The “Galenic Question”: A Solution Based on Historical Sources and a Mathematical Analysis of Texts

: How many different writers authored the huge number of texts attributed to Galen of Pergamum (129~216 Anno Domini (AD)), medical doctor and philosopher, a giant in the history of medicine? The quest to find out which texts were his and which ones were written by others is known as the “Galenic Question”. We propose a “solution” to it through a multidisciplinary approach based (a) on historical research and (b) on a mathematical analysis of the Greek texts. The historical approach considers historical independent sources and anachronisms. The mathematical approach is based on a mathematical theory concerning deep language variables, rarely consciously controlled by any author, and is therefore capable of giving indications on the similarity of texts, with little or no bias. The multidisciplinary approach has convinced us that at least three authors wrote the texts attributed to Galen. The first two were very likely real historical persons: (a) a certain Galen living between the end of the I century Before Christ (BC) and the second half of the I century AD, and (b) the historical Galen of Pergamum (II–III centuries AD). We believe the third (c) to be represented by several unknown authors hiding under the name Galen, but likely living after Galen of Pergamum’s death.


Introduction
Galen of Pergamum-a giant in the history of medicine-is described as medical doctor and philosopher born in Pergamum in the year 129 Anno Domini (AD) and who died in 216.He attended schools of Greek philosophers and medical schools in Pergamum, Smirne and Alexandria.In the year 157, he became the personal doctor of the emperor Marcus Aurelius.Galen reorganized medicine on a unitary basis in which anatomy and the demonstrative methods of Aristotelian and Euclidean origins were central.
Apparently, he was a very prolific writer.Of the 400 writings (in the following, we refer to them as "texts") attributed to him, only 130 are available today, written in Greek, Arabic, Syrian and Latin.However, some texts are not considered authentic but are attributed to a Pseudo-Galen, likely written after his death, and authored by unknown persons who exploited his fame (Fichtner 2023).
What we know about his life comes only from his writings since he is practically unknown to contemporary scholars.His reference to Aristotelian philosophy and monotheism made him accepted by Judaism, Fathers of the Christian Church and Islam.His works were, therefore, adopted as textbooks in Medieval Universities and his authority was practically indisputable until the Renaissance.and commentator on ancient texts, who surpassed all his predecessors.This great praise seems to fit more with an author of the past rather than a contemporary.
Alexander of Aphrodisias, a scholar who lived in Athens around 200 AD, in his comments on Aristotle, cites Galen as a "famous philosopher" together with Plato and Aristotle (Alexander of Aphrodisias,Comm. Arist. Topic,8,5).This statement sounds curious for a writer who should be his contemporary, known above all as a medical researcher and physician, whereas here, he is associated with the great philosophers of the past.
Gargilius Martial in Medicina ex oleribus et pomis, III century AD, mentions only physicians of the I century AD, and among them, he mentions Galen several times in connection with Dioscorides (Gargilius Martial, Medicina ex oleribus et pomis, VI) who lived under the emperor Nero.
Eusebius (IV century) reports that in a work written very likely at the beginning of the III century against the heresy of Theodotus, the heretics almost adored Galen (Eusebius, Historia Ecclesiastica, V, 28, 13-14), seen as a philosopher and logician together with Euclid, Aristotle and Theophrastus.Thus, Galen is considered among the great writers of antiquity, and he does not appear to be contemporary with the heretics who "worshipped" him.
Simeon Metaphraste, in his histories of saints and martyrs, mentions the passio of Carpus, Paphilus, Agathodorus and Agathonyx (Migne 1899), physicians martyred under the emperor Decio (249-251 AD).During their interrogatory, the names of Hippocrates and Galen are mentioned.Now, mentioning Galen together with Hippocrates indicates that he was already so famous as to be associated with Hippocrates, and, therefore, this should place Galen in a very ancient epoch.Moreover, these martyrs are also mentioned by Eusebius of Caesarea (Historia Ecclesiastica, IV, 15, 48) who, however, set their martyrdom under the emperor Lucius Verus (161)(162)(163)(164)(165)(166)(167)(168)(169).Modern scholars share this latter dating because the Greek language is that of the epoch of Marcus Aurelius (Sordi 1961).
These are a few examples of contradictory historical information regarding Galen of Pergamum that raise the question about the existence of more than one author under Galen's name, very likely persons who lived in different epochs.An exhaustive review of these points can be found in (Scarborough 1981).
Several other studies on Galen, even if they do not contradict the current dating, are quite critical and capture the complexity of the contents of his writings, compared to the intellectual nature of his (presumed) epoch (Pietrobelli 2019;Mewaldt 1909;Walzer 1949;Nutton 1995;Mattern 2008;Gill et al. 2009;Clivaz 2011;Totelin 2012;Abbou Hershkovits and Hadromi-Allouche 2013;Favaretti Camposampiero and Scribano 2022;Vegetti 1999).There are many texts attributed to him which, due to their style and arguments, scholars consider to likely be written by other authors-namely a Pseudo-Galen (PG)-after Galen of Pergamum's death (Fichtner 2023).On the contrary, other writings seem to belong to an epoch before Galen of Pergamum.In conclusion, several authors could hide under the historical figure of Galen of Pergamum, living either before or after Galen of Pergamum's epoch.
