Posted by Phil Price:

A blogger (can’t find his name anywhere on his blog) points to an article in the medical literature in 1994 that is…well, it’s shocking, is what it is. This is from the abstract:

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

Yes, that’s right, this guy has rediscovered the trapezoidal rule. You know, that thing most readers of this blog were taught back in 11th or 12th grade, and all med students were taught by freshman year in college.

The blogger finds this amusing, but I find it mostly upsetting and sad. Which is sadder: (1) That this paper got past the referees, (2) that it has been cited dozens of times in the medical literature, including this year, (3) that, if the abstract is to be believed, many medical researchers DON’T use an accurate method to calculate the area under a curve.

Things gets reinvented all the time. I, too, have published results that I’ve later found were previously published by someone else. But I’ve never done it with something that is taught in high school calculus. And — I’m practically spluttering with indignation — if I wanted to calculate something like the area under a curve, I would at least first see if there is already a known way to do it! I wouldn’t invent an obvious method, name it after myself, and send it to a journal, without it ever occurring to me that, gee, maybe someone else has thought about this already! Grrrrrr.

It has been cited 137 times, according to goog scholar. 137 times. And consider that each of those citing papers has multiple authors, this is probably more than 500 people with PhDs who are crediting Tai as a founder of calculus, or something. Better than any of my papers.

Of course, all these blogs linking to the paper probably aren't doing it any harm either.

Hmm, but thinking about his method, I wonder if anyone has ever tried using rectangles with the height determined by the function value at the midpoint. That could save all that difficult use of geometric formulae for triangles (though I think there is a paper from the late 80s that figured out that problem…).

What is surprising to me is that he did not apply for a patent on this.

Yes, the one-eyed man is king in the land of the blind

But perhaps we should not be overly negative about that

This paper may have improved medical care for patients as well as provided some motivations for better math education more widely.

Also, being statisticians we are very likley not to do that with math but perhaps just as likely to do that in philosophy or other areas we have had less formal schooling in.

And admittedly I did something very similar with envelope rules in my thesis. These are almost as simple as trapezoidal rules, but much less well known and I did include references.

Rambling on – simplicity (and obviousness) is in the mind of the beholder.

K?

p.s. love the rational animation in the post before this one – now should I forward it to my teenage son???

One of the comments on the original post points to the PubMed listing for the article (http://www.ncbi.nlm.nih.gov/pubmed/8137688?dopt=Abstract). Apparently there were four response articles published in what I can only assume is the next issue of the same journal. One of them has the apt title "Tai's formula is the trapezoidal rule."

Perhaps an interesting line of discussion is why people continue to cite Tai despite the corrections. It suggests a problem with academic publishing that once something is in a journal, it's hard to take it out even when it's wrong. (For reference, that correction has only been cited 3 times.)

In partial mitigation, one of the four responses/letters to the paper is titled "Tai's formula is the trapezoidal rule". I imagine the other three say more or less the same thing.

Hopefully some of the people citing it were using it as an example of why you need basic maths knowledge.

Like those WinBugs examples with too wrong random effects priors that Andrew pointed out a few years back.

re: "it's hard to take it out even when it's wrong"

Nicely though, on the WinBugs site the "too wrong" priors are commented out rather than removed – which signals an earlier less than ideal state.

K?

I have heard apocryphally that in one lab, before ubiquitous desktop computers, the areas under curves were determined by plotting the curves, cutting them out with scissors and weighing the paper bits on a lab balance.

I don't know of a methods article for that though.

Peter, I've used that method myself (a long time ago, of course) in an analytical chemistry lab.

When I saw the term 'abscissas,' I knew something was awry.

The comments were quite entertaining (as was the author's reply), and are worth a trip to the medical library to read in full. Here are a few snippets.

Commenter 1: "Tai describes a method to determine total area under metabolic curves. However, what is exaggeratedly called 'Tai's mathematical model' is nothing but a simple geometrical formula, well known for many years as the trapezoidal rule. . . . [Her] validation of the formula by means of comparison with a 'true value' [by placing the curve over a piece of graph paper and counting up the little squares] is useless and contains several fallacies."

Commenter 2: "[Tai] uses the trapezoid rule, a basic geometrical concept, which is that the area of a trapezoid is the mean of the length of the two parallel sides times the width. This method has been used . . . for many yeas, and, in my opinion, does not need a new name."

Commenter3: "We were disturbed to read the article by M. M. Tai . . . The author seems to claim 'Tai's formula' as a new method of computing area under a curve. The formula given is simply the trapezoidal rule, published in many beginning calculus texts . . . Although we do not have a first reference, it is our understanding that the trapezoidal rule was known to Isaac Newton in the 17th century."

Full text of the letters to the editor and the author's response are here.

Interestingly, many of the more recent papers that cite it appear to split the difference and say "we use the trapezoidal rule [Tai94] to calculate glucose absorption…" or something like this. So citing this paper persists even after people understand what's going on.