John Napier

John Napier

Logarithms, tables, and calculating rods · 1550–1617 · Scotland

Napier's work illustrates two different ways to reduce arithmetic labor: change the representation of a problem, or precompute small results that can be assembled into a larger result. Both ideas remain useful in computer science. Neither requires claiming that earlier people could not multiply, or that modern processors ordinarily represent numbers as logarithms.

What the historical evidence supports

Napier matriculated at St Andrews in 1563. His subsequent education is poorly documented. He lived at Gartness after his marriage and later returned to Merchiston. His religious writing included a Protestant interpretation of Revelation published in 1593. These facts do not establish a psychological explanation for his mathematical inventions. MacTutor biography

His Mirifici Logarithmorum Canonis Descriptio appeared in 1614. The explanation of the construction followed posthumously in 1619. Henry Briggs visited him in 1615 and 1616; the later development of common logarithms involved both men. Napier's original tables used a scaled system associated with sines, not simply the modern function ln(x). A scholarly historical discussion also dates Kepler's first encounter with the published work to 1617, so a story placing him with Napier's tables in Prague in 1615 is unsafe. Gibson, The Methods of Construction of the Earliest Tables of Logarithms

The 1617 Rabdologiae describes calculating aids including numbered rods. Its digitized edition is an identifiable primary source, although the arithmetic below is an original example rather than a transcription of its instructions. ETH Library edition

Worked mechanism: distribute the multiplication, then carry

To multiply 678 by 7, place rods for 6, 7, and 8 side by side. Each rod contains the small products of its digit. Read row seven:

6 × 7       7 × 7       8 × 7
 4 | 2       4 | 9       5 | 6

Each pair separates tens from units. The diagonals collect contributions with the same place value. From the right:

1. Units: write 6.
2. Tens: 5 + 9 = 14; write 4 and carry 1.
3. Hundreds: 4 + 2 + 1 = 7; write 7.
4. Thousands: write 4.

The result is 4746. The rods have not eliminated addition, carrying, or the possibility of reading the wrong row. They have replaced repeated single-digit multiplication with a small reusable table.

The proof is positional arithmetic:

7(600 + 70 + 8)
= 4200 + 490 + 56
= 4000 + (200 + 400) + (90 + 50) + 6
= 4746.

The diagonal arrangement groups equal powers of ten. A carry replaces ten units of one position with one unit of the next, preserving the represented value. That invariant explains the method more securely than memorizing a diagram.

A different mechanism: change multiplication into addition

For a modern logarithm with base b > 0, b != 1, and positive inputs, log_b(xy) = log_b(x) + log_b(y). This follows because if x=b^u and y=b^v, then xy=b^(u+v). The identity needs a domain: the real logarithm of zero is undefined, and this treatment does not admit negative inputs.

Using modern base-ten tables rounded to four decimal places:

log10(12) ≈ 1.0792
log10(35) ≈ 1.5441
sum       = 2.6233
10^2.6233 ≈ 420.049

The exact product is 420. This is an approximation, even if a calculator displays many digits after the inverse transformation. Each rounded logarithm has error at most 0.00005. Their sum therefore has error at most 0.0001, before any additional rounding in the inverse table. The reconstructed product differs from the true product by a factor between 10^-0.0001 and 10^0.0001, approximately a 0.0231% bound on the larger relative deviation.

Napier's historical construction should be studied on its own terms. The translated Constructio explains the scaled quantities and table-building procedure; its editor also discusses numerical errors. The modern derivation here is an explanation of the transformation principle, not a claim that Napier wrote these equations or used these particular rounded values. Napier, Constructio, Macdonald translation, 1889

The computer science connection

A lookup table trades preparation and storage for less repeated work. A representation change trades one set of operations for another, possibly introducing conversion cost and error. These are separate design decisions. Ordinary binary floating point stores a significand and an exponent; it does not generally store the logarithm of the number. Specialized logarithmic number systems exist, but their existence does not establish a direct historical genealogy for every processor.

Exercises

1. Use the rod method for 587 × 8. Show each diagonal and carry. Explain why simply concatenating the three row entries fails.
2. Suppose two base-ten logarithms are independently rounded to three decimal places. Derive a worst-case multiplicative error bound for their product, assuming exact addition and exact inverse evaluation. Does statistical independence improve a guaranteed bound?
3. A report says, "The logarithm table gives 310046 exactly because the rounded logarithms of 374 and 829 add to 5.492." Identify two distinct problems. Then explain why 12 = 1.5 × 2^3 is not a logarithmic representation of 12.

Compare your reasoning with the worked solutions

1. The row entries are 40 | 64 | 56, where each pair denotes tens and units. From the right, the units digit is 6. The tens diagonal is 5+4=9. The hundreds diagonal is 6+0=6. The thousands digit is 4. Thus the answer is 4696. Concatenating 40, 64, and 56 would ignore that the first partial product represents hundreds and the second tens. Check: 4000+640+56=4696.
2. Each error has absolute value at most 0.0005, so their sum lies in [-0.001,0.001]. The product ratio lies in [10^-0.001,10^0.001]. The upper relative error is about 0.2305%. Independence can support a probabilistic error model, but it does not exclude both errors having the same sign. It therefore does not reduce this worst-case guarantee.
3. Rounded logarithms cannot justify an exact integer result without further reasoning, and 10^5.492 is approximately 310456 rather than 310046. The actual multiplication gives 374×829=310046; its rounded-log approximation is a different quantity. In 1.5×2^3, both a significand and an exponent describe the value. The exponent 3 alone is not log2(12), which lies between 3 and 4. Confusing an exponent field with a logarithm loses the significand.