This is an analysis of the proportions between the various dimensions of a quadruped body as a function of size. Small animals such as cats have a thin, lithe shape with relatively thin legs whereas elephants have a boxy shape with relatively thick legs. The legs whose strengths are proportional to their cross section have to hold up a weight which is proportion to the body volume. If animals had the same shape then there would be a conflict between the weight increasing with the cube of the scale whereas the strength of the legs increasing only with the square of the scale. But animal shape is not constant and changes with scale.

Length of the Torso of the Body

Consider a quadruped's torso as being made up of a horizontal square prism held up by four vertical square prisms (the legs).

The most convenient scale parameter to use for the analysis is the height amd width of the body prism, which will be denoted as λ. (The analysis would be essentially the same if the cross section of the body were taken to be circulars rather than quare. In the analysis the exponents are the crucial elements; the coefficients are not crucial.)

The cross section area of the square prism is then λ². The amount of weight per unit length the spinal column must support is then D=ρgλ², where ρg is the weight density of the body material.

The maximum stress occurs at the midpoint of the spinal column between the support elements above the front and rear legs. Let B be the length of the spinal column between these supporting structures. Let x be the distance from the nearest support. The moment at the midpoint is

M = ∫₀^L/2Dxdx = D(1/2)(B/2)²
= DB²/8

This moment has to be counteracted by the stress in the spinal column. Assume the spinal column has a square cross section of X units on each side. The horizontal spinal column of a quadruped is subjected to compressive stress at the top and tension at the bottom. The strain and hence the stress is proportional to the distance z from the midlevel of the cross section. Thus the stress is equal to kz and the moment due to the stress at z is kz². The area of the infinitesimal element at z is Xdz, where X is the width of the column. The moment generated by this stress is

∫_-X/2^X/2(kz)zXdz = 2Xk(X/2)³/3
= kX⁴/12

The constant k has to such that the moment generated by the stress in the spinal column counterbalances the moment generated from the load of body weight on the column; i.e.,

kX⁴ = DB²/8
so
k = 3DB²/(2X⁴)

The maximum stress is at z=X/2 so

T_max = k(X/2) = (3DB²/(2X⁴))(X/2) = 3DB²/(4X³)).

The length B must be such that the maximum stress in the spine does not exceed some maximum allowable stress T. This means that

B² = (4/3)DT/X³

But D is proportional to λ² and X is proportional to λ so B² is proportional to λ and hence

B is proportional to λ^1/2 .

This means that if body thickness is doubled the body length increases not by 100% but by 41% instead. Thus the animal shape get boxier as the scale increases. Another way of stating this is in terms of the ratio B/λ:

B/λ is inversely proportional to λ^1/2
B/λ = α/λ^1/2.

Leg Thickness

The volume and hence the weight, which is proportional to Bλ², is proportional to λ^5/2. Therefore the cross section of the legs has to be proportional to λ^5/2 and hence the thickness of the legs W has to be proportional to λ^5/4; i.e.,

W = βλ^5/4
and hence
W/λ = βλ^1/4

Thus larger scale quadrupeds' legs are relatively thicker.

Although the thickness of the body is the most convenient parameter for analysis people more commonly think of a body length, such as B, as the measure of animal size. In terms of B the thickness of a quadruped's leg is proportional to B^5/2. This relationship captures the perception of the thickness of a quadruped's legs as a function of quadruped size.

But total body length is the sum of body length B plus neck length N plus head size H. The analysis for neck length N is a bit more complicated than that for body length B because the stress created at the shoulders depends not only on the weight distributed along the neck but also on the weight of the head.

Neck Length

The moment created by weight at a distance x from the shoulders is Exdx, where E is the linear density of weight along the neck and is porportional to λ². The total moment due to this neck weight is the integral from 0 to N; i.e., EN²/2. The moment created by the weight of the head is the head weight, which is proportional to H³, times the lever arm for the head, which is N+H/2. The stress parameter k in the spine at the shoulders has to satisfy the equation

kX⁴/12 = EN²/2 + (N+H/2)ρgH³

where ρg is the weight density of body material. Again the maximum stress is kX/2 so the neck length N has to be such that the maximum stress is equal to the allowable stress T; i.e.,

[3EN² + 6(N+H/2)ρgH³]/X³ = T.

Since E is proportional to λ² and X and H are proportional to λ the equation to be satisfied by N is of the form

c₀N²/λ + c₁N + c₂λ - T = 0.

where the c_i are coefficients independent of λ. This equation has solutions of the form

N = [-c₁ ±(c₀² - (4c₀/λ)(c₂λ-T))^1/2](λ/(2c₀))
which reduces to
N = λ[-(c₁/2c₀) ± (c₁²-4c₀c₂ + 4c₀T/λ)^1/2/(2c₀)]

This means, roughly, that

N = d₀λ±d₁λ^1/2
and
N/λ = d₀ ± d₁/λ^1/2

with d₀ being more important if the neck is thin and d₁ more important if the neck is thick and the head relatively small.

There are two solutions and it appears that the one involving a negative sign is the empirically relevant one. Therefore

N = d₀λ−d₁λ^1/2

This means the neck becomes relatively shorter as scale increases.

Leg Length

The other element of animal shape is the length of the legs relative to the body size. Some of the factors which influence leg length are the need for providing clearance of the underside of the body above vegetation and the need for compatability of leg length and neck length for ground grazing. But longer legs also create stability problems. For ground grazing the leg length L (from the ground to the bottom of the body) has to be such that

L + λ = N + H.

This would mean that leg length would be of the form

e₀λ−e₁λ^1/2
and hence
L/λ = e₀−e₁/λ^1/2 .

Thus leg length is relatively shorter with increases in scale.

Leg Thickness Again

The previous analysis derived the relationship between leg thickness and scale under the assumption that the legs had only to support the torso body weight. When the weight of the neck and head are also taken into account the relationship will be more complex in form but still will indicate that the legs must be relatively thicker for a larger scale quadruped.

Alternative Shapes

There are obviously alternative viable shapes. Some animal shapes involve a shorter neck and the forgoing of ground grazing for bush and tree grazing as in the case of elephants. The elephan trunk provides a means of compensating for restrictions of a relatively shorter neck. The giraffe represents another strategy concerning grazing.

The display below depicts the effect of scale on animal shape. The first silhouette represents an animal of the size of a cow. The second reddish silhouette represents an animal whose scale parameter of body thickness is one half that of the first, roughly the size of a large dog. The grayish third silhouette is for an animal whose body thickness is twice that of the first, roughly the size of a small elephant.

The second silhouette shows the relatively elongated shape and the relatively thinner legs. The third shows the relative boxiness of the body and the relatively thicker legs.