Indeed, as already mentioned, in old age, Galen of Pergamum wrote a couple of books, in which he lists which texts were his own.This fact means that at that time, there was already widespread confusion between his texts and others attributed to a homonym Galen, confusion that was not uncommon for ancient writings.The need to list works attributed to him may have led Galen, whose memory had been lost, to declare numerous texts written by others as his own.The most important Galen scholars agree that extracting autobiographical truth from his writings is impossible, because he constructed a partially imaginary autobiography (Boudon-Millot [2012] 2016, p. 12).
To search for a possible solution to this question, in our study, we examined 57 works written in Greek, with sufficiently large number of words (at least 1500) to allow for reliable statistical results.Modern editions of Galen's texts, updated with respect to (Kühn 1821(Kühn -1833)), can be found in (Singer and Rosen 2024).However, textual differences, due to the updated editions, do not change the deep language variables discussed in Section 4.Moreover, either adding or erasing a few sentences in a sufficiently long text does not significantly affect the average values used in Section 4. Obviously, all the texts are dated according to the current chronology (129-216 AD), except the Pseudo-Galen's texts.
As already stated, some of the texts in Table 1 should refer to different authors.There is a substantial consensus among scholars on the fact that the works numbered from 31 to 42 (PG) were really written after the death of Galen of Pergamum (Fichtner 2023), by the so-called Pseudo-Galen.
Galen appears to have been a compiler of and commentator on previous works, often reported verbatim, but this is hardly noticeable because he does not mention sources (Temkin 1973).The alleged autobiographical passages could also have been inserted later.In fact, Galen's works remained ignored until around 360 AD, and only then did his alleged literary production explode into a myriad of manuscript copies, not to mention compilations, summaries, interpolations and translations into multiple languages, falsely attributing to him the writings of others.These texts have generally been collected by scholars under the name of Pseudo-Galen (Fichtner 2023).
Other works seem to be written by another author who lived before Galen of Pergamum (Scarborough 1981) because of confusions and contradictions which have constantly drawn a series of criticisms of Galen, summarized in a recent book with the significant title Contre Galien.Critiques d'une autorité médicale de l'Antiquité à l'âge moderne (Pietrobelli 2020).
In this regard, it should be highlighted that the biography of an author named Galen but living at the beginning of the I century AD is mentioned in several sources, although generally either ignored or neglected by scholars.In fact, in addition to the autobiography of Galen of Pergamum, there is a second biography in Arabic texts in which Galen is a philosopher and physician living between the end of the I century BC and the second half of the I century AD, until the times of Nero and Vespasian (Ibn Juljul 1992;Vanoli 2012;Musitelli 1984Musitelli -1986)).Furthermore, Arab doctors (IX-XII century) knew Galen's works that were unknown in the West, and many details of his life.Hunayn Ibn Ishāq (IX century), translator of many of Galen's medical texts, wrote that Galen was a contemporary of Christ, and that he died in the year 88 AD at 87 years old, according to the authority of Yahyā the Grammarian, i.e., John the Grammarian.For Sulayman ibn Hassān (i.e., Ibn Juljul, X century, from Cordoba), Galen lived at the time of the emperor Nero and died in Sicily (Ibn Juljul 1992;Vanoli 2012;Musitelli 1984Musitelli -1986)).
In other words, the claim that Galen lived in the epoch of Jesus Christ should be considered seriously because the first Arab translators of Galen had access to lost Byzantine biographies, and their information on the contemporaneity of Galen and Christ confirms the doubts about Galen and the existence of a "Galenic Question" (Scarborough 1981).Thus, the number of authors hiding under Galen's name is at least three.Therefore, in the following, we refer to three "Galen".
We first conjecture a Galen philosopher and physician who lived before Galen of Pergamum, between the late I century BC (Before Christ) and the epochs of Nero (54-68 AD) and Vespasianus (69-79 AD): we refer to him as Galen-1.Secondly, we refer to the philosopher and physician living from 129 to 216 AD, i.e., the historical Galen of Pergamum, as Galen-2.Thirdly, we refer to the authors of texts written after Galen of Pergamum's death, authored to exploit his fame, as Galen-3 (Pseudo-Galen).
The mathematical analysis of the deep language structure of all the writings reported in Table 1, discussed Section 4, will aim either to confirm or to deny our conjecture of the existence of at least three authors.Before this mathematical analysis, in the next section, we examine the works that might have been written by Galen-1.

The Likely Writings of Galen-1
In the following, we examine which texts from Table 1 may have been written by Galen-1 because of possible historical anachronisms or suspicious information in them.

De Sanitate Tuenda (Text 21)
The first mention of Galen in independent texts is in Deipnosophistae-i.e., The Dinner Sophists-by Athenaeus of Naucratis (III century).Galen-present at the banquet narrated by Athenaeus in which figures of the past are present regardless of chronology-is described as a physician, a philosopher, and an expert of wines of Italy, listed with their medical properties, and of bread and flour.According to Athenaeus, Galen wrote philosophical and medical texts of quality and importance well above his predecessors.This is great praise if given by a contemporary of Galen's, who had just died when Athenaeus started writing his work (Boudon-Millot [2012] 2016, p. 10;Nutton 1984).Very likely, he refers to another Galen, a figure of the past who participated in the banquet with other famous figures who lived in different ages.
There are several clues that support this claim.First, the wines described by this Galen (Deipnosophistae, I, 48, 26c-27d)-produced in Lazio and in Campania, also according to writers of the I century BC such as Dioscorides (V, 6) and Pliny the Elder (XIV, 60)-were known only from the I century BC to the first half of the I century AD because of the crisis of agriculture in Italy and the import of wines from other provinces (Carandini 1989), after the destruction of Pompeii, due to the Vesuvius eruption (79 AD).In other words, it is curious that Galen of Pergamum (II century AD), in his work De sanitate tuenda, prescribes the therapeutic use of wines that were no longer produced.For example, the wine Falerno, one of the most famous wines of Campania, is mentioned, but other authors of the II century AD never mention it.Therefore, the Galen mentioned by Athenaeus is probably a figure living before the Vesuvius eruption of 79 AD.Secondly, when this Galen summarizes what physicians have written on bread, flour and cakes, he mentions only ancient physicians living from the IV to the II centuries BC (Athenaeus, Deipnosophistae, III, 83, 115c-116a) (Jacob 2001).In conclusion, De sanitate tuenda should have been written by Galen-1.

