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A Country's Extent as the Area of a Circle, the Weight of Its Tax as the Direction of a Dotted Line: Playfair's The Statistical Breviary, Chart 1

Sun, 12 Oct 2025 00:00:00 +0900

In 1801, in Chart 1 of The Statistical Breviary, William Playfair (1759–1823) drew the extent of the countries of Europe as the area of circles. Population and revenue are shown by vertical lines standing on either side of each circle, measured on the same scale. The tops of the two vertical lines are joined by a dotted line. In his text, Playfair writes that which way the dotted line rises shows whether taxes are heavy in proportion to population.

The sources are the book’s preface, pp.13–16, where he explains how to read the chart, and the country-by-country tables. I read the text in the Bodleian Library copy on the Internet Archive. Because the foldout plates were not captured correctly in that scan, I show the chart from the 1802 French translation. How the circles are divided is covered in an article comparing the two circles of 1801 and 1805, and the 1805 chart is covered in an article on the 1805 United States pie chart.

Photograph of the plate of Chart 1 in The Statistical Breviary

Chart 1 in the French translation. Circles are arranged in order of size, from the large Russia at the far left to the United States at the far right. Red and yellow vertical lines stand on either side of each circle, with their tops joined by a dotted line

The whole plate, original drawing by William Playfair, engraved by Adam (French translation, 1802). Source: David Rumsey Map Collection, David Rumsey Map Center, Stanford Libraries / Internet Archive (CC BY-NC-SA 3.0)

Four Charts Born from the Supplement to a Table

According to the preface, it began with helping with someone else’s book. The editor of the English edition of Boetticher’s statistical tables asked him for a table supplying the changes since publication. Partway through that work, he writes, he realized the following (preface p.3).

it occurred to me, that tables are by no means a good form for conveying such information

In short: tables are not at all a good form for conveying this kind of information. On p.4 he says that when proportion and magnitude are at issue, appealing to the eye is the best and quickest way.

The book appeared in London in 1801. According to Chris Pritchard, it runs to 64 pages, most of them tables by country, with four plates. The contents of the four are listed on p.14.

Chart 1: Europe before the French Revolution
Chart 2: Europe as projected by the peace of Lunéville
Chart 3: The populations of the capitals of Europe
Chart 4: The countries of Hindustan

The original wording for Chart 1 is “Exhibits the powers of Europe as they were previous to the French Revolution.” What is drawn is Europe before the Revolution, not Europe as of 1801. 1801 was the middle of the French Revolutionary Wars, and Chart 2 draws the picture after the Treaty of Lunéville in February (Pritchard 2021).

Charts built from lines were not new for Playfair. On p.15 he writes that about fifteen years earlier he had applied lines to commerce and finance; this refers to The Commercial and Political Atlas. The Statistical Breviary came out in Paris the following year, 1802, in a French translation by D.-F. Donnant, and the plates were re-engraved with French text.

The Reading Instructions Are in Two Pages of the Text

Chart 1 has no legend. How to read it is set out in the text, pp.13–14. The key passage about the vertical lines is this.

The yellow lines on the right of each nation represent the revenue in millions of pounds sterling, measured also upon the same divided scale with the population.

The yellow line on the right represents revenue in millions of pounds, measured on the same scale as population. Just before this he writes that the red line on the left represents population in millions, and that the scale, one division per million, is numbered “at both ends.”

Population and revenue are measured on one and the same scale. The numbers appear at the left and right edges of the chart so that either can be read from both sides. It is not a dual axis, with the left for population and the right for revenue.

Turning the text’s explanation into a table of data and appearance gives the following. The example is France (before the Revolution), with figures from the table on p.33.

