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Formam primam corporalem, quam quidam corporeitatem vocant, lucem esse arbitror.Lux enim per se in omnem partem se ipsam diffundit, ita ut a puncto lucis sphaera lucis quamvis magna subito generetur, nisi obsistat umbrosum. Corporeitas vero est, quam de necessitate consequitur extensio materiae secundum tres dimensiones, cum tamen utraque, corporeitas scilicet et materia, sit substantia in se ipsa simplex, omni carens dimensione. Formam vero in se ipsa simplicem et dimensione carentem in materiam similiter simplicem et dimensione carentem dimensionem in omnem partem inducere fuit impossibile, nisi seipsam multiplicando et in omnem partem subito se diffundendo et in sui diffusione materiam extendendo, cum non possit ipsa forma materiam derelinquere, quia non est separabilis, nec potest ipsa materia a forma evacuari. Atqui lucem esse proposui, cuius per se est haec operatio, scilicet se ipsam multiplicare et in omnem partem subito diffundere. Quicquid igitur hoc opus facit, aut est ipsa lux, aut est hoc opus faciens in quantum participans ipsam lucem, quae hoc facit per se. Corporeitas ergo aut est ipsa lux, aut est dictum opus faciens et in materiam dimensiones inducens, in quantum participat ipsam lucem et agit per virtutem ipsius lucis. At vero formam primam in materiam dimensiones inducere per virtutem formae consequentis ipsam est impossibile. Non est ergo lux forma consequens ipsam corporeitatem, sed est ipsa corporeitas. | THE first corporeal form which some call corporeity is in my opinion light. For light of its very nature diffuses itself in every direction in such a way that a point of light will produce instantaneously a sphere of light of any size whatsoever, unless some opaque object stands in the way. Now the extension of matter in three dimensions is a necessary concomitant of corporeity, and this despite the fact that both corporeity and matter are in themselves simple substances lacking all dimension. But a form that is in itself simple and without dimension could not introduce dimension in every direction into matter, which is likewise simple and without dimension, except by multiplying itself and diffusing itself instantaneously in every direction and thus extending matter in its own diffusion. For the form cannot desert matter, because it is inseparable from it, and matter itself cannot be deprived of form.--But I have proposed that it is light which possesses of its very nature the function of multiplying itself and diffusing itself instantaneously in all directions. Whatever performs this operation is either light or some other agent that acts in virtue of its participation in light to which this operation belongs essentially. Corporeity, therefore, is either light itself or the agent which performs the aforementioned operation and introduces dimensions into matter in virtue of its participation in light, and acts through the power of this same light. But the first form cannot introduce dimensions into matter through the power of a subsequent form. Therefore light is not a form subsequent to corporeity, but it is corporeity itself. |

