Homepage of Rüdiger Heinzerling

 main page

July 1998
Review of the article:

Vetus Testamentum 48 (1998) 196 - 213

With the article of Prof. Colin J. Humphreys a further attempt lies before us to give a literal meaning to the numbers in Numbers 1 and 26. Among other things Humphreys presents a quantitative analysis of the distribution structure of these numbers, yet without assuming that their formation had a special purpose. In the last analysis his article is a variant to the approach of Mendenhall , but with a smart mathematical argumentation and with the numbers having reference to the time of the Exodus instead of the time of the kings.

Humphreys' contribution is characterized by the fact, that with the aid of some additional suppositions he constructs mathematical equations, which are supposed to allow for the estimation of the true size of the tribes of Israel at the time of the Exodus as well as the evaluation of the plausibility of already presented interpretations of the numbers in Numbers. In the following I will not report on his article (it is lying in front of you), but only examine critically some selected points:

A) The basic assumption
B) Is the number 273 'likely to be correct' ?
C) Is the proposed interpretation of the numbers consistent with the census of the serving Levites (Num. 4)?
(17th April 2000: Correction to my summary of this review in Vetus Testamentum)
D) Statistical peculiarities
E) Abstract

A) The basic assumption
(3. The meaning of 'lp, p.199/200) 

Humphreys' basic assumption is, that a scribe (several scribes?) misunderstood the Hebrew word 'lp in some places (when copying the text of Numbers) in the meaning of 'thousand' instead of 'troop' . According to his misunderstanding the scribe re-interpreted the text in more than 40 places in Numbers that are concerned and altered it where necessary. Humphreys substantiates the necessity of these changes with reference to a hypothetical precursor-text. In this text  - according to Humphreys - the word 'lp had side by side the two meanings 'thousand' and 'troop' in compound combinations of numerals (p.206/7).

Humphreys tries to explain the mistake of the ancient scribe by comparison with allegedly similar mistakes by present bible translations: "If bible translators can misinterpret 'lp as 'thousand' (...) when 'clan' is intended (...), there is clearly scope for a scribe copying an early Hebrew text to interpret 'lp as 'thousand' (...) when 'troop' was intended." (p.200)

But this logic has to be exactly reversed: If even the scribes of that time made such a mistake, then even more today's translators, for it is clear: the then native Hebrew speakers understood the text far better than translators living milleniums later. But thus the argument is of no avail to make plausible the supposition, that the scribes made such a mistake in the first place.

So the question remains open if they made this mistake and this problem shall now be discussed more precisely. How do we have to imagine the origin of the mistake?

1) Was there a single scribe who - isolated from all others - single-handedly produced the single copy, so to speak the mother of all copies that were ever passed on to us, and which all show the same (allegedly wrong) interpretation of the word 'lp in the book of Numbers?
2) If so, how did his personal and his language isolation come about, so that he forgot which meanings the word 'lp can take on and simultaneously produced a copy with the 'wrong' notation of the Numbers-text, which determined the whole coming textual tradition in spite of his isolation? If not so, how did even a whole group of scribes get into this kind of isolation from the outside world?
3) How did the scribe come to assume, all his predecessors had failed to notice what he - the one remaining genius - realised only now: That the notation of  the numbers - which every scribe mastered off the cuff - was wrong exactly in the book of Numbers and had now finally to be corrected?
4) Why didn't he hesitate and examine, if the notation that (allegedly) lay before him was self-consistent, since the numbers for the troops, back-calculated by Humphreys, not accidently amount to exactly 598 (Table 2, p.212, correspondingly correct the sum of men)? If the 598 stood there as Humphreys states, how did the scribe fail to notice this number to be the exact sum of the troops?
5) And if he did not fail to notice: How probable is it for him, being a native Hebrew speaker, not to get the same idea as today's authors?
6) Is it at all possible to convert the numbers as Humphreys proposes to do, without first understanding their original meaning? And if not so, if the scribe first had to understand them in order to convert them, why should he convert them at all?

