THE
VOLUME
F. W.
OF
AN
EGG
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.
RECENTLYI have receivedseveralinquiriesas to how one calculatesthe
volumeof an egg from its "dimensions,"and by dimensionsis meant the
length and the maximum breadth. The answer is that it cannot be done
with any real accuracyon the basisof only two measurements.I have
shown(Preston1969) that the "shape,"i.e. the longitudinalcontour,of an
eggand its sizecan be describedwith a high order of accuracyby meansof
four parameters,length,breadth,asymmetry,and bicone,and that it cannot usuallybe describedwith less. The contourdeterminesthe volume,and
hencevolumecannotbe estimatedfrom two measurements
only.
In crosssectionan eggis remarkablycircular. It is thereforelegitimate
to consideran egg as a "surface of revolution," and this assumptionis
alwaysmade. An egglies betweentwo simplegeometricalfigures,a cylinder and a true bicone.
In Figure 1A, we showa cylinder of length L (= 2b) and diameter B
(----2a). Its volumeis (,r/4) ßLB 2 or 2,rba2
An ellipsoidalegg of length L and diameter B would lie entirely inside
the cylinder,touchingthe centersof both endsand making contactwith
the cylindrical surfaceon a circle.
In Figure lB, we showa bicone. Its volumeis (,r/12) ßLB 2. Our egg
would lie entirely outsidethe surfacesof the bicone,and, if symmetrical,it
would passthroughthe apicesof the two conesand would touch the bases
of the conesall the way round.
If our eggis asymmetrical,it would touchthe samepointsand placesof
the biconein Figure 1C, whosevolumeis still (?/12) ßLB 2.
We may surmisethereforethat in real eggs,which lie betweenFigures
1A and 1C, asymmetry(the extent to which one end is biggeror blunter
than the other) makeslittle differenceto volume,but biconemakes a
great deal, the coefficientof LB 2 varying somewhatbut lying between
(,r/4) and (*r/12).
The preliminary assumptionof the early writers is that the egg is, to a
first approximation,an ellipsoidof revolution. If so, its volume would be
(,r/6) ßLB 2. The approximationis sometimesquite good, and taking
ßr = (22/7), the formulabecomes
132
The Auk 91: 132-138. January 1974
January 1974]
Volume o] an Egg
133
• ELLIPSE
//
Figure 1. A, a cylinderof length L (• 2b) and diameterB (----2a) may be regarded
as circumscribingany egg,even a hummingbird's.B, a bicone (two conesbaseto base)
will lie inside any egg, even a tinamou's,which will circumscribeit. C, the two cones
do not needto be identicalin height,though they must have the basein common. D, a
circle circumscribesan ellipse, touching only at the ends or poles. The "eccentric
angle" defines a parameter in terms of which the x and y coordinatesof the ellipse, or
of the oval, can be expressed.
11
v =-
21
ßLB 2
(1)
a formulausedby someearlier writers.
However the approximationis sometimesnot good. The hummingbirds
(see figure in Preston1969) lay blunt-endedeggshalfway betweenthe
ellipsoidand the cylinder,so the coefficientof LB 2 is muchhigher than
11/21, while the grebesand tinamouslay eggsbetweenthe ellipsoidand
134
F.W. Pa•s•o•
[Auk, Vol. 91
the bicone,i.e. they are pointed at both endslike an American or Rugby
football, so the coefficientis lessthan 0r/6) (= 11/21). For a more de-
tailed discussion
of the inadequacies
of variousproposedformulae see
Barth (1953).
The questionthereforeis, in the formulaV = k LBs, what is the appropriate value of k? Obviouslythere is no one value that will fit all species.
Indeeda singlevaluewill not fit all species
of a singlefamily, and in some
clutchesI haveseenit will not fit all eggsof a singleclutch.
BelowI givea mathematical
treatment
in moredetail.
As in Figure 1D, draw a circle circumscribing
the egg,i.e. touchingthe
two endsof the egg. From the center of the circle draw any radius of the
circle,makingan angle(90ø- 0) with the long axisof the egg,till it meets
the peripheryof the circle at a point P• (shownbut not marked in Figure
1D). From P• draw a horizontal line to meet the oval or ellipseat point Pa
(again, shownbut not marked in Figure 1D). Then the coordinates(x, y)
of Pa can be specifiedin termsof the angle 0, which is called the "eccentric
angle," (see Preston 1953) and of the length and (equatorial) breadth of
the egg.
DIGEST
The parametric equation of the longitudinal sectionof an egg may be
taken (v. Figure 1D) as
y==abcos
x
sin
00(1+c•sin
0+casin
a0+etc.)
} (see
Preston
1953)(2)
where 0 is the "eccentricangle," a is the semidiameterat the true equator
(i.e. halfwaybetweenthe two endsof the egg), b is the half-lengthof the
egg, c• and ca are coefficientsthat vary from egg to egg and have to be
found experimentally,and the terms labelled "+ etc." can usually be
neglected.c• and ca are usuallyquite small,so that c•a, c2a, and c•cacan
be neglected.
Slicingthe eggparallel to the equator (perpendicularto the long axis of
the egg) into small thicknessesdy, givesus variouselementsof volume
dV = •rxa ß dy
and the total volume of the egg is
V=f_•r
x•dy
Ignoringtermsthat includenegligiblecoefficients,we have
xa ----aacos
a 0 (1 + 2c•sin0 + 2casin
a 0)
(3)
January 1974]
Volume o! an Egg
135
and of course we have
dy--bcos0'd0
So
V:
•r a2b
.cosaO(l+2qsinO+2c2sin20)
r/2
dO
(4)
•'-•r/2
The completeintegral from •/2 to +•r/2 of the middle term vanishes
(being-cos4 O) and the integral reducesto
V = • a2b
(cosa 0 + 2c2cos
a 0 sin2 O) d 0
r/2
(4a)
• -•/2
and by writing cosa 0 = cos0( 1 - sin2 0), this integratesto
2 )
(S)
If the length of the eggis L (: 2b) and its equatorial (not necessarily
maximum)breadthis B (= 2a) this equationtakesthe form
V=5'LB
2(2)
1+•c2
6
(5a)
If c• is zerothis reducesto V = (•/6) ßLB 2, the volumeof an ellipsoid
of revolution,and it doesnot dependon c• at all, providedwe were justified in assumingc• is comparativelysmall and q2 negligible. c2 can be
eitherpositiveor negative. With mostspeciesand individualparents,c2 is
negative,so the volume of the egg is less than the volume of the circumscribingellipsoid. But •th hummingbirdsand someothersit is positive,
and the volume is then more than that of the ellipsoid.
EFFECTOFUSINGBmaxINSTEADOF Bequatorial
x= b
a cos
0 ( 1 + qsin 0 + c2sin
20 }
inthe
parametric
equation
y:
sin
0
the maximumvalue of x is obtainedwhen dx/dy (not dy/dx) is zero, or,
what is just as good,when dx/d0 = 0.
Let 0mbe the value of 0 that makes x a maximum. If cx = c2 ----0, the
equationof an ellipse,we get
dx
--=
dO
-a sin 0, and this is zerowhen 0----0:
(6)
and this is a correct solution.
Now let c2 = 0 but let c• be non-zero. Then, rememberingthat sin 0mis
assumedsmall and thereforethat cos0mis very near unity, we get
136
F.W. PRESTON
[Auk, Vol. 91
-1 +•/1 + 4cxa
sin 0m•---
(7)
Rememberingthat cx is much lessthan unity, the squareroot term is
very nearly (1 + 2cx2),so that
sin 0m= csvery nearly.
(7a)
If cxandcaarebothnon-zero,
but cos0mis verynearunity (sin0mbeing
small), we get a cubicequationfor sin 0 as follows:
c2sin
s 0m•- cssin
• 0m+ (1 -- 2c2)sin0m- C1= 0
(8)
This canbe solvedexactly,but if sin0mis small (it is often lessthan 0.1),
we can neglectthe term in sins 0mand get a simplequadratic,whosesolution is
sin 0m=
-(1 - 2c2)+ •/(1 - 2c2)2 + 4%2
2c•
(8a)
We may note that when c• and c2 are small (and cs tends to average
about0.1 and c2about-0.1)
sin 0mtendsto be aboutcs,independentlyof the valueof c2.
(9)
This locatesthe positionof the maximumdiameter. The value of that
diameter
is
Bmax
= 2Xmax
----2a COS
0n•(1 + cssin0m+ c•sin• 0m)
= 2a ßX/1- cs'• ß (1 + cs2 + C2C12)
or
Bmax
B -(1 _ C12•
ignoringc2c•2 as very small,and so,remembering
our previouscommenton
the square root term,
Bmax
= 1+c12
B
2
(10)
For instanceif c½= 0.1, Bm•xexceedsB by about one-halfof one percent.
ERRORS
IN ESTIMATING
VOLUMEEROMTHE Two DIMENSIONS,
L and B•nax
If an experimentermeasures
the lengthand maximumdiameterof an egg
and calculatesits volumeby the ellipsoidalformula as
V = -•
ßLB2m•x
6
January 1974]
Volume of an Egg
137
he will overestimatethe volume in the ratio (Braax/B)2, that is (from
equation 10) in the ratio
(1q-c•-)
2or1q-cx
2approximately
(11)
andif cx= 0.1, the erroris 1%, whileif cx: 0.2 (a valueit reachesin only
a few families) the error will be 4%.
But this is not the most serious error.
A moreimportant error is that due to the biconeterm. Sincethis is usu-
ally negative,it will againproducean overestimate,
of value (2/5)c2.
Thus if c2--- 0.1, the overestimateis 4%. In the caseof the Oystercatcher
of Preston1953 the error wouldbe 7%. In the caseof the hummingbirdor
albatross
of Preston1969 (p. 259), the errorwouldbe a very substantial
underestimate.
Worth (1940) considersthat it is closeenoughto assume,for any egg,
that its volumeis 15%lessthan that givenby the ellipsoidalformula,and
this I cannot concede.
It is no use trying to refine the estimate of the volume of an egg by
modifyingarbitrarily the coefficientk in the expression
V = k LB2raax
from its ellipsoidalvalue of •-/6. Errors of 5% or more can readily occur.
They arisefrom assumingthat the shapeof an eggis simplerthan it really
is. Thereare not enoughdataor parameters
in the equation.
THE
INTERNAL
AND EXTERNAL
VOLUMES OF AN EGG
Somewritersare concerned
with the overall (or external)volumeof an
egg (i.e. includingthe shell) and someare concernedmorewith the internal
volume,excludingthe shell.
Stonehouse(1966) is concernedwith the total, external, volume, and
with how far it departsfrom the ellipsoidalvolumeV ----0.524LB •. For a
numberof seabirds'eggshe found an averageof V = 0.51LB 2, and just
about the same for the Australian Black Swan, Cygnus atratus. This is
about $% ]essthan the ellipsoidalvalue, and is due chiefly to the "negative bicone,"the eggsbeing "somewhatmorepointed at the ends" than a
true ellipsoid.Even in a singlespecies,the swan,the degreeof pointedness
varied and the deficiencyof volumeof coursevaried with it.
Cou]son(1963) was interestedin estimatingthe internal volumefrom
the externalmeasurements,
and concludedthat the internal volume averagedabout 0.487 LB •, which is about 9.3% lessthan the (external) ellipsoidalformula wouldgive. If we assumethat the biconeaccountsfor about
3%, there remainsabout 6% that must be due to the shell thickness.
138
F.W. PRESTON
[Auk, Vol. 91
It is easily shown that if the egg shell thicknessis t and the average ex-
ternal diameterof the eggis d, whered = x•/LB2, then the internalvolume
falls short of the external volume by approximately600 t/d percent.
I measuredone egg of the Gallus gallus, the domesticfowl, and found d
to be about 1.9" (= 48 mm) and the shellthicknessaverageabout 0.015"
(= 0.38 mm) so that 600 t/d is about 4.7%.
I also measuredone eggof Numida meleagris,the crownedor helmeted
guineafowl, a specieswhoseeggsare notoriouslythick-shelled.This was a
domesticspecimen,the eggapparentlya trifle lessin breadth, thoughnot
in length,than the averagewild eggin SouthAfrica. I foundd to be about
1.62" (41 mm) and t averagedabout0.022" (0.56 mm), so600 t/d = 8.2%.
Coulson'sKittiwake eggswere thereforeintermediate,in relative shell
thickness,betweenGallus and Numida.
Coulsonwasinterestedin estimatingthe age compositionof a colonyof
gullsby noting that olderbirds tendedto lay bigger,that is more voluminouseggs,but I think he couldhave used the breadth as effectively as the
volume.
Worth (1940) wasinterested
in the problemas to whetheronecould
estimatethe lengthof the incubationperiodif given the volumeof the egg.
Lack (1968) discussed
thispointandconcluded
that the correlationis poor,
and this agreeswith my own, lessextensive,computations.
ACKN'OWLr. DG:•v•EN'TS
I am indebted to A. Robert Short for a preliminary check on the algebra, and in
particular for checkingby actual solution of the equations that, assumingc•----O.1 and
c2-----O.1, the cubic equation (8) and the quadratic (Sa) give virtually identical
answers. I am indebted also to J. L. Glathart for looking over the paper. •' did not
ask either of them to check every detail. •' am also in the debt of Lloyd F. Kiff and
Charles H. Blake for several commentsand suggestions.
L•Tr•ATURE
BART• E.K.
bation.
C•TrD
1953. Calculation of egg volume based on loss of weight during incuAuk 70: 151-159.
CouLso•, J. C. 1963. Egg size and shape in the Kittiwake (R]ssa tridactyla) and
their use in estimating age composition of populations. Proc. Zool. Soc. London
140:211-227
LacK, D. 1968. Ecologicaladaptationsfor breedingin birds. London, Methuen &
Co., Ltd.
P•zsToN, F.W.
1953. The shapesof birds' eggs. Auk 70: 160-182.
P•STON, F. W. 1969. Shapesof birds' eggs: extant North American families. Auk
86: 246-264.
SToN•ous•,
Wo•Ta, C.B.
B.
1966. Egg volumes from linear dimensions. Emu 65: 227-228.
1940. Egg volumesand incubationperiods. Auk 57: 44-60.
Box 49, Meridian Station,Butler, Pennsylvania16001.Accepted15 May
1973.