MMAP DRAWING AND FOLIATIONS OF THE SPHERE
ATHANASE PAPADOPOULOS
Abstract.
I will consider some questions related to Euler’s work oncartography and its consequences, in which the foliations of the sphereby meridians and parallels play important roles.AMS classification: 01A50, 5103Keywords: mathematical geography, map drawing, history of carto-graphy, perfect map, Leonhard Euler, Joseph-Nicolas Delisle, foliation. Introduction
I will consider some questions related to Euler’s work on the drawing ofgeographical maps, in which the foliations of the sphere by meridians andparallels play an important role. This article can be regarded as a sequel tomy article [14] in which I talked about the works of Euler and Chebyshev ongeography. It is also the occasion to straighten out a statement often madein the literature concerning Euler’s contribution to cartography.2.
On Euler’s “perfect” mappings
I will start with a few remarks on Euler’s memoir
De repraesentatione su-perficiei sphaericae super plano (On the representation of spherical surfaceson a plane) [6], presented to the Saint Petersburg Academy of Sciences onSeptember 4, 1775, and published in the 1777 volume of the
Acta AcademiaeScientarum Imperialis Petropolitinae .This memoir is very poorly quoted in the literature. On the one hand, it isreferred to for a result which is attributed to Euler, although this result wasknown since Greek antiquity, and its proof follows immediately from resultsin spherical geometry that were known in that period. On the other hand,there are interesting results that Euler proved in this memoir which, to myknowledge, are never mentioned in the papers or books on mathematicalcartography.The result that is usually attributed to Euler in relation with the paper[6] says that there is no “perfect” map from a subset of the sphere to theEuclidean plane. A confusion is entertained by the fact that the word “per-fect” was used by Euler in his paper without a proper definition. In the(relatively large number of) papers in which Euler’s memoir is quoted, it issaid that Euler proved in this memoir that there is no map from a subsetof the sphere onto the Euclidean plane that preserves distances up to scale.The reason for this situation is that, as it often happens in “historical” pa-pers, authors get their information from papers written by other authors on
Date : 4th September 2020. a r X i v : . [ m a t h . HO ] S e p ATHANASE PAPADOPOULOS the subject without bothering to look into the original sources and try tounderstand them. Let us give a few examples of such quotes.T. Feeman, in his book
Portraits of the Earth , discusses the existence ofa map from a portion of the sphere onto the plane which has a fixed scale,that is, as the author puts it, a map such that the the quantitydistance between two points on the globedistance between their imagesis constant, i.e. independent of the chosen pair of points. The author writes[10, p. 25] (2002): “Over the years various attempts by cartographers tosolve this problem resulted in some ingenious, if flawed, maps. Finally, in1775, Leonhard Euler (1707–1783), the leading mathematician of his day andone of the most important mathematical figures of all time, presented to theSt. Petersburg Academy of Sciences a paper entitled
On representations ofa spherical surface on the plane in which he proved conclusively that sucha map could not exist.”In the article
Dallo spazio come contenitore allo spazio come rete , C. Cor-rales Rodriganez writes [3, p. 125]: “The mathematician Leonhard Euler, inhis article
De representatione superficiei sphaericae super plano , publishedin the eighteenth century, proved that no part of the Earth can be repro-duced over a plane surface without deformation. Euler’s theorem says thatthe perfect map does not exist.” Incidentally, the author adds that “Euler’stheorem has pushed the mathematical cartographers to study spherical geo-metry and trigonometry as a subject in itself, independent of Euclidean geo-metry.” It is not clear to what mathematical cartographers the author isreferring to, but Euler himself was a cartographer, and he had published sev-eral works on spherical geometry and spherical trigonometry several decadesbefore he wrote this memoir. Furthermore, long before Euler, Ptolemy (2ndc.A.D.), one of the most famous mathematical cartographers of all times, wasalso thoroughly involved in spherical geometry and spherical trigonometry.As a matter of fact, the field of spherical geometry remains poorly knownto historians of mathematics. In the book
Portraits of the Earth which wealready mentioned, the author claims (p. 25) that spherical geometry can bedeveloped axiomatically as a Euclidean geometry in which Euclid’s fifth pos-tulate is replaced by a postulate saying that any two lines intersect (in twopoints). This statement is not correct. The confusion is probably caused bythe fact that the axioms of hyperbolic geometry (which, together with Euc-lidean and spherical geometry forms the three “classical” geometries, or thegeometries of constant curvature) are precisely those of Euclidean geometrywith the fifth postulate replaced by its negation. Spherical geometry may [Il matematico Leonard Euler] nel suo articolo De representatione superficiei sphaer-icae super plano , pubblicato nel Settecento, ha provato che nessuna parte della Terra pu`oessere riprodotta su une superficie piana senza deformazione. Il teorema di Eulero diceche la carta perfetta non esiste. Il teorema di Eulero ha spunto i cartografi matematici a studiare la geometria sfericae la trigonometria come materie a s´e, indimendenti dalla geometria euclidea. An edition of Euler’s and his collaborators on spherical geometry will appear in thebook [1].
