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Our post on John Napier and logarithms tells how Napier invented logarithms and how he and Henry Briggs agreed on base 10. It leaves a practical question open: how do you actually work out log 2 to fourteen decimal places with a pen? Briggs did exactly that, for tens of thousands of numbers. This post is about Briggs himself and his method, which you can repeat on a calculator in five minutes.
Who was Henry Briggs?
The MacTutor biography records his birth in February 1561 at Warley Wood near Halifax, Yorkshire, and his death on 26 January 1630 in Oxford. He studied at St John's College, Cambridge, and in 1596 became the first professor of geometry at the newly founded Gresham College in London, a post he held for 23 years. In 1619 Henry Savile asked him to take the new Savilian chair of geometry at Oxford.
Before logarithms, Briggs was already a maker of tables, for finding the height of the pole (1602) and for navigation (1610). He was also working on eclipses, which meant heavy calculation. On 10 March 1615 he wrote to his friend James Ussher that Napier "hath set my head and hands a work with his new and admirable logarithms", adding: "I never saw a book which pleased me better or made me more wonder."
What did Briggs publish?
MacTutor lists *Logarithmorum Chilias Prima* (1617), the logarithms of the first thousand numbers, and *Arithmetica Logarithmica* (1624), which gave the logarithms of 1 to 20,000 and 90,000 to 100,000 to 14 decimal places. In the 1624 book he suggested that the missing numbers could be computed by a team and even offered to supply specially printed paper. Adriaan Vlacq filled the gap in 1628.
MacTutor also notes that although base-10 logarithms with log 1 = 0 first appear in print in Briggs's work, Briggs himself gave Napier credit for the idea, which came out of their discussions. Isaac Barrow later called Briggs the man who made logarithms a vital tool for all scientists.
How did Briggs compute logarithms by hand?
Denis Roegel's reconstruction of the Arithmetica Logarithmica tables (2010) explains the method. Start with log 10 = 1. Taking a square root halves the logarithm, so log √10 = 0.5, log ∜10 = 0.25, and so on. Briggs took the square root of 10 fifty-four times, working with about 32 decimal places, until the root was 1.00000 00000 00000 12781... and its logarithm 1/2⁵⁴.
He noticed that once a number is very close to 1, say 1 + x, its logarithm is almost exactly proportional to x. Dividing the last logarithm by the last x gave him the constant 0.43429 44819..., which we now write as 1/ln 10. Roegel reports that Briggs's value was correct to 16 places.
To find log 2 he used a shortcut. 2¹⁰ = 1024, so 1024/1000 = 1.024 is already close to 1. Roegel records that Briggs took 47 square roots of 1.024, five fewer than starting from 2 would have needed, and obtained log 2 correct to 19 decimal places.
Worked example: find log 2 the Briggs way
You need a calculator with a square root key. The numbers below are rounded.
Start with 1.024 and press √ ten times. You get 1.0000231609, so x = 0.0000231609.
Use log(1 + x) ≈ 0.43429 × x: 0.43429 × 0.0000231609 = 0.0000100586.
Each square root halved the logarithm, so undo ten halvings by multiplying by 2¹⁰ = 1024: log 1.024 ≈ 0.0103000.
log 1024 = log 1000 + log 1.024 = 3 + 0.0103000 = 3.0103000.
log 1024 = 10 log 2, so log 2 ≈ 0.30103. The true value is 0.30102999...
With only ten roots instead of 47 you still get five correct decimals. More roots make x smaller and the approximation in step 2 better.
What mistakes do students make with this?
Forgetting to multiply back. After k square roots, the logarithm must be multiplied by 2ᵏ. Leaving it out gives log 1.024 ≈ 0.00001, which is far too small.
Using the approximation too early. log(1 + x) ≈ 0.4343x is good only when x is tiny. With no square roots at all, it gives log 1.024 ≈ 0.0104, already wrong in the third significant figure.
Rounding x too soon. x is the small difference between two nearly equal numbers. Keep every digit your calculator shows.
The same small-x idea runs through calculus and physics; our post on the binomial approximation in physics uses it for (1 + x)ⁿ. If logarithm rules themselves still feel shaky, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing.
What can a student take from Briggs's way of working?
MacTutor describes him as modest and fair in giving credit to others. He was also patient in a way that paid off for centuries: he reduced a hard problem to one simple operation, a square root, and repeated it with care. When a calculation looks impossible, look for a step you can repeat, and check it on a case you already know.
Try it now: use the same method on 3⁴ = 81 and 81/80 = 1.0125 to estimate log 3, given log 2 = 0.30103.