De placitis Hippocratis et Platonis (Text 15) and Administrationes Anatomicae (Text 17)
The available manuscripts attributed to Galen are dated within the V-VI centuries, except a few older ones, written on papyri (Manetti 2019).The oldest papyrus with a text by Galen (De placitis Hippocratis et Platonis) is dated to approximately 250 AD (Manetti 2019).In it, Galen states that philosophy and medicine are interdependent disciplines (Nutton 1995;Manetti 1981).However, in De ordine librorum suorum, Galen inexplicably does not mention either it or De usu partium, even if these two works are related because both are dedicated to a certain Flavius Boethus (Groag 1943).The text Administrationes anatomicae is also dedicated to Flavius Boethus.Therefore, the three texts De placitis Hippocratis et Platonis, De usu partium and Administrationes anatomicae should be related to the same author, who should not be Galen of Pergamum.
In De placitis Hippocratis et Platonis, Galen's scientific and cultural "clock" stopped in the III century BC, with the sole exception of Posidonius (1st century BC) (Vegetti 2011).Usually, the physicians that Galen cites as his contemporaries are not confirmed in the historical sources, and both their names and their theories are suspect.The external confirmation of physicians is found in the I century BC and at the beginning of the I century AD, concerning the physicians of the Julio-Claudian court.Usually, Galen-a name of Greek origin-is cited alone (Solin 2003), but the full name is Claudius Galenus, as byzantine sources indicate (Alexandru 2021).The term Claudius may indicate that this Galen obtained roman citizenship, very likely during the reign of emperor Claudius (41-54), because he was particularly interested in surrounding himself with physicians (Marasco 1998).In conclusion, this Galen should be Galen-1, not Galen of Pergamum (Galen-2).

De Naturalibus Facultatibus (Text 18)
The work De naturalibus facultatibus is also found in one of the oldest manuscripts, dating back to the 4th-5th century, and in it, Galen presents himself as philosopher, commentator, interpreter of texts and hermeneutic-as he was described by Athenaeus-qualities that could be attributed to Galen-1.

De Usu Partium (Text 7)
We have already evidenced that De usu partium is not mentioned in De ordine librorum suorum, i.e., in the list of his books compiled by Galen of Pergamum himself.Moreover, two fragments of papyrus, from a codex, coming from Egypt and preserved in Florence are dated, because of the paleographic characteristics of the writing, to the second half of the I century AD or, at most, to the first half of the II century AD (Comparetti 1908).Two new studies (Manfredi 1974;Manetti 1985) clarify that the text comes from a commentary on the treatise De Alimento (part of the Corpus Hippocraticum).In this commentary, there are passages, parallels and similarities with De usu partium.Now, if the papyrus belongs to the second half of the I century AD, then the author should be, at most, from the I century AD; therefore, he cannot be Galen of Pergamum.In conclusion, De usu partium should be attributed to Galen-1.

De Atra Bile (Text 28)
In De atra bile, Galen cites Rufus of Ephesus as one "among the most recent" persons who have recently studied the bile.Now, almost all sources agree in dating Rufus between the I century BC and the beginning of the I century AD, and also in identifying him as Cleopatra's doctor (e.g., Tzetzes, Chiliades, VI, 44, 300).The physicians active at the time of Claudius and Nero present him as an undisputed authority on medicines (Abou-Aly 1992).If Galen had written De atra bile in the I century AD, he could have certainly said that Rufus was a physician "among the most recent" physicians, but this statement would sound anachronistic at the end of the II century AD, i.e., if attributed to Galen of Pergamum.
Moreover, in De atra bile there is another element that does date the work.Speaking of the doctor Erasistratus (about 250 BC), Galen recalls one of his cures for madness and also the more ancient cure of the mythical Melampus, who cured the daughters of Proetus of madness.Galen specifies that the cure of Melampus was known not only for two or three hundred years, like that of Erasistratus, but for much longer (Kühn 1821(Kühn -1833, vol. V, pp. 132-33), vol. V, pp. 132-33).Therefore, the interval between Erasistratus and Galen spans only two or three hundred years.Adding three hundred years to 250 BC, we arrive at most at 50 AD, therefore indicating that the Galen author of De Atra Bile should be Galen-1, not Galen of Pergamum.

