Why do the graduations of old French bows so closely approximate a mathematical model?

Over the years numerous articles have been written about the observation that the graduations of old French bows closely approximate a mathematical model. These various studies propose solutions which are more or less technical, suggesting that the old French makers used one mathematical system or another to generate the graduation. But none of these adequately solve the question, but only seem to add to its mystery.

These kinds of explanations fail to take into account that these men were artisans, and not technicians. They were equipped with skills, tools and methods suited to working with highly diverse biological products, producing an acoustic instrument, i.e. the bow shaft, intended to satify an aesthetic demand. Art by its nature does not lend itself to precise quantifiability; neither can the methods which produce it be so.

If you make a graph of the graduations of these bows, the generated curve comes very close to the shape of a parabola. Remembering our high school algebra, a parabola is defined by the equation y= ax2+ bx +c. If you take a gun-barrel view (looking from the head down toward the frog) of one of these old shafts, the sides of the shaft will appear to be two straight and parallel lines. Yet they are curved, as demonstrated by the graph! Why is  this so? To explain the observation, we have to call on the differential calculus we learned in college.

The human eye perceives the size of an object as the inverse square of the distance to the object. That is, if the object is twice as far away, it appears one-fourth the size. Taking a gun barrel view of a shaft is the physical equivalent of taking the first derivative of the ax2+ bx+c function. The first derivative of the parabolic function is y=2ax+b; we recognize this as the equation for a line. The parabolically-curved surface of the sides of a shaft appears to us as two straight lines.

Using a skilled hand and eye, the old French bow makers tapered the shaft so that the two sides of the shaft appeared to be parallel lines. It so happens that the parabolic shape so generated is also an efficient acoustic body, contributing to the acoustic success of these early bows.

When I am trying a new bow, how much attention should I pay to weight?

Let me stress that above all else,  we are concerned with the kind of sound that comes from the you, your violin, and the bow  you will be playing. The best weight is that which sounds the best on your instrument. The best weight for a bow depends somewhat on the violin you have, and somewhat on the player. A bow weighing as little as 58 gm. can work well, but rarely over 63 gm is helpful. This overall weight can be strongly affected by the type of winding, the material  the frog is made from, and what kind of metal is used to mount the frog. Gold is substantially heavier than silver; a faux whalebone or  spun silk winding will weigh 2 or 3 gm. less than an ordinary silver wire winding. An ivory frog will weigh two or three gm. more than an ebony frog.

More important than weight, of  course is  the balance of the  bow, i.e., how heavy it feels in the hand. Depending on how the shaft is mounted, the finished weight can vary as much as 5 gms, yet the feeling  of balance will remain nearly the same. It is therefore, largely the weight of the shaft that gives the bow the balance which you experience; the shaft weight should be  between 35- 39 gm. The feeling of balance is best determined by what I call the  ‘head weight’. I use a system that accurately measures the head weight. For most players it is 16.5- 18 gm. More advanced players tend to like a heavier balance.

All that being said, the flexibility of the shaft may be even more important than the precise weight. To sum up, any bow that weighs between 58 to 63. gm. has the potential to optimize your performance. For a viola, 68 to 73 gm; for cello, 78 to 83gm.