The extinct,giant giraffid Sivatherium giganteum: skeletal reconstruction and body mass estimation

Sivatherium giganteum is an extinct giraffid from the Plio–Pleistocene boundary of the Himalayan foothills. To date, there has been no rigorous skeletal reconstruction of this unusual mammal. Historical and contemporary accounts anecdotally state that Sivatherium rivalled the African elephant in terms of its body mass, but this statement has never been tested. Here, we present a three-dimensional composite skeletal reconstruction and calculate a representative body mass estimate for this species using a volumetric method. We find that the estimated adult body mass of 1246 kg (857—1812 kg range) does not approach that of an African elephant, but confirms that Sivatherium was certainly a large giraffid, and may have been the largest ruminant mammal that has ever existed. We contrast this volumetric estimate with a bivariate scaling estimate derived from Sivatherium''s humeral circumference and find that there is a discrepancy between the two. The difference implies that the humeral circumference of Sivatherium is greater than expected for an animal of this size, and we speculate this may be linked to a cranial shift in centre of mass.     

Downsizing a giant: re-evaluating Dreadnoughtus body mass

Estimates of body mass often represent the founding assumption on which biomechanical and macroevolutionary hypotheses are based. Recently, a scaling equation was applied to a newly discovered titanosaurian sauropod dinosaur (Dreadnoughtus), yielding a 59 300 kg body mass estimate for this animal. Herein, we use a modelling approach to examine the plausibility of this mass estimate for Dreadnoughtus. We find that 59 300 kg for Dreadnoughtus is highly implausible and demonstrate that masses above 40 000 kg require high body densities and expansions of soft tissue volume outside the skeleton several times greater than found in living quadrupedal mammals. Similar results from a small sample of other archosaurs suggests that lower-end mass estimates derived from scaling equations are most plausible for Dreadnoughtus, based on existing volumetric and density data from extant animals. Although volumetric models appear to more tightly constrain dinosaur body mass, there remains a clear need to further support these models with more exhaustive data from living animals. The relative and absolute discrepancies in mass predictions between volumetric models and scaling equations also indicate a need to systematically compare predictions across a wide size and taxonomic range to better inform studies of dinosaur body size.     

Stature and body mass estimation from skeletal remains in the European Holocene

Techniques that are currently available for estimating stature and body mass from European skeletal remains are all subject to various limitations. Here, we develop new prediction equations based on large skeletal samples representing much of the continent and temporal periods ranging from the Mesolithic to the 20th century. Anatomical reconstruction of stature is carried out for 501 individuals, and body mass is calculated from estimated stature and biiliac breadth in 1,145 individuals. These data are used to derive stature estimation formulae based on long bone lengths and body mass estimation formulae based on femoral head breadth. Prediction accuracy is superior to that of previously available methods. No systematic geographic or temporal variation in prediction errors is apparent, except in tibial estimation of stature, where northern and southern European formulae are necessary because of the presence of relatively longer tibiae in southern samples. Thus, these equations should bebroadly applicable to European Holocene skeletal samples.     

Estimating the body mass of extinct ungulates: a study on the use of multiple regression

The correlation between body mass and both skeletal and dental measures in living mammals has enabled paleontologists to obtain reliable estimates of body size for extinct species, usually using log-transformed bivariate least-squares regression equations. Multiple regression, however, has rarely been used for estimating the mass of extinct species, although this technique can clearly improve the predictive equations compared with those adjusted by simple regression. However, the use of multiple regression is problematical, because even those functions explaining a high percentage of the variance of the dependent variable (i.e. body mass) can show a rather limited predictive power. After analyzing which factors determine the predictive ability of multiple regression equations, we propose a new set of algorithms that allow the estimation of the body mass of extinct ungulates. These algorithms are finally applied to three Miocene ungulate species, Dinohippus leidyanus , Stenomylus hitchcocki and Aletomeryx scotti .     

The Postcranial Skeleton of an Exceptionally Complete Individual of the Plated Dinosaur Stegosaurus stenops (Dinosauria: Thyreophora) from the Upper Jurassic Morrison Formation of Wyoming,U.S.A.