Hoc itaque modo completo corpore primo, quod est firmamentum, ipsum expandit lumen suum ab omni parte sua in centrum totius. Cum enim sit lux perfectio primi corporis, quae naturaliter se ipsam multiplicat a corpore primo, de necessitate diffunditur lux in centrum totius. Quae cum sit forma tota non separabilis a materia in sui diffusione a corpore primo, secum extendit spiritualitatem materiae corporis primi. Et sic procedit a corpore primo lumen, quod est corpus spirituale, sive mavis dicere spiritus corporalis. Quod lumen in suo transitu non dividit corpus per quod transit, ideoque subito pertransit a corpore primi caeli usque ad centrum. Nec est eius transitus, sicut si intelligeretur aliquid unum numero transiens subito a caelo in centrum hoc enim forte est impossibile, sed suus transitus est per sui multiplicationem et infinitam generationem luminis. Ipsum ergo lumen a corpore primo in centrum expansum et collectum molem existentem infra corpus primum congregavit; et cum iam non potuit minorari corpus primum, utpote completum et invariabile, nec potuit locus fieri vacuus, necesse fuit, ipsa in congregatione partes extimas molis extendi et disgregari. Et sic proveniebat in intimis partibus dictae molis maior densitas, et in extimis augmentabatur raritas; fuitque potentia tanta luminis congregantis et ipsa in congregatione segregantis, ut ipsas partes extimas molis contentae infra corpus primum ad summum subtiliarent et rarefacerent. Et ita fiebat in ipsis partibus extimis dictae molis sphaera secunda completa nullius impressionis ultra receptibilis. Et sic est complementum et perfectio sphaerae secundae: lumen quidem gignitur ex prima sphaera, et lux, quae in prima sphaera est simplex, in secunda est duplicata. | When the first body, which is the firmament, has in this way been completely actualized, it diffuses its light (lumen) from every part of itself to the center of the universe. For since light (lux) is the perfection of the first body and naturally multiplies itself from the first body, it is necessarily diffused to the center of the universe. And since this light (lux) is a form entirely inseparable from matter in its diffusion from the first body, it extends along with itself the spirituality of the matter of the first body. Thus there proceeds from the first body light (lumen), which is a spiritual body, or if you prefer, a bodily spirit. This light (lumen) in its passing does not divide the body through which it passes, and thus it passes instantaneously from the body of the first heaven to the center of the universe. Furthermore, its passing is not to be understood in the sense of something numerically one passing instantaneously from that heaven to the center of the universe, for this is perhaps impossible, but its passing takes place through the multiplication of itself and the infinite generation of light (lumen). This light (lumen), expanded and brought together from the first body toward the center of the universe, gathered together the mass existing below the first body; and since the first body could no longer be lessened on account of its being completely actualized and unchangeable, and since, too, there could not be a space that was empty, it was necessary that in the very gathering together of this mass the outermost parts should be drawn out and expanded. Thus the inner parts of the aforesaid mass came to be more dense and the outer parts more rarefied; and so great was the power of this light (lumen) gathering together-and in the very act of gathering, separating-that the outermost parts of the mass contained below the first body were drawn out and rarefied to the highest degree. Thus in the outermost parts of the mass in question, the second sphere came into being, completely actualized and susceptible of no further impression. The completeness of actualization and the perfection of the second sphere consist in this that light (lumen) is begotten from the first sphere and that light (lux) which is simple in the first sphere is doubled in the second. |

Sicut autem lumen genitum a corpore primo complevit sphaeram secundam et intra secundam sphaeram molem densiorem reliquit, sic lumen genitum ex sphaera secunda sphaeram tertiam perficit et infra ipsam sphaeram tertiam molem adhuc densiorem congregatione reliquit. Atque ad hunc ordinem processit ipsa congregatio disgregans, donec complerentur novem sphaerae caelestes et congregaretur inter sphaeram nonam infimam moles densata, quae esset quattuor elementorum materia. Sphaera autem infima, quae est sphaera lunae, ex se etiam lumen gignens, lumine suo et molem infra se contentam congregavit et congregando partes eius extimas subtiliavit et disgregavit. Non tamen fuit huius luminis potentia tanta, ut congregando partes eius extimas disgregaret ad summum. Propterea remansit in omni parte molis huius imperfectio et possibilitas receptionis congregationis et disgregationis. Et pars suprema molis huius disgregata non ad summum, sua tamen disgregatione ignis effecta, remansit adhuc materia elementorum. Et hoc elementum ex se lumen gignens et molem infra se contentam congregans eius partes extimas disgregavit, minori tamem ipsius ignis disgregatione; et sic produxit ignem. Ignis vero ex se lumen gignens et molem infra contentam congregans eius partes extimas disgregavit, minori tamen ipsius disgregatione; et sic aerem produxit. Aer quoque ex se corpus spirituale vel spiritum corporalem generans et intra se contentum congregans et congregando exteriora eius disgregans aquam produxit et terram. Sed quia in aqua plus remansit de virtute congregante, quam disgregante, remansit etiam ipsa aqua cum terra ponderosa. | Just as the light (lumen) begotten from the first body completed the actualization of the second sphere and left a denser mass below the second sphere, so the light (lumen) begotten from the second sphere completed the actualization of the third sphere, and through its gathering left below this third sphere a mass of even greater density. This process of simultaneously gathering together and separating continued in this way until the nine heavenly spheres were completely actualized and there was gathered together below the ninth and lowest sphere the dense mass which constitutes the matter of the four elements. But the lowest sphere, he sphere of the moon, which also gives forth light (lumen) from itself, )y its light (lumen) gathered together the mass contained below itself and, by gathering it together, thinned out and expanded its outermost parts. The power of this light (lumen), however, was not so great that by drawing together it could expand the outermost parts of this mass to the highest degree. On this account every part of the mass was left imperfect and capable of being gathered together and expanded. The highest part of his mass was expanded, although not to the greatest possible extent. Nevertheless by its expansion it became fire, although remaining still the matter of the elements. This element giving forth light from itself and drawing together the mass contained below it expanded its outermost parts, but not to as great an extent as the fire was expanded, and in this way it produced air. Air, also, in bringing forth from itself, a spiritual body or a bodily spirit, and drawing together what is contained within itself, and by drawing together, expanding its outer parts, produced water and earth. But because water retained more of the power of drawing together than of the power of expanding, water as well as earth was left with the attribute of weight. |

