Surveying the Types of Tables in Ancient Greek Texts

1.1 Unruled Columnar Lists: Accounts

We do not find ruled tables for administrative purposes in the epigraphical documents or papyri. Greek temple accounts are either prosaic lists – that is, new items do not start new lines – such as the Delian inventories (IG XI 2), or unruled columnar lists, in which the numbers are situated at the right of the column of text, such as in this unprovenanced papyrus (P. Lund IV 4):

λυχνίαι χαλκ(αῖ)	α
ποτήρια χαλκ(αῖ)	ε
. . .
bronze lampstands	1
bronze jars	5
. . .

This is the most typical format of the Greek account.^[14] There is no grid and no column headers. The content is understood to be self-explanatory. In epigraphical records such as the lists of Athenian revenues from taxes on allied polis, we may find the numbers on the left column.^[15] We may have up to two numerical columns: for example, in the inscriptions from the Athenian agora detailing properties confiscated for mutilating herms, one column displays the sales tax and another the total amounts.^[16]

1.2 Unruled Columnar Lists: Mathematical Texts

Columnar lists were used in texts dealing with numbers. It is thus not surprising that they appear several times in the works of Claudius Ptolemy. To begin with, we can consider the list of phases of the fixed stars that forms the central part of Phases a format in between the prosaic and the columnar list. Here we have four kinds of information: the day number within the month, the latitude for which the concerned phase is meant (expressed as the maximum hours of daylight), the information on the astronomical phase, and meteorological information for the day. The situation is complicated by the fact that for a single day we might have phases for more than one latitude, as in the entry for the third day of month Thoth in the passage I copy below (Heiberg 14). In such circumstances, it would make little sense to strictly divide the text in different columns apart from the day number.

ὑποτάξομεν ἤδη τὴν ἀναγραφὴν ἔχουσαν οὕτως·

ΘΩΘ

1. ὡρῶν ιδ <· ὁ ἐπὶ τῆς οὐρᾶς τοῦ Λέοντος ἐπιτέλλει. Ἱππάρχῳ ἐτησίαι παύονται. Εὐδόξῳ ὑετία, βρονταί, ἐτησίαι παύονται.

2. ὡρῶν ιδ· ὁ ἐπὶ τῆς οὐρᾶς τοῦ Λέοντος ἐπιτέλλει, καὶ Στάχυς κρύπτεται. Ἱππάρχῳ ἐπισημαίνει.

3. ὡρῶν ιγ <· ὁ ἐπὶ τῆς οὐρᾶς τοῦ Λέοντος ἐπιτέλλει. ὡρῶν ιε· ὁ καλούμενος Αἲξ ἑσπέριος ἀνατέλλει. Αἰγυπτίοις ἐτησίαι παύονται. Εὐδόξῳ ἄνεμοι μεταπίπτοντες. Καίσαρι ἄνεμος, ὑετός, βρονταί. Ἱππάρχῳ ἀπηλιώτης πνεῖ.

We will now append here the register in the following way:

Thoth

1. 14 ½ hours: the star on the tail of Leo rises. According to Hipparchus, Etesian winds stop. For Eudoxus, rain, showers, Etesian winds stop.

2. 14 hours: the star on the tail of Leo rises, and Spica disappears. For Hipparchus, a change of weather.

3. 13 ½ hours: the star on the tail of Leo rises. 15 hours: the so-called Western goat rises. For the Egyptians, Etesian winds stop. For Eudoxus, the winds change. For Caesar, wind, rain, showers. For Hipparchus, Eastern wind.

Even if the structure is only vaguely tabular, Ptolemy seems to have sought a certain standardization in that, unlike in other calendars, for every day in every month there is at least one latitude containing predictions, thus not leaving any day of the year empty. We do not have ancient copies of this work, but the oldest manuscripts (Vat. gr. 1594, 9th c. and BNF 2390, 13th c.) consistently show different entries for different days and the day numbers separated from the text quasi forming a different column. In fact, in Vat. gr. 1594 the separation is reinforced by placing the day numbers alternatingly on the left and on the right throughout the successive columns of text.

