The Magnificent History of Indian Mathematics

The Magnificent History of Indian Mathematics: A Timeless Legacy

Introduction

"Indian mathematics" is a cornerstone of global intellectual heritage, with its innovations spanning thousands of years. From the "Vedic period" to the modern age, Indian scholars have pioneered revolutionary concepts that laid the foundation for many scientific advancements. This article delves into the rich history of Indian mathematics, exploring its contributions, key figures, and its impact on the modern world.

1. The Foundations: Mathematics in the Vedic Period

• Vedic Texts and Numeracy
  The ancient Vedic texts (Rigveda, Yajurveda, Atharvaveda) showcase early mathematical applications. Rituals and constructions relied on geometric precision, marking the origin of mathematical thought in India.
• Shulba Sutras: Ancient Geometry
  The Shulba Sutras, a part of the Vedas, contained rules for constructing altars and fire pits, emphasizing geometry. They introduced principles of the Pythagorean theorem long before its discovery in Greece.

2. The Invention of Zero: A Revolution in Mathematics

• Aryabhata’s Contributions
  Aryabhata, the father of Indian mathematics, developed the concept of zero as a placeholder. His work, Aryabhatiya, revolutionized numerical systems.

Aryabhata: The Pioneer of Indian Mathematics and Astronomy

Aryabhata (476 CE–550 CE) is one of the greatest mathematicians and astronomers of ancient India. His groundbreaking contributions laid the foundation for modern mathematics and astronomy, earning him a place among the most influential scholars in history. Born during the classical age of Indian learning, Aryabhata's works have inspired generations of mathematicians and scientists across the globe.

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Early Life and Background

Aryabhata was born in 476 CE in a region that is today part of Bihar, India, possibly in the city of Kusumapura (modern-day Patna). Some accounts also associate his birthplace with Kerala. He studied at the Nalanda University, one of the world's oldest centers of higher learning. Aryabhata was deeply influenced by the rich intellectual tradition of ancient India, which emphasized mathematics and astronomy for practical purposes like agriculture, navigation, and timekeeping.

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Major Works

Aryabhata's most famous work is the Aryabhatiya, a compendium of mathematics and astronomy written in Sanskrit. Composed in 499 CE, this seminal text is divided into four sections:

1. Gitikapada: This section deals with cosmology and explains time measurement, including the length of a day, month, and year.
2. Ganitapada: Focused on mathematics, this section introduces arithmetic, algebra, and geometry, including innovative methods for solving problems.
3. Kalakriyapada: This section discusses the concept of time and planetary motion.
4. Golapada: Focused on spherical astronomy, it explains the movements of celestial bodies.

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Contributions to Mathematics

Aryabhata's mathematical achievements include:

1. The Concept of Zero
   While Aryabhata did not explicitly use zero as a symbol, his works laid the foundation for its formal development. He used a positional number system, which depended on the concept of zero as a placeholder.

2. Decimal System
   Aryabhata played a key role in the development of the decimal numeral system. His system of numeration greatly influenced modern arithmetic.

3. Approximation of Pi ()
   Aryabhata calculated the value of as 3.1416, remarkably close to the actual value. He described it as the ratio of the circumference of a circle to its diameter, stating that it was an irrational number.

4. Algebra and Trigonometry
   Aryabhata introduced innovative methods in algebra, including solutions to quadratic equations. He also pioneered trigonometry by defining the sine function (jya in Sanskrit) and providing sine tables for practical use.

5. Area and Volume Formulas
   He formulated equations for calculating the areas of triangles and circles, as well as the volumes of spheres and other geometric shapes.

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Contributions to Astronomy

Aryabhata’s astronomical theories were groundbreaking and far ahead of their time:

1. Heliocentric Model
   Aryabhata proposed that the Earth rotates on its axis, challenging the prevailing geocentric view. He explained that the apparent movement of stars is due to the Earth's rotation.

2. Eclipse Predictions
   Aryabhata accurately described the causes of solar and lunar eclipses, stating that they occur due to the shadows cast by the Earth and the Moon, respectively. This explanation was revolutionary, as it rejected superstitious beliefs.

3. Planetary Motion
   He developed mathematical models to explain the elliptical orbits of planets and their movements relative to the Earth.

4. Length of a Year
   Aryabhata calculated the length of a solar year as 365.358 days, remarkably close to the modern value of 365.256 days.

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Legacy and Global Influence

Aryabhata's contributions had a lasting impact on both Indian and global science:

• His works influenced later Indian mathematicians like Brahmagupta, Bhaskara I, and Bhaskara II.
• Arab scholars translated Aryabhatiya into Arabic, spreading his ideas to the Islamic world and eventually Europe.
• His work formed the basis for advancements in navigation, astronomy, and mathematics in the medieval period.

