The article examines the figure of the architect at work in Renaissance Italy, when a major change occurred in the practice of design with the spread of arithmetic. This deep scientific, technical, methodological, and cultural shift involved the image of the architect and his profession, his relationship with the patron, as well as the cultural conception of architecture.

The essay, crossing disciplinary boundaries, analyses some technical aspects of architectural design in early modern Italy only marginally investigated. If proportional systems and architecture’s theoretical questions have been amply studied, the practical culture, the daily professional practice and its working tools, such as the operative arithmetic actually known to architects, have been only sporadically analysed.

During the Renaissance, especially in Italy, an important development of mathematics occurred and arithmetic was clarified and simplified so to allow its diffusion, but at the same time those disciplines remained essentially despised by aristocratic and intellectual elites. What was the architects’ role in this moment of deep change? Which was the arithmetic usually employed by them in the design process? When did Hindu-Arabic numbers and fractions became familiar in the field of architecture? In the secular battle between geometry and arithmetic, which system was used in what professional cases?

The essay illustrates how architects with different backgrounds responded to this change, through a comparative analysis of all the architectural drawings containing numbers and calculations made by Michelangelo Buonarroti (1475–1564), Baldassarre Peruzzi (1481–1536), and Antonio da Sangallo the Younger (1484–1546).

This article examines the figure of the architect at work in Renaissance Italy, when a major change occurred in the practice of design with the spread of arithmetic. This deep scientific, technical, methodological, and cultural shift, during which competing models coexisted over many decades in the fight between tradition and innovation, involved the image of the architect and his profession, his relationship with the patron, and the cultural conception of architecture. As is well known, proportional systems are an essential part of the design process in this period, and have been, along with architecture’s theoretical questions, amply studied (see, for instance,

In the Renaissance, especially in Italy, there was an important development in the study of mathematics, and algebra was clarified and simplified enough to allow for its wider diffusion (

Hindu-Arabic numerals were developed in India, and arrived in the West in the 12th century through Middle Eastern authors, such as the Persian al-Khwarizmi.

Particularly in Siena, applied science was notably developed by well-known masters and directly funded by the municipality as early as the 13th century (

Bearing in mind the profound difference between the speculative mathematics of the scientists and the operative arithmetic commonly employed by professionals, did architects have an active role in the development and diffusion of algebra? What degree of confidence did they have in numbers, and how much arithmetic did they know and regularly employ in daily practice? In the century-old battle between geometrical models and numerical values, which system was most frequently adopted, and under what circumstances?

The systematic analysis of Peruzzi’s drawings, examined alongside those of Michelangelo and Sangallo, shows that his are by far the most rich in numbers and calculations: in Sangallo’s case, they appear occasionally, while in Michelangelo’s they are practically nonexistent. This observation indicates immediately the varying degree of confidence each architect had in the use of numbers. A drawing, for the plan of San Domenico in Siena (Fig.

Baldassarre Peruzzi, Plan for the church of San Domenico in Siena, Firenze, Gabinetto dei disegni e stampe degli Uffizi, U 545Ar, detail.

Moreover, it is worth noting that he used Roman numerals exclusively in presentation drawings, and a good example of this habit is the well-known folio U 368Ar for Palazzo Massimo alle Colonne.

A more in-depth observation of Peruzzi’s calculations allows an evaluation of his arithmetic ability and demonstrates how he moved with agility through long sums and single-digit multiplication. In a typical case for an architect-surveyor, such as calculating the surface of city walls — here, those of Orbetello (Fig.

Baldassarre Peruzzi, Survey of the fortifications of Orbetello, Firenze, Gabinetto dei disegni e stampe degli Uffizi, U 361Av, detail.

Together with the abundant calculations in the Peruzzi’s drawings, easy operations also appear, such as additions and elementary subtraction like 120 – 6 = 114 in the aforementioned folio, which seem to indicate a clear preference for written rather than mental calculation.

More difficult single or double-digit multiplications, solved the same way as we do today, that is, with the placement-system, are found in many drawings, such as U 453Av, containing the plan of a house and various calculations. Here, again, operations that would seem to be automatic, like 100 × 83, are solved at length, and there is even an error, perhaps due to distraction or Peruzzi’s failing eyesight: in multiplying 4183 × 100 he mistook the 8 for a 6. The operations get more complicated when he needed to multiply with fractions, but Peruzzi managed to do it, even if some calculations were done with approximations. In Figure

Baldassarre Peruzzi, Studies for Saint Peter’s in Rome, Firenze, Gabinetto dei disegni e stampe degli Uffizi, U 629Av, detail.

