James Joyce

The system of weights and measures that governs our daily lives carries, like the water that shaped it, the memory of an ancient order. To pour water into a container, to weigh flour, or to measure the height of a growing child is to invoke units like the pint, the pound, the foot and the inch. It's often assumed these units are all arbitrary conventions, which have somehow been improved over time. Yet the evidence suggests otherwise. When explored carefully, many traditional measures show not confusion, but a remarkable internal consistency, spanning centuries, empires, and civilisations. This is not the story we are typically told. In most accounts, the ancient world was imprecise and fragmented, its units parochial and based on arbitrary body parts. Yet when we examine the ratios preserved in older metrological texts, a different picture emerges.

A recent article on this website explored the calendar reforms of Numa Pompilius, who adjusted Rome’s lunar calendar seemingly without understanding its embedded astronomical harmonies, as the changes he made tell us. It is possible to inherit a structure while remaining unaware of its internal logic. In the modern age, we have updated our systems of measure, redefining the metre in relation to light speed, or the kilogram to Planck’s constant, yet these highly precisely determined values are not necessarily the correct ones, in terms of the overall system, and its foundations. Many of the original relationships have slipped away in the process. The coherence of the older systems remains largely unrecognised, despite their lingering presence. The measures handed down to us carry the imprint of a worldview in which measures of mass, volume, time, and length were interrelated, and in which the proportions of the human body, and nature, and the size of the Earth all echoed one another. It’s as if we changed the words to an old poem to make them easier to read, only to realise later we have sabotaged the rhyme, the rhythm, and the meaning. The difference between information and technology on the one hand, and knowledge and understanding on the other, can help understand how we have dealt with this ancient system. We can learn about cubits being two and a half feet, or a foot being 144 lines, but we also need to be able to read between the lines. In addition, our contemporary disconnection from nature allows us to think it's perfectly natural not to find in the system of measures we have inherited a strong connection to the natural world.

The legal theory of the British system of weights and measures is (a) the standard yard, with all lineal measures and their squares and cubes based upon that; (b) the standard pound of 7000 grains, with all weights based upon that, with the troy pound of 5760 grains for trade purposes; (c) the standard gallon (and multiples and fractions of it), declared to contain 10 pounds of water at 62 F., being in volume 277·274 cubic inches, which contain each 252·724 grains of water in a vacuum at 62°, or 252·458 grains of water weighed with brass weights in air of 62° with the barometer at 30 in. Somewhere within these numbers there is a logic, a deeply reasoned system, tied to water density, various properties of the barley corn, the size of the earth, geometry, and even astronomy, which was eroded over time by reforms, times of political turmoil, and simplifications.

The survival of these fragments of a system at once mathematical, natural, and cultural allows us to guess at a system whose origins stretch back thousands of years. This article will look at modern definitions of units of volume and weight in the imperial system with its UK and US variants, and the irony that this ancient coherence, once tied to natural constants, was clumsily redefined in the modern age of science in the name of precision.

The UK and US Pints and Gallons: What's the Difference?

  The pint is a unit of volume in both the Imperial (UK) and US customary systems, defined as one-eighth of a gallon in each. However, the Imperial pint is about 20% larger than the US liquid pint, while the US dry pint (≈ 551 mL) is slightly larger still. The reason for this difference goes back to choices made in the 18th and 19th centuries. By the mid nineteenth century, two quite different philosophies of measurement were in place either side of the Atlantic. In the United States, the gallon was defined geometrically. Based on a measure associated in Britain with wine, it was equivalent to the measure of a cylinder seven inches in diameter and six inches high, and consequently contained 231 cubic inches. In Britain, this measure had been called the Queen Anne wine gallon. Quarts, pints, and gills based on this were each then dependent on this simple geometry, rather than the properties of any particular liquid or grain under any particular condition. No matter what was being measured by volume, a quart was precisely 57.75 in³ (1/4 of the gallon), a pint 28.875 in³, and a gill 7.21875 in³, and of course this is still the case in the US today. This method, simply based on geometry, reflected a preference for stability and ease of reproducibility: any maker could construct a cask or measure and reproduce the same volume anywhere. The legal definition relied simply on the inch, and without considering the density or temperature of water, or any other substance.

In the early 19th century, in the UK there were three types of gallon in existence. For centuries, gallons had been defined as fixed volumes, the wine gallon (231 cubic inches), the ale gallon (282 in³), the corn gallon (268.8 in³).  Parliament defined the Imperial gallon as:

The Queen Anne Wine Gallon (1707), from which the U.S. gallon descends, was legally defined as 231 cubic inches of wine. But Parliament never specified the weight of that wine, only the volume of the cylinder (7 in. diameter × 6 in. height). By the early 14th century, English statutes showed a direct effort to formalise volume and weight relationships. A pint was defined as a measure that could contain a pound and a half-ounce of wheat by weight, or 6000 troy grains. This defined the Winchester gallon as 8.5 troy pounds, or 272.25 cubic inches. The bushel, comprising 8 gallons, was then 2 178 cubic inches, and the standards continued in binary multiples.

Volume was subordinated to weight: the gallon, quart, and pint were tied to the pound of 7000 grains, and the density of water under specific laboratory conditions to match that weight. The pound was defined in terms of the weight barley grains, and the inch was defined in terms of the length of barley grains, so that this grain was the foundation of the entire British (and American) metrological system. With such solid foundations, basing the units of volume on the weight of water seemed sensible. What could possibly go wrong?

For the sake of simplicity, only one type of gallon was really required. Rather than just pick one, the authorities decided, in their wisdom, to scrap them all and start again. In 1824, the imperial gallon was legally defined as the volume of 10 pounds of distilled water at 62 °F, amounting to approximately 277.274 cubic inches.

