1. Biomechanics Vol. 23, No. IO, pp. 953-965, Printed in Great Britain

@X2-9290/W $3.00+.00 Pergamon Press plc

1990.

NIELS STENSEN’S GEOMETRICAL THEORY OF MUSCLE CONTRACTION (1667): A REAPPRAISAL TROELS KARDEL* Orthopedic Biolhechanics Laboratory and History of Medicine Library, Mayo Clinic/Mayo Foundation, Rochester, Minnesota 55905, U.S.A. Abstract-From reading the Elementorum Myologia Specimen of 1667 by Niels Stensen (Steno), I assert that the text and illustrations contain an observation-based theory on the mechanics of muscle contraction: (1) Based on the study of the structure and motion of several muscles in different animals and in man, Stensen described the contraction of parallel equally long motor fibers formated as uni- or mukipennate structures, each forming a parallelepipedon between parallel tendon plates. The parallelepipedon was used as a model allowing Stensen to apply mathematical methods in the argumentation. When the motor fibers contract, the tendons move in parallel planes, the muscle shortens, but the distance between the tendon planes does not change. There will appear a swelling, even if the volume of the model remains the same. Therefore, the swelling observed during contraction, according to Stensen, is no argument for an increase in muscle bulk and no argument against contraction without any change of muscle volume. (2) In the first century after its proposal, different arguments were published against Stensen’s theory: in 1680 by Borelli (De Motu Animalium), 1694 by Bernoulli (De Motu Musculorum), 1743 by Boerhaave (Praelectiones), and 1762 by Haller (Elementa Physiologic). When read today, these arguments are irrelevant, erroneous, or without scientific documentation. However, by the end of the 18th century, Stensen’s theory all but disappeared from the science literature. (3) Anatomical and biomechanical studies published after 1980 show that the foundation and applicability of Stensen’s theory are still valid. (4) While earlier considered to be perhaps Stensen’s weakest work, arguments are presented to reappraise Elementorum as one of Stensen’s significant publications and as a significant work in the biomechanical sciences.

INTRODUmION

The Danish scientist Niels Stensen (Steno) (1638-86) (Fig. 1) formulated an early theory of the mechanics of muscle contraction including an observation-based mathematical model. Stensen gave a short description and a sketch of a unipennate muscle (Fig. 2) in a letter from L&den, dated 30 April 1663, to Thomas Bartholin (1616-80), his teacher in Copenhagen. The theory was provisionally expanded with a geometrical description in the book De Musculis et Glandulis Obseruationum Specimen (Stenonis, 1664). It was presented in its final version with illustrations at Florence in 1667 as Elementorum Myolo& Specimen, seu Musculi Descriptio Geometrica, which means A Specimen of Elements on the Knowledge of Muscles, or a

that the heart is only a muscle (Willius and Keys, 1941). Reactions to the later Elementorum Myologie Specimen (Stenonis, 1667b) were more subdued, with even a tendency on the part of commentators to question the author’s scientific judgement. This, in spite of Stensen’s remark in Elementorum: ‘It is even a frequent experience that what displeases other people, may be the very things of which the authors are themselves most fond’, indicates that although the theory on muscles had been attacked from the very beginning, it ranked in the author’s own opinion above his earlier celebrated research on the glands and on the brain.

Geometrical Description of Muscles. While De Musculis created much debate with its

OBSERVATION-MODEL-THEORY

Three main elements can be distinguished in Stensen’s analysis of muscle. Together they form a first description of the muscle fiber as the first principle theory of the mechanics of muscle contraction. The of motion and original statements like ‘the shorter is basic element is the observation of the structure of the flesh in a long belly, the stronger is the force different muscles. Next is a geometrical representation because the number of fibers is greater’, it was in or model, ‘mensura’, of the structure used for validgeneral well received and provided an entry to the ating the third element, a main theorem, ‘propositio’. scientific world for the twenty-six-year-old author The geometrical representation enabled Stensen to (Kardel, 1986). De Musculis is now considered to be a introduce mathematical methods in the argumenclassic work in cardiology because of its description tation. Stensen stated that the muscles consist of pennate structures, each of which is composed of equally long Received in final form 8 March 1990. muscle fibers forming a parallelepipedon between *Address for correspondence: Troels Kardel, M. D., 0sterbrogade 54A, DK-2100 Copenhagen, Denmark. Fax (45) parallel tendons (Fig. 3), and that muscle fibers are separated by transverse bands of tissue connected with 32 96 01 12. 953

