In India, formulae have long been known which give the approximate time of day from the length of the shadow cast by a gnomon or by a person standing upright. In the morning, 'time of day' means the time passed since sunrise, and in the afternoon it means the time left until sunset.

Let 'half length of daylight' mean half the time from sunrise to sunset. Then Sphujidhvaja's formula (3rd cent. A.D.) is [1]

where the unit is the height of the gnomon. This is correct at noon and when the sun is at low altitude.

Brahmagupta's formula (7th cent. A.D.) is [2]

The errors of Brahmagupta's formula are sometimes greater and sometimes less than the errors of Sphujidhvaja's formula.

Much earlier works give the time of day for certain specific shadow lengths. Brahmagupta's formula agrees with the data in the *Arthasastra* [3] and, except possibly for one item, the Jaina *Surya Prajnapti* [4].

At an astronomical institution in Ahmadabad, Gujarat, a member of staff demonstrated for me the method of walking heel-to-toe to measure one's shadow in units of one's own foot. Sometimes one person stands with their back to the sun, while someone else marks the end of the shadow [5].

The unit of time in these Indian formulae is the *ghati*, equal to 24 minutes.

One formula has the general form

where C may be 120 [6] or 100 [7]. There is a built in personal equation here, as the ratio of height to foot varies from near 6 to near 7, in general.

In the astrological work *Bhuvana Dipaka* [8] on queries, there are variations based on the time and date of measuring the shadow:

where C=144 from winter solstice to summer solstice, C=135 from summer solstice to winter solstice, and 1 is subtracted in the morning and added in the afternoon. Since the source is a book on queries, these variations are introduced for astrological purposes.

I saw it! - but only on TV [9]. On an island in Lake Tana, a Coptic monk measured his shadow with his own foot. This was to tell the time for their daily activities. I used to wonder whether this was related to the Indian method - having overlooked the fact that the answer was in the reprint of a paper by the late Otto Neugebauer [10].

There are Ethiopian shadow tables for each month, based on the assumption that the shadow of a person, measured in *feet*, increases by 1, 2, 3, 4, 10 feet in the successive hours after noon (or, equally, before noon), where these hours are 1/12 the length of daylight on the day in question.

This is a different system from the Indian one, and Neugebauer traces it to ancient Greece of the 4th or 5th century B.C. In his definitive work, Neugebauer [11] states that several of the ancient source texts give the rule that a man standing upright should mark the end of his shadow on the ground and measure its length by setting one foot in front of the other; so that 'foot' in the texts means the man's own foot length. This is the Ethiopian practice as shown by the TV program.

The only thing to consider is the origin of measuring the shadow with one's own foot - the 'one's own foot' method, say. The whole system has clearly not been transmitted between India on the one hand and Ethiopia or other countries dealt with by Neugebauer on the other hand. In theory, the mere idea of using the one's own foot could have been so transmitted. Or, it could have arisen independently, as a natural procedure. If there was transmission, one can only say that no evidence is known for the use of one's own foot in India in the 4th or 5th centuries B.C.

B1. A.P. Stone, 'Ancient Indian shadow formulae for fractions of the day'. *Ganita Bharati, Bull. Ind. Soc. Hist. Math.* **4** (1982), 90-99 (back to B1)

B2. A.P. Stone, 'Indian shadow formulae using the foot as unit'.*Ganita Bharati, Bull. Ind. Soc. Hist. Math.* **7** (1985), 1-12 (back to B2)

1. David Pingree, *The Yavana Jataka of Sphujidhvaja* (Harvard Oriental Series, 48). I,3; II,415 (back to [1])

2. *Brahmasphuta Siddhanta*, 12.52 (back to [2])

3. *Arthasastra*, 1.19.6- (back to [3])

4. *Surya Prajnapti*, 9 (back to [4])

5. See photo in B.V. Raman, 'Astrology, what it is and what it is not', *The Illustrated Weekly of India*, Sept.7-13, 1980, p.10 (back to [5])

6. Commentary on *Bhuvana Dipaka*, 55 (see below) (back to [6])

7. Private communication from a Malayali gentleman (back to [7])

8. *Bhuvana Dipaka*, 55 (Also known as *Grahabhava Prakasa*) (back to [8])

9. Channel 4 (UK) Wednesday January 15, 1985 (back to [9])

10. O. Neugebauer, 'Notes on Ethiopic astronomy', *Orientalia* **33** (1964) 49-71 (back to [10])

11. O. Neugebauer, *A History of Ancient Mathematical Astronomy* (Springer, Berlin Heidelberg New York, 1975) p.738 (see also pp.737-740) (back to [11])

Last updated: 04 January 2005