Gregory St. Vincent

"Liber hic fere Lemmaticus est, quemadmodum & alter, qui de Circulorum variis proprietatibus tractat. Porrò quo magis materia Lectori admanum sint, omnem in tres partes dividere placuit."

"Grégoire de Saint-Vincent, a Jesuit, born in Bruges in 1584 and died in Ghent in 1667, discovered the expansion of log(1+x) in ascending powers of x. Although a circle squarer, he is worthy of mention for the numerous theorems of interest which he discovered in his search after the impossible... He wrote two books on the subject [1647, 1668]... the fallacy in the quadrature was pointed out by Huygens. In the former work he used indivisibles. An earlier work entitled Theoremata Mathematica, published in 1624, contains a clear account of the method of exhaustions, which is applied to several quadratures, notably that of the hyperbola."

"Grégoire de Saint-Vincent, the most gifted pupil of Clavius... received a sound grounding in Greek mathematics and was... acquainted with the works of Stevin and Valerio. The integration methods which he devised, probably... 1622-9, constituted an extension of Archimedes and [was] in no sense a development of the indivisible techniques of Galileo and Cavalieri. Unfortunately... the original manuscript was lost for many years and not [published] until 1647. Even so, the Opus geometricum attracted attention... it contained... [an] attempt to square the circle, but also... the systematic approach to volumetric integration developed under the name ductus plani in planum. ...geometric series played a significant part [in the integration method] and we are indebted to Grégoire for the clearest early account of the summation of geometric series. ...He goes on to consider... the paradox of Achilles and the tortoise, Zeno, he notes... had failed to recognize that the time intervals were in falling geometric progression, and... although the number of such intervals is infinite, their sum is finite."

"Grégoire... was primarily concerned to illustrate by reference to the ungula that volumetric integration could be reduced, through the ductus in planum, to a consideration of geometric relations between the lines of plane figures. ...Unfortunately, the delayed publication of the Opus geometricum prevented it from receiving... attention... In 1647, ten years after the publication of Descartes' La Géométrie, algebraic methods were rapidly gaining ground and the form and manner of presentation of Grégoire's work was not such as to make easy reading. ...Amongst those who gained much from the Opus geometricum... [was] Blaise Pascal whose Traité des trilignes rectangles et de leurs onglets is based essentially on the ungula of Grégoire. Huygens recommended the section on geometric series to Leibniz who later came to make a thorough study of the entire work. Tschirnhaus... found in the ductus in planum a valuable foundation for the development of his own algebraic integration methods."

"Gregory St. Vincent is the greatest of circle-squarers, and his investigations led him into many truths: he found the property of the arc of the hyperbola which led to Napier's logarithms being called hyperbolic. Montucla says of him, with sly truth, that no one ever squared the circle with so much genius, or, excepting his principal object, with so much success."

"No one ever squared the circle with so much ability or (except for his principal object) with so much success."

"Grégoire de Saint-Vincent... was a Jesuit, taught mathematics in Rome and Prag (1629-1631), and was afterwards called to Spain by Phillip IV as tutor to his son... He wrote two works on geometry [Principia Matheseos Univerales (1651); Exercitationum Mathematicarum Libri quinque (1657)], giving in one of them the quadrature of the hyperbola referred to its asymptotes, and showing that as the area increased in arithmetic series the abscissas increased in geometric series."

"Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another... [i.e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other. ...The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals. The second idea to emerge from the work of the projective geometers is that of transformation and invariance. To project a figure from some point and then take a section of that projection is to transform the figure to a new one. The properties... of interest are those that remain invariant under transformation. Other geometers of the seventeenth century, for example, Gregory of St. Vincent... and Newton, introduced transformations other than projection and section."