The mean length of a violin bow as fixed by Tourte is from 74 to 75 centimetres (29.134 to 29.528 inches English); that of a viola bow is 74 centimetres (29.134 inches), and a 'cello bow 72 to 73 centimetres (28.347 to 28.740). Many people imagine that the plates of silver or gold with which the nut of a bow is inlaid are nothing more than mere ornamentation. But their first purpose is distinctly one of utility, which is as it should be in a work of art; superfluous decoration has no beauty for an artist. It is by means of these metal "loadings" at the heel that the weight of the head is counteracted and the exact point of equilibrium determined. The centre of gravity in a violin bow should be at 19 centimetres (7.48 inches) from the nut; in a 'cello bow at 175 to 180 millimetres (6.89 to 7.087 inches) from the nut. Concerning the geometric proportions of the Tourte bows, I cannot do better than quote Bishop's able translation of the explanation given by Fetis in his notice of A. Stradivarius. [Illustration: FIG. 33.] "The medium length of a bow, to the head exclusively, is 0^m, 700 (27.56 inches). "The bow comprises a cylindrical or prismatic part of uniform dimensions, the length of which is 0^m, 110 (4.33 inches). When this portion is cylindrical, its diameter is 0^m, 008-6/10 (.34 inch). "From this cylindrical or prismatic portion the diameter of the bow decreases up to the head, where it is reduced to 0^m, 005-3/10 (.21 inches). This gives a difference of 0^m, 003-3/10 of a millimetre (.13 inch) between the diameters of the extremities; from whence it follows that the stick comprises ten points where its diameter is necessarily reduced by 3/10 of a millimetre (.012 inch) reckoning from the cylindrical portion. "After proving by a great number of Tourte's bows that these ten points are not only found always at decreasing distances on the same stick, but also that the distances are perceptibly the same, and that the situations of the points are identical on different bows compared together, M. Vuillaume sought to ascertain whether the positions of the ten points could not be obtained by a geometrical construction, by which they might be found with certainty; and by which, consequently, bows might be made whose good condition should be always settled _a priori_. This he attained in the following manner. At the extremity of a right line A B, equal to 0^m, 700 (27.56 inches), that is to say the length of the bow, raise a perpendicular A C, equal to the length of the cylindrical portion, namely 0^m, 110 (4.33 inches). "At the extremity B of the same line, raise another perpendicular B D, of the length 0^m, 022 (.866 inches) and unite the upper extremities of these two perpendiculars, or ordinates by a right line C D, so that the two lines A B and C D, may lie at a certain inclination to each other.