If the history of mathematics interests you, I highly recommend Jacques Sesiano's book “An Introduction to the History of Algebra“ (AMS, 2009). I particularly enjoyed the first chapter on ancient Babylonian algebra. There's one thing that I was really missing though... images of the original text. Digging just one of these images up sent me on something of a journey and sadly I can only use them for educational, non-profit purposes (which is why this is on my blog). The word "text" might give readers the wrong idea, so let me just throw this out there:

Photo credit: © Staatliche Museen zu Berlin – Museum of the Ancient Near East | Photo: Olaf M. Teßmer, May 2010 | USED WITH PERMISSION |

What you're looking at is tablet

**VAT 8389**from the Museum of the Ancient Near East in Berlin, Germany. The tablet contains the solution to system of linear equations that today we'd write like this:
x + y = 1800

⅔∙x - ½∙y = 500

Such representation is anachronistic to say the least.

Today I'm going to pull out everything I know about this tablet: transliteration, literal and adapted translation, mathematical interpretation, and comparison with modern mathematics.

The Babylonians didn't have lined paper, so I've corrected that here:

Photo credit: © Staatliche Museen zu Berlin – Museum of the Ancient Near East | Photo: Olaf M. Teßmer, May 2010 | USED WITH PERMISSION | Edited by me in PowerPoint |

**Transliteration**.

The following transliteration appears in Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (Springer, 2013) by Jens Høyrup. I am no expert in ancient pronunciation of ancient Babylonian cuneiform (my background is in chemical engineering...). This was hard to type and I am not positive I got all the characters right.

