Tag Archives: bad mathematics

Really, I expect better of these guys

There was no blog post yesterday because I was largely on the road back from seeing Hawkwind’s 50th anniversary tour, which practically counts as medieval history itself. Today, however, like quite a lot of the UK academy, I am on strike, and so I have time to make up for that omission. Indeed, if I do it right, I should have time to do some serious blog catch-up work, though if you are in the Leeds area, you may be interested to know that, with my colleague Dr Francesca Petrizzo, I am participating in the local University and College Union’s teach-out at the Quaker Meeting House on Woodhouse Lane on the 28th November, at 14:00-15:00, and that is open to the public, so you could come along and learn from us about ‘The Medieval Mediterranean: Race and Religion’. Maybe see you there! But if not, here is a blog post of a more normal kind, and more will hopefully follow.

UCU pickets during the 2018 strikes at Leeds

UCU pickets during the 2018 strikes at Leeds

So this post got stubbed while I was redrafting the article which became ‘Outgrowing the Dark Ages’ back in May 2016.1 I have written here before about the footnote that you have slaved over, that has grown far too big because it is really a tangent from the article or chapter and, in the final redraft, even as you edit it you know will, in the end, have to be cut. This is one of those. In the end, it did survive in a form, but much truncated.2 The problem of the article, as you may already have seen, is that people have generally misapplied the few numbers we have for agricultural productivity in the early Middle Ages, and that the person who did this with most success, in as much as he has been replicated all over the place, was Georges Duby. But he was not alone in doing bad maths with agricultural figures, and that’s where we come in… (The footnotes I have added; I don’t go quite as far as having footnotes in my footnotes. Not yet.)

“Of course, not everything that has been badly calculated about early medieval crop yields can be placed at the door of Georges Duby. Just as there is good reason to doubt his figures on the basis of experiments in Catalonia, so also there are Catalan attempts at such arithmetic that likewise fail to be justifiable.3 In a study of the ninth-century foundation and refoundation of the Pyrenean monastery of Sant Andreu d’Eixalada and then Sant Miquel de Cuixà, Ramon d’Abadal i de Vinyals noticed in the will of the monastery’s major patron, Protasi, a bequest of all the cereals in the monastery compound, enumerated as 365 modios.4 Courageously, he assumed that this was close to or actually the yield from the probably-recent harvest, and then combined these figures with an earlier donation of an estate by Protasi where annual renders are given, ‘4 houses and a courtyard and 6 orchards and 12 vineyards, and the 30 quinales of wine that go out from there and there are 8 tonnae and 30 modii of corn’. Using this to establish a basic render figure of 7.5 modii of corn per house, Abadal then used the monastery total figure to estimate the house’s total landed endowment. This ingenious operation involved not just the assumption about proximity to the harvest and a myriad of other assumptions, some silent and some supplied in a lengthy footnote, about how much grain was needed to sow a modiata of land and the yield a modiata should produce, all supplied by late nineteenth-century figures from the same area based on a modern calculation of the area of a modiata (‘a little less than half a hectare’). Even if one cared to accept all these assumptions and patches, the essential uselessness of the figures thus obtained should have been apparent to Abadal’s readers when he explained, halfway through the sums, ‘Since we must think, however, that an important part of the harvest relating to these 365 modios of wheat should have corresponded to the direct cultivation of the monastery and not to that of its tenants, if we compute that part at a half…’ For this guess, immediately halving Abadal’s result, there is not even an anonymous nineteenth-century basis and it shows us, again, quite how much needs to be known, when performing such arithmetic, but is not.

Map of the estate of Sant Miquel de Cuixà in 1812

Map of the estate of Sant Miquel de Cuixà in 1812, probably closer to the situation that Abadal described than the ninth-century one. Image by ClaudefàTreball propi, licensed under CC BY-SA 4.0, via Wikimedia Commons.

“Abadal, however, did not use these numbers to achieve a crop yield ratio: how could he have, when he had already supplied that part of the sum from his unnamed nineteenth-century source? This did not, however, stop Manuel Riu i Riu, in an article about metrology and terms for units, referring to this study as if it contained ninth-century figures for both seed sown and crop yielded.5 The former he based on the equation between the land unit modiata and the modius supposedly required to sow it; the latter he got from Abadal’s own figures, not apparently noticing that these were modern patches for the data lacking in the documents. As it happens, the figures that he gave provide a healthy yield figure of 6.25:1, but they are, of course, founded on absolutely nothing of meaning.”

Now, this is not the first time that we have caught Manuel Riu, superb archaeologist and excellent builder of the scholarly community of medievalists in Catalonia but not always quite as critical in his reading of texts as he needed to be, in a slip, but he was famous for his quantitative work and study of medieval units and measures, and he knew Abadal well, and I’d have hoped that he would read him more carefully; but then I’d also have hoped that Abadal wouldn’t have been quite so creative in his invention of his own data. The whole thing is further proof that if you invent numbers in historiography, people will quote them whatever they rest upon, even when they really shouldn’t. I don’t hope to change that as a whole trend, but it would be nice if I could make people more careful about it in this specific area…


1. Jonathan Jarrett, “Outgrowing the Dark Ages: agrarian productivity in Carolingian Europe re-evaluated” in Agricultural History Review Vol. 67 (Reading 2019), pp. 1–28.

2. Ibid., pp. 20-21 n. 77.

3. For the Catalan reasons to doubt Duby, see my older blog post or indeed Jarrett, “Outgrowing the Dark Ages”, pp. 22-25.

4. Ramon d’Abadal i de Vinyals, “Com neix i creix un gran monestir pirinenc abans de l’any mil: Eixalada-Cuixà” in Analecta Montserratensia Vol. 8 (Montserrat 1954), pp. 125–337 at pp. 160-161, is the relevant source; the reprint in Abadal, Dels Visigots als Catalans, ed. Jaume Sobrequés i Callicó, Estudis i documents 13-14 (Barcelona 1969), 2 vols, I, pp. 377–484, doesn’t have the documentary appendix so lacks this bit.

5. Manuel Riu, “Pesos, mides i mesures a la Catalunya del segle XIII: Aportació al seu estudi” in Anuario de Estudios Medievales Vol. 26 (Barcelona 1996), pp. 825–37, reprinted in Immaculada Ollich, Montserrat Rocafiguera and Maria Ocaña (edd.), Experimentació arqueològica sobre conreus medievals a l’Esquerda, 1991-1994: arqueològia experimental. Aplicació a l’agricultura medieval mediterrània (DGICYT PB90-0430) (Barcelona 1998), pp. 77–82.

