Jewish Kabbalistic writings often construct elaborate systems to assist their metaphysical speculations on the divine realm. Occasionally, these systems are presented through diagrams that map out the structure of divine potencies and the dynamic relationships between these potencies and the created world. The best-known examples are the numerous variations of thetheosophical Sefirotic Tree, whose branching structure has come to epitomize the dynamic order of divine entities and powers, (sefirot). Yet the early 13th century Kabbalists also drew on other types of geometric diagrams that were readily available in the scientific and theological environments of the time, namely, the concentric spherical diagram as generally informed by Ptolemaic astronomy. These diagrams, which consisted of ten spheres—the 7 traditional planets, the sphere of fixed stars, the diurnal sphere (Primum Mobile), and, in some cases, the Universal Intellect—were integrated and further modified by Jewish theological and Kabbalistic doctrines of creation.
Kabbalistic texts illustrate how dynamic and adaptive these cosmological models were. So much so, that Kabbalists often integrated elements from markedly different systems, mainly theological or cosmogenic, thereby reconstruing the nature, logic, and order of the cosmic diagrams of the time. One notable case appears in Ginnat Egoz (The Garden of the Nut), a cosmic-Kabbalistic work composed in 1274 by the Castilian Kabbalist Joseph Giqatilla. His text includes a spherical diagram that serves as the structural skeleton of his cosmology (Figures 1–2).
As the diagram suggests, Giqatilla reconfigured the concentric model by integrating the first ten Hebrew letters as ciphers which stand for the ‘parts of the cosmos’. In the manuscripts of Ginnat Egoz, the diagram usually appears as two concentric circles—an inner and an outer sphere. On the inner circumference are inscribed the first ten Hebrew letters, ordered counterclockwise: א, ב, ג, ד, ה, ו, ז, ח, ט, י. These letters, and the Hebrew alphabet more generally, played a significant role in ciphering complex cosmic structures, also due to their numerical value as established in early Rabbinic tradition. The numerology of the Hebrew letters was constructive tool for recasting the relationship between the spheres and parts of the cosmos, as ciphered by the first ten Hebrew letters:
The ten letters correspond to the “ten parts of the universe”, a term which refers to both the number of spheres and to the 10 cosmic qualities that the Hebrew letters carry together with their respective numerical values. The latter is pivotal for the construction of a dynamic cosmos that operates by the qualities epitomized by these linguistic principles. Another pivotal addition is the symbol יא, hovering above the inner concentric letter-arrangement. The numerical value of this symbol is 11, or 10 + 1 (י+א). In Jewish Neoplatonic literature יא often represents the transcendent One in relationship to the tenfold cosmos. In Giqatilla’s diagram it takes on an additional function, namely, the primary principle of divine motion which sustains and governs and spheres.
The categorical distinction between the ten cosmic parts and the principle of divine motion likely prompted later copyists of the Garden of the Nut to render Giqatilla’s cosmic diagram more schematically. Thus, a sixteenth-century manuscript in (Figure 3) arranges the ten letters on one side of the sphere, directly opposite the symbol יא on the other. Each side bears a heading: ḥelqe ha-galgal (“parts of the sphere”) and tenuʿah יא (‘motion Y‘), respectively. Whereas the diagram in the Paris and London manuscripts can be considered integrative (in the sense that the hovering symbol יא is situated in dialogue with the running alphabetic circle), the diagram in the Munich manuscript is pronounceably schematic.
The idea that the universe consists of the principles embedded in the Hebrew alphabet is central to many Jewish texts, and The Garden of the Nut marks another important moment in this rich speculative tradition. But it also affords us an opportunity to better assess the role that diagrams play in the intersection of cosmology and theology. Particularly, Giqatilla’s letter-cosmography stresses the question whether Kabbalistic diagrams served a goal beyond the mere pedagogical illustration of complex ideas? Addressing this question is also instructive for assessing the manuscript tradition of Giqatilla’s Garden of Nut which includes a markedly distinct rendition of the alphabetic-spherical diagram.
Abraham Ibn Ezra’s 12th century Arithmetic Cycle
Let us begin by noting that Giqatilla’s diagram has a history. It bears striking allusions to a diagram presented by the Andalusian polymath Abraham Ibn Ezra (12 c.), one of Giqatilla’s major influences. In his Sefer ha-Mispar (“Book of Numbers”), composed over a century prior to The Garden of Nut, Ibn Ezra offered perhaps the earliest systematic Hebrew introduction to the decimal number system. He prefaced it with a brief meditation on the symbolic qualities of the nine numbers and their analogy to the nine spheres encompassing earth:
[The Hebrew word] Sfar refers to the nine numbers, since nine is the end of any reckoning. You should know that the nine are the true numbers which stand against the nine spheres and all the ensuing numbers are assimilated to them— Abraham Ibn Ezra, Preface to Sefer ha-Mispar, trans. Shlomo Sela (excerpt; adapted).
The first nine Hebrew letters (א–ט) represent the numbers 1–9 and the nine celestial spheres surrounding the sublunary realm. The sequence proceeds counterclockwise, with ṭet (ט, 9) at the apex. Ibn Ezra assumes also the additional symbol 0 (‘void’), functioning as a placeholder within a decimal system. Ibn Ezra is not explicit about the cosmic analogue of the 0, though one might wonder if he had the sublunar realm in mind.
While not figured in the diagram, the letter yod (י, 10) is implied as the radix of the decimal system, rather than one of its counted elements. What is significant about Ibn Ezra circular diagram is its arithmetic mechanism which demonstrates the harmony of the 9 letters and, consequently, of the spheres: Multiplying 9 by any descending integer yields products whose digits are positioned as diametrically opposite pairs.
