Isaac Elishakoff
email: elishako@fau.edu

Who developed the so-called Timoshenko beam theory?

Keywords: Bresse-Rayleigh-Timoshenko-Ehrenfest, beam theory, history, priority

The use of the Google Scholar produces about 78,000 hits on the term ''Timoshenko beam.'' The question of priority is of great importance for this celebrated theory. For the first time in the world literature, this study is devoted to the question of priority. It is that Stephen Prokofievich Timoshenko had a co-author, Paul Ehrenfest. It so happened that the scientific work of Timoshenko dealing with the effect of rotary inertia and shear deformation does not carry the name of Ehrenfest as the co-author. In his 2002 book, Grigolyuk concluded that the theory belonged to both Timoshenko and Ehrenfest. This work confirms Grigolyuk's discovery, in his little known biographic book about Timoshenko, and provides details, including the newly discovered letter of Timoshenko to Ehrenfest, which is published here for the first time over a century after it was sent. This paper establishes that the beam theory that incorporates both the rotary inertia and shear deformation as is known presently, with shear correction factor included, should be referred to as the Timoshenko-Ehrenfest beam theory.

Introduction

At first glance, the question in the title may appear self-evident. Indeed, the title contains the term Timoshenko beam theory. Therefore, one may conclude, naturally, that it must have been S.P. Timoshenko who authored it. Indeed, in the words of Laura, Rossi, and Maurizi [START_REF] Laura | Vibrating Timoshenko beams, a tribute to the 70th anniversary of the publication of Professor S. Timoshenko's epoch making contribution[END_REF], ''the publication, by Stephen Timoshenko, of his now classical theory of vibration of beams, whereby shear and rotatory inertia effects are taken into account, constitutes one of the most remarkable events in the development of the structural dynamics of the twentieth century. Together with the Timoshenko-Mindlin theory of vibrating plates it has influenced the mathematical analysis of the quasi-infinite variety of dynamics of continuous media and structural acoustics problems from bridges and machine elements to surface, underwater, and space vehicles passing through the prediction of the behavior to electronic packages, bioengineering systems, etc.'' These authors describe Timoshenko's contribution as ''epoch making.'' Archibald [2, p.61] characterized S.P. Timoshenko as ''the patron saint of the American engineering.'' Pisarenko [3], the author of the biographical book on Timoshenko, uses the concept of ''Timoshenko phenomenon.'' Terman -Provost Emeritus of Stanford University -in his congratulatory letter [4] to Timoshenko, in conjunction with latter's 90th birth anniversary, wrote, on December 2, 1968: ''I am pleased to report that the 'Timoshenko Legend' continues to flourish undiminished on the Stanford campus.'' Koiter [START_REF] Koiter | Some comments on the so-called Timoshenko beam theory[END_REF][START_REF] Koiter | Comment on ''Timoshenko beam theory is not always more accurate than elementary beam theory[END_REF], in the late 1970s, chose to disagree with the assessment of the above type. In his words: ''What is generally known as Timoshenko beam theory is a good example of a basic principle in the history of science: a theory which bears someone's name is most likely due to someone else [detailed discussion of the 'principle' can be found in section 5 -I.E.]. In fact, additional deflections due to transverse shear occur already in the fourth edition, dated 1868, of W.J.M Rankine's [START_REF] Rankine | A manual of applied mechanics[END_REF][START_REF] Rankine | Miscellaneous scientific papers[END_REF] well-known book Applied Mechanics. The conventional formula for the shear factor k, based on energy considerations, according to the author's definition with the value k = (6/5)(1 + v) for a narrow rectangular cross section, also occurs already in the literature of the nineteenth century.'' Simmonds [START_REF] Simmonds | In support of A.L. Goldenweiser's approach to refining classical plate and shell theories[END_REF] noted: ''.shear deformation effects were first introduced by Rankine [START_REF] Rankine | A manual of applied mechanics[END_REF] and rotary inertia effects by Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF]. In his often-cited paper of 1921, Timoshenko [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF] without explicit reference to either Bresse or Rankine combined these effects to create what is now almost universally referred to as the Timoshenko equations.'' Indeed, the rotary inertia effect was apparently discovered independently of Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] by Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF]. There was, however, one more contributor in developing the theory that takes into account both effects. This was Paul Ehrenfest who moved in 1907 to St Petersburg from Vienna, Austria. The first appearance of the beam theory, which takes into account both shear deformation and rotary inertia, was not made in Timoshenko's often cited paper of 1921, but already in 1916, namely in his book, in Russian, Course in Elasticity (second volume) where S.P. Timoshenko [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF] (figure 7) explicitly writes, in the footnote, that the derivation as made together with P. Ehrenfest. It is a mystery, why the subsequent publications of Timoshenko on this topic omit the name of Ehrenfest, but mention him again in the footnote of his 1922 paper [START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF] (figure 3), now in English, that the work in the paper was performed by Ehrenfest and him. It appears that a more justifiable name of the above theory ought to be Bresse-Rayleigh-Timoshenko-Ehrenfest beam theory or Timoshenko-Ehrenfest theory in short, because the theory as we know it today was put by these scientists in collaboration.

Grigolyuk [15, p.39] (see also [START_REF] Grigolyuk | Timoshenko: life and destiny[END_REF]) writes in his book S.P. Timoshenko: Life and Destiny: ''At that time he solved the problem of principal importance on the effect of shear stresses during the small vibrations of the beams; equation of small vibrations was obtained with shear deformation and rotary inertia taken into account. Then S.P. Timoshenko together with physicist Ehrenfest obtained the exact solution on transverse vibration of the beam of rectangular cross section. Usually it is considered that they constitute the content of works [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF][START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF]. These results are presented in the paper that appeared in Zagreb in 1920 in the journal Glasnik Hrvatskoga prirodsalovnoga drusˇtva [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] (emphasis added -I.E.) (figure 10). Let us remark that these results first were published in 1916 in the Course of Theory of Elasticity. Part II. Beams and Plates ( § §38-39, pp.200-213) [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF].'' As is clearly seen, Grigolyuk attributes the so called Timoshenko beam theory exposed in works [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF][START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF][START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] to both Timoshenko and Ehrenfest, and thus, should be called Timoshenko-Ehrenfest beam theory.

Topic of priority: Timoshenko and Ehrenfest

We provide herewith the details of the above cooperation. Soderberg [18, p.329] writes: ''During this period he happened to meet Paul Ehrenfest, with whom he formed a lasting friendship. Ehrenfest had come from Germany to Russia in the hope that academic freedom there would offer him better opportunities. He was not successful in establishing himself in Russia, but the two met frequently; Ehrenfest would then enlarge upon the current ideas abroad in physics -relativity and quantum mechanics -new to Timoshenko at the time'' (see also Boag et al. [19]).

Pisarenko [20, p.98] writes, in the free translation from Russian: ''In August 1911, S.P. Timoshenko moves to St. Petersburg, in hope to find in the capital city, with concentration of majority of institutions of higher learning, some work. He was unable to find any full time job. He had to look for adjunct appointments, though these were associated with a lower level of remuneration, than a professional job. In order to support his family, he collected about 20 hours of work in various ends of the city. Still, in 1911-1912 he was able to write papers on the action of transverse impact on the beam and effect of shear deformation on transverse vibration of beams. The former work was later translated into German, whereas the latter-into English. Both works in the future served as spring points for many scientists.'' Then Pisarenko [20, p.98] mentions the meeting with Paul Ehrenfest. Let us give the podium to Pisarenko [START_REF] Pisarenko | [END_REF] himself: ''In Petersburg S.P. Timoshenko rented a place on the Aptekarskiy Island, just opposite of Electrotechnical Institute. His close neighbor turned out to be Paul Ehrenfest, their acquaintance coming through Professor Abram Ioffe [also spelled Joffe]. German [Pisarenko making a mistake here, for P. Ehrenfest was an Austrian and later a Dutch citizen -I.E.] physicist [P. Ehrenfest] gave lectures, with pleasure, on the meetings of the Russian Physical-Chemical Society, where not a small number of participants could be found. Ehrenfest was a natural lecturer, who was in constant need of audience. Once they made an acquaintance, scientists [Timoshenko and Ehrenfest] somehow agreed to meet in the botanic garden during the morning strolls. On a clean snow surface one could draw needed figures and write formulas. By this means, before the spring of 1912 Timoshenko listened to Ehrenfest's lectures on the theory of relativity and quantum theories. In summer of 1912 Timoshenko and his wife went abroad in lieu of the D.I. Zhuravsky Prize that he got.'' This was due to the fact that S.P. Timoshenko was able to cover the expenses associated with the trip from the price which was equal to annual salary. ''In September 1912, S.P. Timoshenko returned to St. Petersburg, to continue his pedagogical activity,'' writes Pisarenko [20, p.103].

