On the smallest stage in the world, the double-step
revolves
again and
19Weber, 238 [172].
19Weber, 238 [172].
Kittler-Drunken
It is high time to check whether the so-called naturalists, the immediate contemporaries of Du Bois- Reymond and Claude Bernard, did not in fact write his tirade or mandate into literary deeds without further ado.
In any event, the
great physiologist himself did not do so; instead, Du Bois-Reymond, after he bid farewell to Goethe's "Word"and "Language"in the name
of science, turned to the realm of the imaginary, which Goethe had celebrated as the "image in the truest sense. " In 1890, a small written piece on "Natural Science and Fine Art" ("Naturwissenschaft und bildende Kunst") issued forth from the pen of the Berlin Emeritus.
If there is ever to be something like a history of media studies, Du
Bois-Reymond's almost forgotten writing should appear in the canon of its holy texts next to Ernst Kapp's "Principle Characteristics of a
Philosophy of Technique. " Du Bois-Reymond's text writes the history of art from a perspective which has only recently been occupied again
by Samuel Edgerton or Jonathan Crary. The phenomenality of appearances, as it occurs in canvasses and statues, in painting and
plastic art, is everything but an unmediated beginning. In order for
styles and works of art to even appear, epistemological knowledge must first have established the field of their colors and forms. Du
Bois-Reymond cuts across, however provisionally it may be, the history of specific sciences in order to couple fifteenth-century geometry with the possibility of paintings in linear perspective, or to
3 The Friedrich Wilhelm University was renamed as the Humboldt University in 1949.
? 640 FRIEDRICHKITTLER
couple sixteenth-century anatomy with the possibility of muscle's becoming marble, and finally, the physical theories of color from Newton through Helmholtz with the history of painterly coloritoas a whole. At the end of his essay, however, this course of a history of science, not yet curbed by any modern artistic impulse, makes the transition from static to moving pictures, which is a brand-new idea in 1890. Five years before the Lumi&re brothers will raise Edison's
playful peep-box theater to the status of a mass-image medium, Du Bois-Reymond presents, if you will, the very first history of film. This
history has the unique feature of not describing moving pictures as animation or the surpassing of Daguerre's half-hour exposure times, as has become commonplace and self-evident since Hugo Miinster-
berg's scarcely less-forgotten psychotechnique of film. Du Bois-
Reymond's history does not begin at all with diorama painters or magic lantern players, but rather on a really elementary threshhold:
with the scientific history of moving. In the process of elaborating the thesis that without modern geometry there would have been no
paintings in linear perspective, Du Bois-Reymond's brief history of art ends with the statement that there would be no moving pictures
without the analysis of modern mathematics.
Today, one could make just the reverse statement: one could say
that without moving scientists there would be no modern analysis.
Geometry first became analytic, and that means cinematic, when an
officer on leave, who happened to be a burgeoning philosopher, first dreamed of movement as such in his winter quarters on the Donau.
"Larvatusprodeo"-masked, I go forth-remained Descartes' device, even when he no longer wore the uniform. But what has hindered
empirical movements from achieving mathematical and philosophi- cal honor in the same way seems to be precisely the steady motion of
Descartes' subject which he conceived of as a geometrical point and which analytic geometry since Descartes transposed into the ceaseless movement of a curve-defining point in a field of coordinates. Furthermore, empirical movements remained a domain of dance- instructors and their choreographies, or of the drills of officers educated in Hessen-Nassau and Hessen-Orange to whom the young
Descartes owed his drilling- and fencing-skills. 4Mathematical analysis, on the other hand, moved those dimensionless points, to which the
4 See Wolfgang Schaffner, "K6rperim Takt. Zur Geschichte symbolischer Maschinen im 16. und 17. Jahrhundert" in Kaleidoskopien3 (2000), 188-205.
? MLN 641
swinging pendulum or flying granades were reduced for the sake of simplicity, through every possible adventure of maxima and minima,
of turning points and null-points, of velocity moments and forces of acceleration. Only in the truly novelesque case of that lying baron,
who had the privilege of performing the analysis with his own body, did a cartesian subject ride along on the parable of a granade's trajectory. Once again the text of the novel, in Miinchhausen's case as
in Descartes', relied upon the certainty-unshakeable because imag- ined in the Donau winter quarters-that dreamed motion or faked
flying remained as inescapably true as only the Cogitoitself otherwise could be.
This blessed time of the founding fathers in which consciousness
and analysis, thinking and differential calculus were one and the
same ended bitterly. The reason was simple: not all trajectories of
motion in physical empiricism were permitted to be ascribed to the
cinematics of a single point. What Huygens had achieved in the case
of the swinging pendulum already failed in the case of a vibrating
string whose innumerable points are elastically coupled and thus, in other words, enjoy uncountable degrees of liberty. Not even cartesian
subjects would have been in a position of being or thinking all points of the string at the same time in order to discover the secret of a
single tone from the violin. All that remained was for mathematical analysis to bestow this secret unto a new, no less mysterious theory: to the partial differential equations in brazen opposition to the usual ones. Newton's usual differential equations still sufficed for solar systems, since the stars which were part of those systems do not enjoy a single degree of freedom; but not everything that moves is as simple as in astronomy. In the case of violin strings and bell tones, drumskins and water surfaces, and even windstorms and electromagnetic vibra-
tions, only partial differential equations successfully modelled the countless parts moved in all their dimensions.
Thus the course of modern science finally arrived at physiological, day-to-daymotion. In 1836, a monograph from Wilhelm and Eduard Weber appeared which was almost without precedent: the "Mechan-
ics of the Human Walking Apparatus. An Anatomical-Physiological
dermenschlichen Eine anatomisch- Investigation" (Mechanik Gehwerkzeuge.
physiologischeUntersuchung). E5duardWeberhadatthattimejustbeen
5Weber, Wilhelm and Eduard, UberdieMechanikdermenschlichenGehwerkzeugen,ebst
der eines Versuchsiiberdas des aus der im Beschreibung Herausfallen Schenkelkopfes Pfanne
? 642 FRIEDRICHKITTLER
summoned to the University of Leipzig as professor of medicine; his brother Wilhelm, on the other hand, was already a professor of physics in G6ttingen since 1831, where he had just two years later, together with none other than GauB,made the first electric telegraph run. That alone suggests that the medical subtitle of their joint book is pure understatement. All the anatomical and physiological findings
concerning knees, hips, leg muscles and joints gathered in the first
part only serve the higher purpose of founding a mathematical physics of legs in just as strict a sense as Newton had demanded for
the physics of celestial bodies.
