The Operational Theories Used In Jain Karmic Theory
The operations were of eight types.
i. Addition (samkalana)
The word samkalana denotes the addition of one set to another set. Whatever to be added was written above the set, and the addition was denoted by a tail (pucchadi) E.g. lac increased by unity as . Universe cubed point set increased by infinite was given as . When two or more sets were to be added, then the two or more sets were written above the original set. E.g. when the time fluent added with universe point set and then with aether and anti-aether fluent then the addition was written as. A set slightly greater than its value was deoted by placing a vertical bar above the set.E.g.the set slightly greater than infinity was denoted as, in the Trilokasara the word dhana was used for the operation of addition. In the printed books the tail had been reduced to the horizontal bar.
ii. Subtraction (vyavakalana)
When the set to be subtracted was written above the set with a tail and a small circle.E. g. the crore decreased by unity was written as . Sometimes the quantity to be subtracted was written below and a small circle was placed with it. E.g. the lac decreased by unity was denoted as —–.
Further the symbol was reduced by horizontal bar placed in between the original set and the set to be subtracted from it, in the printed books. When two or more sets to be subtracted from an original set, two or more horizontal bars were placed after the set. E.g.the set of transmigrating bios when reduced by two three and four sensed bios and further five sensed bios set, the remainder to be one sensed bios set which was denoted as.
sometimes crescent shape was also used in between the two sets for subtraction as shown below. In the Dhavala texts, the +symbol called the foot of crow (kakapada)appeared for denoting subtraction. The same symbol appeared for addition in the Bakhali manuscript. In Tiloyapannatti rina word had been used for subtraction. The symbol from Dhavala which was compiled later, must be from Bramhi style. The printed books had removed the total tail symbol by horizontal bar with small circle or cresent.
iii. Multiplication (gunakara)
The multiplier was written after the multiplicand. E.g.pit multiplied by numerate could be written as .the lac multiplied by five as .when this same number again multiplied by five then the number would become.
iv. Division (bhagahara)
The word frequently was used bhajana or bhaga or for division. The word apahrta was used for division through the rational of subtraction and viralana through distribution. The dividend is placed above the divisor which is placed below the former. E.g. a crore is divided by five was denoted by . The pit divided by the numerate was given by . The infinetisimal part of the set of all bios could be denoted as . Similarly the numerater and the denominator were written , the poition of the former being above the latter.e.g. one third was written as . The combination of an integer and a fractional part were written in two ways.e.g. one and one half was written as or.
v.Square (varga or vargita)
the quantity multiplied by itself was meant by vargita and it was denoted by writing the term twice.e.g.the square
of five was denoted by .
The square of badala was written as .
The square of infinite was written as .
Similarly the square of fraction was denoted.e.g.the sqare of the innumerate part of a finger(suchyangula) as a
set of points was written as .
vi. Cube (Ghana)
The symbol for the operation of cubing was denoted by writing the number or the set three times, one following
the other in a row. E.g. the cube eight could be written as 8/8/8.
The cube of pannatti was denoted as ——————-.
The cube of innumerate was written as ———-.
The cube of the numerate part of the finger cubed(ghanangula) could be written as ——-.
The world line (jaga sreni ) was denoted by a horizontal line as _ . The world surface (jagapratara) was denoted
by placing another horizontal bar below it as =. The geometrical figure so obtained could be denoting a set of
points contained in a square of which one side was a measure of a world line (jaga sreni)and the value of its
length was seven ropes (rajju) Similarly the cube of the world line was denoted by placing one more line below
the two lines and the figure so obtained was =. The figure so obtained was called cube universe (hanaloka).
vii. Square root(varga mula)
The symbol for the extraction of square was —-.If the operation was performed one time, then it was represented
as ——and if the operation would be carried out twice then it was represented as——and called second root
(dvitiya mula) The process might be continued for many times. The number of such operations of extracting
square root was a functional operation denoted by placing the number ahead the symbol.
Cube root (Ghana mula)
Usually, factorization method might have been used for treating the operation.e.g. in the treatment of divergent
sequences which were in dyadic form was found out along with cube and non cube forms.
All the above was called parikarmastaka.(from the number of operations.)
3.5.b. The place value operators
A small circle(sunya or bindi) was used to denote the place value or the vowel was attached with a consonant.
Dr.Ojha mentioned about a small inscription of pre-Asokan period found at a village Barali, Ajamera (Rajputana
Museum,Ajamera).It is a piece of an insription of a pillar, in which in the first line “vira(i)ya bhagava(ta)”, and in
the next line “cuturasitiva(sa)”is written. The insciption seems to be related with Jaina’s Vira nirvana samvat,
(527B.C. or still earlier). It contains the accented alphabet ——of Viraya. Thus the mixed symbol — has a deep
significance in so far as writing of zero as well as negative sign is concerned.
In Mahabandha unusual style of an abbrivation operator is found. E.g. Transpriction:
Veuvvyamussa o- pancana o navadamsa o micchatta o solosaka o bhayodugam o oraliya-tejaka o vanna o 4 agu o
4badara-pajjatta-patteya o nimina-titthayara pancanta o jahannu o anto o / sesana sadadinari jaha o ega o ukka
o anto/ Translation:
In the transforming mixed (vaikriyaka misra) volition, the minimal ,maximal bond, nine vision, screening illusion
sixteen affections, fear, aversion, macro phosphorescent-karma finishing body, color, non gravity levity, gross,
developed, every, growth, ford founder, five interferences are obtained within an inter-muhurta. The bond period of
the remaining configurations sata etc is an instant as minimal and an inter muhurta is maximal.
The symbols of Logarithmic values
In the Gommatasara and Labdhisara, symbolism appears in abundant form. The values of logarithms to the base
two (ardhacchada)and the logarithms of logarithms to the base two were given in their commentaries.
The examples are given in the table IV.
The theory of measure
Two types of measures can be found in the Labdhisara. The first type is number measure (samkhya
pramana), they may exceed up to numerate, innumerate and infinite.
The other type of measure is the simile measure (upama pramana) has been defined through construction sets,
the point sets, instant sets and various types of existential sets. In the Labdhisara universe (loka) appears as
point set and pit (palya) asinstant set.
Series and sequences
This aspect can be obtained in major portion of the texts. The state matrices when transforms over to new
transient matrices according to instant forms discrete series under various operations. For defining the trans
science of such matrices is a complex problem which will be studied in detail in the separate points. An
oscillating sequence type phenomenon is also observed known as six-stationed cascade (sat sthana patit
vridhi) changing through geometric sequences with different ratios.
The matrix theory
In order to represent various types of operators, operands and their transformations, various forms of matrices
can be found out for the addition of fluents or up traction or down traction (utakarshana, apakarshana).