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the membrane wrapping the muscle. There are illustrations of the structure of the following muscles: gastrocnemius and semimembranosus (unipennate); biceps of the arm and the abductor and adductor in the Astacus lobster’s claw (bipennate); masseter (complex bipennate); deltoid (multipennate); and muscles of a shark (multipennate with curved myosepts). The muscle contraction was simulated in a geometrical model, a parallelepipedon. The purpose was described as follows: ‘Many have supposed it to be a fact beyond doubt, that during contraction the mass of the muscle increases, because there can be felt a swelling many places in the human body during the contraction. These people then reason, that the muscle is like a bladder becoming shorter the more it is filled. To make it clear, that a swelling alone is not sufficient reason to make one assume that new material makes its way into the muscle during contraction, I will demonstrate: in all muscles that contract there is a swelling, even though the volume of the contracted muscle remains as it was in the non-contracted muscle. To this end it is necessary in the following to make some suppositions and prove them.’ The model (Fig. 4) was based on 44 definitions. In definitions 3 and 33, the shape was determined as a parallelepipedon. The distance between the tendon surfaces, FR, is the height; DK, the width; and CS, the thickness of the muscle. The muscle contracts when the length of the muscle, AE or BD, becomes shorter. The movements of the model were determined by five suppositions: ‘1. When the muscle is contracted, the tendon edges do not change. 2. When the muscle is contracted, the individual parts of flesh remain within the same plane and in continuity. 3. When the muscle is contracted, the individual parts all along the flesh contribute equally in the movement and the shortening. 4. When the flesh is shortened, its width remains the same. 5. The contracted muscle has equally the same volume as the non-contracted muscle.’ Among the suppositions, Stensen held the first three to be certain. Concerning supposition 4, Stensen wrote that the width at least did not increase. In the case of a diminished width, this would only reinforce the arguments. Supposition 5 was made in anticipation of a definite approval by experiments, which followed with increasing accuracy over the following 200 yr. From the definitions and suppositions of the model it follows that during the uniform contraction of equally long motor fibers, tendon plates of constant length move in parallel planes, the muscle shortens, but the distance between the two tendon surfaces does not change (Fig. 5). Using Euclidean mathematical argumentation with six ‘lemmas’, one with a corollary, Stensen reached the main theorem that even ifaccording to supposition i-there is no increase in

muscle volume, contraction of the motor fibers causes a swelling of the muscle by an increased thickness, ‘crassitudo’, i.e. in Fig. 5 by a longer distance from MC to the surface QPDK during contraction than from MC to surface IFDK when in relaxation. The geometrical argumentation was repeated for unipennate muscles with extreme long and short fibers. The legitimacy of using a model was considered: ‘However I am far from trying to impose on my reader that I have examined all muscles in all different animals and that I am more than certain that the position of the surfaces will, in every case, be like which I have described. For the time being, however, I affirm that having seen the position of surfaces in so many cases, I can confidently assert, that it is not without reason that I take this simple and regular structure of the flesh as a model for all others.’ Stensen refrained from explaining the cause of the contraction and stated ‘the explanation offered by the majority . . . , contraction by the inflation by spirits, . . . is built on a shaky foundation’. Apart from the geometrical theory, EIementorum contains reflections and observations on the motor fiber in the contraction of the heart and skeletal muscle and the subdivision of fluids in the muscle (F&on and Wilson, 1966). Further examples are what to take into consideration when comparing the relative strength of muscles with different geometry; and probably for the first time in the literature, Stensen observed red and white muscle fibers in an extremity of a rabbit. Elementorum has been translated in Italian by Marzollo (1968; Stenone, 1986), and excerpts in German by Scherz (1963). Editions in Danish and English are being prepared. THE FIRST CENTURY