At least the anatomical and physiological investigations-or in
other words, the experiments both on corpses' legs in Leipzig and on
living legs in G6ttingen-make it clear from the startwhat the degree of freedom is, which a system of hips, knees and heels enjoys. Every
joint, as writers of computer animations would formulate it, or rather,
program it today, is a three-dimensional transformative-matrixwhose
rotations in turn transform the next subordinate joint. At the end of
the day, however, the matrix-product from all these turns and
displacements produces such variable values, or leg positions, that one is threatened by the appearance of sheer incalculability. "Itcan
perhaps be questioned," so the Weber brothers admit, "whether a
theory of walking and running can be provided at all since we are not walking machines, and these movements are altered in many waysby our free will. "6That merely appears to be the case, however. In 1836, no cartesian subject inhabits the curves or functions of his own limbs. On the contrary, "man binds his movements to certain rules even if he cannot express these rules in words. These rules are based totally on the structure of his body and on the given external conditions. They thus can be deduced from both structure and external conditions. "7
All the preliminary anatomical-physiological investigations have the single goal of reconstructing a per definitionem "unconscious"
walking machine first of all through measures and series of tests on the body, before the triumphant end to the book can go and present
luftverdiinntenRaume,in Wilhelm Weber's Werkee,d. Friedrich Merkel und Otto Fischer (Berlin: Konigliche Gesellschaft der Wissenschaften in Gottingen 1894), 322 [transl. by Maquet, Paul and Ronald Furlong, Mechanicsof theHuman WalkingApparatus(Berlin, Heidelberg, NewYorketc. :Springer-Verlag1992), 226].
6 Ibid. , 3 [VI]. 7 Ibid. [VII].
? MLN 643
the mathematical theory of walking and running. Everything rests on the fact that legs can provide for their own account, so that in the end
they can be in the books as a differential system. In 1836, long before Marey and Muybridge, the attainment of the differential system was at
first possible only in stasis, as if Daguerre's long-term exposures had found a scientifically parallel maneuver. In order to make agility at
least visible as latent motion, however, the brothers cut up the corpses' hips at the precise location where "the balls of the upper-legs lie in their sockets. " Just as in the drawing of a steam engine which
would explain the interplay of parts in the moment of explosion, admiring readers can glimpse the interplay between the ball of the
thigh bone and the socket of the joint.
The femoral head or so-called sphere was sawn perpendicularly from in front backwards and the section was printed on the paper. The curvature of the the two parts here cannot be altered by the drawer. The picture is indeed the imprint of the object itself. The bone thus divided can be used
as a wood-engraving for print. 8
Plate IX
Pelvis
? Fig. 1.
8Ibid. , 14 Fn. [10].
Femur
? 644 FRIEDRICHKITTLER
The work of the Weber brothers makes anatomy leave its proud old amphitheaters, where the dissection resulted in findings that made their way to woodcuts only belatedly. Instead, just as Henry Fox
Talbot's heliography did four years later, they were put onto the printed page as nature's imprint of itself. 9Thus the free will of human drawing-hands in scientific visualization remains just as excluded as it is in the mechanics of human legs. On this worldly scene of all religions and dances of the dead, the skeleton appears on the stage of
knowledge and points no longer to allegories of death, but rather to nothing more than its own animation.
In order to transpose the latent movements that occur around leg
joints, or degrees of freedom, into actual movements, the Weber brothers introduce two simple axioms. First of all, the individual
bones whose interplay comprises a step should swing according to the
law of the pendulum. The differential equation of every partial movement can thus already be accounted for by falling back on
Christian Huygens and his pendulum clock equation from 1656.
Secondly, however, the usual human gait should be a periodically
iterated interplay of all these partial movements, whereby a period is
clearly only counted after a completed pair of steps. '0 In any event, a summation of partial differential equations only appears as total
movement according, first of all, to the three dimensions of space,
and secondly, according to time. Having made these assumptions, the Weber brothers simply go ahead and copy the formula which
Poisson's Traite de mechaniquein the second, recently published edition from 1838 anticipated for such cases. ll With this fact, the misfortune which occupies all of our computer monitors today has come into existence: instead of "word, language and image in the truest sense," "symbol and number" have taken over human gait. Cinematics in the French sense of the word, which as of 1830, and at Ampere's instigation, calculates transversal and rotational move- ments of steam engines, becomes cinematics in the modern sense: it calculates virtual movements in virtual, that is, visualizable spaces.
For the sake of their animation, however, both the leg and Poisson's "general"differential system must make a few sacrifices. "To
9 See also Susanne Holl, "Ob oculos. Andreas Vesals 'De humani corporis fabrica' und der Buchdruck" in Kaleidoskopien3 (2000), 334-357.
0Weber, 191 [128]. " Ibid. , 217 [152].
? M LN 645
simplify the problem we imagine that: 1) Instead of the path there are
only some fixed points (similar to stones laid on the path) along a horizontal line placed as far apart as the length of a step. [. . . ] 2) The
leg is replaced by a straight solid line standing with its lower extremity on one of these points and is retained there by friction. 3) Instead of
being distributed over the entire length of the leg, the mass of the leg is concentrated in one middle point of the solid line. 4) The body is replaced by a mass concentrated in the upper point of this solid line.
This mass flies horizontally at an even velocity. "12In place of the legs and their three joints, a single center of mass around the knee
appears in order to swing henceforth perpetually around a dimen- sionless point which for mathematical reasons has formed itself "instead of the body. "As Christian Morgenstern correctly remarks: "A knee alone goes through the world. A knee alone, and nothing else. "
In fact: this 'nothing' is comprised of pure trigonometric functions. Sine and cosine are the two angle functions which cause this nothing
to step forward; g as gravitation and p as the measure of the circle are the two constants which maintain the steps forward into a period. With knee, g6nu, early Greek mathematics indicated the word for angle;'3 from the angle, gonia, arabic and modern mathematics indicated their trigonometric and thus periodic functions. At the end of this short story of sine and cosine stands the lapidary sentence:
The equation
d2X g sin X=- rl-
of Ch. 128 is replaced by an approximation,as occurs in analysingthe pendulum:
14 d8x
s = - rld ~_rldu2
The freedom of mathematical penstrokes thus finally conquers trigonometry itself. Because neither legs nor pendula are true wheels, that is, they simply do not create angles of any size, the periodic term
12Ibid. , 217 [152].
