EXECUTIVE SUMMARY
When calculators and computers fail — and NZ’s stock of electronic devices is finite, not domestically manufacturable, and degrading — engineers cannot check structural designs, navigators cannot compute positions, and surveyors cannot triangulate without printed mathematical tables. These tables — logarithms, trigonometric functions, square roots, reciprocals, and physical constants — were standard published references for centuries until electronic calculators displaced them in the 1970s–1980s.1 In a recovery scenario where electronic devices are depleted, they become essential infrastructure again.
NZ’s AI inference facility and any surviving computers should
generate these tables during their operational window. The tables
require only a working Python installation with its standard
math library and can be extended to arbitrary
precision and range. The cost of generation is negligible; the
cost of not having them is that engineers, navigators, and
surveyors lose the ability to perform calculations beyond basic
arithmetic.
The Recovery Library includes a precomputed set of mathematical reference tables (linked above) covering the most commonly needed functions. This document explains why these tables are needed, what they contain, how they were generated, and how to extend them.
Contents
COMPUTED DATA: MATHEMATICAL REFERENCE TABLES
View the Mathematical Reference Tables → — Logarithms, trigonometric functions, square roots, reciprocals, and physical constants, precomputed and ready for printing.
View the generation script → — Python source code for regenerating and extending these tables.
1. Why Printed Mathematical Tables Matter
1.1 The dependency
Any technical activity requiring calculation beyond basic arithmetic depends on mathematical tables or an electronic calculator. This includes:
- Navigation: Sight reduction requires trigonometric functions (sine, cosine, inverse tangent). Without tables, celestial navigation is impossible. See Docs #10 (Nautical Almanac), #11 (Sight Reduction), #139 (Celestial Navigation).
- Engineering: Structural calculations, electrical circuit design, hydraulic engineering, and thermodynamics all require logarithms, trigonometric functions, and physical constants. See Doc #17 (Engineering Tables).
- Surveying: Land measurement and mapping require trigonometric functions for triangulation. See Doc #16 (Topographic Atlas).
- Construction: Roof pitch calculations, load-bearing estimates, and material quantity calculations use trigonometric and logarithmic functions. See Docs #164 (Timber Construction), #165 (Plumbing/Water Systems).
- Scientific work: Chemistry, physics, agricultural science, and medical dosage calculations require logarithms, exponentials, and statistical distributions.
1.2 The gap
Electronic calculators have a finite lifespan. Battery-powered scientific calculators may last 5–15 years with careful use; solar-powered models longer, but their LCD displays degrade.2 Once the calculator stock is depleted and no replacements are manufactured, printed tables are the only option.
Slide rules provide an alternative for multiplication, division, and basic trigonometry, but only to approximately 3 significant figures — sufficient for rough estimation, but not for navigation sight reduction (which requires 4 figures to achieve 0.1 nautical mile accuracy) or structural engineering load calculations. Slide rules also lack scales for statistical distributions, Bessel functions, and other advanced functions, which require printed tables.3
1.3 The historical precedent
Published mathematical tables were a standard reference from the early 17th century (Napier’s logarithm tables, 1614) through to the 1980s. Every engineering office, navigation bridge, and school had them. The infrastructure for using printed tables — interpolation techniques, error estimation, table layout conventions — is well established and documented in any pre-1980 mathematics textbook.4 NZ practitioners of waka hourua (double-hulled voyaging canoe) navigation hold memorised angular measurement methods — star altitude by hand span, bearing by stellar azimuth — that provide instrument-independent cross-checks on trigonometric table accuracy; similarly, traditional Māori construction proportional systems for wharenui (meeting houses) and waka encode structural engineering ratios refined over centuries that function as pre-computed reference values.5 6
2. What the Tables Contain
The precomputed mathematical reference tables include:
| Section | Content | Precision | Primary users |
|---|---|---|---|
| Common logarithms (base 10) | log₁₀(N) for N = 1.00–9.99 | 4 decimal places | Engineers, scientists |
| Natural logarithms (base e) | ln(N) for N = 1.00–9.99 | 4 decimal places | Scientists, statisticians |
| Trigonometric functions | sin, cos, tan for 0°–90° | 4 decimal places | Navigators, surveyors, engineers |
| Square roots | √N for N = 1.0–99.9 | 4 decimal places | Engineers, construction |
| Reciprocals | 1/N for N = 1.00–9.99 | 5 decimal places | General calculation |
| Physical constants | Fundamental constants | Full precision | Scientists, engineers |
For most practical recovery applications, 4-figure tables (4 decimal places) provide sufficient accuracy. Navigation sight reduction requires 4-figure trigonometric tables; structural engineering typically requires 3–4 figures; only precision scientific work requires 5+ figures.7
3. How to Generate and Extend the Tables
3.1 Using the generation script
The tables were generated by a Python script
(scripts/generate_math_tables.py) using only the
Python standard library. The script can be run on any computer
with Python installed:
python3 scripts/generate_math_tables.py > tables-mathematical.mdThe script is self-contained and requires no external libraries.