A Theory under Siege

You may recall that some time ago I recorded that I had been reading David Bachrach’s Warfare in Tenth-Century Germany as part of my ongoing reworking of my paper about military service in tenth-century Catalonia.1 There are ways in which that was really useful and ways in which we are just talking about different things, but a thing that we both talk about but take quite different views on is the normal size of early medieval armed forces. Now, this is an old controversy, which goes back far beyond this book and involves names like Halsall, Sawyer and not least, Bernard S. Bachrach, and in some ways I would prefer not to get into it, not least as several of these people including Professor Bachrach (the younger) have been known to read this blog, but I can’t avoid it.2 I would confine myself to an argument that the situation I see in Catalonia, of guardposts and border-raiding and no serious armies except when an Andalusi one comes calling, is just very different from the full-on imperial warfare of Ottonian Germany, and in the article I expect that I will. But still, there are premises to the large army argument, as set up in this book at least, that I find hard to accept, and this post is an attempt at a critical examination of one of them. Where most openly stated, it goes like this:

“Ottonian military operations were consistent with warfare throughout the medieval millennium, which was dominated by sieges, particularly in the context of campaigns that were intended to conquer territory. Contrary to the long-established narrative that the Ottonian kings fielded small armies of a few hundred to a few thousand heavily armed mounted fighters (Ritterkrieger) led by warrior aristocrats, the siege operations that dominated warfare in the tenth century required very large armies, composed predominantly of foot-soldiers.3

One is, initially at least, left to assume the premises behind this. It’s easy enough to come up with some: walled cities are naturally quite large, and need a lot of defenders to keep a whole perimeter secure. If you’re attacking them, you must need more attackers than defenders, right? As it turns out, in fact, you allegedly need quite a lot more: once you get through to p. 226 you learn that, “in order to storm a strongly held enemy fortress an attacker required four to five men for every defender.” The justification for that, however, is farmed out to an article by Bernard Bachrach and Rutherford Aris.4 Now, if you actually get hold of that article, that doesn’t seem to me to be what it says: instead, by virtue of some extremely hypothetical probablity mathematics, it says that a charge by a Viking warband at a typical Anglo-Saxon burh defended by archers of the number implied by the Burghal Hidage (1 to every 1.3 m of wall) would probably have resulted in one to two attackers in every four being hit before they reached the wall.5 Even if that mathematics were somehow realistic, it’s quite a specific situation and one calculated on the basis that Vikings attacking a fortress would only have ladders to deploy so needed to make that approach.6 I don’t see how it can be transportable to a large-scale military operation with siege engines such as (David) Bachrach thinks the Ottonians were able to mount.7 Moreover, it doesn’t actually provide the numbers that he employs, only an implication that if you were to outnumber your opposing force once atop the walls you needed to allow for a fifty per cent casualty rate when planning your attack. That seems like awful odds that no commander would have risked to me, but it’s obviously not what an Ottonian planner would have been facing.

Interior view of the restored Romanesque city walls at Worms

Interior view of the restored Romanesque city walls at Worms, one of the fortress cities Bachrach considers and possibly not too much unlike this in the Ottonian period? CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=167751.

But let’s take one step further back. These are figures based on the idea that victory at a siege is achieved by storming and carrying the enemy’s defences, and indeed so is Bachrach’s account of Ottonian warfare against fortifications more generally.8 But this is, surely, not the only way. More conventional, if much slower, would be simply to starve the defenders out. Now, in that case, your army requirements drop radically. Whereas before you perhaps had to have men all round the fortification, now you really only need to guard the points of access. Possibly someone could resupply a city by hauling sacks up the walls in the dead of night but not, you might think, in any real quantity, especially if you as attacker are sending patrols around every twenty minutes or so, which I assume you, as a thinking tactician, probably would be. Their supply will still not be equal to their demand. This makes siege warfare a much less demanding effort in terms of numbers; one must still be able to supply the besiegers for a prolonged period of time, but that’s easier if there’s fewer of them needed.

Illustration of a (small) siege from the fourteenth-century Codex Manesse

Even more anachronistic, but far too good a picture not to use, an illustration of a (small) siege from the fourteenth-century Codex Manesse

Of course, you still need to have more men than the defenders do, otherwise they could just sortie out and squash your forces. But wait a minute. How many more men than you do they have to have to do that before it would mean leaving the walls partly undefended, giving your troops access to the city and thus losing the war for the sake of a battle? There’s more balance here than might immediately appear, because both sides need their forces at least partly dispersed to hold their positions. The various ways that concentration of force could play out here seem to me to be very hard to calculate with and I’m not at all sure it reduces to any simple arithmetic beyond the basic logic that the more troops whom you can feed you have available, the more tactical options you have. But in a siege situation, that proviso about feeding is quite important, often for both sides.

Crop-marks clearly showing a fortress, supposedly the Slavic fortress of Gana, at Hof-Stauchitz

A different (and more usual?) scale of opponent, crop-marks clearly showing a fortress, supposedly the Slavic fortress of Gana, at Hof-Stauchitz

I should say that I am not, a priori, against the idea that the Ottonians could sometimes field quite large armies, meaning in the double figures of thousands of men, although whether such were necessary for all their campaigns I rather doubt. When they were, though, they had a whole empire to draw upon and Bachrach has here a whole book full of details about how such things might have been organised, resourced, supplied and led which almost make his argument for him; one can reasonably assume that the Ottonians would have wanted to raise large armies and Bachrach shows us how they could have. But he argues it the other way around, from the necessary existence of large armies, through a reading of the sources which illustrates that, to the techniques for their provision and operation, and if these are the starting premises, they seem very shaky to me.


1. D. S. Bachrach, Warfare in Tenth-Century Germany (Woodbridge 2012, repr. 2014).

2. In reverse chronological order, more or less, Guy Halsall, Warfare and Society in the Barbarian West, 450-900 (London 2003); Bernard S. Bachrach, Merovingian Military Organization 481-751 (Minneapolis 1972), what I think of as the wellspring of the maximalist argument, and Peter H. Sawyer, The Age of the Vikings (London 1962, 2nd edn. 1971), which is the place where I first met a sustained attack on the numbers which early medieval sources used for army size. In all these cases, albeit Sawyer least and Bachrach most, references could be proliferated; a lot of Bachrach’s most relevant works are collected in his Warfare and Military Organization in Pre-Crusade Europe, Variorum Collected Studies 720 (Aldershot 2002).

