Douglas Henderson

Ihsane Bichoutar

Bernard Moxham

Virginie Faidherbe

Odile Plaisant

Dr Alexis Guédon
email: alexis.guedon@aphp.fr

Descriptive and functional anatomy of the Heschl Gyrus: historical review, manual labelling and current perspectives

Keywords: Heschl Gyrus, Primary Auditory Cortex, Temporal Lobe, Auditory System

Purpose

The Heschl Gyrus (HG), which includes the Primary Auditory Cortex (PAC), lies on the upper surface of the superior temporal gyrus (T1). It has been the subject of growing interest in the fields of neuroscience over the last decade. Given the considerable interhemispheric and interindividual variability of its morphology, manual labelling remains the gold standard for its radio-anatomical study. The aim of this study was to revisit the original work of Richard L. Heschl, to provide a broad overview of the available anatomical knowledge and to propose a manually labelled 3D digital model.

Methods

We reviewed existing works on the HG, from Heschl's original publication of 1878, Dejerine neuroanatomical atlas of 1895 to the most recent digital atlases (Julich-Brain Cytoarchitectonic Atlas, the Human Connectome Project). Our segmentation work was based on data from the BigBrain Project and used the MRIcron 2019 software.

Results

The original publication by Heschl has been translated into French and English. We propose a correspondence of previous nomenclatures with the most recent ones, including the Terminologia Neuroanatomica. Finally, despite the notable anatomical variability of the HG, clear and coherent segmentation criteria allowed us to generate a 3D digital model of the HG.

Discussion and Conclusion

Heschl's work is still relevant and could impulse further anatomical and functional studies. The segmentation criteria could serve as a reference for manual labelling of the HG. Furthermore, a thorough, and historically based understanding of the morphological, microstructural and functional characteristics of the HG could be useful for manual segmentation.

Introduction

The Heschl Gyrus (HG), which includes the Primary Auditory Cortex (PAC), lies on the upper surface of the superior temporal gyrus (T1). It has been the subject of growing interest in the fields of neuroscience, neuroradiology, psychology and psychiatry over the last decade. Given the considerable interhemispheric and interindividual variability of its morphology, manual labelling remains the gold standard for its radio-anatomical study.

Due to the vast body of literature devoted to this structure, originating from various disciplines, dating back to 1878 and Richard L. Heschl's princeps article, it is not easy to encompass all the knowledge related to this structure. Moreover, the macroanatomical descriptions of the HG are sometimes heterogenous and do not always use the latest rigorous official neuroanatomical terminology, constituting a potential confounding factor in studies relying on manual segmentation.

The aim of this study was to clarify its morphology and functional connectivity, to revisit the original work of Richard L. Heschl, to provide a broad overview of the available knowledge related this structure, to propose coherent and reliable segmentation criteria and a 3D manually labelled digital model.

Biography

Richard Ladislaus Heschl (1824 -1881) received his medical doctorate in 1849 from the University of Vienna, where he trained under the surgeons Franz Schuh (1804 -1865) and Joseph Wattmann (1789 -1866). The following year he became the first assistant to the pathologist Carl von Rokitansky (1804-1878) at the same university. He then pursued his career across the Austrian Empire, which became Austria-Hungary in 1867, being appointed professor of anatomy in Olmütz in 1854, professor of pathological anatomy in Krakow in 1855, in Graz in 1863 and finally in Vienna in 1875 as successor to Rokitansky. He died of pulmonary consumption (most likely tuberculosis) in Vienna on May the 26 th 1881, aged 56 years old [START_REF] Gurlt | On the superior temporal gyrus by R.L. Heschl: English translation of "Über Die Vordere Quere Schläfenwindung Des Menschlichen Großhirns[END_REF] (Fig. 1).

As Amédée Dechambre stated in his "Encyclopedic Dictionary of Medical Sciences" of 1888, Heschl left behind "une foule d'articles et de mémoires dans des recueils périodiques" (a series of articles and memoirs in periodicals), many tomes, including the "Compendium of general and special pathological anatomy" of 1855, the "Sections-Technik" of 1859, and his "On the anterior transverse temporal gyrus" of 1878 that described the gyrus named after him [START_REF] Dechambre | Dictionnaire encyclopédique des sciences médicales[END_REF]19]. He also founded a pathological anatomy museum in Graz.

Fig. 1 Portrait photography of Richard L. Heschl in Graz, October 2 nd , 1872, at the age of 48 (courtesy of A.

Guédon).

Materials and methods

Recent and historical literature

A literature review was conducted in databases such as Pubmed, Google Scholar, Web of Science using keywords such as "Heschl Gyrus", "Primary Auditory Cortex", "Temporal Lobe", "Auditory System". We also used different historical (Brodmann, Von Economo and Koskinas) and recent digital anatomical atlases (Julich-Brain Cytoarchitectonic Atlas, Human Connectome Project's multimodal cortical parcellation 1.0). Richard L. Heschl's historical description [19] is available on Google Book (https://books.google.fr/). An English translation has been recently published [20]. We have done our own translation of the whole article into French and of the most salient passages (the anatomical description of the HG) into English.

Imaging

The radiological illustrations are from the Neuroradiology Department of Lariboisière Hospital, AP-HP, Paris with a three-dimensional T1 gradient echo sequence (MPRAGE) on a 3T MRI (Siemens Skyra, Siemens Healthineers, Germany) and a 3D CT angiogram performed with a biplane angiography system (Artis Zee, Siemens Healthineers, Germany).

Portrait photography

Théophile Le Comte, member of the Malacological Society of Belgium, naturalist living in Lessines, gathered in 1872/73, a beautiful album regrouping the European naturalists of which an unpublished photograph of Richard Ladislaus Heschl dating from 1872.

Manual labelling

The BigBrain Project, devised by the Montreal Neurological Institute and the German Forschungszentrum Jülich (https://bigbrain.loris.ca/main.php), provides a high-resolution 3D digital model of the human brain. It is based on a total of 7404 coronal brain sections of a 65-year-old woman that have been stained according to the Merker method and then digitalized [START_REF] Amunts | BigBrain: An Ultrahigh-Resolution 3D Human Brain Model[END_REF].

For the present study, using MRIcron 2019 software (available at nitrc.org), the left HG on 100 consecutive sections of the 400μm reconstruction (full16_400um_2009b_sym.nii) were manually labelled. The 3D reconstruction was performed using Mango version 4.1 software (http://rii.uthscsa.edu/mango/).

Atlases

The Julich-Brain Atlas [START_REF] Amunts | EBRAINS -Whole-brain parcellation of the Julich-Brain Cytoarchitectonic Atlas (v2.9)[END_REF] relies on histological sections of 23 post-mortem brains. Those tissues were fixed in formalin, MR-imaged, embedded in paraffin, serially cut, stained, digitized, and registered to stereotaxic reference coordinate spaces (the MNI Colin 27 and / or the MNI ICBM 152). The Human Connectome Project Multi-Modal Parcellation [START_REF] Van Essen | The Human Connectome Project: A data acquisition perspective[END_REF] relies on topographic, myelin, cortical thickness, connectivity, and functional maps.

Results

Anatomy of the Heschl Gyrus (HG)

Following the terminology of the 2017 version of the Terminologia Neuroanatomica (TNA) [15], most recent works [START_REF] Amunts | Chapter 36 -Auditory System[END_REF][START_REF] Da Rocha | TASH: Toolbox for the Automated Segmentation of Heschl's gyrus[END_REF][START_REF] Nieuwenhuys | The human central nervous system : a synopsis and atlas[END_REF] describe Heschl's Gyrus/Gyri (HG), or "transverse temporal gyri", as a circumvolution that originates from the depth of the lateral sulcus (of Sylvius) in the retro-insular region, that extends forwards, downwards and laterally along the upper surface of the superior temporal gyrus (T1) towards its lateral bulge. As described by Ten Donkelhaar et al. [START_REF] Ten Donkelaar | Toward a Common Terminology for the Gyri and Sulci of the Human Cerebral Cortex[END_REF], the HG separates the Planum polare (PP) anteriorly from the Planum temporale (PT) posteriorly on the upper surface of T1. In the left cerebral hemisphere, dominant for language, the PT includes Wernicke's area that is involved in the comprehension of written and spoken language. Nevertheless, both MRI and postmortem studies show that there are considerable interhemispheric, and interindividual variations in this transverse temporal gyrus [START_REF] Sluming | Heschl gyrus and its included primary auditory cortex: Structural MRI studies in healthy and diseased subjects[END_REF].

In accordance with TNA 2017, the anteromedial limit of the HG is defined by the anterior transverse temporal sulcus (ATTS) that unites with the circular sulcus of the insula (of Reil). The posterolateral limit is determined by the posterior transverse temporal sulcus (PTTS) (also known as Heschl's sulcus), which lies almost parallel to the ATTS. HG's posteromedial border is formed by a line drawn from the medial ends of the PTTS and the ATTS, while its anterolateral edge is limited by the visible ending of the circumvolution [START_REF] Sluming | Heschl gyrus and its included primary auditory cortex: Structural MRI studies in healthy and diseased subjects[END_REF]. Finally, the inferior boundary of the HG is formed by a line joining the bases of the anterior and posterior TTS (Fig. 2). Owing to the considerable anatomical variations, an intermediate transverse temporal sulcus (ITTS) (of Beck) has been described when this sulcus' length is comparable to those of the anterior and posterior TTS. The ITTS is parallel, between the ATTS and PTTS, in about 40% of the cerebral hemispheres of healthy living subjects [START_REF] Marie | Descriptive anatomy of Heschl's gyri in 430 healthy volunteers, including 198 left-handers[END_REF].

The ITTS separates 2, 3, 4 and up to 5 gyri, with the frequency of the gyrification pattern decreasing with increasing number of gyri, a 5-gyri pattern being extremely rare [START_REF] Amunts | Chapter 36 -Auditory System[END_REF]. In the case of a duplication, the TNA 2017 defines an anterior transverse temporal gyrus and a posterior transverse temporal gyrus.

On sagittal or coronal planes, the HG has been described as having an omega shape; alternatively shaped as a mushroom, a heart, a bow tie or a two separate Ss [START_REF] Sluming | Heschl gyrus and its included primary auditory cortex: Structural MRI studies in healthy and diseased subjects[END_REF]. With regard to what is the minimal length of the ITTS when there are two HGs, the most consensual view in the literature is of an ITTS with a length of at least half of the HG, a trait associated with a depth comparable to those of the TTS [START_REF] Penhune | Interhemispheric Anatomical Differences in Human Primary Auditory Cortex: Probabilistic Mapping and Volume Measurement from Magnetic Resonance Scans[END_REF]; a shorter ITTS is commonly associated with a reduced depth.