Although Stegosaurus is one of the most iconic dinosaurs, well-preserved fossils are rare and as a consequence there is still much that remains unknown about the taxon. A new, exceptionally complete individual affords the opportunity to describe the anatomy of Stegosaurus in detail for the first time in over a century, and enables additional comparisons with other stegosaurian dinosaurs. The new specimen is from the Red Canyon Ranch Quarry, near Shell Wyoming, and appears to have been so well preserved because it was buried rapidly in a pond or body of standing water immediately after death. The quarry is probably located in the middle part of the Morrison Formation, which is believed to be Tithonian in age in this area. The specimen is referable to Stegosaurus stenops based on the possession of an edentulous anterior portion of the dentary and elevated postzygapophyses on the cervical vertebrae. New information provided by the specimen concerns the morphology of the vertebrae, the iliosacral block and dermal armor. Several aspects of its morphology indicate the individual was not fully skeletally mature at the time of death, corroborating a previous histological study.     

New body mass estimates for Omomys carteri,a middle Eocene primate from North America

We report new body mass estimates for the North American Eocene primate Omomys carteri. These estimates are based on postcranial measurements and a variety of analytical methods, including bivariate regression, multiple regression, and principal components analysis (PCA). All body mass estimation equations show high coefficients of determination (R²), and some equations exhibit low prediction errors in accuracy tests involving extant species of body size similar to O. carteri. Equations derived from PCA-summarized data and multiple regression generally perform better than those based on single variables. The consensus of estimates and their statistics suggests a body mass range of 170–290 g. This range is similar to previous estimates for this species based on first molar area (Gingerich, J Hum Evol 10:345–374, 1981; Conroy, Int J Primatol 8:115–137, 1987). Am J Phys Anthropol 109:41–52, 1999. © 1999 Wiley-Liss, Inc.     

Postnatal long bone growth in terrestrial placental mammals: Allometry,life history,and organismal traits

The ontogenetic allometry of long bone proportions is poorly understood in Mammalia. It has previously been suggested that during mammalian ontogeny long bone proportions grow more slender (positive allometry; length ∝ circumference^>1.0), although this conclusion was based upon data from a few small‐bodied taxa. It remains unknown how ontogenetic long bone allometry varies across Mammalia in terms of both taxonomy and body size. We collected long bone length and circumference data for ontogenetic samples of 22 species of mammals spanning six major clades and three orders of magnitude in body mass. Using reduced major axis bivariate regressions to compare bone length to circumference, we found that isometry and positive allometry are the most widespread patterns of growth across mammals. Negative allometry (i.e., bones growing more robust during ontogeny) occurs in mammals but is largely restricted to cetartiodactyls. Using regression slope as a proxy for long bone allometry, we compared long bone allometry to life history and organismal traits. Neonatal body mass, adult body mass, and growth rate have a negative relationship with long bone allometry. At an adult mass of roughly 15–20 kg, long bone growth shifts from positive allometry to mainly isometry and negative allometry. There were no significant relationships between ontogenetic long bone allometry and either cursoriality or basal metabolic rate. J. Morphol. © 2012 Wiley Periodicals, Inc.     

Body mass and stature estimation based on the first metatarsal in humans

Archaeological assemblages often lack the complete long bones needed to estimate stature and body mass. The most accurate estimates of body mass and stature are produced using femoral head diameter and femur length. Foot bones including the first metatarsal preserve relatively well in a range of archaeological contexts. In this article we present regression equations using the first metatarsal to estimate femoral head diameter, femoral length, and body mass in a diverse human sample. The skeletal sample comprised 87 individuals (Andamanese, Australasians, Africans, Native Americans, and British). Results show that all first metatarsal measurements correlate moderately to highly (r = 0.62-0.91) with femoral head diameter and length. The proximal articular dorsoplantar diameter is the best single measurement to predict both femoral dimensions. Percent standard errors of the estimate are below 5%. Equations using two metatarsal measurements show a small increase in accuracy. Direct estimations of body mass (calculated from measured femoral head diameter using previously published equations) have an error of just over 7%. No direct stature estimation equations were derived due to the varied linear body proportions represented in the sample. The equations were tested on a sample of 35 individuals from Christ Church Spitalfields. Percentage differences in estimated and measured femoral head diameter and length were less than 1%. This study demonstrates that it is feasible to use the first metatarsal in the estimation of body mass and stature. The equations presented here are particularly useful for assemblages where the long bones are either missing or fragmented, and enable estimation of these fundamental population parameters in poorly preserved assemblages.     