3. D. S. Bachrach, Warfare in Tenth-Century Germany, p. 1, which is a good place to state your axioms after all.

4. B. S. Bachrach & R. Aris, “Military Technology and Garrison Organization: some observations on Anglo-Saxon military thinking in light of the Burghal Hidage” in Technology and Culture Vol. 31 (Baltimore 1990), pp. 1-17, on JSTOR here, repr. in B. S. Bachrach, Warfare and Military Organization, III, cit. D. Bachrach, Warfare in Tenth-Century Germany, p. 226 n. 2. A much fuller version of the argument here is however to be found in B. S. Bachrach & David Bachrach, “Early Saxon Frontier Warfare: Henry I, Otto I and Carolingian Military Institutions” in Journal of Medieval Military History Vol. 10 (Woodbridge 2012), pp. 17-60, which was presumably not available to cite in D. S. Bachrach, Warfare in Tenth-Century Germany, and would have made that a much longer book to reprise there. I engage with the version in the book here, which I assume to be self-standing.

5. B. S. Bachrach & Aris, “Military Technology”, pp. 5-10 with an appendix for the mathematics pp. 14-18.

6. Ibid., pp. 3-5, on the not unreasonable basis that the Burghal Hidage uses the same ratio for number of defenders required for places with eight-foot thick Roman stone walls (Winchester) and earthen ramparts with wooden palisades (Wareham), so the writers obviously weren’t thinking in terms of attacks actually upon the defences.

7. D. S. Bachrach, Warfare in Tenth-Century Germany, pp. 151-167, covers Ottonian siege equipment, including ladders but also mantlets, rams, ballistas, catapults and mobile towers, largely by providing detail from Vegetius and anchoring it in less-detailed reports from Ottonian sources.

8. Ibid., pp. 12-13 and repeatedly exemplified in his narrative of Henry I’s and Otto I’s campaigns provided pp. 14-69; see also Bachrach & Bachrach, “Early Saxon Frontier Warfare” for a fuller statement of the position.

A way publishers might make Open Access work (but probably wouldn’t)

It is an established trope of this blog that academic publishing is in trouble because of the Internet, and much like the music industry has yet to come up with a really viable alternative to a doomed defensive protectionism. This was already true before the Open Access movement started gaining velocity and a logo and so on, but that has greatly complicated things because, like so many radical movements, its ideological position seems to involve everyone doing more for free, and you can’t run large-scale quality control and distribution on no money, it needs full-time staff. About this time last year, however, some new pieces of the puzzle fell into place that seemed then to represent a possible way forward, and I stubbed a post to speculate about it. It turns out, on writing it up, that really this is two posts, one about how publication is paid for and one about how the work academics put into publishing is compensated. You’d think these were the same issue but it is, of course, the state that we’re in that it’s not. So here’s the first post, about making open access journals (and by extension other forms of open-access publication) work.

I should say straight away, by the way, that as usual with posts about Open Access this is really only a concern for a UK audience for the moment. Those interested in such issues elsewhere may still want to know what hoops the peculiar UK situation is making us jump through, however. My thinking process this time started with a blog post by Natasha White of the publishers Wiley, addressing the particular economic flaw of gold-standard open access in which a grant or an institution (hopefully) pays for an article to be published, and then has to pay again to buy the journal in which that article actually appears, because not everything else in it will be open-access. This is far from the biggest problem with gold OA, which is in any case basically irrelevant to the humanities due to its costs and our funding, but every little step towards a new model for the Academy at large could help, right? Ms White hits the kernel of the issue here:

A number of publishers, including Wiley, have introduced policies to adjust subscription prices for any shift from subscription-funded articles to pay-to-publish open access articles. Journals publishing more open access articles will see price decreases because the publication costs for those articles have already been met. Since non-UK authors don’t have the same type and level of funding to pay open access fees, the majority continue to choose to publish under the subscription model, keeping this the predominant publishing model. This means subscription prices haven’t decreased significantly and UK institutions continue to pay for journal subscriptions to obtain international research. So the UK is seeing an increase in publishing costs as they pay for both open access and subscription fees.

In short, as long as a journal is not entirely gold-standard OA (which would involve barring contributors who can’t do that, i. e. most of the world and, once again, also of the humanities), there will be a subscription price to pay for that journal, which even those publishing under gold OA must pay. Now, however, Jisc have come up with the idea of OA contributions also buying an institution credits, with which they can offset their journal subscriptions with a given publisher according to how much of its publication costs they have already paid. It seems fair as an idea, and represents a creditable willingness on the part of the publishers to make less money for a principle. Of course, an awful lot depends on the exact pricing, not least because nothing seems to require that the compensation actually equal the cost of the subscription and there’s a whole host of issues about who gets the compensation: the author’s university’s library, the grant-making body that paid for the research… So let’s have a play with some hypothetical numbers.

The publisher’s interest seems to be the crucial one here, so I did a bit of back of the envelope maths. Springer helpfully put their journal prices online, unlike most publishers, and the median price for 2014 was €715. The journal in the list charging closest to that was Pituitary, and that year they had a €2,200 open access article processing charge and published 87 articles that year. These are maybe not typical figures (especially for the humanities!), but they are at least middling ones that a real commercial publisher is or was actually using. So, simple arithmetic: if every one of those 87 articles had been published gold OA, it would have brought Springer in €191,400, which is to say, just under 268 subscriptions. I have no idea how many places do in fact subscribe to Pituitary, but there were about 22,000 universities in the world in 2013, so you’d think that Springer could certainly hope for more than 268 of them buying in.