AP DRAWING AND FOLIATIONS OF THE SPHERE 3 be developed axiomatically, but such a set of axioms cannot be so simplyobtained from those of Euclidean geometry.Let us continue with our citations of Euler’s result.R. Osserman, in his paper
Mathematical mapping from Mercator to themillennium [13, p. 234] (2004) attributes the following theorem to Euler:
Itis impossible to make an exact scale map of any part of a spherical surface .By an “exact scale map”, Osserman means a map that preserves distancesup to scale, and he refers again to Euler’s paper [6].P. Robinson, in a paper titled
The sphere is not flat [16] (2006), writesthe following: “The theorem of our title asserts that there is no isometric(that is, distance-preserving) function from the sphere (or indeed from anyof its nonempty open subsets) to the Euclidean plane; more generally, thereis no isometry to any Euclidean space. This theorem may be traced back toEuler, in his
De repraesentatione superficiel sphaericae super plano of 1778.”The same poor attribution to Euler occurs in the chapter titled
Curvatureand the notion of space in the book
Mathematical Masterpieces (2007) by A.Knoebel, J. Lodder, R. Laubenbacher and D. Pengelley [9, p. 163], wherethe authors write: “In the paper presented to the St. Petersburg Academyof Science in 1775,
De repraesentatione superficiei sphaericae super plano ,Euler proved what cartographers had long suspected, namely, the impossib-ility of constructing a flat map of the round world so that all distances onthe globe are proportional (by the same constant of proportionality) to thecorresponding distances on the map.”J. Gray, in his book
Simply Riemann [11] which appeared in 2020, says thefollowing, about Euler’s work on cartography: “Euler used all his analysisto prove that every cartographer suspected: that there could be no map ofthe Earth’s surface onto a plane that is accurate in every respect. Somemaps send curves of shortest length on the sphere to straight lines in theplane; there are maps that send equal angles to equal angles, and there aremaps that scale all areas by the same amount. But there can be no mapthat does all of these at once.”Naturally, popular science authors get their information from mathem-aticians’ writings when they understand them: In the Spanish daily news-paper
La Vanguardia , on 26 March 2017, in an article titled
Un mundo, tresmapas , the author, A. Molins Renter, writes: “Passing from a spherical geo-metric form to a plane support, two-dimensional and usually of rectangularshape, results in the fact that something is always lost in the translation,as the Swiss mathematician and physicist Leonhard Euler already demon-strated in 1778, in his work
De repraesentatione superficiei sphaericae superplano .” One could give are many other examples.In fact, the statement attributed by all these authors to Euler was obvi-ously known to him, but it does not convey the slightest idea of the resultsobtained in the memoir quoted, which are much stronger and much more Pasar de una forma geom´etrica esf´erica a un soporte plano, bidimensional y normal-mente con forma rectangular provoca que algo se pierda siempre en la translaci´on, comoya demostr´o el matem´atico Leonhard Euler en 1778, en su obra
De repraesentatione su-perficiei sphaericae super plano . ATHANASE PAPADOPOULOS interesting than what all these authors claim. Furthermore, as I said, theresult that they quote was known since the 1st-2nd century A.D., since itfollows as a corollary from several results contained in Menelaus’
Spherics on the geometry of spherical triangles. For instance, it is an immediate con-sequence of the result saying that the angle sum in a spherical triangle isalways greater than 2 right angles (this is Proposition 12 in [15]), or fromthe comparison result saying that in any spherical triangle
ABC , if D and E denote the midpoints of AB and B respectively and if DE is the shortest arcjoining them. Then DE > AC/ perfect map , the meaningof this sentence should be understood in context, that is, by following thearguments that lead to it.Recently, C. Charitos and I. Papadoperakis wrote a paper titled