There is a question posed by Walter J. Houston (originally concerning the appropriation of poetic material in biblical prose) which fits the basic assumption of a mis-interpretation of the word 'lp: "...are the authors, as some have alleged, insensitive bunglars making a botch of material they did not understand, or is it their critics who lack understanding?" [In: "Misunderstanding or midrash? The prose appropriation of poetic material in the Hebrew Bible (Part I)", ZAW 109 (1998) 342 - 355 (343).] 
Finally let us pay attention to this: This is not about a gematrical, astronomical / astrological or other meaning with a similar artificial relation to the numbers, which in fact can easily fall into oblivion. It is a matter of the daily used semantics of numerals: There should have been no time in ancient orient when military units were completely abolished. So the designations of these units were being used ... and simultaneously "forgotten" (p.206) ? In my opinion the basic assumption of a mis-interpretation of the numeral 'lp by a scribe is unfounded.

B) Is the number 273 'likely to be correct'?
(4. A new mathematical analysis, p.201-203)

At the beginning of his analysis Humphreys poses the supposition, that the number 273 is striking as 'entirely reasonable' among those mostly very large numbers of the book of Numbers: "This very precise figure is reasonable not only because it is small, but it is also likely to be correct because redemption is involved, which would be taken very seriously." (p.201)
Now it can be discussed if this reason contributes significantly to set out this number to be "reasonable". However, it is already certain, that it does not show 273 to be "likely correct". For if we also take Humphreys reading of the word 'lp serious when this number is being derived (Num. 3,39-51), then we have to imagine the firstborn of Israel to be concentrated in 21 troops or teams (having together 1273 men). With regard to that there is no historical evidence whatever, as far as I know. 

Furthermore there is a decisive problem with regard to the alleged precision of the number 273:

If we write down the number of Levites as Humphreys does (Table 3, p.213): 
21 teams having 1000 men in all, then the number of firstborn is:
21 troops having 1273 men in all.
Here it has to be (21 and) 1273, not (20 and ) 2273 and not (22 and) 273, because otherwise the difference 273 to the 1000 Levites would not result (equation (1) p.201):
   If - L = 273   =>  1273 - 1000 = 273.

Therefore originally a number Iwith 4 (!) decimals precision must have been in the text, while - and this point now is decisive - the number of Levites obviously was precise only with regard to 1 decimal, 2 at maximum. The reason is: The 1000 Levites result from the addition of 3 groups whose size is given only with an uncertainty of plus/minus 100 resp.: 500 + 300 + 200. If one takes serious Humphreys supposition the numbers to be the result of a census of the Levites, then these numbers must have been rounded, because it is exceptionally improbable, that the number of all male Levites older than 3 month of all 3 divisions should accidently be divideable by 100!

This means: Calculating the difference 1273 - 1000 = 273 yields an error of plus / minus 100 at minimum and so the number 273 is not precise, also it is not "likely to be correct" but most probably wrong, in any case unreliable. Especially it is not suited as pivot of an equation system expected to decide on the interpretation of near to all other numbers of the book of Numbers.

At this point it presents itself to discuss immediately an obvious solution proposal (Logically it is not to be found in the article of Humphreys because the precision of the number 273 is there being assumed):
Can the problem be solved by the assertion, that the supernumerary firstborn were counted once again separately? No, because in the text it is explicitly referred to the sum totals being determined (Num. 3, Afterwards the difference between these numbers is calculated. How then should a separate counting look like, without making reference to the already existing sum totals, i.e. without calculating this difference?

However, the underlying problem is running considerably deeper, for the rounding uncertainty does not only concern the number of Levites, but also the number of Israelite men over 20 years old. So we ask ourselves: How is it, that the number of firstborn Israelites unexpectedly is given by a factor from 100 to 1000 more precise? Furthermore: How can this be, that in a real census the number of troops is given with 2 decimals precision (e.g. 46, 59, 45, 74, ...), but the sum totals of ablebodied men is routinely given with only one decimal precision (e.g. 500, 300, 650, 600, ...)? 
It is a relative easy task to determine the sum total even of a group of 500 beduins on horseback with a resolution (although not necessarily with a precision) of plus / minus 1, if they are militarily concentrated e.g. in 50 troops. Why has this information been discarded and the numbers rounded? Actually this fact should have prevented Humphreys from a historical interpretation, because he explicitly holds the view, that: "The prime purpose of a census is to produce accurate numbers..." (p.199). This is not the case in a precursor-text as he envisages it.

