### Mathematical and cultural connections in ancient mathematics: The case of bow-figure

by Professor R. C. Gupta, Ph. D. (Hist. of Math.)

The recent excellent book by Jöran Friberg (see References at the end) clearly exposes some significant connections between mathematics of ancient Egypt and Babylonia. Earlier the cultural unity of the popular antique Surveyor’s Rule

$Area= (1/2)(a+c) \cdot (1/2)(b+d)$ …(1)

for the area of a quadrilateral of sides a, b, c, d, has been well illustrated by the present author (Gupta, 2002). The case of the bow-figure (circular segment up to the semicircle) is discussed in this article.

Fig. 1

Let a segment (Fig. 1) of a circle (of diameter d = 2r) be bounded by the chord PQ (of length c) and arc PVQ (of length s). Let h be the length of the arrow VM which is usually called the height of the segment. The exact formula

$c^2=4h(d-h)$ …(2)

can be found by using the so-called theorem of Pythagoras in the triangle obtained by joining the midpoint M and P (or Q) to the centre of the circle and was known in ancient times. Given c and h, the segment is uniquely defined and d can be found from (2). Here we will describe some empirical rules for finding the arc s and the area of the segment as used in various ancient cultural regions. Such rules continued to be used in practical geometry for quick and approximate mensuration even once trigonometry became available and yielded better results.
For rectification of the arc s, the Babylonians used the simple empirical formula

$s=c+h$ …(3)

and was discovered by the present author (2001) from the calculations mentioned in the mathematical text BM85194, which is dated about 1600 BCE. There are claims that this Babylonian text used (3) to find h from the given s = 60 and c = 50.

Fig. 2

According to the Old Babylonian (about 1800 to 1600 BC) table of constants, the relation between the circumference C and the transversal or diameter d of a circle was

$C=3d$ …(4)

which obviously implies the simple approximation $\pi{} = 3$. For a semicircular arc, the expression (3/2)d is found in BM85210. Thus the length of the arc s of the semicircle (Fig. 2) takes the form

$s=d+r$ …(5)

or
arc PVQ=(base PQ)+(cross-line VM) …(6)

This relation was used in the ancient Greek mathematical text P. Vindob G. 26740 (about 3rd century BCE) to find s (from PQ = 30 and VM = 15) and then the rule

Area = $s^2/3$ …(7)

was applied to find the area of the full circle. Note that (7) comes from the antique rule

Area= $C^2/12$ …(8)

which was popular in Old Babylonian texts, e. g., YBC7302 and YBC11120 (and, of course, C = 2s).

The ancient Babylonians treated the segment in analogy to the semicircle with which the former resembles. Due to this similarity, the relation (6) was interpreted and applied analogously to the segment (Fig. 1) thereby resulting in (3).

Fig. 3

A tradition similar to (3) is found in a peculiar case in India. Mahāvīra in his Gaņitasāra-sańgraha VII.21 (about 850 CE) roughly rectifies the elongated circle (or ellipse of axes 2a and 2b) (Fig. 3) by treating it as a double circular segment. The Gaņitasāra-kaumudī III.49 of Thakkura Pherū (about 1300 CE) gives equivalent of (3) in the form

$c=\sqrt{4(s- \frac{s+h}{2})^2}$ …(9)

which is, otherwise, simply c = s – h.

A significant use of (3) is in providing an easy derivation of the ancient simple formula

$A=(c+h) \cdot h/2$ …(10)

for the area of the segment. The derivation uses the rule

$A_0=(p \cdot w)/4$ …(11)

which gives the area of a generally round plane figure (e. g., circle) of perimeter p and typical width w. The result was known in almost all ancient cultures. The passage from (11) to (10) becomes clear by considering double segment (Fig. 4) for which p = 2s, w = 2h, and $A_0=2A$. In fact, on substituting these in (11) we get

$A=(s \cdot h)/2$ …(12)

which leads to (10) by using (3).

Fig. 4

The formula (10) is explicitly found and used in the Egyptian demotic papyri called P. Cairo (3rd cent BCE). One problem deals with an equilateral triangle (of side 12 divine units) enclosed by a circle (Fig. 5). The height h of the segment PVQ is correctly taken as 1/3 of the height (=$\sqrt{10}$) of the triangle. Its area is found by (10).

