Saturday, May 29, 2004

English tonnage calculations

Sorry for all the math, but it is unavoidable in this sort of analysis. Before the Restoration, the burden calculation used until the end of the age of sail was formulated. That formula was:

LK = length on the keel

LGD = length on the gundeck

B = Beam

D = Depth in hold

Burden in tons = LK x B x B/2 x 1/94

This is often simplified as:

Burden in tons = LK x B x B / 188

The "Beam/2" is a normalized depth, instead of using a true depth in hold, as had previously been the custom. The next step after that, was to use a calculated length of keel, based on the length on the gundeck and the beam:

LK = LGD - 2/5 x B

Now there were both a normalized length on the keel and a normalized depth. Only the length on the gundeck and the beam determined the burden. I believe that eventually, the real length of keel was used, after 1688.

To look at the history of calculations, we have some useful data. There is a list of ships from 1590-1591 that has dimensions and burden in tons. Let us use the famous Ark Royal as our example:

LK 100 ft
Beam 36 ft
Depth 15 ft
Burden 540 tons

So let us look at the burden calculation:

540 tons = 100 ft x 36 ft x 15 ft x 1/100

This calculation holds true for all the ships in the list.

Now, let us look at the Prince Royal, as built in 1610:

LK 115 ft
Beam 43.5 ft
Depth 18 ft
Burden 1200 tons

So let us look at the burden calculation:

1200 tons = 115 ft x 43.5 ft x 18 ft x 4/3 x 1/100 (rounded from 1200.6 tons)

The only variation for this formula formula was that the burden might be rounded to one or two digits, rather than giving an exact number (such as 700 tons or 650 tons). For example, the Constant Reformation was nominally 750 tons, but the actual product was 752.6 tons.

Given our knowledge of formulas, we can now estimate the dimensions of ships for which we only have burdens. For many of the hired or purchased ship from 1642 to 1660, we only know burdens. Let us estimate the dimensions for the English ship, the Cygnet, purchased in 1643.

233 tons = LK x B x D x 4/3 x 1/100

Using some factors from examples, we can estimate each of the dimensions:

LK = LGD x 0.8

B = LGD x 0.25

D = B x 0.4

For the Cygnet, that gives the following:

Burden = LK x B x D x 4/3 x 0.01 = (LGD x 0.8) x (LGD x 0.25) x (B x 0.4) x (4/3) x 0.01 =

Burden = (LGD x 0.8) x (LGD x 0.25) x ((LGD x 0.25) x 0.4) x (4/3) x 0.01 =

Burden = LGD x LGD x LGD x (0.8 x 0.25 x 0.25 x 0.4 x 1.333 x 0.01) =

Burden = LGD x LGD x LGD x 0.000267

This simplifies to:

LGD = (Burden / 0.000267)1/3

So for the Cygnet, the LGD = (239 / 0.000267)1/3 = 96.37ft or 96ft-4in (about)

The LK = LGD / 1.25 = 96.37 / 1.25 = 77.096ft = 77ft-1in (about)

The Beam = LGD / 4 = 96.37 / 4 = 24.09ft = 24ft-1in (about)

The Depth = Beam x 0.4 = 24.09 x 0.4 = 9.636ft = 9ft-8in (about)

So let's test the result: Burden = 77.096 x 24.09 x 9.636 x 4/3 x 1/100 = 238.6, which is finally right!

Anyway, that is my system for estimating. If you change the proportions, the factors change, but the system stays the same, for this earlier period. For the later period, we are reduced to looking at example ships, as the formula doesn't tell us what the real depth was. (I kept making the error of using the LGD in the last calculation, and couldn't figure out why it came out wrong. I just figured out what I had done)

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