*Éric Vilquin. Demography: Analysis and Synthesis. Graziella Caselli, Jacques Vallin, Guillaume Wunsch. Volume 4. Amsterdam: Elsevier, 2006.*

The birth certificate of demography (and of descriptive statistics and epidemiology) is generally considered to be a small book of roughly 100 pages published in London in 1662: Natural and Political Observations Mentioned in a following Index, and made upon the Bills of Mortality. By John Graunt, Citizen of London. With reference to the Government, Religion, Trade, Growth, Ayre, Diseases, and the Several Changes of the said City. The author, John Graunt, was a prosperous London merchant who happened in the Parish Clerks’ Hall on the bills of mortality compiled over decades by the roughly 130 parishes of London. Graunt’s intellectual curiosity led him to undertake a useful, intelligent compilation of the disparate accumulated information. The set of methodologic tools he fashioned for the purpose was called Political Arithmetick by his friend (and possible collaborator) William Petty (1686, 1687). The term démographie was coined only 2 centuries later, by Achille Guillard in his Elements of Human Statistics or Comparative Demography (1855), after which demography and statistics would be increasingly recognized as separate disciplines.

Following John Graunt, the pioneers of scientific demography made significant advances in methods for the observation, measure, and analysis of demographic phenomena. Focusing primarily on mortality for over 2 centuries, they really ventured into the area of fertility only on the eve of the 20th century and into migration even later. However, the measurement of population size and the description of its socio-demographic structures have historical antecedents that considerably predate the advent of demographic analysis.

## Demography Before 1662

Since ancient times, data collection has been undertaken using procedures that are part of the history of demographic science: enumerations and registers. Although the purely demographic objective of such operations was often secondary or even nonexistent alongside military, taxation, and legal purposes, they are nonetheless ancestors of the modern census and vital registration system.

__Ancient Enumerations__

The earliest known enumerations were carried out in Egypt, China, India, Japan, and the Near East. These were generally initiatives by central authorities that aimed to establish a quantitative and qualitative inventory of the country’s human resources. Interpretation of the statistical results is difficult because we must always suspect that the figures were arranged for political or religious reasons. However, precise information on the enumeration methodology is sometimes available. In regions with relatively strong, stable central authority, periodical census taking is part of the art of governing. This generally involves establishing in each district of the territory an inventory of hearths, houses, or individuals, divided into a small number of simple categories based on their relative importance to the state. For instance, if the purpose is to raise an army, the enumeration could identify adult males, without counting women, children, and elderly males, and an enumeration for tax collection purposes could count heads of taxpaying families (ignoring family size), without counting paupers or rich members of tax-exempt categories. Ancient enumerations often cover only a specific subset of the population.

The best-documented case is China. Numerous Chinese texts describe the administrative structures and procedures devised to enumerate the population; these began at least 3000 years ago and were frequently changed. Combined with a system of continuous registration of certain categories of the population (e.g., conscripts, persons subject to corvée labor, various categories of taxpayers, families granted imperial land), the process generally involved a massive counting operation based on the registers, with the total results forwarded in steps along strictly hierarchical channels from the base to the center. At the local level, the registers sometimes included highly detailed sociodemographic information on individuals. From the early 7th century to the early 10th century, the census was based on a family register (i.e., a list of household members specifying their sex and age, as well as the family’s social, economic, and tax status); the register was supposed to be exhaustive, updated once each year by the village or district chief, and sent to the central government once every 3 years. It is difficult to determine to what extent this system functioned properly, but it subsequently underwent considerable changes. The highest degree of refinement occurred under the Ming Dynasty (1368-1644), with the use of Yellow Registers. These were permanent, standardized lists of persons and property that were kept in duplicate; compilations were periodically carried out at every level of the tax administration (i.e., li-chia orlijia system). Every 10 years, the registers were reconstituted family by family and the results tallied to form the census. The imperial government attached great importance to homogeneity and synchronization of the count throughout the territory, as well as to error correction and proper storage of the basic documents. This extremely sophisticated, bureaucratic system has left plentiful statistical data, but critical analysis is problematic. Later transformations and degradations of the system made the Chinese census an instrument for police control and tax administration, but virtually unusable for demographic purposes, from the 17th through the 19th centuries. In China, as elsewhere, the tax function of registration and enumeration almost inevitably caused a deviation, because the exhaustive list of individuals gave way to an increasingly fictive count based on notional tax units (e.g., hearths, men, mouths).

