Mathematical Errors in the Bible

• The Value of π
• Mistakes in Sums
• Mistakes in Counting

I Kings 7:23-26 (II Chronicles 4:2-5) 23Then he made the molten sea; it was round, ten cubits from brim to brim, and five cubits high. A line of thirty cubits would encircle it completely. 24Under its brim were panels all around it, each of ten cubits, surrounding the sea; there were two rows of panels, cast when it was cast. 25It stood on twelve oxen, three facing north, three facing west, three facing south, and three facing east; the sea was set on them. The hindquarters of each were towards the inside. 26It's thickness was a handbreadth; its brim was made like the brim of a cup, like the flower of a lily; it held two thousand baths.

Nobody knows what the molten sea was for. It could be a container for water used in various rituals. [1] Whatever it is, based on the description given above, the drawing below (not drawn to scale) shows what the molten sea could have looked like:

Here is another nail in the coffin of biblical inerrancy-it got one of the most basic ratio of mathematics (and nature) wrong.

Of course fundamentalists and evangelicals hardly ever take anything lying down! They have attempted (and are continuing to attempt) many explanations to avoid this very obvious error. We will look at two main (mutually contradictory!) apologetic strategies to explain away this error. Basically they involve asserting that the ratio in I Kings 7:23-26 is either :

• A Good Enough Representation of π
  or
  
• An Extremely Accurate Representation of π

A Good Enough Representation of π?

Another line of defense would be that since we are talking about something that was written approximately two and a half millennia ago, it would be unfair to judge it by modern standards. An example of such a defense is from the fundamentalist Robert Mounce in his book Answers to Questions about the Bible wrote:

The rough measurements of antiquity do not have to conform to space age requirements. In the culture of that day the measurements were not only adequate but also inerrant. In our determination of what constitutes an error we must judge the accuracy of the scripture according to the prevailing standards of the time. [2] [Emphasis mine-PT]

Note the audacity here. Mounce admits that the figure of π is less than accurate by modern day standards but still insists that the measurement is inerrant because it was �inerrant� for its day. Thus he wants to be able to continue calling the bible inerrant despite this inaccuracy because it was "inerrant" given the knowledge of the culture of its day. Here is a case of having your cake and eating it too!

Mounce wrote that we should "judge the accuracy of scripture according to the prevailing standards of the time". However even when we judge the ratio with the "prevailing standards of the time", we find that the value comes up short. For we know of cultures that were contemporaneous with the Hebrews that were able to come up with values for π far superior to the one given in I Kings. The table below gives these figures: [3]

Date	Culture & Source	Value of p	% Error Compared to Modern Value
1650 BCE	Egyptian: Ahmes Papyrus	3.16049	0.6%
1600 BCE	Babylonian: Susa Tablet	3.125	0.5%
800-500 BCE	Indian: Sulbasutras	3.09	1.6%
550 BCE	Hebrew: I Kings 7:23-26	3.0	4.5%
250 BCE	Greek: Archimedes	3.14163	0.001%

The table clearly shows just how far off the mark the biblical figure is compared to the other cultures. Comparing the percentages of error, the ratio in I Kings is three times less accurate than the Indian, eight times less accurate than the Egyptian, nine times less accurate than Babylonian and more than three thousand times less accurate than the Greek figures! So when Mounce insisted that we judge the accuracy of the biblical ratio by the "standards of the time", our judgment is obvious-it is hopelessly inaccurate!

Another variation of this defense is for the apologist to simply shrug his shoulders and to assert that 3.0 is a "good enough" approximation for π. Thus we have the fundamentalist apologist Gleason Archer stating in his Encyclopedia of Bible Difficulties:

While it is true that the more exact calculation of pi is essential for scientific purposes...the use of approximate proportions or total is a familiar practice of normal speech, even today...The Hebrew author here is obviously speaking in the approximate way that is normal practice even today.[4]

Archer had taken to simply asserting that even if it is inaccurate, it is still okay because "everyone else is doing it".

This defense is hollow for a couple of reasons. Firstly it seems that the best defense of the apologists is that this supposedly inerrant work contains a gross inaccuracy (which is what it is). Secondly it is by no means true that anyone who speaks in a "approximate way today" would use 3.0 as an approximate value for π. Indeed everyone I know who is familiar with the concept of π would use 22/7 or 3.14 as approximations rather than 3.0.

