## 1. Life



Nicole Oresme was born around 1320 in the diocese of Bayeux in
Normandy, possibly in the village of Allemagne (now Fleury-sur-Orne)
on the outskirts of the Norman city of Caen (Burton 2007, 6). By
1341/42 he had obtained his master of arts at the University of Paris
and was probably teaching philosophy there (Courtenay 2000, 544;
Burton 2007, 7). In 1348 his name appears on a list of graduate
scholarship holders in theology at the College of Navarre at the
University of Paris. Oresme became Grand Master of the College in
1356, so he must have completed his doctorate in theology before this
date. Oresme held this position till 1362 and was teaching master in
the faculty of theology during that time (Burton 2007, 10). A detailed
account of Oresme's career as a scholar has been provided by
Lejbowicz (2014).



From 1362, when he left the university, until his death in 1382,
Oresme served Charles, the dauphin of France, who was regent during
his father's captivity (1356-1364) and was crowned King
Charles V on his father's death (1364) (Burton 2007,
11-13). Oresme was appointed canon (1362) and later dean (1364)
of the Cathedral of Rouen and also canon at Sainte-Chapelle in Paris
(1363) (Clagett 1974, 223). Oresme was elected bishop of Lisieux in
1377 and was consecrated in 1378. He died on July 11, 1382.


## 2. Teachings


### 2.1 Ontological Status of Accidents



One of the most interesting features in Oresme's Commentary on
Aristotle's Physics is his view on the ontological status of
accidents. Characteristic of Oresme's view on accidents is that
he does not consider them accidental forms, but only so-called
*condiciones* or *modi* *(se habendi)* of the
substance. But this does not mean that Oresme identifies accidents
with the substance in the way that Ockham had identified the quantity
of a substance with the substance itself. Rather, Oresme regards
accidents as distinct from the substance, but he assigns them a lower
ontological status than the commonly accepted accidental forms. For
Oresme motion, being-in-a-place (*esse in loco*), the quantity
of a substance, its *esse tantam*, and qualities (such as the
*esse album* of a substance) are such *condiciones* or
*modi* (Celeyrette/Mazet 1998; Caroti 2000; Caroti 2001; Caroti
2004; Mazet 2000; Kirschner 1997, 52-61, 73-76, 121,
141-142; Kirschner, 2000b, pp. 263-272). As Panzica has
solved the complex question of Oresme's authorship of his Commentary
on Aristotle's Meteorology by identifying two different redactions
(Panzica 2015; Panzica 2017), we can now be certain that Oresme took
the first steps in establishing his condicio theory of accidents in
the first redaction of his Meteorology Commentary (Kirschner 2010),
which is probably his earliest work (Panzica 2017, 27-33).



To a certain degree Oresme's theory of the ontological status of
accidents is reminiscent of Adam Wodeham's (c. 1298-1358)
and Gregory of Rimini's (c. 1300-1358) theory of complexly
signifiables (*complexe significabilia*) (Adam de Wodeham 1990,
dist. 1, qu. 1, 180-208; Gregory of Rimini 1981, Prologus, qu.
1, artic. 1-3, 1-40; Nuchelmans 1973, 227-242; Biard
2004; Conti 2004; Gaskin 2004). On some occasions Oresme explicitly
uses the expression 'complexe significabile' in the
framework of his ontology of accidents. However, despite these
similarities between Oresme's ontology of accidents and the
theory of complexly signifiables it seems that this connection is of a
secondary nature. For Oresme the determination of the ontological
status of accidents is in the foreground, while Adam Wodeham and
Gregory of Rimini had quite another aim. They wanted to determine what
the object of knowledge is. Their solution that the *significatum
totale* or *significatum adaequatum* of a conclusion or
proposition is the object of knowledge is not confined to conclusions
or propositions concerning accidents, such as 'homo est
albus', whose *significatum totale* is *hominem esse
album* (which itself is a *significabile per complexum*).
Rather, their solution applies to any kind of conclusion or
proposition: for instance, 'Deus est' or 'homo est
animal'. Neither Adam Wodeham nor Gregory of Rimini made any
attempt to identify accidents with the correspondent complexly
signifiables such as 'hominem esse album'. Oresme does not
derive the ontological status of accidents from their being complexly
signifiables. Instead, the converse is true: given that Oresme rejects
the traditional view that accidents are accidental forms, he must
avoid the use of substantives in denominating the ontological status
of accidents and thus quasi-automatically arrives at formulations that
are characteristic of the theory of complexly signifiables.


### 2.2 Non-Aristotelian Concepts of Place, and Space



There were only very few authors since late antiquity - that is,
since Simplicius (b. c. 500, d. after 533) and his contemporary John
Philoponus - who rejected Aristotle's definition of place
as the innermost surface of the surrounding body. In his Commentary on
Aristotle's Physics Oresme argued for the non-Aristotelian
position that the (physical) place of a body is the space filled or
occupied by the body (Kirschner 1997, 101, 116-123; Kirschner
2000a, 146-159). Prior to Oresme Geraldus Odonis (c.
1290-1349) (Bakker/de Boer 2009; Robert 2012, 85-90),
Walter Chatton (c. 1290-1343) (Robert 2012, 83-85) and
William Crathorn (fl. 1330s) (Robert 2012, 90-94) held the same
view. It remains unknown whether their arguments exerted a direct
influence on Oresme. Another anti-Aristotelian theory of place was
propounded by Petrus Aureoli (c. 1280-1322). Aureoli held that
place is the determinate position of the located body in the universe
(Schabel 2000, 126-138; Robert 2012, 79-82).



In stating that the place of a body is the space filled or occupied by
it Oresme does not merely revive an opinion of Greek antiquity; he can
also be regarded, together with philosophers like Gianfrancesco Pico
della Mirandola (1469-1533) (Grant 1981, 275-276, n. 63),
Francesco Patrizi (1529-1597) (Grant 1981, 201; Schmitt 1967,
143), and Giordano Bruno (1548-1600) (Grant 1981, 186-187;
Schmitt 1967, 142-143), as a precursor of Newton
(1643-1727) in the theory of place and absolute space. None the
less, despite the close resemblance that Oresme's view of place
and space bears to Newton's, there are characteristic
differences concerning the ontological status of space.



For Oresme space is neither substance nor accident. It is nothing that
can be signified by a noun or pronoun, but only by adverbs like
'here' and 'there'. That means that space is
not absolutely non-existent, but it has by no means the high
ontological status accorded to it by Newton. While for Newton space
approaches more nearly to the nature of substance than of accident,
for Oresme it ranks on a far lower ontological level than an accident
(Kirschner 1997, 103-104; Kirschner 2000a, 163-164).



Another central conclusion that Oresme draws in his Physics Commentary
in his discussion of the nature of place is that beyond the world,
that is, outside the last sphere, there exists an infinite void space.
Oresme's conception of an extracosmic infinite void space is
well known from other works by him (*Le livre du ciel et du
monde*, *Questiones super De celo*) (Kirschner 2000a,
164-168). Besides Oresme there were few medieval philosophers
who assumed the existence of an infinite void space beyond the world.
The Jewish philosopher Hasdai Crescas (c. 1340-1410/11) (Crescas
1929, 189; Grant 1969, 50, n. 50; Grant 1981, 271, n. 33, 321, n. 5),
Thomas Bradwardine (c. 1290-1349) (Grant 1969, 44-47;
Grant 1981, 135-144), Robert Holkot (d. 1349) (Grant 1981, 350,
n. 130) and William Crathorn (fl. 1330s) (Robert 2012, 77, n. 2) may
be mentioned among them.



Oresme also speaks of the immensity which is outside the Heavens and
identifies this immensity - by which he undoubtedly means the
extracosmic void space - with God himself. This identification
of the infinite void space with God is a characteristic feature of
Oresme's natural philosophy or theology (Kirschner 1997,
105-106; Kirschner 2000a, 168). According to Wolfson (1929,
123), Crescas did not identify the infinite void outside the world
with God's immensity; nor does Bradwardine seem to have held
such a view (Maier 1966, 315, n. 18; Grant 1981, 142). The same
applies to Robert Holkot (Grant 1981, 350, n. 130) and William
Crathorn.


### 2.3 Non-Aristotelian Concept of Time



What has just been said about Oresme's rejection of Aristotelian
tenets on the nature of place is also true for his theory of time.
Aristotle defines time as the number (that is, the measure) of motion
with respect to before and after. Thus he deduces the existence of
time from the existence of motion, which means that time is nothing
independent of motion. In contrast to Aristotle, Oresme in his Physics
Commentary defines time as the successive duration of things
(*duratio rerum successiva*, also *duratio successiva
rerum* or *rerum duratio successiva*), that is, the
duration of the actual existence of things. By deducing his concept of
time from the duration of things, which duration is prior to and
independent of motion, Oresme clearly deviates from Aristotle's
point of view and the conventional manner in which this topic was
discussed among medieval scholastics (Kirschner 2000a, 171-176;
Zanin 2000, 257-259; Caroti 2001).



Among the few who opposed Aristotle's doctrine of time was
Petrus Johannis Olivi (b. c. 1248, d. 1298), who criticized Aristotle
for regarding motion, rather than the actual existence of things, as
the subject of time (Maier 1955, 110-111). The Franciscan
Gerardus Odonis (c. 1290-1349) was also a proponent of the
independence of time from motion (Maier 1955, 134-137).
Oresme's theory of time foreshadows that of classical physics to
a certain degree, but, as is true of place and space, there are
certain differences concerning the ontological status of time
(Kirschner 2000a, 176-178).



Oresme states that the duration of things without any succession is
eternity, which he defines as *duratio rerum tota simul*. As
with the relation between God and extramundane space, Oresme
identifies eternity with God himself (Kirschner 2000a,
178-179).


### 2.4 Theory of Motion



In his commentary on Aristotle's Physics Oresme presents a
detailed and elaborate discussion of the ontological status of motion,
one of the fundamental problems of medieval natural philosophy. His
theory of motion is highly specific and turns out to be an application
of his characteristic *condicio*-theory of accidents (Caroti
1993; Caroti 1994; Kirschner 1997, 52-78; Kirschner 2014).



For Oresme motion is a *fluxus*, a successive entity of its own
that exists in addition to the mobile and the things that are acquired
during motion. This is a clear departure from the nominalistic
position. Concerning its ontological status this *fluxus* is
not considered a separate accidental form but only a *modus*
(*se habendi*) or a *condicio* of the mobile. Thus
Oresme circumvents the difficulties of a pure nominalistic approach,
while simultaneously avoiding the problems that occur if one assigns
the *fluxus* the ontological status of an accidental form, as
Buridan did in the case of local motion in his Physics commentary
(*ultima lectura*) (Buridan, *Questiones super octo
Phisicorum libros Aristotelis*, Qu. III.7, f. 50ra-51ra).
Oresme's own concept of *fluxus* is easily applicable to
all types of motion, be they alterations, changes in quantity or local
motions. Such a uniform concept of motion was one of his major aims.
Unfortunately, posterity does not seem to have appreciated his
endeavour. Thus, at the University of Vienna in the second half of the
15th century it apparently became common to equate Oresme's
concept of motion with Ockham's view (Kirschner 2014).


### 2.5 Cosmology, Astronomy, and Opposition to Astrology



In his *Livre du ciel et du monde* and in other works
(*Questiones super De celo*, *Questiones de spera*)
Oresme brilliantly argues against any proof of the Aristotelian theory
of a stationary Earth and a rotating sphere of the fixed stars.
Although Oresme showed the possibility of a daily axial rotation of
the Earth, he finished by affirming his belief in a stationary Earth
(Clagett 1974, 225; Di Liscia/Shapiro 2020). Similarly, Oresme proves
the possibility of a plurality of worlds, but ultimately keeps to the
Aristotelian tenet of a single cosmos (Clagett 1974, 224-225;
Harvey 2011).



Oresme was a determined opponent of astrology, which he attacked on
religious and scientific grounds. In *De proportionibus
proportionum* Oresme first examined raising rational numbers to
rational powers before extending his work to include irrational
powers. The results of both operations he termed *irrational
ratios*, although he considered the first type commensurable with
rational numbers, and the latter not. His motivation for this study
was a suggestion of Thomas Bradwardine that the relationship between
forces \((F)\), resistances \((R)\), and velocities \((V)\) is
exponential (Grant 1966, 24-40; Clagett 1974, 224). In modern
terms:

 

\[ \frac{F\_2}{R\_2} = \left( \frac{F\_1}{R\_1} \right)^{(V\_2 /V\_1)} \]


Oresme then asserted that the ratio of any two celestial motions is
probably incommensurable (Grant 1971, 67-77). This excludes
precise predictions of successively repeating conjunctions,
oppositions, and other astronomical aspects, and he subsequently
claimed, in *Ad pauca respicientes* (its name derives from the
opening sentence "Concerning some matters ..."), that
astrology was thereby refuted (Grant 1966, 83-111). In his
*Livre de divinacions* and his *Tractatus contra
astronomos* Oresme tries to show that astrology is "most
dangerous to those of high estate, such as princes and lords to whom
appertains the government of the commonwealth" (Coopland 1952,
51). As with astrology, he fought against the widespread belief in
occult and "marvelous" phenomena by explaining them in
terms of natural causes. Oresme's writings against astrology and
magic were due to his concern over the king's and his
court's addiction to these practices. Interestingly, in the
first redaction of his Questions on Meteorology Oresme does not yet
turn out to be the resolute opponent of astrology as we know him from
his later works, but, quite the contrary, adheres to its principles
(Panzica 2021, 2).