The Roman Empire’s quinquennial census has left few traces, and little information is available. What was surely a partial and relatively crude count disappeared with the collapse of the empire. Europe would then remain without a census for 1000 years. Christianity inherited Judaism’s distrust of the census (mysteriously associated with terrible curses in the Biblical tradition), and the West would hesitate for a long time before implementing one. The rare attempts to reinstate a nation-wide census (e.g., the English Domesday Book in 1086 or the French État des Paroisses et des Feuxin 1328) proved to be short-lived. Locally, there are abundant lists and inventories of all sorts, often precise and detailed, but their lack of homogeneity and continuity generally precludes any useful statistical analysis.

During the Renaissance, virtually all theorists of the art of governing (including Niccolo Machiavelli, Jean Bodin, and Antoine de Montchrestien) argued energetically for an exhaustive, periodical, nominative enumeration, detailing its benefits, and generally arguing that accurate measurement of needs, resources and changes would allow the economy and society to be administered in an informed manner, rather than blindly. Rulers sometimes took these arguments seriously: the 16th century has left traces of numerous censuses, particularly in smaller city-states, but this was not a reinstatement of the tradition of the Roman census.

__Recording Demographic Events__

Nominative registration of births and deaths, in connection with ancestor worship, has existed for at least 2000 years in China within the most powerful clans. This may be a private vital register for strictly genealogic purposes, but it is a gold mine for historical demographers. Vestiges of similar registration systems exist in other parts of the world.

In the 14th century, the Christian West slowly introduced into its administrative and legal practices a change that was to have incalculable consequences: the replacement of oral testimony by written evidence. The Church prescribed that certain important events in the life of individuals, hitherto regulated by custom or by faith, should be recorded in writing. Henceforth, parish registers would provide documentary evidence that a person had received the sacraments of baptism, confirmation, or marriage; that the person was alive or dead; and evidence of his or her marital status. It took 2 or 3 centuries for the rules and practices of church registration to move toward homogenization and generalization. Roughly, the idea was definitively established that each church would register the christenings, marriages, and burials of its faithful in the 16th century in some dioceses and in the 17th century throughout Christendom (both Catholic and Protestant).

Church and civil authorities then relayed one another in imposing registers and specifying their content. Royal ordinances in England (1538) and France (Villers-Cotterêts, 1539) requiring that parish registers be kept predated the first general legislation of the Roman Catholic Church (the Council of Trent, 1563). The mass of regulations attests to difficulties in implementing registration and hesitations by authorities. Gradually, after successive reforms, a rigorous, homogeneous system was put in place. In France, this was accomplished with the royal declaration in 1736. These turbulent changes, combined with historical events (marked by the disappearance and destruction of documents), justify the precautions now taken by historical demographers in the statistical analysis of this prodigious mass of individual data on births, marriages, and deaths.

From 1670 onward, the parish registers in some large towns were systematically counted and compiled into more-or-less detailed, periodical tables. As early as 1736, Sweden experimented with what would become in 1746 the Tabellverket, the first national system of demographic statistics and analysis based essentially on the exhaustive extraction of data from parish registers.

After the Revolution in 1789, France “secularized” vital registration by creating the état civil (1792), a civil register modeled on church registration, but which removed the selection criterion of religious affiliation. Several European countries followed suit in the 19th century.

