Gaëlle Boudry
email: gaelle.boudry@inrae.fr

Ilario Mennella

Olivia Ménard

Régis Janvier

Isabelle Nogret

Ashkan Madadlou

Armelle Cahu

Laurence Le Normand

Olivia Menard

Eric Bobillier-Chaumont

Rosalia Ferracane

Paola Vitaglione

Didier Dupont

David Val-Laillet

Development of a functional dairy snack containing oleoylethanolamide that reduces food intake in normal-weight and obese minipigs

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The development of nutritional solutions for the prevention and/or treatment of obesity represents a direct answer that food companies could provide to support "health by food" for body weight and food intake control. In this context, oleoylethanolamide (OEA), a safe bioactive lipid, has attracted much attention from the food and pharmaceutical industries [START_REF] Brown | Oleoylethanolamide: A fat ally in the fight against obesity[END_REF]. Indeed, OEA demonstrated strong anorexigenic properties in mouse and rat models [START_REF] Brown | Oleoylethanolamide: A fat ally in the fight against obesity[END_REF], either after intraperitoneal injection [START_REF] Fu | Oleylethanolamide regulates feeding and body weight through activation of the nuclear receptor PPAR-α[END_REF][START_REF] Gaetani | Modulation of meal pattern in the rat by the anorexic lipid mediator oleoylethanolamide[END_REF][START_REF] Azari | Vagal afferents are not necessary for the satiety effect of the gut lipid messenger oleoylethanolamide[END_REF][START_REF] Lan | GPR119 is required for physiological regulation of glucagon-like peptide-1 secretion but not for metabolic homeostasis[END_REF][START_REF] Proulx | Mechanisms of oleoylethanolamide-induced changes in feeding behavior and motor activity[END_REF][START_REF] Rodríguez De Fonseca | An anorexic lipid mediator regulated by feeding[END_REF] or orally [START_REF] Oveisi | Oleoylethanolamide inhibits food intake in free-feeding rats after oral administration[END_REF]. In humans, research studies highlighted correlations between endogenous plasma OEA and measures such as hunger, satiety or food palatability [START_REF] Grosshans | Oleoylethanolamide and human neural responses to food stimuli in obesity[END_REF][START_REF] Mennella | Oleic acid content of a meal promotes oleoylethanolamide response and reduces subsequent energy intake in humans[END_REF][START_REF] Monteleone | Responses of peripheral endocannabinoids and endocannabinoid-related compounds to hedonic eating in obesity[END_REF][START_REF] Rigamonti | Anticipatory and consummatory effects of (hedonic) chocolate intake are associated with increased circulating levels of the orexigenic peptide ghrelin and endocannabinoids in obese adults[END_REF]. Moreover, a positive effect of oral OEA supplementation twice a day for 8 or 12 weeks on the dietary intakes of obese patients was also reported [START_REF] Laleh | Oleoylethanolamide increases the expression of PPAR-А and reduces appetite and body weight in obese people: A clinical trial[END_REF][START_REF] Payahoo | Investigation the effect of oleoylethanolamide supplementation on the abundance of Akkermansia muciniphila bacterium and the dietary intakes in people with obesity: A randomized clinical trial[END_REF][START_REF] Tutunchi | Oleoylethanolamide supplementation in obese patients newly diagnosed with non-alcoholic fatty liver disease: Effects on metabolic parameters, anthropometric indices, and expression of PPAR-α, UCP1, and UCP2 genes[END_REF].

OEA can be obtained directly from food sources, yet at very low levels [START_REF] De Luca | Food database of N-acylphosphatidylethanolamines, N-acylethanolamines and endocannabinoids and daily intake from a Western, a Mediterranean and a vegetarian diet[END_REF]. Thus, the average N-acylethanolamines (NAE), which include OEA but also linoleylethanolamide (LEA) and palmitoylethanolamide (PEA), daily intake from food sources can be estimated around 0.08 mg per day for Western diet consumers and 0.25 and 0.28 mg per day for Mediterranean and vegetarian diet consumers, respectively [START_REF] De Luca | Food database of N-acylphosphatidylethanolamines, N-acylethanolamines and endocannabinoids and daily intake from a Western, a Mediterranean and a vegetarian diet[END_REF]. OEA can also be endogenously formed from the omega-9 monounsaturated fatty acid, oleic acid, and an oleic acid-enriched diet has been shown to increase OEA levels in several tissues both in animal models and humans [START_REF] Bowen | Oleic acid-derived oleoylethanolamide: A nutritional science perspective[END_REF]. Finally, relatively pure OEA is commercially available and has been Abbreviations: 2-AG, 2-arachidonoylglycerol; AEA, N-arachidonoylethanolamine; EC, endocannabinoid; FAAH, fatty acid amide hydrolase; LEA, linoleylethanolamide; NAE, N-acylethanolamine; NEFA, non-esterified fatty acid; OEA, oleoylethanolamide; PEA, palmitoylethanolamide. incorporated into capsules for clinical trials and eventually used as dietary supplement, designed to be taken in addition to daily food consumption. Another way of consuming OEA would be to incorporate it as an ingredient in a specific tailored functional food. One question to answer first in the development of such an OEA-containing functional food is that of OEA bioavailability and timing of delivery within the organism. Indeed, in rats, the effect of OEA intraperitoneal injection is short-lasting (between 0.5 and 2 h after injection [START_REF] Gaetani | Modulation of meal pattern in the rat by the anorexic lipid mediator oleoylethanolamide[END_REF][START_REF] Azari | Vagal afferents are not necessary for the satiety effect of the gut lipid messenger oleoylethanolamide[END_REF][START_REF] Proulx | Mechanisms of oleoylethanolamide-induced changes in feeding behavior and motor activity[END_REF][START_REF] Rodríguez De Fonseca | An anorexic lipid mediator regulated by feeding[END_REF] but when delivered orally through capsules, the OEA effect on food intake was observed only after 5 h in the same species [START_REF] Oveisi | Oleoylethanolamide inhibits food intake in free-feeding rats after oral administration[END_REF]. Likewise, in mice, one study reported a long-lasting effect (5 h) of OEA intraperitoneal injection [START_REF] Fu | Oleylethanolamide regulates feeding and body weight through activation of the nuclear receptor PPAR-α[END_REF].

One easy way to modulate dietary compounds bioavailability is to change food structure, which is known to modulate gastric emptying as well as intra-luminal pH and digestive enzyme accessibility to nutrients, thus modifying the whole digestive process [START_REF] Bornhorst | Gastric digestion in vivo and in vitro: How the structural aspects of food influence the digestion process[END_REF]. Noteworthy, milk structure, either liquid or semi-solid, e.g. rennet-gel, strongly impacts digestion kinetic up to 6 to 7 h after ingestion [START_REF] Barbé | The heat treatment and the gelation are strong determinants of the kinetics of milk proteins digestion and of the peripheral availability of amino acids[END_REF]. The aim of our work was therefore to compare dairy-based OEA-containing functional foods with two distinct structures (liquid vs. semi-solid) in their efficacy to modulate eating behavior in a relevant animal model for nutrition research, the Yucatan minipig. We first developed an innovative methodology to encapsulate OEA, incorporated these OEA microcapsules in dairy-based liquid or semi-solid snacks, whose effects on eating behavior were tested in normal-weight and obese minipigs. Plasma analyses were further performed. to document the physiological correlates of these behavioral effects. In parallel, dynamic in vitro digestions were performed to study the effect of the snack structure on OEA release.

Materials and methods

Fabrication of OEA-loaded zein-alginate composite particles

The OEA-loaded zein-alginate particles were fabricated through the desolvation method as already described [START_REF] Ge | The thermal resistance and targeting release of zein-sodium alginate binary complexes as a vehicle for the oral delivery of riboflavin[END_REF][START_REF] Jiang | Fabrication and characterization of zein-alginate oligosaccharide complex nanoparticles as delivery vehicles of curcumin[END_REF] with some modifications. Briefly zein (Sigma-Aldrich Z3625) was dissolved (7 % w/v) in 85 % aqueous ethanol solution, followed by dissolving (2.3 % w/v) OEA ((Bertin technologies OEA-d4, OEA 82 %, LEA 2 %, PEA 0.05 %) in the solution. Subsequently, sodium alginate (Sigma-Aldrich W201502) solution (3 % w/v) in distilled water was added drop by drop into the zein-OEA aqueous ethanol solution at a volume ratio of 2.5:1. Ethanol was eliminated using a rotavapor (30 min., 100 rpm) combined with a water bath at 40 • C. The generated OEA-loaded particles were washed by distilled water and the suspension was centrifuged for 20 min. at 3000 rpm. The sedimented particles were dried at 37 • C for 12 h. The film obtained was milled in a powder (OEA+) and then used in the test food. An alginate-zein complex not including OEA (OEA-) was prepared following the same procedure and used in the control food.

Snack preparation

Four snacks (semi-solid containing OEA-or OEA+, liquid containing OEA-or OEA+) were prepared as following: 100 g of skim milk powder were dissolved in 200 g of distilled water, OEA+ or OEA-powder was also dissolved and then 180 µL of rennet (deactivated in case of the liquid food) were added and the food kept at 37 • C for 15 min. The amount of the OEA+ powder to be added was calculated for each animal in order to guarantee 25 mg of OEA per kg of body weight. Consequently, the same amount of OEA-was added in the control food. Skim milk was used to avoid any potential presence of OEA in the lipid fraction of milk.

In vitro digestion experiments

All chemicals and enzymes used for digestions were purchased from Merck, France, Sigma Aldrich (Saint Quentin Fallavier, France), unless otherwise stated.

The two snacks, dairy-based liquid and semi-solid, containing OEA were submitted to the gastrointestinal digestion system DIDGI®, INRAE, France. The computer-controlled system was set up to simulate the gastrointestinal digestion at the adult stage using parameters adapted to simulate either dairy-based liquid or semi-solid meal [START_REF] Egger | Digestion of milk proteins: Comparing static and dynamic in vitro digestion systems with in vivo data[END_REF][START_REF] Macierzanka | Colloidal transport of lipid digesta in human and porcine small intestinal mucus[END_REF]. Before starting the digestion, 24 mL of simulated gastric fluid adjusted at pH 2 were introduced in the artificial stomach to simulate the fasted state. The gastric and intestinal emptying followed an exponential equation, as described previously [START_REF] Elashoff | Analysis of gastric emptying data[END_REF]. The gastric emptying half-time were set at 35 and 70 min and the β coefficient at 1.15 and 2 for liquid and semi-solid, respectively for the gastric phase [START_REF] Minekus | A Multicompartmental dynamic computer-controlled model simulating the stomach and small intestine[END_REF]. The intestinal phase was longer for the semi-solid than for the liquid meal; the intestinal emptying half-times were 85 and 180 min and the β coefficient 1.4 and 1.6 for liquid and semi-solid meals respectively [START_REF] Malagelada | Measurement of gastric functions during digestion of ordinary solid meals in man[END_REF].

The evolution of gastric pH for the liquid (pH = 1.68+ 3.82 (-t/65) ) and semi-solid (pH = 1.68+ 4.32 (-t/65) ) snacks followed exponential equations where t represented the time in minute after ingestion [START_REF] Malagelada | Measurement of gastric functions during digestion of ordinary solid meals in man[END_REF]. The pepsin solution was prepared to cover 2000 U/mL of gastric content and the amount of pepsin needed was calculated based on the activity measured in the powder. The intestinal pH conditions, pH, bile and pancreatin concentrations were set up, as previously reported by Minekus 1995. Intestinal pH was maintained at 6.7 by addition of NaHCO 3 1 mol/L.