Data	Appearance	Example: France
Extent of the country (square miles)	Size of the circle; arranged from left in descending order of size	Circle
The extent figure	Number inside the circle	149,000
Population (millions)	Height of the red vertical line to the left of the circle	26
Revenue (millions of pounds)	Height of the yellow vertical line to the right of the circle, on the same scale as population	19
Weight of tax relative to population	Which way the dotted line joining the tops of the two vertical lines rises	Falls to the right
Population density (people per square mile)	Number directly above the circle	174
Maritime or land power	Color of the circle; green for maritime powers, pale red for land powers	—

The number above each circle is population density, not revenue. The text itself gives examples: “5 over Russia, and 14 over Sweden.” If you read France’s 174 as revenue, it is wildly at odds with the table’s £19 million.

In the French translation’s plate, the circles are colored blue and red. France, Spain, Britain and others are blue; Poland, Prussia and others are red. The maritime powers that the English text calls “green” can be read as having become blue in the French translation.

Only the circles for Russia and Turkey are divided inside. In Russia the inner circle is European territory and the outer ring is Asian territory; Turkey is split into sectors by continent. The text does not explain this division. Here the colors stand for continents rather than maritime or land power. It can be read as the colors carrying two different meanings within a single chart.

The circles of the Russian Empire and the Turkish Empire

Making Countries’ Shapes into the Same Circle

Why draw countries as circles? Playfair gives his reason on p.15.

for where forms are not similar, the eye cannot compare them easily nor accurately. From this circumstance it happens, that we have a more accurate idea of the sizes of the planets, which are spheres, than of the nations of Europe which we see on the maps

Where forms are not similar, the eye cannot compare them easily or accurately; so we have a more accurate idea of the sizes of the planets, which are spheres, than of the nations of Europe that we see on maps. In the preface, p.6, he also writes that when the eye sees similar forms it cannot help comparing their sizes.

On how the size of a circle is determined, the text does not say “proportional to area.” The preface’s phrase is “proportional circles.” I measured the plate of the French translation (the image above, 2000px wide): Russia’s circle has a diameter of about 410px and France’s about 68px. Taking France’s diameter as 1, area-proportional circles would give Russia about 5.6, and diameter-proportional circles about 31. The measured ratio is about 6, so the diameter is roughly proportional to the square root of the extent. Even allowing for measurement error, this is a value consistent with area proportionality.

The circles are arranged from left to right in order of size. On p.7 Playfair writes that on seeing this chart one will be struck by the size of Russia and Turkey. He continues that one will also be struck by the smallness of the nations that have moved the world these five hundred years.

He also decides what not to draw. On p.15 he writes that he combined extent, population, and revenue in one chart because they are the foundation of a power that can be accurately measured. He goes on to say that the form of government and the virtue of individuals also affect a nation’s strength, but cannot be shown in a chart.

The Direction of the Dotted Line Divides at “One Pound per Person”

The explanation of the dotted line spans p.13 to p.14.

The dotted lines drawn between the population and revenue, are merely intended to connect together the lines belonging to the same country. The ascent of those lines being from right to left, or from left to right, shews whether in proportion to its population the country is burdened with heavy taxes or otherwise.

The dotted lines are there to connect the two lines belonging to the same country. And which way a dotted line rises shows whether the country is burdened with heavy taxes in proportion to its population. Read only the first half, and the dotted line is merely a connector. It is the second sentence that gives the dotted line meaning.

Because population and revenue are on the same scale, a dotted line rises to the right when the revenue figure exceeds the population figure. That is, it is a country where tax per person exceeds one pound. Let us check against the tables in the text (£1 = 20 shillings).

Country	Population (millions)	Revenue (millions of pounds)	Tax per person in the table	Dotted line
Russia	25	7.3	6s 8d	Falls to the right
France	26	19	14s 8d	Falls to the right
Spain	11	14	£1 5s 5d	Rises to the right
Britain and Ireland	14	27	£1 18s 3d	Rises to the right

The figures are from the tables on pp.19, 33, 35 and 37. In the chart, the dotted line rises to the right for only four countries: Spain, Britain and Ireland, Portugal, and the United States. Those are also the only four countries whose tax per person in the tables exceeds one pound.

