Aristotle’s Three Logical Figures: A Proposed Reconstruction

 

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AbstractBased on the evidence of the likely near-contemporary mathematical practice of diagrams, this article proposes a possible reconstruction of Aristotle’s three figures as introduced in Prior Analytics 1.4–6.Keywords: Aristotle; Prior Analytics; reconstruction
Already Einarson 1936 pointed out the debt Aristotle’s logic owed to the mathematics of his time. Smith 1978 may have been the first to point out specifically that the alphabetic labels used in Aristotle’s logic—the many A, B, C— must have reflected diagrams in the manner of Greek mathematics.1 Indeed, (1) Aristotle was deeply aware of the mathematics of his time;2 (2) there was a significant burst of mathematical activity in the fourth century, much of it by authors related to Plato’s circle and so directly familiar to Aristotle;3 and, finally, (3) the use of lettered diagrams—indeed, of illustration in general—was, throughout antiquity, specific to mathematics and extremely rare in any other context.4 We can safely say: when Aristotle refers to letters, he almost certainly refers to diagrams in the Greek mathematical manner.
That Aristotle’s logical diagrams are now absent from the manuscripts of the Prior Analytics is disappointing but not surprising. There are cases where the Aristotelian corpus invokes diagrams, internal to the text, which were then lost to the textual transmission,5 and other cases where the transmitted Aristotelian text expected its readers to refer to diagrams external to the text itself.6 Either could obtain here. Smith himself did not try to reconstruct Aristotle’s logical diagrams but a number of proposals have been made in the past. Already in Late Antiquity, commentators produced visual pedagogic tools to help readers navigate Aristotle’s syllogisms. Modern scholars identify the first such use in Ammonius, early in the 6th century CE, with examples that clarify, for instance, statements in modal logic by the use of particular examples and a visual setting out of the relations between the terms:7
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The evidence from silence, prior to Ammonius (if this is indeed by Ammonius and not by later scribes) is not meaningless—Aristotle’s logic is among the most annotated texts from antiquity and the absence of such ancient figures, prior to Ammonius, is telling. It was likely in the same era as Ammonius, for instance, that the famous ‘Porphyry’s tree’ was first drawn, perhaps in the context of medical education in Alexandria.8 Diagrams such as those (perhaps) introduced by Ammonius or some other late ancient authors, as well as Porphyry’s tree, turn a conceptual relation into a two-dimensional structure, emphasizing the person of the teacher as the reader’s guide into a bookish world of learning: the typical move of late ancient and medieval commentary.
Many modern authors took such medieval diagrams as their inspiration for the reconstruction of Aristotle’s own lost diagrams. So, for instance, Kneale and Kneale 1962, 71–2:

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Rose 1968, 113–16 was overall cautious and explicitly noted the limits of relying upon the medieval tradition, and yet seems to have endorsed (cautiously) the following version:

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Such proposals are usually made in passing and without undue confidence. The only author I am familiar with to devote considerable effort to this reconstruction is Wesoly, in a series of spirited articles from 1996 onwards (followed by Englebretsen 2020). Wesoly’s diagrams possess much greater detail, with precise two-dimensional arrangements as well as arrows so that the visual annotation, as such, becomes in some sense logically sufficient. It seems that this logical efficacy is Wesoly’s main criterion for his historical reconstruction. So, Wesoly 2012, 198:
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What is absent from such modern reconstructions is an effort to locate Aristotle in the context of the mathematical practice that, after all, must have inspired him. Let us turn to this inspiration.
There are several contexts in which Greek mathematical texts refer to non-spatial entities, with some or all of the entities referred to, in the text, by single letters. It so happens that all of them use the very same diagrammatic principles, so that, if indeed these are Aristotle’s inspiration, the form of his logical diagrams becomes very clear.
The main extant sources are the treatment of numbers, in Euclid’s Elements Books 7–9; the treatment of general magnitudes in proportion, in Euclid’s Elements 5; and the treatment of notes, in the ps.-Euclidean Sectio Canonis. The comparison to music is the most obvious (and was the one emphasized by both Einarson 1936 as well as Smith 1978). Greek musical theory often proceeds by considering two concepts: single notes, and the intervals defined by them. This is reminiscent of the way in which, in Aristotle’s logic, one works through two main objects: single terms (such as ‘animal’, ‘horse’) and two-term premises (such as ‘Animal belongs to all horse’). In fact, one finds a similar structure in general proportion theory, with magnitudes—and their ratios. And since the discussion in Euclid’s arithmetical books, as well, usually concerns the ratios between numbers, we find a large, coherent field of applications, all reminiscent of Aristotle’s logic, studying non-spatial terms and their dyadic relations.
In the extant manuscripts, there are two major ways in which the terms belonging to those fields are represented.9 In the simplest case, the individual terms are represented as single lines, each set separately, all parallel to each other; next to each, its alphabetic label. Sectio Canonis 1 is a simple example:10