3JohannesLohmann, Musikeund Logos. Aufsdtzezur griechischePnhilosophiuend
MusiktheorieA,nastasios Giannaras,ed. (Stuttgart:MusikwissenschaftlicherVerlag 1970), 20.
14Weber,223 [159].
? 646 FRIEDRICH KITTLER
for a differential equation can be transposed into a linear one. If need be, the sine of c isjust c itself. For the sake of simplicity,15the Weber brothers only need to posit three further variables of their
general leg-swinging equation as constants of one or zero, and paragraph 128, the "Introduction to the Illustration of Walking
Figures," almost writes itself. Before computers with their explicit purpose of crunching systems of differential equations-at least
numerically-have
come into existence, scientific visualization can
only occupy itself with mercilessly simplified formula.
In its own progress, the "Establishment of a theory of walking and
running" achieves its goal when "the position of both legs" can be
constructed "at the various phases of each step" and "according to
these laws" or equations in order to enable "the artist" to judge
"whether he sees this representation corresponding to the actual
attitude of the legs during walking. ""6The new graph-producing science thus appears to subject itself to the judgement of a visual art.
It ends with an oppulant appendix of tables and figures which portray two lonely legs or knees in all individual phases of walking, running and jumping in order truly to resynthesize the unity of a pair of steps.
? Table XV
Fig. X
~k . ' ; ''
~ . . . i . 2
Fig. 2
5Ibid. , 238 [172]. 16Ibid. , 209 [147].
? Fig. 3
p-t).
! (i<
M LN
647
? Table XVI
? Tables XV and XVI from Weber and Weber, Mechanicsof theHuman Walking Apparatus.
However, what is and remains decisive as regards scientific visualiza- tion is the fact that it is no longer subordinate to the pronounced judgments of traditional arts. The Weber brothers have three good
reasons that claim as much. First of all, their entire pride rests on the
achievement of having constructed "the various ways of walking" "in
the various positions of every step according to theory and without
drawing upon experience. " Already, the fact that a movement cap- tured in differential equations dictates those laws of movement
already elevates such a theory discernably over the belatedness of
literary descriptions or painterly representations. Secondly, cinemat- ics and painting relate to each other no differently than early modern geometry and linear perspective once did. Having said as much, the Weber brothers had already brought forth Du Bois-Reymond's argu- ments, even in a more polite fashion.
? 648 FRIEDRICH KITTLER
The positions of the different parts of the body change too quickly during
walking and running to be completely imprinted on the senses and in the memory instantaneously. To capture them one needs certain instruments
and indirect methods which are not available to the artist. [. . . ] It is the same as for the science of perspectives which is so helpful to the artistwhen dealing with architectural pictures. One should not assert that the inner feeling of the artist is sufficient. In perspective this is never the case. [. . . ]
During walking and running one obtains no clear sensory perception of the simultaneous positions of the trunk and limbs because they pass so
rapidly. Therefore, the most capable artist lacks the means directly to perceive in Nature, without any help from science, the true circumstances
as they actually take place. Using perspective gives us the appearance of the truth by representing the distances in space and the positions of the
body. Similarly, study of walking and running brings to our pictures the appearance of the truth in the movements of life which are depicted. 17
Thirdly, it is decisive that artists, not to mention people in general, cannot perceive the mechanics of the human legs at all. This fact distinguishes cinematics since 1836 from linear perspective, which
has been promoted to the very principle of phenomenality and thus of visibility at the latest in the work of Johann Heinrich Lambert. 18
Like the vibrations of the violin's string, the phase pictures of walking
pass by too quickly to fall into perceptual times. What people can no longer see or hear, however, calls for technical media. Only "the
application of the theory of walking and running," the use therefore of"certain instruments and indirect methods," "can bring the appear-
ance of truth from the portrayed living movement into our images. "
Thus let it be-for the second time after Emil Du Bois-Reymond-
solemnly pronounced: it was Wilhelm and Eduard Weber, and neither Marey nor Muybridge, neither Edison, nor the Lumiere
brothers, who programmed the program which goes by the name of
film. In "Poggendorffs Annalen der Physik"-in 1836, global physics could still be found in German technical periodicals-both of them,
or perhaps really just Wilhelm Weber alone, stumbled on a "known technical device" of Faraday, which for a change did not have
anything to do with their common field of research, electro-magnetic induction. Instead, it has to do with "stroboscopic disks" as they had been "used by Joseph] Stampfer" in order to display "these draw-
17Ibid. , 6 [3].
18For further information on Lambert, see Friedrich Kittler, Phdnomen, Perspektive,
Medium. VonJ. H. Lambertzu G. W E Hegel(forthcoming).
? MLN 649
ings" before one's eyes "in such a succession that they gave the
impression of a figure walking quite normally. "'9In other words: that "certain instrument," which "brings the appearance of the truth of
the copied living movement into our images" is quite simply called a
stroboscope. To the great delight of Emil Du Bois-Reymond, the "mechanics of human legs" ends with how-to instructions that go far
beyond the state of affairs which scientific books and their attached tables could achieve. The how-to directions of Wilhelm and Eduard Weber are designed-for the first time in the history of science, as far as I can tell, for the visualization of partial differential equations.
It is interesting to illustrate the space and time data determined
absolutely according to the theory, by building and drawing the position of the limbs at each moment of walking and running
regularly. There is a means of convincing oneself that this construc-
tion matches experience as well as possible despite the imperfect measurement of the body on which one must rely. This means consists
of drawing a number of pictures representing the man in his successive positions during two steps. The internal surface of a
cylinder or of a drum is divided into as many facets as there are pictures plus one. The succesive pictures are glued one on each facet in proper order so that each picture on its facet appears as more displaced forwards than the previous one. The middle picture seems to be exactly one step further than the first. The length of the con- struction must be equal to the length of a double step. The drum is rotated at an even speed during the time of a double step. The figures are observed through slits opposite in the wall of the drum. They
appear as if walking or running. Their movements show a surprising
similaritywith the movements of a man actuallywalking or running. If one had never seen a man walk or run, and if one only knew the
relations between his limbs, one could with the help of the theory
acquire a representation of these movements closely matching experi-
ence and predict what would happen during these displacements. 20 With this grand prediction, the book ends. On its long path, past
Poisson and Faraday to the Weber brothers, Euler's mathematics of rotational movements has arrived in a rotating machine which serves as the "means" or medium of physiological-physical animation.