The section with Poland, France, Spain, and Britain and Ireland. France’s dotted line falls to the right, those of Spain and Britain rise to the right, and Britain’s rises most steeply

Detail cropped from Chart 1, original drawing by William Playfair, engraved by Adam (French translation, 1802). Source: David Rumsey Map Collection, David Rumsey Map Center, Stanford Libraries / Internet Archive (CC BY-NC-SA 3.0)

It matters that Playfair speaks only of the direction in which the line rises. On a common scale, direction works correctly as a binary split at “one pound per person.” The text’s explanation can be read as promising that much and no more.

The steepness of the slope is a separate matter. The two vertical lines stand touching the left and right edges of the circle, so the horizontal gap between them is the circle’s diameter. The slope of the dotted line is the difference between revenue and population divided by that diameter. Russia has a large difference, but its circle is large, so its dotted line is gentle. Britain has a difference of 13 and a small circle, so its dotted line rises steeply. The country’s extent gets mixed into the steepness of the slope.

How Third Parties Saw the Chart

The David Rumsey Map Collection’s catalog entry for the French translation quotes a comment by Akerman and Karrow. Their view is that the circles for countries’ extent, the sectors of Turkey, and the lines for population and taxes are “a creative combination of different visual forms.”

It was Pritchard (2021) who criticized the slope. He sums up the novelty as three things: the area-proportional circles, the division of Russia and Turkey, and the vertical lines with the sloping lines joining them. He then writes:

a direct comparison of these lines, one country to another, is not possible because each gradient is affected by the radius (and hence the area of the country)

The point is that because the slope depends on the circle’s radius, that is, on the country’s extent, it cannot be compared from one country to another. Jonathan Sachs makes the same point in a note, naming Funkhouser (1937) and others. Incidentally, Pritchard writes that population is read on the left scale and the budget on the right, which conflicts with the text’s “the same divided scale.”

Sachs’s main argument is in the history of ideas. He gives weight to the replacement of irregular national territories with “uniformly scaled circles.” If wealth is measured as revenue, a quantity, then wealth is not the same as the extent of territory. He reads this chart as a rebuttal to those who held that territory and population are wealth.

Juliana Adelman (2018) of The Irish Times focuses on intent. She writes that what Playfair most wanted to show was the advantage of the United Kingdom, which raised large revenue from a small territory and a small population. She then adds that overseas colonies are not counted.

Uses

There are four things to take away from Chart 1.

The first is to put things into the same shape before comparing sizes. On a map, countries’ extents are hard to compare. Playfair replaced them with identical circles and left the work of comparing to the eye.

The second is that putting two quantities on the same scale lets you read the relationship between them. By putting population and revenue, which have different units, on one numerical scale, he made a dividing line of “one pound per person” appear on the chart.

The third is that if you show a derived quantity by slope, you should keep the horizontal gap constant. In Chart 1 the gap is the circle’s diameter, itself data, and the country’s extent got mixed into the weight of tax. If you want to show tax per person, drawing that value directly is more reliable.

The fourth is to put the instructions for reading beside the chart. In Chart 1, the reading instructions exist only in the text. If you cut out the chart alone, you could easily misread the population density above a circle as revenue.

On p.16 Playfair writes that the greatest criterion is what remains in the mind of someone who looks at it for the first few minutes. By my count, Chart 1 tried to show six things in those few minutes: extent, population, revenue, population density, maritime or land power, and the weight of tax. The slope of the dotted line can be read as the awkward part that this cramming produced.

