The Epistemic Test

Epistemic test-2: trigonometry

Toledo translations ca. 1125

• in the late 11th century the Muslim-ruled taifa of Toledo in Spain
• a part of the earlier Umayyad khilafat of Qurtuba (Cordoba)
• fell to invading Christian pre-Crusaders.

• Like many Muslim kingdoms, it had a huge Arabic library.
• This library was mass-translated from Arabic to Latin in 1125CE.
• The resulting Latin books were used as the first textbooks
• in the first Christian universities such as those of Cambridge, Oxford, Paris etc.

• This is when the term sine was born due to translation errors.
• Word "Sine" from sinus=fold from Arabic jaib (जेब) = pocket (OED).
• What has trigonometry to do with POCKETS?

• From Sanskrit term for it ardh-jyā
• (half-chord) or जीवा
• rendered in Arabic as jībā (no v sound in Arabic).

• Written as consonantal skeleton "jb" (without nukta-s) like "pls" in SMS.
• Misread by Mozharab/Jew 12th c. Toledo mass translators
• as common word "jaib" = जेब = pocket.😊

• translated as sinus.
• Sine is taught as part of trigonometry
• Word "trigonometry" involves a conceptual error: it is about circles not triangles.
• Hence my pre-test question: what is \(\sin 92^∘\)?
• (In a right-angled triangle there cannot be any angle of \(92^∘\).)

Epistemic test-3: Arithmetic

History of arithmetic

Early Greeks and Romans were BACKWARD in arithmetic

• Names of small numbers similar in Greek, Latin, Persian and Sanskrit
• obvious e.g. सप्त (September), अष्ट (October), नव (November), दश (December)
• More details

But NO large numbers in Greek and Latin

• Greek and Latin numbers stopped at a myriad (10000).
• Though this is a puny number,
• in the English language
• it still connotes an infinitely large number!😀

But Sanskrit numbers go on

• till a trillion (परार्ध, \(10^{12}\)) in the Yajurveda 17.2
• and तल्लक्षण (\(10^{53}\)) and परमाणुरजःप्रवेशानुगता (\(10^{108}\) in Lalitavistara sutta [Life of Buddha] chp.12).
• (Buddha was asked to name numbers after 100 crores,
• Greeks, Romans would have flunked.😀)

Greeks learnt Indian number-names

• due to the Persian (Achaemenid) conquest of Greece.
• (No Aryans, no common origin, read my book on Aryan Race Conjecture, won't go into it today.)
• The Dara (Darius) Vase shows how Greeks learnt arithmetic from Persians
• to pay taxes to them.!

• But Persians too stopped at a myriad
• ="beavan" in Avesta (Ervad Rooyinton P. Peer, personal communication 25 July 2021)
• Same small numbers BUT no large numbers = Greeks learnt from Indians via Iran
• but learnt little.

Primitive pebble arithmetic

• WHY did Greek and Roman arithmetic lack large numbers?
• there was a STRUCTURAL reason.
• Bcoz Roman/early Greek arithmetic was PRIMITIVE, done using pebbles
• and a counting board (called an abacus) as depicted in the Dara (Darius) Vase $-4$th c. CE.

The Roman calculus 😜

• The Latin word "calx" means "pebble" or "gravel stone"
• calculi are thus little stones (used as counters)
• In Greek the pebbles were called psḗphoi and
• the word for "to calculate" is psēphizein (literally, "to pebble")🤣

Graeco-Roman numerals

• Note: In early Greek, the letters used above are the initial letters of
• pente(ΠΕΝΤΕ), deca (ΔΕΚΑ), hecaton (HECATON), xilioi(ΧΙΛΙΟΙ), mypio (ΜΥΡΙΟΙ).
• Hence, described as acrophonic by a Roman historian Herodian of Syria (+3rd c.) until which time they were clearly in use.
• In the early Greek system of writing (till +6th c. CE), only CAPITAL letters were used as in this +4th c. Greek Bible.

Beware

• Western history-cheaters confound it with Minuscule developed in late Byzantine Greek (Turkey) mid-9th c.
• Hence, very wrong to attribute 9th c. knowledge to early Greeks like Archimedes.

• (Greek) Attic numerals used from earliest Greeks times.
• Attic is a dialect of Greek in which donor lists and amounts were written on stone tablets in Athens from about the −5th c. CE.
• So, this system of "Attic numerals" prevailed in Greece from about −5th c. CE to +3rd c. CE (the time of Herodian).
• i.e., until Roman conquest of Greeks.
• This should be clearly separated from

later Byzantine Greek texts.

