Why is it s-l-o-w?

In 1900, when my grandma was born ino a affluent household in Melbourne, Australia,  electricity was new-fangled,  and there were no cars or radio let alone aeroplanes, satellites and computers. Then technology developed at an increasing pace. During the Second World War, the pace was astounding. German engineers developed the V1 and V2 rockets in only a few years, the British made big advances in radar and cryptography and the Americans developed the atomic bomb. Technology is accelerating on a spike, we are told. But some innovations seem to take forever...

The bicycle derailleur – 80 years too late

By comparison, it took about 80 years to develop and optimise the rear derailleur on a bicycle, an apparently simple parallelogram mechanism with five hinges and three springs. The tortuous history (of good ideas, missed ideas, buyer prejudice, and many commercial failures) is covered in huge detail in The Dancing Chain (Berto, 2005).The last major innovation was the invention of the slant-parallelogram by Suntour in 1964. This makes the jockey-wheel move downward as well as inward, so it follows the profile of the freewheel more closely, giving a smoother, quieter gear-shift.

Typical derailleur - the body has 5 hinges and 3 springs

The inelegance of the SI (or Ten Things I Hate About SI ;-)

Our globalised world needs a single, user-friendly measurement system so we can all communicate clearly about our physical world (health, food, energy, environment, infrastructure, transport, etc).

The International System of Units is conceptually elegant, however its notation is a historical mess, inherited from 150 years of irregularities in the metric system. It is also an administrative mess: the units and notation in the SI Brochure 8 (2006) (the primary scientific authority) ('SI8' here) and ISO 80000-1:2009 (the primary engineering authority) are inconsistent.

Consequently, the SI's clarity, usability and utility are impaired:

• the subtleties and irregularities of the SI notation are difficult to understand and remember
• SI expressions are unnecessarily difficult to write on a keyboard
• SI expressions are problematic to display in a web page and transfer between computers.
• SI expressions are syntactically ambiguous, so cannot be parsed by a software style-checker or  mathematical processor

These problems result in the SI being misused by writers/journalists, which contributes to public misunderstanding and more poor SI usage. The SI specification and notation should be made as simple and clear as possible, if only to encourage the citizens of the USA to adopt it! Here are the problems in the recent versions of the SI (SI6/7/8 and ISO 1000/80000), trivial and significant; some are fixable, for some it is too late. [Suggested fixes are in italics]

0. The two international SI specifications are inconsistent
SI8 and ISO 80000:
(a) imply different interpretations about expressing a quantity
SI8: "the value of a quantity is the product of the number and the unit, the space being regarded as a multiplication sign" [i.e. the space is explicitly the number-unit product symbol]
ISO: "The symbol of a unit shall be placed after the numerical value in the expression of a quantity, leaving a space between the numerical value and unit symbol" [i.e. the space is a formatting convention]

(b) specify different unit-unit product symbols
SI8 5.1: "Multiplication must be indicated by a space or a half-high (centred) dot (⋅), since otherwise some prefixes could be misinterpreted as a unit symbol."
ISO 80000 7.2.2: illustrates the allowed product symbols as "N ⋅ m" (with spaces), "N m" and "Nm", the latter being an implicit multiplier that can be used in certain combinations.

(c) specify different "non-SI units allowed for use with the SI" -
SI8: min, h, d, °, ',", ha, L/l, t, eV, u/Da, ua
ISO: min, h, d, °, ',", L/l, t, eV, u

What on Earth do ISO TC/12 and BIPM CCU think they are doing by running separate unit fiefdoms and versions of the mother of all international conventions?
[Fix: these specification are reproduced and interpreted in many secondary documents and in legislation - ISO 80000 should be made identical with SI8 urgently. ]