Hollestelle [START_REF] Hollestelle | Worstelingen met de moderne wetenschap, 1912-1933[END_REF] writes: ''During his studies, he went to Go¨ttingen for a longer period of time, to study with the famous mathematician Felix Klein, whose ideas also had a lasting influence on Ehrenfest. Here he also got to know his future wife, Tatiana Affanassjewa -a Russian mathematics student. After Ehrenfest finished his dissertation, the couple married.'' Ehrenfest moved to St. Petersburg after his wife got a job offer there. ''The circle of physicists in St. Petersburg welcomed Ehrenfest enthusiastically. That, however, did not lead to a job for him.'' Moreover, ''Once in St. Petersburg, Ehrenfest tried to make up for his shortcomings as a non-Russian as best as he could. He tried to fit into Russian society, almost symbolically, by growing a beard and long hair'' (Figure 1).

It is preferable to listen to Timoshenko's [23, pp.122-123] (see also [START_REF] Hollestelle | Worstelingen met de moderne wetenschap, 1912-1933[END_REF][START_REF] Huijnen | Paul Ehrenfest's rough road to Leiden: a physicist's search for position, 1904-1912[END_REF][START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF][START_REF] Timoshenko | Vospominanya (Remembrances)[END_REF]) own testimony: ''I took lodging on Aptekarkiy Island, across for the Electrical Engineering Institute. My nearest neighbor was a physicist, Ehrenfest, to whom Ioffe had introduced me. After finishing at the University of Vienna, Ehrenfest had gone to Go¨ttingen, and there worked successfully in theoretical physics. he found it difficult to build an academic career in Germany, so [he] decided to come to Russia, to Petersburg, where -after the introduction of ''Academic Freedom''-the interest in physics had markedly increased. He did not succeed in obtaining professorship, or even as a teacher. Yet he loved to teach and explain things, and we arranged to meet mornings in the Botanical Garden, when usually there were no visitors, and where on the clean surface of the snow one could make the necessary drawings with a stick. There he delivered to me a number of lectures on matters then new, such as relativity and quantum theory. In the spring of 1912 the snow in the Botanical Garden melted, and the lectures stopped. Ehrenfest had obtained a professorship and left Russia forever.'' Timoshenko [START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF] does not mention in his book As I Remember (see also its editions in Russian [START_REF] Timoshenko | Vospominanya (Remembrances)[END_REF][START_REF] Timoshenko | Vospominaniya (Remembrances)[END_REF][START_REF] Timoshenko | Vospominaniya (Remembrances)[END_REF] and in German [START_REF] Timoshenko | Errinerungen-eine Autobiographie[END_REF]) their scientific cooperation during this period. He neither submits a paper on combined effect of rotary inertia and shear deformation to any Russian or non-Russian Let us return to Timoshenko's book [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF] (see also its reissue [START_REF] Timoshenko | A course in theory of elasticity[END_REF]). In the beginning of his discussion of the combined effect of rotary inertia and shear deformation: ''We jointly with Prof. P. Ehrenfest obtained also the exact solution of the problem of layer of rectangular cross-section.'' The important word here appears to be ''also.'' Thus, Timoshenko informs, that not only he and Ehrenfest obtained the equations for what is now usually referred as Timoshenko beam theory, but they also, obtained some other problem's solution! The pertinent question arises why they did not publish these results immediately? Why did Timoshenko include results obtained with someone else, namely Ehrenfest, in his own book? Did he ask the permission from Ehrenfest? Or did neither Ehrenfest nor Timoshenko attach any importance to what they did together, and hence Timoshenko did not feel obliged to get permission? Is it possible to assume that they thought that the incorporation of rotary inertia and shear deformation effects was unimportant? The answer to the latter question must be an unequivocal no. Timoshenko must have known that Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] and Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF] already took into account rotary inertia. Perhaps Timoshenko did not know French and could have not read the book by Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF]? It should be noted that Timoshenko in 1913 published a paper in French [START_REF] Timoshenko | Sur la stabilitate´des systems e´lastiques application d'ura nouvelle me`thode a`la recherche´de la stabilited e certaines parties constitutives des pons[END_REF]; thus he either knew French or asked someone to translate his manuscript into French. Perhaps Timoshenko did not publish any papers in 1912 and 1913, so he likewise refrained from publication of his results with Ehrenfest? Again, we must answer this inquiry with no. In 1912, Timoshenko published two papers, and in 1913 he published six papers! Many years later, when Timoshenko's [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF] book was republished in the Russian language [START_REF] Timoshenko | A course in theory of elasticity[END_REF] the scientific editor of the re-publication Professor Eduard Ivanovich Grigolyuk contacted Timoshenko and asked him about his comment made in 1916, that ''we together with Prof. P. Ehrenfest obtained also the exact solution for the problem for rectangular cross section'' (figure 3).

What does this expression mean? There are several possibilities. It may mean, and this is possible when considering the different professions of the two persons, that Timoshenko presented the approximate beam theory to Ehrenfest, who helped to find an exact frequency relation for calibration. On the other hand, one should remember that Ehrenfest's Ph.D. dissertation was in classical mechanics; he only later changed his scientific interests. Such interpretation seems to be dismissed by Grigolyuk (see [28, p.338]) who writes: ''This result, obtained by S.P. Timoshenko with P. Ehrenfest, was not published jointly. In his letter to the editor of current edition, dated 2 July 1970, S.P. Timoshenko comments that 'this work was conducted in 1913, when Ehrenfest lived in St. Petersburg. Later on I did not work with him.''' Here following clarifying comment is needed. It may appear impossible that the joint work of Timoshenko and Ehrenfest could have been done in 1913, for Ehrenfest left Russia in 1912. Ehrenfest arrived to Leiden in October 1912, delivering customary inaugural lecture on December 2, 1912. Thus, Timoshenko [28, p. 338] might appear as misremembering the year when he and Ehrenfest obtained the equations describing rotary inertia and shear effect, as well as the elasticity solution for rectangular cross-section beams. Indeed, the year he mentions to Grigolyuk may be inferred as incorrect also from the autobiographical book of Timoshenko [START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF][START_REF] Timoshenko | Vospominanya (Remembrances)[END_REF][START_REF] Timoshenko | Vospominaniya (Remembrances)[END_REF][START_REF] Timoshenko | Errinerungen-eine Autobiographie[END_REF][START_REF] Timoshenko | Vospominaniya (Remembrances)[END_REF] As I Remember (with a Russian edition appearing in Paris, earlier, in 1963 [START_REF] Timoshenko | Vospominanya (Remembrances)[END_REF]; see also [START_REF] Timoshenko | Vospominaniya (Remembrances)[END_REF][START_REF] Timoshenko | Errinerungen-eine Autobiographie[END_REF][START_REF] Timoshenko | Vospominaniya (Remembrances)[END_REF]) where he mentions that ''in the spring of 1912 the snow in the Botanical Garden [of St. Petersburg -I.E.] melted, and the lectures [of Ehrenfest, delivered to Timoshenko in the Botanic Garden -I.E.] stopped. Ehrenfest had obtained a professorship and left Russia forever.'' Here too Timoshenko makes a mistake. Although Ehrenfest left Russia in 1912, he did return to Russia several times, and was elected to the Academy of Sciences of the former USSR, as its Corresponding Member, in 1924, three years prior to Timoshenko being elected to an analogous position.

In his 1968 book As I Remember [START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF], Timoshenko does not mention that the effect shear deformation and rotary inertia were studied by him in cooperation with P. Ehrenfest. As stressed above, this cooperation is mentioned in three places by him, namely in his 1916 book [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF] (figure 4) on theory of elasticity in Russian, in its 1972 republication [START_REF] Timoshenko | A course in theory of elasticity[END_REF], again in Russian, and in his 1922 paper [START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF] (figure 3), in English. However, he did not mention Ehrenfest in either his 1920 paper [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] or in his 1921 paper [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF]. The mentioning of Ehrenfest in the 1922 paper was probably influenced with the fact that he met Ehrenfest in Germany, Jena in 1921; by this time Ehrenfest ''had become a famous professor and now held the chair of physics'' [23, p.223]. Perhaps Timoshenko refrained from mentioning Ehrenfest due to the fact that he did not mention the details of scientific work in the autobiographical [START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF] rather than scientific book? The answer to this inquiry is negative. On page 119 of his autobiography [START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF], Timoshenko writes: ''In the winter of 1911-1912 I wrote a paper on the effect of transverse impact on a beam and investigated the influence of shear stresses on the transverse vibrations of beams. Both these articles were translated, one into German, the other into English, and subsequently served as a point of departure for the research of many authors.'' So, in his autobiography he does mention his scientific works; at the same time, he omits mentioning the cooperation he had with Ehrenfest on this topic.