Both Roman and early Greek

• are adapted to this pebble arithmetic ( गणित)😀
• or coin-counter arithmetic; e.g. XVII=10+5+1+1.
• This already tells you WHY Graeco-Roman arithmetic was limited to small numbers.
• 10000 pebbles is a lot of pebbles!)😀

Indians used the superior place-value system

• as clear from the quotes from Yajurveda 17.2 or Lalita vistara sutta:
• the numbers are all in geometric progression.
• This is also how the place value system is described in Indian ganita texts. e.g. Aryabhatiya (5th c.), Ganita-sara-sangraha (8th c.) Lilavati (12th c.),.
• Place value makes large numbers very easy

Important note on zero

• Place value system needs a place-holder: e.g. write 603. What will you put in 2nd place?
• Place holder was zero as in Aryabhata's "ख द्वि नवके" = two nines of zeros (for 18 places).
• Hence, zero existed in India since place value arithmetic, i.e., Vedic times
• contrary to foolish western claims of its later origins.
• But let us return to the main topic.

India had extensive trade with Roman empire

• but Romans could not say what a myriad myriad (\(10^5 \times 10^5 = 10^{10}\) = 10 billion is.
• Surprising that Romans never learnt arithmetic from India!
• or how to represent large numbers.
• Moral: Ignorant people (duffers?) take much time to understand their ignorance!
• Note: Should I say White Europeans were an inferior RACE (as they did about us)? But I am NOT doing that right now.

Interim summary

• Graeco-Roman arithmetic was backward because it was PEBBLE ARITHMETIC
• Hence could not represent large numbers
• as Indian arithmetic with place value easily did from Vedic times.
• Cartoon summary.

Backward pebble arithmetic persisted in Christian Europe

• Backward Roman and Greek arithmetic
• continued into Christian Europe (later called "Holy Roman Empire")
• To begin with it continued for 1500 years after Dara, until 10th c.

The learned but foolish pope

• Then, a very learned Christian European, Gerbert (Pope Sylvester II, died 1002)
• finally understood the backwardness of Roman-Christian arithmetic.
• Wow! Took 1500 years!😀

Gerbert imported Indian arithmetic

• thus admitting the inferiority of Roman Christian arithmetic,
• he imported Indian arithmetic from Muslim Cordoba (Umayyad Khilafat)
• This was not an easy step,

• because the church hated Muslims
• and fought a long religious war with them
• (pre-Crusades from mid-11th c., and Crusades from end-11th to 16th c.)

Indian arithmetic reached Cordoba

• via 9th c. al Khwarizmi of Baghdad (Abbasid Khilafat)
• who wrote a book Hisab al Hind
• But Gerbert called this arithmetic "Arabic numerals". 😀

Two terms: "Arabic" and "numerals"

• "Arabic" wrong, but excusable since Gerbert got it from Arabs.
• "Numerals" a major blunder by Gerbert
• as if what had changed was only the way of writing numbers
• when what changed was the whole way of doing arithmetic.

Gerbert's apices

• Apices were Gerbert's striking innovation!😀
• That is, instead of having, say, 7 pebbles,
• he had one pebble with the number 7 written on it,
• using Arabic notation, in 976 CE.

Gerbert's achievement

• Large numbers could at long last be
• written in Roman-Christian arithmetic
• by copying the Indian place value system.
• Note: place-value system existed elsewhere (e.g. among Maya) but Europeans learnt it from India via Muslims.

Gerbert's blunder

• He made an abacus for Indian arithmetic.
• bcoz he wrongly assumed that all arithmetic needs an abacus.
• This blunder came naturally to Gerbert
• for he had written a book on the abacus

• BUT abacus subverts the EFFICIENT PROCESS ("algorithms")in Indian arithmetic
• by which large numbers are generated.
• Abacus an inferior and inefficient way to do arithmetic compared to algorithms.
• Let us see why.
• (Note: "algorithm" from al Khwarizmi's Latin name Algorithmus or Algorismus.)

Inefficient Graeco-Roman pebble arithmetic-1: Writing numbers

• Q. Can you write 1788 in Roman numerals?
• A. MDCCLXXXVIII
• Requires 12 symbols 😀 instead of 4 on the superior Indian decimal place value system.
• Moral: Graeco-Roman numerals a VERY inefficient and INFERIOR way of writing numbers.

• Challenge: can you write तल्ल्क्षण (\(10^{53}\)) in Roman numerals?
• Term billion not standardised even in my college days!
• English billion = million million, American billion = 1000 million (America won.)
• Cartoon summary.

