The Logic of Planetary Combination in Vettius Valens

1. Introduction

The Anthologies of Vettius Valens is the most extensive practical astrological text which survives from Greek antiquity.² The Anthologies consists of nine books, which include a huge quantity of information on astrological methods as well as catalogues of predictions. In total, the Anthologies contains around 130 partial or complete horoscopes ranging from 37-188 CE, some of which are used multiple times.³ These horoscopes suggest that the majority of the work was written between 152 and 162 CE, and the Anthologies has been in almost constant use since.⁴ As a resource for ‘literary’ horoscopes (i.e. those not from an original text, such as papyri) it is unparalleled: Neugebauer and Van Hoesen noted there would only be five other extant literary horoscopes dating from before 380 CE if the Anthologies had not survived. ⁵ Although we have little external biographical information about Vettius Valens, one of these horoscopes was suggested by Pingree to be that of Valens himself.⁶ From this horoscope and the associated details Valens provides, we can infer that he was born on 8 February 120 CE in Antioch, travelled widely, including to Egypt, and spent the majority of his working life as an astrologer in Alexandria.⁷⬈#p1

Valens is often contrasted with his near-contemporary Ptolemy (c.100–c.170 CE), also associated with Alexandria. Although both wrote on astrology, their styles are markedly different. In particular, Valens’ work is often characterised as practical, in opposition to the more theoretical work of Ptolemy’s Tetrabiblos. According to Neugebauer, the Anthologies and the Tetrabiblos were written almost simultaneously but exhibit a strong difference in style and purpose: Valens’ work seeks to ‘confirm and refine growing [astrological] doctrine’, whereas Ptolemy’s seeks to form a ‘consistent theory of… a universal science of life.’ (Neugebauer 1954, p. 67). Riley makes a similar case, characterising Ptolemy’s work as ‘axiomatic’ in its method: Ptolemy ‘lays down principles that are universally true, then he applies them in individual cases’; whereas Valens is ‘quasi-exhaustive’, constructing ‘listings of configurations’ that go on ‘at great length’ (Riley 1987, p. 249).⬈#p2

Given this contrast, if Ptolemy is a systematic philosophical thinker it may be tempting to treat Valens as his opposite: a miscellanist, churning out regurgitated screeds of astrological lore. This picture may also be influenced by our knowledge of Ptolemy’s wide variety of other interests, primarily in astronomy but also in geography, harmonics, and optics (the latter two in lost works).⁸ Indeed, even within the Tetrabiblos Ptolemy considers wide areas of astrological inquiry which are absent from the Anthologies, devoting large sections to ‘general’ astrology, which pertains to predictions that affect ‘whole races, countries, and cities’⁹ and including geographical material and predictions of the weather. By contrast, the Anthologies is presented by Valens himself as a handbook for practical astrology.⬈#p3

Scholars have not been impressed by Valens as a thinker: Cumont (1937) called him ‘narrow-minded and devoid of originality’ (p. 18)¹⁰ and Riley (1987) notes that when Valens attempts ‘empirical’ proofs of his craft, these are little more than the retrofitting of observed events to his own previously cast horoscopes (p. 248).¹¹ It is certainly true that Valens does not attempt—or seem to show any interest in—either a detailed philosophical account of the physics of the heavens or a wider investigation of the implications of astrology. This broad contrast between practical and theoretical often carries a (sometimes explicit) value judgement in Ptolemy’s favour (Riley 1987, p. 236). Nevertheless, it does not follow from this contrast in method and aims that Valens’ work was free of conceptual underpinnings. Nor does it follow that it was unsystematic or unscientific – a suggestion implicit in Riley and others.¹² In this article we make the case that Valens’ lists of planetary combinations show a greater degree of cohesion than has been previously noted. We analyse the descriptions quantitatively to show that they are not merely miscellaneous assortments of associations, but exhibit a strong internal logic. Despite the divergent methods and aims on display on the textual surface of Ptolemy and Valens, Valens’ general understanding of the underlying structure of astrology may be closer to Ptolemy than usually assumed.⬈#p4

