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ELEMENTARY PROBLEMS AND CHINESE MATHEMATICS
1) MATHEMATICAL RECREATIONS |
In the Middle Ages there developed a new form of puzzle problem, one suggested by the later Greek writers and modified by oriental influences. Problem of Metrodorus.
So far as the Greek were concerned the source book for this material is the Greek Anthology. This contains the arithmetical puzzle, supposed to be due to Metrodorus about the year 500.
Polycrates speaks : " Blessed Pythagoras ( c 540BC), Heliconian scoin of the Muses, answer my question: How many in thy house are engaged in the contest for wisdom performing excellently?"
Pythagoras answers: " I will tell thee, then, Polycrates. Half of them are occupied with belles lettres; a quarter apply themselves to studying immortal nature; a seventh are all intent on silence and the eternal discourse of their hearts. There are also three women, and above the rest is Theano. That is the number of interpreters of the Muses I gather round me."
The following problem relates to a statue of Pallas ( about the time of such Christian scholars as Capella and Cassiodorus, 460 AD and 502AD) lusty poets. Christians gave half of gold, Thespis one eighth, one tenth, and one twentieth, but the remaining nine talents and the workmanship are the gift of."
The following relates to the finding of the hour indicated on a sundial
and still appears in many, modified to refer to modern clocks:
The following problem has more of an Oriental atmosphere
(unknown author and time):
One of the remote ancestors of a type frequently found in our algebras appears in the following form:
Such problems seem more Oriental than Greek in their general form, but if we could ascertain the facts we should probably find that every people cultivated the somewhat poetic style in the recreations of mathematics.
The Chessboard Problem
One of the best- known problems of the Middle Ages is that relating to the grain being put on the first square, two on the second, four on the third, and so on.
In geometric progression, the total number being
A Dutch arithmetician, Wilkens ( 1669), in Europe, takes the ratio in the chessboard problems as three instead of two, and considers not only the number of grains but also the number of ships necessary to carry the total amount, the value of the cargoes, and the impossibility that all the countries of the world should produce such an amount of wheat.
2) Commercial Problems
Equation of Payments
Tablets of Nineveh as old as the 7th century BC. have the following records:
Interest in Ancient India Bahaskara (1150 AD.)
If the interest of a hundred for a month be five, say what is the interest of sixteen for a year.
If the interest of a hundred for a month and one third be five and one fifth, say what is the interest of sixty- two and a half for three month and one fifth.
3) Applications of Algebra
The second general application is geometric, and this characterizes the works of the Greek writers, with the exception of Diophantus.
The third general application is to fanciful problems relating to human affairs, and this is essentially Oriental in spirit.
The fourth type is characterized by the attempt to relate algebra
actually to the affairs of life. Ahmes ( 1550 B.C) gave numerous
problems like :
Simultaneous Linear Equations. Problems involving simultaneous linear equations were more numerous in the Orient in early times than they were in Europe.
For example in India, Mahavira ( 850 AD) as follows:
His rule for the solution is similar to the one used today for
eliminating one unknown. Another of his problem which attributed to
Euclid, is as follows:
In the Greek Anthology( 500 AD), there are two problem as follows:
2. The second ( xiv, 144) is a dialogue between two statues:
4) Magic Squares
The eight triagram
The original in the book described the numbers of the Magic square as the faces of the dice. The author included that the Magic square was written upon the back of a tortoise which appeared to Emperor Yu ( 2200 B.C) when he was embarking on the Yellow River.
Magic squares and Magic circle in Japan.
Show only a half of a Magic square as given in Hoshino Sanenobu's Ko-ko-gen Sho,1673. The magic square numbers were written in Japanese language, and included the Magic circle of 129 numbers which from Muramatsu kudayu mosei's jinko- ki, 1665. The numbers in each radius add to 524, or with the center1. India in Magic square, It first appears in a jaina inscription in the ancient town of Khajuraho, India (unknown author, 870 - 1200 ).
This Magic square display a somewhat advanced knowledge of the
subject, for it has each of the four minor squares has a relation to the
The three triads ( which locates around the center of the wheel of life) made up of the nine digits:
The first had the following relation:
1 = gold = the sun: symbol circle ( O ) with a dot at the center
2 = silver = the moon symbol looks like a lune .
3 = tin = Jupiter symbol looks like the hand grasping the thunderbolt
The second triad was as follows:
The third triad was as follows:
The outer of the Triad, which is still on the circle, show the wheel of life by representing twelve types of animals to indicates the time in day and the year cycle. One year cycle is twelve years.
AT the bottom of number Eight ( West ) starts with :
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