From a microanatomical perspective, the main structure of interest embodied in the HG is the primary auditory cortex (PAC), initially described by Alfred Walter Campbell (1868 -1937) in 1905 as an "auditory-sensory area". This area is also commonly known as Area 41, or "Area temporalis transversa interna" according to Korbinian Brodmann (1868Brodmann ( -1918) ) in 1909 [START_REF] Brodmann | Vergleichende Lokalisationslehre der Grosshirnrinde[END_REF], or as Area Te1 after Morosan et al. [START_REF] Morosan | Human Primary Auditory Cortex: Cytoarchitectonic Subdivisions and Mapping into a Spatial Reference System[END_REF] (Table 1 and Fig. 3). The grey matter of the neocortex is traditionally described as a superposition of six layers, numbered from the surface (I) to the depth (VI). Histological investigations make it possible to characterise, at the cellular and molecular levels, specific areas of the neopallium linked to particular functions. Thus, the synthesis of cyto-, myelo-, chemo-, and receptor-architectonic studies gives us an overview of the main microstructural characteristics of the PAC: a koniocortex surrounded by pro-and para-koniocortices, a high density of granular cells, small pyramidal cells in layer III, a well-developed and highly myelinated layer IV, a low cell density in layer V, a strong expression of acetylcholine esterase in the neuropile and a high concentration of cholinergic muscarinic M2 receptors [START_REF] Sluming | Heschl gyrus and its included primary auditory cortex: Structural MRI studies in healthy and diseased subjects[END_REF][START_REF] Amunts | Chapter 36 -Auditory System[END_REF] (Fig. 3E).

The inclusion of the PAC in the HG is widely understood. In the case of a duplication of the HG, the PAC most commonly tends to occupy the anterior HG [START_REF] Sluming | Heschl gyrus and its included primary auditory cortex: Structural MRI studies in healthy and diseased subjects[END_REF]. However, macroanatomical landmarks do not seem able to delineate with high accuracy the PAC. For instance, the PAC has been described as extending beyond the PTTS.

The lack of consistent correspondence between the macroanatomical landmarks of the HG and the microanatomical features of the PAC should be considered in the light of the considerable interhemispheric, and interindividual variations for each of these structures, and the significant heterogeneity of the methods and techniques used in the literature [START_REF] Morosan | Human Primary Auditory Cortex: Cytoarchitectonic Subdivisions and Mapping into a Spatial Reference System[END_REF][START_REF] Rademacher | Probabilistic Mapping and Volume Measurement of Human Primary Auditory Cortex[END_REF]. Indeed, descriptions of HG are based on anatomical, morphological and radiological studies, whereas the PAC is identified using cyto-architectonical approaches, connectivity and functional imaging techniques.

Based on historical data from Oskar Vogt (1870 -1959) -Cécile Vogt-Mugnier (1875 -1962) laboratory and from studies by Adolf Hopf (1923 -2011), [START_REF] Nieuwenhuys | A map of the human neocortex showing the estimated overall myelin content of the individual architectonic areas based on the studies of Adolf Hopf[END_REF] have recently published a high precision myeloarchitectonic map of the human neocortex (180 areas). This describes the PAC as covering the anterior and posterior HGs, and defines 13 densely myelinated areas (145 to 157) with a core of high density located on the medial third of the anterior gyrus (146, 148 and 151) [START_REF] Nieuwenhuys | A map of the human neocortex showing the estimated overall myelin content of the individual architectonic areas based on the studies of Adolf Hopf[END_REF] (Fig. 3D).

Table 1 Main microanatomical descriptions of the PAC included in the HG, from the early 20th century to Translator's note: there are no bolded sections in the original article, they have been used to indicate key points.

" […]

After opening the Sylvian fissure and removing the meninges, three surfaces can be observed. A superior surfacethe inferior surface of the operculum (translator's note: most likely the operculum frontale/parietale) with its transverse gyri, -the lateral surface of the Insula, ample and prominent, showing 5 to 7 convolutions; and an inferior surface which corresponds to the superior surface of the left temporal lobe. The latter usually presents a triangular shape with curved edges; the lateral edge is convex towards the outside, corresponding to the superior surface of the superior temporal gyrus (inferior bank of the Sylvian fissure); the medial edge surrounds the Insula at mid-length, forming a short and protruding circular arc; and the posterior edge, shorter than the two others, connects the two posterior ends of the two previous ones and its curvature varies from one brain to another.

After exposing the triangular surface described above, whose most acute angle points anteriorly, one promptly notices that one or more gyri lay on the posterior half and expand transversely from the anterolateral part to the posterior-medial part of the surface. The anterior convolution is always found, but very often one can also observe a second and often a third, sometimes even a fourth or a fifth convolution, either deploying like a fan or lying in a more parallel manner on the upper surface of the temporal lobe.

The first convolution (or anterior transverse temporal gyrus) emerges from the middle of the lateral edge of the superior surface of the temporal lobe, namely from the first temporal gyrus (T 1 according to Ecker); it is the longest of these gyri, merging quite often with the second transverse temporal gyrus. It ends either on its own or merging with the second, in the most posterior angle of the Sylvian fissure, about 1 cm from the entrance of the inferior horn (Translator's note: cornu temporale or cornu inferius) while the other convolutions end more laterally.

[…]

I termed these gyri as a whole, lying on the superior surface of the temporal lobe, (superior) transverse temporal gyri. Regarding the most anterior of these 3 to 4 gyri, which is always found, although its shape may vary, I refer to it as T 1 .

I have looked for it in no less than 1087 documented brains, and in hundreds of others, not less explicitly described and have found it every single time as a well-characterised gyrus of consistent shape. Its length varies with the depth of the Sylvian fissure: its height ranges from 4 to 12 mm, its thickness is usually 12 to 15 mm.

Sometimes the gyrus lies as anteriorly as posteriorly, and sometimes only posteriorly. It is delimited from the superior surface of the temporal lobe by a 6 to 8 mm deep sulcus, in such way that it rises with a 3 to 4 mm wide foot and overlaps it, showing a mushroom shape (on an axial section)."

The French translation is unpublished to the best our knowledge, see Supplementary material 1.

Segmentation criteria

To ensure the reliability of the manual segmentation of the HG, it seems essential to us, on the basis of Heschl's princeps article, our study of the literature, and our subjective experience, to follow two principles from which we can deduce the correct order of identification of the relevant sulci, lobes, plana and gyri.

-Principle 1: Larger structures should be identified first -Principle 2: Sulci should be identified before lobes, plana and gyri

In his 1878 article, Heschl first describes the dissecting instruments, methods, and steps to be followed to identify the HG. He proceeds from the largest to the smallest structures and explains that first the brain must be separated from the cerebellum, then both hemispheres, the meninges removed, the lateral sulcus opened, the fronto-parietal operculum, the Insula, T1, the circular sulcus of the Insula identified. Also, Heschl argued for the importance of the study of gyri and sulci in describing the anatomy of the brain and its embryological development: "Each circumvolution inevitably matches two sulci, and each sulcus two circumvolutions." [19].

From these two principles, we can now deduce the steps for achieving a reliable segmentation of the HG (Table 2).

The widest and most constant sulcus to be identified first is the lateral sulcus, separating the temporal lobe and its superior aspect, the superior temporal gyrus, from the frontal and parietal lobes. Next, the circular sulcus of the Insula defines the Insula and the medial border of T1. At this stage, there is no decisive argument as to whether we should identify the ATTS or the PTTS first. However, Heschl mentions in his paper: "a very deep posterior sulcus is always found, with which the first temporal sulcus (Translator's note: ATTS) merges directly (fissura parallela, t 1 according to Ecker), and ends in the crevices, the most medial and posterior faces of the Sylvian fissure." Dejerine describes it as "deep temporal sulcus" [11] (Table 3). We therefore choose to identify the PTTS, also called Heschl's sulcus [15], before the ATTS. As the Planum temporale, Planum polare and HG/HGs are clearly visible to us, we then look for one or more ITTS. Finally, we identify the inferior border (which is not a sulcus but an artificial border) to complete the segmentation of the HG. Allowing the completion of the segmentation of the HG The 6 steps described in Table 2 should be applied to the coronal, axial and sagittal digital views. In addition, as manual segmentation requires good hand-eye coordination, preparatory drawings on the blackboard with chalk can be very useful (Fig. 5).

3D digital model of the HG and terminology

We generated a 3D digital model of the HG based upon the BigBrain Project data with manual labelling of this structure. As shown in Fig. 6A, the length of the ITTS is 30% of the length of the HG. De facto, we cannot describe two separate HGs but instead a single HG with a partial duplication at its anterolateral third (also defined as common stem duplication). Moreover, the ITTS is not as deep as the ATTS and the PTTS (Figs. 6A andB). Research Group (TEPARG, https://teparg.com/), it will be in accordance with the latest international anatomical terminology of the Terminologia Anatomica (TA, 1998) and of the Terminologia Neuroanatomica (TNA, 2017)

[15] (Table 3).

Functional anatomy of the PAC included in the HG

The ascending pathway of the auditory system involves a multi-synaptic network. It starts in the cochlea, diverges as it reaches the anterior and posterior cochlear nuclei, then projects bilaterally with a contralateral dominance as it continues through the superior olivary complexes and the nuclei of the lateral lemnisci. From the lemnisci, the pathway merges at the contralateral inferior colliculus. Subsequently, the pathway projects to the contralateral auditory cortex through the medial geniculate body (Fig. 7A).

The descending pathway shows even greater complexity. It is also characterised by contralateral dominance, but not all projections cross the middle line. A direct pathway from the auditory cortex to the medial geniculate body and the inferior colliculus has been described. It then diverges, projecting ipsilaterally to the nucleus of the lateral lemniscus, the posterior cochlear nucleus and the medial periolivary nucleus, and contralaterally to the posterior cochlear nucleus. Axons of neurons of the medial periolivary nuclei and the lateral periolivary nuclei on both sides then reach the anterior and posterior cochlear nuclei. The contralateral cochlea is connected to the contralateral medial and lateral periolivary nuclei, and to the ipsilateral medial periolovary nucleus [START_REF] Amunts | Chapter 36 -Auditory System[END_REF] (Fig. 7B). The physical organisation of the PAC neurons is described in terms of their response to sound frequency.

According to this tonotopic organization, cells at the anterior end tend to be more responsive to low tones (corresponding to the apex of the cochlea) and those at the posterior end to high tones (corresponding to the base of the cochlea) [START_REF] Gazzaniga | Cognitive Neuroscience The Biology of the Mind[END_REF].