In Hippocratis Librum Primum Epidemiarum Commentarii (Text 54)
In the work In Hippocratis librum primum epidemiarum commentarii (Kühn 1821-1833, vol. XVII-1, p. 21), Galen underlines that in his time, "many peoples" had accepted the Julian calendar, adopted in Rome in the interval between Julius Caesar and Augustus, and soon used throughout the empire.This observation is more pertinent if written in the early decades of the empire, namely in the middle of the I century AD; therefore, this text was written by Galen-1.

Protrepticus (Text 3)
The first XVI century editors of Galen's works, Ludovico Belisario, Giovanni Battista da Monte and Giovanni Battista Rasario, already supported the hypothesis of two writers both named Galen, and this hypothesis was shared by other scholars in the Bibliotheca Graeca of Fabricius, in which we find at least five authors named Galen (Fabricius 1726).In fact, oddities, contradictions and differences in the texts attributed to Galen have made some Renaissance scholars think that there were two authors named Galen, both eminent physicians and philosophers: the first one son of Menodotus and author of Adhortatio ad artes (also called Protrepticus, text 3) and other works; the second one son of Nikon-born in Pergamum and author of everything else (Argenterio 1566;Tiraquellus 1584).
Protrepticus is a rhetorical exhortation to philosophy, and for this reason, it has often been compared to De usu partium (text 7), a text of philosophy, not of anatomy, despite its title.In it, in describing the parts of the body, Galen raises almost a religious hymn to the divine Providence, to the Creator Demiurge who assigned a specific function to each part (Petit 2018).Moreover, in some of Galen's texts, of which only the Arabic versions survived, there are fragments in which Galen mentions Christians, highlighting their faith (Nutton 1995).Conversely, in more specific medical works, Galen of Pergamum professes agnosticism.Therefore, the author of De usu partium seems to believe in God, differently from Galen of Pergamum.Thus, he seems to be a different person.Indeed, we find a Creator Demiurge-already a Platonic concept, and hence, developed before the Christian faith-only in De usu partium, and not in other Galen writings.

De Theriaca ad Pisonem (Text 4)
Galen, in De Theriaca ad Pisonem, mentions Andomachus, physician of the emperor Nero and creator of the medicine called Theriaca.The work De Theriaca is dedicated to an important Roman character named Piso, who could be Gaius Calpurnius Piso, member of a family at the top of the Roman aristocracy until the time of Nero (Groag 1936); therefore, this text belongs to Galen-1.Also, Boudon-Milot argues that this writing cannot be an authentic Galen of Pergamon treatise (Boudon-Millot 2017).

De Methodo Medendi (Text 8)
There are also other testimonies on dating Galen to the epoch of Nero, in particular, a Western medieval text called "Rapularius", a sort of late medieval encyclopedia attributed to Heinrich Toke, which specifically mentions two doctors named Galen (Hölzel-Ruggiu 2002): "the first Galen lived at the time of the emperor Nero, to whom he dedicated the first six books of the De ingenio sanitatis-i.e., the De methodo medendi-, as he himself attests in the seventh book of the De ingenio.(. ..)And the second Galen, also a physician of great fame, lived in the time of the emperor Antoninus Pius".In fact, in the XII century translation from Arabic to Latin of De ingenio sanitatis, at the beginning of the first book, there is a dedication to Nero, who would have incited Galen to write it (Kibre and Kelter 1987).

De Sectis (Text 20)
According to Photius, a Byzantine scholar of the IX century, some of Galen's works are well written and understandable in terms of vocabulary and syntax, but in others, the stylistic quality is very poor.In particular, speaking of De Sectis, Photius says: "It may be that this book is not predominantly medical, but of a rather philosophical nature, and serves as an introduction to medicine.Furthermore, it is clear that, in terms of vocabulary and syntax, it is pure and clear.These are qualities for which Galen has constant attention.However, in many of his writings, he overloads his books with unnecessary arguments, digressions and overlong periods.In this way he upsets and obscures the meaning of what he has written, fragments the discussion and, due to the length and verbosity, leads the reader to boredom.But the book we are talking about is free of such defects" (Photius, Bibliotheca, C 164).Thus, Photius confirms that strong stylistic differences are evident in the works attributed to Galen of Pergamum, and De Sectis, written without verbosity, and very clear in terms of vocabulary and syntax, could be attributed to Galen-1.

