In the previous article of the series, we explored the undeniable links between Indian and Chaldean astronomies and the strong possibility that migrations of Indian peoples to Mesopotamia may have occurred in the past.
In this article, we look at a wonder of Indian Astronomy, namely, the Indian Sine-Table.
Wonder #2 – The Indian Sine Table
The Table of Sines found in ancient Indian mathematics has long been the subject of discussion and debate. It has aroused great interest among scholars, over the past 200 years, for the following reasons:
- It’s a curious structure
- It’s accuracy
- It’s compact-ness
- An ancient algorithm that is supposed to generate the Table, but, which has, till today, failed all attempts at decipherment
- The Western claim that the Indian Sine-Table was borrowed from the Greeks
Let us examine these points.
But before we jump into all that, let us take a quick look at what a Sine-Table is.
The Sinusoidal Function
Most of us are probably familiar with the Sine Function, which we learned in Middle School Math.
In a right-angled triangle, as shown in Figure 1, the Sine of the angle θ (theta) is defined as the ratio of the opposite side to the hypotenuse, that is, a/c.
As a practical example of its usage, say you are walking up an incline, as shown in Figure 2. If you walked a distance of ‘d’ on the incline, you will have gained an altitude ‘h’ given by d x Sin (θ).
Note that the Sine is a ratio, and therefore it has no units. It is just a number.
The Sine actually belongs to the larger set of six Circular Functions, named so because they describe the position of a body moving in a circle. The other five circular functions are the Cosine, Tangent, Co-Tangent, Secant, and Co-Secant.
Consider a body moving in a circle of radius 1 unit, as shown in Figure 3.
Recall that the Sine is the ratio of the opposite side to the hypotenuse. Since hypotenuse = 1, Sin(θ) = AB/1 = AB. In other words, the altitude (AB) is the Sine itself.
When θ = 0, AB is zero and Sin(θ) = 0. Similarly, when θ = 90°, AB=OB=1 and Sine(θ) = 1.
With that basic understanding, let us now see how the ancient Indians viewed the Sine Function.
The Bow, the Bowstring, and the Arrow
As shown in Figure 4, the ancient Indians imagined in the circular model the figure of a Bow (arc CAD), a Bowstring (segment CD), and an Arrow (BA).
In the figure, the Sine of angle θ, that is CB, is clearly half of the Bowstring (CD).
A Bowstring is called JYA in Sanskrit. And therefore, the Indian Sine is called JYA-ardha (half-bowstring).
You might say, “Well, I agree it’s kind of cute that the ancient Indians named their Sine as ‘half-bowstring’, but it’s hardly anything to get excited about, is it?”
But do wait a bit, because it gets more and more curious from here.
In addition to the regular Sine (CB), the Indians also defined a second Sine – the arrow (BA), called Utkrama-JYA (Inverted Sine). The Inverted or Versed-Sine is unique to Indian mathematics. No other ancient culture made use of it or even knew of it. The Versed-Sine is used in modern times for some special purposes.
Next, look at the Indian radius (R). The length of the radius is set arbitrarily – it can be anything you like.
So, one would expect to see a nice round number, say 100, or 60, or 360. The Greeks, for example, chose a radius of 60 units. The ancient Indians, however, chose a very strange number (3438) for their radius. Why?
Going further, there are 360 degrees in a circle, as we know. Modern Sine Tables usually contain 360 rows which provide the Sine for each degree.
The Indian Sine-Table, however, contains only 24-rows, and, very curiously, provides Sines only in the range 0-to-90 degrees. What about the 90-360 range?
And that brings out another curious feature. In this shortened range of 0-to-90 degrees, Table increments in steps of 3.75 degrees, and not 1-degree. Why this odd step-size of 3.75 degrees?
As Alice in Wonderland said, “curiouser and curiouser”.
For a grand finale to the whole mystery, the ancient Indians provide us with a formula to generate their Table of 24-Sines, so that you don’t need to memorize the Table or carry it around. But, till today, nobody has cracked the formula. All attempts by top scholars over the past 200 years have failed.