Going full-on gold OA could thus cost Springer a substantial sum of money in that model. Article processing charges also have to be paid on articles that get rejected, however, and I don’t know what the rejection rate is. This suddenly makes a lot of difference, and if we are setting up a model in which publishers encourage us to give them money for being considered for publication, and then only the successful get compensation, it looks rather more sinister, doesn’t it? Imagine a humanities version of this, a fictional Exclusive Journal of Medieval Studies, publishing 16 articles a year fully open access, gold all the way, and charged for that at the same rate as Pituitary, but rejecting nineteen for each one it published. This is obviously not typical for the humanities, but as we’ve already said, the humanities ecosystem can probably support very few such organisms, if any, so the extreme is where we have to start. That high level of rejection would be to the journal’s advantage in terms of reputation, but it would also be greatly to the publisher’s financial benefit, because as long as they accepted their 16 articles annually the rejected submissions would be almost pure revenue. 19 times 16 is 304; 304 Pituitary-sized APCs is €668,800, which is 935 Pituitary-sized subscriptions. Suddenly it looks a lot better than capping your revenue at €191,400, doesn’t it? This might nearly make up for the shift of publishing model, and if it didn’t, prices could presumably be changed so that it would. And the more people who wanted to get into this highly exclusive journal but failed, the better those sums would look for the publisher.

So, now consider a halfway house in which this invented journal is still publishing half its articles on a conventional subscription model and the other half on gold OA. Let’s also say that Wiley’s prediction is accurate and that their subscription price remains the same in this world, but they compensate the gold OA authors or their institutions with a pay-out equivalent to the subscription. (Multiple authors obviously makes this model a lot more complex but let’s leave that for now and assume all authors are lone scholars.) Now, the publisher’s revenue comes only from subscriptions or the charges for the OA articles that it rejects. Think about what that means for its would-be authors and the open access agenda for just a moment… To me it seems that while the publisher would have every incentive to encourage open access submissions, that incentive would also pressure them to accept as few as possible.

This would mean, of course, encouraging an awful lot of lower-quality submissions somehow while still only publishing good ones. It’s hard to see such a policy working for long—why would you bother submitting unless you were pretty sure your work was excellent?—but even if it did, good authors might come to prefer to publish somewhere less notoriously keen to gather in APCs without return, because they would probably justly fear what the effect of that on the editorial agenda would be, and then the journal would only have less good work to publish and its reputation would drop. So there is probably a long-term cost to taking this path. But all this is to assume a lot of awareness among authors, which surveys about open access in the UK and the reactions to Elsevier actually enforcing the agreements its authors had signed suggest we don’t have! Nonetheless, even this dally with made-up figures should show that it really does matter what the figures actually are, and so the news that Jisc has basically negotiated a separate (and presumably private) agreement with every one of the publishers who’s so far playing may not be as good as it initially sounds…

The other thing it implies, of course, is a huge pool of reviewers willing to read a lot of bad work, which is probably the real reason this scenario couldn’t arise. But then, technology is also offering ways around that problem, and that’s where the second part of this post was originally meant to go. But this is already long enough, and so that can wait for a couple of days. Meanwhile your thoughts or corrections would as ever be very welcome!

The kind of maths we should not do

A lot of the problems any historian of the early Middle Ages faces are about how typical any given piece of evidence is. When so little survives, can we generalise from the few fragments we have across the great spaces where we simply know nothing? I came up against this while writing the post some time back about widow warlords, where as you may remember I wound up trying to argue for a level of social occurrence that could be common enough to be frequent while still being statistically unusual. The question remained then: how unusual? And this led me to thinking about the best evidence I have for female presence in local society, the good old Vall de Sant Joan hearing, and then the temptation stole upon me to do some very bad maths.

Aerial view of the Vall de Sant Joan de les Abadesses

Aerial view of the Vall de Sant Joan de les Abadesses. I know I use this picture a lot but I find it really hard to get tired of. However, I can no longer find where I got it from, so if you happen to own it do let me know…

Y’see, the Vall de Sant Joan hearing seems to be really good evidence for population size, at least by our starvling early medieval standards. We do not know the whole population of the area, but we think we know how many households there were in it, and we know what size it was: 269, by my count, and about 7 km2.1 Now, we could just multiply up, because the Vall de Sant Joan is in some sense a jurisdictional term and we know how many of those there were in the tenth-century county of Osona, give or take a few for changes, and it’s thirty-seven. If each contained this many households, tenth-century Osona would have been a county of nearly ten thousand households.

The town of Sant Joan de les Abadesses, viewed from across the Pont Vell

There probably aren’t that many households in the Vall de Sant Joan now, for a start, though I wouldn’t mind going back again to look (albeit this time with a car). Image by Espencat (Own work) [Public domain], via Wikimedia Commons

In fact, that is really unlikely to be true, because this was a frontier county and we’re counting its very inward corner, where we can document, more or less by the fact that we can document anything but also by the nature of the actual documents, that an ambitious lordship interest was moving people in here and encouraging settlement that is quite unusually dense.2 Such a figure is likely to be a massive over-estimate. So what should we do instead? Now, here the words of my old colleague Ted Buttrey come back to me with force:

“What should we do? We should do nothing. Nothing can be done. There is no solution to this problem, beyond inventing new data to push the inquiry into the realm of the fanciful. This is uncomfortable but it is true. If we allow ourselves, in our frustration, to confect the missing data, we will to that extent have destroyed our own purpose. To create quantitative studies built of imaginary data, to force an answer by assuring ourselves and others that we know what we do not, and cannot, is to compromise everything that we hold important. Each of us builds, and others build upon us: when we dress up guesses as data we do permanent damage to our scholarship, and to the scholarship of others.”3

He is right, of course, I know he’s right. He is also right that bad guesses get out there and get used even when they are explicitly qualified as such.4 So I must not, I must not attempt to correct the above error by breaking the data down, down to the level of households per villa (which would be 12·2 NO STOP IT), and then multiplying up by the number of villae in Osona. I should not do that not least because we don’t know with any certainty how many villae there were in Osona around the year 913, which is when this data would be comparable, probably not even in total for the tenth century which would add many more than there then were and would fail in any way to counter for the factor of population change over that century; I should not do that because, again, villae in the Vall de Sant Joan were probably over-many and over-stuffed compared to other areas and though those two errors might tend in opposite directions, we cannot know that they would cancel each other out; I should not do it because any operation involving multiplying up a small number to obtain a large one necessarily multiplies the error in that number just-as-many-fold; and I should not do it for many other good solid reasons of mathematical rigour. And in fact I will not. But it is sorely tempting, just because it’s hard to rid myself of the idea that if I could allow for enough factors, this would actually be a better basis for early medieval population figures than we currently have anywhere else.5 But every one of those corrections would be another piece of fiction, an error to be multiplied up. Ted again has the correct admonitions:

“When we enter on these kinds of calculation, we can be confident of two things. First, the answer will be wrong. Whatever it is, it will be wrong, since it cannot be right—once you are guessing, the number of possible permutations is gigantic. Worse, where the errors lies, and how serious they are, cannot be determined… Secondly, we can be confident of something else: when we publish this sort of thing, no matter that it be all set about with caveats and qualifications, the very fact that we thought it worth publishing will give it credibility.”6

And that is of course exactly the pain of it; there are figures that are thought credible abroad already that I feel must be wrong, because the person who put them together on the evidence we don’t have made his own set of assumptions about how the lack of evidence should be countered, and now I prefer my assumptions to his and would like to put into circulation alternative figures that are no more verifiable but feel more likely to me. But this will not make things better. Ted can have the last word, albeit he gives it to someone else:

“We should take to heart the dictum of a character in Umberto Eco’s novel, Foucault’s Pendulum, who explains, ‘For every complex problem there is a simple solution; and it is wrong.'”7


1. The reason we assume that the document, which is a vast parchment recording the names of people who swore that Abbess Emma of Sant Joan de Ripoll had been given the valley to settle by her father Count guifré after he expelled the Saracens from it, records households is because about half of its signatories are female, and mostly appear with a male partner. This looks like an attempt to implicate all the conjugal pairs of the valley in what was in fact a political fiction (see Jonathan Jarrett, “Power over Past and Future: Abbess Emma and the nunnery of Sant Joan de les Abadesses” in Early Medieval Europe Vol. 12 (Oxford 2005 for 2003), pp. 229-258, DOI: 10.1111/j.0963-9462.2004.00128.x), but since there are others who aren’t in pairs, it must also be more than that. Hence, households seems likely. The argument is made most thoroughly in Gaspar Feliu i Montfort, “Sant Joan de les Abadesses: algunes precisions sobre l’acta judicial del 913 i el poblament de la vall” in S. Claramunt and M. T. Ferrer i Mallol (edd.), Homenatge a la memòria del Prof. Dr. Emilio Sáez: aplecs d’estudis dels seus deixebles i collaboradors (Barcelona 1989), pp. 421-434. The count of these households I just redid from a spreadsheet I constructed when writing the thesis that lies behind Jonathan Jarrett, Rulers and Ruled in Tenth-Century Catalonia: pathways of power (Woodbridge 2010), where you can find more detail at pp. 35-51. The area I estimate from the map in Jordi Bolòs & Victor Hurtado, Atles del Comtat d’Osona (798-993) (Barcelona 2001), pp. 94-95 at p. 94. Thus my doubtless inaccurate estimation is already one basic source of error!

2. This is the basic story of Jarrett, Rulers and Ruled, as above and also pp. 57-64.

3. Theodore V. Buttrey, “Calculating Ancient Coin Production: facts and fantasies”, The President’s Address in Numismatic Chronicle Vol. 153 (London 1993), pp. 335-351 at p. 351.

4. My best example is another numismatic one, an article by Warren Esty, “Estimation of the size of a coinage: a survey and comparison of methods” in Numismatic Chronicle Vol. 146 (London 1986), pp. 185–215, which pitted a range of statistical techniques then in use to reason up to ancient currency sizes from current surviving evidence against each other by means of a randomly-generated virtual hoard, and concluded that all were more or less rubbish but a combination of two the least rubbish way to do this, the result of which has of course been that his least-worst method is now the standard among those who do such things…

5. I look here with especially narrowed eyes at Georges Duby, The Early Growth of the European Economy: warriors and peasants from the seventh to the twelfth century, transl. Howard B. Clarke (London 1974), pp. 11-13, which does exactly the trick Ted decries (Buttrey, “Calculating ancient coin production”, pp. 349-350) of surrounding the data with all kinds of cavils and conditions and then rhetorically building on it just the same.

6. Buttrey, “Calculating ancient coin production”, pp. 349-350.

7. Ibid. p. 339.

Working for San Salvatore III: what they got out of it

I have now gone on at great length about the polyptych of Santa Giulia di Brescia without really talking about my actual purpose in reading it, so it’s time to do that.1 You may remember a long time back that I had a go at the idea, repeated in textbook after textbook, that agriculture in the Carolingian period ran at yields hardly more than the grain that was sown.2 This is self-evidently ridiculous if you are familiar either with actual growing of crops (which I am only second-hand) or can do basic maths, but it persists, and the reason it persists, like many another medieval cliché, is Georges Duby.3

Georges Duby

The late Georges Duby

This is not entirely Duby’s fault. He wrote a couple of textbooks in the 1960s and 1970s that somehow remain the world standard for any history of the early medieval economy that actually contains agriculture, and he used the best thinking available and sources known at the time.4 He did a pretty good job of synthesis on that, and though one might wish he’d thought about it a bit harder, it’s really not just him who’s failed to do so, and those that have thought about it haven’t really looked hard enough at his evidence.5 That was, in large part, the Carolingian estate survey of the fiscal centre at Annapes preserved in the text known as the Brevium Exempla, and some time ago already now I gave a paper at Kalamazoo in which I showed that Duby had in fact read the text wrong, or rather failed to read all of its data, as had all those he used, even, I’m sorry to say, Philip Grierson, and I considered that dispatched and proceeded to writing it up.6 But Annapes was not Duby’s only source that seemed to support these awfully low yields, and so I needed to see if the same tricks could be performed with the others too, and you will by now have guessed or maybe already know that one of them was the polyptych of Santa Giulia di Brescia.

Santia Giulia di Brescia from the air

Santa Giulia di Brescia as it now stands, from the air

Duby dealt with the figures from Santa Giulia only in summary fashion. In Rural Economy and Country Life he works Annapes over extensively, coming up with output figures of between 1·5:1 and 2·2:1, and then goes on:

“We must not, of course, generalize from one set of figures obtained from a single source. But it is possible to find elsewhere some other traces of output, somewhat higher than that which can be derived from the Annapes inventory, but even so representing a low yield and a derisory rate of profit when compared with the value of the capital in land and seed corn. One significant fact is that compilers who visited the farms (cours [apparently left in French from Latin ‘curtes’]) of the abbey of San Giulia of Brescia in 905-906 to compile a polyptych found there reserves of grain in the barns which were barely higher and sometimes lower than the quantity needed for sowing. Thus at Prozano where the fields could take 300 muids of seed corn, the stocks in the estate barn amounted to only 360 muids of which 140 were of millet (mil). At Canella 90 muids were needed for sowing and 51 were in the barns; at Temulina 32 and 37.”