## The Birth of Demographic Analysis in 1662

Alongside the parish registers, in the 16th century, a number of large European towns began occasionally to produce bills of mortality. This was a rudimentary system for local monitoring of plagues (the generic and Biblical term for major fatal epidemics); in London, for instance, every Thursday, the parish clerk posted on the church door a bill reporting the number of deaths in the parish during the previous week and the number of deaths attributed to the plague. This was done to inform the population; if the number and proportion of deaths attributable to the plague increased week after week, it was time to take protective measures.

In London, starting early in the 17th century, the practice had become routine and continuous. The parish clerks gradually extended their bills to include the number of christenings, the breakdown between males and females, and the cause of death. (The age at death was added in 1728.) The Company of the Parish Clerks of London (representing 109 parishes in and adjoining London around 1600 and 130 around 1660) received and stored all the bills, and it published an annual report with a summary account at the end of each year.

John Graunt (1620-1674), a wealthy London merchant who served on the Common Council, was an exponent of the experimental science movement that was then in full expansion. Happening on those records in 1660 or 1661, he subjected them (apparently with no preconceived intent) to an enormous amount of critical examination, analysis, and synthesis. The most elementary mathematical tools (what he called his Shop-Arithmetique) proved sufficient for him to forge the tools needed for a synthetic, cautious, and pertinent description of a large mass of analogous information; in so doing, he invented the statistical approach and the principal techniques of descriptive statistics.

Faced with a long series of data on systematic, continuous observations of a natural and social phenomenon (mortality), he sought to identify laws, in much the same way as his astronomer and physicist friends. With a subtle combination of method and intuition, he identified statistical regularities in an area where previously one had seen only the unfathomable secrets of Providence, including the sex ratio at birth, differential mortality by cause and location, comparative demographic dynamics between London and Paris, and the ratio of annual births and deaths to the population. The ratios, proportions, and disproportions that abound in Graunt’s work are the ancestors of our modern proportions, means, rates, and ratios. Despite his many computational errors and some mistaken assumptions, he indicated for the first time the actual magnitude of demographic parameters.

One of his concerns was to evaluate the population of a town or country in the absence of any enumeration. After observing the stability of the numerical ratios between the population and the number of births and deaths per annum, he used the relationship to calculate the population from parish statistics. This multiplier technique remained the principal method of evaluating population size until censuses were restored and continued to compete with censuses through the early part of the 19th century.

Graunt did not have what was needed to construct a life table, but he is nonetheless the inventor. With no information on the age at death, he adopted an arbitrary (and unrealistic) distribution of deaths by age groups to develop the theoretical model of the agespecific distribution of deaths that he suspected. Thirty years later, Edmund Halley (1693) relied on John Graunt’s model to construct the first life table based on an actual distribution of deaths according to age.

Graunt was also a pioneer in the critical analysis of data. Every concept and figure was subjected to inquisitorial scrutiny, after which he corrected or rejected the data. With scrupulous caution and modesty, he always tried to check calculations by several independent means and gave only limited credit to the numeric value of his results—but that does not prevent him from being affirmative and categorical when certain of having disproved generally accepted ideas. Graunt the precursor left an impressive set of tools, which his successors perfected while preserving their initial spirit.

## Direct Measurement of Population

Tax registers naturally provided a census frame that, although incomplete, was frequently brought up to date. Tax registers were the basis for most population estimates in the 16th and 17th centuries. There were significant methodologic problems, however, and the resulting estimates remained highly inaccurate because some levels of society were not included on the registers and because the registers often provided only a list of hearths (households), with the aggravating circumstance that in several regions the hearth was a conventional tax unit unconnected to demographic reality.

In the absence of a simple, homogeneous method, many attempts to conduct local or regional enumerations in the 16th and 17th century (in Spain, Italy, and France) proved to be short-lived. Around 1680, Sébastien de Vauban, the French administrator and military engineer, developed a “general and easy” (générale et facile) counting method, which he personally tested in numerous towns and regions and then placed at the disposal of the French government, which first applied it in the colonies (Vilquin, 1975; Vauban, 1686). Vauban’s innovation was to print blank tables that the census agents completed by following strict instructions. There is one table per parish and one line per house; each line recorded the composition of the household based on the main categories of sex, age, and social status.

