Roman Camps and Orientations

A follow-up to The Orientation of Roman Camps and Forts. This is an applic­a­tion of the Binomial Distribution test that I’m using in my own work applied to the data from the ori­ginal paper, which is why you may have the impres­sion you’ve already read this recently. My ana­lysis may not be cor­rect, so I’m put­ting it up on iScience and sub­mit­ting that to Carnival of Mathematics and Tangled Bank to see if people think the maths is wrong. I’m also put­ting it up on Revise and Dissent where it will get sub­mit­ted to the History Carnival and Four Stone Hearth to see if it’s intel­li­gible and sounds reas­on­able to Historians and Archaeologists.

Roman Camps and their Orientations reconsidered.

There is cur­rently a debate in the pages of the Oxford Journal of Archaeology on the ori­ent­a­tions of Roman camps and forts. Richardson (2005:514–426) argues that the ori­ent­a­tion of these camps is non-random and relied on some form of astro­nom­ical obser­va­tion. He presents data which he argues sup­ports his case. Recently Peterson (2007:103–108) has argued this relies on a flawed use of the Chi-squared test. I accept Peterson’s find­ings that Chi-squared is not a use­ful method. However examin­ing the camps as a bino­mial dis­tri­bu­tion would be feas­ible and would make expli­cit the archae­olo­gical and astro­nom­ical assump­tions made in the argument.

What is a Roman Camp?

The sites being examined are Roman camps and forts in England. One of the major advant­ages that the Roman army had over the nat­ive oppos­i­tion when occupy­ing new ter­rit­ory was their organ­isa­tion. The Roman army was effect­ively a pro­fes­sional army tak­ing on ama­teurs. Their camps reflect this organ­isa­tion. Typically their early camps a ditch sur­roun­ded by a bank in a playing-card shape. They fol­lowed a set design. The rationale for this was if there were attacked by sur­prise equip­ment and people would be in the same place at each camp, min­im­ising the effects of the surprise.

Wallsend fort
A Roman fort at Wallsend. Photo from Google Earth

The ancient sources give some detail on how to lay out a Roman camp. The main gate should face the enemy, or the line of advance (Vegetius 1.23, Hyginus 56). The rear gate should be on the higher ground to aid sur­veil­lance. Sites over­looked by hills were con­sidered a bad idea, as were sites near wood­lands which would allow the enemy to sneak up on the camp. The basic lay­out of the camp could be set up quickly by sur­vey­ors using gro­mae, sur­vey­ing tools for lay­ing out lines at right angles. Hyginus (chapter 12) states that you set up your groma at the junc­tion at the centre of the camp and lay out your roads to the gates from there.

This would appear to be an effi­cient method of lay­ing out a camp. Were obser­va­tions to ori­ent­ate the camp also part of the method? It doesn’t seem neces­sary, but Richardson (415,422–23) provides quotes from ancient sources which sug­gest this is plaus­ible hypo­thesis in some cir­cum­stances.

Testing the camps for astro­nom­ical significance

To test his hypo­thesis Richardson has examined plans and meas­ured the angle the long axis of a camp makes to the meridian, which yields an azi­muth from true north. The accur­acy of these meas­ure­ments is ques­tion­able, not through Richardson’s work but in that they rely on an accur­ate mark­ing of north. A big­ger dif­fi­culty is that such meas­ure­ments do not include declin­a­tion data. If the local hori­zon is flat then an align­ment will point to a dif­fer­ent part of the celes­tial sphere than if it is moun­tain­ous. Finally Richardson notes that some Roman camps, being square, lacked a long axis. He states “…these were alloc­ated to the nearest con­veni­ent group.” He then applies the Chi-squared test to his data. This is iden­ti­fied by Peterson as the weak point of Richardson’s ana­lysis, so this should be examined closely.

Richardson’s hypo­thesis is that there is a delib­er­ate ori­ent­a­tion towards the car­dinal points. There are 360 degrees in a circle and four car­dinal points. A camp can point dir­ectly at a car­dinal point. It can be within one degree of the tar­get, within two degrees, within three degrees and so on. However, by the time you get to forty-five degrees you’re work­ing closer to the next car­dinal point. Richardson there­fore says there are forty-five pos­sible prob­ab­il­ity bins into which a camp can be placed. If there are sixty-seven camps in the sample, you would expect (67/45 = 1.49) 1 and a bit camps in each bin. Between one and two then. What Richardson finds that some bins have con­sid­er­ably more. For the Chi-Squared test what you do is go in each bin and sub­rac­ted the expec­ted value from observed value. You take the answer and square it, and then divide it by the expec­ted value. You do this for all the bins and add up the res­ults and call this final value chi-squared. Confusingly the Greek let­ter χ — chi looks like an X so you’ll often see it writ­ten X2.

