Senior,Birnie AccuratelyEstimatingVesselVolumeFromProfileIllustrations

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Accurately Estimating Vessel Volume from Profile IllustrationsAuthor(s): Louise M. Senior and Dunbar P. Birnie, IIIReviewed work(s):Source: American Antiquity, Vol. 60, No. 2 (Apr., 1995), pp. 319-334Published by: Society for American ArchaeologyStable URL: http://www.jstor.org/stable/282143.

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ACCURATELY ESTIMATING VESSEL VOLUME FROM PROFILEILLUSTRATIONSLouise M. Senior and Dunbar P. Birnie III

The highly fragmented nature of most archaeological ceramic assemblages makes whole or reconstructible vesselsvaluable and rare finds. Vessel volume has rarely been systematically quantified because convenient reconstructionmethods dealing with sherds and partial vessels have been lacking. Now, with the method presented in this paper,highly accurate volumetric capacities of fragmented vessels can be calculated from carefully prepared vessel profileillustrations. The profile is digitized using a small number of points per vessel (20 to 30 points are usually sufficient).These data are then converted to a volumetric measure using a computerized algorithm based on the geometry ofstacked bevel-walled cylinders. This method of determining vessel volumes was tested and shown to be highly repeatableand accurate. Quantifiable sources of error are generally limited to less than one percent per vessel, with the finalaccuracy limited chiefly by the quality of illustration. With this computerized technique,fragmented vessels no longerneed to befully reconstructed in order to obtain volumetric information.Los hallazgos de vasijas completas o reconstruibles son muy raros debido a la gran fragmentacion de la mayoria delos conjuntos cerdmicos arqueologicos. El volumen de las vasijas no ha sido cuantificado sistemdticamente debido ala falta de mdetodos adecuados que utilicen vasijas parciales o fragmentos de estas mismas. Ahora, con el metodopresentado en este reporte, capacidades volumetricas exactas pueden ser calculadas para vasijas parciales o frag-mentadas utilizando ilustraciones cuidadosas de los perfiles de las vasijas. El perfil es digitalizado usando pocospuntos para cada perfil (20 o 30 puntos son usualmente suficientes). Los datos son convertidos a un estimado decapacidad volumetrica usando un algoritmo computarizado que se basa en la geometria de cilindros con paredesinclinadas, montados uno encima de otro. Este metodo de determinacion de volumenes de vasijas ha sido puesto aprueba; el metodo es preciso y exacto. Fuentes de error cuantificable son limitadas a menos del uno por ciento porvasija, con la exactitud final dependiente, en su mayor parte, de la calidad de la ilustracion del perfil de la vasija.Con esta tecnica computarizada, la vasijas fragmentadas no necesitan ser completamente reconstruidas con elproposito de obtener informacion volumetrica.

W hy should we examine pot volumes?Prehistoric vessels were made to putthings in, yet most archaeological analysesstudy only what is on the vessels. Prehistoricvessels were manufactured as containers;whether for cooking, storage, or serving, ves-sel volume (capacity) is an essential attributethat is tied directly to use and function ofpottery. Most methods used to determinevessel capacity depend on the use of whole,or reconstructed, pots. Because of this, mostanalyses of prehistoric pottery, worldwide, arerestricted to surface decoration techniques("style") rather than pottery function sincesuch studies can be made from the large frag-

mented assemblages generally encounteredarchaeologically.1 Techniques of quick, ac-curate volumetric determination from vesselprofiles now exist; thus, this attribute (ca-pacity) should no longer be neglected in ce-ramic analyses.Vessel volume generally relates to size,weight, and transportability of ceramic ves-sels; thus it is significant in exchange and use-life models based on ceramics (Deal 1983:155-6; DeBoer 1985; Graves 1985:23; Lon-gacre 1981:63-64, 1985; Rice 1987:293-306;inter alia). Design variability can also be re-lated to variability in vessel size, as has beendocumented in the Kalinga ethnoarchaeolog-

Louise M. Senior * Department of Anthropology, University of Arizona, Tucson, AZ 85721Dunbar P. Birnie III * Department of Materials Science and Engineering, University of Arizona, Tucson, AZ 85721American Antiquity, 60(2), 1995, pp. 319-334.Copyright ? 1995 by the Society for American Archaeology

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ical assemblage by Graves (1981, 1985). De-sign variability within an assemblage may alsobe partitioned by vessel size because of thevaried length of time that vessels were used;thus small pots may have more up-to-datedesign motifs than large vessels if sampledsynchronically as was observed by DeBoer(1984:557) among the Shipibo-Conibo.Vessel volume is also an attribute fromwhich artifact function can be deduced, al-though multiple lines of inference are neededto adequately assess systemic function fromvessel shape (e.g., Hally 1986; Smith 1988).Vessel capacity measurements have some-times been interpreted as reflecting house-hold size and thus been incorporated intomodels of regional settlement, subsistence,and paleodemography (e.g., Blinman 1986a:86ff.; Blinman 1986b:602-607; Nelson 1981;Smith 1988; Turner and Lofgren 1966).Systematic vessel volume studies may re-veal emic categorization of vessels by sizeand/or capacity (e.g., suggestions by Longa-cre et al. 1988; Miller 1982). Relative stan-dard units of volume or multiples of suchunits may be expressed and disclosed throughvessel capacity measurements, and thus aportion of the metrological scheme of a pastculture can be deduced (e.g., Gelb 1982;Mainkar 1984; Nelson 1985; Powell 1989;Rottliinder 1967; Turner and Lofgren 1966).In addition, the standardization hypothesis(Balfet 1965:163; Costin 1991; Feinman etal. 1984:299; Rice 1981:220-221; Rice 1991;Sinopoli 1988:586) frequently invoked as agauge of relative craft specialization in ce-ramic studies may be more appropriately ap-plied to vessel capacities than to other metricattributes of pottery. Most studies of stan-dardization implicitly assume that produc-tion efficiency is a byproduct of craft routin-ization; further examination of why stan-dardization is important or linked to morecomplex levels of sociopolitical organizationis frequently left unstated.2The "reason" that vessel sizes are so stan-dardized in complex societies may be morerooted in concern for specific, market-relatedvolumetric measures than in motor skill rou-

tinization alone. The interpretation of UrukBevel Rim bowls as ration vessels, althoughcontroversial, is partly based on their stan-dardized capacities (Alden 1973; Beale 1978;Chazan and Lehner 1990; Johnson 1973; LeBrun 1980; Miller 1981; and Surenhagen1975). Ration vessels are noted in Assyriol-ogical literature(e.g., Gelb 1965; Powell 1989)and the Akkadian concern for accurate men-suration undoubtedly led to standardizationof vessel capacities by potters (Senior andWeiss 1992).As suggested by the brief overview above,study of ceramic vessel capacity will contrib-ute very significant data to our knowledge ofancient material culture. Application of thetechnique described in this paper will greatlyincrease the ability of researchers to obtainvolumetric information from their frag-mented ceramic assemblages. In turn, thiswill begin to redirect the study of ceramicsfrom what is "on" pottery to how specificpots functioned.Previous Estimations of Vessel Capacity

