r ,\ "

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UNIVERSITY OF MISSOURI COLLEGE OF AGRICULTURE

AGRICULTURAL EXPERIMENT ST�TION

RESEARCH BULLETIN 115

GROWTH AND DEVELOPMENT lVil1l Special Reference 10 Domeslic Animals'

XI. Further Investigations on Surface Area With Special Reference to Its Significance in Energy Metabolism

(Publicatioll authorized Jan. 12, 1928)

COLUMBIA, MISSOURI

MARCH, 1928

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UNIVERSITY OF MISSOURI COLLEGE OF AGRICULTURE

Agricultural Experiment Station EXECUTIVE BOARD OF CURATORS.-F. M. McDAVID, SllringficlJ; MERCER

ARNOLD, loplin; H. J. I3I.ANTON, Paril

ADVISORY COUNCIL-THE r.,·flSSOURI STAh: BOARD Ofo' AGUleUI.TUItE

STATtoN STAFF, MARCH, 1928 STRATTON DUI.UTH BROOKS, A.M., LL.D., Pre,ide"t

•

F. B. lv1UI\U'ORD. 1'1'1.5., D.Agr., Director S. B. SHiRKY, A.M., Alit. 10 Director

MISS ELLA PAl-lfI.·IEIER, Secretary

AGRICULTURAL CHEMISTRY A G. 1-IOCAtI, Ph. D. L. D. HIlICII, Ph. D. W. S. RI"I'CIII�, Ph. D. A. R. It..LL, B. S. in Arr. ). E. IiUNT!!", A. M. C. I.. S.Ul:w,wuU', A. B. F... W. COWAI., A. fI..f. RUIIUT Boucllu. Jr., A. B.

AGRICUI.TURAL ECONOf..-IICS O. R. JOUI<$OI<, A. ,.,.1. S. D. G�o"I! •. A. M. Rtl< H. FIAOII(, A. Mt F. 1..1'''001511:11, Ph. D. G. n. Tuou<I!, It S. in Agr. PMESTOtl RU:;U.l.kU5 , B. S. iu Agr.

)08111'11 It. RdwfI.I., B. S. in A/lr.

AGIUCUL TUnAL ENGINEERING J. C. WOOl.".,., M. S. M .. cK M. JONKS, r.,·f. S. R. R. r .. MKS, n. S. in Agr. Eng.

ANIMAL HUSBANDRY E. A. TIIOW�M'''GK, B. S. in Agr.

�\'. �'. W��::�I>:�: 5.;11 Agr.

}-'. n. M"",VOMU, M. S. D. W Clllrn:,IIIi,;N, A. M. M. T. fOSTU, A. M. tUTU E. C .. sao ... A. M. II. C. f.,'lovrt:n, n. S. iu Agr. j .. E. CO'Ifon, n. S.ill Agr.

BOTANY AND PIIYSIOLOGY W.). RObbiNS, I'h. D. I. T. SCOTT. I'h. D.

DAlitY HUsnANDRY A. C. R,,(l�D ..... r. M. S. \V.I. II. E. R�IIl, A. M. S ..... )tL nIlO".,., A. M. C. W. TUIII<�., Ph. D. C. W. WEBU, 11. S. in Agr. rr. C. EI.TING, A. tI'l. W .. UKN GIFfO.D, A. M. rr. R. G .. IIMISON, B. S. ill Asr.

ENTOMOLOGY l."ON .. kl) IIASH ... N, Ph. D. K. C. SUI.I.IV"N, PI .. D.

"I" I�n'ice of U. S. [)ejlHlmenl of AGriculture

fo'IELD CROI'S W. C. ETIIUI"CI!, Ph. D. C. A. Ih:LIot. A. M. L.). S"UDLU, Ph. D. R. T. K.ur .. Tklc .... , A. M. 11. M. KU'G, A. M. "'·hn CLA"" Fun., M. S .•

HOME ECONOMICS MIU M""EL C""�UIlLL, A. M. f..·"ss h S SIIi AtlCIi C UNIl, ,\. M .• A. n. Mill lh , 1l1'u" K. \V1II�I'LIl, M. S. M.n M .. kc"Mt;T C. lIEULEk, Ph. D. ""hn E. C" .. MLo·rn,: IlOCK", A. M.

�::::: 8:�':G��' f·C�'���,D

n�s.S,

HORTICULTURE T.). T .. LIIl!R"r, A. M . If. D . lloouM, Ph. D. 11. G. SW .. RTWDIIT, A. M. j. T. QUII-lN, A. M. A. E. Ml)MNI!a:K, Ph. D.

I'OULTllY IIUSBANDRY H. L. KHlrSTI!k, n. S. ill A�r. E .. lu. W. H�NO�kSON, A. M.t E. M. FUNK, A. M.

RURAL SOCIOI.OGY E. L. MOllc"N, A. M. ��I�::ld�1���Ti�'.;"'��;01ll!, Ph. D.

SOILS M. F. MILLU, M. S. ,\. U. II. KlUSUOH. A. M. W. A. AURI!cln, I'h D. RlclI�"" BUOV'�I.", "h. D.I R. E. UII .... N!), A. M. F. L. D .. vls, B. S. in All', II .. HS hNNY, Ph. D. G�o. Z. Doous, A. M.

VrrTEIUNARY Scn:NCE J. w. Cu, ...... w ...... D. V. M., ",.1. D. O. S. CIt15l.U, D. V. M. A.}. DU .... HT, A. M., D. V. M. ANOIt�W U.tN, D. V. /1,.1. R. 1.. CIIOUC ... A. IJ., B. S. in Med

OTHER OFFICERS R. n. P�ICl!, B. I.., TreHluer ;.E�"'K COW"", B. S., Sec'y of Unrvcrlity A. A. Jl!;fFMI!Y. A. B . . Agri"IIIIIul EdilOr J. F. B ... " .. ", PhOlOsul,>ller M,n j .. NE FMOIISIL ... I, I.lbrarian E. E. BIIOWN, nll.intu MaoallCf

tOn leave of abltlLCC

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GROWTH AND DEVELOPMENT l,pith Special Reference to Domestic AnimalJ

XI. Further Investigations on Surface Area With Special Reference to Its Significance in Energy Metabolism

SAMUEL B RODY, JAMES E. CO .... 'FOKT, JOHN S. MAn'HEWS

PREFATORY NOTES AND ACKNOWLEDGMENTS

This hulletin includes JIITjtlU ill/tgTtllor measurelllerllS of surrace area of 482 dairy c;utle, 341 beef cattle, II horses, 16 swim:. (The surface inte�roHor and the /lie I hod of its lise are described in Research Oulletin 89 of this Surion.) The origin,,1 dOlt;1 cllrds include Ihe following entries: a�e, height, condition of fleshiness, area, weight, height at withers, circumference of chest, distance from shoulder to hips, width of hips. For financial reasons, it is nor practicable LO publish the nUIlH:ricOII data. For the use of those who lIIay desire 10 eXluniue the origillal data, the data cards will be kept on file at the Missouri Agricultural Experimcnt St:ltioll.

The dairy cattle were measured by). S. MatlhewS under Ihe supervision of Professor A. C. Ragsdale. The hed caltle, horses, and swiuc Were measured by J. E. Comfort under the supervision of Professor E. A. Trowbridge. Professor D. W. Cllittcntlen cooperated wi til the measurements of the horses.

A ponion of the expenses involved in this work was paid from a grallt of the N:ltionOiI Researcll COllncil for the Natiollal Live Stock and Meaf Board Fellow­ship FUlld. Gl"iitefLiI ;Icknowledgmellt is madt to the Council and to the Chairmall of the COllllllittee of National l.ive Stock and l'vh:at BO:lrd F'ellowships, National He­sellrch Council, (or recommending tlH! grant for this and rdated work in growth of domestic anim;lls.

ABSTRACT

I. Data are presented on d.le relation of surrace area LO body size o( 4111 dairy catlle (189 Holsteins, 154 AyrsJllres, 96 Jerseys, 43 Guernseys); 011 341 heef cattle (Shorthorns, 145 females, 54 males, '10 steers; Herefords, 69 (cm:des, 38 111:l1es, 7 steers; Angus, 8 steers), on I I horses, and 011 16 swine.

2. A 1ll00thematic:d (graphic:d) analysis is presented of these ori�inal data, as well as of the available publishtd data on the relation of IIrea to body size auJ on the relalion of hCHI production 10 body sizto

3. I t is showli that thc IIlIllleric;d value of the pOWer in tht pOwer function relating surface area to body weight vOirits frolll ahOllt 0.'1 to about 0.7 depend.ing: �II tin: relative variatiolls ill the linear siZe of the animals as compared 10 vari;llIoliS III hody weight. III this COllneCtiOn the formulae of rVleeh, of Du Bois :lI1d Du Bois, and of Cowgill and Drabkin are sut.jected to critical analysis; as is also the so-c;dled Surface Area Law of Huhncr. It is concluded, 011 IIlnthelllntical and on biologic:II grounds, that while it may be more convenient, and perhaps more enliglltening, to relale heHl production 10 surface area, it is simpler to relate heat production directly to hody si"e raised to sOllie power by 11 method explained ill detail ill die text.

l. FORMULAE RELATING SURFACE AREA TO BODY SIZE, WITH

SOME APPLICATIONS

This bulletin is a continuation of Missouri Research Bulletin 89 to which the reader is referred for the introductory notes.

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4 MISSOURI AGR. Exp. STA. RESEARCH BULLETIN lIS

Perhaps we hold Letter slale at the outset that we shnll make much lise of the (ollowing equations

or

and

or

A � CHmW" (1)

A� CW" (2)

E� CW" in which A is surface area and E is heat production (or body weight /F, and height H.

Equ:ltiolls (.I) and (2) are closely related, (or if ill equation (I), 111 = 0, then Il'" = I, and equation (1) becomes equation (2).

Equation (2), known to the mathematician as a power Junc/iou, was used iJy Mech (1879) to represent the relation hetween surface area and weight in man. Meeh assigned the value 7:3 to IT ill equation (2) 011 the basis of the assumption that the areas of bodies of similar shape are directly proportional to the squares of their linear size, and the volumes are directly proportional to the cubes of their linear size. \Vhen 11 = 7$, equation (2) is known as the Meeh surface-area formula.

Du Bois criticised the Meeh formulae in its application to lllall in the following words: "It is obvious that a tall, thin man may have exactly the same weight as a short, fat man, yet have a Illuch larger surf�lce :lrea." This objection finally led Du Bois and Du Bois (1916) to adopt equation (1) thus taking into consideration the height as well as the weight of the individual. Equation (1) is for this reason often spoken of as the Du Bois height-weight formub, that is, when 111 is taken to have the value 0.725, and II = 0.425.

Briefly, Du Buis and Du Bois (1915) employed the following reason­ing in arriving at the numerical values of 11/ and J/ for man: Since area is a bidimensional Illeasurement, it follows that in the equation relating area to size, the formula Illust be bidimensional 011 both sides of the equation. Since length is a unidimensional measurement, and weight is a triditllensional measurement, the equations

and A � CW" H (3)

// � CW" H" (4) may be given !"IS examples of formulae which are bidimensional on ooth sides. E(juations (3) and (4) were investigated for this purpose using the aC/Tlally measured (Ly the mold method) areas of nine subjects, and the computed (uy the "linear formula") areas of 33 subjects by solving for the constant, C, the values of m and 11 being assumed as in equations (3) "nd (4).

GlWWTII AND DEVELOPMENT xr 5

Since equations (3) and (4) were found to give errors opposite in sign, other values of the exponents between the two sets given in equations (3) and (4) were tried. In order to keep the formula bidimensional, it was put ill the for III

A = CW'hH'/b (5) and care taken that the Stllll of 1/ a and 1/b remained 2 as in equations (4) and (5). After trying several combinations of a and b, Du Bois and Du Bois concluded that the best agreements between observed and computed values are obtained when (l = 2.35 and b = 1.38. The final equation of Du Bois :llld Du Rois (1916) is therefore,

A = CTf7Ih.1f>HI/U8 � CIF"" H·m (6)

The average value of C was found to be 71.84. The well-known Du Bois prediction chart W:lS then prepared on the basis of equation (6).

Cowgill and Drabkin (1927) concluded that in equation (1) 111 = I . They arrived at this conclusion by the following reasoning. They related area to the product of body weight and a correction factor for body ouild (or state of nutrition) oy the formula

in which dis surface area

A = crr/n N", I'lob8 (or weight TV, and

(7)

N" the nutritional state N-· Ob8

Ntn represents the ratio of the cube root of weight to the linear measure-. W"

Illcnt (that IS, of T) for the most obese individual observed. NOb' rcp-

resents this ratio for the animal Hnder consideration.

