Śulba-sūtra geometry: can we teach it in school today?

C. K. Raju

शुल्ब सूत्र-s (string aphorisms) are manuals for masons
to build fire altars.
(The most famous is shaped like a falcon.)

However, my interest today is not in the altars
but in the underlying way of doing geometry.

Recall that most present-day school math
arithmetic, algebra, "trigonometry" and calculus, probability and statistics
went from India to Europe between 10th to 16th c. CE
for its practical value.

Modified versions were returned to us through colonial education
and we uncritically accepted those modifications as superior.

Note that the above list excludes geometry
which Greeks got from Egypt.

As pointed out in the preceding lecture, Plato tied geometry to religious beliefs about the soul.
That is, Greeks took mystery geometry (रहस्यवादी ज्यामिति) from Egypt.

But Egyptians also had a practical geometry.
As depicted on the east wall of the tomb of Djeserkaseneb in Luxor.

Interestingly, this also used a cord or rope as shown
for land surveying etc.
and the people were called "cord-stretchers" or harpedonaptae in Greek.

However, nothing is known about how they did it
and the only surviving documents for that are the शुल्ब सूत्र-s.

We often hear abotu them in the context of the "Pythagorean theorem".
E.g., in 2015, Harsh Vardhan said, "India gave the Pythagorean theorem to the West".

Harsh Vardhan's point was, first, that
a similar statement is found in various शुल्ब सूत्र-s.
This is indeed true. E.g. Baudhayana 1.12, Apastamba, 1.4, Katyayana, 2.7.

Secondly, his point was that the शुल्ब सूत्र-s have been dated by some to -800 CE
This was before the supposed date of Pythagoras (-500 CE)
Ergo, Pythagoras got it from India.

Among those who contested Harash Vardhan was Prabir Purkayastha
in an article, "Mumbo jumbo as science in the Science Congress", .
He pointed out that Pythagorean triples (e.g. 3, 4, 5) are found also in Iraq and Egypt.
This is true, though there is no general statement.

Purkayastha's main point, however, was this.
He said, "Pythagoras had a proof"
Therefore, regardless of chronology, credit must go to Pythagoras.

This is also exactly the stand taken by our Class IX NCERT math school text.
They say, what Indians did was "practical".

"But in civilisations like Greece, the emphasis was on the reasoning behind why certain constructions work. The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning" NCERT, class IX, Math, pp. 78-79).

Purkayastha is wrong: Indians did have a proof of the Pythagorean theorem
which I pointed out long ago:
- CKR, "Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the YuktiBhāsā",
- Philosophy East and West, 51:3 (2001) pp. 325–362. http://ckraju.net/papers/Hawaii.pdf.

Here is the proof.
However, this is an empirical proof.

As explained in the previous lecture
talk of "reason" in our school text
hides the prohibition of the empirical.

It refers to the church method of reasoning without facts.
(The church connection is hidden by reference to mythical Greeks.)

As also explained in the previous lecture
the belief that Pythagoras or any other Greek gave such (pure deductive) proofs is also pure myth.

No evidence for existence of any Pythagoras or Euclid.
No evidence that Pythagoras gave any proofs.
No one knows what proof he gave.

Westerners under church hegemony believed all sorts of silly myths
and taught us to do the same.

Class IX school text repeats "Euclid" 63 times
Class X school text repeates "Pythagorean theorem" 32 times.
A billion idiots believe without checking that there must be some truth in it.

If we ask what proof the Greeks gave, we will be directed to the book Elements (purportedly written by Euclid).
It has NO deductive proofs.
Repeat: what our school text says is false: the Greeks gave no deductive proofs of any geometrical result.

A quick recap:

Greeks never used axiomatic reasoning

Greeks gave axioms and proofs, separately BUT
no axiomatic proofs (i.e., proofs without using the empirical).

E.g. First proposition of the Elements gives an empirical proof.
One sees the two arcs intersecting.
But they may or may not have a point in common.

Also the 4th proposition (Side-angle-side theorem).
Original proof was by putting one triangle appropriately on top of the other to see that the two triangles are equal.
(But anything one can see is NOT formal math, therefore, today we teach SAS postulate.)

4th proposition needed for the proof of the penultimate ("Pythagorean") theorem.
Hence ALL known Greek proofs of the Pythagorean theorem involve the empirical.