Hopefully, dear reader, you are by now impressed with the wonder and mystery that surrounds the Indian Sine-Table.
Before we look at the Table itself, we need to understand one other item, the R-Sine.
The standard circular Sine (AB), as we saw, varies between 0 and 1. However, this assumes a radius of 1 unit. What if the radius were something other than 1? Well, in that case, we must multiply the standard Sine by that radius.
The Indian radius, as mentioned, is 3438 units. Therefore, the Indian Sine = Standard Sine x 3438.
This Indian Sine, which varies between 0 and 3438, is called an R-Sine, meaning ‘Radius multiplied by Sine’.
The 24-Row Indian R-Sine Table
The Table is shown in Figure 5.
The actual Table contains only two columns (in orange heading). I have inserted three additional columns in the middle (in brown heading) for instructional purposes.
The second column shows the angle. As mentioned, this angle increases in steps of 3.75 degrees, going from 0-to-90 degrees in 24 steps.
The third column shows the standard (modern) Sine for this angle. As the angle varies from 0-90 degrees, the Sine varies from 0 to 1.
In the fourth column we multiply the standard Sine (of the third column) by 3438, to get the modern R-Sines.
In the fifth, we show the Greek R-Sines, as calculated from the Greek Table of Chords (discussed later).
Finally, in the sixth column, we have the Indian R-Sines, as given in ancient Indian astronomy.
From a quick look at the Table, we make a couple of general observations:
1) The calculated Greek R-Sines match perfectly the actual R-Sines (4th and 5th columns).
2) The Indian R-Sines (6th column), though quite accurate, do not match the actual R-Sines perfectly. There are 5 cases in error (shown in yellow highlight).
You may point out that the error in these 5 places is hardly noticeable, being only (1/3438) in each case. And you would be right.
However, the important point here is not the error itself, but the fact that Indian and Greek values for the R-Sine are not exactly the same. This small but vital difference deals a fatal blow to the Western claim that the Indians borrowed their Sine-Table from the Greeks.
While on the subject of the Greeks, let us take a quick look at their Table of Chords.
The Greek Chord-Table
You may be wondering why the heading says ‘Greek Chord-Table’ and not ‘Greek Sine-Table’. Well, dear reader, that is because the Greeks did not know of the Sine Function at all!
Yes, you read that right. The Greeks had no knowledge of the Sine.
No doubt this leaves you a little perplexed. Aren’t the Westerners claiming that the Indians borrowed their Sine-Table from the Greeks?
Let us examine this.
As shown in Figure 6, the Greek mathematicians worked with the full-bowstring (CD). They called it the Chord (Khorde in Greek). They had no concept of the Sine (half-bowstring) at all.
Their Table of Chords has 2 major columns: 1) the arc CAD = θ, and, 2) the corresponding chord length = CD. That’s all.
So, to find the Sine of any angle, say 15 degrees, they would first look up the entry in their Table for an arc twice that amount, that is, 30 degrees, and read off the chord value. Then they would halve that chord, which would be the required Sine for 15 degrees.
The Greeks never made the intuitive leap of considering half-the-bowstring as a separate entity. Thus, they were burdened with cumbersome calculations, which the astute Indian mathematician bypassed by inventing the Sine.
Compared to the compact 24-Row Indian Sine-Table the Greek Chord-Table is huge. It has 360 rows, covering an arc from 0-180 degrees in steps of half-a-degree.
And their chosen radius was 60 units.
And, unlike the Indians, they did not supply any formula to build their Table from scratch. So, in addition to having cumbersome calculations with their Chords, the Greek astronomers and mathematicians had to keep their Chord-Table handy at all times.
Ok, having understood the Greek Table of Chords, let us get back to the Indian Sine-Table, and try to explain some of its curious features.
The Versed Sine, that is, the ‘arrow’ in Figure 4, is called Utkrama-JYA in Indian astronomy. It is called the versine in modern times. It has several related functions like the co-versine and the ha-versine.