Hoc igitur modo productae sunt in esse sphaerae 13 mundi huius sensibilis: novem scilicet caelestes, inalterabiles, inaugmentabiles, ingenerabiles et incorruptibiles, utpote completae, et quattuor existentes modo contrario, alterabiles, augmentabiles, generabiles et corruptibiles, utpote incompletae. Et patens est, quoniam omne corpus superius secundum lumen ex se progenitum est species et perfectio corporis sequentis. Et sicut unitas potentiâ est omnis numerus sequens, sic corpus primum multiplicatione sui luminis est omne corpus sequens. | In this way, therefore, the thirteen spheres of this sensible world were brought into being. Nine of them, the heavenly spheres, are not subject to change, increase, generation or corruption because they are completely actualized. The other four spheres have the opposite mode of being, that is, they are subject to change, increase, generation and corruption, because they are not completely actualized. It is clear that every higher body, in virtue of the light (lumen) which proceeds from it, is the form (species) and perfection of the body that comes after it, And just as unity is potentially every number that comes after it, so the first body, through the multiplication of its light, is every body that comes after it. |

Terra autem est omnia corpora superiora aggregatione in se luminum superiorum. Propterea ipsa est, quae a poëtis Pan dicitur id est totum; et eadem Cybele, quasi cubile, a cubo id est soliditate nominatur, quia ipsa est omnium corporum maxime compressa, hoc est Cybele mater deorum omnium, quia, cum in ipsa superiora lumina sint collecta, non sunt tamen in ea per operationes suas exorta, sed possibile est educi ex ea in actum et operationem lumen cuiuscunque sphaerae volueris; et ita ex ea quasi ex matre quadam quivis deorum procreabitur. Media autem corpora in duabus se habent habitudinibus. Ad inferiora quidem namque se habent sicut caelum primum ad omnia reliqua; et ad superiora, sicut terra ad omnia cetera. Et sic modis aliquibus in quolibet eorum sunt omnia reliqua. | Earth is all the higher bodies because all the higher lights come together in it. For this reason earth is called Pan by the poets, that is 'the whole,' and it is also given the name Cybele, which is almost like cubile, from cube (cubus) that is, a solid. The reason for this is that earth, that is to say, Cybele, the mother of all the gods, is the most compact of all bodies, because, although the higher lights are gathered together in it, nevertheless they do not have their source in the earth through its own operations, but the light (lumen) of any sphere whatever can be educed from it into act and operation. Thus every one of the gods will be begotten from it as from a kind of mother. The intermediate bodies have a twofold relationship. Towards lower bodies they have the same relation as the first heaven has to all other things, and they are related to the higher bodies as earth is related to all other things. And thus in a certain sense each thing contains all other things. |

