‎"TINSEAU (D'AMONDANS, CHARLES de). - GENERALIZATION OF THE PYTHAGOREAN THEOREM.‎
‎Solution de quelques Problêmes relatifs à la Théorie des Surfaces courbes, & des Courbes à double courbure. (Présenté en 1774).‎

‎(Paris, Moutard, Panckoucke, 1780). 4to. Extract from ""Mémoires fe Mathematique et de Physique, Présentés à l'Academie des Sciences par divers Savans"", Tome IX. Pp. 593-624 and 2 folded engraved plates. Clean and fine.‎

Reference : 44931

‎First appearance of an importent papwer in the history of analytic geometry.""In this article Tinseau gave an interesting generalization of the Pythagorean theorem for space of three dimensions: the square of the area of a plane surface is equal to the sum of the squares of the projection of this surface upon three mutually perpendicular coordinate planes....To Tinseau it appears that the use of the word ""conoid"" in the modern sense is due.""(Boyer ""History of Analytic Geometry, p. 207).""Two of the three memoirs that constitute Tinseau’s oeuvre deal with topics in the theory of surfaces and curves of double curvature: planes tangent to a surface, contact curves of circumscribed cones or cylinders, various surfaces attached to a space curve, the determination of the osculatory plane at a point of a space curve, problems of quadrature and cubature involving ruled surfaces, the study of the properties of certain special ruled surfaces (particularly conoids), and various results in the analytic geometry of space. In these two papers the equation of the tangent plane at a point of a surface was first worked out in detail (the equation had been known since Parent), methods of descriptive geometry were used in determining the perpendicular common to two straight lines in space, and the Pythagorean theorem was generalized to space (the square of a plane area is equal to the sum of the squares of the projections of this area on mutually perpendicular planes). (DSB). Although Tinseau published very little, his papers are of great interest as additions to Monge’s earliest works. Indeed, Tinseau appears to have been Monge’s first disciple.‎