3.2 Extending the tables
The generation script can be modified to produce:
- Higher precision: Change the rounding parameter from 4 to 5 or 6 decimal places.
- Finer increments: Reduce the step size (e.g., from 0.01 to 0.001 for logarithms) for more detailed tables. This increases page count proportionally.
- Additional functions: Inverse trigonometric
functions, hyperbolic functions, and exponentials are available in
Python’s standard
mathlibrary. Bessel functions, statistical distributions (normal, chi-squared, F, t), and other advanced functions require installing thescipylibrary — a third-party package that must be downloaded or pre-cached before network access is lost. - Specialised tables: Compound interest tables, unit conversion tables, wire gauge tables, pipe flow tables — any tabulated function that engineers or tradespeople use regularly.
3.3 Without a computer
If no computer is available, mathematical tables can be computed by hand using known series expansions and recurrence relations. This is how the original tables were created. A team of 3–5 human computers, working from published algorithms, could produce a set of 4-figure tables in approximately 2–4 person-months — tedious but entirely feasible.8 Cross-checking (having two people compute independently and comparing results) ensures accuracy.
4. Printing Priority
Mathematical tables should be among the first items printed during the digital-to-print window (Doc #132, Digital-to-Print Priority Schedule). They are:
- Small: The core set (logarithms, trigonometric functions, square roots, reciprocals, constants) fits in approximately 50–80 pages.
- Universal: Used by every technical discipline.
- Permanent: Mathematical functions do not change. A set of tables printed in 2026 is valid indefinitely.
- Irreplaceable without computers: Unlike most reference data, which can be observed or measured, mathematical functions can only be computed.
Recommended print run: 500–1,000 copies for national distribution to schools, workshops, navigation stations, engineering offices, and libraries.9 The per-copy cost at this volume is negligible given the 50–80 page length and standard single-colour printing.
5. Related Documents
- Mathematical Reference Tables — the precomputed tables themselves
- Generation script — Python source code
- Doc #10 (Nautical Almanac) — uses trigonometric tables for sight reduction
- Doc #11 (Sight Reduction Tables) — companion navigation tables
- Doc #17 (Engineering Tables) — engineering-specific reference data
- Doc #132 (Digital-to-Print Priority Schedule) — printing priority schedule
- Doc #135 (Computer Construction) — a functioning computer would generate tables at vastly greater speed and volume
Electronic calculators displaced printed mathematical tables during the 1970s–1980s as pocket scientific calculators (notably the HP-35 in 1972 and the Casio fx series from 1974) became affordable and widely available. The last major new edition of Chambers’s Mathematical Tables was published in 1979.↩︎
LCD display degradation: Liquid crystal displays lose contrast over time, particularly in fluctuating temperature environments. Solar cells degrade at approximately 0.5–1% per year. A well-maintained solar calculator might remain usable for 20–30 years, but this is not a planning assumption for a multi-generational recovery.↩︎
Slide rule limitations: A standard 25 cm slide rule provides approximately 3 significant figures. Specialised scales exist for trigonometric and logarithmic functions, but statistical distributions, Bessel functions, and other advanced functions require tables.↩︎
Standard references: Chambers’s Mathematical Tables (various editions, 1870s–1980s); Castle’s Five-Figure Logarithmic and Other Tables; Abramowitz and Stegun, Handbook of Mathematical Functions (1964). Any of these, if available in NZ libraries, should be preserved.↩︎
Waka hourua navigation: Polynesian navigation methods, including those practised by Māori, involved systematic angular observation of stars above the horizon, which directly corresponds to trigonometric altitude calculations. See David Lewis, We, the Navigators (1972) for detailed reconstruction of Pacific navigational methods. For Southern Hemisphere star paths specifically relevant to Aotearoa, consult living practitioners through the networks described in Docs #137, #139, and #141 (celestial navigation and traditional wayfinding).↩︎
Wharenui and waka proportional systems: Māori master builders (tohunga whakairo and tohunga waka) employed proportional rules governing structural dimensions. These are described in ethnographic and architectural literature, including Hirini Melbourne’s work on traditional Māori construction, and are held as living practice by tohunga and senior craftspeople. Recovery programmes engaging these practitioners (see Doc #160) may access practical structural ratios not recorded in Western engineering literature.↩︎
Precision requirements: Navigation sight reduction to 0.1 nautical mile accuracy requires 4-figure trigonometric tables. Structural engineering calculations for timber framing typically require 3 significant figures. Chemical engineering stoichiometry requires 3–4 figures. Only precision metrology and advanced physics regularly require 6+ figures.↩︎
Human computation: The original logarithm tables (Briggs, 1617–1628) were computed by a small team using finite difference methods. The WPA Mathematical Tables Project (1938–1943) employed hundreds of human computers to produce extensive function tables. A small team with known algorithms can produce basic 4-figure tables relatively quickly.↩︎
Print run estimate: Based on NZ having approximately 2,500 schools, 16 regional council areas, an estimated 200–400 engineering workshops and technical facilities, and approximately 50 public libraries with reference collections. 500–1,000 copies provides one copy per 5–10 major facilities; higher runs are desirable if printing capacity permits.↩︎