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

The methods for calculating the volume of an egg based on its dimensions, specifically length and maximum breadth. The author, F.W. Preston, explains that the shape of an egg and its size cannot be accurately described with less than four parameters: length, breadth, asymmetry, and bicone. He also discusses the assumption of an egg as a surface of revolution and the impact of asymmetry and bicone on the egg's volume.

What you will learn

  • How can the volume of an egg be calculated from its length and maximum breadth?
  • What is the significance of the shape of an egg in determining its volume?
  • How does asymmetry and bicone affect the volume of an egg?

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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
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THE VOLUME OF AN EGG

F. W. PRESTON

IN MEMORIAM, F•A•c•s JosEPIt ASlIBY, 1895- 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 can- not usuallybe describedwith less. The contourdeterminesthe volume,and hencevolumecannotbe estimatedfrom two measurementsonly. In crosssectionan eggis remarkablycircular. It is thereforelegitimate to consideran egg as a "surface of revolution," and this assumptionis alwaysmade. An egglies betweentwo simplegeometricalfigures,a cylin- der 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,rba 2 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

January 1974] Volume o! an Egg 135

and of course we have

So

dy--bcos0'd

V: •r a2b •'-•r/2 r/2.cosaO(l+2qsinO+2c2sin20) dO (4)

The completeintegral from •/2 to +•r/2 of the middle term vanishes (being-cos4 O) and the integral reducesto

V = • a2b • -•/2 r/2(cosa 0 + 2c2cosa 0 sin2 O) d 0 (4a)

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 6 2 1+•c2 (2) (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 justi- fied 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 circum- scribingellipsoid. But •th hummingbirdsand someothersit is positive, and the volume is then more than that of the ellipsoid.

EFFECTOFUSINGBmaxINSTEADOF Bequatorial

intheparametricequationy:bx = a cos0 ( 1 + qsin 0 + c2sinsin 0 20 }

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:

c2sins 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,whosesolu- tion is

-(1 - 2c2)+ •/(1 - 2c2) 2 + 4% sin 0m= (8a) 2c•

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 COS0n•(1 + cssin0m+ c•sin• 0m) = 2a ßX/1- cs'• ß (1 + cs2 + C2C12) or

Bmax _ C12• B -(

ignoringc2c•2 as very small,and so,rememberingour previouscommenton the square root term,

BmaxB = 1+c12 2 (10)

For instanceif c½= 0.1, Bm•xexceedsB by about one-halfof one per- cent.

ERRORSIN ESTIMATINGVOLUMEEROMTHE Two DIMENSIONS,L and B•nax If an experimentermeasuresthe lengthand maximumdiameterof an egg and calculatesits volumeby the ellipsoidalformula as

V = -• ßLB2m•x 6

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 volumi- nouseggs,but I think he couldhave used the breadth as effectively as the volume.

Worth (1940) wasinterestedin the problemas to whetheronecould

estimatethe lengthof the incubationperiodif given the volumeof the egg. Lack (1968) discussedthispointandconcludedthat 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 C•TrD BART• E.K. 1953. Calculation of egg volume based on loss of weight during incu- bation. Auk 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- 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•, B. 1966. Egg volumes from linear dimensions. Emu 65: 227-228. Wo•Ta, C.B. 1940. Egg volumesand incubationperiods. Auk 57: 44-60. Box 49, Meridian Station,Butler, Pennsylvania16001.Accepted15 May