EUROPEAN SYSTEMS SCIENCE UNIO
ISA RESEARCH COMMITTEE 51 ON SOCIOCYBERNETICS
6TH EUROPEAN CONGRESS ON SYSTEMS SCIENCE, PARIS, FRANCE, SEPTEMBER
19-22,2005
_________________________
THE IDENTITY OF THE “PLACE IN LA MANCHA” IN DON QUIXOTE: A
SYSTEMIC APPROACH
F. Parra-Luna
ABSTRACT
A multidisciplinary research team using systemic methodologies to
analyze the novel The ingenious hidalgo Don Quixote of La Mancha has
discovered an underlying mathematical structure in Cervantes’ book: an
architecture emerging when the riding times between actual geographic
places, expressed in days and nights on horseback, were systemically
related to the respective real distances expressed in kilometers, assuming
different riding speeds. Building on this mathematical structure, two facts
were determined: first, the ground that – according to the novel –
Rosinante and Sancho’s donkey could cover in a day’s time; and second,
the identity of Cervantes’ mysterious “place in La Mancha”, indisputably
located in the region known as “Campo de Montiel”, and verified within a
reasonable degree of scientific certitude to be the village of Villanueva de
los Infantes, in the Spanish province of Ciudad Real.
INTRODUCTION
In the four centuries since Cervantes published his novel, no-one has identified the
“place in La Mancha” whose name the author cared not to remember. The Argamasilla de
Alba legend, based on the famous farcical ode written by Cervantes himself to the members
of the Argamasilla (or perhaps the Calatrava) Academy, was later nourished by Fernández
de Avellaneda (1614), along with some of the novel’s earliest commentators such as
Mayans (1737), de los Rios (1782), Fernández Navarrete (1819) and Clemencín (1833).
That premise was later definitively disproved, however, by both Astrana Marín (1948) and
Rodríguez Marín (1949). More recent papers proposing new places by authors such as
Terrero (1959), Ruiz de Vargas (1976), Perona (1988), Ligero Móstoles (1991) and Muñoz
Romero (2001), to name a few, likewise lack scientific substantiation. In fact, there is no
known attempt to rigorously apply scientific method to determine the exact location of Don
Quixote’s “place in La Mancha”.
And yet, today such an approach is perfectly possible thanks to the methodological
paradigms proposed in Systems Theory, among others. Indeed, the systemic approach is
generally characterized by providing for the best possible selection of variables that not
only represent the object studied as a whole, but are themselves inter-related in complex
ways. In the present case, the analysis involves 24 possible villages, 96 decisive distances
and 21 different assumptions about the speed at which Don Quixote and Sancho Panza
were able to ride Rosinante and the donkey, all of which must be taken into account in the
systemic whole. All of this precludes simplistic approaches and conclusions obtained on the
basis of a single village; rather, the solution should be found by optimizing plausible
assumptions (minimization of differences) and taking account of all the potential
candidates.
The present paper will show that the existence of a certain mathematical structure in
the novel provides: firstly, confirmation of the hypothesis put forward in an earlier study
that the “place in La Mancha” is Villanueva de los Infantes; and secondly, to determine the
exact speed sustained by Rosinante and Sancho’s donkey (a matter that has become highly
controversial).
From the vantage of the first objective, the present study entails “revisiting” research
conducted at the Complutense University of Madrid and published under the title E L
QUIJOTE COMO UN SISTEMA DE DISTANCIAS/TIEMPOS: HACIA LA
LOCALIZACIÓN DEL LUGAR DE LA MANCHA (The novel Don Quixote as a
time/distance system: locating the place in La Mancha), Ed. Complutense, Madrid, 2005,
referred to hereafter as the “parent paper”. The reply implicit in this article reformulates
some of the initial assumptions on which that research was based for the sole purpose of
verifying its core hypothesis (location of the “place in La Mancha”), while pursuing a
higher degree of certitude and objectivity, whatever the new findings show. The existence
of a tacit mathematical structure in the work is instrumental to verification or refutation of
the prior research.
THE DISTANCE / RIDING TIME SYSTEM
But is the existence of a mathematical structure in Don Quixote, of any sort, let alone
able to attain the above two objectives, even imaginable? In fact, it is both imaginable and
verifiable, thanks to the facts that Cervantes himself provides:
First, the point of departure: Campo de Montiel, a historic/geographic region with a
total of 24 villages at the time the novel was written, all at empirically measurable distances
from one another.
