Key Concepts

Logarithms, Napier's logarithms, Briggs' logarithms, slide rule, base of logarithm, geometric sequence, arithmetic sequence, exponential function, scaling, linear approximation, tables of logarithms, computation, division to subtraction, multiplication to addition, powers to multiplication, roots to division.

The Fundamental Property of Logarithms

The core concept is that logarithms transform multiplication into addition and division into subtraction. This is demonstrated by selecting values on the x-axis of a curve (implicitly an exponential curve) and mapping them to corresponding y-values. For example, 2 * 4 = 8 on the x-axis corresponds to 1 + 2 = 3 on the y-axis. Similarly, 8 / 4 = 2 maps to 3 - 2 = 1. This property is the foundation for the utility of logarithms.

Historical Context: Navigation and Napier's Logarithms

The video illustrates the practical application of logarithms using a navigation problem: determining the course from Dover, England, to Dunkirk, France. Before logarithms, this involved long division and looking up values in trigonometric tables (e.g., Regiomontanus's tables). John Napier's invention of logarithms in 1614 revolutionized this process. Napier's tables allowed navigators to convert division into subtraction, significantly simplifying calculations. The example shows how dividing 5.61 by 46.6 (representing distances) is replaced by subtracting their corresponding logarithm values from Napier's table. The result is then looked up in the table to find the angle.

Napier's Method and its Impact

Napier's logarithms transformed powers into multiplication and roots into division. He spent 20 years computing these values, which were essentially the y-values of points along a curve. His initial term for them was "artificial numbers," but he settled on "logarithms," meaning "ratio numbers." The East India Company quickly adopted Napier's method, recognizing its efficiency and accuracy.

The Slide Rule: A Physical Embodiment of Logarithms

Edmund Gunter and William Oughtred created the slide rule by placing tick marks on rulers spaced according to the y-values (logarithms) of the curve but labeled with the x-values. This device allowed for rapid calculations based on the principle of adding or subtracting lengths corresponding to logarithms.

Napier's Geometric Sequence and the Definition of Logarithms

Napier defined logarithms based on the motion of two points: point B moving at a constant velocity (arithmetic sequence) and point beta slowing down in a specific way (geometric sequence). The position of point B is defined as the logarithm of the position of point beta. This means logarithms count the number of multiplications of a base number required to reach a given number.

Modern Notation and the Underlying Principle

The modern notation log(a) + log(b) = log(a * b) is presented as a way to rewrite problems in terms of common bases. The logarithms are the exponents. Division is similarly simplified using log(a / b) = log(a) - log(b).

Briggs' Base-10 Logarithms and Computational Challenges

Henry Briggs recognized the potential of Napier's logarithms but proposed using a base of 10 instead of Napier's base of 0.9999999. Base-10 logarithms simplified scaling (multiplying or dividing by 10), making them easier to work with. However, computing base-10 logarithm tables was more complex.

Briggs' Method: Repeated Square Roots and Linear Approximation

Briggs used an ingenious method involving repeated square root extractions to "zoom in" on the curve. He started with the fact that square root operations on the x-axis are equivalent to division by two on the y-axis. By repeatedly taking square roots (up to 54 times, calculated to 33 decimal places), he found a region where the curve approximated a straight line. This allowed him to use a linear formula: log(1 + r) ≈ α * r, where α is the slope of the line.

Computing the Logarithm of 2

Briggs computed the logarithm of 2 by first raising 2 to the 10th power and dividing by 10 cubed to get closer to 1 (1.024). He then took 47 square roots of 1.024 to reach a value very close to 1. Using his linear formula, he computed the logarithm of this value and worked backwards to find the logarithm of 2.

Completion of the Tables and Their Widespread Use

Briggs computed logarithms for numbers from 1 to 20,000 and 90,000 to 100,000 to 14 decimal places. Adrien Vlacq filled in the remaining values from 20,000 to 90,000. These tables became the backbone of human calculation for three centuries.

Slide Rule Mechanics: Adding Lengths, Multiplying Values

The slide rule's tick mark spacing follows the logarithmic curve. Multiplying numbers on a slide rule involves adding lengths corresponding to their logarithms. The sliding action adds the number of steps, and the logarithmic spacing converts this addition into multiplication.

The HP-35 and the End of an Era

In 1972, the HP-35 electronic calculator replaced the slide rule. HP engineers even cited Briggs' Arithmetica Logarithmica as prior art in their implementation of logarithm functions.

Conclusion

Logarithms, despite the obsolescence of tables and slide rules, remain crucial in science, mathematics, and engineering. They play a vital role in machine learning, data visualization, and other advanced applications. The video emphasizes the historical significance and enduring importance of this mathematical concept.