Inefficient Graeco-Roman pebble arithmetic-2: Addition

• Q1. Can you do \(89 \times 89\) in Roman arithmetic?
• Q2. Can you even do 89+89 in Roman arithmetics.
• Note: NOT allowed to convert to decimals, do sum, and convert back. Do it as Romans did.
• Let us answer Q.2 first.

Addition in Graeco-Roman pebble arithmetic

• Step 1: write out LXXXIX in its full form: LXXXVIIII.
• Step 2: Use counters (coins) for each of L, X, V, I (and C)
• Step 3. Pool together all counters used for LXXXVIIII and LXXXVIIII.
• Step 4: Simplify

• Step 4.1: 8 I = 1 V and 3 I. (Operation: remove 5 I’s replace by 1 V)
• Step 4.2: 3 V's = 1 X and 1 V. (Remove 2 V replace by X)
• Step 4.3: 7 X's = 1 L and 2 X (Remove 5 X, replace by 1 L)
• Step 4.4: 3 L's = 1 C and 1 L (Remove 2 L, replace by 1 C)

Final result (Whew!)

• What is left: 1 C, 1 L, 2 X, 1 V and 3 I,
• namely CLXXVIII or 178.
• If we count removal and replacement as one operation each
• we need a total of 6+3+6+3 = 18 operations.

Inefficient Graeco-Roman pebble arithmetic-3: Multiplication

• Now to Q. 1. Do \(89 \times 89\) as Romans did.
• Roman multiplication = repeated addition, hence even more inefficient.
• So you must add 89 to itself 89 times
• needs at LEAST \(18 \times 89 = 1602\) operations
• (ignoring intermediate large numbers which will hence need more operations)

In contrast usual (school) algorithm for

• \(89 \times 89\) needs just 8 operations
• So, Graeco-Roman arithmetic was at least 200 TIMES more inefficient and time consuming
• compare to Indian arithmetic
• Western historians NEVER told you that.
• Why not?

Bcoz it is a major issue

• Q. If early Greek pebble arithmetic was so inferior
• how could the early Greeks have done any science?
• They did not, but West won't admit it: will stand current history of science on its head.
• hence, also, the colonized monkey won't accept it for he FEARS commonsense.

Interim summary

• Graeco-Roman abacus arithmetic VERY inefficient compared to Indian arithmetic.
• Hence, learned Gerbert made a FOOLISH mistake
• in making an abacus for "Arabic numerals" (= Indian arithmetic).
• Cartoon summary

Fibonacci's foolishness

• It took another 2 centuries for Europeans to grasp
• that there was more to Indian arithmetic than the ability to represent large numbers.
• A Florentine merchant understood its efficiency gave a competitive advantage in commerce.

Florence

• was among the richest city states in Europe
• Because Florentines traded with rich Muslims, in Africa

Fibonacci grew up partly in Africa

• in a city he called Bugia which means "lie" 😀 in Italian per Google.
• Perhaps the Algerian port city of Béjaïa.
• In Africa he studied Indian arithmetic through
• al Khwarizmi's Hisab al Hind

Liber Abaci

• In 1202 he published Liber Abaci on Indian arithmetic
• In it he repeatedly speaks of "the art of the nine Indian figures" (p.15),
• on "the recognition of the nine Indian figures and how all numbers are written with them" (p. 16) etc.
• Never uses the word "Arabic" for them.

• Anyway, we need a term to compare "Hindu-Arabic" with
• so I will use the term "Roman-Christian pebble arithmetic".

Fibonacci's 13th c. book is a dumbed down version of

• 9th c. Mahavira's Ganita Sara Sangraha
• Note: Mahavira was a Jain, hence naturally emphasized commercial applications (not astronomy)
• in which Fibonacci was most interested.
• Compare Mahavira's table of contents with Fibonacci's

Why is Fibonacci a dumbed down version?

• Note how square-roots and cube roots
• at the beginning of all Indian ganita texts
• are postponed to the end in Fibonacci's book.
• since these problems were beyond Roman-Christian pebble arithmetic.
• Topics in Mahavira, such as permutations and combination, area of an ellipse are completely missing.

Fibonacci's blunder

• Most importantly Fibonacci did not fully understand subtraction
• Hence, NEGATIVE numbers are missing in Fibonacci.
• Mahavira's TOC mentions positive and negative numbers
• Fibonacci's TOC says that only a SMALLER number can be subtracted from a LARGER number.😀

Fibonacci's blunder was natural

• in Roman-Christian pebble arithmetic
• subtraction means removing pebbles from a given bunch of pebbles
• so, one cannot remove more pebbles than are there!
• There are no negative pebbles.