In the words of these authors, formula (7) "is essenti:llly the J\·leeh.Rubnet ex· pr�ss!ol1 rnllltiplie�1 by � factor c.orrectint;: for the nutritive state of tile suuject. Tile pnrlClp!e l1ilo.1\ .wl11ch t!l1S fa�t?� 15 �ased .15 ,the same :'s lIlat eml�lored h}' "on Pirquel (1917) III lItrl":111g :�t 1115 f!r"�JjI., I'or Ullit IIlc�eases.1!1 leng�h (Slttlll!j; height or stem leng�h) ther� IS.Ulllt CXpal1Sl0n In the three dImenSions WI11Ch determine volume. If ' I,eclfic grav!ty IS :l;�srIHlcd to ue const:lnt, weight inste:ld of volu1I1e may be taken ns t.le f;,c�or WIllI whIch to COUlpare lengdl. Inasmuch as weight is a fUllctio'l of three d1ll1enSI?IlS, lhc cuue .root 0/ weight is the proper unit with which to compare unit chang,e In le.llgtlt. Jt IS ohvlOus tI!:lt tile v:lluc.of tile ratio should ue practically con. st:lllt III suhJccts of t,h� same species ,of approxlInately the S:lrlle :lge aud dlnr:lCteriz. ed by the salllC nutrrtlvc states. Jot IS also obvious that the values for this mtio will be higher ill obese individuals than in drill sub.iecrs."

The value of the exponent of IF, namely 0.70, was arrived at by a method of trial.

Cowgill and Drabkin point out that while their formula, equation (7), call ue simplified to yield the power function (I) (in which case m = 1), it is preferable to write the formula in the form of equation (7) "because of this advantage: its dt:-tracter as a combination of Meeh­Rubner-Dreyer ideas with a correction factor for the nutritive state of the subject is thus made apparent."

6 :r."fISSOURI AGR. Exp. STA. RESEARCH B U LL 1£TIN 115

1 t is evident that the proposed equations of Meeh, of Du Bois and Ou Bois, and of Cowgill and Drabkin may all be reduced [0 the generaliz­ed eq un rioll (I). ] n l'vreeh's formula, the value of JJl in (I) is zero; in Cow­gill and Orabkins formula, the value of 111 in (I) is I; in the formula of DuBois and Du Bois, m in (1) is .725.

After reading these papers, the first r,roblem that called for solution is the eVilluarion of m in equation (I); is it 0, I, or .725?

The solution of the problem is embodied in rig. I. This figure was prepared 011 the basis of the following considerations.

Equation (I) may be written III the (orm A

fi'" elf/n

Taking logarithms, we obtain the equation A 100 - � 11 log W + log C o 1-/'"

(Ra)

(8b)

which inciic.nes that plouing the logarithms of the ratios of ilrea to height, raised to the power 111, against the logarithms of weight, should give a straight line of slope 11. It is these ratios for di(lerent values of III that are r.epresented in Fig. I.

The lowest curve in Fig. I represents the ratios of area to height (that is, 1JJ = I) plotted against weight on a logarithmic grid (equivalent

to plotting log t against log "1/). The curve is seen to be made up of

three faidy distinct segmelHs. The equations for each of the rhree segments are given on the curve. Since accol-ding to eqllHion (I), the

.4 plot of log 1-1 against IF should result in a straight line while Fig. 1

shows that it is not a straight line; therefore, either equation (I) does not represent the data, or the value of 1JJ is not I; in other words this curve rules out the v.llue of m as assumed by Cowgill and Drabkin.

The second curve from the bouolll represents a similar plot when III = .725 (i. e., the value of m as assumed by Du Bois and 011 13ois). ''''hen the value of m is 0_725, the distribution of the datil points ap­proaches Illore nearly a straight line; but it is not straight; that is to sa)'� the value of 0.725 (assumed by Du Bois and Du Bois) is too high.

In similar manner we a)sumed the value ofm to he .5,.4, .3, and 0, ·1

and the resulting values of Hlfi were plotted against If/. Looking over

the chan we concluded by inspection that the best agreement between ohserved and computed values is obtained when 1Il = OA. We ;ncludecl in rhese computations the data by BI-adfield (1927) , Dli Bois and Du Bois (1915), [<rontali (1927), Lissaller (1903), Takahi ra (1925) > and Worner

Fi". I.-The rclatiou between surface area, .'\. lJOdy wcilCllt, W, :m,t 5t:llldi"l( he:i!!I'I, II, in man. 1\150 Ihe rc:i:dion hc;lweeu hCJI productioll, I�, 10 W nut! H. The:

ratios of MC;' tu llci!!ht, II, raised 11, <lifTe:re:,,\ l'OWCI"5, M, arc IllulIed a!!ainSI W";lIht, \V. Nute:1 tl"'t fIJI" aa:l, /\, the strail:hleSI li"t: is o\,laiu· e:t! wile" the: value: of the power. M, is 4. When the value "f the eXllonenl. M, is increas­cd, the curve tends 10 hrcak "I' into three seg­ments. These: breaks arc II',ile ,Ji�li"cl wheu the eXpollent, M. is I (valuc L1se,1 t.y Cowgill an.1 Drahkin) :111<1 _725.

(valne used t,y Du Dois)_ Note also Ihat while for arca, A. a strai!!ht line: is ohtained when the tX-1'(>'''Cllt of hdJotht is 4, >10 such strai\lht line is "htaincd for the he:LI ,"0· tluctillll curvc. Tile �Iar rel're:sellts /lIn. McK (Sec DII Boi� aud Du Bois) wci),:lIi"lI 204 1,0Iln,ls �nd h"villl: tI,e Iodl:hl of :1 12·yc�r-"I,1

\lil"l_ Note: Ih�1 while tile star cullIe On d,e linc whtre the: cl<l'onent of I,cight is !, or _7. it ;s hdllw tht lillc in the olitcr curves. Iful wheu Ihe t''IOUIICllt is I. or .7, a straight line �3n not he used to represent lhc normal data_ Note lhat whtn the: exponent, M, "equals �ero, tI,at is, whe" Ihc hti�ht dilllllll is nOI u:;e,l at all, dala !,oints arc distrilmtc.l ahOl1l a tllitrahly snaighl Jille.

GROWTH AND DEVELOPMENT XI

z ]() : .. ', [ .. '

I��

7

I

8 MISSOUIU Awl. Exp. STA. RI�SI�"'!CI-I BULLETIN 115

(1923); a total of IJ� me,lsurements, and with two exceptions, nil the data points arc within the 10 per cent of the average. This is satis­factor}' considering that the datn of LiSS;lucr included dead infants ill very emaciated condition.

lncidcntally one can determine whether heat production is directly proportional to surface nrcn during growth by merely plotting the ratios of heat production to height raised to the power 0.4 against weight. J ( heat production were proportional to surface area, during growth , then the resulting curve would have the stlllle slope as the curve representing the plot of the ratios of nrea to height raised to the power 0.4. The curve (or heat production is shown by the uppermost set or data (open circles). The data were taken rrom Benedict and co_workers (Carnegie publica­tions 279 and 302) including all the data ror males. I t appears that the curves ror he:tt produClion and ror area are not parallel: the heat pro­duction per unit area changes with incrcasing weight during the period or growth. But this is well known.

The results in rig. 1 lead to the conclusion that ror individuals or "normal" build the equation relating area �o weight and to height is

. 4

/I � 140 "·"IV·" (9)

I(�I \G�'L-!.J-!-J-p,1-l--J,-l,\.l.!-lj,lbl.-J-*--L*.Li;;-Y�,H;'!,;";� W,I.,ht

Fill. 2.-Tle r�hlion bel ween Utl/Rce u.,� �nd weigh I wl,.,n 'he dua by Meel, and "I"tlndler �.e included. By i"eI .. dinll thele d�, .. the devialion. ho," tI.e �"�Ulle were i"c,uled I.om 10 10 1U •• er cenl (fill'. I). The ""me.d.1 ,· .. I"e' of Ihe contU"It "'e.e i"cHued 10mcwhH. Compa.e 10 fig. 1 In. ell"'c .n _ 0,

I ,

•

"

f-- " 200 1-:'1 . �

. � 'I. OO�·

00 ....

-w �

a -

1-f-:-I-;'

• I� .� � -

100 00 60 50 40 30 f-VI'S a

GROWTI-I ANU DEVELOPMENT Xl

-. . ,.. . '�;. � . I ;� r-: . \"- I- I- - I.i !� r-- -:'t . -1--

§ 0 0 ·o� ... o 0 'm , .. :

I-I- F-l-. I-

1= 1= . � I- ,1:0: f-- 1= � i= � F .... 1-1= . 1- 1-.. I-I-I-

-00 � 0 �, 0 0 , • 0

o 0 � = 0 f:::: = I-I- � l- Ie \151 II-2 4 6 0 10

Age

9

n .

f-II" m • 'l' o·

AV �h Iw I-

I"

to: �

in which A is surrace area in square centimeters, H height in centi­meters, and /1/ weight in kilogr,lll1s.

In ,ilH_li\

�idu,ds o� a 1l00.'lllal huild, it is not absolutely necessary to

take hClght Intu cunslderatlon, As shown in Fig. I, the rormula A = 1000 11/·6�5 (20)

r�presel1ts the d;�ta ill a f�irl� satisractory manner, as, widl two excep­tions? all data POilits are wltllIll 10 per cent or the average values. (/I is area III square centimeters ror weight, If/, ill kilugrams.) VVe l1lay say

"'� ---. �--===".---.... -.- - -

10 IvflSSDURI AGR. Exp. 5T:\, RESEARCH BULLETIN 115

that we have not included the measurements of l'vTeeh (1879) anel of I'faundler (l916) because theil" values appear to be high as compared to other measurements. This statement is indicated in Fig. 2.

It should also be pointed out that we have not given the slightest attention to the matter of bidirnensionality on both sides of equations (I) and (9), as we believe that the equati,nll relating area to body size can not be anything but purely empirical.:

It seems that Dti Bois and Du Bois took special pains to get theda­tllm (or Mrs. McK. (represented by a star in rig. (I» on theirllttedcurve. Now til is subject had the height of an average girl of 12 years, and she weighed 93 kilograms, that is, 205 pounds! This evidendy is an extreme­ly unusual case, and it does not seem reasonable to assume that any simple formula representing the relationship for average individuals could also represent the relationship between area and size of a subject of the C)'pe of Mrs. McK.

4 @ <t

3 ,

;" , '

, ' , c

, . I ',' [11500 !XX) 2\0 500

� 0 1500 1000 100

, 'l!j<. -i .. ,' (\.1_ 'L �

, . 2000 2500 1500 2000 200 300 0 1(0)

wtnxnzq

: '

11:

"

, , , ,

2!lOO <iXJ 2000

ti� , . t !o-1

500 = <JOOO

Fill. 4�.-TI,e (eb.inn nf ou r/ace .rca, ,1.10(1) Wcilllli. /Y. and h.illltl,lI; (2) weigh I anuciteOiginlt, C: (3) weighl alHi body lengdl, 1.; (4) wci�hl, hei"I'I, .tld girth. The broken linel .epre.cnl 5 % dcvia­I;onl from Ihe .verase CUrVe. Com""r. wilh Fis. �b and note 11,,,t the lowell average error, . .J, £,(1.86)

;. obtained whenlhe linear rncauHe,nentl arc nOI include.1 in Ihe fonuula. In this d'Ht the arc. ilplolted ag.illll the p.odueil nf weight, II', and Icvcull'near "'euU'em.,,", X, Z; each ui.eu 10 10Ule power (m, II, q). Th'l fignre, U allo Ihe follow;"g (ibl, i. l;ucd on Ihe data for Hohte,n cal11e giv"" in I<el.

lIul. 89 "f Iltillerica.

GnOWTIl .",ND DEVELOPMENT XI II

EfJuation (2) is particularly useful for relating area to body size in domestic animals (cattle, horses, swine, sheep) during growth hecause the relative growth in height as compared to weight is less for animals than it is for man, and, as shown particularly in Fig. 16, introducing height ItS a datum in the formula by no means increases the agreement bel"ween observed and computed values. This relative amount of growth in weight alld in height for man is indicated in Fig. 11 b, Research Bulletin IO�I of this series. The relative growth in weight and in height for cnttle is indicated ill Figs. 5b, 6a, and lOa, Research Bulletin 103. The relative

growth in weight, height, and area is also shown in pig. 3.

1 I o I 1/� , KQ. 30 � 50 60 /lO 100 200 300 400 500 6CO I$JJ 1000

Weight Fig. �l;.-The .cI�tioll ofatc� to weisht (lowcrc","vc) aud of area lO WCiHhl anJ 10 Itci"ltl �I wilJ.<,�

(upper curve) . Sec curve .uJ ICllcud in Fill. 4a.

The fact that including the height datum in the formula relaling area to body size ill cattle does not materially increase the agreemenr between observed and computed values may be shown ill a more precise manner by the following considerations: Let liS determine the rela!ion

1 2 M ISSOURI AGIC Exp. ST,... . H I�SEARCII BULLETIN 1 1 5

between area and any lillear size, such as height at wi thers, body length, or chest girth by plotting area against any of these linear measurements 011 a logari thmic grid and by determining the C0l15tallts for each linear measurement. As a m a tter of fact we have clone this in Research Bulletin 89. The resulting equations, including equations relating area to weight, area to height, etc. may then be combined in to one equation by simply dividing the exponents by a !lI.llnbcr equal to the number of equations which it is desired to combine i n to one equation.

