Volume of a sphere in Ancient China and India

Professor R. C. Gupta, Ph. D. (Hist. Of Math.), Ganita Bharati Academy, R-20, Ras Bahar Colony, JHANSI-284003, India

The famous Greek mathematican Archimedes (died 212 BCE) had already found the correct formula

$V_{o} = (\pi /6) d^3 = (4 \pi /3)$ (1)

very nearly       $=0.5236d^3$        (2)

for the volume of a sphere of diameter d (=2r). On the other hand, the popular ancient Chinese work Jiu Zhang Suan Shu (JZSS) which reached its present form in the first century CE, contains two numerical problems in which d is calculated by using the rule

$d=(16v/9)^{(1/3)}$ (3)

This clearly implies the formula

$v=(9/16) d^3$ (4)

Suppose the sphere is inscribed in a right circular cylinder whose height is d and base is also of diameter d. Also let the cylinder be circumscribed by a cube of side d such that their bases and tops lie in the same planes. It was known that

$\frac{\text{Vol. of the cylinder}}{\text{Vol. of the cube}} = \frac{\pi}{4}$ (5)

Now let

$\frac{\text{Vol. of the sphere}}{\text{Vol. of the cylinder}} = k$ (6)

From these two equations we see that

Vol. of sphere =  $(\pi k/4) d^3$            (7)

The mathematically exact value of k is easily seen to be 2/3.

The JZSS uses the simple approximation $\pi = 3$  for calculating the area of a circle. Therefore a suggested use of the very high value $\pi=27/8$ for getting (4) from (1) is rejected as unlikely. In fact during the first century of CE and in the time of Zhang Heng (78-139 CE), the ratio $\pi /4$ in (5) was taken to be ¾ in China. At that time, it was thought that the ratio k also has the same value $\pi /4$ or ¾. Thus the early Chinese mathematicians were led, by (7), to the empirical rule

Vol. of a sphere, $v=(\pi ^2/16) d^3$            (8)

Or,  $=(9/16) d^3$                                (9)

which is indeed the JZSS formula (4). This formula already greater than the true value of the volume but the calendrist Zhang Heng thought otherwise and made it even worse by taking

$V_1=(9/16) d^3+(1/16) d^3$ (10)

$=(5/8) d^3$ (11)

However, this, compared with (8), leads to $\pi = \sqrt{10}$ which is better than $\pi = 3$ and better even than another of Zhang Heng’s values $\pi =92/29$.

A comment, attributed to Liu Hui (3rd cent. CE), on the JZSS sphere problem shows that (4) implies (6) with $k= \pi /4$. According to Li Chunfeng (602-670), Zu Geng (=Zu Xuan), son of Zu Chongzhi (429-500), had stated that Liu Hui (like Zhang Heng) took (6) with $k= \pi /4$. There is some indication to show that Liu Hui considered this assumption to be wrong although he was unable to derive the correct rule.

In India, Āryabhaţa (born 476 CE) in his work Āryabhaţīya (II,7) gives the rule for the volume of sphere as

exactly  $V_2=A \sqrt{A}$                        (12)

where A is the area of a central section of the sphere (i.e. $A= \pi r^2$). His commentator Bhāskara I (629 CE) quotes a rule (in Sanskrit) which gives a formula equivalent to (4). The rule was regarded empirical and so Āryabhaţa attempted to find a better one.

Mahāvīra (about 850 CE) in his Gaņita-sāra Sańgraha (=GSS, VIII, 28) has also given the same rule and clearly stated it to be practical (i.e. not exact or accurate). The same rule also appears in the Tiloyasāra (gāthā 96) of Nemicandra (about 975 CE), and in the Gaņita-sāra (V, 25) of Ţhakkura Pherū (14th cent.) who gave it the form

$d^3 \cdot (1-1/4)(1-1/4)$ (13)

A detailed discussion of all these Indian rules show that (4) should be interpreted as (8) with $\pi = 3$.

