**Philippe de la Hire** was a French mathematician who worked on conic sections and on geodesy and astronomy.

- He became a professor at the Academy of Painting and Sculpture.
- The most notable mathematician who was frequently in their home was Girard Desargues.
- La Hire was educated as an artist and became skilled in drawing and painting.
- La Hire set off for Venice in 1660 and there spent four years developing his artistic skills and learning geometry.
- The interest in geometry arose from his study of perspective in art, but soon he was finding his mathematics classes more enjoyable than painting.
- Returning to Paris in 1664, La Hire was a wealthy man and able to pursue his interests without the need to seek employment.
- He continued to paint but his serious studies were devoted to geometry.
- Bosse was an artist who was much older than La Hire, but had attended classes on geometry by Girard Desargues from 1641.
- Bosse had published a series of works developing the geometric ideas that he had learnt from Desargues and had established his own school of art in 1661.
- Much influenced by the work of Desargues, both directly and through his friendship with Bosse, La Hire worked on conic sections which he treated projectively.
- He published his first work Observations sur les Points d'Attouchement de Trois Lignes Droites qui touchent la Section d'un Cone Ⓣ(Observations on the touching points of three straight lines which touch the section of a cone) in 1672, followed by his famous treatise Nouvelle méthode en géometrie pour les sections des superficies coniques et cylindriques Ⓣ(New methods in the geometry of conic sections and cylindrical surfaces) in 1673.
- It is a comprehensive study of conic sections by means of the projective approach, based on a homology which permits the deduction of the conic sections under examination from a particular circle.
- This treatise was completed shortly afterwards by a supplement entitled 'Les planiconiques' which presented this method in a more direct fashion.
- The 'Nouvelle méthode' clearly displayed Desargues' influence, even though La Hire, in a note written in 1679 ..., affirmed that he did not become aware of the latter's work until after publication of his own.
- Yet what we know about La Hire's training seems to contradict this assertion.
- Furthermore, the resemblance of their projective descriptions is too obvious for La Hire's not to appear to have been an adaption of Desargues'.
- Nevertheless, La Hire's presentation, which was in classical language and in terms of both space and the plane, was much simpler and clearer.
- Thus La Hire deserves to be considered, after Pascal, a direct disciple of Desargues in projective geometry.
- This assessment may be a little harsh on La Hire who was an extremely honest and meticulous person.
- It is possible that he knowledge these ideas of Desargues came through Bosse rather than directly from Desargues, so his statement that he did not know of Desargues' publications until after his own had been published could still be true.
- He began with their focal definitions and applied Cartesian analytic geometry t the study of equations and the solution of indeterminate problems; he also displayed the Cartesian method for solving certain types of equations by intersections of curves.
- Although not a work of great originality, it summarises the progress achieved in analytical geometry during half a century and contained some interesting ideas, among them the possible extension of space to more than three dimensions.
- On 26 January 1678 La Hire was elected to the Académie des Sciences.
- Of course, often someone deserving of admission to the Academy would enter the section in which a vacancy occurred rather than be forced to wait, perhaps for many years, for a vacancy in a more appropriate section.
- Election to the Academy was a great honour for La Hire, but it also meant a change in life style.
- The Academy was a working organisation so election meant that be was no longer a man of leisure.
- Jean-Baptiste Colbert, the French Minister of Finance, had been instrumental in founding the Academy in 1666 and he now assigned La Hire to assist Jean Picard in the surveying work he was undertaking with the ultimate aim of producing more accurate maps of France.
- Together La Hire and Picard undertook surveying work in Brittany in 1679 and in Guyenne in 1880.
- La Hire then went, without Picard, to survey around Calais and Dunkirk in 1681 and the coast of Provence in 1682.
- We note that La Hire's maps of the Earth were made with the centre of projection, not at the pole, but at r/√2r/√2r/√2 along a radius produced through the pole (where rrr is the radius of the Earth).
- By this time La Hire's work for the Academy was closely linked to the Paris Observatory which, like the Academy, had been founded largely due to Colbert.
- The director was Giovanni Cassini, and the Observatory had published the Connaissance des temps Ⓣ(Understanding time) in 1679 which was the world's first nautical almanac.
- The La Hires had only two bedrooms, a room where Philippe worked, a kitchen and the use of a cellar.
- To some extent the overcrowding was eased by the fact that, at least in the early years of their marriage, La Hire was often absent undertaking work for the Academy.
- He continued the surveying work for the French atlas, but after the death of Colbert in 1683, he was directed by his successor François Michel le Tellier, Marquis de Louvois.
- The Royal Court had moved into the Palace of Versailles in 1682 and La Hire was given surveying projects relating to the supply of water to the new Palace.
- The minister habitually used to tear them up without looking at them, and have the sums sent in rounded up figures.
- In December 1682 he was appointed to the chair of mathematics at the Collège Royale which had remained vacant following the death of Gilles Roberval in October 1675.
- Courses he lectured included astronomy, mechanics, hydrostatics, dioptrics, and navigation.
- If his other commitments stood in his way, his eldest son, Gabriel-Philippe, lectured on his behalf.
- Despite his interests across a whole range of scientific disciplines, La Hire remained fascinated by geometry.
- In 1685 he published a comprehensive work on conic sections Sectiones conicae Ⓣ(Conic sections) which contained a description of Desargues' projective geometry.
- In 1708 he calculated the length of the cardioid.
- He published Traité de méchanique Ⓣ(Treatise on mechanics) in 1695.
- Other topics to which he made important contributions included astronomy, physics and geodesy.
- He also produced tables giving the movements of the Sun, Moon and the planets which he published in 1687, publishing further such tables in 1702.
- La Hire became involved in experimental work in many different scientific areas.
- For example for surveying, which one of his major tasks, he designed an instrument to find the level at a site and studied instruments to compute slopes and elevations.
- We should also mention La Hire's contributions in editing the works of Jean Picard, Edme Mariotte, Gilles Roberval, and Frenicle de Bessy.
- A precise and regular observer, he contributed to the smooth running of the Paris Observatory and to the success of the different geodesic undertakings.
- Although his rejection of the infinitesimal calculus may have rendered a part of his mathematical work sterile, his early works in projective, analytic, and applied geometry place him among the best of the followers of Desargues and Descartes.

Born 18 March 1640, Paris, France. Died 21 April 1718, Paris, France.

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Architecture, Astronomy, Geography, Geometry

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive