
The Carey-Stewart-Warntz conceptualized "gravity model", of the so-called "social physics school" of geographical research, one of the four schools of quantitative geography, according to human geography historian Ron Johnston, in which population potential maps, or distributions of populations divided by distances between cities, are central. [7]

The Carey-Stewart-Warntz conceptualized "gravity model", of the so-called "social physics school" of geographical research, one of the four schools of quantitative geography, according to human geography historian Ron Johnston, in which population potential maps, or distributions of populations divided by distances between cities, are central. [7]

In hmolscience, population potential, “potential of population” or “potential population by gravity model”, is the attractive force felt at a distance away from a social aggregate; quantified by mass of the aggregate, typically the population of a city, divided by the distance away from the aggregate; the gravitational potential models of Joseph Lagrange (1773) applied to people and cities moving around each other like planets.

Early theorists
Early thinkers to theorize about social phenomena in gravitational terms include: George Berkeley (1713), Francesco Algarotti (1737), and Thomas Carlyle (1837).

Carey
In 1858, American sociologist Henry Carey introduced the concept of social gravitation, namely the existence of a hypothetical force of attraction, similar to the gravitational force, existing between aggregates of people, i.e. cities, proportional to the number of people in each aggregate and inversely proportional to the distance between the aggregate; to quote a noted selection:

“The great law of molecular gravitation: man tends of necessity to gravitate toward his fellow man. Gravitation is here, as everywhere else in the material world, in the direct ratio of the mass and in the inverse one of the distance. The greater the number collected in a given space, the greater is the attractive force there exerted, as is seen to have been the case with the great cities of the ancient world, Niniveh and Babylon, Athens and Rome, and is now seen in regard to Paris and London, Vienna and Naples, Philadelphia, New York, and Boston.”
— Henry Carey (1858), Principles of Social Science (pgs. 42-43); quoted in: Stark (1962); Tocalis (1978); Barnes (2014) [1]

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Stewart
In the late 1930s, American physicist John Q. Stewart was ruminating on how to study social aggregation as a force proportional to mass inversely proportional to distance of separation phenomena, in the sense of Lagrange-Newton potential models. The following is a retrospect summary of these thoughts from Stewart’s 1947 “Distribution and Equilibrium of Population” article: [6]

“The evident tendency of people to congregate in larger and larger cities represents an attraction of people for people that turns out to have a mathematical as well as merely verbal resemblance to Newton’s law of gravitation. Lagrange in 1773 found that where the attraction of several planets at once was under consideration, a new mathematical coefficient, not used by Newton, simplified the calculations. This coefficient amounted to a measure of the gravitational influence of a planet of mass m at a distance d, and it was as simple as possible, merely m/d.”

Stewart, in short, was ruminating on some type of social potential equation of the following form:

$P_s = \frac{M_s}{d} \,$

where Ps is sociological potential felt in the vicinity of a given social mass Ms in a given distance d from that mass.

In 1939, Stewart, supposedly either aware or unaware, at that point, of the earlier social gravitation work by Henry Carey (though by 1953 he was for sure), began to use a potential equation of the following form—notation use according to English geographer Rich David (Potential Models in Human Geography, 1980):

$v_{i}={\frac {P_{j}}{D_{{ij}}}}\,$

where vi is the "population potential" at location i, in respect to a given population at location j, Pj is the size of the population at j, and Dij is the distance between locations i and j. Stewart's population potential equation states that population influence of location j on location i is proportional to the size of the population at j doing the influencing, and inversely proportional to the distance between the two locations. [2] Stewart's equation, supposedly, is based, in some way, on either the concept of universal law of gravitation (Newton, 1686) or “potential” (Leibnitz, 1773), or its synonym gravitational potential. [3]

In Dec 1939, Stewart, during his Christmas Holiday break, first applied this potential equation to the geographical distribution of what he referred to as the “Princeton Family,” that is, the 20,438 members listed in the 1940 edition of Princeton University’s Alumni Directory. Specifically, Stewart, with the assistance of a Philip Wilkie, calculated the population potential of every state in relation to New Jersey, Princeton’s home state. From that he worked out the expected number of Princeton alumni for every state, and compared that number to the actual count from the Alumni Directory. Stewart found that expected and actual numbers of alumni “more or less” lined up. It showed him that potential models applied not only to objects in the heavens but also to those down on earth within the social field. [4] Stewart, in other words, found that people graduating from a given university have the "potential" to migrate outward in radius to new cities, cities being akin to a clustering of several planets, wherein the "Stewart potential" represents the social gravitational attraction towards each outward location.