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Commemorations

• The first Indian satellite, launched in 1975, was named Aryabhata in his honor.
• Numerous institutions, awards, and landmarks in India bear his name, celebrating his contributions to science and learning.

• Brahmagupta’s Definition
  Brahmagupta was the first to treat zero as a number and provided rules for its arithmetic operations in his seminal text, Brahmasphutasiddhanta. This innovation transformed global mathematics.

Brahmagupta: The Trailblazer of Indian Mathematics and Astronomy

Brahmagupta (598 CE–668 CE) was a renowned Indian mathematician and astronomer whose pioneering work laid the foundation for many key mathematical concepts. He is best known for formalizing the rules of arithmetic operations involving zero, which revolutionized mathematics. His contributions to algebra, geometry, and astronomy have made him a celebrated figure in the history of science.

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Early Life and Background

Brahmagupta was born in 598 CE in the region of Bhillamala (modern-day Bhinmal in Rajasthan, India). He lived during the Gupta dynasty, a golden age of Indian culture and science. As the head of the astronomical observatory in Ujjain, a prominent center for learning and research, Brahmagupta had access to a wealth of knowledge that shaped his work.

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Major Works

Brahmagupta’s most influential works are Brahmasphutasiddhanta (The Opening of the Universe) and Khandakhadyaka. These texts cover a wide range of topics, including mathematics, astronomy, and geometry.

1. Brahmasphutasiddhanta (628 CE)

This monumental text is divided into 25 chapters and includes mathematical principles, astronomical observations, and methods for calculations.

It is particularly famous for introducing the concept of zero as a number and defining rules for arithmetic operations involving zero.

2. Khandakhadyaka

This treatise focuses on astronomical calculations and the use of trigonometry to predict celestial events.

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Contributions to Mathematics

1. The Formalization of Zero

Brahmagupta was the first to define zero as a number and provide rules for its arithmetic operations, including addition, subtraction, multiplication, and division.

For example, he stated that the result of subtracting a number from itself is zero and described how zero interacts with other numbers in calculations.

2. Algebra

Brahmagupta introduced methods for solving quadratic equations, including both positive and negative solutions.

He provided formulas for solving indeterminate equations, a significant contribution to algebra.

3. Geometry

He developed formulas for the area and perimeter of geometric shapes.

Brahmagupta's famous formula for the area of a cyclic quadrilateral (a four-sided figure inscribed in a circle) is:

A = \sqrt{(s-a)(s-b)(s-c)(s-d)}

4. Number Theory

Brahmagupta worked extensively on properties of integers, including concepts related to divisors, multiplication, and the use of negative numbers.

5. Arithmetic Progression

He provided formulas for the sum of squares and cubes of the first natural numbers, which are foundational in arithmetic progressions.

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Contributions to Astronomy

1. Planetary Motion and Eclipses

Brahmagupta developed methods to calculate the positions of planets and predict solar and lunar eclipses.

His work included corrections to earlier astronomical calculations, enhancing the accuracy of celestial predictions.

2. Earth’s Shape and Gravity

He described the Earth as a sphere and acknowledged the force of gravity, stating that objects fall due to a natural attraction.

3. Calendars

Brahmagupta’s work was instrumental in refining the Hindu calendar. He proposed methods for determining the length of a year and calculating planetary conjunctions.

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Philosophical Insights

Brahmagupta combined mathematics with philosophy, often blending logical reasoning with metaphysical ideas. His ability to interweave scientific precision with abstract thought highlights the depth of his intellectual pursuits.

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Legacy and Influence

1. Impact on Indian Mathematics

Brahmagupta’s contributions influenced later mathematicians like Bhaskara I and Bhaskara II. His texts became foundational learning material for centuries.

2. Global Reach

Arab scholars translated Brahmagupta’s works into Arabic, introducing his ideas to the Islamic world. From there, they reached Europe, significantly influencing Western mathematics and astronomy.

3. Modern Recognition

His formulae and methods are still taught in mathematics today, reflecting their timeless relevance.

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Commemorations

Several Indian institutions and awards are named after Brahmagupta, celebrating his contributions to science and mathematics.

His pioneering work on zero is often highlighted as one of the greatest contributions to global mathematics.

3. Indian Numeral System and Its Global Impact

• Decimal System Development
  The decimal numeral system, integral to modern arithmetic, was first developed in India.
• Spread to the West
  Through Arab scholars, the "Hindu-Arabic numeral system" reached Europe, becoming the universal standard.

