A Mathematical Contribution to the Study of Old Maps

IMAGO ET MENSURA MUNOI

ATTI DEL IX CONGRESSO INTERNAZIONALE DI STORIA DELLA CARTOGRAFIA

A cura di Carla Clivio Marzoli con la collaborazione di

Giacomo Coma Pellegrini e Gaetano Ferro

ISTITUTO DELLA

ENCICLOPEDIA ITALIANA FONDATA DA G. TRECCANI

©

PROPRIETA ARTISTICA E LETTERARIA RISERVATA Copyright by

Istituto della Enciclopedia Italiana, fondata da Giovanni Treccani, Roma

Edito dall'Ufficio Attivita Culturali dell'Istituto della Enciclopedia ltaliana

In redazione, Luciana Buccellato

13515-7 - Stabilimenti Tipolit~grafici «E. Ariani» e «L'Arte della Stampa» dell a S.p.A. Armando Paoletti - Firenze

Indice del tomo secondo

Parte Quarta - ATLANTI E GLOBI

Stanislaw Alexandrowicz, Contribution des cartographes polonais a !'evo­lution de la cartographie de l'Europe centrale et orientale au XVI" et au XVIIe siecle .. ... ...... ........ ... .. .. . ................ .

Dirk de Vries, Atlases and Maps from the Library 0/ Isaac Vossius (1618-1689) . . ....... . .... .... . . ...... . ... . ....... ... . .. .. ... .. .

Oswald A. W . and Margaret S. Dilke, Italy in Ptolemy 's Manual 0/ Geography ...... ........... . .... ...... . ... ....... .. ... . .. .

Theodore Nicholas Foss, The Editing 0/ an Atlas 0/ China. A Compari­son 0/ the Work 0/ I-B . d'Anville and the Improvements 0/ John Green on the Jesuit /K'ang-hsi Atlas ... ... ... ................ .

Ivan Kupcik, Collections 0/ Old Maps, Atlases and Globes in Bohemia

Elio Manzi, William Henry Smyth, l'atlante coro-idrogra/ico siciliano e i rapporti con la cartogra/ia ufficiale delle Due Sicilie ............ .

Monique Pelletier, L'Amerique Septentrionale du Globe de Louis XIV

Bruce B. Solnick, Christopher M. Klein, Cartography and Colonization: the British in the West Indies after 1763 ...... .............. .

Avelino Teixeira da Mota, The Last Century 0/ Portuguese Overseas Car-tography (1875-1975) ..... .. . . .... . ....... ... . .......... ... .

Francisco Vazquez-Maure, Analyse et evaluation de l'Atlas de l'Escorial

Parte Quinta - PROBLEMI DI CARTOGRAFIA STORICA E SCIENTIFICA

Germaine Aujac, La symbolique des representations du monde en Grece ancienne .. .. .... . ........... .. ... ... ... .................. .

Eugenia Bevilacqua, La lenta /ormazione del paesaggio agricolo nella Ter-ra/erma veneziana .... . . ................. .................. .

Antonio Coppola, Un contributo per la storia della tecnica cartogra/ica: inquadramento geodetico per la Gran Carta del Regno di Napoli .

Pag. 327

» 335

» 353

» 361

» 377

» 385

» 393

) 409

» 415

) 423

» 433

) 443

) 449

323

Robert W. Dixon-Gough, An Examination opments Influencing the Portrayal of Maps up to the Nin eteenth Century.

of the Technological Devel­Terrain Representation on

Paolo Fancelli, Gra/ica per la conservazione del paesaggio . . ......... .

Paul D. A. Harvey, The Spread of Mapping to Scale in Europe, 1500-1550

Lamberto Laureti, Origini e sviluppo del tematismo nella cartogra/ia ita-liana tra il Settecento e I'Ottocento con particolare rt/erimento al Mezzogiorno ... ...... ..... . .. ... ........ .. ........... ..... .

Giorgio Mangani, La cartogra/ia storica come fonte per la ricostruzione dell"idea ' di una regione .... ....... . .. . .... . ... . .......... .