Samples were collected after intestinal emptying and kept on ice. Representative aliquots of each hour of digestion (two aliquots per period) were collected up to 4 h for the liquid meal and 6 h for the semisolid meal. No inhibitors were added and samples were frozen at -80 • C for latter endocannabinoid (EC) and N-acetylethanolamide extraction.

Animal experiment

Ethical considerations

Animal experiments were conducted in accordance with the current ethical standards of the European Community (Directive 2010/63/EU), Agreement No. D3527532. The Rennes Ethics Committee in Animal Experiment and the French Ministry of Higher Education and Research have approved and authorized the entire procedure described in this paper (project number APAFIS #11984-2017103009069432_V2).

Housing conditions and diet

Sixteen 20-month-old male (8) and female (8) Yucatan minipigs from the INRAE UE3P animal facility (St-Gilles, France) were housed in individual stainless-steel pens (surface 110 × 80 cm, height 110 cm) equipped with metal chains to play. Bars spaced 8 cm apart allowed contact between animals in adjacent pens. Temperature was maintained at 22 • C and artificial lighting provided a 15:9 light/dark cycle. Each animal had ad libitum access to water. Eight (normal-weight group) minipigs received a standard diet (1.74 kcal/g, 73 kcal/kg body weight 0.75 ) throughout the experiment, while the eight others (obese group) received a high-fat and sucrose diet (2.58 kcal/g, 100 kcal/kg body weight 0.75 ) throughout the experiment (see [START_REF] Gautier | Western diet, obesity and bariatric surgery sequentially modulated anxiety, eating patterns and brain responses to sucrose in adult Yucatan minipigs[END_REF] for diet composition). They received their feed as a unique meal at 1:00 pm every day and were kept on their respective diets for 8 weeks before starting the experiments. Body weight was measured weekly throughout the protocol. One week before the onset of experiment, they were trained to eat the four snacks (on separate days) at 9:00 am within less than 15 min.

Surgery

A long catheter (1219.13 Leadercath; Vygon, Ecouen, France) was inserted into the jugular vein under light anesthesia (5 µg/kg ketamide, i.m. and isoflurane inhalation (Aerane 100 mL, Baxter SAS, France) through a facial mask) two weeks before experiments.

Experiment 1: Impact of snacks on short-term eating behavior (Fig. 1A)

Our objective was to design a functional food that, when consumed in the morning after an overnight fast, would reduce food intake at lunch. Thus, the day of the experiment, overnight fasted minipigs were offered one of the four snacks (semi-solid OEA-or OEA+, liquid OEA-or OEA+) at 9:00am, which was consumed in less than 15 min. The amount of snack offered to the animals was calculated to deliver 25 mg of OEA per kg body weight and 440 kcal. After 4 h during which they did not receive any other feed, minipigs were exposed to their usual diet ad libitum. Eating tests were performed for 1 h using in-house pig-dedicated meal pattern recording troughs as already described [START_REF] Ochoa | Effects of chronic intake of starch-, glucose-and fructose-containing diets on eating behaviour in adult minipigs[END_REF]. Minipigs received each of the four snacks in a.

Experiment 2: Impact of liquid snacks on endocannabinoid and anorexigenic and orexigenic factor plasma profiles (Fig. 1B)

We next evaluated different plasma profiles in the 4 h that followed the liquid snacks consumption. We focused on the liquid snacks only since the OEA+ liquid snack exhibited promising anorexigenic properties. The day of the experiment, minipigs were offered one of the two liquid snacks (OEA-or OEA+) at 9:00am, which was consumed in less than 15 min. Blood (10 mL) was drawn from the intra-jugular catheter either 20 min before or 30, 60, 120, 180 and 240 min after the snack consumption. The animals received the two snacks in a random order with a minimum of 2 days between trials. Blood was immediately placed in tubes containing either EDTA (final concentration 8 mM) for later EC, glucose, non-esterified fatty acid (NEFA), β-hydroxybutyrate and insulin analysis or EDTA (final concentration 8 mM) and anti-DPPIV (Millipore, USA,1:100 v/v) for later GLP-1 analysis, or anti-protease (aprotinincontaining tubes (BD Life Science, France)) for later ghrelin analysis. Blood was immediately centrifuged (2500g, 10 min., 4 • C) and plasma aliquoted and stored at -80 • C until analysis. For ghrelin analysis, HCl 1 N (1:20 v/v) and phenylmethanesulfonyl fluoride (1:100 v/v) were added to plasma before storage at -80 • C.

Extraction and analysis of endocannabinoids and nacetylethanolamides

ECs and NAEs were extracted from EDTA plasma samples and from in vitro digestion samples. A solid-phase extraction according to the method described by Marczylo and colleagues [START_REF] Marczylo | A solid-phase method for the extraction and measurement of anandamide from multiple human biomatrices[END_REF] was carried out using Oasis HLB cartridges (1 cc/30 mg, Waters). After extraction, samples were dried under nitrogen flow and kept at -80 • C until analysis. Acetonitrile/water (50:50) 100 µL were used to reconstitute samples just before LC-MS/MS analysis performed following the method previously described [START_REF] Mennella | Oleic acid content of a meal promotes oleoylethanolamide response and reduces subsequent energy intake in humans[END_REF].

Plasma analysis

Plasma glucose, NEFA and β-hydroxybutyrate concentrations were assessed using an automated spectrophotometric method (Konelab 20i, Thermo Fisher Scientific, Illkirch, France) using specific kits (Thermo Fisher Scientific for glucose and β-hydroxybutyrate, Wako for NEFA). Plasma insulin concentrations were determined using a commercial immunoassay kit (ST AIA-PACK IRI) and the AIA-1800 device (Automated Immunoassay Analyzer; TOSOH Bioscience, Tokyo, Japan). Plasma GLP-1 concentrations were measured using a porcine-dedicated GLP-1 active ELISA kit (Millipore). Finally, plasma active (octanoyled) ghrelin concentrations were determined using dedicated radioimmunoassay kit (Phoenix Pharmaceuticals).

Statistical analysis

Data are presented as means ± standard error of the mean. Normalweight and obese minipig data were analyzed separately. Sex effect and its interaction with other tested parameters was first tested. In lean minipigs, sex was significant for plasma LEA (M > F, P = 0.02), AEA (M > F, P = 0.001), insulin (F > M, P = 0.003) and glucose (F > M, P = 0.04). In obese minipigs, sex was significant for plasma OEA (F > M, P = 0.02), 2-AG (F > M, P = 0.001) and AEA (F > M, P = 0.02). However, for all these parameters, there was not interaction with presence of OEA in the snack. Thus, male and female data were pooled for further analysis. Meal pattern parameters were analyzed using a 2-way analysis of variance testing the snack structure (semi-solid or liquid) and OEA presence (OEA+ or OEA-) and the interaction between these two factors. Plasma parameters were analyzed using a 2-way analysis of variance testing OEA presence (OEA+ or OEA-), time and the interaction between these two factors. Post-hoc analyses were performed using Sidak test when appropriate.

Results

Liquid OEA-enriched dairy snack reduced short-term feed intake

To investigate the impact of OEA-enriched dairy snack on feed intake, we designed OEA-enriched (OEA+) or control (OEA-) snacks with different structures, either in a liquid form (i.e. milk) or a semi-solid form (i.e. curd obtained after rennet coagulation). Overnight fasted normal-weight and obese minipigs were offered the four snacks in a random order and their eating behavior investigated 4 h after snack consumption (Fig. 1A). Irrespective of the type of snack and weight status, all the minipigs displayed a hyperphagic behavior with immediate eating onset and a very long (average duration 2920 s [min: 1168 s -max: 3599 s], i.e. nearly 49 min) and large (average amount 1712 g [min: 560 g -max: 3034 g]) first meal.

The amount of feed was significantly reduced after minipigs had consumed the OEA+ snack in the liquid form compared to the liquid OEA-snack, irrespective of the minipig weight status (OEA+ vs. OEAliquid: normal weight minipigs -14 %, P = 0.016, obese minipigs -%, P = 0.018, Fig. 2A). In addition, in obese minipigs, the amount of feed ingested was greater after OEA+ semi-solid snack compared to OEAsemi-solid snack ingestion (+27 %, P = 0.018), although this was primarily due to a low feed intake after the OEA-semi-solid diet consumption compared to the other control conditions. Duration of the first meal was not different after the OEA-and OEA+ snacks, irrespective of the minipig weight status or the snack structure (Fig. 2B).

Endocannabinoid and related NAEs release from the snack was greater for the liquid than the semi-solid snack in in vitro digestion experiments

To understand why OEA+ snacks with different structures induced different eating behaviors, we evaluated EC release from the liquid and semi-solid OEA+ snacks in in vitro digestion experiments. Luminal samples were taken every hour for 6 h (semi-solid snack) or until complete emptying of the gastric compartment, i.e. after 4 h (liquid snack). Endocannabinoid and related NAEs (OEA but also LEA, PEA and AEA since minor percentages of these compounds were present in the initial OEA powder) concentrations were analyzed in these samples and cumulative EC concentrations calculated for each snack. The concentrations of OEA, LEA and PEA increased with digestion time for both types of snacks but concentrations were largely increased (x 357, 152 and for OEA, LEA and PEA, respectively after 4 h of digestion) for the liquid compared to the semi-solid snack, starting to plateau after 3 h of digestion (Fig. 3 A-C). The concentrations of AEA were below detectable levels. Thus, the difference in eating behavior after liquid or semi-solid OEA+ snacks ingestion could account for the difference in EC release during the digestive process in viv.

Presence of OEA in the dairy liquid snack reduces 2-AG plasma level in normal-weight but not obese animals

To further understand the effect of the liquid OEA+ snack, we next investigated plasma EC profiles during the 4 h following the liquid OEAor OEA+ snack ingestion. We focused on the liquid snacks only since only this snack structure exhibited anorexigenic properties (Fig. 1B). There was no significant difference in fasted EC plasma levels (2arachidonoylglycerol (2-AG), LEA, PEA, N-arachidonoylethanolamine (AEA) and OEA) between normal-weight and obese minipigs, except for a tendency for greater AEA plasma level in obese animals (Table 1). After liquid snack ingestion, the levels of OEA, 2-AG, LEA and PEA, but not AEA, significantly dropped in normal-weight minipigs (Fig. 4A,C,E,G,I). On the contrary, no variation of plasma levels was noticed for these ECs in obese minipigs (Fig. 4B,D,H,J), except AEA levels, which dropped after the liquid snack ingestion (Fig. 4F). There was no difference in EC levels after OEA-or OEA+ liquid snack ingestion, irrespective of the animal weight status, except for 2-AG plasma level, which was lower 4 h after the liquid OEA+ snack compared to the OEA-one in normal-weight minipigs (-52 %, P = 0.034, Fig. 4C).

Liquid OEA-enriched snack altered plasma anorexigenic factors differently depending on the minipig weight status

We next investigated whether liquid OEA+ snack modified various known anorexigenic and orexigenic factors such as GLP-1 or plasma glucose, insulin (anorexigenic factors) and active ghrelin (orexigenic). Obese minipigs exhibited higher glycemia, insulinemia and plasma β-hydroxybutyrate level in the fasting state than normal-weight minipigs but no difference for plasma active ghrelin, GLP-1 or NEFA was observed (Table 1).

In normal-weight minipigs, plasma GLP-1 levels did not vary after liquid snack consumption, irrespective of the presence of OEA (Fig. 5A). On the contrary, in obese minipigs, consumption of the liquid OEA+ snack resulted in a lower plasma GLP-1 level after 4 h compared to basal value (P = 0.049, Fig. 5B). Consumption of the liquid OEA-snack did not affect plasma GLP-1 levels in obese animals. No significant difference in GLP-1 levels between the two snacks was observed at any timepoint.