Fig. 5

The rule (10) is found in the Chinese Jiu Zhang Suan Shu (1st cent. CE) and was known to Zhang Qiujian (about 840 CE). It was used by Shen Gua to derive

$s=c+2h^2/d$ …(13)

which is found in his Menggi Bitan (“Dream Pool Essays”) (1086 CE). Heron (Greek; 1st cent. CE) attributed (10) to “the ancients” and mentioned its modified forms. An improved form is

$A_1=(c+h) \cdot h/2+(\pi{}-3)c^2/8$ …(14)

The form with $\pi{} = 22/7$ is found in the work of the Roman Columella (62 CE), in the Hebrew Mishnat ha-Middot (about 150 CE) and in Chinese Siyuan Yujian (1303 CE) of Zhu Shiji who took $\pi{} = 157/50$ also in (14).

Rule (10) is said to be based on $\pi{} = 3$ for which it gives an exact result in the case of a semicircle. As a rough rule, this is also found in Mahāvīra’s Gaņitasāra Sańgraha VII.43, in Nemicandra’s Trilokasāra, 762 (about 980 CE), and others. It was also modified in India as

$A_2=(c+h) \cdot h \cdot \pi{}/6$ …(15)

The form with $\pi{} = \sqrt{10}$ is found in the Triśatikā, 47 of Śrīdhara (about 750 CE) and in other works up to late times. Forms with $\pi{} = 19/6, 22/7,$ and 63/20 are also found in India.

Fig. 6

Analogy as a method of proof has been quite common through the ages. For deriving (10), several analogies exist. Similarity of segment with semicircle is quite natural. With the approximation $\pi{} = 3$, the area $A_3 (=3r^2/2)$ of the semicircle (Fig. 6) can be expressed as

$A_3 = (1/2) (2r \cdot r) + 2(r^2/4) = (\mbox{area of } \Delta{} PVQ) + 2 \cdot (VM)^2/4$ …(16)

where $(VM)^2/4$ represents the shaded area of each sub-segment.

Fig. 7

Hence, by analogy, the area A4 of the segment (Fig. 7) can also be taken as

$A_4=(\mbox{area of } \Delta PVQ)+2 \cdot (VM)^2/4$ …(17)
$=(1/2)ch+2(h^2/4)$ …(18)

which is the desired result (10). In this context, the Chinese Liu Hui (263 CE) noted an interesting analogy in a regular polygon of 12 sides. Let P, P1, P2, V, P3, P4, Q be the vertices of upper half of the polygon (Fig. 8). The area of this half figure can be easily seen to be exactly $(3/2) r^2$ and Liu Hui found that the area of the shaded subtrapezoid on each adjacent side of $\Delta{} PVQ$ will also be exactly $r^2/4$ (in a semicircle these expressions represent only approximate values).

Fig. 8

Fig. 9

Instead of inscribed triangle, the circumscribed rectangle (enclosing the figures) may be considered. Here we have, for the semicircle (Fig. 9)

$A_5=2r \cdot r-2 \cdot (r^2/4)$
$=(\mbox{area} PEFQ)-2(VM)^2/4$ …(19)

where (VM)^2/4 now represents the (extra) area of each curvilinear corner triangle. So by analogy, the area of the segment (Fig. 10) will be

$A_6=Ch-h^2/2$ …(20)

which is indeed found in the Babylonian text BM85194 according to one interpretation. The Babylonians might have used a simpler concept.

Fig. 10

Fig. 11

Area of each curvilinear triangle can be taken equal to that of an isosceles right triangle at each corner. Together, the two such right triangles (PEG and QFK) (Fig. 11) form a square of area $h^2$. So the area of the segment will be

$A_7=Ch-h^2$ …(21)

One thing to note is that the correction term in (18) or (20) is numerically same (namely $h^2/2$). So by taking the average of $A_4$ and $A_6$, we get the empirical rule

$A_8=(3/4)Ch$ …(22)

for the area of the segment. Interestingly, this rule follows also by directly taking the mean of the areas of the inscribed triangle and the circumscribed rectangle. Adjusted to use the general value of $\pi{}$, it becomes

$A_9=(\pi{}/4) \cdot Ch$ …(23)

The form of (23) with $\pi{} = \sqrt{10}$ was regarded as accurate in the Jaina School in India and is found in various works. These include, for example, the Tiloya Pannatti, IV. 2401 of Yativssabha (before 607 CE) and Brhatksetra Samāsa I. 122 of Jinabhadra Gani (607 CE).