Paola Sereno, Note sull'origine della topogra/ia militare negli Stati Sabaudi

Vladimiro Valerio, A Mathematical Contribution to the Study of Old Maps .. . . ...... ... .... .. ... . .... ... . .. . . ........... .. .. . . .

Tullio Viola, Silvio Manzoni, Maria Teresa Navale, Problemi geometrici applicati alle tecniche costruttive e rappresentative. L 'esempio del tunnel di Samo e un'ipotesi di triangolazione topogra/ica nel VI seco-0~C .................................... ..... ........ .

Helen Wallis, Th e Role of the Painter in Renaissance Marine Cartography

James A. Welu, Cartographic Self Portraits ..... . ..... . ......... .. . .

David Woodward, New Tools for the Study of W atermarks on Sixteenth Century Italian Printed Maps: Beta Radiography and Scanning Densi-tometry ... .......... ... ..... .. ........... ..... ......... .. .

Gli Autori ..... . . ....... .... . . .. . . ... ...... . .......... . ...... .

324

Pag. 455

» 463

» 473

» 479

» 487

» 491

» 497

» 505

» 515

» 525

» 541

» 553

Parte Quinta

Problemi di cartografia storica e scientifica

Problems of Historical and Scientific Cartography

Vladimiro Valerio

A Mathematical Contribution to the Study of Old Maps

The aim of this paper is to make an epistemological contribution to the study of old cartography by analysing one of the components in the map-making process: the geometric structure of the map itself.

The starting point is the assumption that the formal aspect of a map presupposes a specific geometric intention: this may be intuitive, or based on the principles of either euclidean or projective geometry. It may in addition make use of the unifying factor constituted by analytical representation in terms of real functions, by which the plane representation of the sphere was formulated in a general way in the eighteenth and nineteenth centuries.

However, it remains true that any map - even the least pretentious - has at least a topological structure: it represents two-dimensional regions as two­dimensional regions, preserves mutual relationships of contiguity and avoids lacerations and superimpositions. Even certain early maps, traditionally regarded as devoid of geometric structure, can in fact be analysed from the topological point of View.

The pictorial, artistical and biographical aspects connected with old maps draw quite often the leading interest in the historical studies on cartography. According to me we feel the lack of critical studies on the aspects and meanings of the map-making process. This lack forced the scholars to adopt and adapt tools already developped in other oldest disciplines. Skelton's words sound still living to me:

dispite various promising beginnings - it [the two world wars period] failed to provide the subject with a firm general base, secure lines of communication and an accepted methodology (1).

Map as human artefact is the result of many experiences, since the study of old maps is an evident expression of interdisciplinary field of research. Nevertheless I believe there exists a peculiar aspect belonging to the map-making process which enables us to recognize an artefact as a map. The study of the underlaying or

( ') R. A. Skelton, Maps. A Historical Survey of Th eir Study and Collecting, Chicago 19752, p. 92.

497

metahistorical structures should constitute the main effort and the most important contribution to the growth of an autonomous discipline. I think we need providing for a body to the history of cartography, before attending interdisciplinary themes of research.

In the present work the geometric approach, viewed as a contribution to the study of old maps, is applied to the Maritime Atlas projected and constructed between 1781 and 1792 by Giovanni Antonio Rizzi-Zannoni, during his long period of activity in Naples. This choice permits the demonstration that, even in the case of 'so called' well-known works, the contribution of structural geometric analysis is not limited merely to the numerical aspect: such an analysis is of considerable historical importance too, providing as it does reliable evidence of the date, the sources and the results of a given cartographic output.

It is also worth noting that, of all the elements present in a map, the geometric structure is the most stable and, in successive elaborations, the least susceptible to alterations of a stylistic nature. The lucky choice further demonstrates the not-necessary coincidence between geometric structure and cartographical projection: they can live alone or together but, in the last case, they do not necessary correspond to each other.

The Maritime Atlas 0/ the Kingdom 0/ Naples

It is well known that the interest taken by John Acton, War Secretary at the neapolitan court during the last decades of the eighteenth century, was a determining factor to the rapid conclusion of the kingdom charting.

