Euclidean geometry
Rajju Ganita-Part 2

C. K. Raju


Before colonialism we taught Rajju Ganita

  • after colonial education we started teaching Euclidean geometry
  • assuming it is superior.
  • But is it?
  • We never earlier publicly compared the two, and our "experts" refuse to do so today.

Yesterday we made a public comparison for the first time

  • we started by addressing the purva paksha of Euclidean geometry
  • we found that there are many falsehoods and obscurities in the Euclid story in the school text.
  • Lie 0: No evidence for any Euclid. (The story of Greeks, such as Euclid, was concocted by the church during the Crusades.)

Lies about proof

  • Lie 1. School text lies that Indians had no notion of proof.
    • (Indians had a well-developed notion of proof.)
  • Lie 2. School text relates axiomatic proof to the Euclid myth.
    • (But no axiomatic proofs in Euclid book.)

Lie 3

  • School text lies that deductively/axiomatically proved propositions are true.
    • Theorems are at best relative truths.
    • "Pythagorean theorem" is not true anywhere in the real world.
    • No "approximate truth" in formal math.

Lie 4

  • School text lies that: Axiomatic proofs are "superior" to proofs based on pratyaksh
    • (1) Formal math NOT superior for science, or for everyday applications of math (which all involve the pratyaksh),
    • (2) But, rejection of empirical of immense value to the church, since church dogma is immediately refuted by appeal to pratyaksh.

– Hence, rejection of empirical was glorified by the church. (Aquinas’ theorem)

  • (3) Actually, axiomatic/deductive proofs are MORE fallible than empirical proofs or inductive proofs.

Many other lies (not in the school text)

  • E.g. "Math has beauty"
    • (not true of formal math, millions find it ugly)
  • How the church appropriated and reinterpreted reason (from Muslims) for its political convenience, during crusades, etc.

Obscurities of formal math

  • Since the empirical is prohibited
  • points are invisible
  • lines are infinitely thin
  • straight lines not properly defined, etc.

Obscurities of formal math teaching

  • Because formal math is about metaphysics (= unreal)
  • teaching only formal math would expose its uselessness.
  • Therefore, formal math (for propaganda)
  • is mixed with normal math (for practical applications)

But this hotch-potch is self-contradictory like church theology

  • Disallowing superposition led to axiomatic geometry
  • but compass box geometry involves superposition (measuring length with a scale)
  • and the empirical (dots and lines must be visible).

But the "colonially educated" ignorant don't see it

  • But, the ignorant don't see the contradiction in this hotch-potch
  • they get confused by the mixture of propaganda and practical value
  • and acquire the "it works" superstition.

The correct approach

  • is to teach only one type of math
  • Obviously, that has to be normal math
  • since we teach math for its practical applications
  • (and not for some imagined beauty etc.)

The question today is which kind of normal math?

  • That is, if we do only normal math
  • are the instrument in the compass box adequate?
  • What about measurement errors?

Question now is: do we have a theory of "approximate" truth?

  • Also, as we saw, formal mathematical theorems, such as the Pythagorean theorem
  • are, at best, approximate truths (and at worst complete nonsense)
  • Do we have a theory of approximate truth (zeroism)?
  • We see this in Rajju Ganita.

Rajju Ganita

The basic approach is to accept the empirical

  • Points are dots
  • Lines are visible.
  • Infinitely long lines are not needed
  • finite line-segments suffice.

Note that school text definition of an infinite straight line (class VI text) is faulty

  • Moving in one direction on the surface of the earth
  • (direction fixed by compass or line joining two stars)
  • does NOT generate a straight line, but a loxodrome (curved line) (see teacher's manual)
  • Believing this (that moving in a fixed direction meant moving in a straight line) created a major navigational problem in 16th c.

Length of a line is measured by superposition

  • therefore, superposition is quite explicitly accepted in RG
  • (That throws axiomatic geometry out of the window.)
  • However, the great new advantage is that a string is flexible
  • therefore the length of a curved line (and, in particular, a circle)too may be measured.

Angle: a different definition

  • Current school text/Euclidean geometry is based on straight lines
  • that results in a clumsy definition of angle
  • as something to do with two straight lines.

OED/Google definition

The indefinite space included between two meeting lines or planes

  • School text definition (class VI, p.78)

An angle is made up of two rays starting from a common end point.

When you measure an angle what do you measure?

  • I was given the barbaric OED definition of angle as a child (in Class II perhaps)
  • It left me confused: how to measure and angle? How do you measure the "space" between two lines?
  • in the picture, do you measure the line BC or DE?
  • The person who was trying to teach me could not resolve the confusion.

A year or so later (in class III perhaps)

  • I was told to use a protractor to measure an angle.
  • I thought my answer was different from that of another classmate
  • just because our protractors were of different sizes!
  • Why doesn't the measure of an angle depend on the size of the protractor? My teacher assured me but did not explain

Can you build a protractor?