Insulinemia sharply increased 30 min after the liquid OEA-snack consumption in normal-weight minipigs (P = 0.002, Fig. 5C) but not after the liquid OEA+ snack consumption. Insulinemia had returned to normal value at 60 min. In obese animals, liquid snack consumption induced a greater and longer increase in plasma insulin than in normalweight minipigs, with a significant increase of insulinemia compared to basal level 30 and 60 min after liquid OEA+ snack consumption (P = 0.01 and 0.004, respectively, Fig. 5D).

In parallel to insulinemia, plasma glucose level dropped 30 min after both liquid OEA-and OEA+ snack consumption, (P = 0.003 and 0.005, respectively, Fig. 5E) in normal-weight minipigs. It then increased to levels higher than normoglycemia at 120 and 180 min only after the liquid OEA-snack ingestion (P = 0.04 and 0.01, respectively) while it had returned to a normal level as soon as 60 min after liquid OEA+ snack ingestion (Fig. 5E). Contrary to what happened in normal-weight minipigs, no variations in glycemia after snack consumption was observed in obese minipigs, irrespective of the presence or OEA or not (Fig. 5F).

In normal-weight minipigs, plasma active ghrelin level decreased after liquid snack consumption to reach a significantly different level than the basal one after 4 h (P = 0.001 for both snacks, Fig. 5G), with no difference between OEA-and OEA+ liquid snacks. In obese minipigs, no significant variation of the plasma ghrelin level was observed after liquid snack ingestion, irrespective of the presence of OEA (Fig. 5H). Plasma NEFA levels dropped after both liquid snack ingestion in normal-weight minipigs, irrespective of the presence of OEA (Fig. 5I) but did not vary with time in obese minipigs, irrespective of OEA presence in the liquid snack (Fig. 5J). Finally, plasma β-hydroxybutyrate levels did not vary with time in normal weight minipigs, irrespective of the snack (Fig. 5K). However, it increased in obese minipigs after ingestion of the OEA-liquid snack, reaching a plateau at 120 min while it did not change after OEA+ liquid snack ingestion, resulting in a significant difference in plasma β-hydroxybutyrate levels between the two snacks at 120 and 180 min (Fig. 5L).

Discussion

Our objective was to design a dairy-based OEA-containing functional food that, when consumed in the morning after an overnight fast, would reduce food intake at lunch. Our liquid OEA+ snack exhibited promising anorexigenic properties since it reduced the amount of feed consumed by both normal-weight and obese minipigs while the solid snack did not display such effects, a difference that could be accounted for by the higher EC release observed during in vitro digestion experiments with the liquid OEA+ snack compared to solid OEA+ snack. Yet, the mechanisms of effect of the OEA-liquid snack on feed intake seems to be dependent upon the weight status since in normal-weight mini-pigs, difference in 2-AG level could account for the anorexigenic effect of the OEA+ liquid snack, while in obese minipigs, an effect of the OEA+ liquid snack on ketogenesis could be at play.

Table 1

Fasted plasma endocannabinoid and related NAEs and anorexigenic and orexigenic factors in normal-weight and obese minipigs. To evaluate the effect of OEA-containing snacks, we chose to perform our study in adult minipigs since this animal species is close to humans in terms of digestive physiology, eating behavior and brain functions [START_REF] Roura | Critical review evaluating the pig as a model for human nutritional physiology[END_REF]. This species exhibits a functional EC system (synthesis and degrading enzymes, receptors) both at the gut [START_REF] Toschi | Thymol modulates the endocannabinoid system and gut chemosensing of weaning pigs[END_REF](Toschi et al., , 2021) ) and central [START_REF] Goparaju | Enzymes of porcine brain hydrolyzing 2-arachidonoylglycerol, an endogenous ligand of cannabinoid receptors[END_REF][START_REF] Pirone | Immunolocalization of cannabinoid receptor 1 (CB1), monoglyceride lipase (MGL) and fatty-acid amide hydrolase 1 (FAAH) in the pig claustrum[END_REF] levels. Yet OEA effect had never been tested in vivo in pigs before. We observed a progressive decrease of plasma levels of 2-AG, OEA and PEA after snack ingestion in normal-weight minipigs as described postprandially in normal-weight healthy humans [START_REF] Monteleone | Responses of peripheral endocannabinoids and endocannabinoid-related compounds to hedonic eating in obesity[END_REF]. Altogether, this makes the pig a suitable model to evaluate OEA effects upon eating behavior. Moreover, we evaluated the effect of OEA+ snacks in both normal-weight and obese minipigs. Several reports indicated that OEA metabolic and anorixegenic effects differed between lean and obese status in rodent models [START_REF] Bowen | Oleic acid-derived oleoylethanolamide: A nutritional science perspective[END_REF][START_REF] Brown | Oleoylethanolamide: A fat ally in the fight against obesity[END_REF], which justifies testing the OEA+ snacks in both weight status. Although we did not thoroughly characterize the possible differences in the EC system in these animals, obese minipigs displayed features of altered glucose and lipid metabolism (higher fasted glycemia, insulinemia and NEFA plasma levels) but also a tendency for increased AEA plasma levels as already described in obese patients [START_REF] Engeli | Activation of the peripheral endocannabinoid system in human obesity[END_REF]. Moreover, no decrease of EC plasma levels, except for AEA, was observed in our obese minipigs after the snack ingestion as opposed to normal-weight minipigs and humans ( [START_REF] Monteleone | Responses of peripheral endocannabinoids and endocannabinoid-related compounds to hedonic eating in obesity[END_REF], suggesting altered EC system in these obese minipigs.

OEA has been shown to impact eating behavior by acting both on the homeostatic and hedonic regulation of food intake as it interferes with oxytocin and histamide signaling at the hypothalamic level, dopamide and cannabinoid receptor 1 signaling in central reward centers but also GLP-1 and ghrelin pathways at the gut-brain level [START_REF] Brown | Oleoylethanolamide: A fat ally in the fight against obesity[END_REF]. We tested the effect of OEA+ solid and liquid snacks in a 1-hour ad libitum eating test in almost fasted animals (except the snack eaten 4 h before). Noteworthy, we used the pigs' usual diets (e.g. standard diet for the normal-weight minipigs and high-fat and high-sugar diet for obese animals) to avoid neophobia, which has already been documented in minipigs with this type of diets [START_REF] Coquery | Locomotion and eating behavior changes in Yucatan minipigs after unilateral radio-induced ablation of the caudate nucleus[END_REF]. This experimental set-up was first intended to evaluate the homeostatic eating parameters such as latency to eat (time between feed presentation and eating onset) that has been shown to be reduced by prior OEA intraperitoneal injection in food-deprived rats [START_REF] Gaetani | Modulation of meal pattern in the rat by the anorexic lipid mediator oleoylethanolamide[END_REF][START_REF] Azari | Vagal afferents are not necessary for the satiety effect of the gut lipid messenger oleoylethanolamide[END_REF], meal size (objectivizing satiation) that was also reduced in OEAtreated food-deprived rats [START_REF] Gaetani | Modulation of meal pattern in the rat by the anorexic lipid mediator oleoylethanolamide[END_REF][START_REF] Azari | Vagal afferents are not necessary for the satiety effect of the gut lipid messenger oleoylethanolamide[END_REF], inter-meal interval (objectivizing satiety) and speed of ingestion (amount of feed eaten by unit of time). However, since our minipigs were almost fasted, this eating behavior test was performed on highly feed motivated animals, which might have hidden some of the effects of OEA on homeostatic eating parameters. Nonetheless, we observed a significant effect of the OEA+ liquid snack on meal size in both normalweight and obese minipigs, suggesting an effect on satiation as already described in rodents [START_REF] Gaetani | Modulation of meal pattern in the rat by the anorexic lipid mediator oleoylethanolamide[END_REF][START_REF] Azari | Vagal afferents are not necessary for the satiety effect of the gut lipid messenger oleoylethanolamide[END_REF]. Latency to eat was not impacted by OEA+ snacks, which might be accounted for by the high eating motivation of our pigs. Lastly, our eating behavior test was not designed to properly evaluate food motivation. Indeed, food motivation is usually tested using a progressive ratio operant task that evaluates the rewarding impact of food by measuring the animal willingness to work (e.g. pushing a button repeatedly) to obtain food rewards [START_REF] Ferguson | Female mini-pig performance of temporal response differentiation, incremental repeated acquisition, and progressive ratio operant tasks[END_REF]. Thus, further work is needed to evaluate precisely the impact of OEA+ snacks on motivation for feed using such dedicated tests.

The release of EC from the liquid and semi-solid snacks during dynamic in vitro digestion experiments was drastically different. While the digestion of the semi-solid snack ended up with a limited release of OEA, LEA and PEA, the concentrations of these three molecules released after gastrointestinal digestion of the liquid snack were considerably higher. It must be noted that the presence of LEA and PEA in the in vitro digesta likely results from the fact that the OEA we used was not pure and contained very small amounts of these NAEs. At that stage, only hypotheses can be raised to explain the observed differences. A possible OEA degrading effect of the chymosin present in rennet during the semisolid snack production is unlikely. Indeed, the measured concentration of OEA in the semi-solid snack before digestion (382.7 ± 3.6 µg/mL) was very close to that in the liquid snack (427.3 ± 2.8 µg/mL) demonstrating that OEA has not been broken down by chymosin during semi-solid snack preparation. A low efficiency of the extraction protocol used for the quantification of EC in semi-solid snack digestion samples due to the structure of the food matrix itself can hardly be considered since it would have affected the quantification of OEA in the undigested semisolid snack as well. One can also hypothesize that when arriving in the small intestine, OEA delivered via the semi-solid snack is more prone to a fast hydrolysis by enzymes than with the liquid snack. Fatty acid amide hydrolase 1 and 2 (FAAH-1 and 2) and N-acylethanolamine-hydrolyzing acid amidase are abundant in the small intestine in vivo [START_REF] Bowen | Oleic acid-derived oleoylethanolamide: A nutritional science perspective[END_REF]. However, the proteome of pancreatin that was used in the present study to simulate the small intestinal digestion phase has been recently published [START_REF] Wang | In vitro gastric digestion and emptying of cooked white and brown rice using a dynamic human stomach system[END_REF] and none of these enzymes were identified. Another possibility is that OEA in liquid snack that is rapidly transferred from the stomach to the small intestine is subject to a limited oxidation by the oxygen present in the stomach cavity. In contrast, OEA in the semi-solid snack stayed longer in the stomach due to slower gastric emptying rate resulting in increased oxidation. Thus, further investigations are needed to understand the difference of EC release with snack structure.