4. Pioneers of Algebra

• Bhaskara I’s Achievements
  Bhaskara I built on Aryabhata’s work, introducing solutions for quadratic equations.
• Bhaskara II’s Lilavati
  Bhaskara II’s masterpiece Lilavati combined poetry and mathematics, offering elegant problem-solving techniques.

5. Trigonometry and Astronomical Applications

• Aryabhata’s Sine Tables
  Aryabhata introduced the sine function (jya), foundational for modern trigonometry.
• Kerala School’s Advances
  Madhava of Sangamagrama developed infinite series expansions for sine and cosine functions, predating European calculus.

6. Mathematics in Astronomy

• Eclipse Predictions
  Aryabhata and Varahamihira provided accurate methods for predicting solar and lunar eclipses.
• Calendars and Timekeeping
  Indian mathematicians created precise calendars, vital for agriculture and rituals.

7. The Kerala School of Mathematics

Kerala School of Mathematics: The Pioneers of Calculus and Astronomy

The Kerala School of Mathematics, flourishing between the 14th and 16th centuries CE, was a prominent group of mathematicians and astronomers based in the Indian state of Kerala. This school made groundbreaking contributions to calculus, trigonometry, and astronomy, centuries before similar developments occurred in Europe. The Kerala School stands as a testament to India’s rich scientific heritage and its profound influence on global mathematical advancements.

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Origins and Background

The Kerala School of Mathematics traces its roots to the scholarly tradition of Vedic mathematics, which emphasized precision and innovation. The school's most prominent figure, Madhava of Sangamagrama (c. 1340–1425 CE), is regarded as the founder of this intellectual movement. Located in a thriving cultural and trade hub, the Kerala School benefited from interactions with diverse civilizations, which enriched its research and ideas.

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Major Mathematicians and Their Contributions

1. Madhava of Sangamagrama
   • Madhava is considered the father of the Kerala School. He made revolutionary advances in trigonometry and calculus.
   • Infinite Series: He developed infinite series expansions for trigonometric functions like sine, cosine, and tangent.
     For example:

     \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots

2. Nilakantha Somayaji (1444–1544 CE)

   • Nilakantha expanded on Madhava’s work in trigonometry and astronomy.
   • Aryabhatiya Bhashya: He wrote a commentary on Aryabhata’s work, introducing the “Nilakantha model” for planetary orbits, which came close to modern heliocentric theories.
   • Tantrasamgraha: This text provided detailed explanations of planetary motions and improved mathematical techniques for astronomy.

3. Jyeshthadeva

   • Jyeshthadeva compiled much of the school’s knowledge in his seminal work, Yuktibhāṣā (The Rational Discourse).
   • This was the first text to systematically describe the concepts of calculus, including differentiation and integration, with rigorous proofs.

4. Parameshvara (1360–1455 CE)

   • He contributed to observational astronomy and devised correction techniques for astronomical calculations.

5. Achyuta Pisharati

   • He refined earlier works on planetary motion and made contributions to spherical trigonometry.

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Key Contributions

1. Development of Calculus

   • The Kerala School introduced fundamental ideas of calculus, including:
     • Infinite series expansions for functions like sine, cosine, and arctangent.
     • Preliminary concepts of differentiation and integration to calculate areas under curves and rates of change.
   • These ideas were developed over 200 years before similar concepts were formalized in Europe by Newton and Leibniz.

2. Advances in Trigonometry

   • The Kerala School developed highly accurate sine and cosine tables and explored their infinite series representations.
   • They introduced the concept of "versed sine" (), which simplified astronomical calculations.

3. Astronomical Insights

   • The Kerala School created highly accurate models of planetary motion, eclipses, and conjunctions.
   • Their calculations of planetary positions were based on observations and mathematical rigor, surpassing many earlier methods.

4. Applications of Mathematics

   • Their work extended to practical applications, including calendar reform, timekeeping, and navigation.

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Legacy and Global Influence

1. Anticipation of European Calculus

   • The Kerala School’s ideas on infinite series and calculus predate European advancements by centuries.
   • Historians suggest that their works may have influenced European mathematicians indirectly through the transmission of knowledge via trade and colonial connections.

2. Preservation of Knowledge

   • Despite their groundbreaking work, the Kerala School’s contributions remained largely unknown outside India until the 20th century, when scholars began analyzing texts like Yuktibhāṣā.

3. Modern Recognition

   • The Kerala School is now celebrated for its pioneering spirit and for advancing mathematical thought in a way that bridges ancient and modern science.