While understanding the cortico-cortical connections of the PAC still requires further investigations, especially in humans, strong connections with both the anterior and posterior superior temporal gyri have been implicated, relying both on serial and parallel processing [START_REF] Amunts | Chapter 36 -Auditory System[END_REF]. As part of the auditory cortex, the PAC has also been termed BA41 (Brodmann area 41) or A1 or "core area". PAC is also related to the secondary auditory cortex (termed BA42 (Brodmann area 42) (Figs. 3 A andB) or A2 or "belt area") and to the more lateral parabelt area [START_REF] Gazzaniga | Cognitive Neuroscience The Biology of the Mind[END_REF]. Short and long-range connections pass beyond the temporal lobe to reach parietal and frontal regions and also the PAC of the opposite hemisphere. Recent tractography reconstruction has shown that auditory processing pathways can be visualised in vivo and information from the HG appears to be projected to the inferior parietal and posterior frontal gyri through the arcuate fasciculus. A ventral pathway also passes information "from posterior to more anterior temporal and frontal area through the longitudinal fasciculus, the uncinate fasciculus, and the inferior fronto-occipital fasciculus" [START_REF] Maffei | Chapter 16 -Imaging white-matter pathways of the auditory system with diffusion imaging tractography[END_REF] (Fig. 8).

Fig. 8 In vivo tractography reconstruction of the auditory processing pathways from Maffei (2015) [START_REF] Maffei | Chapter 16 -Imaging white-matter pathways of the auditory system with diffusion imaging tractography[END_REF]. White dots: arcuate fasciculus; Black dots: ventral pathway; Black star: the Heschl Gyrus Providing reliable and accurate neuroanatomical atlases nowadays faces considerable challenges because of discrepancies arising from the variety of techniques employed to study the brain, including morphological, cytoand myelo-architectonical investigations and study of functional and cortico-cortical connections. Various descriptions of HG and PAC could be overcome using a multi-modal parcellation approach that merges multiple deterministic and probabilistic maps.

containing probabilistic maps of cortical areas (Fig. 9A). The Julich Brain Atlas also proposes connectivity data from the "1000 brains study" by Caspers et al. [START_REF] Caspers | EBRAINS -1000BRAINS study, connectivity data (v1.1)[END_REF], thus allowing to visualize an estimation of the connection strength from each region to all other areas (Fig. 9B). Given the fast-growing body of neuroimaging data, Yarkoni et al. [START_REF] Yarkoni | Large-scale automated synthesis of human functional neuroimaging data[END_REF] propose a large scale, high quality, automated meta-analysis approach based upon the functional magnetic resonance imaging literature and available online at neurosynth.org (Fig. 11).

Discussion

In our study, we examined the morphology, microstructure and functional connectivity of the HG and its included PAC, providing a broad overview of the available knowledge about this structure. We have revisited the original work of Richard L. Heschl, presenting an unpublished French translation. We proposed segmentation criteria and a manually labelled 3D digital model. We illustrated the challenges that modern neuroanatomical atlases face due to the considerable development of neuroimaging and probabilistic approaches in the last decade and how this relates to the HG.

Comparative anatomy and an evolutionary perspective

Despite the complexity of language and speech in humans, the structure of the PAC (BA41) for non-human primates is similar to that in humans [START_REF] Kaas | Subdivisions of auditory cortex and processing streams in primates[END_REF]. From a microanatomical perspective, these similarities include the properties of sensory area, there being a high density of granular cells in layers III and IV of the cortex, a strong expression of acetylcholinesterase, parvalbumin, cytochrome C oxidase, and a significant thalamic input [START_REF] Kaas | Evolution of Nervous Systems[END_REF]. It has been suggested that structural differences in the PAC of humans are primarily quantitative in nature and could relate to the way language diverged from the communication abilities of earlier primates. This change could beconcomitantly with the increase in brain volume-alongside more complex communicative and social interactions.

A further consideration is the strong involvement of BA47 (the orbital area of the prefrontal cortex) in language processing. The human prefrontal cortex is involved in the processing of syntax, relying on a different connectivity of the ventral pathways for complex sounds processing (including Broca's area or BA 44/45) compared with nonhumans. However, there is great similarity in the extreme capsule (a long bidirectional association pathway involved in language), as demonstrated between humans and macaques.

Differences in humans with respect to the anatomical and functional interhemispheric variability and the language and speech related cortical areas of the left hemisphere could be related to levels of linguistic skills. Differences in auditory processing among species have also been described. For instance, the auditory cortex of the rhesus macaque has been shown to show species-specific vocalisations and to respond more to based-passed noises (BPN) than to pure tone [START_REF] Tian | Functional Specialization in Rhesus Monkey Auditory Cortex[END_REF]. With regard to the areas associated with the PAC (core area), namely the belt, parabelt and lateral belt areas, the degree of similarity between humans and non-human primates appears to be very high. Hence, differences seem to concern high levels of auditory processing rather than the structure of the PAC [START_REF] Kaas | Evolution of Nervous Systems[END_REF].

Clinical considerations

It has been shown that the HG is linked to acoustic processing [START_REF] Warrier | Relating Structure to Function: Heschl's Gyrus and Acoustic Processing[END_REF] and the morphology and size of the HG have been linked to linguistic and musical abilities. Some authors have even considered the HG provides neuroanatomical markers for these abilities [START_REF] Turker | When Music Speaks": Auditory Cortex Morphology as a Neuroanatomical Marker of Language Aptitude and Musicality[END_REF][START_REF] Wengenroth | Increased Volume and Function of Right Auditory Cortex as a Marker for Absolute Pitch[END_REF]. It has been reported that duplication of the HG in the right cerebral hemisphere is correlated to higher scores in foreign languages and musicality tests.

In terms of language skills, bilinguals have larger HGs and multiple HGs in the right hemisphere and increased grey matter volumes are correlated to language aptitude in children and teenagers [START_REF] Turker | Auditory Cortex Morphology Predicts Language Learning Potential in Children and Teenagers[END_REF]. Dyslexia has been linked

to a large duplication of the left HG [START_REF] Leonard | Anatomical Risk Factors for Phonological Dyslexia[END_REF]. Moreover the anatomy of the HG appears to be related to handedness and its gyrification pattern to speech-listening hemispheric lateralization [START_REF] Tzourio-Mazoyer | Heschl's gyrification pattern is related to speechlistening hemispheric lateralization: FMRI investigation in 281 healthy volunteers[END_REF].

In terms of musicality, an increased volume in the right cerebral hemisphere is linked to the ability to identify each sound by its name/frequency ("absolute pitch") [START_REF] Wengenroth | Increased Volume and Function of Right Auditory Cortex as a Marker for Absolute Pitch[END_REF]. Multiple HGs, or an increased grey matter volume for this structure, have also been more frequently identified in both professional and amateur musicians than in nonmusicians [START_REF] Benner | Prevalence and function of Heschl's gyrus morphotypes in musicians[END_REF]. The morphology of HG has additionally been linked to musical instrument preference and multiple HGs, and an increased volume in the left hemisphere, have been found in patients with hypermusia associated with Williams Syndrome (microdeletion 7q11.23).

In deaf persons, the volume of the HG is not modified, although the grey/white matter ratio increases in favour of grey matter. A reduction of the volume of the HG has also been correlated with tinnitus and, in the rare case of a bilateral infarct of the HG, a cortical deafness can be observed [START_REF] Narayanan | A Case of Cortical Deafness due to Bilateral Heschl Gyrus Infarct[END_REF]. Indeed, the ascending pathway of the auditory system projects bilaterally to the primary auditory cortical areas, thus explaining why a unilateral acute cerebral ischemia does not result in deafness, whereas an infarct of Broca's area (BA 44/45) is more reliably associated with aphasia.

Concerning psychiatric disorders, several studies have linked the morphology of the HG to auditory hallucinations in patients with schizophrenia or schizophrenia spectrum disorders, one of the most recent being described by Takahashi et al. [START_REF] Takahashi | Heschl's Gyrus Duplication Pattern in Individuals at Risk of Developing Psychosis and Patients With Schizophrenia[END_REF]. Furthermore, the white and grey matter volumes in female patients suffering from this psychiatric disorder seem inversely correlated to the symptom's severity [12] but, regarding the grey matter, this result is also found in male patients. Paradoxically, auditory hallucinations in patients with schizophrenia could cause, over time, an increase in HG volume.

Beyond the above anatomical, functional, and pathological considerations, a major point to be elucidated is the part played by neural plasticity in converting the subject's experiences and the part of innate genetic factors.

Eventually, further histological and molecular studies are required to understand the mechanisms connecting descriptive anatomy, pathology and linguistic or musical abilities [START_REF] Zatorre | Plasticity in Gray and White[END_REF].

Strengths and limitations

Our study has several strengths. The literature review we propose is comprehensive, including historical sources than are often cited but not discussed in detail in most studies. Given the diverse backgrounds of the authors, we provide an overview of knowledge about the HG that merges perspectives from different relevant fields, including neuroradiology, psychiatry, medical anatomy, neuroscience, cognitive science, literature, and music.

Nevertheless, our study also has several limitations. The segmentation criteria are mainly based on subjective experience, pedagogical tradition, and a review of the literature. The 3D digital model we propose has not been evaluated by an independent panel and the criteria have not been tested in a prospective, single-blind, comparative cohort study. They have a low level of evidence by current methodological standards. Such an experiment could be conducted with first, second and third year medical students in European countries, for example, although establishing standardised conditions and achieving statistical significance may be difficult due to the number of participants required. By comparison, Dalboni da Rocha et al [START_REF] Da Rocha | TASH: Toolbox for the Automated Segmentation of Heschl's gyrus[END_REF] in the TASH study use a single expert to set the gold standard (manually labelled segmentation) to compare and validate their automated segmentation method.

Perspectives

There is no doubt that new technological breakthroughs in the fields of functional neuroimaging, probabilistic neuroimaging data processing, and high-performance computing in a broad sense will both challenge and expand our current knowledge of the HG. As discussed by Amunts et al [START_REF] Amunts | The coming decade of digital brain research -A vision for neuroscience at the intersection of technology and computing[END_REF], the construction of reliable digital twins of the brain may be the next important and disruptive step in achieving clinically relevant applications.

