A General View of Mathematics

A General View of Mathematics
before 1000 B.C.

Where did Mathematics Start?

We have considered some very early examples of counting. At least one dated to 30,000B.C. Counting is but the earliest form of mathematics. It was first a simple device for accounting for quantity. However, this is so basic, even primitive, that it cannot be considered as either a subject or a science.

We are looking for sources of mathematical thought in human activity. These sources come in the form of carvings, inscriptions or manuscripts. Evidence of this kind have four countries of origin, all dating to similar times. They are:

• India
• Egypt
• Mesopotamia
• China

What are the commons features of all?

• Each claims a high degree of antiquity.
• Each claims to have been a pioneer in mathematical development.
• Each had a relatively warm climate, with fertile lands.
• Each flourished along major rivers.
• Each had a strong centralized government and a strong religious life.

CHINA

Even though the Chinese Emperor Shï Huang-ti (259 B.C. - 210(1) B.C.) ordered all books burned and scholars buried in 213 B.C., it is unlikely that such an extensive order could be carried out. Even if so, the contents of these books would have been carried in the memories of many surviving scholars.

Therefore, we can assume with reasonable assurance, that the first authentic mathematics text, the Chóu-peï, dates from about 1105 B.C. (The reason is that it cites the Emporer that died on that date.) Its author is unknown. The Chóu-peï contains its mathematics in the form of several dialogues. One for example relates number mysticism, mensuration, and astronomy. Here are a few extracts:

• The art of numbers is derived from the circle and the square.
• Break the line and make the breadth 3, the length 4; then the distance between the corners is 5.
• Forms are round or pointed; numbers are odd or even. The heaven moves in a circle whose subordinate numbers are odd; the earth rests on a square whose subordinate numbers are even.

The greatest of the Chinese classics in mathematics is the K'iu-ch'ang Suan-shu, or Arithmetic in Nine Sections. Its author and date are unknown. However, after the great book burning, there appeared a mathematician by the name of Ch'ang Ts'ang. He collected great works and appears to have edited the K'iu-ch'ang Suan-shu. This book contains nine sections.

1. Squaring the form. Surveying, with correct formulas for areas of triangles, trapezoids, and circle ( and )
2. Calculating the cereals. Percentages and proportions.
3. Calculating the shares. Relating to partnership and the Rule of Three.
4. Finding length. Finding the sides of figues and including square and cube roots.
5. Finding volumes.
6. Alligation. Relating to motion problems.
7. Excess and deficiency. Relating to the Rule of False Position.
8. Equation. Solving simultaneous linear equations, with some notion of determinants.
9. Right triangle. The Pythagorean triangle.

INDIA

Early Hindu mathematics was produced by a very much different type of people. The Hindus were generally highly imaginative, and their mathematics developed along such lines as the theory of numbers, geometry, and astronomy. However, the Hindu mind was primarily occupied with the arithmetical.

The history of Hindu mathematics may be resolved into two periods: First, the Súlvasutra period which terminates not later than 200 A.D., and the astronomical and mathematical period, extending from 400 A.D. to 1200 A.D. The term Súlvasutra period means the ``rules of the cord", and originally explained the construction of sacrificial alters. The Súlvasutras were composed sometime after 800 B.C. Their aim was primarily not mathematical but religious. Mathematical parts refer to geometrical ideas and mensuration.

The dating and origin of early Hindu mathematical works is even less certain than the Chinese. Some claims are preposterous. For example, the first edition of the Surya Siddhanta, or Knowledge from the Sun, of the Swami Press at Meerut, claims the work was compiled 2,165,000 years ago. Other works are dated even earlier. In fact this famous work was probably composed in the 4th or 5th century of our own era.

About all we can say is that there is some evidence from ancient literature that in very early times India was cognizant of calculations, of astronomy, and of geometry. Judging by the nature of their archtecture, the must have been some considerable body of ``applied arithmetic".