Timoshenko could have combined writing about his scientific work on page 119 with mentioning Ehrenfest on page 123, but somehow, inexplicably, he did not do so. Perhaps by year 1963 when Timoshenko wrote the book [START_REF] Timoshenko | Vospominanya (Remembrances)[END_REF], in Russian, he didn't remember the above cooperation? Indeed, the English version [START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF] of his book is called As I Remember, implying perhaps that not everything was remembered half a century later. In their foreword [START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF], James. M. Gere and Donovan H. Young write, in March 1967: ''We have had the full support of Professor Timoshenko during the translation, and he has carefully checked this edition.'' This ''carefully checked'' edition still leaves many questions unanswered.

Timoshenko does not explain why work that was done with Ehrenfest in winter of 1911-1912 waited until 1916 book [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF] publication with only a footnote about Ehrenfest. Why did Timoshenko wait till 1916 to publish results? Why did neither of the three Timoshenko's papers in English [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF][START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF][START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF]] bear Ehrenfest's name alongside that of his? Perhaps perplexed by this, Grigolyuk [START_REF] Timoshenko | A course in theory of elasticity[END_REF] inquired about this as the editor of the second edition of Timoshenko's [START_REF] Timoshenko | A course in theory of elasticity[END_REF] (figure 5) book on elasticity. Timoshenko's response [START_REF] Timoshenko | A course in theory of elasticity[END_REF] (Figure 5) was terse, noting, ''This work was conducted in 1913, when Ehrenfest lived in St. Petersburg. Later on I did not work with him.'' (Figures 5 and9).

It appears surprising is that the book Collected Papers by S.P. Timoshenko [START_REF] Timoshenko | The Collected Papers[END_REF] reproduces his paper On the Transverse Vibrations of Bars of Uniform Cross-Sections [START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF], but in contrast with the original publication, does not mention Ehrenfest anymore (figure 6). The persistent question arises as to whether this took place, inadvertently, via the editors, E.L. Eriksen, J.N. Goo d i e r ,J . P .D e nH a r t o g ,L . S .J a c o b s e n ,R . E .P e t e r s o n , and D.H. Young. Or perhaps a typist omitted the name of Ehrenfest by mistake? Omitting Ehrenfest's name apparently contradicts the editors' statement on page V, which states: ''the papers themselves. are printed as they were originally published, any corrections or additions appearing in the form of footnotes.'' We must then assume that S.P. Timoshenko made these corrections and additions, though the editors do not explicitly mention this. This fact appears in contradiction with statements made by Timoshenko [START_REF] Timoshenko | The Collected Papers[END_REF] (see also [START_REF] Timoshenko | On the buckling of deep beams[END_REF][START_REF] Westergaard | One hundred fifty years advance in structural analysis[END_REF][START_REF] Pister | Trends in computational structural mechanics. After dinner talk delivered at the Trends in Computational Structural Mechanics Conference[END_REF]) about priority in publications mentioning Ehrenfest. One cannot refrain from asking if Tomoshenko himself deleted or asked secretary or editor(s) to delete mentioning Ehrenfest.

Of course, no clear answer can be given to questions of this type on the basis of the presented facts, as perplexing these facts may be. But these facts cannot be avoided. Also, when judging about publications that were written the first decades of the twentieth century, one should keep in mind that the manners of publishing and citing were possibly different in that period, the latter moreover having been affected by the terrible events of World War I.

In the chapter 22, titled ''We Move to Petersburg,'' Timoshenko [23, pp.119-123] informs us that in August 1911, after being fired from the Dean's job at Kiev Polytechnic Institute, he went to St. Petersburg to look for an adjunct position, since he ''could not get a regular position in the schools (universities -I. E.).'' On page 119 of his autobiography Timoshenko [START_REF] Timoshenko | As I remember: the autobiography of Stephen P. Timoshenko[END_REF] writes ''Although little time was left for science, for which anyway I lacked the necessary peace of mind, I did manage to get something done. In the winter of 1911-12 I wrote a paper on the effect of transverse impact on a beam and investigated the influence of shear stresses on the transverse vibrations of beams. Both these articles were translated, one into German, the other in to English, and subsequently served as a point of departure for the research of many authors.''

The results of the study on the impact on the beam were published in his papers [START_REF] Timoshenko | Erzwungene Schwingugen prismatischer Sta¨be[END_REF][START_REF] Timoshenko | Zur Frage nach der Wirkung eines Stosses auf einen Balken[END_REF], in German. Indeed, the careful reading of the above passage of Timoshenko book reveals, that about the work on ''effect of transverse impact on a beam'' he uses the statement ''I wrote a paper''. However, about his work on ''the influence of shear stresses on the transverse vibrations of beams'' he uses the verb ''investigated.'' This causes questions to arise: ''In the winter of 1911-1912 Timoshenko investigated the influence of shear stresses on the transversal vibrations of beams. Did he write a paper on this topic? If yes, when?'' The first inquiry is answered positively from Timoshenko's own statement that ''both those articles were translated, one into German, the other into English, and subsequently served as a point of departure for the research of many authors.'' Timoshenko fails to inform that he refrained to publish the material of ''the influence of shear stresses on transverse vibrations of beams'' in a journal article form more or less immediately after completing the investigation. He waited a staggering five years, until the work was reported in his book Course in Theory of Elasticity [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF] (figure 4) which was republished [START_REF] Timoshenko | A course in theory of elasticity[END_REF] in 1972 -the year of Timoshenko's death -by the Ukrainian Publishing House ''Naukova Dumka.'' Later, in 1920, the material was published in the Croatian journal Glasnik Hrvatskoga Prirodaslovnoga Drusˇtva (Herald of the Croatian Nature Association) [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF], in English (Figure 10). It should be stressed that this very article is not mentioned in the section ''Articles in English, French, and German'' in the list of publications given by Timoshenko's [23, pp.421 -427] autobiography that appeared in Russian. A year later, in 1921, the same article was published in the British Journal Philosophical Magazine [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF], without mentioning that it was already published in Russian [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF] and in English [START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF]. Is there any difference between Timoshenko's articles dated 1920 [START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF] and 1921 [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF]? Essentially none in the content, except that the 1921 paper [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF] does not reference work by Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF] who took into account the effect of rotary inertia on beam vibrations; additionally, the papers [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF][START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF] use different shear correction factors.

Timoshenko [START_REF] Timoshenko | Vibration problems in engineering[END_REF] attributes the term (Ir=g)(∂ 3 w=∂x∂t 2 )dx in the differential equation for the lateral vibration of the prismatic bar to Lord Rayleigh [12, paragraph 18b]. Leissa [39,p.314] writes: ''In 1877 Lord Rayleigh showed how the addition of 'rotatory' (in the language of his day) inertia effects, to those of classical transitional inertia affected the flexural vibration frequencies of beams.'' Timoshenko [START_REF] Timoshenko | Vibration problems in engineering[END_REF], followed by numerous investigators, also cites Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF] in his second exposition of the effects of rotary inertia and shear deformation [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF]. He fails to mention either Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] or Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF] in his subsequent paper [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF]. The latter paper is essentially identical to his first publication [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF] on this topic except different shear corrections factors, as mentioned above.

Overlooking contribution of Bresse

The third edition of the book Vibration Problems in Engineering, written by Timoshenko in 1955 (p.329), in collaboration with D.H. Young, again, mentions Lord Rayleigh's [START_REF] Rayleigh | The Theory of Sound[END_REF] contribution to the accounting of rotary inertia. Authors refrain from mentioning Bresse's [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] work although the book, by Timoshenko and Young published in 1955, was published after Timoshenko's 1953 [30, p.151] book on the history of strength of materials. This is where Timoshenko specifically mentioned that Bresse [START_REF] Rankine | A manual of applied mechanics[END_REF] was the first investigator who introduced the rotary inertia!

The fourth edition of the book Vibration Problems in Engineering, now co-authored by three authors namely by Timoshenko, Young, and Weaver in 1974, also contains reference to Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF]. The latest edition (fifth), co-authored by Weaver, Timoshenko, and Young, in 1990 (p.435) appeared after Timoshenko's and Young's death, in 1990. It does not mention Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] or Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF] but only Timoshenko's [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF] paper, with the note that the experimental verification of the shear effect was conducted. Strangely, neither of these editions contain reference to the first author who took into account the rotary inertia effect, namely, Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF].