Conclusion

We were able, by means of a thorough review of the literature, to define clear, coherent and reliable criteria for manual labelling of the HG. These criteria could be used as groundwork to ensure the coherence of the various

Fig. 2 1 : 3 : 7 :

 2137 Fig. 2 Anatomy of the Heschl Gyrus. A: Coronal 3D T1 MPRAGE MRI section; B: Axial section; C: Sagittal section. 1: Heschl's Gyrus (HG); 2: posterior temporal transverse sulcus (PTTS) (also known as Heschl's sulcus); 3: anterior temporal transverse sulcus (ATTS); 4: Insula; 5: Frontoparietal region; 6: lateral sulcus (of Sylvius); 7: superior temporal gyrus (T1); 8: intermediate temporal transverse sulcus (ITTS) (of Beck); 9: Planum Temporale (PT); 10: posteromedial boundary; 11: inferior boundary

Fig. 3

 3 Fig. 3 The Primary Auditory Cortex included in the Heschl Gyrus represented in several atlases, from 1909 to nowadays. A and B: Area 41 from Brodmann (1909) [7] reproduced with permission from Elsevier (Morosan, 2001) [26] C: Areas Tc and Td from Von Economo and Koskinas (1925) [14] reproduced with permission from Elsevier (Scholtens LH, 2018) [33] D: Areas 145 to 157 from Nieuwenhuys (2017) [28] reproduced with permission from Springer Nature E: Areas Te1.2, Te1.0, Te1.1 from Morosan (2001) [26] reproduced with permission from Elsevier

Fig. 4

 4 Fig. 4 Arterial vasculature of the Heschl Gyrus. A, C, E: Axial, right parasagittal and coronal 3D T1 MPRAGE MRI sections showing Heschl's gyri along the upper surface of the superior temporal gyri (T1), the left Heschl gyrus is partially duplicated. B, D, F: 3D CT angiogram of the two internal carotid arteries showing the vascularisation of Heschl's gyri (*) with the characteristic loops.

Fig. 5

 5 Fig. 5 Preparatory drawing on the blackboard with chalk. A: Topographic superior view of the upper surface of the right T1. PT: Planum temporale; HG: Heschl Gyrus; PP: Planum polare; INS: Insula; ATTS: Anterior temporal transverse sulcus; PTTS: Posterior temporal transverse sulcus (also known as Heschl's sulcus) B: Single HG, partial and complete duplication (right hemisphere). Red: ATTS; Blue: PTTS; Green: ITTS. ATTS: Anterior

Fig. 6

 6 Fig. 6 Manual labelling and 3D modelisation of the left HG from the BigBrain Dejerine Digital Atlas [31]. A: The left HG from the BigBrain Dejerine Digital Atlas. B: Manual labelling of the HG using MRIcron 2019. Data were used with permission from the BigBrain Project [3].

Fig. 7

 7 Fig. 7 Ascending (A) and descending (B) pathways of the auditory system on a posterior view of the brain stem (without the cerebellum). (Courtesy of the Anatomy Laboratory of Paris Cité University). ACN=anterior cochlear nucleus; PCN=posterior cochlear nucleus; SOC=superior olivary complex; LLN=lateral lemniscus nucleus; IC=inferior colliculus; MGB=medial geniculate body; LPON=lateral periolivary nucleus; MPON=medial periolivary nucleus; AC=auditory cortex

Fig. 9 Fig. 10

 910 Fig. 9 Cytoarchitectonic parcellation and connectivity data from the Julich Brain Atlas. A: Area Te1.1 right from Amunts (2021) [4]. B: Connectivity profile of Area Te1.1 right (red=high, deep blue=low) from Caspers (2021) [8]. Views from the siibra-explorer website: (https://atlases.ebrains.eu/viewer/)

Fig. 11

 11 Fig. 11 Representation of the Heschl Gyrus from Neurosynth.org. This representation is based upon an automated meta-analysis of all the functional magnetic resonance imaging studies in the Neurosynth database whose abstracts include the term "Heschl Gyrus" at least once (85 studies for the HG) (https://neurosynth.org/analyses/terms/heschl%20gyrus/) (Yarkoni et al. 2011) [43]

Figures 3 Figure 6 :Figure 9

 369 Figures 3 ABCE are reproduced with permission from Elsevier

Figure 11

 11 Figure 11 is reproduced with permission from Neurosynth

Table 2

 2 Manual labelling of the Heschl Gyrus: a step-by-step methodology

	Steps Sulci

Table 3

 3 Terminology used for the description of the HG, Heschl (1878),[START_REF] Dejerine | Anatomie des centres nerveux[END_REF] andTNA (2017)

	Heschl (1878) [19]	Dejerine (1895) [11]	TNA (2017) Latin and English [15]
	vordere quere Schläfenwindung	Circonvolutions temporales	(2062) Gyri temporales transversi -
		transverses de Heschl / plis de	Transverse temporal gyri
		passage temporo-pariétaux /
		région rétro-insulaire de Broca
		(Tp)
	sylvische Spalte	Scissure de Sylvius	(2003) Sulcus lateralis -Lateral
			sulcus
	Insel von Reil	Insula	(2074) Insula -Insula
	Klappdeckel		(2043) Operculum parietale -
			Parietal operculum
			(most likely)
	erste Schläfenwindung (T1 nach	première circonvolution	(2057) Gyrus temporalis superior -
	Ecker)	temporale (T1)	Superior temporal gyrus
			(2067) Sulcus temporalis transversus
			anterior -Anterior transverse
			temporal sulcus
			(2068) Sulcus temporalis transversus
			intermedius -Intermediate
			transverse temporal sulcus
	tiefe Furche	Sillon temporal profond (tp)	(2069) Sulcus temporalis transversus
			posterior -Posterior transverse
			temporal sulcus
		Circonvolution de	(2079) Sulcus circularis insulae -
		l'enceinte de l'Insula	Circular sulcus of the insula

Another well worth noting work is the population-based 180-areas multi-modal parcellation by Glasser et al.[START_REF] Glasser | A multi-modal parcellation of human cerebral cortex[END_REF] that is based upon the Human Connectome Project (Human Connectome Project Multi-Modal Parcellation version 1.0) (Figs.10A and B).

Acknowledgements

Funding

The authors did not receive support from any organization for the submitted work.

Availability of data and materials

All data are available on request from the corresponding author.

studies devoted to the HG, and to establish, and validate automated segmentation methods. Richard L. Heschl's work is still relevant today and could impulse further anatomical and functional studies in neurosciences.

Declarations: Ethical Approval

Not applicable.

Competing interests

The authors have no conflicts of interest to declare. Heschl's archive was transferred from the University of Graz to the "Narrenturm" (madhouse tower) of the "naturhistorisches Museum" in Vienna. We thank Petra Greeff, Eduard Winter and Gerald Höfler for their valuable help.

Supplementary Information

Below is the link to the electronic supplementary material.


Raouf Abd

Chouikha

INFINITE PRODUCT REPRESENTATIONS FOR THE WEIERSTRASS ELLIPTIC FUNCTION ℘(z) AND APPLICATIONS

Keywords: theta functions, elliptic functions, trigonometric expansions 2010 Mathematics Subject Classication. 33E05, 30-08, 11F32, 11F16, 11F27. 1

In this paper we bring out innite products expansions for the Weierstrass's elliptic function ℘(z) = ℘(z, τ ) with primitive periods (2, 2τ ) and derive some n-order transformations of that function as well as for its derivative ℘ (z). This allow us to provide some new modular relations.

e 1 (τ ) = ℘(1, τ ), e 2 (τ ) = ℘(-1 -τ, τ ), e 3 (τ ) = ℘(τ, τ ).

The Weierstrass's function ℘(z) = ℘(z; ω, ω ) is an elliptic function with two primitive periods (2ω, 2ω ) such that the imaginary part ω ω > 0 which is of order two, has a double pole at z = 0, Recall that ℘(z) -1 z 2 is analytic in a neighborhood of 0 and it is uniquely dened. One then obtains the analytic series representation of ℘(z)

℘(z) = ℘(z; g 2 , g 3 ) = ℘(z, τ ) = 1 z 2 + m,n 1 (z -2mω -2nω ) 2 - 1 (2mω + 2nω ) 2 ,
where τ = ω ω , The prime symbol means that m and n are not simultaneously zero. A direct consequence of the preceding denition is that the Weierstrass elliptic function is an even function ℘(-z; ω, ω ) = ℘(z; ω, ω ). Moreover, this function veries the following homogeneity condition for any complex λ = 0 ℘(λz; λω, λω ) = λ -2 ℘(z; ω, ω ).

For this reason and for the sake of simplicity we will only consider in the sequel ℘(z) with primitive periods (2, 2τ ), simply denoted by ℘(z; 2, 2τ ) = ℘(z, τ ).

The original constructions of elliptic functions are due to Weierstrass and Jacobi [START_REF] Appell | Fonctions elliptiques et applications Gauthiers-Villard[END_REF][START_REF] Whittaker | A course of Modern Analysis Cambridge[END_REF]. Nice approaches on the subject of elliptic functions are the classic book by Watson and Whittaker [12] or the excellent full compilation of Tannery and Molk [START_REF] Whittaker | A course of Modern Analysis Cambridge[END_REF]. Useful reference handbooks with many details on transcendental functions including those used in this paper are provided by Bateman and Erdelyi, [START_REF] Kiepert | Ueber Theilung und Transformation der elliptischen Functionen[END_REF].

Recall some useful facts on Weierstrass elliptic function. Its values at the halfperiods: ω 1 = 1, ω 2 = -1 -τ, ω 3 = τ are These e i obey the relations e 1 + e 2 + e 3 = 0, e 1 e 2 + e 3 e 1 + e 2 e 3 = -g 2 4 , e 1 e 2 e 3 = g 3 4 .

(1)

Finally, when two of the roots e 1 , e 2 and e 3 coincide, the Weierstrass elliptic function degenerates to a simply periodic function.

