Calculating the Volume of an Egg: A Study by F.W. Preston | Study notes Algebra

THE VOLUME OF AN EGG

F. W. PRESTON

IN MEMORIAM, F•A•c•s JosEPIt ASlIBY, 1895-1972

with whom I collected my first birds' eggs June 8, 1906, in central

England, shared my first seafowl eggs from North Wales a little

later, and with whom I went birding and bird's-nesting for 60 years.

RECENTLY I have received several inquiries as to how one calculates the

volume of an egg from its "dimensions," and by dimensions is meant the

length and the maximum breadth. The answer is that it cannot be done

with any real accuracy on the basis of only two measurements. I have

shown (Preston 1969) that the "shape," i.e. the longitudinal contour, of an

egg and its size can be described with a high order of accuracy by means of

four parameters, length, breadth, asymmetry, and bicone, and that it can-

not usually be described with less. The contour determines the volume, and

hence volume cannot be estimated from two measurements only.

In cross section an egg is remarkably circular. It is therefore legitimate

to consider an egg as a "surface of revolution," and this assumption is

always made. An egg lies between two simple geometrical figures, a cylin-

der and a true bicone.

In Figure 1A, we show a cylinder of length L (= 2b) and diameter B

(---- 2a). Its volume is (,r/4) ß LB 2 or 2,rba 2

An ellipsoidal egg of length L and diameter B would lie entirely inside

the cylinder, touching the centers of both ends and making contact with

the cylindrical surface on a circle.

In Figure lB, we show a bicone. Its volume is (,r/12) ß LB 2. Our egg

would lie entirely outside the surfaces of the bicone, and, if symmetrical, it

would pass through the apices of the two cones and would touch the bases

of the cones all the way round.

If our egg is asymmetrical, it would touch the same points and places of

the bicone in Figure 1C, whose volume is still (?/12) ß LB 2.

We may surmise therefore that in real eggs, which lie between Figures

1A and 1C, asymmetry (the extent to which one end is bigger or blunter

than the other) makes little difference to volume, but bicone makes a

great deal, the coefficient of LB 2 varying somewhat but lying between

(,r/4) and (*r/12).

The preliminary assumption of the early writers is that the egg is, to a

first approximation, an ellipsoid of revolution. If so, its volume would be

(,r/6) ß LB 2. The approximation is sometimes quite good, and taking

ß r = (22/7), the formula becomes

132 The Auk 91: 132-138. January 1974

Calculating the Volume of an Egg: A Study by F.W. Preston, Study notes of Algebra

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