Thus, in the aforemen tioned Research Bulletin 89 we have fOlilld the relation between area and weight (or Holstein cattle to be

A � . 1 42W "

(or height a t withers, /1 � .'i04 H'"

Multiplying the two, we obtain

Therefore A

A' � (. 1 42W") X (.404 /-1 " ')

V(. 142W·") X (.404 /-1 " ') � CW·,, /, X II','''' J f desired, the value of C Illay be determined empirically by plotting values (or area against values of "p,67h X 1-/2.4 h on arithmetically co­ordinate paper and determining the in tercept of the resulting curve.

H we :1IS0 wish to include the circumference-of-chest measurement in QlIr formula, we divide the exponents (using the saine reasoning as auove) by 3. Since the relation between area and circumference of chest, G, is

/1 � 9.2G'-"

therefore the relation of area to weight, height at withers, and cIr­

cumference of chest is A = CII/·nh/_{1.AhC I .64h

Similarly, since the relation between area and body length, L, may be represented by the equation

A � 10.4[1·, therefore the relation of area to weight, height a t withers, circumference of chest nnd body length is

A = CIl?·r'l / 4 /-1 u /4G' ·61/4L1.7 /4 ] n the same manner we may include any number of measurements i n our formula relating area t o size o f animals.

As a matter of fact, however, as shown i n Figs. 4a and 4b, i ncrcasillg the number of linear measurements in the formula does not i ncrease the agreement between observed and computed values to a correspond­ing degree, This Sllbstnn tiates the idea that in the vast majority of cases the simplc equation (2) involving weight only as a datum on the right

"

GROWTII AND DEVELOP M E N T X f 1 3

• side of the equatioll, su flices to ,'eprescilt the relation between stu'face area and body size ill domestic animals,

ProptI"lil:J oj 1111: POlIJtrjllllclioll (3):-Taking log:lritlulls of (2) we ot.Jl:l i n · . . log A = log C + " log IV ( 10) IlIdlC:lt1!lg that such a function gives :l straight linc on a logari thmic grid h:1ving slope u.

Differelltiating (10), we obt:1in dA dIV _ = " _ A IV ( I I )

indicating that tile percent:lge cllallge i n d is II times the percentage change in 11/ Solving {or 11, we obtain

II = dA/A · ,/IF/IV (12) Indicating that the ratio between the two pel"celltnge changes is II, the exponent ill the power function (2).

A Ilumerical example will render the above ideas concrete. Let the pOWer func­tion be A = /(/: ( 1 3) and let tb� value. of the fixed multiple for successive v:llues of IV be 2 ; then we obtain the followl11g pairs of values: IV 2 " 1 6

A T "4 T6 64 256 The ratio of the mult iple ill lhe A series lo dIe Illultiple in the 11/ serie.� is seen to he

.i = 2 = 11 2 . . Doubli ng the .successive values of IV qU:1druples the successive valuc.'! of A. I Jlcre:1S_ IIlg IV :I�COf(lrl1g to a geometrical progression, increases A :1ccnnling to a geometric:l] progressIon.

'I:his Ill:l)' be put in other words as follows: changing the successive v:llues in the !V senc.s b}' III ( = 2) c.hanges the values 111 the A series by Ill" ( = 2'), This shows tll:1 t IncreaSing the values 111 the eX'pon�nt, 11, accor.ding to ar� :lrithrnelical progression il1-c.re:lses the v:llues of the multiple In the A senes according to a geometrical progres­SIOIl. 1�1 other words, tile �elation between the eXpollent, lI, of the power fUllclloll,anri the. ratro between succes�rve values of the dependent vfu·iahle (e. g., area), when the r:l 110 between d.le sllcces�rve values of the (.iependell.t variable (e. g., weighd, is 2, m:l y be computed wltll the :lHi of all exponential eqU:lIIOn, as e. g.,

R = t·mn ( 1 4) ill .which R is the s�id ratio, t is the hase of natural log:lrititrns, .693 is the natural log­arrt/lltl of2,.and " Is.tile exponer� t in the power function. Equation ( 1 4) may be plot­ted on :tn :lrrthlog gnd as .shown tn Chart A enabling one to read the v:lluesoftlle ratio, R, [or the depen?ent varrable wh.en the v:l[ue o{ the exponent is known and when tile ratIO for successrve ":1ll1es of 11/ rs 2. · W!len, as.ir� equ:ltion ( 13) , Ille power is 2, i. e., grC:lter than I , Y increases with r ncreasmg rapldlt}" :lnd wilen IV incre:lses 2 times, A incre:lses

t".19J d = 4 times Vi/hell I I = I , then when 1(/ i ncr6:1ges 2 til1)e9, A i ncreases

, ( .• 9J :< 1 = 2 times tll�t is, A i rlCr�:lses at the same pcrcent:lge r:lte as IV; :lnd the powerequ:ltion becomes a IlIlear equatrOll.

�Vhen ". i� less, !han 1 (as when :lrea is rcl:lted to weight), A increases with de­creaSing r:lprdlty. I hilS when 1/ = . 70, when then IV is iucre:lsed 2 limc�. A in­creases I:. 9U � . 70 = 1 .62 times; i. e . wilen IV i ncre:lses by 100 per cent, A i ncteases by 62 pe� cent. . When II = 731 the�l \yllen 11/ is i ncre:lsed 2 times, A is i'lcreased t·I�I � . 17 = 1 .59 trlll�s;.I. e., whcn wClght rs lnct�ased by lOOper cent, nrea is incrensed by 59

per C� llt . . Srml!nrly, when." = ,?6 (:IS 1 1 1 the exponent in the equation relating :Ire;] to wergin In d:ury c:1ttle), Incre:lsllIg IV by 100 per cent incre;lses 1" by t·tu L le "" 1.47 or by .47 per cent. C�l:lrt A will enalrle one to te:1d the v:llues direct ly without tll� neces.':] ty of complI tatlon.

Since nccording to equation (11)

14 Ivf rssoull.I AGIC EX l' . STA. R ESI�A](C I-I IJ U['L I�T1 N 1 1 5

W I ___ I ___ I ___ I ___ I ___ I ___ I ___ I ___ I'VVV

6 5

n

i 30 & 20 10

Cl'aft A.-The ,cla.ioll between 'he chanl!e ill the dependent "ali�bJe (A) when the i ndependent

" :I,ial,lc (W) il increHe..! by 100 lIer cent lor differc"l val"e. 01 d'e n.ponenl " in th e I)ower equation

A _ lI'a, Knowinll the value of 11,. e'I'onenl " i n lhe IH)"'er I,,,,e.ioll Y _ XU,one c�n read from Ihi. el,a,l lh" pcrce"t�lIe ;lIcreale ill r wh ell ihe vailleol X i, do"bled. Th", whell " _ 0.4 . • hell illCrcuill1!

Ihe vallie of X by 100 p�r cenl (I. e. doubling it), increalel lhe value ol I' by a lillie over J2 per Celt I. See le>;1 for lllrlher Clpla'lAtionl.

<;aM

0,

L

..

.� II

GIWWTH A N D DEVELOPMENT XI

� fZ

V,

�

0 0 c WeiQht

· -----.-------

15

I� 'Il 1 70 1 60 ! 50 /40

Fig. S.-A chart for euimating uca (bued on t h e fo.rnub Givcn on th e chan) wl,en weight and

1'.;11101 atC known, (Note: when weight only ;. known, ,h. U t a . may be .uiulated from fig. I , or froon Fig. 2.) '1'1,. agree men' buw.c .. the arca u utimHeJ from Ihi. chul a n d the Itue arca, can nol b e

"ruler t h a n t h e agreement between ti,. "VCUKC curVe "nd t h e "I".n'cd d a t a u indiuted in Fig, I . ]" othcr word., d,. diilerenc,. belween predicted and true afCU Ulay be a l great ' U 1 0 per CenL Ac_ cordins to our conception the percentage of crror may be Itill greater when II.ing Ibe fonouh of Du

1.I0i. a. iudicalcu in I'ig. I , the lecood curve froUl t h e 1>0((0'0.

1 6 M ISSOURI AGR. Exp. STA. RESEARCH BULLETIN 1 1 5

dA ,I/V 7\ = 11 11' (II)

the percentage change i n A equals the percentage change i n 1// times the value of the exponent 11, the question might b e raised why 1 10t simply multiply the percent­age clHll1gC i n VV (weight) by the c)(poncnt ", i n order to obtain the percentage change on A (nrca), i. e., why not simply say t h a� the percCllt;lItC challge in A is /I limes the pcrCcrHagc chnnge i n /V. The answer to this question is 111 :lt equation ( I I) is concerned wilh limiting \I;,IIIC5; it cont,lins the terms d/V alld ,fA which rep"('sent Ull"y small (i. e. limiting) changes in IV and ill A, toosrnall for their use in practical problems. I t call not lot used for relatively large percentage changes (because tile slope of the power functiOIl is not constant). When, 11Owever, it is possiole by some ll1 :1thernatical de­I,ice to tr:1nslate our question into the form o( an t.>':pol1(lIlia/ equation, as w e have done i n the case o f equation (14), then it becomes possible to lind the relation be­tween percentage cllanges of A and vV (area and weight) n o matter how large the l'ercent:1ge changes. Thispossiuility arises {rorn the {;lct thal, 015 i n e1l llalion ( 1 4), we are comparing percentages i. e., logarithms, in a function which plots as a straight l ine on a logaritlllllic (i. e., relative, which is e(l llivalel1t to a percentage) sC:1le, so that the percentage change r1t allY· point is a"IJ(I)'J proportional to the 1':1iue of the ordinate a t t h a t point; and when the base t is used, it is equivalent to the v:1111e of the ordinate for rhe given percentllge change. This, b y the W,IY, explains i n part a t least the wide lise of natural logarithms when de:lling with relative T:ltes (the rate of increase of the exponential {unction, Y = (b, at anr point is always pToponional to the value of the function at that poin!.)

The preceding analysis of the properties of equation (2), makes i t clear that i t i s easy to prepare a prediction chart Lased on equation (9) including height and weight. \Vhen height rellHtins constant, then equa­tion (9) is reduced to the same form as equation (2), ;]nd therefore when area is plotted against -weight 011 atl arithlog grid a straight l ine should result of slope .53, and i n tercept 240 ,·{AO (or .024 H·40 when dealing i n square meters). Fig. 5 shows sLich a weight_height_area prediction chut for mall.

II. THE SIGNIFICANCE OF SURFACE AREA IN ENERGY

METABOLISM

During the course of the present investigation questions have come lip concerning the theoretical i n terest and the practical u t i l i ty of this work on surface area. This section is presented i n order to save similar misgivings i n others who may undertake to reinvestigate the relation Letween surface area and body size. Thc polemic between Benedict ;llld Lusk, Dl.I Bois and their followers is well known. For this reason it wi l l l10t be considered here.

Before reviewing the situatimt :ls we see it, it m a y be usefu l to recall the fact that fhe degree of proportionality between heat production and ;lrea (or weight) may be determined Ly plotting heat production against body weigh t Oil a logari thmic grid, then detcrmin ing the slope of the curve. I f the slope is I , then heat production is directly proportional t? weight; if the slope is of the order of .% (similar to the slope rela ti ng area to weigh t) then heat production is directly proportional to area. Employing this simple procedure for testing the degree of proportionality between sur­face area :lnd heat production we obtain the following f;lctS:

�� '.

GROWTn ANI) DEVELOPMENT Xl 1 7

I . Fig. 6 shows t h a t t he weight o f the kidney, t h e weight o f the l i ver, and, practically, the weight of the lung, blood, stomach, and in­testine increase direcdy with botly weight at the same relative rate as dnes surface are:1. Sil,ce the kidney, liver, lungs, blood, stomach, and i n testine proLably Lear no lcss important rebtiollship to body-weight than does surface area, then the question naturally suggests i tself, why

HI�I'I'\

i " 'l'llZ f'fY � (n. SIllC

y.t3 1'" 1<"

:1ttH

� 0'1

V-I,

0"

I:�I 1m",

F'g. 6.-The relAtion of ""f .• ce HeR to body we'ght. �nd of the we,�ht of vario".orgon. � " d .ylte",' to hody weiRhl. NOle ,hot the Ilumerical value, of the .Iope. are app"'linutely lhe u m e for all tu ,,'e • .

not relate heat production to any one of these organs or systems rather than to surface area ? The. relation between the weight of these organs and Lody-weight can be determined with as much ease-certainly i n animals other than man-as for surface area.

2. Figs. 7a to 7e show that the area of the aortic cross section, the area of the tracheal cross section, Llood volume, vital capacity, a l l

18 M ISSOURI AGR. Exp. STA. RESEARCH BULLETIN U S

JM! I

20 � I/� / V I .... V /p /.; � '>/ (' Va" �� �' • V

10 /' >C 1,/ 0 Min 6 I' III 'Y

5 nl:llll 17 /' l,>'j /" rtf}' x At � .... 3 V , ..... VI� ./ /. [/ x

/ "'

2 114 -JI 11 � i.; � >C I � ji" -�

" " �� ... /

. . � / �f:;)

• l /; .6 , filL"

Gm 30 40 00 60 00 100 200 300 J1OO, 5OOtro BOO ml 2000 :000 WB1Qht Figs. 7a t� 7e.-TI�e area of the aortic croaa acction, the area of the tracheal croll acction, blood

volu �c, and vital capacity all appear to increalc with increalinB body weight a t approximately the IA me relative rate at docs aurface area. Plotted data by Dreyer a lid co-workers.