Although it was provided with a number of good illustrations and examples, the Elementorum was little understood by Stensen’s contemporaries. The theory was too remote from the generally held belief-handed down from antiquity with great authority by RenC Descartes (15961650) in De Homine (first published 1662 in I_.eiden)--that a substance, ‘the animal spirit’, entered from the brain through hollow nerves to make the muscles swell and contract (Fulton, 1926; Bastholm, 1950; Wilson, 1961; Lindeboom, 1979). In 1675 Stensen took holy orders and abandoned the study of natural sciences for the rest of his lifetime (for a short biography, see Dorozynski, 1988). He therefore left the muscle theory undefended to gradually disappear’ from the attention of researchers (Pedersen, 1987). From the beginning the theory was barely understood, and later it became subject to gross misunderstandings. Stensen’s muscle theory was not adopted more by British anatomists than by any others, but Stensen was indeed respected by the British and his publications, including Elementorum, were eagerly discussed.

Fig. 1. The portrait of Niels Stensen painted as scientist. Uffizi Gallery of Florence, Italy.

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Fig. 6. The muscle structure according to Borelli’s De Motu Animalium (168%

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Fig. 2. Stensen’s drawing of a unipnnate

muscle from a letter in 1663 to T. Bartholin. Printed in 1667 Copenhagen.

Fig. 3. The three-dimensional structure of muscle fibers forming a parallelepipedon between parallel tendon plates. Woodblock print from Elemenrorum (second edition, Amsterdam, 1669).

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3 Fig. 4. Stensen’s parallelepipedon. The definitions are given in the text. From Elementorum.

Fig. 5. The contraction of muscle fibers forming a parallelepipedon between parallel tendon plates. From Elementorum. is evident from surviving correspondence, from the minutes of e.g. the meeting of 16 January 1667 in the newly-founded Royal Society of London, and from accounts of Stensen’ssojourn at Montpellier in the autumn of 1665, where he met a number of travelling British scientists (Rome, 1956; Scherz, 1956; Wilson, 1961; Poynter, 1968; Faller, 1985). It is conceivable that Stensen knew a work from 1658 by Walter Charleton (1619-1707) (Gosch, 1873). Based on geometrical reasoning, Charleton suggested that muscle contraction took place without increase in muscle volume. Charleton’s view had little impact and was opposed by later English authors like Thomas Willis (1621-75), William Croone (1633-84), and John Mayow (1641-79), who built their contraction theories on intumescence from influx of fluid, or from effervescence like a kind of fermentation or explosion by gunpowder, (Wilson, 1961; Hierons and Meyer, 1964). However, Stensen made no reference to Charleton. Stensen has mentioned 1678 in a letter to Gottfried Wilhelm Leibniz (1646-1716) how his understanding of the muscle structure and contraction came This

suddenly during the dissection of the leg of a rabbit at Leiden, probably in 1662-63 (Scherz, 1952; Kardel, 1986, note 27): ‘In the first muscle and by the first incision there became unveiled, what to this moment had been unknown to man, the fabric of the muscles, turning upside-down the system of Descartes’. In The Netherlands the most important investigator of muscle contraction at that time was Jan Swammerdam (1637-80), Stensen’s colleague at Leiden, to whom he refers in Elementorum as ‘my friend Swammerdam’. Swammerdam carried out experiments on frog muscles placed inside glass tubes and showed the constancy of muscle volume during contraction, thereby providing crucial experimental evidence for Stensen’s muscle theory. Swammerdam’s work on muscle was known by Stensen earlier, but it was concluded in written form after 1669, the year of the publication of the second edition of Elementorum, specifically quoted several times as the only reference. However, Swammerdam’s work on muscles was not to be published until nearly seventy years later by Boerhaave in 1737 (Fulton and Wilson, 1966; Schulte, 1968; Lindeboom, 1975). In Denmark, Stensen maintained friendly relations with the influential professor Thomas Bartholin, who, in spite of marked differences with his own opinion on muscle contraction (Bartholin, 1668), in 1667 published Stensen’s aforementioned letter from 1663 on muscles with the first draft of the muscle theory entitled Nova Musculorum et Cordis Fabrica (Stenonis, 1667a). The direct opposition, which Stensen has described in Elementorum, he may have encountered from Cartesian scientists in France, where he stayed in 1665, and in Italy, although in both countries he maintained most cordial relations with his patrons, Melchisedek Thtvenot and Grand Duke Ferdinand0 II, evidenced by a vivid correspondence (Scherz, 1952). Just 28 yr old when he arrived in Florence in 1666, Stensen concluded and published his geometrical analysis of muscles within a year. Like Giovanni Alfonso Borelli (1608-79), 30 yr older, who until 1667 lived at