   The history of the gallon and the pint illustrates the evolution of scientific metrology. The system of weight that the UK pint was originally based on is the Avoirdupois pound, this contrasts with the Troy system, used for precious metals, in which a pound equals 5 760 grains. The Avoirdupois pound's grain-based foundation links it directly to ancient volumetric measures, whereby ten pounds of water equal exactly 70 000 grains, marrying a base-10 water-weight logic to the duodecimal structure of pints, quarts, and gallons (1 gallon = 8 pints = 4 quarts). The density of water at 62°F is such that: 1 US gallon of water = 8.337 Avoirdupois pounds = 58 359 grains, and 1 US pint of water = 1.0421 pounds = 7 295 grains (approx.) In all systems of measurement, weight and volume are fundamentally different kinds of units, which align only under specific conditions, such as when measuring water at a defined temperature, or when weight is used to define a volume.

   At the time, this appeared to be a more scientific foundation, anchoring the gallon to a natural constant: the weight of water. The US system, by contrast, does not produce whole-number relationships between water, pounds, and pints, being based instead on volume defined by geometry.

In 1795 the United States fixed its gallon as exactly 231 cubic inches, the same as the English wine gallon of Queen Anne’s reign (1707), defined as a cylindrical volume 7 inches in diameter and 6 inches deep. This definition was entirely geometric: no reference to water, temperature, or weight was necessary. As such, it proved robust. The American gallon has remained unaltered since its inception, insulated from later debates over the physical properties of water. The US system preserves a continuity with ancient measures that is remarkable in the present age. In this respect the United States are quite unique, as they preserved a unit that had already been in continuous use for centuries and that can be traced, in form if not in name, back to some of the very earliest surviving measures. Most nations either abandoned pre-metric units entirely or, as in Britain, recast some of their measures in the 19th century, breaking the historical chain. Despite these trends, the United States confidently kept faith in the older form, even with the units of volume such as the pint and gallon. In doing so it became, perhaps alone among modern nations, the custodian of a volumetric unit with truly ancient roots.

   Enthusiasts for the so-called imperial system often present it as an ancient artefact, in contrast to the metre, which is sometimes portrayed (in Britain) as a modern intrusion. Indeed, the inch, the foot, the yard, the mile, and many other elements of the system are indeed very old, much more so than is generally suspected. However, the present UK pint and gallon are in fact even more recent than the metre itself. The metric system was brought in to replace the old units of measure in France in the 1790s; Britain’s imperial gallon and pint were only created in 1824. The metre, inch and foot are all ancient.

The transatlantic split crystallised in this way, with the American system embracing geometric simplicity, and ensuring long-term consistency, and the British system seeking natural precision through water’s weight, a standard that increasingly sophisticated technology, and a lack of self-belief, would eventually destabilise. Consequently, American pints and gallons remained unchanged over centuries, whereas their British counterparts were later redefined, first in terms of water, then, by the 1960s, in terms of the metric litre. The difference is reflects a philosophical divide between volume anchored in geometry and volume anchored in physical substance.

All Done With Smoke and Mirrors

Gilbert K. Chesterton

Volume can be defined either in terms of linear lengths (so the barley corn, via the inch, would have lent its length for this purpose, an inch being three barley corns placed length to length) or by a particular weight of something, which fills up a corresponding particular volume (so the barley grain, via the pound, would have lent its weight for this purpose). After all, in the previous century, the French had officially linked their main volumetric unit, the litre, to their main unit of weight, using water, though in the opposite way: the unit of weight depended on the unit of volume.

   Indeed, the kilogram was originally defined in 1795 as the mass of one cubic decimetre of water at the temperature of melting ice, 0 °C. However, this was not quite accurate, so that very soon, the legal definition did not seem to reflect the reality. The litre itself was designed like the Queen Anne wine gallon: geometrically. A litre is the volume of a cube 10 centimetres × 10 centimetres × 10 centimetres. Water, however, is a tricky substance to measure, depending not only on temperature and density, but also on atmospheric pressure, and purity. Also, the very process of freezing or melting introduces instability (ice forming on the container walls, air bubbles, isotopic differences).

   In practice, scientists quickly realised that the mass of one cubic decimetre of water was difficult to pin to the kilo, especially melting ice, so that any laboratory realisation of the kilogram based on water could only approximate the intended value unless all these conditions were rigorously controlled. Under precisely defined circumstances, for example, pure water at 4 °C and standard atmospheric pressure, a cubic decimetre of water does indeed weigh one kilogram, but even tiny deviations produce measurable differences. At 0 °C, for example, water is less dense, so a cubic decimetre would weigh slightly less than one kilogram. In practice, these discrepancies were tiny, on the order of a few parts in 10 000, but in a system built on ideals of rationality and exactness, even small imperfections were glaring. The litre was a stable geometric volume, but the kilogram rested on fluid foundations. The litre depended only on the metre, but the kilo depended on agreeing on how to correctly and precisely determine the weight of water and was therefore unreliable as a standard. The authorities quietly sidestepped the problem. Rather than define more precisely the conditions under which a litre of water would match a kilo, and leave it at that, a prototype kilogram of platinum (later platinum–iridium) was constructed and adopted as the true kilogram in 1799. The water definition remained a historical principle, but in practice the metric system was anchored to a physical artefact, not to water itself.

   The repercussions in France, then, were more conceptual than practical. The ideal of a natural, universal standard, with water as nature’s yardstick, had failed. Once water’s sensitivity to varying conditions was acknowledged, the French solution was essentially to pretend the theory still held while relying on a metal prototype instead. This was the strange backdrop to the decision to scrap the three gallons in existence in Britain in the early 19th century, which had long been established, and build a system of units of volume based on the weight of water.

   Indeed, in the UK, a gallon of water may have legally been equivalent to the volume of 10 pounds of distilled water at 62 °F, but even under the prescribed conditions, it actually weighed closer to 10.012 pounds, illustrating a slight disconnect between law and physical reality.