1.1 Quantitative methods in the history of astrology

The modern mathematical study of Greek astrology owes much to the twentieth century historian Otto Neugebauer, who trained as a mathematician.¹³ While Neugebauer acknowledged the value of astrology for social history–its insight into ‘the daily life, religion and superstition... and cosmogonic ideas’ of the past (Neugebauer 1951)–he emphasised that the primary justification for studying it was because of its value for the history of astronomy and the exact sciences. He argued that astrological texts offer a window onto the mathematical knowledge of their period: ‘[t]he only hope of obtaining a few glimpses of the astronomical methods of the time of Hipparchus rests in the painstaking investigation of wretched writers like Vettius Valens’ (Neugebauer 1951).¹⁴ As late as 1942, Neugebauer could observe in passing that ‘one of the most important sources on Hellenistic-Oriental astronomy, Vettius Valens…is practically unexamined, as far as information about exact astronomy is concerned. This problem very much deserves serious consideration’ (Neugebauer 1942, p. 239).⬈#p5

Until the mid-twentieth century it was unknown whether the horoscopes in the Anthologies referred to real and contemporary planetary configurations. In a pioneering work, Neugebauer and Van Hoesen (1959) showed how to ascertain the exact date of even partial literary horoscopes. Their method was manual and required aligning the motions of planets on long sheets of graph paper.¹⁵ In most cases this permitted them to identify a precise date to an accuracy of around a day. Overall, this successfully led to the identification of dates for almost all horoscopes in the Anthologies, and put the empirical basis of Valens’ work beyond question. Although it would have been easy to invent horoscopes for pedagogical purposes, Valens did not do so.⬈#p6

The modern availability of accurate ephemerides such as the Swiss Ephemeris (Koch and Treindl 2021) together with computational libraries for manipulating horoscopes means that reproducing Neugebauer and Van Hoesen’s work is relatively straightforward. However, our focus here is not the dating of individual horoscopes, but the use of quantitative techniques for aiding textual analysis. Lynn Thorndike applied a similar approach to Firmicus Maternus, manually counting occurrences of terms to introduce a ‘quantitative element’: ‘one naturally assumes that those matters to which Firmicus devotes most space and emphasis are the most prominent features of his age’ (Thorndike 1913, p. 417). We see this as a precursor to the analysis we present here.¹⁶⬈#p7

We are not suggesting that this quantitative analysis reveals details which cannot be discovered by other methods, nor that it is more sophisticated: it is a blunter instrument than close reading. But quantitative textual analysis is a historical method well-suited to uncovering the broad features of a body of text, and the inter-relationships between its sections, which may not be apparent from a linear reading.¹⁷ This is particularly true for a writer like Valens: it is difficult to know whether an unfocused style may be obscuring a greater underlying cohesion in content than is apparent. We would like to suggest that these methods have wider applicability to the investigation of astrological texts; however, as an initial case study we apply them narrowly to a situation where one would expect to uncover relationships: Valens’ descriptions of the qualities of the planets and of their combinations.⬈#p8

1.2 The planets and their qualities

The hierarchy of planets in astrology seems to have been established very early in its history and persisted:⬈#p9

Saturn, Jupiter, Mars, Venus, Mercury.⬈#p10

This order is found across astrological texts from Babylon, Egypt and Greece¹⁸ and continued to be accepted.¹⁹ The reason for this consistency may have pragmatic origins: it matches their distances from the Sun (in a heliocentric model), from furthest to nearest, and so corresponds to their speed of movement over the year through the zodiac.²⁰ This order of the planets is largely consistent.²¹⬈#p11

In classical astrology, ‘planets’ also included the two luminaries, the Sun and Moon, making seven classical planets. The order including the luminaries is more variable. While Valens uses the standard ordering of the five non-luminary planets, he tends to place the luminaries at the start when describing a horoscope,²² proceeding through the planets in this order but mentioning other planets when they occupy the same sign of the zodiac. In this article, when discussing configurations of planets we use the more normal astrological ordering as accepted by Ptolemy²³⬈#p12

