Fétis and Vuillaume: A mathematical approach to bow making
Simple mathematical problems can be intriguing, and since I have a bit more time on my hands than normal, I decided to see what sense I could make of a mathematical description first published in English in 1864.
Many bow makers will be familiar with the brief section at the end of François-Joseph Fétis’s book ‘Notice of Anthony Stradivari,’  in which he describes a method for graduating violin bows, told to him by Jean-Baptiste Vuillaume. The method works on the idea that there are points along a round violin bow stick  at which the diameter successively decreases by 0.3mm, and that the distance between these points can be calculated mathematically.
A numerical sequence where the difference between any two successive numbers gives the same amount is known as an arithmetic series. And that is part of the situation we find here. We have a series of points where the diameter of the bow stick is reduced by 0.3mm, starting with 8.6mm. If we take any two successive values from this sequence, eg 8.6mm and 8.3mm, or 8.3mm and 8.0mm, or any others, we find that the difference between them is always 0.3mm (as the author has already stated).
At the same time, we have another type of series at work, the geometric series. This is a sequence of numbers in which each is separated from its previous number by a constant factor, in other words, each successive number is multiplied by a constant rather than added to or subtracted from. This geometric series, Fétis’s book states, reveals the distances between those points at which the diameter is reduced by 0.3mm.
Well, for those of you braving this out, first of all, congratulations. Personally, I find it quite a neat little set of conditions, and although I have never found a violin bow that perfectly matches these criteria, it is still an interesting problem to investigate, and the measurements I have calculated from it approximate fairly well with many bows.
Fétis not only draws a diagram to aid his explanation but also gives a mathematical equation. The diagram and the equation give different results, and that is why I’m writing this. What I want to show is that the diagram seems to hold good, and I want to provide a few figures to show how it all works mathematically.
Here we have the well-known diagram Fétis gives in his book. The horizontal line is representative of the length of the stick, from the end of the handle all the way along to the narrowest part of the stick, immediately behind the head. The distance between these two points is 700mm (not marked on the diagram).
He says that the handle portion, measuring 110mm, has a constant diameter of 8.6mm. In reality, this portion, when finished, is never of a constant diameter and Fétis statement has to be ignored except to state that this portion of the stick does not conform to the same tapering as the rest.
In other words, we have a length of 700mm, for which the first 110mm is exempt from the arithmetic series. Nevertheless, this section is key to our investigation because it constitutes the first value of our geometric series. And this is why the vertical line on the left hand side of Fétis’s diagram has a length of 110mm.
At the the right-hand end of the diagram there is another perpendicular line 22mm long. A sloping line joins the top of the left-hand perpendicular line to the top of the right-hand perpendicular line. (Why 22mm, you ask? Well, it has nothing to do with bows. It just so happens that if we were to continue the geometric series beyond the twelve values that add together to make our stick length of 700mm, the next number in the series would be approximately 22mm.)
The diagram demonstrates graphically the geometric progression. If a pair of compasses is used to draw an arc from the top of the 110mm perpendicular line down to the baseline, the point where the arc intersects the baseline is the point where a second perpendicular line, representing the start of the second value in our geometric series, may be drawn.
The length of this second line is found by drawing the perpendicular line from the baseline up to the sloping line. Now, if the compasses are used to transmit the length of the second upright line down to the baseline, we have the starting point for the next length in the geometric series. And so on, as shown in the diagram with lines EF, GH etc.
In studying the diagram, it is easy to mistake the sloping line as implying a tapering thickness for the entire length of the stick. It does not - as I said before, the first 110mm does not taper in this way. The upper boundary of the diagram just provides us with a handy slope which defines the limits of any perpendicular lines drawn up from the baseline.
Well, we can do quite a good job of producing the values of each number in the geometric series simply by drawing out the diagram full size and using our compasses to produce the values. And, as far as bow making is concerned, this would give a fair rule of thumb to work to. But let’s do some maths and see if we can use numbers to show what’s going on.
We need to establish the relationship between two successive vertical lines. Let’s call the first one, measuring 110mm long, l₁, and the next one, l₂.
It can be stated that: l₂= l₁- (l₁- l₂)
In other words, l₂is equal to l₁, minus the difference between l₁and l₂. If that sounds obvious, good!
Now, the difference between l₁and l₂can be found by trigonometry. Let’s consider the slope of the line linking the top of l₁with the top of the 22mm line. The angle this line produces can be found by making a triangle where the base is 700mm and the height is (110-22) = 88mm. This gives an angle whose tangent is 88/700. We will call this angle 𝝰.
So, if we go back to our statement that: l₂= l₁- (l₁- l₂)
We can put this another way: l₂= l₁- (l₁tan 𝝰)
= l₁( 1 - tan 𝝰)
= l₁(1 - 88/700)
= l₁(1 - 22/175)
l₂= l₁x 153/175
So, we have our constant factor of 153/175. Not a very convenient figure, I agree. (If you use 7/8 for practical purposes, you will be out by less than 0.1 percent!) 
So, each successive distance is equal to the previous distance multiplied by 153/175. Thus we have found our geometric progression.
Now we can make a list of the twelve values of this geometric series which are of interest to us (for the series goes on beyond that, but that’s not important here).
Here is the list of values for lengths l₁to l₁₂, and they correspond to the perpendicular lines marked on Fétis’s diagram above. I have included the alphabetical labels he uses:
Just to be sure, we can check the sum of the terms from l₁to l₁₂using the following formula:
Sn = a ( 1 - rⁿ) / (1 - r)
Where: S denotes a sum
n is the number of terms
a is the first term
r is the common ratio
So we have:
S₁₂= 110 x ( 1 - (153/175)¹² ) / (1 - (153/175))
And this gives us S₁₂= 700.48 mm
So, we can be confident that our geometric series is correct!
This explanation appears in several later books, but seems to have been copied (with varying accuracy) from Fétis’s book so I will refer to his book alone.
The cross sectional shape is irrelevant provided it is constant.
The difference between 153/175 and 7/8 equals the difference between 1224/1400 and 1225/1400.