Body size and body shape in early hominins - implications of the Gona Pelvis

Discovery of the first complete Early Pleistocene hominin pelvis, Gona BSN49/P27, attributed to Homo erectus, raises a number of issues regarding early hominin body size and shape variation. Here, acetabular breadth, femoral head breadth, and body mass calculated from femoral head breadth are compared in 37 early hominin (6.0-0.26 Ma) specimens, including BSN49/P27. Acetabular and estimated femoral head sizes in the Gona specimen fall close to the means for non-Homo specimens (Orrorin tugenesis, Australopithecus africanus, Paranthropus robustus), and well below the ranges of all previously described Early and Middle Pleistocene Homo specimens. The Gona specimen has an estimated body mass of 33.2 kg, close to the mean for the non-Homo sample (34.1 kg, range 24-51.5 kg, n = 19) and far outside the range for any previously known Homo specimen (mean = 70.5 kg; range 52-82 kg, n = 17). Inclusion of the Gona specimen within H. erectus increases inferred sexual dimorphism in body mass in this taxon to a level greater than that observed here for any other hominin taxon, and increases variation in body mass within H. erectus females to a level much greater than that observed for any living primate species. This raises questions regarding the taxonomic attribution of the Gona specimen. When considered within the context of overall variation in body breadth among early hominins, the mediolaterally very wide Gona pelvis fits within the distribution of other lower latitude Early and Middle Pleistocene specimens, and below that of higher latitude specimens. Thus, ecogeographic variation in body breadth was present among earlier hominins as it is in living humans. The increased M-L pelvic breadth in all earlier hominins relative to modern humans is related to an increase in ellipticity of the birth canal, possibly as a result of a non-rotational birth mechanism that was common to both australopithecines and archaic Homo.     

Predicting euarchontan body mass: A comparison of tarsal and dental variables

Objective: Multiple meaningful ecological characterizations of a species revolve around body mass. Because body mass cannot be directly measured in extinct taxa, reliable body mass predictors are needed. Many published body mass prediction equations rely on dental dimensions, but certain skeletal dimensions may have a more direct and consistent relationship with body mass. We seek to evaluate the reliability of prediction equations for inferring euarchontan body mass based on measurements of the articular facet areas of the astragalus and calcaneus. Methods: Surface areas of five astragalar facets (n = 217 specimens) and two calcaneal facets (n = 163) were measured. Separate ordinary least squares and multiple regression equations are presented for different levels of taxonomic inclusivity, and the reliability of each equation is evaluated with the coefficient of determination, standard error of the estimate, mean prediction error, and the prediction sum of squares statistic. We compare prediction errors to published prediction equations that utilize dental and/or tarsal measures. Finally, we examine the effects of taxonomically specific regressions and apply our equations to a diverse set of non‐primates. Results: Our results reveal that predictions based on facet areas are more reliable than most linear dental or tarsal predictors. Multivariate approaches are often better than univariate methods, but require more information (making them less useful for fragmentary fossils). While some taxonomically specific regressions improve predictive ability, this is not true for all primate groups. Conclusions: Among individual facets, the ectal and fibular facets of the astragalus and the calcaneal cuboid facet are the best body mass predictors. Since these facets have primarily concave curvature and scale with positive allometry relative to body mass, it appears that candidate skeletal proxies for body mass can be identified based on their curvature and scaling coefficients. Am J Phys Anthropol 157:472–506, 2015. © 2015 Wiley Periodicals, Inc.     

Variation in the ontogenetic allometry of horn length in bovids along a body mass continuum

Allometric relationships describe the proportional covariation between morphological, physiological, or life‐history traits and the size of the organisms. Evolutionary allometries estimated among species are expected to result from species differences in ontogenetic allometry, but it remains uncertain whether ontogenetic allometric parameters and particularly the ontogenetic slope can evolve. In bovids, the nonlinear evolutionary allometry between horn length and body mass in males suggests systematic changes in ontogenetic allometry with increasing species body mass. To test this hypothesis, we estimated ontogenetic allometry between horn length and body mass in males and females of 19 bovid species ranging from ca. 5 to 700 kg. Ontogenetic allometry changed systematically with species body mass from steep ontogenetic allometries over a short period of horn growth in small species to shallow allometry with the growth period of horns matching the period of body mass increase in the largest species. Intermediate species displayed steep allometry over long period of horn growth. Females tended to display shallower ontogenetic allometry with longer horn growth compared to males, but these differences were weak and highly variable. These findings show that ontogenetic allometric slope evolved across species possibly as a response to size‐related changes in the selection pressures acting on horn length and body mass.     