The 17th and 18th century European governments were eager for qualitative and quantitative information, but they deluged local officials with such wide-ranging demands that the answers were inevitably vague, if not mere conjecture. One such enterprise, which was better prepared than most, was a genuine success. This was the French Enquête des intendants pour l’instruction du duc de Bourgogne between 1697 and 1700. With collaboration from Vauban, the Duke of Burgundy’s tutors undertook a vast survey with multiple objectives in every province of France. After considerable prodding for information and adjustments, they obtained a complete series of reports that, despite enormous failings, were used pretty much as official statistical yearbooks (often without being updated) by French ministries throughout the 18th century. Their interest and utility were such that the bookseller Saugrain published them in 1709 (Dénombrement du royaume par généralités, élections, paroisses et feux) and even brought out a second edition in 1720 (Nouveau dénombrement du royaume) (Saugrain, 1709, 1720). For the purposes of the survey, a genuine direct census of the population was taken in some provinces and towns, more or less in keeping with Vauban’s approach; however, in most cases, earlier estimates of uncertain provenance were reproduced, or a rapid count using the tax registers was performed.

Genuine nominative, comprehensive censuses have been conducted since the 17th century in nearly every European country, in limited territories. The first modern nationwide censuses were those taken in Iceland in 1703 and Sweden in 1755. The case of Sweden is distinctive. All Swedish parishes had maintained detailed, up to date registers of their population since the late 17th century. In the 1750s, the government ordered a periodic compilation and analysis of the registers. Pehr Wargentin and Edouard Runeberg (1772), who were in charge of the initial work by the Tabellkommissionen, set very high standards at the outset, and for a long time Sweden maintained a half-century lead over the rest of Europe in the area of collection and analysis of national demographic statistics (for Wargentin and Runeberg, see Dupâquier, 1977).

In France, the welter of censuses and surveys ordered by an array of Revolutionary bodies were typically carried out by incompetent, overworked personnel, and almost never reached the processing stage. The creation of a Statistical Bureau in 1799 introduced some order, but the Empire soon made statistics a matter of state secrecy and eliminated the Statistical Bureau, which would definitively reemerge only in 1834. Meanwhile, Austria, Prussia, Russia, the Netherlands and Belgium set up their own statistical bureaus, followed in the third quarter of the century by the other nations of Europe.

One hundred fifty years after Vauban, the ideal of a uniform, thorough, and simple census method was taken up again in the 19th century by the Belgian mathematician and astronomer Adolphe Quetelet. The censuses in the Netherlands (1829 to 1830) and Belgium (1846) allowed him to test his ideas regarding standardization of procedures for collection and analysis (Quetelet, 1847). From defining concepts to training census agents, his systematic approach worked wonders, so the organization of the Belgian census was seen as a model by neighboring countries. However, his dream of every country in the world simultaneously taking identical censuses would never be realized.

## Political Arithmetic

John Graunt’s invention found its first enthusiast in his friend William Petty, who used the term political arithmetic to describe his resolutely quantitative approach to social phenomena (Petty, 1686, 1687). Petty used Graunt’s formulas to build arguments in support of his political theories, but was largely unconcerned with methodologic improvements. Their successors, often astronomers or clergymen, would advance theory and techniques in two main areas: indirect population estimation and the life table.

