Tai's model

{{Short description|Rediscovery of the trapezoidal rule in 1994}}

{{Use mdy dates|date=May 2025}}

File:Trapezium2.gif approximation improves with more divisions.]]

In 1994, nutrition scholar Mary M. Tai published a paper in the journal Diabetes Care entitled "A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves". In the paper, Tai puts forth her discovery of "Tai's model", a method of estimating the area under a curve by dividing the area into simple polygons and summing their totals.{{cite journal |last1=Tai |first1=Mary M |title=A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves |journal=Diabetes Care |date=1 February 1994 |volume=17 |issue=2 |pages=152–154 |doi=10.2337/diacare.17.2.152 |pmid=8137688 |url=https://diabetesjournals.org/care/article/17/2/152/17985/A-Mathematical-Model-for-the-Determination-of |access-date=14 June 2025 |issn=0149-5992|url-access=subscription }} Apparently unbeknownst to Tai (or her peer reviewers and publisher), her "discovery" was in fact the trapezoidal rule, a basic method of calculus whose use dates back to Babylonian astronomers in 350 BCE.{{Cite web |date=2025-03-11 |title=In 1994, A Paper Claimed To Invent A Key Mathematical Rule Established Centuries Ago |url=https://www.iflscience.com/in-1994-a-paper-claimed-to-invent-a-key-mathematical-rule-established-centuries-ago-78368 |access-date=2025-05-05 |website=IFLScience |archive-date=March 21, 2025 |archive-url=https://web.archive.org/web/20250321013744/https://www.iflscience.com/in-1994-a-paper-claimed-to-invent-a-key-mathematical-rule-established-centuries-ago-78368 |url-status=live }}{{Cite web |last=Chasteen-Boyd |first=David |title=Why Future Physicians Should Study Math - Inquiro - Journal of Undergrad Research |url=https://www.uab.edu/inquiro/issues/past-issues/volume-10/why-future-physicians-should-study-math |year=2016 |access-date=2025-05-06 |website=www.uab.edu}}{{Cite book |last=Orlin |first=Ben |title=Change is the only constant: the wisdom of calculus in a madcap world |date=2019 |publisher=Black Dog & Leventhal Publishers |isbn=978-0-316-50908-4 |edition= |location=New York |chapter=XXII. 1994, The Year Calculus was Born}}

Several mathematicians replied to the paper in letters to the journal, objecting to the naming of "Tai's model" and the treatment of a method "used in undergraduate calculus courses" as a novel discovery in the field of diabetes care. A letter entitled "Tai's Formula is the Trapezoidal Rule" pointed out errors in Tai's representation of the underlying mathematics (such as referring to a count of square units below the curve as the "true value" of the area, against which to measure the accuracy of Tai's model) and problems with the method's applicability to glucose tolerance curves, which are already approximations.{{Cite journal |last1=Monaco |first1=Jane |last2=Anderson |first2=Randy |date=1994-10-01 |title=Tai's Formula Is the Trapezoidal Rule |url=https://diabetesjournals.org/care/article/17/10/1224/18808/Tai-s-Formula-Is-the-Trapezoidal-Rule |journal=Diabetes Care |volume=17 |issue=10 |pages=1224–1225 |doi=10.2337/diacare.17.10.1224 |pmid=7677819 |issn=0149-5992|url-access=subscription }}{{Cite journal |last1=Allison |first1=David B |last2=Paultre |first2=Furcy |last3=Maggio |first3=Carol |last4=Mezzitis |first4=Nicholas |last5=Pi-Sunyer |first5=F Xavier |date=1995-02-01 |title=The Use of Areas Under Curves in Diabetes Research |url=https://diabetesjournals.org/care/article/18/2/245/19165/The-Use-of-Areas-Under-Curves-in-Diabetes-Research |journal=Diabetes Care |volume=18 |issue=2 |pages=245–250 |doi=10.2337/diacare.18.2.245 |pmid=7729306 |issn=0149-5992|url-access=subscription }}

Tai responded to the letters, saying that she had derived the method independently during a session with her statistical advisor in 1981—noting that she had a witness to the model's originality.{{Cite journal |last=Tai |first=Mary M |date=1994-10-01 |title=Reply From Mary Tai |url=https://diabetesjournals.org/care/article/17/10/1225/18706/Reply-From-Mary-Tai |journal=Diabetes Care |volume=17 |issue=10 |pages=1225–1226 |doi=10.2337/diacare.17.10.1225b |issn=0149-5992 |archive-date=March 16, 2025 |access-date=May 5, 2025 |archive-url=https://web.archive.org/web/20250316152124/https://diabetesjournals.org/care/article/17/10/1225/18706/Reply-From-Mary-Tai |url-status=live |url-access=subscription }} She explained that Tai's model was only published at the request of her colleagues at the Obesity Research Center, who had been using her model and calling it "Tai's formula". Tai's colleagues wished to cite the formula, she explained, but could not do so as long as it remained unpublished, and thus she submitted it for publication.

Tai continued to refer to "Tai's model" as distinct in her rebuttal, arguing that she had worked out a design that presented the trapezoidal rule in a way that can be easily applied. Mathematicians Garcia and Miller pointed out in 2019 that "every calculus book in existence presents the trapezoidal rule in a manner that can easily be applied!"{{Cite book |last1=Garcia |first1=Stephan Ramon |title=100 years of math milestones: the Pi Mu Epsilon centennial collection |last2=Miller |first2=Steven J. |date=2019 |publisher=American Mathematical Society |isbn=978-1-4704-3652-0 |series=AMS Non-Series Monographs |location=Providence, Rhode Island |pages=435–436}} Tai disagreed that Tai's model is simply the trapezoidal rule, on the basis that her model uses the summed areas of rectangles and triangles rather than trapezoids. A follow-up letter by the authors of "Tai's Formula is the Trapezoidal Rule" pointed out that each contiguous rectangle–triangle pair in Tai's construction forms a single trapezoid.

"A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves" has been cited over 500 times as of March 2025. Forbes and IFLScience say that most of these citations are probably made in jest by researchers using the trapezoidal rule.{{Cite web |last=Knapp |first=Alex |title=Apparently, Calculus Was Invented In 1994 |url=https://www.forbes.com/sites/alexknapp/2011/11/10/apparently-calculus-was-invented-in-1994/ |date=November 10, 2011 |access-date=2025-05-06 |website=Forbes |archive-date=June 23, 2023 |archive-url=https://web.archive.org/web/20230623064729/https://www.forbes.com/sites/alexknapp/2011/11/10/apparently-calculus-was-invented-in-1994/ |url-status=live }}

The episode has been cited as an illustration of the slower-than-expected spread of knowledge in certain contexts.{{Cite book |last=Arbesman |first=Samuel |title=The Half-Life of Facts: Why Everything We Know Has an Expiration Date |date=27 August 2013 |publisher=Penguin Publishing Group |isbn=978-1591846512 |pages=63–64}} It has been discussed as a failure of peer review.{{Cite web |last=Schütz |first=Helmut |date=2025-02-08 |title=Trapezoidal Rules |url=https://bebac.at/articles/Trapezoidal-Rules.phtml |access-date=2025-05-16 |website=Helmut Schütz |language=en}} Garcia and Miller call it a cautionary tale in verifying the originality of one's work before publishing it.