THE JACKSON RATIO- A MEASURED VIEW (letter in reply to a query)

Dear Chris
Thanks for your interest in our use of the derivation from Jackson's graph as a mathatical ratio.

I have spoken to Robin Bone, who first introduced us to this method, but his research is unpublished, and typically for him, lost at this time. He can site however, work from other researchers who back up his adoption of the wt (g) divided by the length cubed (cm) as an accurate formula which replaces and indeed accurately matches Jackson's graph.
In the Veterinary Record (1995) 136, 566-568 Blakey and Kirkwood explore this avenue with RETs, box tortoises, Hermann's and Spurs. Paragraph five quotes them as follows: "between bodies of constant shape volume is related to the cube of linear dimensions. Amonst these animals weight was approximately proportional to length cubed...."

Not having read this five years ago, I simply took a practical physics approach to this, and put the Jackson graph to some simple tests:
I took 12 points on the graph along the "average line", and calculated the weights and lengths, based first on the weight divided by length, which is widely quoted as "The Jackson Ratio", and then using wt div by L cubed, as promoted by Robin Bone.
The results to me (a lapsed physics teacher) were startling:

 weight(g) Length(mm) wt(g)/L(mm) wt(g)/L3(cm) 300 110 2.72 0.22 400 120 3.33 0.23 600 136 4.41 0.24 800 149 5.33 0.23 1100 168 6.54 0.23 1300 180 7.2 0.22 1500 111 7.8 0.21 1650 200 8.25 0.20 1900 214 8.87 0.19 2200 230 9.6 0.18 2400 240 10.00 0.17 2500 256 10.1 0.17

My mathematical friends are quite clear on this issue. Jackson's graph is for most of its length a straight line, and therefore can be expressed as a single ratio- i.e. a formula which when mathematically calculated, defines any point on the graph with the same resulting number. It is clear that the simple calculation of weight divided by length does not meet the definition of a mathematical ratio.
The weight divided by length cubed (i.e. a stylised calculation of density) bears a closer look. Indeed, when one thinks about it, and bears in mind the fact that the proportion of bone in a smaller animal is going to be a greater than in a larger one (this is a surface area compared to volume calculation), this calculation of the ratio is as perfect as one could wish for a tortoise. Clearly, it is a much more effective mathematical definition of the graph than Jackson's "weight divided by length". (This is of course a simplification of his work anyhow- his definition actually defined that the logarithim of the weight should be divided by the logarithm of the length)

Practically, of course, out at MOTs (health checks), this has even more significant effects. Jackson's wt/L calculation is weight specific, not a generalised ratio, and therefore has little use when examining a specific tortoise.
Let us consider an example:
Two tortoises are presented for weighing prior to hibernation. One is going to be dangerously underweight, the other, OK.
Tort1 weighs 1300g and measures 180mm
Tort2 weighs 1700g and measures 235mm
Which should not be hibernated:

Using weight divided by length:
Tort1 is calculated to be: 7.2
Tort2 is calculated to be: 7.2

Using weight divided by length cubed:
Tort1 is calculated to be: 0.22
Tort2 is calculated to be: 0.14

The first calculation is of no help. The second quite clear- the lower the value, the more serious the situation. Indeed, if these measurements are plotted on Jackson's graph, one is on the "average" line, one on the "low" line. Only the "Bone" method picks this up without the use of the graph itself.

The value of this method for other species is significant. Without the dammning nature of the graph, with its "above and below" lines, ratios can be observed, and given, without panicking the owner. Each individual tortoise can be tracked over time by its ratio, giving a clear indication of weight gain/loss, regardless of growth. This gives a much clearer impression of an individual tortoise, and factors in such things as gender, body shape, natural obesity etc. Obviously, the exact range of safe ratios for different species are not yet specified in any detail, and are left to the judgement of our MOT teams, but, if this method became BCG "law", then MOTs could swiftly provide more data for standardisation than Blakey, Kirkwood or even Jackson had ever dreamed of!

Paul Coleman