The units are part of the coordinate

The international foot is 0.3048 metres exactly. The US survey foot was 1200/3937 metres exactly, which works out at 0.304800609601219…

The two differ by two parts per million. Both were called the foot, both were legal, and until the end of 2022 both were in use in the United States for different purposes in the same state, sometimes on the same job.

The shape of the error #

Two parts per million applied to a coordinate is displacement. Two parts per million applied to a dimension is nothing.

A state plane coordinate runs to a million or two feet from its false origin, because the false origin was put a long way off precisely so that no coordinate would carry a sign — the reasoning of a grid has an origin that is not there. Two parts per million of a million and a half feet is three feet, or about ninety centimetres.

A building on that site is three hundred feet across. Two parts per million of three hundred feet is 0.18 millimetres.

So the same mistake produces an error five thousand times larger in the position than in the shape, and every check a practitioner performs is on the shape. Dimensions close. Diagonals check. Angles are exact. The building is simply somewhere else.

The arithmetic, on one state plane coordinate #

A concrete case makes the shape of the error obvious in a way the ratio does not.

Take a state plane easting of 1,500,000 survey feet. Read as international feet it is $1{,}500{,}000 \times 0.3048 = 457{,}200.00$ metres. Read as survey feet it is $1{,}500{,}000 \times 1200/3937 = 457{,}200.91$ metres.

Ninety-one centimetres apart, from a number that is identical in both readings.

Now take a dimension on the same site: 300 feet between two building corners. International, 91.4400 metres; survey, 91.4402 metres. Two tenths of a millimetre.

The same conversion, applied to two numbers that differ by a factor of five thousand, produces errors that differ by a factor of five thousand — which is the definition of a proportional error and is exactly why the false origin makes it dangerous. Had the coordinate been measured from a nearby origin, the displacement would have been millimetres too. The convention that removed the minus signs magnified this error by however far away the origin was put.

Why this is not the usual unit error #

Confusing feet with metres is a factor of 3.28, and nobody carries such an error very far — it produces coordinates in the wrong country and drawings that will not fit on the paper.

What makes the two feet dangerous is that the ratio is 1.000002. There is no magnitude check that catches it, no plausibility check, no visual inspection. A coordinate in the wrong foot is a coordinate that looks completely normal and is wrong by an amount comparable to the width of a road.

This is the same signature as three other failures in this collection, and the family resemblance is the point:

• a datum shift moves a point a hundred metres and preserves its neighbourhood’s shape exactly, which is what a coordinate refers to;
• a uniform scale error in a traverse leaves the figure closing perfectly, which a traverse must close measures;
• an epoch twenty years stale moves the ground half a metre with nothing in the number to say so, which is the epoch is part of the coordinate.

All four are errors that survive every internal consistency check, because all four are transformations that preserve internal consistency. That is not a coincidence; it is a definition. An error a survey can find is an error that breaks a closure, and all four of these are similarities.

Britain’s version of the same problem #

The United States is not alone, and the British case is worth setting beside it because the resolution went the other way.

The British National Grid was computed on the Airy 1830 ellipsoid, whose semi-major axis Airy determined in feet — 20,923,713 of them — and which was converted to metres afterward. Which foot was used for that conversion matters at the same two-parts-per-million level, and the answer is that the modern definition of the Airy ellipsoid is stated in metres and the foot value is historical. The ambiguity was closed by making the metric value authoritative rather than by keeping both.

That is the general remedy and it is available to any authority willing to accept a one-off discontinuity: pick one definition, declare it authoritative, and accept that the other reading of the old data is now simply wrong. The United States chose the opposite in 1959 — preserve the old reading, name it separately — and got 63 years of two feet.

Neither choice is obviously right. Britain’s makes historical documents slightly wrong; America’s made new documents ambiguous. What is clearly wrong is the third option, which is to do neither and let practice sort it out, and that is what happened between 1959 and the mid-1980s before the National Geodetic Survey’s guidance hardened.

Where 1200/3937 came from #

In 1893 the Mendenhall Order defined the US yard as 3600/3937 metres, tying American length to the international prototype metre. That gives a foot of 1200/3937 metres, which is not a round number in either system and was never meant to be — it is the metre-to-yard ratio the Office of Weights and Measures had adopted, expressed as a fraction.