In his *De visione stellarum* Oresme departs from the standard
view of earlier authors in optics such as Ptolemy (2nd
century), Ibn al-Haytham (965-c. 1040), Roger Bacon (c.
1214-c. 1292), and Witelo (c. 1230/35-after 1275), who all
held that refraction can only occur at the interface of two media of
differing densities and thus would not take place in a single medium
of uniformly varying density. He states - more than 300 years
before Robert Hooke (1635-1703) and Newton - that
atmospheric refraction occurs along a curve and proposes to
approximate the curved path of a ray of light in a medium of uniformly
varying density, in this case the atmosphere, by an infinite series of
line segments each representing a single refraction (Burton 2007,
33-64).



In the first redaction of his Commentary on Aristotle's Meteorology
Oresme stated the supralunar location of comets, a unique opinion for
a medieval philosopher (Kirschner 2000c; for the identification of the
anonymous manuscript as Oresmian see Panziza 2015). Oresme's view of
the celestial nature of the Milky Way was also unconventional,
otherwise only held by Albert the Great and Henry of Langenstein.


### 2.6 Mathematics



Oresme's main contributions to mathematics are contained in his
*Questiones super geometriam Euclidis* and his *Tractatus de
configurationibus qualitatum et motuum*. In these works Oresme
conceived of the idea of using rectangular coordinates
(*latitudo* and *longitudo*) and the resulting geometric
figures (*configurationes*) to distinguish between uniform and
nonuniform distributions of various quantities, such as the change of
velocity in relation to time or the distribution of the intensities of
a quality in relation to the extension of the subject. In the
discussion of motions the base line (*longitudo*) is the time,
while the perpendiculars raised on the base line
(*latitudines*) represent the velocity from instant to instant
in the motion. Thus uniform acceleration is represented by a right
triangle. Oresme even extended his definition to include
three-dimensional figures (Clagett 1974, 226-228). Thus, he
helped lay the foundation that later led to the discovery of analytic
geometry by Rene Descartes (1596-1650).



Furthermore, Oresme used his figures to give the first proof of the
Merton theorem, discovered at Oxford in the 1330's: the distance
traveled in any given period by a body moving under uniform
acceleration is the same as if the body moved at a uniform speed equal
to its speed at the midpoint of the period (Clagett 1974,
225-226; Smorynski 2017, 216-222). Some scholars believe
that Oresme's graphical representation of velocities was of
great influence in the further development of kinematics, affecting in
particular the work of Galileo (1564-1642).



Further noteworthy achievements in the field of mathematics are
Oresme's geometric proofs of the sums of certain converging
series, particularly in his *Questiones super geometriam
Euclidis* and his *Tractatus de configurationibus qualitatum et
motuum* (Clagett 1974, 228). Most interestingly, Oresme seems to
have given a general rule of how to find the sum of all convergent
series of the form:

 

\[ a + \frac{a}{m} + \frac{a}{m^2} + \frac{a}{m^3} + \ldots + \frac{a}{m^n} + \frac{a}{m^{n+1}} + \ldots , \]


with *a* being any quantity (*aliqua
quantitas*) and *m* any natural number greater
than or equal to 2 (cf. Murdoch 1964; Mazet 2003). He informs us that
we have to take the difference of two successive terms, that is
\(a/m^n - a/m^{n+1}\), and divide it through the first of these terms,
that is \(a/m^n\), so that we get:

 

\[ \frac{a/m^n - a/m^{n+1}}{a/m^n} = \frac{m - 1}{m}.\]


The reverse of this fraction, i.e. \(\frac{m}{m-1}\), will be the
proportion of the sum of the whole series to the first term of the
series, *a*. Thus, if we have the series



\[ 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots \frac{1}{3^n} + \ldots, \]


to use Oresme's own example, the sum will be 3/2 (Oresme,
*Questiones super geometriam Euclidis*, H. L. L. Busard (ed.),
2010, Qu. 2, ll. 48-57). If *a* is 2, the sum
will be 3. Furthermore Oresme was the first to prove that the harmonic
series

 

\[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots \frac{1}{n} +\ldots \]


is divergent by arguing that this series consists of an infinity of
parts that are greater than \(1/2\), so that the whole is infinite.
His demonstration rests on the fact that the third and the fourth term
taken together \((1/3 + 1/4)\) are greater than \(1/2\), which also
applies to the sum of the fifth through the eighth term \(1/5 + 1/6 +
1/7 + 1/8\), which is greater than \(4 \times 1/8\) and the sum of the
ninth through the 16th term and so on (Oresme, *loc.
cit.*, Qu. 2, ll. 58-68).