内蒙古和林格尔县土城子遗址战国时期人群的体质量

体质量推算作为体质人类学的一项重要研究方法,为衡量古代居民体型提供了新的途径,同时也为解决考古学、历史学问题提供了新的研究思路。本文通过"生物力学"和"形态测量学"两种方法对内蒙古和林格尔县土城子遗址2005年出土的34例战国时期人骨标本进行了体质量推算。统计分析结果表明,该组男性居民的平均体质量约为67.21kg,女性居民的平均体质量约为54.63kg,男女两性之间的差异显著。通过与战国时期内蒙古长城地带井沟子居民的比较分析,土城子男性居民在平均体质量上明显大于井沟子男性居民,并且体质量值整体分布偏高,而女性组则差异不显著。结合考古学、历史学相关研究材料,该数据为土城子居民为戍边军士的身份提供了进一步的证据。     

Estimation of stature and body mass from the skeleton among coastal and mid-altitude Andean populations

Adult stature and body mass represent fundamental biological characteristics of individuals and populations, as they are relevant to a range of problems from assessing nutrition and health to longer term evolutionary processes. Stature and body mass estimation from skeletal dimensions are therefore key to addressing biological and social questions about past populations. Anatomical reconstruction provides the most direct proxy for living stature but is only suitable for well-preserved remains. Regression equations for estimating stature from bone lengths are therefore extremely useful, though it is well recognized that differences in body proportions limit the cross-application of equations between samples. Here, we assess the accuracy of published stature estimation equations from worldwide and New World groups applied to archaeological samples from the central Andean coast and highlands of South America. As no existing equations are clearly appropriate, new sample-specific regression equations are presented. Anatomical stature reconstruction is further complicated by artificial cranial modification (ACM) influencing cranial height in Andean samples, so this problem is investigated in the current sample. Although ACM has minimal impact here, the possibility should be explored in other samples before anatomical stature estimation is attempted. Recommendations are also made for estimating body mass from femoral head diameter. The mean of three previously published equations is shown to offer minimal bias and the most reliable estimate of body mass in the study samples.     

Body size prediction from juvenile skeletal remains

There are currently no methods for predicting body mass from juvenile skeletal remains and only a very limited number for predicting stature. In this study, stature and body mass prediction equations are generated for each year from 1 to 17 years of age using a subset of the Denver Growth Study sample, followed longitudinally (n = 20 individuals, 340 observations). Radiographic measurements of femoral distal metaphyseal and head breadth are used to predict body mass and long bone lengths are used to predict stature. In addition, pelvic bi-iliac breadth and long bone lengths are used to predict body mass in older adolescents. Relative prediction errors are equal to or smaller than those associated with similar adult estimation formulae. Body proportions change continuously throughout growth, necessitating age-specific formulae. Adult formulae overestimate stature and body mass in younger juveniles, but work well in 17-year-olds from the sample, indicating that in terms of body proportions they are representative of the general population. To illustrate use of the techniques, they are applied to the juvenile Homo erectus (ergaster) KNM-WT 15000 skeleton. New body mass and stature estimates for this specimen are similar to previous estimates derived using other methods. Body mass estimates range from 50 to 53 kg, and stature was probably slightly under 157 cm, although a precise stature estimate is difficult to determine due to differences in linear body proportions between KNM-WT 15000 and the Denver reference sample.     