Let there be an interval BC, and let B be a multiple of C, and let B be to D as is C to B. I assert then that D is a multiple of C.
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A somewhat more complicated case arises where one needs to sum up, or subtract, terms. Now, in music, one note added to another does not make a third note, but, instead, it makes an ontologically distinct object—the interval. (Occasionally, however, notes are considered strictly as numerical values, and then they may be summed up or subtracted.). A magnitude, however, added to a magnitude, does give rise to a third magnitude, hence representations such as those of Euclid’s Elements 5.1:

Let any number of magnitudes whatever AB, CD be respectively equimultiples of any magnitudes E, F equal in multitude. I say that, whatever multiple AB is of E, that multiple shall AB, CD also be of E, F…. Let AB be divided into the magnitudes AG, GB equal to E …11
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In the course of Elements 5.1 one considers AB as a composite of AG, GB; hence the magnitudes AG, GB are each defined by two labels. The magnitudes E, F do not participate in such compositions and therefore each is labeled by a single letter. In this case, then, we still have lines set in parallel, but some of the lines have a single letter next to them while others have three or more letters, referring to two or more elements, each bounded and labeled by the combination of two letters.
The examples from music, proportion theory and arithmetic are not only widespread, and coherent, but are even close in time to Aristotle’s (in general, it seems clear that mathematics directly inspired by music theory was especially dominant in the fourth century BCE12). It is worth noting that the practice is found in later sources as well: so, for instance, Archimedes’ Sphere and Cylinder 1.2, solving a problem in geometrical proportion theory, where AB, D are set out as general magnitudes (see the figure on the next page).13
In short, there is no ambiguity in the historical evidence, and to the extent that we believe that mathematics was indeed Aristotle’s inspiration, we know precisely which diagrams it was that inspired him: parallel lines, with letters set next to them.
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The text of the Prior Analytics rather infrequently refers to the combination of two labeled terms (the very first introduction of labeled terms, 25a14, is one such exception: hē AB protasis, ‘the premise AB’, and there are a few similar uses later in the treatise. These, as noted, are the functional equivalent of ‘intervals’ in music). Much more often, the text refers simply to single terms taken in isolation. And, indeed, such terms are like musical notes, not like mathematical magnitudes. A ‘horse’, added to an ‘animal’, does not make a new, third term. It can only make an ontologically distinct category—a premise. To the extent that Aristotle is inspired by mathematics, then, his diagrams should take the form of parallel lines, with a single letter next to each.
Fortunately for us, Aristotle does briefly discuss the spatial structure of the figures as they are introduced. So, with the first figure (25b35–7):14

I call that the middle which both is itself in another and has another in it—this is also middle in position—and call both that which is itself in another and that which has another in it extremes.
Obviously Aristotle uses the word ‘middle’ primarily in a logical sense, but he notes that, in the figure, the object standing for the logical middle is in fact in the spatial middle of the figure (‘in position’); this helps to clarify that ‘extremes’, too, are so-called because they are at the two spatial extremes of this arrangement which we can now understand as three parallel lines. Aristotle later on refers to one of those extremes as ‘greater’ and to the other as ‘smaller’. In the usage of Aristotle scholars those adjectives become ‘major’ and ‘minor’ and scholars have got used to think of those terms as marking a logical position in the syllogism, but it is extremely likely, given the presence of a figure with straight lines, that Aristotle in fact refers to the sizes of the lines.15 In this case of the first figure, then, relations of size are replicated as relations in space: the line in the middle is also middle in size.
So far, then, we have established that we have three parallel lines marked (in Latin transliteration) from A to C and arranged in that order, such that A>B>C in length. There are in principle two degrees of freedom, depending on the direction by which we wish to move from A to C: left to right or right to left, top to bottom or bottom to top. (It would be very surprising, given the mathematical examples or indeed given general cognitive considerations, to have anything other than a single horizontal or vertical arrangement).
The second figure is discussed in slightly greater detail (26b35–39):