GROWTH AND DEVELOPM ENT XI 19

weloht HIt k< 4,0 � V

.. � 10 // A A �. w .... V .�

o r! V 4 'l

�oo I r� tD Gm ill M tllr

V V

m> Kg. 20

200 lXl 400 500 EOO \Wlght

Fig. i'b.

� rs

�� IA�

""'� �l ,1'1 � ,,�, V �r �g� e� / t:i ",n

30 40 flO 00 10 00 00 100 �yWelQlit

Fig. ld.

'. ) � GmlO 15 20 25 30

Cc 25

; "I.-� � 15

7' .. �2C Xl -4:xJ 5Xr 00 tIXI IXD

8 �r 1/ \00 90 eo

'CO 7r."\·" Q) .... .... " .,

1lC� 40 m. 000 MXlO � � 4CXXl

200 2m ?AD """ loW

?OJ 100 / 160 LhlO

/'

Weight Fig. 7c.

V 17 [7.

........:: w

lHu

-:/

ICIl

111 IJHY

.; -'" lMVYI

-:fD)

3CXX)

hscJt _i2'i1l1l 120 140 160 too 200

WeIQht Fig. le.

20

-, •

�

Iv I I SSOURI AGR. Exp. STA. RESEARCII BULLETIN 1 1 5

1'<1 S; ,:: 1''-' t'S ,�� � "-

."

I j;-I-o-�'n6'qaJ )J anWA OOruaAV ",,1ID asau:JU1

.' '"

uoudumslJ(() '0

1 �

GROWTH A N I) DEVELOPMENT Xl 21

increase with body-weight at approximntely the same relative rate as does surface area. This has been poinled out before by a diAerent tech­nique by Dreyer and co-workers.

3. Fig. 8 indicates that in the flounder, heat production is (prac­tically) directly proportional to body weight and not to surface area.

4. Fig. 9 i ndicates that during pregnancy, heat production increases even more rapidly than body weiglH.

5. Fig. 10 shows that during fasting, heat production decreases

"t'-IMati 1iI111s 100

90

40 DaysO

om .� 4

•

'r!( 1 \ .

� • •

I) 1

I, 6 t2

V! iJ() OO

70

� 100 00

1\ 1\ 00

b t 70 16 DsD

00 r.-'['1.'. 00 � T" L'\ . '

u �" , 1'\ • •

�fF o'�� 1 • " lila ..

1 !l-o G U , 0\ � t 0 4. 0

e��1 1

�I> L400 f'" !'} " 01 e ILt . . ,I� -• ea '1[" 1 00 H

� III

t-!-1-9

B ..Dr.: " 0

0 [1>, 7 St!!! DsO K g

o 16 III

• )k, I";� I ''tl/;"" .., J:':.lo

• Srf,e 16 'l4 32Y � 45 55 65 DsD Fasting I"LQlit o 16

rilt_ IO.-The tth!iv� decline in body WeiNfll .. "I i" Iteu IHOduclion du.inR 'anjn�. Ch . . , I ,ep­relcnll heat production plolled �B�i"ot weillht nu 101l"";I II,,,i.: eoo.din�le rUt>". (du� II,. I�enerlict on

Leunlin. u lIi"en in Pllblic�lion 2OJ, Cunesi" I nOlilll1ion of Wuhington). The uluel of Ihe UIIO­n"nl. n. in Ihe power fu"ction ('0"11111)", 1.9101 hut p.odu"tion when ulee". 2.2 hcH l!fodnCI,on on Rw�k.

e,,;,,� in the morning. 2.3 IOU I heu p.oduction) i,,<lieu., IhH the puctnla8" ch�"Re in he.t prodnction i. ahOlil twice n B,e� 1 �I the percentaG",chanRe in body weillhl. I" I l lh" ume dau '"'� plolled on a n

" ,i lhloR grid. The utio o f I h " .lol'e" " gi'<e. 1(1111101)' t h e "'me v�llIe u t h e ,'�llIe of t h e e�l'onen! ( ,O il .0067 - 1.8) in I . I l h � n d I l lb p,elen! • • i",ifar �o",pHi,ol' for lI�en C "".I D du,;"g fao'ing

(I�e Benedici �ud nit�m�n C�rneill:ie I'ubiic";o" 377, pp. 166 �"d 168, Aplil 18 10 "hy I, fall), The .OJ.l9 ulio 0' the .lopu i. rOllllhly 2 ( ..

. ii078): that i., the percellugc decline in he"t I"odllction i. �"out t"'tce

�. lI.eat II the percenl�ge decline in body "'ci"hl. I l l b ,eptetenlS hen l>loduClion plollNI "I!�inlt d,y. IlIlins II h repreoenl. body 'High! plolled ."�i,,lt day. of futin". Simil'r c:o'npuilon' ale ",�de fo,

Ihe ubbi! (VI), �nd dOli (V), �nd lIuine� pi" ( I V). In !he dOli u,d the T�bhil Ihe pe.eenUle decline in hUI produclion il. roulhly, dOllble Ihal lor Ihe ptrCcullle decline in wcight. In the J!"inc, pig, thc

percenl�le decline in hut production il the Utile " Ih� pc,cenUle decline in body '¥eiltht. IV, V, �lId VI ,ep'Clent diU by Rllhne. ucited hy VOlt (l. flinl. 1901, X L I , 1 1 3). II il cc.tain tb,! the decline in

h •• t producl;on d".inlll: fuling, "nd pI •• " " .. bly the ;nncole in hcol production ,h"i"l1 le .• limentAlion

(it i. cllrioul th�t ... hile 10 much lou been done On huinll, no .uch Htelllion lou been given 10 the procn. of ,c·.lirncllution), dou not vuy directly wilh hody I,ubec.

---_. � ", .. - --

22 tv1 bSOU1l1 AGR. E x !'. STt\. R ESEAllC I l n l Jl..LETIN 1 1 5

:WBB ""l1'lo1 MJ " � � � � �

" 1--=-". '., , =jl - -::=- -"". � 'i! . . � a� .. I'� 1-.& � -� � ( �

j � � f\> . J--l:c< f,f;; I� r--b::-� r--. . - � .s ·1-� � - - I- j- -

��� !r �- <CI. � to .., "'" '" �

SU!"IDlld arn.o ",q!l�!a

I'"

� I "" 08 � '< l�li � M' I �

� \. '� Ii! II�I'<I ,

" h......" r" � 10 1 0 i' l'--Gl lil

t I I

GHO\VTII ANn DEVELOPM ENT X I

more rapidly than I.>ody weight (dming re-alimcntation the lieHt pro­duction presumably increases more rapidly than oody weight).

6. Figs. I I a and I I I.> indicate that (ood consumption during growth tends to increase directly with hody weight (01' body weight tends to increase directly with (ood consumption).

7. r.igs. 1 2a and 1 2 b show that during the first year, heat pro­ducrion in children i ncreases directly with body weight or even somc­what more rapidly than body weight. Fol lowing the age o( one year, hea t production increases variollsly (rom the .56 to the .73 power o( body weight, while we have found that surface area increases directly wi th the .68 power of I.>od y weigh t.

8. Finally, we do not quite see the logic involved first in relating area to body weight (thus introducing one set o( errors); then computing area (rom body weight (thus in troducing a se,ond set o( errors); and finally, relating heat production to the computed area. 'Why not relate heat production to hody weight directly in die first place with the aid o( equation (2), or simply wil'h the aid of charts sllch as arc given in f.ig. 1 2 ?

'vVe do not CJuite see how the statement "heat production varies directly with body sl.lrfa(;t�" is any better than the statemellt "heat producl-ion varies directly with the .56 01' with lhe .67 power of body weight". Any one who understands the meaning o( the power function (2), as explained in the preceding section, understands one of these statements as well as rhe other.

I t must be granted, however, that since it is customary to express heat production per unit area, this expression gives a (eeling o( definite­ness ;u1(1 concreteness which is rather lacking in a mere (ormu l a when one does not understand it.

The central thought of the presenc discussion is that thert is no lIet'essily for bringing area into the problem o(heat production when olle measures not area but weight, or weight and height. ]f one desires a "norm" or standard for basal mewbolisl11 then why nOt use a chan, SUell as Fig. 1 2, 01' the uppermost curve i n rig. I , or Figs. 39, 40, 4 1 , ,md 42 ?

24

. .

l'vI ISSOURI AGR. Exp. STA. RESEARCII DVLLETIN 1 15

Weight F'�'. 12� ."d 1 2 b.-The ,ebt;oll of heu produclion 10 bod)' weight. Note that. followi"g

the �ge of one yu,.the value of the exponent va,i.,. r.on, ,56 to ,7J. l'.ecedi"F!' one reo •• he.t l"odllClioll iIlCt�,"e. di,ecll), with bod)' wei,lllh!. Thi. direct prol.ortiollolil)· bet"·een heot 1'.01l1lCtiOIl . nd bod)' weight precedinll o"e yeJr il U,bnolHi'led by F'R. 12b . • poge 25. ,n ... hich heat p'DductiD n ; ' plotted ag.;ntl body ",e;ght FDr ;"r."tl of the UIl.e . g e (n<:",·bo.1I

i"I'"I') .

·1 I I I

GROWTH AND DEVELOPMENT X l

RatloEEHEEffiillRlEm1:ffi1ffi1'l

r+t-I-jl-Iot tt-t- I-t-tii-l-i HtlYrtrl:ttltllllO !'j 1.5 , '

•

,tl+��t�I+t-mnJ1W j H;lii!'·n.Ii-lotttti11111t 140 tl

t;"'I-I7r-t-l � , H;-f-r-J<1. �-,

• + _ � .. __ _ l . ' . 120

Fig. 1 2b.-Th� rdotion (of he.H produetion to body weight (lower cu,ve) �nd to the woduct of height ."..1 we;�ht (upper cun'el of new born ;lIbntl. DH. by ilelledict on:l Talbot.

III. THE NEW DATA ON SURFACE AREA

25

Regardless of our opinion concerning the significance of surface area i n metabolisll1, we have cOIl$cientiously investigated the new data on surface area (which, as poin ted Ollt, includes 482 dairy cattle, 43 1 beef cattle, 1 2 horses, 1 2 swine). We Ilave also examined the data on 47 young women rneasurcd ;It this S t:ltioll by Mrs. Bradfield (Missouri Agr. Exp. St<l. Research l3,ldletill 109).

Fig. JJ includes all (34 1 ) rneasurements on beef c<lttie, including :111 :1ges, :1nd all degrees of Acshi ncss. The equation is givcn on the chart. The slope is seen to be .56 (not .67) . All data points are within 10 per cent of the average curve.

1 t should be noted that ill the C:lse of dairy cattle, careful ly measured under the very best conditions, the data points fall within 5 per cellt of the average as sh'(f\\'11 in Fig. 14. The data on beef cattle were obtained under extremely unfavorable conditions. The animals were willi, not used to oeing handled, very excited, <llld anlloyed oy Aies i/l the hotlest part of the Slimmer. The huge number of data points were obl:lined i n

26

• �

"!'

NhsSOURI AGIC Exp. STA. n l�Srrt\nCB B U LLETI N 1 1 5

I!.

:

.,<

...�� P;" �: .

l\a;: . ' . . :.jl

t;;:; t, · ·

W'ig�'

. I'W"'I" :; I? �;.

,

>

'1.'».11

'1'!'i'

Fig. 1J.-TI,c ,d�tion between .urface HU and hody wcil!hl in beel cude, indudiuG bod, Icxu

and all llalC. of nutrition, Thi. cbut rel'r"I",nl J 1 1 a nimal •.

order to compensate for tilt: t:rrors in the measurement's due to the UIl­favorable conditions under which the measurements were made.

Additional charl"S for heef cattle, divided according to breed) sex, etc., are given i n the appendix.

Fig. 1 5a presenrs , ti l the new data on dairy cattle (measured by Mr. Matthews). The data pu bl ished in Res. Bul. 89 (measure men ts by Elr­ing) were also incillded in this chart. The difference in slope between

,-hese two sets of data consists in the fact that Elting measured the area of the tai l) while M a uhews did not. l Sb shows tht.! new da ta plotted by

breeds. Additional clHlrts represerll"ing IVfa tthews' rneHsurements are given.

in Figs. 1 3 and 14, Res. Bul . 96 of this series, and in the appendix of the present bLllletin.

The new data 011 horses are included i n Fig. 1 6 ; dIe new data on swine are included in Fig. 17.

.\

b;,

GIWWTH AND ])I�VELOPMENT XI 27

lncluding a l inear datum i n the formula relating area to body size did nOt improve the degree of ;'lgreemenr between observed and computed values.

