The simplicity desired cannot however be attained unless all have a common starting-point, in the definition of similar polyhedrons. The best course is assuredly to consider that theory in the point of view in which it is employed in the arts, especially in sculpture; i.e. to conceive the given system of points, M, N, P, . . . . to have lines passing from them through a point S, the _pole of similitude_, and prolonged beyond it to M’, N’, P’, . . . . so that SM’, SN’, SP’, . . . . are proportional to SM, SN, SP, . . . . . Then the points M’, N’, P’, . . . . form a system _similar_ to M, N, P, . . . . . The areas and volumes of the cylinder, of the cone, and of the sphere must be deduced from the areas and from the volumes of the prism, of the pyramid, and of the polygonal sector, with the same simplicity which we have required for the measure of the surface of the circle, and for the same reasons. It is, besides, the only means of easily extending to cones and cylinders with any bases whatever, right or oblique, those properties of cones and cylinders,--right and with circular bases,--which are applicable to them. Numerical examples of the calculations, by logarithms, of these areas and volumes, including the area of a spherical triangle, will make another sheet to be presented to the examiners. PROGRAMME OF GEOMETRY. 1. OF PLANE FIGURES. Measure of the distance of two points.--Two finite right lines being given, to find their common measure, or at least their approximate ratio. _Of angles._--Right, acute, obtuse angles.--Angles vertically opposite are equal. _Of triangles._--Angles and sides.--The simplest cases of equality.--Elementary problems on the construction of angles and of triangles. _Of perpendiculars and of oblique lines._ Among all the lines that can be drawn from a given point to a given right line, the perpendicular is the shortest, and the oblique lines are longer in proportion to their divergence from the foot of the perpendicular. _Properties of the isosceles triangle._--Problems on tracing perpendiculars.--Division of a given straight line into equal parts. Cases of equality of right-angled triangles. _Of parallel lines._ Properties of the angles formed by two parallels and a secant.--Reciprocally, when these properties exist for two right lines and a common secant, the two lines are parallel.[1]--Through a given point, to draw a right line parallel to a given right line, or cutting it at a given angle.--Equality of angles having their sides parallel and their openings placed in the same direction. [Footnote 1: It will be admitted, as a postulate, that only one parallel to a given right line can pass through a given point.] Sum of the angles of a triangle.