Now what does this value tell us? What we need to do is look up the value in a χ2 table. Richardson has argued there are 45 dif­fer­ent bins, so we look up the χ2 value with 44 degrees of free­dom. This will give us a range of val­ues. As χ2 gets higher, so it becomes more and more improb­able. In Richardson’s case he argues that this value of χ2 in would only hap­pen less than one time in a thou­sand by chance. The prob­lem is how robust are these results?

This is the key cri­ti­cism of Peterson, who argues that Chi-squared tests can­not reli­ably work on small samples. Worryingly he argues that the min­imum expec­ted value in any bin should be five, which means the min­imum sample size for such a test should be 5x45= 225 camps. As a gen­eral rule this sample size is simply lar­ger than could reas­on­ably be hoped for.

There may be an altern­at­ive method. There is a prob­ab­il­ity that the camps will or will not sat­isfy cer­tain cri­teria. This would appear to be suit­able for exam­in­a­tion as a bino­mial dis­tri­bu­tion. A bino­mial dis­tri­bu­tion is the set of cir­cum­stances that an event either will or will not occur in n attempts, where n is the sample size.

Binomial dis­tri­bu­tions

The easi­est example is toss­ing a coin. It either will or will not land on heads. If you tossed sixty coins at the same time, then on aver­age they would have heads thirty times. But not every­one would have thirty heads. There would be a nor­mal dis­tri­bu­tion which means that there would be a spread of res­ults some people would have thirty-one heads, oth­ers only twenty-nine. The for­mu­lae gov­ern­ing a bino­mial dis­tri­bu­tion are simple.

n is the sample size
p is the prob­ab­il­ity the event will hap­pen between 0 — abso­lutely will not hap­pen — and 1 abso­lutely will hap­pen.
q is the prob­ab­il­ity the event will not hap­pen, which will be 1-p

The aver­age res­ult will be np. In out case of sixty coin tosses, the aver­age res­ult will be 60x0.5=30 heads.

The stand­ard devi­ation will be √(npq)

The reason you would want the stand­ard devi­ation is that it tells you how sig­ni­fic­ant a res­ult might be. If you toss sixty coins in this dis­tri­bu­tion 2/3 of the time you will have 30 +/- 1 stand­ard devi­ation heads. Ninety-five per cent of the time you will have 30 +/- two stand­ard devi­ations. Ninety-nine per cent of the time you will have 30 +/- three stand­ard devi­ations. Usually in the social sci­ences people get inter­ested after 2 stand­ard deviations.

Coins Graph
Graph of prob­able res­ults of toss­ing 60 coins. Red area is one stand­ard devi­ation. Red and Orange two stand­ard devi­ations. Click for full size.

This method doesn’t require the prob­ab­il­ity to be 50:50. Take a stand­ard six sided die. The prob­ab­il­ity of throw­ing a six is one in six. If you throw sixty dice you should expect about 10 sixes. Sometimes you’ll have more other times less. Would it be unusual to throw twelve. In this case the for­mu­lae can show the answer.
Average = np = 60*1/6 = 10
Standard Deviation = √(npq) = √ (60x1/6x5/6) = 2.886751
So two thirds of the time you should expect to throw between eight and twelve sixes.

Chance of throwing sixes
Chance of throw­ing sixes with sixty dice. Red area is one stand­ard devi­ation. Red and Orange two stand­ard devi­ations. Click for full size.
The Binomial dis­tri­bu­tion of Roman camps

What has this got to do with Roman camps? The Roman camps have a prob­ab­il­ity asso­ci­ated with them. They could by chance line up they do. So to assign a value to p we need to openly state what the prob­ab­il­ity they align as they do by chance is. This is an expli­citly archae­olo­gical ques­tion. To return to the camps, how to you align a camp to North?