Ptolemy’s calendar can be conceived as a deluxe version of stone parapegmata, the public calendars of star-phases and meteorological predictions in which holes were drilled for each entry, for the purpose of marking the current day by moving a peg.^[17] See, for example, the Miletus parapegma (I reproduce the beginning of the part corresponding to Aquarius):

• ἐν ὑδροχόω[ι ὁ] ἥλιος

• [.....] ἑῶιος ἄρχεται δύνων καὶ λύρα δύνει

• •

• ὄρνις ἀ[κ]ρόνυχτος ἄρχεται δύ[νων]

• • • • • • • • •

• ἀνδρόμεδα ἄρχεται ἑῶια ἐ[πι] τέλλειν

• The sun is in Aquarius

• [.....] begins to set in the evening and the Lyre sets

• •

• Cygnus begins to set in the evening

• • • • • • • • •

• Andromeda begins to rise in the morning

This is the standard form of the epigraphical parapegma. It has in common with Ptolemy the fact that the holes for the days (here unnumbered) are situated at the left, but it has even less of a tabular structure, since quite frequently several continuous days have no information and the holes are just placed horizontally one after the other, breaking the quasi-columnar structure. Parapegmata that are not designed for public exhibition, such as the one on papyrus P. Hibeh 27, do not even start at different lines for different days. On the other hand, Geminus’ parapegma, which has been preserved in the manuscript tradition, has different rows for each entry, but the text is completely prosaic (I reproduce the beginning of Leo):

Τὸν δὲ Λέοντα διαπορεύεται ὁ ἥλιος ἐν ἡμέραις <λα>.

Ἐν μὲν οὖν τῇ α^ῃ ἡμέρᾳ Εὐκτήμονι Κύων μὲν ἐκφανής, πνῖγος δὲ ἐπιγίνεται· ἐπισημαίνει.

Ἐν δὲ τῇ ε^ῃ Εὐδόξῳ Ἀετὸς ἑῷος δύνει.

The sun travels Leo in 31 days.

In the first day according to Euctemon Sirius is shining, it is stifling hot; change of weather.

In the fifth day according to Eudoxus Aquila sets in the morning.

In the Geography, Ptolemy undertakes the huge enterprise to provide coordinates for every significant city or natural accident in the known world. The large central portion of the work, covering books II–VII, consists in the list of such places with their coordinates, interrupted by titles. The contents are grouped first by region, then by type of place, for example the Greek mountains here (III 14.11):

Ὄρη δὲ εἰσὶν ἐν τῷ εἰρημένῳ τμήματι τό τε Καλλίδρομον ὄρος, οὗ τὸ μέσον
ἐπέχει μοίρας	μθ	λη δʹ
καὶ ὁ Κόραξ ὄρος	μθ γʹ	λη
καὶ ὁ Παρνασσὸς ὄρος	ν γʹ	λη
καὶ ὁ Ἑλικὼν ὄρος	να	λζ <ʹ δʹ

There are mountains in the said region: mount Kallidromos, whose center has the
coordinates	49	38 ¼
and mount Korax	49 1/3	38
and mount Parnassus	50 1/3	38
and mount Helicon	51	37 ¾

The features of this list are different. In contrast with the case of the calendar in the Phases, the structure here is completely regular: name of the place and coordinates (longitude and latitude). Furthermore, there is a special feature which allows for ulterior improvements, namely the ample space left in between, which is not only evident in the manuscripts, but also signaled in Ptolemy’s presentation (II 1.3):

Διὸ καὶ τὰς παραθέσεις τῶν μοιρῶν ἐφ′ ἑκάστου τοῖς ἐκτὸς μέρεσι τῶν σελιδίων παρεθήκαμεν κανονίων τρόπον, προτάσσοντες μέντοι τὰς τοῦ μήκους τῶν τοῦ πλάτους, ὅπως, ἐάν τινες ἐμπίπτωσι διορθώσεις ἀπὸ τῆς πλείονος ἱστορίας, ἐνῇ ἐν τοῖς ἐχομένοις διαλείμμασι τῶν σελιδίων ποιεῖσθαι τὰς παραθέσεις αὐτῶν.

So we have written beside in the extreme part of the columns, in the manner of tables, the coordinates of every place, putting the longitude before the latitude, so that, if anyone finds corrections from a deeper investigation, it would be possible to write their coordinates.

Here “columns” (σελίδια) refers to the papyrus columns. Ptolemy’s observation ensures that the manuscripts reflect the original shape of these columnar lists. It is worth noting that Ptolemy does not say that he puts the information in tables (κανόνια, κανόνες) as elsewhere when ruled tables are used, but “in the manner of tables” (κανονίων τρόπον).^[18] The disclaimer likely refers to the absence of rulings (as we have it, again, in the manuscripts). It would make little sense to draw rulings for such simple presentation which, in addition, takes the most part of this large work. Contrary to Riggsby’s requirement of symmetry, in this case the table has an extremely predominant vertical format.