References

Primary sources

William Playfair, The Statistical Breviary; Shewing, on a Principle Entirely New, the Resources of Every State and Kingdom in Europe, London, 1801 (Bodleian Library copy, digitized by Google). Internet Archive / Full text / Google Books
William Playfair (trans. D.-F. Donnant), French translation (Paris, 1802), Chart 1, “Carte statistique représentant l’étendue, la population et les revenus des principales nations de l’Europe”. David Rumsey Map Collection. Copy on the Internet Archive

Secondary sources

Chris Pritchard, “Life of Pie: William Playfair and the Impact of the Visual,” Mathematical Association, Presidential Essay No.4, July 2021. PDF
Jonathan Sachs, “1786/1801: William Playfair, Statistical Graphics, and the Meaning of an Event,” BRANCH. Article
Juliana Adelman, “The man with a finger in many pie charts,” The Irish Times, 2018-03-05. Article
Catalog entry of the David Rumsey Map Collection for Chart 1 of the French translation (quoting Akerman & Karrow). Copy on the Internet Archive

Dividing the Size of the United States with One Circle: Playfair's 1805 "Divided Circle"

Sun, 12 Oct 2025 00:00:00 +0900

In 1805, William Playfair translated D. F. Donnant’s (Denis-François Donnant) statistical account of the United States into English and added a figure that divides the area of the states and territories into sectors of a single circle. Beneath the figure he had engraved “This Newly invented Method is intended to shew the Proportions between the divisions in a Striking Manner.” In other words, the method is newly invented, and it is meant to show the ratios between divisions in a striking way.

The book’s title is Statistical Account of the United States of America (London: Greenland and Norris, 1805). “Statistical Representation of the United States of America,” which is often cited as if it were the name of the work, is the title of the plate, not of the book. What follows draws on the images of the copy held by the Lehigh University Libraries and on the text of the Harvard copy at the Internet Archive.

The plate of the 1805 circle of the United States

A Translation with Just One Added Figure

Playfair himself describes how the book came about, in his preface and in the translator’s preface.

In 1801, Playfair publishes the Statistical Breviary.
Donnant in Paris translates it into French and adds an account of the United States.
Donnant sends the French translation to Playfair with an English letter dated November 1802.
Playfair translates the United States part into English, adds the figure, a dedication, and a treatise on trade at the end, and publishes it in 1805.

The title of the French translation is Éléments de statistique (1802). This comes not from the preface but from the bibliography of Bellhouse 2022.

In the translator’s preface, Playfair says the translation is not literal and that he has supplemented the explanations with a divided circle. He also writes that Donnant, who has approved of his own way of showing statistics through geometry, will not take this as an alteration.

Title page of D. F. Donnant, tr. William Playfair, Statistical Account of the United States of America (1805)

Source: Title page of D. F. Donnant, tr. William Playfair, Statistical Account of the United States of America (London, 1805), held by Lehigh University Libraries Special Collections / from Lehigh Library Exhibits (No Copyright – United States)

The title page is printed with “ILLUSTRATED BY A DIVIDED CIRCLE.” It goes on to explain that the engraved, hand-colored figure shows the ratio of the extent of the states, the eastern territories, and the newly acquired Louisiana by a new method. The subtitle is “With an Addition on the Trade to America, for the use of Commercial Men.” The advertisement at the end lists this figure first in the table of contents and describes it as “a Method entirely new, distinct, and EXPEDITIOUS.” The readers he had in mind were people in business.

The dedication is addressed to President Thomas Jefferson. Playfair writes that Jefferson, while in France, approved of his method of using lines for finance, and he urges the government to put effort into collecting statistics. Twenty-five copies were sent to Jefferson (Brinton 1939, Pritchard 2021).

How the Circle Was Divided

Hand-colored figure assigning the areas of the states and territories of the United States to the angles of sectors of a single circle. The states are pink, Florida is white, and the Western Territory and Louisiana are green

Source: William Playfair, “Statistical Representation of the United States of America” (folding plate), in the same book, held by Lehigh University Libraries Special Collections / from Lehigh Library Exhibits (No Copyright – United States)

The figure is a folding plate, and the creases show in the image. The correspondence between data and appearance is as follows. The numbers were read from Lehigh’s high-resolution image, and the spellings are kept as they appear in the figure.