Et species et perfectio corporum omnium est lux: sed superiorum corporum magis spiritualis et simplex, inferiorum vero corporum magis corporalis et multiplicata. Nec sunt omnia corpora eiusdem speciei, licet a luce simpla vel multiplicata fuerint profecta, sicut nec omnes numeri sunt eiusdem speciei, cum tamen sint ab unitate maiori vel minori multiplicatione collecti. | The form (species) and perfection of all bodies is light, but in the higher bodies it is more spiritual and simple, whereas in the lower bodies it is more corporeal and multiplied. Furthermore, all bodies are not of the same form (species) even though they all proceed from light, whether simple or multiplied, just as all numbers are not the same in form (species) despite the fact that they are all derived from unity by a greater or lesser multiplication. |

Et in hoc sermone forte manifesta est intentio dicentium "omnia esse unum ab unius lucis perfectione" et intentio dicentium "ea, quae sunt multa, esse multa ab ipsius lucis diversa multiplicatione." | This discussion may perhaps clarify the meaning of those who say that 'all things are one by the perfection of one light' and also the meaning of those who say that 'things which are many are many through the multiplication of light itself in different degrees.' |

Cum autem corpora inferiora participant formam superiorum corporum, corpus inferius participatione eiusdem formae cum superiore corpore est receptivum motus ab eadem virtute motiva incorporali, a qua virtute motiva movetur corpus superius. Quapropter virtus incorporalis intelligentiae vel animae, quae movet sphaeram primam et supremam motu diurno, movet omnes sphaeras caelestes inferiores eodem diurno motu. Sed quanto inferiores fuerint, tanto debilius hunc motum recipiunt, quia quanto fuerit sphaera inferior, tanto est in ea lux prima corporalis minus pura et debilior. | But since lower bodies participate in the form of the higher bodies, the lower body because it participates in the same form as the higher body, receives its motion from the same incorporeal moving power by which the higher body is moved. For this reason the incorporeal power of intelligence or soul, which moves the first and highest sphere with a diurnal motion, moves all the lower heavenly spheres with this same diurnal motion. But in proportion as these spheres are lower they receive this motion in a more weakened state because in proportion as a sphere is lower the purity and strength of the first corporeal light is lessened in it. |

Licet autem elementa participent formam caeli primi, non tamen moventur a motore caeli primi motu diurno. Quamquam participant illa luce prima, non tamen oboediunt virtuti motivae primae, cum habeant istam lucem impuram, debilem, elongatam a puritate eius in corpore primo, et cum habeant etiam densitatem materiae, quae est principium resistentiae et inoboedientiae. Putant tamen aliqui, quod sphaera ignis circumrotetur motu diurno, et significationem ipsius ponunt circumrotationem cometarum, et dicunt etiam hunc motum derivari usque in aquas maris, ita ut ex eo proveniat fluxus marium. Verumtamen omnes recte philosophantes terram ab hoc motu dicunt esse immunem. | But although the elements participate in the form of the first heaven, nevertheless they are not moved by the mover of the first heaven with a diurnal motion. Although they participate in that first light, they are not subject to the first moving power since that light in them is impure, weak, and far removed from the purity which it has in the first body, and also because they possess the denseness of matter which is the principle of resistance and stubbornness. Nevertheless, there are some who think that the sphere of fire rotates with a diurnal motion, and they take the rotating motion of comets to be an indication of this. They say also that this motion extends even to the waters of the sea, in such a way that the tide of the seas proceeds from it. But all sound philosophers say that the earth is free from this motion. |

Eodem quoque modo sphaerae, quae sunt post sphaeram secundam, quae fere secundum computationem in sursum facta nominatur octava, quia participant formam illius, communicant omnes in motu suo, quem habent proprium praeter motum diurnum. | In this same way, too, the spheres that come after the second sphere which is usually called the eighth when we compute from the earth upward, all share in the motion of this second sphere because they participate in its form. Indeed this motion is proper to each of them in addition to the diurnal motion. |