The survey was undertaken by G. A. Rizzi-Zannoni in 1781 at the very beginning of his arrival in Naples from Padova. The historian Diodati tells us that

[ ... ] furonvi spediti architetti, astronomi, disegnatori, piloti e diversi altri per prendere le misure, nonche girar tutto illitorale e i luoghi mediterranei del regno (2).

The Maritime Atlas was projected into 23 imperial sheets plus an index and a frontispiece . Evep if there is no index table which allowed the user to have an overall view of the work, the author must have been drawn it before measuring and scaling the map. The existence of such a document, so far nowhere to be found, is confirmed by the fact that contemporaries already knew the exact number of the sheets many years before its completion (3) .

Further no old source tells us about the projection adopted in the construction

(2) L. Diodati, Vita dell'abate Ferdinando Galiani, Napoli 1788, p. 75 «< [ ... ] architects, astronomers , map-makers, pilots, and many other people travelled all over the coast and the islands of the Kingdom getting measures of them» ).

(3) A manuscript kept in the National Library of Naples (Ba 5D68) has been recently proposed as draft of the Maritime Atlas: see E. Manzi, La bozza dell'atlante marittimo delle due Sicilie, in «Rivista Geografica Italiana» LXXXII (1975), pp. 471-478. I cannot believe this manuscript is a draft drawn within 1781-1785 as the author tries to say. First we must consider that the observations in Puglia took the years 1786-1787, secondly the coast-line of the Gulf of Naples as it is drawn in the so called 'draft' was only established in 1792-1794. On the contrary the manuscript was conceived as a reduced chart taken from the Atlas , and constructed not before 1792-1794.

498

of the whole map, while it is known in full detail for the Land-Atlas of the kingdom by the same cartographer (4).

That is why I was forced to drawn an index table in order to study the map as a whole - beyond the single sheet (Fig. 1).

The graphical index, obtained by reducing the scale of each map, enabled me to recognize the grid of meridians and parallels, from which the projection adopted is easily understandable. The grid is rectangular: parallels and meridians are all straight lines cut at right angles at equal intervals. These conditions are satisfied by a conventional cylindrical projection called 'rectangular' or 'simple cylindrical' (plate carree in french, piana rettangolare in italian) (5). The projection does not preserve either bearings (or azimuths) or the areas, and it is an aphylactic projection; but it preserves length along the meridians and along two parallels called 'standard parallels' .

Under the assumption of the sphere-like earth, which we know was adopted by Rizzi-Zannoni, the equation of the map - unless the scale factor - is:

1) x =Rc:p y = RA cos c:po,

where x and y, c:p and A are respectively the plane and the geographical co-ordinates; R is the dimension of the radius of the earth, c:po is the latitude of the standard parallel.

Determined R and chosen c:po the 1) gives a one-to-one function between each point of the earth surface and a corresponding point on the scaled map; thus using the function 1) we have a representation of the sphere on the plane.

The earth dimension was calculated by Rizzi-Zannoni making use of the difference in latitude between Rome and Santa Maria di Leuca (6). Since the observations and after the due calculation it followed that the measure of a degree along the latitude was 57,000 french toises, equivalent to 111,093 metres. The corresponding radius of the sphere to such a figure for the latitudinal degree is

R = 6,365,160 m .

According to the degree of latitude the scale of the map is 1 : 90,099. To such a scale the uniform distance between the meridians drawn in the map is 77,755 m,

(4) Contemporary references to the Maritime Atlas are very few ones. For the first time A. Blessich, Un geografo italiano del secolo XVIII: Giovannt Antonio Rizzi·Zannoni (1736·1814), in «Bollettino della Societil Geografica Italiana », 1898, speaks about the projection, he observes «of course it is plane ». On the contrary, E. Manzi, L'atlante marittimo del Regno di Napoli, in «Rendiconti dell'Accademia dei Lincei. Classe di Scienze Morali, Storiche Filologiche», VIII S., XXIX (1974), p. 265, says with unproved certainty: «[ ... ] /u adoperata la proiezione di Cassini, ossza la cilindrica in versa, per zone aventi ciascuna un proprio meridzano medio» (<< [ ... ] it was adopted the Cassini projection, that is the inverse cylindrical, shared in parts with mean meridians»). This is no longer true, the author makes confusion between the Maritime and the Land Atlases, but even in the last one there is no use of 'mean meridians'. We cannot understand and I ignore the sources used by E. Manzi.