Even though the attribute "vessel capacity"has been relatively neglected in archaeolog-ical analyses of ceramics, varied methods thatdetermine vessel volume exist in the litera-ture. Most of these techniques rely on the useof whole or reconstructed pots; thus they havebeen useful only for those relatively rare finds.Other techniques, if based on specific keymeasurements of vessels (e.g., height, diam-eter, etc.), are relevant only when the assem-blage analyzed is highly standardized in termsof vessel form (e.g., Ericson and DeAtley1976; Fitting 1970:174-175; Fitting and Hal-sey 1966; Hagstrum and Hildebrand 1990;Michael, Grantz and Maslowski 1974;Mounier 1987). The primary techniques thathave been applied to archaeological finds arediscussed below.Researchers should realize that vessel ca-pacity can be defined in at least two ways. Itcan be measured as "total possible capaci-ty"-that is, capacity up to the meniscus ofthe vessel rim. Obviously, this is probablynot a practical capacity for the vessel, but it

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is easily replicable between researchers. The"effective volume" of a vessel is frequentlycited in archaeological literature (e.g., Hally1986); this is a measure of capacity up to thepoint of greatest constriction, or to some pointat which the investigator believes the vesselwas considered full. Though perhaps morerealistic than the total capacity, estimationsof effective volume are subject to observererror. There is also danger inherent in Hally'spoint that vessel orifice is related to vesselcapacity; though somewhat true in a broadsense, this factor varies by specific assem-blages. Either the total possible capacity orthe effective volume can be measured withthe methods listed below.Fluid Volume MethodMeasurements of the water contained in avessel is one of the most obvious and expe-dient ways to ascertain comparative volumesof vessels. It is also advantageous because itdoes not require any special equipment orprograms for calculation. There are four pri-mary limitations to the accuracy of this meth-od. First, it is possible for water to be ab-sorbed by the vessel walls, making a slightchange in the apparent volume; this can beremedied by lining the vessel with plastic be-fore water is added, or by measuring the ca-pacity after the vessel walls are fully saturat-ed. Second, it is limited by the true accuracyof the measuring container. For example,most common laboratory glassware is ratedonly to within five percent of its nominalvolume. Third, use of water to measure ca-pacity limits the analysis exclusively to wholeand reconstructed vessels; fragments are notuseful in any way. Finally, many museumsand other institutions will not allow use ofwater after the objects are accessioned intocollections for fear of damage. Water, or oth-er materials placed inside vessels, may phys-ically damage and contaminate fragile paint-ed decorations or residues on pots.Dry Volume MethodThis method can be performed with any kindof free-flowing solid (lentils, rice, bird, or

mustard seeds, etc.), but many museums al-low only the use of lightweight polystyrenepacking material ("styritos") since these arebelieved to be stable, inert substances thatwill not alter the vessel in any way. The meth-od is essentially similar in all respects to thewater method detailed above; vessels are filledwith a solid substance, and then the quantityof this substance is carefully measured.This method may not be as precise or re-peatable as the water method: Styritos tendto pack with a lot of air space between them.When styritos are measured with beakers orother measuring devices, only a fairly roughestimate of quantity is calculated. Also, thepoint at which the volume is "leveled off,"such as at the rim of the vessel (or measuringdevice), is somewhat arbitrary and subject tothe investigator's eye (whereas water wouldrun out of the vessel if it exceeded the ca-pacity, even slightly). This method also re-quires complete or reconstructed vessels;profiles of vessels cannot be measured. Shouldfree-flowing solids be used in volumetricmeasurements, researchers should try to usematerials with very small radii (such as mus-tard seed), since these are not as subject toerrors of heaping and packing. Heaping caneasily be remedied by leveling the moundedsolid with a straight rod drawn across themouth of the vessel (see Mainkar 1984:148).Density MethodsVessel capacity can also be estimated by mea-suring the mass of material (of known den-sity) filling the vessel. This may eliminateadditive errorsaccumulated when measuringdevices are repeatedly used (such as measur-ing a 30-liter vessel with a one-liter contain-er). Mainkar (1984:148) summarizes thismethod:

Take well-sifted sand. . . allow the sand to flow undergravity into the pot and when a heap of sand has beenformed on the mouth of the pot, stop the flow. Levelthe heap with a straight rod or plate so that the sandis filled up to the top of the mouth of the pot. Weighthe pot and the sand. Remove the sand and weigh theempty pot. Determine the density of sand in a suitablemanner. The volume of the pot can be calculated fromthe weight of the sand and its density.

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Note that vessel capacity may also be mea-sured this way using water, which has a moreeasily standardized density than sand. Thismethod, though eliminating some error dueto repeated use of laboratory glassware, isdependent on both the accuracy of the scaleused and the utilization of complete vessels.Methods Using the Assessment ofGeometric SolidsSeveral investigators have employed for-mulae derived from geometry to assess thegeneral shape of pottery studied, and thusderive their respective volumes using keymeasurements from the pots in these for-mulae. Primary examples of these methodshave been discussed by Castillo Tejero, andLitvak (1968), Ericson and Stickel (1973) andJohnson (1973). Although these methods cangenerally render a useful value for vessel ca-pacity, they are usually not very accurate ifthe assemblage is varied in form and littlestandardized. Care must be taken to com-pensate for vessel wall thickness. Because ex-terior pot dimensions are most commonlytabulated, the volume could be overestimat-ed, even if the geometry is well-suited to thevessel form.As discussed by Rice (1987:220-222), theCastillo Tejero-Litvak method assigns nu-merical values to vessel silhouettes and toreference points on the silhouettes. Vesselsare thus described as sections or combina-tions of the geometric forms described bythese points (e.g., a rectangular solid minusan ovaloid of specific dimensions). Thismethod can be used to calculate capacity. Italso has the benefit of not disturbing the in-terior contents of the vessel (since it is basedentirely on observation of vessel dimen-sions), and thus will not disturb prospects forresidue analyses.One drawback to this method is that ityields only an idealized representation of thevessel volume. The accuracy of the geometricrepresentation depends on how smoothly theinside surfaces of the vessel mimic the set ofshapes used to render the volume estimate.Thus it may only be useful for vessels that

