Monday, March 5, 2012

The Curve of Conversion

Of all the world’s religions none has spread with such speed and force as Islam.  Racing across vast stretches of land, its expansion was so rapid and complete even remote pockets of Europe still inhabited by pagans were overtaken.  Many of these cultural outliers would have become Muslims before having ever been fully Christian irrespective of Christianity’s 600-year lead time.[1]  Within ten years of the death of Muhammad history was witness to the fall of the great power of Sassanid Persia and very nearly Byzantium as well.  Lands from the Aral Sea to the Persian Gulf and from the Pyrenees to the Hindu Kush would all succumb to this new force. 
Certainly the military conquests of early Islam were exceptional in their speed, scope and destructiveness – as wars tend to be – yet these remain secondary to the more subtle process of conversion and acculturation which followed.  Dwelling on the overt aspects of these invasions is a great and often made mistake.  Images of fanatical, turbaned warriors riding out to wreak havoc may garner attention and excitement, yet also serves to obscure and distract from a far more meaningful conquest – the conquest of hearts and minds.
It may at first seem antithetical to attach greater significance to subsequent processes of acculturation than to the physical act of conquest itself.  However, history has shown us that stunning military successes are not exceptional; great powers rise and eventually fall.  The Arab conquests, although impressive, are easily matched and even surpassed in speed and scope by others such as Alexander the Great, Genghis Khan, or European overseas empires.   The distinguishing feature of the Arab conquests is that today many of these conquered people proudly call themselves Arabs, speak their language and practice their religion without a trace of enmity towards their former conquerors.  The same cannot be said of other empires: residents of Shanghai and Samarkand do not proclaim themselves Mongols, neither do those of Islamabad or Karachi title themselves Englishmen.
Let us turn to the example of ancient Iran, which is notable among the early Arab conquests because the people, although Muslims, have managed to preserve their non-Arab identity.  It is necessary to emphasize that Persia was not Islamized either for lack of pride or will to resist.  At the time of the Arab conquest it was the heart of the Sassanid Empire, the incarnation of yet another powerful Persian dynasty whose cultural heritage stretched back some 1,200-years to Cyrus the Great.  Even after defeat to Arab invaders anti-Islamic revolts continued sporadically for nearly two hundred years and in the Alburz Mountains of the north even longer.[2]  Denigrating Arabs was common practice among certain Persian authors:   

          Our fathers [Persians] gave you [Arabs] your kingdom, but you showed no gratitude for our
          benefactions. Return to your county in the Hijaz, to eat lizards and to graze your sheep...[3]

Clearly Persians of the time were the inheritors of a strong and proud cultural tradition, one that should have weathered the storm of military defeat just as they had endured earlier defeats at the hands of Macedonians and Asiatic nomads.  While such fervent patriotism may have helped preserve Persian identity as non-Arabs, it was not enough to prevent the steady and inexorable process of Islamization.  The end result was simply astounding; within two and a half centuries the majority of Persians had converted to Islam.  Those same Persian authors who so roundly despised Arabs, had none-the-less adopted their religion.[4] 

Source: Richard W. Bulliet, Conversion to Islam in the Medieval Period: An Essay in Quantitative History (Cambridge: Harvard University Press, 1979), p. 23.  