On the smallest stage in the world, the double-step revolves again and
19Weber, 238 [172]. 20 Ibid. , 294 f. [218].
? 650 FRIEDRICHKITTLER
again, removed as it is from all empiricism. Progress stands still, because it is completed. As in the differential system, the sine of 0 and 2 x p are one and the same.
At precisely the position of such completion, positivists like Du
Bois-Reymond literally forget their part. His report on Weberian
"mechanics"simply glosses over the "surprisingagreement" between simulated and empirical walking, in order to fade in a prehistory of
film in its place. After Marey's flying birdwings and Muybridge's
galloping horse legs, everything is read in a completely different way. In 1890, what emerges out of the eternal return of poesy from a
limited number of images of the stroboscope is the endless prose called film.
"The peculiar effect presented itself," Du Bois-Reymond writes about the Weber brothers, "the figure portraying the beginning and end of the step, where man rests for a short time on both feet,
certainly looks completely as painters have alwaysalready portrayed walking people, except that in the middle of the step, where the so-
called moving leg swings past the standing leg, the most strange and even ludicrous sight appears: like a drunken town-musician, man
seems to trip over his own feet, and no one has ever seen a walking man in such a position. "21 Science thus becomes the alienation effect which strips quotidian and artistic perception of the fiction of totalization, in order to reveal the naturalistic truth of the drunken town-musician behind the aesthetic appearance of human walking.
One may be astonished: that Du Bois-Reymond is not exactly writing about drunks at the town fair or stroboscope exhibitors is already a
wonder. Indeed, Norbert Wiener began after 1920 to equate his utterly non-differentiable functions expressly, albeit metaphorically, with the pace of a drunken sailor in order to smuggle concepts like prose, series of chance, and drunken behavior into mathematics. Under such prosaic conditions, science becomes the courage to
tolerate "the strangest, most ludicrous sight" of mathematical-synthe- sized movement long enough until empirical, that is, prosaic media techniques like film rush to the rescue. Eduard Weber, as Du Bois-
Reymond continues to say,unfortunately passed awayin 1871, that is,
scarcely one year before Muybridge's unexpected sequential horse- leg pictures, "HerrWilhelm Weber has however seen that after almost
four centuries, instant photography declared him and his brother to
21 Emil Du Bois-Reymond, "Naturwissenschaftund bildende Kunst"in Reden(Leipzig: Veit 1912, expanded edition), 407.
? MLN 651
be completely correct. Mr. EadweardMuybridge in San Francisco first applied it in 1872 at the encouragement from Mr. Stanford, in order
to capture the sequential positions of horses in various gaits. The same thing was then shown as had been seen in the Weberian
schematic drawings: images appeared as no one could have imagined seeing them in reality. Oriented toward street scenes, processions,
and such things, the camera frequently captures images of people in just as strange positions, as those which the Weber brothers had
assigned to them for theoretical reasons. The same could be said about the amazing series of images of a flying bird during the beat of a wing which Herr Mareyaimed for through his photographic flint. "22
I am coming to the end. Surely, it would be nice to know whether Wilhelm Weber, who died in G6ttingen in 1891, could have chatted with Du Bois-Reymond about horses' legs. The ancestral gallery of our technical images would have had one more forefather, and the stage of knowledge one more hero's role. And the boredom, that
every year new books appear on Jules-Etienne Marey as the scientific origin of cinema, would be ended. 23Out of pure nostalgic love that
celebrates film like Eurydice, in proportion to its disappearance, the differences between prose and poesy, between Marey'sgraphic method
and Wilhelm Weber's movement equations, disappear too. Appar- ently, historians of media do not want to admit even today that augurists of virtual motion are always already in advance of the forerunners of cinema. It is not Marey's"animal machine," but rather Weber's "mechanics of human legs" that casts its shadow up to the
present day.
For it is the case that, of the two goals of media techniques that the
Weber brothers presented in good platonic fashion as "ideas for a theory of walking and running," they only achieved the first. 24In the wheel of life known as the stroboscope, the cartesian subject is turned
as a machine. According to Wolfgang Schaffner, the drill-regiment of Moritz of Orange is finally sublated into a mathematical concept.
That is however the reason why the Weber brothers expressly forego
the further elaboration of completely possible "applications"of their differential equations "toward military science. "25 The second, or
22 Ibid. , 407f.
23Michel Frizot, "Comment ca marche. De l'exactitude dans l'instant" in La Revuedu Museed'Orsay4 (1997), 74-83.
24Weber,205 [144]. 25 Ibid. , XII.
? 652 FRIEDRICHKITTLER
rather highest, point of their goals is no longer a living being, but rather the complete and "successful"26formulation of all models for
the construction of machines "which, like man, are carried by two
supports and move forwards by alternating support and swing. "27The fact that their mathematics reduces man's two legs to simple pendula
or connecting rods thus has good reasons. Certainly, "the human spirit of invention has most recently achieved the construction of
steam engines which, without being pulled by animals, move with great speed and power. "But steam engines or "railroadtrains"give up on soft or steep ground, or when faced with "desert"or "rivers. "28 The mechanics of the human legs would therefore only be completed
when they were implemented as robotics, whereby it would no longer play a role whether "a machine moved through steam" would come
"on two, four, six or more legs. "29As far as one hears, the Japanese infantry,because of their well-known recruitment problems, is at least the most advanced at this time when it comes to two-legged fighting robots.
With this last argument, however, the inquiry of the Weber brothers
is reversed just as dramatically as it is consequently. No further equation is sought in order to calculate a path from given lines and
angles which once had been known as legs. Quite the opposite is the
case: what is sought is an equation that derives the corresponding leg-
work from the given beginnings and endpoints of a spatial displace- ment. From the cinematics of 1836 have emerged the inverse
cinematics of today, and that means cinematics as determined by
computers. There is no computer graphicist who does not-accord- ing to Euler, Poisson, or Weber-have to solve differential equations
of a spatial displacement. All computer animations which make their
way onto our monitors by walking, flying or even shooting are products of a "prediction"from 1836: "Ifone had never seen a man
walking and running, and only knew the proportions of his limbs, then one could with the help of theory create an idea of these
movements that fits experience very well and that could predict what
happens through these movements. " On computer monitors, if I may continue and conclude, science steps onto the stage.