Appearance	Data it encodes	Notation in the figure
The whole circle	Total area of the United States	“Total Extent 1,929,000 Square Miles or 1224 Millions of Acres”
Central angle of a sector	Area of each division (square miles)	Virginia 169,000, Georgia 150,000, and so on. The six small states have names only
Order of the sectors	States in descending order of area	Counterclockwise from Virginia at upper right to Delaware, followed by Florida, Louisiana, and the Western Territory
Color (hand-colored)	Kind of division	Pink = states, white = East & West Florida “not yet united”, green = Western Territory and Louisiana
Lettering along the arc	Grouping of regions	“WESTERN TERRITORY” on the green at right; “Newly Acquired (in 1804)” and “LOUISIANA” on the green at bottom

The Lehigh University Library exhibit explains the three colors as distinguishing states, territories, and the Spanish-held East and West Florida. The Louisiana Purchase treaty was signed on April 30, 1803, but the figure says “in 1804.”

In the 1801 Breviary, what Playfair drew was a row of circles, one per country, to be compared by size. The notation of that figure is covered in a separate article, and the difference between comparing by circle area and dividing by sector angle is covered in the article on the two kinds of circles. This article is about the 1805 circle, which divides a single whole into sectors.

The Line under the Figure, and a Passage in the Text

Two lines of lettering are engraved under the title of the plate. The first is the following sentence, and the second is the line giving the total area.

This Newly invented Method is intended to shew the Proportions between the divisions in a Striking Manner

It means that this newly invented method is meant to show the ratios between divisions in a striking way.

On page 17 of the text, Playfair explains the figure as follows.

The divided circle which represents the surface of the country, has been constructed for the purpose of shewing, in an easy and accurate manner, the proportional extent of the different provinces; and it is scarcely possible to look at it without a few important observations occurring to the mind.

In short, it was made to show the ratios of the extent of the regions easily and accurately, and on looking at it, it is hardly possible for a few important thoughts not to come to mind.

Examples of those thoughts follow. The differences in the size of the divisions are too great, so rearrangements will occur once the population evens out. The western territory will in time be divided into separate states. The ratio between Virginia and Delaware is absurd, and cannot stand as it is. He means the largest state at the upper right of the figure and the state at the far left that is a mere thin line.

The text calls the figure “easy and accurate,” while the engraved line says “Striking.” This can be read as Playfair asking both accuracy and impact of the same figure. The treatise on trade at the end says that the West was three-fifths the size of the United States before the Louisiana Purchase and is now two-thirds, and proposes a company to send British goods to the West. The green sector, which takes up a little over half of the circle, can be read as also serving to make this argument visible.

Checking the Numbers

In 2022, the statistician David R. Bellhouse pointed out errors in this figure. New York and New Jersey are confused. Compared with a map from around 1805, the states’ areas do not match. For each state, Playfair multiplied the length and width that Donnant gave to get an area, ignoring the fact that the states are not rectangles, Bellhouse says.

Left side of the figure. Sectors for Massachussets, South Carolina, and Maryland, six thin states with names only, and the white East & West Florida

Source: An enlargement of the left side of the same plate. Lehigh University Libraries Special Collections / from Lehigh Library Exhibits (No Copyright – United States)

I also measured the sector angles on the Lehigh image. I picked up the radial boundary lines from the center, so an error of about 1° is possible. The table below sets the angle obtained by dividing each printed value by the total area and multiplying by 360° next to my measurement.

Division	Printed in the figure	Angle from the printed value	My measurement
Virginia	169,000	31.5°	about 32°
Georgia	150,000	28.0°	about 27.5°
Kentuckey	115,000	21.5°	about 21.5°
New Jersey	105,000	19.6°	about 19.5°
North Carolina	84,000	15.7°	about 15.5°
Massachussets	73,000	13.6°	about 13.5°
South Carolina	25,000	4.7°	about 8°
Maryland	14,740	2.8°	about 5°
East & West Florida	none	—	about 12.5°
Louisiana	725,000	135.3°	about 111°
Western Territory	450,000	84.0°	about 84°

The six large states and the Western Territory are close to proportional to the printed values. The ones that deviate are South Carolina, Maryland, and Louisiana. Louisiana is about 25° smaller than its printed value, and working back from the angle gives roughly 590,000 square miles.