On thenon-existence of a perfect map from the 2-sphere to the Euclidean plane [2] inwhich they give a precise statement of Euler’s result and provide a detailedproof of it.To state Euler’s result correctly, we call a map f from a region S of the2-sphere to the Euclidean plane perfect if every point in the domain has aneighborhood on which the following two conditions hold:(1) f sends meridians and parallels to two fields of lines that make mu-tually the same angles;(2) f preserves distances infinitesimally along the meridians and theparallels.Thus, a perfect map sends the meridians and parallels to two line fields thatare orthogonal. Furthermore, a perfect map preserves globally the lengthelement along the meridians and the parallels. One should note here thaton the spherical globe, the meridians are geodesics but the parallels arenot. The fact that distances are preserved infinitesimally along the meridi-ans implies immediately that the distances between points on these linesare preserved. It also follows, although not so immediately, that distancesbetween points on parallels is preserved by a perfect map.The idea of Euler’s proof was to translate these geometrical conditionsinto a system of partial differential equations and to show that this systemhas no solution.Furthermore, Euler, in his paper, after showing the non-existence of aperfect map, proves several other results. He declares that since perfectmaps do not exist, one has to look for best approximations. He writes: “Weare led to consider representations which are not similar, so that the sphericalfigure differs in some manner from its image in the plane.” He then examinesseveral particular projections of the sphere, searching systematically for thepartial differential equations that they satisfy. He considers several classesof maps: conformal maps (which he calls “similitudes on the small scale”),area-preserving maps, and maps where the images of all the meridians are AP DRAWING AND FOLIATIONS OF THE SPHERE 5 perpendicular to a given axis while those of all parallels are parallel to it.He gives examples of maps satisfying each of the above three properties andin each case he studies their distance and angle distortion.3.
On the action of a geographical map on the foliations byparallels and meridians
An important feature of Euler’s memoir [6] and of the other memoirs thatwe shall consider below is that most of the properties of the geographicalmaps that are requested are formulated in terms of how these maps trans-form the two geographically most famous foliations of the sphere, namely,the foliations by parallels and by meridians.Let us recall that when the surface of the Earth is considered to be asphere, the parallels are the family of circles that are equidistant from theequator. The latter is the great circle that is perpendicular to the rotationaxis of the Earth, the one that separates the Northern hemisphere fromSouthern one. In the geometry of the sphere, the parallels are geometriccircles, that is, equidistant points from a center, which is either the Northor the South pole (a circle on the sphere has two centers). Furthermore,the parallels are small circles, that is, intersections of the sphere with planesthat do not pass through the center, except the equator itself, which isalso considered as a parallel (at zero distance from itself) and which is agreat circle, the intersection of the sphere with a plane passing through thecenter. The difference is important because on the sphere great circles aregeodesics whereas small circles are not. This foliations has two singularpoints, situated at the North and South poles.The second foliation that is used in the paper [6] is the foliation by me-ridians , whose leaves are the great circles perpendicular to the equator,or, equivalently, the great circles that pass through the North and Southpoles. Unlike the parallels, the meridians are all great circles, and thereforegeodesics. It has two singular points, situated at the North and South poles.Since the time of ancient Greek geography, the foliations by parallels andmeridians play an important role in map drawing, for representing regionsof the Earth, but also for maps of the celestial sphere. Figure 1 is a repro-duction of a representation of a celestial globe dating from the 1st centuryA. D. on which the foliations by parallels and meridians are drawn. Thesame picture could serve for the representation of the Earth.The properties of the images of the meridians and parallels are importantfactors in several known projections. For instance, under a stereographicprojection centered at the North pole, the parallels are sent to concentriccircles centered at the image of the South pole, while the meridians aresent to straight lines meeting at the North pole. The projection from thecenter of the sphere to a plane tangent to the South pole, known as thegnomonic projection, and which was used since the times of Thales, has asimilar property: parallels are sent to circles centered at the South pole andmeridians are sent to straight lines passing through this pole.The foliations by parallels and by meridians are perpendicular. Theirimages by a geographical map are usually drawn on the map, see e.g. Figures2 and ?? . ATHANASE PAPADOPOULOS
Figure 1.
