# Decimal measurement of area

Martin Vliestra looks at key steps in the search for a simple system of measuring area.

One of the earliest practical applications of decimal numbers was the invention in 1620 by Edmund Gunter of what is now known as “Gunter’s chain”.  Traditionally land surveyors used metal chains of a known length with links of a constant size to measure distances.  Gunter’s chain (now known simply as a “chain”) is 22 yards in length and has 100 links, each of 7.92 inches.  Although these measurements might look arbitrary, they certainly helped the land surveyor because:

• There are 100 links in a chain
• There are 10 chains in a furlong
• There are 10 square chains in an acre

There had been a number of developments in mathematics in the years preceding Gunter’s invention. Although Fibonacci had introduced the Hindu-Arabic numeral system (which, unlike the Roman numeral system had a symbol for zero) into Europe in the thirteenth century, its adoption had been slow. In 1585 Simon Stevin showed how to extend this system to describe decimal fractions. This was particularly useful since the system of representing fractions using Roman numerals was very cumbersome and limited

In 1614 John Napier published the book “Mirifici Logarithmorum Canonis Descriptio” in which he described the use of logarithms in multiplication and division. This, coupled with the developments in trigonometry at the same time, made it possible for surveyors to calculate the areas of irregularly-shaped fields.  One technique is to divide the field up into triangles and for each triangle to apply the classic formula which is (or should be) known by every schoolboy and girl:

Area = ½ A.B.sin theta

Unless A and B (the sides of the triangle) are quoted using decimal numbers, the calculation is extremely complex. Since Gunter developed his chain so that each link was 0.01 chain, it was relatively easy to measure the side of a field to two decimal places by merely counting the links. Although the final answer came out in square chains, it was not difficult to convert this to acres, roods and perches to keep the lawyers happy.

Does this mean that we should retain the chain and the acre? Before the UK’s metric changeover in the 1970s, it was normal to quote the areas of land in acres, roods and perches (40 perches = 1 rood and 4 roods = 1 acre). In the case of farms this was manageable, though if roods and perches were also used for added accuracy, the quantities became unwieldy.  The use of acres for measuring the areas of counties resulted in unwieldy numbers – for example Lincolnshire is about 1.72 million acres.  At the other end of the scale, domestic plots of land would be measured in perches (if they were measured at all) which does not easily convert to acres and are not that well understood.  Square yards would be far more sensible than roods, but there is an inconvenient factor of 5 ½ when converting between yards and poles, rods or perches.  In addition, if one is measuring the footprint of a building, one would use square feet rather that square yards.

It goes without saying that square metres are a sensible unit of measure for expressing the areas of house footprints and domestic plots of land. Once the numbers start to get large, square metres transform seamlessly into hectares (1 hectare = 10 000 square metres), a suitable unit in which to express the area of a farm and thence into square kilometres (one square kilometre = 100 hectares), a suitable unit to express the area of a county.

I know that for many people, the statement that Lincolnshire has an area of 6,959 square kilometres is pretty meaningless, but if this figure is quoted to me, I will mentally work out that the square root of 6959 is a bit over 80 (80 x 80 = 6400), so the area of Lincolnshire is equivalent to a square with sides of 80 kilometres (80 km is the approximate distance between London and Brighton). There is no easy way to visualise 1.72 million acres!

## 9 thoughts on “Decimal measurement of area”

1. Daniel says:

It is odd that Gunter chose 22 yards for his chain and not a more rounded 20, such that today, it is equal to about 20 m almost exactly with only 117 mm difference. That chain today could as well be considered to exactly 20 m with minimal noticeable difference. Who knows, that 117 mm difference may have been a different amount in times past as the foot has never been constant and varied over the centuries.

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2. John Steele says:

It is not quite as easy but there are 640 acres in a square mile, and the same process indicates Lincolnshire is equivalent to a square just under 52 miles on a side. The 640 acre per square mile figure is pretty well known in the US because most of the country was surveyed in one square mile sections under the Public Land Survey System in the 1800s, then further subdivided. Modern surveying is decimal feet here, we abandoned the chain a long time ago. Only the acre survives as 43560 ft². Home lots are in square feet and larger properties in acres.