In the final analysis a census of about 500 people makes no sense at all if it does not come up to a precision of only a few units, because a crowd of 500 people concentrated in 50 groups can be estimated on to plus / minus 100, a census is completely unnecessary. But this means, the reading of the numbers as proposed by Mendenhall and Humphreys reduces the sense of the narration to absurdity.
Eventually: In the traditionally accepted reading of the numbers they come up to a resolution of about 3 decimals, which is a reasonable quantity (even if the absolute size of the numbers seems to be incredible). Though the precision of the number 22273 as well as the difference 273 remains suspect. And it is exactly this: their alleged accuracy and obvious unreliability as result of a census, which gives reason to think about a completely different interpretation of the numbers as a whole. What came out of such an analysis will be possible to be read here.

C) Is the proposed interpretation of the numbers consistent with the census of the serving Levites (Num. 4)?

The consistence check of the numbers of Levites in Num.3 , which Humphreys carries out in passage 5.2 (p.204/5) with regard to the other numbers of his mathematical approach, seems to deliver a very satisfying result.

Since he also applies his (resp. Mendenhall's) interpretation of the notation of the numbers to more distant places of the Pentateuch like Num.25,9 ; Ex. 12,37 ... (p.209/10), it is certainly adequate, in fact necessary, to check the consistence of the numbers in Num.4 as well, since they are directly connected with the numbers in Num.3.
The Levites serving at the sanctuary were solely men between 30 and 50 years old. Their number according to Humphreys' notation:

Kohat:        2 'lp and  750 men
Gershon:      2 'lp and  630 men
Merari:       3 'lp and  200 men
Total:        7 'lp and 1580 men

So the number of serving Levites between 30 and 50 years old is about 50% greater than the number of Levites in all, which is assumed to have been 1000 (Table 3, p.213, Conclusion vi p. 211). If the estimation of 50% of the population being under 20 years old (equ. 3, p. 202) is fairly reliable, then the number of people between 30 and 50 years old can be estimated by about 1/4 of the total population. But in this case the number of Levites should have been about 4 * 1580 = 6160 , instead of 1000. So the number of Levites is about 6 times too large, especially greater than the sum total of all available Levites.

Let us compare in detail. The sizes of the teams in Num.3 are:

Kohat:        8 'lp and  300 men
Gershon:      7 'lp and  500 men
Merari:       6 'lp and  200 men
Total:       21 'lp and 1000 men


For Kohat applies: Even if we prefer the reading 600 (instead of 300), the 750 serving Levites are more men than Kohat can make available. The same holds for Gershon (630 out of 500). For Merari (200 out of 200) one would have to suppose, that all male Levites had been between 30 and 50 years old, which is equally impossible.

So a fundamental inconsistency exists. Can it be removed?

To rescue the approach of Mendenhall and Humphreys we first have to consider another point of criticism. In his article Humphreys does not care about the possibility of there having been numbers of men for single tribes in the text, that exceeded 1000. Why for instance should be applied to Reuben: 46 'lp and 500 men (7. 'The correct interpretation ...' , p. 206), why not e.g.  45 'lp and 1500 men or 44 'lp and 2500 men? Of course this would have some effect on the largeness of troops which would rapidly increase. But this would not necessarily be 'bad'. But now one could assume a similar thing for the counting of the Levites in chapter 3, for example:

Kohat:        7 'lp and  1300 men
Gershon:      6 'lp and  1500 men
Merari:       5 'lp and  1200 men
Total:       18 'lp and  4000 men

[ 17th Apr. 2000: Correction to my summary of this review in Vetus Testamentum: 

Yesterday I discovered an odd mistake in my presentation of this ad hoc example in my summary in VT. I wrote that the "'lp numbers in the right column", i.e. the numbers of the serving Levites in chapter iv, could be split up in order to 'raise' the number of men. Of course it has to be the left column, i.e. the numbers for the Levites in all in chapter iii, as I correctly wrote here on my homepage in 1998 (see above). Unfortunately I even inserted the wrong table showing the splitted up numbers for the serving Levites. - The cause for this mistake must have been the one year distance between writing this review and that summary... plus a certain lack of concentration while working on the latter. Sorry! - However, the two main arguments (as explained here) that I selected for the summary remain unchanged. ]