We next sought to unravel the mechanism by which the liquid OEA+ snack reduced food intake in our minipigs. We first established plasma EC profiles during the 4 h following the liquid snack ingestion and preceding the meal tests. Despite the presence of 20 mg/kg of BW of OEA in the OEA+ snack, no increase in plasma OEA levels was noticed, irrespective of the minipig ponderal status. On the contrary, plasma OEA levels decreased after snack ingestion in normal-weight minipigs, as well as those of PEA, 2-AG and LEA, as already described post-prandially in humans [START_REF] Monteleone | Responses of peripheral endocannabinoids and endocannabinoid-related compounds to hedonic eating in obesity[END_REF]. Absence of increase in plasma OEA levels following OEA ingestion or stimulation of OEA endogenous synthesis has already been described, leading some authors to consider that exogenous OEA is poorly absorbed and that plasma OEA is a spillover from OEA-producing organs [START_REF] Brown | Oleoylethanolamide: A fat ally in the fight against obesity[END_REF]. Noteworthy, ECs, including OEA, are degraded by FAAH, an intra-cellular enzyme that is highly present in the intestinal epithelium [START_REF] Katayama | Distribution of anandamide amidohydrolase in rat tissues with special reference to small intestine[END_REF]. Thus, the absence of increase of OEA plasma levels might account for by a high rate of degradation in the intestine. We also established the plasma levels of OEA relative compound (2-AG, LEA PEA and AEA) after liquid snack ingestion. Interestingly, plasma 2-AG was lower 4 h after liquid OEA+ snack ingestion than after OEA-ingestion in normal-weight minipigs. 2-AG has opposite effect compared to OEA as it increases food intake due to its affinity for CB1 receptors [START_REF] Lau | Endocannabinoid modulation of homeostatic and non-homeostatic feeding circuits[END_REF]. Thus, the lower plasma 2-AG level in normal-weight minipigs 4 h after liquid OEA+ snack ingestion compared to the liquid OEA-snack (i.e. at the moment of the meal test) might have a role in the reduced food intake observed with this OEA+ snack in these animals, although this warrants further investigations.

We also speculated that other mechanisms might be involved than a direct effect of circulating ECs on the brain. Endogenous OEA has been shown to bind to GLP-1 receptors, potentiating GLP-1 effect [START_REF] Cheng | Modulation of glucagon-like peptide-1 (GLP-1) Potency by endocannabinoid-like lipids represents a novel mode of regulating GLP-1 receptor signaling[END_REF]. We therefore measured plasma GLP-1 levels after liquid OEA+ and OEA-snack ingestion, which did not vary in normal-weight minipigs and even decreased 4 h after the OEA+ snack ingestion in obese minipigs, probably because of the low energy and/or nutrient composition of this snack. To evaluate the potentiating effect of OEA upon GLP-1 receptor, we investigated the GLP-1 incretin effect which is mediated by GLP-1 receptor at the pancreatic level [START_REF] Reed | Recent advances in understanding the role of glucagon-like peptide 1[END_REF], hypothesizing that the OEA+ snack would increase plasma insulin levels and/or reduce plasma glucose levels more efficiently than the OEAsnack. However, variations in glycemia and insulinemia after snack ingestion were similar between OEA+ and OEA-snacks, ruling out the hypothesis of OEA-mediated enhancing of GLP-1 action at the central level. Noteworthy, the variations in glycemia and insulinemia were very small, probably due to the low glycemic index of milk that composed the snack and in agreement with a recent human study showing the postprandial hormonal response following milk consumption in healthy subjects [START_REF] Tagliamonte | Milk protein digestion and the gut microbiome influence gastrointestinal discomfort after cow milk consumption in healthy subjects[END_REF]. We also investigated the ghrelin pathway since OEA had been shown to reduce ghrelin levels in fasted rats [START_REF] Serrano | Oleoylethanolamide: Effects on hypothalamic transmitters and gut peptides regulating food intake[END_REF] and that OEA and ghrelin levels were correlated in a palatable food anticipatory and consummatory test in obese humans [START_REF] Rigamonti | Anticipatory and consummatory effects of (hedonic) chocolate intake are associated with increased circulating levels of the orexigenic peptide ghrelin and endocannabinoids in obese adults[END_REF]. Yet, no difference in plasma ghrelin levels was observed after liquid OEA+ or OEA-snack ingestion, also excluding this hypothesis.

Lastly, we explored the potential effect of OEA on lipid metabolism. Indeed, OEA is a high-affinity endogenous ligand of the nuclear receptor peroxisome proliferator-activated receptor alpha (PPAR-α) and as such stimulates lipolysis and fatty acid β-oxidation, both known to act as anorexigenic signals [START_REF] Langhans | Ketone kinetics and D-(-)-3hydroxybutyrate-induced inhibition of feeding in rats[END_REF]. Circulating NEFA and ketone bodies, enhanced during fasting to provide energy to energydemanding organs, usually decrease after eating, mainly due to the inhibitory action of insulin on lipolysis and the concomitant stimulation of lipogenesis, which, in turn, inhibits fatty acid β-oxidation. This decrease in plasma NEFA after snack ingestion was observed in normalweight animals but not obese ones, showing one again alteration of metabolism in obese minipigs. Yet, NEFA plasma level did not differ between OEA-and OEA+ liquid snacks in both normal weight and obese minipigs, ruling out an effect of OEA on lipolysis in our case. Interestingly, in obese animals, β-hydroxybutyrate, a marker of ketogenesis, increased after the OEA-liquid snack consumption but not the OEA+ one. In metabolically healthy animals, ketone bodies contribute to inhibition of eating [START_REF] Langhans | Ketone kinetics and D-(-)-3hydroxybutyrate-induced inhibition of feeding in rats[END_REF]. Thus, this lower plasma level of β-hydroxybutyrate after OEA+ liquid snack ingestion in obese minipigs and the reduced food intake in the eating behavior test are counterintuitive. Yet, the impact of ketone bodies on appetite regulation in obese individuals is poorly described. Noteworthy, fasting plasma β-hydroxybutyrate level was higher in obese than normal-weight minipigs, indicating altered ketogenesis in obese animals, as already observed in mice chronically fed a high-fat diet [START_REF] Sunny | Progressive adaptation of hepatic ketogenesis in mice fed a high-fat diet[END_REF]. Increased ketogenesis after the OEA-liquid snack ingestion also probably reflects aberrant metabolic responses in obese animals, that seems to be counteracting by the presence of OEA in the snack. Further investigations are therefore needed to evaluate if the OEA+ liquid snack effect on food intake in obese animals is mediated by an effect on ketogenesis.

This princeps preclinical trial in Yucatan minipigs demonstrated the ability of an OEA supplemented functional dairy liquid snack to decrease food intake during a meal test performed 4 h later in fasted and highlymotivated normal-weight and obese individuals. A semi-solid snack (i.e. curd) did not produce such effects on feed intake, nor the significant liberation of OEA, LEA and PEA observed with the liquid snack in in vitro digestors. We must acknowledge that the OEA powder we used in our study was not pure OEA and that, even if they were present in very small proportion (<2%), other NAEs could also have played a role in the observed anorexigenic effect of the liquid snack. Moreover, our eating behavior tests were performed in highly feed motivated animals that prevented us from properly evaluate satiation and satiety effects of our snacks. We also did not evaluate the long-lasting effects of the snacks on eating behavior. Finally, our mechanistic investigations point towards an effect of the OEA+ liquid snack on plasma 2-AG level in normalweight animals and on ketogenesis in obese animals to explain this anorexigenic effect. Yet, further experiments are needed to firmly conclude on the effect of our functional snack, which might modulate both homeostatic and hedonic brain circuits involved in the control of food intake and motivation.

Fig. 1 .

 1 Fig. 1. Overview of the animal experiment design.

Fig. 2 .

 2 Fig. 2. Impact of OEA-enriched snack on short-term eating behavior in normal-weight and obese minipigs. Overnight fasted normal-weight and obese minipigs were fed the OEA enriched (OEA+) or control (OEA-) snack in liquid or solid form. After 4 h, they were allowed access to feed. The amount of feed ingested (A) and duration (B) of the first meal were recorded. All pigs received the four snacks (liquid OEA-, liquid OEA+, solid OEA-, solid OEA+) in a random order with a minimum of 2 days between tests. Values are means ± SEM, * P < 0.05, n = 8 normal-weight and 8 obese minipigs.

Fig. 3 .

 3 Fig. 3. Concentrations of endocannabinoids and related N-acylethanolamines in the intestinal luminal compartment during snack in vitro digestion tests. OEA+ -liquid and solid snacks were submitted to in vitro gastro-duodenal digestions to measure endocannabinoid (A: oleoylethanolamide (OEA), B: linoleylethanolamide (LEA), C: palmitoylethanolamide (PEA)) concentrations in the intestinal luminal compartment.

Fig. 4 .

 4 Fig. 4. Plasma endocannabinoid and related acylethanolamides profiles following OEA-enriched or control liquid snack ingestion in normal-weight and obese minipigs. Plasma ECs and related NAEs (A-B: oleoylethanolamide (OEA), C-D: 2-arachidonoylglycerol (2-AG), E-F: arachidonoylethanolamine (AEA), G-H: linoleylethanolamide (LEA), I-J: palmitoylethanolamide (PEA)) levels were determined in normal-weight and obese minipigs after ingestion of the OEA-(open circles) or OEA+ (solid circles) liquid snacks. Values are means ± SEM, * P < 0.05 OEA-vs. OEA+, § P < 0.05 T0 vs. other timepoints for OEA-snack, & P < 0.05 T0 vs. other timepoints for the OEA+ snack. N = 8 normal-weight and 8 obese minipigs.

Fig. 5 .

 5 Fig. 5. Plasma anorexigen and orexigen factor profiles following OEA-enriched or control liquid snack ingestion in normal-weight and obese minipigs. Plasma GLP-1 (A-B), insuline (C-D), glucose (E-F), ghrelin (G-H), non-esterified fatty acid (NEFA) (I-J) and β-hydroxybutyrate (K-L) levels were determined in normalweight and obese minipigs after ingestion of the OEA-(open circles) or OEA+ (solid circles) liquid snacks. Values are means ± SEM, § P < 0.05 T0 vs. other timepoints for the OEA-snack, & P < 0.05 T0 vs. other timepoints for the OEA+ snack. N = 8 normal-weight and 8 obese minipigs.

Acknowledgment

Authors would like to thank Alain CHAUVIN, Mickaël GENISSEL, Julien GEORGES and Francis LE GOUEVEC for taking care of the animals and Sylvie GUERIN for her help during the whole experiment.

Funding

IM was funded by Region Bretagne (SAD2016, #16007892), Université Bretagne Loire) and AgreenSkill+ (grant application n • 1155).

Data availability

Data will be made available on request.

Author contribution GB, IM, DVL: designed and directed the project, carried out experiments, drafted the manuscript; PV, DD: discussed resulted, reviewed the manuscript; OM, RJ, IN, AM, AC, LLN: carried out the experiments and performed the analysis, reviewed the manuscript, EBC: designed and manufactured equipment (eating microstructure troughs), reviewed the manuscript.

Ethic statement

Animal experiments were conducted in accordance with the current ethical standards of the European Community (Directive 2010/63/EU), Agreement No. D3527532. The Rennes Ethics Committee in Animal Experiment and the French Ministry of Higher Education and Research have approved and authorized the entire procedure described in this paper (project number APAFIS #11984-2017103009069432_V2).

CRediT authorship contribution statement

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Daniel De Smet

Keywords:

Doxographies reporting the opinions and sayings of the most famous sages from Antiquity were a popular genre in medieval Arabic literature 1 . However, with the exception of the Placita philosophorum attributed to Plutarch or to Aetius, the Arabic doxographies are generally not translations of known Greek texts, but rather genuine compilations, based on ancient sources, readapting and transforming the borrowed material in a Neoplatonic and, above all, Islamic perspective. At first glance, one gets the impression that the (often anonymous) authors of these doxographies had only a superficial knowledge of ancient philosophy and ascribed at random their own doctrines to renowned authorities of the past. A closer study however, reveals that their interpretations are far from arbitrary as they join the general trend of

previous Christian and Neoplatonic authors to present the main philosophers of ancient Greece as true monotheists, believing in the one and unique God who is the ultimate cause of the production of the world2 . This is in particular the case with the Arabic Pythagoras, whose alleged doctrine of tawḥīd, the profession of God's oneness, is deeply rooted in pre-Islamic (neo)-Pythagorism. In the following pages, we will try to follow the subsequent steps of the islamisation of Pythagoras' theory of numbers in the Arabic doxographical tradition.