Conclusions
Based on the above discussion, we conjecture that of the 57 texts listed in Table 1, only the 13 texts listed in Table 2 can be attributed to Galen-1, an author who lived between the end of the I century BC and the second half of the I century AD.Table 3 lists the texts attributed to Galen-2 (Galen of Pergamum).The other texts from Table 1 are likely attributed to Galen-3.This proposed subdivision will be tested mathematically in the next section.

Deep Language Parameters and Vector Representation of Texts
Let us consider a text and its subdivision in disjoint blocks long enough to give reliable average values (Matricciani 2019).For each text block, let n S be the number of sentences contained in it, n W the number of words, n C the number of characters contained in the n w words and n I the number of punctuation marks (interpunctions) contained in the n S sentences.
In the present study, we divided each text from Table 1 into disjoint blocks of approximately 300 words, so that the statistics of linguistic variables need not be weighted by the length (in words) of the block texts, as done in (Matricciani 2019;2023a;2023b;Matricciani and De Caro 2019), where the blocks considered are chapters of novels.In all cases, all other alphanumeric characters, notes and titles were deleted, leaving only words and interpunctions, so as to obtain as much as possible the plain text written by the author.
For each text block, we computed the following variables (Matricciani 2019): (1) (2) We refer to them as the deep language variables (Matricciani 2019).In other words, C P gives the number of characters per word; P F gives the words per sentence; I P gives the words per interpunctions I P (this parameter is also referred to as the "word interval" (Matricciani 2019)); and M F gives the interpunctions per sentence (this variable gives the number of I ′ P s contained in a sentence).Very likely, these four linguistic variables are rarely consciously controlled by any author; therefore, their statistics can give useful indications on the similarity of texts with little or no bias.Notice that they also reveal readers' (and writers', as well) short-term memory characteristics (Matricciani 2024) and the readability index of the text (Matricciani 2023c) Table 4 reports the mean values of these four deep language variables for each text from Table 1.Specifically, let m be the number of samples (i.e., M disjoint blocks), and the mean value < P F >, for example, is given by: The variables defined in Equations ( 1)-( 4) allow us to study the variances in texts of similar length, scatterplots of the variables and the linguistic channels (Matricciani 2019(Matricciani , 2023a(Matricciani , 2023b(Matricciani , 2023c(Matricciani , 2024;;Matricciani and De Caro 2019).
Notice that In other words, the mean value < P F > is not given by the total number of words W divided by the total number of sentences S, unless all text blocks are of an equal number of words and sentences, which never ooccurs.The same discussion applies to all other variables.For example, for text 7 (De usu partium), W = 194, 985, S = 5845; therefore, W/S = 33.36,while < P F > = 35.05(Table 4).
The values reported in Table 4 can be used to represent texts in Cartesian coordinates (Matricciani 2019).This geometrical representation supports, as we show next, our alleged attribution of the texts from Table 1 to Galen-1, Galen-2 and Galen-3.In this Cartesian plane, two texts share a common mathematical structure if their relative Pythagorean distance is small, i.e., if the vectors show close endings.In other words, a small distance means that texts are mathematically similar, a feature that authors very likely do not consciously control.
The geometrical representation is based on defining the six vectors of the indicated components of deep language variables (Matricciani 2019 → R 6 = (< I P >, < C P >) and their resulting vector sum: From a vector analysis, the two orthogonal components of → R are given by x = ∑ 6 k=1 x k and y = ∑ 6 k=1 y k , which can be represented as single points in the first Cartesian quadrant.Notice that the choice of which variables represents the xand y-components is not irrelevant because, once the choice is made, the numerical results will depend on it, but not the relative comparisons and general conclusions.Moreover, to avoid different ranges in the x-and y-axes, we use the following normalized variables: In Equations ( 7) and ( 8) the maximum and minimum values are those obtainable from Table 4.A scatterplot of the resulting normalized coordinates is shown in Figure 1: green for the texts attributed to Galen-1; red for the texts attributed to Galen-2; and blue for the texts attributed to Galen-3 (Pseudo-Galen).The blue and the green regions have a negligible intersection, therefore indicating, very likely, that the texts that fall in these regions were written by different authors.Next, we calculate some probabilities to further pursue this topic.