Whose blunder?

• One can discuss whose blunder this was.
• Fibonacci went by al Khwarizmi (not Mahavira, directly)
• al Khwarizmi did not accept negative numbers!
• But it was nevertheless a natural blunder for Fibonacci and numerous Europeans after him.

Note

• Fibonacci was a trader, not a mathematician
• Fibonacci numbers are copied from India but not attributed
• bcoz Western historians invariably want to credit Christians.

Interim summary

• Fibonacci understood that the efficiency of indian arithmetic
• offered a competitive advantage for commerce.
• But he blundered, and FAILED to understand SUBTRACTION and NEGATIVE NUMBERS,
• bcoz he was stuck to his paradigm of primitive Roman-Christian pebble arithmetic
• which lacks negative pebbles.

Zero as nothing = "no pebble"

• Zero glorified by Bollywood, Manoj Kumar, Purab Paschim.
• Actually, because of Fibonacci "Arabic" "numerals" (=Indian arithmetic) spread in Florence, Venice etc. (not beyond)
• BUT, to understand negative numbers one must also understand zero.
• Alas, accustomed to their paradigm of primitive Roman-Christian pebble arithmetic
• many other Florentine merchants failed to understand both place-value and zero.

Roman-Christian numerals are "additive"

• This is the first point to understand
• X represents 10 pebbles and I represents one pebble (or coin)
• Therefore, XXII means 10+10+1+1=22

On this system, Florentines understood as "nothing"

• or "no pebble"
• exactly as in Gerbert's apices.
• This is one possible meaning of zero.
• However, on the place-value system this is not the ONLY meaning of zero.

Recall Aryabhata and his खद्विनवके

• or "2 nines of zeros"
• as 18 placeholders for numbers up to \(10^{18}\)
• 000000000 000000000
• A blank entry leaves only the symbol 0.

But, this is not the only possible meaning of 0

• on Indian arithmetic numbers are leftists 😀
• अन्कानाम वामतो गति:
• the places are filled from right to left.
• Therefore, a 0 at the beginning of a number does mean nothing: 011 = 11.

But this is NOT true for 0 at the end or in middle of a number

• 110 ≠ 11 ≠ 101 😀
• This sort of thing also happens for the position of a word in an English sentence
• as in of the change of meaning from the position of the word "only"
• in the following sentence

"He said that he loves her"

• "ONLY he said that he loves her" (0111111)
• "He ONLY said that he loves her" (1011111)
• "He said ONLY that he loves her" (1101111)
• "He said that ONLY he loves her" (1110111)
• "He said that he ONLY loves her" (1111011)
• "He said that he loves ONLY her" (1111101)

That is, 0 means nothing ONLY at the beginning of a number 😀

• But in Roman Christian arithmetic number was pebble or coin or counter
• a "thing" in short
• so Florentines thought its meaning must be the same
• regardless of its position!

Florentine law against zero

• Hence, Florence in 1299/1300 passed a law
• that any financial contract written in "Arabic numerals"
• Should also be written in words.
• We still follow that practice in writing cheques.

Interim summary

• In Indian place value arithmetic zero has a double role
• as (1)the number 0, and (2) as a placeholder
• Therefore, the meaning of zero changes with its position.
• This confused people accustomed to Roman Christian arithmetic.
• Cartoon summary

Confusion about negative numbers in modern times

• Thus, West was backward in elementary arithmetic
• from early Greek and Roman times until 10th c.
• Then, Europeans imported Indian arithmetic ("Arabic numerals")
• but had great difficulty in understanding it.

Did Europeans overcome their difficulty with arithmetic by 13th c?

• No! Their confusion about negative numbers etc. persisted until 20th c.
• Whole story too long to tell here
• Will give just two examples.

Euler: two kinds of \(-1\)?

• Wallis, Euler etc. argued that \(-1 < 0\) but \(\infty < -1\) 🤣🤣
• Proof? \(-1 < 0 <1\), hence \(\frac {1}{1} < \frac {1}{0} = ∞ < \frac {1}{-1} = -1\)
• Euler cheated! Euler method to solve ODEs copied from Aryabhata/Nilakantha
• His soln. of Fermat challenge problem a solved exercise in Bhaskar II (Bijaganita 87).(Wrote on Indian calendar in 1740.)
• Cartoon summary.