… I call such a figure the second. In it, I call that term the middle which is predicated of both, and call those of which it is predicated extremes; the major extreme is the one lying next to the middle, while the minor extreme is the one farther from the middle. (The middle is placed outside the extremes and is first in position).
Previously, Aristotle noted the coincidence that the middle in size was also the middle in position. He does not repeat this now because the relations of position are now altered. Unless Aristotle actively seeks to confuse his readers, we are meant to understand that the relations of ‘greater’, ‘middle’ and ‘smaller’ are now strictly the relations of size between the lines. The positions, on the other hand, are explicitly spelled out: the middle is first in position, followed by the greater and then finally followed by the smaller. Aristotle’s reference to ‘first’ and ‘farther’ strongly imply that he is referring to a figure organized from left to right. Had the figure been organized from the top to bottom, one would have expected terms such as ‘above’ and ‘below’; it is hard to believe that ‘first’ does not mean, here, ‘leftmost’, with the three terms arranged in the order of writing, from left to right.16
Finally, here is the description of the third figure (28a13–15):

[in the third figure,] by major extreme I mean the one farther from the middle and by minor the one closer. The middle is placed outside the extremes and is last in position.
Here we have the greater line, followed by the smaller line, with the middle line appearing last.
In the preliminary passage of chapters 1.4–6, as he introduces the figures, Aristotle makes several comments and produces several examples for each. When discussing the first figure, the terms are then always (in Latin transliteration) A, B, C; when discussing the second, they are M, N, X; when discussing the third, they are P, R, S. The alphabetical order from M to X, and from P to S, is assigned according to the spatial, rather than the logical arrangement: thus, in the second figure, M is the middle while, in the third figure, S is the middle. The consistent reference to the same labels in those chapters, 1.4–6, powerfully suggests that Aristotle refers, throughout, to a single set of material figures. Further, the explicit reference to ‘second’ and ‘third’ figures suggests that the figures are located together and marked by a word or, perhaps better, a numeral. Indeed, the main reason to think that the figures are organized together on a single diagram is the very choice to mark each with a separate set of labels. Later on in the treatise, when Aristotle discusses the second and third figures, he generally speaking reverts to A, B, C: there is no deep connection, in his mind, between the second figure and the letters M, N, X and between the third figure and the letters P, R, S. The choice to label each with different letters makes sense, however, if the three figures are considered as part of a single drawing, so that A, B and C are simply not available for the second and third figures. Aristotle’s ‘three figures’ are therefore, likely, to begin with, a single diagram with, indeed, three figures in it, which we are now in a position to reconstruct:
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If nothing else, then, this is helpful in the basic terms of Aristotelian hermeneutics. The passages quoted above have all been a thorn in scholarship’s side, the references to the relative locations taken in various metaphysical or syntactic senses,17 but they become straightforward as soon as the visual reference is understood. For this reason alone, I am profoundly perplexed by the fact that the argument of this paper—which is, as far as I can see, nothing more than the plain reading of Aristotle’s text—has not yet been made. Perhaps this is yet another example of past humanists’ focusing on text and ignoring image (even where the text plainly speaks about images.) More to the point, a number of past scholars, as noted, did suggest possible reconstructions of the figures and what is noticeable is that all of them looked for efficacious diagrams: that is, diagrams whose very structure would provide material help in judging the validity of logical claims. This trend in the scholarship in part reflects the influence of the transmission: it is hard to look for Aristotle’s figures without thinking about the significant accumulation of late ancient and medieval visual tools which are, indeed, generally speaking, more eloquent than those reconstructed above. But this is unfair: modern scholars know full well that this scholiastic tradition begins not earlier than Late Antiquity. The main reason past scholars looked for efficacious diagrams must have been that, otherwise, it was less clear to them why Aristotle would go through the trouble of drawing them in the first place.
Why indeed have the figures? To begin with, the figures reconstructed above are not entirely unhelpful. One thing which I think they are good for is for showing that it is indeed reasonable to have precisely three figures. A moment’s combinatoric reflection shows that there are six ways of arranging three lines of unequal length and that these six fall into three symmetrical pairs. (One gets all six arrangements by taking the three figures reconstructed above, and then flipping each along a vertical axis, so that the decreasing first figure A-B-C becomes an increasing mirror figure C-B-A, etc.) Clearly, the symmetrical arrangement is not considered to be a separate case, since one studies a particular relative order between the middle and the extremes; hence three figures are sufficient.
Further, the figures are clearly useful as, so to speak, mnemonic tools: once we become familiar with the meaning of length and sequence, we can use them to read off the overall structure of the figure. A syllogism has two premises and a conclusion. Each of these states a predication, which is spatially organized from left to right. The premises are the two predications involving the middle; the conclusion is the one involving the greater and the smaller alone. Thus, in the first figure, the two premises are that the greater is predicated of the middle and the middle is predicated of the smaller. In the second figure, the two premises are that the middle is predicated of both greater and smaller; in the third, that both the greater and the smaller are predicated of the middle. In all cases, the conclusion is that the greater is predicated of the smaller. (That the predication runs from the greater to the smaller and not vice versa is simply Aristotle’s way of choosing his three figures out of the six available).