It ought to be emphasized that Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF] did not refer to the work by Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF]. The partial explanation could be attributed to the circumstances in which his seminal book The Theory of Sound, came into being. Crandall [START_REF] Crandall | Rayleigh's influence on engineering vibration theory[END_REF] writes: ''In 1871 at the age of 28 he married Evelyn Balfour. Six months later, he fell seriously ill with rheumatic fever and nearly died. While on recovery, he was advised not to spend the first winter after his illness in England and so the young couple spent the three winter months on a houseboat traveling on the Nile River in Egypt. In the spring, they toured in Greece and Italy, returning to London in May 1873. A month later, John's father, the second Baron Rayleigh, died. Thus at the age of 30, John William Strutt assumed the title and the responsibilities of Lordship.'' To a large extent the discipline of vibrations as we know it today was born during those 3 months on the River Nile. According to Rayleigh's son's account, ''It was a monotonous life but it suited them. My father began his well-known treatise on the Theory of Sound, and worked at it all the morning in the cabin. He was able to write much of the earlier part of the book without access to a large library.In the mornings it was difficult to persuade him to land, even to see the most enchanting temple.'' The treatise was eventually completed 4 years later and published in two volumes: Vol. 1, on vibrations, in 1877, and Vol. 2 on acoustics, in 1878. Major additions were made in the second editions, in 1894 for Vol. 1 and in 1896 for Vol. 2.'' It is quite possible that Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF] did not know the work by Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF]. Alternatively, he may have thought that the extension of the Bernoulli-Euler beam theory to include rotary inertia was a minor issue. Naturally, during his tour on the Nile he could not have searched for references.

Several authors refer to the so-called Timoshenko beams as Bresse-Timoshenko beams. The encyclopedia of vibration [41, p.239] uses the term ''Bresse-Timoshenko theory,'' noting: ''Credit for including the rotatory inertia effect is given to Rayleigh in the 1870s, while Timoshenko added the shear flexibility effect in 1921.' ' Bert and Gordaninejad [START_REF] Bert | Transverse shear effects in bimodular composite laminates[END_REF][START_REF] Bert | Forced vibration of Timoshenko beams made of multimodular materials[END_REF] note that the effect of ''rotatory inertia dates back to the work of Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF], which predates.Timoshenko's [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF] shear deformable beam theory.'' They also note: ''A beam model taking into consideration the shear deformation effect is usually known as the Timoshenko beam model.'' Then, they add: ''A beam model taking into consideration the shear deformation effect is usually known as the Timoshenko beam model.'' Hagedorn and DasGupta [44, p.144]. write about ''some.aspects of the Timoshenko (or Bresse) beam.'' Laurenson and Yuceoglu [START_REF] Laurenson | Advances in aerospace structures and materials[END_REF] state: ''Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] in 1859 was the first to include both the rotatory inertia and shear flexibility. However, today this theory is referred to as the Timoshenko beam theory.'' Pascariu [START_REF] Pascariu | An improved theory for studying mechanically and thermally induced vibrations of cylindrical shells with transverse shear dependency on frequency[END_REF] stresses: ''Bresse is the first who is attributed with corrections of rotatory inertia and shear.'' Likewise, Beyer [START_REF] Beyer | Sounds of our times: two hundred years of acoustics[END_REF] states: ''It is of some historical interest that the rotatory inertia correction, usually attributed to Rayleigh (Theory of Sound, [12, pp.744-746]), is given by Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] in his Cours de Mechanique Applique (1859, p.126), which was largely unknown when Timoshenko proposed his model in 1921.'' Jahsman [START_REF] Jahsman | Elastic wave propagation and vibrations in plane stress and flexure[END_REF] wrote: ''Bresse was the first to develop the beam equations corrected for rotatory inertia and shear.'' Naturally such a beam model that takes into account rotary inertia ought to be referred to as a Bresse beam, or as Bresse-Rayleigh beam. Ironically, there is only a single work that refers, in the title, to Bresse-Rayleigh beam. This is the paper by Jackson and Oyadiji [START_REF] Jackson | Free vibration analysis of rotating tapered Bresse-Rayleigh beams using the differential transformation method[END_REF]. Other authors, when discussing the rotary inertia alone, mention Rayleigh's name. The Google Scholar website found, in March 2017, 390,000 hits on the term Rayleigh's beam; this number is not precise since numerous results contained the concept of Timoshenko beam.

In addition to the pioneering work by Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF], Timoshenko [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF][START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] did not cite additional work in his two papers, published in the West, on the effect of rotary inertia. This work is done by Searle [START_REF] Searle | The effects of rotatory inertia on the vibration of bars[END_REF], and published by a universally accessible journal Philosophical Magazine. This is the very journal in which a year after his first publication [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF] in English, his second publication [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] in English appeared, of the material first exposed in his book [START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF]. It should be stressed that Timoshenko's [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF] second publication was a close copy of the first, 1920 publication in the Yugoslav, namely Croatian, journal [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] (figure 10). The only difference between two papers consists in the fact that Timoshenko [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] cites Lord Rayleigh's contribution, whereas the later paper [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF] does not mention Rayleigh's work; likewise, he adopts different shear correction factors in these two papers.

The Google Scholar website produces over 78,000 hits on the term Timoshenko beam. Hence it is not surprising that the work by Laura, Rossi, and Maurizi [START_REF] Laura | Vibrating Timoshenko beams, a tribute to the 70th anniversary of the publication of Professor S. Timoshenko's epoch making contribution[END_REF] designates this method as ''epoch making'' in their review some 70 years after the theory was enunciated. These authors, as many others in the West, did not know that the method made a maiden appearance in 1916 (figure 4), and not in 1921, as it is uniformly thought.

These authors cannot be blamed, as it were, owing to the fact that the paper by Timoshenko [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] is not well known. In Timoshenko's autobiography [23, p.422], the 1920 paper [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] is not mentioned in the list of Timoshenko's papers. Timoshenko might have not been proud of the double publication he committed. Note that ref. [START_REF] Timoshenko | On the differential equation for the flexural vibrations of prismatical rods[END_REF] is not mentioned in the list of his publication dedicated to Timoshenko's 65th Birth Anniversary (figure 8). Nevertheless, we cannot blame Timoshenko either; it was a turbulent time in his life: he desperately wanted publicity in the West, in order to find a professorial job. As someone remarked, ''without publicity there is no prosperity.'' While at Zagreb Polytechnic, Timoshenko, by all probability, was looking elsewhere; thus, he probably considered the publication of his works, in then-famous journal Philosophical Magazine, as a gateway to a better position. The present writer is not aware of the publication etiquette at that time. Nowadays, multiple publications of the same paper, are not welcomed, to state mildly, and often are retracted.

The differential equation that takes into account both shear and rotary inertia effect reads:

EI ∂ 4 w ∂x 4 + rA ∂ 2 w ∂t 2 À rI 1 + E k 0 G ∂ 4 w ∂x 2 ∂t 2 + r 2 I k 0 G ∂ 4 w ∂t 4 = 0 ð1Þ
This equation is usually referred to as the Timoshenko equation. Timoshenko had predecessors, namely Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] and Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF], as far as the rotary inertia is concerned; and in relation to the shear deformation effect, albeit in statics, Koiter [START_REF] Koiter | Some comments on the so-called Timoshenko beam theory[END_REF][START_REF] Koiter | Comment on ''Timoshenko beam theory is not always more accurate than elementary beam theory[END_REF] pointed at others' contributions. It appears that one ought to refrain from using the uniformly known term Timoshenko beam theory, and utilize more informative terms. We will return to this topic later. It should be remarked that Mindlin [START_REF] Mindlin | An introduction to the mathematical theory of vibrations of elastic plates[END_REF][START_REF] Mindlin | An introduction to the mathematical theory of vibration of elastic plates[END_REF], Bert [START_REF] Bert | A critical evaluation of new plate theories applied to laminated composites[END_REF], and Elishakoff and Lubliner [START_REF] Elishakoff | Random vibration of a structure via classical and nonclassical theories[END_REF] refer to these sets as Bresse-Timoshenko equations. They felt, as Mindlin and Deresiewitz [START_REF] Mindlin | Timoshenko's shear coefficient for flexural vibration of beams[END_REF] did, that the historical justice demanded that at least the name of Bresse should have being reinstated when one discusses the effect of rotary inertia and shear deformation. Neither of these authors knew that the modern theory that S.P. Timoshenko advanced was developed with Paul Ehrenfest. It appears therefore fully justified to call it Bresse-Rayleigh-Timoshenko-Ehrenfest theory or BRTE theory if one wants to include all participants in the development of this theory. A shorter name would be Timoshenko-Ehrenfest beam theory. Indeed, Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] and Rayleigh [START_REF] Rayleigh | The Theory of Sound[END_REF] derived the rotary inertia effect but the contribution of the shear deformation is often more important than that due to rotary inertia. Now, Bresse [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF] derived, according to Artobolevsky, Bobrovnitsky, and Genkin [START_REF] Artobolevsky | Introduction to acoustic dynamics of machines[END_REF] (see also Challamel [START_REF] Challamel | Personal communication[END_REF][START_REF] Challamel | Personal communication[END_REF]) the effect of shear deformation but he did not utilize the shear coefficient. Thus the modern theory of beams, as it stands now, taking into account both rotary inertia and shear deformation incorporating shear correction factor, is due to two scientists: S.P. Timoshenko and P. Ehrenfest, in a personal jaw-droping testimony of Timoshenko -one of the two co-authors. This appears to be relatively straight forward case of establishing priority. This is why hereinafter we will adopt the name Timoshenko-Ehrenfest beam theory.

