## Rouché and de Comberousse's Treatise on Geometry

Eugène Rouché and Charles de Comberousse published their classic two volume work

*Traité de géométrie élémentaire*in 1964-1866. The textbook went through many editions and what we present below is the contents of the 7th edition published in 1900. We note that at this time Rouché was still active in revising and augmenting the work but de Comberousse had died three years before this 7th edition was published. The book was written in French but we present below an English translation of the contents:Eugène Rouché and Charles de Comberousse

Treatise on Geometry

**7th edition, revised and augmented**

Volume I : Plane Geometry

Volume II : Geometry in space

With a note from Henri Poincaré: On Non-Euclidean Geometry

Gauthier-Villars, Paris, 1900.

### Volume I : Plane Geometry.

**Book I. The straight line.**

1 - Angles.

2 - Triangles.

3 - Perpendiculars and obliques.

4 - Parallel lines.

5 - The sum of the angles of a polygon.

6 - The parallelogram.

7 - Symmetrical figures.

2 - Triangles.

3 - Perpendiculars and obliques.

4 - Parallel lines.

5 - The sum of the angles of a polygon.

6 - The parallelogram.

7 - Symmetrical figures.

**Book II. The circumference of a circle.**

1 - Arcs and cords.

2 - Tangent to the circle. Mutual positions of two circumferences.

3 - Measurement of angles.

4 - Construction of angles and triangles.

5 - Drawing of parallels and perpendiculars.

6 - Problems on tangents.

2 - Tangent to the circle. Mutual positions of two circumferences.

3 - Measurement of angles.

4 - Construction of angles and triangles.

5 - Drawing of parallels and perpendiculars.

6 - Problems on tangents.

**Appendix**.

- Problem solving. Method of successive substitutions. Analysis and synthesis.

- Method by intersection of locus; examples; the Simson line.

- Auxiliary constructions: translation, reversal, etc .; Polygon billiards; construct a polygon knowing the perpendiculars through the midpoints of the sides.

- Equal polygons of the same sense.

- Displacement of a figure in its plane, instantaneous centre of rotation. Point where a moving line touches its envelope.

- Equal and opposite polygons.

- Method by intersection of locus; examples; the Simson line.

- Auxiliary constructions: translation, reversal, etc .; Polygon billiards; construct a polygon knowing the perpendiculars through the midpoints of the sides.

- Equal polygons of the same sense.

- Displacement of a figure in its plane, instantaneous centre of rotation. Point where a moving line touches its envelope.

- Equal and opposite polygons.

**Book III. Similar figures.**

1 - Proportional lines.

2 - Proportional lines in the circle.

3 - Similarity of polygons.

4 - Metric relationships between different parts of a triangle.

5 - Problems relating to proportional lines.

6 - Regular polygons.

7 - Problems on regular polygons.

8 - Measurement of the circumference.

2 - Proportional lines in the circle.

3 - Similarity of polygons.

4 - Metric relationships between different parts of a triangle.

5 - Problems relating to proportional lines.

6 - Regular polygons.

7 - Problems on regular polygons.

8 - Measurement of the circumference.

**Appendix.**

- Principle of signs. Rectilinear segments and angles; formulas for change of origin.

- Theory of projections; generalization of the fundamental formula of rectilinear trigonometry.

- Transversals in the triangle. Theorems of Menelaus and Jean de Ceva. Applications; trigonometric relationships.

- Properties of the complete quadrilateral.

- Cross-ratio of four points on a straight line; it is projective; its trigonometric expression.

- Cross-ratio of a pencil of four straight lines, of four points, or four tangents of a circle.

- Fundamental properties relating to two pencils which have a common homologous ray and an equal cross-ratio, or to two rectilinear series of four points which have an equal cross-ratio and a common homologous point.

- Homological triangles.

- Hexagons of Pascal and Brianchon.

- Harmonic divisions and families, fundamental properties and formulas; harmonic mean; polar relative to an angle.

- Pole and polar in the circle; autopolar triangle.

- Method of reciprocal polars; transformation of descriptive properties and metric properties; applications to inscribed and circumscribed polygons.

- Homothety. Properties of homothetic figures. Centres and axes of homothety of three homothetic figures two by two and, in particular, of three circles.

- General definition of similarity. Double pole of two similar figures.

- Method of similar figures and method by reversal; inscribing in a quadrilateral a quadrilateral similar to a given quadrilateral; combination of the rotation method with the construction of a homothetic figure; problem of the reason section.

- Power of a point with respect to a circle. Radial axis of two circles; centre of three circles; common axis to three circles, other properties of the complete quadrilateral. Homological points of a system of two circles, properties of the systems of two circles affected by a third.