On the other hand, the Weierstrass function ℘(z, τ ) is connected to the Jacobi theta functions

θ i (v) = θ i (v, τ ), i = 1, 2, 3, 4 where v = z 2 : ℘(z) = ( 1 2 ) 2 [-4η - d 2 logθ 1 (v) dv 2 ] η = η(τ ) = - 1 12 θ 1 (0) θ 1 (0) = π 2 2 [ 1 6 + n≥1 1 (sin nπτ ) 2 ].
(2)

We have also

℘(z + τ ) = ( 1 2 ) 2 [-4η - d 2 logθ 4 (v) dv 2 ].
Therefore,

e 3 (τ ) = ℘(τ ) = -η(τ ) + π 2 k≥0 1 1 -cos(2k + 1)πτ , e 3 (τ ) = ( π 2 ) 2 θ 1 (0) 3θ 1 (0) - θ 4 (0) θ 4 (0) .
By the same way

e 2 (τ ) = -η(τ + 1) + π 2 k≥0 1 1 + cos(2k + 1)πτ = ( π 2 ) 2 θ 1 (0) 3θ 1 (0) - θ 3 (0) θ 3 (0) . e 1 (τ ) = η(τ ) + η(τ + 1) -2π 2 k≥0 1 (sin(2k + 1)πτ ) 2 = ( π 2 ) 2 θ 1 (0) 3θ 1 (0) - θ 2 (0) θ 2 (0) .
Notice also the following theta function identity [START_REF] Tannery | Elements de la theorie des Fonctions Elliptiques[END_REF][START_REF] Whittaker | A course of Modern Analysis Cambridge[END_REF] deriving from (1)

θ 1 (0) θ 1 (0) = θ 2 (0) θ 2 (0) + θ 3 (0) θ 3 (0) + θ 4 (0) θ 4 (0) .
1.2. The Weierstrass' sigma function.

The Weierstrass' sigma function is an entire function dened by

σ(z) = σ(z, τ ) = z m,n (1 - z m + nτ ) exp z m + nτ + z 2 2(m + nτ ) 2 .
The prime symbol means that m and n are not simultaneously zero. This function is also connected to ℘(z)

℘(z) = σ 2 (z) -σ(z)σ (z) σ 2 (z)
We obtain the analogous for a connection with the sigma function

℘(z) -e i = - σ(z + ω i )σ(z -ω i ) σ 2 (z)σ 2 (ω i ) = σ i (z) σ(z) 2 , i = 1, 2, 3,
where

ω 1 = 1, ω 2 = -1-τ, ω 3 = τ.
Then we have the zeros of ℘ (z) are e i = ℘(ω i ).

Notice that the function ℘(z) -℘(ω i ) is an elliptic function of order two, then it has only poles and zeros and hence the function

[℘(z) -℘(ω i )] 1 2 = σi(z) σ(z)
is a single valued function.

The aim of this paper.

The literature on various representations of the function ℘(z) is notably abundant. Several types of representations whether in the form of analytic or trigonometric series are widely described [START_REF] Appell | Fonctions elliptiques et applications Gauthiers-Villard[END_REF][START_REF] Erdelyi | Higher transcendental functions[END_REF][START_REF] Tannery | Elements de la theorie des Fonctions Elliptiques[END_REF][START_REF] Whittaker | A course of Modern Analysis Cambridge[END_REF]. In this work we will be particularly interested in the development in innite products of this function. However, note that this approach seems to have been little exploited. This paper is organized as follows.

First of all we exhibit a representation in innite products for the Weierstrass elliptic function ℘(z) = ℘(z, τ ) with two primitive periods 2, 2τ where z, τ are complex numbers such that the imaginary part of τ is positive, τ > 0 and z < 2 τ, :

℘(z) -e 1 = (π cot πz 2 ) 2 4 k≥1 cot(kπτ -πz 2 ) cot(kπτ + πz 2 ) [cot(kπτ )] 2 2 .
On the other hand, we will considered the n-order odd decomposition as innite products of the Weierstrass sigma functions or for their quotient

σ i (z, nτ ) = exp(z 2 P 1 ) σ j (z) n-1 r=1 σ j (z + 2r n )σ j (z -2r n ) σ 2 ( 2r n )
,

where

P 1 = n-1 r=1 ℘( 2r
n ), as well as

σ j σ (nz, nτ ) = n-1 m=0 σ j σ (z + 2mπ n , τ ) n-1 m=0 σ σ j ( 2mπ n , τ ), j = 1, 2, 3.
We refer for that for example to Tannery and Molk [10, T.2, p.215,246] as well as to H. Schwarz [9, p.6,36].

The last decomposition allows us to deduce n-decomposition for the elliptic Weierstrass function ℘(z, τ ) :

℘(nz, nτ ) -e 1 (nτ ) = 4 π 4 n-1 θ 2 3 (0, nτ )θ 2 4 (0, nτ ) [θ 2 3 (0, τ )θ 2 4 (0, τ )] n n-1 m=0 ℘(z + m n , τ ) -e 1 (τ ) ,
where e 1 = ℘(1, τ ), as well as for its derivative :

℘ (nz, nτ ) = 4 π 4 n-1 θ 2 1 (0, nτ ) θ 2 1 (0, τ ) n n-1 m=0 ℘ (z + m n , τ ).
Finally, we will also consider the function ξ βγ (u) introduced by Tannery-Molk [10, T2, p.168]. They are dened by

ξ βγ (u) = σ β σ γ (u) = ℘(u) -e β ℘(u) -e γ
where βγ = 0, 1, 2, 3. Here σ 0 (u) = σ(u) and e j are the zeros of ℘ (u).

We then explore the n-decomposition of the logarithmic derivative for these functions.

As a consequence of all the above, for z < 2 τ we derive modular identities which seem to be new in the literature:

k =0 1 sin(2knπτ + nπz) = n-1 m=0 k =0 1 sin(2kπτ + π(z + 2m n )) = 2 1-n k =0 n-1 m=1 1 sin(2kπτ + π(z + 2m n )) = θ 2 2 (0, nτ ) [θ 2 2 (0, τ )] n n-1 m=0 k =0 1 sin(2kπτ + πz + mπ n )
.

To prove that we will use the above n-decomposition of ℘(z, τ ) as well as the n-transformations of

℘ (z) ℘(z) -e 1 = - 2π sin πz + 2π k≥1 1 sin(2kπτ -πz) - 1 sin(2kπτ + πz) .

Weierstrass's function ℘(z) and infinite products

As we have seen in 1.1, the Weierstrass's function ℘(z) = ℘(z; ω, ω ) is an elliptic function of order two with two primitive periods (2ω, 2ω ) verifying the condition

τ = ω ω , τ > 0.
Recall throughout this paper we will take ω = 1, ω = τ is imaginary. In order to avoid any ambiguity ℘(z) always denotes ℘(z, τ ).

Innite product representations of ℘(z).

We have seen above in 1.2, the connection between ℘(z) and theta functions. The following is well known for

v = z 2ω = z 2 : ℘(z) = e i + 1 4
θ i+1 (v) πθ i+1 (0) θ 1 (0) θ 1 (v) 2 , i = 1, 2, 3.
These relations allows us to derive variant innite products expressing the Weierstrass's function Theorem 2-1 The Weierstrass's function ℘(z) = ℘(z, τ ) with primitive periods 2 and 2τ veries the following identities

℘(z) -e 1 = (π cot πz 2 ) 2 4 k≥1 cot(kπτ -πz 2 ) cot(kπτ + πz 2 ) [cot(kπτ )] 2 2 = (π 2 θ 3 (0)θ 4 (0) cot πz 2 ) 2 4 k≥1 cot(kπτ - πz 2 ) cot(kπτ + πz 2 ) 2 ,
where e 1 = ℘(1), and Imz < 2 Imτ . We obtain the two other innite products by permuting the e i

℘(z)-e 3 = π 2 2 sin πz 2 2 k≥1 sin((k -1 2 )πτ -π z 2 ) sin(kπτ -π z 2 ) sin((k -1 2 )πτ + π z 2 ) sin(kπτ + π z 2 ) 2 sin(kπτ ) sin((k -1 2 )πτ ) 4 , ℘(z)-e 2 = π 2 2 sin πz 2 2 k≥1 cos((k -1 2 )πτ -π z 2 ) sin(kπτ -π z 2 ) cos((k -1 2 )πτ + π z 2 ) sin(kπτ + π z 2 ) 2 sin(kπτ ) cos((k -1 2 )πτ ) 4 .
Proof of Theorem 2-1 Starting from (as seen above 1.2)

℘(z) = e 1 + 1 4 π θ 2 (v) θ 2 (0) θ 1 (0) θ 1 (v) 2 ,
and by [2, Corollary 3-5] which asserts

θ 1 (v, τ ) π(sin πv) θ 1 (0, τ ) = 1 - sin πv sin kπτ 2 = k≥1 sin(kπτ -πv) sin(kπτ + πv) [sin(kπτ )] 2 , θ 2 (v, τ ) (cos πv) θ 2 (0, τ ) = 1 - sin πv cos kπτ 2 = k≥1 cos(kπτ -πv) cos(kπτ + πv) [cos(kπτ )] 2 .
Then,

θ 2 (v) θ 2 (0) πθ 1 (0) θ 1 (v) = cot πv k≥1 cos[kπτ + πv] cos[kπτ -πv](sin[kπτ ]) 2 sin[kπτ + πv] sin[kπτ -πv](cos[kπτ ]) 2 = cot πv k≥1 cot(kπτ + πv) cot(kπτ -πv) (cot kπτ ) 2 .
Moreover, by the relationship with the elliptic Weierstrass's sigma function

℘(z) -e 1 = σ1z
σz 2 , we have for v = z 2 (see for example Schwarz [9,p. 8,36]) (2 sin πv) π k≥1 sin(kπτ -πv) sin(kπτ + πv)

σ 1 z = e
[sin(kπτ )] 2 .
We then derive the expression

σ 1 z σz = (π cot πv) 2 k≥1 cot(kπτ -πv) cot(kπτ + πv) [cot(kπτ )] 2 ,
and deduce

℘(z)-e 1 = (π cot πz 2 ) 2 4 k≥1 cot(kπτ -πv) cot(kπτ + πv) [cot(kπτ )] 2 2 = (π cot πz 2 ) 2 4 k =0 cot(kπτ -πz 2 ) [cot kπτ ] 2 .
By permutation of the e i , i = 1, 2, 3 we also obtain the other expressions (see Schwarz [9, p.36])

σ 2 z = e ηv 2 2 k≥0 cos((k -1 2 )πτ -πv) cos((k -1 2 )πτ + πv) cos((k -1 2 )πτ ) 2 , σ 3 z = e ηv 2 2 k≥0 sin((k -1 2 )πτ -πv) sin((k -1 2 )πτ + πv) sin((k -1 2 )πτ ) 2 ,
and then deduce analog innite products for

℘(z) -e 2 = σ 2 z σz 2 ℘(z) -e 3 = σ 3 z σz 2 .
Remark 2-2 : (i) This expansion as innite product of elliptic functions may be dierently proved. Indeed, starting from the innite product [9, p.8] noticed that the sigma function may also be written

sin( πu 2ω ) = πu 2ω n≥0 1 - u 2nω e u 2nω n≥0 1 + u 2nω e -u 2nω H.A. Schwarz
σz = 2ω π sin(πv)e 2ηωv 2 n≥1
sin(nπτ -πv) sin(nπτ + πv)

(sin nπτ ) 2 = e 2ηωv 2 2ω π sin(πv) n≥1 1 - (sin πv) 2 (sin nπτ ) 2 , where v = z 2ω , η = π 2 2ω 1 6 + n≥0 1 (sin nπτ ) 2 .
(ii) Other relations and descriptive properties of ℘(z) may be derived from Theorem 2-1. For example, We may write :

℘(z + 1) = e 1 + (π tan π z 2 ) 2 4 k =0 tan(kπτ -π z 2 ) cot(kπτ ) 2 .
We then derive the following well known expression ([4, 13.13, p.333]) to the periods (2, 2τ ). We then have

(℘(z + 1) -e 1 ) (℘(z) -e 1 ) =
℘ (z) ℘(z) -e 1 = - 2π sin πz + 2π k =0 1 sin(2kπτ -πz) = - σ(2z) σ 2 (z)σ 2 1 (z)
, where σ(z), σ 1 (z) are the Weierstrass elliptic functions (as dened above in 1.3).