The Doxography of Pseudo-Ammonius and the Refutatio omnium haeresium

The Book of Ammonius concerning the opinions of the philosophers and their different doctrines about the principles and the Creator is an Arabic doxography whose purpose is to show that the elite among the ancient philosophers, although disagreeing about details, share a common belief in the oneness of God, who created the world ex nihilo 3 . Compared with his colleagues, philosophers as famous as Thales, Xenophanes, Empedocles, Democritus, Anaximenes, Anaxagoras, Heraclitus or Parmenides, Pythagoras holds a most prominent position: "the doctrine of Pythagoras, the sage from Samos, is not based on personal opinion (ra'y), but is the result of spiritual contemplation (iṭṭilā' rūḥānī) of the worlds and their contents" (Ps.-Ammonius, XIV, 1). As the Prophet Muhammad will experience much later, by his celestial ascent (mi'rāj), Pythagoras rose through the sky and travelled from the world of Nature to the world of Soul and then, upwards, to the world of Intellect, seeing with his own eyes the forms and the lights contained in these worlds (XVI, 5) 4 . Thus, he discovered that the universe has a harmonic structure, based on numbers and melodies (luḥūn), and that the different realms of reality (Nature, Soul, Intellect) are organized in a hierarchical order, increasing in perfection, beauty and brightness (XIV,(7)(8)(9)(10)(11)(12)(13)(14)XVI,(7)(8)(9)(10). But his ascension stopped at the level of the Intellect, which is however not the ultimate cause of the existence of the universe. He realized then that the path leading to God remains blocked forever.

In Italy (Īṭāliya), where he died, Pythagoras proclaimed that "the Creator (al-bāri') is one (wāḥid) and that He cannot be perceived neither by the Intellect nor by the Soul; as He transcends the highest spiritual attributes, His essence (huwiyya) remains imperceptible". In the realms of Intellect, Soul and Nature, God can only be perceived through the traces (āthār) He leaves in each of them (XIV,(2)(3)(4)(5)(6). In other words, God is hidden in His essence, but reveals Himself by His actions.

Pythagoras' doctrine of tawḥīd -the attestation of God's onenessis the result of his speculations about the numbers of the decade. The first number, one, is male, whereas two is female; subsequently, three is male and four female. These two pairs, coupled together [(1+2)+(3+4)], generate the number ten, the completion and achievement of all numbers.

Similarly, the numbers one to four correspond respectively to the elements of fire (male), air (female), water (male) and earth (female); mixed together, they generate the world of nature in its perfection, reflecting the perfect number ten. In his books, Pythagoras clearly explained that the first two numbers are created (mubda'a), their Creator being the One, "the ultimate first above model used by the Muslim Arabic compiler and the date of composition remain open questions (Rudolph 1989, 11-15). It is likely that the work, in its actual form, dates from the second half of the 9 th century; the first author who quotes from it was the Isma'ili Abū Ḥātim al-Rāzī (d. 934); see De Smet 2014. 4 According to the Ikhwān al-Ṣafā ' 1957, I, 137-138, such an experience is quite common, as Ptolemy, Hermes, Aristotle (in fact Plotinus, as the author is referring to the Arabic Theology based on the Enneads but transmitted in the Arabic tradition under the name of Aristotle), Pythagoras, Jesus and Muhammad were all granted with a celestial ascension. whom there is no first" (awwal al-awā'il wa lā awwal fawqahu) (XV,(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(20)(21)[START_REF]See Carmella Baffioni's contribution in this[END_REF](30). This means that God, as Ultimate reality, represents a unity transcending the number one.

A disciple of Pythagoras went to Persia and taught this subtle doctrine to the Mages, who misunderstood its meaning completely. A certain FLNJS, also called MRĪŪS 5 , identified the male principle, the monad, with light and the female dyad with obscurity; mixed together, they generated the universe. Another Mage, Zardusht (Zarathustra), claimed that the first two principles are two creating Gods, thus perverting Pythagoras' monotheism into dualism (XV,(1)(2)(3)(4)(5)19,(25)(26)[START_REF]Of course, in the Islamic tradition, Ḥarrān was very famous for its pagan "Sabeans" and their esoteric, Hermetic religion[END_REF](28)(31)(32)(33)(34)(35)(36)(37)(38). A second disciple of Pythagoras, an Italian named Kalanos, was more successful in India. He found in Brahman an intelligent and noble pupil, preaching the purification of soul and body. Unfortunately, after Brahman's death, the doctrine was corrupted and gave rise to two competing sects, preaching an anthropomorphic conception of God and an excessive form of asceticism, even promoting suicide (XVI,(1)(2)(3).

A main source for Ps.-Ammonius is the Refutation of all heresies, traditionally attributed to the bishop Hippolytus of Rome (first half of the 3 th century) 6 . According to the Christian author, Pythagoras, a Samian exiled in Italy, identified God with the monad (μόνας). This first monadthe number one, which has no limit and cannot be comprehended 7is not generated (ἀγέννητον) and acts as the principle (ἀρχή) of all numbers; being male, he generates them as a father. The second number, the dyad generated by the monad, is female and functions as the mother of all subsequent beings. The first four numbers produce the perfect number ten (Ref.,I,2,(1)(2)(6)(7)(8)VI,23,(1)(2)(3). The author of the Refutatio further explains that Pythagoras met Zaratas (Zarathustra) and learned from him the distinction between the monad and the dyad; he once mentions in his chapter on the Brahmans the gymnosophist Kalanos as a contemporary of Alexander the Great (Ref.,I,2,(12)(13)24,7). The meeting between Pythagoras and Zarathustra, and the figure of Kalanos as an Indian Pythagorean are common issues in ancient literature 8 . However, the idea that Zarathustra and Kalanos were direct disciples of Pythagoras may be an 5 These names appear in a corrupted form in the manuscript. The passage is quoted in al-Shahrastānī's Milal, where the names could be read as Philonikos and Marzinūsh (Marinus?); see Al-Shahrastānī, II, 82; Jolivet -Monnot 1993, 211. 6 New edition of the Greek text with English translation in Litwa 2016; for the question of authorship, see ibid., XXXII-XL. The parallels with Ps.-Ammonius are indicated in Rudolph 2016, 91-94, 98. 7 It seems here that the author of the Refutatio makes no distinction between the monad (μόνας) and the number one (ἕν); cf. Litwa 2016, 13, note 21. However, this distinction is clearly present in Ps.-Ammonius, when he has Pythagoras declare that the unity of God transcends the first of numbers. On this distinction, see below. 8 Rudolph 1989, 170-173, 176. invention of the Arabic Ps.-Ammonius, in order to show that Persian dualism and Indian anthropomorphism are perversions of Pythagoras' sound monotheism.

Aetius arabus

The account concerning Pythagoras in the Refutatio shares a number of elements with the Placita philosophorum attributed to Plutarch or Aetius9 . The Arabic translation of the Placita by Qusṭā b. Lūqā (d. ca. 912) is a further contribution to the islamisation of Pythagoras' thought.

Here we learn that the Samian, settled in Italy, was with Thales one of the founders of philosophy, and even the « inventor » of the term falsafa10 . He described the monad (al-wāḥida) as "the special efficient cause (al-'illa al-fā'ila al-khāṣṣa), which is God Most-High and the Intellect" (wa hiya Allāh 'azza wa jalla wa-l-'aql)11 . The dyad is a receptive principle, identified with matter ('unṣur), receiving the influences from the monad and producing the sensitive world (Aetius arabus, I, 3, 8)12 . As God and Intellect, the monad is Goodness (al-khayr), whereas the dyad, called « demon » (dhāmun), represents evil (al-sharr) and contains the plurality of the material world (Aetius Ar.,I,7,18). This dualism reminds us about the erroneous interpretation of Pythagoras' theory of numbers by the Mages according to Ps.-Ammonius. Moreover, Pythagoras is said to believe in the generation of the universe: "the world is generated and it is God Most-High who generated it" (al-'ālam mukawwan wa Allāh 'azza wa jalla kawwanahu).

Being corporeal, the world is subject to corruption, but thanks to God's governance (siyāsa) and protection (ḥifẓ) it is kept into existence (Aetius Ar., II, 4, 1-2).

Pythagoras, leader of the alchemists: the Turba philosophorum

The Refutation of all heresies also served as a source for the author of the Turba philosophorum, an alchemical treatise composed around the year 900, and thus roughly contemporary with Ps.-Ammonius. The work, transmitted in a Medieval Latin translation, opens with 9 sermones in which Presocratic philosophers are discussing, under the guidance of Pythagoras, about God and the creation of the universe. Rudolph has discovered that the lecture given by Pythagoras himself (sermo 8) has only a remote connection with Pythagorean doctrine, as it is mainly based on Refutatio X, 32-33, which gives a short summary of Christian theology according to the author's interpretation 13 . This means that the Turba presents us a Christian, monotheistic Pythagoras, emphasizing the creation of the world by God. Thus, Pythagoras declares before the assembly of philosophers and alchemists that God existed before everything, and that no being is coexistent with Him. He created (creavit) first the four elements (fire, air, water and earth), and then, He created from them all things, both in the higher and the lower worlds 14 .

A fragment of the Arabic original of the Turba, which is however lacking in the Latin version, gives an Islamic adaptation, including references to the Qur'ān, of Pythagoras' theory of numbers as exposed in Ref. I, 2, 6-8: "Pythagoras said: I say to you, oh sons of learning, that God is one and alone; He has not generated and is not generated, and nothing is with Him (Allāh wāḥid fard lam yalid wa lam yūlad wa lam yakun lahu shay'un) [cf. Qur. S.112]. When He created [the imperative] 'be' (kun) 15 , He created as first thing the one (al-wāḥid). This one is alone (fard) and masculine, being the beginning of calculation; the root (aṣl) of all things is one and nobody is able to attain knowledge about this one or about its root. Here, the erring nations are confused. But you, assembly of natural philosophers, you have to profess God's unity (tawḥīd Allāh) and to avoid doubts [about it]. Know that the root and beginning of calculation is the one, male and alone, and that all creatures proceed from this one. As for the two, it comes after the one and it is female.

Both [the one and the two] form a composed pair" 16 . 13 Rudolph 1990, 108-114, 121; on the Turba, see Rudolph 2005, 164-167. 14 Latin text with German translation in Plessner 1975, 72-73, 75. 15 According to the Qur'ān (for instance S. 2: 117), if God wants to create something, He says kun (the imperative of the verb kāna, "to be") and the thing comes into existence (fayakūnu, "and it is"). Hence, God's order (amr) is often considered in the Islamic tradition as the first created being. 16 The Arabic text was published, with a German translation, in Ruska 1931, 297, 300-301;cf. Rudolph 1990, 116. The Arabic Turba, following Ps.-Ammonius, makes a clear distinction between the uncreated unity of God and the created unity of the number one, which is the principle of all further numbers and beings in the universe.

The Oxford manuscript Marsh 539 and the Tehran doxography

A similar distinction is implied in a passage ascribed to Pythagoras in the anonymous doxography transmitted in the Oxford manuscript Marsh 539, which Elvira Wakelnig qualifies as a "philosophy reader" compiled in the circle of Miskawayh (d. ca 1030) 17 . Pythagoras mentions here four divine numbers (a'dād ilāhiyya), which are all good and devoid of evil; they are identified as the intellect, the soul, the [first] form, and "the beneficent power which brings all beings to perfection". They are opposed to the four numbers of nature (a'dād al-ṭabī'a), which are devoid of good; it are nature, matter (hayūlā), the second form, and time with its motions.

