HomeThe following article was published in The Bowyers Journal in June 2005:
Finding the Pike for a Longbow Using Formula
Introduction
This article aims to simplify the process of piking a longbow with the aid of a mathematical formula. It is aimed at bowyers, both amateur and professional that may have difficulty using the existing explanations. It is a fact that those who have constructed many bows, will have their own method of judging how much to pike a bow to bring it to the desired weight and this article is not designed to convert them from their own methods and is aimed at those without a set method. It is probably that many bowyers make a bow that is longer than the desired length of the finished bow. In order to come up with a final bow weight, the bowyer will have to calculate how much to take off the bow in order to estimate the final weight of the bow. There are several explanations as to how to do this but these are not easy to work with in practice, especially with amateur or starter bowyers.
It is generally accepted that with a 1% of bow length reduction, there will be an increase In the weight of the bow by 5%. Bickerstaffe explains it thus,
‘As you shorten a bow, for each 1% of the length that you remove the weight will increase by 5%. So if you were to have a bow at 75” long and 40lbs draw weight, 1% would be equal to 0.75” and 5% of 40lbs is 2lbs, therefore, to shorten the bow by three inches would increase its weight by 8lbs leaving us with a 72” finished bow at 48lbs draw weight…’.
This explanation seems relatively simple until you are in the workshop and are attempting this calculation whilst working on the bow. It is unwieldy and difficult to complete. It I also not very easy to carry out the calculation the other way around, that is, you wish to finish a bow of a particular length and wish to find what weight the bow will become if piked to that length. This is a less usual problem than the latter calculation but it can and does occur.
The Traditional Bowyer’s Bible has a similar and equally complex explaination.
'Here is a simple formula. For each 1% of the length cut off, the weight will go up by approximately 5%. Divide the weight you want by the weight you have. The figure to the right of the decimal is the percentage increase you want. Divide this decimal figure by five to yield the percentage the bow should be shortened. Multiply this figure by the bow’s length to give the amount to be cut off. Take half of this amount from each end.'
It then goes on with an example that doesn’t really clarify things.
‘For example, let’s say a bow is 66” and pulls 52 pounds, but was intended to be 58 pounds. Dividing the weight we want by the weight we have yields a figure of 1.1153. The figure to the right of the decimal, .1153, is the percentage increase in weight we want. Dividing this figure by 5 gives the percentage the bow should be cut off, or .023. Multiplying .023 times the bow’s length, 66”, shows that 1.5” should be cut from the bow, or ¾” from both tips.’
This explanation is mathematically flawed and over complicated. Again, attempting to use this calculation, it would be very difficult to work out the weight of a bow for a particular length.
Both of these explanations are complicated to interpret and in the workshop environment, complicated to apply.
The problem
The problem here is not one of investigation but of explanation. The mathematics are straightforward but their application in an empirical environment requires clarity. As a non-mathematician, I decided to simplify the problem of interpretation. A mathematical problem does not always work best as an explanation in words and sometimes a mathematical equation is the simplest solution. Both of the explanations above are unwieldy and can much better be placed into a mathematical context. Both explanations describe the same thing but avoid what many people fear most, mathematics. In mathematical terms, both Bickerstaffe and Hamm are attempting to explain the following equation:
At first sight, this formula may seem impenetrable but it is actually fairly straightforward. Each letter represents a dimension that is already known and the pike (p) is the unknown. If the pike is known and it is the desired weight increase that is unknown then some mathematical manipulation is required. We will deal with that later.
The following letters represent the following dimensions:
p = pike – the total amount in inches that has to be taken from the bow.
l = length – the total length of the bow to be piked in inches.
w = weight – the current draw weight of the bow.
i = increase – the desired weight increase.
How to calculate the pike
Whilst working with the formula, it does not matter what unit of measurement is used as long as all of the measurements are consistent. The result will be in whichever unit is chosen. For the purposes of this paper, I will be using inches and imperial pounds, as this is the traditional unit of measurement used in bowery. To use the formula, it is simply a case of substitution and the use of a scientific calculator. For example, to use Bickerstaffe’s model where he uses a 75” bow that has a draw weight of 40lbs and wishes to increase this to 48lbs. We substitute the relevant letters for the known measurements; the length, weight and increase into the formula thus:
Using our scientific calculator, type in <75 ÷ 40 x 8 ÷ 5> and the calculator will do the rest. If you use a non-scientific calculator, the result will be wrong as it will not calculate the equation in the correct order. In this case the result will be 3, which means that p = 3” or the pike is three inches, 1½” off either limb. Strangely this is the same figure that Bickerstaffe came up with.
The other calculation we can complete from the formula is to find the weight increase if a known amount is taken off the length of the bow. However, it is much easier in practical terms to manipulate the formula. This allows the non-mathematician to enter the data into the scientific calculator and not have to complete any equation manipulation, which can be daunting. For this calculation, the equation we require is:
The letters in this formula correspond with the same measurements that were used in the previous calculation. For this example, take a 76” bow with a draw weight of 50lbs and we wish to find the increase in weight if we take 4” inches off the total length of the bow making the final length 72”. The unknown variable here is the increase in weight. Insert the known variables into the formula; the pike, the bow length and the bow weight:
From here we simply enter the calculation into the scientific calculator <4 x 50 ÷ 76 x 5> and we will come up with the answer 13.157. This represents the weight in pounds that the bow will increase if four inches are taken off the bow in total. So the total weight increase to the bow is 13.2lbs approximately. This calculation works every time with any unit of measurement.
Conclusion
In my workshop, I keep this formula on an A4 sheet covered with sticky back plastic and when I require to carry out the calculation, just fill in the blanks with a washable felt tip and enter it into the calculator. Annex A and Annex B show the templates of the formulae that would be covered with sticky back plastic and the measurements written with washable felt tip pen over the top of the relevant letter and then entered into the scientific calculator. For me, this works very well. Others may have their own method, which will work equally well. This method for finding the unknown variable can be used by anyone with a scientific calculator.
©Pete Davidson
11st April 2004