In his book on history of strength of materials, Timoshenko [53, p.418] provides his own take on what became to be known as Timoshenko beam theory: ''Lateral vibrations of shafts and beams core also of great practical importance. The simplest cases of vibration of prismatic bars were their solutions included in books on acoustics. In applying these solutions to practical beams in which the crosssectional dimensions are not very small in comparison to the length, or to cases where higher modes of vibration are of importance, it has become necessary to derive, more complete differential equation, in which the effect of shearing forces on deflections is considered.'' 4. Did Ehrenfest refuse his name to appear as that of the co-author?

In S.P. Timoshenko's paper ''On Transverse Vibration of Bars of Uniform Cross-Section'' (Philosophical Magazine and Journal of Science, Series 6, 1922, Vol. 43, pp.125-131), on page 127 it is remarked that the work was conducted by P. Ehrenfest and by Timoshenko jointly but solution (i.e. its numerical implementation of the frequency equation, belonged to S.P. Timoshenko. Ehrenfest's name, however, does not appear as one of the authors. A nagging thought persists: Did Ehrenfest refuse to be the co-author of the paper on the effect of shear deformation and rotary inertia with Timoshenko? This possibility cannot be totally discarded, since Ehrenfest's approach to science is best illustrated by what he wrote to Robert Oppenheimer in the summer of 1928, after Oppenheimer invited himself for an extended stay in Leiden, ''If you intend to mount heavy mathematical artillery again during your coming year in Europe, I would ask you not only to not come to Leiden, but if possible not even to Holland, and just because I am really so fond of you and want to keep it that way. But if, on the contrary, you want to spend at least your few months patiently, comfortably, and joyfully in discussions that keep coming back to the same few points, chatting about a few basic questions with me and our young people -and without thinking much about publishing (!!!) (emphasis by present writer -I.E.)why then I welcome you with open arms!'' (see [START_REF] Monk | Oppenheimer: a life inside the center[END_REF][START_REF] Monk | Oppenheimer: a life inside the center[END_REF]).

As stressed above, in his 1968 book As I Remember Timoshenko [START_REF] Pisarenko | [END_REF] does not mention that the effect shear deformation and rotary inertia were studied by him in cooperation with P. Ehrenfest. As emphasized above this cooperation is mentioned in three places by him, namely in his 1916 book on theory of elasticity in Russian [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF], in its 1972 republication [START_REF] Challamel | Personal communication[END_REF], again in the Russian version of that book on the theory of elasticity, and in his 1922 paper [START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF]. However, he did not mention Ehrenfest in neither his 1920 paper [START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF] nor in his 1921 paper [START_REF] Rankine | Miscellaneous scientific papers[END_REF] or in three editions of the vibration text book.

Fortunately, Paul Ehrenfest found his fame in the Netherlands. He tragically committed suicide in 1933. After some years, the grateful Dutch created the Paul Ehrenfest Archive at the Museum Boerhaave in Leiden. Moreover, Bruce Wheaton [START_REF] Wheaton | Catalogue of the Paul Ehrenfest Archive at the Museum Boerhaave Leiden[END_REF] published the Catalogue listing all available documents in the Ehrenfest Archive. Two pages in the Catalogue mention a letter and postcards received from S.P. Timoshenko. When the present author applied to the Archive with the request to receive copies of the correspondence, the Archive was in the process of digitization, and hence the correspondence was not available. However, the present author obtained later on the of the additional material, specifically the letter and postcards received by S.P. Timoshenko while Ehrenfest was still in St. Petersburg. These correspondences were, by all probability, considered important by Ehrenfest since he brought these with him to Leiden. Only one letter [START_REF] Timoshenko | A course of elasticity theory[END_REF] (figure 2) is associated with the beam theory that takes into account the rotary inertia and shear deformation. Over half a century later, Timoshenko [START_REF] Hollestelle | Worstelingen met de moderne wetenschap, 1912-1933[END_REF] composed his autobiography titled Remembrances, which appeared in Russian, in Paris. In 1968 the English version appeared [START_REF] Pisarenko | [END_REF] titled As I Remember, translated from the Russian edition by Robert Addis. In their foreword [START_REF] Pisarenko | [END_REF], James M. Gere and Donovan H. Young write, in March 1967: ''It is only natural that the School of Engineering at Stanford University, where Timoshenko taught for so many years, has arranged for the English translation and publication of this book.'' To digress, let's note that the Timoshenko's book translation was a highly unusual gesture of good will and respect on the behalf of Stanford University. In Chapter 22 titled ''We Move to Petersburg,'' Timoshenko [20, p.119] writes: ''In the winter 1911-12 I wrote a paper on the effect of a transverse impact on a beam and investigated the influence of shear deformation on the transverse vibrations of the beams. Both these articles were translated, one into German, the other into English, and subsequently served as a point of departure for the research of many authors''. Later on, in the same chapter, Timoshenko [20, pp.122-123] tells us the story of his meeting with Paul Ehrenfest, and [START_REF] Timoshenko | The Collected Papers[END_REF]. The footnote on Paul Ehrenfest appearing in his original paper [START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF] (figure 3) is missing. It is unknown who deleted the footnote appearing on the original paper. The footnote and manner of writing differ from original, meaning that the paper was re-typed, with the footnote about P. Ehrenfest was deleted. reproduces the picture taken in winter 1912 (figure 1). In light of the fact that in his book on the theory of elasticity (Timoshenko, 1916, p. 206) (tittle page in figure 4) mentioned that the theory was developed with Ehrenfest, Timoshenko could have combined writing about his scientific work on page 119 of autobiography with mentioning the cooperation with Ehrenfest. However, he delayed mentioning Ehrenfest till later pages, namely 122 and 123, without mentioning their cooperation and the beautiful theory that was a result of this joint work!

General comments on priority and conclusion

One correspondent wrote to this writer: ''.The questions of priority were discussed by many authors who noted that it is difficult to expect a fair judgement in this field.'' For example, discussing history of mathematics, Borodich [START_REF] Borodich | The Hertz-type and adhesive contact problems for depth-sensing indentation[END_REF] wrote: ''There is even a joke that the fundamental law of the history of mathematics says: ''Any proposition named after someone traces its origin back to another one'' (translation from German:''Ein Satz, der einen Namen tragt, stammt von einem Anderen'') [START_REF] Go¨rtler | Zur Geschichte des P-Theorems[END_REF] (note the similarity with W.T. Koiter's statement above). It is interesting to note that Sir Michael Berry referred to a similar statement as the ''theorem of Arnold.'' Arnold's [START_REF] Arnold | Newton and Hooke: pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals[END_REF] statement reads: ''Integration had already been encountered with Archimedes, and differentiation with Pascal and Fermat; the connection between these operations was known to Barrow. What did Newton do in analysis? What was his main mathematical discovery? Newton invented Taylor series, the main instrument of analysis'' might be brought as an illustration of the ''law.''

Another correspondent wrote ''The Gibbs phenomenon was discovered and described in detail 50 years before Gibbs by Wilderham.. One of the physicists, with annoyance said, if you did something worthwhile, predecessors immediately show up! This phenomenon was studied in detail by van Dyke [START_REF] Van Dyke | Nineteenth-century roots of the boundary-layer idea[END_REF] as applied to boundary layer theory. He noted that some of the ideas of this theory were used by many scientists long before Prandtl. However, these works can be compared with grains that have not sprouted, and Prandtl's work is rooted from which the new theories gave grown up.''

To reiterate in this particular case Timoshenko himself writes in 1916 that the theory was developed by him together with Ehrenfest. Moreover, in his 1922 paper [START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF] Timoshenko writes that the theory was developed by Ehrenfest (noting his co-author as the first one!) and him, and that solution, i.e. enumeration of results, was conducted by Timoshenko.