I-1 i-na
bùr 4 še.gur am-ku-us

I-2 i-na
bùr ša-ni-im 3 še.gur am-ku-us

I-3 še-um
ugu še-in 8,20 i-ter

I-4 garim
ĝar.ĝar-ma 30

I-5 garim
en.nam

I-6 30
bu-ra-am ĝar.ra 20 še-am ša im-ku-sú ĝar.ra

I-7 30
bu-ra-am ša-ni-am ĝar.ra

I-8 15
še-am ša im-ku-sú

I-9 8,20
ša še-um ugu še-im i-te-ru ĝar.ra

I-10 ù 30
ku-mur-ri a.šà garim.mes ĝar.ra-ma

I-11 30
ku-mur-ri a.šà garim.meš

I-12 a-na
ši-na ḫe-pé-ma 15

I-13 15 ù 15
a-di si-ni-šu ĝar.ra-ma

I-14 igi 30
bu-ri-im pu-tur-ma 2

I-15 2 a-na
20 še ša im-ku-su

I-16 íl 40
še-um lul a-na 15 ša a-di ši-ni-šu

I-16a ta-aš-ku-nu

I-17 íl 10
re-eš-ka li-ki-il

I-18 igi 30
bu-ri-im ša-ni-im pu-tur-ma 2

I-19 2 a-na
15 še-im ša im-ku-sú

I-20 íl 30
š-um lul a-na 15 ša a-di ši-ni-šu

I-20a ta-aš-ku-nu
íl 7,30

I-21 10 ša
re-eš-ka ú-ka-lu

I-22 ugu 7,30
mi-nam i-ter 2,30 i-ter

I-23 2,30 ša
i-te-ru i-na 8,20

I-24 ša še-um
ugu še-im i-te-ru

II-1 ú-sú-uḫ-ma
5,50 te-zi-ib

II-2 5,50 ša
te-zi-bu

II-3 re-eš-ka
li-ki-il

II-4 40
ta-ki-ir-tam ù 30 ta-ki-ir-tam

II-5 ĝar.ĝar-ma
1,10 i-gi-a-am ú-ul i-de

II-6 mi-nam
a-na 1,10 lu-uš-ku-un

II-7 ša 5,50
ša re-eš-ka ú-ka-lu i-na-di-nam

II-8 5
ĝar.ra 5 a-na 1,10 íl

II-9 5,50
it-ta-di-kum

II-10 5 ša
ta-aš-ku-nu i-na 15 ša a-di ši-ni-šu

II-11 ta-aš-ku-nu
i-na iš-te-en ú-sú-uḫ

II-12 a-na
iš-te-en sí-im-ma

II-13 iš-te-en
20 ša-nu-um 10

II-14 20 a.šà
garim iš-te-at 10 a.ša garim ša-ni-tim

II-15 šum-ma
20 a.šà garim iš-te-at

II-16 10 a.šà
garim ša-ni-tim še-ú-ši-na en.nam

II-17 igi 30
bu-ri-im pu-tur-ma 2

II-18 2 a-na
20 še-im ša im-ku-sú

II-19 íl 40
a-na 20 a.šà garim iš-te-at

II-20 íl 13,20
še-um ša 20 a.šà garim

II-21 igi 30
bu-ri-im ša-ni-im pu-ṭur-ma 2

II-22 2 a-na
15 še-im ša im-ku-sú íl 30

II-23 30 a-na
10 a.šà garim ša-ni-tim

II-24 íl 5
še-um ša 10 a.šà garim ša-ni-tim

II-25 13,20
še-um a.šà garim iš-te-at

II-26 ugu 5
še-im a.šà garim ša-ni-tim

II-27 mi-nam
i-ter 8,20 i-ter

**Literal Translation**

Next is a word-for-word translation. It's worth mentioning that Babylonian numbers were written in sexagesimal (base 60). For clarity I've included the place value of numerals which the Babylonians inferred by context.

I-1 From a
bur 04. kur of grain I have collected.

I-2 from a
second bur 03. kur of grain I have collected.

I-3 grain
over grain, 08,20. it went beyond

I-4 My
plots I have accumulated: 30,00.

I-5 My
plots what?

I-6 30,00. the bur, posit. 20,00. , the grain which he has collected, posit.

I-7 30,00. ,
the second bur, posit.

I-8 15,00. ,
the grain which he has collected

I-9 08,20. which the grain over the grain went beyond

I-10 and
30,00. the accumulation of the surfaces of the plots posit:

I-11 30,00.
the accumulation of the surfaces of the plots

I-12 to two
break: 15,00.

I-13 15,00.
and 15,00. until twice posit:

I-14 Inverse
30,00. , of the bur, detach 00.00,02

I-15 00.00,02 to 20,00. , the grain which he has collected

I-16 raise,
00.40 the false grain; to 15,00. which until twice

I-16a you have
posited,

I-17 raise,
10,00. may your head hold!

I-18 Inverse
30,00. , of the second bur, detach 00.00,02

I-19 00.00,02 to 15,00. , the grain which he has collected

I-20 raise,
00.30 the false grain; to 15,00. which until twice

I-20a you have
posited, raise 07,30. .

I-21 10,00.
which your head holds

I-22 over
07,30. what goes beyond? 02,30. it goes beyond

I-23 02,30.
which it goes beyond, from 08,20.

I-24 which
the grain over the grain goes beyond

II-1 tear
out: 05,50. you leave

II-2 05,50.
which you have left

II-3 may
your head hold!

II-4 00.40,
the change, and 00.30 the change,

II-5 accumulate
01.10 . The inverse I do not know.

II-6 What to
01.10 may I posit?

II-7 which
05,50. which you head holds gives me?

II-8 05,00.
posit. 05,00. to 01.10 raise.

II-9 05,50.
it gives to you.

II-10 05,00.
which you have posited, from 15,00. which until twice

II-11 you have
posited, from one tear out,

II-12 to one
append:

II-13 The
first is 20,00. , the second is 10,00.

II-14 20,00.
is the surface of the first plot, 10,00. is the surface of the second plot

II-15 If
20,00. is the surface of the first plot,

II-16 10,00.
the surface of the second plot, their grains what?

II-17 Inverse
30,00. , of the bur, detach 00,00.02 .

II-18 00,00.02 to 20,00. , the grain which he has collected,

II-19 raise
00.40 to 20,00. . The surface of the first plot,

II-20 raise,
13,20. . The grain of 20,00. , the surface of the meadow

II-21 Inverse
30,00. , of the second bur, detach
00.00,02

II-22 00.00,02 to 15,00. the grain which he has collect, raise, 00.30

II-23 00.30 to 10,00. , the surface of the second plot

II-24 raise,
05,00. the grain of the surface of the second plot.

II-25 13,20.
the grain of the surface of the first plot

II-26 over 05,00.
the grain of the surface of the second plot

II-27 what
goes beyond? 08,20. it goes beyond.

**Adapted Translation**

Between the strange grammar, the use of sexagesimal, lack of units (which if written, would have been long outdated [says the guy from the US...]), and ancient methodology, I can't make much of this without differing expertise. The following adapted translation is based on that found in Jacques Sesiano's book: “An Introduction to the History of Algebra“ (AMS, 2009).

"Bur" and "sar" are units of field area. 1 bur = 1800 sar. A sar is about 36 square meters.