   It is true that the weight of 7000 grains of barley, the basis of the avoirdupois pound, may vary somewhat, especially at the levels of precision required in the world of science. But crucially, there was agreement on how much it weighed in law, and how much an individual grain weighed. The 7000 grain pound weight had been in use for so long in Britain that it seemed like an unshakeable foundation. In France, since the revolution, however, the unit of weight depended on the shifting properties of a fluid. Only by freezing the standard into platinum did the kilogram survive as a usable unit. The irony is striking: the French system was conceived as a triumph of natural order, yet its permanence rested on a man-made cylinder that was only reasonably close to the weight of a litre of water. Yet the system survived because it could be re-anchored: first in platinum, later in definitions tied to the metre and second, and now (since 2019) in fundamental constants of physics. Yet here too the ground is less firm than it seems: the metre is now defined by the speed of light, but the speed of light itself is fixed by the definition of the metre (and the second). What was meant to be the ultimate escape from arbitrariness turns out to be a closed loop, a system no less circular, and arguably no less contrived, than the attempt to use water as a foundation and then choosing an arbitrary value instead. Both in France in the 1790s and in Britain in the 1820s, the reformers claimed to be grounding their systems in the natural weight of water. Yet when agreement on determining the precise conditions under which to weigh water proved elusive, the authorities simply fixed upon a convenient approximate value and declared it definitive. In effect, the standard was no longer the water itself but some other thing that seemed more stable at the time. The result was a kind of circular reasoning: the system was said to rest on nature, but in reality it rested on a legal fiction, cloaked in the language of natural law. The weight of a quantity of water would then be measured and reported in relation to the already-chosen anchor value. This process amounted to fixing a measure at some convenient value, and then treating it as an immovable bedrock. From that point forward, reformers and technicians piled ever greater refinements upon what was, at root, a compromise, an approximation, a random value, uprooted from the ancient system's foundations. The more precise the definitions became, the more they served to distract from the fact that the standard itself was arbitrary.

  Another option, be it for the French in 1795 or the British in 1824 and 1963, might have been to anchor the value of the weight of a particular quantity of water to exact physical conditions, ensuring reproducibility, if not easily, then at least precisely. That would have preserved not only the coherence of the system but also the intellectual integrity of those tasked with maintaining it. It would have avoided the circularity of a unit being defined in relation to a natural constant which itself was defined in relation to that same unit.

 To be fair, in the age of science, beginning with the 17th century, precision was essential, and so no unit of measure escaped scrutiny. The pound of 1824 was still defined as 7000 grains, a continuity reaching back into medieval England and beyond. What changed was the metric value of the grain. When Britain and the United States agreed in 1959 to fix the pound at exactly 0.45359237 kilograms, this locked the grain to 0.06479891 g. But earlier, when the grain was derived from the weight of a cubic inch of water or from late Egyptian standards, its value hovered around 0.0648–0.0650 g. Thus, while the count of 7000 grains to the pound never wavered, the true weight of the grain slid slightly across history, depending on whether it was tied to water, to the yard, or later to the kilogram.

   This slight drift in the grain’s value mirrors the fate of the water standard more broadly. At modest levels of accuracy, a cubic inch of water at a given temperature, or a grain defined through the pound, appeared perfectly stable. But once measurement entered the realm of parts per million, decisions on how to define the units were needed. Ever greater scientific precision revealed an ever greater array of conditions to agree upon. The consequence of indecision was to throw the baby out with the bathwater, and just accept some value that already was backed by widespread agreement, which for the imperial gallon and pint happened to be the cubic decimetre, or litre.

   By 1963, the British had abandoned any pretence that the imperial gallon and pint were anchored in natural constants such as the weight of water. Instead, they redefined the gallon as exactly 4.54609 litres, with the pint following as one-eighth of this volume, preserving the old divisions at least, with the use of the number 8. In doing so, the UK did not simply adopt the metric system; it attached its units of length, weight and volume to a system that had had to distance itself from the physical things it had been legally defined in relation to water, and even, arguably, the polar circumference. The litre was at least clearly defined in relation to a linear unit, the metre (or decimetre). If the British had kept the Queen Anne wine gallon, as had been done in the States, they could have had a unit of volume defined in relation to a simple linear unit, the inch. But instead, both the long historical use of the imperial units of volume, and the claim to a natural standard, had been relinquished. The inch itself, though continuous in practice with its historical usage, was redefined in the mid-20th century so as to align exactly with the metric system: 1 inch = 2.54 centimetres (since 1959). The inch, and the measures that depend on it, such as the foot and the yard, as well as the pint, and gallon, and other measures of volume which depend on them, are all directly dependent on the metric system. The metric system itself is dependednt on a unit of length derived from a particular measurement of the earth´s polar circumference, which was accepted at a certain time in history, but is now considered outdated. The illusion of an ancient pedigree, of independence from French measurements, is quite convincing, but the gallon and pint in the UK today are as much British as they are French.

Perhaps that's as it should be. After all Britain and France have a long history of similar measures. Before the metric system was introduced, the units of volume in France were similar to the US gallon, the ancient Queen Anne wine gallon. They even used pints in France, as Jon Bosak has noted, which were very close in measure to the US pint today:

Historical associations of the US Gallon and Pint

Compared to the modern British pint and gallon, the American gallon stands out as remarkably stable. Its Queen Anne wine-gallon origin, defined geometrically in terms of 231 cubic inches, links back seamlessly to older English measures, and to even older Babylonian and Egyptian ones. The US gallon preserves centuries of practical and symbolic relationships between length, volume, and weight, a system both intelligible and enduring.

1. Connecting the US pint to historical measures of volume

   These words of Livio Stecchini's are the result of a detailed study of ancient measures. They tell of a very long standing tradition. Many ancient civilisations, including the Babylonians, Egyptians, and Romans, used interrelated systems of length, volume, and weight. These systems were not arbitrary but designed with reference to geodetic values, and based on elements of the natural world, such as water and barley, and also the seven heavenly bodies. The study of units of length, weight and volume provides an intriguing connection between us and remote times. As Frank Skinner points out:

   Jon Bosak has documented the long continuity of the US measures, including measures of volume, all the way back to Egypt and Babylon. He notes that one of the earliest surviving artefacts of volumetric measure, a silver vase attributed to King Entemena of Lagash (c. 2400 BC, now in the Louvre), has a carefully measured capacity of 471 mL (28.74 cubic inches). This corresponds almost exactly to a US pint, of 28.875 inches, one eigth of the Queen Anne wine gallon used in early modern England. Over a millennium later, a Neo-Babylonian vessel from Persepolis (c. 600 BCE) was found to contain 944.9 mL, aligning closely with the modern U.S. quart of 946.3 mL. These survivals suggest that the pint–quart–gallon family is a very old inheritance, rooted in geometry, whose coherence was recognised across successive civilisations.