__Demographers__

The first demographers were English and Dutch: Edmund Halley, Matthew Hale, Gregory King, Charles Davenant, Louis and Christian Huygens, and Jan De Witt. Their shared concerns included evaluating population worldwide and in selected countries and towns; the long-term population trend in Europe and worldwide; and the actuarial computation of annuities. Some demographers also addressed the distribution of population by sex or age group, and even ventured population projections. In 1696, Gregory King computed ratios for births, deaths, and nuptiality by dividing the population by the annual numbers of births, deaths, and marriages (King, 1696/1973); it was not until 80 years later that the corresponding rates were computed, when Jean-Louis Muret (1766) reversed the numerator and denominator. The Dutch actuaries Willem Kersseboom and Nicolas Struyck, starting in 1738, quarreled over the determination of multipliers; in doing so, they unwittingly laid the bases of stationary population theory, and more importantly, they systematically compiled statistical surveys of demographic data from various sources (Kersseboom, 1970; Struyck, 1912, cited by Dupâquier and Dupâquier, 1985, p.162-166). Gregory King and Nicolas Struyck are doubtless the most important figures in the first half of the 18th century because they improved on the methods of their predecessors, and they methodically weighed the relevance and validity of the data and techniques. Nicolas Struyck was the first to investigate the disturbing role of migrations in demographic estimates, to be wary of computations based on small numbers of subjects, and the first who did not set out to prove at all costs that all is for the best in Creation.

The German pastor and mathematician Johann Peter Süssmilch (1707-1767) was the first demographer to achieve fame and the first to popularize the new science. His principal work, The Divine Order, published in 1741 and revised in 1761-1762, enjoyed genuine success. The book provides a synthesis of the state of political arithmetic on the two publication dates, and its application to an impressive amount of data. Süssmilch was a judicious critic but not really an innovator. His purpose was to accumulate scientific evidence to support his religious convictions.

In France, Antoine Deparcieux (Essay on the probabilities of the length of human life, 1746, and Addition to the essay …., 1760) made a notable improvement to the measurement of the death rate. Louis Messance (1766) conducted a methodical demographic analysis of three French provinces. The Abbé Expilly (1762-1770) collected astronomic quantities of demographic data. Jean-Baptiste Moheau published a veritable treatise on demography (Researches and Considerations on the population of France, 1778), which can be compared with The Divine Order. His arithmetic, though brilliant, remained classic, but he stands out from most of his peers by devoting several chapters to interpreting his results, evidencing a systemic vision of the sociodemographic dynamics of a nation.

Each European country had its political arithmeticians. Among dozens of others, two who stand out were the Swede Pehr Wargentin (1717-1783) and Switzerland’s Jean-Louis Muret (1715-1796). Wargentin carried out in the 1750s an exemplary analysis of the data produced by the admirable Swedish statistical system (see Dupâquier, 1977), and Muret (Muret, 1766), who performed many differential analyses (by sex, by region, and by period), can be called the true inventor of the crude rates of births, deaths, and marriages.

In the 18th century, the political arithmeticians knew and corresponded with one another, and submitted their discussions to a learned audience by publishing in the first scientific journals (i.e., the published proceedings of the academies). Their methodologic problems came to the attention of mathematicians, including the Bernouillis, d’Alembert, Euler, Lagrange, and Laplace. The mathematicians cautiously advanced the initial mathematical expressions for the political arithmeticians’ empirical parameters, and used their observations (e.g., the stability of the sex ratio at birth or the probability of survival from one age to another) to formulate the standard problems in the calculus of probabilities. For instance, at the request of Johann Peter Süssmilch, Leonhard Euler developed around 1760 the empirical recipe for computing the doubling time of a population, which had hardly progressed since John Graunt), and established the algebraic relationships between population size, structure and movement. Pierre-Simon de Laplace in 1778 called on probability theory to demonstrate that the sex ratio at birth differs significantly from 1 and to measure the ranges of variation of certain parameters (Laplace, 1781).

Kersseboom applied the life table to the average annual number of births to determine the population pyramid for the (stationary) population. The size of the (stationary) population is then equal to the annual number of births multiplied by the average lifetime.