Indeed, that result follows from Theorem 2-1 and the identity

d cot x dx 1 cot x = - 2 sin 2x
.

Then

℘ (z) ℘(z) -e 1 = - 4π 2 sin πz + 4π 2 k≥1 1 sin(2kπτ -πz) - 1 sin(2kπτ + πz) = k 2π sin(2kπτ -πz)
.

We may expressed it otherwise (for example : see Lawden [7, p.161]) by mean the Weierstrass's elliptic zeta function dened by ζ (z) = -℘(z)

℘ (z) ℘(z) -e 1 = 2 σ 1 σ 1 (z) - σ σ (z) = -2 σ 2 (z)σ 3 (z) σ(z)σ 1 (z) = - σ(2z) σ 2 (z)σ 2 1 (z) = 2ζ(z+1)-2ζ(z)-2η.
Concerning the sigma functions we derive the following results (here η =

σ σ (z) = ηz + π 2 cot( πz 2 ) + π 2 k≥1 cot(kπτ + πz 2 ) -cot(kπτ - πz 2 ) = ηz + π 2 cot( πz 2 ) + π 2 k≥1 sin πz -cos 2 ( πz 2 ) + cos 2 (kπτ ) , σ 1 σ 1 (z) = σ σ (z + 1) = ηz - π 2 tan( πz 2 ) - π 2 k≥1 sin πz -sin 2 ( πz 2 ) + cos 2 (kπτ ) , σ 2 σ 2 (z) = ηz - π 2 k≥0 tan((k - 1 2 )πτ + πz 2 ) -tan((k - 1 2 )πτ + πz 2 ) = ηz - π 2 k≥1 sin πz -sin 2 ( πz 2 ) + cos 2 ((k -1 2 )πτ ) , σ 3 σ 3 (z) = σ 2 σ 2 (z + 1) = ηz + π 2 k≥1 sin πz -cos 2 ( πz 2 ) + cos 2 ((k -1 2 )πτ )
.

Corollary 2-5 The elliptic sigma functions and their derivatives relative to periods (2, 2τ ) verify the identities

℘ (z) 2(℘(z) -e 1 ) = σ 1 σ 1 (z) - σ σ (z) = π k≥1 1 sin(2kπτ -πz) - 1 sin(2kπτ + πz) , σ 2 σ 2 (z) - σ 3 σ 3 (z) = π k≥0 1 sin((2k -1)πτ -πz) - 1 sin((2k -1)πτ + πz) . σ 1 σ 1 (z) + σ σ (z) = 2ηz + π cot(πz) + π k≥1 [cot(2kπτ + πz) -cot(2kπτ -πz)] , σ 2 σ 2 (z) + σ 3 σ 3 (z) = 2ηz + π k≥0 [cot((2k -1)πτ + πz) -cot((2k -1)πτ -πz)] .
Next results provide other expressions for the Weierstrass function as well as for its derivative The Weierstrass function relative to periods (2, 2τ ) veries the identities

℘(z, τ ) -e 2 (τ ) = π 2 k≥1 1 sin(2kπτ + πz) k≥1 1 sin( -2kπ τ + πz) , ℘(z, τ ) -e 3 (τ ) = π 2 k≥1 1 sin(2kπτ + πz) k≥1 1 sin( -2kπ τ +1 + πz) , ℘(z, τ ) -e 1 τ ) = π 2 k≥1 1 sin( 2kπ τ +1 + πz) k≥1 1 sin( -2kπ τ + πz) .
Indeed, one has since [4, p.368] e 1 ( -1 τ ) = e 3 (τ ) and e 1 ( -1 τ +1 ) = e 2 (τ ) then

℘ (z) ℘(z) -e 3 = -2π k≥1 1 sin( -2kπ τ + πz) , ℘ (z) ℘(z) -e 2 = -2π k≥1 1 sin( -2kπ τ +1 + πz) .
Moreover, since

℘ 2 (z) (℘(z) -e 1 )(℘(z) -e 3 ) = ℘ (z) (℘(z) -e 1 ) ℘ (z) (℘(z) -e 3 ) = 4 [℘(z) -e 2 ]
it easily follows the rst identity of Corollary 2-6. On the other hand, since τ ) we then obtain the two other identities.

e 2 (1 + τ ) = e 3 (τ ), e 1 ( -1 τ +1 ) = e 2 (
Corollary 2-7 The derivative of Weierstrass's function

℘ (z) = ℘ (z, τ ) = d℘(z) dz , ℘ (z) = d 2 ℘(z)
dz 2 with primitive periods 2 and 2τ veries the following identities

℘ (z, τ ) = -2π 3 k≥1 1 sin(2kπτ + πz) k≥1 1 sin( -2kπ τ + πz) k≥1 1 sin( -2kπ τ +1 + πz) . ℘ (z) ℘ (z) = -π k≥1 1 sin(2kπτ -πz) + 1 sin( -2kπ τ + πz) + 1 sin( -2kπ τ +1 + πz) . ℘ (z) ℘(z) -e 1 = 4π 2   k≥1 1 sin(2kπτ -πz)   2 +π 2 k≥1   1 cos 2 (kπτ -π z 2 ) - k≥1 1 sin 2 (kπτ -π z 2 )   .
Consider the second derivative of the Weierstrass function ℘ (z, τ ) = d 2 ℘(z) dz 2 . We have the following which may be easily deduced from the classical properties

2 ℘ (z) ℘ (z) = ℘ (z) ℘(z) -e 1 + ℘ (z) ℘(z) -e 2 + ℘ (z) ℘(z) -e 3 .
Thus we derive

℘ (z) ℘ (z) = -π k≥1 1 sin(2kπτ -πz) -π k≥1 1 sin( -2kπ τ + πz) -π k≥1 1 sin( -2kπ τ +1 + πz)
.

On the other hand, consider the derivative

d dz ℘ (z) ℘(z) -e 1 = ℘ (z) ℘(z) -e 1 - ℘ (z) ℘(z) -e 1 2 = -2π 2 k cos(2kπτ -πz) sin 2 (2kπτ -πz) = π 2 k≥1 1 cos 2 (kπτ -π z 2 ) - 1 sin 2 (kπτ -π z 2 )
.

We then derive

℘ (z) ℘(z) -e 1 = 4π 2   k≥1 1 sin(2kπτ -πz)   2 +π 2 k≥1 1 cos 2 (kπτ -π z 2 ) - 1 sin 2 (kπτ -π z 2 )
.

On the other hand, by [9, p.13, (1.)] here is another connection between the Weierstrass function ℘(z, τ ), and the sigma function

- σ(u + v)σ(u -v) σ 2 (u)σ 2 (v) = ℘(u) -℘(v).
Since σ(u) = 2 πθ 1 (0,τ ) e ηu 2 /2 θ 1 ( u 2 , τ ), then

℘(u) -℘(v) = (πθ 1 (0, τ )) 2 θ 1 ( u+v 2 , τ )θ 1 ( u-v 2 , τ ) θ 1 ( u 2 , τ )θ 1 ( v 2 , τ ) 2 .
In the limit case when v → u after dividing both sides by v -u, we nd

℘ (u) = - σ(2u) σ 4 (u) = -[πθ 1 (0, τ )] 3 θ 1 (2u, τ ) [θ 1 (u, τ )] 4 .
However, using again innite products of theta functions we may express ℘ (u) as innite product.

Theorem 2-8 The derivative of Weierstrass's function

℘ (z) = ℘ (z, τ ) = d℘(z) dz
with primitive periods 2 and 2τ veries the following equality

℘ (u) = - sin 2πv (sin πv) 4 k≥1
sin(kπτ + 2πv) sin(kπτ -2πv)(sin(kπτ )) 6 [sin(kπτ + πv) sin(kπτ -πv)] 4 .

Proof of Theorem 2-8 Indeed, this expression may be deduced from [2, Cor.

3-5]

θ 1 (v, τ ) (π sin πv) θ 1 (0, τ ) = k≥1 1 - sin πv sin kπτ 2 = k≥1 cos 2πv -cos 2kπτ 1 -cos 2kπτ , [ θ 1 (v, τ ) (π sin πv) θ 1 (0, τ ) ] 4 = k≥1 1 - sin πv sin kπτ 2 4 = k≥1 cos 2πv -cos 2kπτ 1 -cos 2kπτ 4 .
Thus sin(kπτ + 2πv) sin(kπτ -2πv)(sin(kπτ )) 6 [sin(kπτ + πv) sin(kπτ -πv)] 4 .

θ 1 (2v, τ ) θ 1 (0, τ ) (θ 1 (0, τ )) 4 [θ 1 (v, τ )] 4 = -π 3 ℘ (z) =

The functions ξ(u, τ )

Following Tannery [10, T.2, p. 168] it is often appropriate to introduce the quotients that can be formed by means of two functions σ(z, τ ) relating to the same variable and the same periods. Thus the four functions σ, σ 1 , σ 2 , σ 3 generate twelve functions. Indexes α, β, γ are selected according to the convention : they are dierent and be chosen between the values 0, 1, 2, 3

ξ α0 (u) = σ α σ (u) = ℘(u) -e α , ξ 0α (u) = σ σ α (u) = 1 ℘(u) -e α , ξ βγ (u) = σ β σ γ (u) = ℘(u) -e β ℘(u) -e γ .
We will simply write ξ βγ (u) instead ξ βγ (u, τ ) when there is no ambiguity. These functions are even or odd depending on whether they contain the index 0 or not. They are algebraic function of ℘(u) and have a single pole as simple singularity. Moreover, these functions are doubly periodic and verify many algebraic relations see [10, T.2, p. 168-187]. Observe that we may deduce from the above :

℘ (u) = -2 σ 1 σ (u) σ 2 σ (u) σ 3 σ (u) = -2ξ α0 (u) ξ β0 (u) ξ γ0 (u).
These relations yield in particular, [10, T.2, p. 171]

ξ α0 (u) = ℘ (u) 2 ℘(u) -e α = -℘(u) -e β ℘(u) -e γ = -ξ β0 (u) ξ γ0 (u), ξ 0α (u) = ξ βα (u) ξ γα (u), ξ βγ (u) = -(e β -e γ )ξ 0γ (u) ξ αγ (u), ξ β0 (u) ξ γ1 (u) = -℘ (u) 2(℘(u) -e 1 )
.

