Mathematical data for bibliographic descriptions of cartographic materials and spatial data

Jan Smits, Koninklijke Bibliotheek

Original release: 1996
Last update: January 7, 2014

Map projections
Map projection is "the process of systematically transforming positions on the Earth's spherical surface to a flat map while maintaining spatial relationships. This process is accomplished by the use of geometry or, more commonly, by mathematical formulas. Map projection can be best visualized by imagining a light bulb placed at the centre of a transparent globe and having its lines of longitude and latitude cast upon either a flat sheet of paper or a sheet of paper rolled into a cylinder or cone placed over the globe." (from Atlas of Canada: map projection).

A good text for beginners to consult the text concerning scale and map projection from Arthur H. Robinson's et al. book Elements of cartography (6th ed., New York, 1995).
For a more sophisticated approach one can use the unit on , which is part of Brian Klinkenberg's GIS and Cartography Online Resources with the University of California at Santa Barbara.

For those converting analogue to digital the following publications are available:

Map projections used by the U.S. Geological Survey / by John P. Snyder. - 2nd ed. - Washington : United States Government Printing Office, 1984. - 313 p. : ill. ; 23 cm + map. - (Geological Survey bulletin ; 1532)

A more recent online edition of this publication with a zipped file which contains the entire text of USGS Bulletin 1856, Bibliography of Map Projections, edited by John P. Snyder with Harry Steward and published in 1988. John Snyder has since corrected, supplemented, and renumbered the text in 1994 and 1996. It is also converted from a coded file, which can be printed with all the diacritical marks in the various languages on an Epson printer using a homemade word processor, to HTML codes to permit reading of all diacriticals allowed on the Internet. The exceptions are diacriticals used only in Eastern European languages, which are removed and the letter shown without a diacritical, except that the Hungarian double accent acute is made an umlaut. He also has the Bibliography in a Microsoft Word file, so that all Eastern- and Western-European diacriticals, as well as new insertions of Russian Cyrillic following the transliterations already included, may be displayed in the printed form or on the screen.
The USGS upkeeps the site Map projections publications, which contains many papers with descriptions and visualisation of the main projections, a summary of projection properties, and a summary of areas suitable of mapping with projections.
Some of these projections are also illustrated on Zbigniew Zwolinski's 'The Great Globe Gallery'.

The most recent publication in this field is:
Map projection transformation : principles and applications / Qihe Yang, John P. Snyder, Waldo R. Tobler. - London : Taylor & Francis, 2000. - xv, 367 p. : ill. ; 21 cm. - ISBN 0-7484-0667-0 (Hard cover); ISBN 0-7484-0668-9 (pbk.).

A more sophisticated site with actual mapprojection and their algebraic formulae can be found on the Map projection-page of Wolfram Research.

Somewhat older, simpler and less extensive publications are:

Still older and more specialized is the Dutch publication with projections concerning charts:

And in English

Going back in time I have found a German booklet as number 30 in the series Bibliothek zur Erd-, Länder- u[nd] Völkerkunde aus der Sammlung Göschen:

Even older in time are the following publications:

With the computerization of cartography the amount of projections proliferated and fortunately also the Internet-resources available. The following sources are multiple sources with many hyperlinks to other documents or web-sites.

At this map projection homepage you will find a collection of information relating to map projections. This Home Page was inspired by a seminar in map projections in the Geography Department, Hunter College, City University of New York, led by Dr. Keith C. Clarke , Geography Department, UCSB.

Other extensive home pages are the Map Projection Overview by Peter H. Dana of the Department of Geography, University of Texas at Austin, and the European Map Projections by Stefan A. Voser of the Institut für Geodäsie, Universität der Bundeswehr München.

Bar scale values
Scale is "A ratio representing the relationship between a specified distance on a map and the actual distance on the ground. For example, at the scale of 1:50 000, 1 unit of measurement on the map equals 50 000 units of the same measurement on the ground. Map scale is frequently expressed as a representative fraction and graphically as a bar scale" (from : Scale).

Herman Wagner (1840-1929) gives a large historical exposé concerning scales in "The mapscale" (Der Kartenmaßstab. In: Zeitschrift der Gesellschaft f¨r Erdkunde zu Berlin. 1914. pp. 1-34, 81-117), where he connects the use of the scale with the projection used. Knowing this one must always be aware that a certain scale (being it a scale bar or a representative fraction) only gives true values on but a small part of a map. Depending on the kind of projection the deviation will be larger or smaller, also keeping in mind whether it is a large scale or small scale map one is viewing.

To calculate the distance between two cities using the great circle method (as the crow flies) one needs to know latitude and longitude of the two places. Bali and Indonesia on the net provides a distance calculator using geographic placenames. It does the arithmetics based on the 'PROJ' system available from the U.S. Geological Survey, when necessary supported by a locational map, and a travel map with driving directions. It also shows the compass headings between the two cities.
Another easy to use programme is the Great Circle Calculator. Here one should, however, fill in the right geographical co-ordinates for latitude and longitude. The result will be a distance in miles or kilometres. There are no auxiliary services. For those interested in these calculations a query on distance "great circle" on the search-engine Google will result in 21,600 hits.

