Hi there! Have you ever wondered how mathematicians created logarithm tables before computers existed? It‘s an incredible history spanning centuries of diligent number-crunching by people known as "computers".
In this article, we‘ll explore the life of one such computer – the statistician Dr. Alexander John Thompson – who astonishingly calculated logarithms by hand to create the most precise logarithm table ever made. Let me explain what logarithms are first and why anyone would undertake something so labour-intensive…
What are Logarithms and Why do We Care?
Simply put, logarithms are functions that transform difficult multiplications into easier additions. For example:
log(10) + log(100) = log(1000)See how the multiplication was reduced to peaceful coexistence? Such math proxies enabled essential calculations in astronomy, navigation and science for centuries.
The first logarithm tables were published in 1614 by Scottish mathematician John Napier, who relied on tedious manual computations. Then in 1624, Henry Briggs improved them to the base-10 system we still use today, storing logs for the first 20,000 numbers calculated by hand!
Over the next 300 years, scholars published increasingly precise tables – reaching 8 decimal digits of accuracy by the 1800s. But that‘s when our friend Alexander Thompson joined the fray to take it to the next level…
| Table | Creator | Year | Numbers | Accuracy |
|---|---|---|---|---|
| Arithmetica Logarithmica | Henry Briggs | 1624 | 1-20,000 | 14 decimal places |
| Vlacq‘s Tables | Adriaan Vlacq | 1628 | 1-100,000 | 10 places |
| French National Tables | French Academy | 1798 | 1-200,000 | 8 places |
| Logarithmetica Britannica | Alexander Thompson | 1934-52 | 10,000-100,000 | 20 places |
As you can see, Thompson expanded the scope while doubling the precision. But how did he pull this off without computers? That‘s the fascinating part…
From Statistics to Mathematical Tables
Born in 1885 in Essex, England, Alexander John Thompson was a bright student who studied statistics at the University of London. After graduating, he joined the British civil service analyzing census and registry data.
This work cemented Thompson‘s stats skills while exposing him to the practical need for accurate technical tables. By 1922, Thompson embarked on crafting a new gold standard logarithm reference – his quest for numeric perfection had begun!
The Ultimate Logarithm Quest
Thompson aimed to extend Briggs‘ original 1624 logarithm table from 20,000 to 100,000 numbers, calculated to a mighty 20 decimal places – dwarfing the 8 digit French table made with an army of mathematicians.
This would require calculating over 400 million digits with no electronic aids. So Thompson designed ingenious mechanical devices to churn through the numerical onslaught.
Let‘s looks at how his crazy contraption worked!
The Difference Engine: An Analog Beast for Analog Computing
Thompson‘s system exploited the fact that while logarithmic tables span millions of numbers, the gaps between successive values form smooth, well-behaved progressions. By mechanically integrating these gaps, huge calculations get disentangled into manageable chunks.
His machine consisted of 4 main sections:
1. Input Mechanism
- Set dial sliders for each number‘s leading digits
2. Integrating Unit
- Adds successive numbers entered in unit 1
3. Differencing Unit 1
- Finds difference between successive integrated sums
4. Differencing Unit 2
- Takes differences of differences from unit 3 to yield nice polynomial trends
Higher order statistical estimation techniques were then applied to accurately approximate logs from these smooth difference tables.
So rather than direct calculation, the engine "integrates" errors away by harnessing mathematical structure – differences of logs behave very nicely!
This astonishing system was completed in 1927 after almost 5 years of development. But issues still plagued Thompson…
The Publishing Odyssey
Generating tables with the engine was slow and finicky – it produced one number‘s log every 5-10 minutes! And dust or shocks could halt the fragile gears. So progress advanced in fits and starts:
- 1924: 90,000-100,000
- 1927: 80,000-90,000
- 1928: 40,000-50,000
…
Just when tables were nearing completion in 1937, WWII paper shortages choked output. Not discouraged, Thompson persisted with worn equipment until final publication in 1952 – a 30 year scientific marathon!
The Bittersweet End
At last, Thompson‘s 20-place masterpiece was complete. But by 1952 electronic computers were already threatening to usurp human computers. Yet Logarithmetica Britannica remains a towering testament to diligence and ingenuity – one person with the right tools can achieve wonders!
So next time you casually access logarithms on your smartwatch, I encourage you to pause and admire structures like Thompson‘s monumental achievement. Our technological comforts rest on efforts of such numeric heroes!
I hope you enjoyed reading about this logarithm labor of love as much as I did researching it! Let me know if you have any other questions.
