Introduction:

John Thompson was a fine example of an 18th-century philomath and easily measured up to its definition as "a lover of learning; a student, especially of mathematics, natural philosophy, and the like".2 He formed his life around an attraction to mathematics which became a defining thread to his energies and activities.

Historical information about Thompson is meager with some of the better sources being five popular magazines of the arts and sciences for the period: Martin's Magazine, Ladies' Diary, The Gentleman's Diary, The Gentleman and Lady's Palladium, and the Palladium Extraordinary. Here is a record over the 25 year window from 1742 to 1767 where he provided answers to various questions and submitted his own questions to these publications as a challenge for others. His contributions covered a wide range of mathematical categories: fluxions, land surveying, geometry, and astronomy - works supportive of calling Thompson a philomath or using a term from another reader: "mathematical sportsman".3

Adding to our knowledge about Thompson is The John Thompson Collection, twenty three mathematical and philosophical instruments at the Museum of the History of Science in Oxford.4 A majority of these items are practical instruments useful in land surveying and lend material assistance for making connections between capabilities of an instrument and Thompson's interests and skills.

Mathematical recreations and land surveying joined when Thompson submitted a question to the 1766 Gentleman's Diary connected to the Atherstone Enclosure of 1765. His proposed problem was more than just a mathematical challenge for readers because he also publicly took to task the original surveyors over their error in measuring the Atherstone common fields. Beyond the civic and social issues concerning this survey, this engaging problem required mathematical abilities in astronomy as well as trigonometry in order to be solved.

John Thompson:

John Thompson was born March 1, 1720 in Witherley, a town located near the southwestern border of Leicestershire, England. He was introduced to reading by his mother who was said to be "a literary woman"5 and, against the wishes of his father, she purchased books to augment his education obtained from the nearby Atherstone school. His father wanted him to enter the family profession of farming and grazing, but his strong interest and ability in mathematics caused him to "lett his paternal estate with the sole view of being at liberty to follow his favourite studies".6 Clearly, the mathematical arts were these 'favourite studies' which were a source of life-long passion in Thompson who, initially self-taught, became known beyond the local confines of Witherley.

By the age of 22, Thompson was already active in mathematics. A 1742 letter from Anthony Thacker7 (fl. 1720-46) of Birmingham Free School contained "an invitation to come over to Birmingham" to discuss "any thing in Fluxions or Trigonometry."8 Thacker's note also made the request to "please to bring the Second Vol. of Colin Maclaurin's Fluxions with you",9 suggesting Thompson had the beginnings of a mathematics library with this well-known book on fluxions.

Thirteen years later, John Thompson's personal notebook dated 1755 holds 270 pages of detailed problems and exercises on mathematical subjects of fluxions and conic sections.10 One intricate problem found in this notebook is a geometry question involving the Sun, Earth and Moon, summarized as what is "the Nature and Description of the curve generated by the center of the Moon" given that the Moon revolves around the Earth and the Earth revolves around the Sun with certain orbits and periods.

During his life, it was reported he wrote on the "mathematical, philosophical, and astronomical sciences" 11 though none of his writings survive today. A Gregorian telescope of 2-inch aperture and an early 3-draw 9-foot telescope, both from the John Thompson Collection, indicate he had more than just mathematical interest in astronomy. Furthermore, he taught his sons Ralph (1759-1839) and Samuel (1763-1831) land surveying with Ralph known to have been an active surveyor, a fact determined when four of Ralph's maps were located during my research.

From land surveying to geometry to fluxions, editors praised Thompson's work with comments like 'an elaborate and elegant method of solution' and described him as a 'curious and careful computer, who always commanded our esteem'.12 However, at the age of 62, after "being too fond of a studious life" which "brought on the rheumatism",13 John Thompson died February 25, 1783. The event was marked in the Leicester and Nottingham Journal as the passing of a person "eminently distinguished in his most early years by his attachment to the study of Mathematics and Philosophy."14

Land Surveying and the Enclosure of Atherstone Common Fields:

In 1765, the common fields belonging to the town of Atherstone in the county of Warwick were enclosed16 as authorized by a Private Act of Parliament.17 This act specified that 100 acres of land in one or two parts of not less than 40 acres each were to be surveyed and allocated to the cottager's.18 However, the cottager's, who had opposed the enclosure act, felt that the survey allotted too much land for their "Cottager's Piece"19 and they requested John Thompson re-survey this acreage. The result was Thompson did find an extra three acres had been assigned by the original surveyors.