Additionally, based upon the response by Timoshenko, Grigolyuk [START_REF] Grigolyuk | Timoshenko: life and destiny[END_REF][START_REF] Grigolyuk | Timoshenko: life and destiny[END_REF] -who was a fellow Ukrainian and a fan of Timoshenko's -unequivocally concluded in the biographical book on Timoshenko that the theory belonged to Timoshenko and Ehrenfest (see the quote of Grigolyuk [START_REF] Grigolyuk | Timoshenko: life and destiny[END_REF][START_REF] Grigolyuk | Timoshenko: life and destiny[END_REF] in the end of section 1).

The foreword to Arnold's [START_REF] Arnold | Mathematical understanding of nature: essays on amazing physical phenomena and their understanding by mathematicians[END_REF] book mentions so-called ''Arnold's law (implied by statements in his many letters disputing priority.): Discoveries are rarely attributed to the correct person.'' In words of Sir Michael Berry [START_REF] Berry | Private communication[END_REF] ''Arnold's law applies to itself. I have heard it referred to by other names.'' Indeed, it appears perhaps earlier as the Stigler's [START_REF] Stigler | Stigler's law of eponymy[END_REF] law of eponymy, stating ''No scientific discovery is named after its original discoverer,'' which probably is an exaggerated maxim.

In this particular case, the judgment on priority is much simpler than in numerous other cases in the history of ideas: Timoshenko was one of the co-authors of the now celebrated beam theory; the other was Ehrenfest. This work confirms the conclusion made by Grigolyuk [START_REF] Grigolyuk | Timoshenko: life and destiny[END_REF][START_REF] Grigolyuk | Timoshenko: life and destiny[END_REF] on this matter: This study shows that theory of beams that incorporates shear deformation and rotary inertia, ought to be referred to as the Timoshenko-Ehrenfest beam theory.

The extensive evidence provided in this article, and especially Timoshenko's own revealing testimonies in [START_REF] Bresse | Cours de me´canique applique´e-re ´sistance des mate´riaux et stabilite´des constructions[END_REF][START_REF] Timoshenko | On the correction for shear of the differential equation for transverse vibrations of prismatic bar[END_REF][START_REF] Challamel | Personal communication[END_REF] and his 1913 letter to Ehrenfest, show that adopting the name Timoshenko-Ehrenfest beam theory will serve both the justice and truth.

In this respect, it appears apropos to quote Psalm 85:12: ''Truth will sprout from the Earth.'' In the words of A.J. Heschel (1907-1972) -theologian and philosopher -''truth lies buried, stifled in the grave, yet it remains alive. Truth wants to emerge, but man does not permit its appearance. Man's structure stands like a mausoleum on the grave of truth, preventing it from raising its head.'' Indeed, the search conducted by the present writer might strongly remind the archeological digging and the construction of the entire picture by sealing pieces of the puzzle together. Once I learned that Timoshenko and Ehrenfest developed the beam theory together, this idea consumed me, in light of human responsibility for searching for truth and bringing it to the community of readers who are advised to make their own conclusions.

provided subtle discussions on the ''Arnold law.'' The author is thankful to Mechanics and Computation Department at Stanford University and especially Professor Wei Cai, for my appointment, in June 2019, as the S.P. Timoshenko Visiting Scholar. Their warm and kind hospitality as well as the invitation to deliver the seminar on the topic of this paper are greatly appreciated. The author will be most grateful to any individual who could communicate additional information and/or documents on the topics touched upon in this study, at the electronic address, elishako@fau.edu.

Figure 1 .

 1 Figure 1. Paul Ehrenfest (left) and Stephen Timoshenko in St. Petersburg, published by permission of Museum Boerhaave, Leiden, The Netherlands

Figure 2 .

 2 Figure 2. Copy of Timoshenko's letter to Ehrenfest, apparently dated January 10, 1913, including the equation taking into account shear deformation and rotary inertia (Courtesy of Prof. Dirk van Delft Director of Museum Boerhaave, Leiden, The Netherlands).Full letter is reproduced in ref.[START_REF] Elishakoff | Handbook on Timoshenko-Ehrenfest beam and Uflyand-Mindlin plate theories[END_REF]

Figure 3 .

 3 Figure 3. ''Copy of the page in ref. [14] Timoshenko, 1992. The footnote states "This ''frequency equation'' was found by Prof. P. Ehrenfest and myself in collaboration. The solution given in this paper is my own''.

  The top right corner contains the date ''January 10;'' the handwriting, apparently by Bruce Wheaton, below the date indicates the year as 1908. But this letter couldn't have been written in 1908 for then Timoshenko and Ehrenfest didn't work together. Moreover, Timoshenko was in Kiev whereas Ehrenfest was in St. Petersburg. The very next page, however, which is not reproduced here, reveals that the letter was written in 1913. Indeed, Timoshenko informs Ehrenfest: ''After you left [St. Petersburg I.E] I did not have a chance of undertaking walks.'' Ehrenfest left St. Petersburg in 1912; thus the letter must have been written in 1913. Indeed, Timoshenko mentioned the year 1913 in his response to Grigolyuk. Just before the equation, Timoshenko writes [58]: ''Let us summarize the obtained results.'' Since Timoshenko does not say what the equation signifies, that shows that they both knew what they were discussing, clearly indicating that this equation was known to both of them, not only to Timoshenkothe composer of the letter.

Figure 4 .

 4 Figure 4. Title page of S.P. Timoshenko's book [13] A course in theory of elasticity. Part II. Beams and plates, Petrograd, 1916, in the Russian language.

Figure 5 .

 5 Figure 5. Copy of page 338 from S.P. Timoshenko's [28] book A course in theory of elasticity, second edition, 1972. The footnote mentioning Ehrenfest, was included along with the comment of the book's editor Prof. E.I. Grigolyuk, S.P. Timoshenko responded that the work was conducted when P. Ehrenfest lived on St, Petersburg, ''later, i didn't work with him," wrote S.P. Timoshenko''.

Figure 6 .

 6 Figure 6. Copy of the page from S.P. Timoshenko's article On the Transverse Vibrations of Bars of Uniform Cross-Section [14] included in The Collected Papers of Stephen P. Timoshenko[START_REF] Timoshenko | The Collected Papers[END_REF]. The footnote on Paul Ehrenfest appearing in his original paper[START_REF] Timoshenko | On the transverse vibration of bars of uniform cross sections[END_REF] (figure3) is missing. It is unknown who deleted the footnote appearing on the original paper. The footnote and manner of writing differ from original, meaning that the paper was re-typed, with the footnote about P. Ehrenfest was deleted.

Figure 7 .Figure 8 .

 78 Figure 7. The copy of page 206 of Timoshenko's elasticity book[START_REF] Timoshenko | A course of elasticity theory. Part 2: rods and plates[END_REF]. After exposing the new beam theory, S.P. Timoshenko mentions, in the footnote 2: ''We jointly with Prof. P. Ehrenfest, obtained also an exact solution for the layer of rectangular crosssection.''

Figure 9 .Figure 10 .

 910 Figure 9. An extract from page 338 of the second edition of Timoshenko's [58] book A course in theory of elasticity published in 1972 with E.I. Grigolyuk serving as editor. In square parentheses Grigolyuk comments: ''The discussion concerns the transverse vibrations of the beam with rectangular cross section based on theory of elasticity. This result, obtained by S.P. Timoshenko with P. Ehrenfest, was not published [by them] jointly. In the letter to the editor of the current edition, S.P. Timoshenko notes that this work was performed in 1913, when Ehrenfest lived in Petersburg. Later I did not work with him.''

Acknowledgements

I express gratitude to numerous researchers with whom I had a chance to discuss the beam theory that incorporates the effects of rotary inertia and shear deformation. Sincere thanks are expressed to Prof. Dirk van Delft, of the Boerhaave Museum of Leiden, the Netherlands for allowing incorporation of Figures 1 and2 into this essay. Author thanks the erudite reviewers who

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Isaac Elishakoff

https://orcid.org/0000-0002-1723-7771


Sean Chorney
email: sean_chorney@sfu.ca

Canan Gunes
email: cgunes@sfu.ca

Nathalie Sinclair

Multiplicative reasoning through two-handed gestures

Keywords: Two-handed gestures, multiplicative thinking, touchscreen technology

This exploratory study focuses on two-handed gestures when using an innovative iPad application called TouchTimes. We suggest that the use of the two hands is relevant to multiplicative reasoning as it provides a way of expressing multiplication as a coordination of two quantities, rather than simply as repeated addition. We report on the use of TouchTimes by two third-grade girls; we first exemplify the designed two-handed gestures the girls used and then identify some emergent gestures that we see as being relevant to multiplicative thinking, particularly in terms of the relation between the multiplicand and the multiplier in a multiplication expression.

