A modern trigonometric table is a list of right triangles with hypotenuse 1 and approximations to the side lengths sin θ and cos θ, along with the ratio tan θ = sin θ/ cos θ . We propose that P322 is a different kind of trigonometric table which lists right triangles with long side 1, exact short side β and exact diagonal δ– in place of the approximations sin θ and cos θ. The ratios β/δ or δ/β (equivalent to tan θ) are not given because they cannot be calculated exactly on account of the divisions involved. Instead P322 separates this information into three exact numbers: a related squared ratio which can be used as an index, and simplified values b and d for β and δ which allow the user to make their own approximation to these ratios.

If this interpretation is correct, then P322 replaces Hipparchus’ ‘table of chords’ as the world’s oldest trigonometric table — but it is additionally unique because of its exact nature, which would make it the world’s only completely accurate trigonometric table. These insights expose an entirely new level of sophistication for OB mathematics. We present an improved approach to the generation and reconstruction of the table which concurs with Britton, Proust and Shnider (2011) on the likely missing columns. We present the generally accepted reconstruction of the table both in sexagesimal form as P322(CR) and also in approximate decimal form as P322(CR-Decimal8). We show that in principle the information on P322(CR) is sufficient to perform the same function as a modern trigonometric table using only OB techniques, and we apply it to contemporary OB questions regarding the measurements of a rectangle.

We then exhibit the impressive mathematical power of P322(CR-Decimal8) by showing that P322 holds its own as a computational device even against Madhava’s sine table from 3000 years later. This is a strong argument that the essential purpose of P322 was indeed trigonometric: suggesting that an OB scribe unwittingly created an effective trigonometric table 3000 years ahead of its time is an untenable position.

The novel approach to trigonometry and geometrical problems encapsulated by P322 resonates with modern investigations centered around rational trigonometry both in the Euclidean and non-Euclidean set-tings, including both hyperbolic and elliptic or spherical geometries (Wildberger, 2005, 2013, 2010). The classical framing of trigonometry and geometry around circles and angular measurement is only one of a spectrum of possible approaches, so perhaps we should view angular trigonometry as a social construct originating from the needs of Seleucid astronomy rather than a necessary and intrinsic aspect of geometry.--D.F. Mansfield, N.J. Wildberger (2017) "Plimpton 322 is Babylonian exact sexagesimal trigonometry,"Historia Mathematica, 2. [Ht Dan Schneider & Sam Rickless]

About a decade ago, I was contemplating an argument that suggested economic ideas needed to be tested on data-sets far removed, temporally and culturally, from the present (see here for the fruits of my research; and especially here). I googled, and stumbled on a dissertation, *The Bourse of Babylon*, by Alice Slotsky (now twenty years old). To simplify, because all dissertations are complex entities, it presented and interpreted ancient tables, going back many millennia, with price data alongside astronomical data (the latter being more familiar to historians of science). These price data alone has inspired quite a bit of fascinating research by economic and political historians. But none of the research I found (except for Slotsky's) really addressed the question why the ancient scribes of the temple of Marduk would put sophisticated astronomical data alongside detailed market data in the very same tables. While we cannot completely rule out that the tables were used to generate or predict omens, I think part of the answer has to do with the fact that these very same tables also include natural phenomena important to agriculture. That is, I thought, and still think, these tables represent a basket of goods (useful both for comparing purchasing power as well as for thinking about insurance purposes) that [A] allowed the ancient Babylonians to try to find, or manipulate, patterns in prices related to natural (astronomical and agricultural) cycles and, perhaps, political disturbances. Let's call [A] a hypothesis about Babylonian scientific mind-set in which earthly and celestial patterns may have an intricate connection. Of course, it is unclear what exactly is the content of [A] until the clay-tablets are studied with an open mind. (When I was a PhD student I got hints of the excitement surrounding these tablets from Noel Swerdlow.)

Now, Aristotle was familiar with [A] because in the *Politics* he reports a version of [A] as a fable about Thales:

"so the story goes, because of his poverty was taunted with the uselessness of philosophy; but from his knowledge of astronomy he had observed while it was still winter that there was going to be a large crop of olives. So he raised a small sum of money and paid round deposits for the whole of the olive-presses in Miletus and Chios, which he hired at a low rent as nobody was running him up; and when the season arrived, there was a sudden demand for a number of presses at the same time, and by letting them out on what terms he liked he realized a large sum of money, so proving that it is easy for philosophers to be rich if they choose, but this is not what they care about. Thales then is reported to have thus displayed his wisdom, but as [20] a matter of fact this device of taking an opportunity to secure a monopoly is a universal principle of business; hence even some states have recourse to this plan as a method of raising revenue when short of funds: they introduce a monopoly of marketable goods.." (Aristotle,

Politics 1;1259a)

What's interesting about Aristotle's response is that he offers an *unmasking* of the story. Aristotle shows that Thales exploits a universal mechanism (local monopoly) that drives up prices. Aristotle is not interested in exploring either (a) *an error theory*, that is, why would people have thought this about Thales nor (b) the possibility that the story is, in fact, a* true report* of Thales's *beliefs*. (In both cases, Aristotle's unmasking can still be the true explanation of the observed price rise.)+ In other research, I have explored the significance of Thales as the original exemplar of philosophical wisdom (including Aristotle's rejection of the Socratic idea -- recall *Theaetetus*, [174a] --that Thales contained secret, esoteric wisdom in *Nicomachean Ethics*, 6.7.5)[recall this post] and Hume's insistence that Thales is the origin of Spinozism (recall). Both (a) and (b) take the historical significance of [A], the Babylonian mind-set seriously in a way that Aristotle's move rules out.