Greek arithmetical tables preserved on papyrus show the same format: several columns of text divided in subcolumns, without rulings. Specifically, the two numbers that are to be added or multiplied are written very close together at the left, and the result is given at the right.^[19] See for example P. Mich. inv. 5663:

It is clear that the papyrus columns are not conceived as the columns of a table, since the numbers do not correspond with each other across columns: for example, the first preserved column begins by 5 × 6, while the second column at the same level has 8 × 4. In fact, my rendering is not completely faithful, since the lines of column I and those of column II are not aligned. This disposition clearly reflects the act of speech, and likely derives therefrom, that is, from the recitation of the times table from top to bottom. For comparison, the Mesopotamian and the Egyptian arithmetical tables are structured in the same unruled format by entries. It is difficult to assess the extent to which these textual traditions might have influenced one another: the influence of Egyptian arithmetic in Greek arithmetic is well-known in concrete aspects such as the use of unit-fractions, but the columnar-list formatting seems too general to be an indicator of a possible transmission.^[20]

Even if this simple kind of table found many uses, we have the impression that it was little employed in scientific treatises. Thus, we have found some unruled columnar lists in Ptolemy, but they are sometimes explicitly avoided in some places of his corpus where it would probably seem natural to use them. For example, in his astrological treatise Ptolemy refuses to provide a list for what he calls the Chaldean terms, on the grounds that his explanation of their working is sufficient (Tetrabiblos I 21.12 Hübner):

Ὁ δὲ Χαλδαϊκὸς τρόπος ἁπλῆν μέν τινα ἔχει, οὐχ οὕτως δὲ αὐτάρκη […], ὥστε μέντοι καὶ χωρὶς ἀναγραφῆς δύνασθαι ῥᾳδίως τινὰ διαβάλλειν αὐτούς.

The Chaldaean method involves a simple sequence, although not self-evident […], so that, nevertheless, one could easily understand them even without any recording.

There were several different systems for this doctrine. Τhe astrological terms were subdivisions within each of the zodiacal signs ruled by the planets; within each sign intervals corresponding to each of the planets were defined comprising each of them a different number of degrees. The unruled columnar list for what Ptolemy calls the “Chaldean terms” would have looked like this:

Aries	Jup 8	Ven 7	Sat 6	Mer 5	Mar 4
Taurus	Ven 8	Sat 7	Mer 6	Mar 5	Jup 4
Gemini	Sat 8	Mer 7	Mar 6	Jup 5	Ven 4
Cancer	Mar 8	Jup 7	Ven 6	Sat 5	Mer 4
. . . (from here on for the next signs the structure repeats itself)

Here the structure is repetitive enough for us to reconstruct the list with Ptolemy’s indications, but for the more typical system of terms, which Ptolemy calls the “Egyptian,” he has no other option than to register them in full, since they did not seem to follow a simple set of rules (a circumstance that Ptolemy finds fault with): here the manuscripts contain a ruled table comprising the signs as columns, and showing partial sums of the terms, which were useful to astrologers for discovering to what term-ruler belonged a concrete degree. However, more probably Ptolemy displayed the terms in unruled format without the partial sums, as in the table reconstructed above for the “Chaldean terms.” Otherwise, considering his practice elsewhere, we would expect him to mention the partial sums and the ruled format. This is supported by the fact that in the oldest Arabic translations of the Tetrabiblos this table appears as an unruled columnar list, without the partial sums.^[21]

1.3 Unruled Columnar Lists without Numbers: Philosophy and Grammar

Aristotle used the columnar list for non-numerical contents. He has a 3-column list in the Eudemian Ethics (1221a), containing affects (πάθη) of the soul ordered according to the categories of excess, deficiency, and adequacy:

ὀργιλότης	ἀναλγησία	πραότης
θρασύτης	δειλία	ἀνδρεία
ἀναισχυντία	κατάπληξις	αἰδώς
. . .

irascibility	lack of feeling	gentleness
foolhardiness	cowardice	bravery
shamelessness	shyness	modesty
. . .

He calls this layout ὑπογραφή, the same word that he applies to an anatomical lettered diagram which appears in History of animals (510a). In De interpretatione 22a he uses the same term^[22] to refer to a similar columnar list for his theory of opposites in four columns, which is shown in its original form only in some manuscripts:^[23]

δυνατὸν εἶναι	ἐνδεχόμενον εἶναι	οὐκ ἀδύνατον εἶναι	οὐκ ἀναγκαῖον εἶναι
δυνατὸν μὴ εἶναι	ἐνδεχόμενον μὴ εἶναι	οὐκ αδύνατον μὴ εἶναι	οὐκ ἀναγκαῖον μὴ εἶναι
οὐ δυνατὸν εἶναι	οὐ ἐνδεχόμενον εἶναι	ἀδύνατον εἶναι	ἀναγκαῖον εἶναι
οὐ δυνατὸν μὴ εἶναι	οὐ ἐνδεχόμενον μὴ εἶναι	ἀδύνατον μὴ εἶναι	ἀναγκαῖον μὴ εἶναι

possible to be	admissible to be	not impossible to be	not necessary to be
possible not to be	admissible not to be	not impossible not to be	not necessary not to be
not possible to be	not admissible to be	impossible to be	necessary to be
not possible not to be	not admissible not to be	impossible not to be	necessary not to be

Even if he does not use the same word, Aristotle may have been influenced by the layout of what he calls the συστοιχίαι of the Pythagoreans, literally “elements belonging together” that were probably structured in two columns. For example, in Metaph. 986a we read:

ἕτεροι δὲ τῶν αὐτῶν τούτων τὰς ἀρχὰς δέκα λέγουσιν εἶναι τὰς κατὰ συστοιχίαν λεγομένας, πέρας ἄπειρον, περιττὸν ἄρτιον, ἓν πλῆθος.