Saturn, Jupiter, Mars, the Sun, Venus, Mercury, the Moon⬈#p13

The order of the five non-luminary planets is consistent. But what of their qualities? Because of the continuous popularity of astrology from ancient times to our own, many of the characteristics of the planets in Greek astrology remain familiar to us. The idea of Mars as war-like, or Saturn as cold and elderly, were, in Greek thought, deeply mythically rooted. Indeed, Riley notes that similarities in the mythic characteristics ascribed to planets and signs in Valens are also accepted by Ptolemy, suggesting their ubiquity (Riley 1988, p. 68). Such widespread acceptance coupled with a modern familiarity has probably occluded further study of planetary characteristics.²⁴ Yet it has been established that alongside the basic characterisations, Babylonian, Egyptian, and Greek sources give a varying series of associations and characteristics for the planets. Some of these are repeated and popular; some appear only rarely. Commenting on this tendency in the case of Saturn, Bouché-Leclercq wrote that, from such a vast store of examples, ‘astrologers chose according to their liking how to compose the type of this powerful and dreaded star’.²⁵⬈#p14

Partly due to these sorts of mythic characterisations, the view of the qualitative aspects of astrology as whimsical or pedantic often still prevails. There has been little systematic effort to investigate the internal conceptual consistency of Greek astrological interpretation – even of particular astrologers, with the exception of Ptolemy. Any focus on consistency in the scholarly literature is usually on the mathematical or theoretical aspects. This is not surprising, and is likely to be the result of two main trends. First, the interpretative claims of astrology still carry weight, and, whether implicitly or explicitly, most scholars seek to distance themselves from the central claim that the stars can cause or signify human events.²⁶ Second, even with Greco-Roman divinatory methods that are more straightforwardly of the past, study of the precise claims of predictive methods is grounded by a scepticism which discourages taking particular interpretative claims too seriously – if the claims are false, then why suppose that they must be conceptually cohesive? The Ptolemaic approach placing planets within the scheme of Aristotelian physics is the exception, because of the clear link to wider Greek science. To modern readers it often appears a refreshing contrast to arbitrary lists of associations due to its physical basis.⬈#p15

However, this distinction must not be overstated. Overall, there is a complex relationship between the general mythic characteristics of the planets, more narrowly-defined astrological traditions about them, and the physical claims made by astrologers about the planets’ natures and characteristics.²⁷ Riley recognises the first and third of these categories, noting that sometimes in Ptolemy’s Tetrabiblos there is a slightly uneasy relationship between the two, and maintaining that Ptolemy distinguished between them, even if not in the terminology of ‘mythical’ vs ‘physical’ (Riley 1988, p. 68). For Ptolemy, the physical characteristics of the planets fit into an Aristotelian system of balanced opposites: characteristics included physical qualities like hot and cold, wet and dry, as well as biological characteristics like male and female, and temporal characteristics like diurnal and nocturnal.²⁸ The physical characteristics of the planets have an effect on the Earth: ‘Venus adds heat to the prevailing conditions, Saturn cold, Jupiter moisture, Mars dryness, Mercury motion and wind.’²⁹⬈#p16

Ptolemy also designates some secondary characteristics which are easy to categorise, including each planet’s status as broadly beneficent or maleficent. It is never explicitly stated whether this primarily means beneficent towards human beings; Ptolemy does seem to have in mind also the relationship of the planets to one another. In Ptolemy’s system the benefics are Jupiter, Venus, and the moon; the malefics are Saturn and Mars; the neutrals are the Sun and Mercury. While Ptolemy attributes these categorisations to ‘the ancients’ (οἱ παλαιοί) he also gives an Aristotelian physical explanation: those planets which have predominantly cold or dry natures are malefic.³⁰⬈#p17

