The career as a painter of Mutio Oddi (1569–1639), born in Urbino, was cut short when it was discovered that he suffered from defects in his vision. Turning to mathematical studies and military service, Oddi served as court architect to Francesco Maria II della Rovere, Duke of Urbino (1549–1631). A series of unfortunate escapades – including a nude bath in the river Metauro, a fight with the ducal secretary, and an incriminating letter – led to Oddi's imprisonment and banishment from the duke's kingdom. He later found employment at Milan as a mathematics tutor and public lecturer and then as chief fortifications engineer in the Republic of Lucca. Oddi wrote various treatises on instruments and practical mathematics and, at the end of his life, returned to Urbino, where he erected memorials to the city's renowned artisans, renovated the house of Raphael (1483–1520), and served as Urbino's first public lecturer in mathematics.

Oddi's eclectic career, Marr argues, provides a window into the mathematical culture of late Renaissance Italy. Marr's aim is to call attention to individuals, like Oddi, whose modern-day obscurity often leaves them marginalized by historians of art and science. In this sense, the title of the book, Between Raphael and Galileo, is somewhat of a misnomer, for Marr does not intend to portray Oddi as a link between Raphael and Galileo. Instead, though Oddi did see himself as continuing a tradition of Urbino uomini illustri (the most famous of whom was Raphael), his relationship to Galileo was more ambiguous. Oddi, for Marr, represents the ‘ordinary’ mathematician of late sixteenth-century Italy, one who inhabited the same space as Galileo but did not manage to be included amongst our century's list of uomini illustri.

The volume begins by considering the importance of place in Oddi's career, in particular the way in which Urbino's distinctive style of mathematics, one which merged mathematics and craft, shaped Oddi's approach. Part II relies on the sources of historians of both science and art – in particular, rare teaching records and a painting made by one of Oddi's pupils – to probe the content and context of Oddi's teaching in Milan. Finally, in Part III, Marr explores Oddi's work as an author of instrument books and a maker and broker of instruments, concluding with a final chapter on Oddi's views on the proper relationship between disegno, mathematics, and their associated practices.

Marr's more general conclusions, which place Oddi as an intermediary in a mathematical culture that united the work of the mind and the hand, are perhaps less important than the particulars he draws out from his evidence. Especially enlightening are the archival materials Marr has found detailing Oddi's expenses for the printing and distribution of his 1625 book on surveying, Dello squadro trattato, as well as those pertaining to Oddi's business with the Urbino workshop in the construction and circulation of its instruments. Equally compelling is the way in which Marr integrates the history of art and the history of science. In his analysis of the role early modern mathematicians and artisans assigned to disegno (arts of design), he shows how Oddi collected and traded drawings done by, among others, Raphael and Leon Battista Alberti, in order to further his own study of disegno, but also as part of an emerging gift economy in which the collection of drawings and discussions of their style and iconography allowed non-professionals to display their own erudition. Analysis of two paintings, the Oddi–Linder double portrait and the Linder Gallery Interior, provides insight into Oddi's pedagogy and his attitudes towards the proper relationship between mathematics and material culture. Marr reads the latter picture, for example, as a coded critique of Galileo's reliance on telescopic observations. Furthermore, he argues, this interpretation reveals that a conservative, Aristotelian, non-Galilean approach could coexist with what may appear to us, in Oddi's work, as a ‘modern’ celebration of the material culture of mathematics and instruments.

By delving into Oddi's material and mental universe, Marr offers an engaging perspective on the early modern mathematical practitioner, one familiar with art, war, practical mathematics and the ancient classics, and one who negotiated a complicated network of aristocratic patrons, artisans and printers. In an effort, perhaps, to make Oddi's ‘ordinary’ life seem more relevant to disciplines often obsessed with larger-than-life figures, Marr emphasizes throughout how Oddi's career compares with those of his better-studied contemporaries, Galileo (primarily) but also Christoph Clavius and others. One only wishes that Marr had not felt so compelled to justify his treatment of Oddi in this way, for, in doing so, Oddi is often reduced to a mere exemplum, relevant only because he demonstrates how the experiences of Galileo (and others) can stand in for their less-illustrious counterparts. Instead, Marr's new monograph should be celebrated for what it is: a much-needed study of the material, textual and artistic world of the sixteenth-century mathematician.