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6 The Uses of Mathematics in Opus maius, Parts IV-VII: Science Interpreted Moraliter.

6 The Uses of Mathematics in Opus maius, Parts IV-VII: Science Interpreted Moraliter.

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J. Hackett

Aristotle says in the sixth book of the Metaphysics, mathematics, natural and

divine, the mathematical is of no small importance in grasping the knowledge of

the other two parts, as Ptolemy states in the first chapter of the Almagest’ (Opus

maius, Part IV, 98–99).. Bacon covers a wide area in his application of mathematics, physics, astrology, theology, church history, geography.

For the purposes of this chapter, however, I will focus on the section in Opus

maius IV dealing with Astrology/Scientia Experimentalis: the reason for this is to

indicate that Bacon may well be the one condemned in regard to the charge of

‘Astral Determinism’ in the Parisian Condemnations of 1270, 1277. While Bacon is

not named explicitly, it is clear that some of the propositions cover his doctrine of

universal univocal causation based on the effects of the heavenly bodies on human

bodies and on temperament. And while Bacon does attempt to save freedom of the

Will, it is also clear that he does believe in an astral determination of individual

personality and in the astral determinism of sects and religions. Or better, he clearly

attempts to save Astronomy-Astrology for Statecraft and for the Church by means

of argument and authority: the matter is so important to him that he treats of it in

many places, most notably in Opus maius IV, Opus maiux VI, part three, the

Introduction and Notes to the Secretum secretorum, De secretis operibus naturae

et de nullitate magiae.

Writing on Astrology in the Middle Ages and Renaissance, the renowned

authority on the subject, Graziella Federici Vescovini states:

Roger Bacon was one of the staunchest defenders of the theory of the birth of religions brought

about by the great conjunctions. He Bacon agrees with the astrological theory of religions and

while he strongly advocates Freedom of the Will, he does so at the cost of opposing the latter to

a more strictly deterministic world based on the influence of the stars. Proposition 68, 70,

76, 94 deal with this issue of Fatalism based on ‘Astral Determinism’ (Vescovini 2011, 2014).

I have argued that Proposition 101–107 beginning with the claim that no agency is

open to alternatives; indeed, all agency is determined to one outcome clearly find

warrant in Roger Bacon’s works (Hackett 2000). The diversity of heavenly positions

determines one’s fate. Further, not only is one’s physical complexion is determined

but so also one’s spiritual well-being is determined. Health, infirmity, life and death

are determined by the stars such that the heavenly bodies influence human destiny.

Bacon scientia experimentalis, Part III, and his Moralis philosophia, Part IV,

which Bacon calls the most important part of his Moral Philosophy, are a major

attempt to save Mathematics (Astronomy-Astrology) in regard to contingent

changes on earth and in regard to the human organism. It is the basis for his theory

of prediction of the future and for his astro-sociology of world-religions. A related

approach to Bacon can be found in the works of his younger contemporary, and I

believe his helper, Master Peter of Limoges. This is clear from the sermon De

antichristo (Be´riou 1986).12 Both John North and Graziella Federici Vescovini


See Be´riou (1986). This work reference to a De antichristo by Pierre de Limoges, who was a

Master in Arts at Paris, and then, a theologian, in the 1260s indicates that he had with Roger Bacon

a common concern with astronomy/astrology.

7 From Sapientes antiqui at Lincoln to the New Sapientes moderni. . .


have documented Bacon’s use and dependence of Abu’Mashr’s Liber

conjunctionum, and I have argued for his additional uses of Abu’s Mashr’s Introduction to Astronomy, Pseudo-Ptolemy, Centiloquium, Pseudo-Ptolemy, De disposition sphaerae.