Ipsae autem caelestes sphaerae, quia completae sunt, non receptibiles rarefactionis aut condensationis, lux in eis non inclinat partes materiae a centro, ut rarefaciat eas, vel ad centrum, ut condenset. Et propter hoc ipsae sphaerae caelestes non sunt receptibiles motus sursum aut deorsum, sed solummodo motus circularis a virtute motiva intellectiva, quae in sese aspectum corporaliter reverberans ipsas sphaeras corporali circulat revolutione. Ipsa autem elementa, quia incompleta, rarefactibilia et condensabilia, inclinat lumen, quod in eis est, aut a centro, ut rarefaciat, aut ad centrum, ut condenset. Et propter hoc ipsa sunt aut sursum aut deorsum naturaliter mobilia. | But because the heavenly spheres are completely actualized and are not receptive of rarefaction or condensation, light (lux) in them does not incline the parts of matter either away from the center so as to rarefy them, or toward the center to condense them. On this account the heavenly spheres are not receptive of up or down motion but only of circular motion by an intellectual moving power, which by directing its glance upon them in a corporeal way revolves the spheres themselves in a circular corporeal motion. But because the elements are incompletely actualized and subject to rarefaction and condensation, the light (lumen) which is in them inclines them away from the center so as to rarefy them, or toward the center so as to condense them. And on this account they are naturally capable of being moved in an upward or downward motion. |

Amplius: formam primam corporalem formis omnibus sequentibus digniorem et excellentioris et nobilioris essentiae et magis assimilatam formis stantibus separatis arbitrantur sapientes. Lux vero omnibus rebus corporalibus dignioris et nobilioris et excellentioris essentiae est, et magis omnibus corporibus assimilatur formis stantibus separatis, quae sunt intelligentiae. Lux est ergo prima forma corporalis. | Furthermore, the first corporeal form is, in the opinion of the philosophers, more exalted and of a nobler and more excellent essence than all the forms that come after it. It bears, also, a closer resemblance to the forms that exist apart from matter. But light is more exalted and of a nobler and more excellent essence than all corporeal things. It has, moreover, greater similarity than all bodies to the forms that exist apart from matter, namely, the intelligences. Light therefore is the first corporeal form. |

In supremo autem corpore, quod est simplicissimum corporum, est reperire quattuor, scilicet formam, materiam, compositionem et compositum. Forma autem, utpote simplicissima, unitatis obtinet locum. Materia autem propter duplicem potentiam ipsius, susceptibilitatem scilicet impressionum et earundem receptibilitatem, et etiam propter densitatem, quae radicaliter est ipsius materiae, quae primo et principaliter accidit binario, binarii naturam merito sortitur. Compositio vero ternarium in se tenet, quia in ea patet materia formata et forma materiata et ipsa compositionis proprietas, quae a materia et forma alia et tertia reperitur in unoquoque composito. Et quod est compositum praeter haec tria proprium, sub numero quaternario comprehenditur. Est ergo in primo corpore, in quo scilicet virtualiter cetera corpora sunt, quaternarius, et ideo radicaliter numerus ceterorum corporum non ultra denarium invenitur. Unitas namque formae et binarius materiae et ternarius compositionis et quaternarius compositi, cum aggregantur, denarium constituunt. Propter hoc est denarius numerus corporum sphaerarum mundi, quia sphaera elementorum licet dividatur in quattuor, una tamen est participatione naturae terrestris corruptibilis. | The highest body, which is the simplest of all bodies, contains four constituents, namely form, matter, composition and the composite. Now the form being the simplest holds the position of unity. But matter on account of its twofold potency, namely its susceptibility to impressions and its receptiveness of them, and also on account of its' denseness which belongs fundamentally to matter but which is primarily and principally characteristic of a thing which is a duality, is rightly allotted the nature of a duality. But composition has a trinity in itself because there appears in it informed matter and materialized form and that which is distinctive of the composition, which is found in every composite as a third Constituent distinct from matter and form. And that which is the composite proper, over and above these three constituents, is classed as a quaternary. There is, therefore, in the first body, in which all other bodies exist virtually, a quaternary and therefore the number of the remaining bodies is basically not more than ten. For the unity of the form, the duality of the matter, the trinity of the composition and the quaternity of the composite when they are added make a total of ten. On this account ten is the number of the bodies of the spheres of the world, because the sphere of the elements, although it is divided into four, is nevertheless one by its participation in earthly corruptible nature. |