The graph shows the rate of conversion as percentage of population on the Y-axis, and dates according to Islamic and Gregorian calendars respectively are given along the X-axis. The shape of the data is unmistakable as a classic bell curve, a shape common to natural processes such as innovation diffusion. 
Clearly there is something unusual about Muslim conquests and in particular the Arab invasions of the early medieval period.  To delineate exactly what this is let us briefly review the normative model of conquest and rule:  A certain group is able due to a variety of complex factors, to attain and exploit a disparity in military power over its neighbors, whom it promptly conquers growing in size until it eventually reaches a critical mass called ‘empire.’  This empire does not last eternally, it experiences the familiar life cycle of rise, decline and fall.[5]  It is during the latter stages when an empire begins to weaken that formerly conquered people inevitably seize the opportunity to rise up, regain their long lost independence and strike back at their former conquerors for past indignities.  Usually this period also includes a cultural revival on the part of the subject peoples, perhaps of traditions that had been repressed during the time of foreign domination.  However, when we turn to the Arab conquests the results diverge dramatically from the predicted outcome.  The subject peoples do not seem to clearly remember their former identities or fully assert their independence; they remain fundamentally changed. 
The ability of Islamic powers to overcome the hostility normally engendered between a conqueror and his subjects must be recognized for the exceptional quality that it is.  Time and again experience has shown that military victory may come easily, but the identity of conquered people will always outlast the period of their subjugation provided they are allowed to persist undisturbed in their traditional modes of living.  When an imposed foreign power inevitably begins to falter, this natural hostility quickly makes itself felt as former subjects rise up to restore their position and redress perceived humiliations.  The ruins of Nineveh sacked by a coalition of subject peoples never to rise again, bears witness to the depths of this animosity.  This is the age-old dilemma of conquest and rule which has forever plagued nations and empires­- conquest is ephemeral, identity lasting.  It is towards this eventuality that Islamic rule provides an ingenious solution: change the identity of the conquered people to match your own and they become less of a liability and more of an asset.  This process is referred to, knowingly or not, as Islamization and it is this process, not martial valor which has secured conquered lands as permanent fixtures within Dar al-Islam, the House of Islam.    
Viewed within this framework Islamic conquest is unique in the sense that it brings with it a methodology for consolidating what would otherwise be purely transitory military gains.  This methodology is so unique and it’s after effects so pervasive that it is more appropriate to think of Islamization as an innovation - much the same as carburized steel or hybrid corn.  The implications of understanding Islamization and its ensuing rise as an advantageous innovation are far reaching.  Logically it permits an analysis of the rise of Islam in less relativistic terms.  Innovations are measured based on provable utility, not improvable theology.  Furthermore, Islamization’s propagation can be modeled as a type of spatial diffusion, the same process whereby technological innovations are disseminated throughout a population.      
Studies of spatial diffusion are not limited solely to technological innovations.  The concept’s application extends to other expansive phenomena such as epidemics, forest fires, pollution and so forth.  Here it will be applied to Islamization.  There are generally two types of diffusion recognized: expansion diffusion and relocation diffusion.  A forest fire is an example of relocation diffusion; as the fire spreads previous areas are vacated and the cumulative effect is negated.  By contrast expansion diffusion, as its name implies, continues to expand without relinquishing prior coverage.[6] 
We find that in regard to its self-perpetuation Islamization is configured to act in a manner most resembling the process of expansion diffusion.  For an expansive ideology, nothing grates against the gears of its existence quite like the loss of followers or territory, and in these instances we find the most vehement reactions among Muslims.  The mandating of apostasy as a crime punishable by death is a central tenet of Sharia law, not the product of misguided fanatics.  In other words it is a proper feature of Islamic rule.  The effect of rules such as these naturally ensures that adherents to Islam are gained and rarely lost, thereby skewing all permissible changes in religion towards Islam.  When military reverses result in the diminution of Muslim-held territory these ‘lost’ lands are never ideologically surrendered.  Centuries later we witness their loss mourned as though torn away by a recent catastrophe.[7]  Arabic poetry abounds with mourning for such nakbah, many of which stem from the loss of Muslim Spain.[8]  It seems areas once Muslim are forever considered a part of Dar al-Islam; those still lost await reincorporation through future reconquest. 
            Working from the assumption that Islamization is an expansive type of spatial diffusion allows a reconstruction of the spread of Islam throughout the Near East utilizing statistical concepts of normal distribution and the cumulative distribution function (the S-curve).  This is the central thesis of the American historian Richard W. Bulliet’s work, Conversion to Islam in the Medieval Period: An Essay in Quantitative History.  Through the tabulation of name changes in medieval genealogical dictionaries, Bulliet was able to trace indigenous conversion rates in what are today Iran, Iraq, Syria, Egypt, Tunisia and Spain.  Essentially the method assumes that where a name change occurs from a traditional native name to a clearly Muslim one, a conversion to Islam has also occurred.  A simple example of this would be a lineal series of Persian or Syriac names where an Arabic name such as Abdallah, meaning “slave of Allah,” suddenly appears.  In practice detecting these name changes is not always so straight forward especially in regions already speaking a Semitic language.[9] 
            When considering Islamization as a process of innovation diffusion the fundamental question naturally arises as to why any religion should predominate over another in a manner analogous to the displacement of an inferior technology by a superior one? To Muslims such dominance is undoubtedly evidence of the divine truth of their faith, while for those of us of a different persuasion the question is a poignant one. Bulliet notes:

 An iron plow may be superior to a wooden plow in an absolute
sense that would make its adoption an obvious act for anyone
who had the means to obtain it and to whom its superiority was
clearly demonstrated. One religion is not in the same sense
clearly superior to another. Without prejudgment of the matter
of revealed truth or moral worth, it may safely be said that
whatever superiority one religion may have to another, it is
not clearly demonstrable in a practical or mechanical sense.
Inevitably, some individuals reject the arguments that are put
forward by their converted neighbors and cling to their ancestral
     In many instances a practical superiority of one religion over
another may be created, often for brief lengths of time but
sometimes permanently. Such phenomena as persecution,
differential taxation based upon religion, and direct or indirect
financial rewards for converts obviously have an effect upon the
rate of conversion. Yet these selective factors are frequently
unrelated or only peripheral to the doctrines of the religion
involved; they may change from one period to another.[10]

While Bulliet has attempted to distance any particular religion from consistent use of these “selective factors,” he has in fact laid bare the forces that create a “practical superiority” of Islam vis-à-vis other religions.  Payment of the jizyra, political exclusion for non-Muslims and the pervasive pressure of Sharia law are not exceptional instances of Muslim hegemony, they are the norm.  The prescripts of Sharia pervade into every conceivable aspect of daily life and exert a constant and steady conformist influence.  Outside of the discriminatory and impositional nature of Islamic rule there is simply no reasonable explanation why an individual’s otherwise subjective choice of religion should systematically gravitate towards Islam.[11]  As will be demonstrated, the unparalleled expansion of Islam is the quantitative proof of its coercive and discriminatory nature.

Conversion Case Studies for Iran, Iraq, Syria, Egypt, Tunisia and Spain

Source: Richard W. Bulliet, Conversion to Islam in the Medieval Period: An Essay in Quantitative History (Cambridge: Harvard University Press, 1979), p. 44.  

Bulliet’s first case study tracking the rise of Islam as a form of innovation diffusion is that of Iran. The slope of the graph starts slowly when conversion was rare and most converts were slaves or social outcasts, although some members of the upper class, who having the most possessions to lose, sought to secure their wealth by ingratiating themselves to the new rulers.[12]  As time progressed and Islam became a more familiar and accepted mode of life the rate of conversion increased peaking in the 9th century.  The sharply accelerating slope of the curve is due to the contagion effect; as the number of Muslims in a population increases, the probability of interaction with non-Muslims correspondingly increases the likelihood of conversion.  Bulliet refers to this tendency as “social conversion,” a conversion based on level of exposure and personal motivation rather than an acceptance of theological arguments.[13] 
Examples of these types of conversions can be found among Ottoman kisve bahasi convert petitions, which reveal 19-percent were widows, orphans, or single parents, 9-percent were children of single parents included with their guardians, 19-percent sought a government career upon conversion and 80-percent were single adults.[14]  These numbers indicate that individuals under duress from fairly normal life circumstances were more likely to convert, as were single adults who had not yet fully invested themselves in their communities.  This may be considered the softer side of conversion to Islam and most academics seize upon this as evidence of tolerant Islamic rule.  Needless to say this is an absurd conclusion.  The reason these individuals are converting is because the Muslim-ruled society in which they live allocates greater resources and opportunities to Muslims.  This is the very definition of religious discrimination not the exemplification of tolerance.   
Dividing the normal distribution of the bell curve into percentages corresponding to standard deviations allows the conversion process to be partitioned logically.  Bulliet has labeled the first 2.5% of converts as “innovators,” the next 13.5% as “early adopters,” the next 34% as “early majority,” the next 34% as “late majority,” and finally the remaining 16% as “laggards.”  Each of these categories exhibits discernable traits.[15]  As mentioned earlier, the first group, the so-called innovators, were mainly prisoners of war and individuals who were already socially ostracized.  Later groups were more respected, more powerful, and more numerous.  The graph for Iran displays events from Iranian history along the cumulative distribution curve (the S-curve) and reveals that the motivations and timing of various Iranian movements are closely related to the process of mass conversion.  Midway through the curve anti-Islamic revolts give way to inter-Muslim disputes as the population becomes overwhelming Muslim.
This same pattern of conversion is repeated throughout the other medieval Islamic domains with only minor differences.  It is noteworthy that five of the six founding members of the modern Arab League are from these same regions of early conquest- a testament to the lasting success of the conversion process not only in religious terms but also ethnic identity.  For those regions such as Tunisia and Spain conquered at a later date, the curve of conversion obviously has a later starting point.  If however, the curves themselves are aligned relative to their starting points, their shape and timing closely coincide to one another.[16]  Each curve begins inconspicuously then accelerates dramatically before finally tapering off.  Anti-Islamic conflicts continually disappear past the midpoint of conversion as the non-Muslim population fades into obscurity.  
Some interesting regional differences do appear, most noticeably the rate (slope) of conversion is far more rapid in Iran than the other regions reaching the “late majority” stage by about 875, a full century (30%) faster than the other countries.[17]  Bulliet reasons that differing population densities along with a host of other potential factors may be responsible for this aberration.[18]  It seems eminently more likely that Zoroastrian Iran’s lack of a sufficiently attractive alternative to Islam was the primary cause for this.  Other lands from Syria to Spain were largely Christian at the time of their conquest and it is reasonable to deduce that the absence of an Iranian Church reduced resistance and eased the conversion process.[19]  Although variations do exist, the consistent emergence of a distinct curve of conversion across various cultures and regions furnishes compelling evidence for the transmittance of Islam as a powerful act of innovation diffusion.

