Da Vinci and the Square Man

The Roman architect Vitruvius was the first to state some simple proportions supposed to exist in "the" human body. Some of his claims are blatantly wrong, and his rule "foot length = 1/6 of body height" was changed by da Vinci into the more reasonable 1/7. Some other claims, like "arm span = body height", seem very appropriate though, and many renaissance artists besides da Vinci have provided illustrations of the homo ad quadratum. The woodcut below is by Cesare Cesariano (1521 Vitruvius translation). 

Being unambiguous and easy to measure, the arm-span claim is perfect for investigation and, for once, we had an easy task finding scientific sources to quote from. Many well-documented studies are directly available on the internet—drag for "arm span, height'—, because there is some medical interest in deducing body height from other measurements. Arm length does prove to be the most reliable measurement to use, but the correlation turns out not to be 1:1.

Let's first agree that the claim is a statistical one. No one believes that "people" have feet 1/7 of their body height, because there are people with small feet and people with large feet. Likewise, there are people with short or long legs, with short or long arms. Sure enough, actual experiments result in a cloud of data, not in the platonic straight line y=x which would follow from exact and universal equality between the two measures. Moreover, due to the intrinsic limitations on precision, data are not points ("is equal to this number") but intervals ("lies between this number and that number"). Here (p. 40) an example from 2011, dealing with some 300 Nigerians, between 20 and 50 years old:

The experiments, which have been done worldwide, over decades, and for different sexes, races, ages and even for some very specific populations, agree on this: 

statistically speaking, arm span exceeds body height, to a degree which is highly dependent on sex and race, and somewhat dependent on age.

As early as 1987, a study of young women of two races revealed that, on average, arm span exceeded height by 1.8 cm for the white sample, and by 8.3 cm for the black sample (here and here). Meanwhile everybody agrees that a universal correlation is of no use, and that different formulas are required for the sexes and the races. A nice overview is to be found here (Table 1). There we also find the latest, and best documented, of the correlation formulas. For white males between 20 and 80 years old, body heigh H, arm span AS (both in centimeters) and age A (in years) are correlated by the formula

H = 54.1 + 0,70 AS - 0.08 A,

found to be in very close agreement (well, as good as it gets) with the actual measurements. (If Vitruvius were right, the formula would simply be H = AS.) 

To qualify as a model for da Vinci's Square Man, one must satisfy 

H = 54.1 + 0,70 H - 0.08 A,

i.e.

H = 180.33 - 0.26 A.

Plugging in A=20 and A=80, we find that only those between 1.59 m and 1.75 m stand a chance, and they have to apply at a very specific age: those of 1.75 m as early as 20, while professional models of 1.59 m have to wait until they're 80. All others would shamefully see their fingertips protrude from the square!

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