Of course, a reasonable person looking at this passage will simply conclude that it is not a big deal. The Hebrew culture was obviously a pre-scientific one and as such as a value of 3.0 for π, while grossly inaccurate, is to be expected. After all we have no reason to believe that the Hebrew authors of the Bible were mathematically literate. Of course it also means that the Bible is a human document with all its inherent flaws.

However to Biblical inerranists this view is totally unacceptable, since behind these authors is God's supposed inspiration. Since, as we have seen above, the "close enough" defense does not work, many apologists are turning the tables around and are asserting that the value given in I Kings 7:23-26 is extremely accurate after all! We will look at these set of defenses next.

An Extremely Accurate Representation of π??

We will look at these attempts below.

1. The measurements are rounded off
2. The measurement are accurate but the circumference is meant for a different place on the outside of the molten sea
3. The measurements are accurate but the circumference is meant for a different place on the inside of the molten sea
4. The measurements are approximations but there is a mystical hidden correction factor in the word for "line" which gives the correct result

We will look at one given in the fundamentalist website Answers in Genesis. According to this explanation the way measurements (with a line with each cubit length marked on it) are done then would mean that all dimensions are "cubitised", i.e. those less than half a cubit are rounded down to the nearest cubit while those more than half a cubit would be rounded up. It is therefore claimed that the measurements given in I Kings 27:3 are "rounded off" numbers and that the actual dimensions would give π of a more acceptable value. They suggest, as examples, a circumference of 30.32 cubits and diameter of 9.65 cubits. This gives viola a ratio of 3.142, or π !

This argument carries no weight for two very simple reasons.

Firstly, the idea that the lengths are "cubitised" (measured in quantums of whole cubits) in the Bible is pure nonsense. In fact we have seen other places in the Bible where more detailed measurements ("non-cubitised") are given. As an example, the passage below is part of the instructions for the construction of the Ark of the Covenant:

Exodus 25:17 Then you shall make a mercy seat of pure gold; two cubits and a half shall be its length, and a cubit and a half its width.

Thus if we use the "exact" numbers suggested above and round them up to half cubits (which was certainly possible), the circumference would be 30.5 cubits and the diameter would have been 9.5 cubits. This would give a value of πof 3.211 or about 2.2% off the value of p. The actual value given, 3.0 (or 30/10), is about 4.5% off the mark. Thus, even allowing for the coarseness of the measuring unit available (1/2 a cubit), the writers of I Kings still did not get the most accurate available result.

Secondly, where did they get the numbers (30.32 and 9.65) from? A magician's hat! It was merely guessed that these could be the numbers. But guesses prove nothing. If one is allowed to simply come up with any numbers in place of the supposedly "rounded off" ones then why couldn't it have, say, a circumference of 29.65 cubits and the diameter to be 10.45 cubits, thus giving π as 2.837. This is even more inaccurate than 3.0 as an approximate of p, thus showing that the Biblical authors were even more mathematically inept than once thought. If the believer cries foul at this simple picking numbers from nowhere, then he gets the idea of how a skeptic would view the first explanation.

2. "The Measurement for the Circumference is Not Taken at the Brim"

This explanation accepts that the circumference is given accurately as 30 cubits but it is a measurement for the body of the molten sea, not the brim circumference. In the same posting as above (given as a second alternate explanation) it is "explained" that the circumference is of the body below the rim, however since the body is smaller than the brim the diameter must be less than 10 also. They therefore give 30/9.55 which comes up to 3.142! There are many problems with this explanation.

Firstly note that this explanation is contrary to the first explanation above-the first one assumes (rightly) that the natural unforced reading of verse 23 would mean that the dimensions for diameter and circumference refer to the same circle (at the brim). This second one says that the natural reading is not to be used and that it is the circumference of the body of the molten sea below the brim that is meant. Thus if the first explanation is true, the second has to be false and if the second explanation is true, the first has to be false (it does not follow, of course, that if one of false the other must be true. These are not contradictory statements, merely contrary statements.) The very fact that it is given in the same posting means that the apologists wouldn't mind whichever the reader choose as long as his faith is saved.

Secondly note that the number 9.55 cubits came from nowhere in the passage! It is merely used as the number, when used as the denominator with 30 as the numerator, it gives 3.142! In other words, the apologist has worked backwards, just to get the result as π

Thirdly a natural reading of the passage very obviously shows that it is the dimension of the brim that is being talked about. It mentioned it was five cubits high (to be brim), 10 cubits wide (brim to brim) and thirty cubits in circumference. Only after these dimensions were given (verse 24) did the passage began to speak about other parts of the molten sea; for it then starts with (7:24) "Under its brim...". It would be bending the meaning of the passage tremendously to take that the third measurement does not now mean the dimension at the brim and that it means measuring below the brim.