Source: Richard W. Bulliet, Conversion to Islam in the Medieval Period: An Essay in Quantitative History (Cambridge: Harvard University Press, 1979), p. 83.  

Source: Richard W. Bulliet, Conversion to Islam in the Medieval Period: An Essay in Quantitative History (Cambridge: Harvard University Press, 1979), p. 97.  

Source: Richard W. Bulliet, Conversion to Islam in the Medieval Period: An Essay in Quantitative History (Cambridge: Harvard University Press, 1979), p. 109.  

Perhaps the most misunderstood aspect of Bulliet’s curve is the manner in which it is depicted rising up near the 100-percent threshold indicated on the Y-axis.  In fact, the curve of conversion is not intended to represent the total population.  Remember that Bulliet compiled his data from Muslim genealogical dictionaries and worked backwards tracing the path of conversion among those who had become Muslims.  Thus the cumulative S-curve is tracking the rate of conversion only among those who eventually did convert.  When the 100-percent threshold is reached, it does not mean absolutely all individuals in a given society have converted, it simply means that everyone who is amenable to conversion has done so and the process is largely complete.[20]  Iran and other countries continue to have and always will have those who remain entirely resistant to conversion. 
If we return to the example of an iron plow versus a wooden plow, we may safely assume that adoption of the iron plow will not be met with ideological animosity.  The superior technology will spread fluidly amongst the population so long as it is demonstrably beneficial and readily available.  It is helpful to think of Islamization as similarly propagated in a smooth curve so long as ideological dissenters are excluded.  The use of Muslim genealogical dictionaries in compiling the data elegantly removes these resistant individuals from the data revealing the characteristically smooth curve of conversion. Conscious human antipathy is not a normal feature of innovation diffusion and there exists no model for predicting the tempo of conversion among those who are adamantly set against it.   However, the continual disappearance of anti-Islamic revolts halfway through the S-curve supports the notion that the vast majority of any given society is indeed amenable to conversion.  Bulliet’s theorem remains invaluable in charting the flow of Islamization in any given population, so long as it is under Islamic rule and disproportionate benefits accrue to Muslims thereby incentivizing conversion.    
In Iran Bulliet suggests something on the order of twenty percent of the total population as a resistant residual.[21]  Therefore in the year 890, roughly halfway along the curve, Iran as a whole would have been 40-percent Muslim and by the late 10th century 80-percent, based solely on extrapolated rates of conversion.[22]  In actuality emigration of non-Muslims spurred by discrimination and sometimes violence would have already taken a significant toll, boosting the actual percentage of Muslims even higher.  Remnants of this non-dissuaded element continue to exist in Iranian diaspora communities, most notably the Parsi of India who remain Zoroastrians in culture if not actual belief.  In Iran it seems that those most resistant to Islamization ultimately chose to emigrate rather than convert resulting in the polity today that is quite nearly one hundred percent Muslim.  Other more common residual groups are the scattered Christian communities in the Near East including: Maronites in Lebanon, Assyrians in Iraq, Copts in Egypt, and Syriac Christians in Syria.  The prevalence of Christian communities as centers of resistance brings us to our final and most important chapter in the path of Islamization – that of medieval Iberia.    
The Iberian Peninsula is an exception among the regions of early Islamic conquests for the obvious reason that today it is devoutly Catholic rather than Muslim.  In purely military terms Spanish resistance was a debacle.  Fighting was sparse, only two major battles are recorded and almost all cities and towns surrendered without a fight. Within three years the invasion that had begun essentially as an amphibious raid in force had overrun almost the entire peninsula.  The apparent ease of victory concealed a deep seated enemy in the form of an unconquered indigenous identity which would later give rise to the reconquista, an unparalleled renunciation of Islamic culture and rule.  Here again we have evidence for the primacy of identity rather than short-term military success as the final arbiter of conflict.  The reason the Daylamite tribesmen of Iran never swept south from their mountain strongholds to reconquer their nation as did the Asturians and Aragonese of Spain, was not due to a comparative lack of martial élan.  It was due to a lack of ideological impetus and the subsequent extinguishment of their non-Muslim identity through conversion.
Spain, like all other Islamic lands, experienced the curve of conversion.  However, here the unconverted segment of the population was quite large; entirely unconverted in the independent Spanish kingdoms of the north and estimable at nearly forty percent in the areas of Muslim rule.  If we compare the situation in Catholic Spain to Zoroastrian Iran, we observe that in Spain the process of conversion was approximately 70-percent slower and the portion of the population entirely resistant to conversion was nearly twice as large.[23]  Large Christian populations persisted in regions far removed from the central Muslim authority of Cordoba in northerly locales such as Toledo and Coimbra.  These in turn were offset by areas of dense Muslim settlement along the Guadalquivir and Guadiano Rivers where Christian populations of less than thirty percent were likely ca. 1105.[24]  These general estimations are in line with that of Ottoman Bulgaria whose population remained approximately 45-percent Christian after the completion of the conversion process in the early 18th century.[25]  In Spain these remaining Christian communities proved critical to sustaining an indigenous identity and provided demographic reinforcement as the reconquista pushed southward.