• Calculus Precursors
  Scholars like Madhava anticipated concepts of differentiation and integration centuries before Newton and Leibniz.
• Infinite Series
  The Kerala School explored infinite series for pi, sine, and cosine, demonstrating remarkable foresight.

8. Geometry and Architecture

• Temple Construction
  Mathematical precision guided the construction of temples, demonstrating an intricate understanding of geometry.

Urban Planning
Ancient cities like Mohenjo-Daro showcased mathematical planning in their layout.

Mohenjo-daro: Contributions to Mathematics

Mohenjo-daro, a key city of the ancient Indus Valley Civilization (c. 2600–1900 BCE), represents one of the earliest centers of urban planning and engineering in human history. The mathematical understanding displayed in the construction, measurement systems, and urban design of Mohenjo-daro reflects the advanced knowledge of the civilization's people. While direct mathematical texts or manuscripts from this period have not been discovered, archaeological evidence reveals significant contributions to the field of mathematics.

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1. Standardized Measurement System

One of the most remarkable mathematical contributions of Mohenjo-daro is its standardized system of weights and measures.

• Weights: Archaeologists have unearthed carefully shaped and polished stone weights in binary ratios (e.g., 1:2:4:8:16) and decimal multiples. These weights demonstrate an early understanding of proportionality and systematic measurement.
• Scales and Cubes: Balances and uniform cube-like objects were likely used for trade, ensuring fairness and consistency in commercial transactions.
• Accuracy: The precision of these weights suggests advanced mathematical skills and a deep understanding of ratios and proportions.

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2. Urban Planning and Geometry

Mohenjo-daro's urban layout reflects sophisticated geometric principles:

• Grid System: The city was designed on a grid pattern, with streets running at right angles. This reveals knowledge of geometry and spatial organization.
• Uniformity of Bricks: The baked bricks used in construction had standard dimensions (ratio approximately 1:2:4), ensuring structural stability. This consistent ratio indicates a clear understanding of geometry and arithmetic.
• Proportions in Buildings: The sizes of houses and public structures were proportionally planned, indicating the use of mathematical calculations in architecture.

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3. Advanced Drainage Systems

The drainage systems of Mohenjo-daro, celebrated as a marvel of ancient engineering, also exhibit mathematical precision:

• Slope Calculations: The drains were designed with gradients to allow the smooth flow of water, implying an understanding of slope and measurement.
• Uniformity: The widths and depths of drainage channels were consistent, showcasing an organized approach to civil engineering.

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4. Proto-Mathematical Symbols and Trade

Seals and inscriptions found at Mohenjo-daro suggest the use of proto-mathematical symbols:

• Numerical Notation: Some seals contain linear marks that may represent numerical symbols or units used for accounting and trade.
• Commerce and Mathematics: The thriving trade network of the Indus Valley likely required arithmetic for inventory management, trade agreements, and taxation.

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5. Astronomy and Calendrical Knowledge

Although direct evidence of astronomical studies is limited, the city's alignment and potential use of early calendars suggest some awareness of celestial movements:

• Seasonal Planning: The city's agriculture-based economy likely depended on mathematical knowledge of seasonal cycles, aiding in the development of rudimentary calendars.

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Legacy and Influence

The mathematical practices of Mohenjo-daro laid the foundation for later developments in Indian mathematics:

• The standardization seen in weights and measures influenced subsequent civilizations in the Indian subcontinent.
• The Indus Valley's emphasis on geometry and proportionality likely informed the architectural and mathematical advancements of later periods, including the Vedic era.

9. Mathematics in Jain and Buddhist Traditions

• Jain Mathematics
  Jain scholars explored infinity, permutations, and combinations in their texts.
• Buddhist Influence
  Buddhist mathematicians contributed to number theory and geometry, enriching Indian mathematical thought.

10. Legacy of Indian Mathematics in Modern Times

• Global Recognition
  India’s mathematical contributions form the backbone of modern computing and engineering.
• Inspiration for Today
  Indian mathematicians continue to inspire researchers worldwide, emphasizing the timeless relevance of this heritage.

11. Contributions of Women in Indian Mathematics

• Unrecognized Scholars
  Women in ancient India often contributed anonymously, yet their impact resonates through mathematical texts.

Modern Trailblazers
Mathematicians like Shakuntala Devi highlight the ongoing legacy of women in the field.

Shakuntala Devi: The Mathematical Marvel

Shakuntala Devi (1929–2013), celebrated worldwide as the "Human Computer," was an Indian prodigy whose unparalleled mental calculation abilities left an indelible mark on mathematics. Her incredible speed and accuracy in solving complex numerical problems without the aid of any mechanical devices set her apart as a global phenomenon. Beyond her mathematical genius, she was an author, thinker, and pioneer in various fields, making her a multi-faceted personality.

