The Invention That Saved Science

Key Concepts

Logarithms, geometric progression, arithmetic progression, exponents, Rudolphine Tables, Kepler's Laws of Planetary Motion, ellipses, prosthaphaeresis, red numbers, black numbers, ratio numbers, Napier's logarithms, Briggs' logarithms, base 10 logarithms, natural logarithms, slide rule.

The Challenge of Astronomical Calculation

In the early 17th century, astronomical calculations were extremely difficult and time-consuming, hindering scientific progress. Emperor Rudolph II tasked his astronomers with creating a perfected map of the heavens, a task made arduous by the sheer volume of computations required. Scientists spent months on basic calculations like multiplication and division of seven-digit numbers.

Kepler and Tycho Brahe

Johannes Kepler, a young astronomer, sought access to Tycho Brahe's precise planetary measurements to validate his theories. Brahe, known for his meticulous data collection over 20 years, challenged Kepler to map the orbit of Mars. Mars' orbit was particularly problematic due to its complex loops and varying speed. Kepler initially estimated he could solve the problem in 8 days, but it took him years. He employed a trial-and-error approach, repeatedly guessing parameters and performing calculations, a process he repeated 70 times in a year.

The Invention of Logarithms

The invention of logarithms provided a computational power analogous to the invention of the computer in the 20th century. Logarithms allowed scientists to transform multiplication and division problems into addition and subtraction, significantly speeding up calculations.

Michael Stifel and Geometric Progressions

Michael Stifel, in his 1544 book Arithmetica Integra, demonstrated the relationship between geometric and arithmetic progressions. He created a table showing powers of two and assigned "exposed numbers" (exponents) to each value. This concept allowed multiplication to be performed by adding exponents and division by subtracting them. However, the table's usefulness was limited because it only contained exact numbers, and the spacing between numbers increased as they grew larger.

Jost Bürgi's Approach

Jost Bürgi, a skilled mechanic and mathematician, developed a more precise method for creating tables that could turn multiplication problems into addition. He used a common ratio close to one (e.g., 1.00001) to create a table with closely spaced values. He called the top numbers "red numbers" and the lower numbers "black numbers." Bürgi's method involved scaling steps, similar to scientific notation, to keep values within the table's bounds. He calculated that using a base of 1.00001 would give him up to 9 digits of accuracy.

John Napier's Logarithms

John Napier, a Scottish nobleman and amateur astronomer, independently developed logarithms. His approach involved a complex analogy of motion along two lines: one a line segment of a defined length (10 million units) where an object moves at a velocity proportional to the remaining distance, and the other an infinitely long ray where an object moves at a constant speed. Napier showed that equal ratios on the first line corresponded to equal distances on the second line, thus turning multiplication into addition. He called his system "ratio numbers" or "logarithms."

Napier's Tables and Their Impact

Napier created tables of logarithms of sine values, tailored to the needs of astronomers and navigators. His tables had one entry per arc minute from 0 to 90 degrees (5,400 entries). The East India Company commissioned an English translation of Napier's logs for its navigators, marking the beginning of the age of the logarithm.

Kepler's Use of Logarithms

Kepler obtained Napier's logarithms around 1617 and found them to be a revelation. They helped him discover patterns in the universe, particularly the relationship between a planet's orbital period and its distance from the sun. By taking the logarithms of these values, Kepler found a simple linear relationship: the square of the orbital period is proportional to the cube of the semi-major axis. This became Kepler's third law of planetary motion, published in 1619.

Henry Briggs and Base 10 Logarithms

Henry Briggs, an English mathematician, improved Napier's logarithms by setting the logarithm of 10 equal to one, creating base 10 logarithms. This simplified calculations by making each integer part of the logarithm value correspond to a shift of the decimal point. Briggs collaborated with Napier on this improvement, and Briggs did the new calculations himself over the course of the next 8 years.

The Rudolphine Tables

Kepler used logarithms to complete the Rudolphine Tables, which were published in 1627. The tables were 30 times more accurate than previous tables and could be used to calculate planetary positions from 4000 BC to years into the future. They became the defining resource for astronomers and navigators for the next century.

Legacy of Logarithms

Logarithm tables and slide rules were used for calculations by mathematicians and scientists for over 350 years, until the introduction of the handheld calculator in 1972. The computational algorithms used in early calculators were based on the same methods that Briggs had used to calculate base 10 logs.

Conclusion

The invention of logarithms by Napier and Bürgi, and their subsequent refinement by Briggs, revolutionized scientific calculation and enabled breakthroughs in astronomy and other fields. Kepler's use of logarithms to discover his third law of planetary motion and complete the Rudolphine Tables exemplifies the transformative power of this mathematical tool. Logarithms remained essential for scientific and engineering calculations for centuries, leaving a lasting legacy in the history of science and technology.