LA HIRE, Philippe de.

Nouvelle Méthode en Géométrie pour les Superficies coniques et cylindriques [- Les Planiconiques]. [Bound with:] De Cycloide Lemma.

Paris; Paris: chez l’autheur et Thomas Moette; [n.p.], 1673 [-1674]; 1676.

First edition, extremely rare, of these two works, of which the Nouvelle Méthode is the first substantial treatment of projective geometry, explicating the Brouillon Projet of Girard Desargues (1591-1661). “The BrouillonProjet (1639) was published in an edition of only 50 copies and won very little support. In fact, its reception was generally hostile, and Desargues was engaged in a pamphleteering battle for years with his detractors. At first his only supporters were Pascal, most of whose work on projective geometry is also lost, and the engraver Abraham Bosse. Desargues became discouraged by the attacks on his work and left the dissemination of his ideas up to Bosse, who was not really mathematically equipped for the task. Projective geometry secured a place in mathematics only with the publication of a book by Philippe de Hire (1673). It seems quite likely that La Hire’s book influenced Newton [see below]” (Stillwell, p. 153). Michel Chasles, writing in 1837, noted that La Hire’s work is “extrêmement rare” (Chasles, p. 128). “The treatise of 1673 is where De La Hire shows himself to be truly original and innovative, and which leads us to regard him as one of the founders of modern geometry” (translated from ibid.). “La Hire certainly read the Rough Draft on Conics [Brouillon Projet] thoroughly; for a long time the only known copy of the work was one made by La Hire himself in 1679. Possibly he made his own handwritten copy of the Rough Draft on Conics from a printed copy belonging to his father, the painter Laurent de la Hire (1606-1656), a pupil of Desargues and a friend and colleague of Abraham Bosse at the Académie. Philippe de la Hire had by then written his first book on geometry, his Nouvelle Méthode en Géométrie … It too has become extremely rare, and he was later to write that his new method had been found difficult because it involved planes and solids” (Field & Gray, p. 37). La Hire’s point of view in his Nouvelle Methode was entirely projective. He regarded all conics as projections of circles, and used the harmonic division of four points, which he showed was projectively invariant, to obtain theorems about poles and polars. The work is in two parts. In the first (pp. 1-72), La Hire treats conics as sections of the cone which are then projected onto the plane of the base of the cone; this approach was later expanded in his Sectiones conicae (1685). In the second, Les Planiconiques (pp. 73-94), he treats conics by entirely planar methods; Chasles notes that this part, which Chasles regards as the more original, “offered the first sufficiently general method for transforming figures of one kind into other figures of the same kind”. The Planiconiques was published one year after the Nouvelle Méthode and was not added to all copies (the Lyon and Marburg copies of the Nouvelle Méthode end at p. 72). The second work in the present volume, De Cycloid Lemma, is even rarer than the Nouvelle Méthode. It presents a geometrical construction of the tangent at any point of the cycloid – the method was discovered by Descartes and Fermat but they did not publish it. OCLC lists 12 copies of the Nouvelle Méthode worldwide (Columbia only in US), and 6 copies of De Cycloide Lemma, of which three are bound with the Nouvelle Méthode (including Columbia) and three are bound separately (BL, Erfurt & Lyon, the first two of which hold the Nouvelle Méthode). It is unclear how many of the copies of the cycloid pamphlet are complete, as several lack the plate (e.g., the BNF copy). RBH lists only the Macclesfield copy of the Nouvelle Méthode since 1961 (bound with De Cycloide Lemma but lacking its plate), and no other copy of the cycloid pamphlet.

“When Desargues circulated fifty copies of his Brouillon Project d'une Atteinte aux Événemens des Rencontres du Cone avec un Plan (Rough Draft of an Essay on the results of taking plane sections of a cone) in 1639, he was contributing to a lively contemporary study of geometry. Descartes’s novel algebraic methods had been published two years before, and in 1639 Mydorge published a more classical treatment of the conic sections. The classical authors themselves were increasingly well studied. Desargues had available Commandino’s Latin edition of Euclid’s Elements, published in 1572, as well as his Latin edition of the first four books of Apollonius’ Conics, published in 1566 with extensive commentaries by Eutocius, Pappus and Commandino himself. The last four books of the Conics were unknown in Desargues' time” (Field & Gray, p. 1).

“The Brouillon Projet on conics, of which he published fifty copies in 1639, is a daring projective presentation of the theory of conic sections; although considered at first in three-dimensional space, as plane sections of a cone of revolution, these curves are in fact studied as plane perspective figures by means of involution, a transformation that holds a place of distinction in the series of demonstrations. But the use of an original vocabulary and the refusal to resort to Cartesian symbolism make the reading of this essay rather difficult and partially explain its meager success.

“Although he praised the unitary conception that inspired Desargues, Descartes doubted that the use of geometry alone could yield results as good as those that a recourse to algebra would provide. As for Fermat, he reserved his judgment, and the only geometer who really comprehended the originality and breadth of Desargues’s views was the young Blaise Pascal, who in 1640 published the brief Essay pour les Coniques, inspired directly by the Brouillon Projet. But since the great Traité des Coniques that Pascal later wrote has been lost, Desargues’s example survived only in certain of the youthful works of Philippe de La Hire and perhaps in a few essays of the young Newton” (DSB).

