Hyperbole vs Hypotenuse – Did the Greeks Really “Not” Develop Trigonometry?

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Raphael’s iconic School of Athens

One of the best things about mathematics is that it makes life easier. When tired of doing so much work, don’t divide your working time. Instead, multiply! – by zero. Sure, you’ll be unemployed, but think about that unrestricted rest. Be a positive integer! And one of the best moments in life is discovering things that make mathematics easier. Learning that fingers can be used as numerical markers for counting and not just an apparatus for picking noses is one of the most memorable moments of childhood.

Of course, there’s no denying that mathematics can also make life difficult. “Fresh hell for fresh high-school students” is an almost universal theme. My high-school physics teacher, ever so cogent, saw early sines of struggle among us students, so she introduced to us the mnemonic SOH-CAH-TOA. We breezed through life after arming ourselves with that knowledge. Basically, it’s a phrase meant to help one remember how to solve for functions sine (SOH: Sine = opposite/hypotenuse), cosine (CAH: Cosine = adjacent/hypotenuse), and tangent (TOA: Tan = opposite/adjacent). Neat stuff.

This was really handy in solving vector problems that involved triangular diagrams. Hence, knowledge of trigonometry was vital – it can either be your angle or yuor devil. To choose the side of angels required familiarity with one theorem that is so basic, yet contain so much potency in solving various trigonometric problems: the Pythagorean theorem. What is it? It defines the relationships of the three lines of a right triangle. When explained in basic mathematical terms, the theorem states that the square of the hypotenuse (or the longest line of the triangle, the line opposite the right angle [90°]) is equal to the sum of the squares of the other two lines.

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Visual representation of the Pythagorean theorem with an accompanying equation to boot. Thanks Wolfram Alpha!

The theorem bears the name of Pythagoras, a Greek mathematician-cum-philosopher that lived in the 6th-5th BCE. Debates and disagreements still abound whether Pythagoras was the one who actually postulated the theorem, or if it was one (or a group) of his followers (called the Pythagoreans) and had the theorem attributed to him. No one can know for sure. What is underlined by these discussions is that Pythagoras inspired a Greek intellectual movement that developed trigonometry.

But all that is about to be demolished. The reputation of the Ancient Greeks as a wellspring of mathematical thinking – along with my childhood admiration for them – is about to go down the drain. On 24 August 2017, the UK based media outfit The Telegraph boldly declared in an article that:

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So the Greeks did not develop trigonometry. The Pythagoreans were fraudsters. Has my life been a lie?

To acquaint myself with the blatant historical falsity that I have been fed with as a schoolboy, I perused the article to see what the actual arguments and facts were. After doing so, my frown turned into a smile. My initial hurt has been healed. To the relief of my aching heart, the headline was just an outright lie.

The teetotal point of the article can be summarized by its opening paragraph:

A 3,700-year-old clay tablet has proven that the Babylonians developed trigonometry 1,500 years before the Greeks and were using a sophisticated method of mathematics which could change how we calculate today.

So it really isn’t about the Greeks not developing trigonometry, but it is about the Greeks not “inventing” trigonometry because the Babylonians did it before them. A point that is entirely different from what the headline announces. There is nothing in the article that presents the case that the Greeks did not develop trigonometry. This guile attempt to denigrate the Greeks to bolster the Babylonians, through the shrewd use of hyperbole betrays the basic definitions of the word develop in order to heighten the clickbait nature of the article. The court calls on Merriam-Webster to testify.

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Screenshot of the entry for “develop” from the Merriam-Webster Dictionary android app.

Telegraph seems to think that develop only means the third definition, that is, to create something. If that is the case, then the Greeks certainly did not develop trigonometry since it was created before them by the Babylonians. However, that does not mean that the Greeks did not cause trigonometry to grow and flourish, which also means to develop it.

Hipparchus is the Greek mathematician mentioned in the article used to prove their silly point. Since the discovered Babylonian tablet, Plimpton 322, predates the work of Hipparchus, Telegraph arrived at the puerile point that the Greeks did not develop trigonometry. But little do they know that the mere the mention of Hipparchus already exposes their dumb drivel. In his A History of Greek Mathematics, Sir Thomas Heath acknowledges the fact that Hipparchus is not the first to discover trigonometry, but “even if he did not invent it, Hipparchus is the first person of whose systematic use of trigonometry we have documentary evidence.” What this means is that archaeological artifacts show that Hipparchus advanced trigonometry during his time. Ergo, Hipparchus, a Greek, developed trigonometry.

But Hipparchus is just one name in a list of many Greek contributors.