Figure 1.A scatterplot of the resulting vector given by Equation ( 6), in normalized coordinates.The texts are indicated according to the order number reported in Table 1.The texts attributed to Galen-1 are indicated by green circles; the texts attributed to Galen-2 are indicated by red circles; the texts attributed to Galen-3 (Pseudo-Galen) are indicated by blue circles.The dashed lines contour regions containing the three sets.
Let us first calculate the a priori probability that a text falls in the green region of Figure 1.The probability that, by chance, a text attributed to Galen-1 falls in the green Figure 1.A scatterplot of the resulting vector given by Equation ( 6), in normalized coordinates.The texts are indicated according to the order number reported in Table 1.The texts attributed to Galen-1 are indicated by green circles; the texts attributed to Galen-2 are indicated by red circles; the texts attributed to Galen-3 (Pseudo-Galen) are indicated by blue circles.The dashed lines contour regions containing the three sets.
We notice the following facts: The blue and the green regions have a negligible intersection, therefore indicating, very likely, that the texts that fall in these regions were written by different authors.Next, we calculate some probabilities to further pursue this topic.
Let us first calculate the a priori probability that a text falls in the green region of Figure 1.The probability that, by chance, a text attributed to Galen-1 falls in the green region is given by the ratio between the area delimited by the green dashed line and the total area, i.e., the area delimited by the red dashed line.This probability is p = 0.15.Now, the probability that, by chance, the 13 texts attributed to Galen-1 all fall in the green area can be calculated with the binomial distribution.
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.In the following, the binary outcome is represented by a text either inside or outside the green region.To evaluate joint probabilities, there is the so-called probability mass function (Papoulis 1990), given by Equation ( 9).This formula describes the probability of observing exactly n successes in N trials, given by the binomial coefficient, multiplied by the probability of success raised to the power of the number of successes, multiplied by the probability of failure raised to the power of the number of failures.
Therefore, the joint probability that n points out of N are in the green region is given by (Papoulis 1990): Setting p = 0.15 and N = n = 13, the mean value and the standard deviation of the binomial distribution are given, respectively, by < N > = p × N = 1.95 and s = p × (1 − p) × N = 1.29.Now, we can explicitly calculate the probability that n = 13 texts fall in the green region, out of N = 13 First (t-test), we calculate the t-value: The probability of having t values greater than 8.6 with N = 13 degrees of freedom is p ∼ = 1 × 10 −6 (Papoulis 1990).Therefore, we can exclude that the clustering of the points attributed to Galen-1 is due to chance.
We also observe that 6 red points (11,30,43,50,52,55) out of N = 32 also fall in the green area.Now, the probability that n = 6 fall in the green area can be calculated in the same way.Since now, t = 0.3, the probability that t > 0.3 with N = 32 degrees of freedom is p = 0.77; therefore, these 6 texts could have fallen into the green area by chance, because their distribution covers the entire red area.
In the next section, based on these results, we merge the three sets of texts into three single texts and study them in the vector plane.

Deep Language Parameters of Galen-1, Galen-2 and Galen-3
In this section, we consider the three sets of texts of Section 4 as three single texts.In other words, we consider three different alleged authors who wrote three long texts.This new analysis shows that there were definitely at least three authors.
Table 5 reports the mean value and standard deviation of the mean (in parentheses) of P F , I P , C p and M F , for the three authors.At a glance, these values already show significant differences between the three authors, which are clearly evident in the vector plane shown in Figure 2, with normalized coordinates so that Galen-3 is set at the origin of the coordinates, at point (0, 0), and Galen-1 is at (1, 1).
Table 5. Mean value and standard deviation of the mean (in parentheses) of the deep language variables P F , I P , C p and M F , calculated for the three sests of texts referred to as Galen-1, Galen-2 and Galen-3.6) for Galen-1 (green dot and 3-sigma green circle), Galen-2 (red dot and 3-sigma red circle), Galen-3 (blue dot and 3-sigma blue circle).