To sum up: Bacon defends a radical opposition between inviolate Freedom of

the Will, and physical laws of nature that are determinate and necessary. Further, all

influence of species in medio are physical (Hackett 2011). There are no spiritual

species in nature. In this, Bacon believes the vulgus in philosophy and theology is

just confused. Bacon believes in Astral Determinism both in respect of physical and

psychological reality. This has been shown in regard to the doctrine of species by

Yael Kedar in her fine Ph.D. work at Haifa University (Kedar 2009). The importance of all of this for the education of the prince as seen in the Pseudo-Aristotle,

Secretum secretorum, Part VII of the Opus maius is connected with the defense of


To appreciate how toxic Bacon’s ideas may have appeared to some contemporaries one need only turn to the Errores philosophorum of Aegidius Romanum

(Giles of Rome, OESA 1944). Written c. 1268–1271, this work is a head on attack

on Greco-Arabian necessetarianism. I have already noted the same reaction to

Bacon’s concerns by the Master-General, Bonaventure.13

Giles of Rome’s work was written between 1269 and 1272 and is exactly

contemporary with Roger Bacon’s writings for Pope Clement IV. Condemned

here are the natural philosophy and metaphysics of Aristotle, Avicenna, Averroes,

al-Ghazali, al-Kindi and Maimonides. With the exception possibly of Maimonides,

all of these thinkers are defended by Roger Bacon. When one turns to al-Kindi and

his theory of universal determinism, radiation and magic, one notices the extent of

Roger Bacon’s dependence on al-Kindi. Aegidius claims (1) ‘al-Kindi erred in

asserting that the future depends simply and without qualification upon the state of

the super-celestial bodies. Hence in this same book in the chapter on the rays of the

stars, he states that “one who knew fully the state of the heavenly bodies would have

complete knowledge of both past and future.”’ (2) ‘Again, he erred in believing that

the effects of all causes in the world extended to every individual.’ (3) Ulterius,

erravit credens omnia de necessitate contingent.

These errors of the philosopher al-Kindi could be directed entirely against Roger

Bacon, and in the context of 1270 they were probably directed against Bacon and

his circle, especially since this circle took the work of mathematics so seriously and

because by means of the scientia experimentalis, the older Bacon and the younger

Magisters Peter of Maricourt (Picardus) and perhaps Magister Petrus de Limoges,

attempted to defend this new form of scientific study in the context of the more

verbal and dialectical education in the arts. The latter had interests similar to Bacon

in regard to the importance of astronomy/astrology, and to their uses in Theology.


See note 34 above.



J. Hackett

The Applications of Mathematics: Perspectiva

Richard Newhauser has demonstrated the significant use by Master Peter of Limoges

of Roger Bacon’s moral uses of the study of Perspectiva. He shows the extent to

which this Master drew heavily on the third part of Bacon’s Opus maius, Part V, the

Perspectiva (Newhauser 2012).14 In Roma magistra mundi, the essays in honor of

Fr. Leonard E. Boyle, I provided reasons for thinking that Paris MS BN Lat. 7434

contains a version of the Perspectiva that pre-dates the text of the Perspectiva in the

Opus maius. Further, I believe that it contains the oldest text of the Perspectiva.15

Just recently, A. Mark Smith has shed new light on the nature of Bacon’s

Perspectiva. Smith comments as follows:

Although Grosseteste’s effort to submit the physics of light to geometrical analysis was

only partly successful, he inspired Roger Bacon to bring that effort to fruition. It is perhaps

no exaggeration, in fact, to say that in sharing the same Augustinian theological leanings,

the same desire for broad learning, and the same enthusiasm for applying mathematics to

the analysis of natural philosophy, Bacon was, in a sense, Grosseteste’s alter ego. But

Bacon had one clear advantage over Grosseteste: he could draw on a much wider array of

sources in carrying out his program (Smith 2015).

As Bacon notes in the Opus tertium, there was no knowledge of Perspectiva at

Paris in the 1260s; the subject had been taught but twice, and that was at the

University of Oxford. Smith does a great job in situating the three parts of the

Perspectiva. He devotes more care to part one on the relation of optics and the

psychological process of perception than Lindberg did. Of course, this was the big

bone of contention between both scholars: Was Bacon primarily interested in providing a geometrical theory of vision after the manner of Ibn al-Haytham or was the

concern with optics subordinated to providing for a more comprehensive and better

theory of perception and knowledge. In regard to Roger Bacon, I believe that Smith

has now shown that the latter position is the correct one in the case of Bacon. He

shows that this is not so in the case of Witelo. Here, much more than in Bacon and in

Pecham, one is dealing primarily with a detailed mathematical treatment of optics.