have convenient geometric outlines. A mod-ification of this method was applied by Senior(1990).The method employed by Ericson andStickel (1973) incorporates both the geomet-ric form(s) representative of the vessel andthe appropriate measurements (in millime-ters) of the vessel in volumetric formulae.Advantages and drawbacks to this methodare essentially the same as those stated abovefor the Castillo-Tejero-Litvak method.Johnson (1973:135) used measured attri-butes of Uruk straight-sided bevel-rim bowlsto ascertain volumes geometrically. Thesequickly made vessels were hypothesized tobe ration bowls; thus their capacity (and itsrange of variation) was deemed interesting.Because the bowls all have the same generalshape, a single formula was used. The ad-vantages of this method are that it can beapplied to fragmentary as well as whole ves-sels; all that is needed is a complete vesselprofile so that all the pertinent attributes couldbe measured. Disadvantages are that themethod is specific only to straight-sided, openvessel forms.Calculus Method-Stacked CylindersNelson (1985:312-131) estimates vessel vol-ume through the calculus method of "stackedcylinders," which envisions the vessel as di-vided horizontally into a series of slices (seeFigure 1). The interior diameter of the vesselis measured for each slice, from mouth tobase; such slices represent the diameters ofvery thin cylinders (greatly exaggerated inFigure 1). Stacked one on top of another, thesecylinders represent the entire vessel's volume(Rice 1987:221-222). Several researchers inJapan have used this method for makingmeasurements from scale drawings (e.g., Fu-jimura 1981; Kobayashi 1992).Smith (1983; 1985:262) also determinesvessel capacity with calculus. Pot capacity iscalculated "by integration as the volume ofthe solid of revolution formed by revolvingthe profile curve around the x axis" (Smith1985:262). Smith first derives a polynomialexpression to fit each vessel profile, but this

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Figure 1. Schematic representation of the stacked cyl-inder technique for calculating the volume of a vessel.is essentially the stacked cylinder calculusmethod. These polynomial expressions caneasily be used to assess vessel volume fromintegration of its revolution (Smith 1985:291-293). Smith's polynomials are derived so thathe can assess other vessel variables besidescapacity from the expression with his com-puter-simulation technique (SHAPE). Smith'sintegration of vessel profile polynomials ishighly accurate if the polynomial truly ren-ders the pot shape; however, it is still only asgood as the theoretical limit of the calculusintegration, wherein the "true" value of thevessel volume is calculated from addition ofthe areas of an infinite number of "slices" orstacked cylinders.Data from a vessel profile, if accuratelyrendered, can be utilized when working withcalculus-based capacity measures; thus, re-construction or dependence on whole vesselsis not necessary. This opens up a much largersample for analysis since the majority of ar-chaeological finds are highly fragmented andeven if whole vessels are found, full recon-struction is very time consuming and thuscostly. This method can also yield fairly ac-curate volumetric data if enough points aremeasured (i.e., if enough "cylinders" are cal-culated).The limitation of this method lies in therough boundary formed by the cylinders ofdifferent diameters stacked on each other(Figure 1). Also, accurate measurement of thevarious cylinders is somewhat difficult andtime consuming to procure; this could be fa-

bR1H1

R2H2

R3 /Figure 2. Schematic representation of the use of bevel-walled cylinders for integrating the volume of a vessel.The illustration uses the same vessel but fills it with edge-matching beveled cylinders. The small outlined box (topright) defines the enlarged area, which shows exactly howdifferent radii and heights are defined. R1, R2, and R3are radii measured at sequential positions along the ves-sel interior. H, and H2are the heights of successive slices.

cilitated by taking measurements from well-executed scale drawings of the vessel profiles(see new method, below). Because the cal-culations required in Nelson's "stacked cyl-inder method" are fairly time consuming, thismethod would be greatly expedited throughuse of a computer spread-sheet program, eventhough this is not mentioned in the publishedwork (Nelson 1985:312-313). Such a pro-gram would quickly process the computa-tions and store the estimated capacities sothat they could later be statistically com-pared.

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New Method of Vessel Capacity EstimationVessel volumes can now be quickly measuredfrom scale drawings.3The cross sections (pro-files) are digitized and fed into a computerwhere an integration is performed to yieldseveral characteristic features: volume, max-imum diameter, rim diameter, and height.This method, like others using calculus, as-sumes vessel symmetry around the axis ofthe illustrated profile; it is thus not suitablefor dramatically asymmetric pottery. In-depthdescriptions of the algorithm, its implemen-tation, and utilization are presented in fol-lowing sections.AlgorithmThe new method uses an improved stacked-cylinder method. The algorithm, rather thanbeing restricted to vertical-walled cylinders,incorporates the calculus for cylinders withslanted edges.4 The edges match as they arestacked, and thus provide a smoother, morecontinuous, rendition of the vessel profile.Figure 2a shows the use of slanted cylindersfor modeling the vessel capacity (comparewith Figure 1 drawn with the same vessel).When using the stacked cylinder methodabove, each cylinder has a radius measuredat half of its height, so that the overlappingcomers will yield some canceling of errorsand approximate the true volume. Figure 2ashows how the same vessel would be modeledby slanted cylinders. This gives a shape thatis much smoother than that given by the ver-tical-walled stack represented in Figure 1.The vessel is first represented by a list ofcoordinates that are recorded directly fromthe outline of the vessel (to get the vessel'scapacity the outline must be taken on theinterior wall as presented in the projection ofthe cross section). This defines a sequence ofpoints where the radius is known. Theseknown radius positions then define the topsand bottoms of beveled cylinders, which,when summed, determine the volume of thevessel.Each beveled cylinder has a volume thatis determined by the height of the cylinderand the top and bottom radii (defined as H,

R1, and R2, respectively). The individualbeveled cylinder volume is:V= (R2+ RR2 + R2)3 2 2 (1)This formula was used by Johnson (1973) toestimate volume of straight sided vessels; forhis application, R1 is the rim diameter, R2 isthe base diameter, and H is the total height.In the present algorithm, the total vesselvolume is simply the sum of the volumes ofall N cylinders that have been defined by trac-ing the outline of the vessel interior:

V= - '(R2 + RRi+RI+ R2 1) (2)i=1

When writing a computer program to im-plement this summation, care must be takento ensure that the outline is defined in a par-ticular order around the vessel. It is usuallymost expedient to start at one rim and workdirectly around the vessel. Each cylinder isdefined by the points used to define the inputoutline. Adjacent outline points define suc-cessive radii (Ri and Ri+ ) by calculating thehorizontal distance from the center line. And,the vertical separation between the two pointsdefines the height (Hi). Figure 2b illustrateshow the successive radii and heights are se-lected. This close-up segment from Figure 2ashows two beveled cylinders being used tomodel part of the volume. The upper beveledcylinder has R, as the top radius, R2 as thebottom radius, and H1 as the height. AfterEquation 1 has been evaluated for this slice,then the summation moves to the next slicein sequence. To force the beveled cylindersto match smoothly the bottom radius fromthe previous cylinder is used as the top forthe next one; thus R2, R3, and H2 will be usedfor the next calculation, and so on.Since the height calculation is based on asequentially traced contour, any reversals indirection will yield a negative height that willresult in a negative volume for that slice. Thisassures that concave-bottomed vessels areaccurately rendered using this method, be-cause the dimpled base region actually re-duces the capacity of the vessel.

