Log Tables (extra bit) - Numberphile

Key Concepts

• Logarithms (base 10 and other bases)
• John Napier's method for creating log tables
• Base of logarithms (Napier's base vs. base 10)
• Iterative subtraction method
• Error accumulation in manual calculations
• Charles Babbage's Difference Engine

John Napier's Method for Creating Log Tables

The video explains how John Napier created the first logarithm tables between 1594 and 1614, before the advent of calculus and the work of mathematicians like Newton, Leibniz, and Gauss. Napier's method involved significant manual calculation.

Base of Logarithms

• While modern log tables often use base 10, Napier's original method used a different base.
• The speaker uses an example with a base derived from the number a = 1 - 0.1 = 0.9 to the power of 10 to illustrate the process.

Iterative Subtraction Method

• Napier's method involved iterative subtraction to calculate values for the table.
• Example:
  • a^0.1 = 1 - 0.1 = 0.9
  • a^0.2 = 0.9 - (0.9 * 0.1) = 0.81
  • a^0.3 = 0.81 - (0.81 * 0.1) = 0.729
• This process is repeated to build up the table.

Napier's Actual Base

• Napier used the number 1 - (1/10⁷) to the power of 10⁷ as his base for greater accuracy.
• This involved much more complex calculations than the simplified example.

Transition to Base 10 and Briggs' Contribution

• Henry Briggs recognized that logarithms to base e (or Napier's base) could be converted to base 10.
• This made the tables more accessible and commercially viable since people were more familiar with base 10.
• Briggs created a simple transformation to convert from Napier's base to base 10, which helped popularize the use of logarithms.

Error Accumulation and Babbage's Difference Engine

Errors in Manual Calculation

• Performing long calculations with many decimal places (e.g., 8 or more) leads to errors due to manual subtraction and truncation.
• These errors accumulate over time, affecting the accuracy of the tables.

Impact on Scientific Applications

• Inaccurate log tables could lead to discrepancies between theoretical predictions and experimental results in fields like astronomy.
• Astronomers relied on these tables for accurate measurements and calculations.

Babbage's Difference Engine

• Charles Babbage attempted to build a "Difference Engine" to automate the calculation of mathematical tables and eliminate errors.
• The goal was to ensure the accuracy of the tables used in scientific and engineering applications.
• Babbage was ultimately unsuccessful in completing the Difference Engine.

Synthesis/Conclusion

The video details the laborious process John Napier undertook to create the first logarithm tables using iterative subtraction with a base different from the now-common base 10. While Napier did the foundational work, Briggs' conversion to base 10 made the tables more practical. The video also highlights the challenges of manual calculation, including error accumulation, and the motivation behind Babbage's Difference Engine to automate the process and improve accuracy. The key takeaway is an appreciation for the immense effort required to create these tables and the importance of accurate mathematical tools for scientific advancement.