Maria Concepcion Villanueva

Tito De Luis

Jean-Yves Morais

Jacques Weigel

Moreau

Jacques Moreau Weigel

An

Senegal ) Chavance

Ba Mamadou

Gascuel Didier

; Vakily

Jan Michael

Pauly Daniel

Luis Tito-Oe-Morais

Jean-Yves Weigel

Jacques Moreau

L Trro-De-Morais

J.-Y Weigel

Imprimé Sur

Papier Blanchi

Sans Chlore

Pierre Chavance

Moctar 8â

Didier Gascuel

Daniel Pauly

An Ecopath model of the Sine-Saloum Delta Biosphere Reserve (Senegal)

Keywords:

come   L'archive ouverte pluridisciplinaire

L E DELTA du Sine-Saloum (Sénégal) présente un gradient de salinité inversé avec hyper-salinité à l'amont. Il offre une importante ressource piscicole aux populations locales. En vingt ans, les captures totales sont passées de trente mille à quinze mille tonnes et la diversité des captures a également baissé. Ethmalosafimbriata, un Clupéidé pélagique, est devenu progressivement dominant. Des pratiques et un aménagement des pêches inadaptés ont amené ces déclins, également dus en partie à la dégradation de l'environnement qui fait courir à l'écosystème des risques écologiques réels. Le logiciel Ecopath avec Ecosim a été employé ici pour analyser les relations trophiques au sein de cet écosystème et pour explorer les possibilités d'un aménagement écologiquement durable de la ressource.

INTRODUCTION 407 W EST Ati-ican estuaries, characterised by a high species diversity, are highly productive (ALBARET & DIOUF, 1994). Sine-Saloum (fig. 1) is located about 100 km south of Dakar (Senegal) and covers an area of 543 km 2 • It is one of the country's most populated regions, with 10 per cent of th is area inhabited by around 16 per cent of the country's total population man y dependent on the natural resources of the area.

The inverse hyper salinity of the Sine-Saloum is an important factor affecting its biodiversity. Bousso, 1996;[START_REF] Diouf | Les peuplements de poissons des milieux estuariens de l 'AFique de l'Ouest: l'exemple de l'estuaire hyperhalin du Sine Saloum[END_REF]. Ellimalosa jimbriala (fig. 2) now represents 80 per cent of the actual catch in 2000 (DEME el a!., 2000;[START_REF] Deme | Effort de pêche, captures spécifiques et valeurs économiques des débarquements de la pêche continentale dans le neuve Sénégal et au Sine Saloum, rapport scientifique[END_REF]. Postcolonial fishing practices contributed significantly to the decl ine of catches both in numbers and di-versity due to overexploitation, mostly by immigrating fishers (DIOUF el al., 1998).

The aim of this contribution is to construct a preliminary food wcb model for the Sine-Saloum ecosystem, to describe its complex trophic structure and dynamics, and how their variations can affect the tisheries.

MATERIALS & METHOOS

E COPATH with Ecosim (EwE), dcscribed by CHRISTENSEN el al. (2000), and several contributors in this volume, was used to identify the trophic relationships and community structure in the Sine-Saloum estuary. Groups were mainly based on data by ALBARET and DIOUF (1994).

Existing data l'rom field samples and various publications (UNESCO, 1982;1985;[START_REF] Bousso | Exp/oitation des Stocks dans « L'estuaire)) et les B%ngs du Sine-Sa/oum Evo/Ulion depuis 20 ans[END_REF]ALBARET & DIOUF, 1994;[START_REF] Bousso | Ressource halieutique et système d'exploitation dans l 'esiliaire du Sine-Saloum (Sénégal), rapport scientifique chercheur associé[END_REF][START_REF] Diouf | Les peuplements de poissons des milieux estuariens de l 'AFique de l'Ouest: l'exemple de l'estuaire hyperhalin du Sine Saloum[END_REF]; DIOUF el a!., 1999; DEME el a!., 2001) on this ecosystem were utilised in order to build the mode!. A total of 38 groups were created (table 1), based on species abundance and the structu rc of thc trophic system (ALBARET & DIOUF, 1994;Bousso, 1996;[START_REF] Diouf | Les peuplements de poissons des milieux estuariens de l 'AFique de l'Ouest: l'exemple de l'estuaire hyperhalin du Sine Saloum[END_REF]. Quantitative diet composition were not available, and thus qualitative information was used taken mainly from DIOUF (1996), Fishbase (FROESE & PAUL Y, 2000) and colleagues' personal communications. This preliminary Ecopath model was designed to investigate possible consequences to ecosystem health of further changcs in fishing patterns and intensities.

The data base used here, as weil as the sources of any inputs, is avai lable upon request from the first author (see also www.ecopath.org).

RESULTS & DISCUSSION

T HE system's primary production/respiration ratio of 1.084 indicates that it is rich in organic materials from demineralisation and other active microbial processes [START_REF] Diouf | Les peuplements de poissons des milieux estuariens de l 'AFique de l'Ouest: l'exemple de l'estuaire hyperhalin du Sine Saloum[END_REF].

The estimated biomass for each group (table 1) seem to be reasonable but it should be emphasised that further studies are needed to improve Ihe mode!. Estimated total annual biomass of tish groups considered is 37.7 l.km•' (table 1).

The low system omnivory index of 0.146, with a conneclanee index of 0.184, is due to feeding speeial isation by various fish groups. The tish assemblages are dominated by groups preying mainly on zooplankton, especially during Ihe juvenile stages. Generally, estuarine food webs are driven by phytoplankton or detrilal production [START_REF] Blaber | Tropica/ ESlllarine Fishes : Ec%gy, Exp/oitation and Conservation[END_REF].

For Sine-Saloum, the latter seems ta be the case, due 10 Ihe low primary production. The low primary productivity can be attributed to water turbidity, inefficienl nutrient cycling and environmental degradation leading 10 reduced dissolved oxygen concentrations in the estuary (UNESCO, 1985;BLASER, 2000).

The trophic chain is short and flows mainly from detritus to higher trophic levels. The relatively high transfèr efficiency (16 per cent) associated with these flows can be attributed to the low abundance of phytoplanklon and of organisms grazing il.

The transfer efficiencies increase at higher lrophic levels, which is related to the higher quality of the food being transferred. (2001). This may be due to unreported capturcs of less economically important animais or discards of non-marketable species not included in annual catches, as weil as unrecorded cxport of substantial amounts to neighbouring countries su ch as Gambia. 133 ln most simulation runs \Vith Ecosim, increascd fishing pressure using cast nets cause a marked dccline of the MIIgilidae. Increasing fishing activity using any gill or shrimp nets showed negative effects to the biomass of Sciaenidae exccpt for P elollgatlls (table Ill). Increasing fishing pressure by a factor of 2 led to a dramatic decrease in the biomass of Pseudotolithlls spp., Elops spp. and S. tri-tor. Olher groups such as Sphvraena spp., Arius spp., Dasvatis spp., Chloroscombrus and Hell7icarallx spp. E. jill/brio/a, as weil as the bivalve and gastropod group showed progressive decrease in thcir biomass. The biomasses of Brac17vdeuterlls ol/ritlls, Galeoides decadoctyllls, Sardillella lI1oderensis, Psettodes and Citharvclllhys spp., Ili.l'ha olricalla as wcll as the cephalopods increased due ta decrease of predation pressure and inter-specifie competition among these groups. Fish groups specialised in consuming molluscs, shrimp and other invertebrates showed negative responses in biomass only when the effort of gears capturing their preferred preys was increased. Various scenarios were simulated using the fishing policy routine of Ecosim. Criteria considered were based on requirements outlined by DIOUF el al.

(1998) regarding sustainable management strategies in agreement with the current tisheries legislation. The policy search routine, when set to favour ecosystem structure and net economic value suggestcd that tishing with gill nets heavily exploiting clupeid spccies, as weil as targeting of shrimps and molluscs would lead to the best compromise.

CONCLUSION

T HE trophic structure in the Sine-Saloum estuary is changing mainly from decline of l'ish stocks due to intensification of fishing activity and environmental degradation. Depletion of some groups due to increased catch and predation rates may allow consequential increasc of biomass among other groups but will eventually lead to threats to ecosystem stability and resources sustainability.

Future management policies should consider resource conservation as such system serve as nursery grounds for many commercially exploited resources.

It should be noted that this is only a preliminary study, and that fùrther investigations will bc conducted in the near rùture. However comments will be grcatly appreciated. *

FIG. 1 .

 1 FIG. 1. -Map showing location of the studied area and the zone considered as Biosphere reserve of the Delta of Sine-Saloum. (Sources: http://www.au-senegal.com_and Encarta).

  loum: L'environnement aquatique. Les ressources haliel/tiques et leur e.\ploitatian, Union mondiale pour la conservation de la nature (U.I.C.N.), 107 p. FROESE (R.) & D. PAULY (ed.), 2000. -Fishbase 2000: Concepts. Design and Data Sources, Los Banos, Philippines, ICLARM, 344 p. TEUGELS et al. (ed.), 1994. -"Biological Diversity in African Fresh-and Brackish, Water Fishes", Ann. Mus. R. Afr. Centr.. Zool., 275. UNESCO, 1982. -Atelier d'étude des mangroves et de l'estuaire du Saloul11, Sénégal, rapport final. Unesco, Sénégal, 175 p. UNESCO, 1985. -L'estuaire et la mangrove du Sine Saloum: Résultats d'un Atelier régional Unesco-Comar tenu à Dakar, Sénégal, Unesco, Sénégal, 139 p.