HumboldtUniversity,Berlin JocelynHolland, translator
26 Ibid. ,210 [147]. 27 Ibid. , 209 [147]. 28 Ibid. , 4 f. [2].
29 Ibid. , 4 [2].
great physiologist himself did not do so; instead, Du Bois-Reymond, after he bid farewell to Goethe's "Word"and "Language"in the name
of science, turned to the realm of the imaginary, which Goethe had celebrated as the "image in the truest sense. " In 1890, a small written piece on "Natural Science and Fine Art" ("Naturwissenschaft und bildende Kunst") issued forth from the pen of the Berlin Emeritus.
If there is ever to be something like a history of media studies, Du
Bois-Reymond's almost forgotten writing should appear in the canon of its holy texts next to Ernst Kapp's "Principle Characteristics of a
Philosophy of Technique. " Du Bois-Reymond's text writes the history of art from a perspective which has only recently been occupied again
by Samuel Edgerton or Jonathan Crary. The phenomenality of appearances, as it occurs in canvasses and statues, in painting and
plastic art, is everything but an unmediated beginning. In order for
styles and works of art to even appear, epistemological knowledge must first have established the field of their colors and forms. Du
Bois-Reymond cuts across, however provisionally it may be, the history of specific sciences in order to couple fifteenth-century geometry with the possibility of paintings in linear perspective, or to
3 The Friedrich Wilhelm University was renamed as the Humboldt University in 1949.
? 640 FRIEDRICHKITTLER
couple sixteenth-century anatomy with the possibility of muscle's becoming marble, and finally, the physical theories of color from Newton through Helmholtz with the history of painterly coloritoas a whole. At the end of his essay, however, this course of a history of science, not yet curbed by any modern artistic impulse, makes the transition from static to moving pictures, which is a brand-new idea in 1890. Five years before the Lumi&re brothers will raise Edison's
playful peep-box theater to the status of a mass-image medium, Du Bois-Reymond presents, if you will, the very first history of film. This
history has the unique feature of not describing moving pictures as animation or the surpassing of Daguerre's half-hour exposure times, as has become commonplace and self-evident since Hugo Miinster-
berg's scarcely less-forgotten psychotechnique of film. Du Bois-
Reymond's history does not begin at all with diorama painters or magic lantern players, but rather on a really elementary threshhold:
with the scientific history of moving. In the process of elaborating the thesis that without modern geometry there would have been no
paintings in linear perspective, Du Bois-Reymond's brief history of art ends with the statement that there would be no moving pictures
without the analysis of modern mathematics.
Today, one could make just the reverse statement: one could say
that without moving scientists there would be no modern analysis.
Geometry first became analytic, and that means cinematic, when an
officer on leave, who happened to be a burgeoning philosopher, first dreamed of movement as such in his winter quarters on the Donau.
"Larvatusprodeo"-masked, I go forth-remained Descartes' device, even when he no longer wore the uniform. But what has hindered
empirical movements from achieving mathematical and philosophi- cal honor in the same way seems to be precisely the steady motion of
Descartes' subject which he conceived of as a geometrical point and which analytic geometry since Descartes transposed into the ceaseless movement of a curve-defining point in a field of coordinates. Furthermore, empirical movements remained a domain of dance- instructors and their choreographies, or of the drills of officers educated in Hessen-Nassau and Hessen-Orange to whom the young
Descartes owed his drilling- and fencing-skills. 4Mathematical analysis, on the other hand, moved those dimensionless points, to which the
4 See Wolfgang Schaffner, "K6rperim Takt. Zur Geschichte symbolischer Maschinen im 16. und 17. Jahrhundert" in Kaleidoskopien3 (2000), 188-205.
? MLN 641
swinging pendulum or flying granades were reduced for the sake of simplicity, through every possible adventure of maxima and minima,
of turning points and null-points, of velocity moments and forces of acceleration. Only in the truly novelesque case of that lying baron,
who had the privilege of performing the analysis with his own body, did a cartesian subject ride along on the parable of a granade's trajectory. Once again the text of the novel, in Miinchhausen's case as
in Descartes', relied upon the certainty-unshakeable because imag- ined in the Donau winter quarters-that dreamed motion or faked
flying remained as inescapably true as only the Cogitoitself otherwise could be.
This blessed time of the founding fathers in which consciousness
and analysis, thinking and differential calculus were one and the
same ended bitterly. The reason was simple: not all trajectories of
motion in physical empiricism were permitted to be ascribed to the
cinematics of a single point. What Huygens had achieved in the case
of the swinging pendulum already failed in the case of a vibrating
string whose innumerable points are elastically coupled and thus, in other words, enjoy uncountable degrees of liberty. Not even cartesian
subjects would have been in a position of being or thinking all points of the string at the same time in order to discover the secret of a
single tone from the violin. All that remained was for mathematical analysis to bestow this secret unto a new, no less mysterious theory: to the partial differential equations in brazen opposition to the usual ones. Newton's usual differential equations still sufficed for solar systems, since the stars which were part of those systems do not enjoy a single degree of freedom; but not everything that moves is as simple as in astronomy. In the case of violin strings and bell tones, drumskins and water surfaces, and even windstorms and electromagnetic vibra-
tions, only partial differential equations successfully modelled the countless parts moved in all their dimensions.
Thus the course of modern science finally arrived at physiological, day-to-daymotion. In 1836, a monograph from Wilhelm and Eduard Weber appeared which was almost without precedent: the "Mechan-
ics of the Human Walking Apparatus. An Anatomical-Physiological
dermenschlichen Eine anatomisch- Investigation" (Mechanik Gehwerkzeuge.
physiologischeUntersuchung). E5duardWeberhadatthattimejustbeen
5Weber, Wilhelm and Eduard, UberdieMechanikdermenschlichenGehwerkzeugen,ebst
der eines Versuchsiiberdas des aus der im Beschreibung Herausfallen Schenkelkopfes Pfanne
? 642 FRIEDRICHKITTLER
summoned to the University of Leipzig as professor of medicine; his brother Wilhelm, on the other hand, was already a professor of physics in G6ttingen since 1831, where he had just two years later, together with none other than GauB,made the first electric telegraph run. That alone suggests that the medical subtitle of their joint book is pure understatement. All the anatomical and physiological findings
concerning knees, hips, leg muscles and joints gathered in the first
part only serve the higher purpose of founding a mathematical physics of legs in just as strict a sense as Newton had demanded for
the physics of celestial bodies.