Author
Table 5. Mean value and standard deviation of the mean (in parentheses) of the deep language variables  ,  ,  and  , calculated for the three sests of texts referred to as Galen-1, Galen-2 and Galen-3.The ending points of the vectors shown in Figure 2 are computed from the values of Table 5 according to Equation (6); the variance of the x-and y-coordinates is calculated by summing the variances of the variables of each coordinate in Equation ( 6), whose square root is reported in Table 5.The result of this calculation is reported in Table 6.

Author
From these latter values, we calculate the normalized coordinates of the ending points drawn in Figure 2 and the 3-standard-deviation (sigma) circles.For example, the 1sigma radius of Galen-1 is given by 0.079 + 0.371 ) = 0.379.Therefore, the 3-sigma circle of Galen-1 has a center in (41.724, 112.922) with a radius of 3 × 0.379.Then, these values are normalized so that Galen-3 is at (0, 0) and Galen-1 at (1, 1).
We can see that the probability of mistaking one author for another is practically zero, because the 3-sigma circles are very distant from each other.6) for Galen-1 (green dot and 3-sigma green circle), Galen-2 (red dot and 3-sigma red circle), Galen-3 (blue dot and 3-sigma blue circle).
The ending points of the vectors shown in Figure 2 are computed from the values of Table 5 according to Equation (6); the variance of the xand y-coordinates is calculated by summing the variances of the variables of each coordinate in Equation ( 6), whose square root is reported in Table 5.The result of this calculation is reported in Table 6.Table 6.Mean value and standard deviation of xand y-coordinates of texts referred to as Galen-1, Galen-2 and Galen-3, used for drawing Figure 2 with normalized coordinates.From these latter values, we calculate the normalized coordinates of the ending points drawn in Figure 2 and the 3-standard-deviation (sigma) circles.For example, the 1-sigma radius of Galen-1 is given by (0.079 2 + 0.371 2 ) = 0.379.Therefore, the 3-sigma circle of Galen-1 has a center in (41.724, 112.922) with a radius of 3 × 0.379.Then, these values are normalized so that Galen-3 is at (0, 0) and Galen-1 at (1, 1).

Author
We can see that the probability of mistaking one author for another is practically zero, because the 3-sigma circles are very distant from each other.
In conclusion, Figure 2 says that there is no overlapping among the three sets of texts when considered as single long texts.In other words, Galen-1, Galen-2 and Galen-3 are three different "authors", the first two very likely real persons, the third an unknown number of authors.
(a) The texts allegedly attributed to Galen-1 fall into the region delimited by the dashed green line.(b) The texts attributed to Galen-2 fall into the region delimited by the dashed blue line.(c) The texts allegedly attributed to Galen-3 (Pseudo-Galen) fall in the large region delimited by the red dashed line which includes all texts.
(a) The texts allegedly attributed to Galen-1 fall into the region delimited by the dashed green line.(b) The texts attributed to Galen-2 fall into the region delimited by the dashed blue line.(c) The texts allegedly attributed to Galen-3 (Pseudo-Galen) fall in the large region delimited by the red dashed line which includes all texts.
Table 1 lists the Latin titles of these texts with the presumed epoch of writing, according to scholars.

Table 1 .
List of texts written in Greek, attributed to Galen of Pergamum, considered in the present paper, with total number of words W > 1500 and total number of sentences S. PG refers to Pseudo-Galen (or Galen-3 in present paper).

Table 4 .
The mean values of the deep language variables P F , I P , C p and M F , calculated from samples of about 300 words in each text.