TABLE 1

 1 Basic inpu/ and es/illla/ed parame/ers ofthe Ecopath model for Sine-Salollm. TL is the /rophic level. Habita/ area/or each group is /heji-ac/ion of/he /otal area where biomass ofspecies are concen/ra/ed. Biomass A pertains /0 biomass in /his habita/ area and B /0 biomass in /otal area. PIB is the produc/ionlbiomass ratio, QIB the consllmp/ionlbiomass ratio Basic estimates calculated bv the mode! are indica/ed in italics Structure trophique du delta du Sine-Saloum telle qu'analysée par Ecopath : TL est le niveau trophique;« Habitat area » désigne la fraction de la surface totale où le groupe considéré est concentré,

	la biomasse A est celle pour cet habitat et B celle comme une moyenne pour tout l'écosystème.
	P/B est le rappon Production/Biomasse, et Q/B est le rappon consommation/biomasse.
	Seules les valeurs en italiques ont été calculées par le logiciel
	GROUP NAME	TL	HABITAT AREA	BIOMASS A BIOMASS B (T. Ktvr') (T.KM•')	P/B (yi)	Q/B (yi)	EE	GE
	Scomberomorus IrilOr	3.4	0.50	0.009	0.004 2.520	12.60 0.950 0.200
	Elaps spp.	3.3	0.50	OJJ21	0.010 1.990	9.95 0.950 0.200
	Spftyraena spp.	3.3	0.25	8.688	2.172 1.700	8.50 0.176 0200
	Arills spp.	3.2	0.50	2.544	1.272 2.655	17.70 0.312 0.150
	Polvdac/yllis qlladrifilis	3.5	1.00	1.272	1.272 1.540	no 0.079 0.200
	Pselld%lilhlls elongallis	3.3	1.00	1.800	1.800 1.680	840 0.026 0.200
	Olher Sciaenidae	3.3	1.00	0.089	0.089 J .650	8.25 0.950 0.200
	PsellodeslCilharichlhys	3.3	0.50	0.166	0.0166 1.980	Il.00 0.950 0329
	Caleoides decadaClyllis	3.2	0.50	2.150	2.150 1.515	10.10 0.950 0.150
	Dosl'{lIis spp.	31	0.50	0.075	0.037 1.650	8.25 0.950 0.100
	Il Brachydelllerus al/rilliS	2.7	0.50	0019	OJJl 0 2.870	28.70 0.950 0.150
	PleclorhinchllslPomodasvs	3.2	0.75	1.558	1.168 1.270	847 0.950 0.150
	lIisha ofricano	2.9	0.50	2.444	1222 2445	16.30 0.950 0.150
	Cynoglosslls spp.	3.1	0.50	9.224	4.612 1 .200	8.00 0.950 0180
	ChloroscombrllslHemicaranx	33	0.25	0.066	0.016 3.3 12	1740 0.935 0150
	Carongidae (ben tho-pelagie)	3.3	0.50	0.718	0.359 1.335	8.90 0.950 0.150
	Cerreidae	2.9	0.50	0.128	OJJ64 3.075	20.50 0.950 0.180
	Other Corangidae	3.2	0.50	0.224	0.112 2407	13.3 7 0.440 0.150
	Ephippidae	3.0	0.75	OJJ13	0.010 2.130	14.20 0.950 0.150
	Elhmolosafimbriolo	2.7	0.75	23.543	17.657 2.685	17.90 0.543 0100
	21 Sordillello moderensis	2.8	0.50	0.203	0.102 3.390	33.90 0.950 0.050
	Mligilidae	2.4	0.50	2.750	1.3 75 1.386	27.72 0692 0.051
	Sorolheradon melonOlheron	2.3	0.60	2.100	1.260 1.275	25.20 0.856 0050
	Tilopio gl/ineensis	2.3	0.50	1.800	0.900 1.030	20.60 0.613 0.250
	Pelagie Shrimps	2.3	0.75	4.674	3.505 4.500	18.00 0.950 0.250
	Litloral shrimps	2.3	0.25	26.853	6.713 4.500	18.00 0.950 0312
	Pelagie erabs	2.7	0.75	7.409	5.557 2.500	8.00 0.950 0.312
	Littoral erabs	2.1	0.25	3.576	0.894 2.500	8.00 0.950 0.312
	Cephalopods	3.0	0.50	29.892	14.946 2.500	8.00 0.950 0.333
	Bivalves and Gastropods	2.0	0.50	21320	10.600 3.000	9.00 0.950 0.320
	31 Annelids and Polyehaets	2.0	0.25	26.530	6.633 8.000	25.00 0.950 0.314
	Zoobenthos	2.1	0.25	32.853	8.213 22.000	70.00 0.950 0.333
	33 Pelagie zooplanklOn	2.1	0.50	7.221	3.611 50.000	150.00 0.950 0.333
	Littoral zooplankton	2.1	0.50	4.104	2.052 50.000	150.00 0.950 0.333
	Benth ie algae	1.0	0.25	50.000	12.500 36.798		0.950
	Pelagie phytoplanklOn	1.0	0.50	6.810	3.405200.000		0.950
	Litloral phytoplankton	1.0	0.50	1.968	0.984365.000		0.950
	Detritus	/.0	1.000	1.000				0.863

  Captures annuelles (l.km-2) estimées par groupe pour chaque engin. Le total est estimé à 32,34 t.km-2 Le niveau trophique moyen des captures est de 2.63

			is slightly higher than what has been observed
			(15,370 t) by
	Sardil1ella maderel1sis. Sarotherodol1 melal1otherol1,
	shrimps and crabs (table II). As a whole, the fish-
	ery is operating at a trophic level of about 2.63.
	GROUP\ CATCH Total annual catch is 32.34 t.km" (17,560 t) which DUGOUT BI:ACH SEINE	ENCIRCUNG	CAST NETS	FIXED GILL	DERIVED	TRAWL	SIIRIMP	TRAI'S	OTHER	TOTAL
				GILLNETS		NETS	GILL NETS	UNES	NETS		GEARS	CATCH
	1 Scomberolllorolls tritor			0.005			0.005					0.01
	2 Elops spp.										0.010	0.01
	3 SphyroelJa spp.		0.200	0.100		0.070	0.260					0.63
	Arius spp.	0.1 la	0.120	0.010		0.140	0.140	0.040				0.56
	5 Polydac/ylus qlladrijilis		0.060	0.015		0.010	0.050		0.005			0.14
	Pseud%li/hlls elolJga/lIs		0.010			0.010	0.050					0.07
	7 Other SciaelJidae spp.		0.030			0.025	0.070		0.005			0.13
	8 Psel/odes/Ci/haric:h/hys										0.010	0.01
	Ga/eoides decadac/yllls					0.009			0.001			0.01
	la Dm)'a/is spp.										0.010	0.01
	Il Brachydell/erlls auri/liS										0.010	0.01
	Plec/orhinelllls/Pomada,)'s				0.005	0.003			0.002			0.01
	Ilisho ofi'iC:OlJa								O.OIIJ			0.01
	(l'noglos.l•us spp.		0,005			0.003			0.001	0.001		0.01
	Clllomscombl'1ls//-Iemic:aranx spp.			0.020								0.02
	CaralJgidae (ben/ho-pelagic)			0.010			0.060					0.07
	Gerreidae spp.		0.010									0.01
	Other CaralJgidae spp.		0.015	0.005								0.02
	Ephippidae spp.		0.010									0.01
	Ethmalosafimbria/a		0.560	22.250	0.015	0.300	0.050		0.005	0.010		23.19
	21 SardilJella maderelJsis		0.006	0.002	0.002							0.01
	Mugilidae spp.		0.375	0.010	0.150		0.040		0.005	0.010		0.59
	Saro/herodon lIIelanotherolJ		0.180		0.030	0.030	0.010			0.090		0.34
	Tilapia guilJeensis		0,070		0.020	0.030	0.005		0.005	0.010		0.14
	Pelagie Shrimps								1.050			\.05
	Lilloral shrimps								0.350			0.35
	Pelagie erabs										0.010	0.01
	Littoral erabs									0.100		0.10
	Cepha lopods										0.010	0.01
	Bivalves and Gaslropods									4.800		4.80
	31 TOTAL CATCH	0.110	1.651	22.427	0.222	0.630	0.740	0.040	1.439	5.021	0.060	32.340
	TROI'IIiC LEVEL	3.18	2.74	2.75	2.40	2.92	320	3.18	2.28	2.01	3.02	2.63

TABLE Il

Estimated catch (t.km-

2

) pel' group/in each geaI'. Total annual catch estimated was 32.34 (t.km-

2

), operating at atrophie level 0

(2.63

Various fishing gears, though mostly non-specilic, are uscd to exploit faunal resources in the estuary, Sphyrael1a spp., Arills spp.. Ethmalosa jimbri(({a, DEME et al.

TABLE " 1

 " Evollltiol1 of biomass ofa[fected grollps lvhenfishing ef/àrt is il1creased by a factor of1lI'0, ail gears combined Évolution de la biomasse des groupes dont l'abondance est affectée par un doublement généralisé (sur 10 ans) de l'activité de pêche, tous engins confondus

	GROUP Nt\ME	Biomass (Start)	Biomass (End)	13 iomass (SIE)	Catch (Start)	Catch (End)	Catch (SIE)	Value (Start)	Value (End)	Value (SIE)
	Scomberomorous Irilor	0.004	0.000	0.00	0.011	0.000	0.00	0.006	0.000	0.00
	E/ops spp.	0.010	0.002	0.22	0.011	0.004	0.36	0.002	0.001	0.36
		2.150	1.683	0.78	0.714	0.916	1.28	0.750	0.961	1.28
		1.256	1.13 1	0.90	0.633	0.934	1.47	0.253	0.373	1.47
	Po/vdacly/us q/ladrijilis	1.268	1.341	106	0.160	0.277	1.73	0.271	0.470	1.73
	Pseudololi/hus e/ongallls	1.797	1.681	0.94	0.080	0.123	1.53	0.015	0.023	1.53
	Other Sciaenidae	0.085	0.001	0.01	0.142	0.002	0.02	0.027	0.000	0.02
	Psellodes/Cilarichl!J.vs	0.083	0.102	1.23	0.011	0.023	2.01	0.002	0.004	2.01
	Ga/eoides decadaclylus	1.076	1584	1.47	0.011	0.028	2.41	0.002	0.005	2.41
	Dasyalis spp.	0.037	0.022	0.60	(J.OI!	0.011	0.99	0.002	0.002	0.99
	Brachydelllerus aurilllS	0.009	0.02!	2.22	(l.O Il	0.041	3.63	0.002	O.OOS	3.63
	Pleclorhinchus/Pomadasys	1.167	1.010	0.87	0.011	0.016	1.42	0.002	0.003	1.42
	llisha afi'icana	1.222	1.356	1.1 1	O.(JII	0.021	1.82	0.002	0.004	1.82
	Cvnog/ossus spp.	4.614	4.973	1.08	0.011	0.020	1.76	0.002	0.004	1.76
	Ch/oroscombrus/Hemical'(Jnx	0.016	0.012	0.73	0.022	0.026	1.19	0.004	0.005	1.19
	Caral1gidae (ben tho-pelagie)	0.357	0.531	1.49	(J.(J80	0.194	2.24	0.015	0.037	2.44
	Gerreidae	0.064	0.067	1.05	(J.OII	0.020	1.72	0.002	0.004	1.72
	Olher CalYlngidae	0.112	0.148	1.33	(J.(J23	0.050	2.17	2.068	0.009	2.17
	Ephippidae	0.010	0.011	1.12	0.011	0.020	1.84	0.002	0.004	1.84
	Elhma/osa jimbriata	17.021	10.685	0.63	25.58(J	26.314	1.03	2.068	2.127	1.03
	Sardillella maderensis	0.\02	0.150	1.48	0.0\ 1	0028	2.42	0.002	0.005	2.42
	Mugi/idae	1.356	0.740	0.55	0.666	0.595	0.89	0.173	0.155	0.89
	Sarolherodon melanolheron	1249	1.062	0.85	0.386	0.537	1.39	0.145	0.202	1.39
	Ti/apia guineel1sis	0.896	0.809	0.90	0.160	0.236	1.48	0.060	0.089	1.48
	Pelagie shrimps	3.48 ,	3.511	1.01	1.194	1.972	1.65	1.492	2.465	1.65
	Liuoral shrimps	6.710	6984	1.04	0.401	0.683	1.70	0.501	0.854	1.70
	Pelagie erabs	5.593	Il.210	2.00	0.012	Om8	328	0.002	0.007	3.28
	Littoral erabs	0.892	0.909	1.02	0.114	0.191	1.67	0.022	0.036	1.67
	Cephalopods	15.047	23.919	1.59	0.012	0.030	2.60	0.002	0.006	2.60
	Bivalves and Gastropods	10.528	8.532	0.81	5.427	7.205	1.33	1.03 ,	1.369

Sphyrael1a spp. Arius spp.

Actes du symposium international, Dakar (Sénégal), 24-28 juin 2002

Pêcheries maritimes, écosyslèmes & sociélés en Afi'ique de "Ouest : Un demi-siècle de changement

Pêcheries maritimes, écosystèmes & sociétés en Afrique de "Ouesl : Un demi-siècle de changemenl

ACKNOWLEDGEMENTS

413

T HE authors wish to thank Jean-Jacques ALBARET, Jean-Marc ECOUTIN and Guy VIDY (ail with IRD, Dakar, Senegal), especially for scientific comments and suggestions and the rest of the staff involved in the project Rap (Réponses adaptives des poissons), spearheaded by Raymond LAi:. We also thank Moustapha DEME (Researcher-CRODT), Omar SARR and Papa Samba DIOUF (Head-WWF, Senegal) for supplementary information.

Mode/ oflhe Sine-Sa/ol/111 De/la Biosphere Reserve (Senega/


Athanase Papadopoulos
email: athanase.papadopoulos@math.unistra.fr

Norbert A ' Campo
email: norbert.acampo@gmail.com

Transitional Geometry

Keywords: AMS classification: 01-99, 53-02, 53-03, 53A35 transformation group, projective geometry, elliptic geometry, spherical geometry, Lobachevsky geometry, non-Euclidean geometry, transitional geometry, coherent element, family, deformation

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

In this paper, we relate the three geometries of constant curvature by constructing a continuous transition between them. We construct a fiber space E → [-1, 1], where the fiber above each point t ∈ [-1, 1] is a twodimensional space of constant curvature which is positive for t > 0 and Date: November 1, 2014. negative for t < 0 and such that when t tends to 0 from each side, the geometry converges to the geometry of the Euclidean plane. Making the last statement precise will consist in showing that segments, angles, figures and several properties and propositions of the two non-Euclidean geometries converge in some appropriate sense to those of the Euclidean one. We say that the geometries transit. At the level of the basic notions (points, lines, distances, angles, etc.) and of the geometrical properties of the figures (trigonometric formulae, area, etc.), we say that they transit in a coherent way. This is expressed by the existence of some analytic sections of the fiber space E which allows us to follow these notions and properties. The group-theoretic definitions of each of the primary notions of the geometries of constant curvature (points, lines, etc.) make them transit in a coherent way.

The idea of transition of geometries was already emitted by Klein in §15 of his paper [START_REF] Klein | Über die sogenannte Nicht-Euklidische Geometrie Mathematische Annalen IV[END_REF] (see also the comments in [START_REF] Campo | On Klein's So-called Non-Euclidean geometry[END_REF]). We note however that the abstract theoretical setting of transformation groups was not really developed at the time Klein wrote the paper [START_REF] Klein | Über die sogenannte Nicht-Euklidische Geometrie Mathematische Annalen IV[END_REF]. The notion of "classical group", which is probably the most convenient setting for the description of these groups, was developed later, in works of Weyl [START_REF] Weyl | The Classical Groups: Their Invariants and Representations[END_REF], Dieudonné [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF] and others.

The construction we describe here is also developed in Chapter 9 of our Notes on hyperbolic geometry [START_REF] Campo | Notes on hyperbolic geometry[END_REF]. We review the basic ideas and we add to them some new results and comments. Beyond the fact that it establishes relations between the three classical geometries, this theory highlights at the same time the important notions of families and of deformations.

The fiber space E

We work in a fiber space over [-1, 1] where the fibers above each point are built out of the groups of transformations of the three geometries. Each fiber is a geometry of constant curvature considered as a homogeneous space in the sense of Lie group theory, that is, a space of cosets G/H 0 where G is a Lie group -the orthogonal group of some quadratic form -and H 0 the stabilizer of a point. This is an example of the classical concept of Klein geometry, see e.g. [START_REF] Frances | Transformation groups in non-Riemannian geometry[END_REF] in this volume.

When the parameter t ∈ [-1, 1] is negative, the homogeneous space is the hyperbolic plane of a certain constant negative curvature. When it is positive, the homogeneous space is the sphere (or the elliptic plane) of a certain constant positive curvature. When the parameter is zero, the homogeneous space is the Euclidean plane. We can define points, lines and other notions in the geometry using the underlying group. For instance, a point will be a maximal abelian compact subgroup of G (which can be thought of as the stabilizer group of the point in the ambient group). The space of the geometry is the set of points, and it is thus built out of the group. A line is a maximal set of maximal abelian compact subgroups having the property that the subgroup generated by any two elements in this set is abelian. Other basic elements of the geometry (angle, etc.) can be defined in a similar fashion. This is in the spirit of Klein's Erlangen program where a geometry consists of a group action. In this case, the underlying space of the geometry is even constructed from the abstract group. The definitions are made in such a way that points, distances, angles, geometrical figures and trigonometric formulae vary continuously from hyperbolic to spherical geometry, transiting through the Euclidean, and several phenomena can be explained in a coherent manner.

To be more precise, we consider the vector space R 3 equipped with a basis, and we denote the coordinates of a point p by (x(p), y(p), z(p)) or more simply (x, y, z).

We recall that the isometry groups of the hyperbolic plane and of the sphere are respectively the orthogonal groups of the quadratic form

(x, y, z) → -x 2 -y 2 + z 2

and

(x, y, z) → x 2 + y 2 + z 2 . Introducing a nonzero real parameter t does not make a difference at the level of the axioms of the geometries (although it affects the curvature): for any t < 0 (respectively t > 0) the isometry group of the hyperbolic plane (respectively the sphere) is isomorphic to the orthogonal group of the quadratic form (1)

q t : (x, y, z) → tx 2 + ty 2 + z 2 .
We wish to include the Euclidean plane in this picture.

The first guess to reach the Euclidean plane is to give the parameter t the value 0. This does not lead to the desired result. In fact, although the orthogonal groups of the quadratic forms q t , for t > 0 (respectively for t < 0) are all isomorphic, they are not uniformly bounded in terms of t, and when t → 0, their dimension blows up. Indeed, when t → 0 from either side, the quadratic form q t reduces to (x, y, z) → z 2 whose orthogonal group is much larger than the isometry group of the Euclidean plane.

Thus, an adjustment is needed. For this, we shall introduce the notion of "coherent element". This will make the Euclidean plane automorphism group (and the Euclidean plane itself) appear in a continuous way between the hyperbolic and spherical automorphism group (respectively the hyperbolic plane and the sphere).

We now define the fiber space E, equipped with its fibration

E → [-1, 1].
The fiber E t above each t ∈ [-1, 1] will be a space of constant positive curvature t 2 for t > 0 and of constant negative curvature -t 2 for t < 0, such that when t converges to 0 from either side, E t converges to the Euclidean plane.

We explain this now. It will be useful to consider the elements of the space E t as matrices, and when we do so we shall denote such an element by a capital letter A. [START_REF] Campo | Notes on hyperbolic geometry[END_REF] be the fibration whose fiber J t above t is the subgroup of GL(3, R) consisting of the stabilizers of the form q t . Finally, let I ⊂ J be defined by taking fiberwise the connected component of the identity element in the orthogonal group of the quadratic form q t , for each t ∈ [-1, 1].

Let J ⊂ GL(3, R) × [-1, 1] → [-1,
In other words, for every t = 0, I t is the orientation-preserving component of the orthogonal group of the quadratic form q t . It is a Lie group of dimension 3. However, as we already noticed, I 0 is a Lie group of dimension 6. It is the matrix group

I 0 = {A = (A ij ) ∈ GL(3, R) | A 3,1 = A 3,2 = 0, A 3,3 = 1, det(A) > 0}.
This is the group of matrices of the form

  a b e c d f 0 0 1  
with ad -bc > 0. The fact that the determinant is positive is a consequence of the orientation-preserving assumption. Thus, I 0 is the usual matrix representation group of the orientation-preserving group of affine transformations of R 2 . Equivalently, it is the group of matrices preserving the (x, y)-plane in R 3 . (One can deduce this fom the fact that the affine group is the subgroup of the group of projective transformations that preserve a hyperplane in projective space.) This shows again that I 0 is not isomorphic to the group of orientation-preserving Euclidean motions of the plane. We shall reduce the size of I 0 , by restricting the type of matrices that we consider.

Definition 2.1 (Coherent element). An element A of I 0 is said to be coherent if for all i and j satisfying 1 ≤ i, j ≤ 3, there exists an analytic function A i,j (t) such that for each t ∈ [-1, 1] the matrix A t = (A i,j (t)) belongs to I t , and A 0 = A.