The Indian word for arrow is ‘sara’, from which comes the Arabic word ‘sahem’ for Versed-Sine. The Latin term is similar (Sagitta = arrow).
The haversine appears in the so-called haversine formula, which is used in navigation to estimate the distances on a spherical object, such as the earth, given the longitude and latitude.
As mentioned, the ancient Indians were the first to define and make use of the Versed-Sine.
An Ingenious Shortcut – the Radius of 3438
Next, let us figure out why the Indians used this strange radius of 3438 units.
Angular measurements, as we know, are made in degrees, minutes, and seconds. This can be a bit confusing for the novice because minutes and seconds are also units of time.
To distinguish the angular minute/second from the temporal (time) minute/second, the prefix ‘arc’ is often used for the former.
Thus, 1 degree = 60 arc-minutes, and, 1 arc-minute = 60 arc-seconds.
Now, how wide exactly is 1-degree in the sky?
If you held up your middle 3 fingers together at arm’s length (with the back of your hand facing you), that would be about 5 degrees wide in the sky. In drawings and paintings, the full-moon is often depicted as a fairly large object.
The novice astronomer is always surprised to learn that the full-moon is only half-a-degree in width and that a single finger, held at arms-length, will easily cover the moon completely. Check it out next time you see the moon.
How about 1 arc-minute? Taking the moon again, the full moon is 30 arc-minutes wide. So, one arc-minute is 1/30 of the full moon.
Finally, how about 1 arc-second? How wide is that?
Imagine trying to observe a small coin placed 4 kilometers away! That is how small 1 arc-second is. It is 1/3600 of a degree – far too small for the human eye to perceive. Measurements in the arc-second range fall in the realm of powerful telescopes.
Ok, now that we know about the degree, arc-minute and arc-second, how about the human eye? How small a width, or angular measure, can it perceive?
It is generally accepted that the smallest angular width the human eye can discern is 1 arc-minute, that is, 1/60 of a degree, or 1/30 of the full moon.
Now, given the fact that the precision of the human eye is 1 arc-minute, would it surprise you to know that all measurements and calculations in Indian astronomy are done in arc-minutes? The practical-minded Indian astronomer did not push his calculations to finer than 1 arc-minute.
After all, why bother to calculate the position of a heavenly body to the arc-second, when your observational data is only in arc-minutes?
In this regard, it comes as a surprise to learn that the Greeks did not have this practical mindset. They took their astronomical calculations down to the arc-second, and sometimes even finer than that!
Getting back to our topic, there are 360 degrees in a circle as we know, which amounts to 360 × 60 = 21600 arc-minutes. That is, there are 21600 arc-minutes in a complete circle.
Figure 7 shows a comparison of Greek and Indian models in the calculation.
The Greek model uses a radius of 60 units, which results in a circumference of 377 units. Thus, the Greek planetary positions are obtained in 1/377 of a circle.
On the other hand, the Indian radius, being 3438, results in a circumference of 21600 units. And thus, Indian planetary positions are obtained as 1/21600 of a circle, or one arc-minute. What an ingenious idea!
While Indian calculation results can be directly compared to observational data, which are in arc-minutes, the Greek astronomer is burdened with further computations to convert his 1/377 units into degrees and minutes.
In summary, the circular model of radius 3438 is a very convenient shortcut, which would make perfect sense to a practical astronomer. Kudos to the ancient Indians!
Compactness – Range of 0-90 degrees
Next, let us figure out why the Indian Sine-Table is given only for the range 0-90 degrees and not 0-360.
It can be seen in Figure 8, which depicts a typical sinusoidal curve, that the sine values from 0-90 degrees are repeated in the rest of the curve from 90-360 degrees.
From 90-180 degrees, the sine values are exactly those from 0-90, but in reverse. Similarly, sine values from 180-270 degrees are exactly those from 0-90, but with a –ve sign. And lastly, sine values from 270-360 degrees are those from 0-90 in reverse and with a –ve sign.