1. The implicit-product rule in ISO 80000 is confusing and widely misunderstood
ISO 80000 7.2.2: illustrates the allowed product symbols as "N ⋅ m" (with spaces) and "N m" and notes "The latter form may also be written without a space, i.e. Nm, provided that special care is taken when the symbol for one of the units is the same as the symbol for a prefix. This is the case for m, metre and milli, and for T, tesla and tera."
This clause is unclear (what does "take special care" mean?) and incorrect since only the leading unit needs to be unique (e.g. the symbol "hW" is ambiguous; the symbol "Wh" is unambiguous). It should simply state "except when the symbol for the first unit is the same as the symbol for a prefix". Consequently, the rule is widely misunderstood or ignored, resulting in many ambiguous SI expressions, such as "mAh" used to designate the capacity of household batteries.
It is also bad practice to illustrate the multiplier symbol "N ⋅ m", instead of describing it - from Acrobat Reader or a print copy, it is impossible to tell if it contains spaces, and this has a large impact on the parsability of an SI quantity expression.
[Fix: in any case, the ISO 80000 implicit multiplier should be discarded, as per SI 8, allowing humans/software to unambiguously identify both the prefix and unit symbols in a derived unit. The middle-dot multiplier should be explicitly stated]

2. The implicit-product rule in ISO 80000 causes ambiguities
The implicit product symbol combining two unambiguous unit symbols, can still form an ambiguous symbol; e.g. "lm" is ambiguous, representing both lumen and litre-metre.
[Fix: use an explicit multiplier to disambiguate this: lm is lumen, l·m is litre-metre.]

3. The unit-unit product symbol has too many options
The choice of three product symbols in ISO 80000 is bound to confuse users.
In SI 8, the allowed symbols are "N · m" and "N m", which reduces some of the confusion.
[Fix: there should be an explicit single representation of the product symbol, analogous to the SI's single representation of the divisor ("/"), allowing humans/software to unambiguously identify both the prefix and unit symbols in a compound unit. ]

4. The number-unit product symbol is not explicitly stated in ISO 80000
The representation of the multiplier between the numerical value and the unit is not explicitly specified in ISO 80000. Clause 7.1.4 states "Unit symbols...shall be placed after the complete numerical value in the expression for a quantity, leaving a space between the numerical value and the unit symbol." This reads more like a formatting suggestion than the international convention of quantity calculus. Consequently journalists often incorrectly concatenate the number and unit. The explicit specification of a number-unit multiplier would reveal the first axiom of quantity calculus (i.e. that a quantity is the product of a number and a unit).
[Fix: ISO 80000 should simply and explicitly state that the number-unit product symbol is a space.]

5. Some SI symbols are duplicated.
The characters "d", "h","m" and "T" are used both as unit symbols (day, hour, metre, tesla) and prefix symbols (deci, hecto, milli, tera). This can cause confusion, but does not cause ambiguity in SI 6/7/8 and in ISO 80000 if rule 7.2.2 is followed (i.e. if these symbols are not used before an implicit product symbol).
[Fix: none, it's too late]

6. The space and full-stop are ambiguous symbols
Since these symbols have other meanings in a SI expression, this makes the computer parsing of SI expressions problematic. For example in the ISO 1000 expression "1.234 567 x 103 N.m/s", the space represents a digit separator, two scientific notation separators and the number-unit multiplier; the full-stop represents a decimal point and unit-unit multiplier.
[Fix: the space should be reserved as a separator, and the full-stop reserved as a decimal point. ]

7. A prefix symbol plus unit symbol forms another unit
E.g. in SI6/7 where “Pa” can be “petaare” or “pascal”, and ISO 1000/80000 and SI 6/7/8 where “cd” can be “centiday” or “candela”)
[Fix: change the symbol for day to "day"]

8. A prefixsymbol  plus unit symbol forms another prefix
In SI 6/7 where “da” can be “deciare” or “deca”).
[Fix: make all prefix symbols single-character - change "da" to "D"]

9. Four SI symbols are not on a U.S. keyboard
Three of the 49 SI prefix/unit symbols (μ, Ω ,°C) are (inconsistently) not Latin characters. None of these three symbols, nor the multiplier symbol "half-high dot" (·) are represented on the U.S. or Western-European keyboards. As the vast majority of scientific writing is in English/European languages, this creates unnecessary delays and difficulties for authors in generating these characters in different software by the use of special codes, key combinations or menus. For example, in this HTML document, the required "character entity" codes are #mu, #Omega, #deg, #halfdot; this is hardly intuitive or simple for an author. If the symbols are copied from a Microsoft Word document - a common work-around - they are usually rendered incorrectly by non-Microsoft browsers, e.g. Mozilla FireFox renders "mu" as "m" and omega as "W". How does your browser render "m" and "W"?
[Fix: rename these to "u" and "Ohm" . Use a keyboard character as the multiplier symbol - the asterisk is the universally recognised keyboard character for multiplication.]