__The Life Table__

Even before the construction of the first true life table by Edmund Halley (1693), the brothers Christian and Louis Huygens, in 1669 (Huygens, 1669/1920), were attracted by the model developed by John Graunt. They developed the concepts of life expectancy and the probable length of life at any given age, as well as the graphical representation of the parameters of a survival table. However, this was not reported to contemporaries. Around 1680, Gottfried Wilhelm Leibniz outlined a theoretical life table (Leibniz, 1680?/1866), but failed to follow through with his work. It therefore fell to Edmund Halley (1656-1742), the British astronomer, who used the first statistics of deaths according to age, from Breslau in 1687 and 1688, to construct the first table based on actual data. Taking some liberty with the crude data, he computed a table of survivors by age, starting with 1000 infants younger than 1 year. The ages are expressed in complete years at the start of the table, and subsequently by age at birthday. However, lacking information on the age structure of the population of Breslau, he had to assume that the distribution of deaths by age was the same as the age structure of the living population (the cumulative deaths method), which (as he was aware) is the case only for a closed, stationary population. He did not calculate death probabilities or life expectancies.

Life table methodology made slow, modest progress in the 18th century. Kersseboom, Struyck, Smart, and Simpson eliminated the need to assume a closed population and stationarity. Before introducing the technique into France, Antoine Deparcieux (1746) analyzed his predecessors’ work in detail and definitively distinguished between the mean length of life and the median length of life (i.e., probable lifetime). Deparcieux gave the life table its current form, breaking with the cumulative deaths method. Disposing of good life history data on small populations of rentiers or religious communities, he could establish the ratio of deaths at any age to the total population exposed to the risk of dying at that age. Because he had no data on children, he surveyed mothers from various social backgrounds at the end of their reproductive lives regarding the survival of their children. Pehr Wargentin was the first to dispose of the age-specific distributions of the population and of deaths for the same period (see Dupâquier, 1977).

Early in the 19th century, Emmanuel Duvillard de Durand (1806) used an essay by Daniel Bernoulli (1760) as the basis for calculating the gain in life expectancy that would result from the elimination of a cause of death (i.e., smallpox).

As rulers and savants showed growing interest in statistics, the number of national life tables increased. There were the first timid steps in also establishing tables for subpopulations (e.g., males and females, town and country). Around 1850, on the strength of the exemplary work of Adolphe Quetelet (1854, 1872) in Belgium and William Farr (1885) in England, the life table was definitively anchored to the corresponding population pyramid derived from the census.

__Indirect Methods of Population Estimation__

The old procedure for population estimation based on an enumeration of hearths did not disappear quickly, and the multiplier technique discovered by John Graunt seems to have been used little before 1750. Gregory King tried both methods in 1696 but did not publish his results. Spurred on by the first French demographers (e.g., Messance, Moheau), enthusiasm for Graunt’s method spread in the second half of the 18th century. The political arithmeticians, focusing on small portions of the territory, conducted exhaustive enumerations of the population and counts of christenings in the parish registers: the ratio of these two numbers was the birth multiplier, which could be used in any town or region to compute the total size of the population on the basis of the average annual number of births registered. Similar multipliers based on the annual number of deaths or marriages were tested. Fascinated by the stability—a relative stability, but astonishing nonetheless—of the numerical ratios they found, demographers cherished the secret ambition of determining the universal multiplier, but the quest was gradually abandoned as they recognized the need to calculate specific multipliers for different contexts (e.g., town/country, plains/mountains, north/south). The most extravagant enterprise in this area was conducted by the Abbé Expilly, who spent years working on the whole of France. He had baptisms, marriages and burials in the registers of every parish in France counted for the periods 1690-1701 and 1752-1763 (to identify the direction of population change). Funds ran out as he neared the goal, but he still managed to publish statistics for more than 15,000 parishes, in six volumes; the seventh was never released (Expilly, 1762-1770).

French ministers (Turgot and then Necker) ordered that the computations be redone on regional samples, while scientists (Moheau, Laplace) were refining the theory. However, the French Revolution chose to return to a Roman census.

The rare rudimentary attempts at demographic projections, as early as the 17th century (e.g., Petty, King, Vauban), use exponential extrapolation and are based on fixed, crude assumptions: Vauban (no date/1845), for instance, assumed that marriage is universal, that each couple produces four children on average, and that generations succeed one another every 30 years.