Second, “theoretical” distances (expressed in riding time): between the enigmatic
“place in La Mancha” (hereafter abbreviated “L”) and three places (Puerto Lápice, Sierra
Morena and El Toboso) explicitly cited by Cervantes, and one (Munera) implicitly
mentioned; and all of these are real geographic sites with their respective distances to the
region’s 24 villages. Moreover, these theoretical riding times can be quantified to a
reasonable degree of accuracy. These data were structured on the basis of these premises
and the information gathered during the research effort described above. The parent paper
2
announced some of the quantitative keys implicit in the novel: among others, the so-called
topological solution, which is the one reconsidered here under more certain and verifiable
assumptions. But, outside the data and calculations that led to advancing the hypothesis that
Villanueva de los Infantes is the “place in La Mancha”, certain problematic issues remained
outstanding, namely: to know if the hypothesis of 31 and 36 kms/day were correct; and to
verify with new data if the “place of la Mancha” is Villanueva de los Infantes.
a) The speed of 31 km/day assumed in the above study was obviously too slow a pace
to travel from Villanueva de los Infantes to El Toboso in the time allotted by the
author. It was found (Chapter 8, “Model Ethics”) that if a speed of 36 km/day for
that particular journey (so different from the preceding two) had been assumed
instead, no discrepancy would have arisen between the two days and one night that,
according to Cervantes, it took to travel that far, and the distance of approximately
90 km between the two villages.
b) Cervantes himself – albeit implicitly - established a speed of 31 km/day for the
journey from Puerto Lápice to Sierra Morena on the occasion of Don Quixote’s
second departure from home. Eminent Cervantine experts have proved (see
Casasayas, 1999) that Don Quixote left home the first time on Friday, 28 July (one
assumes in 1589) and returned two days later. His respite lasted for a further 17.
Slightly reinterpreting Casasayas’ calculations, Don Quixote (this time in the
company of Sancho Panza) would have set out for the second time on the night of
15 August. It took the travelers two days to reach Puerto Lápice, having spent the
night of 17 August in a shepherd’s hut just outside the village. By 22 August they
reached Don Quixote’s place of penance, in the very midst of Sierra Morena (in the
region of Andalusia, eight leagues from Almodóvar del Campo and more than thirty
from El Toboso), after traveling what translates to approximately 141 km (from
Puerto Lápice). Since they rode this distance in five, or more precisely, four and a
half days, their speed, calculated from Cervantes’ literal account, was 31 km/day. It
was gratifying for the research team to find – when they had nearly completed the
parent study - that the speed of 31 km that they had adopted from the outset based
on a series of calculations and estimated averages, concurred with Cervantes’ own
assumptions.
c) As discussed in Annex 4 to the initial paper, the speed of 31 km/day deduced for the
distance from Puerto Lápice to Sierra Morena was reached despite the couple’s
sundry and on occasion physically traumatic adventures (peddlers from Yanguas,
sheep, a corpse, fulling mills and so on). It was consequently found reasonable to
increase that pace by ten to twenty per cent (fifteen was ultimately adopted) for less
eventful journeys.
d) Hence the use in the aforementioned Annex 4 of a speed of 36 km/day to locate the
inn where Don Quixote and Alvaro Tarfe met, and on those grounds conclude that
in all likelihood their encounter took place in the Munera, a village located in what
is now the Spanish province of Albacete.
MUNERA, ALBACETE AS THE HYPOTHETICAL “TARFE POINT”
Moreover, the identification of Munera as the venue for that meeting is further
sustained by the following evidence: firstly, the name of that village appeared more often
than any other as the match for the connection defined in terms of the distance between
“Tarfe point” and “place in La Mancha”. Secondly, it is the most logical place for the three
3
characters to have parted company, considering that Don Alvaro Tarfe was heading
towards Granada and Don Quixote/Sancho towards their home in Campo de Montiel. And
thirdly, it is highly unlikely that the “Tarfe point” would be in any nearby village, given
Munera’s relative importance and cross-roads location.
The establishment of the location of the “Tarfe point” at Munera with an acceptable
degree of certainty provides the fourth geographic point of reference needed to determine
the distance/riding time system and the resulting mathematical structure in the novel.