Ptolemy’s post hoc rationalisation of existing astrological lore shows that certain overarching conceptual associations for planets were so widely accepted as to be indisputable. These broad meanings of each planet were likely known even to non-astrologers in Valens’ time.³¹ Yet despite this they were of limited use for astrology, since in practice any horoscope contains a specific configuration of planets. Due to the length of the periods of planets, many configurations never occur in the lifetime of any given astrologer. Deciphering the configuration’s precise meaning is a complex art. This is part of the paradox of astrology: whereas the meaning of each component part may be intuitive, and (assuming a horoscope can be accurately cast) the data all on display, the overall meaning can remain hopelessly obscure. Notably, while Ptolemy discusses the characteristics of the planets, he avoids any discussion of real horoscopes or even of planetary combinations. Indeed, he excuses himself from any responsibility to discuss planets in combination at all, writing that ‘[i]t is of course a hopeless and impossible task to mention the proper outcome of every combination’.³²⬈#p18

In contrast, Valens is far more concerned with the real situations an astrologer will encounter. As stated above, he provides real horoscopes throughout the Anthologies. And at the beginning of Book 1 he gives a list of the characteristics of each planet (1.1ff). After this, almost as a footnote, he explains that although there are ‘benefic’ and ‘malefic’ stars, their effects can be enhanced or mitigated depending on how they are disposed. Valens does not specify which are benefic and malefic, probably because he assumes these designations are known to his reader. He then proceeds, in contrast to Ptolemy, to enumerate the characteristics of combinations of two and three planets. It is worth considering the reason for this. Conceptually, it is common to separate the practice of astrology into the computing of horoscopes with astronomical methods and then their interpretation.³³ But the interpretation of a new astrological chart is an integrated practice, drawing on a combination of knowledge: of course the practical computation of horoscopes and calculation of angles between planetary positions, but also on established interpretations of commonly observed patterns. That is, the qualitative lore of astrology is not only conceptual but inherently practical. The raw materials of astrology are not only the computed positions of planets and their individual ‘meanings’ but also a knowledge of the effects of these planets in combination, as well as their shifting valency in specific zodiac signs or ‘places’ of the chart. However, the combinatorial possibilities are huge, making iterating them in a pre-computer age a hopeless task. So the novice astrologer cannot be instructed in the pre-defined meaning of every chart. Viewed this way, it is understandable that astrologers naturally built up methods for progressive ‘chunking’ of horoscopes into important and repeated features that themselves had a body of ‘pre-computed’ associations. Valens does not just aim to provide his readers with the foundational principles of astrology. By providing a compendium of the characteristics of combinations of two and three planets, he provides a practical basis for interpretation once any given horoscope is broken down into intermediate component parts. Modern computers make combinatorial problems tractable. As historians, we can therefore bring new techniques to bear on this combinatorial ‘data’ to better understand the logic and dynamics of astrological interpretation.⬈#p19

2. Planetary Combination

The seven planets and their ‘combinations’ (συγκράσεων) are among the first things discussed at length in Book 1 of the Anthologies. It seems reasonable to take them as of primary importance for Valens. In this section, we first consider the significance of this term, before conducting a practical calculation of the actual possible occurrences of combinations to ground the analysis that follows.⬈#p20

2.1 The concept of synkrasis

It is important at this point to demarcate exactly what Valens means by synkrasis (σύγκρασις). Philological evidence shows that the term was used beyond astrology, with the LSJ giving ‘a mixing together, commixture, blending, tempering’ as the first sense (p.1666). The majority of uses of σύγκρασις are theological or ontological.³⁴ Among astrological writers σύγκρασις -εως appears in Valens,³⁵ Ptolemy,³⁶ Hephaestio of Thebes,³⁷ ps-Manetho,³⁸ Paulus Alexandrinus,³⁹ in a commentary by Olympiodorus on Paulus Alexandrinus,⁴⁰ and occasionally in critiques of astrology, e.g. Origen of Alexandria.⁴¹ All of these uses refer to the mixing of the effects of planets, with some uses making more abstract reference to the planets in general (ps-Manetho, Hephaestio), and some uses referring more specifically to the mixing of particular planets. A typical use is Paulus of Alexandria referring specifically to the mixing of the effects of Mars and Mercury in the place of Good Fortune, i.e. the fifth house.⁴²⬈#p21