Trying to give scales for pre-1800 maps implies always 3 to 4 measurements and should result in phrases as 'Scale varying from [ca. 1:7,400] to [ca. 1:8,400]' when derived from measurements on modern maps. Or 'Scale [ca. 1:7,900], measurement derived from scale bar (900 rods = 33 mm)'. When the scale bar is not used in this way its mention should be relegated to the notes.
I advise curators and editors of facsimiles to be careful with scales and never to use one scale denominator when the map does not have a geometrical basis based on triangulation. When the cataloguer is not sure it is better to state 'Scale unknown' and give a scale-bar note than giving a quizzing approximation with which nothing can be proved or which creates confusion.

When the calculation of a scale is dependent on a grid of geographic co-ordinates one should measure the distance between two succesive parallels (1° = 111.11 km or 60 nautical miles, 1’ = 1.85 km or 1 nautical mile) using a meridian, when possible in the middle of the map.

For those not used to calculating scales Terry Reese has created the site Scale calculator, which allows for American standard, metric, and miscellaneous conversions.

In Petermanns geographische Mitteilungen (1855-2004), a famous German geographical journal, almost every map contains a scale denominator as well as a scale bar. The scale bar denominates a certain value per 1° longitude at the equator. The longitudinal measurement of 1° longitude at the equator is 111,324 kilometres or 60 nautical miles.
As there are some very exotic local scale bars which might be unknown the following table gives the values (in order of precision, as used in Petermann) ordered by the English name of the country in which the value is used. It may be that a bar scale is wrongly attributed to a certain country or area as they have to be interpreted from the German or do not have any explication of their origin.
Some values are only related to specific maps [i.e. 4,000 pied = 20 mm] and thus do not give any objective measure. They are included, however, to show their existence. Numbers in Bold under the heading '1 degree' are most used on the maps.
Only verbatim statements from Petermann are used and in no way are measures recalculated.

The table is updated till and including annual 1945.

PETERMANN'S BAR SCALE VALUES
Country Name 1 degree Remarks
GENERAL Geographical mile 15 1 = 7,420.44 m
Kilometre 111; 111.11; 111.3; 111.301; 111.3066**; 111.307; 111.31
111.324 with an equator of 40,076.60 km
Nautical mile 60 In 1874 'geographical miles' are used in Stieler's Schulatlas
1,000 geometrical paces****
Nautic league, Sea league**** 20
AFRICA Pack-camel hours 30; 31; 33 1 = 3.7 km
1 = 3.6 km
1 = 4.415 km, 69-74 camel paces per minute (1 metre = 1.022 pace)
Travel-camel hours 17 1 = 6.5 km
(Caravan) hours 25
travel hours (on land) 22.6; 30 1 = 5 km = 1 hour on horse
1 = 4.8 km
1 daytrip of 10 hours of 4 km = 80 mm
travel hours (by boat) 1 = 6 km
ARABIA Great miles**** 50
ARGENTINA Legua 21.42; 21.5
ASIA Parasang 1 = ca. 5.2 km
AUSTRIA Post (or Polizey) Meile 14.67 1 = 4,000 Wiener Klaften = 24,000 Wiener Fusse
Wiener Klafte 58,683
Wiener Zoll 1 = 500 Wiener Klaften
BELGIUM mijl 20
BRAZIL Legua 18 1 = 6,000 m
CHILE Legua 20.0; 24.6
CHINA Li 193; 193.4**; 199.9; 200; 250
Great li or Chinese furlongs**** 200
COLOMBIA Legua Granadinas 22.15 1 = 6,280 Varas = 5.024 km
By Law of May 25, 1836
CUBA Legua regular antigua 20 = 60 mm
DANMARK Mile* 14.77; 14.79***
League**** 13.5
FRANCE Heure 25
Lieue 25
Lieue marine 20
Lieue metrique 28
Pied 4,000 = 20 mm
GERMANY (Geographische) Meile 15 1 = 1[0],000 Schritt
1 = 4 nautische Meilen
1 = 1,972.25 Rheinl. Ruthen
1 = 7.42 km
Baierische Chauss&eaigu;ee Meile*** 15.009
Kleine Böhmische Meile*** 16.12; 17.3****
Jewish mile** 100.80
Norddeutsche Meile** 14.84
Nürnberger Meile** 13.10
Preussische Meile 14.77; 14.776*** 1 = 2,000 Rh. Ruthen = 10,000 Schritt
(Reise)Stunde 25 1 = 1,1182.15 Rheinl. Ruthen
GREECE Miles employed in the Archipelago**** 95.5
Miles employed in Turkey**** 87
Olympic stadi * 600 1 = 184.7 m
1 = 184.18 m /1 pletron = 100 feet /1 foot = 0.3 m
Strabonic stadi 625 1 = 180 m
Royal stadi 111,307 1 = 1 km
HUNGARY Mile** 13.30
ICELAND Pingmannaleidir 2.995 1 = 5 danish miles = 60,000 el
INDIA Cosses of Hindoostan**** 42
Carnatic cosses**** 37.5
INDONESIA Javaanse palen 59.09; 60; 73.8; 73.86
IRAN Adschmi 17.4
Farsak 10 = 78 mm
Mile** 22.50
IRAQ Adschmi 17.4
Common miles of Piemonte**** 50
IRELAND Mile**** 54.4.34.2
Common miles of Piemonte**** 50
Great miles of Piemonte**** 45
Miles of Milan and Tuscany**** 67.2
Roman mile* 75 600 = 49 mm [= 872 km]; 1=7,000 Napolitan palms****
JAPAN Ri 28.3; 28.32 1 = 36 Tcho
LITHUANIA Mile**** 20
MEXICO Legua 26.56; 26.6
Millas 79.7
NETHERLANDS uren** 19.67
mijl 111.307
NORWAY [pace] 3,000 = 21 mm
Miles* 9.85
OTTOMAN EMPIRE
see also: Turkey
hours 25
PERSIA Common pasarangs**** 17
Legua or Great pasarangs**** 50
POLAND Mile** 20
PORTUGAL Legoa 18; 22.26; 17.5**** 1=7.572 varas****
Legua maritima 20
RUSSIA Werst 104; 104.16; 104.2; 104.3; 104.33; 104.34; 105* 6.96 = 1 geographical mile
20 = 1 Zoll; 1=500 sazen****
Werst fixed by Peter The Great**** 90
SPAIN Legua nueva 16.6; 16.64; 16.65; 17.66; 17.5**** 1 = 8,000/7.572**** varas
Castilian legal league**** 26.5 1 = 5,000 varas
SPANISH AMERICA Legua (maritima) 20 Probably a Spanish measure; 1 = 5,000 m, 1 = 5,770 m
SWEDEN Miles* 10.41 (12; 10.5****)
League**** 12.5
League used in Lapland**** 21
SWITZERLAND Ruthe 2.000 = 88 mm
Stunde 23.15; 23.18; 20.67*** 1 = 16,000 Swiss feet = 4,800 metres
TURKEY Agat 22.26; 22**** 1 = 3 Berri
Berri** 66.67
[feet] 200 = 63 mm
[hour] 25
UNITED KINGDOM Statute mile 69.1; 69.12; 69.13; 69.15; 69.16; 69.164 Also called English, British or American miles
Geographical mile 60