In this way we would get space to 'place' the Levites serving at the sanctuary. Though, the whole thing still looks rather strange. While with Kohat 750 / 1300 = 58% of the men are serving, with Merari there are only 200 / 1200 = 17% . However, let us assume this to be correct as well, after all given the traditional reading the numerical proportions are not entirely convincing, too (a poor argument, I know). Then we have to imagine, that the scribe who is supposed to have changed the hypothetical precursor-text, had 18 'lp and 4273 instead of 21 'lp and 1273 as numbers for the firstborn Israelites in the text before him, because otherwise the difference 273 would not result. By analogy larger quantities for the number of men in the census in chapters 1 and 2 should now be put up (correspondingly lower the numbers for the troops), so that their sum total too becomes larger by a factor of 4. All these numbers the scribe would have to have altered according to Humphreys theory, without understanding them, and even for the very reason because he did not understand! How credible is that? In near to all numbers now there is a double 'lp written, because there are real thousands to be notated as well. In chapter 2 several thousands have to be notated in the sum totals for the 4 camps, similar to the over all sum total as given by Humphreys (603.550 = 598 'lp and 5 'lp and 550 men, p.207). In the example this could look like the following for the east camp:

Judah:     73 'lp and 1 'lp + 600 men
Issachar:  53 'lp and 1 'lp + 400 men
Zebulun:   56 'lp and 1 'lp + 400 men
Total:    182 'lp and 4 'lp + 400 men

This sum again would have to be added to the other sums to give the over all sum total. And the scribe should not have realised, what was going on here?

I think, this ad hoc example to rescue the hypothesis of Mendenhall and Humphreys has failed as well and there is no point in further searching for a historical interpretation. There are still many questions that could be discussed, e.g.:
  • Why do the numbers of men per troop fluctuate so strongly within one and the same census?
  • How do we interpret - according to the proposed reading - the numbers in Num.31, where 675,000 sheep and 16,000 human beings appear in one and the same calculation?
  • Why are the quantities for the largeness of troops in Num. 1 and 26 only in between 200 and 730, why are there no quantities above 1000 and especially no quantities with a hundred digit of 0, 1, 8 or 9?

  • At least this last question shall be shortly discussed yet:

    D) Statistical peculiarities

    The statistical peculiarities of the numbers in Numbers I will only discuss very shortly, since this phenomenon is extensively dealt with in my article.

    There are several statistical phenomena which are to be explained by any interpretation of the numbers in Numbers. Among them:

    - The double numbers 40,500 and 53,400.
    - Comparable frequencies of the digits in the least significant decimal.
    - Six pairs of numbers, whose single constituents have two digits in common.
    - The symmetrical distribution of these pairs among Num. 1 and 26.

    These characteristic features must be statistically evaluated and be used for the interpretation, if possible. A historical approach to the exegesis, taking the numbers as a result of a real census, cannot accomplish this. A simple inspection of the distribution of the decimal digits already shows, that the numbers do not originate in a simple "census of the people". One should realize for instance, that the digits 0, 1, 8 and 9 do not appear in the least significant decimal, and that neither in Num. 1 nor in Num. 26 . This requires an explanation, which is not delivered by the approach of Mendenhall and Humphreys.

    E) Abstract (Review 1: "The Number of People in the Exodus from Egypt: Decoding Mathematically the Very Large Numbers in Numbers i and xxvi" by Colin J. Humphreys)

    Prof. Colin J. Humphreys tried to show "that if there were '273 first born Israelites who exceed the number of Levites' (Num. iii 43), then the total number of Israelite men aged over 20 in the census following the Exodus was about 5000, not 603,550 as apparently recorded in Numbers." (p.196) This attempt of a historical interpretation of the numbers in Numbers implies a translation of the Hebrew word 'lp as 'troops' instead of 'thousand', which must be rejected in this case because of four main reasons: 
    A) The basic assumption that a scribe misinterpreted 'lp as 'thousand' where 'troop' was intended and altered the text according to his misunderstanding is unfounded and implies a chain of improbable assumptions. 
    B) The assumption that the number 273 is "likely to be correct" (p.201) can be shown to be wrong, because this number is defined as the difference between a precise (1273 first born Israelites) and an obviously rounded number (1000 Levites), which can never yield a correct number in the sense intended by Humphreys or be used to evaluate the majority of other numbers in this biblical book. In addition, the numbers of men are clearly rounded, provided they should be taken as a result of a census at all. Therefore the strangely precise number 273 (and 22273 resp.) is not "entirely reasonable". 
    C) The proposed reading of the numbers in Num. 1 - 3 is not consistent with the same reading of the numbers of the serving Levites in Num. 4 , because the number of serving Levites is greater than the number of male Levites in all. Even an extended ad hoc variant of this approach fails to work convincingly. 
    D) The striking statistical characteristics of the numbers (as explored in my article on this subject, cp. 'Akzent 7') are not explained by a historical interpretation.