10. These four symbols are not represented in common character sets
These symbols are not represented in the foundational 7-bit ASCII. The Greek character omega "Ω" also has no character in the ubiquitous ISO-8859-1 (Latin-1). One day, all operating systems and application programs will use that mother-of-all-character-sets Unicode; meanwhile these omissions can cause errors for software in rendering the symbols across different computer platforms.
[Fix: adopting fix #1 will ensure these are in the widely-recognised ASCII character set]

11. The kilogram problem
The base unit of mass (kg) has a prefix, so one cannot form multiples by adding another prefix, e.g. "kkg" for tonne.
[Fix: many pedants suggest using the original French name Grave, or Giorgi or Galli, symbol "G"]

12. The prefix names are inconsistent.
Eight of the ten multiple prefixes are suffixed with -a, except for kilo and hecto. Seven of the ten submultiple prefixes are suffixed with -o, except for deci, centi and milli.
[Fix: rename these prefixes to kila, hecta, deco, cento, millo]

13. The prefix symbols are inconsistent.
Nineteen of the twenty prefix symbols are single-character, except for "da" which has two characters. This make it difficult/impossible for software to parse and implement SI prefixes.
[Fix: change the symbol for deca to "D"]

14. The prefix symbols have mixed case.
One would expect in a consistent and elegant system of units, the prefixes for multiples would be UPPER CASE and the prefixes for submultiples would be lower case. No such luck. The prefixes kilo (k), hecto (h) and deca (da) are all lower case.
[Fix: change the symbols to "K", "H" and "D"]

15. The prefix symbols are not systematic.
One would expect in a really consistent and elegant system of units, the prefix for a multiple would be the UPPER CASE of the corresponding submultiple. This would reduce the number of prefixes to be learnt to 10. Belatedly in 1991, the 19th CGPM partially adopted this obvious idea and declared zetta (Z=1021)/zepto (z=10-21) and yotta (Y=1024)/yocto (y=10-24).
[Fix: none, it's too late]

16. Uneven distribution of letters
There is a very uneven distribution of initial letters used for unit symbols/prefixes. "S/s" (s, sr, S, Sv), "C/c" (C, °C, cd, c), "M/m"(m, min, mol, m, M) and "K/k" (kg, K, kat, k) are overloaded, which is a recipe for human confusion. The letters "b, D, e, g, i, I, j, o, O, q, Q, R, t, u, U, v, w, x, X" are unused.
[Fix: none, it's too late]

17. The principal symbol for the unit-unit multiplier is unknown outside science
The half-high dot (·), the principal symbol for multiplier in a compound unit, is a specialised mathematical symbol unknown to the news media and magazines (e.g. who typically write "MWh" for electricity consumption), and so is unfamiliar to non-scientific users.
[Fix: change the symbol to an asterisk, the universal keyboard character for the multiplier operator]

18. ISO 1000 is inconsistent regarding alternative symbols
The ISO 1000 specification illustrates the multiplier in a compound unit as "N · m" and notes this may be "written with a dot on the line (N.m) instead of a half-high dot for systems with a limited character set". This inconsistent concession is of limited use since no such allowance is made for the Greek characters or degree symbol.
[Fix: adopting fix #4 means that alternate multiplier symbols are unnecessary]

19. The specification omits - use of prefixes
There is no explicit statement about which of the “units accepted for use with the SI” takes prefixes. ISO 1000 clause 7.2 states "prefixes given in table 4 may be attached to some of the units given in table 5 and table 6". SI 8 is similarly vague.

20. The specification omits - numbers in a compound unit
An axiom of quantity calculus is that base units may be algebraically combined to form derived units, but the SI specifications do not explicitly prohibit the use of numbers in a derived unit, although this is surely the intention. Consequently, there is widespread incorrect usage of certain conveniently-sized compound units (e.g automotive fuel consumption as L/100 km.