THE OBJECTIVE DISTANCES BETWEEN VILLAGES AND THE FOUR POUINTS
OF REFERENCE
Measuring the distances between villages was an exercise fraught with difficulties due
to the inability to objectively establish what would have been the most logical routes – and
respective distances - in the late sixteenth century (differences between maps, new roads,
short-cuts, paths and so on) and indeed, the want of a more verifiable system of distances
left the researchers with an uncomfortable sensation of uncertainty. Based on the old adage
that “La Mancha has no roads, for it is nothing but a road” (plain that it is), this problem
was circumvented by measuring the straight-line distances between villages on a validated
map (Michelin 2004) to reach a consensus on the question with the highest possible degree
of objectivity. Table 1 shows then, exactly, straight-line distances (scale: 1mm.=1 km.)
HYPOTHESES ON KILOMETERS TRAVELLED PER DAY
This new approach forms a part of the mathematical strategy presented hereunder, which
purports not only to avoid the subjectivity involved in choosing one of several alternative
routes and taking problematic measurements, but especially to free the analysis of any
dependence on different hypotheses about the ground that the two mounts could cover in a
day’s time. Therefore, all the reasonable hypotheses about speed were tested to find the one
that reduced to a minimum all the discrepancies in the distances between the 24 villages in
question and the presumed “place in La Mancha”. The aim was to reduce the differences
between theoretical riding time and actual distance from each village to each of the four
points of reference, to calculate a speed necessarily acceptable for each and every one of
the villages that would consequently minimize the sum of the discrepancies for all of them
as a whole.
The first step was to estimate the riding time (expressed in kilometers) from the
unknown place in La Mancha (L) to each of the four “points of reference”. The criteria
considered, all of which can be found in different chapters of the novel, were as follows:
Distance to Puerto Lápice
*Don Quixote and Sancho rode almost a whole night (from 1 to 6=5 hours=3,1x5=15,5
kms).
*The next day, they were estimated to have traveled 1 kilometer less than normal because
of the windmill episode (31-1=30 kms/day).
*On the second day they “discovered” Puerto Lápice at three o’clock in the afternoon.
Assuming that they started at approximately eight o’clock in the morning (Don Quixote
woke Sancho when the sun was already up) they rode for seven hours, or 3,1x7=21,7 kms.
(Chapters XII to XXIV, Part I).Summarizing, they were estimated to rode a little more than
two days (63 kms).
4
Distance from Sierra Morena
Two days, or 62 kms (Chapter 37, Part I)
Distance to El Toboso
*one night
* two days (Chap. 9, Part II). Summarizing, 15,5+31+31=77,5 kms..
Distance from Munera (“Tarfe point”)
*Distance from the inn where they met Don Alvaro Tarfe to the place where they parted
company (half a league), i.e., 2.75 km.
*They rode for a night and a day.
Since when they reached the village there were children playing in the vegetable patches
outside the town, it must have been around ten in the morning, from which it may be
deduced that they rode another four hours, or 4x3,1=12,4 kms. (Chap. 72, Part II).
Summarizing, 2,75+15,5+31+12,4=61,65, or 62 kms.
THE MATHEMATICAL STRUCTURE
In table 1 are the distances between the 24 villages of Campo de Montiel until the four
reference points given by Cervantes: Puerto Lápice (PL), Sierra Morena (SM), El Toboso
(ET) and Munera (M).
Let us consider this table as the matrix (aij) i = 1,…,24 j = 1,..,4.
Equally the vector d = (d1,d2,d3,d4) where dj represents the number of days spent by D.
Quijote and Sancho to arrive from the “place of La Mancha” (L) until the four reference
points.
d1 = days to arrive until PL.
d2 = days to arrive until SM.
d3 = days to arrive until ET.
d4 = days to arrive until M.
According with the information given by Cervantes in the novel, it can be estimated the dj
(j =1,..,4), thus,
d = (2.03, 2, 2.5, 2)
Let us call v the speed by day in kms, and define now the matrix (xij):
xij = v dj for all i
Let us consider now the following matrix
(_ij) = (_ xij - aij _)
which represents the discrepancies between the xij , (theoretical distances) that are function
of v, and Aij (real distances).