There are the Pythagoreans who were already mentioned, whose theorem remains fundamental in trigonometry. (Fact: the word hypotenuse comes from the Greek hypoteinousa.)  Over in Egypt, the Greco-Roman Ptolemy used trigonometry in many of his diagrams and equations in his work Syntaxis Mathematica (popularly known as The Almagest). Menelaus of Alexandria also contributed to the growth of trigonometry (and geometry in general) by pioneering the study of spherical triangles.

Telegraph needs to be called out for their carelessness. It is worth noting that the writer of this article, and perhaps the one also responsible for the horrible headline, is their science editor. Hyperbolic headlines are cheap sensationalist tactics to gain quick clicks and readership. A science editor willing to abandon accuracy and veracity for the sake of catchpenny journalism does not deserve to be a science editor, nor at the very least be an editor.

And why is that so? It feeds and tolerates a modern crisis in social media wherein a vast number of readers don’t actually read the contents of an article, but immediately base their judgments on what headlines say. Forbes reports that 59% of social media users simply share articles without actually reading the contents (it would be nice to actually read the full paper, but it is behind a paywall). This number is alarming especially for scientific concerns, as it shows that misinformation can run rampant easily. This is what the Telegraph is guilty of in this gaffe. Take a look at the number of shares their article has on Facebook.

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188,925 shares!

Using 59% of 188,925 as a rough measure (since the study was based on Twitter, not on Facebook engagements) to estimate the number of times it has been shared without content checking, the article has had around 111,466 unread shares (if that’s the proper term). That’s more than a hundred thousand misinformed people thanks to Telegraph! Alarm quickly turns to hilarity once you read this entry in their About Us section on their website:

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Congratulations, Telegraph!

Here’s to hoping that in a few thousand years, robot archaeologists won’t find anything that may lead cyborg journalists to write “3,700 year old British article rewrites history of social media – and shows that The Telegraph did not become the no.1 quality news brand in the UK.”

They might bring a bit of credence to that credit if and only if they rectify their error. Unfortunately, they still haven’t bothered to remedy their recklessness. It’s important to urge them to do so, and pester them until they do.

Why?

Because what they did, aside from disqualifying the Greeks from any part in the development of trigonometry, is indirectly forward and support the false historical view that science is a series of “discoveries”, with one person or group beating others to the punch. What that disregards is the real movement of knowledge across human history and geological space. The reality being a series of “developments” and “improvements” upon older foundations of knowledge and thought – with episodes of discoveries and revolutions. Modern mathematics stands on the shoulders of Babylonian, Egyptian, Greek, Islamic, Indian, and Persian giants: each borrowing, learning, and developing from one another.

Taking away what other people have done for the advancement of mathematics doesn’t forward knowledge at all. Like mathematics, knowledge moves through the operations of addition, subtraction, division, and multiplication. And in all these operations, to arrive at a correct solution, all variables must also be correct. That’s something Telegraph doesn’t seem to know, I highly suspect.

So let’s thank the Greeks for developing trigonometry! Special appreciation should also be extended to the Babylonians for doing it way before the Greeks! For it is with their works that our modern existence is blessed with amazing technologies that make life easier. All these buildings, gadgets, and machines that we enjoy today have all benefited from the mathematics laid down by the ancients.

We all just want to make mathematics easier anyway, right? So why not just help each other out. The Babylonians made Plimpton 322 to make calculations easier. And unbeknownst to many, this Babylonian gizmo has a modern equivalent, proving the continuity of knowledge in our history. Don’t we also use the same gadget to help us out when we’re stuck with a problem?

Hey Apple, here’s an idea for your new project: the iPlimpton tablet.

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The Plimpton 322 tablet. Photo by Christine Proust©. Taken from: http://isaw.nyu.edu//exhibitions/before-pythagoras/items/plimpton-322/

 

EPILOGUE: Some will say that I am just inflating a minor pedantic issue. A hyperbole, in-itself. After all, don’t I also use clickbaity titles in some of my articles (like here, here, and here)? True. They do contain pathetic clickbait elements. But they are just modern spins of past predicaments; for example, the use of the word “prank” to describe al-Haytham’s ridiculous proclamations that got him incarcerated. These, however, are different from bold pronouncements that withhold people of their achievements and works. Pronouncements that contain no shred of truth. Recognizing the achievements of men and women of merit for their contributions to science and mathematics is the subject of this blog, and the Telegraph headline hits close to home.

References:

Frankopan, Peter. The silk roads: a new history of the world. Bloomsbury Publishing Plc: England, UK. 2016.

Heath, Thomas. A history of Greek mathematics: from Thales to Euclid. (Vol. 2). Dover Publications: New York, USA. 1981.

 

One comment

  1. […] And one of the most distinct theorems associated with the triangle is Ancient Greek by export, though arguably not exactly by origin. Everyone who has had the most basic lessons in trigonometry is sure to have […]

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