21. The specifications omit - combining multipliers
There is no rule about combining different multipliers in a compound unit (e.g. is "kg m²/(s²·K mol)" acceptable for molar entropy?)

22. The specifications omit - typography of spaces
The font for quantities is italic, and for units is upright (e.g. 'm' means 'mass'; 'm' means 'metre'). The typography of spaces (i.e. the space width between between groups of numbers, between numbers and units, and between units) is unspecified. In the font used in ISO 80000, the multiplier "half-high dot" appears to be separated by spaces while the alternate symbol, the full stop, is not. The SI brochure (5.3.4) specifies a "thin space" as a separator between groups of three digits whereas ISO 80000 does not.

23. Both specifications misname the product symbol
In ISO 80000 and SI 8, the unit-unit product symbol is referred to as "the half-high dot". There is no such character in ASCII, ISO-8859-1 or Unicode. Presumably this is the Unicode character "middle dot" (U+00B7); or is it the Unicode "dot operator" (U+22C5)?

Summary of fixes to build a syntactically unambiguous SI-notation
1. ISO 80000 should be made identical to SI8
2. Change the unit symbol 'd' to 'day'
3. Explicitly require that derived units containing numbers (e.g. fuel consumption 'L/100 km' ) are written in standard mathematical  notation, i.e. 'L/(100 km)' or  'L/100/km' or 'L/km/100'.

Summary of fixes to build a more user-friendly SI-notation

1. Specify a single unit-unit product symbol, the keyboard symbol "*"
2. Change the prefix symbols "da, h, k" to "D, H, K"
3. Change the Greek symbols μ and Ω to (Latin) keyboard symbols "u" and "Ohm"


[drafted 2008-05-22; rev 2011-04-14]

Precedence in unit-parsing

Operators in mathematical expressions are conventionally parsed left to right in this order of precedence:
1. Parentheses
2. Factorial
3. Exponentiation
4. Multiplication and division
5. Addition and subtraction

But what about when a mathematical expression includes a quantity expression (i.e. the product of a number and a unit)? Here are some considerations, applied to the International System of Units (SI).

1. The binding of the prefix to the unit has the highest precedence This is expressed in the SI Brochure 3.1 as "The grouping formed by a prefix symbol attached to a unit symbol constitutes a new inseparable unit symbol (forming a multiple or submultiple of the unit concerned) that can be raised to a positive or negative power and that can be combined with other unit symbols to form compound unit symbols".

2. When a unit expresion has more than one divisor, how should it be expressed?
The SI Brochure 5.1 incorrectly states "In forming products and quotients of unit symbols the normal rules of algebraic multiplication or division apply...A solidus must not be used more than once in a given expression without brackets to remove ambiguities." This means that the previously commonly-written quantity expression for the acceleration due to Earth gravity (9.81 m/s/s) is not acceptable SI usage, although this meets the mathematical precedence rules and is not ambiguous.The SI uses the example:
"kg/(s³ A), or m kg s^–3 A^–1, but not m kg/s³/A, nor m kg/s³ A", even though the first three have the same meaning. Examples 3 and 4 may be ambiguous to some humans, but they are not mathematically ambiguous. The authors of the SI Brochure should stick to established mathematics, and not modify it to suit their view of human capacity.

3. A number rather than a prefix is used with a unit
The common metric unit of fuel consumption is 'L/100 km', in which the number '100' acts somewhat like a prefix. It is a dog of a unit, since
(a) the SI Brochure doesn't allow numbers in a unit expression (although it doesn't explicitly deprecate it)
(b) it evaluates left-to-right as 'L/100*km' which has dimension [length]⁴.
In SI style, it can't be written with a prefix, since 100 km is 10⁵ m, and there is no prefix for 10⁵. NIST writes it as 'L/(100 km)' - which parses correctly, but is still not a properly-formed SI unit expression.

This problem can be fixed with a new precedence rule:
The binding of the number to the unit has a precedence above multiplication/division

4. It is meaningless to factorialise a quantity expression.

5. Addition/subtraction operators are not used in quantity expressions.

So the precedence order for quantity expressions is:
1. Prefix-unit binding
2. Parentheses
3. Exponentiation
4. Number-unit binding
5. Multiplication and division

The Google search engine incorporates a basic calculator which is unit-aware and which abides by the order above: try googling '0.4 gal/mile in L/100 km', '5 km^2 * 0.4 L/ha' or '5 mph in m/10 s'. The Wolfram Alpha 'computational knowledge engine' , which does a pretty good job of parsing quantity expressions, chokes on '0.4 gal/mile in L/100 km', and even '0.4 gal/mile in L/(100 km)', so needs to have its precedence rules tweaked.