*

From: Brockhaus' Conversations-Lexikon, 13. Aufl., 1882-1887.
** From: Mass und Gewicht / Hans-Joachim v. Alberti, 1957 (see below).
*** From: Stieler's Karte von Deutschland in 25 Blätter
**** From: An untitled English atlas, published 1790-1798

MILE

Miglio (Italian), mijl (Dutch), mile (English), milha (Portuguese), milla (Spanish), mille (French), all deriving from the Latin mille = thousand. Measure in ancient Rome as milia passuum, later miliarum = 1,000 steps (paces) of 5 Roman feet = 1,478.7 m or 8 stadia (the latter according to classical authors).
In general 1 sea mile = 1 nautical mile = 1 geographical mile = 1 minute latitude or 1 minute longitude at the equator. (info: Maura O'Connor)
A closely related measure derived from the mile is the old Gaulish measure leig, in Latin leuca or leuga, later league (English), lega (Italian), legua (Spanish and Provencal), legoa (Portuguese), lieue (French), which usually was equivalent to 3 miles.
Both denominations seem to have been rather common in western Asia and Europe.

The following measures derive from: Lexikon der Münzen, Maße, Gewichte, Zählarten und Zeitgrößen aller Länder der Erde / be arbeitet und herausgegeben von Richard Klimpert. - 2. Vielfach verb. Und verm. A ufl. - Berlin : Regenhardt, 1896. [metres Lex.] or
Grand dictionnaire universel du XIXe siècle / par Pierre Larousse. - Paris : Administration du Grand Dictionnaire Universel, 1874. [metres Dict.].
Though the mile and league seem to be common names sources do not totally agree as to their value!