Other pieces of evidence include the tomb of Hesy in Egypt, dating from around 2650 BC, within which murals show life-sized measures of volume. There are copper vessels in two sets, on based on 28.8 cubic inches, and another based on 21.6 cubic inches. The US pint today measures 28.875 cubic inches, and 8 pints of 28.8 inches create a gallon of 230.4 cubic inches. This is the volume which produces the Saxon foot of 13.2 inches, and which is itself the product of multiples of 12, with 28.8 being 12 x 12 x 2 / 10 and 230.4 being 12 x 12 x 16 / 10. The 21.6 cubic inch measure is itself 6 x 6 x 6 / 10 cubic inches, and as Jon Bosak points out:

It was Petrie who measured these units of measure and established their basis in multiples of 6 and 12 inches. As Bosak remarks:

Bosak also identifies a unit of 230.6 cubic inches from Persepolis, which also corresponds to a US gallon, or very nearly, as it is just slightly under the value derived from a cylinder of base radius 3.5 inches and height 6. If using a value for pi equivalent to the one in a calculator instead of 22/7, it is just slightly over the value derived from a cube with sides 12, 12 and 16, divided by 10.

Another unit of volume identified by Jon Bosak is from Olympia, dug up in 1954, and is 28.99 cubic inches, which is very close to the US pint. (15) Jon Bosak also presents evidence ranging from the Hebrew bath to the gudda or cuddy of Arabia, all of which are closely related to the US gallon. As Jon Bosak explains, "1 U.S. fluid ounce of wine weighs 1 tower ounce. And this means that the kotyle, and the 21.6 in³ unit found in Hesy’s tomb, are the same as our 12-ounce can or beer bottle. Next time you’re holding a 12- ounce can of soda, note the weight; that’s a tower pound."

2. The Saxon Foot

   
   

      As Jon Bosak (10) has noted, if we take the Queen Anne gallon (231 in³), then ten such gallons form the cube of a Saxon foot, of 13.2 or 13.22 inches. This is an amazing insight: the same unit of volume can be arrived at by constructing 10 cylinders, each with a base diameter of 7 inches and a side of 6 inches, or by constructing a cube with sides of 13.22 inches. This length of 13.2 or 13.22 (or 13.125) inches is a key unit in the history of metrology, called Saxon, Northern, Drusian or Sumerian foot. Three of these feet make a wand or ancient metre of 39.375, 39.6 or 39.623 inches, close to the modern metre of 39.3700787 inches. In other words, embedded in the very definition of the US gallon is a hidden cube that points straight to a foot and yard long known in Europe, and one that shadows the metric unit with surprising accuracy.

   Units of length can be linked up via units of volume. For example, as Jon Bosak points out, the Sumerian foot of 330 mm can be connected to the Egyptian cubit, as 20.624 inches. One cubic Egyptian cubit of 20.624 inches is equivalent to 4 cubic Sumerian feet of 330 m.

   This 13.2-inch foot is widely attested in pre-modern practice. Sir Flinders Petrie, in his article Weights and Measures for the 1911 Encyclopaedia Britannica, wrote, referring to it as Belgic:

   This is an interesting aspect of the Saxon foot, and by extension to the US gallon: that it has possible ties to a decimal system of measure, and to a unit close to a third of a metre.

     John Neal agrees with Flinders Petrie that the Belgic foot is essentially one third of a metre, "commonly known as, a Belgic foot, i.e. nine to eight of the Roman foot, as established by Nero Claudius Drusus, the Roman governor of Gaul." (8) Neal adds however:

   But whatever the denomination, a unit of 13.2 or 13.22 inches is linked to the US gallon. A third of the modern metre is 13.12336 inches, a third of an ancient metre of 39.375 inches (used in the 19th century by metrologists such as C. Mauss also) is 13.125 inches, and a third of a Saxon wand is 13.2 inches. Flinders Petrie used a slightly different metric-imperial ratio, writing in 1911: in his article for the Encyclopaedia Britannica, he gives 39.37 inches for the metre in the United States and 39.3700113 inches in Great Britain, which would yield a slightly shorter foot of 13.12334 inches.

   Flinders Petrie doesn't state what he believes the Belgic foot to measure exactly, but he equates it a series of measures close in value found in various sites, close to 13.3 inches or 13.22 inches. Flinders Petrie explains:

   What remains of the predecessor of the imperial system in Britain is indeed decimal: a mile is 8 furlongs, but was once 10 furlongs. A furlong is 10 chains, a chain is 10 fathoms. 4800 feet of 13.2 inches fit into a mile. A foot of 13.2 inches gives a yard of 39.6 inches, which is 54 x 0.733333 inches. A unit of about 40 inches was in use, called the yard and full hand, and a third of that would have been 13¹/³ inches. Flinders Petrie goes on to say:

Petrie also finds a measure 26.43 inches, twice 13.215, in Egypt, in a cubit rod at Kahun. (12)

     As Jon Bosak has observed, if the Queen Anne wine gallon were slightly smaller than it is now, instead of 231 cubic inches, 230.4 cubic inches (12 x 12 x 16 / 10), then the cube root of that would be 13.2077 inches, which is an interesting connection to the other main value observed for the Saxon foot. A linear measure of 13.2 inches is therefore the cube root of a volume equivalent to a cuboid of 12 x 12 x 16 inches. This volume, however, would be equivalent to 10 cylinders with a base of 7 inches diameter and a height of just under 6 inches (5.98).

   In addition the cubit of 21 inches may be linked to the Saxon foot of 13.22 inches, via a measure of volume, as 40 Queen Anne wine gallons make 21³ inches, approximately. So 40 cylinders with a base diameter of 7 inches, a height of 6 inches, and with pi as 22/7 are 9 240 cubic inches and the cube rot of that is 20.984 inches.

   It´s interesting to note, also, that a 400th part of a US gallon, when computed based on the volume of a cylinder with base raddius 3.5 and height 6 inches, and with pi as the value given by a calculator, not 22/7, is 0.577268 cubic inches, and this value is very close to 1/√3. That is to say that the US gallon can also theoretically be understood as 400 / √3 cubic inches (230.9401).