(5) The map constructed in this projection are called 'plane'. See M. Fiorini, Le proiezioni delle carte geogra/iche, Bologna 1881, p. 337; and A. Cagnoli, Notizie astronomiche adattate all'uso comune, Parma 18512, p. 225. See also A. Steers, An Introduction to the Study 0/ Map Projections, London 1970, p. 135.

(6) C. Firrao, Sull'Ojficio Topogra/ico di Napoli. Origine e vicende, Napoli 1868, p. 7.

499

10 11 12 13 14 . 15 16 17

43

~ 1\ ~ ~.

43

J I'" L ~ ~~, /!'" /1' ... /1,,' ' ><--,

~ .. , j

42 42

(1/' 11/ ~

~ ~l\" . ~ ~ . '\

, ( " '" ," "< " . '-. ... ~ ~

~ ~,.~, ~ ~ " ... ~'~ . ~~ i ~ "

~ ~ ,.p~ VI'VN f'\ ~ .AT<. t /[':: v-VI'VI----"': /

~ -* t7 ." 1"'1/1"'1/ oq", /1'" /" " I" /

41

40

41

40

~ 'L ~v 1" /

l/ZI " "'~ i2:'>< e C>< ~ t7~ ~

--::s A

~~ ,c~ ~ Quodro d 'U"I. ,.. ~ {.. . ~ R.,icoloto '_""of/co

,., / tI. II'

~W# l~ ATLANTE WAR/TT/1II0 delle -;/ - DUE SIC/LIE

(17 •• _/7 •• ) M · jJ~

!: . , . " .. - .. .. , .

VI"'/I' V I"""/" I

39

38

39

38

10 '11 12 13 14 15 16 11

Fig. 1 - Index table and cartographical grid of the Maritime Atlas. (Drawn by V. Valerio. )

500

which gives the dimension of one degree along the longitude at the latitude of the standard parallel. Thus the latitude of the standard parallel is

CjJo = 45° 34' 50" N (and S) .

The choice of a standard parallel so far from the mean latitude of the map - that is 40° 30' circa - focuses a first anomalous aspect of the mapping, further emphasized by the analysis of both linear and superficial deformations.

As the meridians and parallels cut at right angles the directions of the higher and smaller linear deformations coincide along them. The value of the coefficients of deformation, called n and m, IS

2) n = 1

along the meridians - smce the name of equidistant projection - and

2') cos CjJo

m=---cos CjJ

along the parallels: obviously varying with the latitude CjJ. Since CjJ > CjJo follows m > 1, and vice-versa. Further we observe that CjJ = CjJo implies m = 1, thus the map is also equidistant along the standard parallel.

The coefficient of deformation related to the areas IS

3) M = n . m

and it coincides with 2'):

3') M= cos CjJo

cos CjJ

Since 2') and 3') it follows that m and M increase whenever CjJ > CjJo, and decrease when CjJ < CjJo.

At the Nand S extremities of the map the latitudes are lower than the standard parallel's one, thus according to 2') and 3') the representation appears to be E-W compressed: the higher the distance from the standard parallel the bigger will be the deformation.

I have computed the variation of m (the same as M which coincides on) and of the scale in the following latitudes:

CjJ = 42° m = 0.94182 CjJ = 40° m = 0.91377 CjJ = 38° m = 0.88819

1 : 95,664 1: 98,612 1 : 101,44l.

As the above table shows in Reggio Calabria, near the very South of the peninsula, the percentage of diminution of the scale is 1l.2 % - referred to the scale along the standard parallel, while the scale along the meridians remains unaltered.

The bearing deformation called d is the difference between a given angle on the map, say a, and its corresponding' true value, say a':

4) d=o.-o.'.

501

The highest deformation in bearing has given by the formula (1)

tg dmax = 1/2 (vnTm - vmFiJ. At the latitude of Reggio Calabria we have

dmax = 3° 24'.