In 1959 an international agreement redefined the yard as exactly 0.9144 metres, making the foot exactly 0.3048. Every English-speaking country adopted it.

The United States adopted it too, with one exception: existing geodetic survey data, expressed in feet, would have shifted if reinterpreted, so the older foot was retained for that purpose and named the US survey foot. The exception was meant to be temporary and lasted 63 years. It was finally withdrawn on 31 December 2022, and the coordinates measured in it did not go anywhere.

What the withdrawal did and did not settle #

The definition is gone. The data is not.

Every state plane coordinate published before 2023 in a state that used feet is in survey feet, and it is still the legal description of a great many property boundaries. A withdrawal changes what new work must use; it cannot reach into a century of records.

So the practical situation after 2022 is worse rather than better in one specific respect: there are now three possibilities for a coordinate labelled “feet” — international, survey, and unlabelled-and-therefore-unknown — where before there were effectively two and a strong regional convention about which. A file’s date is now evidence about its units, which is a thing no coordinate system should ever require.

What it looks like when it happens #

The failure has a recognisable signature and it is worth being able to name.

A site is set out from coordinates, the work is checked internally, and everything closes. Then somebody ties into a neighbouring parcel, or a utility survey done by another firm, or a control mark, and finds a discrepancy of a few tenths of a metre in a consistent direction. The discrepancy does not vary across the site. It does not grow with distance within the job. It has no rotation in it.

That constancy is diagnostic. A datum problem produces a constant offset too, but of a different size and usually a different direction; a scale problem produces an offset that grows across the job; a rotation problem produces one that swings. A constant offset that does not vary across a site and does vary between sites in proportion to their distance from the grid’s origin is a unit problem and almost nothing else.

Two feet in one file #

The worst case is not a file in the wrong foot. It is a file in both.

That happens when a dataset is assembled from sources of different vintages — some pre-1959 records in survey feet, some modern work in international feet, both labelled “feet” — and merged without conversion. The result is a dataset whose internal consistency is broken by an amount that varies with which source each point came from, which is invisible in the coordinates and nearly impossible to unpick afterwards.

Nothing in this collection’s machinery can measure that case, because it is not a property of a coordinate system but of a provenance. It is recorded here because it is the actual failure mode that made the National Geodetic Survey’s guidance so emphatic, and because it is the reason the 2022 withdrawal was worth doing even though it cannot repair anything already written.

Why a proportional error is the hardest kind #

It is worth being precise about why this error class defeats checking, because the reason generalises well beyond units.

A survey’s internal checks are all comparisons of differences: a closure is a sum of differences, a diagonal check is a difference of differences, a bearing is a ratio of them. A transformation that multiplies every coordinate by $1+\varepsilon$ multiplies every difference by $1+\varepsilon$ too — so every check compares two quantities that have both been scaled, and the scaling cancels.

Formally: the wrong foot applies a similarity transformation about the false origin. Similarities preserve every ratio, every angle and every shape. The only quantities they do not preserve are absolute lengths and absolute positions, and a self-contained survey measures neither — it measures lengths relative to its own instrument, which has been scaled too.

So the check has to come from outside, and the amount of outside needed is exactly the amount that fixes the similarity: one point pins the translation, a second pins the rotation and the scale. Two known points defeat the wrong foot completely, and that is a cheaper remedy than it sounds and is the standing recommendation of every guidance note on the subject.

What was computed, and how #

The ratio, from the two exact definitions: $(1200/3937)/0.3048 = 1.000002000004$, giving 2.000004 parts per million. The digits past the first are the arithmetic’s own and are quoted because the ratio is exact — this is one of very few numbers in this collection that is a definition rather than a measurement, and the collection’s habit of computing rather than quoting applies to it in reverse.

The displacement against coordinate size, at forty points out to two and a half million feet.

The dimension error across a site, at three hundred feet.