Their ultimate principle is "[The] One, which is self-subsistent (qā'im bi-dhātihi). It is the first goodness from which appear the four numbers; from their movements, [the One] generates (kawwana) the other four [numbers]. The first One is the head (ra's) of the eight numbers, their mover (muḥarrik), their organizer (mudabbir), their sustainer (muqawwim), their protector (ḥāfiẓ) and their creator (bāri')" 18 .

A second passage attributed to Pythagoras in the same collection confirms the distinction between the First (al-awwal), which is simple, one, pure, different from all things and devoid of every form of duality, and the intellect in this world, which is pure, separate, and acting as the agent (al-fā'il) for all that is contained in it 19 .

Wakelnig discovered recently in a Tehran manuscript a doxography entirely devoted to the sayings of the ancient sages concerning God's unity and uniqueness: The most precious words of the philosophers professing God's oneness and of the authorities of the past (Nawādir min 17 Wakelnig 2014, 1-7. 18 Arabic text and English translation in Wakelnig 2014, 66-67. 19 Wakelnig 2014, 78-79. kalām al-falāsifa al-muwaḥḥidīn wa-l-a'lām al-māḍiyyīn) 20 . The author claims that ancient sages such as Hermes, Pythagoras, Empedocles, Socrates and Plato have professed God's oneness (tawḥīd); they accepted Allāh as the cause ('illa) of the seen and the unseen, the creator of all things (mubdi' al-kull) and their organizer (mudabbir) 21 . In particular, Pythagoras developed an apophatic theology, as according to him God's essence cannot be perceived by the intellect, an idea we found already ascribed to Pythagoras in Ps.-Ammonius. His argument is that "God Most-High is the Creator and the Maker of the intellect (mubdi' al-'aql wa ṣāni'uhu); if the intellect would be able to obtain knowledge of God's essence, then the knower would be superior to the known object, which is impossible in the case of God. Hence, the intellect can only grasp the traces (āthār) of His wisdom in His creatures; God can only be known through His actions 22 .

Nevertheless, Pythagoras, by a sound reflection of his intellect on his own essence, came to the insight of the existence and oneness of God, his Creator. God's essence being out of reach for the intellect, he concluded that the only thing we can state about Him is al-huwiyya (the he-ness or ipseity), "as we say: 'he'" (ka-qawlinā huwa) 23 .

The prophetic origin of Pythagoras' monotheism according to al-'Āmirī and the Ikhwān al-

Ṣafā'

In his Book on the Afterlife (Kitāb al-Amad 'ala l-abad), al-'Āmirī (d. 992) used the Doxography of Ps.-Ammonius while presenting the teachings of the ancient sages. According to al-'Āmirī, Empedocles, Pythagoras, Socrates, Plato and Aristotle all agree on "the affirmation of the Creator, the unity of His essence and the negation of rivals and antagonists (andād wa aḍdād)

to Him" (Amad, III, 14) 24 . He also reveals how those sages acquired such a sound conception of monotheism, an explanation lacking in Ps.-Ammonius, at least in the version that came down to 20 On this work, its contents, its possible sources, along with the edition of the Arabic text and an English translation, see Wakelnig 2015, in particular 223-245. 21 Wakelnig 2015, 234-235. 22 Wakelnig 2015, 240-241. 23 Wakelnig 2015, 234-237. 24 Rowson 1988, 76-77. Al-'Āmirī attributes to Pythagoras a cosmology based on a hierarchy of twelve ranks: the four elements compose the sublunary world; the seven planetary spheres and the Throne enveloping them form the celestial world; at the top, there is a luminous world imperceptible to the intellects. The purified human souls can gradually ascend these ranks and finally contemplate what proceeds from the divine Wisdom (Amad, IV, 6-7; Rowson 1988, 80-83). This reminds Pythagoras' spiritual ascension described in Ps.-Ammonius. us. Empedocles was a companion of the sage Luqmān in Syria, a contemporary of the prophet Salomon mentioned in the Qur'ān (S. 31:12) (Amad, III, 1-2) 25 . Pythagoras, travelling in Egypt, learned geometry from the local priests, whereas the companions of Salomon, son of David, who lived there as exiles from Syria, initiated him in the sciences of nature and metaphysics (al-'ulūm al-ilāhiyya). He brought these three sciences -geometry, physics and "the science of religion" ('ilm al-dīn)to Greece, where he discovered by his own thinking the science of melodies and numbers. "He claimed that he acquired these sciences from the niche of prophecy" (mishkāt alnubuwwa) (Amad, III, 3)26 .

The puzzling affirmation of the Brethren of Purity (Ikhwān al-Ṣafā'), claiming that "Pythagoras was a monotheist sage (ḥakīm muwaḥḥid), one of the people of Ḥarrān in Syria" 27 , is probably not alien to the supposed prophetic origin of his philosophy, as Ḥarrān is mentioned in the Bible as a place where Abraham stayed 28 . And indeed, the text continues with the exposition of Pythagoras' theory of numbers, in particular his speculations about the number one in relation to God's oneness 29 . For our present purpose, it is important to note that the Ikhwān's Pythagoras associates the Creator with the number one, the Intellect with the number two, the Soul with the number three, and so on 30 . Apparently, he makes no distinction between an uncreated monad, corresponding to the Creator, and the created number one, corresponding to the first created being, the Intellect.

Al-Shahrastānī's synthesis

In the long chapter about Pythagoras in his famous Book of religions and sects (Kitāb almilal wa-l-niḥal), al-Shahrastānī (d. 1153) compiled the different accounts he could find in Arabic doxographical sources, in the first place Ps.-Ammonius, which he used extensively 31 . About the oneness of the Creator, he wrote: "The Creator Most-High is one, but not as [the number] one, as He does not fall under the numbers. He can be perceived neither by the intellect, nor by the soul, as intellectual thought is not able to grasp Him and the language of the soul is unable to describe Him, being elevated above the spiritual attributes. He is imperceptible from His essence, but can only be grasped through His traces, operations and actions. Hence, each of the worlds perceives Him by way of the traces left in it by His operation, so that He is qualified and described according to the operation that is specific for Him [...]" 32 .

Thus far, al-Shahrastānī followed Ps.-Ammonius' elaborations about God's incomprehensiveness and unity, transcending the first of the numbers. Next comes a long reflection about different kinds of unity, also attributed to Pythagoras:

"He [Pythagoras] further said: unity (al-waḥda) can be divided into unity which is not acquired from something elseit is the unity of the Creator Most-High, a unity encompassing everything, a unity governing everything, a unity from which proceed the unities and the plurality in the existentsand unity received from something else, which is the unity of the creatures.

Sometimes he said: unity in general can be divided into unity prior to eternity (dahr), unity simultaneous with eternity, unity posterior to eternity but prior to time, and unity simultaneous with time. The unity prior to eternity is the unity of the Creator Most-High; the unity simultaneous with eternity is the eternity of the first Intellect; the unity posterior to eternity and prior to time is the unity of the Soul; the unity simultaneous with time is the unity of the elements (al-'anāṣir) and the composed beings. Sometimes he had still another division of unity, saying: unity can be divided into unity by essence and unity by accident. Unity by essence only belongs to the Originator (mubdi') of the universe; from Him the unities proceed, which are in the numbers and in the numerable things. Unity by accident can be divided into what is the principle of the number without being comprised in the number, and what is the principle of the number, 31 Al-Shahrastānī, II, 74-83. For a survey of the sources, see the notes to the French translation of Jolivet -Monnot 1993, 201-212; for the use of Ps.-Ammonius in particular, see Rudolph 1989, 24, 29. 32 Al-Shahrastānī, II, 74; cf. Jolivet -Monnot 1993, 201. being comprised in it. The first [member of this division] is like the unity of the Agent Intellect (al-'aql al-fa''āl), as it is neither comprised in the number, nor in the numerable thing. The second [member of the division] can in turn be divided into what is comprised in [the number], as a part of itfor instance, two is composed of two ones; similarly, every number is necessarily composed of ones; as far as the number increases, its relation with the unity decreases -, and into what is comprised in it, as something that is inherent in it, and not simply as a part of it. This means that every number or numerable thing always has a unity, which is inherent in it; for instance, two and three, in so far as they are two and three, are one. In the same way, the numerable things, be they composed or simple, are one, either by genus or species or individually; for instance substance, in so far as it is substance in general, man in so far as he is man, and a given individual, such as Zayd, in so far as he is this specific individual, [all these are] one. Thus, unity is never dissociated from the existents. This unity is acquired from the unity of the Creator Most-High, being inherent in all beings, even if they are multiple in their essences.

The nobility of every existent depends on the predominance of unity in it; all what is more removed from plurality is nobler and more perfect"33 .

Starting from Ps.-Ammonius' distinction between the unity of the Creator and the unity of the numbers, al-Shahrastānī's Pythagoras develops a complex theory about unity, which can be summarized as follows. Unity can be divided in three ways, which are partly overlapping. ( 1 In other terms, the unity of the Creator, which is the ultimate principle of all unity in the universe, remains hidden and inaccessible; it is dissociated from the unity of His creatures by the intermediate level of the Intellect, the first created being whose unity is the direct cause of all numbers in the corporeal world, beginning with the number one. However, further on in the chapter, Pythagoras associates the Intellect with the number two, the Soul with three and Nature with four, implying that the Creator corresponds to the first of numbers35 . This inconsistency with what precedes is the result of al-Shahrastānī's compilation of different conflicting sources.

The transcendent One and the first of numbers

In later Antiquity, a certain consensus arose, both in pagan and in Christian sources, about the fact that Pythagoras' theory of numbers has to be understood in a monotheistic and even creationist way. As the principle of numbers, the one or unity produces all subsequent numbers; numbers being the principles of what exists in the universe, all existents ultimately proceed from the first principle, which is one and unique. Here, two conflicting opinions seem to have coexisted: (1) the ultimate unity and first principle is the first number, the number one; (2) the ultimate unity, as principle or creator of the first number, transcends numerical unity. Both positions are attested in the Arabic Pythagoras.

The first position is common in Christian apologetic literature, trying to convince the pagan reader that his philosophy is only a pale imitation of the revealed truth and, hence, that it is better to convert to Christianity. Cyril of Alexandria (d. 444), for instance, claimed that Pythagoras openly professed the unity of God, the ultimate principle and mover of the universe, and thus that he is far removed from the rough polytheism of his adversary, the Emperor Julian (Contra Julianum,I,42). This passage was adapted in Syriac in the 6 th century by an anonymous Christian author aiming at converting the pagan community in Ḥarrān 36 . Similarly, we already have met with the idea that the one and unique God, creator of the universe, corresponds to the number one, according to the account about Pythagoras in the Refutatio omnium haeresium, whose aim it is to show that Christian heretics are only Christian in appearance, as they in fact rely on pagan philosophers. The Refutatio is a main source for the Arabic Pythagoras, in particular Ps.-Ammonius, but here, as we have seen, there is a shift from the first to the second position (the One transcending the number one).

However, the first position is also ascribed to Pythagoras in pagan sources. We found it in the Arabic Placita philosophorum, but it is also attested, for instance, in Diogenes Laertius (VIII, 25) 37 . By far the most remarkable text in Arabic reflecting this position is a commentary on Pythagoras' Golden Verses, attributed to Proclus and supposedly translated by Ibn al-Ṭayyib (11 th century). This translation is quite unique, as it makes no attempt to eliminate the references to pagan religion and polytheism. Here we learn that, according to Pythagoras, "the first of numbers refers to God Most-High (awwal al-'adad ishāra ilā Allāh ta'ālā), because He possesses pure unity (al-waḥdāniyya al-khāliṣa)", whereas the subsequent numbers of the decade, composed by the addition of one, refer to the secondary divinities (al-āliha al-thawānī), as the unity is only reserved to Zeus (Zā'ūs). In fact there are four types of numbers: divine (the units), intellectual (dyads), psychic (triads) and natural (tetrads), Zeus being the one unique father (Zā'ūs al-ab almutawaḥḥid) of all 38 .