There is a surprise when one moves on to division. I wondered why I could not find anything that resembled modern long division. The absence of arithmetic symbols, introduced only at the end of the century and not regularly used when making calculations as personal notes, does not help. The reason is that during this period the most commonly used system to illustrate division was either the ‘galley’ or the ‘boat’ (‘a galera’ or ‘per battello’), as Sfortunati teaches us (

Baldassarre Peruzzi, Project for the dome of Siena’s Cathedral, Firenze, Gabinetto dei disegni e stampe degli Uffizi, U 494Ar, detail.

The drawing for the fortifications of Orbetello (Fig.

Another classic geometry problem, constantly found in architecture, is the calculation of the circumference and, therefore, the value of π. The drawing for the cupola of the Duomo in Siena (Fig.

Compared to Peruzzi’s drawings, as noted above, those by Antonio da Sangallo the Younger contain only a few calculations. It is important, in this respect, to point out the difference between the presence on a drawing of numbers and measurements, and that of actual arithmetical operations. A basic calculation can be found on drawing U 294Ar, where there is a survey of the perimeter of the walls of Castro and a long column addition problem to estimate the total length of it, according to the method already seen in Peruzzi’s work, which is typical of an architect-surveyor. One can see in other documents how Sangallo too operated with a certain ease with addition and subtraction, as well as with fractions. Double-digit multiplication, as in the plan for San Francesco in Castro (U 736Ar), also makes an appearance, and is solved with the placement system. In the drawing of the walls of Parma (U 799Ar) there is proof that Sangallo was capable of solving double-digit multiplication with fractions, too.

If, overall, there are not many calculations, there are, instead, many drawings with geometrical studies. In the case of Figure

Antonio da Sangallo the Younger, Geometrical studies, Firenze, Gabinetto dei disegni e stampe degli Uffizi, U 851Av, detail.

Antonio da Sangallo the Younger, Calculations, Firenze, Gabinetto dei disegni e stampe degli Uffizi, U 856Ar, detail.

Circumference calculations are also present in Sangallo’s drawings, and one can take, as an example, Figure

Antonio da Sangallo the Younger, Project for the model of Saint Peter’s in Rome, Firenze, Gabinetto dei disegni e stampe degli Uffizi, U 87Av, detail.

Later, in the text inscribed in U 267Ar (still for the model of Saint Peter’s dome), Sangallo also illustrated the proposal for the cupola’s profile as semielliptical (transcription in

Geometrical construction of a semielliptical profile according to the method illustrated in Dürer’s treatise. Image created by author.

The case of Michelangelo Buonarroti is profoundly different from that of the two architects already considered, although the three were roughly contemporaries, and his procedures in arithmetic are even further removed from our current practices than theirs (see

Michelangelo treated architectural drawings essentially like figure drawings, working on the page more or less freehand, without preliminary geometric construction, although sometimes with a few lines drawn with a stylus or lead pencil. He also showed great liberty regarding proportions, which he determined empirically with

With respect to addition, Michelangelo used a rather laborious method that one can define as ‘accumulation’, whereby he indicated each unit with a sign and then proceeded to do the sum by counting all the signs (

Michelangelo Buonarroti, ‘Accumulations’ and architectural sketches, Firenze, Casa Buonarroti, AB, I, 155, 276v, detail.

Then, when he had to multiply, Michelangelo resorted to an equally muddled visual-geometric system, constructing a grid with sides of as many units as there were values to multiply and then counting the number of quadrants: he thus transformed multiplication into accumulation. An example can be seen in Figure

Michelangelo Buonarroti, ‘Accumulations’ and architectural studies, Firenze, Casa Buonarroti, CB 75Av, detail.

There is no general consensus among scholars on this hypothesis.

Michelangelo Buonarroti and a second hand, Architectural studies and calculations, Firenze, Casa Buonarroti, CB 75Ar, detail.

Michelangelo Buonarroti, Architectural studies, Firenze, Casa Buonarroti, CB 76Ar, detail.