## Development of Modern Demographic Analysis in the 19th Century

__Death Rate__

To construct his hypothetical life table, John Graunt implicitly used a mathematical function expressing the age-specific risk of dying, but because this was not indicated in his book, the idea was not taken up immediately. After several attempts—more complicated but not more convincing—to mathematically model mortality by age in the 18th century (De Moivre, Euler, Duvillard, Laplace), the Englishman Benjamin Gompertz, in 1825, made the assumption that the number of survivors decreases exponentially as age increases; this was corrected in 1860 by the German William Makeham, who incorporated in the equation a term to represent deaths that were not dependent on age. This initial mathematical law provides a close fit to the death rate for adults, but not for infants and children. In the meantime, William Farr (1843/1885) provided the definitive formulation of the concepts of age-specific rates (including the infant mortality rate), death probabilities, the probability of survival, and standardized rate. Louis-Adolphe Bertillon (1866) discovered how the crude death rate is dependent on the age structure of the population. In 1868, Georg Friedrich Knapp, in analyzing the classification of deaths by years of age and year of birth, distinguished between longitudinal analysis and cross-sectional analysis.

The age-old dream of a mathematical law expressing mortality as a function of age was shattered in 1897 by Karl Pearson, who showed that the statistical curves of mortality by age must be decomposed into several mathematical laws that are juxtaposed or superposed.

__Fertility__

If the political arithmeticians focused more on the factors of fertility than on measuring fertility, it was because the necessary data were totally unavailable. Births were first classified by mother’s age in Sweden in 1775, and the example was not imitated elsewhere until after 1850. Before then, considerations on differential fertility (between town and country, between monogamy and polygamy, and between seasons) were based on economic or moral arguments; the rare calculations were based on rough assumptions regarding “prolificacy” (e.g., that an average marriage produces four children). The nearly universal “populationism” among 18th century Europeans, including demographers, provided less incentive to measure fertility than to seek ways to maximize fertility. This explains the considerable persistence of extremely crude assumptions.

Even before it became possible to study age-specific fertility, a number of significant steps forward were made. The first fertility estimator was the birth multiplier (number of inhabitants per birth). The political arithmeticians all devoted numerous pages to this, as they sought to determine whether it was stable or variable, and the best way of evaluating it. After Gregory King, but independently, Jean-Louis Muret transformed the multiplier into a crude birth rate by reversing the ratio (number of births per 100 inhabitants). This use would not become widespread before the 19th century.

Some considered it more logical to take the ratio of births to marriages, and this parameter enjoyed considerable favor. The general legitimate and illegitimate fertility rates were developed in 1815 by Joshua Milne (1815). Even toward the middle of the 19th century, Quetelet believed the best measure of a population’s fertility was the ratio of births to marriages and that the analysis could be further refined by classifying couples based on the age difference between spouses and age at marriage (Quetelet, 1835).

The breakthrough came from Scotland. Dr. James Matthews Duncan (Fecundity, Fertility, Sterility, and Allied Topics, 1866) took advantage of an exceptional statistical source, which decomposed births in Edinburgh and Glasgow by the mother’s age and the duration of marriage; he used this to measure, among other things, the intervals between successive births, parities by age and by duration of marriage, and completed fertility. However, these methodologic innovations would be disseminated 30 years later, when the national statistical offices grasped the need to collect highly detailed data on fertility. Similarly, the clear distinction he introduced between fertility and fecundity was not entirely understood at the time, and was generally accepted only 50 years later. Because he conducted a cohort analysis of married women who had neither died nor become widows before the age of 45, mortality did not interfere with his estimates of completed fertility.

__General Models__

Adolphe Quetelet had met Thomas Malthus in 1833, and the two had briefly corresponded. Quetelet admired the English clergyman’s theory but considered that it remained to be cast in the bronze of mathematics. He assigned a young colleague, the brilliant mathematician Pierre-François Verhulst, the task of finding the best mathematical model for Malthus’ theory of population growth over time. The main difficulty was finding an exponential function (Malthus’population increases in a geometric ratio) that progressively generates a slowdown in the rate of growth (i.e., that incorporates Malthus’ checks). After an initial attempt left Quetelet unsatisfied, in 1847 Verhulst discovered that thelogistic curve was an excellent way to model Malthus’ theory and that it provided the theory with a complement, if not the crowning touch, in that the curve tended toward a horizontal asymptote: the maximum population. Verhulst’s work was published but passed unnoticed, and the logistic model was rediscovered only in 1920 by the Americans Raymond Pearl and Lowell Reed.