Millions, billions, trillions and the SI prefixes

The SI prefixes are rarely adopted by the news media for expressing large quantities, typically sums of money, populations and geological time (quantities of widespread discussion!). Journalists worldwide abbreviate thousand as 'K' or 'k', million as 'M' or 'm', US billion as 'B' or 'b', and US trillion as 'T' or 't'.

The Australian Government Style Guide (AGPS 2006, p.174) recommends:

"Milllions of dollars may be expressed by placing 'm' (unspaced and without a full-stop) after the number (e.g. $2.751m)"

A web search of news websites shows that the most common currency abbreviations are:
thousand k = 52% (K = 48%)
million m =77% (M = 23%)
billion B = 58% (b = 42%)
trillion T = 88% (t = 12%)

It looks like the AGPS meme has prevailed with 'm'; but it's a bad idea - 'm' is the SI symbol for 'metre'. Why would we want to invent another four abbreviations, when we have four perfectly-good internationally-agreed prefix symbols (k , M, G, T)? Thankfully, the AGPS gives no guide to currency abbreviations for thousands, billions and trillions.
Luckily, the abbreviations 'k', 'M' and 'T' match the SI prefix symbols for thousand, million and trillion, so at least headlines which state TV audience of 65M or 2008 US budget of $2.9T are aligned with the SI.

It's unlikely that media articles will ever use the agreed international representation of a physical quantity (i.e. the product of a number and a quantity, e.g. '40 G$') to represent a sum of money, so perhaps the CGPM should surrender to common usage, and replace G with B ('biga') as the prefix for 10⁹. Then the headlines of the $40B Enron trial will also align with the SI!

Metrication fiascos #75: "L/100 km" is NOT a unit of anything

There is no definition of a quantity or unit for automotive fuel usage in the International Standard ISO 80000 Quantities and Units, so we should use national standards. The most common measures are the reciprocal quantities 'fuel economy' and 'fuel consumption':
1. The US Government mandates a new-car label stating 'fuel economy' with a unit of 'mpg'.
2. The European Union and Australia mandate a new-car label stating 'fuel consumption' with a unit of 'L/100 km'.

The latter is not a unit, by the international definition. It breaks the rule for forming compound units in ISO 80000 Part 1: General, 3.2 Combinations of symbols for units, which states that compound units are formed by multiplication and/or division of units (no mention of numerical factors). There is no other unit in scientific or public use which uses a numerical factor and breaks this rule. Furthermore, this 'unit' is invariably written 'L/100 km', which when parsed with standard mathematical precedence, yields a meaningless expression 'L/100*km' of dimension [length]⁴.

When petrol bowsers and car odometers were metricated and this 'unit' was introduced to Australia in 1976, I queried the Metric Conversion Board. The reply, in essence, said:
"1. The Europeans use this unit, so we should harmonise.
2. Most cars have a fuel consumption of 12 - 20 L/100 km, so their fuel economy (5 - 8.5 km/L) has unacceptable rounding errors when expressed with one decimal point."

I replied then, and I say now:
1. Adopting someone else's incorrect convention does not make it correct.
2. With improvements in technology, the average car fuel consumption will reduce, and one day it will be in the range 5 - 8 L/100 km, so the rounding error will then apply to this measure.

So what to do? Fuel consumption is the better quantity to use for comparing the fuel usage of two vehicles, since it can be subtracted to give their relative fuel usage. For the benefit of consumers, the world (including the USA, which is just about to revise its fuel economy label) should adopt it. We should write the unit as 'L/(100 km)' (as per NIST Special Publication 811 Appendix B5) so it can be parsed correctly by future 'unit-aware' software.

Commodity Obfuscation

On ABC NewsRadio this morning, I heard world prices for rice in 100 lbs, oil in barrels, soybeans in bushels, wheat in metric tonnes and soybean meal in short tons. I'm vaguely interested in the price of fish, and how it relates to my living expenses and world hunger, but I was none the wiser.

