About Gaspar Lax

Region	Western philosophy, Spanish philosophy
Era	Early modern philosophy, Renaissance philosophy
Name	Gaspar Lax
Birth Date	1487
Death Date	1560
Birth Place	Sariñena, Crown of Aragon
School Tradition	Scholasticism, Nominalism
Alma Mater	University of Paris
Main Interests	Logic, Mathematics

Gaspar Lax was an Aragonese philosopher, logician and mathematician. He was born in Sariñena (Huesca) in 1487, the son of Leonor de la Cueva and Gaspar Lax, a physician, and had two brothers and four sisters¹.

After a brief stay at the Estudio de Artes in Zaragoza, he went to Paris at the age of 17.

Period in Paris

By 1507-1508, he was already teaching at the Collège de Calvi, known as the “little Sorbonne,” and later moved to the Collège de Montaigu, where he alternated teaching with his own education under John Major².

In Paris, he was known as the “Prince of Sophists,” and his works and lessons were highly praised³.

He remained at Montaigu until 1516, leaving a lasting impression on his students, including Juan de Celaya, Juan Dolz del Castellar, Francisco de Vitoria, and Luis Vives⁴.

Return to Spain

In 1516, Lax returned to Spain, possibly due to a decree expelling foreigners from the University of Paris⁵.

He returned to Zaragoza in 1516 with great prestige and was hired at the Sertorian University of Huesca, earning 1,000 Jaquesan salaries and receiving special treatment and additional benefits. There, he taught arts from 1516 to 1520, the same year in which he obtained a degree in Theology from that university.⁶

He settled at the Estudio Mayor de Artes in Zaragoza, where he taught logic and mathematics. He was appointed Vice-Chancellor and Master, equivalent to Rector. His death in 1560 was followed with great interest by the members of the University, and he was buried in the Church of San Nicolás de Bari⁷.

At the University of Alcalá, some of Lax’s works were studied. For example, the “Tractatus terminorum una cum prima parte Summularum” (Paris, 1512) was reprinted by the University itself (1512) and recommended to students by the professor of logic, Tomás García de Villanueva. In the dedication, Lax is called “nostrae Hispaniae decus” (pride of our Spain)⁸.

He continued in Zaragoza with great recognition in the academic world. In 1559, he suffered from blindness and gout problems, and finally passed away on February 23, 1560, after having developed a prestigious career in France and Spain⁹.

Index of Works and Detailed Table of Contents

Tractatus exponibilium propositionum (Parisius 1507)

Littera minutissima

De resolutione — Page 5
De propositione exponibili — Page 6
De universalibus — Page 7
De exclusivis — Page 17
De exceptivis — Page 61
De reduplicativis — Page 77
De immediate — Page 82
De incipit et desinit — Page 89
De differt — Page 125
De ita et sicut — Page 137
De comparativis — Page 143
De superlativis — Page 166
De contingenti — Page 168
De signis collectivis — Page 171
De ly ‘infinitum’ — Page 178
De ly ‘semper’ ly ‘aeternum’ — Page 183
De ly ‘totus’ — Page 184
De ly ‘essentialiter’ et ‘intrinsece’ — Page 184
De terminis numerabilibus — Page 186
De ly ‘adaequate’ — Page 188
Expliciunt exponibilia magistri Gasparis Lax aragonensis — Page 189

Insolubilia (Parisius 1508)

Littera minutissima

Circa insolubilium propositionum difficultates…
Prima quaestio: An aliqua propositio possit se falsificare aut impossibilitare aut propter reflexionem significare aliter esse quam est — Page 3
Secunda quaestio: An in se falsificantibus propositionibus duo contradictoria simul vera vel simul falsa reperiantur — Page 14
Tertia quaestio: Utrum in propositionibus insolubilibus aliqua propositio vera et falsa, possibilis simul et impossibilis aut contingens et necessaria in eodem sensu reperiatur — Page 20
Quarta quaestio: Utrum aliquo casu possibili posito stet eundem simul decipi et non decipi seu errare et non errare — Page 33

Tractatus syllogismorum (Parisius 1509)