Scholars have argued that the concept of synkrasis was an important one in astrology. For example, in his notes on the Corpus Hermeticum, Festugière notes that the astrological doctrine of synkrasis is well known (bien connue) and the term is used to designate both the combination of elements (or elementary qualities) in the body and the combination of influence of the stars.⁴³ Ptolemy uses the word in both senses in the Tetrabiblos (the body: 1.2.18 H11; the stars: e.g. 2.9.18 H142) suggesting a strong analogy between his understanding of how planetary influences combine in an Aristotelian framework. Valens’ use of the word in the Anthologies applies only to planets. As we show below, quantitative analysis of the characteristics he assigns to planetary combinations suggests Valens is also using a (less explicit but nevertheless present) underlying conceptual framework.⬈#p22

The use of synkrasis suggests that Valens is not talking specifically about conjunction (the occurrence of two or more planets in the same sign of the zodiac) but more generally about how the astrologer should take the influences of two or more planets into account together in a horoscope. These influences must be modulated by the particular aspect of the planets. How exactly the influences of planets combined was a practical question. In astrology, one of the most important possible combinations was conjunction: the occurrence of two planets in the same sign of the zodiac. Planetary combinations were discussed using the term ‘aspect’: the geometrical relationship between the positions of planets in the zodiac which translated into a conceptual schema of combined influence. While conjunction was often thought of as its own category, it is sometimes counted alongside the other aspects too.⁴⁴ Other important aspects include the benefic aspects of trine and sextile, and the malefic aspects of opposition and square.⁴⁵ When Valens discusses synkrasis, we believe he has all of these aspects in mind, and is providing a framework for the astrologer to modulate the predicted outcomes appropriately according to whether the aspect in question is benefic or malefic and the relative strength of each planet’s influence.⬈#p23

2.2 Practical limits on planetary combination

Whether or not astrology is viewed as a science, it rests upon a foundation of empirical data about the sky. It is therefore useful to consider the constraints on planetary combination that arise in real astrological data. First, the calculation of combinations of the seven planets and luminaries; in his work, Valens considers both combinations of two planets (‘doubles’) and combinations of three planets (‘triples’). The number of combinations of n non-repeated elements from seven elements is given by $\left(\genfrac{}{}{0ex}{}{7}{n}\right)$ (‘seven choose n’). There are thus $\left(\genfrac{}{}{0ex}{}{7}{2}\right)$=21 possible doubles and $\left(\genfrac{}{}{0ex}{}{7}{3}\right)$=35 possible triples. In the Anthologies, the descriptions of planetary synkrasis omits two doubles (Mars Sun; Mars Moon), and three triples (Saturn, Mercury, Sun; Saturn, Mercury, Moon; Saturn, Sun, Moon). Although omission in the original cannot be ruled out, it would be surprising for a handbook aiming to be a comprehensive and practical guide, and we therefore suggest it may have arisen from a simple transcription error.⁴⁶⬈#p24

Importantly, not all combinations of planets are realised, particularly once one brings in the signs of the zodiac as well. A scheme outlined by Boxer (2020) can be used as a simplifying way to discuss the historical occurrences of conjunctions. In brief, the twelve signs of the zodiac can be represented in hexadecimal notation.⁴⁷ The current zodiacal sign of each the seven planets (in the order Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon)⁴⁸ can then be converted into a digit in hexadecimal notation, so a configuration of planets can be represented as a seven-digit code. We follow Boxer in referring to this as a Z-code (from ‘Zodiac-code’).⁴⁹ For example, the Z-code⬈#p25

means that Saturn is in Aquarius (A), Jupiter in Aquarius (A), Mars in Taurus (1), the Sun in Aquarius (A), Venus in Capricorn (9), Mercury in Aquarius (A), and the Moon in Aries (0). Z-codes offer a convenient way to simplify historical astronomical data to investigate the planetary combinations realised within the resulting horoscopes.⁵⁰ This is appropriate for Valens, since he usually quotes only the zodiacal signs for each planet.⁵¹⬈#p26

