## Introduction

Mathematics has passed through ages and developed through people such as Archimedes, Gauss and Lagrange. These mathematicians have proved a lot and introduced many theories and equations that are applicable in day-to-day activities. There were also prominent women who surpassed their traditional roles and injustices to become world-renowned mathematicians. Even though up to the twentieth century most women didn’t have access to higher education and for those who were able were discouraged from pursuing science-based courses. But, there were few remarkable women who flourished during the eighteenth century. People like Sophie Germain changed the thinking of people like Gauss and Lagrange through their problem-solving techniques and abilities. In this paper a brief history of women mathematicians and their developments in mathematics-related world is seen. The paper will see the contribution of Sophie Germain and Maria Agnesi to the world of mathematics.

## Maria Agnesi

Maria Agnesi was born on 16 May, 1718 and died on 9 January, 1799. She was a daughter of Italian origin who was wealthy and also a mathematician. She was a mathematician, philosopher and a linguist. She is the first person and woman to write a calculus book on integral calculus and differential. She was also an honorary member in the faculty of mathematics at the University of Bologna. Maria is considered as among the first mathematicians who have contributed a lot to the modern time mathematics. When her father was ill in 1750 she chaired natural philosophy and mathematics at Bologna, after she was appointed by Pope Benedict XIV. To her honor a crater on Venus was named after her (Kramer, 1996 pp.123).

Her contributions to mathematics included the writing of the first book on calculus and the following equations.

y = 1

X² + 1 where a is a non zero constant.

All this came at a period when the Europeans didn’t think about any consideration for the education of women. She took to her consideration to educate her siblings which she then created textbook. The textbook which took approximately ten years and it is the first surviving book. It had around 1000 pages and was a two-volume with elementary and advanced mathematics. The first volume of the book talks on algebra, arithmetic, analytic geometry, calculus and trig (Kramer, 1996, pp.134-139). The second volume talks about infinite series and differential equations. She is also linked to the John Colson, Witch of Agnesi curve, which was a misname of the name ‘Persiaran.

Author C. De Brosses said of her: “It is marvelous to see a person of (Agnesi’s) age so conversant with such abstract subjects.”

## Sophie Germain

She was born in 1776 and died 1831 and was a French mathematician. She made contributions to the fields of number theory and differential geometry. At the age of 13 she read a document on Archimedes and was motivated on how someone went to an extent of being killed because of concentration on mathematics. She was interested in Lagrange’s teaching where she submitted papers using a pseudonym “Monsieur Le Blanc”.

After the teacher was impressed by the writings, the real Germain was forced to reveal herself. Using the same pseudonym in 1804 she presented a paper to Carl Friedrich Gauss after she read Gauss’s writings; Disquisitiones Arithmeticae. She later revealed herself after she learned that Gauss was at the same risk as Archimedes. Frequent letters with Gauss ended after Gauss became a professor at University of Gottingen and he shifted to applied mathematics (Larson & Robert, 2003 pp. 130-134).

Her main contribution to mathematics was x⁵ + y⁵ = z⁵ where x, y and z are integers. This she proofed that all the integers (x, y and z) were divisible by five. This solution then restricted the works of Fermat’s last theorem. She also contributed to the prime number theory 2p + 1 is also prime. Her famous formula was the “*Sophie Germain’s Identity*” and using two integers x and y, then;

x⁴ + 4y⁴ = (x² + 2y² + 2xy) (x² + 2y² – 2xy)

Example: Prove Sophie Germain’s Identity equations using positive integers.

The integers that are used are 2 and 3: x = 2 and y = 3; then,

From the equation x⁴ + 4y⁴ = (x² + 2y² + 2xy) (x² + 2y² – 2xy)

The Left side of the equation x⁴ + 4y⁴ = 2⁴ + 4(3)⁴ = 16 + 4 (81) = 340

The right side = x² + 2y² + 2xy) (x² + 2y² – 2xy = (2² + 2(3)² + 2*2*3) (2² + 2(3)² -2*2*3) = (34) (10) =340.

## Sophie Germain’s Theorem

“Let n be an odd prime. If there is an auxiliary prime p with the properties that

- xn + yn + zn = 0 mod p implies that x = 0 mod p, or y = 0 mod p, or z = 0 mod p, and
- xn = n mod p is impossible for any value of x,

Then Case I of Fermat’s Last Theorem is true for n”.

From these we see that the development of mathematics was influenced by women. Women like Maria Agnesi influenced the writing of the first calculus book after taking into consideration her responsibilities of being the firstborn in the family. She also wanted to be a philanthropist through her assistance to the church but the idea of mathematics was behind her mind. Sophie Germain on the other hand gave morale and inspired famous mathematicians; Gauss and Lagrange on her writings about calculus and proving that part of Fermat theorem was not substantial. Sophie equation on prime numbers even though there is still criticism on the truth behind it; there is no seen prove (x⁵ + y⁵

= z⁵) of this equation from her. (Mazzotti, 2007, pp. 42- 89)Through the experience of these women and their determination the mathematical view of women has changed but still requires a lot of motivation.

## Works Cited Source

Kramer, Edna. “Agnesi, Maria Gaetana”. New York: Charles Scribner’s Sons. 1996.

Larson, R., & Robert P., Calculus of a Single Variable: Early Transcendental Functions (3rd edition). New York: Houghton Mifflin Company. 2003.

Larry Riddle, Aghes Scot College, Web.

Mazzotti, Massimo. The world of Maria Gaetana Agnesi, mathematician of God. Baltimore: Johns Hopkins University Press. 2007.