Littera minuta

Tractatus syllogismorum feliciter incipit — Page 9
Prima pars — Page 9
Secunda pars — Page 13
Tertia pars principalis — Page 32
Quarta pars principalis — Page 33
De Barbara — Page 34
De Celarent — Page 68
De Darii — Page 82
De Ferio — Page 88
De modis primae figurae concludentibus indirecte — Page 92
De Celantes — Page 100
De Dabitis — Page 105
De Fapesmo et Frisesomorum — Page 107
De secunda figura — Page 117
De Camestres — Page 126
De Festino — Page 132
De Baroco — Page 134
Tertia figura — Page 140
De Felapton et Ferison — Page 149
De Disamis — Page 154
De Bocardo — Page 156
De potestatibus syllogismorum — Page 163
Quinta pars principalis — Page 165
De syllogismis expositoriis — Page 165
Sexta et ultima pars principales — Page 179
De syllogismis qui solent fieri in terminis divinis — Page 179

Tractatus de materiis & de oppositionibus in generali (Parisius 1511)

Littera minuta

De materiis — Page 4
De oppositionibus in generali — Page 36
De contrariis — Page 6
Lex sive natura contrariarum — Page 15
Subcontrariae — Page 27
Lex sive natura subcontrariarum — Page 28
Contradictoriae — Page 33
Lex sive natura contradictoriarum — Page 40
Subalternae — Page 87
Lex sive natura subalternarum — Page 88
De aequipollentiis — Page 92
Prima regula — Page 92
Secunda regula — Page 93
Tertia regula — Page 96

Exponibilia (Parisius 1512)

Add index to this book

Insolubilia (Parisius 1512)

Circa insolubilium propositionum difficultates…
Prima quaestio — Page 4
Secunda quaestio — Page 24
Tertia quaestio — Page 35
Quarta quaestio — Page 66

Obligationes (Parisius 1512)

Litterae minutae

Prima pars obligationum — Page 12
Secunda pars principalis — Page 16
Prima regula — Page 16
Secunda regula principalis — Page 24
Tertia regula principalis — Page 33
Quarta regula principalis — Page 38
Tertia pars principalis — Page 66
Secunda principale — Page 67
Tertium — Page 68
Quarta et ultima pars principalis — Page 73
De positione complexa — Page 80
De positione disiunctivae — Page 81
De positione convertibilium et similium et dissimilium — Page 83
De depositione secunda specie obligationis — Page 95
De depositione complexa — Page 99
De mixtionibus istarum specierum obligationis — Page 108
Finis obligationum — Page 115

Termini. Secundo revisi (Parisiis 1512)

Terminus prout ad propositum sufficit capitur dupliciter — Page 7
De significare — Page 10
De divisionibus huius termini ‘terminus’ — Page 19
De alia divisione de ly ‘terminus’ — Page 31
De alia divisione termini — Page 36
Parvae divisiones terminorum — Page 42
De alia divisione terminorum — Page 50

Tractatus de oppositionibus propositionum categoricarum in speciali, et de earum aequipollentiis (Parisius 1512)

De contrariis — Page 6
Lex sive natura contrariarum — Page 15
Subcontrariae — Page 27
Lex sive natura subcontrariarum — Page 28
Contradictoriae — Page 33
Lex sive natura contradictoriarum — Page 40
Subalternae — Page 87
Lex sive natura subalternarum — Page 88
De aequipollentiis — Page 92
Prima regula — Page 92
Secunda regula — Page 93
Tertia regula — Page 96

Impositiones (Parisius 1513)

Prima quaestio: Utrum quilibet terminus… possit converti et ipsi in sic significando subordinari — Page 4
Secunda quaestio: Utrum quilibet terminus… possit converti et ipsi in sic significando subordinari — Page 58
Tertia quaestio: An quilibet terminus… possit converti et ipsi in sic significando subordinari — Page 68
Quarta quaestio: Utrum quilibet terminus… possit converti secundum illam significationem… — Page 77
Quinta quaestio: Utrum possit, illo ultimo modo tacto, fieri impositio tam absoluta quam dependens, cadens et renascens — Page 82

Tractatus syllogismorum (1514)