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nearby Pisa as professor of mathematics, Stensen was entertained by the Medici Court. In spite of sharing interests in muscle research and mathematical analysis (Foster, 1901; Franceschini, 1951), Borelli and Stensen may not have had an easy time together (Scherz, 1987). Borelli, regarding Stensen as a rival in his area of science (Vugs, 1968), already shortly after Stensen had arrived in a letter to Malpighi expressed disregard of Stensen and questioned his motives (Adelman, 1975). Stensen and Borelli made no named reference to the other’s scientific theories, and they did not correspond-at least no letters have survived. Borelli’s richly-illustrated two-volume life-work on muscles and motion, De Motu Animalium (Borelli, 1680-81), was largely finished already at the time of Stensen’s visit (Scherz, 1967), but was not to be published until thirteen years after Elementorum, in 1680-the year after its author’s death. Borelli was clearly of the opinion that contraction takes place by a sudden inflation of the muscle (Fulton and Wilson, 1966). Concerning the structure and function of muscle, Borelli’s propositions 4 and 5 (out of 457 propositions) are of particular interest. Borelli’s own opinion is expressed in his proposition 4 with reference to his Figs 1,2,3 and 4 of Tabula I (Fig. 6). Borelli described-like Stensen-that only the fleshy filaments are shortened, whereas the tendons retain the same length: ‘only the flesh fibers AB, CD, EF, GN, and C exert a force by contracting when they carry considerable weights’, whereas the tendons serve ‘like a handle to which the fibers are attached’. However, from the illustrations mentioned above, it is evident that Borelli’s tendons were shaped out of speculation rather than observation. Then, Borelli’s proposition 5 is an ‘examination of the structure of muscle, as recently discovered, and of its mode of action’. With reference to his Figs 5, 6, 7

and 8 of Tabula I (Fig. 6), Borelli rendered the arrangement of muscle by ‘these famous men’. When considering what is written and illustrated in proposition 5, ‘they’ can be none other than Stensen. Borelli offered several arguments against the unipennate arrangement, or rhomboidal instrument, of muscle shown in the figures. Borelli’s arguments can now be studied in full in a 1989 English edition of De Motu Animalium translated by Dr Paul Maquet (Borelli, 19893;most of this article’s Borelli quotations are taken from this book. Borelli found that ‘the structure and arrangement of the muscle or of the rhomboidal instrument appear to be very inappropriate to raise a weight R’, and finished proposition 5 by stating ‘in conclusion, such single muscles are not seen normally and do not act in the way these famous authors think they do. This mode of action can occur only with some muscles composed of several rhomboids as we shall see later’. According to Borelli, the force of contraction was provided by series of small rhombs (Wilson, 1961): ‘Therefore they will be rather similar to rhombs, enclosed by threads, which may be dilated by the