Like the first explanation, we can dismiss this second one as without any foundation and is a purely ad hoc one.

3. "The Circumference Being Referred to is the Inside Circumference of the Molten Sea"

This explanation (again mutually exclusive with the other two above), given by another website, attempts to save the day by stating that the circumference is accurate but it refers to the inner circumference. Thus the ID would be 10 cubits less double the width of the brim, which is supposedly given in I Kings 27:26 as "a handbreadth thick". He then goes on to give a cubit as 20 inches and a handbreadth as 4.5 inches. Thus the inner diameter is now [(10 x 20) - (2 x 4.5)] 191 inches. With the circumference as 600 inches (20 x 30), the value of the circumference over the diameter is now 600 divided by 191 which gives 3.14136! Which is a lot closer to π than 3.0!

The problem with this explanation is simple, it is demonstrably wrong.

Firstly it made an unwarranted assumption that the width of the brim is equal to the thickness of the wall of the molten sea. Note that verse 26 is talking about the thickness of wall of the molten sea not the width of the brim. To assume that these two are identical is certainly not called for. Look at the line drawing above again, the width of the brim in no way has to be the same as the thickness of the molten sea.

Secondly the inner circumference as the meaning of the passage is certainly a forced reading of the passage. The verse above "A line of thirty cubits would encircle it completely." (taken from the NRSV) clearly indicates it is the external dimension that is being talking about. How would an inner circumference "encircle" the molten sea completely? Other translations give the same meaning:

I Kings 7:23 (King James Version) [A] line of 30 cubits did compass it round about.

I Kings 7:23 (New King James Version) [And] a line of 30 cubits measured its circumference.

I Kings 7:23 (New International Version) It took a line of 30 cubits to measure round it.

In none of the above is the inner circumference even suggested by the passage. In fact the Biblia Hebraica Stuttgartensia (the Leningrad Codex of the OT used by most Bible translation as "the" Hebrew Bible) gives the word here as saviv. This is given in Holladay's A Concise Hebrew and Aramaic Lexicon of the Old Testament as "all around", "from all sides" or the "surrounding environs" or "neighborhood" [5]. All these words give a very explicit meaning of something outside surrounding what is being mentioned.

Thus the argument fails for these reasons. The identity of the width of the brim, with the thickness of the molten sea is not warranted. The verse clearly implies a dimension (in this case the circumference) external to, or on the outside of, the molten sea.

4. "There is a Hidden Correction Factor Within the Text"

This attempt at salvaging biblical inerrancy is probably the silliest of the lot, but for completeness sake, we will present it over here.

One example of how such a defense is done goes like this. The word translated as "line" (in I Kings 7:23) is actually (qof-vav-he). [b] The normal way for the word "line" to be written is without the final (he), thus it would be (qof-vav). In the Hebrew massoretic text, notes are given in the margins when the word written (called kethiv) in the text is not what the scribes thought should be there, they give in the margin what should be read (called qere) in its place. In this case, then, the main text (kethiv) is given as (qof-vav-he), while in the marginal note is given the qere, (qof-vav). The presence of this kethiv/cere apparatus hints of a deeper meaning, which in this case means that the word should be treated as a mathematical formula.

According to this explanation, the ancient Hebrews gave numerical values to their alphabets. Thus (qof) represents 100, (vav) represents six while (he) represents five. , the normal way the word "line" is written gives a total value of 106 (100 + 6). While , gives a total value of 111 (100 + 6 + 5). This gives an adjustment of 111/106 for the circumference. This gives a value of π of 3.141509 [(111/106) x (30/10)]. This gives a final value very close to the modern value of π (3.141592).

The critique of this argument is simple, given enough ingenuity and with no strict rules to abide by, any kind of numerical co-incidences or numerical value can be "discovered". For instance, why must the value of "line" be considered a correction factor? And why is it 111/106? Why not 106/111, 111+106 or 111-106 or even 111 x 106? The answer is simple, these wouldn't give the desired result! There are no rules, the derivation is merely driven by the need to get the value to come close to 3.14159. Thus as a defense of biblical inerrancy it represents another ad hoc defense. [c]

That is not the only thing wrong with the defense.