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ImplementationThis technique has been implemented usingan IBM AT-compatible personal computerand a Summagraphics "Summasketch" dig-itizing tablet. A powerful computer is not re-quired to run this program efficiently andquickly; it has also been used with a dual-floppy portable PC computer connected tothe same digitizer. The computer and digi-tizer are attached through the parallel com-munications port (using RS232 protocol). Thepresent program for communicating with thetablet and for later integrating the volume iswritten in PASCAL, although any languagewould suffice.The digitizing tablet is the key element be-cause it enables very easy data collection fromscale illustrations. The "mouse" (or "puck")has a cross-hair that can accurately point atsequential positions along contours, and theirlocation is specified simply by pressing a but-ton on the mouse. This arrangement allowsa large number of points to be accuratelymeasured from a scale drawing of the vesselprofile.

Two versions of the program have beenimplemented. The first works with full-potillustrations; the second version uses half-potillustrations but requires the specification ofthe location of the midline of the pot (sincethere are not two sides to average). The half-pot program is favored by these authors forthree reasons: (1) interior vessel profiles aregenerally only illustrated on one side of avessel following common illustration con-ventions; (2) it is faster to digitize only halfof the vessel; and (3) there is no noted dif-ference in accuracy between the two methodsso the slightly faster version is preferred.ProceduresAs with all techniques, reasonable care mustbe taken when measuring the volume. Theillustration must be carefully sized and ori-ented to guarantee accurate data collection.As an example, we describe the measurementof a half-pot cross section illustration sincethis is slightly more complex and, as notedabove, it is our preferred method.

The scale drawing is first taped to the dig-itizing tablet. The entire drawing must be dig-itized without removing it; thus, the illustra-tion must be proportioned to fit within theactive area of the digitizer (usually about onefoot square). This may sometimes requirephotocopy reductions to make the drawingfit. As long as the illustration's scale is re-duced by the same ratio, then the potentialdistortion of the photocopying process willhave little impact on the final measured vol-ume.5Care must be taken when affixing the draw-ing to the tablet; it must be directly verticalsince the X-Y position of the cross-hairs isused to define the sequential positions alongthe contour. To assure this, the illustration'scenterline should be aligned such that it isperpendicular to the edge of the digitizer'sactive area. This can easily be done with adrafting triangle or T-square. After attach-ment, the illustration scale is digitized bypointing to each end of the scale marker andthen entering the measured length valuethrough the computer keyboard. Next, pointafter point along the vessel profile is enteredinto the program. Each point is entered bycarefully moving the cross-hairs so that it isdirectly on the contour. Once it is in position,one of the mouse buttons is pressed to registerits location. Then the mouse is moved to thenext position along the contour, thus tracingthe outline. Measured points can be takenwith arbitrary spacing along the contour; theprogram does not require that they be evenlyspaced to produce a capacity measurement.

There is no specific guideline on how close-ly the data points should be spaced along thecontour. It is usually better to gather as manydata points as patience will allow, but thepoints can usually be spaced somewhere be-tween a millimeter and a centimeter apartand still yield excellent volumes, as discussedbelow. The spacing of the data points can alsobe adjusted by the user when entering thedata; portions of the cross section that havesharp curvature, carinations, or other oddshapes will require more closely spaced inputpoints. With practice, most vessel profiles can

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be digitized in less than two minutes usingthis program.After the half-contour is completely spec-ified, then the centerline is specified. Whenthese data have been collected, the programquickly goes about integrating the sequentialvolume elements and presenting the final ves-sel characteristics. These calculations arecompleted within only a few seconds.Discussion

This procedure offers distinct advantages overmany other methods for determining volumebecause whole vessels are not required. Thelevel of accuracy afforded by this techniqueis examined in close detail below. Quanti-tative comparisons with other techniques arealso made. Many Computer-Aided Design(CAD) programs offer software that can per-form similar functions to the procedure de-scribed in this article; however, such pro-grams are usually very expensive and maynot be feasible for all archaeological projects.The sources of error described below will beroughly equivalent whether a CAD programor the "Senior-Birnie" method is applied toestimate vessel capacity from scale drawings.Over the last 30 years, efforts to identifyand reduce various sources of sampling, ex-planation, and measurement errors in ar-chaeological investigations have increased.Assessment of measurement error is especial-ly important when developing new methodssuch as that discussed in this article becauseerror in measurement can be magnified whenusing highly sensitive digitizing equipmentand thus be a significant source of variationin archaeological observations. Such contrib-uted variability must not, however, be con-fused with the variation present in the ar-chaeological record.6Before making any in-depth analysis of thesources and magnitudes of error found fordifferent techniques, it is useful to define var-ious terms that relate to measurement errorin general. The most important distinctionto make is between accuracy and precision.Accuracy defines how close the measuredvolume is to the "true" volume. Accuracy isthe fundamental evaluation of measurements

that have been made; we desire our mea-surements to be as accurate as possible. Pre-cision, on the other hand, is an assessmentof the quantitative quality of the measure-ment itself and is not a relative comparisonwith the "true" value. Two of the key ele-ments that play a part in precision are (1) thenumber of significant digits that a measure-ment has, and (2) the repeatability of a par-ticular measurement.7 The number of signif-icant digits is usually the number of decimalplaces that can be obtained from a particularmeasuring instrument (ruler, tape measure,beaker, etc.). Note that most measuring de-vices have relatively low instrumental pre-cision.8 For instance, most metric rulers havegraduated scales at increments of 1 mm, forc-ing measurements to have only about onesignificant digit at the centimeter level.Repeatability is related to the number ofsignificant digits, but it also incorporates somelevel of random error that might cause vari-ation in repeated measurements using theparticular instrument. The repeatability cannever be better than the precision of the mea-surement, but it could be far worse if therandom error contribution is large. Randomerrors are usually assumed to be normallydistributed, which is the basis for using themean and standard deviation as quantifica-tion of the variability of measured values.Even after a set of repeated measurementshas been made (yielding a mean and standarddeviation), there may not be certainty thatthe resultant values represent an accurate an-swer. In fact, many measurement techniqueshave systematic error contributions to eachmeasurement. An example of systematic er-ror would be a ruler that had changed lengththrough thermal expansion or had its endabraded through use, thus reducing its length.These errors give an average bias in one di-rection or another, causing the result to berepeatable, but inaccurate. These errors areusually much more difficult to identify andquantify than random measurement error. Insummary, accuracy is limited by all sourcesof error, including instrumental precision,random measurement error, and systematicerror in measurement. Because of these def-

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initions, accuracy can never be better thanprecision, although precision may easily bebetter than accuracy for any given measure-ment.The sources of error found in this methodfall into three basic categories: those inherentto creating a digitized illustration, those fromthe digitization process, and those that arepart of the mathematical integration of vol-ume elements. These various contributionswill be discussed here; wherever possible, at-tempts have been made to quantify thesecontributions.Error in Illustration PreparationThe image digitized is usually a photocopyof a higher quality illustration. This illustra-tion (and its photocopy) have built-in errorcontributions that come from several factors:

1) Vessel profile reconstruction from theexisting sherds may be inaccurate;2) Illustration of the vessel profile may beinaccurate;93) Rendering of the scale marker may beinaccurate; and4) Photocopying may stretch or distort theillustration. 10

Whenever fragmented vessels are recon-structed (especially for partial reconstruc-tions), the possibility exists that the sherdswill not be glued into exactly their originalplace, shape, or configuration. If only vesselprofiles are being reconstructed, this couldcontribute error to the production of the pro-file sketch. Great care should be taken in ex-actly matching the existing sherds during re-construction; best results will probably be ob-tained by experienced curators. There will bemore error with large vessels than small onessimply because more sherds are being refit,and potentially ill-fit, into the vessel profile.Ironically, it is generally easier to produce anaccurate interior profile rendering from afragmented pot than an unbroken one, so thismethod is probably best when used with frag-mented remains, despite possible distortionsdue to reconstruction.Because the present method relies heavilyon the use of an accurately rendered scale

illustration of the vessel profile after recon-struction, production of this illustration isprobably the greatest potential source of errorin the method. In many regions of the world,highly accurate illustrations of all vessel pro-files are routinely prepared by professionalillustrators; because these illustrators pro-duce so many sherd drawings, their work isassumed to be very consistent. It is very dif-ficult, however, to assess the real accuracy ofthis step.Vessel profile renderings are rarely esti-mated from smaller fragments;when they are,illustrators routinely signify such guessti-mates with dotted lines. For obvious reasons,use of such illustrations should be avoidedsince any error in the estimation will be dra-matically compounded when the metrics areconverted to volumetric units.When applying this (or any) method to alarge comparative sample, it is ideal that alldrawings used be prepared by the same il-lustrator.1' This will reduce random varia-tions between illustrators, and any systematicerror will not interfere with comparisonsmade between various vessel assemblages.Any possible error in rendering the scaleon a drawing is particularly insidious becauseit will magnify or compress the entire illus-tration by the amount of error in the scale.Since we are calculating volumes directlybased on this linear scale marker, such errorwill be amplified by the mathematical oper-ations used. This will typically triple the ef-fective error from scale marker to final mea-sured capacity because length is taken to thethird power to find volume.12Photocopying of published works or illus-trations (which might be necessary to allowattachment to the surface of the digitizingtablet) can also contribute error.To minimizesuch error, it is critical that the illustration'sscale be included on the image copied. In thisway, the scale will be subjected to the samepossible distortions as the rest of the image.It is assumed that photocopying error is arelatively uniform stretching or reductionspecific to each machine used. For most cop-ying machines this effect is rather minor andwould only become apparent if copies were

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made sequentially from earlier copies, am-plifying the effect.Mechanical Errors in DigitizationThe digitization process itself also has limi-tations. These limitations have two sources:mechanical and human. Mechanical limita-tions can best be overcome if the illustrationdigitized is at least a 1:1 rendering of the pot;however, this may not always be feasible be-cause of the expense of extra-large digitizersneeded for over-sized drawings. The more anillustration is reduced, the greaterthe amountof error introduced by slight differences inlocation of digitized points. It is desirable touse as much of the active area of the digitizingtablet as possible. This increases the resolu-tion of the measurements being made.Even though there are limits to the preci-sion of the numbers obtained from the dig-itizing tablet, no constraint is placed nor-mally on the accuracy based on this effect.Digitizing tablets usually measure X-Y po-sitions with an absolute accuracy of about 1/20of a millimeter; this significantly exceeds theaccuracy of most illustrations or photocopies.Human Error in DigitizationThe manual dexterity, "eye," and general pa-tience of the individual digitizing an illustra-tion can also contribute to, or influence, thehuman error in implementation of this meth-od. The actual choice of digitized points is acrucial factor in this method.The subjective choice of digitized pointsimpacts three areas: (1) defining the outline,(2) defining a centerline for a half-vessel il-lustration, and (3) defining the length of theillustration scale. These three categories aredifficult to separate from each other. Instead,the aggregateerrorfor these contributions canbe estimated by having a single user measurethe same drawing several times and calcu-lating the coefficient of variation in the re-sultant volumes.A measurement of these human error ef-fects was conducted using the geometric ves-sel profile shown in Figure 3. This illustrationhas an idealized shape that allows the exact

volume to be calculated directly from theknown dimensions used to plot it (again usingEquation 2). Thus, comparison between themeasurement and the true value is allowed.Figure 3 was measured 10 times; each newmeasurement was performed after removingand reattaching the illustration to the activearea of the digitizing tablet. Any human errorinvolved in accuracy of illustration orienta-tion, placement, and attachment was there-fore included in this process.The vessel volume from these 10 mea-surements is 24,690 ? 105 cc. This comparesvery favorably with the true calculated valueof 24,609 cc. The coefficient of variation involume measurements was .004. Since thesemeasurements included all steps of the pro-cess, the coefficient of variation directlyquantifies the aggregate contribution of aim-ing errorbecause of variations in manual dex-terity when "aiming" the digitizer puck. Thisextremely small error contribution is be-lieved to be significantly smaller than errorsderived from illustration or reconstructionprocesses.Good results also depend to a large extenton whether the user is familiar with the dig-itizing equipment, the program, and the pro-cess of performing the measurement. Thequality of the technique, even when appliedby beginners, is illustrated by another com-parison: the variation in volumes as mea-sured by many different novice users.In sampling 24 individuals, many of whomhad never touched any digitizing equipment,we found a coefficient of variation of .029.13A real pot illustration, rather than an ideal-ized one, was used in this study. This indi-cates that even novice users can achieve aprecision of closer than three percent Webelieve that any of these users could reachprecision levels closer to .004 with only lim-ited practice.Error in Mathematical AlgorithmFinally there may be some error in the math-ematical method for estimating volume. Thisis consistently true for calculus-based tech-niques. All of these techniques become more

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I I I I I I0 10Figure 3. Cross section of a hypothetical geometricalobject for testing the human error accuracy of the digi-tizer volume-measurement technique. The true volumecan be found by summing two beveled disk volumes usingEquation 2.accurate as the size of the increment becomessmaller. In the limit of infinitely small incre-ments, then the answer reaches the true val-ue. Such a mathematical limit is never ac-tually achieved. The mathematical errorsaccumulated when integrating the vessel pro-file illustrations can be simulated by digitiz-ing a curved shape as shown in Figure 4. Thisperfect hemisphere also has a volume thatcan be geometrically calculated. When it wasmeasured 10 separate times (with careful re-moval and reattachment to the digitizing tab-let), the volume was 16,470 + 86 cc, givinga coefficient of variation of .005. The truevolume was calculated to be 16,755 cc, basedon the geometry of the drawing. The digitizedvolume is approximately 1.7 percent smallerthan the true volume, significantly beyondthe random error of the technique. This is aresult of the concave hemispherical shape:

I I I I I I0 10Figure 4. Cross section of a hemisphere for quantifi-cation of summation errors resulting from the use of bev-eled disk volume elements in calculating capacity.each digitized layer is approximated as hav-ing a straight wall, which erroneously trun-cates a tiny bit of the inside of the volume.Still, this error contribution is extremelysmall.Comparison with Other MethodsFinally, we have compared all of the abovetechniques with our computer-based meth-od. For this comparison we have chosen twovessels that were available to us both as wholevessels and as cross-sectional illustrations.Figure 5 shows the cross-sectional views ofthe two vessels. The left and right vesselswere drawn to the same scale and will bereferredto as "A" and "B," respectively. Forboth of these pots we measured their volumeusing the liquid capacity (with water)and theirdry capacity (using rice and "styritos"). Eachof these measures was made three times foreach substance in each vessel; these valueswere averaged and are presented in Table 1.The volume was then calculated using asimple ellipsoid model using:

DrD2H6 (3)

where D is the largest diameter and H is theheight. Next, the volume was calculated bythe method of stacked cylinders, using incre-ments of 1 cm and measuring the radius atthe center of each slab. Finally, the illustra-

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tion was digitized using the new method. Theresultant volumes (in cc) are given in Table1. The last row of the table gives our bestestimate of the absolute accuracy of eachtechnique.If capacity measured with rice grains orwater is considered the "true" capacity ofvessels, then the Senior-Birnie method de-scribed in this paper compares favorably asshown in Table 1. Rice and water are shownto yield approximately the same figures forvessel capacity. Rice could give slightly largervalues than water because it can be "heaped,"whereas water will drain out.'4 These mea-surements are our most accurate direct mea-surements of volume. "Styritos" are less ac-curate and are not recommended for furtheruse in volumetric studies. The inaccuracy ofpolystyrene packing material is caused by (1)the inability to pack them tightly and (2) theinability to "level" off the measuring device.They are also difficult to work with since stat-ic electricity makes them cling to vessel walls.The geometric solid method may be good forthese vessels since their form is nearly ellip-soidal.The Senior-Bimie method is mainly lim-ited by the accuracy of the illustration in thiscomparison, since the method was proven tobe highly repeatable (precise) in the tests onFigures 3 and 4. This method has been dem-onstrated to be highly accurate and fast; oncethe illustration is created, this method is thefastest of all methods tested.Future DirectionsThe technique described in this paper reliesvery heavily on an accurate illustration; infact, it can be said to be only as good as theillustrator involved. It may thus seem morefruitful to digitize vessel photographs to elim-inate the rendering error. However, this willonly partially alleviate errors in volumetricestimations for at least four reasons. First,great care must be exercised during the photoshoot so that the vessel is photographed as atrue profile without any included distortion.Second, even if the vessel is in the exact pro-file position, parallax distortion around the

edge may hinder exact rendering of the profilewith the digitizer. Third, care must be takento place the scale for the photograph in ex-actly the same plane as the pot to avoid ex-aggeration of errors based on the scale. Fi-nally, most vessels photographed this waywill be whole pots; therefore the capacity es-timated will be based only on the exteriorprofile of the vessel, and will equal the vesselcapacity plus the clay used in vessel produc-tion. Were these factors addressed, then useof photographs rather than scale drawingscould be expeditious.The digitization of photographs is an im-portant future direction for applications ofthe Senior-Birnie method since many ar-chaeologists routinely use photographs ratherthan illustrator's renderings. Parallax distor-tion may be an easily resolvable problem; thephotogrammetry and remote sensing litera-tures should be consulted for algorithms thatcompensate for the distortions of differentkinds of lenses. Vessels should be photo-graphed with the longest possible lenses andwith the largest format cameras practicableso as to minimize distortion. Future researchdirections should also include estimates thatcompensate for vessel wall thickness whenusing photographs of exterior vessel profiles.Conclusions

A new technique for measuring accurate ves-sel capacities from profile illustrations hasbeen developed. It is based on the use of adigitizing tablet to quantify the shape of avessel cross-sectional illustration and a com-puter to integrate the volume numerically.Whole or reconstructed vessels are not need-ed, although this method will also work oncomplete artifacts. Additionally, capacity ofsymmetrical archaeological features can becalculated using this method if accurate fea-ture sections are digitized.Future investigations might also use pho-tographs, rather than vessel renderings, forproduction of profile information. New il-lustration techniques developed by HarrisonEiteljorg and Nancy Wilkie may also greatlyfacilitate use of this method of capacity es-

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Figure 5. Ethnographic vessels from Paradijon, Philippines, collected by M. Neupert (illustrated by MasashiKobayashi). Note that the ridges shown on the interior of the lower vessel are rills remaining from manufacture;they are not illustration flaws or products of computer-scanning technologies.

timation: pottery profiles can now be accu-rately drawn using AutoCAD 12 and a dig-itizing light pen (Eiteljorg 1994).Advantages of this technique are that thisprogram runs very quickly on a standard per-sonal computer (PC). Most vessels can beaccurately measured from scale drawings inonly a couple of minutes. The brief time re-quired for this method allows investigatorsto average several readings for the best pos-sible assessment of the vessel capacity. Mostarchaeological projects are now equipped witha computer adequate for this computation;additional costs of obtaining a digitizing tab-let currently run approximately $250 to $300.

The method described herein provides a lowbudget alternative that supplies very fast, ac-curate, and repeatable measurement of vesselor feature capacities using scale illustrations.Public Access to Pot-Volume Program

The authors will provide the compiled PAS-CAL program on diskette as "shareware." NoMacintosh version is available. Send your ad-dress, disk density, and size preference plusU.S. $5.00 (to cover postage and disk pur-chase) to the authors. Please do not requestthis material through American Antiquity.Acknowledgments. The comments on this manuscriptby the American Antiquity editorial staff, the anonymous

Table 1. Comparison of Volumes Measured Using Several Different Techniques.Geometric Stacked StackedMeasurement Solid Vertical Bevel-walledMethod Water Rice Styritos Ellipsoids Cylinders Cylinders

Vessel A 1350 1400 1650 1640 1580 1515Vessel B 2400 2400 2550 3385 2280 2440Estimated Accuracy 5% 5% 5% + ?% . % ?%+ Ren% 2% + Ren%Note: The estimated accuracies have been assigned based on quantifiable sources of error discussed in the text.All three of our direct volume measurements used measuring devices with about 5% basic accuracy. When usingstyritos, however, an unknown amount of additional inaccuracy occurs due to problems with packing and measuringthe larger pieces. Note that the measurement of Vessel A is about 20% larger than that found using water or rice.For the geometric shape representation method we have not quantified the error contribution, but again Vessel Ayields a much larger value, suggesting error in the range of 15 to 20%. Of course, this depends on how well the potmatches the geometric representation. Finally, both stacked cylinder methods derive their measurements fromillustrations of the vessels rather than directly from the vessels. Rendering error from the illustration process hasnot been quantified but is included in these two columns as "Ren%." We believe that experienced illustrators willhave accuracies at about the 5% level. (As explained in the text, this level of error would arise from linearmeasurement and rendering error at a level of less than 2%.) For the vertical-walled cylinder method we have notquantified the error contributions because of the mathematical summation; however, Vessel A gave a value about15% larger than the direct measurement. In the case of the bevel-walled cylinder method, the 2% estimated errorcontribution is based on the 1.7%difference between calculated and measured values found when testing Figure 4.This includes all contributions due to illustration attachment, pointing error, and repeatability.