Details on the methods and instruments used by the original surveyors are not known. Also lacking, and of greater importance, is the means to identify what was the nature of the error which produced these extra three acres. One clue is provided from Thompson's criticism of these surveyors as "unmathematical bunglers",20 ,21 leading to a suspicion that the error was connected to surveying methods which required higher mathematical skills and knowledge and possibly a set of corresponding complex instruments such as a theodolite for the taking of angles.

In contrast, details suggesting the methods and instruments employed by Thompson come from two sources. First, his methodological leanings in land surveying are likely compatible with those found in the 1770 book "A Treatise on Mensuration" by Charles Hutton which lists as one of the book's subscribers22 "John Thompson, Wetherley, Leicestershire", and leads one to believe John Thompson agreed with the book's approach towards surveys using the Gunter's chain and plane table as compared to utilizing a theodolite. 23 Second, a set of his surviving instruments that include Gunter's chains, a plane table,24 but not a theodolite are found in The John Thompson Collection at the Museum of the History of Science in Oxford.

From this information, it seems viable Thompson employed the mathematically undemanding plane table and Gunter's chains for his surveying, but coupled this with his strength and knowledge as a mathematician to work in a very accurate manner. Yet it leaves the question of exactly why he was more accurate in his surveying of this particular parcel, and this makes the mathematical nature of his proposed "Question 290" in the 1766 Gentleman's Diary based on this survey even more inviting.

Question 290:

Based on details from the original Cottager's Piece survey of 1765, Question 290 was John Thompson's mathematical submission for readers of the Gentleman's Diary published in 1766.25 The challenge was to compute the total area of the two trapezia (see diagram below) using the provided information and obtain the same (incorrect) value as the "unmathematical bunglers" of the first survey. By publicly referring to first surveyors in this manner, Thompson showed contempt for their results and was clearly displaying a not-so-small level of sportsman-like mathematical posturing.

Thompson's question itself is nicely mathematical and has been both simplified and complicated for the submission to a general audience; simplified so the two trapezoidal areas overlook the irregular shapes of actual fields, rough boundaries of hedges and streams found in the Atherstone common fields, and complicated by the limited amount and type of information provided which forced the reader to use difficult trigonometrical formulas to calculate an answer.

The data permitting the reader to solve this problem included land surveying, geographical, and astronomical elements. It included six lengths measured in chains and links,26 four angles given in degrees and minutes,27 the latitude of the location in question, and indicated directions of sunrise/sunset along with the sun's declination and azimuth at rising. My own answer of 103.1 acres was obtained after several weeks effort. When the answers from four readers were published the following year, there were four different solutions using four very different approaches, so even mathematicians as well as surveyors had difficulty finding conformity regarding this Cottager's Piece.

Did John Thompson's own survey yield an accurate answer as compared to the original survey (on which this question's data is based) because the final solution was calculated with better mathematical methods?28 Did he measure angle data linearly in chains rather than degrees?29 These queries probably touch near his reasons for making his "unmathematical bungler's" statement, that his knowledge and skills for mensuration of irregular shapes let him handle the uneven edges found in the real world of land surveying better than the original surveyors. Unfortunately, this question proposed to the Gentleman's Diary showed land boundaries as perfectly straight lines and lacked details which would assist in resolving these particular questions.

Two additional points are worth discussing here. First, each of the two areas shown in the question exceed the 40 acre requirement as specified by the Enclosure Act (my computed answers are 45.57 and 57.53 acres). This helps substantiate the reality of the question when compared to the actual surveyed lands. Second, the question's actual (and desired) answer was known to John Thompson,30 and with its publication, he must have been very sure of his data as well as accurate in his calculations. Additional verification of this land surveying data is seen when astronomical aspects of the question are examined next.

How Thompson's question affected the outcome of the Atherstone Enclosure is not known. One text on the history of Atherstone does not mention his name or role in the six pages covering the enclosure episode.31 Nevertheless, Thompson was noted to enjoy "great satisfaction"32 from the answers submitted by readers.