You could align hold up a plumb line to the North star and let it drop. If you do this then there’s a prob­lem. Polaris isn’t exactly on the North Celestial Pole. In AD 250 it was at 89° 17′. So there’s a one and a half degree spread on either side of the North Celestial Pole. If you factor in at least some human error then the min­imum accur­acy you can argue for is 2/360. If you bring in evid­ence from the his­tor­ical data then you could point out these camps were built dur­ing the day when Polaris wasn’t vis­ible. This could make the align­ments less accur­ate and any­thing +/- five degrees might have been thought to have been aligned north, in which case p is 10/360. Others might argue for other accuracies, but to get a value for p, you have to be expli­cit about which archae­olo­gical or his­tor­ical evid­ence you’re using. Once you have a value you can plug it in, and if you like use a few other val­ues too to see how robust your res­ults are.

In the case of Richardson’s data he gives the data in 10° wide blocks between 0° and 180° from north. This gives a camp a 1/18 chance of being in any bin and allows us to plot this prob­ab­il­ity graph. The aver­age is four. I’ve shaded to six green and to eight orange. In real­ity the stand­ard devi­ation is 1.9, so the shad­ing should stop just short, but his res­ult of six camps facing within ten degrees of north does not look sig­ni­fic­ant enough to be troubled with.

Camps orientated to north
Camps aligned to north. Click for full size.

His other hypo­thesis is that the camps were aligned within 10 to car­dinal points. In this instance this makes p equal to 4/18. This is an inter­est­ing res­ult. On aver­age we should expect fif­teen camps to match. The stand­ard devi­ation of 3.4 means that 15+/-6.8 between nine and twenty-two camps should face car­dinal points. There is an ele­ment of doubt. If you throw enough tests at the data then you’d expect about one in twenty to have sig­ni­fic­ance at two stand­ard devi­ations. Nonetheless as a fil­ter to ask if an idea is worth invest­ig­at­ing fur­ther it is very useful.

Camps orientated to north
Camps aligned to car­dinal points. Click for full size.

Part of the prob­lem is that it’s clear that local con­sid­er­a­tions, like the pres­ence of a hos­tile force or dir­ec­tion of travel took pre­ced­ence in plan­ning. With closer sur­vey it would be pos­sible to elim­in­ate some of these known non-astronomical align­ments. The easi­est cat­egory to remove would be those whose align­ments matches the align­ments of local roads as it seems highly likely that the army would have stopped here while mov­ing along this route. The next step would be to sur­vey the sites and see if other lim­it­a­tions like local topo­graphy define the ori­ent­a­tions of camps. If pre-defined ori­ent­a­tions can be elim­in­ated then it may be pos­sible to show astro­nom­ical align­ment was used as a guide when there wasn’t a local enemy to face, or a route to follow.

There are a couple of advant­ages in using a bino­mial test for stat­ist­ical sig­ni­fic­ance. One is that it is com­par­at­ively sim­pler to use than Chi-squared. Chi-squared may not be arcane math­em­at­ics, but the sug­ges­tion by Peterson that help needs to be sought can be given to most archae­olo­gists. Even in cases where Chi-squared is appro­pri­ate, if archae­olo­gists can­not fol­low the argu­ment, then per­haps another method is needed. Hopefully the above method is slightly more transparent.

The other advant­age is that it can be applied to smal­ler samples. If the sample is too small to be stat­ist­ic­ally sig­ni­fic­ant then this will become appar­ent. Take for instance toss­ing two coins. If they both come up heads is that sig­ni­fic­ant? np states the expec­ted aver­age is one, two stand­ard devi­ations would be 1.4, so we would expect 95% of samples to fall between 1 +/- 1.4. Given the max­imum res­ult will be 2, we will always be in this range. The test tells us the sample is too small to be helpful.

Also because the test is about eval­u­at­ing the prob­ab­il­ity some­thing will hap­pen it can be used for non-spatial data. For example recently there have been claims that the Iron Age peoples of Fiskerton have been able to pre­dict lunar eclipses using Saros cycles. This is data dis­trib­uted in time, but non­ethe­less is should be pos­sible to use the archae­olo­gical record and the data to argue for a value of p for the prob­ab­il­ity of pre­dict­ing an eclipse in a Saros cycle within a given time­frame. That would prob­ably be the other example used if I expand this into a full paper.