Source: Richard W. Bulliet, Conversion to Islam in the Medieval Period: An Essay in Quantitative History (Cambridge: Harvard University Press, 1979), p. 124.  

As these graphs repeatedly demonstrate, Islamization is a monumental force whose expansion is readily observable as a type of innovation diffusion.  No society or nation has proven immune to the curve of conversion, although the degree of resistance can vary greatly.  Extended periods of Islamic hegemony in Europe have shown that the West also is not exempt from the effects of Islamization, though in the end it has been successful in forestalling and eventually ejecting Islamic powers.  The foremost reason for the West’s continued success in maintaining its identity is that it frequently possesses military superiority over its Muslim neighbors and thereby thwarts the imposition of Islamic rule.  Muslim military successes in Europe are mostly traceable to the collapse of the Roman Empire.  Invasions in Spain and Sicily were made possible by the disorder following the collapse of the western half of the Roman Empire and centuries later the decline of the eastern half, Byzantium, enabled Ottoman successes.  Even during these times of Muslim dominance Judeo-Christian heritage has afforded a sufficiently resistant alternative to ensure sizable numbers of the population remain unconverted, biding their time until the tide of power turns back in their favor.  This phenomenon is not exclusive to Spain alone; similar narratives appear in Sicily, the Balkan states, and even the Eurasian steppe where Christian populations also gained independence from Muslim rule and ultimately restored demographic supremacy.   
Today in the West the process of Islamization has returned, this time sanctioned by Western governments who often act as the surrogate for what has historically been Islamic rule.  Although policies of multiculturalism do provide ample opportunity for Islamization, the situation remains a far milder version than past experiences of direct Islamic rule when the full weight of Sharia law could be brought to bear.  This difference should prove debilitating enough to prevent a reenactment of the accelerated S-curve rate of conversion outlined previously.  In each of those cases implementation of Islamic rule over a subject population was the prerequisite for advancing the curve of conversion. Islamic rule is the platform by which becoming Muslim is made the premium for social advancement.  Without this condition we should not expect history to repeat itself exactly.  A more likely scenario today is the emergence of many disparate curves of conversion at a localized level where Muslims are able to dominate particular cities or neighborhoods. Because the curve builds quite slowly it may yet be too early to confirm this trend.   
While personal experiences may vary let the reader understand: the slowly mounting pressure, the intrusions into daily life and the accompanying sensation of rising tension are all symptomatic of experiencing first-hand that age-old force, that inexorable push along the curve of conversion.  As a social process Islamization knows no bounds and all aspects of daily life are disputed territory.  The occupation of public spaces for prayer, refusal to remove the headscarf and the endless attempts to insert Islamic modes of living wherever possible- these all are methods of exposure and acclimatization intended to drive a population along the curve of conversion, until they abdicate their identity and enter Dar al-Islam.  