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3. Daniel says:

It isn’t a surprise that the Americans abandoned the chain. The chain is decimal based an Americans loathe decimals. They prefer fractions. Yes, the money system is decimal based, but I’m sure it was foisted on the population without a vote. If there was a vote they would chose a fractional system. In fact the coinage in the US and Canada is not decimal, but fractional. Instead of coins being in the 1:2:5 series, they are in a mixed decimal-fractional series. The lower coins (1, 5 & 10 ¢) are in a decimal format but rarely if ever called by their cent names, instead are called penny, nickle and dime. The penny and nickle are marked as “one cent” and “five cent”, but the 10 ¢ coin is marked as “one dime”.

There is no 2 and 20 ¢ coin, instead there is quarter dollar and half dollar instead of 25 and 50 ¢ piece. The half dollar is not used even though coins are minted and can be found at some banks. In summary, the Americans have gone to a great effort to fractionalise their decimal coinage as best as possible.

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4. John Steele says:

American surveyors abandoned the chain in favor of decimal feet to two decimal places. 0.01 ft is approximately one eighth of an inch ( 0.12 in, exactly), substantially higher resolution than the 7.92 in link, so abandoning the decimal chain doesn’t quite ring true.

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5. @John Steele: May I remind you that the precesion to which one can measure depends on the instrument used rather than the system of units chosen. In the case of digital instruments, one should check what the design value of one bit is. For example, my digital thermometer was designed such that one bit equalled 0.1 degrees Celsius and the Fahrenheit display is calculated from the Celsius reading, not measured separately. Such a calculation CANNOT add extra accuracy, it can only give you the same accuracy or it can degrade the accuracy.

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6. John Steele says:

@Martin Vliestra
True enough, but the chains I have seen as examples are not subdivided below the link, and results were reported in chains and links. Long steel tapes are marked to 0.01 feet (US surveyors use decimal feet, not inches), the plat for my lot (and the whole subdivision) is to the hundredth and even the irregular lots seem to close to better than one part in 10000, the urban surveying standard.

I suppose one could scratch subdivisions on the links of a chain, but I have never seen such chains, or results from surveys dated to the 1800s. There are plenty of errors in the original PLSS surveys in the US, but “the monuments prevail,” and large properties are generally sold by reference to the section monuments, and the acreage suffixed with “more or less.” Smaller property require remeasurement of the pertinent section monuments, then the surveyed from that starting point.

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7. Tim Bentley says:

Don’t forget that the chain, ie the measuring device was made into a solely metric device and widely used by Land Surveyors in the 1960’s and 70’s. At 20m in length it was the main method of carrying out ‘chain surveys’ up until the mid to late 70’s when the method was replaced by EDM’s Electronic Distance Measuring.
At Nottingham City
Engineers Department where I worked from 1970 we had four teams of three surveyors who spent most of their time doing chain surveys.
The chains were laid out in line between the ‘trig’ points and a steel tape was used to measure the ‘offset’ at right angles to the point that needed to be shown on the final drawing. The chain was the only practical instrument to use especially over rough ground. We recorded our measurements in decimetres to the nearest 5cm so all our measurements were therefore either whole numbers or 0/5. Note that we used:/ instead of a decimal point as a pencil dot may have washed off the paper in wet weather.
We got quite skilled at throwing the chain down the line as we moved on and we stood on the handle to line up by eye and gave hand signals to the person on the other end who marked the end either with a chaining arrow or with a crayon if on roads or pavements.
The worst part of the job was folding/wrapping without gloves the freezing cold chain at the end of the session as it would often have dog mess on it as this was many years before dog owners routinely picked -up after their pets did their business!

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8. Tim Bentley says:

Martin
“Before the UK’s metric changeover in the 1970s, it was normal to quote the areas of land in acres, roods and perches (40 perches = 1 rood and 4 roods”

In my long experience of land surveying and a large historic rural estate, I have only very rarely come across the fearsome roods and perches.
Their general use disappeared long before the 1970’s and believe it to be from 1880’s onwards. The Ordnance Survey 1:2500 ( 25 inch to 1 mile) County Series Sheets that covered Britain showed land parcels areas in acres and decimals.
These 1::2500 OS Sheets formed the most important tool in rural land management for a hundred years. They were so universal that 1:2500 became the main metric scale from the 1970s.
In urban areas 1:1250 became the main OS large scale map but the OS didn’t go as large as 1:500 as they did a hundred years before for some urban areas.
For anyone interested in these fine old OS look at the National Library of Scotland Side-by-Side viewer where hundreds of scanned maps are free to view online. See maps.nls.uk

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9. BrianAC says:

@ Tim Bentley