3. The Greek foot

   Another line of connection runs through the Greek foot of 12.15 inches, a fundamental unit in metrology. If we take the Queen Anne wine gallon, the direct ancestor of the American gallon, as a cylinder six inches high and seven inches across, so a volume of π × 3.5² × 6 ≈ 231 in³, and adjust it by the factor 70/9, the cube root of the resulting volume comes out very close to 12.15 inches, the Greek foot. Using the value for π given by a calculator, the result is 12.155 inches; with the ancient approximation 22/7 it is 12.157 inches. In either case the agreement is intriguing, and shows that the gallon defined in Britain in 1707 and inherited unchanged by the United States also preserves a link back to the canonical Greek foot. The American gallon therefore stands not only on the Saxon foot but also on this classical foundation, binding together northern and Mediterranean traditions in a single coherent measure.

The number 230.4 equals 8 × 28.8, and 28.8 cubic inches is one of the “Hesy vessel” measures documented by Flinders Petrie in the tomb of Hesy, alongside 21.6 cubic inches. In other words, a gallon of 230.4 in³ sits naturally on the same Egyptian volumetric ladder. The statutory 231 in³, by contrast, achieves its elegance only by way of a fractional approximation of π (22/7), since the Queen Anne wine gallon was defined as a cylinder of radius 3.5 inches and height 7. This formula reproduces 231 exactly, but sacrifices the neat factor structure.

   If we treat 230.4 as the intended gallon, and then multiply by 70/9, the result is 1792 cubic inches. 1792 = 64 × 28. Why multiply the gallon by 7? The septenary pattern runs like a hidden thread through the history of weights and measures. In England, the avoirdupois pound was fixed at 7 000 grains, and from there multiples of 7 cascade through the system: the 14-lb. stone, the 28-lb. quarter, the 56-lb. sack, even the 91-lb. “quarter wool sack” preserved in bronze weights at Winchester. A set of the weights from 1340 have been preserved at Winchester, embossed with the Plantagenet arms, and the series is the denominations being 7, 7, 14, 28, 56 lb. of A.D. 1340 and 91 lb. (the Quarter Wool Sack) of A.D. 1357. Taking the cube root of 1792 cubic inches gives 12.146 inches, which is the value of the classical Greek foot (≈ 12.15 inches). This volume could also be conceived as a cuboid with sides of 8, 14 and 16 inches. A volume of 1792 in³  is 5/6 of a Winchester bushel, 10/3 pecks, and 20/3 dry gallons. And, ∛17923 =12.146 in, essentially the Greek foot (≈12.15 in). Thus, from Hesy’s 28.8-inch³ rung to the Greek foot and into the American dry measures, the same scaffolding reappears. The legal gallon of 231 in³ can then be seen as a pragmatic use of 22/7, as an approximation of π, while the underlying 230.4 in³ preserves the older harmonic lattice.

   .

The inch is key

All the units of measure of volume considered so far, with the exception of the litre, rely on geometries expressed in integer numbers of inches. The inch is a key ancient measure.

It seems the US gallon can be understood as either derived from a cylinder, with base diameter 7, height 6, in which case the volume is 229.6 cubic inches, with the value for pi being the one in a calculator, or 231, with pi being 22/7, or from a cuboid, giving a volume of 230.4 cubic inches. Saxon feet of 13.22 and 13.2 inches derive from these alternative geometries, as the cube roots of the volumes created. As Bosak notes, the volume of 10 US wine gallons is equivalent to the cubic Saxon foot, but as a length of 13.22 inches, but with a unit of 230.4 cubic inches, aligned with the number 12, then the resulting Saxon foot, is 13.2 inches

Cubic root of a cuboid with sides 12, 12 and 16 inches:

∛(12 x 12 x 16) = 13.2077

Cubic root of 10 US gallons, based on the cylinder with radius 3.5 and height 6 inches:

∛(3.5² x 6π x 10) = 13.21739 (calculator value for pi)

∛(3.5² x 6π x 10) = 13.21739 (pi as 22/7)

The inch may descend from a forgotten but precise geodetic standard used to harmonise human measures with natural laws. The metre is linked to the polar circumference of the earth, and the inch to the equatorial circumference. Indeed, it can be estimated at 365.242199 x 4 320 000 inches, which is the number of days per year, and a key harmonic number, linked to cycles of time, notably in the Indian tradition.

How much does a cubic inch of water weigh?

The 1911 Encyclopædia Britannica entry for the Avoirdupois pound (c0-written by Flinders Petrie) is as follows:

In the late nineteenth century, precision science had caught up with the standard. Physicists such as Guillaume, Chappuis, and Mendeléeff demonstrated that the density of water varied minutely with temperature, barometric pressure, dissolved gases, and even isotopic composition. Tables of values proliferated, showing cubic inches of water weighing 252.854 grains at 4 °C, 252.568 grains at 62 °F, or 252.407 grains at 68 °F. A few hundredths of a degree, or a shift in air pressure, was enough to change the value. By the middle of the twentieth century, the belief in an water standard had collapsed, as nobody could agree on what conditions to define the weight of water by. When, in 1824, the UK had deliberately sought to anchor its measures in nature, specifically water, for the measures of volume, the attempt had ultimately been in vain. The high sensitivity of water to various conditions made it simply too tricky to use the weight or volume of water a standard, in a world in which scientific accuracy was becoming ever greater. In the end, a value for the UK gallon was chosen almost at random, corresponding to unspecified conditions in a laboratory, to anchor the rest of the system of volumes and weights to. But perhaps this commendable ideal of using water, as barley had also been used, to base the system in, had not been without precedent.

The grain is a unit of weight, and is a foundation of the entire system. 7000 grains make a pound. The imperial gallon was defined until 1963 as containing 10 pounds of water at 62° Farenheit, equivalent in volume to 277.274 cubic inches, and each cubic inch was defined as containing 252.724 grains of water in a vacuum at 62°, or 252·458 grains of water weighed with brass weights in air of 62° with the barometer at 30 in. The number of grains per gallon of water doesn't seem, on the face of it, very promising in terms of finding connections between the weight and volume of water. But what if the conditions under which water is measured, which caused so much trouble to the French in the 18th century and the British in the 19th century, were tweaked somewhat? This was Stephen Dail's insight.