Hope the implication is clear.
The Sine values for the range 90-360 degrees can be easily obtained from the smaller range of 0-90 degrees by simple manipulation. So why carry around the extra baggage?
This brevity is typical of Indian texts, which usually present data in as compact a form as possible.
The Odd Step-Size of 3.75 degrees
Moving on to the next puzzle, we ask, in a little bewilderment, why the Indian R-Sine table has this odd-looking step-size of 3.75 degrees, and not a regular step of, say, 1-degree or 2-degrees?
Related to that is another question. The Indian Sine-Table provides only 24 angles. How does one calculate Sines for in-between values?
For example, the Table gives R-Sines for 3.75 and 7.5 degrees. How does one compute the R-Sine of 5 degrees?
For in-between angles, the ancient texts recommend that you linearly interpolate between the two nearest values of the angle. So, for 5 degrees, you would interpolate between 3.75 and 7.5 degrees.
Naturally, this linear-interpolation will introduce a small error. How much error is that? The answer is shown in Figure 9.
As seen in the figure, as we move from 0-90 degrees, the error due to interpolation is seen to increase. However, the maximum positive error is only 0.7/3438, and the maximum negative error only -1.7/3438, both very small quantities.
Thus, the maximum positional error in Indian calculations caused by the interpolation will be only 1.7 arc-minutes, which is within the range of acceptable tolerance, since observational measurements with the naked-eye have a precision of 1 arc-minute.
In summary, the compact 24-Row Indian Sine-Table does not lose out to the much larger 360-Row Greek Chord-Table as far as accuracy is concerned.
That still leaves the question of why the Indians used 3.75 degree-steps.
It’s a vexing puzzle; one that has had scholars scratching their heads for a couple of hundred years. There are no good answers as yet, but only a tantalizing hint or two. From these hints, it appears that the choice of 24-Sines (and step-size of 3.75 degree) may have something to do with the Sine formula the ancient Indians used to generate the Table, as discussed in the next section.
The Mysterious Sine Algorithm of the Ancient Indians
As mentioned, the ancient texts provide a curious algorithm to generate the 24-Row Table of Sines from scratch. Needless to say, there has been much discussion and debate and many articles written on the topic.
Before we get to the algorithm itself, let us lay some background.
Figure 10 below shows us the variation of the Sine (vertical red line) as we advance from 0 to 90°, in steps of 15°.
It can be readily seen that though the total Sine increases with angle, the increment (in red) decreases at each step, with increasing angle. In mathematics, this is called the ‘first-difference’. So, we say the first-difference of the Sine Function decreases with angle.
What about the second-difference? That is the difference of the difference.
Figure 11 shows us the interesting fact that while the first-difference decreases with angle, the second-difference increases with angle.
The Indian Algorithm apparently teaches us how to calculate this second-difference for each angle. Then, it instructs how to use these second-differences to calculate first-differences, and from that the Sine itself.
However, nothing is crystal clear. The language is obscure and convoluted, and the entire algorithm is compressed into a single verse.
Scholars over the years have made various interpretations – all unsuccessful. Nobody has, as yet, been able to replicate the Table exactly as given in the text, though they have come close.
See the Appendix below for the actual algorithm.
Comparing Indian and Greek Tables
Having examined in some detail the Indian and Greek Tables, we are now in a position to compare the two.
- The most standout item, quite obviously, is the fact that the Greeks knew only the Chord. They had no idea of the Sine or the Versed-Sine. These Sines are purely Indian inventions. (Round-1 to the Indians).
- As regards accuracy, the Chord-table is a little more accurate than the Indian Sine-Table. The discrepancy, however, amounts to only 1/3438 in the 5 error cases. (Round-2 to the Greeks).
- Considering Ease of Use, the 24-Row Indian Sine, written in poetic verse, is easy to memorize. The Greek mathematician, apart from having to do cumbersome calculations to extract the Sine from the Chord, was also forced to carry around his 360-row Table of Chords. (Round-3 to the Indians).