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motive faculty and as a result, the fibers may be contracted. Consequently, it must be conceived that the finest threads or fibers of the muscles are series of little mechanisms, porous or rhomboidal in form, like a chain composed of threads in rhombs’ (from proposit0 114). The idea that small rhombs, Macinula rhomboidales, were the functional basis of muscle contraction had its origin in early microscopic examinations of muscle by e.g. Swammerdam (1737) describing a globular structure. Two-dimensional observations may have been misinterpreted by deducing a threedimensional globular pattern from the rounded forms of muscle cells seen in cross-section. Borelli shared this conception with William Croone, author of De Ratione Musculorum (Croone, 1664). With limited success, Stensen at Montpellier had told Croone that muscles did not need any increase in volume to make them swell during contraction (Wilson, 1961). Years later Croone expressed that the publication of ‘the long expected Work of Borelli . . . has made me have a better opinion of this Thought [the bladder expansion explanation] than else I should have had (Croone, 1681; Fulton and Wilson, 1966, Needham, 1971). For the next 300 yr Borelli’s work was considered the basic work in muscle mechanics, even being named as one of the hundred most famous books in science (Horblit, 1964), while Stensen’s work on muscles was almost forgotten. How can this be? From a review in 1873 by Gosch, and from the literature, I have recorded a number of objections perpetuat+ against Stensen’s theory. As mentioned earlier, the first objections are found in Borelli’s work: that Stensen’s theory made no provision for volume expansion, and that the theory did not work because of the proposed structure of muscle. Conflicting paradigms were at issue in 1694 when Johannes Bernoulli (1667-1748), to become a leading member of the Swiss family of mathematicians, published a thesis De Motu Musculorum at Basel. Bernoulli praised Borelli’s work and elaborated mathematically ‘the curve following which the motor fiber is expanded’. Bernoulli referred to Stem&s idea of muscle contraction without access of new material as ‘ridiculous’. Stensen’s theory was discarded because it disobeyed the (Aristotelian) rule that any movement must be caused by an external force: ‘Omne quod movetur, movetur ab alio’. The latter objaction was repeated in 1761 in the posthumous anatomical works by the French anatomist Joseph Duvemey (16481730). Borelli’s work became re-issued at L&den in 1710 by the influential Dutch professor Hermann Boerhaave (1668-1738), now to include an additional chapter, Meditationes mathematics de motu musculorum, written by Bernoulli. In 1703 Boerhaave gave a lecture on the use of mechanical methods in medicine praising no less than thirty-eight authors, but even when Boerhaave said ‘what is a muscle but a composition of the very

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finest . . . threads’, he did not mention Stensen. In his 415th lecture, published posthumously (Boerhaave, 1743), Boerhaave attempted to evaluate Stensen’s geometries: ‘All muscle fibers form oblique parallelograms with two tendons. Hence, in a contraction of the muscle, the fibers are drawn toward the beginning, the angles are changed, the figures become shorter, more like rectangles, and decrease as much in length as they increase in width. So this most brilliant man [Stensen] thought that, by these phenomena, he had sufficiently proved what they indicated-the contracted muscle becomes shorter and more swollen. In truth he had not paid attention to another theorem from mathematics, in which we teach that among parallelograms the rectangle includes the greatest area, when any figure of them have equal interchangeable lines. Therefore the greater space is contained within the same perimeter . . . . For if he had been mindful of this well known theorem, he would easily have seen, according to the proper hypothesis, that when the parallelogram becomes shorter, it is the result of the new material which fills in the increasing area.’ Boerhaave clearly misinterpreted Stensen’s model, as his statement implies that during contraction the parallelogram (or parallelepipedon), rises up, so to speak, with an increased height (the distance between the tendon plates) and volume, but without any change of the fiber length. Thus, more than half a century after Stensen’s description of the muscle fiber, the leading commentator of physiology apparently had not quite understood its ability to shorten. Like Borelli, Boerhaave discarded Stensen’s model because no material was provided for volume expansion, and this opinion was adopted likewise in 1762 and 1774 by the great compiler of physiology, Albrecht von Haller (1708-77) of Bern. Haller, a former student of both Boerhaave and Bernoulli, commented on Stensen’s muscle structure and made the last objection: ‘This fabrication is not true. It is rare to be offered muscles of this sort’. However, Haller did not offer arguments to support this statement or to counter Stensen’s documentation. And Haller repeated earlier misrepresentations: ‘Nor does the illustrious Stenonius furnish a provision of fluid matter, which according to his hypothesis should be poured into the space of fibers unless he wishes them to become emptied. For while the rhombs change into quadrates, at the same time their areas are enlarged and become more capacious. Therefore it is necessary to replenish the rhombs’. Otherwise, Haller is recognized as the first having reported to have seen muscle fibers contract under a microscope (Fulton, 1926). The unquestioned authority of Haller a century later is well documented by the following quotation from a foot note in a physiological report from Berlin: ‘The original work was not accessible to me, and I am therefore quoting it [Elementorurn] from Haller’ (Schiffer, 1869; Kardel, 1986, note 46).