Water reaches its maximum density at approximately 4°C (39.2°F), and this is the temeprature used to define the gram. 1 cm³ of water at 4°C weighs approximately 1 gram, and 1 in³ of water = 252.88679 grains, one cubic foot of water weighs 436,988.37 grains, or 62.42691 Avoirdupois pounds (7000 grains to the pound).

At slightly higher temperatures, water becomes less dense. Around 80°F, 26.67°C, a cubic inch of water weighs 252 grains, and one cubic foot of water weighs closer to 62.208 pounds. This is 12⁵ / 4000, or 12⁴ x 3 / 1000, and an intriguing connection to a duodecimal way of counting. The mean circumference of the earth is given by John Michell as 12⁵ /10 miles, and it is indeed very close to the modern estimate. Stephen Dail´s insight that 400 cubic feet of water at 26.67 C weigh 12⁵ /10 pounds is very significant.

It seems that from the outset, the cubic inch, the barley grain, and the pound were linked through water. If there are 252 grains per cubic inch of water, then, many of the apparent inconsistencies in ancient units of length, volume, and weight fall away. Stephen Dail demonstrates that the ancients had built a metrological framework on just such a foundation. Drawing also on Jon Bosak´s research into the density ratios of Water to Barley being 8/5 and Barley to Wheat being 4/5 making Water to Wheat as 32/25, Stephen Dail was able to place "all of those previously separate weight and volumetric units within into a coherent singular system, with many dynamic parts that made perfect sense. Seeing how these same ratios would play out over a variety of different aspects such as the Babylonian Mina of 7776 grains being in ratio 8/5 to the cubic inch of Gold at 4860 grains, or the Roman land area Jugerum unit of 27,225 sq. English Ft. x 8/5 = the Saxon acre of 43,560 sq. ft." (14) The Roman Congius (400,000 grains) aligns precisely with such a system:

• 400,000 grains ÷ 252 grains/in³ = 1,587.3 in³

• Taking this as a cube yields a side of 11.736 inches

• A Roman foot of 11.736 inches (the Pes Monetalis) is in line with archaeological data

This coherence is lost when one uses grams and litres.

 Stephen Dail also proposes that the true Egyptian cubic cubit had a volume of 8 750 cubic inches. This is significant because it relates to the avoirdupois pound as 5:4 (8750 ÷ 7000 = 5/4), a simple harmonic ratio. From here he derives strikingly coherent grain totals:

• 8750 in³ × 252 gr/in³ = 2 205 000 grains

• Divided into 300 hon = 7350 grains per hon

• Deben = 1470 grains (via 15 000-part division)

These values generate totals compatible with the later Avoirdupois pound (e.g. 315 lb in all), connecting the Egyptian system to the modern pound.

Traditionally, the “talent” was an abstract weight. Dail reinterprets it as volumetric: 2000 in³ × 252 gr/in³ = 504 000 grains = 72 lb. He also reintroduces the Nilometer cubit (17.71 in = 5555.6 in³ = 1 400 000 grains = 200 lb), the Portuguese pound (7078.5 gr, from 8640 in³ × 196.875 gr/in³), and others.

This raises the question: are we dealing with units defined both by geometry and by water-weight? On one side we see simple geometric forms — cubes, cylinders — chosen for their neat integer volumes when computed with approximations such as π = 22/7. On the other, those same volumes are anchored to the grain-water standard (≈252 grains per cubic inch), so that length, volume, and weight were instantly convertible in practice. The repetition of a small core of integers and ratios — 252, 108, 7 776, 54 000, 7 000 — looks precisely like the hallmark of a system built for divisibility, testability, and coherence across trade, architecture, and geodesy.

The grain sits at the heart of this design. The modern grain is legally 0.06479891 g, defined as 1/7000 of the avoirdupois pound of 0.45359237 kg. Historically, the grain as a unit of weight varied slightly, 0.0648 g, 0.0650 g, but the 7000-grain pound remained constant. In Dail’s framework, the grain is not arbitrary at all: it is one 252nd of a cubic inch of water.. Bosak has shown that the density relationships between water, barley, and wheat fall into simple integers:

• Water : Barley = 8 : 5

• Barley : Wheat = 4 : 5

• Hence Water : Wheat = 32 : 25

These ratios reverberate through early standards. Stephen Dail has shown that the Babylonian mina of 7776 grains stands in 8/5 ratio to the cubic inch of gold at 4860 grains, and the Roman land unit jugerum of 27,225 sq ft × 8/5 yields the Saxon acre of 43,560 sq ft. In English law, a pint of wheat weighed a tower pound, while a pint of wine weighed a merchant’s pound, precisely in proportion to their densities. Thus trade goods, weight systems, and volumetric standards were interlocked not by coincidence, but by calibration to these density ratios. At the core is the inch, itself defined in English law as the length of three barleycorns laid end to end, but not confined to the English or imperial system.

When modern science rejected water as a standard, only to replace it with increasingly abstract definitions (first metal bars, then wavelengths, then the speed of light), an ancient system that had been built on grain and water was partially abandoned. In fact, by redefining its standards in terms of blocks of metal or speeds, it could be argues that in effect the ancient system has been abandoned in all but names. Water, at the right temperature, formed the very backbone of a coherent system. In the interplay between barley, wheat, water, grain, inch, foot, pound, and Earth measure, we glimpse an ancient metrology that was not only practical but cosmological: a metrology where to weigh a pint of wheat, or a cubic foot of water, was also to measure the world. Weight, volume, and length were engineered together, not created in isolation.

From the weight of a single barley grain to the size of the Earth, ancient systems of measurement reveal a level of integration that is both practical and profoundly systematic. The pint, the grain, the pound, the cubic inch are parts of a metrological language that once spoke fluently across domains of agriculture, astronomy, and geometry. That so much of this structure survived, often in customary use, suggests not merely tradition, but enduring function. In restoring the avoirdupois pound and related units to this broader framework, it becomes clear that the past did not lack precision, only our understanding of its underlying structure.

