A mathematical formula for child proportion
The one proportion which appears to be common to the adult and the child is the proportion of the trunk to the total height: in simple whole number fractions, about 3/8.
However, at birth the head is about a quarter of the total height, whereas in Leonardo's ideal man the head is an eighth of the total height.
It follows that in Leonardo's adult male the head plus the trunk makes up 1/8 + 3/8 = 1/2 of the total height, leaving 1/2 for the lower limbs. In the baby, the head plus the trunk make up 1/4 + 3/8 = 5/8, leaving only 3/8 for the lower limbs.
In the example I have drawn here, the head is a little less than 1/4 and the lower limbs a little more than 3/8, corresponding to a typical putto of about 3 years old rather than a newborn baby.
Thus, by keeping the trunk at 3/8, making the head smaller, and correspondingly lengthening the lower limbs, we can make our child look older as much as we please. We may consider whether there is a formula allowing us to calculate the proportions of ages in between.
Why have such a formula? One may point out that real people's body proportions vary enormously, and that the representation of the ideal has fallen out of fashion. This is true. However, in learning a standard one has an instrument by which to measure the reality. For example, it is a common error in drawing a face to make the forehead not deep enough. If one has a formula that says that the ridge of the orbit just above the eyes lies about half way between the chin and the top of the head, one can measure ones drawing to see if it is way out on this account. One would also measure the person sitting in front of one, to see to what extent their proportions vary from the expected ones. Some artists prefer to rely on measurement only, but this too can be inexact and misleading. Why not use an additional tool that may be of some service?
Notes to the assumptions:
By assumption ii, we may now feed these data into a standard linear equation, as follows:
Let H be the total height in number of heads, and
let a be the age in years.
The general formula for any linear equation is:
y = mx + c
where m is the gradient and c is a constant.
Our y is the height in heads H, and our x is the age of the child a. Thus our general equation becomes: H = m.a + c
To find c we set a=0, to obtain:
H = m.0 + c
c = H when a = 0
Now we already said that the height H of a baby aged 0 years is 4 heads (assumption i), so we get:
c = 4
(Note that c is a constant that applies whatever the values taken by the variables: for the technically minded, it represents the intercept on the y-axis.)
Now we substitute this value for c back into the equation, this time for the case of the 18-year-old. So we have a=18 and H=8 (by assumption iii above), thus:
8 = m.18 + 4
Using the usual Islamic methods (so who do you think invented algebra?) we rearrange to get:
m = (8 - 4) / 18
which simplifies to:
m = 2 / 9
Substituting this back into our original equation: H = m.a + c
H = 2a/9 + 4
So for example, if the age of a child a is 9, the child will be H= 2x9/9+4= 6 heads in height.
Once again applying the cunning of Muhammad ibn Musa al Khwarizmi, we may rearrange the equation so as to be able to estimate the age of a child in a painting. We count the number of heads H that would make up the total height, and substitute into the formula to get the age a:
a = 9(H - 4)/2
to Human Proportion I
to Child Proportion 1 the constancy of the trunk measurement and its relation to the divine ratio.
to Child Proportion 3: application of the formula to calculating the age of the Infanta Margarita Teresa in one of Velasquez's portraits of her, and constructing a convincing nude study.
to Margarita Teresa: work in progress on a study for a copy of Velásquez's painting.
The author asserts intellectual property rights regarding images text and content 2003 Martin Dace