The space of coherent elements

We denote by E ⊂ I ⊂ GL(3, R) × [-1, 1] the set of coherent elements. For each t ∈ [-1, 1] we denote by E t the set of coherent elements that are above the point t.

From the definition, we have E t = I t for all t = 0. We now study E 0 . The coherent elements of I 0 form a group and they have important features. We start with the following:

Proposition 3.1. If A ∈ I 0 is coherent, then det(A) = 1.
Proof. For t = 0, we have det(A) = 1 since A is an orthogonal matrix of a non-degenerate quadratic form. The result then follows from the definition of coherence and from the fact that the map t → det(A t ) is continuous.

We also have the following: Proposition 3.2. An element A ∈ I 0 is coherent if and only if A is an orientation-preserving motion of the Euclidean plane.

Proof. Let A ∈ I 0 be a coherent element. From the definition, we have A = lim t→0 A(t), with A(t) ∈ I t , t > 0. Thus, the 9 sequences A ij (t), for 1 ≤ i, j ≤ 3, converge and therefore they are bounded. For t = 0, since A(t) preserves the quadratic form q t , therefore it also preserves the form 1 t q t . For i = 1, 2, 3, let A i (t) denote the i-th column of A(t). We show that for i, j < 3, the 1 t q t -scalar product of A i (t), A j (t) is the i, j-th coefficient of the Kronecker delta function.

We have

1 t q t (A) = A 11 (t) A 21 (t) A 31 (t)   1 0 0 0 1 0 0 0 1 t     A 11 (t) A 21 (t) A 31 (t)   = A 11 (t) 2 + A 21 (t) 2 + 1 t A 31 (t) 2 .
Since A(t) preserve 1 t q t , we get A 11 (t) 2 + A 21 (t) 2 + 1 t A 31 (t) 2 = 1 for all t = 0. Since the sequences A ij (t) converge, we obtain lim t→0 A 31 (t) = 0, therefore the coherence property imposes A 31 (0) = 0. Now we use the fact that the section A 31 (t) is differentiable (recall that by coherence, it is even analytic). This gives

A 31 (t) = 0 + λt + O(t 2 ), therefore 1 t A 31 (t) 2 = λ 2 t + O(t 3 ), which implies lim t→0 1 t A 31 (t) 2 = 0. We get A 11 (0) 2 + A 21 (0) 2 = 1.
In the same way, we prove that A 12 (0

) 2 + A 22 (0) 2 = 1 and A 11 (0)A 12 (0) + A 21 (0)A 22 (0) = 0.
It follows that the first principal 2-bloc of A is an orthogonal matrix, which implies that E 0 is a direct isometry group of the Euclidean plane. In other words, the 2 × 2 matrix

A 11 A 12 A 21 A 22
preserves the standard quadratic form on R 2 . Thus, it is an element of the Euclidean rotation group SO(2).

Note.-In the proof of the preceding proposition, we used the fact that the coherence property imposes that the sections A ij are differentiable. There is another proof in [START_REF] Campo | Notes on hyperbolic geometry[END_REF] (Proposition 2.2) which only uses the existence of sections that are only continuous.

The set of coherent elements of I 0 , being the group of orientation-preserving motions of the Euclidean plane, is the group we are looking for.

From now on, we use the notation I 0 for the subgroup of coherent elements instead of the previously defined group I 0 . Thus, in the following, for each t ∈ [-1, 1], any element of I t is coherent.

The space of coherent elements (for variable t) is not a locally trivial fiber bundle (for instance, the topological type of the fiber I t is not constant; it is compact for t > 0 and noncompact for t < 0). However this space has nice properties which follow from the fact that it has many continuous (and even analytic) sections.

The algebraic description of points and lines

For every t ∈ [-1, 1], we define E t as the set of maximal abelian compact subgroups of I t . An element of E t is a point of our geometry. Note that any maximal abelian compact subgroups of I t is isomorphic to the circle group SO [START_REF] Campo | On Klein's So-called Non-Euclidean geometry[END_REF]. With this in hand, a coherent family of points in the fiber space (E t ) is a coherent family of maximal abelian compact subgroups of I t . This is a family depending analytically on the parameter t. We can also consider coherent families of pairs (respectively triples, etc.) of points. This defines the segments (respectively triangles, etc.) of our geometries. We can study the corresponding distance (respectively area, etc.) function of t defined by such a pair or triple, in algebraic terms.

For each p ∈ E t , we denote by K p ⊂ I t the maximal subgroup that defines p. Since K p is a group isomorphic to the circle group SO(2), for each p ∈ E t , there exists a unique order-two element s p ∈ K p . We shall make use of this element. In the circle group SO [START_REF] Campo | On Klein's So-called Non-Euclidean geometry[END_REF], this corresponds to the rotation of angle π. We call s p the reflection, or involution in I t , of center p. In this way, any point in E t is represented by an involution. For each t = 0, an involution is a self-map of the space of our geometry (the sphere or the hyperbolic plane) that fixes the given point, whose square is the identity, and whose differential at the given point is -Id.

This algebraic description of points in E t as involutions has certain advantages. In particular, we can define compositions of involutions and we can use this operation to describe algebraically lines and other geometric objects in E t .

A line in E t is a maximal subset L of E t satisfying the following property:

(*) The subgroup of I t generated by the all elements of the form s p s p ′ , for p, p ′ ∈ L, is abelian. In the the case t = 0, we ask furthermore that the group is proper. (For t = 0, this is automatic).

In other words, each time we take four points s p 1 , s p 2 , s p 3 , s p 4 in E t represented by involutions, then, s p 1 s p 2 commutes with s p 3 s p 4 .

Given two distinct points in E t , there is a unique line joining them; this is the maximal subset L of E t containing them and satisfying property (*) above.

The group I t acts by conjugation on E t and by reflections along lines. With these notions of points and lines defined group-theoretically, one can check that the postulates of the geometry can be expressed in group theoretic terms: any two distinct points belong to a line, two lines intersect in at most one point. (For this to hold, in the case t > 0, one has to be in the projective plane and not on the sphere.)

We now introduce circles. Let s be a point represented by a subgroup K p and an involution s p . Any element ρ in K p acts on the space of points E t by conjugation:

(ρ, K p ) → ρK p ρ -1 .
A circle of center p in our geometry is an orbit of such an action. For t > 0 and for every point p, among such circles, there is one and only one circle centered at p which is a line of our geometry.

A triangle in our space is determined by three distinct points with three lines joining them pairwise together with the choice of a connected component of the complement of these lines which contains the three given points on its boundary. The last condition is necessary in the case of the sphere, snce, in general, three lines divide the sphere into eight connected components.

Let us consider in more detail the case t > 0. This case is particularly interesting, because we can formulate the elements of spherical polarity theory in this setting. We work in the projective plane (elliptic space) rather than the sphere. In this way, the set of points of the geometry E t (t > 0) can be thought of as the set of unordered pairs of antipodal points on the sphere in the 3-dimensional Euclidean space (R 3 , q t ).

To each line in E t is associated a well-defined point called its pole. In algebraic terms (that is, in our description of points as involutions), the pole of a line is the unique involution s p which, as an element of I t , fixes globally the line and does not belong to it.

Conversely, to each point p, we can associate the line of which p is the pole. This line is called the equator of p. There are several equivalent algebraic characterizations of that line. For instance, it is the unique line L such that for any point q on L, the involution s q fixes the point p. In other words, the equator of a point s p is the set of points q = p satisfying s q (p) = p. 1This correspondence between points and lines is at the basis of duality theory.

If two points p and q in E t are distinct, the product of the corresponding involutions s p and s q is a translation along the line joining these points. More concretely, s p s q is a rotation along the line in 3-space which is perpendicular to the plane of the great circle determined by p and q. This line passes through the pole s N of the great circle, and therefore the product s p s q commutes with the pole s N (seen as an involution).

Given p ∈ E t , the equator of p is the set of all q ∈ E t whose image under the stabilizer of p in I t is a straight line.

In spherical (or elliptic) geometry, a symmetry with respect to a point is also a symmetry with respect to a line. This can be seen using the above description of points as involutions. From the definition, the involution corresponding to the pole of a line fixes the pole, but it also fixes pointwise the equator. The pole can be characterized in this setting as being the unique isolated fixed point of its involution. The set of other fixed points is the equator. Thus, to a line in elliptic geometry is naturally associated an involution.

We can use polarity to define perpendicularity between lines: Consider two lines L 1 and L 2 that intersect at a point p. Then L 1 and L 2 are perpendicular if the pole of L 1 belongs to L 2 . This is equivalent to the fact that the polar dual to L 2 belongs to L 1 . This also defines the notion of right angle.

In what follows, we shall measure lengths in the geometry E t for t > 0. We know that in spherical (or elliptic) geometry, there is a natural length unit. The length unit in this space can be taken to be the diameter of a line (all lines in that geometry are homeomorphic to a circle), or as the diameter of the whole space (which is compact). We can also use the correspondence between lines and poles and define a normalized distance on E t , by fixing once and for all the distance from a point to its equator. We normalize this distance by setting it equal to π 2 √ t . (Remember that each space E t , for t > 0, is the sphere of constant curvature t 2 .) After defining the distance on E t , we can check that the segment which joins a pole to a point on its equator is the shortest path to the equator.

Transition of points, lines, distances, curvature and triangles

Before we continue, let us summarize what we did. We chose of a fixed basis for R 3 . We defined the space E = (E t ) t∈[-1,1] of coherent elements as a subset of a space of matrices, equipped with a map onto [-1, 1]. A point p = (x, y, z) in R 3 equipped with a basis gives a group-theoretically defined point in E t for every t ∈ [-1, 1] with t = 0, namely, the stabilizer of the vector whose coordinates are (x, y, z) in the group of linear transformations of R 3 preserving the form q t . Using the coherence property, we extended to t = 0, and for any t ∈ [-1, 1], we defined a point A t in E t as a stabilizer group under the action of the group I t , namely, a maximal abelian subgroup K t of I t . The family A t , t ∈ [-1, 1] is a coherent family of points. This grouptheoretic definition of points allows the continuous transition of points in the space E. Likewise, the group-theoretic definition of lines makes this notion coherent transit from one geometry to another. Now we introduce the notion of angle. We want a notion that transits between the various geometries. This can also be done group-theoretically in a coherent manner. There are various ways of doing it, and one way is the following. We start with the group-theoretic description of a point as a maximal compact abelian subgroup (the "stabilizer" of the point). Such a subgroup is isomorphic to a circle group. But the circle group is also the set of all oriented lines starting at the given point. It is equipped with its natural Haar measure, which we normalize so that the total measure of the circle is 2π. This gives a measure on the set of oriented lines. From these measures, we obtain the notion of angle at every point. It is also possible to give a coherent definition of an angle by using a formula, and we shall do this in §7.