A Question of Astronomy:

Question 290 submitted by John Thompson required determination of areas for two trapezia, labeled ABDE and CFGH (see earlier diagram). ABDE could be solved using the provided length and angle data, but CFGH required finding an angle determined by a pair of sunrise/sunset shadow lines in order to be solved. This angle added a level of difficulty which required knowledge beyond the basic mathematics of flat surface area calculation and could only be solved by those who understood both spherical geometry and the astronomical terminology used.

My efforts to solve this problem foundered over the terms and framework, mostly due to the fact I was working in unfamiliar mathematical territories. After a week of struggle, I began engaging the problem with the help of historical texts which connected well to the original terminology of the problem. This decision caused me to pause and reflect on how this 233 year old question would be approached by a student of today and what level of student could master such a question.33

Providing the astronomical framework and solution of spherical triangles was Edmund Gunter's 1624 volume "Use of the Sector, Crosse-Staffe, and Other Instruments".34 Here were examples using the language of the times, such as "To find a fide by knowing the bafe and the other fide",35 to solve problems. One helpful diagram showed an arc of the declination from the sun to the equator at the beginning of Taurus,36 a date which turned out to be surprisingly close to the date computed for Question 290 (see following page).

All that remained was to locate the astronomical definitions used in Thompson's question, and this was solved by employing Charles Hutton's "A Mathematical and Philosophical Dictionary" published in 1796.37 Here the terms 'declination' and 'azimuth' were defined in forms close to what Gunter had utilized:

At this point in my efforts, it helped to review solutions to this question published in the Gentleman's Diary from the following year (1767). One solution was very helpful such that it provided key information: azimuth and amplitude complement each other.40 Now, with this additional detail, an understanding of how to use the provided latitude value, and the formulas of solution for right spherical triangles,41 an answer was obtained for the angle BFH of 56º 24". Now the question can be solved with a final answer of 103.1 acres for the area of the two trapezia.

With the experience and knowledge of my solution, I continued my research on this question. Curiously, I found the provided latitude of 52º 40" to be slightly in error: Atherstone is listed today as being latitude 52º 35".42 Why a 5" difference? One possible answer is provided when the question is reworked with this value, which now returns an answer of 104.0 acres. This result mismatches with the known 3 acre error (103 acres = 100 acres) for the "unmathematical bunglers" of the first survey, which makes it possible Thompson adjusted this latitude value to obtain a desired answer. It should be said that latitude was one of the question's two astronomical values Thompson could safely alter since he couldn't change any of the land surveying data involved.43

Another potential fault emerges when considering the sun shadow angle usage in the question, that the provided sunrise and sunset angles allow establishing a direction for North and thus turning the question data into a "map" for a portion of the Atherstone fields. Aligning this map with the likely actual geographic location of the enclosed common fields permits comparison of these two North directions, and now a 20° difference can be measured.44 Again, this discrepancy suggests John Thompson adjusted the layout of his question to permit using sun shadow angles and was not due to any error in land surveying.

Both of these "adjustments" seem appropriate in creating a challenging question for the readers of the Gentleman's Diary, and what must be considered plausible is Thompson wanted to display his mathematical abilities not only to the readers of the Gentleman's Diary but to the "unmathematical bunglers" too.

Thompson's infusion of astronomical data into a land surveying problem is to be admired. Rather than use this shadow line data to define a point in the question as a land surveyor might do to establish a fixed location, Thompson lays down two lines which then form an angle for part of the land in question. This lay out gives a sense of being similar to a sundial where shadow lines and not points are required for time readings. Also, sufficient data is supplied for computing a date for this problem (my calculations indicate a date on or about April 30th).

Certainly what John Thompson accomplished was the making of a well crafted mathematical problem for the recreation of many 18th century readers (and a 20th century graduate student). Question 290 displays ample evidence of his mathematical knowledge in astronomy and land surveying and easily confirms the reasons for the praise he received during his life as a mathematician.

Final Comments:

John Thompson was a good example of an 18th century philomath as seen from this question published in 1766, a work which readily documents his knowledge and abilities in mathematics, astronomy, and land surveying. Additionally, fuller understanding of Thompson was gained when his surviving scientific instruments were examined, underlining the valuable role these historical items can perform.