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reviewers, and Barbara Mills were greatly appreciatedby the authors, and we thank them for significantly im-proving the final product. Terry Majewski and JanetWalker greatly facilitated submission of this article withtheir fine editorial skills: thank you very much We alsowish to thank the "EGGS" of the University of ArizonaAnthropology Department (particularly Andrea Free-man, Charles Tompkins, and Helga W6cherl) for initialinspiration and challenge in developing this method. NinoAimo kindly translated our abstract. Many thanks toCarol Kramer and William Longacre, as well as to MarkNeupert, for the use of their ethnographically collectedvessels in testing this method. Kramer graciously al-lowed use of her excellent illustrations, as well as fielddata collected on vessel capacities. Masashi Kobayashiallowed use of his vessel illustrations produced for hisPh.D. dissertation on Philippine ceramic ethnoarchaeol-ogy. Diane Dittemore allowed Senior access to Kalingavessels stored in the Arizona State Museum and gra-ciously supplied a seemingly endless volume of "styri-tos" for use in vessel capacity measurements. We heartilythank you all for your help and support. A preliminaryreport and test of this method appeared in Senior andGann (1992).

References CitedAlden, J. R.1973 The Question of Trade in Proto-Elamite Iran.Unpublished Master's thesis, Department of An-thropology, University of Pennsylvania, Philadel-

phia.Arnold, P. J.1991 Domestic Ceramic Production and Spatial Or-ganization: A Mexican Case Study in Ethnoar-chaeology. Cambridge University Press, Cam-bridge.Balfet, H.1965 Ethnographical Observations in North Africaand Archaeological Interpretation: The Pottery ofthe Maghreb. In Ceramics and Man, edited by F.Matson, pp. 161-177. Viking Fund Publications inAnthropology No. 41, New York.Beale, T. W.1978 Bevel Rim Bowls and Their Implications forChange and Economic Organization in the LaterFourth Millennium B.C. Journal of Near EasternStudies 37:289-313.Blinman, E.1986a Additive Technologies Group Final Report.Chapter 2 in Dolores Archaeological Program: FinalSynthetic Report, compiled by David A. Breternitz,C. K. Robinson, and G. T. Gross, pp. 633-661. U.S.Department of the Interior, Bureau of Reclamation,Engineering and Research Center, Denver.1986b Technology: Ceramic Containers. In DoloresArchaeological Program: Final Synthetic Report,compiled by David A. Breternitz, C. K. Robinson,and G. T. Gross, pp. 595-609. U. S. Departmentof the Interior, Bureau of Reclamation, Engineeringand Research Center, Denver.Castillo Tejero, N., and J. Litvak1968 Un Sistema de Estudio Para Formas de Vasi-

jas. Technologia 2. Departamento de Prehist6oria,Instituto Nacional de Antropologia e Historia, Mex-ico City.Chazan, M., and M. Lehner1990 An Ancient Analogy: Pot Baked Bread in An-cient Egypt and Mesopotamia. Paleorient 16:21-35.Coombes, G. B.1979 The Archaeology of the Northeast Mojave Des-ert. Bureau of Land Management, Cultural Re-sources Publications, Archaeology.Costin, C.1991 Craft Specialization: Issues in Defining, Doc-umenting, and Explaining the Organization of Pro-duction. In Archaeological Method and Theory, vol.3, edited by M. B. Schiffer, pp. 1-55. University ofArizona Press, Tucson.Daniels, F., J. H. Mathews, J. W. Williams, P. Bender,and R. A. Alberty1956 Experimental Physical Chemistry. McGraw-Hill, New York.Deal, M.1983 Pottery Ethnoarchaeology Among the TzeltalMaya. Ph.D. dissertation, Simon Fraser University,Burnaby, British Columbia.DeBoer, W. R.1980 Vessel Shape from Rim Sherds: An Experi-ment on the Effect of the Individual Illustrator.Jour-nal of Field Archaeology 7:131-135.1984 The Last Pottery Show: System and Sense inCeramic Studies. In The Many Dimensions of Pot-tery, edited by S. Van der Leeuw and A. C. Prit-chard, pp. 527-571. University of Amsterdam, Al-bert Egges van Giffen Instituut voor Prae-en Pro-tohistorie, Amsterdam.1985 Pots and Pans Do Not Speak, Nor Do TheyLie. In Decoding Prehistoric Ceramics, edited by B.Nelson, 347-357. Southern Illinois University Press,Carbondale.Dibble, H. L., and M. C. Bernard1980 A Comparative Study of Basic Edge AngleMeasurements. American Antiquity 45:857-865.Eiteljorg, H. II1994 AutoCAD for Pottery Profiles. CSA Newsletter6(4):5-9. Center for the Study of Architecture, BrynMawr, Pennsylvania.Ericson, J. E., and S. P. DeAtley1976 Reconstructing Ceramic Assemblages: An Ex-periment to Derive the Morphology and Capacityof Parent Vessels from Sherds. American Antiquity41:484-489.

Ericson, J. E., and E. G. Stickel1973 A Proposed Classification System for Ceram-ics. World Archaeology 4:357-367.Feinman, G., S. Kowalewski, and R. Blanton1984 Modelling Ceramic Production and Organi-zational Change in the Pre-Hispanic Valley of Oa-xaca, Mexico. In The Many Dimensions of Pottery,edited by S. Van der Leeuw and A. C. Pritchard,pp. 297-337. University of Amsterdam, Albert Eggesvan Giffen Instituut voor Prae-en Protohistorie,Amsterdam.Fish, P. R.1978 Consistency in Archaeological Measurementand Classification: A Pilot Study. American Antiq-uity 43:86-89.