In 1947, Stewart then calculating total population potential for the US, in 1940, as a whole, via the following equation:

$V_i = \sum_{j=1}^N \frac{P_j}{D_{ij}} \,$

where Vi is the total population potential for location i, meaning that for any given location i, total population potential is equal to the sum of all the individual potentials created by populations found at every location j, summed through the total number of locations N. With this data, now knowing the total population potentials for all locations, then constructed an isopleth map, i.e. “equipotential” map, wherein each line of equipotential connects all the places with the same population potential:

Somewhere in this "isopleth" map argument, Stewart, to note, devotes a small section to what he calls "demographic energy" or "energy of interchange", arguing along the lines that just as in physics, the potential is the energy in the field of a unit of mass, so to is the population potential the "demographic energy" in the social field around a given unit mass of population.

Stewart's isopleth map shows that population potentials are high in the northeastern states, indicating considerable influence of population on any given location within that region. Stewart also gave the following population potential equipotential maps: [6]

Stewart then gave US isopleths as the changed over a century:

Warntz
In 1964, American economic macrodemographer William Warntz made

Warntz then made the following American and Canada isopleth map:

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See also
● Gibbs landscape

References
1. (a) Carey, Henry C. (1858-59). The Principles of Social Science (Vol I , Vol II, Vol III). J.B. Lippincott & Co.
(b) Stark, Werner. (1962). The Fundamental Forms of Social Thought (pg. 143). Routledge & Kegan Paul.
(c) Tocalis, T.R. (1978). “Changing Theoretical Foundations of the Gravity Concept of Human Interactions”, in: Perspectives in Geography 3: the Nature of Change in Geographical Ideas (editor: Berry G.J.L.) (pgs. 67-124; quote, pg. 68). Northern Illinois University Press.
(d) Barnes, Trevor J. and Wilson, Matthew W. (2014). “Big Data, Social Physics, and Spatial Analysis: the Early Years” (pre:pdf) (abs) (pdf) (pg. 3), in: Big Data & Society, Jun.
2. (a) Rich, David C. (1980). Potential Models in Human Geography: Concepts and Techniques in Modern Geography, Volume 26, Study Group in Quantitative Methods, Institute of British Geographers, University of East Anglia, Norwich: GeoBooks.
(b) Barnes, Trevor J. and Wilson, Matthew W. (2014). “Big Data, Social Physics, and Spatial Analysis: the Early Years” (pre:pdf) (abs) (pdf) (pgs. 5-6), Big Data & Society, Apr-Jun:1-14.
3. (a) Newton’s law of universal gravitation – Wikipedia.
(b) Gravitational potential – Wikipedia.
4. (a) Stewart, John Q. (1940). “The Gravity of the Princeton Family,” Princeton Alumni Weekly (pgs. 409–10), Feb 9; in: Box 3, folder 9, John Q. Stewart Papers, Rare Books and Special Collections, Princeton University.
(b) Barnes, Trevor J. and Wilson, Matthew W. (2014). “Big Data, Social Physics, and Spatial Analysis: the Early Years” (pre:pdf) (abs) (pdf) (pg. 6), Big Data & Society, Apr-Jun:1-14.
5. (a) Stewart, John Q. (1947). “Empirical Mathematical Rules Concerning Distribution and Equilibrium of Population” (abs) (pdf), Geographical Review, 37(3):461-85.
(b) Barnes, Trevor J. and Wilson, Matthew W. (2014). “Big Data, Social Physics, and Spatial Analysis: the Early Years” (pre:pdf) (abs) (pdf) (pg. 6), Big Data & Society, Apr-Jun:1-14.
6. Stewart, John Q. (1947). “Empirical Mathematical Rules Concerning Distribution and Equilibrium of Population” (abs) (pdf), Geographical Review, 37(3):461-85.
7. (a) Johnston, Ron J. (1997). Geography and Geographers: Anglo-American Human Geography Since 1945 (pgs. 62-73). London: Arnold.
(b) Holt-Jensen, Arild. (2009). Geography: History and Concepts (social physics + John Q. Stewart, pg. 87; gravity model, pg. 88). Sage.

Further reading
● Warntz, William. (1964). “A New Map of the Surface of Population Potentials for the United States, 1960” (abs), Geographical Review, 54(2):170-84.

External links
● Demographic gravitation – Wikipedia.