And with that he moved onto Saint-Germain-des-Prés near Paris and pulled a similar trick there.7 And in the slightly later and much shorter Early Growth of the European Economy he didn’t even give that much detail (or a reference to the primary source), limiting himself to dealing again with Annapes and then adding:

“The Lombard monastery of St Giulia of Bréscia [sic], which consumed some 6,600 measures of grain annually, would have 9,000 sown to cover its needs, which means that the return normally available to the lord was being estimated at 1·7 to 1.”8

The best way to see what is wrong with this is to look closely at how the compilers of Santa Giulia’s polyptych were using their figures, figures that I’ve already argued here they were receiving in a standard format. And doing so shows firstly that Duby, and Luzzatto before him, were again wrong in assuming that these figures mean what they wanted to mean, and in fact that using them to calculate yield is impossible except in one single case where the formula was bent, and in that case it comes out at at least 4·25:1 and probably rather higher. Don’t believe me? Watch this! Continue reading

Seminar CIX: where’s the money in early Anglo-Saxon England?

I think I’ve decided to run with this theme for the moment, though some things may get shifted around for greater elegance. I’ll advertise any major changes, though. Thankyou all for comments. Now, back to the backlog! It probably won’t surprise you, now that you know that the Winton Institute for Monetary History were being medieval in their seminar series last year, that I was back there before long, on this occasion because Michael Metcalf was speaking, and he is a man whose work I’d read quite a bit of as an undergraduate and heard much of during my time at the Fitzwilliam Museum, but never actually met. He could best be described as Mark Blackburn‘s opposite number in Oxford, but started rather earlier and has perhaps said some more, um, adventurous things, so I was interested to see what would transpire.1 His paper, presented on the 16th November 2011, was entitled, “Thrymsas and sceattas and the balance of payments”.

Reverse of a silver penny probably struck in the Thames Valley between 730 and 745, Fitzwilliam Museum CM.1815-2007, De Wit Collection

A sceatta (pronounced 'she-att-er'), to wit, a silver penny probably struck in the Thames Valley between 730 and 745, Fitzwilliam Museum CM.1815-2007, De Wit Collection. This is the reverse; it's really about a centimetre across.

It used, as Professor Metcalf began by saying, to be thought that the monetary economy began in the High Middle Ages, but ever since metal-detecting got big this has got steadily harder and harder to maintain. The take-off point keeps getting earlier and earlier as more coin is discovered. Especially prevalent have been finds of the earliest coinages of early medieval England and its neighbours, the coins usually known as sceattas (though Mark preferred ‘early pennies’) and their rarer gold predecessors, thrymsas. These have multiplied to such an extent that everyone is now agreed that the old classification of them makes no more sense but very few people have dared to risk putting a foundation down in the ongoing flow of evidence to start a new one, so the coins remain somewhat poorly understood.2 Once it was clear that the high medieval proponents were wrong, anyway, the next paradigm came from none other than Richard Hodges, who in the early 1980s suggested that these coins were in use only by élites and that the average peasant never saw them.3 This was defensible at the time of writing but is an especially hard-hit casualty of the increase in evidence; there’s just too much now, too widely-spread, for any sensible reconstruction of how much there once was to fit such an idea. So, it’s necessary to rethink what coinage in early Anglo-Saxon England was actually doing.

Reverse of Anglo-Saxon gold shilling of King Eadbald of Kent (616-640) struck at Canterbury, Fitzwilliam Museum CM.778-2002

A thrymsa, or shilling, in this instance of King Eadbald of Kent (616-640) struck at Canterbury, Fitzwilliam Museum CM.778-2002, the reverse again, and even tinier than the previous

This paper had a go, then, at doing this by analysing the distribution of finds of these coinages. We are especially able to get at these now because of electronic resources like the venerable Corpus of Early Medieval Coin Finds at the Fitzwilliam and the rather newer NUMIS in the Netherlands. Playing with these quickly reveals the one-sidedness of finds of the (probably) earliest sceattas, of which many were made in both the Netherlands and England but English ones of which are only found in the latter, whereas the Low Countries ones get everywhere.4 This suggests to Professor Metcalf a balance of payments, and he suggested therefore that wool was a key export even this early, since the coins are in fact found most thickly in the Cotswolds and Yorkshire Wolds, wool-producing areas of note. That also suggests that the goods were shipped direct with no trade on the way! So, that might be élite, if that’s how you see English wool production even as early as the seventh century,5 though it might also not, but the distribution of finds of locally-minted gold coins of the seventh century is basically uniform, so it seems quite unlikely that the good stuff was being concentrated by any such interest. The élite hypothesis does, therefore, seem to have to go.

Replica of an Anglo-Saxon coin die found at Cripplegate, York, with modern UK twopence piece for scale

Replica of a (much later) Anglo-Saxon coin die found at Cripplegate, York, with modern UK twopence piece for scale

A lot of the paper rested on estimates of the sizes of coinage, an area in which Professor Metcalf has become famous.6 For example: we can now identify nine hundred dies used in the striking of the surviving corpus of seventh-century thrymsas. There are various well-established means for multiplying these figures up towards an estimate of the whole coinage, which when applied here reasoned for three million plus coins total, on a multiplier of five thousand coins per total extrapolated dies, and more probably something like a million in circulation at once.7 Of the gold. If we use modern parishes as a guide to how many villages there were (and you see here what I meant by ‘adventurous’), we might then expect there to be 300-odd gold coins in any given village at once! Now, I am pretty dubious about this kind of arithmetic, as you will know, although even if you halve these figures and double the number of ‘villages’ (a thing that didn’t really exist in the seventh century but let’s just assume it means ‘district’ or ‘area’ and that’s fine8—and one point that came up in questions that I’d never considered is that one thing that must be missing from distribution maps of coin finds is settlements, at least where they have continued, because you can’t metal-detect in towns!) that is still quite a lot of gold to spread out. All the same, even if the actual numbers are rubbish, one point is still true: doing the same maths with the same multipliers for later Anglo-Saxon England nets you much much less. Unless there was something specifically weird about the way money was produced in one or other period (and there certainly was about the later period, given how widely and in what small quantities it might be minted, but that ought to exaggerate the later figures, not shrink them), England was more monetised in the seventh century than it was even in the eleventh.