Finally in 1779 the curator of the Pisa Academy of Science, Angelo Fabroni (1732-1803), once again repeated Boerhaave’s erroneous representation of Stensen’s geometry: ‘One marvels that Steno rejected the best known theorem in geometry, the impossibility of an obliqueangled parallelogram’s changing into a rectangle without receiving new material with which the greater part of the space is filled.’ Thus, observations in disagreement with Stensen’s descriptions were hardly presented, and the arguments used to falsify the theory were either erroneous or irrelevant. Nevertheless, with verdicts by such prominent scientists, Stensen’s geometrical theory of muscle contraction became side-tracked from the mainstream of science. Were there no advocates of Stensen’s geometrical muscle description? Two reviews in scientific journals are listed in the Stensen bibliography (Jensen, 1986); a few favorable comments in letters to third persons are recorded by Scherz (1987), but there are no later scientific comments of importance. IN THE HANDS OF HISTORIANS

In 1873 the zoologist and diplomat Christian Carl August Gosch (1832-1913) summarized Stensen’s scientific works in Danish and described the muscle theory and its fate, so full of misunderstandings. While Gosch’s research of the old literature is fundamental to the present work, I wish to question Gosch in his conclusion that the drawback to Stensen’s theory was not flaws in the geometrical analysis, but simply that the model-according to Haller-did not agree with the actual structure of the muscles. This view was maintained in 1910 by the editor of Stensen’s collected scientific work, the physiologist and medical historian Vilhelm Maar (1871.-1940), who considered Elemenrorum to be Stensen’s perhaps weakest work; and more recently by Moe (1988). The physiologist and medical historian John F. Fulton (1899-1960) found that Stensen ‘laid the foundation of Muscular Mechanics as we know it now’ (Fulton, 1926). In 1950 the general practitioner Eyvind Bastholm (1904-89) summarized Stensen’s work on muscle in an excellent thesis The history of muscle physiology (Basthelm, 1950) at the University of Copenhagen. Bastholm credited Stensen that in his work ‘he laid a stimulating and fundamental foundation of muscular mechanics which others could proceed with, even if they did not accept his mathematical conceptions in detail’. Bastholm repeated Gosch’s view that Stensen’s theory had enjoyed only a brief life in science, not because of faulty calculation, but ‘solely because he had become fascinated by an ingenious idea whose correctness he lacked the means sufficiently to verify’. More recent comments are one from 1971 by the muscle biochemist Dorothy Needham (18961987), who in a great historical review of the biochemistry of

Muscle theory of Niels Stensen

muscle characterized Stensen’s muscle theory as ‘complicated’, and one by myself calling it a ‘trial of the reader’s patience’ (Kardel, 1986, note 3 1). The Stensen biographer Gustav Scherz (1895-1971) appropriately considered Elementorum to be Stensen’s most controversial work (Scherz, 1987). REAPPRAISAL

Not until the 1980s was the dormant theory to be reappraised, when various observers noted that the structure of several muscles was close to that of Stensen’s description, while other researchers linked observations and models of muscles in contraction. Studies by Pfuhl (1937), Benninghoff and Rollhiiuser (1952), and Gans and Bock (1965) had elucidated the mechanical principles of pennate muscles from a theoretical point of view. Gans and Bock indicated the advantage in unipennate muscles as regards the intramuscular pressure during contraction (Fig. 7). In 1981, two studies of normal human anatomy described the parallelogram formed by muscle fibers of equal length between parallel tendon plates. Muscles around the elbow were described by An et al. (1981) and muscles in the lower arm and hand by Brand et al. (1981), both studies quoting Stensen. The constancy of fiber length and pennation angles within the same muscle was demonstrated in muscles of the lower limb in man by Wickiewicz et al. (1983), and in muscles of the hind leg of rabbit-the same species and extremity as studied by Stensen in 1663-by Lieber

Fig. 6. Diagram demonstrating that an evenly pinnate muscle will not develop any internal preeeure during contraction unless the distance between attachment surfaces shortens