In sum, the hand that had performed the calculation in Figure

Further proof that Michelangelo usually did not use Hindu-Arabic numerals can be found in AB, I, 127, 240r–241r, in which blocks of marble are drawn in red pencil with measurements in numbers. If the only unmistakable difference in the handwriting of the numbers is the round and horizontally rotated 2, the handwriting of the letters is still completely different from Michelangelo’s unique and easily distinguishable hand. Suffice it to note the presence of an ‘l’ with the loop and the ‘b’ with two loops and a flourish, elements that are completely absent from the master’s handwriting. Also, the columnar addition in Figure

Therefore, one can conclude that Hindu-Arabic numerals and algorism remained completely unfamiliar to Michelangelo, and something he used with insecurity, if at all, preferring instead to transform the problems into a visually controllable system. What is sometimes called Michelangelo’s ‘lack of interest’ in numbers and measurements — in line with the idea that he approached architecture as a figure artist, concentrated on the perfection of the ‘gesture’ more than the process (

Michelangelo’s avoidance of preliminary line and square constructions and proportional procedures, the usual practice in all Renaissance architecture, can also be read as a way of reinforcing his image as an artist, trying to distance himself as much as possible from the figure of the artisan, tied to practical, utilitarian activity. Michelangelo pursued this idea with increased stubbornness over the years. One of the most renowned and significant episodes in this respect dates to 1547, when he wrote to his nephew: ‘I would like you to send Giovan Francesco [Ughi?] to measure the height of the cupola of Santa Maria del Fiore, from where the lantern starts down to the ground, and the height of the whole lantern, and send them to me; and mark on the letter the length of a third of the Florentine arm’ (‘vorrei che per mezzo di messer Giovan Francesco [Ughi?] tu avel’ altezza della cupola di Santa Maria del Fiore, da dove comincia la lanterna insino a terra, e poi l’altezza di tucta la lanterna, e mandassimela; e mandami segniato in su la lectera un terzo del braccio fiorentino’).

Throughout his life, Michelangelo tried to reinforce his noble status in innumerable letters, especially in his later years and in correspondence with his hard-pressed nephew Lionardo, in which he underlined his membership in the aristocracy. In July 1540, for example, he wrote, ‘I received three shirts with your letter, and I was amazed that you had sent them to me, because they are so large, and there is no peasant here’ (‘I’ ò ricievuto con la tuo lectera tre camice, e sonmi molto maravigliato me l’abbiate mandate, perché son sì grosse che qua non è contadino nessuno’). On April 14, 1543 he extended this theme: ‘When you write to me, don’t write: ‘Michelangelo Simoni’ or ‘sculptor.’ It is enough to say ‘Michelangnol Buonarroti,’ as I am known here’ (‘quando mi scrivi, non far nella sopra scricta: ‘Michelangelo Simoni‘, né ‘scultore‘. Basta dir: ‘Michelangnol Buonarroti‘, ché così son conosciuto qua’). And finally, on February 1, 1549: ‘It is well known that we are ancient Florentine citizens and nobles as much as any other house’ (‘gli è noto che noi siano antichi cictadini fiorentini e nobili quante ogni altra casa’,

This strategy was put in practice, first, from the material point of view, as Michelangelo became enormously rich,

I propose that his avoidance of learning and using Hindu-Arabic numerals and mercantile arithmetic, which were already ignored and even scorned by the intellectual elite and aristocratic culture of the time, was part of Michelangelo’s broader strategy to align himself with the nobility. During his years of training, in fact, first as an apprentice painter with Ghirlandaio and then as a sculptor under the guidance of Bertoldo di Giovanni in the Giardino di San Marco, it seems likely that he did not attend an abacus school, which was normally frequented after primary education for learning reading and writing, education he received at Settignano among marble and stone cutters.

