Let _s__{1}, _s__{2}, ..., _s__{_n_} be the number of sides of the various faces, and _f_ the number of faces. Now since the sum of the angles of a polygon of _s_ sides is (_s_ - 2)180 deg., therefore the sum of the angles of all the faces is (_s__{1} + _s__{2} + _s__{3} + ... + _s__{_n_} - 2_f_)180 deg.. But _s__{1} + _s__{2} + _s__{3} + ... + _s__{_n_} is twice the number of edges, because each edge belongs to two faces. [therefore] the sum of the angles of all the faces is (2_e_ - 2_f_)180 deg., or (_e_ - _f_)360 deg.. Since the polyhedron is convex, it is possible to find some outside point of view, _P_, from which some face, as _ABCDE_, covers up the whole figure, as in this illustration. If we think of all the vertices projected on _ABCDE_, by lines through _P_, the sum of the angles of all the faces will be the same as the sum of the angles of all their projections on _ABCDE_. Calling _ABCDE_ _s__{1}, and thinking of the projections as traced by dotted lines on the opposite side of _s__{1}, this sum is evidently equal to (1) the sum of the angles in _s__{1}, or (_s__{1} - 2) 180 deg., plus (2) the sum of the angles on the other side of _s__{1}, or (_s__{1} - 2)180 deg., plus (3) the sum of the angles about the various points shown as inside of _s__{1}, of which there are _v_ - _s__{1} points, about each of which the sum of the angles is 360 deg., making (_v_ - _s__{1})360 deg. in all. [Illustration] Adding, we have (_s__{1} - 2)180 deg. + (_s__{1} - 2)180 deg. + (_v_ - _s__{1})360 deg. = [(_s__{1} - 2) + (_v_ - _s__{1})]360 deg. = (_v_ - 2)360 deg.. Equating the two sums already found, we have (_e_ - _f_)360 deg. = (_v_ - 2)360 deg., or _e_ - _f_ = _v_ - 2, or _e_ + 2 = _v_ + _f_. This proof is too abstract for most pupils in the high school, but it is more scientific than those found in any of the elementary textbooks, and teachers will find it of service in relieving their own minds of any question as to the legitimacy of the theorem. Although this proposition is generally attributed to Euler, and was, indeed, rediscovered by him and published in 1752, it was known to the great French geometer Descartes, a fact that Leibnitz mentions.[91] This theorem has a very practical application in the study of crystals, since it offers a convenient check on the count of faces, edges, and vertices. Some use of crystals, or even of polyhedrons cut from a piece of crayon, is desirable when studying Euler's proposition. The following illustrations of common forms of crystals may be used in this connection: [Illustration]