MILE measurements td>
Country Area/name metres Lex. metres Dict. 1 degree Remarks
AUSTRIA Bohemia 6,910
Malachia 7,848.5 4,000 klafter
Postal mile 7,585.937 14.65 4,000 klafter = 24,000 feet
DANMARK 7,532.485 14.77 Prussian mile
FRANCE Lieue vieil 4,451.9 4,444 25 Picardie, Normandie, Champagne
Lieue moyenne 5,008.4
Lieue marine 5,564.9 5,555 20
Lieue d'Artois / Maine / Perche / Poitou 3,964 28
Lieue de Beauce/ Gotinais 3,268 34
Lieue du Bourbonnais 4,826 23
Lieue de Bourgogne 5,121 21.5
Lieue de Bretagne / d'Anjou 4,581 24.25
Lieue de Paris / Sologne / Touraine 3,933 28.25
Lieue de Provence / Gascogne 5,849 19
GERMANY Baden 8,889 12.50
Baltic provinces 7,467.5 14.879
Bayern 7,420,438 7,426 15
Böhmen 7,498.5 14.821 12,600 el
Braunschweig 7,419.42
Bremen 1,852
Geographic mile 7,420.438 15
Gotha 7,421.125
Hamburg 7,532.485 7,500 Prussian mile
Hannover 7,419 7,532 15.002 24,000 Rheinland feet
Hessen-Darmstadt 7,500
Kurhessen 9,206.37 12.07
Lippe-Detmold 9,264.42
North German Bund 7,500 7,407 14.84
Nürnberg 13.10
Oldenburg, police mile 8,876.37 1,500 ruthe
Oldenburg, geographic mile 7,419.86
Prussia 7,532.485 7,407.407 14.754 24,000 feet
Rhine 7,783
Saxonian mile 7,500 14.84
Saxonian police mile 9,064 32,000 feet
Schleswig-Holstein 8,803.48
Tirol = Innsbruck 10,691.111
Weimar 7,363.026 6,798
Württemberg 7,448.748 14.67
GREAT BRITAIN League 5,569.339
sea-league 3 nautical miles or 5.556 km (info Victor Prescott)
London mile 1,523.986 73.0308 5,000 feet or 8 furlongs
Statute mile 1,609.3295 69.16 Since the change of the statute in 1593 this is 5,280 feet
Nautical/geographic mile 1,854.965 60 6,085.898 feet
HUNGARY 8,353.6 13.30
IRELAND Irish mile 2,240 yards or 6,720 feet (Gazetteer of the British Isles, J. Bartholomew Sons, 1966)
IRAN4,946
ITALYLombardia1,784.808
Napels1,855.11060
Piemonte2,466.0768
Rome1,487.93474.675
Toscane1,635.67
Venice1,738.6751,834
Venice nautical mile1.852
LITHUANIA8,95428,530 Rheinland feet
NETHERLANDSMijl5,5655,85720.2020,629 Rheinland feet (Dict.)
Nautical mile5,55620
NORWAY11,295.4811,1399.8536,000 feet
PERSIA. see: IRAN
POLAND7,420.43820 [!]
PORTUGALLegoa6,179.740
Legoa nova5,00022.26
Milha2,065.653
RUSSIAWerst1,067104.3
SPAINCommon league5,606.569
Legua maritima = Legua legal5,565.3296,365
Legua nueva6,687.2416.64
Legua regular antigua5,572.7
Royal league7,066.375
Milla1,4131,000 paces
SWEDENMil10,688.43610.4136,000 feet
SWITZERLAND4,8084,48023.15hour
TUNESIALand mileappr. 1,500
Nautical mile1,806.756.673,700 Draa
TURKEYMile1,607
Nautical mile1,479

Geoff Armitage's Conversion table of measurements
Geoff Armitage of the British Library Map Library in the past years has created a conversion table into millimetres of measures appearing in the British Library map catalogues. I am greatly indebted to him to be able to enrich this document with his table.