Moreover, ξ α0 is solution of the dierential equation The derivatives of the Weierstrass functions yield

3℘(u) = ξ 2 α0 (u) + ξ 2 β0 (u) + ξ 2 γ0 (u), ℘ (u) ℘ (u) = 2ξ α0 (u) + 2ξ β0 (u) + 2ξ γ0 (u).
Notice for

ω 1 = 1, ω 2 = 1 + τ, ω 3 = τ we derive ξ α0 (ω β ) = e β -e α , ξ βγ (ω α ) = √ e α -e β √ e α -e γ .
The periods are

k = ξ 21 (ω 3 ), k = ξ 23 (ω 1 ).
Many other interesting properties concerning the functions ξ βγ (u) (homogeneity, variations, growing,...) can be found in [10, T.2, p. 170-190]. In [START_REF] Tannery | Elements de la theorie des Fonctions Elliptiques[END_REF] we will also observe a rigorous and systematic study of these functions as well as their periodicity, their relationships with other elliptic functions and connections with theta functions of Jacobi. The following is particularly interesting Theorem 3-1 The function ξ α0 (u, τ ) = ξ α0 (u) verify the identities Notice under the action of the modular group Γ 0 the permutation between the e j does not change the Weierstrass function ℘(z, τ ). Indeed by changing τ into τ + 1 or into -1 τ it yields [4, p. 365]:

1 .

 1 The elliptic Weierstrass's function ℘(z).

ηv 2 2

 2 (cos πv) k≥1 cos(kπτ -πv) cos(kπτ + πv)

k≥1 1 16 (cot kπτ ) 8 = (e 1 -e 2 )(e 1 -e 3 (e 3 -e 2 )(e 3 -e 1 ) = k≥1 1 16 cot kπ τ 8 , (e 2 -e 1 )(e 2 -e 3 2 . 2 .

 18121332318212322 Logarithmic dierentiation of ℘(z) -e 1 . Taking the logarithmic dierentiation in Theorem 2-1 we derive the following Corollary 2-3 Let ℘ (z) the derivative of the Weierstrass function relative

1 (

 1 sin nπτ ) 2 ) using their innite product expansions [10, T.2, p.246] and their logarithmic derivatives Corollary 2-4 The Weierstrass's elliptic sigma functions and their derivatives relative to periods (2, 2τ ) verify the identities

4 = π 3 4 = π 3

 4343 sin 2πv (sin πv) 4 k≥1 [cos 4πv -cos 2kπτ ][1 -cos 2kπτ ] 3 [cos 2πv -cos 2kπτ ] sin 2πv (sin πv) 4 k≥1

dy du 2 =

 2 (e α -e β + y 2 ) (e α -e γ + y 2 ).

  may write for any integer m ∈ [0, n -1]℘(z+ m n , τ )-e 1 (τ ) = (π cot(πv + mπ n )) 2 4 k≥1 cot(kπτ + πv + mπ n ) cot(kπτ -πv + mπ n ) (cot(kπτ ) -e 1 (τ ) = 4 π 2 [℘(nz, nτ ) -e 1 (nτ )] .Turn now to the other cases j = 2, 3, by the same way we deduce analog decomposition of ℘(nz, nτ ) -e 3 (nτ ), ℘(nz, nτ ) -e 2 (nτ ) from Theorem 2-1 since℘(z) = e 3 + π22 sin πz

e 3

 3 (τ ), e 3 (1 + τ ) = e 2 (τ ).

Proof of Theorem 3-1 This theorem is a direct consequence of Corollaries 2-3 and 2-5.

Transformations of order n

Then the n-transformation theory of functions deals with the relations between functions belonging to dierent pairs of primitive periods : (2, 2τ ), (2, 2nτ ).

If we suppose n is not a prime number it will be the product of two or more odd primes, and the transformation will break up into distinct transformations each of which may be separately considered. We therefore now assume n an odd prime: the modular equation is in this case an irreducible equation of the order n + 1. Then, it is convenient to restrict the attention to the case n an odd number. Thus, we are going to study the transformations whose order is odd and positive, transformations which are reduced to those that we change the period (2, τ ) into ( 2 n , τ ) without changing τ . This corresponds to changing v to nv and τ to nτ . We shall always assume

Observe by Lawden [7, p.252], Enneper [4, p.240] or Roy [9, p.89] that any transformation of order n > 1 may be represented as a product of transformations of rst order and of transformations of higher order with matrix

Moreover, any transformation τ = nτ can be separated into a product when n has prime factors. Therefore, we only study the case of transformation when n is a prime and limit our study for this type of matrix.

4.1. Transformation of ℘(z, τ ). Theorem 2-1 provided an expansion of the Weierstrass function ℘(z, τ ) relative to periods 2, 2τ ) as innite product

where e 1 = ℘(1). It allows to derive in particular a n-order transformation formula for i = 1, 2, 3.

That permits to deduce various identities as

where j = 1, 2, 3,

More precisely, one gets the following connection between ℘(nz, nτ ) and ℘(z, τ ) derived from Theorem 2-1 Theorem 4-1 Let n be an odd integer and consider the Weierstrass's function ℘(nz, nτ ) with primitive periods 2 and 2nτ , then the following identity holds

where e j (τ ) are the zeros of ℘ (z, τ ), i = 1, 2, 3 and z < 2 τ .

Proof of Theorem 4-1 Prove at rst for i = 1. Notice by Theorem 2-1

We start from the classical trigonometric product formulas valid for n odd integer

Thus we derive the expression

Therefore, we nd analog expansions for ℘(nz, nτ )-e 2 (nτ ) and ℘(nz, nτ )-e 3 (nτ ).

then by Remark 2-2 (ii) we may deduce the following n-decomposition of ℘ (nz, nτ ) .

Corollary 4-2 Let n be an odd integer, then the following identity holds

Among others interesting equalities, we may derive the following identities

From Theorem 4-1 we also get Corollary 4-3 Let ℘(z, τ ) be the Weierstrass function and ℘ (z, τ ) its derivative relative to the periods (2, 2τ ), then the following identities hold for any odd n and j = 1, 2, 3

.

Indeed, (i) may be deduced from Theorem 4-1 and also by the n-order transformation of the quotient of sigma functions (see also [10, T.2, p.215], or [11, ex 9 p.456])

(ii) is deduced from (i) since

Moreover, since n is odd then the quotient (ii) over (i) yields (iii).

Corollary 4-4 Let ℘(z, τ ) be the Weierstrass function and ℘ (z, τ ) its derivative relative to the periods (2, 2τ ), then the following identities hold for any odd n and j = 1, 2, 3

.

Moreover, one gets for j = 1

.

Indeed, (i) is derived from Theorem 4-1 and Corollary 4-2 and using the quotient

℘(nz,nτ )-e1(nτ ) . Corollary 2-3 implies (ii).

4.2.

The n-transformations of the functions ξ(u). We start from the n-transformations of the sigma functions. In [10,T.1, p.234] Tannery-Molk proved

, for j = 1, 2, 3 we obtain the quotient

.

We then deduce [10, T.2, p.215]

.

This shows that ξ α0 (nu, nτ )) is a rational function of ξ α0 (u), as well as ξ βγ (nu, nτ ) is a rational function of ξ βγ (u, τ ).

We then derive the identity

.

We obtain the modular relations of the periods

The zeros become

.

By the same way if we replace τ by τ n one obtains the formulas

.

More generally ([10, T.2, p. 217]), if we replace τ by τ +2p n for any odd n and integer p one gets

Other applications may be derived from the decomposition of ℘ (nz,nτ ) ℘(nz,nτ )-e1(nτ ) as the following modular identities Corollary 4-5 For 0 < (τ ) < z and for any odd integer n the following identity holds

.

To prove that, take the logarithmic dierentiation of ξ 10 (nz, nτ ) see [10, T.2, p.215] :

, which yields

.

Therefore, Corollary 2-3 implies Corollary 4-4.

Corollary 4-6 For 0 < (τ ) < z and for any odd integer n the following identities hold

.

To prove Corollary 4-6, recall by Corollary 2-3 we have We then obtain an equality allowing us to invert the sum and the product : .


PANEL: CONFUCIANISM (pre-modern history)

Chosǒn Confucian scholars' attitudes toward the Laozi KIM Daeyeol (INaLCO, Paris) Is it shocking to hear that the Chosǒn literati were radically or rigidly Neo-Confucian?

Probably not, at least for the general public. This idea is so surreptitiously inculcated in the mind of most of us. On the contrary, it might be relatively surprising to hear that Korean Confucian literati of this time widely read, wrote about and even accepted intellectual, ideological, religious or cosmological traditions other than Confucianism. It seems to me that the cultural plurality in Chosǒn society has been often passed over, though it is also known that different cultural traditions coexisted and even functioned in mutual complementary.

Confucianism, as a whole system of culture, was incontestably one of the dominant currents in Korean society and provided models in "almost" every field of culture for Chosǒn period. However, its cultural landscape was more complex and Confucian religious or ideological community membership -the determination of which often depended on circumstances -was, in a sense, rather "soft". As a way to understand the cultural plurality, we can focus on and interpret people's more ultimate concerns and consider different traditions as cultural models or means given to realize them, insofar as a cultural pattern has a role in relation with meaning or further purpose which man aspire to attain to. In working on a cultural model, we have to keep in mind that a model is useful for something and not always the final or sole destination. This sort of reflexion can arise in various fields and in particular we can think it regarding moral thought and religious behaviour, in which man throws his sight over the way of thinking, expressing, or seeking something more fundamental or more ultimate. This point of view also demands to take into account the ways in which a dominant tradition is imposed in relation to other cultural factors and contents. In my today's talk, I would like to give an example by presenting some cases of Chosǒn literati whose attitudes toward the Laozi are interesting in this respect.

