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Cultural Unity of Ancient Mathematics: the example of the Surveyor’s Rule
According to the theory of the Diffusion of Culture, various
widespread practices and beliefs are the product of certain special circumstances.
The theory tends to regard similar ideas and practices as evidence of historical
connection. For instance the universal use of weekdays in the same order and
with similar names indicate that the special idea of using a week of seven days
must have originated at a single place and then diffused over the globe. In
mathematics, an interesting case is that of the Surveyor’s Rule
(1)
for the area of a quadrilateral ABCD of sides AB = a, BC = b, CD = c, DA = d
as shown in Fig.1.
I first mention the universal popularity of the rule (1) as follows.
(i) Mesopotamia
The rule (1) was supposedly used in some Mesopotamian metro-mathematical texts
from Uruk IV (towards the end of the 4th millennium BC). It was used in the
pre-Babylonian or Sumerian texts (3rd millennium BC), and in the Old Babylonian
mathematical texts (c.2000-c.1600 BC) YBC 4675 (in which the area of a figure
of sides 17, 290, 7 and 310 is computed, the answer being 3600) and YBC 7290.
In the late Babylonian period (3rd century BC), the formula (1) was the standard
method for computing relevant areas.
(ii) Egypt
In the famous Ahmes (or Rhind) Mathematical Papyrus (c.
1650 BC), Problem 52 seems to have been solved by rule (1). Later on Egyptian
inscriptions (dated c. 100 BC) in the Horus temple at Edfu do explicitly employ
(1) e.g. for computing the area when the sides are 16, 4, 15, and 3 1/2, the
answer given being 58 1/8.
(iii) Europe
The rule (1) was used in some ancient Greek works such as the Liber geëponicus
and the pseudo-Heronian Geometrica. It was used by the Roman surveyors
and is also found in the Propositiones ad acuendos juvenes of Alcuin
(c.735-840 AD) and in the Geometry of Gerbert (940-1003) who became Pope
Sylvester II in AD 999. Similar rules were used quite late in Germany (c.1400)
and Russia.
(iv) China
The anonymous Chinese work Wu Tshao Suan Ching or Wucao suanjing
("Computational Canon of the Five Administrative Departments") of
about 400 AD computes the area of a quadrangular field of sides 35, 25, 45 and
15 paces by using (1). The same rule is used in another Chinese work called
Xiahou Yang suanjing ("Xiahou Yang’s Computational Canon").
(v) India
The first explicit statement of (1) in India is found verbally in the Brahmasphuta
Siddhanta (XII, 21) of Brahamgupta (AD 628). He clearly says that the rule
gives gross (sthula) area. For accuracy he gives
Area = SqRt (s – a)(s – b)(s – c)(s – d)
where s = (a + b + c + d)/2
Sridhara (c.750AD) in his Patiganita (verse 112) has quoted Brahmagupta’s
verbal rule for (1) only to criticise it. Anyway, as a practical formula, the
rule (1) is found in many subsequent Indian works such as Ganitasara-sarigraha
(VII, 7) of Mahavira (c.850 AD).
(vii) Arabia
The rule (1) was known to some Arab authors during medieval times. For instance,
it is found in the Ghunya al-Hussab ("Reckoner’s Wealth")
of Ahmed ibn Thabat (c.1200 AD). Abu’l Wafa (10th century) also knew it.
Thus we find that the use of the antique Surveyor’s Rule
(1) was quite widespread among various ancient and medieval cultural areas.
Some other relevant remarks may also be made as follows:
* Historically speaking, the rule (1) was also used to find the area of a triangle
by assuming it to be a quadrilateral in which one side is zero, i.e. by taking
d = 0 say. So, for a triangle of sides a, b, c, we have
area = (a + c).b/4 (2)
The area obtained in (2) is always in excess of the true area. Moreover, the
result is not unique, as we may also derive the expressions (b + c).a/4 and
(a + b).c/4 for the area of the triangle by similar argument.
* Mathematically, the rule (1) gives exact area only in the case of a rectangle
(including the square). In all other cases it yields a higher result. For we
have (see Fig. 1)
.
.
* Paradoxically, the rule (1) will yield the area of non-existent
figures. E.g. if a = 13, b = 4, c = 5, d = 2, we get S = 27 by (1), yet the
figure cannot be drawn (why?).
*Practically, the Surveyor’s Rule (1) is a good formula because it is based
on averaging. It continues to be used even today to approximate the area of
plane quadrangular fields by just measuring the four bounding sides. Otherwise,
a mere four sides are not enough to fix or define a quadrilateral uniquely.
So it is not theoretically possible to find any formula for exact area in terms
of four sides alone.
References
R. C. Gupta, "The Process of Averaging in Ancient and Medieval Mathematics",
Ganita Bharati, Vol.3 (1981), pp. 32-42
R. C. Gupta, "Primitive Area of a Quadrilateral", Ganita Bharati,
Vol.19 (1997), pp. 52-59
R. C. Gupta, "Something is Better than Nothing", Ganita Chandrika,
New Series, 2(3) (2001), pp. 22-25
R. C. Gupta
Jhansi, India
(Note: When putting the above article on the website, I had great difficulty with the mathematical symbols. The web software didn't provide many of the characters needed. Please forgive me if I did not accurately reproduce this article. Thank you. Karen Dee Michalowicz)
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