First conceived as a technological aid to astronomers by Napier, the logarithmic tables were a kind of printable analog computer. The logarithms are now 400 years old, and this is a good time to ponder about their role in the history of science.

It was not until 1614 that Napier’s first work on this subject, Mirifia logarithmorum canonis descriptio (known as the Descriptio), was published. In addition to tables of logarithms the Descriptio also contains an account of the nature of logarithms and a number of examples explaining their use. The East India Company was so impressed by Napier’s Descriptio that it asked Edward Wright, a Cambridge mathematician and expert in navigation, to translate it into English for the benefit of the Company’s seafarers. From the very beginning of logarithms their utility to navigators has been of supreme importance in their development. (Graham Jagger, The Making of Logarithmic Tables)

The early tables by Napier, Briggs and others depended heavily on the book and printing press as a mechanical technology, and this device would later be fashioned into slide rules.

While the tables certainly made calculation simpler, their own construction - to begin with, was extremely difficult.

Laying Down The Tables

The conceptual idea behind the tables was, for any calculation - replace the process of multiplication by addition, which was much simpler for a human being. This is possible because, recall the law of logarithms: log (a*b) = log(a) + log(b). You just compute the sum on the right and look up the tables to arrive at the value for a*b.

Graham Jagger notes in the same essay, “Realising that astronomical and navigational calculations involved primarily trigonometrical functions, especially sines, Napier set out to construct a table by which multiplication of these sines could be replaced by addition: the tables in the Descriptio are of logarithms of sines. They consist of seven columns, and are semi-quadrantally arranged.”

Because the early tables were far from perfect, subsequent innovators such as Henry Briggs heavily modified the architecture and design (or shall we say “user interface”?), and even theory - to simplify them further. Thus it was that a large body of new mathematics flowed simply from the problems involved in the very act of making logarithmic tables.

How would the constructor of tables ensure that they contain no errors? Not unlike computer programmers, would he have to devise his own techniques of debugging and error-correction?

Here is an example of the painstaking calculations involved in the process:

To find the logarithm of 2, Briggs raised it to the tenth power, viz.1024, and extracted the square root of 1.024 forty-seven times, the result being 1.00000 00000 00000 16851 60570 53949 77. Multiplying the significant figures by 4342 … he obtained the logarithm of this quantity, viz. 0.00000 00000 00000 07318 55936 90623 9336, which multiplied by 2^47 gave 0.01029 99566 39811 95265 277444, the logarithm of 1.024, true to 17 or 18 places. Adding the characteristic 3, and dividing by 10, he found (since 2 is the tenth root of 1024) log 2 = .30102 99956 63981 195. Briggs calculated in a similar manner log 6, and thence deduced log 3. (Encyclopaedia Brittanica, 2013 )

A Fountain Of Discovery

What if a certain number was absent from the table and the averages of two values had to be taken? This becomes especially troublesome in the case of trigonometry, as this remark about William Oughtred’s 1657 tables will reveal:

Following the tables there is an appendix of some ten pages which gives rules of interpolation in the trigonometrical tables for angles where linear interpolation is inappropriate; that is, near 0° for the logarithm of sines and tangents, and near 90° for the logarithm of tangents. In these regions the rates of change of the tabulated functions are large and highly non-linear.

The design of logarithmic tables, hence directly propelled - some major advances such as the Lagrange three-point interpolation formula. H.H. Goldstine describes this eloquently:

…the ideas of Napier and Briggs spread rapidly across Europe, and we shall see Kepler calculating his own tables as soon as he heard of Napier’s idea. From this point onwards the theory of finite differences was to be further developed with great artistry by such men as Newton, Euler, Gauss, Laplace, and Lagrange, among others. In fact we shall see that virtually all the great mathematicians of the seventeenth and eighteenth centuries had a hand in the subject.

Another fascinating aspect is Napier’s original description of logarithms in terms of the relative motion of points along a line, (as opposed to any kind of exponents or roots):

For Napier, a logarithmic table is “a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space…" Rafael Villarreal-Calderon interprets this as saying: "In a calculus sense, Napier’s logarithms could be seen as measures of “instantaneous” velocities."

This suggests that John Napier may have imagined his humble logarithms as something far more colossal that would bloom much later in the great minds of Newton and Leibniz: the calculus.

References:

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