If we notice that nautical charts should be orthomorphic (i.e . d = 0) the value of dmax is not well satisfactory. Nevertheless in the practice it does not seem to be a considerable drawback due to the fact that the chart should be used only for a little coasting trade. The problem of misleading routes would have arised in tracing longer geographical courses.

But we must bear in mind that all the above questions could have been avoided simply by choosing the standard parallel near to the central latitude of the region. I can 't understand why Rizzi-Zannoni had chosen cpo so far from the map centre ! A plausible interpretation could be found in the necessity of drawing the rumbs for navigation purposes (Fig. 1). From one hand they can be easily constructed within a square with its diagonals, from the other hand such a square ought to be tied to the geographical grid in order to make correct evaluations. The first map of the Atlas bears engraved in a corner that the partition of the latitude in the frame of each map should be used «per misurare le distanze da luogo a luogo» (to measure distances between different places).

It is quite clear the need of tracing a square grid for constructing rumbs and the requirement of referring the grid to the geographical co-ordinates. The grid drawn in the map has the dimension of 7' in latitude by 10' in longitude. The two units used for constructing the rumbs' squares lead to a standard parallel cpo = 45° 35' circa (8) .

But there is something else to be taken into account, and this is the most striking aspect of such a mathematical analysis . In spite of a great deal of observations and measurements taken in many sheets I was not able to notice any appreciable deformation (either m or M) as it was supposed to be found. Further the coast-line of the Maritime Atlas coincides exactly with the shape of the Land-Atlas except for the Salentine peninsula, due to the Cassini projection adopted in the Land-Atlas (Fig. 2). Calabria is quite equal in the two atlases while the highest deformations should be there located.

This matter could be explained in this way: the chart has been constructed in a graphic way not an analytic one, avoiding any calculation of the plane co-ordinates for the geographical points.

Then a geographical grid has been superimposed to the map making use of the already known astronomical co-ordinates of the Castle Sant'Elmo in Naples and the Fortress in Lecce.

The hypothesis of a disconnection between the geometric structure of the map

C) N. Franchi, Elementi di Cartograjia, Firenze 1950 (1 975 ), . p. 27. (8) I tried to construct different squares tied to the latitude and longitude. The unique one which

might be a sostitute for the Rizzi-Zannoni's one measures 10' in longitude and 8' in latitude, and it leads to <Vo = 36°25' circa. Even this value is far from the mean latitude of the Kingdom of Naples.

502

Vl o Vo

14" 14" 15"

I"" IS" IS"

'-x 41"

~~'~~--~-----'\: \ 41°

. [ 40·

I' 400

~ m ro ~ ~ ~ _~

J-;----f I I I J j

Fig. 2 - Superimposition of the Salentine peninsula taken from the Maritime Atlas (slight contour) and the Land-Atlas (dark COntour). (Drawn by V. Vateria.)

and 'the projection grid is also proved by the fact that longitudes of the Maritime Atlas are surprisingly different from the corresponding ones in the Land-Atlas (Fig. 2).

There is an encreasingly error eastward in the chart, starting from 8' in Porto Palinuro, up to 19' in Santa Maria di Leuca. The absence of a corresponding error between the chart and the map - the last brought to a completion 30 years later -confirms the first atlas was not constructed making use of geographical co-ordinates, but only on the base of trigonometrical observations.

The errors in longitude come from the incorrect astronomical observation made in Lecce in 1786·1787, according to which the co-ordinate of the Castle were calculated 19' eastward. Referring to the grid the Maritime Atlas takes a step backward compared to the up to date knowledge collected together in the Encyclopedie (9).

The disgrace of the chart is basically due to the wrong evaluation of longitude and to the incoherence between the geometric structure and the grid of projection, both never pointed out so far.

(9) See R. Bonne and N. Desmarest (eds), Atlas Encyclopedique, 2 vols, Paris 1787 ; quoted from Padova 1789, in particular the Analyse des caries de cel A tlas , vol. I, pp. 1 ff., pp. 13 ff.

504