Tractatum syllogismorum in lucem iam emissurum — Page 4a
De divisionem huius termini ‘syllogismus’ — Page 4a
Pro diffinitione syllogismi categorici — Page 4b
Prima conditio — Page 4b
Secunda conditio — Page 5a
Tertia conditio — Page 5a
Contra istam solutionem… — Page 5b
Contra hoc arguitur dupliciter… — Page 5b
Quarta conditio — Page 6a
Secunda contra istam conditionem… — Page 6a
Ad istud argumentum… — Page 6a
De terminus ‘syllogismus’ dupliciter capitur — Page 6b
Secunda pars — Page 8b
De principiis regulativis syllogismorum affirmativorum — Page 8b
Regulae omnibus figuris generales — Page 9b
De regula ut tactum est… — Page 10a
De syllogismus est bonus… — Page 12b
De similitudine formae acceptionis terminorum — Page 13a
De propositiones — Page 13b
De particularitate medii sequitur — Page 17a
Tertia pars principalis — Page 27b
Conditiones generales omnibus syllogismis — Page 27b
Quarta pars principalis — Page 29a
De syllogismis primae figurae — Page 29a
De Barbara — Page 30a
De mixtione duarum modalium — Page 60b
De Celarent — Page 64b
Celarent secundus modus… — Page 64b
Quinta pars principalis — Page 175b
De syllogismis expositoriis — Page 175b
De syllogismis qui solent fieri in terminis divinis — Page 181
Expliciunt syllogismi Magistri Gasparis Lax — Page 187b
Tractatus de arte inveniendi medium — Page 190
De arte inveniendi medium ad syllogizandum… — Page 192a
De applicatione ad diversa genera syllogismorum — Page 203a
De opinionibus antiquorum circa artem inveniendi medium — Page 203a

Arithmetica speculativa (Parisius 1515)

Argumentorum duodecim librorum arithmetices Gasparis Lax
Primus: Numerorum communes proprietates ostendit — Page 7
Secundus: Proportionum et proportionalitatum communes passiones demonstrat
Tertius: De numero primo et composito… considerat
Quartus: Numerorum continuo proportionalium proprietates discutit
Quintus: Additionem, substractionem et partitionem proportionum docet
Sextus: De numeris quadratis, cubicis… pertractat
Septimus: Numeri paris, imparis… proprietates aperit
Octavus: De commensurabilitate et incommensurabilitate… inquirit
Nonus: Fractionum vulgarium… declarat
Decimus: Specierum proportionum rationalium… enucleat
Undecimus: Trium medietatum principalium… explanat
Duodecimus: Aliarum medietatum minus principalium passiones investigat

Proportiones (Parisius 1515)

Pages 52

Calculationes generales philosophice (Cesarauguste 1517)

Calculationum generalium liber primus — Page 4
Libri primi definitiones — Page 4
De dimensionibus quantitatis — Page 4
Primi libri praepositio prima — Page 5
De demonstrationibus propositionum — Page 8
De fundamentis de uniformibus — Page 12
De uniformiter difformiter longum — Page 15
Calculationum generalium liber secundus — Page 15
Libri secundi definitio prima — Page 15
De ratione maiorari — Page 18
Libri secundi propositio prima — Page 21
De propositionibus positas — Page 41
Calculationum generalium liber tertius — Page 46
Libri tertii definitio prima — Page 46
De ratione privativa — Page 49
De corollaria huius materiae — Page 49
De privativis et definitionem positarum — Page 55
De corollaria deducta ex definitionibus — Page 56
Libri tertii propositio prima — Page 57
Calculationum generalium liber quartus — Page 73
Libri quarti definitio prima — Page 73
De demonstrationes propositionum — Page 78
Libri quarti propositio prima — Page 79
Calculationum generalium liber quintus — Page 88
Libri quinti propositio prima — Page 88

Tractatus parvorum logicalium (Cesaraugustae 1521)

Suppositio est terminus qui accipitur in propositione pro suo significato — Page 12
De suppositionibus improprie dictis… — Page 18
De suppositione proprie dicta et materiali — Page 20
De suppositione proprie dicta et personali — Page 23
De relativis — Page 31
De ampliationibus — Page 38
De restrictionibus — Page 49
De appellationibus — Page 51
De appellatione formali — Page 52
De positione propositionum inesse — Page 54
De modalibus — Page 66
De modalibus divisis — Page 71
De oppositionibus et aequipollentiis modalium divisarum — Page 84
De consequentiis in generali — Page 85
De divisione consequentiae… — Page 98
De regulis generalibus consequentiarum — Page 106
De ascensu et de descensu — Page 109
De divisione descensus et ascensus — Page 116
De conversione simplici — Page 123
De conversion per accidens — Page 133
De conversione per contrapositiontem. Prima Quaestio — Page 134
De conversione per contrapositiontem. Secunda Quaestio — Page 145
De conversione per contrapositionem pertractandum est — Page 150
Finis — Page 160

Tractatus summularum (Cesaraugustane 1521)

