"The medium length of a bow, to the head exclusively, is 0m, 700 (27.56 inches).
"The bow comprises a cylindrical or prismatic part of uniform dimensions, the length of which is 0m, 110 (4.33 inches). When this portion is cylindrical, its diameter is 0m, 0086/10 (.34 inch).
"From this cylindrical or prismatic portion the diameter of the bow decreases up to the head, where it is reduced to 0m, 0053/10 (.21 inches). This gives a difference of 0m, 0033/10 of a millimètre (.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 millimètre (.012 inch) reckoning from the cylindrical portion.
| FIG. 33. |
"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 à priori. This he attained in the following manner. At the extremity of a right line A B, equal to 0m, 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 0m, 110 (4.33 inches).
"At the extremity B of the same line, raise another perpendicular B D, of the length 0m, 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.
"Take the length 0m, 110 (4.33 inches) of the ordinate A C with the compasses, and set it off on the line A B, from A to e: from the point thus obtained, draw another ordinate (parallel to A C and perpendicular to A B), until it meets the line C D.
"Between these two ordinates A C and e f—the latter of which is necessarily less than the former—lies the cylindrical portion of the bow, whose diameter, as before stated, is 0m, 0086/10 (.34 inch).
"Then take the length of the ordinate last obtained, e f, and set it off, as before, on the line A B, from f to g, and at the point g draw a third ordinate g h, the length of which must also be set off on the line A B, to determine thereon a new point i, from which to draw the fourth ordinate, i j: the length of which, likewise, when set off on the line A B, determines the point where the fifth ordinate k l is to be drawn. The latter, in like manner, determines the sixth m n, and so of the others, to the last but one y z.