Ex his patet, quod denarius sit numerus universitatis.perfectus, quia omne totum et perfectum aliquid habet in se sicut formam et unitatem, et aliquid sicut materiam et binarium, et aliquid sicut compositionem et ternarium, et aliquid sicut compositum et quaternarium. Nec contingit ultra haec quattuor quintum addere. Quapropter omne totum et perfectum est decem. | From these considerations it is clear that ten is the perfect number in the universe, because every perfect whole has something in it corresponding to form and unity, and something corresponding to matter and duality, something corresponding to composition and trinity, and something corresponding to the composite and quaternity. Nor is it possible to add a fifth to these four. For this reason every perfect whole is ten. |

His autem manifestum est, quod solae quinque proportiones repertae in his quattuor numeris unum, duo, tria, quattuor aptantur compositioni et concordiae stabilienti omne compositum.Quapropter istae solae quinque proportiones concordes sunt in muscis modulationibus, gesticulationibus et rythmicis temporibus. Explicit tractatus de luce Lincolniensis. | On this account it is manifest that only five proportions found in these four numbers, one, two, three, four, are suited to composition and to the harmony that gives stability to every composite. For this reason these five proportions are the only ones that produce harmony in musical melodies, in bodily movements, and in rhythmic measures. |

Lux ergo, quae est prima forma in materia prima creata, seipsam per seipsam undique infinities multiplicans et in omnem partem aequaliter porrigens, materiam, quam relinquere non potuit, secum distrahens in tantam molem, quanta est mundi machina, in principio temporis extendebat. Nec potuit extensio materiae fieri per finitam lucis multiplicationem, quia simplex finities replicatum quantum non generat, sicut ostendit Aristoteles in de caelo et mundo. Infinities vero multiplicatum necesse est finitum quantum generare, quia productum ex infinita multiplicatione alicuius in infinitum excedit illud, ex cuius multiplicatione producitur. Atqui simplex a simplici non exceditur in infinitum, sed solum quantum finitum in infinitum excedit simplex. Quantum enim infinitum infinities infinite excedit simplex. Lux igitur, quae est in se simplex, infinities multiplicata materiam similiter simplicem in dimensiones finitae magnitudinis necesse est extendere. | Thus light, which is the first form created in first matter, multiplied itself by its very nature an infinite number of times on all sides and spread itself out uniformly in every direction. In this way it proceeded in the beginning of time to extend matter which it could not leave behind, by drawing it out along with itself into a mass the size of the material universe. This extension of matter could not be brought about through a finite multiplication of light, because the multiplication of a simple being a finite number of times does not produce a quantity, as Aristotle shows in the De Caelo et Mundo. However, the multiplication of a simple being an infinite number of times must produce a finite quantity, because a product which is the result of an infinite multiplication exceeds infinitely that through the multiplication of which it is produced, Now one simple being cannot exceed another simple being infinitely, but only a finite quantity infinitely exceeds a simple being. For an infinite quantity exceeds a simple being by infinity times infinity. Therefore, when light, which is in itself simple, is multiplied an infinite number of times, it must extend matter, which is likewise simple, into finite dimensions. |