“Philippe de la Hire (1640-1718) published three treatises on conics, in 1673, in 1679, and in 1685. From the point of view of modern geometry, the treatise of 1673 is by far the most original. Unfortunately, because of its rarity, it did not have a very wide circulation, and today too many historians of science pass over it in silence, concentrating instead on the Latin treatise of 1685, which is merely a development of the first part of the treatise of 1673. Although La Hire, in a note attached to his copy of the Brouillon Projet of Desargues, claims not to have known the treatise of Desargues until after 1673, and not to have been inspired by it, it seems to us, on the contrary, that this inspiration is manifest. A first reason is that La Hire’s father, the king’s painter, was a diligent student of Desargues’s oral lectures, so it would be very surprising if he were unaware of the existence of this treatise and were unable to make its content known to his son. The second, and most important, reason is drawn from the study of the text. It seems that La Hire, knowing at least the spirit of Desargues’s treatise, has tried to derive from it a work using the same principles but avoiding the faults that were the source of the violent criticisms by Desargues’s adversaries.

“Indeed, La Hire seems to want to make a synthesis of the theories of Desargues, while giving them a classical gloss … His method … amounts to deducing the projective properties of an arbitrary conic in space situated on a cone with a horizontal circular base from the properties of the base circle, via the intermediary of the projection of the conic section onto the plane of the base. He uses, in fact, the properties of cylindrical projection (for the passage from the curve in space to its projection) and of homology (for the passage from the conic to the projection of the base circle).

“The method for studying the properties of conic sections ‘in the cone’ led quite logically to another procedure for studying them which consists in deducing directly – in the plane – every conic from a circle by a homology. This is the object of the second part of La Hire’s small treatise, titled Les Planiconiques. This very ingenious method is applied with the aid of several lemmas on the elementary properties of homology. So here, apparently, we have a treatise which, although inspired by the ideas of Desargues, is not afraid to develop them further” (translated from Taton, pp. 204-205).

Chasles (pp. 128-9) describes the method of the Planiconiques as follows. “Suppose that we have in a plane two straight lines parallel to each other, which the author calls the formatrice and directrice, and a point called the pole.Through each point M of a given curve in the plane, one draws, in an arbitrary direction, a transversal; it meets the directrix at a point which one joins to the pole by a line;and the former at a second point through which we draw a parallel to this line.This parallel meets the straight line which goes from the point M to the pole, in a point M', which is said to be formed by the point M.Each point of the proposed curve will thus form a corresponding point of a second curve. The points of a straight line form points belonging to a second straight line, and these two straight lines will intersect on the formatrice.Finally, the points of a circle will form the points of a conic section. Such is the method by which De La Hire studied in the plane, without the need for any solid, nor any other plane than that of the figure, the sections of a cone.This is what he called reducing the cone and its sections to a plane.”

“Whiteside has pointed out (Mathematical Papers of Isaac Newton, VI, 271, n.70) that [the Nouvelle Méthode] received a favourable review, probably from CoIlins, in 1676, and that Newton may have read it, for Hooke wrote to him mentioning it in 1679. There are certainly similarities between ingenious projective transformations described by both men … In Book I of his Principia, [Newton showed] how to solve all of the six different problems of the form: find the conic through k points and tangent to m lines, k + m = 5. In the course of accomplishing this feat he introduced a projective transformation capable, as he remarked, of transforming any conic to a circle … It is strikingly similar to the one given by La Hire in his Planiconiques, which was printed in the same volume as his Nouvelle Méthode” (Field & Gray, p. 37). The Nouvelle Méthode also influenced Leibniz. “The ideas of Desargues and Pascal led Leibniz to a ‘dynamical’ vision of geometry. In the years 1672-76 Leibniz was in Paris, where he greatly increased his mathematical knowledge. In 1673 he was informed on Desargues’ perspective through La Hire’s Nouvelle Méthodeen Géométrie …” (Del Centini & Fiocca).

Determining the properties of the cycloid, the curve traced out by a point on the circumference of a circle as it rolls along a straight line, served as a test bed for the techniques of seventeenth century mathematics. The earliest significant work on the cycloid is due to Gilles Personne de Roberval (1602-75), who obtained many of its properties before 1636, although he kept his results secret and they remained unpublished until 1693. Roberval, in particular, gave a construction of the tangent at a point on the cycloid using his method of ‘composition of movements’, a precursor of the differential calculus. This problem was also solved by Pierre de Fermat and René Descartes, although they also did not publish their work. It was instead first published by La Hire in the first part of his De Cycloide Lemma (pp. 1-4). The remainder of this little pamphlet is devoted to a problem relating to conics, which is perhaps why it is often, although not always, found bound together with the Nouvelle Méthode.

Chasles, AperVu historique des Méthodes en Géométrie, 1837. Del Centini & Fiocca, ‘Boscovich’s geometrical principle of continuity and the ‘msyteies of infinity’, Historia Mathematica 45 (2018), pp. 131-175. Field & Gray, The Geometrical Work of Girard Desargues, 1987. Stillwell, Mathematics and its History, 3^rd edition, 2010. Taton, Le Prehistoire de la ‘Geometrie moderne’, Revue d’Histoire des Sciences et de leurs Applications 2 (1949), pp. 197-224.

4to (201 x 162 mm). [Nouvelle Méthode:] pp. [viii], 94, with 25 folding engraved plates (10 bound before title page, 15 at end of volume, the last 2 referring to Les Planiconiques), woodcut vignette on title, woodcut initials, head- and tail-pieces. [De Cycloide Lemma:] pp. 6, with 13 figures on one plate (cut up with each figure tipped in at the appropriate place in the text).

Item #6349

Price: $9,500.00