GEOFF ARMITAGE'S CONVERSION TABLE OF MEASUREMENTS
Name of Measure Approx. Equivalent in millimetres
Antwerp ruthen 5,736
Aunes 1,143
Baras castellanes 835
Bolognese foot 380
Brabant foot 281
Bracas 2,200
Braccia 600
Brasse 595
Brazos castellanas 1,683
Brazza 595
British fathom 1,828
Cable 219,456
Calemberger foot 292
Calemberger ruthen 4,672
Canne 2,000
Canne anconitane 2,000
Canne napolitane 2,096
Canne romane 2,112
Canne siciliane 2,028
Carmi 2,000
Castilian league 6,350,500
Castilian varas 835
Chain 20,117
Cleffter 2,000
Common league 7,408,900
Dutch league 5,969,990
Dutch mile 1,000,000
English league 4,828,032
Faden 1,629
Fathom 1,828
Florentine braccia 583
Florentine mile 1,778,000
Foot 305
French foot 330
French league 4,448,200
French marine league 5,556,700
French pace 812
French toise 1,949 (post-1812: 2,000)
Genevese toise 2,599
Geometrical foot 337
Geometrical pace 1,524
German mile 7,649,000
Irish perch 6,400
Italian mile 1,852,200
Italian pace 1,500
Kilometre 1,000,000
Klaffter 2,000
Lachter 2,036
League 4,828,032
Leucarum Hispanicarum [= Spanish league???] 6,300,000
Lieue [= league] 4,828,032
Lieue commune de France 4,445,400
Lieue japonaise ???
Lieue marine 5,556,700
Marine league 5,556,700
Marine mile 1,852,200
Metre 1,000
Mexican league 4,190,000
Mexican varas 848
Milanese mile 1,652,600
Mile 1,609,344
Miliarium/milliaria [= English mile] 1,609,344
Mille (itineraire) 1,949,000
Mille marin 1,852,200
Milliaria anglica [= English mile] 1,609,344
Milliaria germanica [=German mile] 7,649,000
Milliaria Italica [= Italian mile] 1,852,200
Milliaria thietm. [= Thietmarsh mile ???] ???
Modenese perch 3,180
Nautic[al] mile 1,852,200
Pace 762
Palmi 250
Palmi genovese 249
Palmi romani 228
Paraguay league 4,190,000
Paris foot 330
Pas [= French pace] 812
Passi 1,500
Pedum [= Foot???] 305
Perch 5,029
Perticarum [= Perch] 5,029
Pertiche ferrarese 4,038
Pertiche modenese 3,180
Pertiche versonese 2,057
Piedmontese mile 1,778,000
Pole 5,029
Rhenish/Rheinland/Rynland - rod/ruthen/roeden 3,766
Rhenish foot 314
Rhenish verge/yard 3,766
Rhine see Rhenish
Rod 5,029
Roden/Danish perches??? (La Rode) 3,138
Roman palmi 228
Russian faden/fathom 1,629
Russian toise 1,604
Rynland see Rhenish
Scala [ignore; note the next word]
Schrit[te] 1,710
Schuh [= German foot???] 290
Scots chain 22,676
Sea league = marine league??? 5,556,700
Sea mile = nautic[al] mile??? 1,852,200
Spanish league 6,300,000
Spanish maritime league 5,566,700
T. [= Toise] 2,000
T[h]oise 2,000
Trabocci/Trabucchi 3,000
Trabocchi of Piacenza 2,819
Varas [Castellanas/Castille/Espanolas/Spanish] 858
Venetian mile 1,738,700
Venetian pasa/pace 1,739
Verge de Rhin[land] 3,766
Veronese mile 1,778,000
Werst 1,066,780
Yard 914

As an aid to research and cataloguing the following table contains publications which concern measures etc. Should there be more than one significant publication in a country they are, when possible, organized from the general to the specif ic.