We used Flatlib, a Python 3 library for traditional astrology⁵² in conjunction with the Swiss Ephemeris to calculate the Z-code for each day in Valens’ century and the century before, from 1st January 1 CE to 31st December 200 CE, taking midday on each day. We used the latitude and longitude of Alexandria 31.22° N, 29 96° E.⁵³ One should expect slightly different results with this approach from what would have been calculated by contemporary astrologers such as Valens at the time due to a different frame of reference.⁵⁴ From these Z-codes, we found the days which included conjunctions. As noted above, this approach treats all cases where planets share the same sign as conjunctions (see n. 50).⬈#p27

One immediate observation is that planetary conjunctions are extremely common: 97.5% of days in this period contain at least one conjunction.⁵⁵ 74.3% contain a double conjunction, 34.4% contain a triple conjunction, and 17.2% contain both a double and a triple conjunction (each in a different sign). Even quadruple conjunctions occur on 7.5% of days, although higher order conjunctions are much rarer (n=5, 0.85%; n=6, 0.07%; n=7, 0.0%).⬈#p28

These results demonstrate that planetary conjunctions occur in the vast majority of possible horoscopes. Even if a concept of synkrasis was only important for conjunction, it would be invoked at least once in the interpretation of almost every horoscope. In fact, when other aspects are taken into account, any horoscope contains at least nine aspectual relationships between planets (see Appendix).⁵⁶ Valens’ notions of planetary combination are therefore of immense value for understanding how practising astrologers conceptualised the combinatorial mainstays of horoscopic prediction.⬈#p29

2.3 The length of Valens’ descriptions

The length of the descriptions alone already gives some indication of Valens’ priorities (Figure 1). Notably, Mercury has the longest description of any planet; as the fastest-moving, it frequently interacts with other planets, so its modifying character is important for Valens. However, particular combinations involving Mercury are not particularly notable since they recur frequently; for example, the Sun and Mercury will be in conjunction in over 50% of horoscopes. While the cumulative number of words devoted to triples is the largest (planets 1,046; doubles 1,010; triples 2,145), Valens devotes more space on average to single planets than to combinations (mean number of words in descriptions: planets 149; doubles 53; triples 67) However, some combinations have longer descriptions than some planets. For example, the triple ‘Saturn Mars Mercury’ has the longest description of any combination, consistent with its conceptual importance: for Valens, the combination of two malefic planets with the modifier of Mercury requires a great deal of exposition.⬈#p30

There is no overall relationship between the frequency of occurrence of a conjunction of planets with the length of the description of the corresponding combination (Figure 1). Although triple conjunctions occur less frequently than double conjunctions, Valens devotes more space on average to describing triple combinations. This may suggest that he attaches more importance to elucidating the characteristics of triples, which involve the mingling of three influences and so are correspondingly more complex combinations than doubles.⬈#p31

3. Sentiment Analysis

As noted above, the planets in Greek astrology had associated characteristics. Perhaps the most fundamental was their categorisation as benefic (positive) or malefic (negative). We decided to therefore attempt to quantify the descriptive terms used by Valens as positive, negative, or neutral, to provide a rough quantitative index as to the overall ‘sentiment’ of a particular planet or planetary combination.⬈#p32

Sentiment analysis is a method for analysing snippets of text for positive or negative valences.⁵⁷ Some types of sentiment analysis use natural language processing software, while others rely on manually-mined data. Some approaches use both, especially with complex texts such as literary works or historical sources.⁵⁸ Here, we used a manual approach to break sentences within descriptions into lexical terms and assess their individual sentiments, and then compute mean overall sentiments for each description.⬈#p33

First, we manually sorted Valens’ descriptions of the planets and the effects of their combinations from the following sections of the Anthologies:⬈#p34

• Anthologies 1.1 - a list of the characteristics and effects of each planet.
• Anthologies 1.19 - a list of the effects of each double combination.
• Anthologies 1.20 - a list of the effects of each triple combination.

3.1 Computing a sentiment index

Sentences in Valens’ passages of planetary descriptions tend to appear in one of four different forms. The initial descriptions of each single planet appear primarily in the following form:⬈#p35

Planet X is A, B, C (where A, B, C) are adjectives).