Others from the same group say that the principles are ten, those that are called “according to the systoichiai”: limit-unlimited, odd-even, one-multiple.

The deduction that these may have been distributed into two columns comes from assertions like Metaph. 1093b (τῆς συστοιχίας ἐστὶ τῆς τοῦ καλοῦ τὸ περιττόν, τὸ εὐθύ, “the even, the straight, belong to the συστοιχία of the good”), in which Aristotle puts together as one συστοιχία the first elements of the pairs. Furthermore, the fact that in many of the manuscripts there is no conjunction between the elements of the pairs suggests that a space was left in between in the Pythagorean writings.

The unruled columnar lists of the Eudemian ethics and of De interpretatione also often appear in prosaic style in the manuscripts.^[24] Such textual evolution reveals how foreign was this format even to ancient Greek philosophical treatises. As for why we find it in Aristotle, it is worth recalling that his surviving treatises were of internal circulation within the Lyceum: thus, it would probably be more admissible for them than for other, more literary texts, to transgress the rigid standards of papyrus-writing etiquette.

This format must have been common, however, in less formal texts, such as grammatical lists. See, for example, two second-century-AD papyri showing two-subcolumn formatting, the first for nominal form and case name, and the second one for Greek verbal form and Latin translation (in Greek letters):

P. Oxy. 86.5538:
τὸ βέλος	ὀρθ[ή
τοῦ βέλος	γεν[ική
τῷ βέλει	δοτ[ική
. . .

P. Oxy. 82.5302:
ζωγράφουσιν	πινγουντ
ζωγράψ[ουσιν]	πινγηντ
ζεύγνυσι	ιουνγιτ
ζεύγν[υμι]	ιουνγω
ζεύξω	ιουνγαμ
. . .

Again as in the case of arithmetical tables, these tables are not ruled, and the pairs of subcolumns do not correspond to each other; rather, the whole papyrus is conceived as a long list of two columns.^[25]

1.4 Ruled Columnar Lists: Astronomical Tables

This type is characterized by having multiple columns (sometimes including subcolumns) typically extending beyond the regular breadth of the papyrus column. Their complex structure thus naturally requires rulings, but similarly to the unruled type, the vertical dimension is predominant and often requires these tables to split into several partitions due to their great number of rows.

Such tables are for the most part astronomical. Out of the roughly 5,500 Oxyrhynchus papyri which have hitherto been published, approximately 100 are astronomical tables which have been edited, translated and commented in a dedicated monograph (they are more than half the total number of published astronomical papyri).^[26] Surviving astronomical tables by known authors are mostly limited to Ptolemy. The production of tables is one of the main goals throughout Ptolemy’s Almagest, and his Handy Tables consist in a more convenient presentation of the tables of the Almagest, without the theoretical development and in more practical formats for astrologers. These tables will be crucial to astrologers from the third century onward for the calculation of the planetary positions.^[27] The end-products of Ptolemy’s Analemma and the Harmonics are, once again, tables. The tables of the Harmonics are of course not astronomical, but it is clear that Ptolemy adapts the format of the astronomical table (along with the sexagesimal-style fractions) to music theory, after having advocated for similar principles for both the study of the motions of the heavenly bodies and harmonics (I 2).