It remains thus (_ij) as a function of speed.
If we could know exactly this speed (f.i. the 31 Km. per day used previously in our
published work), then the matrix of discrepancies (_ij), would be known.
F.i. for i = 1 (Albaladejo), the file would be:
_11 = _100 – 31*2.03 _ = 37
5
_12 = _69 – 31*2_ = 7
_13 = _100 – 31*2.5 _ = 22.5
_14 = _54 – 31*2.5 _ = 8
as it can be checked in the hypothetical figure 1.
.
.
for i = 24 (Villanueva de los Infantes), the file would be
_11 = _77 – 31*2.03 _ = 14
_12 = _58 – 31*2_ =4
_13 = _85 – 31*2.5 _ = 7.5
_14 = _57 – 31*2 _ = 5
In the same way, for i = 2,…,23, the matrix (_ij), is shown in table 2.
The problem now is to compare the discrepancies of the 24 villages, each one represented
by its file in the matrix (_ij). Fig. 2 shows the need of overlapping the circles in an optimum
point (L) by minimization of discrepancies.
But the comparisons between vectors are not easy unless we have access to certain
mathematical methodologies. The idea is to apply the Multicriteria Decision Theory,
defining a function z able to associate to each file of the matrix (vector) a real value,
making then possible to compare real numbers. In principle, the minimum value of them
will correspond to the unknown “place of La Mancha”.
Symbollicaly:
Z
(_i1, _i2, _i3, _i4) _ Z(_i1, _i2, _i3, _i4) = r _ R
Following the Multicriteria Decision Theory, and once analyzed the components of the
vector (criteria), we can set up different functions of value which will give very similar
solutions to the problem. :
(1) Z (_i1, _i2, _i3, _i4) = ∑ni=1 _ij
(2) Z(_i1, _i2, _i3, _i4) = ∑ni=1( _ij )2
(3) Z(_i1, _i2, _i3, _i4) = maxi _ij
With any one of these four functions of value and once each village has been evaluated, the
village achieve the minj Z, will be the “place of La Mancha”.
Repeating this procedure for different hypothesis of speed v (not only for the 31 kms. day),
we will get the optimum villages for each value of v. In other words, from this
methodological perspective the optimum for each village will be determined by the
minimum values of any of the “relative maximin”, “absolute maximin”, “sum of maximin”
and “global discrepancy”. It can be seen in table 3 that there are two minimum ”maximin”
(0,15), two “absolute maximin” (10), two “sum of maximin” (605), but different “global
6
discrepancies” (1377 and 1399). It can said then that this “global discrepancy” can be
taken as the most relevant and sensible explicative tool.
As a consequence of this, we may rise a second problem:
If, instead of consider the _ij constant, we set up them as function of v, _ij(v), then the
problem can be solved by minimizing a convenient objective function related to v.
This will give us, in principle, and according with the function we choose, different
solutions.
Let us take as a function of the global discrepancy, the sum of the total discrepancies. That
means to use the metrique L1. The problem will be:
minvO = minv ∑i ∑j _ij(v) = minv ∑i ∑j _ v dj – aij _
and we will get the polygonal fig. 2
In this case, which correspond to a parabola,, we can get its minimum in an analytic form,
deriving respect to v and equalizing to cero. Thus
dO/ dv = 2 ∑i ∑j( v dj – aij ) dj
which making equal cero
we get
∑i ∑j( v dj – aij ) dj = 0
∑i ∑j( v dj2 – aij dj ) = 0
that is to say
v ∑i ∑j dj2 = ∑i ∑j aij dj
them
vopt = ∑i ∑j aij dj / 24∑j dj2
7
Which can be repeated for the other three metriques L2,L3,L4. (see the adjusted equations
and their correspondent derivatives in table 4) where these derivatives mean the speed of
Rosinante and Sancho’s donkey.
Summarizing: Table 4 shows the four possible adjustments of data in table 3 taking the
speed v as the independent variable (X).
We have then four final results and two main alternatives of interpretation. The first one is
to consider the best result as given by the best adjustment (R2 = 0,9976) and take as the
most valid v=34,678 kms/day. The second, is the weighed average of the four results doing
V= 4∑i=1 vi ei / 4∑i=1 ei..