  • Anyway, if an angle is about two straight lines (or rays)
  • why is a semi-circular protractor needed to measure it?
  • Can you build one? (If it is not supplied ready-made in the compass box.)
  • And how exactly do you DEFINE equal parts of a semicircle? (Axiomatic definition requires calculus, unreal real numbers, and limits.)

In RG these problems disappear

  • because we can speak of the length of curved lines right from the beginning (ab initio)
  • (empirically, using a string,
  • not axiomatically, using something (axiomatic arc length) that most people don't understand

New/traditional definition

  • An angle is defined as the relative length of an arc or चाप
  • when the arc is measured relative to the circumference of the circle, the measure of the angle is in degrees
  • when the arc is measured relative to the radius of the circle, the angle is in radians.

Advantage RG in definition of angle

  • Most students today confused about radians, and by
  • the conversion formula 360° = \(2\pi\) radians
  • which is just the formula relating circumference to radius of a circle.

Traditional degree measure from astronomy

Term चाप (arc) for angle remained in use until 16th c.

  • E.g., famous निहत्य चाप वर्गेण चापं from Yukitidipika, 441
  • used in the Madhava sine series
  • Term चाप was replaced by term कोण only after 1723 translation of "Euclid" by Samrat Jagannath.

Further advantages of RG definition of angle

However, these are relatively minor issues

Calculation vs theorem

From ancient times mathematics was about calculation

  • metaphysical proof (proof which prohibits pratyaksh) based on reason was a political requirement of the Crusading church
  • for which the church concocted the myth of Euclid.
  • In the 20th c., the partial exposure of the "Euclid" myth (no axiomatic proofs in the book) led to the birth of formal mathematics to "save" the myth
  • Formal math imitates the church theology of metaphysical proof, and asserts it is "superior".

But even for the most basic practical applications

  • such as the measurement of length or area
  • formal math is forced to revert to normal math
  • as in the use of a compass box in geometry.
  • The same thing happens in advanced science and technology.

Further, theorems are NOT true

  • as we saw to be the case with the Pythagorean theorem
  • for triangles drawn on the surface of the earth.
  • On RG, metaphyical or axiomatic theorem proving is rejected as an INFERIOR and impractical activity.
  • Of course, proof itself (as in प्रत्यक्ष प्रमाण, अनुमान) is not rejected.

Therefore, in RG, instead of the Pythagorean theorem

  • we have two "Pythagorean" calculations (which, of course, Pythagoras did not do and could not have done)
  • (1) determining the diagonal from knowledge of the two sides of a rectangle
  • (2) determining one side from a knowledge of the diagonal, and the angle it makes with any one side.

While there are no approximate truths

  • (or approximately true theorems)
  • everyone understands an approximate calculation.
  • Both the above calculations necessarily involve approximation.

As stated in the शुल्ब सूत्र

  • e.g. Manava sulba sutra 10.10
  • CALCULATING the diagonal requires square ROOTS.
  • The simplest case of the unit square
  • results in \(\sqrt 2\) called सविशेष (with something remaining = approximation)

In contrast, the formal mathematical way of stating this

  • is to say that "\(\sqrt 2\) is irrational".
  • The proof of this is fraudulently attributed to some unknown Greek, without serious evidence.
  • However, the fact is that Greeks had no algorithm for division
  • and no fractions, though Egyptians definitely knew fractions.

That Europeans actually learnt from Arabs

  • (though chauvinistically attributed to Greeks)
  • is clear from the very term "surd" for \(\sqrt 2\).
  • The OED tells us it comes from the Latin term surdus
  • which comes from Arabic (through Byzantine Greek)

Surd etymology (OED)

[ad. L. surdus (in active sense) deaf, (in pass. sense) silent, mute, dumb, (of sound, etc.) dull, indistinct.    The mathematical sense 'irrational' arises from L. surdus being used to render Gr. ἄλογος (Euclid bk. x. Def.), app. through the medium of Arab. açamm deaf, as in jaðr açamm surd root.]

Why is \(\sqrt 2\) deaf?

  • The Sanskrit term for diagonal is कर्ण
  • which refers to \(\sqrt 2\), as in diagonal of the unit square,
  • but also means "ear" (as in कर्ण of Mahabharata or Hindi कान).

When Indian arithmetic went to the Arabs from the 8th c.

  • the term "bad कर्ण" for \(\sqrt 2\)
  • was mistranslated as "bad ear" = deaf.
  • Europe got its knowledge of elementary arithmetic "algorithms" from Al Khwarizmi (= Algorismus) through
  • Byzantine Greek translations from Arabic (9th c. onward, e.g. Panchatantra, 11th c.)
  • and Latin translations from Arabic (Toledo, 1125).

What is "irrational"

  • Further, few people, today, are able to correctly define "irrational"
  • which involves an metaphysical (=unreal) fantasy.
  • They often wrongly say "an irrational number is a number which is not rational".
  • Huzurbazar anecdote. (Requires "real" numbers.)