It must be remembered that numbers are invariably spelled out in full in the main text of the Hebrew Bible. Thus, for instance, the number "thirty" (for the circumference) is written here as sheloshim. Why is it not written as lamed or kaf-yod (Either of which would be the alphabetical representation of the number 30.)[d]? The answer is simple, the practice of using Hebrew alphabets to represent numbers only started in the Maccabean period, [6] around the mid second century BCE.[e] The Book of Kings reached its final form around the middle of the sixth century BCE, a full four centuries before the Maccabean period! [7] Thus applying the alphabets-as-numerals concept would be grossly anachronistic.

Furthermore, while the Bible does attach significance to numbers (for instance seven means perfection, forty is normally taken to mean a long period of time), it does not contain numerological speculation one finds in the works of the Pythagoreans or in the later Jewish Qabbalah. [8] Thus to apply such numerological explanations actually goes against the grain of the whole Hebrew Bible.

Finally the idea that the presence of a marginal note in the massoretic text, substituting a kethiv with a qere, implies a "hidden" meaning is pure nonsense. As the Oxford Companion to the Bible explains, kethiv is simply an Aramaic term meaning "(that which is) written" and qere "(that which is) to be read". Textual scholars only recognize three uses for the kethiv/qere apparatus: [9]

• Due to the scribal respect for the Hebrew consonantal text, errors in the main text are not changed (i.e. left as the kethiv), instead a note at the margins of the Bible is added giving either the correct spelling or correct vowels (the qere).
• Sometimes words such as the name of God (yod-he-vav-he or YHWH), which are not uttered, are left in the main text. Thus the qere provides another reading which can be pronounced, such as adonai.
• Finally some scholars think that the kethiv/qere method is also used to preserve variant readings when the text of the Hebrew Bible was standardized during the first century CE.

Thus the numerological/mystical argument of a hidden factor fails.

The Biblical Value of p: A Summary

It is also important to note that these "explanations" are mutually exclusive. At best only one could be true. (But, as we have seen, there is no logical reason why they can't all be false.) Some of these are given side by side in the same apologetic work for the believer to pick and choose which "explanation" they prefer. It is obvious that to these apologists the truth of the matter is irrelevant and is subordinated to saving the faith of the believer.

Why this proliferation of mutually exclusive explanations? The answer is simple, because the natural, unforced reading of the passage gives π as 3.0. And 3.0 is not only grossly inadequate by our modern standards but also in comparison with the contemporaneous cultures of Egypt, Babylon, India and Greece.

There is not a hint anywhere in the passage that either the numbers refer to somewhere else on the molten sea or that the numbers were approximate. Thus the apologists are left to their own devices and simply choose the first ad hoc explanation that come to their minds. All the while the natural reading stares upon their faces from I Kings 7:23.

Summarized this way, the defenses are unconvincing and the error still stands in I Kings 7:23, the Bible says π is equal to 3.0!.

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Mistakes in Sums

There is another summation error, this time big enough to give even the �round off� apologetic explanation a �headache� (in trying to come up with an explanation!). The passage Ezra 1:7-11 concerns the articles from the Temple which were returned to the Jews. It was mentioned, in Ezra 7:8, that the chief royal treasury made an inventory and that these are the items returned:

Ezra 1:11 all the vessels of gold and of silver were five thousand four hundred and sixty nine.

Note that Ezra 1:10 does not allow other unlisted items to be assumed since the last portion already mentioned �other vessels� or other articles. Obviously a discrepancy of 2,970 exist between the given total of 5,469 and the actual total of 2,499.

Another mistake in arithmetic which, as far as I can tell, was first pointed out by the American patriot, Thomas Paine (1737-1809), in his anti-Christian book The Age of Reason,[10] comes from the books of Ezra and Nehemiah. In the second chapter of Ezra and the seventh chapter of Nehemiah, the authors gave a list of tribes and families, and the numbers of people of each, that were returning from the Babylonian exile. At the end of this list both Ezra and Nehemiah gave exactly the same total for the total number of people returning:

Ezra 2:64 (Nehemiah 7:66) The whole assembly together was forty two thousand three hundred and sixty.

Thus the number is 42,360 people. The table below gives a breakdown of the list of the number of people given by Ezra and Nehemiah. Note that both Ezra's and Nehemiah's total do not come up to the figure they mentioned. In fact they even contradict one another. Ezra's total was actually 29,818 and Nehemiah's was 31,089. Both the authors can't do simple sums! Ezra missed the final total by 12,542 while Nehemiah missed it by 11,271. This discrepancy is not the only thing wrong with the two lists. The numbers of the individual families given by the two lists also contradict each other in at least 16 different places. For instance, Ezra gives the number of the sons of Arah as 775 while Nehemiah gives it as 652.