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Fitting, J. E.1970 The Archaeology of Michigan. Natural HistoryPress, Garden City, New York.Fitting, J. E., and J. P. Halsey1966 Rim Diameter and Vessel Size in Wayne WareVessels. The Wisconsin Archaeologist 47:208-211.Fujimura, H.1981 Measuring Vessel Volume. Archaeological Re-search (Koukogaku Kenkyu) 28:106-117.Gelb, I. J.1982 Measures of Dry and Liquid Capacity. Journalof the American Oriental Society 103:585-590.Graves, M. W.1981 Ethnoarchaeology of Kalinga Ceramic Design.Ph.D. dissertation, Department of Anthropology,University of Arizona, Tucson.1985 Ceramic Design Variability within a KalingaVillage: Temporal and Spatial Processes. In Decod-ing Prehistoric Ceramics, edited by B. Nelson, pp.9-34. Southern Illinois University Press, Carbon-dale.Hagstrum, M., and J. Hildebrand1990 The Two-curvature Method for Reconstruct-ing Ceramic Morphology. American Antiquity 55:388-403.

Hally, D. J.1986 The Identification of Vessel Function: A CaseStudy from Northwest Georgia. American Antiquity51:267-295.Johnson, G. A.1973 Local Exchange and Early State Developmentin Southwestern Iran. Anthropology Paper No. 51.Museum of Anthropology, University of Michigan,Ann Arbor.Kobayashi, M.1992 Changes in a Formal Assemblage from JomonPottery to Yayoi Pottery. Hokuetsu Archaeology(Hokuetsu Koukogaku) 5:1-34.LeBrun, A.1980 Les Ecuelles Grossieres: Etat de la Question.In L 'Archeologiede L 'Iraq,edited by M. T. Barrelet,pp. 59-70. Centre Nationale Recherche Scientifique(CNRS), Paris.Longacre, W. A.1981 Kalinga Pottery, and EthnoarchaeologicalStudy. In Pattern of the Past, edited by I. Hodder,G. Isaac, and N. Hammond, pp. 49-66. CambridgeUniversity Press, Cambridge.1985 Pottery Use-life Among the Kalinga, NorthernLuzon, the Philippines. In Decoding Prehistoric Ce-ramics, edited by B. Nelson, pp. 334-346. SouthernIllinois University Press, Carbondale.Longacre, W. A., K. Kvamme, and M. Kobayashi1988 Southwestern Pottery Standardization: AnEthnoarchaeological View from the Philippines. TheKiva 53:101-112.Mainkar, V. B.1984 Metrology in the Indus Civilization. In Fron-tiers of the Indus Civilization, edited by B. B. Laland S. P. Gupta, pp. 141-151. Indian Archaeolog-ical Society, New Delhi.Michael, R. L., D. Grantz, and R. Maslowski

1974 Vessel Diameters from Sherds: A Mathemat-ical Approach. Pennsylvania Archaeologist 44(4):42-43.

Miller, A.1981 Straw Tempered Ware. In An Early Town onthe Deh Luran Plain: Excavations at Tepe Faru-khabad, edited by H. T. Wright. Memoirs No. 13,pp. 126-130. Museum of Anthropology, Universityof Michigan, Ann Arbor.Miller, D.1982 Artefacts as Products of Human CategorisationProcesses. In Symbolic and Structural Archaeology,edited by I. Hodder, pp. 17-25. Cambridge Uni-versity Press, Cambridge.Mounier, R. A.1987 Estimation of Capacity in Aboriginal ConoidalVessels. Journal of Middle Atlantic Archaeology 3:95-102.Nelson, B.1981 Ethnoarchaeology and Paleodemography: ATest of Turner and Lofgren's Hypothesis. Journalof Anthropological Research 37:107-129.1985 Reconstructing Ceramic Vessels and TheirSystemic Contexts. In Decoding Prehistoric Ceram-ics, edited by B. Nelson, pp. 310-329. Southern Il-linois University Press, Carbondale.Plog, S.1985 Estimating Vessel Orifice Diameters: Mea-surement Methods and Measurement Error. In De-coding Prehistoric Ceramics, edited by B. Nelson,pp. 243-253. Southern Illinois University Press,Carbondale.Plog, S., F. Plog, and W. Wait1978 Decision Making in Modern Surveys. In Ad-vances in Archaeological Method and Theory, vol.1, edited by M. B. Schiffer, pp. 383-421. AcademicPress, New York.Powell, M. A.1989 MaBe und Gewichte. Reallexikon der Assy-riologie und Vorderasiatischen Archiologie 7:457-517.Rice, P.1981 Evolution of Specialized Pottery Production:A Trial Model. Current Anthropology 22:219-240.1987 Pottery Analysis: A Sourcebook. University ofChicago Press, Chicago.1991 Specialization, Standardization and Diversity:A Retrospective. In The Ceramic Legacy of Anna0. Shepard, edited by R. Bishop and F. Lange, pp.27-279. University Press of Colorado, Niwot.

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Notes' We do not imply that style is easier to infer from sherdsthan function; rather, style is commonly examined inorder to establish the chronological and ethnic affiliationof archaeological assemblages. One could arguethat stylefrom fragmented assemblages is just as difficult to de-termine, if not more so, than function.2 Exceptions to this are the connections made betweenpolitical economy and standardization explicitly dis-cussed by Feinman et al. 1984 and Costin 1991. Notealso Philip Arnold's recent work wherein a high degreeof standardization is evident in the absence of craft spe-cialization in Tuxtla, Mexico (Arnold 1991).3 Use of photographs, rather than scaled drawings, isdiscussed at the end of this article.4 In geometry this shape is called afrustrum (pl.frustra);"truncated cone" is also sometimes used.

5 Photocopy distortion is often greatest around the edgesof the image and drawings should therefore be scruti-nized in these regions for error.6 Various archaeologists have already discussed thisproblem (Coombs 1979:68-70; DeBoer 1980; Dibble andBernard 1980; Fish 1978; S. Plog, F. Plog and Wait 1978:414; S. Plog 1985; Tuggle 1970:86; inter alia).7 Repeatability is frequently called replicability in thearchaeological literature.8 Instrumental precision is also called "sensitivity" inarchaeological, and other, literatures.9 Use of appropriate photographs could renderthis prob-lem moot; however, most photographs would not yieldinterior capacity measurements since the outside of thevessel wall would usually be digitized. Use of photo-graphs, and errors inherent to their use, is discussed laterin this article.10Many photocopy machines are constructed to delib-erately slightly reduce the image copied to approximately99-98 percent of the original size.I If photographs are used, it is also best that they beprepared by the same photographer, or at least one usingthe same criteria for aligning the vessel in true profileposition.12 The effective error is not cubed. During the mathe-matical operation of multiplication, the fractional errorcontributions from the two factors are added. So, whenfinding the volume, each of the three dimensions addsto the error (see Daniels et al. 1956:324ff. for discussionof experimental error calculation).13These individuals participated in this project at theThird Southwest Symposium, Tucson, Arizona, duringthe poster session presentation by Senior and Gann(1992).14See Mainkar (1984:148) for method to avoid heapingof small solids.

Received February 24, 1993; accepted July 20, 1994.

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Documents

Senior,Birnie AccuratelyEstimatingVesselVolumeFromProfileIllustrations