For the descriptions of the effects of planets, either singly or in combination, terms appear primarily in the three following forms:⬈#p36

X causes or signifies D (where D is an abstract noun e.g. ‘death’ or ‘illness’).⁵⁹

or⬈#p37

X causes or signifies [those linked to it to be] Es (where E is a nounthat encodes a practice or profession e.g. ‘murderers’ or ‘goldsmiths’).

or⬈#p38

X causes or signifies Fs to be G (where F is a noun e.g. ‘men’ and G is an adjective e.g. ‘famous’ or ‘wealthy’).

We constructed a database of terms, including the original Greek and an English translation of all nouns and adjectives used in any of these formulations, and which planet or combination of planets they applied to. One of us (Hall) then assigned to each individual term (henceforth referred to as ‘lexical term’) a rating of ‘positive’, ‘neutral’, or ‘negative’. In many cases the assignations were very clear, due to the extremely frequent use of words with obvious sentiments like ‘injury’ or ‘happiness’. In some cases, it was necessary to use cultural knowledge of Hellenistic and post-Hellenistic Greek attitudes to money, sex, certain professions, certain family arrangements and so on in order to reach a judgement on whether a term would be viewed as positive or negative.⁶⁰⬈#p39

This manual approach has several limitations. First, it can be difficult to extract lexical terms from descriptions: should ‘associated with fame and fortune’ be two lexical terms, or just one? Second, even classifying as positive, negative or neutral requires a subjective judgment. Third, these categories are crude and do not capture any nuance of interpretation. Fourth, we only used one assessment rather than multiple expert assessments. However, despite these limitations the resulting sentiment indices are useful for comparing descriptions and looking at relationships between them. We do not put too much weight in the significance and categorisation of individual lexical terms; classifying them in this way is a means to an end, to compute a single index for each description.⬈#p40

An alternative approach would be to use automated sentiment analysis with an existing tool. We explored this approach and ran descriptions (in English) through an existing sentiment analysis tool in the Stanza Python 3 library.⁶¹ This tool is trained on modern English text sources and assigns a sentence as negative, neutral, or positive. We used the mean sentiment of the sentences in a description. We found an approximate correlation (Pearson’s r¿0.6) between our manual assessments and these automatically computed sentiments. On inspection, the automated sentiment analysis appeared inferior to our assessment. As an indicative example, take the following sentence:⬈#p41

Men also derive benefits from women, and coming into possession of estates and land, they become lords. (Saturn, Jupiter, the Moon)⬈#p42

In the context of astrological prediction, this sentence is uncomplicatedly positive. We assessed it as having two positive lexical terms: ‘derive benefits from women’ and ‘coming into possession of estates and land(, they become lords)’. However, this sentence is given a neutral overall sentiment by Stanza’s classifier.⁶² Reliable assessments depend strongly on the cultural context, whether this is by training on a carefully selected training dataset or through expert knowledge. In the absence of an appropriate automated tool for Valens’ context, we therefore chose to use our manual assessment to compute overall sentiment of descriptions.⁶³⬈#p43

3.2 Sentiment index results

For the single planets, the descriptive passages range in length from 99 to 256 words, with a median of 57 lexical terms in a description. Around half of terms in the descriptions were assigned as neutral (188/400, 47%). There is a very slight excess of positive terms compared to negative terms (29% vs. 24%). The malefics (Saturn and Mars) both have strong negative overall sentiment according to this method. It is of interest that the extremity of sentiment (whether positive or negative) corresponds approximately to the ordering of the five non-luminary planets: Saturn and Jupiter are equal and opposite, Mars and Venus are similarly comparable but with reduced magnitude, and Mercury has the smallest sentiment.⁶⁴⬈#p44

From these assignations, we then computed an overall sentiment S for each of the descriptions from their component lexical terms as⬈#p45

where p is the number of positive lexical terms, n the number of negative terms, and m the number of neutral terms. This sentiment S can range from -1 (all lexical terms are negative) to +1 (all lexical terms are positive). Table 1 gives these values for all single planets.⬈#p46

For example, Saturn has 4 positive lexical terms, 21 neutral, and 50 negative. So we have⬈#p47