The established and general position of the Neo-Confucian orthodox school from the beginning of Chosǒn Dynasty is that Daoism is 'heterodox' (idan 異端). However, after about two centuries' evolution during which the Korean Neo-Confucianism developed on its own, the literati started to show diverse attitudes: tolerant, open or even receptive vis-à-vis the 'heterodox' tradition. For example: Daoist hygienic system was highly estimated and adopted in the former chapters of the Precious Mirror of Eastern Medicine (Tong'ŭi pogam 東醫寶鑑) published by a royal order; also, many poems of Confucian literati were inspired by the idea of Daoist Immortals' world.

With regard to Daoist philosophical ideas, several scholars of Chosǒn commented the Laozi and tried to interpret it differently from their predecessors. About twenty scholars also left shorter texts about this Daoist book. Some of them said, for example, that those who considered it as a heterodox text did not correctly understand it. In general, they read it while keeping their Confucian eyes. Still, it would show that, seen from this angle too, the boundaries between Confucianism and other traditions could be ambiguous for some intellectual groups in Chosǒn.

Five commentaries on the Laozi fully transmitted to the present day attract our first attention (see Table 1). Indeed, they are the subjects of very recently published research works (see Bibliography). Excepting Yi Ch'ungik, all of these commentators were well known authors and thinkers by their time, and moreover, they all occupied one of the highest official posts in government at a point of their life. In general, they found out that certain ideas from the Laozi can be in accordance with Confucianism on the level of fundamental thought. Their central interests focused on principles and moral values to which Confucian literati adhered.

One of the pivotal figures in history of Korean Confucianism, Yi I left a commentary on the Laozi. This is characterized by an interpretation of the Laozi in a pure Confucian terms and point of view, as in the example of his reading of the word 'inaction' (muwi 無爲) into governing the people according to the 'decree of Heaven' (ch'ǒnmyǒng 天 命). But, regarding the issue developed in this paper, the most interesting point is that a large part of his commentary selects the passages concerning 'making mind empty' (son In addition to these commentaries, some shorter but not less significant texts for my argument confirm that many Chosǒn Confucian literati read and even appreciated the so-called 'heterodox' writing (see Table 2). These texts appeared from the beginning of the 16 th century on. Before Yi I, Yi Haeng declared that the Laozi deserved esteem, and Sin Hŭm, contemporary with Yulgok, sang that it enlightened him and made his mind empty. The latter also argued that the author of the Laozi didn't consider the Confucian virtues as defects, but that he was distressed about the loss of the Way and the Virtue, and that its profound meanings were distorted by later Daoists and Buddhists. Along with Sin Hŭm, Hǒ Kyun expressed a similar opinion and Chang Yu could agree with Yi

損
Haeng to say that the Laozi should not be neglected. Since Yun Hyu, many of them balanced their criticisms against the Daoist book with the esteem they expressed about it. On the one hand, they underlined its limits : for example, the Laozi concerns the principles of things (mulli 物理) not those of human being (Yun Hyu, Cho Kumyǒng),

or it talks about the Dao without careful consideration. "He (author of the Laozi) was good in beholding the Dao, wrote Im Sangdǒk, but not good in expressing it." On the other hand, they regretted that people, even Confucian literati, did not rightly understand these five thousands words and criticized what could not be criticized (Yi Tǒksu, Im Sangdǒk, Yun Ki). In Yi Tǒksu's opinion, one could easily misunderstand the book's profound meanings about Dao because of its ambiguous and subtle expressions, and one could thus be misled as most of Daoists were; but one should meet ultimate beauty and joy in the idea of frugality and of lack of concern expressed in the book.

To conclude, we could at first underline the fact that most of these literati were high officials in government, that is to say, not marginal persons but central,

  ) or 'frugality' (saek 嗇) and interprets them from the viewpoint of Confucian moral concerns related to 'self-cultivation' (ch'igi 治己) and 'governing the people' (ch'iin 治 人). It raises an interesting question: why did this great Confucian scholar rely on the Laozi to remind the fundamental moral duty of Confucian literati? In the epilogue to his commentary, he deplores that moral persons are too rare in his time, as if he were warning the leading class of Confucian society of his time by professing the moral instructions drawn from the Laozi.As to Pak Sedang who frankly criticized Zhu Xi, he exposed in his commentary Daoist characteristics different from those Confucians but tried to prove that basically these Daoist characteristics are not contradictory to Confucian logic and thought. This attitude is also seen in the commentary of Sǒ Myǒng'ŭng. Considered as one of the pioneers of 'Northern school' (pukhakp'a 北學派), Sǒ Myǒng'ŭng even introduces in his interpretation of the Laozi some proper Daoist cosmological notions (as chǒng 精, ki 氣, sin 神) or practical ideas (as yangsaeng 養生). Moreover, he explains the criticisms in the Laozi against some Confucian ideas ('sense of humane' (in 仁), 'sense of just' (ŭi 義), 'learning' (hak 學), for examples) and tries to clarify the real meaning of these criticisms. For him, this is not contradictory to but even accordant with Confucian fundamental spirit. This kind of attempt is also seen in the commentary done by Yi Ch'ungik. A scholar of the Kanghwa school, a Korean Yangming school, entirely dissociated from orthodox Zhu Xi-ism, he even called the author of the Laozi as a 'mysterious sage' (hyǒnsǒng 玄聖).HongSǒkchu is one of the central figures of the history of Confucian ideas in 19 th century Korea. While being widely interested in diverse intellectual traditions out of Confucianism, he tried to replace the perspective of 'ordering the world' (kyǒngse 經世) in the center of the Zhu Xi-ist Neo-Confucianism. In his close examining of the similarities and the differences between the ideas of the Laozi and Confucianism, he finds out that some ideas of the book are closer to Confucianism, not only than the individualism of later Daoists, who were only looking forward to their individual salvation, but also than the immorality of vulgar Confucian literati, and even than both of the dhyāna school of Buddhism and the Yangming school of Neo-Confucianism who neglect the necessity of accumulating efforts at learning, which Confucianism advocates. While scrutinizing the limits of the Laozi, however he also elucidates valid meanings of this Daoist book expressed in its peculiar ironies, which were employed only to criticize the abuses of it's time disguised with moral virtues.

  active and influential figures in Chosǒn society. This seems to raise an important question in relation with the general issue of our panel: to what degree did the founders of the Zhu Xi-ist school and Korean Neo-Confucian predecessors have authority on later Confucian literati? We can say at least that the attitudes of these literati toward the Laozi are not dogmatic or sectarian. More than to certain canonic texts, schools, or charismatic persons, they referred to comprehensive principles or fundamental values, which could possibly fade the Confucian identity. Further research should be done in order to confirm this interpretation and to understand better these "positive" attitudes regarding the Laozi. The cases analyzed in this paper, however, seem to suggest the idea of an erosion of the border line between orthodox Confucianism and philosophical Daoism, coming from the very inside of Confucian literati group itself. After all, beyond these misty and instable boundaries between Confucianism and the other intellectual or spiritual traditions of Chosǒn society, it may be also useful to ask what were the practical goals or ultimate concerns in the quest of which some groups of Chosǒn literati in their proper circumstances adopted Confucian legacy as one of the available ways.

Table 1 : commentaries

 1

				One of the
				highest
		Name	Date	official posts/ grades	School	Commentarie s
				occupied by
				person
	1	李珥	1536-1584	判書 (正二 品)	西人	醇言

Table 2 : others texts

 2 Lao Tzu in the History of Korean Confucianism). Seoul National University Press. KIM Hak Mok 김학목. 2002. "Chosǒn yuhakchadŭl ŭi Todǒkkyǒng chusǒk kwa kŭ sidae sanghwang -Sunǒn, Sinju Todǒkkyǒng, Chǒngno rŭl chungsim ŭro -" 조선 유학자들의『道德經』 주석과 그 시 대상황 -『순언』 『신주도덕경』 『정노』를 중심으로 -, Tongsǒ ch'ǒlhak yǒn'gu 동서철학연 구, n° 24, p. 115-134. KIM Hak Mok. 2003. "Yǒnch'ǒn Hong Sǒkchu ka Todǒkkyǒng ŭl chusǒkhan mokchǒk" 淵泉 洪奭周가 道德經을 주석한 목적, Ch'ǒlhak yǒn'gu 철학연구, n° 60, p. 5-24. KIM Hak Mok. 2004. "Kanghwa hakp'a ŭi Todǒkkyǒng chusǒk e kwanhan koch'al -Ch'owǒn Yi Ch'ungik ŭi Ch'owǒn tamno rŭl chungsim ŭro" 江華學派의 『道德經』 주석에 관한 고찰 -椒園 李忠翊의『椒園談老』를 중심으로 -, Tongsǒ ch'ǒlhak yǒn'gu 동서철학연구, n° 34, p. 277-299.

		Name	Date	One of the highest official posts/grades occupied by person	School	Texts
	1	李荇	1478-1534	左議政 (正一品)	X	讀老子
	2	申欽	1556-1628	判書 (正二品)	西人 ?	老子吟
						讀道德
						經
					書道德經後
	3	許筠	1569-1618	參議 (正三品)	大北 ?	老子
	4	鄭忠信 1576-1636	兵馬節度使 (從二品武官)	X	讀老子 有感
	5	張維	1587-1638	右議政 (正一品)	?	老子見 道體
	6	尹鑴	1617-1680	判書 (正二品)	X	老子道 德經序
	7	朴世堂 1629-1703	判書 (正二品)	少論	讀老子
	8	任相元 1638-1697	判書 (正二品)	X	讀老子
	9	李德壽 1673-1744	判書, 大提學 (正二品)	X	老子要 解序
	10	鄭來僑 1681-1759	僉知中樞府事 (正三品武官, 中人出身) X	讀老子
	11	林象德 1683-1719	牧使 (正三品)	少論?	老子論
	12	姜再恒 1689-1756	縣監 (正六品)	少論	問禮於 老子
	13	趙龜命 1693-1737	佐郞 (正六品)	老論?	讀老子
	14	徐命膺 1716-1787	判書, 大提學 (正二品)	北學派 讀老子
	15	李匡呂 1720-1783	參奉 (從九品)	?	讀老子
						書蘇子
	16	尹愭	1741-1826	參議 (正三品)	星浩	由老子
						論後
	17	申綽	1760-1828	X	江華學 派	老子旨 略序
						老子道
	18	李圭景 1788-?	X	X	德經辨
						證說
	Selected bibliography

KEUM Jangtae 금장태. 2006. Han'guk yuhak ŭi Noja ihae 한국유학의 『노자』이해 (Interpretations and Commentaries of