PUBLICATIONS CONCERNING BAR SCALE VALUES AND OTHER MEASURES
Country Publication Remarks (LANGUAGE)
UNIVERSAL Dictionnaire des poids et mesures anciens et modernes, contenant des tables des monnais de tous les pays / par Horace Doursther. - 3e éd. - Amsterdam : Meridian, 1976. - IV, 603 p. - ISBN 90-6041-111-0
Original ed.: Bruxelles : M. Hayez, 1840
Reprint: Amsterdam : Meridian, 1965
Based on published works between 1830-1840. All measures are converted to the decimal system and, where necessary, to other universal measures. Measures are usually regionally subdivided to area of origin. Also locally used designations are included with reference to the french designation (FRENCH)
Elsevier's encyclopedic dictionary of measures / Donald Fenna. - Amsterdam (etc.) : Elsevier, 1998. - XXIII, 582 p. ; 25 cm. - ISBN 0-444-50046-4 Some 4,000 terms are identified in familiar English alphabetic order and related to their fellow units within their culture and to corresponding terms of adjacent and other interacting peoples. With index by country. (ENGLISH)
Geographical conversion tables = Tables de conversion géographique = Geographischen Umrechnungstafeln = Geograficeskie tablicy perevoda = Tablas de conversion geográficas / comp. and ed. by D.H.K. Amiran and A.P. Schick. - Chicago : UGI ; Zürich : International Geographical Institute. - XXXVI, 315 p. : tab., maps ; 25 cm. (ENGLISH, FRENCH, GERMAN, RUSSIAN, SPANISH)
Mass und Gewicht : geschichtliche und tabellarische Darstellungen von den Anfängen biz zur Gegenwart [Measures and weights : history and tables from the beginning till the present] / Hans-Joachim v. Alberti. Berlin : Akademie Verlag, 1957. - XX, 580 p. (GERMAN)
Monnaies, poids, mesures et usages commerciaux de tous les états du monde. - 2e éd. - Paris [etc.] : Hachette, 1875. - VIII, 386 p. Arranged geographically by French name. Appendix: Tableaux de conversion des monnaies, poids et mesures d'Angleterre en monnaies, poids et mesures de France et réciproquement. (FRENCH)
NTC's encyclopaedia of international weights and measures / William D. Johnstone. - Lincolnwood (Illinois) : NTC Publishing group, 1966. - 329 p. ; 15 cm. - ISBN 0-8442-0850-7 Section on units of length 57 pp. Includes ancient linear units (ENGLISH)
Spravochnik mer / sostaviteli V.A.Sokolov i L.M.Krasavin ; Nauchno-issledovatel'skij konyunkturnyj institut ministerstva vneshnej torgovli soyuza SSR. - Vtoroe, dopolnennoe izdanie [Handbook of measurements / compilers: V.A.Sokolov and L.M.Krasavin ; Scientific Market Conditions Research Institute of the Ministry of Foreign Trade of the USSR. - 2nd, augmented ed.] - Moskva : Vneshtorgizdat, 1960. - 246 p. Linear measures encountered in the alphabetical listing are in Russian, with the Latin given in parenthesis in those cases where the Russian transliterates differently from the original. Some exotic measures are encountered, like the British 'nail' (5.71 cm) and American 'place' (76.2 cm). Measures arranged by country and alphabetically. (RUSSIAN)
For good measure : a complete compenduim of international weights and measures / William D. Johnstone. - New York : Holt, Rinehart and Winston, [ca. 1975]. - XXII, 329 p. - ISBN 0-03-013946-5 Part one: Units of length (pp. 1-57)
Part five: the metric system and conversion tables (pp. 208-212) (ENGLISH)
CLASSICAL Byzantinische Metrologie [Byzantian metrology] / von Erich Schilbach. - München : C.H. Beck'sche Verlagsbuchhandlung, 1970 . - XXIX, 291 p. - ISBN 3-406-01424-0 P. 13-55: longitudinal measures (GERMAN)
Griechische und römische Metrologie [Greek and Roman metrology] / von Friedrich Hultsch. - 2. Bearb. - Graz : Akademische Druck- u. Verlagsanstalt, 1971. - XIV, 745 p. P. 27-39: longitudinal measures (GERMAN)
ISLAMIC WORLD Islamische Masse und Gewichte : umgerechnet ins metrische System [Islamic measures and weights : converted to the metric system] / von Walther Hinz. - Leiden : E.J. Brill, 1955. - 66 p. P. 54-64: longitudinal measurements (GERMAN)
BELGIUM Oude maten, gewichten en muntstelsels in Vlaanderen, Brabant en Limburg [Old measures, weights and monetary systems in Flanders, Brabant and Limburg] / Paul Vandewalle. - Gent : Belgisch Centrum voor Landelijke Geschiedenis, 1984. - 70 p. Arranged by municipality, refering to a table of 17 geographical entities with their measures. (DUTCH)
DANMARK De gamle danske længdeenheder [The old danish units of distance] / N.E. Nørlund. - København : Munskgaard, 19 44. - 80, [12] p. A history of Danish units of distance (DANISH)
Mål og vægt [Measures and weights] / Poul Rasmussen. - København : Danish Association of Historical Societies, 1967. - 87 p. A handbook of medieval weights and measures (DANISH)
FRANCE French weights and measures before the revolution : a dictionary of provincial and local units / Ronald Edward Zupko. - Bloomington : Indiana University Press, 1978. - XLVII, 208 p. - ISBN 0-253-32408-7 (ENGLISH)
GERMANY Bi-Lexicon alten Masse, Münzen und Gewichte / Helmut I. Kahnt und Bernt Knors. - Leipzig : Bibliographisches Institut, 1986. - 380 p. : ill. - ISBN 3-323-00013-7 (GERMAN)
ITALY Italian weights and measures from the Middle Ages to the nineteenth century / Ronald Edward Zupko. - Philadelphia : American Philosophical Society, 1981. - LXXXIV, 339 p. - ISBN 0-87169-973-8 (ENGLISH)
THE NETHERLANDS De oude Nederlandse maten en gewichten / J.M. Verhoeff. - 2nd ed. - Amsterdam : P.J. Meertens-Instituut, 1983. - XIII, 131 p. - (Publicaties van het P.J. Meertens-Instituut ; Deel 3). - ISBN 90-70389-07-X Contains: Dutch measures for weight, lenght, contents and volume, from the Middle Ages till the present, arranged by area and by name (DUTCH)
Vergelijkingstafels van lengetematen en landmaten / J.H. van Swinden ; uitg. en ingel. door R. Rentenaar. - Wageningen : PUDOC, 1971. - 2 dl. (153 + 170 p.); 30 cm. - Met lit. opg. - ISBN 90-220-0352-3 Contains reprints of the Dutch parts of Van Swinden's Vergelijkings-tafels tusschen de Hollandsche lengte-maten en den mètre en Vergelijkings-tafels tusschen de Hollandsche land-maten en de hectare, (both from 1812), and his notes (DUTCH)
NORTH AMERICA Archaeological metrology: English, French, American and Canadian systems of weights and measures for North American historical archaeology / Lester A. Ross. - [Ottawa, Ont.] : National Historic Parks and Sites Branch, Parks Canada, 1983. - 123 p. - (History and archaeology, ISSN 0225-0101 ; 68). - ISBN 0-660-11336-8 English systems: linear systems (pp. 5-54). Measures are given in two historical tables, 1305-1826 and 1826-present. Metric equivalent is given.
French systems (New France): linear systems (pp. 75-80). This gives the time periods in which the systems were in use, with metric equivalents.
American systems: pp. 87-90.
Canadian systems: pp. 91-100, giving measures used in the Dominion as well as in the provinces.
(ENGLISH)
Métrologie archéologique : systèemes de poids et mesures anglais, français, américain et canadien pour l'archéologie historique de l'Amérique du Nord / Lester A. Ross. - [Ottawa, Ont.] : Direction des lieux des parcs historiques nationaux, Parcs Canada, 1983. - 115 p. - (Histoire et archéologie, ISSN 0227-3551 ; 68). - ISBN 0-660-91044-6 Systèmes anglais: Longeur: pp. 46-50
Systèmes français: pp. 71-76
Systèmes américain: p. 86
Systèmes canadiens: pp. 95-97
UNITED KINGDOM British weights and measures : a history from a ntiquity to the seventeenth century / R.E. Zupko. - Madison : University of Wisc onsin Press, 1977. - 248 p ; 16 cm. - ISBN 0-299-07340-8 88 pp. of tabl es; includes old British and European measures; extensive bibliography and index es (ENGLISH)
A dictionary of english weights and measures from Anglo-Saxon times to the nineteenth century / R.E. Zupko. - Madison : University of Wisconsin Press, 1968. - 224 p. ; 15 cm Dictionary arrangement; extensive bibliography (ENGLISH)
The weights and measures of England / R.D. Connor. - London : Her Majesty's Stationery Office, 1987. - XXVI, 422 p. - ISBN 0-11-290435-1 Includes classical and Celtic measures (ENGLISH)

