**Age 12.**

*Arithmetic. *Four fundamental rules with decimals; changing common fractions into decimals and conversely. Short methods. Simplest cases of percentage, discount, and interest, and other problems of everyday life, with application of the rule of three. Representation of number sequences by means of line segments and areas. Use of tables, especially calculation of mean values and ratios. Applications to the life of the community and the state.

*Geometry.* Intuitional development of fundamental notions; sides and angles of triangles. Simplest triangle constructions. Congruence theorems.

*Geometrical Drawing and Measurement. *Practice in the use of ruler, drawing triangles, and compasses. Drawing of parallels and perpendiculars and of nets of cubes, parallelopipeds, tetrahedrons, and octahedrons. Orthogonal projection of cube and parallelopiped. Construction of the space-diagonals of these bodies. Construction of prisms and pyramids and of nets of these solids. Measurement of segments and angles.

**Age 13.**

*Algebra. *Introduction of operations with letters. Four fundamental operations with integral and with fractional numbers. Representation by means of segments and rays. Calculation of tables from formulas. Simple equations of first degree with one unknown, in connection with operations with rational numbers. Introduction of linear function, if possible. Drawing of curves.

*Geometry. *Completion of the study of triangles. Continued construction of triangles. Theory of quadrilaterals, especially parallelograms and trapezoids. Calculation and comparison of areas (Pythagorean theorem). Extension of geometrical considerations and measurements to space.

*Geometrical Drawing and Measurement. *Projection of points, line segments, and plane figures upon the plane. Angle of inclination of line and of plane. Intersections of two planes whose traces and inclinations are given. Pyramids. Roofs. Measure of segments and angles.

**Age 14.**

*Algebra. *Equations of first degree with one or more unknowns; simple applications, especially from everyday life. Introduction of graphical representation of empirical functions; representation of linear functions (the linear function as straight line) and their use in solving equations of first degree. The function *y* = *mx*. Use of millimetre paper.

*Geometry. *Circles. Further study of space-forms.

*Geometrical Drawing and Measurement. *Simple masses of shrubbery and borders of garden paths. Measurement of segments, angles, and areas.

**Age 15.**

*Algebra. *Powers with positive and negative integral exponents. The function *y* = *x*^{n}, *n* positive or negative, and its graphical representation, especially the graphs of the general parabola and of the rectangular hyperbola. Operations with radicals, and their graphs. Extraction of square root. Exponential and logarithmic functions. Inverse of a function. Four place logarithmic table and the use of logarithms. The slide rule. Quadratic equations with one unknown.

*Geometry. *Equality of ratios between segments; theory of similarity. Application to circle and right triangle. Length and area of circle. Selections from the history of geometrical problems (e.g. quadrature of the circle). Calculation of simplest figures.

*Geometrical Drawing and Measurement. *Tri-rectangular axes; the regular bodies in axonometric representation . Curve drawing in connection with algebra. Approximate constructions (circle). More exact measurements (nonius). Surveying in connection with the theory of similarity.

**Age 16.**

*Algebra. *Simple integral and rational functions. Simple equations, and systems of equations which c an be solved by quadratic equations - numerical and graphical treatment. Arithmetic and geometric series. Infinite geometric series. Compound interest and bonds, with applications from commercial life (political arithmetic). Binomial theorem with positive integral exponent.

*Geometry. *The trigonometric functions. Simple triangle calculations. Goniometry. Notion of periodic function. Geometrical calculations continued.

*Geometrical Drawing and Measurement. *Projection of the circle. Constructions for trigonometric problems, including those whose data are not all in one plane. Curve drawing in connection with algebra. Simple exercises in surveying and levelling.

**Age 17 and Age 18.**

*Algebra.* Introduction to infinitesimal calculus. Definition of differential quotient, its geometrical and its physical significance; its application to rational and, if possible, to trigonometric functions, especially in the calculation of greatest and least values, points of inflexion, inflexional tangents, etc. Simplest exercises in finding areas and volumes with the aid of integral calculus (e.g. sphere, paraboloid, etc.). Construction of the number system, from the positive integer to the complex number. Simple representations by means of functions of a complex variable.

*Geometry.* Review of the curvilinear figures thus far studied and the introduction of analytic geometry as far as the general equation of second degree. Supplementary theorems from stereometry (sphere). Straight line and plane in space. Plane and cone. Fundamental notions of spherical trigonometry (sine theorem and cosine theorem). Applications to geodesy and astronomy.

*Geometrical Drawing and Measurement.* Fundamental problems on the point, line, and plane. Conic sections. Projection of the sphere. Simple astronomical observations with measurements and calculations. Review from historical and philosophical points of view.