Astronomical tables existed at least from the late Hellenistic age, a time of substantial transmission of astronomical and astrological lore from Mesopotamia. The details of such transmission are for the most part obscure to us due to the lack of surviving texts. We know that Hipparchus in the mid-second century BC used Babylonian observations of eclipses for his theory of the motion of the Sun and the Moon.^[28] Hipparchus’ age also witnessed the birth of Greek astrology as we know it from later times.^[29] As for tables, we have several indications suggesting their use by astronomers and astrologers at that time: for example, it is accepted that Hipparchus had constructed a table of chords (an equivalent of a sine table) for similar purposes as Ptolemy’s (Almagest I 11);^[30] and the treatise of Hypsicles on the rising times seems establish the foundations for the construction of the so-called anaphorical table, a table of rising times for the calculation of the ascendant point. An Oxyrhynchus papyrus (POxy 4276, 2nd/3rd c. AD) in fact mentions a table of rising times attributed to Hipparchus (ἀναφορικόν), and Vettius Valens (IX 12) refers to tables for the motion of the Sun and the Moon which he attributes to Hipparchus, Sudines, Kydenas, and Apollonius. It is often supposed that Sudines is the expert on exstipicy (entrails divination) mentioned by Polyaenus as accompanying king Attalus I of Pergamon and that Apollonius is the geometer from Perge (both 3rd c. BC), but historians of astronomy disagree with these identifications, and place these astronomers close to or later than Hipparchus’ age.^[31] This period coincides with a greater Hellenization in Mesopotamia, which no doubt fostered the transmission of Babylonian astronomical practices and ideas.^[32] Of course, some form of Greek astronomy existed before that time, but most theories and methods necessarily involving precise calculations with tables seem to have started by then.^[33]

A specimen from around 100 BC, the Keskintos astronomical inscription, may be indicative of the evolution from simple columnar list to astronomical table. The table is unruled (but this may be due to its epigraphic nature) and it provides period relations of different kinds for the planets, in a manner which is not particularly space-effective and which is, as is often the case in unruled columnar lists, reminiscent of verbal accounts (I reproduce Jones’ translation of the first well-preserved entries):^[34]

Mars	in longitude	zodiacals	15,492	Mars	in longitude	zodiacals	154,920
Mars	in latitude	tropicals	15,436	Mars	in latitude	tropicals	154,360
Mars	in depth	revolutions	4,096x	Mars	in depth	revolutions	401,650
Mars	in rel. pos.	passages	13,648	Mars	in rel. pos.	passages	136,480
Jupiter	in longitude	zodiacals	2,450	Jupiter	in longitude	zodiacals	24,500

In fact, the whole structure consists of two tables side by side showing integer specific periods for each of the planets (four different kinds of periods for each) in 29,140 and 291,400 solar years respectively. Each table, comprising in the present state (if it were complete) 4 × 5 = 20 lines (for the five planets) and four columns, could be reduced to just 7 rows (one for each planet) and 5 columns (headed as: planets, zodiacal periods in longitude, tropical periods in latitude, revolutions in depth; passages in relative position).

A natural question is whether Greek astronomical tables were a Babylonian import. I will argue that this is a fair possibility. Babylonian astronomy had many coexisting branches, comprising texts both in the form of prosaic account and in fully tabular layout.^[35] These are extremely formalized types of texts, the products of a well-established institutional setting.^[36] In the tabular format, a standard set of parameters is often given in columns against different times represented by the rows. For example, planetary ephemerides list in two columns synodic phenomena of the planets (stations, conjunctions, and so on) giving the dates in the first column and the zodiacal position in the second one.^[37] The Greek counterparts, representing a significant subset of Greek astronomical tables, are the so-called planetary epoch tables. These give longitudes and dates of planetary synodic phenomena in practically the same format as the Babylonian tablets, in addition to using the same mathematical procedures (before Ptolemaic tables) – zone and zig-zag functions.^[38] In other cases Greek tables differ widely from their Babylonian predecessors.^[39] In the end, probably the best way to put it is that both arrived together: astronomical theory frequently in the form of astronomical tables.^[40]

2.1 An Arithmetical Table in a Funerary Stele

A table of multiples 10 × 10 engraved on the funerary stele of a teacher (Gen. MAH inv. 27937) may well be the first attested table of this kind. It forms the atrezzo of a learning room along with a cithara hanging on the wall, the teacher, and a small-scaled boy.^[56] The inscription under the relief identifies the deceased as Πτολεμαῖος ὁ γεωμέτρης. The piece is unprovenanced, and it has tentatively been dated to the 3rd century BC, but a more realistic estimate would be between 200 and 50 BC.^[57] In this case we have a full-fledged table with rulings:

Since multiplication tables adopt a different form (see above), I believe that we should consider this specimen a table of multiples. We will see below that Nicomachus presents this same table as a table of multiples. Of course, as a table of multiples it would also serve the purposes of multiplication. It is easy to perceive that the medium here underscores and reflects the square grid. Indeed, the stele is designed as a sort of mise-en-abîme: the whole scene is set in a deeply incised square, and the square table is balanced by another square formed by the teacher’s seat at the other extreme. If the multiplication tables that we commonly find on papyrus reflect the oral, repetitive learning of the operations – in the same way as school kids learn today, the multiplicand is repeated in each row – the inscriptional milieu privileges the synoptic presentation and a symmetric and compact presentation.