The modification of empirical rules by adjusting them to suitable values of π is an ancient practice. For example,

$A=(c+h)h/2$ (14)

for the area of a circular segment (of chord c and height h) was supposed to be based on $\pi = 3$ for which it gives exact value in the case of a semicircle. It was modified or adjusted to $\pi = 3$ in the form

$(c+h)(h/2) \cdot (22/21)$ (15)

In the Mensurae attributed to Heron (1st cent. CE). More interestingly, (14) wa adjusted to $\pi = \sqrt{10}$ (a popular Jaina value) in the form

$(c+h)(h/2) \cdot \sqrt{\frac{10}{9}}$ (16)

In India by Śrīdhara (about 750 CE) in his Triśatikā (rule 47), by Pherū in his Gaņita-sāra (III, 46) and by others. However, if a rule (to be modified) was presumed to involve the square of π or of circular circumference, the adjustment was done by the factor 10/9. Thus we find Mahāvīra giving his accurate rule for the volume of a sphere as (GSS, VIII, 28½)

$V_3=10v/9$ (17)

$=5r^3$ (18)

which happens to be the same as V1 in (11). For Mahāvīra $\pi = 3$ and (4) were rough or practical, and $\pi = \sqrt{10}$ was accurate. It may be pointed out that (18) can also be obtained from Āryabhaţa’s rule (12) by taking $A=3r^2$, and using the usual Indian rule

$\sqrt{(a^2+x)}=a+\frac{x}{(2a+1)}$, with a=1 and x=2.

In China, the exact Archimedean formula (1) was proved by Zu Geng (6th cent. CE). He used it for both $\pi = 3$ and $\pi = 22/7$ for convenience, although he knew even the far better value $\pi = 355/113$ from his father. For the simple value $\pi = 3$, we have simply

$V_4=d^3/2$ (19)

If this is adjusted to a Jaina value $\pi = 19/6$, we get

$V_5=d^3/2 \cdot 19/18$ (20)

which is found in the Triśatikā (rule 56) of Śrīdhara. The value $\pi = 19/6$ is commonly derived from the usual Jaina value $\pi = \sqrt{10}$ (used in their sacred works) by using the approximation

$\sqrt{a^2+x}=a+\frac{x}{(2a)}$, with a = 3 and x = 1.

The formula (20) is also found in the Siddhānta Śekhara (XIII, 46) of Śrīpati (11th cent.), and in the Mahāsiddhānta (XVI, 108) of Āryabhaţa II whose date, following recent research, has been changed from the 10th to the 15th century.

The famous Indian mathematican Bhāskara II (12th cent.) not only gave the correct formulas for the surface and volume of a sphere but also supplied justifications. His Līlāvatī (rule 201) contains

$S_0=4 \pi r^2$ (21)

$V_0=(S_0 d)/6=(\pi /6) d^3$ (22)

The name of Archimedes, and his beautiful proof, is hardly ever met in ancient China or India. Was the ancient orient unaware of Greek achievements, or was it a case of older methods and practices being preferred even where new ones were known? The Jaina Pherū used the square form of the Jainisation adjustment factor 10/9 to improve the ancient formula (for surface of sphere)

$S=C^2/4$ (23)

which can be derived from a rule of Mahāvīra (GSS, VII, 25); here C is the circumference of the great circle. Pherū also used the same factor for V4 or $d^3/2$ which he perhaps picked up from (20). But in this case he was confused because here it is π which is involved and not π2. Had he used the (linear) Jainisation factor $\sqrt{\frac{10}{9}}$, his result would have been far better.

References

1.      L. Berggren et. al, Pi: A Source Book. Springer, New York, 2004.

2.      R. C. Gupta, “Volume of a Sphere in Ancient Indian Mathematics”, Journal of the Asiatic Society, XXX (1988), 128-140.

3.      R. C. Gupta, “Techniques of Ancient Empirical Mathematics”, Indian Journal of History of Science 45 (2010), 63-100.

4.      J.-C. Martzloff, A History of Chinese Mathematics. Springer, Berlin, 1997.

5.      D. B. Wagner, “Doubts Concerning the Attribution of Liu Hui’s Commentary (on JZSS)”, Acta Orientalia, 39 (1978), 199-212.

6.      D. B. Wagner, “Liu Hui and Tsu Keng-chih on The Volume of a Sphere”, Chinese Science III (1978), 59-79.