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Mathematical Sutras: Almost certainly apocryphal

Presents a list of 16 enigmatic sutras, and illustrates how these are to be
used in various mathematical operations.
5
A sutra in the vedic and post-vedic literature is a short, enigmatic
statement, that required considerable commentary to elucidate.
This book presents a set of sixteen sutras, claiming these as Vedic.
However, no vedic source is mentioned in the text.

For example, the sutra  ekAdhikeNa pUrveNa literally means "one more than
the before", from which no hint of mathematics would be obvious.  Bharati
Krishna Tirtha then provides a commentary, in which it is explicated.
One interepretation may be "multiply the one before by one more than it."
This can be applied for squaring numbers ending in 5;  thus, given 35, the
number "before" is 3, and one more is 4.  The product of these two is 12, and
the answer is 12 followed by 25 (square of 5) = 1225.  This holds for all
numbers ending in 5, and is easily proven.

That these formulae could not be vedic in any sense is clear by the
appearance of formulaes for computing structures such as recurrent decimal
fractions (e.g. 1/9, 1/29), which require the notion of decimal fractions, 
and calculus (convergence of infinite series), which would not have been
known even five centuries back.  Perhaps the closest we come is with the
Kerala mathematicians of the 15th-16th c.

The sutras are not from the Vedas

Although the formulas are claimed to be "vedic", with a source in the
Atharva Veda, they appear to have been actually formulated by Krishna
Tirtha maharaj himself.  The famous astrophysicist Jayant Narlikar has
writes in The Scientific Edge (2003):

	K.S. Shukla, a renowned scholar of ancient Indian mathematics...
	recalled meeting Swamiji, showing him an authorized edition of
	Atharva Veda and pointing out that the sixteen sutras were not in
	any of its appendices (parishiShTas).  Swamiji is said to have
	replied that they occurred in his parishiShTa and in no other!  In
	short, Swamiji claimed the sutras to be Vedic on his own authority and
	no other. p.27 [incident narrated by SG Dani of TIFR]

Narlikar goes on to comment that "no one, howsoever exalted, has the right or
privilege to add anything supplementary to the Vedas and claim it is as
authentic as the Vedas themselves, or else there is no authenticity left in
any [original] part of the Vedas.

Sutras are Clever : Urdhva-tiryaka

Narilkar is scathing about the sutras that they do not add anything to
mathematical knowledge (unlike, for example, the diary scribblings of
Srinivasa Ramanujan).

However, the sutras are quite clever and of interest to the average
intelligent person.  Another sutra, Urdhva tiryaka ("up and angled")
shows how you can find the product of two multi-digit numbers in one step
by performing a series of multiplications with numbers at an angle.  For
example,

		   step 1	step 2		step 3		step 4
      423		3	2 3		4 2 3		4 2
    x  36	-->	|  	  				 \ 
   ------		6     	3 6		  3 6 		  3
				diagonals       diagonals	last
				2x6 + 3x3 ;	4x6 + 2x3 	diagonal    
				      	  	+3x[empty]	4 x 3


STEPS:
	     1. 6x3 = 18 :   	       write    8  carry 1
	     2. 6x2 + 3x3 = 21 (+ 1) = 22 :   2    carry 2	
	     3. 3x2 + 6x4 = 30 (+ 2) = 32 : 2      carry 3
   	     4. 3x4 = 12 + 3 =           15  
	     	      	     		 ----------
		==>  So the answer is    15 2 2 8

Recurring decimals

The sutra ekAdhikeNa pUrveNa which we saw before has a second
interpretation.  It is used to obtain the recurring decimal
fractions of the form  1/19, 1/29, etc.
This is also quite clever, but we must realize that the
very notion of recurring fractions does not arise in the original
sources.

Early life of Bharati Krishna Tirathji

Way back in 1921, Jagadguru Sankaracharya of Puri, went to jail.
He did so for upholding what he considered the "Hindu Dharma" of
promoting Hindu-Muslim unity.

The Khilafat Movement against the British for having deposed the last
Caliph was at its peak.
Swami Bharati Krishna Tirath Ji shared platform with the famous Ali
Brothers -- Maulanas Mohammed Ali and Shaukat Ali -- Dr Kitchlew,
Maulana Husain Ahmed of Deoband and others.

Bharati Krishna Tirtha and the consonant place-valud code

The Sanskrit consonants

ka, ta, pa, and ya all denote 1;
kha, tha, pha, and ra all represent 2;
ga, da, ba, and la all stand for 3;
Gha, dha, bha, and va all represent 4;
gna, na, ma, and sa all represent 5;
ca, ta, and sa all stand for 6;
cha, tha, and sa all denote 7;
ja, da, and ha all represent 8;
jha and dha stand for 9; and
ka means zero.

Vowels make no difference and it is left to the author to select a
particular consonant or vowel at each step. This great latitude allows one
to bring about additional meanings of his own choice. For example kapa,
tapa, papa, and yapa all mean 11. By a particular choice of consonants and
vowels one can compose a hymn with double or triple meanings. Here
is an actual sutra of spiritual content, as well as secular mathematical
significance.

	 gopi bhagya madhuvrata
	 srngiso dadhi sandhiga
	 khala jivita khatava
	 gala hala rasandara

While this verse is a type of petition to Krishna, when learning it one
can also learn the value of pi/10 (i.e. the ratio of the circumference of
a circle to its diameter divided by 10) to 32 decimal places. It has a
self-contained master-key for extending the evaluation to any number of
decimal places.

The translation is as follows:

    O Lord anointed with the yogurt of the milkmaids' worship (Krishna), O
    savior of the fallen, O master of Shiva, please protect me.

At the same time, by application of the consonant code given above, this
verse directly yields the decimal equivalent of pi divided by 10: pi/10 =
0.31415926535897932384626433832792.

While this is ascribed to BKT, and indeed, such place value
notation with letters is common in ancient texts (and
well-entrenched by the time of Aryabhata), such an accurate value
of pi seems unlikely.