Geographical co-ordinates
Though Greek philosophers like Pythagoras, Aristoteles, and Erathosthenes already posed that the earth was spherical it was the famous Greek astronomer Hipparchos (ca. 190 - 125 B.C.) who thought to cover this sphere with a grid of meridians and parallels. Following the Babylonian use of dividing circles and angles according to the sexagesimal system he created a grid of 360 lines running from the North to the South Pole and 180 lines running parallel to the equator. The lines running from the North to the South Pole later were called meridians, because when two places had the same time at noon they were on the same meridian, after the Latin 'meridies'.

For a general and mathematical overview there is the Coordinate Systems Overview by Peter H. Dana of the Department of Geography, University of Texas at Austin.

Looking for ways for coordinate conversion and transformation the site Cartographic links for botanists compiled by Raino Lampinen, Botanical Museum, Finnish Museum of Natural History, contains mapping software packages, which have various utilities for coordinate conversion.

For geographic coordinate transformation pertaining to the Dutch grid and vice versa one can use the website Transformatie van RD-coördinaten en geografische coördinaten created by Ed. Stevenhagen. (There is also a Java-script with maps where the location is indicated). Besides it automatically gives the coordinates in WGS84 and the meridian-convergence.

In 1761 John Harrison (1693-1776) solved the longitude problem when his Model No. 4 or "H. 4" chronometer was used on a nine-week trip from London to Jamaica. During this trip his clock only lost five seconds, or about 1.25 minutes of longitude. His "K. 1" clock was successfully tested by James Cook on his second voyage around the world, beginning in 1772. (Boorstin, Daniel J. (1991). The discoverers. Vol. 1, p. 86.).
Connected to the problem of the prime-meridian is that of it's opposite, the date line. From a Western point of view this was always situated somewhere at its antipode, as fictitionally treated by Umberto Eco in his The island of the day before (originally published as L'isola del giorno prima, 1994). A more scientific treatment of this problem can be found on A History of the International Date Line by Robert H. van Gent.

As the position of prime meridians is not always known I reproduce here a table in use with the CCK (Dutch Union Map Catalogue), ammended with information from other sources, among others Cartographic materials : a manual of interpretation for AACR2. The position is given with respect to the (Greenwich) International Prime Meridian, adopted at the 1884 International Meridian Conference at Washington DC, USA.