2.2 Nicomachus of Gerasa and Theon of Smyrna (Late First Century AD)

These authors do not use the words πλινθίς/πλινθίον, but rather the more generic term διάγραμμα.^[58] Nicomachus sets up four tables in his Introduction to Arithmetic, while Theon just one in his only surviving work, On useful things for understanding the mathematics of Plato. I will briefly describe them, together with Nicomachus’ and Theon’s remarks.

1. Nicomachus I 19. A 10 × 10 table presented as a sequence of series of multiples, exactly like the table on the Geneva stele (Section 2.1 above). Nicomachus signals several properties of the table: in the first place, the second row minus the first gives 1 2 3 …, the third minus the first gives 2 4 6 …; etc. Secondly, the relation between the elements of the third and the second column give the so-called sesquialter ratios, which are crucial in music theory (3:2, 6:4, 9:6 …); and those of the fourth to those of the third give the sesquitertian ratios (4:3, 8:6, 12:9 …). Then, the corner cells of the table are all units, and in particular 10 times 10 equals 100; finally, the diagonal starting from 1 is formed by squares.

2. Nicomachus I 10. A 7 × 7 table of odd-times even numbers obtained through multiplication of the numbers of the series of odd numbers from 3 (3, 5, 7 …) by the numbers of the even-times even series from 4 (4, 8, 16, 32 …). In relation to this table, Nicomachus presents as idiosyncratic of odd-times even numbers (that is, numbers obtained through a product of an even and an odd number) characteristics which are in fact properties of arithmetic and geometric progressions. Indeed, the rows are geometric progressions, and the columns are arithmetic; the observations are related to middle elements and their relationship with the extremes.

3. Nicomachus II 3. Two 7 × 7 triangular tables (upper right half with the main diagonal) containing sequences of doubles and triples in the rows, respectively. The first numbers of the first rows in the first table are the following (the subsequent rows are built so that the diagonal is formed by the triples):

   

   Nicomachus mentions the fact that in this table the ratios formed by subsequent numbers in vertical are always sesquialter (3:2 and equivalent). A similar property with the ratio 4:3 is found in the second table, which presents a diagonal formed by quadruples.

4. Nicomachus II 12. A table containing the number of dots (or pebbles) that form polygonal numbers, ordered in rows by type of polygon (triangles, squares, pentagons, etc.). These figures are extensions of the classical tetraktys image of an equilateral triangle formed by four rows of 1, 2, 3, and 4 dots respectively. Then the first row (triangles) is 1, 3, 6, 10, etc. (that is, the number of dots in each triangle, in increasing order). Nicomachus notes a couple of curious relations across different rows of the table.

5. Theon 101 Hiller. Theon constructs a 3 × 3 square containing the first nine natural numbers in order, by columns:

   

He notes that in all straight lines through the center (the second row, the second column, and the two diagonals), 5 is the arithmetic mean of the extremes.^[59]

In all five tables, most of them square, we see that both authors remark on relatively surprising characteristics. Nicomachus even employs the term θαυμαστῶς (“amazing”). It is clear that Nicomachus and Theon attempt to seduce their readers with curious relationships between numbers found in many possible directions, mainly horizontally and vertically, but also diagonally. It is significant, for example, that Nicomachus speaks of στίχοι (usually “rows”) for both rows and columns.^[60]

2.3 Dispute-resolving πλινθίς in a Pseudo-Pythagorean Letter and Related Tables

Several manuscripts^[61] contain this table introduced by a letter allegedly penned by Pythagoras, addressed to his son Telauges and providing instructions for using it in the resolution of disputes between two individuals, beginning in this way:

Πυθαγόρας Τηλαύγῃ χαίρειν   

Πολλὰ παθὼν καὶ πολλὰ πειράσας ἐπέσταλκά σοι τόδε βιβλίον ἔχον ἐν ἑαυτῷ πλινθίδα πάνυ χαριεστάτην· ἐντυχὼν γὰρ εἰς αὐτὴν διὰ τῶν ὑποκειμένων γραμμάτων εἴσεται τά τε ἐνεστῶτα καὶ τὰ προγεγονότα καὶ τὰ αὖθις ἐσόμενα. ὑπέταξα οὖν πλινθίδα ἐννεάδος δοκιμαζομένην τρόπῳ τοιῷδε . . .

Pythagoras to Telauges,

After much toil and effort, I have sent you this book containing a most graceful table. For when you get hold of it you will have access to the present, past and future through its lines. I have placed below the table of the ennead, to be tested in the following way . . .

W, “wins”; acc’r, “the accuser wins”; acc’d, “the accused wins”

In the instructions, the text tells the reader to reduce the name of the contestants to numbers from 1 to 9, by adding up the numbers corresponding to the letters of their names in a modified alphanumerical system (α, ι, ρ = 1; β, κ, σ = 2, etc.) and subtracting so many sets of nine (enneads) as possible. Then we look for the cell in the table featuring the numbers of the two contestants and whoever has the Ν above wins (νικᾷ) (W in our version). In case of equality, the text alternately says the accuser or the accused wins.