De diffinitione — Page 12
De divisione — Page 15
De termino — Page 18
De significare — Page 24
De significare naturaliter et eius divisione — Page 31
De significare ad placitum et eius divisione — Page 28
De divisione termini in mentalem, vocalem, scriptum et mixtum — Page 35
De termino categorematico et syncategorematico — Page 47
De termino primae et secundae intentionis — Page 49
De termino singulari et communi — Page 55
De aliis divisionibus terminorum — Page 64
Capitulum de nomine — Page 66
Capitulum de verbo — Page 75
Capitulum de oratione — Page 79
Capitulum de Propositione — Page 79
Capitulum de divisione propositionis penes substantiam — Page 85
Capitulum de subiecto et praedicato — Page 89
Divisio propositionis categoricae penes quantitatem — Page 92
Divisio propositionis penes qualitatem — Page 96

Summa parvorum logicalium (Cesaraugustae 1525)

(I don’t have this book)

Questiones phisicales (Cesarauguste 1527)

Quaestiones physicales magistri Gasparis Lax feliciter incipiunt
De quantitate — Page 7
De quantitate discreta — Page 35
De toto — Page 38
De maximo et minimo — Page 44
De infinito — Page 79
De moto penes causam — Page 103
De raritate et densitate — Page 135

Summa Syllogismorum (1528)

Tractatum syllogismorum — Page 67c
Secunda pars: De principiis regulativis syllogismorum — Page 69a
In hac tertia parte principali videndae sunt: Conditiones generales omnibus syllogismis — Page 72c
Quarta pars — Page 72d
De prima figura — Page 72d
De Barbara — Page 73a
De Celarent — Page 79d
De Darii — Page 82c
De Ferio — Page 84b
De Fapesmo et Frisesomorum — Page 89a
De secunda figura — Page 90d
De Cesare — Page 91
De Camestres — Page 93c
De Festino — Page 94b
De Baroco — Page 94b
Tertia figura — Page 95c
De Darapti et Datisi — Page 95d
De Felapton et Ferison — Page 97c
De Disamis — Page 99b
De Bocardo — Page 100a
Quinta pars principalis — Page 101c
Ultima pars principalis — Page 103c
De syllogismi in terminis divinis — Page 103c
De syllogismis expositoriis in terminis divinis — Page 104d
Tractatus de arte inveniendi medium — Page 105b
Explicit Summa syllogismorum — Page 109d

Summa parvorum logicalium (Cesaraugustae 1532)

De suppositionibus — Page 3c
De suppositionem improprie dictam et proprie dictam — Page 7d
De suppositione proprie dicta et materiali — Page 8c
De suppositione proprie dicta et personali — Page 9d
De relativis grammaticalibus — Page 14a
De ampliationibus — Page 17b
De restrictionibus — Page 24c
De appellationibus — Page 25c
De appellatione formali — Page 26c
De positione propositionum de inesse — Page 27c
De modalibus — Page 33c
De modalibus divisis — Page 36b
De oppositionibus et aequipollentiis modalium divisarum — Page 43b
Explicit Summa parvorum logicalium — Page 44b

Tractatus Consequentiarum (1532)

De consequentis in generali — Page 45c
Divisio consequentiae — Page 53c
De regulis generalibus consequentiarum — Page 57c
De descensu et ascensu — Page 58c
Explicit Tractatus consequentiarum — Page 66b

References

González Ancín, Miguel & Towns, Otis. (2017). Miguel Servet en España (1506-1527). Edición ampliada, pp. 140-142, 360. ↩
Ayala Martínez, Jorge M. (2000). Pensadores Aragoneses. Historia de las ideas filosóficas en Aragón, p. 271. ↩
English Wikipedia: Gaspar Lax. ↩
Ayala Martínez, Jorge M. (2000). Pensadores Aragoneses. Historia de las ideas filosóficas en Aragón, p. 271. ↩
Ayala Martínez, Jorge M. (2000). Pensadores Aragoneses. Historia de las ideas filosóficas en Aragón, p. 272. ↩
González Ancín y Towns, 2017, pp. 148-150. ↩
Ayala Martínez, Jorge M. (2000). Pensadores Aragoneses. Historia de las ideas filosóficas en Aragón, p. 272. ↩
Ayala Martínez, Jorge M. (2000). Pensadores Aragoneses. Historia de las ideas filosóficas en Aragón. pp. 272-273. ↩
Spanish Wikipedia: Gaspar Lax ↩