LOCATION OF PRIME MERIDIANS
City Country Position
Alexandria Egypt Used by Albert Hermann in 1930 for a reconstruction of a map of Marinus of Tyrus. The meridians are hours west or east of Alexandria
Amersfoort Netherlands E 005°23’
Amsterdam Netherlands E 004°53’01"
Antwerp Belgium E 004°22’50"
Athens Greece E 023°42’59"
Batavia (Jakarta) Indonesia E 106°48’28"
Berlin Germany E 013°23’55"
Berne Switzerland E 007°26’22"
Bogota Colombia W 074°04’53"
Bombay India E 072°48’55"
Brussels Belgium E 004°22’06"
Bucharest Romania E 026°07’
Cádiz Spain W 006°17’42"
Canberra Australia E 149°08’
Capetown South-Africa E 018°28’41"
Caracas Venezuela W 066°55’50"
Celebes, Middle Meridian of Indonesia E 121°48’
Christiana (Oslo) Norway E 010°43’23"
Copenhagen Denmark E 012°34’40"
Córdoba Argentina W 064°12’03"
Ferro Canary Islands W 017°39’46"
Greenwich United Kingdom E 000°00’00"
Genoa Italy E 008°55’
Helsinki Finland E 024°57’17"
Istanbul Turkey E 028°58’50"
Jakarta Indonesia See: Batavia
Julianehaab Greenland W 046°02’22"
Kaliningrad Russia See: Köningsberg
Köningsberg Russia E 020°29’47"
Leningrad Russia See: St. Petersburg
Lissabon Portugal W 009°11’10"
London United Kingdom W 000°05’43"
Madras India E 080°14’50"
Madrid Spain W 003°41’15"
Mexico City Mexico W 099°11’40"
Moscow Russia E 037°34’15"
Munich Germany E 011°36’32"
Naples Italy E 014°15’42"
New York City (Manhattan) United States W 074°00’29"
Oldenburg Germany O 008°12’
Oslo Norway See: Christiana
Padang, Sumatra Indonesia E 100°22’01"
Paris France E 002°20’14"
Peking China E 116°28’10"
Philadelphia United States W 075°08’55"
Pulkovo (St. Petersburg) Russia E 030°19’39"
Quito Ecuador W 070°30’
Rio de Janeiro Brazil W 043°01’21"
Rome Italy E 012°29’05"
Rotterdam Netherlands E 004°29’46"
San Fernando Spain W 006°12’
San Francisco United States W 122°27’
Santiago Chile W 070°41’00"
Singkawang, Borneo Indonesia E 108°59’41"
South Sumatra, Middle Meridian of Indonesia E 103°33’/td>
St. Petersburg Russia E 030°18’59"
Stockholm Sweden E 018°03’30"
Sucre Bolivia W 065°15’
Sydney Australia E 151°12’23"
Tenerife Canary Islands W 016°35’
Tirana Albania E 019°46’45"
Tokyo Japan E 139°44’40"
Washington (D.C.) United States W 077°00’34"

When not taking into account which prime meridian is used the following situation might occur.

HUMOR: Teaching Coordinates

The geography teacher was lecturing on map reading. After explaining about latitude, longitude, degrees, minutes, and seconds, the teacher asked, "Suppose I asked you to meet me for lunch at 23 degrees, 4 minutes, 30 seconds north latitude and 45 degrees, 15 minutes, zero seconds east longitude."

After a confused silence, a voice volunteered, "I guess you'd be eating alone."

(Ken Everard, on Maphist, 8 February 2001)

When the teacher meant GMT as prime meridian he would have been lunching somewhere in the Arabian Desert called Dawasir. Had the teacher meant e.g. the San Francisco prime meridian he would have been lunching on a boat on the Great Bahama Bank near Channel Rock!

Centesimal system of co-ordinates

(derived verbatim from: Cartographic materials : a manual of interpretation for AACR2)
The sexagesimal division of the circle is now virtually universal in cartographic work. However, in the 18th century French scientists, using the metric system, devised the centesimal division of the circle. Today there exist large numbers of maps of France and its former colonial territories based on such a system. It can be quite confusing due to the relative closeness of the values.
The centesimal division of the circle is extremely simple. The entire circle is divided into 400 grads (a right angle in 90° in the sexagesimal system, 100 grads in the centesimal system). Each grad is in turn divided into 100 minutes and each minute into 100 seconds. The centesimal values can be expressed in regular decimal form or as minutes and seconds. The grad is shown as "G" and the centesimal minutes and seconds have the same marks as the sexagesimal ones, but with the slopes of the marks in the opposite direction.

Sexagesimal notation: 37°23’12”"
Centesimal notation: 48G.5734 or 48G57`34``

The process of conversion is very simple.
It is known that 90° equals 100G and that 60 sexagesimal minutes or seconds equals 100 centesimal minutes or seconds. Through a simple proportion multiply the centesimal values by 0.9 to obtain sexagesimal degrees and the remainders are multiplied by 60 to obtain sexagesimal minutes and seconds.

The latitude of downtown Saigon is 11G.9297 N or 11G97`27`` N.

Therefore:
11.9727 x .9 = 10.77543
                            .77543 x 60 = 46.5258
                                                       .5258 x 60 = 31.5480
A centesimal value of 11G.9297 N or 11G97`27`` N equals a sexagesimal value of 10°46’32" N.

A Decimal To Degrees Converter or Degrees to Decimal Degrees Converter is made available by Gary J. Park of the Earth Observation Group.

Equinox
The equinox is one of the two points of intersection of the ecliptic and the celestial equator, occupied by the sun when its declination is 0°. This for most map curators intangible phenomenon has been well described in Cartographic materials : a manual of interpretation for AACR2 paragraph 3D2, p. 62-65. I refer those who are interested to this text as no other source is available to me.