Anchored in Approximation: The Paradox of Exactitude

In a sense, the more precise science became, the less usable water became as a foundation. But the move away from water and grain to base the system of measures on also reveals a lack of vision for a unified, coherent system, which does not rely on speeds, or artefacts held under lock and key. The move towards using the speed of light tells a story of speed understood within space time, but it is difficult to extract from this speed a unit to measure time with, and another to measure space with, as speed depends on both. It appears precise, but the precision is tautological. It tells a story of a science overwhelmed by the precision it was able to wield in measure, and which could not find anything in the natural world that was constant enough to meet the new standards of exactitude. Water and grain, for all their variability, offer a natural unity. A grain of barley can act as a weight, volume, and a length; a cubic inch of water tied weight, length, and volume together. The elegance lay in the way one substance such as the grain or barley corn could link space, matter, and even time (in the Hebrew calendar).

To use a speed as the cornerstone of a system is to collapse the distinction between the two. The old system had a unity: a cubit, a pound, a day were distinct but interlocking. The new system has no such harmony; it relies on abstract artefacts (the caesium hyperfine transition, the Boltzmann constant, Planck’s constant), which are exact by definition but opaque to ordinary intuition. In that sense, the shift away from water and grain was not just about precision, but discarding the quest for a unified, comprehensible system.

In that sense, the history of metrology is one of shedding natural approximations (barley, water, metal bars) in favour of abstractions grounded in invariants (geometry, atomic transitions, physical constants). The cubic inch of water, linked to 252 grains, sits very close to the heart of historical metrology. It was a bridge between length, volume, and weight. The 1824 Imperial gallon carried that idea forward; its abandonment in 1963 marked the end of a millennium-long tradition.

It is one of the paradoxes of the history of measurement that the modern age of science and technology, with its passion for precision, often fixed its standards on foundations that turn out to be approximate or randomly selected.

Stephen Dail has shown that measures interrelate across domains, bound together by harmonic constants such as 875/864, and that the 7000 grain pound is natural, not arbitrary. By showing that water weight in grains, at specific cubic-inch volumes, consistently produces Avoirdupois-like figures, he argues that the pound arose organically from water measurement, rather than by fiat. There is an extraordinary continuity from Egypt to modernity, as Jon Bosak has demonstrated for the ÚS gallon. Units such as the hon, deben, and ro find echoes in Middle Kingdom Egypt and in the English system. What modernity treats as an imperial holdover may in fact be a remnant of an ancient global order.

Most academic historians of measurement interpret the evolution of units as local, adaptive, and pragmatic, subject to error and mutation. It is indeed quite true that over time, in different places, measures were modified slightly. This makes the fact that the original design of the system of measures from the ancient world can still be understood all the more remarkable. In the end, the paradox remains: the modern age, capable of great exactitude, has moved away from the ancient anchor of the system, water, barley, the earth itself, as it could not find a way to measure them precisely enough.

Another intriguing departure from ancient systems lies in the starting point. Today, both the metric and the imperial frameworks derive measures of weight and volume from linear standards. The litre is defined as the cube of the metre divided into tenths, the gram originally the weight of a litre of water; the U.S. gallon emerges from the inch and a cylinder six by seven inches. In each case, the line is the foundation, and weight and volume follow.

By contrast, the evidence of historical metrology suggests that in antiquity it may sometimes have been the other way round. Units of length such as the Greek foot, the Roman foot, the Saxon foot, and even the Egyptian royal cubit can plausibly be reconstructed as secondary, derivatives of volumetric standards. In this view, the cube, not the line, is primary.

Stephen Dail’s “Egyptian cubic cubit” of 8750 cubic inches, multiplied by the weight of water per cubic inch (252 grains), yields 2 205 000 grains. Divided into 300 hon, this produces 7350 grains per hon, and by extension a deben of 1470 grains. The ratio at the heart of this structure, 8750 ÷ 7000 = 5/4, is not just a number but a simple harmonic, the perfect fourth in music.Traditionally the “talent” has been treated as an abstract unit of weight. Dail reinterprets it volumetrically: at 26.67 degrees celsius, 2000 in³ of water weighs 504 000 grains, or 72 lb. Likewise, the Nilometer cubit (17.71 in), with a volume of 5555.6 in³, corresponds to 1,400,000 grains, or 200 lb. Even the Portuguese pound of 7078.5 grains finds its place when derived from 8640 in³ of water at 196.875 gr/in³.

If this reading is correct, the very logic of ancient metrology differs from our own. Modern systems take line and plane as their basis. The ancients, by contrast, may have anchored their measures in volume and its relation to water and grain, a system at once practical, harmonious, and cosmological.

The inch, in this model, emerges not as an arbitrary medieval holdover, but as part of a coherent and deeply symbolic metrological matrix that may have once unified measures of weight, mass, time, and celestial proportion. While modern science has moved toward the metric system, the inch-based system appears to preserve a lineage of thinking in which the structure of nature was mirrored in the act of measurement. The ancient system, once abandoned for convenience and laboratory precision, remains an intellectual artefact of the ancient world, sophisticated, empirically grounded, and remarkably resilient.

At first glance, historical units of measure, suc as pints, gallons, pounds, and feet, appear messy, inconsistent, or purely conventional. Yet a closer look reveals something astonishing: these units interlock with mathematical precision, linked through water, volume, grain, and even harmonic ratios reminiscent of musical tuning. What might seem like arbitrary relics of local tradition are, in fact, surviving fragments of a coherent, ancient metrological system, one that may stretch back to Sumer and Egypt. The Avoirdupois pound, the Saxon foot, and the US and UK gallons and pints are living evidence of a system designed for precision, harmony, and global applicability, hinting that the ancients understood measurement as both a practical and intellectual pursuit. So while we're technologically more advanced, we may have lost the cultural or cosmological unity that older systems preserved.