A celestial globe with its two perpendicular foliationsby parallels and meridians. Wall painting fragment from the firstcentury A.D. Metropolitan Museum of Art, New York, Departmentof Greek and Roman Art. (Photo A. Papadopoulos.)
Two other memoirs by Euler on geography were published the sameyear as his memoir
De repraesentatione superficiei sphaericae super plano ,namely,
De proiectione geographica superficiei sphaericae [7] and
De proiec-tione geographica Deslisliana in mappa generali imperii russici usitata [8]. Iwould like to make a few comments on the latter.The memoir [8] is concerned with a projection from the sphere that wasused by Joseph-Nicolas Delisle who, during several years, was the maingeographer and the director of the astronomical department of the SaintPetersburg Academy of Sciences. He was in charge of drawing new andprecise maps of the Russian Empire. From 1735 to 1740, Euler assistedDelisle in this work until he became himself the head of the geographydepartment of the Academy, after a conflict emerged between Delisle and theAcademy’s administration, in relation with the so-called
Atlas Russicus (the“Russian Atlas”), a project initiated by Peter the Great and of which Delislewas in charge, and which he kept postponing. In 1740, the responsibility ofthe Russian Atlas was taken away from him and given to Euler.In his memoir, Euler starts by reviewing the main properties of a stereo-graphic projection used by the geographer Johann Matthias Hasius. Thelatter had published in Nuremberg, in 1739, a map of Russia known underthe name “Imperii Russici et Tatariae universae tam majoris et asiaticae
AP DRAWING AND FOLIATIONS OF THE SPHERE 7
Figure 2.
A map of the Northern Pacific, with the Eastern partof Asia and the Northern part of America, from Euler’s
Atlas Geo-graphicus (Berlin, 1753) quam minoris et europaeae tabula” (Geographical map of the Russian Em-pire and of Tataria, both large and small, in Europe and Asia). Eulermentions properties of the images of the two foliations by parallels and me-ridians, in particular that these images intersect at right angles (in fact, themap is conformal, that is, it is angle-preserving). Euler then reviews theinconveniences of this projection: length is highly distorted in the large, es-pecially for maps that represent large regions of the Earth and the images ofthe meridians are not evenly curved on the geographical map, even thoughthese lines are circles. In particular, the province of Kamchatka is distortedby a factor of four, compared to another region at the center of the map.In §
5, of his memoir, Euler states the following four properties that arerequired from an ideal geographical map: (i) the images of the meridians arestraight lines; (ii) the degrees of latitudes do not change along meridians; (iii)the images pf the parallels meet the images of the meridians at right angles;(iv) at each point of the map, the ratio of the degree on the parallel to thedegree on the meridian is the same as on the sphere. He then declares thatsince this cannot be achieved one may request, instead of the last condition,that the deviation of the degree of latitude to the degree of longitude at eachpoint from the true ratio be as small as possible (ideally, this error shouldnot be noticeable).
ATHANASE PAPADOPOULOS
Figure 3.
