Joannes della Faille, S.J.: Mathematics, Modesty and Missed Opportunities (review)

This book describes the life of Joannes della Faille (1597-1652) and provides commentary on his mathematical work. Ad Meskens makes meticulous use of earlier scholarship and archival sources, including previously unstudied material held privately by the della Faille family. Meskens seeks to raise awareness of della Faille's mathematical accomplishments, published and unpublished, judging that a proper assessment would lead to recognition in the same company as more celebrated contemporaries such as Blaise Pascal (1623-1662) and Girard Desargues (1591-1661).

Meskens situates the life of della Faille within the civil and religious strife in the Low Countries during the late sixteenth and early seventeenth centuries. Joannes, son of a wealthy Flemish merchant, was born and raised in Antwerp, at that time under Spanish rule and a center of the Counter-Reformation. The young della Faille attended the Antwerp Jesuit College, early exhibiting both an attraction to a religious vocation and a talent for mathematics. His initiation into the Jesuit tradition of mathematical scholarship was accomplished especially through the instruction of Gregorius a Sancto Vincentio (1584-1667), himself a student of Christopher Clavius (1537-1612), a pioneer investigator in infinitesimal calculus.

Della Faille, ordained in 1621, was dispatched to Madrid in 1629, where King Philip IV had encouraged the Jesuits to found the Estudios Reales of the Colegio Imperial in an effort to build up Spanish technical expertise. While in Madrid della Faille published his only book, a short treatise deriving the center of mass of a sector of a circle. This was a tour de force of geometric reasoning, explicated by Meskens in one of his chapters.

Della Faille's skills were recognized at the highest level of the Spanish court. In 1646 he became mathematics tutor to Philip IV's son, Don Juan of Austria, and later accompanied the latter's military expeditions in the Mediterranean. Della Faille died in Barcelona in 1652, shortly after Don Juan had suppressed a rebellion in that city.

Gregorius a Sancto Vincentio asserted that della Faille may have written as many as thirty unpublished mathematical manuscripts. Meskens is unable to confirm such a large number, but has found several such manuscripts, of which he gives special attention to those on conic sections, composed of carefully crafted sequences of theorems without proofs. Meskens has succeeded, with considerable ingenuity, in providing anachronistic proofs by means of analytic geometry, and he expresses his confidence that della Faille was in possession of synthetic geometrical proofs. While acknowledging that della Faille was not so audacious an innovator as Desargues, Meskens is eager to credit della Faille with anticipating later developments in projective geometry, and urges that his name be attached to theorems usually associated with Pascal and William Braikenridge (ca. 1700-1762). These claims on behalf of della Faille's prowess cannot be incontrovertibly supported by the available evidence, but Meskens makes a plausible case.

Meskens concludes by musing provocatively, albeit superficially, on the conditions contributing to della Faille's obscurity in the history of mathematics. Factors mentioned include the relative weakness and isolation of Spanish mathematical activity during della Faille's time, the failure of Don Juan to memorialize properly his tutor, and (as alluded to in the subtitle of the book) della Faille's Jesuit spirituality that urged modesty.

The book is produced in a handsome paperback volume with numerous illustrations, including a cover reproducing the notable portrait of della Faille by Anthony Van Dyck. There are a few typographical errors, awkward phrasings, and missing words, but the most distracting blemish is provided by the exceedingly mechanical right justification of the text, resulting in numerous unconventional hyphenations.