So was the Pythagorean theorem known before Pythagoras in India?
Strangely, BOTH proponents and opponents of the claim are wrong!
There was neither any Pythagoras, nor any deductive proof of his "theorem" before the 20th c.

The "error" about the formal proof of Proposition 1 in "Euclid" was realized in the 19th c.
The "error" abut the formal proof of Proposition 4 (SAS) were pointed out by Russell, Hilbert, etc.

Strictly speaking there are no errors in the book
because its author never intended to provide pure deductive proofs
but only to promote Egyptian/Neoplatonic beliefs about the soul.

In the 1960's the "Sputnik crisis" led to a revamp of US STEM education.
Since then SAS became a POSTULATE
(as we adopted in the 1970's and retain till today).

Confounding distinct geometries

This technique of blindly imitating the West has led to massive confusion in our school texts.
SAS is made into a postulate on the grounds that its empirical proof (by superposition) is disallowed.

But every student today carries a compass box (geometry box).
Measuring the length of a visible line segment is done emirically
by superposing the ruler on the line segment.

Because our sole teaching is blind imitation, we fail to apply commonsense.
If empirical methods are to be rejected, let us reject also the compass box.
If they are to be accepted, then SAS is a theorem, not a postulate (as in the original).

Indeed, Hilbert's geometry is synthetic:
i.e., to avoid superposition, length measurement is disallowed.
But area is still defined (to be able to prove the Pythagorean theorem which is about equal areas).
Defining area without defining length first is beyond school texts.

Math at the class IX level is a compulsory course.
Anyone with a degree is supposed to have qualified in it. (Else return your degree.)
But in a nation of a 1.3 billion people, few understand this simple thing.
And, no one protests.

Finally let us look at the "Pythagorean theorem" in the शुल्ब सूत्र once again.
This time in the Manava शुल्ब सूत्र 10.10.

Note, first that this statement is about a rectangle and its diagonal.
(This is not altogether trivial since, according to Heath, Egyptians did not know what a right-angled triangle was.)

Subtle difference: e.g. Manava sulba sutra 10.10
uses square ROOTS to state "Pythagorean theorem".

That is instead of \(d^2 = l^2 + b^2\)
It states the result in the form \(d = \sqrt{l^2 + b^2}\).

Calculation, not proof

This tells us how to CALCULATE the diagonal.
Calculation NOT available in the Western way
(because the West learnt about square roots very late.)

Inexactitude

If we take \(l=b=1\)
then we get \(d = \sqrt 2\).

The square root algorithm for 2 goes on endlessly.
शुल्ब सूत्र-s accept \(\sqrt 2\) is inexact.

Western religious superstition that math is eternal truth
led to the foolish religious belief that math is exact.

But where is the "Pythagorean theorem" ever exact (in the real world)?

Not on the curved surface of the earth
(as pointed out by Bhaskar 1)
Not in curved space

Rajju Ganit

Therefore, instead of teaching the ritualistic and foolish hotch-potch of geometries
we should revert to teaching शुल्ब सूत्र geometry or Rajju Ganit.

The first result is greater conceptual clarity.
The emirical is accepted.

So, we do not measure things on one principle
and prove things on another.

Points are visible, no more mystery about that.
A stretched cord defines a straight line.
(Zeroism: Doesn't have to be EXACT, nothing in the real world is exact.)

As noted earlier, the word for angle is चाप
meaning the relative length of a curved arc
(relative to the radius, in radians, or the circumference of the circle, in degrees).

If we join the ends of the arc to the centre of the circle we get back the two-straight-line definition.
(This is the real meaning of the little arc used between the two straight lines to denote an angle.)

However, we have gained something.
An angle can be larger than \(360\deg\)
which is impossible on the two straight line definition.
(It can also be negative.)

The need to measure such larger angles arises in astronomy
as does the need to measure negative angles (retograde motion).

The radian measure arises naturally with this definition
and many students who have used only the protractor are confused about it.

As already pointed out a curved arc cannot be measured with the instruments in a compass box.
The need to do so arises, if we want to measure the area of a field with non-straight boundaries.

Some instruments (set squares, divider) are redundant and never used.
But blind imitation is the param dharma of the colonised.

Practical advantage:

children can measure latitutde, longitude,
height of a tree, mountain, size of the earth etc.

Teaching experiments carried out in

Conclusions

Alternatives are available
IF we want our children to have knowledge and commonsense.
(But do we? Does our government? )
गणित eliminates the metaphysics of math hence makes math easy.