Hey mum, erm... I'm studying to become a physicist, not a physician

Apart from the usual incredible memories I have from my trip to Iceland, which you can find pictures of on some other people's blog/Facebook page, I have also brought back two other things. The first is URTI, which I think is medical jargon for "the unknown infection that's stuffing up your nose". A few panadols later, I am up-and-running, ready to serve out some Icelandic humour (that's the second thing I brought back).

Unfortunately for my mum, I am not training to become a physician. I trust that she has figured that out over the years. As I have myself made a host of English language gaffes, I am not entitled to berate her for the unintentional mix up with "physicist". However, and this is no by-product of mathematical snobbery, the following accidental rediscovery of the trapezium rule cannot be forgiven:

A mathematical model for the determination of total area under glucose tolerance and other metabolic curves.

1. M M Tai

1. Obesity Research Center, St. Luke's-Roosevelt Hospital Center, New York, New York.

Abstract

OBJECTIVE--To develop a mathematical model for the determination of total areas under curves from various metabolic studies. RESEARCH DESIGN AND METHODS--In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method (less than +/- 0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin. RESULTS--Tai's model proves to be able to 1) determine total area under a curve with precision; 2) calculate area with varied shapes that may or may not intercept on one or both X/Y axes; 3) estimate total area under a curve plotted against varied time intervals (abscissas), whereas other formulas only allow the same time interval; and 4) compare total areas of metabolic curves produced by different studies. CONCLUSIONS--The Tai model allows flexibility in experimental conditions, which means, in the case of the glucose-response curve, samples can be taken with differing time intervals and total area under the curve can still be determined with precision.

For the record, Dr Tai's mathematical discovery was published in a top medical journal, peer-reviewed by a number of world-leading experts, and, gasp..., cited over 100 times by other researchers. If you have bothered to sift through the abstract, you might have suspected that the confidently-named "Tai's Model" is really the "trapezium rule", which is something we have all learnt in secondary school when the teachers were preparing us for calculus.

I actually found this via some other blog, which had scathing comments about plagarizing integration from Newton. Actually, the blogger was probably being kind. Newton's invention of the calculus certainly required quite a profound insight. Independent discovery of integration, even if it took place centuries after Newton's, might arguably require some cognition. The definite integral roughly refers to the area under a curve, and through the Fundamental Theorem of Calculus, one can relate this geometrical quantity to the slope of the curve, i.e, differentiation That is at least Newton's insight. As for the trapezium rule (or in modern terminology, Tai's Model), the insight is that the area under a curve lies between the following two quantities: (1) the area of the trapeziums that you can squeeze under the curve, but almost couldn't, and (2) the area of the trapeziums that you can't but almost could. Most people will probably have played with one of those toys with holes of different shapes and corresponding pegs for those holes. A pleasant lesson from this toy is that you cannot squeeze a square peg into a round hole, and then you develop the commonsense intuition of area (which, incidently (or incidentally?), may not be quite right since it leads to the Banach-Tarski paradox). Equipped with this, you might discover, as Dr. Tai did, that you can approximate an area by breaking it up into little rectangles and triangles, and then summing up their individual areas. It might be advisable to check if your surgeon is familiar with Tai's Model before your operation.

At the other end of the spectrum, some people in the medical profession do have a wonderful sense of humour: