Logarithms and tables
A numerical table by which multiplication and division are reduced to addition and subtraction: the product of two numbers is equal to the sum of their logarithms.
The Scottish mathematician John Napier printed the first tables in 1614. The Swiss clockmaker Jost Burgi reached the same idea independently but published his progression tables only in 1620: many historians grant him priority in the invention but not in publication.
Kepler at once applied logarithms to the reduction of astronomical observations; the Englishman Henry Briggs together with Napier reworked them into base-ten logarithms. On this rests three centuries of computation by tables.
Why this date
exact: Napier's Mirifici logarithmorum canonis descriptio was printed in 1614; Burgi reached the idea independently (most estimates c. 1588-1600) but published his progression tables only in 1620
What it grew from
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Decimal fractions in computation
influenced · Confidence: high
Napier learned decimal notation for fractions from Stevin and used it with the decimal point in the Constructio, but the 1614 tables themselves are built on integers (whole sine 10^7), so Stevin's fractions were not a physical prerequisite of logarithms.
Dispute: The influence runs both ways: Stevin gave Napier the notation, but according to Roegel it was Napier's logarithms that became the main vehicle of the decimal idea in practice. Moreover, Clavius already used a decimal point in a table of sines in 1593 (Cajori vol. 1, p. 324; Boyer), so Napier is not the only channel from Stevin to modern notation.
Link sources
- Baron M. E. «Napier, John», Dictionary of Scientific Biography (via Encyclopedia.com) (1974) Verified 20 Sep 2026
Another achievement was the effective use he made of decimal notation (which he had learnt of from Stevin) in conjunction with the decimal point.
- O'Connor J. J., Robertson E. F. «Simon Stevin», MacTutor History of Mathematics (2004) Verified 20 Sep 2026
Stevin's notation was to be taken up by Clavius and Napier and it developed into that used today.
- Roegel D. «Napier's ideal construction of the logarithms», LOCOMAT project, LORIA (2012) Verified 20 Sep 2026
Napier was in fact the main introducer of decimal fractions into common practice and his new mathematical instrument became the best vehicle of the decimal idea.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Baron M. E. «Napier, John», Dictionary of Scientific Biography (via Encyclopedia.com) (1974) Verified 20 Sep 2026
What grew out of it
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Mathematical Principles of Natural Philosophy
influenced · Confidence: high
Napier's logarithms let Kepler reduce decades of astronomical observations into the Rudolphine Tables: Kepler calculated eight-figure logarithms and published them with the tables, and answered critics by proving how logarithms work from Euclid's Elements. These reduced observations gave Newton the empirical foundation for the theory of gravitation set out in the Principia of 1687.
Link sources
- O'Connor J. J., Robertson E. F. «John Napier», MacTutor History of Mathematics Verified 27 Sep 2026
It was through the use of logarithms that Kepler was able to reduce his observations and make his breakthrough which then in turn underpinned Newton's theory of gravitation.
- O'Connor J. J., Robertson E. F. «Johannes Kepler», MacTutor History of Mathematics Verified 27 Sep 2026
Kepler calculated tables of eight-figure logarithms, which were published with the Rudolphine Tables (Ulm, 1628).
- O'Connor J. J., Robertson E. F. «Johannes Kepler», MacTutor History of Mathematics Verified 27 Sep 2026
Kepler's answer to the second objection was to publish a proof of how logarithms worked, based on an impeccably respectable source: Euclid's Elements Book 5.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- O'Connor J. J., Robertson E. F. «John Napier», MacTutor History of Mathematics Verified 27 Sep 2026
Node sources
- Baron M. E. «Napier, John», Dictionary of Scientific Biography (via Encyclopedia.com) (1974) Verified 20 Sep 2026
Many Continental historians have accorded him priority in the actual invention, although he certainly did not have it in the publication of his Arithmetische unit geometrischie Progress-Tabulen (1620).
- O'Connor J. J., Robertson E. F. «John Napier», MacTutor History of Mathematics (1998) Verified 20 Sep 2026
Napier's discussion of logarithms appears in Mirifici logarithmorum canonis descriptio in 1614.
- O'Connor J. J., Robertson E. F. «Joost Bürgi», MacTutor History of Mathematics (2000) Verified 20 Sep 2026
Bürgi's method is different from that of Napier and was clearly invented independently.
- Roegel D. «Napier's ideal construction of the logarithms», LOCOMAT project, LORIA (2012) Verified 20 Sep 2026
only Napier and Jost Bürgi (1552-1632) constructed tables for the purpose of simplifying the calculations
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.