- Notions of involution; double points, provisions of the following figure that double points exist or do not exist. Condition for three pairs of points in a straight line to form an involution; properties of the quadrilateral.

- Pencils of circles; points of a pencil of the first kind; fundamental points of a pencil of the second kind; conjugate pencils; new expression of the power of a point with respect to a circle. Orthogonal circles; circles cut diametrically.

- Inversion. Properties of inverse figures; conservation of angles; the inverse of a line or a circle; any pencil of circles of the first kind or of the second kind may be transformed by inversion into a system of concentric circles or a pencil of straight lines; properties of the inverse circles of two tangent circles; the ratio of the tangent common to the geometric mean of the rays is the same for the system of two circles and for the inverse figure.

- Method of transformation by reciprocal vectors. A relation that binds the lengths of the common tangents, when a circle touches four others.

- Isogonal circles.

- Problem of Appolonius; circles tangent to three given circles. Solutions for individual cases. General discussion. Some properties of circles tangent to three circles.

- Circle cutting three given circles at given angles; special cases, general solution.

- Circle of the nine points or Euler circle; its contact with the inscribed and exscribed circles or Feuerbach's theorem.

- Inversors of Peaucellier and Hart.

- Problem of Castillon.

- Problem of Malfatti.

- Transformation by reciprocal semi-straight lines. Properties of semi-straight lines and cycles. Applications of this method employed alone or in combination with the method of transformation by reciprocal vectors.

- Theory of projections; generalization of the fundamental formula of rectilinear trigonometry.

- Transversals in the triangle. Theorems of Menelaus and Jean de Ceva. Applications; trigonometric relationships.

- Properties of the complete quadrilateral.

- Cross-ratio of four points on a straight line; it is projective; its trigonometric expression.

- Cross-ratio of a pencil of four straight lines, of four points, or four tangents of a circle.

- Fundamental properties relating to two pencils which have a common homologous ray and an equal cross-ratio, or to two rectilinear series of four points which have an equal cross-ratio and a common homologous point.

- Homological triangles.

- Hexagons of Pascal and Brianchon.

- Harmonic divisions and families, fundamental properties and formulas; harmonic mean; polar relative to an angle.

- Pole and polar in the circle; autopolar triangle.

- Method of reciprocal polars; transformation of descriptive properties and metric properties; applications to inscribed and circumscribed polygons.

- Homothety. Properties of homothetic figures. Centres and axes of homothety of three homothetic figures two by two and, in particular, of three circles.

- General definition of similarity. Double pole of two similar figures.

- Method of similar figures and method by reversal; inscribing in a quadrilateral a quadrilateral similar to a given quadrilateral; combination of the rotation method with the construction of a homothetic figure; problem of the reason section.

- Power of a point with respect to a circle. Radial axis of two circles; centre of three circles; common axis to three circles, other properties of the complete quadrilateral. Homological points of a system of two circles, properties of the systems of two circles affected by a third.

- Notions of involution; double points, provisions of the following figure that double points exist or do not exist. Condition for three pairs of points in a straight line to form an involution; properties of the quadrilateral.

- Pencils of circles; points of a pencil of the first kind; fundamental points of a pencil of the second kind; conjugate pencils; new expression of the power of a point with respect to a circle. Orthogonal circles; circles cut diametrically.

- Inversion. Properties of inverse figures; conservation of angles; the inverse of a line or a circle; any pencil of circles of the first kind or of the second kind may be transformed by inversion into a system of concentric circles or a pencil of straight lines; properties of the inverse circles of two tangent circles; the ratio of the tangent common to the geometric mean of the rays is the same for the system of two circles and for the inverse figure.

- Method of transformation by reciprocal vectors. A relation that binds the lengths of the common tangents, when a circle touches four others.

- Isogonal circles.

- Problem of Appolonius; circles tangent to three given circles. Solutions for individual cases. General discussion. Some properties of circles tangent to three circles.

- Circle cutting three given circles at given angles; special cases, general solution.

- Circle of the nine points or Euler circle; its contact with the inscribed and exscribed circles or Feuerbach's theorem.

- Inversors of Peaucellier and Hart.

- Problem of Castillon.

- Problem of Malfatti.

- Transformation by reciprocal semi-straight lines. Properties of semi-straight lines and cycles. Applications of this method employed alone or in combination with the method of transformation by reciprocal vectors.

**Book IV. Areas.**

1 - Measurement of polygon areas.

2 - Comparison of areas.

3 - Areas of the regular polygon and the circle.

4 - Problems on areas.

2 - Comparison of areas.

3 - Areas of the regular polygon and the circle.

4 - Problems on areas.