[1]Stephen McKenna, Paganism and pagan survivals in Spain up to the fall of the Visigothic kingdom (The Catholic University of America, 1938), p. 129; also Thomas F. Glick, Islamic and Christian Spain in the early Middle Ages (BRILL, 2005), pp. 350f; in the case of the Pechenegs see Kemal H. Karpat, ed., The Turks of Bulgaria: the history, culture and political fate of a minority (Isis Press, 1990), p. 99; for the Volga-Bulgars see Durmus Arik, “Islam among the Chuvashes and its Role in the Change of Chuvash Ethnicity.,” Journal of Muslim Minority Affairs 27, no. 1 (April 2007): 37-54. p. 39

[2]Richard W. Bulliet, Conversion to Islam in the Medieval Period: An Essay in Quantitative History (Cambridge: Harvard University Press, 1979),  p. 44

[3]Quotation in Göran Larsson, Ibn García's shu’ūbiyya letter: ethnic and theological tensions in medieval al-Andalus (BRILL, 2003). p. 172

[4]Ibid., p. 41

[5]This is an extremely simplified version of the process; see Carroll Quigley, The evolution of civilizations: an introduction to historical analysis (Liberty Press, 1979), pp. 146ff; and also Matthew Melko, The nature of civilizations (Porter Sargent Publisher, 1969), pp. 101ff.

[6]Andrew David Cliff, Spatial diffusion: an historical geography of epidemics in an island community (CUP Archive, 1981), pp 6f.

[7]Ajami, F. (2004, Mar 22). The moors last laugh. Wall Street Journal, pp. A.6-A.6;                 Abdal Hakim Murad, “Plovdiv: Granada of the East - Q-News July 2006, Issue 367,”

[8]See Alexander E. Elinson, Looking back at al-Andalus: the poetics of loss and nostalgia in medieval Arabic and Hebrew literature (BRILL, 2009).  Modern Palestinians commemorate a “Nakbah Day” to mourn the creation of the State of Israel

[9]A detailed discussion of the methodology used to attribute conversions to specific time periods can be found in Richard W. Bulliet, op cit., pp. 22ff; For the process in the Balkans see Anton Minkov, Conversion to Islam in the Balkans: Kisve bahası petitions and Ottoman social life, 1670-1730 (BRILL, 2004), p. 16

[10] Bulliet, op. cit., pp. 30-31

[11]Elesha Coffman, “Secrets of Islam's success: it spread faster than any other religion in history. Here are some reasons why,” Christian History 21, no. 2 (May 1, 2002), pp. 16-18.  

[12] Bulliet, op. cit., pp. 41-42

[13] Bulliet, op. cit., pp. 33-34

[14]Anton Minkov, Conversion to Islam in the Balkans: Kisve bahası petitions and Ottoman social life, 1670-1730 (BRILL, 2004), pp. 170f, 183.

[15] Bulliet, op. cit., pp. 50-51

[16] Bulliet, op. cit., pp. 78-79

[17] (975-875) / (975-646) = 30.4%; Minkov, op. cit., p. 17

[18] Bulliet, op. cit., p. 82

[19]Minkov, op. cit., p. 17

[20] Thomas F. Glick, Islamic and Christian Spain in the early Middle Ages (BRILL, 2005), pp 33-35 makes this mistake as does Ann Rosemary Christys, Christians in Al-Andalus 711-1000 (Routledge, 2002), p 3.  L.P. Harvey rightly senses that Glick’s numbers are unrealistic but wrongly attributes the error by extension to Bulliet’s theorem, see L. P. Harvey, Islamic Spain, 1250 to 1500 (University of Chicago Press, 1992), pp. 8-9  

Some authors have correctly understood this significant nuance, see Bernard F. Reilly, The contest of Christian and Muslim Spain: 1031-1157 (Blackwell, 1995), p. 18 and Richard Fletcher, Moorish Spain (University of California Press, 2006), p. XIV

[21]Bulliet, op. cit., pp. 43-44

[22]50% * 80% = 40%

[23]Spain: 1105-711 = 394 years Iran: 875-646= 229 years; (394-229)/229 = 72%

[24]Bulliet, op. cit., p. 118

[25]The decade of the 1730s would have seen the peak Muslim population of Bulgaria prior to spectacular Christian demographic gains that came later.  See Anton Minkov, op. cit., pp. 59ff.  For Muslim population just before the massacres and expulsions of 1878 see Bilâl N. Şimşir, The Turks of Bulgaria: (1878-1985) (K. Rustem, 1988). p. 16