Three assertions, and the third is the one that matters. The ratio must be two parts per million to within a hundredth; the displacement at a million and a half feet must exceed half a metre; and the error across a three-hundred-foot site must be below a millimetre. The last one is the check that could have failed, and it is the essay: an assertion that only demanded a large displacement would pass on an error that was large everywhere, which would be a detectable error and a different subject.

The one thing that does catch it #

There is a check that finds a wrong foot, and it is the one nobody runs on a small job: compare against a known point outside the work.

The error is a scaling about the false origin, so it grows linearly with distance from that origin and is essentially constant across any one site. Two sites a hundred miles apart have displacements differing by a few centimetres; a site and a national control mark have displacements differing by however far apart they are. Tying a job to published control — rather than to itself — converts an undetectable systematic error into a visible discrepancy.

That is the same argument for the same reason as a published coordinate is a result: the value of an external reference is not that it is more accurate but that it is independent, and independence is the only thing that catches an error which preserves internal consistency.

Where the model stops #

The 2 ppm is exact and the consequences are not. Whether ninety centimetres matters depends entirely on the job, and the answer runs from irrelevant for a topographic map at 1:25,000 to litigation for a boundary. The essay computes the displacement; it does not and cannot rank it.

Nothing here handles the chain rule. A coordinate that has been converted between units more than once — feet to metres to feet, with a rounding at each step — carries an error that is not simply 2 ppm, and reconstructing what happened to such a value is usually impossible. This is the ordinary state of legacy cadastral data.

And the international foot is not the only alternative. The chain, at 66 international feet, and the link, at a hundredth of a chain, are the units a great many older land records are actually in, and a chain defined in survey feet differs from one defined in international feet by the same 2 ppm one level down.

The generalisation #

A unit that has two definitions differing by less than any tolerance in the field is more dangerous than a unit with none, because the discipline that would catch a big discrepancy never engages.

The pattern recurs wherever a standard is revised without renaming the quantity. It is the reason a version number belongs in the data rather than in the documentation, and the reason the datum realisations in the epoch is part of the coordinate are numbered rather than merely dated: WGS84 has had six realisations differing by decimetres, all called WGS84, and a coordinate carrying the bare name has the same problem this essay describes with a different constant.

The general rule that falls out is worth stating: if a redefinition is too small to be noticed, the old and the new must not share a name. Both feet were called feet, and 63 years of ambiguity followed.

The unit a screen adds #

Two definitions of the foot, two parts per million apart, is a unit failure inside the file. Drawing the coordinate adds a third unit that is in neither the file nor the specification: the pixel.

A pixel is a cell of the projected plane whose size depends on the zoom level and whose ground size depends on the latitude as well — 38.2 metres at zoom 12, 23.8 at 51.5° north. Every vertex drawn is moved to one, by up to half a diagonal, and the two quantisations do not commute: storing to five decimal places and then drawing puts a vertex in a different pixel from drawing and then storing, for 13.3 per cent of positions along a parallel at zoom 16.

The file’s own unit and the screen’s do not fail together. East-west the two grids stay in a constant ratio at every latitude, because the same $\cos\varphi$ appears in both; north-south they cross at a computable latitude, which on a six-decimal file at zoom 14 is past the tiling’s own cut — so on that combination the stored coordinate is the finer of the two everywhere a reader can go — the pixel is a place with a size is the same arithmetic from the screen’s side.

Who found it, and when #

The problem was recognised immediately. The 1959 agreement itself carved out the survey foot exception, which means the people making the change knew that reinterpreting existing geodetic data would move it and chose the ambiguity over the movement.

That was arguably right. The alternative — reissuing every coordinate in the national network — would have moved published values by two parts per million for no gain in accuracy, and would have created exactly the same confusion in the opposite direction, with the added feature that the old published values would have become wrong rather than merely differently defined.

What was not right, in retrospect, was leaving both called foot. The National Geodetic Survey’s own guidance from the 1980s onward is emphatic that the qualifier must always be written, which is an admission that the naming was the failure. The 2022 withdrawal fixes the definition and leaves the naming problem in every file written before it.

Where this goes next #

The unit is the fourth of the five declarations. Where a job’s own needs override the grid entirely — scaling it so that a set-out distance equals a ground distance — the result is not a map projection at all: a grid scaled to the ground is not a map.