Where v are speeds and e errors
As we prefere the later, the final result is V=33,5008 kms/day which would be the real
speed of Rosinante and Sancho’s donkey according with Cervantes’ information, and
Villanueva de los Infantes the “place of La Mancha” as it can be seen also in table 3.
CONCLUSION
Cervantes’ novel Don Quixote of La Mancha has an implicit mathematical structure that
emerges when the riding times-distances between the villages in the geographic area where
the action takes place are organized into a formal system. Moreover, this mathematical
structure would reveal the identity of the author’s enigmatic and to date unknown “place in
La Mancha”.
Despite the logical skepticism that the mere enunciation of such a structure in any
literary text – not to mention a novel as famous and universal as Don Quixote - may elicit,
the present study is believed to have effectively proved that a systemic approach can in fact
be used to reach both of the objectives defined at the outset. By relating all the actual
distances between the villages existing in Campo de Montiel and the four points of
reference provided by the author (three explicitly: Puerto Lápice, Sierra Morena and El
Toboso; and one implicitly: Munera) to the “theoretical” distances (or riding times) based
on different hypotheses about the distances that Rosinante and Sancho Panza’s donkey
could travel per day, this approach revealed the formal or mathematical structure
underlying the text, from which the two specific facts listed below could be determined:
First, the overlap among the circles illustrated in Graph 3 coincides precisely with the
geographic area known as Campo de Montiel, putting an end to certain self-serving
speculations and redefinitions in this regard.
And second, the speed of 33.5 km/day is the result calculated from a literal reading of a
series of facts in Cervantes’ text. This rule out any further conjecturing about the much
debated speed at which the novel’s two main characters rode their respective mounts. The
speed implicit in Cervantes’ definition of a riding time of four and a half days from Puerto
Lápice to Sierra Morena was confirmed by the mathematical structure comprising the
system of distances-riding times subsisting in the novel.
Finally, the chief finding of the exercise was the verification of the hypothesis put
forward in the parent paper to the effect that the “place in La Mancha” is Villanueva de los
Infantes, with the difference that here the certainty of the outcome could be scientifically
8
quantified. On the grounds of the fit to the parabolic equations, the error calculated was
….…
REFERENCES
•
•
•
•
•
•
•
•
•
Astrana Marín, L., “Vida ejemplar y heróica de Miguel de Cervantes Saavedra”, I.E.
Reus, Madrid, 1948.
Casasayas, J.M., “Itinerario y cronología en la segunda parte del Quijote”, Anales
Cervantinos, XXXV, 1999.
Ligero Móstoles, A. “La Mancha de don Quijote”, Ayuntamiento de Alcazar de San
Juan, 1991.
Muñoz Romero, J., “La única y verdadera ruta de Don Quijote”, Ledoria, Toledo,
2001.
Parra Luna et al. “El Quijote como sistema de distancias/tiempos: hacia la
localización del lugar de la Mancha”, Ed. Complutense, Madrid, 2005.
Perona D. “Geografía cervantina”, Alba Grupo Espasa, 1998.
Rodríguez Marín, A. “El Ingenioso Hidalgo don Quijote de la Mancha”, por Miguel
de Cervantes, Atlas, tomo IX, Madrid, 1949.
Ruiz de Vargas, L. “El verdadero pueblo de don Quijote”, CIMBRA, Rev. De
Ingeniería Técnica de Obras Públicas, Mayo 1976, num. 107
Terrero J. “La ruta de don Quijote”, Imp. Viuda de Galo Saez, Madrid, 1959.
9
Figure 1:
Hyp. 31 kms/day
ALBALADEJO
El Toboso
100
77
Puerto Lápice
63
100
Munera
L
62
54
69
62
Sierra
Morena
(Sta. Elena)
10
Table 1
Real distances between the villages of Campo de Montiel until
the four referential points
Villages
Albaladejo
Alcubillas
Alhambra
Almedina
Cañamares
Carrizosa
Castellar s
Cozar
Fuenllana
Membrilla
Montiel
Ossa
Puebla P.
Ruidera
S.C. Cañamos
Solana
Terrinches
Torre J. A.
Torres M.
