[3] Numerical examples on the areas and volumes of the round bodies, including the area of a spherical triangle, will be required by the examiners. The calculations will be made by logarithms.
[4] The true distinction between Algebra and Arithmetic is so commonly overlooked that it maybe well to present it here, in the words of Comte. “The complete solution of every question of calculation is necessarily composed of two successive parts, which have essentially distinct natures. In the first, the object is to transform the proposed equations, so as to make apparent the manner in which the unknown quantities are formed by the known ones; it is this which constitutes the Algebraic question. In the second, our object is to find the value of the formulas thus obtained; that is, to determine directly the values of the numbers sought, which are already represented by certain explicit functions of given numbers; this is the Arithmetical question. Thus the stopping-point of the algebraic part of the solution becomes the starting-point of the arithmetical part.
“Algebra may therefore be defined as having for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way Arithmetic may be defined as intended for the determination of the values of functions. Henceforth, therefore, we may call Algebra the Calculus of Functions, and Arithmetic the Calculus of Values.”
[5] The students will apply these reductions to a numerical equation of the second degree, and will determine the situation of the new axes with respect to the original axes, by means of trigonometrical tables. They will show to the examiner the complete calculations of this reduction and the trace of the two systems of axes and of the curves.
[6] They will be deduced from the property, previously demonstrated, of the derivative of the ordinate with respect to the abscissa.
[7] The method of the change of the planes of projection will be used for the resolution of these problems.
[8] Compiled from “Report and Appendix of English Commissioners on Military Education.” 1857.
[9] In an Analysis of the Report of this Commission, see page 97.
[10] The influence exercised in the various branches of study, and consequently in the position of the students in the list classified according to merit, by the professors and répétiteurs on the one hand, and by the examiners on the other, as in the table above.
[11] The students are selected, by a competitive examination, out of a very large number of candidates, as will be seen from the following table, extracted from the yearly calendars:—