Peterson, J.W.M. 2007. ‘Random Orientation of Roman Camps’. Oxford Journal of Archaeology 26(1). 103–108.

Richardson, A. 2005. ‘The Orientation of Roman Camps and Forts’. Oxford Journal of Archaeology 24(4). 415–426.


When he's not tired, fixing his car or caught in train delays, Alun Salt works part-time for the Annals of Botany weblog. His PhD was in ancient science at the University of Leicester, but he doesn't know Richard III.

5 Responses

  1. dca says:

    You should ask what your null hypo­thesis is (test­ing against a uni­form
    dis­tri­bu­tion of dir­ec­tions?). Consult

    Fisher, N. I. (1993)
    fIS­tat­ist­ical Analysis of Circular DatafP
    (New York, Cambridge University Press)

    for an array of meth­ods (I don’t know if the cited art­icles included this reference).

    I do not see that slopes affect whether walls are NS or not.

  2. Brett says:

    I was never a stats wiz, but the meth­od­o­logy seems reas­on­able to me. What is the geo­graphic and/or tem­poral dis­tri­bu­tion of the sample though? Are they all from sim­ilar regions and peri­ods? If not, prac­tices may have changed — that is, the camps would not all be drawn from the same pop­u­la­tion, and that may skew the res­ults. Or maybe the Romans did build camps exactly the same from one end of the Empire to the other over 500 years, I don’t know!

    Another ques­tion I have is, why would one expect NS align­ment any­way? I can’t think of any prac­tical reas­ons to do this uni­formly, or even as uni­formly as pos­sible. You do note that local cir­cum­stances would often have played a part in align­ment — unless there was a pretty good reason for NS align­ment (prac­tical or oth­er­wise), I would have thought the mil­it­ary sur­vey­ors (or who­ever) would have been encour­aged to ori­ent the camp to best advant­age. But then I guess that’s the point of the test — to see if there is any NS align­ment to explain :)

  3. Alun says:

    dca, thanks for the book sug­ges­tion. We don’t have that at Leicester so I’ll have to get an inter-library loan for it. The null hypo­thesis in this case is a ran­dom dis­tri­bu­tion, which I thought was con­sist­ent with using the nor­m­al­ised prob­ab­il­ity curves but I see I don’t men­tion it any­where above. I’ll have to cor­rect that. The Romans liked to have the rear gate higher than the front gate, and it’s likely this would have taken pre­ced­ence over astro­nom­ical con­sid­er­a­tions. If the local topo­graphy slopes from west to east, then it would be expec­ted that the camp’s main gate would be at the east.

    Brett, the camps are all from England but there’s no tem­poral data, which I agree is a prob­lem. The big­ger prob­lem is, as you say, the vari­ab­il­ity of local topo­graphy. There are sev­eral things to look­for like the pres­ence of an enemy, or dir­ec­tion of travel before astro­nomy would become used. This will leave a lot of noise to sig­nal, and again as you say, it isn’t cer­tain there is a sig­nal. Martin Sterry, one of the Roman PhD stu­dents at Leicester, has sug­ges­ted using North African data as a comparison.

  4. dearieme says:

    If you expect an attack at dawn, do you expect the attack­ers to want the advant­age of your hav­ing a low sun in your eyes? If so, will that influ­ence how you lay the camp out e.g. don’t have gates aligned with the rising sun dur­ing the period you expect the camp to be in use.

  5. Alun says:

    This comes back to the prob­lem that Brett has men­tioned, not all camps are equal. Your sug­ges­tion is excel­lent if you’re in an area with known enemy. Yet if you’re camped for a long period or without hos­tile forces per­haps an align­ment towards the east would bring in sun­light sooner and illu­min­ate the insides of the leather tents. Also east is more often the bet­ter dir­ec­tion to face away from rain. Either way it should be pos­sible to have a look to see if east is pre­ferred to west or vice versa. Unfortunately I can’t find that data in the ori­ginal papers, so that might mean that a large scale sur­vey is necessary.

    I sus­pect you’re right in that if there is an astro­nom­ical pref­er­ence, it’s more likely to be for prac­tical reas­ons, rather than high accur­acy reas­ons. However, I’m not cer­tain that I’ve shown there is a stat­ist­ic­ally sig­ni­fic­ant effect to explain.