Similarly, we have seen that, according to the Ikhwān al-Ṣafā', Pythagoras identified God with the first of all numbers. However, this position appears rather scarcely in the Arabic Pythagoras file. Most of the sources we have examined make a clear distinction between the unity of God and the unity of the number one: Ps.-Ammonius, the Turba, the Marsh 539, the Tehran doxography, al-'Āmirī and al-Shahrastānī, all depending in some way or another on Ps.-Ammonius.

And indeed, Ps.-Ammonius seems to be the great promoter of this doctrine, which he explicitly attributes to Anaximenes, who would have stated: "[The Creator] is the One, which is not the one of the numbers, because the one of the numbers can be multiplied, whereas He cannot be multiplied" (Ps.-Ammonius, XI, 6) 39 .

Clearly under the influence of Ps.-Ammonius, al-'Āmirī ascribes a similar teaching to Empedocles: 37 Here it is stated that, according to Pythagoras, the monad is the principle of all things, which generates the dyad, and so on. 38 Linley 1984, 78-79; 92-93; on this text, see O'Meara 1989, 231-232. 39 Rudolph 1989, 45, 86. "[The Creator's] unity is not like the unity of any of the existents in the world, as the worldly existents are subjected to multiplication, either in their parts or in their qualities or in their equivalents, whereas His essence is exalted above this" (Amad, IV, 2) 40 .

At the same time, he acknowledges that Pythagoras agreed with Empedocles on the matter of the Creator's attributes (Amad, IV, 5) 41 .

In fact, the thesis about the superiority of the Creator's unity is widespread in the Arabic Neoplatonic tradition, where it often appears without any direct connection to Pythagoras.

Following Plotinus' claim that the One, the first principle of the emanation of the universe, is distinct from the numerical one (for instance Enneads, V, 5, 4; V, 6, 4; VI, 9, 5-6), the Plotiniana arabica develop this position, identifying the One with the Creator of monotheistic religions. In the so-called Theology of Aristotle, containing the Arabic version of selected passages from the Enneads, we read that "the true, pure and simple One, which encompasses all things, be they simple or composed, which is prior to all plurality, and is the cause of the existence (āniyya) of things and their plurality, is what produces the numbers (fā'il al-'adad)". Contrary to a widespread idea, the author explains, the number one is not the principle of all things 42 . This is virtually impossible, because "The First Creator (al-bāri' al-awwal) is only one (wāḥid faqaṭ) in all respects, and His essence is an originating essence (dhāt mubdi'a) [...]. He originated (abda'a) the world as one; it is necessary that the oneness of the originated (waḥdāniyyat al-mubda') should not be like the oneness of the originator (waḥdāniyyat al-mubdi'), otherwise the originator and the originated, the cause and the effect, would be one thing" 43 . This distinction between the uncreated creative unity of God and the created unity of the first number, which we have found attributed to the Arabic Pythagoras, also occurs in the 40 Rowson 1988, 78-79. 41 Rowson 1988, 80-81; cf. De Smet 1998, 63. 42 Badawi 1977b, 112; Lewis 1959, VIII, § 130, p. 271. 43 Badawi 1977b, 148; Lewis 1959, X, § 93, p. 455. different Pythagorean fractions. Some of the arguments also appear in our Arabic sources, for instance that the transcendent monad is not subject to multiplication, being inalterably one, whereas the numerical one can be multiplied and divided, a position attributed by Ps.-Ammonius to Anaximenes (VI, 1) and by al-'Āmirī to Empedocles 47 . The same lack of consensus among the Pythagoreans is manifest in the chapter "On the Monad" of the Theology of Arithmetics attributed to Iamblichus 48 . It seems thus that the different versions of the doctrine of attributed to the Arabic Pythagoras rely on this kind of sources, sharing authentic but conflicting Pythagorean elements, often integrated into an overall Neoplatonic background.

The creation of a Shi'i Pythagoras

The philosophy of unity, supposedly taught by the Arabic Pythagoras, was not an exotic curiosum fancied by the authors of doxographies for a learned public fond of "alternative" forms of thought. It really influenced Islamic philosophers in their endeavor to give a philosophical legitimation to Qur'ānic monotheism. This happened mainly in the Shi'i tradition, as is clear from the Ikhwān al-Ṣafā', who explicitly claim the heritage of Pythagoras. In the same period, the Isma'ili theologian Ḥamīd al-Dīn al-Kirmānī (d. ca. 1021), without mentioning the name of Pythagoras, elaborates in his Brilliant Epistle on the Meaning of Unification, the Unifier and the Unified (al-Risāla al-Durriyya fī ma'nā al-tawḥīd wa-l-muwaḥḥid wa-l-muwaḥḥad) a complex reflection about the concept of "one", which is highly reminiscent of what we found in the Arabic sources about the philosopher from Samos. According to al-Kirmānī, God cannot be associated with the number one, as this latter is always generated (muḥdath). He then adopts a fourfold division of the term "one", which is very close to the division later ascribed to Pythagoras by al-Shahrastānī: the one in relation to the individual, the one in relation to the species, the one in relation to the genus and, finally, the "absolute one" (al-wāḥid al-muṭlaq). The latter is the numerical one, the first of numbers, whose essence is twofold, being composed of unity and what bears this unity. In its four forms, the one is generated as the result of a preceding cause. The most perfect form of unity is realized in the first Intellect, the first originated being. As God is the Originator of the perfect unity of the Intellect, He transcends unity and oneness: God is neither 47 This interesting passage from Theon of Smyrna's Expositio rerum mathematicarum ad legendum Platonem utilium is edited, translated and commented in Kalvesmaki 2013, 175-180. 48 Waterfield 1988, 36-39.

)

  Unity not acquired from outside = the unity of the Creator; unity acquired from outside = the unity of His creatures. (2) Unity prior to eternity = the unity of the Creator; unity in eternity = the unity of the Intellect; unity between eternity and time = the unity of the Soul; unity in time = the unity of the material beings 34 . (3) Unity by essence = the unity of the Creator; unity by accident (a) as the principle of number exterior to number and numerable realities = the unity of the Intellect; (b) as the principle of number inside number and numerable realities: (α) the unity as part of every number (2=1+1); (β) the unity that unifies the essence of every genus, species or individual.

For a general overview, see[START_REF] Daiber | Doxographie und Geschichtsschreibung über griechische Philosophen in islamischer Zeit[END_REF][START_REF] Daiber | Hellenistisch-kaiserzeitliche Doxographie und philosophischer Synkretismus in islamischer Zeit[END_REF]

On this general trend, seeWakelnig 2015, 205-214.

Transmitted in a single manuscript (Istanbul, Aya Sofya 2450), the text was edited, with an annotated German translation, by[START_REF] Rudolph | Die Doxographie des Pseudo-Ammonios[END_REF]. The identity of its author "Ammonius", the existence of a (Christian) Greek or Syriac

Mansfeld 1992, 22-25 enumerates the similarities and concludes that both texts depend on a common source.

Same idea in Iamblichus' Vita Pythagorica, § 58.

By contrast, we have seen that in Ps.-Ammonius Allāh and the Intellect are clearly distinguished, as are the monad and the number one, the first of numbers.

Arabic text and German translation in[START_REF] Daiber | Aetius Arabus[END_REF]

Rowson 1988, 70-71; cf. De Smet 1998, 38-45.

Rowson 1988, 70-71. Travels in Egypt are a common place in ancient biographies of Pythagoras; Christian

Al-Shahrastānī, II, 74-75; cf. Jolivet -Monnot 1993, 202.

The distinction of these four levels (above eternity, eternity, between eternity and time, and time) and their association with respectively the First Cause, the Intellect, the Soul and the corporeal world derives from the Arabic paraphrase of Proclus' Elementatio Theologica; seeBadawi 1977, 4-5.

Al-Shahrastānī, II, 77; cf. Jolivet -Monnot 1993, 204-205.

Brock 1983, 229-230.

Treatise on divine science (Risāla fil-'ilm al-ilāhī), still another Arabic Plotinus text. The ultimate first Principle cannot be described, as it is devoid of any form of multiplicity: "If the First is not simple (mabsūṭ) and truly one, and outside any description and composition, it is not first at all [...]. The pure, true One is not one in respect of anything, but by itself alone.

Apart from the First, there is nothing purely simple" 44 .

Hence, the One remains outside all numbers: "Although we mention Him and count Him with the number of all things, yet He is above all number and is outside all categories of number. For substantial number (al-'adad al-jawharī) is not applicable to Him, nor is quantitative number (al-'adad al-kammī). Substantial number is that which applies to the essence of a thing and quantitative number is what applies to the amount of the essence of a thing: how much is it?" 45 .

Neoplatonism versus Pythagoreanism?

It is clear from these samples that the Arabic Pythagoras, as depicted by Ps.-Ammonius and al-Shahrastānī, takes the form of a Neoplatonic philosopher. Could we conclude then that the Arabic authors assigned almost at random Neoplatonic doctrines, which they found in sources such as the Theology of Aristotle, to ancient sages? In the case of Pythagoras, this is quite unlikely, as the debate about the relationship between the divine unity and the number one is attested in the Pythagorean School long before Plotinus. Already in the Corpus Hermeticum, Hermes advised his son in the following way: "Oh my son Tat, pray first the Lord, the Father, the Unique, who is not the one but source of the one" (Corp. Herm., V,2). Festugière compiled from the works of Philo (d. ca. 50 CE) conflicting Pythagorean passages, sometimes opposing, sometimes identifying the transcendent monad and the first of numbers 46 . In a long doxographical part of his Mathematics useful for reading Plato, Theon of Smyrna (2 th century) enumerates no less than six different opinions about this question, showing the lack of consensus among 44 Badawi 1977b, 178; Lewis 1959, § 162-165, p. 333. 45 Badawi 1977b, 180; Lewis 1959, § 182-183, p. 345. 46 Festugière 1954, 19-25. aḥad, nor wāḥid, but fard ("single", "unique"), His uniqueness being unknowable even to the Intellect 49 .

In the second half of the 17 th century, Quṭb al-Dīn Ashkevarī presented in his Beloved of the Hearts (Maḥbūb al-qulūb) an ultimate synthesis, piling together most that had been written in Arabic during previous centuries about the sage Pythagoras. He introduced into his chapter long quotations from the famous Sunni mystic Ibn al-'Arabī, along with traditions attributed to the first Shi'i Imams, thus showing how the Samian philosopher was perfectly integrated into the complex network of Iranian thought in the Safavid period. Here, at the eve of modern times, Pythagoras was still a guide to a true and profound understanding of the fundamental tenet of Islam: the unity and oneness of God 50 . 49 See the English translation of this treatise in Hunzai 2008, 200-207; cf. De Smet 1995, 85-86. 50 Terrier 2016, 373-420.


Schéma semi-implicite Discrétisation en temps (Schéma Euler semi-implicite) :

1 Initialisation : U 0 donnée, P 0 approché.

2 Étape de prédiction : Calculer U * la solution de

Nb : On peut avoir BU * ̸ = 0 3 Calcul de la pression : Calculer P ′ la solution de BM -1 B t P ′ = δt -1 BU * , P m+1 = P ′ + P m 4 Etape de correction : Calculer U m+1 la solution de MU m+1 = MU * -δtB t P ′ Dans TrioCFD, ces étapes sont rapides car M est diagonale pour les éléments finis de Crouzeix-Raviart.