Oresme's discussion of the infinite in his Physics Commentary is
another fascinating testimony to the originality of this outstanding
medieval philosopher. Oresme demonstrates by thought experiments that
of two actual infinites neither is bigger or smaller than the other.
Oresme's proof is somewhat reminiscent of Georg Cantor's
(1845-1918) demonstration that certain infinite sets are
equinumerous. Thus Oresme shows by the principle of one-to-one
correspondence that the collection of odd natural numbers is not
smaller than the collection of natural numbers, because it is possible
to count the odd natural numbers by the natural numbers (Sesiano 1996;
Kirschner 1997, 79-83, 88-92).



Oresme was not the first to use the principle of one-to-one
correspondence in discussing the properties of actual infinites.
Bradwardine, whose main aim was to refute Aristotle's opinion
that the world is eternal, applied the principle of one-to-one
correspondence to show that two infinites would be equal or - in
modern terms - that an infinite subset is equal to the set of
which it is a part (Bradwardine 1618, 121C-124C). On the other
hand Bradwardine takes it for granted that an infinite subset is
smaller than the set of which it is a part. Thus he is of the opinion
that under the assumption of an eternal world that has no beginning
the multitude of all human souls, which until now have been created,
must be greater than the multitude of the male or the female souls
alone (Bradwardine 1618, 132E-133A). From this contradiction
- an infinite subset cannot be both smaller than and equal to
the set of which it is a part - Bradwardine draws the inference
that the eternity of the world is impossible (Thakkar 2009,
626-629).



Unlike Bradwardine, Oresme shows that of two actual infinites neither
is greater or smaller than the other. This result is different from
Bradwardine's, because Oresme's result does not
necessarily imply equality between actual infinites. Moreover Oresme
shows that cases can be conceived in which two infinites can be
regarded as unequal, but this unequality is not to be understood in
the sense of 'smaller' or 'greater' (Oresme
does not contradict himself), but rather in the sense of
'different'. Since comparable quantities are either equal
to one another or one is smaller or greater than the other, Oresme
concludes that actual infinites are incomparable: that is, that
notions like 'smaller', 'bigger', and
'equal' do not apply to infinites (Sesiano 1996; Kirschner
1997, 79-83, 88-92). Oresme's treatment of the
infinite was extensively used by Pierre Ceffons when he commented the
*Sentences* in Paris in 1348-1349 (Mazet 2004,
175-182).


### 2.7 Economics



Oresme is generally considered the greatest medieval economist. He
presented his economic ideas in commentaries on the *Ethics*,
*Politics*, and *Economics*, as well as an earlier
treatise, *De origine, natura, jure et mutationibus monetarum*,
the first comprehensive work upon money. In his *De origine,
natura, jure et mutationibus monetarum*, of which he himself made
a French translation under the title *Traictie de la
premiere invention des monnoies*, Oresme argued that coinage
belongs to the public, not to the prince, who has no right to vary
arbitrarily the content or weight (Woodhouse 2017). By clearly
delineating the destructive effects on a nation's economy of a
debasement of the currency, he influenced Charles V's monetary
and tax policies.



Oresme also stated that in a society in which two currencies with the
same designation but of different value circulate, the money of lower
value drives out the money of higher value. This economic law was also
discovered independently of Oresme by Nicolaus Copernicus
(1473-1543), the famous astronomer, who wrote about a reform of
the Prussian coinage, and by Thomas Gresham (1519-1597). Today
it is called Gresham's Law, or sometimes the Law of Oresme,
Copernicus, and Gresham, but its oldest version can be found in
Aristophanes's poem 'The Frogs' (Balch 1908;
Sparavigna 2014).



Oresme's interest and engagement in economic problems should not
be seen as an isolated subject in his intellectual life. As Kaye
(1998, 2) has argued, the preoccupation with measurement, gradation,
and the quantification of qualities, so typical of Oresme and the
natural philosophy of his period, was strongly influenced by the
contemporary rapid monetization of European society.