A computational analysis of limb and body dimensions in Tyrannosaurus rex with implications for locomotion, ontogeny, and growth

The large theropod dinosaur Tyrannosaurus rex underwent remarkable changes during its growth from <10 kg hatchlings to >6000 kg adults in <20 years. These changes raise fascinating questions about the morphological transformations involved, peak growth rates, and scaling of limb muscle sizes as well as the body's centre of mass that could have influenced ontogenetic changes of locomotion in T. rex. Here we address these questions using three-dimensionally scanned computer models of four large, well-preserved fossil specimens as well as a putative juvenile individual. Furthermore we quantify the variations of estimated body mass, centre of mass and segment dimensions, to characterize inaccuracies in our reconstructions. These inaccuracies include not only subjectivity but also incomplete preservation and inconsistent articulations of museum skeletons. Although those problems cause ambiguity, we conclude that adult T. rex had body masses around 6000-8000 kg, with the largest known specimen ("Sue") perhaps ～9500 kg. Our results show that during T. rex ontogeny, the torso became longer and heavier whereas the limbs became proportionately shorter and lighter. Our estimates of peak growth rates are about twice as rapid as previous ones but generally support previous methods, despite biases caused by the usage of scale models and equations that underestimate body masses. We tentatively infer that the hindlimb extensor muscles masses, including the large tail muscle M. caudofemoralis longus, may have decreased in their relative size as the centre of mass shifted craniodorsally during T. rex ontogeny. Such ontogenetic changes would have worsened any relative or absolute decline of maximal locomotor performance. Regardless, T. rex probably had hip and thigh muscles relatively larger than any extant animal's. Overall, the limb "antigravity" muscles may have been as large as or even larger than those of ratite birds, which themselves have the most muscular limbs of any living animal.     

Minimum convex hull mass estimations of complete mounted skeletons

Body mass is a critical parameter used to constrain biomechanical and physiological traits of organisms. Volumetric methods are becoming more common as techniques for estimating the body masses of fossil vertebrates. However, they are often accused of excessive subjective input when estimating the thickness of missing soft tissue. Here, we demonstrate an alternative approach where a minimum convex hull is derived mathematically from the point cloud generated by laser-scanning mounted skeletons. This has the advantage of requiring minimal user intervention and is thus more objective and far quicker. We test this method on 14 relatively large-bodied mammalian skeletons and demonstrate that it consistently underestimates body mass by 21 per cent with minimal scatter around the regression line. We therefore suggest that it is a robust method of estimating body mass where a mounted skeletal reconstruction is available and demonstrate its usage to predict the body mass of one of the largest, relatively complete sauropod dinosaurs: Giraffatitan brancai (previously Brachiosaurus) as 23200 kg.     

Euclidean geometry explains why lengths allow precise body mass estimates in terrestrial invertebrates: the case of oribatid mites

Indirect measures of soil invertebrate body mass M based on equations relating the latter to body length (l) are becoming increasingly used due to the required painstaking laboratory work and the technical difficulties involved in obtaining some thousands of reliable weight estimates for animals that can be very small. The implicit assumption of such equations is that dM/dV=δ, where V is body volume and δ is a constant density value. Classical Euclidean scaling implies that V∝l³∝M. One may thus derive M from l when the latter can provide a good estimate of V and the assumption of a constant δ is respected. In invertebrates, equations relating weight to length indicate that the power model always provides the best fit. However, authors only focused on the empirical estimation of slopes linking the body mass to the length measure variables, sometimes fitting exponential and linear models that are not theoretically grounded. This paper explicates how power laws derive from fundamental Euclidean scaling and describes the expected allometric exponents under the above assumptions. Based on the classical Euclidean scaling theory, an equivalent sphere is defined as a theoretical sphere with a volume equal to that of the organism whose body mass must be estimated. The illustrated application to a data set on soil oribatid mites helps clarify all these issues. Lastly, a general procedure for more precise estimation of M from V and δ is suggested.     

Predicting body mass of bonobos (Pan paniscus) with human-based morphometric equations

A primate's body mass covaries with numerous ecological, physiological, and behavioral characteristics. This versatility and potential to provide insight into an animal's life has made body mass prediction a frequent and important objective in paleoanthropology. In hominin paleontology, the most commonly employed body mass prediction equations (BMPEs) are “mechanical” and “morphometric”: uni- or multivariate linear regressions incorporating dimensions of load-bearing skeletal elements and stature and living bi-iliac breadth as predictor variables, respectively. The precision and accuracy of BMPEs are contingent on multiple factors, however, one of the most notable and pervasive potential sources of error is extrapolation beyond the limits of the reference sample. In this study, we use a test sample requiring extrapolation—56 bonobos (Pan paniscus) from the Lola ya Bonobo sanctuary in Kinshasa, Democratic Republic of the Congo—to evaluate the predictive accuracy of human-based morphometric BMPEs. We first assess systemic differences in stature and bi-iliac breadth between humans and bonobos. Due to significant differences in the scaling relationships of body mass and stature between bonobos and humans, we use panel regression to generate a novel BMPE based on living bi-iliac breadth. We then compare the predictive accuracy of two previously published morphometric equations with the novel equation and find that the novel equation predicts bonobo body mass most accurately overall (41 of 56 bonobos predicted within 20% of their observed body mass). The novel BMPE is particularly accurate between 25 and 45 kg. Given differences in limb proportions, pelvic morphology, and body tissue composition between the human reference and bonobo test samples, we find these results promising and evaluate the novel BMPE's potential application to fossil hominins.     