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and Blevins (1989). Even if these findings do not necessarily say that all of these muscles are shaped and contract like a parallelepipedon. they indicate that this model is a good approximation. As mentioned earlier Stensen expressly stated that the model was a simplification. Where do we then find the simplifications? Firstly, nature does not go by lines but by curves, and motor fibers are not always straight lines as in Stensen’s Tabula I. Secondly, during contraction the individual parts do not remain exactly within the same plane as stated by Stensen in supposition 2, but they may bend or twist according to the local stress and pressure. It should also be mentioned that the structure of some muscles is not at all penniform, like ring muscles and flat muscle sheets, and that some muscles do not have a uniform fiber length, and a few have zero pennation angle. Nevertheless for most extremity muscles Stensen’s model remains a useful approximation. In 1984-85 the static structure became dynamic working muscle models, when Huijing and Woittiez (1984), using a planimetric model, and Woittiez et aI. (1984), using a three-dimensional model, in computer simulations compared generated and observed data of the force-length relation from experiments in rat muscles, testing the validity of models with essentially the same geometry of Stensen’s (Fig. 8), though without reference back to that model. Unlike Stensen, who assumed a constant volume in the muscle model, 317 yr later this was defined in the computer programs. Woittiez et al. (1984) concluded that ‘the

Fig. 7. Sketch to show change in fiber proportions with contraction of a laterally restrained pinnate muscle. After KUMHE~ (1961, Fig. 21)

Fig. 7. Two- and three-dimensional models of muscle contraction. From Gans and Bock (1965) and Kummer (1961).

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Fig. 2. A. A representation of the three-dimensional muscle model. A segment in the constructed muscle is shown. The most proximal and most distal fiber are indicated, as well as tendon plates and tendons. B. The same elements are indicated in a planimetric representation of the constructed muscle. The constructed muscle has lateral symmetry with respect to its central planes. In the planimetric representation the course of fiber arrangment is shown. C. The effect of the increase of the angles of fibers and tendon plates as the muscle shortena. This is shown in the planimetric representation of a constructed muscle. State 1: muscle length = 125%. State 2: muscle length = 75%. Fig. 8. The muscle structure in the model of Woittiez et al. (1984).

muscle model closely approximates the actual muscle form and function’ (Fig. 9). Otten (1985) used a slightly different three-dimensio&l. model, still close to Stensen’s model. Otten reatihed a similar good fit between generated and observed data in a study of the force-length relations of the hind leg of the cat. In 1989 Kaufman et al., quoting Stensen, refined the model of Woittiez et al. (1984) and formulated a descriptive mathematical model of muscle contraction ‘complex enough to represent many of the mechanical and architectural properties of active muscle; yet simple enough that only two parameters (muscle fiber optimal length and muscle belly optimal length) are needed’. While the works of An et al. (1981), Brand et al. (1981), Wickiewicz et al. (1983), and Lieber and Blevins (1989) demonstrate the close similarity between Stensen’s model and the structure of a number of muscles in different species, the works of Huijing and Woittiez (1984), Woittiez et al. (1984), Otten (1985),

and Kaufman et al. (19’89)demonstrate the predictive power needed to consider Stensen’s geometrical description of muscles in terms of today’s science. It is likely that in situ observations, e.g. studies combining morphometry and intramuscular pressure measurements during contraction, with present-day cornputative power and soft tissue imaging techniques, may add to the models mentioned here from an array of algorithms recently evaluated by Otten (1988). Indeed, Otten’s conclusion on why to make muscle models comes close to the way Stensen pioneered modelling in the biological science in 1667: ‘Models of functional architecture of skeletal muscles are not ends, but rather important means in understanding motor control’. Ultimately such an understanding may prove to be beneficial in the improvement of patient care when applicable to the description of movements in man, as exemplified by the study of An et al. (1989). They implemented the model of Kaufman et al. (1989) in an analytical model for the