Torrenueva
Villahermosa
Villamanrique
Villa. Fuente
Villa. Infantes
PL
100
72
61
91
92
69
90
85
79
42
90
75
100
65
95
48
98
93
88
78
84
98
98
77
SM
69
52
70
55
78
66
27
46
64
65
67
92
55
87
64
65
64
44
61
30
69
49
80
58
ET
100
85
68
100
90
75
110
98
84
67
93
65
107
61
100
67
102
105
94
102
85
108
94
85
M
54
64
52
62
41
50
89
68
52
75
50
25
66
36
58
66
57
73
59
88
46
71
42
57
* Source: Map Michelin (España, Portugal) 2004
Scale 1mm=1km
PL=Puerto Lápice (C. Real)
SM=Sierra Morena (Jaen)
ET=El Toboso (Toledo)
M=Munera (Albacete)
11
Table 2
Discrepancies between "theoretical distances" and real distances
among the villages of Campo de Montiel and the four referential
points
Villages
Albaladejo
Alcubillas
Alambra
Almedina
Cañamares
Carrizosa
Castellar s
Cozar
Fuenllana
Membrilla
Montiel
Ossa
Puebla P.
Ruidera
S.C. Cañamos
Solana
Berrinches
Torre J. A.
Torres M.
Torrenueva
Villahermosa
Villamanrique
Villa. Fuente
Villa. Infantes
PL = - 63
SM = - 62
ET = - 77,5
M = - 62
37
7
22,5
8
9
10
7,5
2
2
8
9,5
10
28
7
22,5
0
29
16
12,5
21
6
4
2,5
12
27
35
32,5
27
22
16
20,5
6
16
2
6,5
10
21
3
10,5
13
27
5
15,5
12
12
30
12,5
37
37
7
29,5
4
2
25
16,5
26
32
2
22,5
4
15
3
10,5
4
35
2
24,5
5
30
18
27,5
11
25
1
16,5
3
15
32
24,5
26
21
7
7,5
16
35
13
30,5
9
35
18
16,5
20
14
4
7,5
5
* Source: Map Michelin (España, Portugal) 2004
Scale 1mm=1km
PL=Puerto Lápice (C. Real)
SM=Sierra Morena (Jaen)
ET=El Toboso (Toledo)
M=Munera (Albacete)
12
Table 3
Hypothesis of
speed (v)
Global
discrepancies
Sum of
"Maximin"
Absolute
"Maximin"
Relative
"Maximin"
Village
X
20
21
22
23
24
25
26
27
28
29
Y1
2951
2793
2617
2439
2277
2153
1979
1863
1751
1663
Y2
1124
1087
1036
973
923
889
822
774
722
677
Y3
25
23
21
20
18
16
14
12
10
10
Y4
0,63
0,55
0,48
0,49
0,38
0,32
0,27
0,22
0,18
0,17
La Solana
"
"
"
"
"
"
"
"
Carrizosa
30
31
32
33
34
35
36
37
38
39
40
1575
1507
1445
1399
1377
1391
1413
1460
1499
1567
1649
643
621
609
605
605
617
638
666
697
737
786
10
10
12
10
10
12
14
14
16
18
21
0,17
0,16
0,18
0,15
0,15
0,17
0,2
0,19
0,21
0,23
0,27
Carriz. + Alh.
Alc. + Alh.
V. Infantes
"
"
"
"
Torres M.
"
"
"
13
Figure 2:
3500
3000
2500
Global Discrepancy
2000
1500
1000
2
y = 7,4262x - 514,94x + 10338
2
R = 0,9976
500
0
0
10
20
30
40
50
kms/day
14
Table 4: Final results
Dependent
Derivative
Variable
p
Equation
Error
(v)
Y1 Global discrepancy
Y = 7,4262x2 - 514,94x + 10338
R2=0,9976
34,678
Y2 Sum of “Maximin”
Y = 3,1119x2 - 207,65x + 4088,4
R2=0,9804
33,363
Y3 Absolute “Maximin”
Y = 0,1297x2 - 8,0816x + 135,98
R2=0,9669
31,155
Y4 Relative “Maximin”
Y = 0,2855x2 - 18,853x + 325,65
R2=0,9781
33,018
15
FIGURE 2:
APPROXIMATIONS TOWARDS THE MINIMIZATION OF
DISCREPANCIES BETWEEN “THEORETICAL DISTANCES” AND
REAL DISTANCES.
ET
M
PL
L
SM
16