Distribution map of sceatta finds in England and the Continent

Distribution map of sceatta finds in England and the Continent, from Archaeology in Europe (linked through) not one used by Professor Metcalf, whose maps' detail was rather finer, but I somehow find scrounging other people's handouts onto the web without their okay a step too far

So, you know, what, why and how? The answers are yet to come, but the questions are getting louder and louder. Some answers that did get suggested in questions were, the obvious one perhaps, a consumer class in the peasantry (John Blair), monasteries (also John Blair, you will be shocked to learn) both as consumers and as industrial drivers of the economy, salt and meat being bought in bulk (the latter of which was also John’s suggestion, in fact) and, back from the dead, the élite (Anthony Hotson, though here obviously channelling Chris Wickham, sadly absent), in as much as by promoting commerce and appropriating surplus that people are thus made to produce they are causing production and a market economy… And any of these might be the right answer, or indeed all of them, but none of this is coming from texts, or Henri Pirenne would likely have had the answer eighty years ago. This is one of those instances where the answer really does lie in the soil.


1. Most famously, perhaps unfairly, in one of his earliest papers, “How Large Was the Anglo-Saxon Currency?” in Economic History Review 2nd Series Vol. 18 (London 1965), pp. 475-482, where he used an estimate of 30,000 coins produced by each hypothetical die to produce a maximum figure that somehow immediately became orthodoxy…

2. The basic classification goes back to Stuart Rigold, if I understand rightly—and I really may not!—in his “The principal series of English sceattas” in British Numismatic Journal Vol. 47 (London 1977), pp. 21-30, but this rapidly got liberally beaten about, not least in David Hill and Michael Metcalf (edd.), Sceattas in England and on the continent: The Seventh Oxford Symposium on Coinage and Monetary History, British Archaeological Reports (British Series) 128 (Oxford 1984) and subsequently in D. M. Metcalf, Thrymsas and Sceattas in the Ashmolean Museum, Oxford (Oxford 1993), 3 vols. Lately Tony Abramson has proposed a new classification in his Sceattas – An Illustrated Guide (London 2006), but the profession doesn’t seem to be happy with this and I believe we can expect more work on this soon, fostered not least by Tony’s readiness to get people together and talking about these coins as witnessed in the volume he’s edited, Studies in Early Medieval Coinage 1: Two Decades of Discovery (Woodbridge 2008), which came out of Leeds sessions (a lesson to me in prompt publication).

3. Richard Hodges, Dark Age Economics: the origins of towns and trade A. D. 600-1000 (London 1982, 2nd edn 1989).

4. Presumably this is all available in W. Op den Velde & D. M. Metcalf, The Monetary Economy of the Netherlands, c. 690-c. 715 and the Trade with England: A study of the Sceattas of Series D, Jaarboek voor Munt- en Penningkunde Vol. 90 (Utrecht 2003) and eidem, The monetary economy of the Netherlands, c. 690 – c. 760 and the trade with England: a study of the “Porcupine” Sceattas of series E, Jaarboek voor Munt- en Penningkunde Vol. 96 (Utrecht 2010 for 2009), but I confess I only just now found out about those so I can’t say for sure.

5. My ill-disguised Grierson fandom obliges me to mention Philip Grierson, “The Relations Between England and Flanders Before the Norman Conquest” in Transactions of the Royal Historical Society 4th Series Vol. 23 (London 1941), pp. 71-112, repr. in Richard W. Southern (ed.), Essays in Medieval History: selected from the Transactions of the Royal Historical Society on its centenary (London 1968), pp. 61-92, though even he didn’t try to push the wool link back this many centuries, not then having the last seventy years’ finds to work from.

6. Not just his “How Large was the Anglo-Saxon Currency” above, to which one might like to compare Philip Grierson’s reply, “The Volume of Anglo-Saxon Coinage” in Economic History Review 2nd Series Vol. 20 (London 1967), pp. 153-160, repr. in Grierson, Dark Age Numismatics: selected studies, Collected Studies 96 (London 1979), XXXVIII, but more recently Metcalf, “Some Speculations on the Volume of the German Coinage in the Tenth and Eleventh Centuries” in Thomas Fischer & Peter Ilisch (edd.), Lagom. Festschrift für Peter Berghaus zum 60. Geburtstag am 20. November 1979 (Münster 1981), pp. 185-193, another one I know of rather than know, and Metcalf, “Can We Believe the Very Large Figure of £72, 000 for the Geld Levied by Cnut in 1018” in K. Jonsson (ed.), Studies in Late Anglo-Saxon Coinage: in memory of Bror Emil Hildebrand (Stockholm 1990), pp. 165-176, which I have actually read, although a very long time ago…

7. Since these methods involve at crucial points making up figures, I don’t myself put much trust in them and in this I’m guided not least by Ted Buttrey, “Calculating ancient coin production: facts and fantasies”, Presidential Address in Numismatic Chronicle Vol. 153 (London 1993), pp. 335–351, but also by the careful compromise suggested by Martin Allen, “The volume of the English currency, c. 973–1158″ in Barrie Cook & Gareth Williams (edd.), Coinage and history in the North Sea world, c. AD 500–1200. Essays in honour of Marion Archibald (Leiden 2006), pp. 487-523. There’s a computer simulation test of the various statistical estimators in use published by Warren Esty, “Estimation of the size of a coinage: a survey and comparison of methods” in Numismatic Chronicle Vol. 146 (London 1986), pp. 185–215, which gives some reason for credence in such methods, but ultimately I’m with Ted on this: even if it might just be right, it’s inherent to the method that you can’t know that… That doesn’t invalidate Professor Metcalf’s relative conclusions, however!

8. See Helena Hamerow, “Settlement Mobility and the ‘Middle Saxon’ Shift: rural settlements and settlement patterns in Anglo-Saxon England” in Anglo-Saxon England Vol. 20 (Cambridge 1993), pp. 1-17.