Conclusion

We are not so different from our ancient ancestors. We still drink pints, for example. The pint is one of those units that seems stubbornly ordinary, yet it carries within it the memory of water and barley, the two great foundations of civilisation. Through beer and bread, these measures tied agriculture, sustenance, and sociability into a single fabric of meaning. Yet today, in Britain and Ireland, our pints are poured into glasses defined not by this brilliantly coherent system of water and grain, but by the compromises and shortcuts of modern law, approximations whose neatness conceals their weakness.

The irony is that the metric system, with its obsession with exactitude, is not really any firmer. The metre was meant to be a natural measure, one ten-millionth of the Earth’s quadrant, yet it soon slipped its tether, successively redefined by metal bars, atomic vibrations, and finally the speed of light itself. Each redefinition created ever more precision, but at the cost of coherence. The kilogram, once tied to the density of water, now hangs on Planck’s constant: elegant in theory, but divorced from the rhythms of nature. Our measures have become definitions that float above the world, consistent but abstract, brilliant but detached.

By contrast, the older measures were entangled with the real. The cubic inch of water weighing 252 grains; the pound of 7000 grains, equal to 27.777 cubic inches; the US gallon of 231 cubic inches, resting on a cylinder built with π as 22/7; the Saxon and Greek feet, which reveal themselves not as arbitrary lengths but as echoes of volumes and weights. Research by Jon Bosak has shown the simple elegance of the Queen Anne wine gallon; Stephen Dail has demonstrated the harmonic depth of the 252-grain standard and the 8750-inch cubic cubit. Taken together, they reveal a system not chaotic but deliberate, mathematical and musical, rooted in nature’s own constants.

What should a system of measure be based on? Our ancestors answered: water, grain, and number. By doing so they created a unity between volume, weight, and length that resonated with agriculture, astronomy, and even cosmology. By abandoning that unity, we gained precision but lost vision.

To study the pint, the pound, the foot, and the gallon is to rediscover that older vision. They are fragments of a poem we no longer know how to read, yet still recite daily. They remind us that measures were once bridges between human needs and the larger harmonies of Earth and sky. And perhaps, when we next raise a pint, whether defined by 231 cubic inches or 568 millilitres, we might pause to taste in it not only barley and water, but the long memory of a system that bound measure to meaning.

Notes

1. Livio Stechini, The Origin of Metrics, https://web.archive.org/web/20190913231201/http://www.metrum.org/measures/metrics.htm

2. Jon Bosak: "I’m sorry to say that the quarter-century delay in completing this history means that I’m not the first person to have published this observation. In finishing up the book, I came across a work by Daniel McLean McDonald, a British industrialist who apparently created an institute in order to publish a collection of his papers titled The Origins of Metrology (Cambridge: McDonald Institute for Archaeological Research, 1992), on page 8 of which he notes that (McDonald gives all of his figures to 10 places). McDonald has apparently never heard of the Northern foot and is incorrect, in my opinion, in considering this foot to be the Sumerian foot; in fact, it seems to me that he is incorrect in most of his conclusions. But while I had already noticed the relationship between the gallon and this cubic foot back in the mid-1980s, McDonald gets the credit for publishing first."

3. Jim Wakefield, Rollrights

   6. Jim Wakefield, The Great Pyramid of Giza's length can be interpreted as 687 Saxon feet of 13.2 inches, with 687 being a number derived from the cycle of Mars, in days

   7.Flinders Petrie, M.W. 1911, "Weights and Measures", Encyclopædia Britannica 1911 Encyclopædia Britannica/Weights and Measures - Wikisource, the free online library

   8. Neal, John, August 06, 2003, post on Graham Hancock Message Board,

   Re: AOM anwers to Ilona Masserena - Graham Hancock Official Website

   9 Neal, John, July 2003, "Ancient measurement systems: Their fractional integration", Author of the Month on Graham Hancock website, Ancient Measurement Systems: Their fractional integration - Graham Hancock Official Website

   10 Bosak p 49

   11. Skinner, Frank, 1951, The English Yard and Pound Weight

   12. Flinders Petrie, M.W. 1890, Kahun, Gurob and Hawara, p 27

   13. Bosak p.57

   14. Stephen Dail, personal correspondence

   15. Bosak p 62

   16. Watson, Sir C. M., 1910, British weights and measures as described in the laws of England from Anglo-Saxon times, John Murray, Albermarle Street,

   https://ia601606.us.archive.org/25/items/britishweightsme00watsuoft/britishweightsme00watsuoft.pdf p 23

   17. Bosak p 57

   
   

Bibliography

• Berriman, Algernon Edward. 1953. Historical Metrology: A New Analysis of the Archaeological and the Historical Evidence Relating to Weights and Measures. London: Dent.

• Bosak, Jon. 2013. Weights and Measures: From the Ancient World to the Modern. Self-published research, accessed via http://www.jonbosak.org.

• Dail, Stephen. Various papers on ancient metrology. See e.g. Ancient Metrology: A Rational System of Weights, Measures and Harmonics. Academia.edu

• Encyclopaedia Britannica. 1911. Avoirdupois. Vol. 3. Cambridge: University Press.

• Encyclopaedia Britannica. 1911. “Weights and Measures.” By William Matthew Flinders Petrie and Henry James Chaney. Vol. 28. Cambridge: University Press.

• Mauss, C. LÉglise de Saint Jeremie.

• Neal, John. 2025. The Measures of Gaul. Academia.edu. https://www.academia.edu/129581233/The_Measures_of_Gaul.

• Quibell, James Edward. 1913. Excavations at Saqqara, 1911–12: The Tomb of Hesy. Cairo: Institut français d’archéologie orientale. https://archive.org/details/cu31924028671299/page/n97/mode/2up.

• Skinner, F. G. 1951. “The English Yard and Pound Weight.” Bulletin of the British Society for the History of Science 1 (7): 186. https://doi.org/10.1017/S0950563600000646.

• Stecchini, Livio C. 1971. “The Historical Metrology of the World.” In The Secret of the Great Pyramid, by Peter Tompkins. New York: Harper & Row.

• Zupko, Ronald Edward. 1977. British Weights and Measures: A History from Antiquity to the Seventeenth Century. Madison: University of Wisconsin Press. https://archive.org/details/britishweightsme0000zupk/page/200/mode/2up.