As we can see, the table is a square displaying all the possibilities (9 × 9 = 81). It is also evident that the N’s are perfectly balanced, as if in a chess board, except in the diagonal which represents a sort of parenthesis in which the result is obtained in the mentioned fashion (but even the diagonal itself is balanced).

The date of this document is unknown: we only have the terminus ante quem of Hippolytus (around 200 AD), who explains the numerological procedure in his Refutatio (IV 14); it has been tentatively dated as late Hellenistic because Pythagorean pseudepigraphs are generally thought to have been composed around that time.^[62]

2.4 Tables for the Length of Life by Critodemus (in Valens VIII)

In book VIII of his Anthologies, the astrologer Vettius Valens (second century AD) deals with and reproduces two tables that come from the work of Critodemus (probably end of second/early first century BC) but which were probably appended to Critodemus’ work at a later stage, probably in the early first century AD.^[68] Both tables are enigmatically called κανόνιον καὶ πλινθίον. Indeed, on the one hand they shows the typical structure of an astronomical table, with columns representing the zodiacal signs and the rows corresponding to the 30 degrees of each zodiacal sign. This is, for example, the same structure as Ptolemy’s tables of rising times in the Handy Tables. On the other hand, Critodemus’ tables are structured in vertical sets of numbers (not explicitly but by construction) arranged in a symmetric square (the second table is a variation of the first). They are used to obtain the length of life entering the table in the degree corresponding to the ascendant degree of the concerned person, then multiplying a certain quantity (dependent on the ascendant degree) by the number corresponding to the relevant sign-column.

The numbers in the sign-column can be grouped vertically in sets of 6 numbers, so that in one column there are only 5 different sets (there are 30 degree-rows) and the sets are never repeated in one column or in one row, taking the first 5 signs (then the structure repeats itself). This type of balanced arrangement is called a Latin square (Euler’s denomination). If we designate the five consecutive sets of the first column by A B C D E, then we have:^[69]

2.5 On the Terms πλινθίς and πλινθίον

Now, what are symmetric tables (πλινθία and πλινθίδες)? They are ruled tables mostly displaying simple, integer numbers that maintain some kind of symmetry or balance. They are frequently square, and they rather belong to the genres of arithmology and recreative arithmetic than to science. Some instances recall the famous magic squares that were transmitted from China to the Islamic world via India, but there is no evidence that anything exactly like them (where numbers in all rows, columns, and main diagonals add up the same value) was developed or adopted in ancient Greek culture.^[70]

Where do the terms for them come from? Unlike κανόνες and κανόνια, πλινθία and πλινθίδες could simply refer to grids without contents. Our first datable evidence of the term is the Jewish exegete Philo (early 1st c. AD), who refers to a diagram which he calls πλινθίον in the following way (De opificio mundi 107):

Ἔστι δὲ οὐ τελεσφόρος μόνον, ἀλλὰ καὶ ὡς ἔπος εἰπεῖν ἁρμονικωτάτη καὶ τρόπον τινὰ πηγὴ τοῦ καλλίστου διαγράμματος, ὃ πάσας μὲν τὰς ἁρμονίας, τὴν διὰ τεττάρων, τὴν διὰ πέντε, τὴν διὰ πασῶν, πάσας δὲ τὰς ἀναλογίας, τὴν ἀριθμητικήν, τὴν γεωμετρικήν, ἔτι δὲ τὴν ἁρμονικὴν περιέχει. τὸ δὲ πλινθίον συνέστηκεν ἐκ τῶνδε τῶν ἀριθμῶν, ἓξ ὀκτὼ ἐννέα δώδεκα· (…) τὸ διάγραμμα ἢ πλινθίον ἢ ὅ τι χρὴ καλεῖν . . .

There is such a thing as the source of the most beautiful diagram which is not only perfect, but, so to speak, the most harmonic. This diagram comprises all harmonies: the fourth, the fifth, the octave; and all means: the arithmetic, the geometric, and also the harmonic. The table (πλινθίον) consists of these numbers: six eight nine twelve. (…) The diagram or table (πλινθίον) or whatever we should call it . . .