Now we consider transition of distances.

To measure distances between points, we use a model for the space E t for each t ∈ [0, 1]. We take the unit sphere S t , that is, the space of vectors in R 3 of norm one with respect to the quadratic form q t . This is the subset defined by ( 2)

S t = {(x, y, z) | tx 2 + ty 2 + z 2 = 1} ⊂ R 3 .
We take the quotient of this set by the action of Z 2 that sends a vector to its opposite. Note that the unit sphere S t (before taking the quotient) is connected for t > 0, and it has two connected components for t < 0. After taking the quotient, all spaces become connected. The group of transformations of R 3 which preserve the quadratic form q t acts transitively on S t /Z 2 for each t. For t = 0, the unit sphere (before taking the quotient) is defined by the equation z 2 = 1, and it has two connected components, namely, the planes z = 1 and z = -1.

In this way, the elements of the geometries E t can be seen either algebraically as cosets in a group (the isometry group modulo the stabilizer of a point) or, geometrically, as elements of the unit sphere quotiented by Z 2 . Now we define distances between pairs of points in each E t and then we study transition of distances.

For t = 0, we use the angular distance in E t . To do so, for each t = 0, we let β t be the bilinear form associated to the quadratic form q t , that is,

β t (x, y) = q t (x + y) -q t (x) -q t (y) 2 .
We set, for every x t and y t in E t ,

(3) w t (x t , y t ) = arccos β t (x t , y t ) q t (x t ) q t (y t )

We call w t (x t , y t ) the angular distance between the points x t and y t . For each t = 0, the angular distance can be thought of as being defined on S t /Z 2 , but also on the 2-dimensional projective plane, the quotient of R 3 \ {0}/Z 2 by the nonzero homotheties. As a distance on the space E t , we shall take w t multiplied by a constant c t that we determine below. Note that although Equation ( 2) describes an ellipsoid in the Euclidean orthonormal coordinates (x, y, z), this surface, equipped with this angular metric induced from the quadratic form, is isometric to a round sphere.

For each t = 0, equipped with the normalized distance function c t w t , the point set E t becomes a metric space. Its points and its geodesics coincide with the group-theoretically defined points and lines that we considered in §1. This can be deduced from the fact that the isometry group of the space acts transitively on points and on directions, and that both notions (the metric and the group-theoretic) are invariant by this action. There are other ways of seeing this fact.

We can draw a picture of S t for each t ∈ [0, 1]. In dimension 2, this is represented in Figure 1. To get the 3-dimensional picture, this figure has to be rotated in space around the y-axis.

Any vector in R 3 defines a point in each geometry E t by taking the intersection of the ray containing it with S t (or in the quotient of this intersection by Z 2 ). This is represented in Figure 2, in which the vector is denoted by P , and where we see three intersection points with the level surfaces S t : for t > 0, t = 0 and t < 0. Note that the two points with Cartesian coordinates (0, 0, 1) and (0, 0, -1) belong to all geometries (they belong to S t for any t ∈ [-1, 1]). We shall use this in the following discussion about triangles, where some vertices will be taken to be at these points, and this will simplify the computations.

We now discuss transitions of distances. We want the distance between two points in the geometry E 0 to be the limit as t → 0 (from both sides, t > 0 and t < 0) of the distance in E t between corresponding points (after normalization). The next proposition tells us how to choose the normalization factor c t .

Let A and B be two points in R 3 . We have a natural way of considering each of these points to lie in E t , for every t ∈ [-1, 1]. Let us denote by A t , B t the corresponding points.

Proposition 5.1. The limit as t → 0, t > 0 (respectively t < 0), of the distance from A t to B t in E t normalized by the factor 1/ √ t is equal to the Euclidean distance between A 0 and B 0 .

Using this result, we shall define the distance between points A and B in E 0 in such a way that the distances in Proposition 5.1 vary continuously for t ∈ [-1, 1].

Proof. For t ∈ [-1, 1], let a = (0, 0, 1) ∈ R 3 and consider the coherent family of points

A t ∈ E t , t ∈ [-1, 1] represented by a. For x ∈ R, let B x t ∈ E t , t ∈ [-1, 1]
be the coherent family of points represented by the stabilizer of

x y t > 0 t < 0 t = 0 Figure 1. x y t > 0 Figure 2.
the vector b x = (x, 0, 1) of R 3 . The family [A t , B x t ], t ∈ [0, 1] is a coherent family of segments in E.

We start with t > 0. We compute the limit as t → 0, t > 0, of the distance from A t to B x t in E t , normalized by dividing it by √ t. We have

q t (a) = 1, q t (b x ) = tx 2 + 1 and β t (a, b x ) = 1.
We use the angular distance from A t to B x t , measured with q t , t > 0. By Equation ( 3), we have

w t (A t , β x t ) = arccos ( β t (a, b x ) q t (a) q t (b x ) ) = arccos( 1 √ tx 2 + 1 ) = | arctan( √ tx)| = | √ tx - 1 3 √ t 3 x 3 + 1 5 √ t 5 x 5 -. . . |
Taking c t = 1/ √ t as a factor for distances in E t , the distance c t w t between the two points becomes

(4) D t (A t , B x t ) = 1 √ t arccos( 1 √ tx 2 + 1 ) = |x - 1 3 tx 3 + 1 5 t 2 x 5 -. . . |.
The limit as t → 0, t > 0, of D t (A t , B x t ) is |x|, which is what the distance from A 0 to B x 0 in E 0 ought to be. Thus, we define the distance D t on each E t to be the angular distance w t given in (3) divided by √ t. In this way, the distance function transits. The fact that points and distances transit between the various geometries implies that triangles also transit. We already noted that angles also transit. From this, we get several properties. For instance, since angle bisectors may be defined using equidistance, the following property transits between the various geometries: Proposition 5.2. The angle bisectors in a triangle intersect at a common point.

The reader can search for other properties that transit between the various geometries.

Curvature is also a coherent notion. Indeed, curvature in spherical and in hyperbolic geometry is defined as the integral of the excess (respectively the defect) to two right angles of the angle sum in a triangle (and to a multiple of a right angle for more general polygons). This notion transits through Euclidean geometry, which is characterized by zero excess to two right angles. One can make this more precise, by choosing a triangle with fixed side lengths -for instance with three sides equal to a quarter of a circle -and making this triangle transit through the geometries.

Transition of trigonometric formulae

The trigonometric formulae, in a geometry, make relations between lengths and angles in triangles. They are at the bases of the geometry, since from these formulae, one can recover the geometric properties of the space. Let us recall the formal similarities between the trigonometric formulae in hyperbolic and in spherical geometry. One example is the famous "sine rule", which is stated as follows:

For any triangle ABC, with sides (or side lengths) a, b, c opposite to the vertices A, B, C, we have, in Euclidean geometry : Thus, to pass from the Euclidean to the spherical (respectively the hyperbolic), one replaces a side length by the cosine of the side length (respectively the sine of the length). Since using the sine rule one can prove many other trigonometric formulae, it is natural to expect that there are many occurrences of trigonometric formulae in non-Euclidean geometry where one replaces the circular functions of the side lengths (sin, cos, etc.) by the hyperbolic function of these side lengths, in order to get the hyperbolic geometry formulae from the spherical ones. Examples are contained in the following table. They concern a triangle ABC having a right angle at C:

a sin A = b sin B = c sin C ,
Hyperbolic Euclidean Spherical cosh c = cosh a cosh b c 2 = a 2 + b 2 cos c = cos a cos b sinh b = sinh c sin B b = c sin B sin b = sin c sin B tanh a = tanh c cos B a = c cos B tan a = tan c cos B cosh c = cot A cot B 1 = cot A cot B cos c = cot A cot B cos A = cosh a sin B cos A = sin B cos A = cos a sin B tan a = sinh b tan A a = b tan A tan a = sin b tan A
In this table, the Euclidean formulae in the middle column are obtained by taking Taylor series expansions of any of the two corresponding non-Euclidean ones, for side lengths tending to 0. This is a consequence of the fact that the spherical and the hyperbolic geometries become Euclidean at the level of infinitesimal triangles.

The transitional geometry sheds a new light on the analogies between the trigonometric formulae.

Fisrt, we must recall that the above formulae are valid for spaces of constant curvatures -1, 0 and 1. In a geometry E t , with t = 0, one has to introduce a parameter t in the above trigonometric formulae. For instance, for a right triangle ABC with right angle at C, we have, in spherical geometry (t > 0) : cos √ tc = cos √ ta cos √ tb and in hyperbolic geometry (t < 0) :

cosh √ -ta = cosh √ -tb cosh √ -tc.
Now we can study the transition of this formula, called the Pythagorean theorem. 2 We first study the transition of a triangle. For t ∈ [-1, 1], let a = (0, 0, 1) ∈ R 3 and consider the coherent family of points A t ∈ E t , t ∈ [-1, 1] represented by the vector a. Take a real number x. For any t ∈ [0, 1], let B x t ∈ E t and C x t ∈ E t be the two coherent families of points represented by the stabilizers in I t of the vectors b x = (x, 0, 1) and c x = (0, x, 1) of R 3 . The family ∆ t (x, y) = (A t , B x t , C y t ), t ∈ [0, 1] is a coherent family of triangles. The triangle ∆ 0 (x, y) is a right triangle at A, with catheti ratio x/y. To see that each triangle ∆ t (x, y) is right, we note that if we change B into -B, we do not change the angle value. (Recall the definition of the angle measure as the Haar measure on SO(2).) Therefore there is a transformation of the space E t that belongs to our isometry group and that sends this angle to its opposite. Therefore the angle is equal to its symmetric image, and therefore it is right. We saw (proof of Proposition 2.1) that in the geometry E t , the distance between the two points A t and B x t is given by (5)

D t (A t , B x t ) = 1 √ t arccos( 1 √ tx 2 + 1 )
Likewise, the distance between the two points A t and C y t is equal to

(6) D t (A t , C y t ) = 1 √ t arccos( 1
ty 2 + 1 )
In the geometry E 0 , the distance from from A 0 to C y 0 is equal to |y|. We need to know the distance in E 0 between the points B x 0 and C y 0 . The angular distance from B x t to C y t , measured with q t , t > 0, is

w t (B x t , C y t ) = arccos ( β t (b x , c y ) q t (b x ), q t (c y ) ) = arccos( 1 √ tx 2 + 1 ty 2 + 1 )
= arccos( 1 t 2 x 4 y 4 + tx 2 + ty 2 + 1 ).