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others, Stensen explicitly searched for contradicting evidence from observations and in the literature. Altogether, in the present article, arguments are presented to rank Elementorwn Myologia Specimen both among Stensen’s significant publications, as has been CONCLUDING REMARKS suggested by Snorrason (1983), and among the signiFor a period of no less than a century after its ficant works on models of functional architecture in proposal, Stensen’s geometrical theory of muscle con- skeletal muscle, as has been suggested by Otten (1988), traction was under attack by eminent scientists. and in a broader sense, in the biomechanical sciences Finally it all but disappeared from the science literat- (Fulton, 1926). ure. However, when viewed after two more centuries, Another remarkable feature of Elementorum is the Stensen’s description, based on observations of the program Stensen formulated for the application of muscle structure, is surprisingly close to muscle mathematical methods in the study of the living models currently under investigation in biomechaniorganism (Maar, 1910; Nordensk%ld, 1928; Pedersen, cal laboratories, while the objections, based on tradi1987; Moe, 1988). Until then mathematics had been tion and speculative theories, fall apart. successfully used in physics, optics, astronomy, and Throughout his research Stensen cared more for geography following Galilei’s program from Saggiadescribing the effects than for searching for the causes tore 1623 for the study of the Universe by means of (Portal, 1770), i.e. answering the question how rather mathematical methods. Stensen’s program, which will than why. Thus, Stensen payed more respect to the conclude the present article, was excerpted, probably by teaching of Galilei (1564-1642) than to the teaching of Croone (Poynter, 1968), in the Philosophical TransDescartes (Scherz, 1967). Elementorum MyologiE actions of the Royal Society of 1668: Specimen (Stenonis, 1667b) exemplifies this by a ‘The Author of this Book declareth, that his design careful mathematical evaluation of an observationin composing it was, to shew, that in a Muscle neither based model. In the conclusions Stensen did not go the Parts of it can be distinctly named, nor its Motion beyond what was covered by observations and by duely consider’d, unless the Doctrine thereof become a physical and mathematical rules (Marzollo, 1968). part of the Mathernaticks. And he is of opinion, that Before accepting conclusions of his own and those of there is no other cause of the many Errors, which spoil determination of muscle forces across the elbow joint during isometric loading conditions.

Force (%I

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th(%)

Fig. 9. The match between generated and observed data in rat semimembranosus (SM) and gastrocnemius (GM) muscle in the study of Woittiez et al. (1984).

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the History concerning the Humane Body, than that Anatomy hath hitherto disdaind the Laws of the Mathematicks. And therefore inviteth those that are studious in that part of Philosophy, to consider, that our Body is an Engine made up of a thousand subordinate Engins, whose true knowledge whoever thinks that it can be investigated without Mathematical assistance, must also think, that there is matter without Extension, and Body without Figure.’ Acknowledgements-The basic requirement for this study has been the Danish translation from Latin of Stensen’s Ekmentorum by Lit. Phil. Marianne Alenius. I have benefited from discussions with Associate Professor Kirsti Andersen, University of Arhus, Denmark, for obtaining a better understanding of Stensen’s Euclidean mathematics. Most of the study was carried out in the winter of 1989 during a sojourn as visiting scientist at the Mayo Clinic/Mayo Foundation, Rochester, Minnesota, U.S.A. For stimulating interest and support I thank Professors Edmund Y. S. Chao, Ph. D., Kai-Nan An, Ph. D., and Ronald L. Linscheid, M.D., and Dr Kenton R. Kaufman, Ph. D., of the Orthopedic Biomechanics Laboratory. Throughout the study I have received excellent assistance by History of Medicine Librarian Donald S. Pady, Mayo Foundation Library, and by Advisory Librarian Paul Aagaard Christiansen, University Library 2, Copenhagen, Denmark. Except where other references are made, the Latin quotations were translated by Sister Emmanuel Collins, Ph. D., OSF, Assisi Heights, Rochester, Minnesota. Finally I wish also to express my gratitude to Professor Leonard G. Wilson, Ph. D., University of Minnesota, Minneapolis, and Professor Olaf Pedersen, Ph. D., University of Arhus, for critique and valuable suggestions, and to the Stensen-Scholar Cand. Mag. Harriet M. Hansen, Bjerringbro, Denmark, for advice. Economical support was received from: Aage Louis-Hansen Memory Foundation; Foundation for General Practice in Copenhagen, Frederiksberg, TArnby, and Dragor; Foundation for Education and Development in General Practice in Denmark; and the Danish Medical Research Foundation. Note added in proof--Two articles in recent issues of this journal add to the presented evidence in support of Stensen’s scheme for the structure and function of muscle. Studying the muscle fiber architecture in the human lower limb, Friederich and Brand (1990) describe that within a given muscle the fiber lengths are remarkably similar. Hoy et al. (1990) have developed a musculoskeletal model of the human lower limb applying an actuator, described in their Fig. 1, which is indistinguishable from Stensen’s model (Fig. 3). By model simulation Hoy et al. predicted series of observations of summed actuator moments.

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