# Why logistic (sigmoid) ogive and not autocatalytic curve?

We call our S-shaped response curve a logistic ogive. How did this term originate?

S-shaped curves are also called sigmoid curves or sigmoidal functions, from the Greek "s", sigma. In his "Mathematical Researches into the Law of Population Growth Increase" (Nouveaux Memoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles, 18, Art. 1, 1-45, 1845), Pierre Francois Verhulst (1804-1849), Professor of Analysis at the Belgian Military College, examines population growth in Belgium. He discovers that sigmoid curves are useful for describing population growth. Following Malthus, Verhulst hypothesizes that small populations increase geometrically, because the supply of resources exceeds demand. Then, as supply and demand balance, population growth is constant. Finally, as demand exceeds supply, population growth decreases at the same rate that it had increased. Verhulst describes this process with an equation that enables him to predict when a population will reach any given size (see Verhulst's Figure):

t = log10( p/ ( m/n - p ) ) / m

where, for Belgium, with 1830 population of 4,247,113,
t is the time in years, since the mid-point of the curve (computed to be in 1807)
p is the population count
1/m is a population growth coefficient of 87.885
m/n is the projected maximum population of 6,583,700.

Verhulst writes "We will give the name logistic [logistique] to the curve" (1845 p.8). Though he does not explain this choice, there is a connection with the logarithmic basis of the function. Logarithm was coined by John Napier (1550-1617) from Greek logos (ratio, proportion, reckoning) and arithmos (number). Logistic comes from the Greek logistikos (computational). In the 1700's, logarithmic and logistic were synonymous. Since computation is needed to predict the supplies an army requires, logistics has come to be also used for the movement and supply of troops. Chemists, in their own work, discovered that the sigmoid curve describes an autocatalytic reaction. In autocatalysis, a product of the chemical reaction increases the speed of the reaction. But the reaction ceases when there is no chemical left to react. Consequently chemists called their sigmoid curve, the autocatalytic curve.

Raymond Pearl (1879-1940) renewed Verhulst's terminology. His "Studies in Human Biology" (Baltimore: Williams & Wilkins, 1924) review Verhulst's population growth work. In later work, Pearl exhorts researchers to use logistic curve or function in preference to autocatalytic curve because the latter is tied to a physical process. Pearl's student, Joseph Berkson, popularized the term logistic in his application of the sigmoid curve to bio-assay quantal response.

Berkson coined logit (pronouced "low-jit") in 1944 as a contraction of logistic unit to indicate the unit of measurement (J. Amer. Stat. Soc. 39:357-365). Georg Rasch derives logit as a contraction of logistic transform (1980, p.80). Ben Wright derives logit as a contraction of log-odds unit. Logit is also used to characterize the logistic function in the way that probit (probability unit, coined by Chester Bliss about 1934), characterizes the cumulative normal function. Ogive (pronounced oh-jive) indicates the shape of the logistic curve. An ogive is an architectural shape - a pointed arch. Modern mathematical use of ogive began in an 1875 paper by Francis Galton (Phil. Mag. 49:33-46). Galton's curve is of finite length and comes to a point (see Galton's Figure). Galton's curve has been idealized and its axes rotated to produce smooth shapes like Verhulst's Figure. The name ogive has been retained, even though the curve now resembles the architectural ogee (pronounced oh-jee).

The combination, logistic ogive, to identify the logistic curve is recent. George Udny Yule discusses the logistic curve at length in his 1924 Presidential Address to the Royal Statistical Society (J. Roy. Stat. Soc. 88:1-62, 1925). This address includes tables of the numerical values of the logistic function, and suggests three methods for estimating the parameters of a logistic curve. But Yule does not use the term ogive. W.D. Ashton also fails to mention ogive in his 1972 monograph on the logistic curve, entitled "The Logit Transformation" (London: Griffin). Perhaps it is Rasch's student, Erling Andersen, who introduces the use, logistic ogive (Psychometrika 42:69-81, 1977).

Why logistic ogive and not autocatalytic curve?, J Linacre … Rasch Measurement Transactions, 1993, 6:4 p. 260-1

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