Rule of False Position

According to Needham J (1959) it seems impossible that the "world should ever have been troubled by an equation like ax + b = 0". To solve this sort of equation the old mathematicians used a method known in later Europe as Rule of False Position. This rule took on a form that was called 'Double False'. If we take into consideration the above equation, then the rule can be explained as follows:-

Let g1 and g2 be two guesses as to the value of x and let f1 and f2 be the failures. It would also follow that the values of ag1 + b and ag2 + b would be equal to zero if the guesses were right.

Then........................ag1 + b = f1....................(1)

and.........................ag2 + b = f2....................(2)

hence......................a(g1 - g2) = f1 - f2..........(3)

from (1)..................ag1g2 + bg2 = f1g2

from (2)..................ag1g2 + bg1 = f2g1

therefore.................b(g2 -g1) = f1g2 - f2g1....(4)

dividing (4) by (3)..-b/a = (f1g2 - f2g1)/(f1 - f2)

but.........................-b/a = x

Using the above we can conclude that x can be found.

This Rule of False Position was transmitted to Europe by Arabic mathematicians. It also occurs in the works of al Khwarizmi and several other later writers.

This rule had Chinese origin by Chhien Pao-Tsung followed by Chang Yin-Lin. However, Chang Yin-Lin pointed out that this rule is nothing else than the Chinese methods Ying pu tsu (Too much and not enough) which is actually the title of the seventh chapter of the 'Chiu chang suan shu'.

There was also another Chinese involved in the naming of this rule. Liu Hui called it the thia nu rule. The word thia is taken from lunar movements. Thia means the latest appearance of the waning moon. The word nu means the earliest appearance of the waxing moon.

The Chinese also used the words Chia ling which in translation means 'let us now assume'. These words would be used for the adoption of the first proposal g1, whether it be excess or deficiency. The words ling chih are used for the adoption of the second proposal g2, again whether it be excess or deficiency.

However, the final name for this rule is the Rule of False Position.

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