Body mass estimation in xenarthra: A predictive equation suitable for all quadrupedal terrestrial placentals?

The Magnorder Xenarthra includes strange extinct groups, like glyptodonts, similar to large armadillos, and ground sloths, terrestrial relatives of the extant tree sloths. They have created considerable paleobiological interest in the last decades; however, the ecology of most of these species is still controversial or unknown. The body mass estimation of extinct species has great importance for paleobiological reconstructions. The commonest way to estimate body mass from fossils is through linear regression. However, if the studied species does not have similar extant relatives, the allometric pattern described by the regression could differ from those shown by the extinct group. That is the case for glyptodonts and ground sloths. Thus, stepwise multiple regression were developed including extant xenarthrans (their taxonomic relatives) and ungulates (their size and ecological relatives). Cases were weighted to maximize the taxonomic evenness. Twenty‐eight equations were obtained. The distribution of the percent of prediction error (%PE) was analyzed between taxonomic groups (Perissodactyla, Artiodactyla, and Xenarthra) and size groups (0–20 kg, 20–300 kg, and more than 300 kg). To assess the predictive power of the functions, equations were applied to species not included in the regression development [test set cross validation, (TSCV)]. Only five equations had a homogeneous %PE between the aforementioned groups. These were applied to five extinct species. A mean body mass of 80 kg was estimated for Propalaehoplophorus australis (Cingulata: Glyptodontidae), 594 kg for Scelidotherium leptocephalum (Phyllophaga: Mylodontidae), and 3,550.7 kg for Lestodon armatus (Phyllophaga: Mylodontidae). The high scatter of the body mass estimations obtained for Catonyx tarijensis (Phyllophaga: Mylodontidae) and Thalassocnus natans (Phyllophaga: Megatheriidae), probably due to different specializations, prevented us from predicting its body mass. Surprisingly, although obtained from ungulates and xenarthrans, these five selected equations were also able to predict the body mass of species from groups as different as rodents, carnivores, hyracoideans, or tubulidentates. This result suggests the presence of a complex common allometric pattern for all quadrupedal placentals. J. Morphol., 2008. © 2008 Wiley‐Liss, Inc.     

Evaluation of juvenile stature and body mass prediction

This investigation evaluates the performance of juvenile stature (from tibia and radius lengths) and body mass (from breadth of the femoral distal metaphysis) prediction equations based on the Denver Growth Study sample (Ruff C. 2007. Am J Phys Anthropol 133 698-716). The sample used here for evaluation is an independent sample of juveniles brought to the Franklin County (Ohio) Coroner in 1990-1991. The Ohio sample differs somewhat from the Denver reference sample: it includes approximately 25% African-Americans (rather than all European-Americans), a significant number of right limb bones were measured (rather than all left side), it includes a wider range of economic statuses and it includes individuals who died from disease and trauma. As such the composition and measures of the Ohio sample correspond more generally to that seen in skeletal samples so that the accuracy of the estimates from the present sample should approach those found in practical applications of these methods. Results indicate that both juvenile body mass and stature are estimated relatively accurately. Accuracy of body mass estimates for 1-13-year-old juveniles is similar for African-American and European-American males and females. The least accurate estimates are for individuals in the 8-13 years age class (excluding individuals with body mass indices greater than the age specific 95th percentile): n = 9, +/- 2.9 kg, 95% confidence interval 1.4-4.4 kg. Accuracy of stature estimates for 1-17-year-old juveniles is comparable for the tibia and radius and, as with body mass estimates, are similar for African-American and European-American males and females. For combined age, sex, and ancestry groups average accuracies are in the +/-3.5 to +/-6.5 cm range. Some limitations of the methods are discussed.