Indeed, Smith is quite correct in noticing that with respect to the mathematics, Bacon

does not make advances in the arguments. He simply falls back on Euclid in his

attempt to solve the problem of the angle of incidence. But of course, Bacon was

writing an introductory persuasio to encourage the serious study of optics in the part

of the curriculum of the medieval university, called the Quadrivium.16


See Newhauser (2012) Introduction, xi–xxiii for Peter’s reliance on both Roger Bacon and John

Pecham, but especially, Peter’s use of part three of Bacon’s Perspectiva. Quite significant here is

the fact that Peter nowhere mentions Bacon by name.


See Hackett (1998). I have prepared an edition of this text and am working on a translation of

this text.


Might it not have been the case that it was Bacon’s intention by means of his works for Pope

Clement IV to influence the progress of science at the Papal Studium in Viterbo. As we will see

below, the influence of his works there in the mid-1270s has been acknowledged by David

C. Lindberg.

7 From Sapientes antiqui at Lincoln to the New Sapientes moderni. . .


Indeed, Bacon’s advocacy for optics was very successful. In the next decade,

Bacon’s confrere, John Pecham wrote the textbook for the university teaching of

optics, the Perspectiva communis. It is a masterpiece of educational pedagogy.

Thus, Bacon’s plea for the improved study of science in the medieval university

was not a failure; it was a massive success.

I will argue that Bacon’s intense theological concerns condition the manner in

which he sets up the parameters of his scientific account of optics in Part I and Part

II of the text.


The Third Part of the Perspectiva

One issue has bothered students of Bacon’s optics. Why does Bacon make such a

spirited defense of a combination of an intromission and extramission theory of

vision. Now, of course, one could argue that he is combining elements of Neoplatonic Aristotelianism with the straight geometrical optics of Ibn al-Haytham. There

is some truth in that view. However, a review of the Third part of the Perspectiva,

that is the part which deals with the spiritual uses of optics, the part indeed which so

much appealed to the great Master of Preaching, the theologian at the Sorbonne,

Peter of Limoges as can be seen from his Tractatus de oculo morali indicates

another reason for the combining of intromissions and extramission. Here one can

find Bacon’s rationale for his integration of both an intromission theory of vision

and an extramission theory of vision.

Following his remarks on the need for optics in order to know the natures of

things, Bacon writes about the preservation of the spiritual pupil of the eye, that is,

the soul.

In a comment on this, Bacon lists the seven spiritual gifts, the seven virtues, three

theological and four cardinal virtues, seven gifts of the Holy Spirit and seven petitions of

the Our Father, the eight beatitudes (here, the eyelids provide a convenient eight member to

the seven parts of the eye, ‘so that to the eight spiritual guardians there correspond the same

number of corporeal ones’ (Lindberg 1996).

David C. Lindberg saw Bacon’s combining of the intromission/extramission

theories of vision as a sign of confusion in optical theory. I can see why he would

think so, but I am not so sure that that alone is the issue. Could it not be the case that

Bacon’s theological motivation, namely, the need to find a model for the relation of

grace and free will requires him to have a perceptual theory that unites intromission

and extramission to match what is required in theological doctrine. Is it not the case

that Bacon finds a convenient symbiosis between natural science and the requirements of theology? Indeed, it is the case. That Bacon is motivated by theological

concern is clear from what follows in his text:

It has been said that not only is intromission (of species) required for vision, but also the

extramission and cooperation of its own for vision, but also the extramission and


J. Hackett

cooperation of its own power and species. Similarly, spiritual vision requires not only that

the soul should be the recipient from without of divine Grace and Powers, but also that it

should cooperate by its own power. For consent and the exercise of free will are required,

along with the grace of God, if we are to see and gain the state of salvation (Ibid.) [Text 15].