Introduction

In this paper we report on an exploratory study involving a pair of grade three students working on an iPad application called TouchTimes (TT, [START_REF] Jackiw | TouchTimes (iPad App)[END_REF]) that was designed to offer an accessible model of multiplication that does not rely on repeated addition. TT uses the multi-touch affordance of the iPad to express the relationship between multiplicand, multiplier and product through the use of specifically designed two-handed gestures. It is an extension of TouchCounts, also designed by [START_REF] Jackiw | TouchCounts (iPad App)[END_REF], which enables gesture-based actions for counting, adding and subtracting and has been shown to improve subitizing, the ability to differentiate between cardinality and ordinality [START_REF] Sinclair | Whatever be their number: Counting on the visible, the audible and the tangible[END_REF] as well as basic number sense [START_REF] Sedaghatjou | Exploring cardinality in the era of touchscreen-based technology[END_REF][START_REF] Sinclair | Rhythm in number: Exploring the affective, social and mathematical dimensions of using TouchCounts[END_REF]. TT builds on this TouchCounts research by extending the gesture-based arithmetic. In this research study, we are interested in the way two handed gestures can support young students' multiplicative thinking.

Research on learning multiplication

Multiplication is difficult for many students, yet very important for subsequent higher-grade topics such as proportion and algebra (Siegler et al., 2012). One of the difficulties with multiplication results from students' excessive exposure to multiplication as repeated addition [START_REF] Askew | Multiplicative reasoning: Teaching primary pupils in ways that focus on functional relationships[END_REF]. It is important to develop students' understanding of multiplication in terms of relations, particularly the relationship between the multiplicand and the multiplier. Designed to develop the bi-directional functionality of multiplicand and multiplier, TT allows for both the multiplicand and the multiplier to be changed either during and after a multiplication computation. We hypothesise that this offers not only an empirical environment for experimentation, but also the possibility of noticing the relationship the multiplicand and the multiplier have on the product.

In the work of [START_REF] Clark | Identification of multiplicative thinking in children in grades 1-5[END_REF], multiplication has two levels of abstraction. The shift from the many-to-one and an inclusion relation between multiplicand and multiplier. Multiplication is relational and functional in that both multiplier and multiplicands are abstractions that affect each other. For example, in the case of , if 3 is the multiplicand and 4 the multiplier, 3 to 1 is the abstraction of three becoming one (the unit) and the 'inclusion relation involves a transfer of units in that the multiplicand "3" counts the number of items while the multiplier "4" counts the number of triples of these' (Boulet, 1998, p. 14). In TT the physical use of two hands is meant to enact this dual abstraction. We are thus interested in studying the extent to which students' gestural interactions with TT supports their multiplicative thinking.

Brief description of TouchTimes

The initial screen of TT starts blank, except for a vertical line (Figure 1a). A user can create two kinds of mathematical objects: pips and pods. The first finger(s) down on the screen produces pips, which are coloured discs that appear under each finger. For example, if the user places three fingers on the left side (LS) of the screen, three differently coloured pips appear (Figure 1b). If the user then places four fingers on the right side (RS) while holding the three pips, four 'pods' of three pips appear (Figure 1c). Each pod is a bounded object containing a smaller version of the number, colour and configuration of the LS pips. If a finger is lifted on the LS, then there will be two pips there and each of the four pods will contain two pips. More pips can also be created by placing more fingers on the LS; this will affect the RS pods accordingly). If all the pip-making fingers are lifted, the pods disappear too (multiplication by 0), and the screen goes blank. Additional pods can be created on the RS; once created they remain on the screen even when fingers are lifted (this enables multiplication of numbers bigger than

, where , which is the maximum number of simultaneous touches allowed). The situation can be completely reversed: if the first touch is on the RS, then pips are produced there and touches on the LS produce pods. As soon as there are both pips and pods on the screen, a multiplication expression appears on the top of the screen. For the above situation, the mathematical expression is (Figure 1c). Once created, pods can be dragged anywhere on the screen and can also be trashed. Pips can also be moved, as long as the fingers are not lifted. In the reversed situation, the multiplication expression is 12 = 4

The two hands have different functions, and in this way, multiplication is experienced relationally, rather than in terms of repeated addition.

Theoretical framing

Research into iPads is still relatively new and there have been various frameworks proposed to study how the iPad interrelates with mathematical thinking. Embodied theories have been common since the iPad offers unique kinaesthetic experiences with number, particularly in terms of the gestures required to create and manipulate mathematical objects. [START_REF] Bairral | The use of hands and manipulation touchscreen in high school geometry classes[END_REF] use the idea of embodiment to suggest that the action of the body in any type of activity acts as a grounding for cognition. However, we align with researchers based in a nondualist paradigm (challenging the separation of mind and body), and see embodied action not as a reflection of thought, nor as a grounding of thought, but rather as an aspect of mathematical thinking [START_REF] De Freitas | Mathematics and the body: Material entanglements in the classroom[END_REF].

We employ an embodied framework that sees both the tool (the iPad and application) and the children as a cohesive unit, as an assemblage [START_REF] De Freitas | Mathematics and the body: Material entanglements in the classroom[END_REF]. By seeing TT not as a separate entity from the students but as 'entwined' (Nemirovsky et al. 2013;[START_REF] Arzarello | A cognitive analysis of dragging practises in Cabri environments[END_REF] we can theorize a formation of bodily-based mathematical activity. This form of embodiment shifts from the tool-student duality, suggesting instead that the mathematical thinking is action-with-tools.

Given the design of TT, we are particularly interested in how gestures are indicative of students' mathematical thinking. We follow [START_REF] Sinclair | Number's subtle touch: expanding finger gnosis in the era of multi-touch technologies[END_REF] in seeing gesture on the iPad as epistemic and communicative. [START_REF] Bairral | The use of hands and manipulation touchscreen in high school geometry classes[END_REF] claim that there are "six basic finger actions for input on a touchscreen: tap, double tap, long tap (hold), drag, flick, and multi-touch (rotate)" (p. 2). For TT, we suggest that the last type of gesture, multi-touch, can be expanded and elaborated, particularly in relation to the use of two-handed gestures, on which little research has been conducted in mathematics education. An exception is [START_REF] Duijzer | Touchscreen tablets: Coordinating action and perception for mathematical cognition[END_REF], who study the coordination of action and perception in terms of proportional reasoning in an environment where children move their two hands in the same direction, but at different rates. [START_REF] Bairral | The use of hands and manipulation touchscreen in high school geometry classes[END_REF] address two kinds of two-handed gestures: (1) when one hand is holding and the other is dragging; (2) when both hands are doing the same gesture (as in when two fingers are dragging away or towards each other to expand or reduce an image). In TT each hand performs different movements and different functionalities. One hand acts as the multiplier; the other as the multiplicand. Our research questions therefore are: What are the different types of two-handed gestures used by novice, young learners? How do the two coordinating hands express multiplicative thinking?

Methods

In this study, we examined two students' first contact with TT. We worked with two grade three students in an elementary school in a downtown part of British Columbia. We will call them Jessica and Ava in this paper. This pair of students was chosen by the classroom teacher who thought that they would work well together and to provide Jessica, who often ignores instructions, with a different kind of mathematical experience. We video recorded their interaction with the application. The session lasted for 27 minutes. For the initial seven minutes, the students interacted with the tablet without being given a specific task by the interviewer (third author). After the seventh minute, the interviewer occasionally proposed a mathematical task. We transcribed the video, paying attention to when the girls made two-handed gestures, as well as to their awareness of multiplication. For example, if the girls moved their fingers in such a way that the multiplicand reduced by one, we identified their awareness of changes based on what they said. We distinguished situations in which each girl used a hand, which we call a two-person bi-handed gesture, from situations in which one girl used two hands, which we call a solo bi-handed gesture. We see the former as being highly relevant to exploring the social aspect of joint coordination (see [START_REF] Sebanz | Prediction in joint action: what, when, and where[END_REF].

Analysis

We present our analysis in two parts. First, we use an a priori analysis of the "designed" gestures, that is, the gestures that TT was designed to allow, and describe which ones were observed in the video. We then present two of the "emergent" gestures found, that is, gestures that were not explicitly designed, but that were identified as being relative to the girls' thinking about multiplication.

A priori analysis of designed gestures

Each gesture identified in Table 1 is done with a single hand. The two-handed gesture comes into existence when any two of these gestures (including identical gestures) are done on different sides of the screen and with different hands (though it is technically possible to use one hand to touch both sides of the screen). The number of two-handed gestures then grows to include all the combinations of any two of the gestures in Table 1.