In other words, the figures of the Prior Analytics, as I reconstruct them, are good for specifying the references of the discourse. Of course, diagrams can do much more in geometry, where the more topological relations seen in the diagram, such as inclusion or intersection, can be read off as holding between the geometrical objects themselves.18 A minimal version of this is preserved by some non-spatial diagrams, for instance as seen in Elements 5.1 above: the relation of the labels A, G, B as depicted along a single line in the diagram can be used to show that AG + GB = AB (this is topological inclusion—as algebraic summation). But this is no longer the case in the diagrams of Greek music, arithmetic or proportion theory, where each line has only a single letter standing next to it. Thus, in a diagram such as the first one of the Sectio Canonis, as seen above, the relations of the labels—that these are three lines, in decreasing length from left to right—are not deductively active at all. Once the setting-out verbalizes those relations, the ordering of the three lines in a particular order of size serves, perhaps, as a mnemonic, allowing the reader to pick off from the figure—without the need to go back to the textual statement of the setting out—which is a multiple of which (the greater, of the smaller). But no conclusion flows from the visual ordering of the figure itself, over and above the statements already contained in the setting out. Such, then, were, I argue, Aristotle’s logical figures. Which is evidently the case: after all, the loss of the figures did not prevent us from following any of Aristotle’s arguments. The figures are, indeed, in this sense, deductively inert.
But perhaps all of this misses a more fundamental point. This way of talking about ‘mnemonics’ seems to suggest that Greek mathematicians used line diagrams in their music theory, arithmetic and proportion theory, so that they would have a good mnemonic with which to know what ‘A’, ‘B’ and ‘C’ refer to. This, I think, is misleading and seems to ignore the crucial fact that the use of such diagrams was an absolute requirement. Absent a figure with lines, Greek mathematicians felt that they did not properly set up their references. In other words, the function of the lines is not mnemonic but semiotic. We are so habituated to the use of letters as semiotic tools that we see them as transparently representative of abstract objects such as magnitudes, numbers or notes, but it should be emphasized a shape such as ‘A’—two slanted lines and a smaller horizontal line running in between them—has nothing to do, as such, with, say, the sounds of music. ‘A’ is no more and no less natural as a symbol for a musical note than a vertical straight line. In Greek mathematics, one became habituated to lines as the natural semiotic tools. It is not as if the symbolism of a Greek mathematical text was constituted by letters of the alphabet—which were then displayed, for mnemonic purposes, as lines. Rather, the symbolism of a Greek mathematical text was constituted by points and lines, which were then labelled, for the sake of ease of reference, by letters of the alphabet. In a geometrical context, a line—however badly drawn—stands for a line. In an arithmetical context, it stands for a number, in a musical context for a note. And in the case of Aristotle’s logic, it could stand for a horse.19 It is inconvenient, in geometry as in logic, to keep pointing with one’s index finger—‘this line’, ‘that line’—and for this reason letters of the alphabet were attached as indices allowing easier reference (that could also be directly textualized).20
Two comparisons come to mind. First, in Chrysippus’ logic one no longer uses letters and instead, when the elements of logical demonstrations (which, in Chrysippus’ logic, are propositions) are picked up, they are referred to by ordinal numbers, ‘the first,’ ‘the second,’ etc.21 This is not a matter of innocuously transitioning between equivalent ordinal systems—letters of the alphabet, ordinal numbers—but rather serves to remind us that Chrysippus no longer uses diagrams. Chrysippus’ ordinal numbers do not pick up elements in a spatial array of lines but, instead, pick up elements in the temporal series of natural language. The ‘first’ proposition is the one first asserted, the ‘second’ follows it, etc. So, yet another example of Chrysippus’ turning away from the science of his time.22
Second, we may note claims such as those made by Kneale and Kneale (1962, 61), that one of Aristotle’s great innovations was that he ‘[used] letters as term-variables’. Considered sufficiently abstractly, it is perhaps valid to consider the line, next to which the letter A lies, as a term-variable. (The line may stand for ‘horse’ but it may also stand for ‘animal’). But the letter A is not a variable, because it is not even a symbol but is, instead, simply an index, picking up a reference to a particular line. Perhaps a failure to see that is among the difficulties preventing past scholars from visualizing Aristotle’s figures. It is noteworthy that the reconstructions offered by past scholars are all variations on the idea of enmeshing alphabetic letters within system of curves and arrows, the letters standing for terms and the curves and arrows for their relations. In fact, Greek diagrammatic practice does not use letters as symbolic representations for the terms under discussion: for this, one mostly has to wait for modern algebra. Aristotle himself would have been familiar with a different kind of diagrams, where the letters were tools for pointing at things; a textualized alternative to one’s index finger.
We find ourselves pondering Aristotle’s index finger and so we are drawn to consider the context and materiality of the figures. As noted, the stability of the lettering for each figure for the duration of chapters 1.4–6 suggests the existence of a single set of figures and the most obvious hypothesis is that Aristotle worked in a classroom where he could refer to a fixed drawing (a whiteboard, perhaps?23). I think that this is very likely but that we should also admit our ignorance and consider a wider range of possibilities. As noted above, the lettering of the second figure with M, N, X and of the third with P, R, S is kept only through 1.4–6 and, later on in the treatise, Aristotle usually refers to diagrams with A, B, C even when discussing the second and third figures. This reveals to us two things. First, it emphasizes, once again, that the main semiotic tool are the lines, not the labels. Aristotle evidently feels that the figure consists not in the choice of three labels but in the spatial arrangement of three lines of differing lengths. Second, the text of the Prior Analytics does not emerge just from the context of speaking about a single diagram, on a single whiteboard, on the walls of the Lyceum. Had Aristotle referred throughout the treatise to that single figure, drawn on the wall, he would always use the labels introduced in chapters 1.4–6. But if he does not refer to a single set of figures throughout, and instead relies on different visual aids in different contexts, we are left very much in the dark concerning the details of his practice in any particular context. The best statement in my view is that the environment of Aristotle’s school was awash in a variety of textual artifacts: papyri, whiteboards, wax tablets (among other media).24 Learning and research were done mostly in oral conversation, making reference to such textual artifacts as the need arose. Sometimes, one looked at the documents of the master and of his close associates. Probably multiple copies were made of these, and it is likely that many of those copies accompanied the text itself by the drawings it implied. Sometimes, one would refer to drawings attached to the wall. In particular, it seems likely that at some point Aristotle discussed in oral teaching the basic principles of the syllogistic, and that this discussion was pursued in sight of a whiteboard, containing the three figures. This discussion informed chapters 1.4–6 of the Prior Analytics. Its diagram was then lost; I believe it is now reconstructed.
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NOTES
1

