Chapter Plan for the 18th Century: The History of Numbers
Introduction
- Overview of the 18th Century in Mathematics
- Begin with a broad perspective on how the 18th century served as a bridge between the Scientific Revolution and the Enlightenment in mathematics.
- Highlight the prolific period of mathematical discoveries with emphasis on the key figures such as Newton, Leibniz, Euler, Gauss, Lagrange, Bernoulli, Taylor, and Maclaurin, and their collective impact on advancing calculus, complex numbers, number theory, and the acceptance of theoretical mathematics.
The Evolution of Calculus
- Controversy and Contributions of Newton and Leibniz
- Detail the emergence of calculus in the 17th century, leading to the early 18th-century controversy over its invention.
- Explore the investigation by the Royal Society, the bias accusations, and the eventual recognition of both mathematicians’ contributions.
- Discuss the practical applications of calculus in engineering and its significance in the Industrial Revolution, alongside its reliance on the Cartesian coordinate system.
Complex Numbers: From Scepticism to Acceptance
- De Moivre, Euler, and Gauss’s Contributions
- Narrate the initial resistance to complex numbers due to their abstract nature, leading to their formal acceptance.
- Examine Euler’s formalisation of the imaginary unit () and his groundbreaking formula linking exponential functions with trigonometry.
- Conclude with Gauss’s pivotal proof in 1799 that established the foundational importance of complex numbers in polynomial equations.
Euler’s Pioneering Work in Number Theory
- Theorems and the Introduction of the Totient Function
- Detail Euler’s substantial contributions to number theory, including his work on the infinity of primes, the introduction of the totient function, and generalisation of Fermat’s Little Theorem.
- Highlight Euler’s diverse interests and achievements in number theory, laying the groundwork for future developments.
The Systematic Study of Numbers: Lagrange’s Influence
- Foundations of Modern Number Theory
- Discuss Lagrange’s methodical approach to number theory, marking a departure from empirical methods to analytical rigour.
- Focus on his proof of the four-square theorem and contributions to Diophantine equations and the theory of quadratic forms, illustrating the move towards modern algebraic number theory.
Rational and Beyond: The Acceptance of New Numbers
- Irrational and Transcendental Numbers
- Trace the historical resistance to irrational numbers from the Pythagoreans to their gradual acceptance.
- Introduce transcendental numbers and the challenges they posed to traditional mathematical understanding, highlighting early studies and the proof of their existence.
Infinite Series and the Concept of Convergence
- Significant Contributions and Formalisation
- Delve into the formalisation of infinite series, focusing on Euler’s solutions to the Basel problem and the introduction of Taylor and Maclaurin series.
- Discuss the implications of these developments on the broader mathematical landscape.
Evolution of Mathematical Notation and Technological Impact
- Standardisation of Symbols and the Role of the Printing Press
- Review the evolution of mathematical notation towards a more standardised language facilitated by Euler and others.
- Analyse the impact of improved communication among mathematicians and the role of the printing press in disseminating new ideas and notations efficiently.
Conclusion
- Synthesis and Outlook
- Synthesise the key developments of the 18th century, reflecting on how they set the stage for future advancements in mathematics.
- Consider the lasting impact of the century’s contributions to the history of numbers and the continued evolution of mathematical thought.
This chapter plan aims to provide a comprehensive and detailed exploration of the pivotal 18th century in the history of mathematics, focusing on the evolution of ideas, the resolution of controversies, and the significant contributions of the century’s foremost mathematicians.