**Appendix.**

- Approximate assessment of the area of a figure with a curvilinear contour. Simpson's formula. Formula of Poncelet.

- Upper limit of the difference between the length of an arc of a circle and that of its chord.

- Maximum and minimums of plane figures. Maximum of the area of a triangle in which the base and the perimeter are known, or the lengths of two sides, or the sum of two sides.

- Between all the plane figures with equal perimeters, the circle is a maximum. Maximum of a figure consisting of a line and an arbitrary line. Maximum of a polygon with given sides, or of a polygon whose perimeter and number of sides are given. Application to regular polygons.

- Upper limit of the difference between the length of an arc of a circle and that of its chord.

- Maximum and minimums of plane figures. Maximum of the area of a triangle in which the base and the perimeter are known, or the lengths of two sides, or the sum of two sides.

- Between all the plane figures with equal perimeters, the circle is a maximum. Maximum of a figure consisting of a line and an arbitrary line. Maximum of a polygon with given sides, or of a polygon whose perimeter and number of sides are given. Application to regular polygons.

**Notes.**

1 - Measurement of quantities.

2 - On the impossibility of squaring the circle.

3 - On the recent geometry of the triangle.

4 - On geometrography.

2 - On the impossibility of squaring the circle.

3 - On the recent geometry of the triangle.

4 - On geometrography.

### Volume II. Geometry in space.

**Book V. The plane.**

1 - First notions on the plan.

2 - Straight lines and parallel planes.

3 - Straight and perpendicular plane.

4 - Projection of a line on a plane. Angle of a line and a plane. Shorter distance of two straight lines.

5 - Angles of the dihedral.

6 - Perpendicular planes.

7 - Polyhedral angles.

2 - Straight lines and parallel planes.

3 - Straight and perpendicular plane.

4 - Projection of a line on a plane. Angle of a line and a plane. Shorter distance of two straight lines.

5 - Angles of the dihedral.

6 - Perpendicular planes.

7 - Polyhedral angles.

**Appendix.**

- Left quadrilateral cut by any plane and, in particular, by a plane parallel to two opposite sides.

- Cross-ratio of four planes.

- Fundamental properties of the central projection or perspective. Vanishing point of a straight line. A condition for straight lines to have their perspectives parallel. Vanishing line of a plane; conception of the line at infinity of a plane.

- Cross-ratio of four planes.

- Fundamental properties of the central projection or perspective. Vanishing point of a straight line. A condition for straight lines to have their perspectives parallel. Vanishing line of a plane; conception of the line at infinity of a plane.

**Book VI. Polyhedra.**

1 - General properties and surface area of the prism.

2 - Volume of the prism.

3 - General properties and surface area of the pyramid.

4 - Volume of the pyramid.

5 - Symmetrical figures.

6 - Similar polyhedra.

2 - Volume of the prism.

3 - General properties and surface area of the pyramid.

4 - Volume of the pyramid.

5 - Symmetrical figures.

6 - Similar polyhedra.

**Appendix.**

- General properties of convex polyhedra. Euler's theorem (V + F = E + 2) and its consequences.

- Conditions of equality and similarity of two convex polyhedra; many of the conditions necessary to determine a convex polyhedron.

- Projection of a plane area.

- Centre of proportional distances.

- Centre of gravity: triangle, trapezoid, polygon; tetrahedron, polyhedron.

- Lateral area and volume of any truncated prism.

- Method of demonstration of the projective properties.

- Rule to recognize the projectivity of certain metric relationships; the trigonometric expression of the cross-ratio of a pencil. Homological structures; their origin; their construction; limit lines.

- Metric properties of homological figures. Coefficient of homology, new definition.

- Homology of the projections of two plane figures in perspective, reciprocal.

- Conditions of equality and similarity of two convex polyhedra; many of the conditions necessary to determine a convex polyhedron.

- Projection of a plane area.

- Centre of proportional distances.

- Centre of gravity: triangle, trapezoid, polygon; tetrahedron, polyhedron.

- Lateral area and volume of any truncated prism.

- Method of demonstration of the projective properties.

- Rule to recognize the projectivity of certain metric relationships; the trigonometric expression of the cross-ratio of a pencil. Homological structures; their origin; their construction; limit lines.

- Metric properties of homological figures. Coefficient of homology, new definition.

- Homology of the projections of two plane figures in perspective, reciprocal.

**Book VII. Circular bodies.**

1 - Cylinder of revolution.

2 - Cone of revolution.

3 - First notions about the sphere.

4 - Properties of spherical triangles.

5 - Area of the sphere.

6 - Volume of the sphere.

7 - General information on surfaces.

2 - Cone of revolution.

3 - First notions about the sphere.

4 - Properties of spherical triangles.

5 - Area of the sphere.