Up to higher order terms, this expression is equal to t 2 x 4 y 4 + tx 2 + ty 2 .

2 By extension from the Euclidean case, a Pythagorean theorem, in a certain geometry, is a formula that makes relations between the edges and angles of a right triangle.

We collect the information in the following lemma, using the fact that in the coherent geometry E 0 , distances are obtained as a limit of normalized distances in E t for t > 0 (or t < 0). Lemma 6.1. In the geometry E t , we have

D t (A t , B x t ) = 1 √ t arccos( 1 √ tx 2 + 1 ), D t (A t , C y t ) = 1 √ t arccos( 1
ty 2 + 1 ), and
D t (B x t , C y t ) = 1 √ t arccos( 1 √ tx 2 + 1 ty 2 + 1
).

In the coherent geometry E 0 , we have

lim t→0 D t (A t , B x t ) = D 0 (A 0 , B x 0 ) = |x|, lim t→0 D t (A t , C x t ) = D 0 (A 0 , C y 0 ) = |y| and lim t→0 D t (B t , C x t ) = D 0 (B x 0 , C y 0 ) = x 2 + y 2 .
Using this lemma, we now show that the Euclidean Pythagorean formula in E 0 is a limit of the Pythagorean formula in E t for t > 0.

Let x and y be real numbers and let t > 0. In the triangle ∆ t (x, y), the lengths of the three sides (measured in E t ) are the angular distances √ tD t (A t , B

= 1 × 1,
which is correct but which is not the Euclidean Pythagorean theorem. We obtain a more useful result by taking another limit.

We transform the spherical Pythagorean theorem into the following one:

√ t 1 -cos( √ tD t (A t , B x t )) cos( √ tD t (A t , C x t )) = √ t 1 -cos( √ tD t (C y t , B x t )) .
Writing this equation at the first order using the expansion 1

√ t arccos( 1 √ tx 2 + 1 ) = |x - 1 3 tx 3 + 1 5 t 2 x 5 -. . . |
and taking the limit as t → 0, with t > 0 (use Equation 5) gives At the same time, we can study convergence of angles. For this take a triangle ABC with right angle at C and let α be the angle at A. For each t, the (lengths of the) sides opposite to A, B, C are respectively a t , b t , c t in the geometry E t . Take the side lengths a and b to be constant, a t = a and b t = b. For each t, we have, in the geometry E t ,

D 0 (A 0 , B x 0 ) 2 + D 0 (A 0 , C y 0 ) 2 = D 0 (B x 0 , C y 0 ) 2 ,
cos α t = tan √ ta tan √ tb .
As t → 0, we have cos α t → a/b. Thus, the angle α t in the triangle A t B t C t converges indeed to the Euclidean angle. Note that this can also be used to define the notion of angle in each geometry. The reader can check that this definition amounts to the one we gave in §5.

Finally we consider transition of area.

There are several works dealing with non-Euclidean area. For instance, in the paper [START_REF] Euler | Variae speculationes super area triangulorum sphaericorum[END_REF], Euler obtained the following formula for the area ∆ of a spherical triangle of side lengths a, b, c: . This formula should be compared to the Heron formula in Euclidean geometry that gives the area ∆ of a triangle in terms of its side length. We take the Taylor expansion in t in Formula [START_REF] Gauss | Königliche Gesellshaft der Wissenschaften[END_REF].

Using the formula for the cosine of the double and taking the square, we obtain the formula The first term in this expansion is 1 16 a 2 b 2 s 4 , that is, 1 16 a 2 b 2 t 2 , which is the square of the expression 1 4 abt. Recall now that for the transition of the distance function ( §5), we had to normalize the length function in E t , dividing it by √ t. It is natural then, for the transition of the area function, to divide it by t. Thus, the result that we obtain is 1 2 ab, which is indeed the area of a Euclidean triangle with base a and altitude b.

(
Finally, we note that there has been a recent activity on transition of geometries in dimension 3, namely, on moving continuously between the eight Thurston geometries, and also on varying continuously between Riemannian and Lorentzian geometries on orbifolds. We mention the works of Porti [START_REF] Porti | Regenerating hyperbolic cone structures from Nil[END_REF], [START_REF] Porti | Regenerating hyperbolic cone 3-manifolds from dimension 2[END_REF] and Porti and Weiss [START_REF] Porti | Deforming Euclidean cone 3-manifolds[END_REF] and Cooper, Hodgson and Kerckhoff [START_REF] Cooper | Three-dimensional orbifolds and conemanifolds[END_REF], Kerckhoff and Storm [START_REF] Kerckhoff | From the hyperbolic 24-cell to the cuboctahedron[END_REF] and Dancieger [START_REF] Dancieger | A geometric transition from hyperbolic to anti de Sitter geometry[END_REF]. The ideas and the methods are different from those of the present paper.

As a conclusion, we propose the following problem.

Problem. Extend this theory of transition of geometries to a parameter space which, instead of being the interval [-1, 1], is a neighborhood of the origin in the complex plane C. This is in accordance with a well-established tradition to trying to extend to teh complex world. We recall in this respect the following words of Painlevé [START_REF] Painlevé | Reproduced in OEuvres de Paul Painlevé[END_REF] "Entre deux vérités du domaine réel, le chemin le plus facile et le plus court passe bien souvent par le domaine complexe." (Between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.) 3 We also recall Riemann's words, from his Inaugural dissertation [START_REF] Riemann | Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse[END_REF], concerning the introduction of complex numbers: "If one applies these laws of dependence in an extended context, by giving the related variables complex values, there emerges a regulatrity and harmony which would otherwise have remained concealed."

cos 1 2 a cos 1 2 b cos 1 2

 2 a + cos b + cos c 4 c

  which is the Pythagorean theorem in the geometry E 0 . This is indeed the familiar Pythagorean theorem in Euclidean geometry.It is possible to do the same calculation in the hyperbolic case (t < 0), where the hyperbolic Pythagorean theorem is:

	Proposition 6.2. cosh( √ -tD t (A t , B x t )) cosh( √	-tD t (A t , C y t )) = cosh( √	-tD t (C y t , B x t )).
	7. Transition of angles and of area
	Using the Pythagorean theorem in spherical and hyperbolic geometry, it
	is possible to prove the other formulae in the above table, in particular the
	formulae	cos A t =	tan tan	√ a t √ b t
	in spherical geometry and	cos A t =	tanh tanh	√ a t √ b t
	in hyperbolic geometry.

  In the geometry E t (t > 0), Euler's formula becomes We would like to see that Euler's formula[START_REF] Gauss | Königliche Gesellshaft der Wissenschaften[END_REF] transits and leads to the area formula in Euclidean geometry for t = 0. It suffices to deal with right triangles, and we therefore assume that the angle at c is right. We use the

	Pythagorean theorem
					cos	√	tc = cos	√	ta cos	√ tb.
	Setting s =	√	cos t to simplify notation, we write this formula as 1 2 ∆ = 1 + cos √ ta + cos √ tb + cos √ tc 4 cos √ t 2 a cos √ t 2 b cos √ t . 2 c
	(8)		cos	1 2	∆ =	1 + cos sa + cos sb + cos sc 4 cos s 2 a cos s 2 b cos s 2 c	.

  cos 1 2 ∆) 2 = (1 + cos sa + cos sb + cos sa × cos sb) 2 2(1 + cos sa)(1 + cos sb)(1 + cos sa × cos sb)..

	The degree-8 Taylor expansion of 1 -(cos 1 2 ∆) 2 is
	1 16	a 2 b 2 s 4 + (	1 96	a 2 b 4 +	1 96	a 4 b

2

) + O(s 8 ).

This correspondence between poles and lines holds because we work in the elliptic plane, and not on the sphere. In the latter case, there would be two "poles", which are exchanged by the involution sq associated to a point q on the line.

Painelvé's sentence has been taken taken over by Hadamard in[START_REF] Hadamard | An Essay on the Psychology of Invention in the Mathematical Field[END_REF]: "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one."

The authors were supported by the French ANR project FINSLER, by the GEAR network of the National Science Foundation (GEometric structures And Representation varieties) and by the Erwin Schrödinger Institute (Vienna). The second author was also supported by a TUBITAK 2221 grant for a visit to Galatasaray University (Istanbul).


Keywords: ICT adoption, ICT usage, Ordred Logit Model, New Organizational Practices. JEL Classification: L21, O31, O33

Introduction

The economics literature has emphasized how prior IT investments and new forms of workplace organization may be complementary with investments in IT [START_REF] Brynjolfsson | Complementarity in Organizations[END_REF]. According to this literature, weak or lack of performance within firms that have invested in ICT is directly related to the lack or weak adoption of the complementary NOP especially in the emerging countries context1 .

Generally, two types of conjectures were formulated concerning the behaviour of companies in emerging countries with regard to the adoption and diffusion of IT and NOP. The first relates to the uncoordinated behaviours of adoption of the IT and NOP in emergent countries due to structural weakness of management and of governance of firms. This situation leads to the absence of complementarities between IT and NOP and weak global effect on performance. In some cases this anarchistic adoption process may lead to an internal "disorganization" of the firms and to a decrease of their structural competitiveness [START_REF] Knights | Information technology as organization/disorganization[END_REF].

The second conjecture supposes fast adoption of organizational and technological change dictated by the markets forces (requirement of competitiveness). In spite of their context firms are rather reactive and adopt rapidly new technologies and practices. Complementary adoption pushed by the market forces improves their performance. This is particularly true for firms operating in an international setting and which are export-oriented (Spyros, 2004).