Advances from the concept of marital fertility to that of population reproduction required the modeling of nuptiality. This was done by Louis Adolphe Bertillon (1872). The Germans Richard Böckh (1886) and Robert Kuczynski (1931) could then define the crude and net reproduction rates. However, the concept of population replacement was to become operative through another channel. In 1907 and 1911, Alfred Lotka adopted a purely theoretical, abstract approach to study the renewal of a population that evolves as a function of inputs and outputs (Sharpe and Lotka, 1911; Lotka, 1907). Laying the bases for mathematical demography, he developed the theory of stable populations (Lotka, 1934, 1939) that would revitalize demographic analysis after World War I.

__Graphical Tools__

The first graphs in the history of demography were drawn by Christian Huygens (1669), who used curves to depict the numbers of survivors at various ages in Graunt’s table, and the corresponding life expectancies. However, he showed them only to his brother, and they remained unknown. Setting aside three or four curious but largely impractical attempts at geometric illustration, which failed to gain acceptance, the first to make systematic, frequent use of graphical representations in statistics were Jean-Baptiste Fourier (1821) and Adolphe Quetelet (starting in 1827). Quetelet was virtually obsessed by means and variations, and privileged the representation of deviations between observations and their mean. Through his educational abilities and the international statistics conferences he was the first to organize and to which he contributed indefatigably, he led all European statisticians to use a broken line or curve to represent variations in the phenomena they observed.

The second half of the 19th century saw several ingenious but virtually indecipherable attempts at three-dimensional graphical representation (i.e., stereograms), which was soon abandoned. Another invention that proved more successful was the population pyramid. The initial idea came from General Francis Walker (1874), the Superintendent of the 1870 Census in the United States, who used it liberally to illustrate his analysis of the U.S. censuses of 1850, 1860, and 1870. In 1878, Heinrich Schwabe brought the shape into line with the principles of the histogram, and it has remained unchanged ever since.

In 1874, following up on the intuition of several precursors, three German mathematicians, Georg Friedrich Knapp, Karl Becker, and Wilhelm Lexis, looking for the most efficient way of plotting biographic events in a system with three coordinates (i.e., time, age, and date of birth), devised two alternative versions of what is now called the Lexis diagram. The modern version was developed by Roland Pressat around 1953 (Personal communication. See Vandeschrick, 1992, p. 1249.).

## Conclusions

The origins of demographic science are recent. Only 350 years ago, it could seriously be claimed without incurring scientific refutation that men outnumber women six to one worldwide or that gout is a hundred times more lethal than influenza. John Graunt, in his simple but inspired constructions, planted the seed that has grown into a full array of instruments for measurement and analysis of demographic phenomena. Some developed faster than others. Progress in life table techniques was significantly stimulated by the quest for accurate calculation of annuities. At the same time, the political arithmeticians, eager to use their multipliers to prove that the population was growing (or declining), thought they could eliminate the need for complicated census taking. They also evaluated the constants of what Adolphe Quetelet would call social physics. Mathematicians then joined the fray, testing the concepts and theorems of probability theory on the only continuous statistical series available, vital registration statistics. Fertility had long been the poor relation to demographic research—not for lack of interest, but for lack of the data required to analyze fertility. The development of concepts and tools for demographic analysis has always been closely associated with changes in the production of population statistics, but quite often, the conceptual developments preceded the data. From its primitive, artisan, and utilitarian origins, the progress and results achieved by demography between the late 17th century and the early part of the 20th were remarkable. The considerable work by Alfred Lotka, before and after World War I, would unify demographic analysis and impart a new impetus, supported and amplified by fears of population explosion.