Fig. 12

Lastly, a geometrical unity of most of the above rules may be highlighted. A circular segment PVQP can be nicely approximated (in area) to a suitable trapezoid PRSQ (Fig. 12). The following are easily seen:

(i) When RS =h, we get the popular formula (10);
(ii) When RS = c – h, we get the rule (20);
(iii) When RS = c – 2h, we get (21);
(iv) When RS = c/2, we get (22);
(v) And when RS = r, we get a new rule, namely
$A_{10}=(c+r) \cdot h/2,\mbox{when } r …(24)

The present author found this rule by using the ancient Babylonian rule (3) in the usual relation

Segment PVQP = (Sector OPVQO) – (Triangle OPQ)

where O is the centre (not shown) of the arc PVQ (Fig. 1). Thus apparently different rules stand connected and unified mathematically. We do seek patterns in mathematics as well as in its history.

By: Professor R. C. Gupta, Ph. D. (Hist. of Math.)
R-20, Ras Bahar Colony, P. O. Sipri Bazar, JHANSI-284003 (India)

REFERENCES
Jöran Friberg: Unexpected Links between Egyptian and Babylonian Mathematics. World Scientific, Singapore, 2005.
R. C. Gupta: “Mahaviracarya on the Perimeter and Area of an Ellipse”, Mathematics Education (Siwan), Vol. 8, No. 1 (1974), Sec. B, 17-19.
R. C. Gupta: “On Some Jaina Mathematics Rules”, Ganita Bharati, 11 (1989), 18-26.
R. C. Gupta: “Mensuration of a Circular Segment in Babylonian Mathematics”, Ganita Bharati, 23 (2001), 12-17.
R. C. Gupta: “Cultural Unity of Ancient Mathematics: the Example of the Surveyor’s Rule”, HPM Newsletter No. 50 (July 2002), 2-3.
R. C. Gupta: “Area of a Bow-Figure in India”, Studies in the History of Exact Sciences (Pingree Volume), Brill, Leiden, 2004, pp. 517-532.
R. C. Gupta: “Techniques of Ancient Empirical Mathematics”, Indian Journal of History of Science, 45 (2010), 63-100.
Takoa Hayashi: “Narayana’s Rule for a Segment of a circle”, Ganita Bharati, 12 (1990), 1-9.
T. L. Heath: A History of Greek Mathematics, Vol. II, Dover Reprint, New York, 1981.
W. R. Knorr: Review of Waerden’s book (see below). British Journal of History of Science, 18 (1985), 197-212.
J.-C. Martzloff: A History of Chinese Mathematics, Springer, 1997.
K. Muroi: “The area of a semicircle in Babylonian Mathematics [etc]”, Sugakushi Kenkyu, No. 143 (1994), 50-61.
O. Neugebauer and A. Sachs (ed.): Mathematical Cuneiform Texts. American Oriental Society and ASOR, New Haven, 1945.
SaKHYa (ed. and transl.): Ganitasara-Kaumudi of Thakkura Pheru. Manohar, New Dehli, 2009.
A. Seidenberg: “On the Area of a Semi-Circle”, Archive for History of Exact Science, 9(4) (1972), 171-211.
B. L. van der Waerden: Geometry and Algebra in Ancient Civilizations. Springer Verlag, Berlin, 1983.
D. Wagner: “Translation of the Discussions of the Circle-Mensuration in JZSS and the Commentaries by Liu Hui etc.” Typed manuscript (1973) so kindly made available by him to RCG in 1989.

Note: The layout of this article will be better in the print version, appearing in the next (paper) issue of HPM Newsletter.