    The abstract sight-seeing tour ------> This way, please

    Postscriptum:    (24 Jan. 1999) 

    Having read this review the reader will possibly arrive at the conclusion, that I think Prof. Humphreys' article does not contain really valuable information. But quite on the contrary, I would not have written such a detailed review if I had thought it wasn't worth it! The article of Prof. Humphreys, especially the equation system that he developed, is the smartest approach to interpreting the numbers that I have ever read. 

    As far as I know Prof. Humphreys would like to revise his approach in a way that treats the Levites somewhat differently than the other Israelites, the argument being that it may not be a coincidence that the discrepancy C) concerns the Levites. There are also discrepances between the different language versions of the Old Testament texts concerning the Levites, and the troop sizes he calculated for the Levites are very different from that of the others. So it would be a very interesting alternative if he could show the numbers of the Levites in chapters 3 and 4 together follow some other principle than the numbers of the other tribes in chapters 1 and 2.

    18 Mar '99 :

    I am delighted that the author of the reviewed article, Professor Colin Humphreys, Cambridge, accepted my invitation to write a reply to my review of his article: Thank you very much Professor Humphreys! And I am happy to publish it here on my homepage so the reader can easily make up his own mind by downloading and reading the complete material at once:

    Response to the Review by Ruediger Heinzerling of my article, "The Number of People in the Exodus from Egypt:  Decoding Mathematically the very large Numbers in Numbers I and XXVI" by Colin J Humphreys, Cambridge, Vetus Testamentum 48 (1998) 196-213.

    First, I would like to thank Ruediger Heinzerling for his careful reading of my article and for his detailed comments.  My response to his criticisms are as follows:

    A) The basic assumption

    First I believe there can be no doubt that the Hebrew 'lp had several different meanings, and the meaning was determined by the context.  Thus 'lp could mean "thousand" and it could also mean "troop" (or group or clan, etc).  It was used in both these meanings in the Old Testament and my paper gives references to this.  It is also clear that if a word can have more than one meaning then there is the potential for misinterpretation and confusion.  I believe this is what happened, as described in my paper.

    I suggest that in the original source document for the census numbers in the book of Numbers, 'lp had the meanings I give in my paper.  At a much later date, this source document was misinterpreted and 'lp was given
    the meaning "thousand" in all the census figures.  In his review, Heinzerling asks whether this was a single scribe or a group of scribes who did this, and how they came to make this mistake.  I do not know whether or not it was a single scribe or a group of scribes, but I can suggest two reasons why the mistake may have been made.  First, the scribe may 

    have wished to maximise the number of Israelites and given two alternative interpretations of 'lp, chose the one which maximised the numbers, so that the Israelites were as numerous as "the stars in the sky" or as "the sand on the seashore".  Hence the scribe, or group of scribes, may have maximised the numbers for what they saw as very good reasons.

    A second reason may be that on my analysis the average size of a troop was about 10 men.  Although the El-Amarna tablets show that at the time of the Exodus, a typical troop size was indeed about 10 men (the reference to this is given in my paper), in later times, troop sizes
    were undoubtedly much greater, as indeed they are today.  Hence a scribe, or a group of scribes, copying the original source document many centuries later and faced with two possible interpretations of 'lp, deliberately chose not to interpret 'lp as "troop" because this made no sense to them.  The alternative was to interpret 'lp as "thousand" everywhere, thus yielding the very large numbers we have in the book of Numbers.  I have to say that personally I find this possible explanation
    of what happened to be extremely plausible.

    B) Is the number 273 likely to be correct?

    Heinzerling is critical of my assumption here because he argues that since other numbers, for example the number of Levites being 1000, are clearly rounded numbers, then the number 273 cannot itself be precise. I disagree with this argument.