Apparently the source of this obfuscation is the Chicago Board of Trade.

We have a good-enough International System of Units, can we use it please folks? Let's quote the grains by mass (tonne), and oil by volume (L). Then we're in a better position to compare commodity and consumer prices.

Or is that the very point?

World Year of Physics 2005? So show me Einstein's papers!

[pub 20051207]
During 1905, the physics graduate Albert Einstein, while employed as a patent examiner in Bern, Switzerland, submitted 5 papers to the German journal Annalen der Physik. They all contained breakthroughs in understanding major physics problems of the day. One wonders how much effort he was putting into his day job!

Several years ago, I decided I'd like to read and to try and understand these papers, and searched the WWW, presuming these most significant works would be gathered, translated and annotated on some Einstein homage website. No such luck!

In 2004, UNESCO declared 2005 as the International Year of Physics, and after some brawling , this was endorsed by the United Nations General Assembly. The impetus was to celebrate Einstein's Miraculous Year, so-named after Newton's reference to his own stellar year of 1666. Good, thought I, at least some august body in the physics community will organise translations and a commentary on a web page, to honour the year and promote physics. No such luck! Scientists don't have a clue about self-promotion!

Luckily John Stachel has given us the book "Einstein's miraculous year: five papers that changed the face of physics" (Princeton University Press, 1998). The amazing thing about these papers is that they are more or less comprehensible to someone with high-school physics/math.

Schadenfreude - Stachel notes there are simple math and notation errors in the Special Relativity paper! Einstein was human after all!
For instance:

In Section 7, "for v = -c, f = ∞" should be "for v = -c, f = -∞"

In Section 8,



should be

The Milky Way should be catalogued as NGC 0!

The New General Catalogue of Nebulae and Clusters of Stars was compiled by J.L.E. Dreyer in 1887 with about 8000 objects, and is still in use. 

I was miffed to find it does not contain an entry for our very own galaxy, aka The Milky Way, arguably the most important in the Universe! The first entry, NGC 1, is some 13^th magnitude galaxy in Pegasus. It's not really surprising that the NGC doesn’t include The Milky Way, since we are embedded in it and it subtends 180° of the sky to each observer.

Nonetheless, I hereby propose that The Milky Way, should be granted pride-of-place in the catalogue, as NGC 0, the Centre of the Universe, with right ascension 0:0 hours, declination 0:0 degrees, apparent diameter 10800 arcminutes.

Cavendish mis-reported again?

[revised 2008-10-13]
Falconer (1999) has written an arresting account of Cavendish and his famous gravitation experiment (Cavendish, 1798) and highlights three important aspects of his achievement:

1. Cavendish set out to measure the density of the earth as a ratio of the density of water, not the universal gravitation constant G, a concept which lay about 80 years in the future.

2. In the same way that Newton’s theory of Universal Gravitation linked astronomical motion to local motion (i.e. falling objects), the experiment set an example for investigating natural phenomena in a laboratory.

3. It is a seminal paper in measurement science, focussing more on the estimation and elimination of errors than on the experimental results.

That article is a welcome counter to the widespread misreporting in physics texts, course notes and commentaries, that Cavendish set out to measure G (Gonzalez, 2001). However, the article contains two technical errors, which are discussed below, in the hope that these too don’t get repeated by others.

1. Falconer writes (p.474) that “the essential feature of the experiment consisted of using a torsion balance to find the attraction between a lead sphere 8 inches in diameter and another lead sphere 2 inches in diameter”. (Cavendish calls these spheres the “weight” and the “ball” respectively.) In fact, nowhere does Cavendish mention the diameter of the weight, but in an acknowledgment of the balance’s inventor, states (p.470) “the weights which Mr. MICHELL intended to use were 8 inches diameter”. We can estimate the diameter of the “weight” that Cavendish actually used, from his scale diagram (approximately 11 inches), or calculate it from the given mass of 2439000 grains and the density of cast lead as approximately 11.7 inches.