In other words, Bacon’s fusion of extramission with intromission theory of

vision is motivated by the need to find a natural analogue in vision and perception

for the relation between Grace and human freedom of the will.

Further, citing the relation between the need for proper distance for vision

and spiritual distance, Bacon draws a very tight parallel between perspectival

perception and spiritual perception, in this case drawing closely from Ibn


Since corporeal vision is of three kinds—namely, sense alone [direct impact of the species

of light and color], knowledge of the universal [vague individual] and knowledge by

syllogism—it is likewise necessary for mankind to have a threefold [spiritual] vision. For

by sense alone we gain an insufficient grasp of a few things, such as light and color; and this

cognition is weak, revealing whether things exist and what they are. But by knowledge we

grasp what kind they are and what qualities they possess; whether the light of the sun or the

moon, whether white or black. By syllogism we grasp everything associated with light and

color according to all twenty common sensibles. Therefore, the first cognition is weak, the

second is more perfect, and the third is most perfect. So it is that in spiritual vision; for what

a man knows by his own sense alone is very modest, since he lacks the other two kinds of

cognition, [the first of which is] through teachers, from youth to old age, for we can always

learn them from those who are wiser than ourselves. And [if cognition is by sense alone] we

are also without the third kind of cognition, which occurs through divine illumination


But what is also significant about cognitio per syllogismum is that it has nothing

whatsoever to do with the common Aristotelian use of the term Syllogism as in

formal argument. Bacon is referring to the use of the term in Ibn al-Haytham’s

optics where it refers to direct intuitive perception of singulars. This is the kind of

confirmatory use of intuitive experience that in the Opus maius, Part VI, Bacon

holds is needed to confirm the rational teachings of books and teachers. It has much

significance for epistemology in that it starts a tradition that in Duns Scotus and

William of Ockham and their contemporaries will become central, the intuitive

cognition of singulars (South 2002). This is the source of the doctrine of experience

and intuitive cognition in William of Ockham which has been clearly outlined by

Peter King (2003).17

The remaining two sections deal with the subdivision of vision into direct,

reflected and refracted vision and to its uses in moral persuasion. It also deals

with the application of mirrors to the technology of war.


For the beginning of this account of experience in Roger Bacon, see Hackett (2008–2009).

7 From Sapientes antiqui at Lincoln to the New Sapientes moderni. . .



The Utility of Mathematics: Opus maius VI: De scientia


In this section, I will deal with only one aspect of Bacon’s concerns in his Opus

maius, Part VI. The matter of his relation to the Magister Petrus de Maricourt,

Picardus, has been carefully examined recently by Sylvia Nagel (2012). I will

examine Bacon’s stated concern in the aforementioned text about the nova

translatio of Aristotle’s Meteorologica.

Bacon is quite clear: he does not have time to complete a formal treatise on the

rainbow, halo and related matters. But why in the first place is Bacon so exercised

about the rainbow and halo? What is the big deal? Here we must return to Robert

Grosseteste, and the matter of the translations of Aristotle, especially the translation

of the Book of the Meteorologies of Aristotle.


Roger Bacon’s Criticism of the Translators, Especially

the Translation of William of Moerbecke

Throughout the Opus maius and elsewhere, Bacon talks about the importance of

Book Three of Aristotle’s Metheorologica on the rainbow and halo. Further, the

account of the rainbow and halo provides Bacon with the exemplum for the first

prerogative of his Scientia experimentalis, the confirmation by adequate experiences of the rational claims of the other parts of Natural Philosophy and Optics. But

why pick Book three of Aristotle’s Metheorologica? Surely, there are lots of other

examples from Ibn al-Haythan, Ptolemy and others. What is going on here? Does it

perhaps have to do with difficulties in the translation of Aristotle from Greek into

Latin? (Perspectiva 324–325).