At least the anatomical and physiological investigations-or in
other words, the experiments both on corpses' legs in Leipzig and on
living legs in G6ttingen-make it clear from the startwhat the degree of freedom is, which a system of hips, knees and heels enjoys. Every
joint, as writers of computer animations would formulate it, or rather,
program it today, is a three-dimensional transformative-matrixwhose
rotations in turn transform the next subordinate joint. At the end of
the day, however, the matrix-product from all these turns and
displacements produces such variable values, or leg positions, that one is threatened by the appearance of sheer incalculability. "Itcan
perhaps be questioned," so the Weber brothers admit, "whether a
theory of walking and running can be provided at all since we are not walking machines, and these movements are altered in many waysby our free will. "6That merely appears to be the case, however. In 1836, no cartesian subject inhabits the curves or functions of his own limbs. On the contrary, "man binds his movements to certain rules even if he cannot express these rules in words. These rules are based totally on the structure of his body and on the given external conditions. They thus can be deduced from both structure and external conditions. "7
All the preliminary anatomical-physiological investigations have the single goal of reconstructing a per definitionem "unconscious"
walking machine first of all through measures and series of tests on the body, before the triumphant end to the book can go and present
luftverdiinntenRaume,in Wilhelm Weber's Werkee,d. Friedrich Merkel und Otto Fischer (Berlin: Konigliche Gesellschaft der Wissenschaften in Gottingen 1894), 322 [transl. by Maquet, Paul and Ronald Furlong, Mechanicsof theHuman WalkingApparatus(Berlin, Heidelberg, NewYorketc. :Springer-Verlag1992), 226].
6 Ibid. , 3 [VI]. 7 Ibid. [VII].
? MLN 643
the mathematical theory of walking and running. Everything rests on the fact that legs can provide for their own account, so that in the end
they can be in the books as a differential system. In 1836, long before Marey and Muybridge, the attainment of the differential system was at
first possible only in stasis, as if Daguerre's long-term exposures had found a scientifically parallel maneuver. In order to make agility at
least visible as latent motion, however, the brothers cut up the corpses' hips at the precise location where "the balls of the upper-legs lie in their sockets. " Just as in the drawing of a steam engine which
would explain the interplay of parts in the moment of explosion, admiring readers can glimpse the interplay between the ball of the
thigh bone and the socket of the joint.
The femoral head or so-called sphere was sawn perpendicularly from in front backwards and the section was printed on the paper. The curvature of the the two parts here cannot be altered by the drawer. The picture is indeed the imprint of the object itself. The bone thus divided can be used
as a wood-engraving for print. 8
Plate IX
Pelvis
? Fig. 1.
8Ibid. , 14 Fn. [10].
Femur
? 644 FRIEDRICHKITTLER
The work of the Weber brothers makes anatomy leave its proud old amphitheaters, where the dissection resulted in findings that made their way to woodcuts only belatedly. Instead, just as Henry Fox
Talbot's heliography did four years later, they were put onto the printed page as nature's imprint of itself. 9Thus the free will of human drawing-hands in scientific visualization remains just as excluded as it is in the mechanics of human legs. On this worldly scene of all religions and dances of the dead, the skeleton appears on the stage of
knowledge and points no longer to allegories of death, but rather to nothing more than its own animation.
In order to transpose the latent movements that occur around leg
joints, or degrees of freedom, into actual movements, the Weber brothers introduce two simple axioms. First of all, the individual
bones whose interplay comprises a step should swing according to the
law of the pendulum. The differential equation of every partial movement can thus already be accounted for by falling back on
Christian Huygens and his pendulum clock equation from 1656.
Secondly, however, the usual human gait should be a periodically
iterated interplay of all these partial movements, whereby a period is
clearly only counted after a completed pair of steps. '0 In any event, a summation of partial differential equations only appears as total
movement according, first of all, to the three dimensions of space,
and secondly, according to time. Having made these assumptions, the Weber brothers simply go ahead and copy the formula which
Poisson's Traite de mechaniquein the second, recently published edition from 1838 anticipated for such cases. ll With this fact, the misfortune which occupies all of our computer monitors today has come into existence: instead of "word, language and image in the truest sense," "symbol and number" have taken over human gait. Cinematics in the French sense of the word, which as of 1830, and at Ampere's instigation, calculates transversal and rotational move- ments of steam engines, becomes cinematics in the modern sense: it calculates virtual movements in virtual, that is, visualizable spaces.
For the sake of their animation, however, both the leg and Poisson's "general"differential system must make a few sacrifices. "To
9 See also Susanne Holl, "Ob oculos. Andreas Vesals 'De humani corporis fabrica' und der Buchdruck" in Kaleidoskopien3 (2000), 334-357.
0Weber, 191 [128]. " Ibid. , 217 [152].
? M LN 645
simplify the problem we imagine that: 1) Instead of the path there are
only some fixed points (similar to stones laid on the path) along a horizontal line placed as far apart as the length of a step. [. . . ] 2) The
leg is replaced by a straight solid line standing with its lower extremity on one of these points and is retained there by friction. 3) Instead of
being distributed over the entire length of the leg, the mass of the leg is concentrated in one middle point of the solid line. 4) The body is replaced by a mass concentrated in the upper point of this solid line.
This mass flies horizontally at an even velocity. "12In place of the legs and their three joints, a single center of mass around the knee
appears in order to swing henceforth perpetually around a dimen- sionless point which for mathematical reasons has formed itself "instead of the body. "As Christian Morgenstern correctly remarks: "A knee alone goes through the world. A knee alone, and nothing else. "
In fact: this 'nothing' is comprised of pure trigonometric functions. Sine and cosine are the two angle functions which cause this nothing
to step forward; g as gravitation and p as the measure of the circle are the two constants which maintain the steps forward into a period. With knee, g6nu, early Greek mathematics indicated the word for angle;'3 from the angle, gonia, arabic and modern mathematics indicated their trigonometric and thus periodic functions. At the end of this short story of sine and cosine stands the lapidary sentence:
The equation
d2X g sin X=- rl-
of Ch. 128 is replaced by an approximation,as occurs in analysingthe pendulum:
14 d8x
s = - rld ~_rldu2
The freedom of mathematical penstrokes thus finally conquers trigonometry itself. Because neither legs nor pendula are true wheels, that is, they simply do not create angles of any size, the periodic term
12Ibid. , 217 [152].
