The presentation given today by Kathiann House, after Dr. Barsky's commentary on Cardano (born 9/24/1501), Max Noether (born 9/24/1844), and Lev Genrikhovich Shnirelman (died 9/24/1938), held to the main concept that Medieval India played a large role in the development of mathematics despite the fact that mathematicians from other countries may have done similar problems. In this lecture, the topics covered consisted of Magadha, Chandragupta Maurya, Ashoka, Mohammed Ghori, the Sulvasutras, the Siddhantas, the Aryabhatiya, the Brahmaphuta, and the values of pi.
Among the topics covered, the Siddhantas interested me the most, especially in the area of the development of the trigonometric function of "sin". Kathiann House had mentioned that the Siddhantas primarily included trigonometry and surveying. In her commentary on this subject, the Indians found that the correspondence between half of a cord of a circle will find the function of sin. According to her presentation, the word sin comes from the Sanskrit word jya-ardha (meaning half cord), which in turn has been abbreviated to jya or jiva. The Arabians then came along and called it jiba (meaning breast). Translated into Latin, the word for breast is sinus, which is how we came up with the notion of sin. However, how did the Arabians decide to call it a breast? Either they found it so close to jiva that they decided to call it jiba to be comedians, or they may have found a correlation between the shape of the breast and sin. In chapter seven of a History of Mathematics by Victor Katz, the Arabians found "The Sine Theorem for Spherical Triangles" which stated that in any spherical triangle ABC, sin a/sin A = sin b/sin B = sin c/sin C. They proved this by drawing a line perpendicular to the segment AB. They then produced two spherical triangles in figure 7.23. The diagram shown appears to look very similar to a "breast/bosom". Now this may have came after their decision to call it jiba or before, but I think that because of the shape link of this Arabian sine theorem to the breast, they developed the theorem before they decided to call it jiba. According to the time frame given in this presentation (eighth century B.C. to seventh century A.D.), both of these assumptions are possible.
Today's discussion opened with an announcement that today is the birthday of Girolamo Cardano. Cardano wrote the first Latin treatise devoted solely to algebra. He lectured on mathematics, astronomy, and physics, among other topics, and also achieved notoriety as a medical doctor. He was the first to publish the solution to the reduced cubic equation.
Student lecturer Kathiann House then assumed center stage. Ms. House briefly described the path followed by the social climate and the political situation in India in the first millennium of the common era and for a few centuries after that. She then described some of the influential writings in mathematics from India during this period, including Sulvasutras and Siddhantas. In English, Sulvasutras means "Rules of the Cord," and, fittingly, Sulvasutras discussed rules for construction of right triangles. The Siddhantas were books on astronomy. Ms. House added that most texts in this era were written in poetic fashion.
Ms. House then proceeded to work through a series of problems. The first involved the relationship between half of a chord of a circle and half of the angle subtended by the chord. With a bit of help from Dr. Barsky, a demonstration was given showing how this related to the sine function in trigonometry. The other problems used algorithms for finding the cubes and cube roots of numbers. I found these algorithms intriguing in their simplicity. They involve relatively easy computations and would probably be fairly easy to learn for students in pre-algebra classes.
Ms. House then briefly discussed the evolution of the place value system of numbers, and the use of base ten. She also stated that the first documented acknowledgment of negative numbers came from India.
Katz goes into more detail on these last topics. He diagrams the evolution of modern numbers on page 216, showing that back in the sixteenth century there were numerals being used that are strikingly similar to what we now call "Arabic numerals." Katz also describes rules that were set down for arithmetic with negative numbers and zero. He even quotes Bhaskara as saying that a fraction, n/0, where n is any nonzero number, "is termed an infinite quantity." So apparently they did have a concept of infinity back then.
Kathiann began her presentation by giving a brief description of India's past from the days of Alexander the Great to the 12th Century. Then she spoke of three (sets of) books that came from India which dealt with mathematics. Her mathematical or computational portion was on how to find the cube of a number and to find the cube root of a number.
The computation of finding the cube root of a number was very interesting. It made the computation very easy by allowing variables to represent the digits of the number being sought. Using the resulting polynomial from cubing the tens and ones digits of the mystery number was a fascinating idea. Then, by deducting which portions of the polynomial referred to the number's digit's placements, one could solve for the variables, and thus the number being sought.
This lecture began with a brief overview of the history of India and some of the social and cultural factors that influenced the development of mathematics in India. The caste system developed in India led to the astronomical and mathematical knowledge of the ancient Indians being handed down orally from one generation to the other, which kept their knowledge limited to the select caste of the Brahmins. In the third century BC, Alexander the Great conquered much of India, bringing with him the knowledge of the ancient Greeks. The Muslims who conquered India in the twelfth century also surely brought much of their mathematical knowledge. Since the early development of Indian mathematics was never written down, it seems that it would be hard to determine what influence outside forces had on the development and what had been passed down from the past.
Several early mathematical texts were discussed. The earliest, the Sulvasutras, or "rules of the cord", shows that they had knowledge of the Pythagorean theorem. The Surya-Siddhanta, written about the fourth century, shows the Indian's familiarity with trigonometry, and that they had developed a method of finding the sine of an angle. Two algorithms for finding the cube and the cube root of large numbers were then discussed. These algorithms come from the Aryabhatiya, written about the fifth century BC The method she demonstrated for finding the cube roots of large numbers was particularly interesting.
First Kathiann covered the history of India leading up to Medieval India. India started as warring states in the first millennium. Then they created the caste system. During this time they did not have a written language, but passed on all their astronomy and mathematics through oral tradition. Everything was done in poetic stanzas, making memorization easier. India was reconquered by Chandragupta Maurya, who was succeeded by Ashoka. It was during Ashoka's rule that the first written numbers appeared on pillars. Later the Gupta dynasty developed high culture and learning and spread Hinduism. In the 12th century the Moslem "Sultante of Delhi" was established, and astronomy was studied. There were many ancient mathematical texts written by the Indians. The three versions of the Sulvasutras dealt with rules of the chord and did geometric algebra. The five Siddhantas were systems of astronomy, which studied things like sine of an angle, trigonometry and surveying.
One of the more important texts, the Aryabhatiya, was written by Aryabhata. It covered mathematics, astronomy and cubes after 10. Kathiann showed us how to found the cubes of two digit numbers after 10, and then how to find cube roots. Aryabhata claimed "the 1st thing to do is put down the cube of the 1st digit in a row of four figures in a geometrical ratio in exact proportion subsisting between them". Next you "put down under the 2nd and 3rd numbers just two times the said numbers themselves and add them". We did the 11, 12, 13, 14, and 16. I have tried to do 26 on my own, but can not figure it out. I thought that on the top row you put the tens digit in the first box, then the ones digit in the second box, then the square of the ones digit in the third box, then the cube of the ones digit in the fourth box, but this didn't work. So I was wondering if there was a specific formula for finding the "geometrical ratio" or if it is different for each case? Finding a two digit cube root of a number uses binomial expansion of the equation (e x 10 + f)^3, where e represents the tens digit and f represents the ones digit. Then you take the first term, e^3 x 10^3, and find an e that gets you as close to the thousands part of your number. Next you plug the number you get for e into the first term and subtract it off the original number. Next you plug your e value into the second term, (3e^2 x 10^2 x f), and take that coefficient and divide it into the remainder you got previously. The whole number you get is your f value and the remainder should equal the sum of the third and fourth terms.