In the same way, we assigned lexical terms within the descriptions of double and triple combinations and then computed an overall sentiment for each description (see Supplementary Tables: Table 2 for doubles and and Table 3 for triples). Interestingly, the number of neutral terms was much lower for double and triple combinations, with none in descriptions of doubles and only 29 of 635 terms for triples (< 5%). This suggests a different conceptual categorisation for the qualities of combinations of planets as opposed to the qualities of planets on their own.⬈#p48

There is no relationship between the sentiment of combinations of planets and their frequency of occurrence in conjunction (Figure 2). Most conjunctions occur in at most 10% of horoscopes, with triple conjunctions obviously rarer than double conjunctions. The three most frequent double conjunctions are those involving the Sun, Venus, and Mercury, all of which are present in more than 20% of horoscopes. These doubles are all strongly positive in sentiment. This may suggest a certain degree of pragmatism for the working astrologer who would encounter these conjunctions in more than a fifth of their client-work. Other pragmatic considerations about the ways in which synkrasis would be encountered might also be relevant. For example, Mercury and the Sun cannot ever be in any other aspect other than conjunction.⁶⁵ In a practical sense this might mean that the synkrasis of Mercury and the Sun does not have to be as polyvalent as for most other combinations of planets. The concept of frequency of encounter of an aspect is also worth considering. Although a planet’s speed is irrelevant to the number of horoscopes in which it is in aspect with another, it affects the diversity and duration of those aspects. For example, because Mercury moves quickly, in a time period of a year it will have been in every physically possible aspect with every other planet at some point during the year. In comparison, aspects between two slow-moving planets change less frequently but last for longer when they occur. So one might speculate that any given instance of synkrasis involving Mercury would be more likely to be ‘routine’ for an astrologer, resting on familiar knowledge, whereas a previously unencountered aspect of Saturn might prompt a consultation of existing lore to aid interpretation.⬈#p49

3.3 Building combinatorial models to predict sentiment

By reducing the descriptions of the planets and combinations down to a dataset of sentiment indices, we can perform statistical analysis to predict the sentiment of a combination. There are several options for doing this.⬈#p50

Independent bodies. In the first instance, we used a linear model to predict S with a dummy variable xi for the presence of each body i, which takes the value 1 when body i is present in a conjunction and is 0 otherwise. That is, ⬈#p51

To take a specific example,$S_{SaJu}={\beta }_{Sa}+{\beta }_{Ju}$. We set the intercept term to zero, allowing us to calculate an effect for each dummy variable. Intuitively, this is a model where each planet i contributes a consistent and fixed effect ${\beta }_i$ to all combinations it occurs in. The model allows for no interaction or modulation of this effect by the presence of other planets in the combination. We fitted a model of this type for the double sentiments and the triple sentiments separately, and also compared the effect for each planet to its individual sentiment (Figure 3).⬈#p52

The results show a good concordancy between the double (red) and triple (black) models. The effect of Saturn and Mars in a combination is overall a negative one, whereas Jupiter is the most positive predictor in both models. There are some interesting discrepancies between the sentiment of the planets and their effects in the models. Venus has a positive sentiment (it is benefic) but in fact has no average effect in combinations. The Sun appears positive in doubles, but has no effect in triples.⬈#p53

Mixing models. Rather than assuming that each planet has an overall independent effect when it is present in a combination and calculating this effect, we can work from its single sentiment. In this approach, we assume that the sentiment of a conjunction is related to the mean of the sentiments of its component parts.⁶⁶ For a double, the only possible component parts are single planets. Thus, the predicted sentiment ${\tilde{S}}_{pq}$ of planets p and q in combination is⬈#p54

This predicted sentiment is strongly correlated with the actual sentiment (Figure 4), although there are some discrepancies. For example, the mean sentiment of Saturn and Mercury is negative, but in fact in combination they are broadly positive in effect together. Thus the relationship is not simply additive.⬈#p55

For the case of triples, there are multiple possible component parts. At the base level, a triple is composed of three planets, so⬈#p56

However, one can also consider a triple as a combination of three doubles. In this model, one has⬈#p57

A third possibility is in fact given by taking into account the three possible combinations of doubles and singles:⁶⁷⬈#p58