"Kur" and "sila" are units of grain volume. 1 kur = 300 sila. A sila is about 1 liter.

I-1 The
1st field gave 4 kur for each bur (2/3 sila for each sar)

I-2 The
2nd field gave 3 kur for each bur (1/2 sila for each sar)

I-3 The
1st field gave 500 more sila than the 2nd

I-4 The
sum of the fields' areas is 1 bur (1800 sar)

I-5 What
is the area of each field?

I-6 The
1st field gave 1200 sila for every 1800 sar

I-7 The
2nd field gave 900 sila for every 1800 sar

I-8 ↑

I-9 The
1st field gave 500 more sila than the 2nd

I-10 The
total area of the two fields is 1800 sar

I-11 ↑

I-12 1800/2 =
900

I-13 You now
have two values of 900

I-14 (1800)^(-1)
= (1/1800)

I-15 (1/1800)
× 1200 = (2/3) "false grain"

I-16 (2/3) ×
900 = 600 "hold"

I-16a ↑

I-17 ↑

I-18 (1800)^(-1)
= (1/1800)

I-19 (1/1800)
× 900 = (1/2) "false grain"

I-20 (1/2) ×
900 = 450

I-20a ↑

I-21 600 -
450 = 150

I-22 ↑

I-23 500 -
150 = 350 "hold"

I-24 ↑

II-1 ↑

II-2 ↑

II-3 ↑

II-4 (2/3) +
(1/2) = (7/6)

II-5 (7/6)^(-1)
= unknown

II-6 (7/6) ×
z = 350

II-7 ↑

II-8 300 ×
(7/6) = 350, z = 300

II-9 ↑

II-10 900 +
300 = 1200, 900 - 300 = 600

II-11 ↑

II-12 ↑

II-13 The 1st
field has an area of 1200 sar, The 2nd field has an area of 600 sar

II-14 ↑

II-15 ↑

II-16 What are
the yields of each field?

II-17 (1800)^(-1)
= (1/1800)

II-18 (1/1800)
× 1200 = (2/3)

II-19 (2/3) ×
1200 = 800 "grain of the 1st
field"

II-20 ↑

II-21 (1800)^(-1)
= (1/1800)

II-22 (1/1800)
× 900 = (1/2)

II-23 (1/2) ×
600 = 300 "grain of 2nd field"

II-24 ↑

II-25 The 1st
field produced 800 sila

II-26 The 2nd
field produced 300 sila

II-27 800 -
300 = 500

**Mathematical Interpretation**

To make sense of this I've prepared the following graphic. I've done my best to identify the major events in each part of the procedure.

Image Credit: Self (Power Point) |

The Babylonians solved this problem by figuring out how far the area of each field deviated (steps B-G) from half of the total area (step A). Step H has the answer, and steps I-K verify this answer is correct.

**Analysis of Method**

Working backward from the described method of verification, we can use modern methods to determine the ancient methods. Here we call the deviation from midpoint "z" and reverse engineer its method of determination. How did the Babylonians figure this out? I can speculate until the cows come home... but rigorous answers have been lost to the sands of time.

Image Credit: Self |

**Modern Method**

Here's how the math looks today after filtering through 4000 years of advancement by civilization after civilization, specifically tenth-c. BCE Assyria, the seventh-c. BCE Chaldeans, sixth-c. BCE Persia, fourth-c. BCE Greece, first-c. CE Rome, fifth-c. CE India., ninth-c. CE Arabia, and 11th-c. CE Europe. By the beginning of the scientific revolution (17th-c. CE Europe), the notation and methodology would have been recognizable as something like this:

x + y = 1800

⅔∙x - ½∙y = 500

y =
1800 – x

⅔∙x - ½∙(1800
- x) = 500

⅔∙x -
900 + ½∙x = 500

(7/6)∙x
= 1400

x =
1200

(1200)
+ y = 1800

y = 600

Is our modern method better? Yes. To be fair, it has the advantage of contributions of nearly every great civilization of Eurasia over the last 4000 years. Let us hope these methods survive even the fall of our own civilization.

I like how all of these discovery and ideas of basic Algebra, come down to Galois's theory, in the abstract realm of Modern Algebra. The key contributing factor to Galois's discovery of various equations, all are noted from predecessors of mathematicians, who all played a role in improving and rewriting Algebra. Galois tirelessly refers back to Gauss and Lagrange's works in the process of writing and composing existing equations, and finding it in a simple explanation, which to some may rather be complicated. I'd like how you reference the sources and restructure the ancient tablet to make Algebra more understandable.

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