Augustus De Morgan: Dunce

• A very influential mathematician of the 19th c.
• Professor of math at University College London.
• part of an influential group which rejected negative numbers
• E.g.1 Algebra and Calculus [p. xi]
• E.g.2 Study of Difficulties in math [p. 72]

Interim summary

• Confusion about negative numbers persisted in Europe
• from Fibonacci till Dunce de Morgan end 19th century.
• (in fact in my Algebra school text in 1960's)
• Cartoon summary

Summary

• West was backward in math (elementary arithmetic) from early Greek and Roman times.
• Took 2000 years to understand their inferiority
• then started importing Indian arithmetic 10th c. on
• foolishly called it "Arabic" numerals then or "Hindu-Arabic numerals" today

Summary (contd.)

• But took very long to understand its fundamental differences from Roman Christian pebble arithmetic
• such as place-value, it's efficiency, use of large numbers, zero, negative numbers.
• Those who learnt school arithmetic from us, how can they claim to be superior in math?
• The colonizer left but his lies remain in your mind!

Epilogue: fractions and the Roman Christian calendar (astronomy)

• Bad arithmetic leads to bad science.
• Roman-Christian arithmetic which lacked fractions
• hence led to the shoddy Roman-Christian calendar
• which we accept as our national calendar today.

The Greek calendar

• was excessively bad
• The learned Solon told Croesus, first Lydian ruler of Greeks (source Herodotus)
• that a year has 375 days.😀
• Romans laughed at the Greek calends.(OED)

But the Roman calendar was equally lousy

• The Julian calendar was made by Egyptians,
• hence after the Roman conquest of Egypt)
• needed a corrective year of 445 days! 😀
• Romans laughably didn't even grasp the Egyptian prescription
• that every 4th year must be a leap year.

Augustus Caesar

• For 20 years Romans kept making every 3rd year a leap year 😀
• Then Augustus Caesar corrected it, hence got the month August named after himself,
• and got it increased to 31 days.

So, Roman calendar based on vanity of Roman emperors, Julius, Augustus

• NOT scientific
• Hence, months have varying durations 28, 29, 30, 31 days.
• Nothing to do with any observed motion of the moon. (Word MONTH derives from moon.)
• But we think it is "scientific temper" to adopt this unscientific calendar,
• not the Indian calendar in which every month has EXACTLY 30 tithi-s.

Later it became the official Christian calendar

• First Council of Nicaea (ca. 325 CE) adopted this as the official Christian calendar
• to fix a common date of Easter, to bring about unity among Christians in the Roman Empire.
• So, it can accurately be called the Roman-Christian calendar.
• Gregorian reform of 1582 done by a Pope since this miserable calendar was failing badly for centuries.

All of our colonized historians use this calendar

• they give dates using the terms AD and BC
• so that belief in Christ (as our Lord) is asserted in every date
• though historicity of Jesus is very doubtful.

• The colonized think imitating the West (long dominated by the church) is secular!😀
• Hence, also, our only two secular festivals Independence Day and Republic day
• are defined only on the Roman-Christian calendar
• though that term is never used.

But scientific Indian calendar

• is called Hindu calendar
• although it is clearly used also by Buddhists, Jains, Sikhs

Critical point

• Roman-Christian calendar is bad because of bad Roman-Christian arithmetic
• which lacks fractions
• hence could not depict the (tropical or sidereal) year accurately
• and could not calculate the "synodic" month
• still determined by religious authority of a synod or priest.

Egyptians HAD unit fractions

• as in the famous "Eye of Horus fraction"
• but inferior Graeco-Roman arithmetic had no way to depict general fractions
• Can't write \(\frac{4}{5}\) as \(\frac{IV}{V}\) (then cancel the V?! 😀)
• Hence used the primitive system of leap years.

Christoph Clavius, author of the Gregorian reform

• introduced Indian fractions (direct from India) as practical mathematics in the Jesuit syllabus in ca. 1576
• but due to widespread ignorance of fractions in Roman-Christian Europe then
• the Gregorian calendar continues to use the primitive system of leap years.

Calendar still inferior and unscientific

• Consequently it does not get the tropical year right from year to year
• but only on a thousand year average.
• The Gregorian calendar is still bad an unscientific
• but, for us, aping the West means being superior!

Key point: ALL these cases

• of arithmetic, algebra, trigonometry
• involved various degrees of conceptual incomprehension
• by Europeans while copying from Indians
• but we have declared the duffer as "superior"
• and are playing "follow the leader".