# How did Quetelet discover that the body mass is proportional to the squared height?

The Body Mass Index (BMI) compares body masses on the assumption they scale with height squared, not cubed, an example of allometry. BMI is due to Lambert Quetelet. Why did he settle on this power law? Was it a theoretical case such as conserving pressure, or an empirical exponent?

BMI is now widely used for detecting obesity, but Quetelet's motivation was in defining the characteristics of an ‘average man’. Quetelet was one of the early enthusiasts of what we now call statistical studies and aspired to extend statistical analysis beyond demographic and anthropometric characteristics to other aptitudes, associated with behaviour, and even mind. A good scientific biography is Eknoyan's Adolphe Quetelet (1796–1874) — the average man and indices of obesity.

In 1831–1832, at the start of his career, Quetelet conducted the first cross-sectional study of newborns and children based on height and weight, and extended it to the study of adults. The results were published in the Proceedings of the Academy of Sciences as ‘Recherches sur le poids de l’homme aux différent âges’ (Research on the weight of man at different ages, 1832), where the index is first proposed (he did not call it BMI). The empirical weight to height distribution did not really fit into a Gaussian curve.

Based on his studies, in 1835, Quetelet published a book, A Treatise on Man subtitled Essays on Social Physics , in three volumes (renamed into Social Physics or an Assay on the Development of the Faculties of Man in 1869). Chapter 2 of the second volume explains his motivation for the index, which is clearly empirical:

"If man increased equally in all dimensions, his weight at different ages would be as the cube of his height. Now, this is not what we really observe. The increase of weight is slower, except during the first year after birth; then the proportion we have just pointed out is pretty regularly observed. But after this period, and until near the age of puberty, weight increases nearly as the square of the height. The development of weight again becomes very rapid at puberty, and almost stops after the twenty-fifth year. In general, we do not err much when we assume that during development the squares of the weight at different ages are as the fifth powers of the height; which naturally leads to this conclusion, in supporting the specific gravity constant, that the transverse growth of man is less than the vertical".

The index, known to the experts as the Quetelet Index, remained obscure for over a century, until the relation between body weight and mortality, cardiac disease and diabetes, became a medical concern after World War II. The new studies in the 1960s also suggested that the normal body weight in kilograms was proportional to the square of the height in meters, as Quetelet observed. Keys introduced the name Body Mass Index in a 1972 comparative study of available indices, that confirmed the validity of Quetelet's observation.

• Interesting that the exponent could also be 2.5 depending on the dataset.
– J.G.
Dec 30, 2018 at 9:46