A map of the Russian Empire by Joseph-Nicolas Del-isle (Saint Petersburg, 1745)
He then recalls the construction of Delisle’s map.In this map, one first chooses two outer parallels that contain the regionthat is to be represented. In the case of the Russian Empire, these outermostparallels are chosen to be those at 40 o and 70 o of altitude. Then, to choosestwo other special parallels on which the ratios of the degrees of latitude to thedegrees of longitude will be represented by their exact values. Euler writesthat the question becomes that of choosing these two new parallels in such away that the maximum deviation of the ratios of the degrees of latitude andlongitude over the entire map is minimized. He writes that Delisle foundthat the optimal choice of these parallels is to take them equidistant fromthe central parallel of the map and from the outermost parallels chosen.Besides, distances should be preserved on all meridians, and the maximumof a certain deviation over the entire map must be minimized. Figure 3reproduces a map drawn by Delisle using his method.Starting in §
7, Euler presents a mathematical construction of a family ofstraight lines representing meridians which are at distance one degree fromeach other. He notes that one advantage of Delisle’s projection is that whilemeridians are represented by straight lines, the images of the other greatcircles do not deviate considerably from straight lines ( §
22 of [8]), and hegives some precise estimates of this deviation ( § § § AP DRAWING AND FOLIATIONS OF THE SPHERE 9 to be very large and he concludes that the shortest lines on the map do notdiffer sensibly from straight lines.In conclusion, let us stress once again that in Euler’s treatment of Delisle’sprojection, like in the theorem we stated above on perfect maps, the import-ant requirements concern the behavior of the images of the two foliationsdefined by the parallels and the meridians.Besides the memoir [8] in which he described Delisle’s method of drawinggeographical maps, Euler explained the same method in the
Atlas Geograph-icus omnes orbis terrarum regiones in XLI tabulis exhibens [5] that was pub-lished by the
Acad´emie Royale des Sciences et Belles Lettres de Prusse , inthe year 1753, in Berlin, where he worked for 25 years, between his two staysin Saint Petersburg. The atlas contains 45 maps, and was edited under thedirection of Euler who also wrote its preface, which is dated May 13, 1753.Several projections are used in this atlas, which is concerned only with largeparts of the Earth. In all these projections, the meridians are perpendicularto the parallels.The map in Figure 2 Figure 2 is extracted from this atlas. It is the last onein the series, and it is drawn using Delisle’s method. Euler, in the preface,comments on this method. Euler writes that Delisle’s method seems to himthe most appropriate for a proper representation of these Northern regionsof the terrestrial globe. He recalls that in this representation, the meridiansare straight lines and all their degrees are equal: the images of two meridiansthat are distant apart by one degree converge in such a manner that at twoaltitudes that are chosen in advance, the ratio of the degrees of longitude tothe degrees of latitude are the same ratio as in reality. This is the propertythat he presents in his memoir [8] that we noted above. For the RussianEmpire, after the choice of the two outer parallels at 40 o and 70 o , the twoparallels which are at the same elevation from the extremities of the regionthat is represented as well as from its center are those at 47 o (cid:48) and t 62 o (cid:48) .Under these two altitudes, the ratios between the degrees of longitude andlatitude are accurate on the map. At the other locations, they are almostaccurate (the difference is not noticeable). Besides, in this representation,all the meridians (which are straight lines) merge at a point, although thispoint is not the North pole; it is at a distance which would correspond to 7degrees farther than this pole. From this point as center, the images of theparallels are circles. Euler writes that one should not regard as a shortageof this map the fact that the center in which all the meridians intersect isso far from the pole, nor the fact that on this map the parallels, which formsemi-circles, do not occupy 180 o in longitude, but much more, sometimeseven up to 250 o .Lagrange, whose name is associated to the one Euler in several respects,already stressed in his paper on the construction of geographical maps thefact that the only thing we have to do drawing a geographical map is tospecify the images of meridians and parallels according to a certain rule (see[12, p. 640]). This simple remark was at the basis of the the developmentof modern mathematical cartography. We have tried to convey this idea bymentioning examples from the works of Euler and Delisle, but others may be found in works of Lambert, Gauss, Bonnet and others. The forthcomingbook [1] contains a section on cartography at the epoch of Euler. Acknowledgements.
This paper is based on a talk I gave in Obninsk, on May14, 2019, at a conference commemorating Pafnouti Chebyshev. I would liketo thank the organizer, Valerii Galkin, for inviting me to that conference. Iwould also like to thank Alena Zhukova for her corrections on a preliminaryversion of this paper.
