## 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.

## Reader Comments (1)

What are the most important qualities a hand made bow possesses, in contrast to a cheaper bow which is most probably machine- made?

I am taking violin lessons, and my teacher has told me that she owns and works with a very special bow which costed upwards of $7,000. I know that for a professional violinist, this is quite the norm, and I am wondering what special qualities such a bow possesses to justify a high price. Is there a physics -related angle on the answer as well? When I talk to people, about this, their answers are always vague and very subjective. They talk about how the bow draws the tone out from the violin, and how it brings an instrument to life, but is there a special ingredient or quality brought into the mix by the instincts and hand touch of the bow maker that cannot be accessed by the cheaper models . Is there a vibratory resonance in the stick, that sympathetically vibrates with the instrument, the bridge and strings or sound post? Do the bow hairs transmit directly to the stick which then matches a vibratory response from the instrument's body's sound transmission? Any light you may be able to shed on this would be greatly appreciated. I have switched recently from a cheap $45.00 bow to a $750.00 bow, and I must say , I certainly do notice a difference. I love playing with the latter, and feel I could not go back. I would just love to understand it better, and my mind longs to understand the physics behind it. All the Best!

Wayne