"THE NUMBER OF PEOPLE IN THE
EXODUS FROM EGYPT: DECODING MATHEMATICALLY THE VERY LARGE NUMBERS IN NUMBERS
I AND XXVI"
COLIN J. HUMPHREYS, Cambridge
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
A) The basic assumption
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:
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."
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.
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."
is a decisive problem with regard to the alleged precision of the number
If we write down the number of
Levites as Humphreys does (Table 3, p.213):
Therefore originally a number If with 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):
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, ...)?
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.
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.
2 'lp and 750 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:
8 'lp and 300 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:
7 'lp and 1300 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
73 'lp and 1 'lp + 600 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.:
At least this last question shall be shortly discussed yet:
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.
|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.|
(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:
(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.
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:
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:
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
|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
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
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
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.
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.
(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
Once again I thank Ruediger Heinzerling for his very careful reading of my paper, and for the very interesting comments which he makes.
|... 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.
30th October 2001: There is still an additional article, this time by Prof. G.A.Rendsburg.
to review no. 2)