The four numbers apparently contained in this grid (περιέχει) were used to display the different ratios defining the basic concords of ancient Greek music and the three basic means. They are found, for example, in the diagram of the helikon in Ptolemy’s Harmonics (cf. Section 1.4.2 above). No doubt Philo referred to the same grid that Plutarch mentions in De anima procreatione in Timaeo 2018A, and which was drawn by Aristides Quintilianus in his treatise on music theory (III 4) – the four numbers are found in the sum of small squares in each of the four rectangle-divisions:^[71]

The lexicon by the grammarian Julius Pollux (2nd/3rd c. AD) furnishes another instance, calling πλινθίον the grid on which a game similar to checkers (πόλις) is played.^[72] The terms πλινθία and πλινθίδες are diminutives of the word πλίνθος, “brick”. Did their meaning as grids derive from the fact that the grid resembles a brick construction? This is very likely, especially if we think of opus reticulatum, in which bricks are square-shaped and disposed in an orthogonal grid, in contrast with the other types of masonry techniques where the coincidence of vertical and horizontal joints must be avoided.

Structure of opus reticulatum

This style of brick construction was in vogue between the first century BC/AD in central and southern Italy. This does not mean that the symmetric table was born there and by that time, but it may suggest that in that chronological and geographical context it was more fashionable than elsewhere or at other times. We have seen that the dispute-resolving Pythagorean table was called πλινθίς (Section 2.3 above). In general, the date of the texts attributed to Pythagoras can only be approximately established between the Hellenistic and the early Imperial age, and their provenance is similarly difficult to track down, but Rome is a fair possibility.^[73] As we have seen, Nicomachus, who was probably connected to the Roman ruling class, and who contextualized his arithmetical work in a Platonizing-Pythagorean setting, used this type of tables, too.^[74]

Vitruvius also knew the term πλινθίον (IX 8.1), since he applied it to a type of sundial displaying the heavens as a coordinate grid.^[75] The question is not straightforward here, since we know that πλινθίον also designated a square in the heavens. It was used in this sense several times by Hipparchus in his commentary on Aratus, for single squares defined by constellations (I 5.2, 5.4, 5.5, 5.6, etc.). Ptolemy takes up this terminology in his catalogue of stars in Almagest VII 5 (2.38 Heiberg), and Hippolytus (early 3rd century AD) used it for the four quadrants of the heavens in Refutatio IV 44.^[76] In the pseudoepigraphic letter attributed to the Egyptian astrologer-king Nechepsos (Section 2.3.2 above), it is clear that the author, probably from Egypt, was not thoroughly familiar with the concept as a symmetric table and mixed it with the tradition of πλινθίον as a heavenly quadrant. Therefore, there seem to have existed two different naming traditions, colliding or coinciding in some instances, both derived from the idea of brick: an earlier one emphasizing the idea of spherical rectangle, in the fields of mechanics and astronomy (starting as early as the third century BC); and a later one, perhaps originating in Italy (from the first century BC onward), denoting an orthogonal grid. That the origin of both traditions in the vocabulary is in building and manufacturing should not be surprising, since this field was extremely productive in the creation of terms for mathematical objects and textual layouts: we have already seen the example of κανών, but there are numerous examples: to name just a few, τραπέζιον from the shape of a table (τράπεζα); γνώμων (an L-shaped area) from the carpenter’s square; σελίς (column of text on papyrus) from σέλμα (deck of a ship and hence block of parallel rowing benches).

As for the symmetric tables themselves, their established status as a category of table can be contextualized together with better-known visual devices in Greek culture. One element is the stoikhedon epigraphical style, which was not in vogue anymore by the time these tables appeared, but which must have been in display in numerous surviving inscriptions.^[77] The text of such inscriptions was disposed in a rectangular grid of small square cells each containing one letter. In most cases, especially in Attica, the grid itself was not incised or very shallowly incised, but elsewhere it was often clearly seen.^[78]

Then, we have several archaeological artifacts dated in the early centuries of the Roman Imperial age (thus coinciding with the emergence of symmetric tables), which contain visual games using letters in square grids or in a square frame. In Greek, there are two examples of square grids containing a phrase that can be read jumping from the center in any direction to one of the four corners (without going backwards). One is from the reverse of the Tabulae iliacae (1st c. AD), forming a sentence with the title of the work and the signature of the author.^[79] It may be relevant that the production context of this relatively early instance is Rome. Another square grid of the same type appears in the so-called Moschion stele (2nd or 3rd c. AD), where it is accompanied by bilingual texts in Demotic and Greek, providing instructions to the reader; in the Greek texts the square is referred to as πλινθίς.^[80] A similar kind of crossword game is represented by the famous SATOR and ROMA OLIM squares, in which the text can be read from either corner to the opposite (upper left to lower right or the other way around), be it in rows or columns. Both are found in Pompeii – again, a Roman context.^[81] These kinds of visual wordplays found a continuation in the Latin poems of Publilius Optatianus (early fourth century), which were set up in stoikhedon format, often in the shape of a square, and containing symmetric intexti. The square is not the only shape in which such inscriptional games appear – for example, some of the Tabulae iliacae contain a crossword in a different shape^[82] – but it is undeniable that the square shape was the most frequent one.^[83]