EZRA			NEHEMIAH
Verse	Passage	No. of People	Verse	Passage	No. of People
2: 3 2: 4 2: 5 2: 6 2: 7 2: 8 2: 9 2:10 2:11 2:12 2:13 2:14 2:15 2:16 2:17 2:18 2:19 2:20 2:21 2:22 2:23 2:24 2:25 2:26 2:27 2:28 2:29 2:30 2:31 2:32 2:33 2:34 2:35 2:36 2:37 2:38 2:39 2:40 2:41 2:42 2:58 2:60	sons of Parosh sons of Shephatiah sons of Arah sons of Jeshua & Joab sons of Elam sons of Zattu sons of Zaccai sons of Bani sons of Bebai sons of Azgad sons of Adonikam sons of Bigvai sons of Adin sons of Ater sons of Bezai sons of Jorah sons of Hashum sons of Gibbar sons of Bethlehem sons of Netophah men of Anathoth sons of Azmaveth sons of Kiriatharim... sons of Ramah & Geba men of Michmas men of Behtel & Ai sons of Nebo sons of Magbish sons of other Elam sons of Harim sons of Lod, Hadid & Ono sons of Jericho sons of Senaah sons of Jedaiah sons of Immer sons of Pashhur sons of Harim sons of Hadaviah sons of Asaph sons of gatekeeper servants of temple... sons of Delaiah, Tobiah & Nekoda	2172 372 775 2812 1254 945 760 642 623 1222 666 2056 454 98 323 112 223 95 123 56 128 42 743 621 122 223 52 156 1254 320 725 345 3630 973 1052 1247 1017 74 128 139 392 652	7: 8 7: 9 7:10 7:11 7:12 7:13 7:14 7:15 7:16 7:17 7:18 7:19 7:20 7:21 7:22 7:23 7:24 7:25 7:26 - 7:27 7:28 7:29 7:30 7:31 7:32 7:33 - 7:34 7:35 7:37 7:36 7:38 7:39 7:40 7:41 7:42 7:43 7:44 7:45 7:46 7:47	sons of Parosh sons of Shephatiah sons of Arah sons of Jeshua & Joab sons of Elam sons of Zattu sons of Zaccai sons of Binnui sons of Bebai sons of Azgad sons of Adonikam sons of Bigvai sons of Adin sons of Ater sons of Hashum sons of Bezai sons of Hariph sons of Gibbeon men of Bethlehem & Netophah men of Anathoth men of Bethazmaveth men of Kiriathjearim... men of Ramah & Geba men of Michmas men of Behtel & Ai men of other Nebo - sons of other Elam sons of Harim sons of Lod, Hadid & Ono sons of Jericho sons of Senaah sons of Jedaiah sons of Immer sons of Pashhur sons of Harim sons of Jeshua..of Hodevah sons of Asaph gatekeepers servants of temple sons of Delaiah, Tobiah & Nekoda	2172 372 652 2818 1254 845 760 648 628 2322 667 2067 655 98 328 324 112 95 - 188 128 42 743 621 122 123 52 - 1254 320 721 345 3930 973 1052 1247 1017 74 148 138 392 642
Total		29818	Total		31089

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Mistakes in Counting

I Chronicles 3:22 The sons of Shemaiah: Huttush, Igal, Bariah, Neriah, and Shaphat, six.

Note that there were five sons listed, yet the Chronicler counted six! [For those of you with the NIV, please read note [f] below] We find similar errors in I Chronicles 25:3, where five names were also given and again the chronicler counted six. (One begins to wonder if this is a learning disorder of the chronicler) Again in I Chronicles 3: 19-20 where eight names were given and he counted five! Chronicles is not the only author who can't count, we see the same problem in the book of Joshua:

Joshua 15:33-36 And in the lowland, Eshtaol, Zorah, Ashnah, Zanoah, Engannim, Tappuah, Enam, Jarmuth, Adullam, Socoh, Azekah, Shaaraim, Adithaim, Gederah, Gederothaim: fourteen cities with their villages.

Anyone who can count will see that there are fifteen cities listed above, not fourteen. Again in Joshua 15:21-32 there were thirty-six cities actually listed but the author only counted twenty-nine! Also in Joshua 19:2-6 we have fourteen cities listed but the author said there was only thirteen.

So we have seen that some of the authors of the Bible didn�t know the value of π, some didn�t know how to do simple sums and some couldn�t even count correctly! How are we to define a book as inerrant when its authors lack even these simple skills!

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Notes

References

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