3JohannesLohmann, Musikeund Logos. Aufsdtzezur griechischePnhilosophiuend
MusiktheorieA,nastasios Giannaras,ed. (Stuttgart:MusikwissenschaftlicherVerlag 1970), 20.
14Weber,223 [159].
? 646 FRIEDRICH KITTLER
for a differential equation can be transposed into a linear one. If need be, the sine of c isjust c itself. For the sake of simplicity,15the Weber brothers only need to posit three further variables of their
general leg-swinging equation as constants of one or zero, and paragraph 128, the "Introduction to the Illustration of Walking
Figures," almost writes itself. Before computers with their explicit purpose of crunching systems of differential equations-at least
numerically-have
come into existence, scientific visualization can
only occupy itself with mercilessly simplified formula.
In its own progress, the "Establishment of a theory of walking and
running" achieves its goal when "the position of both legs" can be
constructed "at the various phases of each step" and "according to
these laws" or equations in order to enable "the artist" to judge
"whether he sees this representation corresponding to the actual
attitude of the legs during walking. ""6The new graph-producing science thus appears to subject itself to the judgement of a visual art.
It ends with an oppulant appendix of tables and figures which portray two lonely legs or knees in all individual phases of walking, running and jumping in order truly to resynthesize the unity of a pair of steps.
? Table XV
Fig. X
~k . ' ; ''
~ . . . i . 2
Fig. 2
5Ibid. , 238 [172]. 16Ibid. , 209 [147].
? Fig. 3
p-t).
! (i<
M LN
647
? Table XVI
? Tables XV and XVI from Weber and Weber, Mechanicsof theHuman Walking Apparatus.
However, what is and remains decisive as regards scientific visualiza- tion is the fact that it is no longer subordinate to the pronounced judgments of traditional arts. The Weber brothers have three good
reasons that claim as much. First of all, their entire pride rests on the
achievement of having constructed "the various ways of walking" "in
the various positions of every step according to theory and without
drawing upon experience. " Already, the fact that a movement cap- tured in differential equations dictates those laws of movement
already elevates such a theory discernably over the belatedness of
literary descriptions or painterly representations. Secondly, cinemat- ics and painting relate to each other no differently than early modern geometry and linear perspective once did. Having said as much, the Weber brothers had already brought forth Du Bois-Reymond's argu- ments, even in a more polite fashion.
? 648 FRIEDRICH KITTLER
The positions of the different parts of the body change too quickly during
walking and running to be completely imprinted on the senses and in the memory instantaneously. To capture them one needs certain instruments
and indirect methods which are not available to the artist. [. . . ] It is the same as for the science of perspectives which is so helpful to the artistwhen dealing with architectural pictures. One should not assert that the inner feeling of the artist is sufficient. In perspective this is never the case. [. . . ]
During walking and running one obtains no clear sensory perception of the simultaneous positions of the trunk and limbs because they pass so
rapidly. Therefore, the most capable artist lacks the means directly to perceive in Nature, without any help from science, the true circumstances
as they actually take place. Using perspective gives us the appearance of the truth by representing the distances in space and the positions of the
body. Similarly, study of walking and running brings to our pictures the appearance of the truth in the movements of life which are depicted. 17
Thirdly, it is decisive that artists, not to mention people in general, cannot perceive the mechanics of the human legs at all. This fact distinguishes cinematics since 1836 from linear perspective, which
has been promoted to the very principle of phenomenality and thus of visibility at the latest in the work of Johann Heinrich Lambert. 18
Like the vibrations of the violin's string, the phase pictures of walking
pass by too quickly to fall into perceptual times. What people can no longer see or hear, however, calls for technical media. Only "the
application of the theory of walking and running," the use therefore of"certain instruments and indirect methods," "can bring the appear-
ance of truth from the portrayed living movement into our images. "
Thus let it be-for the second time after Emil Du Bois-Reymond-
solemnly pronounced: it was Wilhelm and Eduard Weber, and neither Marey nor Muybridge, neither Edison, nor the Lumiere
brothers, who programmed the program which goes by the name of
film. In "Poggendorffs Annalen der Physik"-in 1836, global physics could still be found in German technical periodicals-both of them,
or perhaps really just Wilhelm Weber alone, stumbled on a "known technical device" of Faraday, which for a change did not have
anything to do with their common field of research, electro-magnetic induction. Instead, it has to do with "stroboscopic disks" as they had been "used by Joseph] Stampfer" in order to display "these draw-
17Ibid. , 6 [3].
18For further information on Lambert, see Friedrich Kittler, Phdnomen, Perspektive,
Medium. VonJ. H. Lambertzu G. W E Hegel(forthcoming).
? MLN 649
ings" before one's eyes "in such a succession that they gave the
impression of a figure walking quite normally. "'9In other words: that "certain instrument," which "brings the appearance of the truth of
the copied living movement into our images" is quite simply called a
stroboscope. To the great delight of Emil Du Bois-Reymond, the "mechanics of human legs" ends with how-to instructions that go far
beyond the state of affairs which scientific books and their attached tables could achieve. The how-to directions of Wilhelm and Eduard Weber are designed-for the first time in the history of science, as far as I can tell, for the visualization of partial differential equations.
It is interesting to illustrate the space and time data determined
absolutely according to the theory, by building and drawing the position of the limbs at each moment of walking and running
regularly. There is a means of convincing oneself that this construc-
tion matches experience as well as possible despite the imperfect measurement of the body on which one must rely. This means consists
of drawing a number of pictures representing the man in his successive positions during two steps. The internal surface of a
cylinder or of a drum is divided into as many facets as there are pictures plus one. The succesive pictures are glued one on each facet in proper order so that each picture on its facet appears as more displaced forwards than the previous one. The middle picture seems to be exactly one step further than the first. The length of the con- struction must be equal to the length of a double step. The drum is rotated at an even speed during the time of a double step. The figures are observed through slits opposite in the wall of the drum. They
appear as if walking or running. Their movements show a surprising
similaritywith the movements of a man actuallywalking or running. If one had never seen a man walk or run, and if one only knew the
relations between his limbs, one could with the help of the theory
acquire a representation of these movements closely matching experi-
ence and predict what would happen during these displacements. 20 With this grand prediction, the book ends. On its long path, past
Poisson and Faraday to the Weber brothers, Euler's mathematics of rotational movements has arrived in a rotating machine which serves as the "means" or medium of physiological-physical animation.
On the smallest stage in the world, the double-step revolves again and
19Weber, 238 [172]. 20 Ibid. , 294 f. [218].