Est autem possibile, ut aggregatio numeri infinita ad congregationem infinitam in omni numerali se habeat proportione et etiam in omni non numerali. Et sunt infinita aliis infinitis plura et alia aliis pauciora. Aggregatio omnium numerorum tam parium quam imparium est infinita, et ita est maior aggregatione omnium numerorum parium, quae nihilominus est infinita; Excedit namque eam aggregatione omnium numerorum imparium. Aggregatio etiam numerorum ab unitate continue duplorum est infinita; et similiter aggregatio omnium subduplorum illis duplis correspondentium est infinita. Quorum subduplorum aggregationem necesse est esse subduplam ad aggregationem duplorum suorum. Similiter aggregatio omnium numerorum ab unitate triplorum tripla est aggregationi omnium subtriplorum suorum istis triplis respondentium. Et similiter patet de omnibus speciebus numeralis proportionis, quoniam secundum quamlibet earum proportionari potest finitum ad infinitum. | It is possible, however, that an infinite sum of number is related to an infinite sum in every proportion, numerical and non-numerical. And some infinites are larger than other infinites, and some are smaller. Thus the sum of all numbers both even and odd is infinite. It is at the same time greater than the sum of all the even numbers although this is likewise infinite, for it exceeds it by the sum of all the odd numbers. The sum, too, of all numbers starting with one and continuing by doubling each successive number is infinite, and similarly the sum of all the halves corresponding to the doubles is infinite. The sum of these halves must be half of the sum of their doubles. In the same way the sum of all numbers starting with one and multiplying by three successively is three times the sum of all the thirds corresponding to these triples. It is likewise clear in regard to all kinds of numerical proportion that there can be a proportion of finite to infinite according to each of them. |

Si vero ponatur aggregatio infinita omnium duplorum continue ab unitate et aggregatio infinita omnium subduplorum illis duplis correspondentium, tollaturque de aggregatione subduplorum unitas vel quivis numerus finitus, iam subtractione facta non remanebit inter aggregationem primam et residuum de aggregatione secunda dupla proportio; sed nec aliqua numeralis proportio, quia si de numerali proportione per subtractionem a minori extremitate relinquatur alia numeralis proportio, oportet, ut subtractum istius, a quo subtrahitur, sit pars aliquota vel aliquot partes aliquotae. Numerus vero finitus numeri infiniti aliquota vel aliquot aliquotae esse non potest. Subtracto igitur numero de aggregatione subdupla infinita non remanet proportio numeralis inter aggregationem duplam infinitam et residuum de aggregatione subdupla infinita. | But if we posit an infinite sum of all doubles starting with one, and an infinite sum of all the halves corresponding to these doubles, and if one, or some other finite number, be subtracted from the sum of the halves, then, as soon as this subtraction is made, there will no longer be a two to one proportion between the first sum and what is left of the second sum. Indeed there will not be any numerical proportion, because if a second numerical proportion is to be left from the first as the result of subtraction from the lesser member of the proportion, then what is subtracted must be an aliquot part or aliquot parts of an aliquot part of that from which it is subtracted. But a finite number cannot be an aliquot part or aliquot parts of an aliquot part of an infinite number Therefore when we subtract a number from an infinite sum of halves there will not remain a numerical proportion between the infinite sum of doubles and what is left from the infinite sum of halves |

His ergo ita se habentibus manifestum est, quod lux multiplicatione sua infinita extendit materiam in dimensiones finitas minores et dimensiones finitas maiores secundum quaslibet proportiones se habentes ad invicem, numerales scilicet et non numerales. Si enim lux multiplicatione sui infinita extendit materiam in dimensionem bicubitam, eadem infinita multiplicatione duplicata extendit eam in dimensionem tetracubitam, et eadem subduplicata extendit eam in dimensionem monocubitam; et sic secundum ceteras proportiones numerales et non numerales. | Since this is so, it is clear that light through the infinite multiplication of itself extends matter into finite dimensions that are smaller and larger according to certain proportions that they have to one another, namely, numerical and non-numerical For if light through the infinite multiplication of itself extends matter into a dimension of two cubits, by the doubling of this same infinite multiplication it extends it into a dimension of four cubits, and by the dividing in half of this infinite multiplication, it extends it into a dimension of one cubit. Thus it proceeds according to numerical and non-numerical proportions. |