6 - Volume of the sphere.

7 - General information on surfaces.

**Appendix.**

- Guldin theorems on the area or volume generated by the rotation of a line or a plane area around an axis in its plane.

- Theorems on the maximum of figures. The sphere has the largest volume among the bodies of the same surface area.

- Regular convex polyhedra; there are only five; their construction; spheres inscribed or circumscribed.

- Calculation of the dihedral of a regular polyhedron. Calculation of the radii of the spheres inscribed or circumscribed.

- Regular polygons and polyhedra of superior species. There are only four regular polyhedra of superior species.

- Find the species of a regular polyhedron; generalization of the Euler formula. Application to regular polyhedra of superior species, their construction.

- Homothetic figures in space. Centres and axes of four homothetic figures two by two, and in particular four spheres.

- Similarity in space. Homologous figures in Space. Plane at infinity. Principle of the construction of bas-reliefs.

- Pole and polar plane with respect to the sphere. Reciprocal lines

- Radical plane of two spheres; axis of three spheres; radical centre of four spheres; properties of anti-homologous points.

- Supplement of the theory of inverse figures and of the method of transformation by reciprocal vectors; the inverse figure of a plane, a sphere or a circumference; conservation of angles.

- Stereographic projection.

- Sphere tangent to four given spheres, Dupuis's theorem.

- Sphere tangent to four given planes, number of solutions, calculation of radii.

- Figures traced on the sphere: cross-ratio; harmonic ratio; pole and polar with respect to a circle of the sphere; radical axis; centres of similarity; isogonal circles.

- Problems relating to the contact of the circles of the sphere. Circle intersecting three given circles at given angles, and the analogous problem of plane geometry. Sphere cutting four given spheres at given angles.

- Theorems on the maximum of figures. The sphere has the largest volume among the bodies of the same surface area.

- Regular convex polyhedra; there are only five; their construction; spheres inscribed or circumscribed.

- Calculation of the dihedral of a regular polyhedron. Calculation of the radii of the spheres inscribed or circumscribed.

- Regular polygons and polyhedra of superior species. There are only four regular polyhedra of superior species.

- Find the species of a regular polyhedron; generalization of the Euler formula. Application to regular polyhedra of superior species, their construction.

- Homothetic figures in space. Centres and axes of four homothetic figures two by two, and in particular four spheres.

- Similarity in space. Homologous figures in Space. Plane at infinity. Principle of the construction of bas-reliefs.

- Pole and polar plane with respect to the sphere. Reciprocal lines

- Radical plane of two spheres; axis of three spheres; radical centre of four spheres; properties of anti-homologous points.

- Supplement of the theory of inverse figures and of the method of transformation by reciprocal vectors; the inverse figure of a plane, a sphere or a circumference; conservation of angles.

- Stereographic projection.

- Sphere tangent to four given spheres, Dupuis's theorem.

- Sphere tangent to four given planes, number of solutions, calculation of radii.

- Figures traced on the sphere: cross-ratio; harmonic ratio; pole and polar with respect to a circle of the sphere; radical axis; centres of similarity; isogonal circles.

- Problems relating to the contact of the circles of the sphere. Circle intersecting three given circles at given angles, and the analogous problem of plane geometry. Sphere cutting four given spheres at given angles.

**Book VIII. Curves and standard surfaces.**

1 - Basic properties of the ellipse.

2 - Fundamental properties of the hyperbola.

3 - Basic properties of the parabola.

4 - Ellipse considered as orthogonal projection of the circle.

5 - Parabola considered as the limit of the ellipse.

6 - Common origin of the three curves. Planar sections of the cone of revolution.

7 - Basic properties of the helix.

2 - Fundamental properties of the hyperbola.

3 - Basic properties of the parabola.

4 - Ellipse considered as orthogonal projection of the circle.

5 - Parabola considered as the limit of the ellipse.

6 - Common origin of the three curves. Planar sections of the cone of revolution.

7 - Basic properties of the helix.

**Appendix.**

- Homography and involution.

- Second-order curves.

- Theory of surfaces of the second order.

- Study of some surfaces of higher order.

- Second-order curves.

- Theory of surfaces of the second order.

- Study of some surfaces of higher order.

**Notes.**

1 - On the application of determinants to geometry.

2 - Henri Poincaré: On Non-Euclidean Geometry.

3 - On linear and quadratic transformations, the conics associated with a triangle and the systems of three directly similar figures.

4 - On the recent geometry of the tetrahedron.

2 - Henri Poincaré: On Non-Euclidean Geometry.

3 - On linear and quadratic transformations, the conics associated with a triangle and the systems of three directly similar figures.

4 - On the recent geometry of the tetrahedron.