? 650 FRIEDRICHKITTLER
again, removed as it is from all empiricism. Progress stands still, because it is completed. As in the differential system, the sine of 0 and 2 x p are one and the same.
At precisely the position of such completion, positivists like Du
Bois-Reymond literally forget their part. His report on Weberian
"mechanics"simply glosses over the "surprisingagreement" between simulated and empirical walking, in order to fade in a prehistory of
film in its place. After Marey's flying birdwings and Muybridge's
galloping horse legs, everything is read in a completely different way. In 1890, what emerges out of the eternal return of poesy from a
limited number of images of the stroboscope is the endless prose called film.
"The peculiar effect presented itself," Du Bois-Reymond writes about the Weber brothers, "the figure portraying the beginning and end of the step, where man rests for a short time on both feet,
certainly looks completely as painters have alwaysalready portrayed walking people, except that in the middle of the step, where the so-
called moving leg swings past the standing leg, the most strange and even ludicrous sight appears: like a drunken town-musician, man
seems to trip over his own feet, and no one has ever seen a walking man in such a position. "21 Science thus becomes the alienation effect which strips quotidian and artistic perception of the fiction of totalization, in order to reveal the naturalistic truth of the drunken town-musician behind the aesthetic appearance of human walking.
One may be astonished: that Du Bois-Reymond is not exactly writing about drunks at the town fair or stroboscope exhibitors is already a
wonder. Indeed, Norbert Wiener began after 1920 to equate his utterly non-differentiable functions expressly, albeit metaphorically, with the pace of a drunken sailor in order to smuggle concepts like prose, series of chance, and drunken behavior into mathematics. Under such prosaic conditions, science becomes the courage to
tolerate "the strangest, most ludicrous sight" of mathematical-synthe- sized movement long enough until empirical, that is, prosaic media techniques like film rush to the rescue. Eduard Weber, as Du Bois-
Reymond continues to say,unfortunately passed awayin 1871, that is,
scarcely one year before Muybridge's unexpected sequential horse- leg pictures, "HerrWilhelm Weber has however seen that after almost
four centuries, instant photography declared him and his brother to
21 Emil Du Bois-Reymond, "Naturwissenschaftund bildende Kunst"in Reden(Leipzig: Veit 1912, expanded edition), 407.
? MLN 651
be completely correct. Mr. EadweardMuybridge in San Francisco first applied it in 1872 at the encouragement from Mr. Stanford, in order
to capture the sequential positions of horses in various gaits. The same thing was then shown as had been seen in the Weberian
schematic drawings: images appeared as no one could have imagined seeing them in reality. Oriented toward street scenes, processions,
and such things, the camera frequently captures images of people in just as strange positions, as those which the Weber brothers had
assigned to them for theoretical reasons. The same could be said about the amazing series of images of a flying bird during the beat of a wing which Herr Mareyaimed for through his photographic flint. "22
I am coming to the end. Surely, it would be nice to know whether Wilhelm Weber, who died in G6ttingen in 1891, could have chatted with Du Bois-Reymond about horses' legs. The ancestral gallery of our technical images would have had one more forefather, and the stage of knowledge one more hero's role. And the boredom, that
every year new books appear on Jules-Etienne Marey as the scientific origin of cinema, would be ended. 23Out of pure nostalgic love that
celebrates film like Eurydice, in proportion to its disappearance, the differences between prose and poesy, between Marey'sgraphic method
and Wilhelm Weber's movement equations, disappear too. Appar- ently, historians of media do not want to admit even today that augurists of virtual motion are always already in advance of the forerunners of cinema. It is not Marey's"animal machine," but rather Weber's "mechanics of human legs" that casts its shadow up to the
present day.
For it is the case that, of the two goals of media techniques that the
Weber brothers presented in good platonic fashion as "ideas for a theory of walking and running," they only achieved the first. 24In the wheel of life known as the stroboscope, the cartesian subject is turned
as a machine. According to Wolfgang Schaffner, the drill-regiment of Moritz of Orange is finally sublated into a mathematical concept.
That is however the reason why the Weber brothers expressly forego
the further elaboration of completely possible "applications"of their differential equations "toward military science. "25 The second, or
22 Ibid. , 407f.
23Michel Frizot, "Comment ca marche. De l'exactitude dans l'instant" in La Revuedu Museed'Orsay4 (1997), 74-83.
24Weber,205 [144]. 25 Ibid. , XII.
? 652 FRIEDRICHKITTLER
rather highest, point of their goals is no longer a living being, but rather the complete and "successful"26formulation of all models for
the construction of machines "which, like man, are carried by two
supports and move forwards by alternating support and swing. "27The fact that their mathematics reduces man's two legs to simple pendula
or connecting rods thus has good reasons. Certainly, "the human spirit of invention has most recently achieved the construction of
steam engines which, without being pulled by animals, move with great speed and power. "But steam engines or "railroadtrains"give up on soft or steep ground, or when faced with "desert"or "rivers. "28 The mechanics of the human legs would therefore only be completed
when they were implemented as robotics, whereby it would no longer play a role whether "a machine moved through steam" would come
"on two, four, six or more legs. "29As far as one hears, the Japanese infantry,because of their well-known recruitment problems, is at least the most advanced at this time when it comes to two-legged fighting robots.
With this last argument, however, the inquiry of the Weber brothers
is reversed just as dramatically as it is consequently. No further equation is sought in order to calculate a path from given lines and
angles which once had been known as legs. Quite the opposite is the
case: what is sought is an equation that derives the corresponding leg-
work from the given beginnings and endpoints of a spatial displace- ment. From the cinematics of 1836 have emerged the inverse
cinematics of today, and that means cinematics as determined by
computers. There is no computer graphicist who does not-accord- ing to Euler, Poisson, or Weber-have to solve differential equations
of a spatial displacement. All computer animations which make their
way onto our monitors by walking, flying or even shooting are products of a "prediction"from 1836: "Ifone had never seen a man
walking and running, and only knew the proportions of his limbs, then one could with the help of theory create an idea of these
movements that fits experience very well and that could predict what
happens through these movements. " On computer monitors, if I may continue and conclude, science steps onto the stage.
HumboldtUniversity,Berlin JocelynHolland, translator
26 Ibid. ,210 [147]. 27 Ibid. , 209 [147]. 28 Ibid. , 4 f. [2].
29 Ibid. , 4 [2].