Attempting to solve this question myself provided me with a number of benefits. First, it provided a clear sense of the effort required by Thompson to create it in 1765. Second, it helped to guide me though the mathematics required to work out an answer and suggests benefit for teaching history of astronomy using old texts and questions to fully appreciate the material and history involved. Lastly, it allowed me to better comprehend events behind the question and how looking beyond immediate details of a published work (or a scientific instrument for that matter) enriches our understanding of past events.

Regarding the land surveying error of three acres, who was right? At least the land was available to walk again and it seemed Thompson had the last word in this matter. Yet, by taking this dispute to a mathematical public, he positioned his submission in front of a safe audience where the "unmathematical bunglers" would be at a disadvantage. It is unfortunate both sets of land surveying data for the Cottagers Piece are missing since it would be straightforward to determine who was wrong for whatever reason: mathematical error and/or deficient skills.

As mathematical champion for the Atherstone Cottager's, John Thompson was able to simultaneously resolve a serious land surveying conflict and produce a mathematical query (or inquisition) to the apparent satisfaction of all except, of course, the unknown "unmathematical bunglers".

Author information:

Dana A. Freiburger is presently a software engineer with Amdahl Corporation in Sunnyvale, California. His interest in scientific instruments recently took him to the University of Oxford for a year where he completed a M.Sc. degree in the History of Science. He has been notified of his acceptance to the University of Wisconsin-Madison and will begin additional graduate study in the History of Science starting January 2000. Contact e-mail address is: daf@netcom.com.

Footnotes:

1. This poster presents an extension of my dissertation, "18th Century Surveying Instruments of John Thompson", originally submitted September 22, 1998, for a M.Sc. degree at the University of Oxford.
2. Oxford English Dictionary, 2nd edition.
3. John Thompson Collection, MHS, MS Museum 212, letter dated January 6, 1766, from G. Getii of London uses the term 'mathematical sportsman' in asking for Thompson's support on another matter.
4. The collection consists of two telescopes, a plane table and drawing board, a pair of levels, a surveyor's cross, a pantograph, a set of Napier's bones and two Gunter's chains, as well as sectors, scale rulers, slide rules, a parallel ruler and compasses.
5. John Nichols, "The History and Antiquities of Hinckley, in the county of Leicester...", Hinckley and District Museum, 1993, Volume I, Part II, "Additions and Corrections in Vol. IV.", facsimile of the 2nd edition originally published in 1816. Bodleian shelfmark M94.F04547. Pg 157.
6. The Leicester and Nottingham Journal, March 29, 1783. The word "lett" meaning let or lease.
7. E.G.R. Taylor, The Mathematical Practitioners of Hanoverian England 1714-1840, 1966, Cambridge University Press, entry 214, pg 166.
8. John Thompson Collection, MHS, MS Museum 212, letter dated July 30, 1742.
9. Maclaurin's two volumes on fluxions, A Treatise of Fluxions, was published in 1742, and this is the same year as Thacker's letter, showing Thompson had expeditious access to this important work.
10. John Thompson Collection, MHS, MS Museum 210, notebook dated February 21, 1755.
11. John Nichols, "The History and Antiquities of Hinckley, in the county of Leicester...". Pg 157.
12. Gentleman and Lady's Palladium, 1762, pgs 35-36, British Library shelfmark P.P.2477.pb.
13. John Nichols, "The History and Antiquities of Hinckley, in the county of Leicester...". Pg 157.
14. The Leicester and Nottingham Journal, March 29, 1783.
15. T.H. Worth, The Anstey Enclosures: a study of Anstey based on a close scrutiny of the 1761 Acts of Enclosure, and a lifetime spent in walking over the fields of Anstey, self-published, 1978, quote on pg 1. Also found as an anonymous response in 1764 to Sir Charles Pratt's fencing of common land.
16. Enclosure altered ownership and access to a town's commonland and fields by taking most of this land into private hands, but still leaving some portion available for traditional common usage such as grazing.
17. Great Britain, Parliament. Bills 1764-02-24, An act for dividing and inclosing the open and common fields of Atherstone, in the county of Warwick, London, 1764, pgs 58-78.
18. Brenda Watts and Eleanor Winyard, The History of Atherstone, Mercia Publications LTD, 1988, "...cottagers, who worked as labourers or at a trade because they had no land of their own, still retained the right as owners of dwellings in the town to graze animals on the commonland and in the fields.", pg 28.
19. Ordnance Survey Map, Warwickshire Sheet VI.II.23, Atherstone, 1887, Bodleian Map Library Reference C17.70 a.42. This map sheet retains the name "Cottager's Piece" based on this event.
20. Warwickshire Record Office, reference HR 35/53, "An Accurate Survey of & for Cottage Piece taken by Mat. Baker", fieldbook notes and two maps dated July 1764. These maps were for the land owners as part of pre-enclosure allocation layouts, and included an undated handwritten note likely referring to the Thompson re-survey: "another was taken scince to lay the cottage land near to Witherley - per which the land owners are injured by the Com[missioner], giving 3 acres too much...". It is not known if Mat. Baker is one of the 'unmathematical bunglers' Thompson referred to in his question.
21. Gentleman's Diary, 1766, Question 290 posed by John Thompson of Witherley-Bridge, pgs 38-39.
22. Charles Hutton, A Treatise on Mensuration, 1st edition, 1770, Subscribers, pg xiii.
23. Hutton's book depicts Gunter's chains as the "most useful instruments", declares "the plane table is certainly the most excellent", and offers the caution of "the new-improved theodolite is of the most dangerous consequence", thus being clear on his viewpoints for each of these instruments.
24. This undated plane table, made by George Adams Sr., was said to be of Thompson's own design.
25. Gentleman's Diary, 1766, Question 290 posed by John Thompson of Witherley-Bridge, pgs 38-39. Interesting to note Thompson called himself a "Land-Surveyor" in the discussion portion of this problem.
26. Chains indicates the Gunter's Chain, developed in 1620 by Edmund Gunter, and has a length of 22 yards. A chain contains 100 links. For acreage measures, an area one chain square is equal to 1/10 of an acre.
27. The problem used the abbreviations 'gr' for degrees and 'm for minutes: 1gr 3m is 1 degree, 3 minutes.
28. Gentleman and Lady's Palladium, 1762, Question 194, pgs 41-42. The editor described John Thompson's answer as "...(confirming the Truth of the Proposer's Solution) which artist we find to be accurate (as he is elegant) in all his Solutions, and always to be depended on."
29. Proposing this particular question could lead to Thompson himself being considered unmathematical!
30. Thompson must make his case to support his calling the original surveyors "unmathematical bunglers".
31. Brenda Watts, Eleanor Winyard, The History of Atherstone, Mercia Publications LTD, 1988, pgs 31-36.
32. John Nichols, "The History and Antiquities of Hinckley, in the county of Leicester...". Pg 157.
33. Also, I used an electronic calculator and an Excel spreadsheet to help work the data for this problem.
34. Edmund Gunter, "Use of the Sector, Crosse-Staffe, and Other Instruments", 1624, Da Capo Press, New York, Theatrvm Orbis Terrarvm Ltd., Amsterdam. Facsimile reprint 1971. Pgs 74-88.
35. Ibid, pg 76.
36. Ibid, pg 74.
37. Charles Hutton, "A Mathematical and Philosophical Dictionary", 1796, London.
38. Ibid, pg 360.
39. Ibid, pg 178.
40. Gentleman's Diary, 1767, Answer to Question 290 by Mr. E. Smith, pg 43.
41. CRC Standard Mathematical Tables, 1974, 22nd edition, CRC Press, Cleveland, Ohio. Pgs 238-9.
42. The Times Atlas of the World, Vol. 3 - Northern Europe, London, 1955, Index to Gazetteer, pg 3.
43. The other value being the declination/azimuth total of 78 degrees and 27 minutes.
44. This real field location, originally named Windmill Hill, is based on the description of the common fields in Watts and Winyard's The History of Atherstone, and an 1887 Ordnance Survey Map which shows two helpful landmarks: the Priory Cottage and Watling Street (an old Roman Road), and then finding a very well-matched angle to connect and orient these two maps.