The best treatment of Aristotle’s quasi-mathematical passages is a concise section in an article by Vitrac (2002, 248–55).
2

Not that this requires any special evidence, but the ubiquity of mathematical examples in the Analytics itself is relevant: see Ierodiakonou 2002.
3

This claim is elaborated in the second chapter of Netz 2022.
4

Netz 2013, 232–41.
5

The clearest case is the ps. Aristotelian Mechanics, whose extant diagrams were introduced by Byzantine hands—so, evidently, were lost at some point prior to the Middle Ages (van Leeuwen 2016, 97–101).
6

In his biological works Aristotle occasionally asks the reader to consult a separate text, titled the anatomai, ‘dissections’, in order to visualize the contents of the biological text. It is widely believed that the anatomai contained visual images of anatomical structures. See von Staden 2013, 115–17.
7

Image (with my translation) from Wallies 1899, 39. I am not sure that the figures in Ammonius’ text are from his own hand, rather than coming from the hand of a later Medieval accretion, but at any rate no earlier figures of this kind are attested. (Of course, the fact that the earliest author in whose textual tradition those diagrams are found is Ammonius is not dispositive though it does suggest, as the text asserts, a fairly late origin). This particular form of curved lines connecting terms is ubiquitous in Medieval scholia of music and proportion theory and belongs to an entire domain worthy of study on its own right (Lee 2020 is a beginning in this direction of the study of what he calls visual scholia).
8