The total does not close either. The ten divisions with printed values add up to 1,910,740 square miles, only 18,260 short of the total area. Florida and the six small states should fit into that remainder, yet the figure gives them about 23°, the equivalent of about 120,000 square miles. Since a circle closes at 360°, some sector cannot be drawn as printed. In this figure, it can be read as showing up in Louisiana.

There are other discrepancies. The figure names 14 states, and Pennsylvania’s name is missing. Just before the summary table in the text, it says “Since then, the number of States has increased to sixteen.” Louisiana’s 725,000 square miles also differs from the 828,000 square miles that the U.S. National Archives gives for the purchased land.

How Third Parties Saw the Figure

Willard C. Brinton dedicated his 1939 Graphic Presentation to Playfair. In the preface there is a scene in which he leafs through this book at the Huntington Library and exclaims, “This is the first example of a sector chart.” Then he notices the engraved line beneath the one and only figure. On page 81 he writes “one of the first sector charts,” but he also notes that in the 1801 Breviary the Turkish empire’s circle was divided into three, and adds that that one might be the first.

Bellhouse distinguishes the 1801 figure as “An anticipation” and the 1805 figure as a “fully-fledged pie chart.” His view is that in 1801 Playfair was not yet fully aware of what he had done. He cites David S. Moore’s view that a pie chart can be used when emphasizing the relation between parts and a whole, and writes that this figure is an example of that.

Chris Pritchard, in 2021, looks at the figure from the book’s origins and its political context. His assessment is that it shows both the ratios among the states and their ratios to the whole country, and makes plain the effect of the Louisiana Purchase two years earlier. He also writes that one of the 25 copies sent to Jefferson went to Humboldt.

The three accounts have different emphases. Brinton reads the figure as an early sector chart, Bellhouse from the quality of the data, and Pritchard from the context of the purchase.

What to Take Away

Here are the points from this figure that can be carried back into visualization work.

State the whole. With the total area written under the figure, the sectors can be read as quantities relative to the whole.
Decide the order. Going around the circle from the largest state, the gap between Virginia and Delaware shows at a glance.
Use color for kind. The three colors are not about quantity; they distinguish states, unincorporated land, and territory.
Write the numbers on the sectors. Because the values are printed, a reader 200 years later can check them against the angles and find the discrepancies.
Check the total before drawing. If the parts do not add up to the whole, the discrepancy will show up in some sector.

Playfair asked this figure for both “Striking” and “accurate.” The impact comes from color, ordering, and the large green sector. The accuracy cannot be known without checking the numbers one by one. When drawing a pie chart, check these two separately.

References

Primary sources

D. F. Donnant, tr. William Playfair, Statistical Account of the United States of America, London: Greenland and Norris, 1805 (Harvard copy, Internet Archive). archive.org / Full OCR text
Lehigh University Libraries, Special Collections, “Statistical Account of the United States of America.” (Images of the title page and plate from the library’s copy). Lehigh Library Exhibits
Lehigh University Library, “At a Glance: Selected Works in the History of Data Visualization,” Case One: Playfair. Exhibit page
U.S. National Archives, “Louisiana Purchase Treaty (1803).” archives.gov

Secondary sources

Willard C. Brinton, Graphic Presentation, New York: Brinton Associates, 1939 (preface pp. 11–12, p. 81). archive.org
David R. Bellhouse, “William Playfair’s Statistical Graphs,” CMS Notes, 54(1), February 2022. notes.math.ca
Chris Pritchard, “Life of Pie: William Playfair and the Impact of the Visual,” Mathematical Association Presidential Essay No.4, July 2021. PDF