Our article contributes to this line of research and demonstrates that firms operating in open emergent economies are reactive to both technological and organizational change via their international leakages. Adoption of IT and NOP is more related to market dynamics (external environment) than to internal environment (specificities of the given countries). Using a unique questionnaire run in 2004 for 175 firms, we show with an ordered logit model that in an earlier stage of IT and NOP adoption there was strong complementary in the adoption process. We will not explore the effects on performances and focus only on the link between the adoption and intensity of use of IT and NOP.

Our article presents two originalities. First, we investigate the complementarities between IT and NOP by considering three waves of technologies. We make the difference between the technologies based on their age and purpose. We show that the complementarity is a matter of the last waves of IT and less the first waves. Second, we show that the complementarity between IT and NOP adoption occurs in earlier stage of adoption of IT.

The paper is structured as follows. Section 2 surveys the related theoretical and empirical literature focused on the complementarity between IT and NOP. Section 3 presents the sample's characteristics and our econometric methodology. Section 4 discusses the results and section 5 concludes.

Literature Review

Three alternative research strategies were used in order to show such complementarities. The first one consists in examining the correlations between productivity, investment in IT and NPO through an approach by the production function [START_REF] Bresnahan | Information Technology, Workplace Organization and the Demand for Skilled Labor: Firm-Level Evidence[END_REF], [START_REF] Caroli | Skill-biased organizational change: evidence from a panel of British and French establishments[END_REF], [START_REF] Raymond | Antecedents and performance outcomes of advanced manufacturing systems sophistication in SMEs[END_REF]). The organization is approached as a sum of dummy variables showing if the new organizational practice exists or not within the firm. This research strategy examines if links exist rather than to show by which mechanisms go through [START_REF] Ben Youssef | Intra-firm diffusion of innovation: Evidence from Tunisian SME's in matter of Information and Communication Technologies[END_REF]al., 2011, Black andLynch, 2004;Bloom and al., 2010;Brynjolfsson and Hitt, 2004;[START_REF] Caroli | Skill-biased organizational change: evidence from a panel of British and French establishments[END_REF].

The second strategy used by Ichniowski and al. (1997), [START_REF] Athey | An Empirical Framework for Testing Theories About Complimentarity in Organizational Design[END_REF] and formalized by [START_REF] Greenan | A Firm Level Investigation of the Complementarity between Information and Communication Technologies and New Organizational Practices[END_REF] consists to use varied and multiple information on the evolution of computerization and the organization within firms and to assess the complementarities in a dynamic way. The complementarity is revealed by specific indicators showing the nature of the change between two periods.

The third strategy consists in an indirect approach [START_REF] Brynjolfsson | Computing Productivity: Firm-Level Evidence[END_REF]. The authors measure the evolution of the data-processing capital through time (6 to 8 years). This time coincides with life period of computer equipment. Their observations tend to show that the longer time of adoption of the material is, the more the impact on the productivity is high. This impact is revealed by the elasticity of the IT capital on production. Their explanation lies in the fact that, behind this observation, the factor organization hides. The improvement of the productivity would be related to the fact that the firms once they invested in IT adopt organisational practices. The two actions would be sequential and related one to the other.

The optimal combination between the two factors requires time. The longer time is and the more the firms reach a high degree of effectiveness.

We aim at contributing to this research line and to challenge the last research strategy. We advocate that the complementarity between the adoption of IT and adoption of NOP may be revealed by the difference in the extent of the link between different generations of IT and NOP. We show that the complementarity is strengthened when the technology evolves.

Adoption and usage of latest technologies are pushed by the prior adoption of NOP.

Data and empirical method

Data

The data for this study were gathered by a face-to-face questionnaire administrated to a random sample of 175 firms in Tunis, the largest metropolitan cities in Tunisia during the period from June 2004 to February 2005.2 Out of 320 surveys prepared, 205 were partially or totally completed and returned. However, only 175 usable surveys were retained for data analysis because they were totally completed and free from any contradictions, providing a response rate of nearly 55%. The high response rate is a common fact in Tunisia as the Tunisian legislation obligates firms to respond to such type of surveys and the face-to-face procedure also helped to reach such level of response.

Our sample has all the characteristics that make it an ideal candidate to represent the central nucleolus of stable and viable Tunisian firms. As we shall see below, there are no size and industry specific "traits" that would make it difficult to generalise our results. 3 This characteristic makes our study interesting when it comes to investigating IT adoption and usage in the majority of Tunisian firms (and, to some extent, of South Mediterranean firms) regions, and not only in the most advanced ones. Appendix 1, gives more details about the sample and its composition.

Prerequisite conditions for the success of such a survey and to make it comparable to what is practiced in other parts of the world were established. In particular regarding, training of interviewers, the test period. Participants were assured that their answers would be treated with confidentiality and all respondents were chosen from the top management staff.

Ben Youssef et al. ( 2011) identified three waves of IT using the same database. All indicators (adoption level of IT 4 , depth of usage of IT 5 and time required to use particular IT 6 ) show up that there are three waves: The first wave is called "general used technologies" assumed to be relatively widespread (more than 80%), intensively used (between 4 and 5 on Likert scale) and rapidly introduced in all business sectors. These technologies are: fixed phones, telecopy, office, computers and general purpose software. The second wave is formed by "intermediary technologies" with high potential of use. In the mid of the nineties they were named « new » IT: Internet, E-mail, specific software, free software, and mobile phones. The third wave is based on networking technologies. They are among the latest technological generations of IT.

Most of them need to optimize their use, costly investment, know-how, and qualified human resources. These technologies are Intranet, laptops, videoconference (VC) and Electronic Data Interchange (EDI). All firms use the technologies of the first wave. We focused on the second and third waves of these technologies.

Thus, in order to highlight the depth of adoption of IT and its main determinants, we discriminate between these waves of technologies and focus only on the second and third.

Empirical method

In order to study the relationship between IT adoption and usage and their determinants we use an ordered logit econometric model. The aim of the model is to determine the effect of a given factor on the probability of the IT adoption and use by the firm. This method makes it possible to study the impacts of different factors on a multinomial ordered variable. This method is widely used nowadays for similar studies [START_REF] Bocquet | The variety of ICT adopters in the intra-firm diffusion process: Theoretical arguments and empirical evidence[END_REF].

Starting from these assumptions, we use the ordered logit models 7 . The explained variable is subscripted from 0 to 9 for the total adoption's score. These variables are thus discrete and 4 Estimated by the percentage of firms using each technology 5 Evaluated by the average score of intensity of use indicated by respondents for each technology (likert scale) 6 Calculated by the time gap between the creation date of the firm and the first use of each technology 7 The basic variables of our study are binary and qualitative. Firms' answers give us the information on whether they adopt a particular technology or not. For example, a firm indicates if it chooses the adoption of Intranet or not. Since we have various types of binary variables, they are gathered, then, in different types of scores, in order to formulate a total score of IT adoption. This gives us the multinomial character of this distribution (because it ordinate. A logit multinomial model would thus neglect the ordinality of the dependent variable while a linear regression, on an opposite sense, would treat the difference between indices 3 and 4 in the same way that it treats the difference between indices 1 and 2, whereas this corresponds only to one classification. In these two cases, the estimators would be thus biased [START_REF] Greene | Econometric Analysis" Fourth Edition[END_REF][START_REF] Thomas | Econometric of the qualitative variables[END_REF]. Following this method, it's possible to study the influence exerted by series of factors on a multinomial ordered variable [START_REF] Greene | Econometric Analysis" Fourth Edition[END_REF][START_REF] Thomas | Econometric of the qualitative variables[END_REF]. Data were analysed using multiple regression analysis with a stepwise procedure used to determine the relative importance of a set of independent variables in determining the firm's intensity of IT use score. The data were analysed using SPSS 19.0, to the principal component analysis, and STATA, v. 11.0, to the estimation of the models. The data were examined for violation of the assumptions underlying multivariate methods prior to the analysis.

Definitions of the variables

Dependent variables

In our study we considered two models for two different dependant variables. Our dependent variables are obtained starting from the calculation of a total score of ICT adoption and a score of ICT usage for the second and third wave as defined above.

Intensity of ICT adoption

Our data describe five types of IT adopted by firms for the second wave and four types for the third wave. All IT variables are binary, in which value '0' means that the respondents were asked whether their firm had adopted the considered technology and value'1' that it is adopted. Thus, every firm has a score between 0 and 5 for the second wave and between 0 and 4 for the third wave. The variable used here is an ordered multinomial variable characterizing the adoption of the IT (score of adoption). We obtain two IT adoption score: IT adoption score (wave2) and IT adoption score (Wave3).

Intensity of IT use

is composed of various methods) and the ordered character (because it is deduced starting from other binary variables).

Firms were asked to classify themselves, on a 5 points ordinal scale, according to the answer8 that best described their intensity of usage of IT tools. The IT tools list included the tools listed above for Wave 2 and Wave 3. The weighted values were added across all IT tools listed to yield a total intensity of IT use score. Higher scores represent more intense use.

Explanatory variables

The aim of our estimation is to characterize the adoption and usage of IT by looking at two explanatory factors: characteristics of the firm and NOP.

characteristics of the firm

Five explanatory factors impacting new technology adoption were highlighted by empirical literature: size of the firm, firm's seniority, adoption time, employees' skills and, firm's web site 9 .

Size of the firm: is a variable with four response levels which measures a firm's size (from '0' if the firm has less than 25 employees to '3' if it has more than 100).

Firm's seniority: is a variable with four response levels which measures the "Firm Age" (from '0' if the firm aged less than 2 years to '3' if it aged more than 100 years).

Adoption time: it represents time since first adoption of IT tools. This variable summarizes the speed of adoption of the IT tools.

Web Site: is a dummy variable, which equals 1 if the firm has a Web site in 2005 and 0 if not.

Employees' skills: is a variable that measures the skilled labour intensity. It represents share of qualified employees (managers and high qualified personnel in the administrative and technical departments) over the total number of employees.

There are also other important factors that could influence the decision of IT adoption such as belonging to a group and whether the firm is national or multinational. But these variables are not available in our database.

New Organisational Practices (NOP)

Four groups of NOP are considered:

1-Work Organization: comprises Work by project, Modular work and 5S method 2-Quality approach: comprises ISO certification and Total Quality management (TQM); 3-Internal contractual arrangement: including organization in profit center, internal control, etc…; 4-External contractual arrangement: including the just in time delivery system and just in time process system. These four groups are all binary, in which value '0' means that the respondents were asked whether their firm had adopted the considered practice and value'1' that it is adopted.