The late Joseph Brams once observed that Bacon had spoken strongly against

Guillelmus Flemengus, translator, and that his remarks are compromised by ‘le ton

ironique et le gout de l’exagge´ration.’ Yet, he saw Bacon’s remarks as an issue of

some embarrassment for modern scholars. The newly published critical edition and

study of William of Moerbecke’s nova translation of the Meteorologica of Aristotle

in the Aristoteles Latinus series by Gudrun Vuillemin-Diem makes it possible for

the first time to provide a trustworthy evaluation of Bacon’s remarks in Opus maius

VI on the rainbow (Aristoteles Latinus 2008, X, 2.1/2.2, Meteorologica). Further,

her study of Moerbecke’s translation methods, especially in his translation(s) of the

Meteorologica allow us to prove that Bacon’s strong remarks in 1271–1272 about

Moerbecke as a translator were no exaggeration. They expressed a serious concern

with the first two versions of the translation, which did, indeed, have serious


What are the dates of the three successive translations and which scholars at

Paris first used them? G1a the uncorrected version was done in 1260. It is likely that

Bacon knew this version c. 1266–1267, though he may also have known version


J. Hackett

G1b, the second version. This latter version was known to Thomas Aquinas

c. 1269–1271, and GT, the third version, was known at the Papal Court sometime

after 1270. It is the view of Gudrun Vuillemin-Diem that Thomas Aquinas is the

first authentic user of the second version of the translation.18 Or to be more precise,

he may have used G1b (1267), and he certainly knew the third version form 1270.

Unfortunately, we lack the commentary of Thomas Aquinas on Book Three of the

Meterorologica. However, if Thomas Aquinas is the first authentic use of the

second version, G1b, what can we say about Roger Bacon’s use of either the first

or second version? We can state that Roger Bacon knew the translatio nova already

c. 1266–1267, and may have known the second version G1b, thereby preceding

Thomas Aquinas as the first to comment on the new translation of the

Meteorologica. We can also say quite categorically that Roger Bacon, drawing

on his knowledge of Aristotelian commentators from Grosseteste, Adam of

Buekfield and others, wrote on the first prerogative on experimental science as a

correction of Moerbecke’s account of the rainbow: this is what he means by his

appeal to the practice of Robert Grosseteste and the difficulties of the perverse

translations of Aristotle. Like Grosseteste, he intends to use his own experience and

other authors, most notably Seneca and writers on optics to correct and advance the

study of the rainbow. Bacon tries to correct Moerbecke’s Aristotle by a new

treatment of the figure of the rainbow and he shows how an adequate use of

instruments such as the astrolabe one can give precise measurements of the possible

altitude of the rainbow (42 ). He also studies the question of the objectivity of

rainbows and other secondary stars, the role of vision and the nature of the colors.

The question naturally arises concerning Bacon’s move to write on philosophical

topics c. 1260. Did the new translation of Aristotle move him to return to Natural

Philosophy and Logic in order to offer a broad criticism of the new readings of the

text of Aristotle?

But the question remains; could Bacon, the writer of a Greek Grammar and the

commentator in Compendium studii philosophiae of the meaning of Greek words,

have been wrong and just simply prejudiced in a negative manner alone about

Moerbecke; was it a fit of pique that he did not get to translate the Meteorologica?

Was it just a case of corporate professional rivalry? At any rate, according to the

editor of the latter text in the Aristoteles Latinus, it is clear even in version two

(1267), that Moerbecke had trouble with the geometrical figure of the rainbow. A

careful review of all three versions by the editor proves that there are major

problems in Moerbecke’s first two versions: numerous omissions of words, especially syntactical words, mechanical translation from the Greek with assimilation

mistakes, additions by Moerbecke himself, difficulties with unusual (that is, scientific) Greek words, confusion of single words such as those for cloud and fog,

significant failure in reading the text, numerous textual conjectures. Above all,

there was a significant problem with the diagram of the rainbow.


Ibid., Aristoteles Latinus, X, Meteorologica Translatio Guillelmi De Morbeka, 2.1: 349–350.

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6 The Uses of Mathematics in Opus maius, Parts IV-VII: Science Interpreted Moraliter.

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