2. A feature of Falconer’s paper is that it expands some missing steps in Cavendish’s calculation of the ratio of the attraction of the weight on the ball to the attraction of the Earth on the ball. The ball and weight had a centre-centre separation of 8.85 inches. Cavendish introduced (p.510) a fictitious particle (what we now call a “point mass”) and a fictitious sphere of water one foot in diameter, which has mass 1/10.64 that of the weight. He writes “...and therefore [the weight’s] attraction on a particle placed at the centre of the ball, is to the attraction of a spherical foot of water on an equal particle placed on its surface, as 10.64 x (6/8.85)² to 1”. Unfortunately, Falconer confuses the ball for Cavendish’s particle, with the result that her rephrasing (p.475) "Then the attraction, F, of the weight on the ball was F=10.64 x (6/8.85)² times the attraction of the sphere of water on the ball if the ball were on the surface of the sphere", is geometrically incorrect and confusing because the ball has a finite radius of 1 inch, which would require the ratio to be (7/8.85)² if it were touching the sphere of water.

References
Cavendish, H. 1798. Experiments to determine the density of the Earth. Phil. Trans of the Royal Society of London 88, 469-526
Falconer, I. 1999. Henry Cavendish – the man and the measurement. J. Measurement Science and Technology 10, 470-477
Gonzalez, A. 2001. "Weighing'' the Earth: a Newtonian Test and the Origin of an Anachronism. Science and Education, 10, 6, 515-543.

The elegance of the SI

1. The great political foundation of the International System of Units is that it has been developed by international agreement by the General Conference on Weights and Measures meeting four-six yearly since 1875.
2. The great intellectual foundation of the SI is that it is based on a formal quantity calculus (ISO 31:1992), using seven mutually independent base quantities.
   
3. The great physical foundation of the SI is that its base units are mostly rooted in the physical world, defined using various universal physical constants.
4. The great mathematical foundation of the SI is that it is a coherent system of units, so that physical equations like E=mc² don’t require conversion factors. In other words, if you use the base unit for “c” (m/s) and the base unit for “m” (kg), the result by definition is in joules, the SI unit of energy. This is far easier, and therefore less prone to error, than using the foot-pound-second (FPS) system, where a factor is required depending on whether you want the answer in ergs, BTU’s or “megatons of TNT”.
5. The great arithmetical foundation of the SI is that it is decimal-based, so you don’t have to remember how many pounds in a UK or US ton, how many fluid ounces in a UK or US gallon, and all the other complicated diversity of FPS units.
6. The great rigour of the SI is that it has a defined symbol for every unit, as distinct from the plethora of abbreviations invented for many FPS units (e.g. mi/hr, mph; psi, lbf/in² etc).
7. The great scalability of the SI derives from having a defined prefix for each multiple and submultiple of a unit (e.g. n=10^-9 , G= 10⁹), allowing a huge range of values to be expressed concisely.

Metrication fiascos #1 - the Mars Climate Orbiter

NASA's 1998 Mars Climate Orbiter (MCO) had several trajectory correction thrusts applied on its journey, but each time the resulting correction was smaller than ordered. Perhaps some mental alarm bells should have started ringing at Mission Control. Ultimately the MCO arrived 170 km lower than planned. It disappeared, presumably burning up in the Martian atmosphere.

What went wrong? Clearly space missions are very complex projects, and the MCO Mishap Investigation Board found many small oversights of project management and operation. The root cause however was "Failure to use metric units in the coding of a ground software file, 'Small Forces', used in trajectory models". In other words, the thrust required for each trajectory correction was calculated by the ground software in pound-seconds, but was actuated by the spacecraft software in newton-seconds, less than a quarter of the required thrust. It's not surprising that the trajectory corrections didn't work. The Board and many commentators drew this lesson: don’t mix units.

But this is a trivial conclusion; there is a more important lesson to be learned. The ground software  wrote a pure number to a file, which was transmitted to the spacecraft software which interpreted it as a quantity. We need to agree on ways to fully represent units in data, so that a physical quantity ('1234 lbf·s' in this case) can be transmitted wholly and unambiguously between computers, and can be rejected or converted if necessary by the receiver.

This is going to be a growing problem, as we have an increasing number of autonomous computers (trains, mining equipment, cars, robots etc) interacting in the physical world and sharing data.