Iste, ut reor, fuit intellectus philosophorum ponentium omnia componi ex atomis et dicentium, corpora ex superficiebus componi et superficies ex lineis et lineas ex punctis. Nec contradicit haec sententia ei, quae ponit, magnitudinem solum ex magnitudinibus componi, quia tot modis dicitur totum, quot modis dicitur pars. Aliter namque dicitur medietas pars totius, quae bis sumpta reddit totum, et aliter est costa pars diametri, quae non aliquotiens sumpta reddit diametrum, sed aliquotiens sumpta exsuperatur a diametro. Et aliter dicitur angulus contingentiae pars anguli recti, in quo est infinities, et tamen finite subtractus ab eo diminuit illum; et aliter punctus pars lineae, in qua est infinities, et finite subtractus ab ea non diminuit eam. | It is my opinion that this was the meaning of the theory of those philosophers who held that everything is composed of atoms, and said that bodies are composed of surfaces, and surfaces of lines, and lines of points. This opinion does not contradict the theory that a magnitude is composed only of magnitudes, because for every meaning of the word whole, there is a corresponding meaning of the word part. Thus we say that a half is part of a whole, because two halves make a whole. We say, too, that a side is part of a diameter but in a different sense; because no matter how many times a side is taken it does not make a diameter, but is always less than the diameter. Again we say that an angle of contingence is part of a right angle because there is an infinite number of angles of contingence in a right angle, and yet when an angle of contingence is subtracted from a right angle a finite number of times the latter becomes smaller. It is in a different sense, however, that a point is said to be part of a line in which it is contained an infinite number of times, for when a point is taken away from a line a finite number of times this does not shorten the line. |

Rediens igitur ad sermonem meum dico, quod lux multiplicatione sui infinita in omnem partem aequaliter facta materiam undique aequaliter in formam sphaericam extendit, consequiturque de necessitate huius extensionis partes extremas materiae plus extendi et magis rarefieri, quam partes intimas centro propinquas. Et cum partes extremae fuerint ad summum rarefactae, partes interiores adhuc erunt maioris rarefactionis susceptibiles. | To return therefore to my theme, I say that light through the infinite multiplication of itself equally in all directions extends matter on all sides equally into the form of a sphere and, as a necessary consequence of this extension, the outermost parts of matter are more extended and more rarefied than those within, which are close to the center. And since the outermost parts will be rarefied to the highest degree, the inner parts will have the possibility of further rarefaction. |

Lux ergo praedicto modo materiam primam in formam sphaericam extendens et extremas partes ad summum rarefaciens, in extima sphaera complevit possibilitatem materiae, nec reliquit eam susceptibilem ulterioris impressionis. Et sic perfectum est corpus primum in extremitate sphaerae, quod dicitur firmamentum, nihil habens in sui compositione nisi materiam primam et formam primam. Et ideo est corpus simplicissimum quoad partes constituentes essentiam et maximam quantitatem, non differens a corpore genere nisi per hoc quod in ipso materia est completa per formam primam solum. Corpus vero genus, quod est in hoc et in aliis corporibus, habens in sui essentia materiam primam et formam primam, abstrahit a complemento materiae per formam primam et a diminutione materiae per formam primam. | In this way light, by extending first matter into the form of a sphere, and by rarefying its outermost parts to the highest degree, actualized completely in the outermost sphere the potentiality of matter, and left this matter without any potency to further impression. And thus the first body in the outermost part of the sphere, the body which is called the firmament, is perfect, because it has nothing in its composition but first matter and first form. It is therefore the simplest of all bodies with respect to the parts that constitute its essence and with respect to its quantity which is the greatest possible in extent. It differs from the genus body only in this respect, that in it the matter is completely actualized through the first form alone. But the genus body, which is in this and in other bodies and has in its essence first matter and first form; abstracts from the complete actualization of matter through the first form and from the diminution of matter through the first form, |

THE LOGIC MUSEUM II
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