    I am very happy to accept that many of the numbers in the book of Numbers are rounded figures.  Indeed then, as now, if the number of people in a crowd is given as 500, then this is obviously a rounded number.  However, equally the number 273 to my mind is clearly not a
    rounded number, since it is clearly a precise figure.  In addition, although this number is the difference between the number of firstborn Israelites and the number of Levites, and even though when written down both of these numbers may have been written as rounded numbers, I believe the difference between these two figures would have been precise because, as I state in my paper, the number of firstborn Israelites which exceeded the number of Levites was extremely important to the Israelites because of the religious significance of redemption.

    The number 273 stands out from all the other numbers in the census account in Numbers because it is so much smaller than the other numbers and so much more precise.  I therefore believe this number is correct, and it has been correctly transmitted because it does not involve 'lp. This number of course forms the basis of my theory and I believe it to be a sound basis.

    C) The numbers of Levites

    The analysis I give in my paper consistently has the number of Levites at about 1000, and I describe this consistency on pages 204 and 205 of my paper.  Heinzerling refers to some difficulties in the numbers of the three branches of the Levites.  I agree there are some problems here but it is also worth pointing out that it is only for the Levites where the Hebrew and the Septuagint manuscripts do not agree (see my paper page 213).  There may therefore have been some corruption of the numbers in the individual branches of the Levites.

    D) Statistical peculiarities

    (Revision of 22/3/99 inserted on 21/4/99)

    Heinzerling notes, as have others, that there are some curious statistical peculiarities in the very large numbers in Numbers.  This strongly suggests to me that these numbers are not random numbers, in which one would not expect there to be statistical peculiarities, but that these are real numbers, and that statistical peculiarities arise for a real reason.  In particular, Heinzerling notes, as have others, that the digits 0, 1, 8 and 9 do not appear in the least significant decimal, and he states, "This requires an explanation, which is not delivered by the approach of Mendenhall and Humphreys".  Although I did not address this point in my paper, I think I do have a very good explanation, which is as follows.

    The number in the tribe of Reuben (for example) is given as 46,500.  What Heinzerling means by the "least significant decimal" here is the number 5. On my interpretation, the 

    tribe of Reuben had 46 troops comprising 500 men. The number of men in each tribe range from only 200 in the tribe of Manesseh up to 700 in the tribe of Dan. There is no tribe with as few as 100 men, nor as many as 800 or 900 men. I believe there is a very real reason for this which is as follows.  If the 12 tribes did indeed originate from the 12 sons of Jacob, then of the course of time, one would expect each of the tribes to grow at a slightly different rate, but not at a vastly different rate. Hence at the time of the exodus, the smallest tribe Manesseh (which was in any case a half-tribe) had 200 men and the largest tribe had 700 men. No tribe had multiplied at a rate so low as to produce only 100 men, nor at a rate so high as to produce more than 800 men. I believe that this is a very logical explanation of what Heinzerling calls a statistical peculiarity.

    Once again I thank Ruediger Heinzerling for his very careful reading of my paper, and for the very interesting comments which he makes.

    (Professor Colin Humphreys, 

    ... and I would like to thank Professor Humphreys very much for his detailed comments on the review! I am completely sure that the readers of this page will be as grateful as I am to have his original reply at hand!

    As always there are still arguments left on all sides, and I would like to invite experts in this field to write down their own opinion or, even better, new alternative interpretations that could be published in this place as well. 

    At present (March 1999) I myself do not want to extend this discussion, since I have plenty of work in finishing 'Akzent 6' , after that, hopefully, analysing the chronology elaborated by Knut Stenring and Gerhard Larsson, and finally preparing an English variant of my article on the numbers in Numbers.

    October 1999:  An English article about the numbers in Numbers 1 and 26 can now be found here: 'Akzent 7'.

    30th October 1999: A short summary of the arguments presented in my review above is scheduled to appear in the year 2000 in VT.

    17th April 2000: A correction to that summary.

    10th August 2000: I should also explicitly mention the two papers of Prof. J. Milgrom and Prof. M.McEntire resp., concerning Prof. Humphreys' approach, as well as his response  to both of them.

    30th October 2001: There is still an additional article, this time by Prof. G.A.Rendsburg.


            (go to review no. 2)


    © seit  1998  Rüdiger Heinzerling