Savage-Smith 2002, 122–5.
9

There is reasonable room for debate concerning the fidelity of the medieval transmission of diagrams. See Carman 2018, Netz 2020b. However, the question considered here is that of the basic tools of representation and it is a prudent assumption that our manuscript evidence is, at this broad level, correct. (Notice, however, that I do not think we should put much stock in, for instance, the orientation of the lines in the manuscripts—this was certainly transformed, at least occasionally, according to the spatial needs of the scribes; see n. 11 below).
10

von Jan 1895, 150 clearly explains the manuscript evidence concerning the diagrams, which seem to be reconstructed by the editor while keeping closely to the manuscript evidence. Translation from Barker 1989 II, 194.
11

Translation (and first figure) from Heath 1926 II, 138. In fact, in the main Greek manuscripts the lines are vertical, and I provide an example in the second figure, from Vat. Gr. 190 75v. This manuscript stands out, however, in that the diagrams of Book 5 are inset within the main text; usually, they are arranged on the margins and it is possible that this could have influenced the orientation. ‘The most common and apparent external factor guiding diagram shape is its allocated space’ (Lee 2020, 226).
12

Once again, this is discussed in Netz 2022, second chapter. I emphasize the musical context in this paper. It was the main context suggested by the seminal paper in the field of the mathematics of the Prior Analytics, namely Einarson 1936. Netz 2022, chapter 2, serves to explain why this is in fact likely to have been the most salient context. Briefly: no figure loomed as large over the mathematics of the early fourth century as that of Archytas, and it is reasonable to assume that, of his achievements, the foundation of musical theory is the one that mattered most to contemporary philosophers.
13

Netz 2004, 44. As noted there, the lay-out of the figure is the same across the manuscripts.
14

Translations of Prior Analytics are all from Smith 1989.
15

It is perhaps worth mentioning that what is sometimes called, in modern logic, ‘the minor premise’ is, for Aristotle ‘the premise at the smaller extreme’ (e.g. 51a23: my translation).
16

That the natural order of diagrams is, in Greek, according to the order of writing from left to right, is expected and overwhelmingly observed in the evidence. A related consequence is that diagrams are often inverted when translated into Arabic: Lee 2020, 262.
17

For instance Striker 2009, 100, on the second figure: ‘… The labels “major”, “middle”, and “minor” are no longer linked to the extensions of the terms. This has the somewhat paradoxical result that the middle is said to be “placed outside the extremes”.’
18

This argument has been pursued by Manders 1995 (2008) and Netz 1999, and is now widely accepted in the scholarship.
19

This is flippant: the reference of the line is not a horse but more general. Indeed, there is no good account in the literature for the use of the neuter article for the terms of the syllogism such as to A, ‘the A’ (the noun ‘term’, itself, is masculine). It appears that the neuter article takes on, in this specific context, the indefinite meaning of ‘the A <whatever it might be>’. This is a question of Greek grammar rather than of Aristotelian exegesis, but it does appear that at some level of abstraction, neuter could be understood as the default case. (For a suggestive analogous discussion, see Corbett and Fraser 2000, 6–72: why does Russian use neuter in bylo kholodno, ‘it [neuter] was cold’? Who is the neuter subject being cold here? Apparently, ‘things in general’?) Whatever our interpretation of this grammatical question, it is clear the article refers not to the line (which would have required a feminine article) but to the thing represented by the line.
20

The indexical character of the letters used in Greek mathematical texts is the subject of Netz 1999, 42–51. Of course, in the case of mathematics the texts we possess were intended for an audience of readers, distant in place and time, so that indices alone could never suffice; this is less clear for the texts we now know as the works of Aristotle.
21

See e.g. Bobzien 2005, 129–31.
22

Netz 2020, Section 4.4.
23

There was a practice of publishing decrees etc. not only as stone inscriptions but also on wooden boards (made white by being covered with gypsum; presumably the writing itself was charcoal-based). Such temporary inscriptions were referred to as leukōmata (not to be confused with the homonymous eye disease). Since the technology was available, it is assumed—though no direct evidence exists—that it was also used for teaching. A suggestive proposal is that writing on wood would be especially amenable for more elaborate drawings (Thomas 2019, 262–3; for a brief statement on leukōmata in general, see Thomas (no relation) 1992, 83).
24

This is analogous to the way in which the Peripatetic writing project gradually became a manifold of many genres, the argument of Asper 2015.