References [1] R. Caddeo, A. Papadopoulos and A. Zhukova, Spherical geometry in the Eighteenthcentury, book to appear.[2] C. Charitos and I. Papadoperakis, On the non-existence of a perfect map from the2-sphere to the Euclidean plane. Eighteen essays in non-Euclidean geometry, ed. V.Alberge and A. Papadopoulos, Eur. Math. Soc., Z¨urich, 2019, p. 125-134.[3] C. Corrales Ridriganez, Dallo spazio come contenitore allo spazio come rete, In:Matematica e cultura 2000. Papers from the conference held at the Universit`a Ca’Foscari di Venezia, Venice, March 26-27, 1999. Ed. M. Emmer. Collana Matematicae Cultura. Springer-Verlag Italia, Milan, 2000.[4] J.-N. Delisle, A Proposal for the Measurement of the Earth in Russia, Read at aMeeting of the Academy of Sciences of St. Petersbourg, Jan. 21. 1737. by Mr Jos.Nic. de L’Isle, First Professor of Astronomy, and F. R. S., Translated from the FrenchPrinted at St. Petersbourgh, Phil. Trans. R. Soc. Lond. 40, 27-49 (1737).[5] L. Euler, Preface of the
Atlas Geographicus omnes orbis terrarum regiones in XLItabulis exhibens , Acad´emie Royale des Sciences et Belles Lettres de Prusse, Berlin,1753.
Opera Omnia , Series 3, Volume 2, p. 305-317.[6] L. Euler, De repraesentatione superficiei sphaericae super plano, Acta Academiae Sci-entarum Imperialis Petropolitinae 1777, p. 107-132,
Opera Omnia , Series 1, Volume28, pp. 248-275.[7] L. Euler, De proiectione geographica superficiei sphaericae, Acta Academiae Scient-arum Imperialis Petropolitinae 1777, p. 133-142,
Opera Omnia
Series 1, Volume 28,pp. 276-287.[8] L. Euler, De proiectione geographica Deslisliana in mappa generali imperii russici us-itata, Acta Academiae Scientarum Imperialis Petropolitinae 1777, p. 143-153,
OperaOmnia
Series 1, Volume 28.[9] A. Knoebel, J. Lodder, R. Laubenbacher and D. Pengelley , Mathematical Master-pieces: Further chronicles by the explorers, Springer Verlag, 2007,[10] T. G. Feeman, Portraits of the Earth: : A mathematician looks at maps, AMS,Providence, RA, 2002.[11] J. Gray, Simply Riemann, Simply Charly, New York, 2020.[12] J.-L. de Lagrange, Sur la construction des cartes g´eographiques, Nouveaux m´emoiresde l’Acad´emie royale des sciences et belles-lettres de Berlin, ann´ee 1779, Premierm´emoire,
Œuvres compl`etes , tome 4, 637-664. Second m´emoire
Œuvres compl`etes ,tome 4, 664-692.[13] R. Osserman, Mathematical Mapping from Mercator to the Millennium, In: Math-ematical Adventures for Students and Amateurs, ed. edited by D. F. Hayes and T.Shubin, Mathematical Association of America, 2004, p. 233-257.[14] A. Papadopoulos, Euler and Chebyshev: From the sphere to the plane and backwards,Proceedings in Cybernetics (A volume dedicated to the jubilee of Academician Vladi-mir Betelin), 2 (2016) p. 55–69.[15] R. Rashed and A. Papadopoulos,
Menelaus’ Spherics: Early Translation and al-M¯ah¯an¯ı/al-Haraw¯ı’s Version (Critical edition of Menelaus’
Spherics from the Arabicmanuscripts, with historical and mathematical commentaries), De Gruyter, Series:Scientia Graeco-Arabica, 21, 2017, 890 pages.[16] P. L. Robinson, The sphere is not flat, American Mathematical Monthly , Feb., 2006,Vol. 113, No. 2 (Feb., 2006), p. 171-173.
AP DRAWING AND FOLIATIONS OF THE SPHERE 11
Athanase Papadopoulos, Universit´e de Strasbourg and CNRS, 7 rue Ren´eDescartes, 67084 Strasbourg Cedex, France
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