### By Jack Lynch

Napier, John. Mirifici logarithmorum canonis descriptio ejusque usus, in utraque trigonometria; ut etiam in omni logistica mathematica, amplissimi, facillimi, & expeditissimi explicatio: Authore ac inventore, Ioanne Nepero, Barone Merchistonii, &c. Scoto. Edinburgh, 1614.

John Napier discovered the logarithm — at least, he was one of several in the early seventeenth century to understand the principles behind logarithms, and the first to publish the fruits of his research in Mirifici logarithmorum.

It's not easy to understand what Napier is saying, and that's not only because

(a) it's about logarithms and

(b) it's in Latin.

No; we also have to reckon with the fact that (c) his definitions aren't at all intuitive for those who've learned modern definitions of exponentiation and logarithms. His definition, for instance, is geometric, not algebraic:

Linea æqualiter crescere dicitur, quum punctus eam describens, æqualibus momentia per æqualia intervalla progreditur.

It comes with a diagram.

And as one historian describes it:

Logarithms, as Napier first understood them, and even logarithms in the later form agreed upon by him and Briggs, did not appear to their inventor in the light in which we now regard them. The modern exponential notation an was not yet invented, and it was not for more than a hundred years that the idea of a logarithm as the index of the power of the base found a place in works on algebra. Indeed, in the original system of Napier, there is no mention of a base system at all; and in the modified and improved system, though as a matter of fact it does in a sense consist of logarithms to the base 10, no stress is laid upon that point. (Carslaw, “The Discovery of Logarithms,” p. 77)

He also doesn't use the base-10 logarithms that are familiar to those who care about such things today. (Shortly after he published his book, he realized base-10 logs were a better idea, and he published several follow-ups.)

After a theoretical discussion of this new kind of number, he devotes eighty-eight pages to his table (see picture).

It seems exceedingly obscure, but Napier's discovery in pure mathematics was absolutely crucial for advances in applied math for centuries to come. As Edmund Wingate put it in Logarithmotechnia; or, The Construction, and Use of the Logarithmeticall Tables (1635):

For as much as amongst many inventions, that concerne the Mathematicks, none can be found comparable to this of the Logarithmes, the worthy labours of those Learned men which have endevoured [sic] to advance it, are to be prized accordingly.

Pierre-Simon Laplace, the nineteenth-century French mathematician and astronomer, marveled at this “admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.” A nineteenth-century book on logarithms makes their labor-saving value clear: “This method,” James Mill Peirce writes, “has contributed very powerfully to the modern advance of science, and especially of astronomy, by facilitating the laborious calculations without which that advance could not have been made.”

The Mirifici gives the logs of integers up to 1,000 to eight decimal places; he left it to his successor, Henry Briggs, to work out 30,000 more logarithms to fourteen decimal places: if you're dying to read that, ask your local bookseller to order Arithmetica logarithmica sive Logarithmorum chiliades triginta, pro numeris naturali serie crescentibus ab unitate ad 20,000: et a 90,000 ad 100,000 Quorum ope multa perficiuntur arithmetica problemata et geometrica. Hos numeros primus invenit clarissimus vir Iohannes Neperus baro Merchistonij: eos autem ex eiusdem sententia mutavit, eorumque ortum et vsum illustravit Henricus Briggius, in celeberrima Academia Oxoniensi geometriae professor Savilianus (1624).

Or you can wait for the movie.

The best short introduction is an article: H. S. Carslaw, “The Discovery of Logarithms by Napier,” The Mathematical Gazette 8, no. 117 (May 1915): 76–84; no. 118 (July 1915): 115–19. Even more accessible is Jack Oliver, “The Birth of Logarithms,” Mathematics in School29, no. 5 (Nov. 2000): 9–13. Serious, hard-core geeks, though, will want to work their way through the whole of M. Campbell-Kelly, M. Croarken, R. Flood, and E. Robson, eds., The History of Mathematical Tables: From Sumer to Spreadsheets (Oxford: Oxford Univ. Press, 2003). I've not only read it, but have taken ten single-spaced typed pages of notes, just to ensure my nerd credentials are beyond question.

(Posted on You Could Look It Up, presented here by permission of the author.)

### You are interested in rare books on mathematics?

(Posted on You Could Look It Up, presented here by permission of the author.)