I t may he noted inciden t'llly that since in cattle, area is directl y

proportional t o (roughly) the square o f height a t withers, therefore (as explained in section I) when height a t w ithers is included as a datum i n the formu l a relat i ng area to body size, the proper value of the expo­Ilell t for height is I (m = I ) . I n the appendix several charts are shown in which the ratios of area to height are plotted against the correspond i ng weights. The ngreement between observed and computed values is 1 10 better i n these charts than in the ones in which height is not i ncluded as a datum i n the formula; i n case 0f Fig. 1 6, it is much worse.

The poorer agreement between observed and compured values in the

charts based on Matrhews' measurt;l1ltnts (Fig. 1 5) than in thechan based

�.-�����,�--.-������ ��I.I--f--\--\--I- --- --- I-j-l--H-l.,;i(� 5.01---- -I--I-H-l--f-H

-- -- --1--1- --.. ' ,;)!1��' H-H-H - - -- - - ---1--1--1-+';" ,,"''-I' -I-

1--1--1--1-1-- - - - - --4 .0f---I--I-I--I--I-H- - - - - - - - _ .

30'1--f---1--+-1----1-----

201-----

I I ,,/'7 " , ' � -r'1-�--k;(/ -1-1--1-1----: -: --:k;1I-' -I--I-I-H-I-

, '" I---1-- ---�' �:---- -- ---1---

" Jel··ey • Hol'lEl ,

Fig. 1 � .-Cau I� when ea,ciully rncuurcu wil II I he .n, hc� i",cllralOr nuder ideal eondil io." dn nO! dcvi�le by mOre Ih.n S p"re�"l f'olll d,eaver�gc c"rv�. "ven if .1I agcI ."..t twO breed. He ,nd"..t",1 a ' ;'

Ihe CUe with Ihe �nilll�r, on Itoi, cha,1. Thi. chart i. the Urne at I'ig. 1 in Hu. /.luI. &:9. hUI Wtlr, ,he S % l in • • drawn in 10 U to Ihow Ihe jlercentage d�vialion. from Ihe averlHe.

28 j\lfISSOURI AGR. Ex!'. 5Th. RESEARCH DULLETIN l l 5

" I "'" . 7 -----\- -6 W, ' ;!o 5 ,i;t@ I>" 4 t . � � �' I,r 3 - - - � ,[}>V ,,,-I- -

... '" � • • 1 �-&' 2 /��� t---t.o -�� Tor'

15 •

.Y � !l'" ""..,., I� N � UlQ H�� :;; /

Elhnp Hols� lI(lls � ·Y ' hi", 1 9 . , - ( 11 Ix>< .� -/. 1-1-

,- "'" .. " K�S 21) JO "" '" Ell 70 00 90 100 200 JOO 400 500 ((Xl '"' "'" W.'Qht

Fill_ lh-The rel.1lio" between area �",l weiRlot of t h e neW clu� (482 �ni ,".I,) on d�irr e.nle. For

!"Ilher e.Jlla,,�tion. lec le_t.

...... �_ ..... ,�_-.. --- -- -_ . -

..,. ,

+

GROWTH AND DEvELorr.I ENT X l

Sq�(

'-.,

.\§ ]� * �( , .,

I

l�=' Kg �

. ,;.; ':;\""

1.,;;Ji r\<�

f''.''·

. ,

::,#

,.,.", , :H .. '·

�:�it

�;t >'" I : :xl

I'kIQht

� •

•

' .

�Ir

�Jllio Fig. ISh.-The rebtinn helween I",f�(c uu �,,,I body weight in

Ih" """cral b'''cd. of cattle. Iheed difTucncu ... ;(10 rcopeel 10 thi.

, .. htion.hi" aTe inlil'lnifiunt .

29

30 IVI ISSOUIII AGIt. Exp. STA. RESEARCH BULLET I N 1 1 5

j .. .: �5Y.1Jr . --1-_.I-°'+'+'++-H

Fill". 16.-Thc relation oelween lurf��c area anJ body IVcilllu 01 horoel. r ncilidi "II a lincar d3(um in the forrUllla ;uercHe' d,. deviuion. I,etwe"n the uillcrvc<.l ami computed valuCi.

!

! '�f-=-1,'�-�-��1 1�-1��·11·£· ���;·�1: :�I--�I�-� .�t=:I=l±I: .� �'f-+-H+l H--I--+-I A- --- · I··� ff'-' ,,'H-H+tlH--++1 3--

2f-�<-. 1;� 7.r 1e Kgl 2 3 4 5 6 8 1D '10 3Il 40 50 ro w !00 300 lVelllht

Fig, ] 7.-'1'10" rdation bClwcen arca a,,,1 body weighl of ,wine. TI,e croun which rCllre.cut th. neW dHa arc 10 p�r C.III bclow (he average, of I'logan and Skou[,y,

If i

1

GROWTII AND DEVELOPlIlENT x r

. . �

31

on Elting's measurements (Fig. 14) is explained h y the fact that while Elting confined his measurements to animals belonging to [he University of IVI issouri herd. Ma tthews measured animals belonging to several other herds. J t was difricult [0 measure many of these animals, because, unlike college herds, they were not used [0 being handled.

Hannah Stillman Ilradfield of the Home Economics Department of this Station employed the surface integra[Or for measuring the surface

lio

120

10( �

fA N.

2D -

I. 1B

1.1-1.2 ,

/' � � /-1. V .>'

- . , , . , , - ' ,

, . " rz , . , , , /1\:>1 , -V <VI I I

, "

� . /

5.

.

- , l' '7 / , ' ,

07. Iin�

, ..

, ..;

)

.

-. ,

, .� � -,

· , � .

· .

----- -

.

ill . ·

'I · KQ. 25 30 35 tID 50 60 10 00 100 WelQht Fig. ]8a.-The rdatiou bet Weeu IlIrface a rca and hody

weill],t of the IlIbjecl' mcuurcd loy 1\,1(1. Bradfield. The "I'per CUrVe includel height (Ill .. I) at a datum. Note tI,at

the value of the ell'Onenl i • . 55 .nd nOt .68 at i"dicatcd ill Fig. I. NolC allo that the data pointl «19 individnah). ,n' clu.]illil the very heavy iudividual. ;He diuriLutod within the 5 % limi, •. Note alIa ,hal d'e exponent 01 wcillht did nOl

ch.nlle by induding hcillh" I I , al a d�t"", ;n tbe formula .

area of young women (of approximately college age). She measured 47 individuals. Details of her work together with the numeric.!! data are gi v_ en i n a disserta tion (Uni versi ty arM issouri Library) 1927) and i n Research Bulletin I 09 0f this Station ( 1928). We have plotted her data with the

result shown in Fig. 1 8a. The value of the exponent or equation (2) for these data is only .55 as compared to .68 found in fi'ig. I . This low

32 M ISSOURI AGIL Exp. STA. RESEARCH BULLETIN 1 1 5

Fig. 1 8b.-Th� r�13!ive diotrib"lioll 01 1\<1. •. nradfield'. d�{� w h .. n plaited on � ehut 01 wid� unll" on � , to inchule �Il wC;/l101O �ncl .gel. Thi. chart .ho"'. that the .Iope of the line r,,131;u� Hu to \�eigll l io leu lor ", .. !lire individuah (heigh t nearl)' contt�111J th .. " for grow;ng

indi,·idll�l. (weight � I well H height inocHc), The tri�ngl e on the "'.tlfer"e right repu.enu

r.h., McK. Nou Ih.\ �n ;ndi,'iclual hCH';e, Ih3n Mri. flkK .• mcuurctl by )\·1 ". Budfi,,!d. hilt

011 the Avcuge line.

\VclQht Fig. 18t.-The rdalio" b�twee" .u,bee a,e�. A, and bod)' "·ci�ht.

W, and the the , .. 1,,1;0" between area, weight and height, L, ill ,,,�n. Plolted Ir011l data by Froni,li.

value or the exponent is also i ndicated 111 Fig. 1 8 b i n which rvlrs. Brad­field's measurements were plotted together with other measurements i ncluding very young children.

The low values o( the exponent (or Mrs. Bradfield's data arc due

i I

GROWTII A N D DEVELOPMENT X I 33

to the (act that as compared to the range in weight, the r:lngc in height (or these individuals is relatively very low. It has already been explained that when one wishes to combine an equation relat ing area to weight

A - G� (a) with all equation relating area to height

A - C2 11'" (0)

J , IwlQht

5 6 7 8 Fig. I Sd.-The relation of an:a to houy weight. \V,

anu to I,e;�!'t. fl. Plotte\\ froll1 data by LiS53uer.

/. :; -; r: '.t' .-::: )(: ;:. -P

"II HeiQht Fis· 19.-The, ch."1 rthling uea 10 "'ei,!!;ht in

lIIo".cr. "" r be of ;" telut i l l thi. connection. "'hue

d�u "·ere obll;"ed with the ,,,,f.ce ;l1legr�tor in Dr.

Edgor Aile,,', J..boutory of Ihe U"j'·elli1r. Dr. AUe" II13Y publi l!. Ihe "" ",eric.1 d�,a nut! a chHt lund 011

the'n ehcwhue,

thell in the combined eq ua tion I'he val ues o( the exponents are Idlved A = C H",/'l/1/"/'l (c)

, But when the height felBains roughly constant, as ill the case of Mrs. I3radfield's subjects, then /-1'" in (b) above is roughly constant, and consequentl)' C 1-1'" is roughly consttl n t so equation (c) becoilles

/I - C,W"

34 M ISSOURI AGR. Exp. STA. RESEARCH BULLETI N 1 1 5

.

" .

. , , . \ , " , '.

" , "

, , , . ,

'A' 0\ , "

' I,{"

\\' \ ,'.\'. � \\\ \� " ,\� ��,:t.

\ " ' :�: " \\

\:\ \'\ " :� \\.: \

§ 2 � � � �, a

t!: 9 � � �� "" �� :\'\, [\"� r--,I'i'

i'C"[\ 1"[\ I'" 1'\ I, ,� [\� " '\.1\.

-

� .... I� '\. � .'I'\. , ."-1' " -c ro: � "� i' .'1"-�'\ 'I..� � ,'-1"'-

l'\ k"'- 0l'\.. � :\: � � �

� � c;:J :3

" -' 'H"', -:� ,

,\ ' ""

,�'\.\ ,. �,,,\,\ ,��,\ \� , \\

, " � . , ' �. �� t' ?

2 � ...J cq CCl &q � �

! I '

�

� :-.� �I'\. " l'\.' �' � 1\.1\, � � I\� l\

.3 OJ q

8 fa S � 10 " 10 1"""

b

, "'1

GROWTH AND DEVELOPMENT XI 35

J n other words, in equation (1 ) when heigh t a s well a s weight changes (as during active growth), the value of the exponen t 11 is greater than when weight i ncreases and height remains nearly constant. When the height remains absolutely constan t while weight increases, the value of n in equation (c) would be the same as in equation (a) ; when weight remains constant and height increases, then the value of m would be the same in (c) and in (b) ; when both change as duri ng active growth, the values of m and n in (c) are approximately half of the values in (a) and (b).

As a further iIIustration of the influence of the relative constancy of height on the value of the exponen t of weigh t we may plot the areas for various weights, but for constan t heights, on a logarithmic grid, using for this purpose data i n terpolated from the Du B'ois prediction chart. The slope, n, of the l ines is seen in Fig. 20a to be .425 (the value of the exponent for weight as given by the Du Bois formula) . But when heigh t is not included as a datum, then the value of the exponent is not .425, but .68 as shown in Fig. 20b.

Ca10Ples pep daV

600

500

�

200 Kg.

. .. v

6

�

".,

� "" ,,,1 �'l �

�� I:'� �� ·V � .J

�� �...,. -

k V � rO�'

Dog,L1JS� a 10 1�

Weight

Sq.M. 6

3

Fis. 2 I.-A compa rilon of tllC incrc&lc in heal production. and thc incrc&le in lurface area WiLb illcreaains body weisht of dOBI XI X and XXVI I in Luak'i Laboratory, Plotted from dAta cited by Cowsill and Drabkin, Comparc with Fig, Hb.

We may conclude, therefore, that in animals of approximately the same height, the area varies not with the % power of weight, bu t wi th some smaller power of weight.

36 M issouni AGR. Exp. STA. RESEARCH B ULLETI N l t 5

J t may be permissible to poi n t out what "ppcars to t!lC wr,i ter

,as

a fallacy. Fig. 21 represents the i ncrease i n heat production with In­creasing weight of two adult dogs of the same body len�th \Dogs X l X and X.;\VlI i n Lusk's Laboratory). The heat productlcn I S seen to increase directly with the .70 power of the weight, while the sur­(ace area (as computed by Cowgill and Drabkin, page 47) increases with the .37 power of the body weight. There appears to be, tI�ererore, a n error i n assuming that heat production is d irectly proportional to surface. area. I r Fig. 21 represents the siturttion correctly, then he�t pro­duction increases almost 1 .9 times as rapidly with incrensing weIght as does surface area. The fallacy, i f presen t, consisted i n assuming that surface area in animals of cons/an/ length i ncreases directly with the 31 power of the weight. (Figure 2 1 is an illustration fav�ring the inclusion of the height datum in the formula relating to body size).. . . One practical conclusion from this discussion is that Illvestl�ntiolls having for their aim the qunntitative evaluation of heat production per u n i t surface of an individual can not rely on a formula derived fronl data on a popula/ion. Individual variability does not permit the

.applicat'iol� of

averages to inClividual cases. This statement should be eVident to a biol­ogist or even to the layman. I t is the discrepancy betweel� average ex­pectations and individual variations which forms the baSIS of such an eminentlv practical business as l ife insurance.