Single tap (T): Tapping the screen with one or many fingers. In Figure 2, we depict three of the more frequent gestures: T, T; T, H; H&T, H. Below, we provide a brief description of each as it occurred in the video.

Figure 2: (a) T,T;(b) H,T;(c) H&L,H From the students' perspective, in Figure 2a the middle finger of the first hand taps the RS of the screen and the index finger of the second hand taps the LS of the screen. This two-handed gesture was mostly observed at the beginning of the video as the girls were first exploring TT. They stopped using this gesture after a while, either because they realized that this did not create more than one object at a time, or that a holding gesture was necessary to create a pod. In Figure 2b, the first hand holds five fingers on the LS all-at-once, the second hand taps the index finger on the RS. This gesture produces one pod, which is a unit of five pips. In Figure 2c, there are three fingers of one hand on the LS, while lifting the pinky and thumb, while the index finger of the second hand was held on the RS. This gesture was very common and changed the number of pips within each of the pods.

A posteriori analysis of the emergent gestures

We will present two episodes, each of which involved the use of an emergent gesture, which we will refer to as a three-handed gesture and a dancing gesture, respectively.

At the beginning of the first episode chosen, Jessica holds five fingers of her right hand on the RS and Ava holds two pods on the LS (H, H). There were 15 pods on the screen showing . Jessica then used her left hand to make another pip, thereby making six pips in total (Figure 3a), changing the equation to , and changing all the pods to include six pips.

Jessica: What the Ava Wooow (drags index finger, producing 7 pips, as in Figure 3b) Jessica di di di di didi di di di di oh wait I can change the shape When she put her sixth finger down, Jessica brought her second hand into play so that the girls were making a three-handed gesture of hold pips, drag pip and hold pods (Figure 3a). When she said "What the" she seemed to be surprised at the effect of the sixth pip, which changed the configuration of each of the 15 pods, as well as the multiplication statement. Ava then let go of the pods she was holding with her left hand and used her index finger on the RS to create another pip, which she dragged on the screen, saying "wooow". There were now seven pips in each pod and a new equation . This new three-handed gesture now included a hold, a drag and a drag (Figure 3b). As Jessica dragged her sixth finger, the shape of each pod changed to mimic the configuration of her six fingers, which she seemed to realise when she says "I can change the shape". In this episode the girls make two different three-handed gestures. In the first instance, Ava had been busy making pods and it was only when Jessica brought her third hand into play that she seemed to notice that the additional pip created an important change on the screen, prompting her to say "What the". Then when Ava sees what Jessica is doing, she changes from a holding pods gesture to a dragging pips gesture. In the first three-handed gesture, Ava does not need to be holding the pods, so the gesture might feel extraneous to the activity. However, it is perhaps because of the fact that she is holding pods that she noticed the interesting effect produced by Jessica's added pip (the pods would have changed size and configuration right under her fingertips, so to speak). In the second case, Ava and Jessica are both doing drag-pip gestures, and so replicating the same gesture that could have been done by just one hand. However, it is in wanting to mimic Jessica that Ava starts dragging a pip, and so we see how the third hand enables both girls to do the same thing at the same time, together producing a change in the shape of the pods that is more than what each of them could have produced on their own. In the second episode, Ava was holding five fingers on the left screen (which she held for the complete episode; see Figures 4a,4b) and Jessica had just finished creating pods and dragging them around the screen. There were 31 pods on the screen and the multiplication sentence at this point was . Jessica touched one finger on the LS under Ava's hand (H, T) and then lifted it up again (H, L). Once more she tapped in the same spot, down and up. The multiplication sentence changed from to , back and forth. She repeated this a third time and said, "They're dancing again" and both girls giggled. She continued to tap this way for a total of eight repetitions. Jessica, still maintaining one finger on the screen, then adds a second finger, causing each pod to increase to seven pips (Figure 4b). She then began to touch up and down repeatedly with two fingers; the multiplication sentence changing from to , back and forth (Figure 4a,4b). This time, however, Jessica added a secondary rhythm. She sang 'di di didi di di' while tapping; on each 'di' she tapped once' on the 'didi' she tapped quickly, and in between phrases, she did not tap. She continued to tap with two fingers like this through four phases of 'di di didi di di'. We see this 'dancing gesture' as an emergent one that actually combines a repetitive set of (H, T) gestures. It becomes a single gesture for the girls that enables them to make the pods "dance". Aside from the evident visual appeal of this dancing gesture, it also embodies the girls' functional awareness of the relation between the pips and the pods, showing as it does how the number of pips affects the shape and colour of each of the pods. With it, the girls can change the multiplicand very quickly, seeing the effect on both the visual dynamics of the screen, as well as on the multiplication sentence.

Discussion

Our first research question asked, what were the different types of two-handed gestures used by novice, young learners? In our a priori table we identified the designed, that is, the basic gestures supported in TT, which were the gestures that we expected the children to make. The most frequently-used gesture was the (H, T) in which one girl would hold pips (usually Ava), and the other would create pods one finger at a time. The girls rather quickly began to use hold gestures when they realised that lifting their fingers off the pips would clear the screen. They can often be seen holding both pips and pods, especially at the beginning. Dragging was used both to move pips on the screen, as discussed in the first episode above, and also to drag pods to the trash. The use of the gestures in Table 1 provided a starting point for their developing fluency with the tool.

We also identified emergent gestures that seemed to play more important roles in the girls' understanding of how the pips and pods related to each other. In the first episode, we describe two different three-handed gestures made by the girls that helped both to draw their attention to the relation between the number and position of pips on the screen, and to the shape of the pods. The second three-handed gesture seemed to also have an important social component in allowing both girls to do the same gesture at the same time. As with these three-handed gestures, the emergent dancing gesture combined two or more of the gestures from Table 1. The dancing gesture can thus be seen as a bundling of several, sequentially produced (H&T, H) and (H&L, H) gestures. Over the course of the experiment, this gesture occurred multiple times with different numbers of pips and pods. The girls' continued return to this gesture, as well as the identification of its presence by such utterances as, 'they're dancing again', highlight their awareness of the multiplicative relation between multiplicand and multiplier.

Our second research question asked: How do the two coordinating hands express multiplicative thinking? It was in the use of the two emergent gestures that the girls seemed to express and notice the relation between pips and pods. In making the number of pips change, the girls noticed that the number of pips determined the shape and size of the pod, thus observing the effect of the size of the unit on the pods. The fact that the relation between the pips and the pods took some time to notice and control suggests that this is not a simple relation to understand, both in terms of physically coordinating two hands (or more) across two people, and in terms of tracking the bi-directional relation between multiplicand and multiplier. This embodied approach to multiplication seemed to initiate gestures through which the girls could explore/express this multiplicative relation.

Conclusion

In this exploratory study, which looks at how gestures were used in the TT app, we identified 6x6 different "designed" gestures and exemplified three of the most commonly-used ones. We then described two additional emergent gestures that were used several times by the girls and seemed important to their growing fluency with the app and their awareness of the multiplicative relation between the pips and pods (multiplicand and multiplier). We propose that the gesture-based multiplicative actions of the girls offered them novel ways to explore and express multiplicative relationships. In the repeated addition approach, the relation privileges a focus on the multiplier (3+3+3+3+…). In TT, however, both operands can be manipulated with immediate feedback, both in terms of the visual presentation of the pods, and symbolically through the equation. If repeated addition tends to be algorithmic, linear, sequential and unidirectional, TT seems to offer a more interactive, bi-directional and simultaneous instantiation of multiplication.

This research provides insight into the link between action and mathematical reasoning, and has implications for supporting students' multiplicative thinking. In future work, a closer examination of how learners coordinate the gestural expression with symbolic expressions (the equations shown on the screen) should be examined more closely. Also, the collaborative interactions of learners working together, as expressed through the three-handed gestures described here, deserves further attention.

Figure 1 :

 1 Figure 1: (a) Initial screen of TT; (b) Creating 3 pips; (c) Creating 4 pods

  Hold (H): Pressing a screen object continuously without lifting the finger/s. Drag (D): Moving an object with finger/s on the screen continuously without lifting Hold and tap (H&T): While some fingers are holding, other finger/s are tapping Hold and drag (H&D): While some fingers are holding, other finger/s are dragging Hold and lift (H&L): While some fingers are holding, other finger/s are lifting

Figure 3 :

 3 Figure 3: (a) Adding a 6th pip; (b) Adding a 7 th pip

Figure 4 :

 4 Figure 4: (a) Jessica about to tap; (b) Jessica tapping two

Table 1 : Designed gestures in TT

 1