En 3D, pour un schéma aussi robuste qu'en 2D, il faut rajouter des ddls en pression, ce qui augmente la taille du système de solveur de pression. Commissariat à l'énergie atomique et aux énergies alternatives A.PEITAVY short-event 2 / 12

Problème de Stokes

Navier-Stokes isotherme et incompressible

ρ(ut + (u • ∇ )u) -µ∆ u + ∇ p = f dans Ω × R + div u = 0 dans Ω × R + Objectif :
Améliorer la précision du code de résolution thermohydraulique de la DES à un nombre de degré de liberté fixé. Soit un maillage triangulaire T h composé de faces F ∈ F h .

Système semi-discrétisé :

Problème variationnel discret pour le problème de Stokes

Trouver (u h , p h ) ∈ X h,0 × M h tel que a(u h , v h ) + b(v h , p h ) = (f , v h ) 0,Ω ∀v h ∈ X h,0 b(u h , q h ) = 0 ∀q h ∈ M h avec      a h : X h × X h → R a h (u h , v h ) = ν T ∈T h T ∇u h : ∇v h      b : X h × M h → R b(v h , q h ) = - T ∈T h T div (v h )q h
Commissariat à l'énergie atomique et aux énergies alternatives A.PEITAVY short-event 3 / 12

Les éléments finis de Crouzeix-Raviart non conforme

On définit l'espace des vitesses :

X h,0 := {v h ∈ L 2 (Ω), (v h ) |T ∈ P 1 (T ) ∀T ∈ T h et F [v h ] = 0 ∀F ∈ F h }
Plusieurs choix pour la pression : TrioCFD :

Crouzeix-Raviart non-conforme ▶ M h := {q h ∈ L 2 0 (Ω) q h ∈ P 0 (T ), ∀T ∈ T h } ▶ Nb : d composantes
▶ M h := {q h ∈ Q 0,h + Q 1,h } Q 0,h := L 2 0 (Ω) ∩ P 0 (T h ), Q 1,h := L 2 0 (Ω) ∩ P 1 LG (T h )
▶ Encore des modes parasites pour d = 3 , (P 1 + P 0 + P a )

           ν T ∈T h T ∇u h : ∇v h - T ∈T h T div (v h )p h = (∇ ϕ, v h ) 0,Ω ∀v h ∈ X h,0 - T ∈T h T div (u h )q h = 0 ∀q h ∈ M h
La solution est alors (u, p) = (0, ϕ). Or pour les éléments finis de Crouzeix-Raviart, on a

T ∈T h T ∇ ϕ • v h = - T ∈T h T ϕ div v h + ∂T ϕ v h • n = - T ∈T h T ϕ div v h + F ∈F h F ϕ [v h • n] Le terme F F ϕ [v h • n] est nul si ϕ est constante ("Patch test").
Autrement, ce terme devient une "source numérique" pour ν∆u h .

→ Erreur en vitesse qui dépend de la viscosité.

Néanmoins la convergence est garantie. ▶ Exprimer le gradient de la pression à l'aide des inconnues auxiliaires de pression sur les arêtes et celle de la cellule.

p T p S T 1 a 1 p S T 1 a 2 S T 1 S T 2 S T 3 n T 3 n T 2 n T 1 S p T p S 2 p S 1 n 2 n 1 n 4 n 3 Q S T F 1 T F 2 T F 3 T F 4 T G S T = 1 |Q S T | (p S 1 -p T ) • n 1 + (p S 2 -p T ) • n 2 (1)
La formule est exacte sur des simplexes pour des fonctions affines.

▶ Éliminer les inconnues auxiliaires en imposant la continuité du flux du gradient sur les arêtes.

▶ Éléments finis X h pour les composantes de la vitesse (seul la matrice de couplage vitesse/pression est modifiée).

▶ On garde les degrés du bord car on ne peut pas y imposer la continuité des flux.

▶ On applique ce schéma pour discrétiser le gradient de pression. On résout donc le problème de Stokes avec :

b h (v h , q h ) = T ∈T h Q∈Q(T ) Q v h • ∇ h q h n T 4 d n T 4 c p 4 p 1 p 2 p 3 p d pa p b pc S Q S T 3 Q S T 4 Q S T 1 Q S T 2 b a d c
La continuité des flux entre deux triangles T 2 et T 3 , partageant l'arête b, et le sommet S s'écrit

∇ S T 3 p • n b = ∇ S T 2 p • n b
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Système MPFA

Intégration par parties

On n'a pas l'égalité suivante : ▶ Erreur exacte pour une vitesse affine et une pression quadratique pour TrioCFD en 2D, mais plus de ddls (en 3D cela s'accentue).

T ∈T h T p T div v h ̸ = - T ∈T h T ∇ h p T • v h =⇒ B ne discrétise pas la divergence Système considéré   A G int G b D 0 0 G T b 0 0     U P int P b   =   F 0 G T b Ue   Avec : ▶ 0 ≤ i, j ≤ dN F ▶ 0 ≤ k ≤ N T ▶ 0 ≤ l ≤ 2N F b (A) ij := T ∈T h T ∇ ϕ i : ∇ ϕ j (G int ) ki : T ∈T h T ∇ h ψ int k • ϕ i (G b ) li : T ∈T h T ∇ h ψ b k • ϕ i (F ) i := T ∈T h T f • ϕ i (D) ik := T ∈T h T ψ int k div ϕ i ▶ La
▶ Plus de robustesse par rapport à la viscosité cinématique.

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Résultats pour l'équation de Stokes sur Ω = [0; 1] 2 ux = 0, uy = 0 p = sin(2πx)sin( 2πy)

h ∥u -u CR h ∥ 0 ∥u -u Trio h ∥ 0 ∥u -u Mps h ∥ 0 1.00 × 10 -1
3.01 × 10 -3 1.04 × 10 -5 1.44 × 10 -4 5.00 × 10 -2 6.93 × 10 -4 6.52 × 10 -7 1.75 × 10 -5 2.50 × 10 -2

1.63 × 10 -4 4.02 × 10 -8 2.09 × 10 -6 1.25 × 10 -2 4.14 × 10 -5 2.54 × 10 -9 3.91

× 10 -7 τ τ CR u = 2.06 τ Trio u = 4.00 τ MPs u = 2.84 h ∥p -p CR h ∥ 0 ∥p -p Trio h ∥ 0 ∥p -p Mps h ∥ 0 1.00 × 10 -1
1.88 × 10 -1 2.48 × 10 -2 1.99 × 10 -2 5.00 × 10 -2 9.06 × 10 -2 5.49 × 10 -3 4.31 × 10 -3 2.50 × 10 -2 4.40 × 10 -2 1.37

× 10 -3 1.09 × 10 -3 1.25 × 10 -2 2.22 × 10 -2 3.50 × 10 -4 2.78 × 10 -4 τ τ CR p = 1.03 τ Trio p = 2.05 τ MPs p = 2.05
ux = sin(2πy )(cos(2πx) -1), uy = -sin(2πx)(cos(2πy ) -1), p = sin(2πx)sin( 2πy)

h ∥u -u CR h ∥ 0 ∥u -u Trio h ∥ 0 ∥u -u Mps h ∥ 0 1.00 × 10 -1
2.69 × 10 -2 2.45 × 10 -2 2.61 × 10 -2 5.00 × 10 -2 6.92 × 10 -3 5.84 × 10 -3 6.77

× 10 -3 2.50 × 10 -2 1.70 × 10 -3 1.45 × 10 -3 1.66 × 10 -3 1.25 × 10 -2 4.28 × 10 -4 3.65 × 10 -4 4.22 × 10 -4 τ τ CR u = 1.99 τ Trio u = 2.02 τ MPs u = 1.98 h ∥p -p CR h ∥ 0 ∥p -p Trio h ∥ 0 ∥p -p Mps h ∥ 0 1.00 × 10 -1
8.35 × 10 -1 1.14 × 10 -0 1.47 × 10 -0 5.00 × 10 -2 4.06 × 10 -1 6.08 × 10 -1 5.75 × 10 -1 2.50 × 10 -2 1.88 × 10 -1 2.89 × 10 -1 2.28 × 10 -1 1.25 × 10 -2 9.50 × 10 -2 1.45 × 10 -1 1.13 × 10 -1 τ τ CR 9.99 × 10 -1 1.40 × 10 -0 2.59 × 10 -0 10 -1 1.25 × 10 -1 1.40 × 10 -1 2.59 × 10 -1 10 -2 7.66 × 10 -2 1.40 × 10 -2 2.59 × 10 -2 10 -3 7.59 × 10 -2 1.40 × 10 -3 2.59 × 10 -3 ux = sin(2πy )(cos(2πx) -1), uy = -sin(2πx)(cos(2πy ) -1), p = x 5 + x 4 y 3 + x 2 y + y 4 -7/12

p = x + y -1 ν[m 2 s -1 ] ∥u -u CR h ∥ 0 ∥u -u Trio h ∥ 0 ∥u -u Mps h ∥ 0 10 -0 2.
ν[m 2 s -1 ] ∥u -u CR h ∥ 0 ∥u -u Trio h ∥ 0 ∥u -u Mps h ∥ 0 10 -0 2.69 × 10 -2 2.45 × 10 -2 2.61 × 10 -2 10 -1 3.35 × 10 -2 2.45 × 10 -2 2.61 × 10 -2 10 -2 1.98 × 10 -1 2.45 × 10 -2 2.63 × 10 -2 10 -3 1.96 × 10 -0 2.44 × 10 -2 3.48 × 10 -2 ν[m 2 s -1 ] ∥p -p CR h ∥ 0 ∥p -p Trio h ∥ 0 ∥p -p Mps h ∥ 0 10 -0
6.53 × 10 -1 8.97 × 10 -1 2.59 × 10 -0 10 -1 1.45 × 10 -1 8.99 × 10 -2 2.59 × 10 -1 10 -2 1.30 × 10 -1 1.08 × 10 -2 2.70 × 10 -2 10 -3 1.30 × 10 -0 6.13 × 10 -3 7.86 × 10 -3 ▶ L'erreur en vitesse ne dépend plus de la viscosité. Les modes parasites en pression sont nuls.

▶ En 3D TrioCFD : P 1 + P 0 + P a ▶ Cas générale : on est plus robuste par rapport à la viscosité.

  Figure 1 -Solution cavité entraînée avec TrioCFD Commissariat à l'énergie atomique et aux énergies alternatives A.PEITAVY short-event 1 / 12

  Figure 2 -DDLs Crouzeix-Raviart non-conforme, d = 2

Figure 3 -

 3 Figure 3 -DDLs TrioCFD, d = 2 Commissariat à l'énergie atomique et aux énergies alternatives A.PEITAVY short-event 4 / 12

u▶

  = 0, p = x 5 + x 4 y 3 + x 2 y + y 4 -Découper nos triangles en trois quadrangles ▶ Ajouter deux inconnues auxiliaires de pression par arête.

▶

  base pour la vitesse est donnée par(ϕ α F ) α∈{x,y } F ∈F f et pour la pression (ψ int T ) T ∈T h et (ψ b l ) 0≤l≤dN F b▶ La vitesse est imposée sur l'épaisseur du bord avec l'équation G Erreur exacte pour une vitesse affine et une pression affine pour TrioCFD et Mps mais pas pour CR.

▶

  En 3D TrioCFD : P 1 + P 0 + P a ▶ τu ≃ 2 et τp ≃ 1 -> Ordre de convergence attendu. ▶ ∥p -p Trio h ∥ 0 ≳ ∥p -p Mps h ux = sin(2πy )(cos(2πx) -1), uy = -sin(2πx)(cos(2πy ) -1),

▶▶Figure 5 -

 5 Figure 4 -MPFA en 3D

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