Whatever formal mathematics might believe

  • "real" numbers are not real.
  • One can write down \(\sqrt 2\) to fairly high precision, but NEVER exactly
  • as the formal mathematician fantasizes.

Note: Fantasy of real numbers was NEVER used in Indian tradition

  • which used Brahmagupta's sophisticated "non-Archimedean" arithmetic of polynomials
  • (in terms of formalism: any ordered field larger than reals must be non-Archimedean)
  • (a non-Archimedean field has infinities and infinitesimals, but no unique limits).
  • and Zeroism.


  • One can write down various representations of \(\sqrt 2\)
  • 1.4, 1.41, 1.414, … , 1.414213562373095, …
  • Zeroism enables one to treat all of these as different names of an entity:
  • called \(\sqrt 2\) due to a paucity of names.

Zeroism comes from शून्यवाद

  • On Buddhist क्षणिकवाद the world mutates every instant, as do you, with every breath.
  • Therefore, there is no such thing as a unique individual across time
  • but only a procession of distinct individuals,
  • who are all approximately the same
  • and are given a single name due to a paucity of names.

With non-Archimedean arithmetic

  • one has infinities and infinitesimals
  • and instead of unique limits, one discards infinitesimals.
  • This is a very quick account of how Zeroism works not only in RG, but also in calculus.
  • Point is: this is the much needed philosophy of "approximate truth" of a calculation.


The second "Pythagorean calculation" involves what is usually called trigonometry

  • The term "trigonometry" in a wrong Western term
  • for what should be correctly called circle-metry.
  • That the West had a very poor understanding of trigonometry
  • is clear from the very word "sine" for the function.

OED on "sine" = pocket = जेब

[ad. L. sinus a bend, bay, etc.; also, the hanging fold of the upper part of a toga, the bosom of a garment, and hence used to render the synonymous Arab. jaib, applied in geometry as in sense 2. Cf. F. sinus,

The original Indian term was jiva (जीवा) (as in the sine series)

  • which was rendered in Arabic as jiba (no v sound in Arabic)
  • and written without nuktas as the consonantal skeleton "jb"
  • misread by 12th c. Toledo translators (Mozharabs, and Jews)
  • as jaib, since they knew no math,and certainly no "trigonometry".

Colonial education WRONGLY teaches the "triangle" definition

Worse, with the "Euclidean" geometric, or "triangle" definition of sine in class X text

Colonial education imitates the European historical experience

  • (on the principle that "phylogeny is ontogeny")
  • this pushes us into the abyss of historical European ignorance of math
  • Over 1500 years behind our own traditional knowledge!
  • Further, it primarily teaches us to be like obedient dogs, incapable of challenging the master's education system.

Rajju Ganita poses this challenge by

The traditional definition of sine and cosine (कोटि-ज्या, को-ज्या)

However, note that expressions for circular functions involve infinite series

  • exactly like the infinite series for \(\sqrt 2\)
  • which can NEVER be summed exactly in practice
  • regardless of any formal theory of limits.
  • As such, the second Pythagorean calculation, too, must always reamin an approximation

Note that Nilakantha (नीलकंठ) (of Aryabhata school)

Calculating sine values etc.

Further, instead of proving theorems

  • children need to be taught how to calculate sine values
  • such as sin 1°
  • which are required for any practical applications of circle-metry
  • such as determining the radius of the earth
  • both as in RG text)

The current NCERT class X school text talks of practical applications

  • by measuring real-life angles
  • but does not explain what instruments need to be usedfor this
  • (To measure the radius of the earth, as was traditionally done in India
  • it is important to have a precise way to measure small angles less than 1°.)

The school text also keeps mum whether

  • horizontality and verticality are to be decided empirically
  • as Aryabhata explained (e.g. with a plumb line) or done axiomatically.
  • It will involve explaining that
  • ANY practical application of math MUST use normal mathematics and accept the empirical.

Note: a simple way to measure real-life angles

  • is to use the existing protractor
  • with a plumb line.
  • To hammer the point home, this is NOT "Euclidean" geometry
  • but traditional normal mathematics.

However, this instrument is not accurate enough

  • unlike the kamal or rapalagai used traditionally by Indian navigators
  • such as the one who brought Vasco to India.
  • As described in Cultural Foundations of Mathematics
  • the instrument uses a two-scale principle fraudulently called "Vernier" principle.

There was a plan to develop

  • a separate set of instruments for Rajju Ganita
  • but we ran out of time and money.
  • Redevelopment of traditional knowledge requires both sorts of investment
  • which Indians are rarely willing to make ( they prefer to invest in professors from Harvard, etc.)

Value of \(\pi\)

Traditional values

  • Value of \(\pi\) stated in sulba sutra
  • More precise value of \(\pi\) calculated by Aryabhata.
  • this value repeated by al Khwarizmi
  • echoed by Europeans (e.g. Simon Stevin) until

16th c.

Traditional values

This method also explained in Rajju Ganita text