, and his scholarly output remained prolific.
In 1710, Grandi published De infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica, expressing gratitude to the Royal Society for his election. Upon Marchetti’s death, Grandi succeeded him as Professor of Mathematics at the University of Pisa in 1714. In the same year, Newton sent him a copy of the second edition of Principia. Grandi took on additional responsibilities in 1716 as superintendent of water in Tuscany, and in 1717 as Pontifical Mathematician, advising on hydraulics in the Romanga region and participating in a survey of the Po system. He was involved in projects such as draining the Chiana Valley and the Pontine Marshes. Another major project was his collaboration with Tommaso Bonaventuri and Benedetto Bresciani on a 3-volume edition of Works of Galileo Galilei (1718) [1]. Grandi contributed a “Note on the Treatise of Galileo Concerning Natural Motion,” introducing the term ‘versiera’ (from the Latin ‘sinus versus’) for the curve later known as the Witch of Agnesi.
One of Grandi’s most enduring mathematical legacies is his definition of rhodonea curves, or rose curves. He first described these curves in a 1713 letter to Leibniz but published his findings in “Handful or bouquet of geometrical roses” in the Philosophical Transactions of the Royal Society of London a decade later. He further elaborated on these curves in Flores geometrici ex Rhodonearum, et Cloeliarum curvarum descriptione resultantes (1728), where he also defined clelie curves, named after Countess Clelia Borromeo. Clelie curves, initially mentioned in the same 1713 letter to Leibniz, are defined on a sphere by the relationship between longitude ($theta$) and colatitude ($phi$) as $theta = m phi$, where m is a constant. Grandi also applied the term “clelies” to curves defined by trigonometric equations involving sine functions.
Recognizing that Flores geometrici was in Latin, Grandi produced an Italian version in 1729 to reach a wider audience, adding further explanations and proofs. His later years involved extensive travel, practical hydraulic work, and advising Pope Clement XII on calendar reform in Rome. He also undertook biographical writings of religious figures. Mathematics remained a central focus, with publications including an Italian edition 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 accelerated his publication efforts. By late 1740, his cognitive abilities had severely diminished, although he managed to dictate a letter in early 1741. His physical health deteriorated in May 1742, and on June 26, he collapsed in the monastery church. Guido Grandi died approximately a week later and was buried in the monastery church, his tomb adorned with a marble bust by Giovanni Baratta (1670-1747) and an inscription by his student, Father A Forzoni, marking the final chapter in the life of this multifaceted 18th-century scholar.
References
[1] Galileo Galilei, Le Opere di Galileo Galilei (3 volumes, Florence, 1718).
[7] U Baldini, L’attività scientifica di Guido Grandi nel carteggio con Giovanni Bernoulli (1703-1713), Boll. Storia Sci. Mat. 2 (2) (1982), 55-99.
[8] U Baldini, La corrispondenza scientifica di Guido Grandi (1698-1714). Catalogo, indici e regesti, Quaderni di storia e didattica delle scienze (2) (Urbino, 1985), 1-183.
[10] U Baldini and P Nardi, Alle origini del calcolo infinitesimale in Italia. Guido Grandi, i Gesuiti e la controversia sul metodo delle indivisibili (1698-1715), Arch. Internat. Hist. Sci. 38 (120-121) (1988), 5-77.
[13] G Lami, Lettere di uomini illustri del secolo XVIII I (Florence, 1739), 1-64.
[16] M Panza, La ‘vera Metafisica’ del calcolo infinitesimale: l’analisi dei ‘Principj’ di Guido Grandi (1703), in S Simoli (ed.), La filosofia delle scienze di Guido Grandi (Florence, 1994), 119-207.
[17] J Pier, Unbekannte Briefe von Guido Grandi an Johann Bernoulli aus den Jahren 1700-1737 in der Burgerbibliothek Bern, Verhandlungen der Naturforschenden Gesellschaft in Basel 85 (1-2) (1976), 53-67.
[18] J Pier, Die Korrespondenz von Johann Bernoulli mit Guido Grandi, Verhandlungen der Naturforschenden Gesellschaft in Basel 86 (1) (1977), 75-118.
[19] G Sarton, Guido Grandi of Pisa (1671-1742), Isis 37 (3/4) (1947), 198-217.
[29] C S Truesdell, Leonard Euler, supreme geometer (1707-1783), in The tercentenary of Leonhard Euler (Birkhäuser, Basel, 1985), 1-45.