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Early Years and Discovery of Genius

Shakuntala Devi was born on November 4, 1929, in Bengaluru, India, into a modest family. Her father, who worked in a circus, realized her extraordinary talent for numbers when she began solving arithmetic problems effortlessly at the age of three.

Despite limited formal education due to financial hardships, her innate ability to understand and manipulate numbers blossomed. By the age of six, she was performing complex calculations and entertaining audiences with her mathematical feats.

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Mathematical Brilliance

Shakuntala Devi earned the nickname "Human Computer" due to her ability to perform mental calculations that outpaced modern machines.

1. Record-Breaking Feats

   • In 1980, she achieved global fame when she multiplied two 13-digit numbers— and —in just 28 seconds. This incredible feat earned her a place in the Guinness Book of World Records.
   • She was also known for her quick solutions to intricate problems in various fields of mathematics, such as cube roots, powers, and algebraic expressions.

2. Demonstrations of Talent

   • Shakuntala Devi performed around the world, showcasing her mathematical prowess at institutions such as MIT, Stanford University, and the University of Rome.
   • Her demonstrations involved solving highly complex problems posed by professors and students, often leaving audiences astonished.

3. Deep Connection with Numbers

   • Unlike many mental calculators, Shakuntala Devi didn’t just rely on rote memorization. Her connection with numbers was intuitive, and she often described solving problems as a form of art.

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Contributions Beyond Mental Calculation

1. Popularizing Mathematics

   • Shakuntala Devi’s passion for mathematics went beyond personal accomplishments. She authored several books aimed at making math engaging and accessible, including:
     • Figuring the Joy of Numbers
     • Puzzles to Puzzle You
     • Mathability: Awaken the Math Genius in Your Child
   • Through her works, she aimed to demystify mathematics and encourage learners to embrace it as a fun and creative subject.

2. Exploration of Astrology

   • Shakuntala Devi had a keen interest in astrology and wrote extensively about it. Her knowledge in this domain demonstrated her intellectual versatility.

3. Social Advocacy

   • In 1977, she authored The World of Homosexuals, the first book in India addressing the subject of homosexuality. Her progressive views on the LGBTQ+ community showcased her commitment to inclusivity and equality.

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Global Recognition

1. Accolades and Acknowledgment

   • Shakuntala Devi’s talent was recognized globally, and she was invited to perform in several countries, including the United States, Europe, and Asia.
   • Her achievements were widely documented, and her name became synonymous with mental mathematics.

2. Media and Cultural Impact

   • In 2020, a biographical movie titled Shakuntala Devi was released, with Vidya Balan portraying her. The film highlighted her journey, her passion for numbers, and the challenges she faced in balancing her professional and personal life.

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Inspiring Philosophy

Shakuntala Devi believed that mathematics was not a subject to fear but a tool to explore and enjoy. She often emphasized that everyone has the potential to excel in math with practice, curiosity, and the right mindset. Her optimistic and approachable perspective continues to motivate learners to overcome their fear of numbers.

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Legacy

1. Champion of Mathematics

   • Shakuntala Devi’s efforts to popularize mathematics have inspired generations of students and educators.
   • Her methods and books remain a source of learning and enjoyment for math enthusiasts.

2. Role Model for Women

   • As a self-taught mathematician in a male-dominated field, Shakuntala Devi broke barriers and became a role model for women aspiring to make their mark in STEM fields.

3. Timeless Achievements

   • Her Guinness World Record and her contributions to literature, mathematics, and social thought continue to be celebrated globally.

12. Indian Mathematics: A Cultural Perspective

• Integration with Philosophy
  Mathematics in India was deeply intertwined with philosophy and spirituality.
• Mathematical Stories
  Texts like Lilavati used storytelling to teach mathematical concepts, showcasing India’s unique approach.

13. The Future of Indian Mathematics

• Reconnecting with Ancient Wisdom
  Modern education systems are revisiting ancient Indian mathematical texts for inspiration.
• Technological Innovations
  The principles of Indian mathematics are finding new applications in artificial intelligence and data science.

Conclusion: A Legacy That Inspires the World

Indian mathematics is more than a historical achievement; it is a living legacy that continues to inspire innovation. From zero to calculus, the contributions of Indian scholars are a testament to their intellectual brilliance. By preserving and celebrating this heritage, we can draw lessons for future advancements in science and mathematics.!