I should have read this the moment I bought it, III

9780754662549

The third main paper in Davis’s and McCormick’s The Long Morning of Medieval Europe presents two problems.1 One is that to critique it is to speak ill of the dead, because its author Riccardo Francovich died in an accident while the book was in its long press process. The other is that of balancing the fact that what he said is important and deserves notice but was also really quite exaggerated. The basic case that he was making is that although one of the many things we tie up with the much-vaunted ‘feudal transformation’ is that medieval settlement seems from its documents to have become much more nucleated in the tenth and eleventh centuries—for reasons that may include castles, development of the trading economy, violence and insecurity, and so on, the usual problem with causation when so many factors are in play at once—in actual fact, when we do the digging, nucleation often turns out to have been much older than that, often predating the castles or whatever by a few centuries.

The Tuscan hilltown of Monteriggioni; photo by Michael Ferris

The Tuscan hilltown of Monteriggioni; photo by Michael Ferris

Well, OK. Actually I think we knew this in England, where rather than the ‘nucleation of the year 1000’ or whatever we have the mid-Saxon settlement shift.2 This is just that for a while Anglo-Saxon settlement archæology was the field leader, I think, and it’s not surprising to me that other areas show similar things. The problem is with how similar. Although Francovich cited some work from France and Germany too, his evidence is mainly from Tuscany, where he worked most of his life and where indeed he died. There, there is a definite pattern where settlement moves onto the hilltops and indeed after a while fortifies itself, and then later the towns in question develop castles.3 But the Tuscan hilltop towns are not like very many other places! Not least, in good ol’ Catalonia, which is geographically and agriculturally not so different in this period and which may even have seen a small amount of Italian settlement, this doesn’t happen. In fact the nucleation doesn’t even seem to occur till the transformation period, and unlike elsewhere here we have both early enough documents to see the difference and some digging. There certainly are hilltop sites with early medieval phases, but they’re almost always much older than that, and far more often cemeteries and burial grounds than settlements. In England, settlement stays low and by rivers for the most part: that’s largely because cereal agriculture is more important than pastoral there and that obviously implies staying close to the fields, rather than the fascinating multi-level mountainside plots that Francovich described.4 Angeliki Laiou wonders in his summary paper whether the things that Francovich saw are really more than Tuscan phenomena, and his dissonant examples come from Byzantium, where of course a different and older tax system might be expected not to have permitted such a change if previous contributor Joachim Henning be right.5

Figure 3.1b from Francovich, "Beginnings of Hilltop Villages"

But it’s not just that the data is regional. His title recognises that, even if the text forgets it in places. It’s that some of the data is misleadingly handled. Never mind the use of percentages from less than a hundred data points. See this graph.6 It’s lovely and clear but first, you can’t reason back from the graph to the data and secondly, its axis is not steadily spaced. (I know there are some graphs where that can’t be done, logarithmic scales and so on, but really, this isn’t one.) What do I mean? Well, count it up. We’re dealing with a range of what, 800 years? It’s hard to be sure because the categories overlap.7 Are the fourth-century sites at the first graph point with the first-century stuff or are they with the sixth-century stuff halfway along? Similarly with the sixth century indeed. So let’s be generous and assume that the categories don’t overlap, and that really we’re running from years 1-350, 351-550 and 551-700. Those are still periods of 350, 200 and 150 years respectively, so putting them equidistant along the axis is cheating. It makes the figures a lot less surprising too. 2521 divided by 350 is 7·20, 506 divided by 200 2·53 and the last ratio is 1·34, so overall the change is by a factor of 5, not more than 12 as his percentages suggest. How did this get through review? As many times before, I am frustrated by historians’ inability or unwillingness to apply the mathematics they presumably learnt at school.

The castle and church of Montarrenti, Tuscany, centre of one of Francovichs most important projects

The castle and church of Montarrenti, Tuscany, centre of one of Francovich's most important projects

Of course a factor of five is still a factor of five even if it’s not a factor of twelve. That is still a massive drop in settlement density. A big change did happen here, I’m not denying that, and it needs explaining: Francovich’s paper goes a very long way to do that and contains innumerable references of work that proves his local points, as well as interesting tendrils to other places. The analysis of building change and of settlement formation is really valuable and also well illustrated, as is the example that a purely documentary view is doomed to missing things. But not only is this, as with any other explanation of social change in that period, not going to explain everywhere, but we have to read it as if we were marking it in order to know what’s really being presented, and that annoys me. It doesn’t serve his memory well.


1. †Riccardo Francovich, “The Beginnings of Hilltop Villages in Early Medieval Tuscany” in Jennifer R. Davis & Michael McCormick (edd.), The Long Morning of Medieval Europe: new directions in early medieval studies (Aldershot 2008), pp. 55-82.

2. See Helen Hamerow, “Settlement Mobility and the ‘Middle Saxon’ Shift: rural settlements and settlement patterns in Anglo-Saxon England” in Anglo-Saxon England Vol. 20 (Cambridge 1993), pp. 1-17.

3. A more moderate version of the same findings can be found in Richard Hodges, “Size matters: new light on the Italian Dark Ages” in Philippe Sénac (ed.), De la Tarraconaise à la Marche Supérieure d’al-Andalus : les habitats ruraux (IVe-XIe siècle). Desde la Tarraconense a la Marca Superior de al-Andalus: los asentamientos rurales (siglos IV-XI), Méridiennes : Études Médiévales Ibériques 2 (Toulouse 2006), pp. 223-229, but given that Francovich originally presented this material in 2004, and so did Hodges his, and that the first footnote of Francovich’s article includes the words, “My habit of thinking in parallel with Richard Hodges, in a dialogue that has developed over twenty years, has inspired gratitude toward my friend that equals the intellectual stimulation that he provides”, I am mainly sad for Hodges rather than suspicious at the coincidence.

4. Francovich, “Beginnings of Hilltop Villages”, pp. 63-64.

5. A. Laiou, “The Early Medieval Economy: Data, Production, Exchange and Demand” in Davis & McCormick, Long Morning, pp. 99-104 at p. 100; J. Henning, “Strong rulers—weak economy? Rome, the Carolingians, and the archaeology of slavery in the 1st millennium AD”, ibid. pp. 33-53.

6. Francovich, “Beginnings of Hilltop Villages”, p. 65.

7. It should be admitted that the categories may not be his, as the data is coming from a database system via the interpretations of another archæologist, Marco Valenti, L’insediamento altomedievale delle campagne toscane (Florence 2004), acknowledged Francovich, “Beginnings of Hilltop Villages”, p. 64 n. 42. I don’t think that excuses the faults in its presentation.