Results and discussion

This section presents and discusses the results.

The estimates obtained in the case of the discrete variables are shown in Table 1. By observing all the explanatory variables, it can be outlined that almost all variables (except for those related to external contractual arrangement) has a statistically significant effect on the adoption and use of IT and our results confirm most of the theoretical expected effects.

Characteristics of the firm and patterns of adoption of IT

Firstly, our work confirms the rank effect described by [START_REF] Karshenas | Rank, Stock, Order, and Epidemic Effects in the Diffusion of New Process Technologies: An Empirical Model[END_REF] which has also been applied by [START_REF] Bartoloni | The Determinants of Technology Adoption in Italian Manufacturing Industries[END_REF]. We found that firms' adoption and use of IT is a positive function of their size. As it was expected, there is a positive correlation between firm's size and IT capital stock (adoption), showing the existence of scale economies for digital investment (Lawson and al., 2003). Larger firms are more likely to adopt digital technologies because they show lower levels of financial coordination tools (wave 2 and 3).

Secondly, concerning firm's seniority effect, we show that IT adoption and uses differ according to the models. Thus, the seniority of the firm has a more significant and positive effect on the adoption and the usage of the second wave IT then the third. On the other hand, the effect of this variable in the intensity of use's case is more important for aged firms than for the others. Within our sample and especially for the companies that are older than ten years we find a significant effect.

Thirdly, the adoption time variable is significant to explain the adoption and the depth of adoption of IT. We had defined this variable as the speed of adoption of each technology by individual firm that is the inverse of the time taken by each firm to adopt a new technology.

The correlation between the two variables indicates that a propagation effect exist and confirm our first intuition that the intra-firm hypotheses are suitable to descript the adoption phenomena. However, the relation between adoption time and IT adoption is nonlinear, as the effect of the variable "adoption time" on IT adoption increases but remains less than proportionally. The estimation results confirm this non-linearity as the sign of the square of the variable "Adoption time" is negative and statistically significant. We conclude that the IT adoption may be an increasing but concave function (see Appendix 3 for details).

Moreover, when we look at the variable having a website, we find that the correlation is significant with adoption and use of third waves' IT. However, this effect is not significant for the IT tools second wave. Our findings suggest that the contagion dynamics are working and the epidemic effect is confirmed [START_REF] Hollenstein | The determinants of the adoption of information and communication technologies (ICT). An empirical analysis based on firm-level data for the Swiss business sector[END_REF].

Finally, we find that firms that have more qualified personnel are those whose adoption and use of IT are more important. This result is important for the use and adoption of IT tools in the third wave. Thus, IT tools of the third wave require more skilled employees. These findings are supported by previous studies that also demonstrate a positive correlation between IT uses and workers skills (Bresnahan et al., 2002, Arvanitis andLoukis, 2009) and that investment in human capital is the main determinant of IT [START_REF] Mughal | Human capital and the adoption of information and communications technologies: Evidence from investment climate survey of Pakistan[END_REF].

NOP, adoption and intensity of use of IT

Tables 1 and2 shows that NOP have rather different effects on IT adoption and intensity of usage patterns. Firstly, there are asymmetric effects between third and second generation of IT. While the probability of adoption and intensity of use of IT of third is positive and significant when firms adopt NOP (Org gr1). There is no relation between adoption and Intensity of use of IT of second generation of IT and the adoption of NOP (Org gr1). The newest generation of IT are more likely to be adopted and to complement NOP. The latter comer effect is holding.

Secondly, table 3 shows also that NOP (Org gr2) quality management system is strongly related to intensity of usage of IT whether it is second or third generation of IT. However, only the adoption of third generation of IT is linked to NOP (Org gr2). Quality management systems involve an intensive usage of IT. The latest generations of IT seems more appropriate to this kind of NOP allowing better coordination of the entire quality processes. This is in line with the work of [START_REF] Foss | Original and Derived Judgment: An Entrepreneurial Theory of Economic Organization[END_REF], which demonstrated that the interaction among decisionmakers and executives are creating less hierarchical structures.

Thirdly, our results show also that NOP (Org gr3) internal contractualization system is strongly related to intensity of usage of IT whether it is second or third generation of IT.

However, these NOP are linked to the adoption to the second generation of IT and not to the third generation of IT. This kind of NOP seems does less intensively need coordination between team members [START_REF] Forman | Diffusion of information and communication technologies to businesses[END_REF]. This result is related to the one obtained by [START_REF] Martin | The effects of ICT use on employee's motivations: an empirical evaluation[END_REF] ICT usage increases the development of a team spirit between workers.

Fourthly, our results show that the adoption of IT (whether is second or third generation) is not linked to NOP (Org gr4) external contractual arrangement. However, intensity of usage of third IT generation is weakly linked to NOP (Org gr4) external contractual arrangement. This weak effect may be explained by weak adoption of IT of external partners (clients, subcontractors...), especially second generation. However an intensive usage of third generation of IT is linked to the adoption of NOP (Org gr4).

Overall, our results confirm that in an earlier stage of adoption of these technologies, the adoption of NOP by Tunisian firms have pushed them to adopted and use the latest wave of technology. The complementarity between the adoption of IT and adoption of NOP is revealed by the difference in the extent of the link between second and third generation of IT.

While, it is expected that the latest IT are less adopted and used. We show that prior adoption of NOP pushed Tunisian firms to better use and adopt the third wave of technologies.

Conclusion

The central proposition of this study is that the intensity of use of the latest information technologies (at that moment of time) is closely linked to the adoption of NOP of Tunisian firms. Our findings confirm the theoretical conjecture according to which market forces rather than time push firms even in an emergent country (Tunisia) to adopt NOP and IT.

In sum, investing in new IT-based production machinery changes not only the quantity of that input and related inputs, but it also changes a business's competitive strategy as well as the work practices and skill requirements of the workforce needed to implement the new strategy.

Adoption of the NPO was directly pushed by international subcontracting and the insertion of the Tunisian firms in the international market. This way of adoption of NPO preceded the computerization and the use of IT.

In terms of policy implications, the above findings suggest both the use of IT (of latest generation of technologies) and NOP will need to increase the skills of the employees.

Increasing the productivity of emergent economies firms will need huge investment in human capital. Otherwise, the process will be constrained. Upgrading human capital within emergent countries SMEs needs to build an absorptive capacity in matter of IT. It can be achieved by lowering, through different types of policy instruments, the hiring and training costs of educated workers, and especially university graduates.

Table 1

 1 Determinants of IT Adoption and Intensity of Usages

	Explanatory Variables	IT adoption	Intensity of IT use
		Wave 2	Wave 3	Wave 2	Wave 3
		Odds Ratios 10 Odds Ratios	Coef.	Coef.
	A. Internal characteristics of the firm
	Firm size
	10 Odds Ratio>1(respectively Odds Ratio<1) suggests that dependent variable are positively (respectively
	negatively) associated with the explanotry variable.

*: Significant at 10%, **: Significant at 5%, ***: Significant at 1% IT adoption Model results are presented as odds ratios

Table 2 : Scores of ICT adoption and usage: Summary Statistics

 2

					Scores Adoption		Score Usage
		Number of firms	%	total	Groupe 1 Groupe 2 Groupe 3	Total
	Size
	-5 to 25	33	18.86	7.30	3.75	2.84	0.69	40.69
	-25 to 50	39	22.29	8.41	3.76	3.58	1.05	44.84
	-50 to 100	40	22.86	8.60	3.65	3.40	1.55	44.50
	-more than 100	63	36.00	10.87	3.95	4.34	2.57	51.66
	Age
	-less than 2
	years	5	3.38	5.40	2.60	2.20	0.60	34.20
	-2 to 5 years	20	13.51	7.19	3.38	2.80	1.00	38.09
	-5 to 10 years	32	21.62	8.91	3.80	3.86	1.25	45.22
	-+ 10 years	91	61.49	9.72	3.93	3.84	1.93	48.92
	New organizational practices
	1. Work Organization
	-Work by
	project	73	41.71	10.36	3.97	4.19	2.20	51.38
	-Modular work	35	20.00	9.68	3.77	3.62	2.28	48.40
	-5 S method	46	26.29	10.58	3.97	4.28	2.32	52.34
	2. Quality approach
	-ISO
	certification	59	33.71	11.08	3.89	4.38	2.79	53.30
	-TQM	63	36.00	10.61	3.86	4.14	2.61	51.44
	3. Internal contractualization
	-Center of
	profit	64	36.57	10.28	3.98	4.32	1.96	51.40
	-Internal	28	16.00	10.14	3.82	4.00	2.32	50.50
	control
	4. External Contractualization
	-Delivery
	system	56	32.00	9.69	3.89	3.73	2.07	48.33
	-Production	41	23.43	9.26	3.85	3.39	2.02	46.09
	system
	Total	175	100	9.13	3.80	3.68	1.64	46.44

India[START_REF] Lal | Determinants of the adoption of information technology: a case study of electrical and electronic goods manufacturing firms in India[END_REF]; Chili(Fuentes & Gilchrist,

2005); Nigeria[START_REF] Apulu | Drivers for information and communication technology adoption: A case study of Nigerian small and medium sized enterprises[END_REF][START_REF] Lal | Globalization and adoption of ICTs in Nigerian SMEs[END_REF]; Tunisia[START_REF] Ben Youssef | Intra-firm diffusion of innovation: Evidence from Tunisian SME's in matter of Information and Communication Technologies[END_REF]; Golf Arab Countries[START_REF] Nour | The use and economic impacts of ICT at the macro-micro levels in the Arab Gulf countries[END_REF]; Pakistan[START_REF] Mughal | Human capital and the adoption of information and communications technologies: Evidence from investment climate survey of Pakistan[END_REF]; Malaysia[START_REF] Alam | ICT adoption in small and medium enterprises: An empirical evidence of service sectors in Malaysia[END_REF]

The whole number of industrial firms with more than 10 employees in Tunisia is stable and about 5600. More than

30% of them are partially or totally owned by foreign investors.3 Unfortunately the used survey did not include questions concerning ownership issue.

The answers were weighted as (1) null, (2) weak, (3) average (4) important, (5) very important.

There are also other important factors that could influence the decision of IT adoption such as belonging to a group and whether the firm is national or multinational. Unfortunately, these variables are not available in our database.