In (juantitative i nvestigations between heat production and body surface, the individual animal must be measured. The surface i ntegrator described in Research l3ulletin 89 of this Station offers a practical method for measuring the area directly. Quite recently, after the appearance of the aforementioned Research Bulletin 89, Frontali published a photo­graph of an i n tegrator apparently designed many years ag� by Bordier. Frontali's inslrument, LOgether with his table o( data, IS re­produced i n Fig. 22.

SUMMARY AND CONCLUSIONS

1 II addition to the presentation o( the data a n d the charts as out­lined in the abstract, this work may be summarized as follows:

1 . The surface area is directly proportional to some power of the bod)' weight. Meeh assumed the value of the power to be 31· As a mat­ter of fact, the value o( this power varies, in Ollr experience, (rom about .32 to .72 depending on the form of the animal. The change in form o( the animal due to increase in weight during growth is quite diO-erent froll1 the change in (orm of mature animals during .fattening or fast­ing. The lowest value o( the power i s (or fattening; the highest for growth. 1nclllding [I li neal' datu ill ill the (ormula i s helpful i n mall)'

I -4"

\

-T� _ . ..... � ... <. .. _ � _ _ • __ • - � .- -- ---- . ..

GROWTH A N I) DEVELOPMENT X I 37

cases in compensating for differences i n body (orm i n the prediction of surface area.

2. The use of a prediction chart, or a prediction formula (necessarL Iy formulated on the basis of measurelllents of a population) for estimat­ing the surface area of an individual ilia), involve an enor as high as 10 per cent of the true value.

3. Because of the necessarily large error i n volved in applying to all individllal a (ormula based 011 the measurement of a population, i t is suggested that i f it is desired to relate heat production to body surface of an individual, a surface i ntegrator should be used ill actually Illeasur­ing the area. I t is perfectly proper however to lise a forllluia (or large­scale computations on popfl!atiollS.

4. "Vhile the practice of relating heat production to surface area may be j lls�ifiecl by Clistom i t is entirely ullnecessary in princi ple. J t ap­pears that all vi tal organs i n the hody increase directly with some factional power of the body weight, the value of this power being within the limits of the value o( the power for weight, in the equation relating area to weight. (That is to say, the relative i n crease in mass o( the various supporting nnd cOllnecting tissues in the bod)' is greater with increasing body weight than the i ncrease in mass o( the vit:d organs (viscera)). 1nciden tally, this differential nature of growth may be olle-if lIot the chief one- of the factors l imiting the size of the body. This being the case, it is no Illore rational to relate heat production to su rface area than to the weight of the (one, or all) vital organs. ]t is simpler to relate heat production to a power of body weight or to powers of weight and height (especially with the aid o(logarithmic paper) than to relate heat produc­tion to surface area, either by measuring surface area ciirectly, or by using a formula which may i n troduce i n to the result all error as high as 1 0 per cent.

38 r.,iI ISSOURI AGI<, Exp. STA. R I�SEARCII OULLETIN 1 1 5

FiB. 22.-A photograph of Hordier', device for mCaJu.inB area recentty ulcd:by �'tOlluli <made by Julu l�ich:lId. 2S Rue Mdinguc, P�fi'J and I photograph 01 a lulojtCI iIluUUlio.ll lhc llIethod of IIliug the integrato., "'h. GUI Thorn.jo, mcch�"ician Univcuity of /I,'liuoud, lU�Jc the " Ulatt ;"\CIII.lOr ducrilocd iu Ru, Uu!. 89 of Ihil .erici. The dau loy frontali are on IUSC 39

--- ---- -

GROWTI'! A N I) DEVELOPM ENT X I 39

� S�pu6�1_ cal.ol." I 1 " I !.;. J NOMa Eli SI:: ... ::-; li� 1 i� Ou . t�.·� It�,�i t �,- ! I _"..; � _",.. 0 01. ._ In .-.. �9=���,J�� __ ��.J���� __ ;!��-�-���"'"-· __ ,J __ -J"-�· �<cl""��

1 r. G"irolli . ,J g. 24 fiO I 3.30ti 2M2 2213 230S! !!OQO 02·j � Ill. !f,U/',iI . . ' III. I 53 5 ".050 302=' 2(103 2331 1 2,182 0 1 2 3 III . ]JuogtU . , t 111. 1 Y.t 54

" 1 " 1 3 0 3061 2637 2372 Z(,H 0 1 5 . t r. O'lfeddll . . 1 ill. 2 5 7 5: 1 4 0 :1545 30M 2701 3UO I !lS4 5 Ill. Dro . . 1 11I. 2 Vi IiG 4.000 :1·1:1·1 2955 2613, 28!t7 691 6 Ill. Pirl .. . . . . 111. 2 ',. 5 7 5.30ti :HnO 3 1 1 :! 27:18 337S 0:17 7 f. Pellu . . . \ m. 3 5-1 5 4..&5(; 322:1 2773 2·160 2 0 1 0 588 8 m • • gertl/ . . . . m. 3 50

' 1 4 080 3471 2987 2631 2802 1i62 9 I. MUCCioni ' l m. 3 1i8 r.: 1M :\5:.!G 30 :10 2721 :1100 6()7

10 Ill. 1'llylia! . . 11,. ·1 fiO. S, 4 . .jM 3221 2711 2(;2·, :!7:12 0 1 ·, I I Ill. lhmia . . 1lI. ·1 63 0.000 39110 3381 3102 :10103 1167 12 m. I l'Jllardo . . m' '' }i: 62 5 . I M 3519 3M·1 2813 :.I0831 5DS 13 ,". 0illCDbb! '1 LlL. G h 60 fi.7fifi 3822 32RO 20H 33011 li8!) J.I. f. IMolini • . . m. O 62 6.100 311n 3 4 1 9 :1088 34010 M5 1 5 f. ,Dent . . . . 11 Ill. 7 63 5.795 3848 3304 3()!l0 3 1.1011 ,,. 16 In. Ollbriolll . m. 10 65 7.000 4354 3147 3388 3401 .t04 17 1O. 1.A'"ni . . . 111. 1 0 64.5, 7.200 4.137 3 8 1 8 3�09 3505 486 18 f. 1PII1:1I . . . . m. I I 7o. 5I B.ooo 4 1 8 7 3625 3

,'.,", ''''

'''

01 '

,'

,'

, HI iii. UIlII . . . . . m . 12 70 8.326 4798 4206 1 20 Ill . Pi'llno . . . 1 JIl. If i 12.5 9.8:10 (I.I tHI HID7 .1221i �507 467 21 m . • 4mbll . . . ,,,. 20 74.6 !tooo /j 1 4 8 H20 4 1 6 1 4728 1i2/j 22 f. Denoli"i . ' m. 23 85 r lO.200 6591'i 4 8 1 4 4929 1'i00", 400

. 23 In. Seehi . . . . m. 24 85.3' 1 1. :100 51111°1 5 1 5 4 6Ql'I7 IIl13 470 24 r. Porcedda ,". 2 4 86 1 12.060 621161 11384. 5241 6 1 64. 428

6 f. S,ll/nIl . . . m. 29 76.6 7.8001 4680 ·1027 3086 38561 '" 26 f. Oaf/lin m. 3O 83.5 10.660 576111 40M, 4866 6172, 485 27 f. Zam . . . . a. 4 00 1 12.oro 11252 5379! 5Hi3 5623, 466 28 III. Salina 11.. 4. 85.7 12.050 0:,Hi2 5:119 1 5202 5 1 1 0

14.20

'!l f. porcu 11.. 5 Vl 98.61 15.200 72001 6280 1 6371 6260 4 1 1 1 30 f. li'lJnni a . 7 100. 5 1 1.07('1' 7884 6?1I5 , 071'16 7 1 1 8 4 U 1

;� I�;. �;:t:,:ft" :. � t� :�� ��:;�� I �2i� g��:. 16;�� IJ��11 �!� 33 f. Cartfl . . . . 11: 1 2 · 1 2 1 . 5 25.1,00 1014H 80;}3 Dll'I8, 8071'i: 323

H o 2 I(U

no 52.7 5 1 . 7 50.2 4.8.0 1i0.0 53. .1 48.3 IH.!! ·1 fl. 7 (,0.(, 55.6 1i4..6 1i3.2 64..3 5 1 . 1 58.8 6:1.·1 63..1 1i8.8 62.2 00 .• "'.3 6 J . 0 62,4 60.01 63.3 70.8 66.2 74.7 6d.4

( I I I cal.l'.llll. ( 100 crt!.!'.I,!;.

( 100 cal.ll.kg.

(\II crtl. p. kg.

(00 cal. p. kg. (88 cnl. p. kg.

logg. ipolrof.

40 l'vl lssoUIH AGR. Exp. STA. RESEARCII BULLltTl N 1 1 5

REFERENCES CITED "T\VOOU, H., and \'VEAKLEY! C. E.) \'V. Va. Agric. Expt. Sta. Bu!. 1 85, 1924. llAl.l)WIN, 3. T., Univ. lowa Studies ill Child \Velrarc, 1921 , r. No. I , DENIWICT, F. G . and TALBOT, F . B.) Carnegie Institution o( vVashington.

Publications 20 1 , 302; and 279 ,�ith J. A. Harris. BENEDICT, F. G. and Fox, Proc. Am. Philos. Soc., 1 927., XLVI, 5 1 1 . 130l-lR, C., and HJ\SSELBALCH, K . A., Skand. Arch. Physio!., 1903, xiv,389. IlLUN'r, K. ,' ai, J. BioI. Chem. 1 926, LXV J I , 49 1 . I J RAI)r1ELO, H . S., fvlissouri Agr. Exp. Sta. Research Bulletin 109. '";< CA RMAN , G, G. ami MITCHEI,I,) H. 1- 1 . , AIll. J . Physio� ' J 1926,lxxvi,?80.

---�('" COWGI I.I., G. R., and DRABKIN, D. L., Am. J . Physlo!., 1 927 1xXXI,36. DREYER, C. and RAY, \V., Proc. Hoy. Soc. London, 1 9 10, B., lxxxii, 545;

Phil. Trans. B. cci, 133; ccii, 1 9 1 . DREn:R, G., RAY, W., AND \VA LKER, E. \"'. A. , 1'roc. Hoy. Soc. Londoll,

1 9 1 2- 13, B., lxxxvi, 39, 56. DREYER, G., Lancet, 1 9 1 9, ii, 227. Du 1301S, D., and Du BOIS, E. F., Arch. Int. Meei., 1 9 1 6, xvii, 863. FRONTALI, Rivista eli Clinicia Pediatricia, 1 927, xxv, 24 1 . GI I,E'ITE, 1700<\ Allowances for Healthy Children. New York Association

for J lllproving Conditions of the Poor. 1 9 1 7. I luTCIlINSON, cited by Dreyer, 1 9 1 9. HOGAN, A. G., and SKOUIW, C. l. , J . Agric. Iles. 1923, xxv, 4 1 9. .lULl,. M. A.,cited uy l-1. L. Kempster i n Mo. Agric. Exp. Sta. Res. Bul. 96. KUNDE, M. M . , and STEINHAUSE, A . I- L , Am. J . Physiol., 1 926, lxxviii, 1 27. LISSi\UEIt, 'vV., J ahrb. Kinderh., 1 903, lvi i i , 302. MACLEOD, GRACE. Dissertation, Columbia University, 1924. MEEII, K., Z. BioI ., 1 879, xv, 425. MITCHEI.I" I- J . H., el al., II I . Agric. Exp. Stit. l3uls. 278, ( 1 926 alld J. Agric.

Hcs. 1927, xxxiv, 945, 927, and 945; see also M issouri Agric. Exp. S'n,ioll Hes. Bul. 96, 1927.

MORGULlS, S., Am. J. Physiol., 1 9 1 5, xxxvi, 207. rVIUULTON, 'l'RowIlRIDGE, and l-lAIGII data cited by I-lOGAN and SKOU IJ\'

(see above); also i n M issouri Agric. Exp. Station Res. Bul. 1 8. I-'FAUNDI.Ell, M., Z. Kinderh., 1 9 1 6, xiv, 1 . RUBNER, M., Z. l3io!., 1 883, xix, 536. �TEWA llT, G., Am. J. Ph}'siol., 1 9? 1-2, I �i i i , 45.