- As far as Practicality goes, the ingenious Indian idea of using a radius of 3438 units, that straightaway gives planetary longitudes in arc-minutes, which can be directly compared with observational data without further massaging, gets high marks. This is unlike the hapless Greeks, with their radius of 60 units, who were burdened with a lot more calculation before being able to compare their computational results with observational data. (Round-4 to the Indians).
- The 24-Row Indian Sine-Table, though being much smaller than the Greek Chord-Table, does not suffer from a lack of accuracy. Although their compact Table is easy to memorize, the ancient Indians went a step further, and provided a formula to generate the Table from scratch, unlike the Greeks. (Round-5 to the Indians).
And the winner is … (you be the judge, dear reader).
The Western Spin (Politics)
It has always been a matter of great pride for the Western intelligentsia to declare that they derive their civilization from the Greeks. While that is a good thing, no doubt, it has, unfortunately, resulted in two kinds of mischief being perpetrated upon the rest of the world:
1) The suppression of the achievements of other ancient civilizations, and
2) the artificial boosting of the ancient Greeks to superhuman levels.
Such being the situation, the ingenious Indian Sine-Table has expectedly produced mild-to-severe heartburn among Western scholars of the past and continues to do so even today. Here are a couple of typical examples.
(Prof. D. W. Whitney, Translation of the Surya-Siddhanta, 1858.)
“That the second-differences of the Sines were proportional to the Sines themselves was probably known to the Hindus only by observation. Had their trigonometry sufficed to demonstrate it, they might easily have constructed a much more complete and accurate table of Sines.”
(George Rusby Kaye, Indian Mathematics, 1915.)
“They are followed by a table of 24 sines, progressing by intervals of 3.75 degrees, obviously taken from the Greek Table of Chords.”
Rest assured, dear reader, that this is only the tip of the iceberg. There has been a lot more mischief on this topic, which would take too long to detail here, and which would needlessly turn this article acrimonious.
To cut a long story short, here is the Western spin on the Indian Sine-Table in brief:
- The Indians borrowed the Greek Table of Chords. Then, in sheer dumb luck, they hit upon the idea of using the half-Chord instead of the full-Chord.
- They reduced the number of entries from 360-Rows to 24-Rows for some obscure reason.
- They changed the radius from 60 to 3438, again, by sheer dumb luck.
- They started using the Versed-Sine, for some obscure reason.
- They observed (by sheer dumb luck of course) that the second-differences were proportional to the Sines themselves, and thus produced a formula to generate their Sine-Table.
Totally believable, right?
The Case for Borrowing
Interestingly, if anything, circumstantial evidence actually points the other way – it is very likely that the Greeks borrowed a lot of their mathematics, and all of their astronomy, from the Indians.
In the immediate aftermath of Alexander’s invasion of India, by that I mean only 5-10 years later (by 315-310 BC), the Greeks had obtained, seemingly out of thin air, a ton of mathematics (the 13 books of Euclid), and the rudiments of astronomy that were a simplified version of Indian astronomy. Enough said.
The Indian Sine-Table, with its accuracy, practicality, and compactness, is a wonder of ancient science. A mysterious, and, as yet, unresolved formula for its derivation adds to its alluring mystique.
For mathematically-minded readers who would like to take a crack at the 200-year old puzzle, here is the Sine algorithm from two different sources – the Surya-Siddhanta and Aryabhata.
Surya-Siddhanta: Chapter-II, Verses 15 and 16:
“The eighth part of the minutes of a Zodiac Sign (30 degrees) is called the first Sine. That, increased by the remainder left after subtracting from it the quotient arising from dividing it by itself is the second Sine.”
“Thus, dividing the tabular sines in succession by the first and adding to them, in each case, what is left after subtracting the quotient from the first, the result is 24 tabular Sines.”
Aryabhata: Ganitapada, Verse 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.”
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