That is to say, while we find in Plato, especially, occasional nods to older knowledge in Egypt and Mesopotamia and some invitation to explore it, Aristotle's philosophy, with its enormous interest in empirical observation, shows a remarkable tendency to exclude the very possibility of eastern, ancient knowledge and removes [A] from serious consideration. While Newton scholars are not unfamiliar with the fact that the Platonic impulse recurs throughout history, where it gets merged and touched with Hermetic and even Protestant ideas (for a great book that I have been reading on the topic, see Levitin's massive *Ancient Wisdom in the Age of Science*); the touchstone for knowledge are the concepts and ideas familiar from, and developed out of, Greek science and philosophy. (The previous sentence allows, of course, that this development also involves a series of revolutionary rejections of modes of thought found in Greek science and philosophy.)

The significance of this is two-fold: first, our philosophical tradition has an original rejection of what I (and, recall, Justin Smith) have called 'rustic wisdom' (recall here). We can (recall) understand the significance of David Graeber's work as, in part, a reflection on the consequences of this rejection: our inherited, urbane traditions of philosophy fail to recognize that our manners (note the plural) of theorizing are itself conditioned by particular and peculiar circumstances. In particular, these circumstances presuppose the violent eradication of alternative ways of thinking.

[Update: this paragraph inserted:]* The Babylonians represent, of course, an urbane philosophy (not rustic wisdom). What I had not previously grasped is that the original rejection of rustic wisdom was accompanied by a de-mystification and consequently lack of interest in the Babylonian scientific mind-set.

Second, and this reinforces the first, there is, thus, a recurring tendency to trace our origins to a clean slate that involves the starting-point of knowledge and, thus, a denial of other ways of knowing. This move is a recurring temptation at macro and micro levels. Sometimes these other ways are, of course, unimpressive or just crap. But the denial also always entails the genuine possibility of a version of (what we now call) a Kuhn-loss; the historical forgetting of previous knowledge that is, in some senses, superior to ours. Given my commitments, I should not be surprised that this process is even possible in mathematics. But I am surprised by Mansfield & Wildberger's findings.

I leave the final word today with Mansfield & Wildberger interpretation of P322:

Artifacts from ancient civilizations offer us a glimpse at their complex social and scientific achievements. Sophisticated artifacts offer us a larger window, and P322 is undeniably complex. Through P322 we see a forgotten ancient approach to trigonometry that is based on exact sexagesimal ratios.

The approximate nature of modern trigonometry is so culturally enshrined that we give it no second thought. It comes as a complete surprise, then, to find that the OB culture developed a trigonometry that actually only contains exact information, and so any imprecision is a consequence of using the table rather than inherent in the table itself.

P322 should be seen as an exact sexagesimal trigonometric table text...No other explanation has achieved this level of cohesion with the

evidence.

P322 is historically and mathematically significant because it is both the first trigonometric table and also the only trigonometric table that is precise. Irrational numbers and their approximations are seen as essential to classical metrical geometry, but here we have shown they are not actually necessary for trigonometry. If the dice of history had fallen a different way, and the deep mathematical understanding of the scribe who created P322 not been lost, then very possibly ratio-based trigonometry would have developed alongside our angle-based approach.

This new interpretation of P322 significantly elevates the status of Babylonian mathematics, and the vast number of untranslated tablets are likely to contain many more surprises waiting to be found. The discovery of trigonometry is attributed to the ancient Greeks, but this needs to be reconsidered in light of the much earlier, computationally simpler and more precise Babylonian style of exact sexagesimal trigonometry. In addition to being historically significant, P322 also brings the founding assumptions of our own mathematical culture into perspective. Perhaps this different and simpler way of thinking has the potential to unlock improvements in science, engineering, and mathematics education today. (24)

+Update: In private communication, Eric Brown has suggested that it is possible that Aristotle thought that "Thales' monopoly was especially lucrative because of his successful prediction, which rested on astronomical observation." I agree that in this post I have not given reasons for denying this possibility. I think there are two defeasible, considerations against this possibility: (i) in the rest of the paragraph, Aristotle does not suggests that political science/economy would benefit from the knowledge of astronomy; (ii) I know of no evidence that Aristotle thought astronomy could offer such evidence about crops.

*I thank Liam Kofi Bright for alerting me to confusion.

Thanks for a nice digression on an interesting range of topics!

Posted by: Norman Wildberger | 08/25/2017 at 05:40 PM

I've just been reading through your older posts on rustic wisdom. I hope that someday you'll bring all this material together (sophists, Phaedrus, professionalized [urban] philosophy, Kuhn-loss, etc) into one long essay. It's a fascinating and important subject and I appreciate your take on it.

Posted by: Mark Anderson | 08/29/2017 at 01:20 PM

Thank you for your kind words.

I have to admit that I have grown fond of the shorter essay and so increasingly less likely to write the long essay (except for the stuff that I need to publish for my pay-masters). I also fear that a lot of my work that I share on the blog is impossible to publish in ordinary journal-format.

Posted by: Eric Schliesser | 08/29/2017 at 01:30 PM