.. SEU FF'ERT, R. 'vV., and HERTEl" [-., Z. 13101., 1 924-5, lXXXII, 7. TAKAHlltA, H., Report of the Jmperiai Government Institute for Nutri­

tion, 1927, I, No. 1 . Paper 1 1 . THOMAS, K . , Arch. Anat. 1I. Physiol., Physiol. Abt. 1 9 1 1 , p . 9. THUMPSON, H. B., and MENDE L, L. B., Am. J. Physio!., 1 9 1 8, xli, 43 1 :

see also Beard, I-l. I-I., Am. J . Physio!., 1 926, lxxv, 645. \-VANG, G. 1-1., Am. J . Physiol., 1 925, lxxi, 736. WORNEll, [- J., Z. Gesammt. Expel'. Med., 1923, xxxiii, 5 1 0

I

T GROWTH A N I) DEVELOPMENT X l 4 1

APPENDIX

The charts giving the relation between area, weight, and line'lI­size of the original data that could not be i ncorporated conveniently i n the body o f the text arc given i n the appendix.

'fhe original data on surface area as measured with the surface inte­grator having been analysed, i t appeared desirable to annlyze ill similar manner published data 011 surface area. This we did. The resulting charts are prescnted in this appcndix. In this way this bulletin becomes a fairly complete summary of the available data on surface area.

Since the in terest in the surface area problem is a development of the suggestion that heat production is proportional to surface area, it seemed natural to want to examine the heat production c1ata in the same manner as we have examined the data 011 surface area. Accordingly several charts 011 the relation between heat production and bod}' weight have also been prepared, and the resultR are presented in this appendix.

VVe have not a ttempted to list values for heat production per unit area as computed by the new formulae, as the reader can easily compute sllch values for the particular Ret of data in which he is interested. Al l that he wi l i need to do is to read from the chart (or compute from the given formula) the area for the desired weight, then take a similar read­ing for heat production for the desired weight, and divide the value for heat production by the value for surface area for the given weight .

A n a l ternative method, illustrated by the following example on the dog, suggests i tself. Frolll Fig. 36, heat production is related to weight (in the dog) by the evaluation

E = 88.GIV·" = C,W" \Vhell weight is one kilogram, E = 88.6 calories. From Fig. 340, area i s related to weight ( i n the dog) b y the equation

A = . I OW·"· = C,W'" in which A is area ill sq. ll1�ters, and /1/ is weight in kilograms. When weight is one kilogram, A = . 10 sq. meters, and heat produclion per sq. meter is therefore 88.6 X 10. = 886 calories. In other words, heat pro-

d. . C,

lIction pel' un i t area = - . C2 Since this procedure i n volves extrapolation to one kilogram, and

since in man)' cases the law relating area to weight, or heat production to weight, is not known for weight olle kilogram, therefore this method must be used with caution. There is, of course, 110 objection against lIsing this method if one considers the theoretical heat productioll for weight one kilogram aR an empirical constant in the same sense that olle con­siders, for instance, th __ Meeh constant as empirical.

42 !vr ISSOUIU AGIl. Exp. ST.-\. R I�SE.""I( C I l B UI.I.ETlN 115

,

;;::� llit' _/

• -- ;? -� '7 V; 1:/

t7:!: F-,� -_�:';:.:T

I�! ,

-- -"',

""", � w

Fig. 2J.-TI.e reiuioR of un 10 weighl of I [ucford, female,. The numerah on Ihe charI

inJiule lI,e �1Ie1 of the animal, in OIuntl" .

-1\-,

1 ;;, !-�-

�2 '/.-� -.:,-«

f.(;2 f::;:-' I : 1 : I ;

:w lUI '"' lUI

Pi,. H.-The relation of arc- 1 0 \Veillhl, Shortholn re",.I ... Tire "umer�h indi<:lIc lI,e a��. "f o1,� ."i",.I.

GIWWTII AND DEVELOPMENT XI

I - .. ...

7:- 1# .. �, I:;;:; :-:--

� � -1/" 1/ -/;

/,

-

;---,

,

"" .., w.al�t

1--

-[-

i -[-rol--+--I-I-

.- -

1" 11. 2511.-5),onhorn main.

====-1+- +-1--11,,-: ;,: _ _ _ ,"tm,/I.I,."

.\ ' r • . , • 11 t • � I 1 f . ' hi

. �::;:;:::� ;����\ � ! to:�_� - �Iml -�. I--mill

K� :L�"-"-".��7"·:�.: L�_-"'.-.-'C"�_L",,j,-''',,\-',ro"o,.,i;\i''t�f--�==1:=�200il-;::t�lOO�l::; .. ���,,,�, � .. �,jij""t!;j,", \O • .l;JI�t

fiB. 25b.-Hercford main.

43

\

\

44

• �

1

1",xI' . ST,\, RE.SI�,\RCII B U LLET I N 1 1 5 M I SSOURI J\(.jR.

. '

" 1/

. ,."- ,17.-:4 ' : ,;,-:., r: '

"

,I�l*' "" .";

�,,, ""

\9

em. IU 30 00 00 LO"

. • and levu" 01 the linur mUlu,emcntl

Fig. 27.-The ,eI.tlon bet"te" ITt . I f ,II breed. of beel cattle, 10 the

lor lIudord cutle. Thea" ,CIUIt, �'C t),P'CI 0 .

retuln lor the other bleed, ITt onuucd.

" a:

«

GROWTH AND DEVELOPM ENT X I

s.oll-H--t-H-I-'I-1--!--I---1-+++,- , , -I----�-I-I-hI-l-b","Hd+, 4.011---j--\-++-r-l-¥I-tI' -�H--j-H:J,f,!l;fHm11' 400

' - 1-- I--I--+7f'Hri- I­� \ I-+/----v-�f."+-'H+I--- -/� -;3 p --

1--t--+-+-t-p'l-J- til' 3011----!--+-b1!li"l.-

lOV- I--\-I--I-H-I'- /' / 1---

300

200 I i.· �/�kd=d=�=W���� 1"11 fiO·:=���:::=r�� 6()() 501�=+=4=1I=-+I�I�#-U"m f-----H---j--,H-tl:H--'*I+I'+!-' 1500

40

17. 3 V:: �-

// Vi <'\' 2 0V

---LnJ>1° . .

. --

- - _ . , -I---H,-I-I<i1,J,/f-H'

-- , 2 00

00 / IlOOIOO

WeiQ11t

I ,

6()() -- 500

--400

300

r 200 300 «Xl XO Ii!) IlOO roo

45

Fig. 28.-An Ulempt 10 .. been mad" 10 dUe.m;n" the innuence of the dellle" of 'alnUI on the .tope of the curvet for the bed utile. The animal. ... ere acco,dinSly .�parat�d into "thinl" and "flU" and th�ir ar� .. .. ere plotted IA"inlt "ei,ht II Iho .. n: the .. ,iol of Ihe "�H to Ihe hd,hll .. ere alao

plolted. fn interpreling the curve., it .hould be remembered Ih." the data reluelent ,rO .. inA animall: thlt ii, the compui.on il made hcr .... n n you nil h i (0. thin) animall and malu.e fll (or thin) animall. Thil i, tn altogether diffe.enl malic. from comparing the uu. of animtll of the lime aile (and there­fore nf the lime finear lize) durin, futing or fattening. The Itller p.oblem .... IUIII!Clled 10 UI for in_ vel,i,uion hI' Dr. E. 8. Fotbu, IJi.eClor of the I n.tilllte of Anirnal NlItri,ion It SUle College, I'a. We

pl�nncd to caffY it out. lIowever, the UI,ente involved in luch an undertakinll tOilet her .. i,h Ihe rClult. of the ptu�nt invcuillation, whieh ",ake il di flie"lt for u. to .ee the lifilliliullceof the problem, Cluled II . to huiut., in nnde.takin, il. The • .,'Ultl of ,hi. chan lie not eon.iftelli. In lorne cuel Ih., nponcnu are hi,h.,r for fat animal I (Shollho.n, (.,,,,.Ie.), "'I'ile ill other. the nponenta are hill he. for lI,e thin aniruh (lhrdOld ferule.). Open cirde. replCfent the rduion of Ilea, A, t o l'Iei,hl, W. FuJI cirdCl repluen! th., .eI�lion of area, A, to "'eillhl, W. and 10 heillht at witherl. II.

4 6 Mrssoulu ACR. Exp. STA. RESEAnCH DULLETIN l I S

I'i! 51 I,

1�'�,cb--�"�'���R4i�+�---+--�-+--I'-+��'+I'++11++ I

� -- j ' I� , ' , ' 'I h

'"

tOig. 29a.-Thc r.ration Hta, A. to weight, W, anti to l .. ,ighl at wilherl, II.

GHOWTH AND DEVELOI'M ENT XI

fuOO rTTTTT�Tm�--'---�,,--'TTrnTftum 400 r-·-I-jr-rr-t-I Htl-t - I t---t---i-HTi-j�I-i<I'mlfl;tl 350

"

-- . ,

J�e:, ' __

Kg, ?IJ <It) !'O ffJ 00 1".0

o

• t s

I I 100 lOO 400 500 600 600 iXIO

Weight Fig, 1913.-'1'11" rcl�liOIl of Jru to .... eight and 10 height in beef cattle. The bro­

ken linn rCl'rClenl 5 and 10 pcr ccut deviation. from the average.

47

48

f ( ,

I

,IVl ,sSOURI }\GIL Exp. STA. RESEAI{C II BULLETIN I l S

tID tOO w

JOO

250

100 H-

150 OW

"

I'" 1< /r;: / 120

KQ. lO

, , ' ,

. .

:L-' 0 , '

,,' ,. -,

;;!! :/

I)

, - ,� " ,

/..':j /" f' .-� 'F-

;:t ;2-:.,'

::�:: -;.-"� l 0

•

)()

, 0 . . -- .

' , :§�': , ' �:: .

G I

I IIJU

II .< �"., .-

if

3.v 4 6OO lXJO

Fig. lO.-The lelu'oll of HU 10 ,",ciRiIt �lId 10 height of dailY ('{lle.

-'.

, .

GnOWTII ANI) DEVELOPMENT XI

,I�

C,' l?fl l:;>� 450 400 J�-- -

J

50

200 ,

, , 5Or- - - " , , '::� , " 1

-" . 20 JO 40 !iQ 60

,

00

..

Wi �,;� ,�� ;;; '. :;;1,'

bJP �,i 1/

�;f.,,;;'," 0 '

. ,

I �,/ � ;;8 �"! .-, .-, . ,:;, . . -

,/,/t.

, .' , /&Jt�, " , � ,-

0·' . ' o , .-

/'�t.. � o Ier <'Y ?

. or. ey if Da . e l l �

iXJ

Fig. J I.-(Continued).

49

,

. - _ . .

. . . . .

_" A ....... _ ..... �_ • -

11 .. ----�

----- �

Area

t.i. � � � � " _ .. - ""

- c

< �\ I APea .... ""'" '-" '"

c I .l'! � '"

c · � � � ��l; - - " - .... .. . .... .... "" :> .. - .... "" . 0 - 0 ....

� -§' § -r&

'" "" u' _. � -jg ;;::

f, � �' � T .. .., Do ::: -l -: � � � �

s " :' , " g " '\

. ' , , ..

I I

' ,

I o "

;;: � � t;; :: :. '1 � 1 ,.. 2 � � j .. � c: � ;;: ;:;. ;; � , .. ::. [ : � �

" . ', ' .

" <" .'< ' :',

. ' . , ' .. e: " . H� � t B , , . . H;i�n � §: �

. ,

.:.

�{" I I I I ; ;; ;. s · : s . , " s " , . , " � N

. " ' . . ." . . . I ,. . " .. ,.

!!. � .. =- ::: ';- Do ;:: { :>

:. � . � � � � � �' i � �I ;:! �I :?& . ,\' I �i:�: I I II �i [ r

� i. � t " 5 ,,;: � I ' . �. '"

- I " I 4, I>�: . . ... J I I I I !;1 . .

1 I I ;; ;; ."" :;; " " "

; � � � :: c:.. .

-:i' a ;' "§-3 : � [ t :� � �tH

Sq m

1000 �

11100 .. ) �

� fJJJ <: 500

"

�

200

§ I I I I1i I I I I I �I Ieii' I I '" I § • I I I

I I

§ I I I 0 : § §

<' II II I I I " . I I

.' .' " I

.. I 1 � . • 1

: I I

' , .

I ..

-f- -

I I I I 0 0 I I I I

: o 0 I I

1/.

' , ,

I •

I I I , , ,

' .

� "

, ' • � yo; " I I I

1-0 ' I I 1 I , I I

I I I I I I ..-,' I I I II I I " /":�'lllrje 1 1 v'" I I I I

G: i.Xl1OO "" 3 I

Weight

"' ,.. E n

�� t � .... � C" :. § : � :-. � � � ; � � i � HUH

D

�

I II l ' l ---r%f' y" , /. ·· · 1 I

]/(' 1 I I I 0 I

:00

0 0 I I I

I I

' , ' , 0 0 , I

I I I I I I I I

I I I I I I 1

ll?1,tooL 1lXXJ JOOO om:

figi. Ha and Hb.-The reluion between area, A, �nd wdrbt, W, in the dOG. Theu;._ per hall of FiG. Hb OD tbe right bued on the dau b)" Cowgill �nd Dnbkin indicnu thn in the canol animahoI differeDt weight but of the ,arne body length the area iodiren. Iy i, proportional to 1'"ti£"in railed 10 the power .37, Thi, i. lub.untiated by the CurVel paning throu£"h the trianE)ts in tbe lower hali of the figure. The laur curve repreltnu dog, XIX .nd XVI! in LUl.k'i LaboTuoT)" (Quoted by CO\\'giU and Dubkin) both of

which hive approximHeir the aame body lelllth. Wben. however, both length and wtight \·ary. tht arta i. proportional to tht .70 power of tht weigh!. The conclulion il

� I :;' if 1 . -::.-.:e , ,�� .

