Aryabhata: the Aryabhatiya · Chapter 4 of 8

#Sines, shadows and triangles

Verses 11 to 18 of the Ganita give the geometry an astronomer needs: sines, shadows, right triangles and overlapping circles.

#Building a sine table (Ganita 11)

One should divide a quarter of the circumference of a circle (into as many equal parts as are desired). From the triangles and quadrilaterals (which are formed) one will have on the radius as many sines of equal arcs as are desired (Ganita 11).

This is the geometric way to make a sine table: divide the quarter circle, draw the half-chords (sines), and compute them from the triangles. Clark notes frankly that the exact method Aryabhata used to work out his table "is not known" (note to Ganita 11).

#The sine and its name

The Indian jya or jiva — "bowstring", the half-chord of an arc — was taken into Arabic as jiba, written without vowels as jb, misread as jaib ("fold" or "bay"), and translated into Latin as sinus — from which comes the English word sine. The trigonometric sine is an Indian idea travelling under a mistranslated name.

#The rule for the differences (Ganita 12)

By what number the second sine is less than the first sine, and by the quotient obtained by dividing the sum of the preceding sines by the first sine, by the sum of these two quantities the following sines are less than the first sine (Ganita 12).

Here the "sines" are the entries of Aryabhata's table, which are differences (Chapter 2). Their sum so far is the sine reached so far. So, in modern terms: each new difference equals the previous difference minus (the sine reached so far ÷ 225). Starting from the first, 225:

• second difference: 225 − 225/225 = 224; the second sine is 225 + 224 = 449;
• third difference: 224 − 449/225 ≈ 224 − 2 = 222; the third sine is 671;
• fourth difference: 222 − 671/225 ≈ 222 − 3 = 219; the fourth sine is 890;
• fifth difference: 219 − 890/225 ≈ 219 − 4 = 215.

These match Aryabhata's table (Chapter 2). Clark notes that strict use of the rule gives "slight divergences" further on, which the commentators explain by the handling of fractions. The rule says that the change in the differences is proportional to the sine itself — which is exactly true of the sine function. It is, in effect, a way of computing sines from their second differences, a technique that became central to numerical calculation.

#The astronomer's instruments (Ganita 13)

The circle is made by turning, and the triangle and the quadrilateral by means of a karna (hypotenuse); the horizontal is determined by water, and the perpendicular by the plumb-line (Ganita 13).

A circle is drawn by turning a cord round a fixed point; right angles are checked with a diagonal; a level is found with water; a vertical with a plumb-line. Observation needs these tools.

#The gnomon and its shadow (Ganita 14–16)

The gnomon (shanku) is a vertical stick of known height whose shadow is measured. It was the astronomer's most basic instrument: from the shadow, the Sun's altitude, the time and the latitude can all be found.

• Ganita 14. Add the square of the gnomon's height to the square of its shadow; the square root is the radius of the "shadow circle". (This is the hypotenuse of the triangle formed by the gnomon and its shadow.)

• Ganita 15. The shadow cast by a gnomon from a lamp on a post: multiply the gnomon's height by its distance from the post, and divide by the difference between the post's height and the gnomon's height.

  Example. A lamp is on a post 10 units high. A gnomon 2 units high stands 8 units from the post. Its shadow is 2 × 8 ÷ (10 − 2) = 2 units. (Check with similar triangles: the shadow tip is 10 units from the post; 10 : 10 = 2 : 2.)

• Ganita 16. With two shadows of the same gnomon at two positions, one can find the distance to the light and its height — the principle of measuring heights and distances from shadows.

#The right triangle (Ganita 17)

The square of the bhuja (base) plus the square of the koti (upright) is the square of the karna (hypotenuse) (Ganita 17).

This is the theorem known in the West as Pythagoras's. In India it was known much earlier, in the geometry of the Shulba Sutras, used for building fire altars. Aryabhata states it as a working rule; it underlies almost every calculation in his astronomy.

The verse continues, in the second half, with the rule for a chord in a circle: the product of the two parts of a diameter cut by a chord equals the square of the half-chord.

#Overlapping circles (Ganita 18)

(The diameters of) two circles (separately) minus the grasa, multiplied by the grasa, and divided separately by the sum of (the diameters of) the two circles after the grasa has been subtracted from each, will give respectively the sampatasaras of the two circles (Ganita 18).

When two circles overlap, the grasa ("bite") is the depth of the overlap along the line joining their centres. The rule gives how the common chord divides that bite between the two circles. This is exactly the geometry of an eclipse: the Moon's disc and the Sun's (or the Earth's shadow) overlapping. Aryabhata uses it in the Gola (Chapter 7) to compute how much of the Sun or Moon is covered.

#Summary

• A sine table is built by dividing the quarter circle and computing the half-chords (Ganita 11); the exact method of Aryabhata's table is unknown.
• The English word "sine" comes, through Arabic and a misreading, from the Sanskrit jiva.
• Ganita 12 computes each sine difference from the previous one and the sum of the sines — a second-difference method.
• The basic instruments: cord, diagonal, water level, plumb-line (Ganita 13).
• The gnomon and its shadow give heights and distances (Ganita 14–16).
• The right-triangle theorem (Ganita 17) and the rule for overlapping circles (Ganita 18) underlie the eclipse calculations.

#Exercises

1. Use Ganita 12 to compute the sixth difference from the fifth (215) and the fifth sine (890 + 215 = 1,105).
2. A gnomon 3 units high stands 6 units from a lamp-post 12 units high. How long is its shadow?
3. A gnomon 12 units high casts a shadow 5 units long. What is the "radius of the shadow circle" (Ganita 14)?
4. Where does the English word "sine" come from?
5. Why is Ganita 18 important for astronomy?

Answers: 1. 215 − 1,105 ÷ 225 ≈ 215 − 4.9 ≈ 210, as in Aryabhata's table. 2. 3 × 6 ÷ (12 − 3) = 2 units. 3. √(144 + 25) = √169 = 13. 4. From Sanskrit jiva (bowstring), through Arabic jiba, misread as jaib (fold) and translated into Latin as sinus. 5. Because it gives the overlap of two discs — the Moon over the Sun, or the Moon in the Earth's shadow — which is how eclipses are calculated.