In answering the questions raised at the beginning, I have tried to demonstrate that Peruzzi and Sangallo most likely went to an abacus school in their youth, while Michelangelo did not. Even if Italian architects, particularly those who attended an abacus school, were at the forefront of Europe, and 16th-century Italy was the mathematical centre of excellence in general, one cannot claim that Italian architects had an active role in developing the discipline of mathematics; rather, they simply used their operative arithmetic knowledge well. Peruzzi, for example, was a learned architect and came from Siena, celebrated for its tradition of engineering and applied mathematics, but he did not show any theoretical interest in the field, except when it applied to problems faced in the daily design process. The same seems true for Sangallo the Younger: the idea that in the design process both followed a ‘doctrine conceived in strictly technical terms, without any reference to philological-antiquarian culture’ (

The degree of confidence with numbers and operative arithmetic demonstrated by these two architects, though, seems quite high, apart from the issue of division, which appears to have remained mysterious to them.

It is not unlikely that Michelangelo’s initial ignorance and then refusal of algorism displayed a certain level of snobbishness, which made him, also on this matter, closer to the nobility to which he so much wanted to belong.

I presented an earlier version of this research in Ceriani Sebregondi (

This term indicates the practical, basic arithmetic, or art of computing. See Smith (

To evaluate the role this text played among other abacus treatises, see Ulivi (

Al-Khwarizmi lived at the beginning of the 9th century and the term ‘algorithm’ comes from his name.

For an overview on schooling in the rest of Europe, see Grendler (

Struik (

A Florentine contract of 1519 for an abacus teacher, including the syllabus, is illustrated in Goldthwaite (

Some of these problems have been pointed out in Carpo (

Swetz (

For an overview of his life and activity, see Ceriani Sebregondi (

The only other surviving drawings with Roman numerals are U 352Ar, for Palazzo Lambertini in Bologna, and U 355Ar, U 356Ar, U 357Ar, U 358Ar for Palazzo Ricci in Montepulciano.

Elam (

Wallace (

Smith (

This figure was still the value commonly used in professional practice in the 1600s — Francesco Righi, Francesco Borromini’s assistant, uses it for the estimates for Sant’Agnese Church in Rome (

Pagliara and Veronese (

In any case, Bramante, Raphael, and Peruzzi also used pencil: the difference is mostly in Michelangelo’s extensive and prevailing use of it.

On these drawings and their function for the San Lorenzo building site, see Wallace (

See also Wallace (

Tolnay, for instance, defines the Hindu-Arabic numerals in CB 17Av (Tolnay 579v) as ‘calculations in Michelangelo’s hand,’ but, although they are certainly by his hand, they are actually just a list of numbers, not a calculation. Maurer (

Estimates, accounts, and expenses were, in fact, overseen by others in big building programs most of the time. See Wallace (

30.7.1547, Michelangelo to Lionardo (

3.9.1547, Michelangelo to Lionardo (

2.5.1548, Michelangelo to Lionardo (

Also in October 1542, he wrote to an unknown monsignor: ‘I am a noble Florentine citizen, and the son of a respectable man’ (‘sono cittadino fiorentino, nobile, e figliolo d’omo dabbene’,

Michelangelo’s visceral avarice is notorious by now, aimed at assuring the social rise of the Buonarroti family, often achieved with personal privations and by conducting a life of extreme poverty (

This question became one of his main preoccupations, especially in the 1540s and 50s, and his correspondence shows constant concern with Florentine real estate: the Buonarroti lineage was Florentine, and therefore, he had to concentrate his efforts there. See, for example, the letter to Lionardo of December 4, 1546: ‘You must have received the letter I wrote you about buying a respectable house for one thousand five hundred or two thousand ecus, which should be in our Neighborhood, if possible. I say these things because

See the letter of July 1, 1497 to his father from Rome, in which he signed himself ‘Michelagniolo, sculptor in Rome’ (‘Michelagniolo scultore in Roma’,

In his letters from Bologna from 1506–07, one can recognize the second phase of Michelangelo’s youthful handwriting, in which the well-known Michelangelesque characters appear, including the ‘ch’ and ‘c’, elongated below the line; the ‘q’ cut off below the line; and the ‘g’, with its restrained flourish (

On June 28, 1487, at only 12 years of age, he was already documented as Domenico Ghirlandaio’s shop assistant, and April 1, 1488 he was cited as an apprentice painter, while between approximately 1490 and 1496 he was at the Giardino di San Marco at the expense of Lorenzo de’ Medici, becoming a member of the family (

Bardeschi Ciulich (

On the education and training of Antonio, see Bruschi (

Rouse Ball (

It seems that only from the end of the 17th century were architects across Europe comfortable with the use of numbers (

Grendler (