I � §; I I .<,':'.! ' , ,1 I I I . 'o� , I � ",-", 1;;- I I tP'. ",:: ,

.. �. . ' " i " � � I - I I ' . . ' .. .

I . ' 0:-. J t :{'''O< � I I - I I I I . , • I

I I I I · I � §

_. I I .

I 0 0 .

k J

[00 T II U 00 ' , . . /0 ' , ;U . ' o � ' 60 �" :..>- • I I · r p' I !D t ,I ' " I I I ' . A-'j- I IE � �

40 .'

�� r I 11100 ' 0 &OOJ 0

\000

IJIOO " j llXXl [.' :' '

���

I I I I

ffi/ I JJ "

. . o , : �

. I I I

, : �. " I ' 1 I I O ' I I 1 '�""11!> uslld<\j; I I \Jw,c.j,oQI1I D:<j,�

(XI JlJlOO J0C004IlJIOO I!il

,

I.ln. 3:1 <V :;) (l) iO 00 ro ZOO :lOO !.DO fiYJ \lI1i(inL C\PcumfcrellC2 of Qlest am Lengtil

thu tht exponent in tnt cQuuion relating arta to "'eight ;1 dif· ferent for GtowinG �nim�ll, or for animalioi different bod)· weight bUlin tht ume Itlle of nutrition. than for animah of the lime lin· ear lilt bUI in different IUtei of nutrition. Compare with Fig. 21 .

�

s: � '" o " �

>-" "

r-1 X � :.n � >

'" " u. '" > � o:i " r

6 Z

'"

Cl g " �

> Z '"

o '" � " � � '" Z �

x �

u,

52 MISSOUIU ACR. Exp. 5TA. H�ES I�Ar(CIi B UCLET1 N l l S

400 IE -.a! lll­

H--/--l-

- -

o

Fig. 15.-'1'1. .. rel�rion between ue� pud weight in the rat.

[00')= -

1:-= � 1Il= -

--' "

=? k���·�

H u[ 8 10 ]) LV � (J)

Weight Fi�. J6.-Th" .eI�tinn bet"'een I,ut l"OIiucrion �nd uody weight in II,,,

dog. Th'. fi8"'" .ho .... thu heu production i. direclly ])loponian,," to weight railed t o t h .. pOI." • . 65. Fig. J4 .howl lhu in <1<181 VU)';"II in Ii""" .ize �I well u in ""i,M (i. e. in dog. of the urn" bod)' lorm) the �.ea ;. directly proportional 10 weight r�i.ed to the power .70; in anim.h of Ihe nrn" body I/ncllt but of .Ii frerent "" i,lr. (i. c .. diRe ring in body 101 mJ. the "' eo ;. proportion_ .t 10 "'eight rai_e,l lo the pown .J7. The an.wer to Ihe qncnion ... I,elhe. heu p,oduuion i. Of il not p",.,o<l;o".l lo lu,hee .. �. ,I.er�fn,� d�l'�ndl on I .. : ... the �re� v.ried with we;"ht of the animall teprcuntcd in tl,il cI" rt, From LUIIe'I dau (ICC FiR, 2 1 ) it "PI'UII tl,�t he"t I" oduct;oll te"d, 10 he I'rol'ort;onal lO the H power ,,' wej�ht e v e n if Are. ".ries only wi,h the .37 power 01 weight, The u"per curve rel'reoenll tI'e meUUlemenll by Ito.,lmer (188J, 1'. S-12); the lower eUt\'e rep,uen" the Inel.u,ement. by LUlie And by Kunde and Stein. houle H cited by tI'e latter.

� .

-

I

GHOWTH A N U DEVELO P M ENT XI

.2 J A !> li .o LU

53

fig. J7 •. -The v.luu of the e�l'onenll in !l'e /elation between hut produetion and body weight of the chiek emb,),o appear 10 Ya'), from .5 to La.

"'�, LL ' I ,!,: ,

-} I: "

� ,,-,-I> :. '; v..:� -:< ;,;.:; : .

�g �i �

� [ KfI Filt· J7b.-The relation between heal P'OduClioll and bod)' weight in the fowl. I f hut

production were prol'oflion�1 10 lurFaee Iru du.lug tl,e whole period of Siowth Ihe Vllue. of the ��I'one"t, n. would be O.67. The v.luco of the eKpouenU "ar)' from 0,68 10 1.3.

a" o c..

'< ..., �. �. 00

F � � � � 0 ::0 .... c.. �

gi ;.-t5

<;;;) •

0\ ....,

N

,

A!'$

.� �

� � 17I b'1 � r � � rio1 . . .. ,

. ,

l 1 1

�f"'" -..., u. 0" ... b.::� ... t-.. � ......

APea � .---

- U! " ... " -.!::" "

APea

.;::;. � \ I Jg � 1; ;1: 8 8: 8 :;1 8

", "" ,, ..£.'

" , -

� Arte1J BH'ru'?-.9 rr

""

Pl ]g t(Crn) X �8 Q " ,

h,

o '1 ir j �. ft :r i 8 � � � � .�: ... ""' '" i;,:.� '1 \

l'I,'sr. I 1 1-'I'tII-1'I'ltf-H::jo'UJ---1 I ','.�\

I •• ��Ef=3 �. � � � �

:z: : o o· .. ::0

.!"' g, ;" SD 9 ;: .. !' ::0

� c;

IS I"" C� U 8 � :.;0

�

l' I> �

�. � .

' . ,, � �c/"

.. � ...... � .. �i� .�

- 'Q.v.;,," � , "j Fl� � e� ,<iT '" r'" �

� ,,- ... C!i ..5: .... 'I -8 -I

-_ .....

r- --. '�l-- - ._+- - . --t--.

§ i 8: � m :§!

I I I I " " I 1 1 1 1 1 1 1 1 1

I '

,Z ,, � � �l�� " �

, ".:0+ ... � 11f'V)

olll�

,,� , A�

� ' : I " , II Ii "

<<I f o 60 W ro OO lOO HelQht

I

•

Mlll, cf ?LX>

Fig. 39.-The relation of heat production to height. Between birth and one year heat production i s proportional to the cube of the height; between 1 and H yean, heat production i s proportional to height. Following about H years, he;at production is. in normally built individuals. nearly independent of height. Compare this chart with Fig. l Ib i n Research Bulletin 104.

rat�� �. �tI!l\ M

50 "

-

� 8:­� �

uv '

0 '0, ..... '" �

40 -"'9 � � 1> ,

-

J:> - ,i/J _0 1 --I

30 '0 !.. '., :..::> =: nrrrn--�r-�-" ���J�j' ����i

�

ro ,� �

o

1 . 1 I I I lu� �---:::-... ?O

'9 '.

a . .�

� , �,

� "IS,� ........ 0 �

---1

40 " , I!II.. 0. 0

---1

Itt o b-.,. �� -t

30 -.. 1L� " '::' � • ::::::

� -,,: . i"'IMJJl nnrnr----t-i����·��·k�!jl �

2cl ' • ,,�l :-'l d�t:I:-:--

10 '2D �Ytt A 'li:n � :: , c!

30 00 !lO W OO l O -':""'" K� 6 WeiQht Fig. 40.-The heat production per u nit weight for different weights be­

tween ages 1 and 29 yean. Plotted from data by Benedict and alSociatea (Carnegie Publications 302 and 279).

U1 �

� tii en o C � )-C') ?' trJ >< ;t' r..n � � rn m > � (") II: to c:: i' i' j Z .... U1

C) � o � .., :::: > Z t:!

t:1 [:l <: [:l 1; .., a:: [:l z .., � -

U1 U1

� I

,

56

J

M I SSOU1H AGIt. Exp. STA. RESEARCH D U LLET I N I t S

",mlll-l+j_I_H+H

7

I : � . j: � ." �m=mm����=.R �� I1I� IJ' II " ulli l -I--H-I+I - I+ I + I I + I-H+. I-HH-+:+++aiflW

,, �1+17':+-1

o 7 6

\'lelght 00

FiS . • I.-Th� .chtion of hUI production, E, 10 "'ciShl, W, and 10 hei""l. II. (comp�rc to Figl. J9 and 12�), An incidtnul faCI made dear b)' Ihil chart il,hal he�' l"oducI;on b)' � no,,,,,1 ;"ai,ialllli rna)' vary by 20 per cen. frorn the �"e,a8e 10' .hc 1'01' .. "";0". ConrpOlc II,il chart wi.h fig. I l h in Rue." ch 1I"lIe,;n I(H.

I

. .... .

GnOWTII AND DEVELOPMENT Xl

(,' Day

1-.iii I'-:� /' ! 1./) � r:: \ .. �

1m :c 100

K j

Fig. 42.-Thc rt'hlioll 1�lwe .. n heat production. E. anti botly weitht, \Y. of luge wild hirtl�. l'lott .. <1 from data by lJem:dict and Fox (Prot'. Am. Phil. Soc., 1927, I,XVI, 5 1 1 ). The birds ineluded. 311d their weillll18

in kilojtfnms. :ne u fol1o ... · s : Cassow:uy, 17.6: C",uIQt, 10.3; Tnuulleter �wan, 8.9; Javal! adjutant, 5.7; Jabiru. 5.5: Jllack·h:lcked \u:licl1l1, 5. 1 ;

Dc:nded ,."lInrt'. 5.1 : Pa,-"disc c:r�n .. (4.0) ; Sa"dhiU en"' ''' , 3.9; Drown pelkall, 3.5: EUfopean namingo, 3.0; Chilean �e:l eagle, 2.9; Cura5sow, 2.8; UI:lck·necked screamer, 2.6; Purpie gu:!.u, 2.0; Mexic:m blue heron,

1 9 ' Mexic:!.l1 blue heron, 1 .9; Mexicau blue heroll, 1.8; M�Jtic:1II lJIue h�r�n, 1 .7 ; Great horned owl, lAS; Paeilic gull, \.2 1 ; Chile�n sku"-,

0.97; White ibis, 0.94 ; "runita .. bittern, 0.60.

57

. �� .. ' -=-�., __ � _M_ ____ _ ---- �. -- --

58 "MISSOURI AGR. EX1'. 51'A. RESEARCH BULLETIN l i S

-t<­I-P­t-I:' 1:< I� o �IQ t ., aat �\ �Ia 1 • l'l l Pro

10

I-1-1-H- I-"'ict

II �!I'

�a '" 22

� f

6 I--t-�

ol

t-

�·illi. H� and du.-Body weight, \V,and hnt Ilrudu.;tioll, 1I,0f umc individual. "Iolle<l againlt age

011 an a,ill,lulI Slit.!. I n d,art inlet for male. a to,n,.ui,ulI ;. luade buwccIl dediuc in hut I>rod ... uion per "'''' weillht w;ll. illcrcuinll �IIC, �"o.l lhe decll"" ill 1",11. Ule. The j>llhc rate �ppcan 10 decline one­

half u rapidly u the hcH productio", lIetween the a�uol 2 and 15 ),CH' body we;ght incrcucl .Il the

rate of 'J pcr cent I'er yeu (uppcr,:ul",,) I"hilclicH I>fOJuClioll ;"Ctcu., � I 6 per cent pcr ycu (lower

curve). Thu ;. \0 UY. Ihe ;"creuc in hut production i. about .66 ol lhc increllc in loudy weillhl for

Ihe ", .. lu; for the femaln, it ;1 .H at grut. TI,� dHh-aud·dot Curvel repruent the toul euer,,' neeoJ.

GHOWTIl AND DEVELOPMENT X I 59

1M l�

I I !,l, PI> c. .. .S r· 1-:: .. '" rDay

Ir I.. I/L bl 1-- 1--- L" 1M I /v ,.., .C .9-

�

i- I- 3=

h f- lo .'

'" \-

' "

GO M ISSOUIU AGR. Exp. STA. RESEARCH BULLETIN J J 5

n Chut B.-Til;. ch�rti. pruented to ""abl,, an)'one to determine tI,,� 'ormula of a powerfunclion

.dating 3.,,;1 to ",cillh! (or hut production 10 weill 1,1). The expon"nt, ". ill the (""clion Y - ex-

;, the ,!ope of the ti"" nil a IOIl�ritl'1lIic llrid. Tili. U t i he determined with a I)�ir of divide ... The coeffi. cielll, C, ;. the valu" of )' when X .. I. If the value for X _ I i. not 011 the ch.rl, then it may be dctermiucd by reading th" ".Iue of Xd for n _ 10, or X _ 100 from thi. ch.rl B, and lolving for C • .1:.

X"

,

I - -

f{l I

i "