Reconstructing Giqatilla’s Experiment
Ibn Ezra was a polymathic thinker and several of his ideas, in both areas of linguistics and cosmology, became pivotal to Giqatilla. There are grounds to assume that Ibn Ezra’s cosmic diagram was among these adopted ideas, and not simply because of the graphic and doctrinal allusions. If we read Giqatilla’s diagram through Ibn Ezra’s arithmetical logic, its inner workings become clearer. In the discussion following his diagram, Giqatilla introduces various cosmic constructs by manipulating the elements presented in the alphabetic diagram. One of these hermeneutical products is the following fourfold set of letters: “The parts of the sphere [are] אט (AT), בח (BḤ), גז (GZ), דו (DW).”
This set alludes to an established Rabbinic hermeneutical formula, known as אטב״ח (ATBḤ), where specific letters in the Hebrew alphabetic system are interchangeable with their respective counterparts – e.g., the letter ’aleph (א) with tet (ט), bet (ב) with ḥet (ח), and so forth. Giqatilla adopts this hermeneutical device while recasting its function and significance through the logic of Ibn Ezra’s cosmic-decimal system. He does not spell out his methods, but the logic can be construed if we correctly identify the key variables in his diagram while using Ezra’s system as a frame of reference. The same variables are at play in each of the systems:
- The multiplicand – the sequential letters around the circle;
- The multiplier – the letter at the apex (ṭet, 9, in Ibn Ezra; yod, 10, in Giqatilla);
- The radix – the numerical base that determines the system’s internal coherence.
The crucial change lies in this last variable. The compound symbol יא, whose numerical value is 11 (’aleph + yod), stands above the circle as a new counting base Giqatilla’s diagram therefore operates not on a decimal but on an undecimal system. (Giqatilla uses the symbol יא for this undecimal radix, but for the sake of clarity we may use the letter A as a placeholder for the radix 11, by which logic the number 10 is the last of the counted number: 1-10.)
This small adjustment transforms the arithmetic while recasting the parts of the universe. Where Ibn Ezra’s 9 × 9 produced 81 (אח), Giqatilla’s multiplication (within an undecimal system) yields 73 (גז). Each of the ten letters, multiplied by 10, produces a value recalculated according to base 11. The decimal radix and units’ places are redefined: 10² = 9(A) + 1, where A signifies the new radix 11 and, 1 represents the remainder. Similarly, the product 9 × 10, which equals 90 in decimal terms, becomes 82 in the undecimal system [8(A) + 2]. The letter ח׳ represents the undecimal grouping [8(A)] and 1 constitutes the remainder of the unit digit.
Under this paradigm, Giqatilla reconstructs the ATBḤ letter-pairing formula. From his modified arithmetic emerges the four pairings: אט (AT), בח (BḤ), גז (GZ), דו (DW). Each represents a structural correspondence between cosmic parts as sustained by divine motion, echoing but transforming Ibn Ezra’s earlier decimal pairs (אח, בז, גו, דה). Following Ibn Ezra’s logic, the digits constituting the resulted value stand symmetrically in relationship to the apex (the multiplier), while diametrically opposing each other: אט (AT), בח (BḤ), גז (GZ), and דו (DW). Similarly, the addition of the two digits/letter of each pair—e.g. א+ט—amounts to the apex (10) and thus completes the order of numbers. Diverging from Ibn Ezra, Giqatilla’s apex is the letter י׳ (rather than ט׳) and, more importantly, the radix, that is, the basis of the system in toto, is the combine letters יא (rather than י׳).
The addition of יא to the cosmic diagram is quite instructive. It reveals how established systems are adopted and further modified by new metaphysical and theological ideas, on the one hand, and with the aid of traditional hermeneutical formulas, on the other. Like other 13th century Kabbalists, Giqatilla developed a distinct cosmology which he based on the active principle of the יא, the latter signifying the One as the direct and active and cause of cosmic motion and its sustainability. No less significant is what Giqatilla’s adopted diagram teaches us about how original theological systems take form. Giqatilla’s model borrows from the available systems of his time while reworking their elements and, at the same time, ‘reinventing’ a new cosmology whose principles and essence assume a different and perhaps more radical conception of creation. Finally, this process of adaptation also shows us why graphic precision is important. The schematic version of the diagram (Figure 3) loses the entire logic of the spherical letters in relationship to the apex, and their diametric opposition in relationship to each other. I shall conclude this interesting spherical diagram journey, from Ibn Ezra to Giqatilla, with the afterlife of Giqatilla’s Garden of Nut. This work occupies a unique and important place in Kabbalistic literature, and some of its ideas and themes left a noticeable mark on subsequent Kabbalistic developments. One of these themes is the spherical diagram which the fourteenth century Kabbalist Menaḥem Ṣiyyoni incorporated and further reworked in his commentary on the Pentateuch.
Difference and shifts in the process of copying was not unique to The Garden of Nut and here, too, we find interesting variations among manuscripts.
The diagram is modified considerably in a Munich manuscript. (Figure 9) Not only does it reconfigure the diametric order of the letters, running now clockwise, it repositions the letter yod (י). This letter now initiates a new inner circle consisting of the Hebrew letters י, צ, כ, פ, ל, ע, מ, ס, נ the latter forming the pairs יצ, כפ, לע, מס (These pairs complete the constituents of some version of the Rabbinic ATBḤ hermeneutic formula.)
Giqatilla’s ATBḤ sphere provided the basis for further cosmic-theological diagrams. Some of these were modified and integrated into more complex theosophical systems of divine potencies (structured vertically) while borrowing further elements from the hermeneutical ATBḤ formula (Figure 10).
All in all, the journey from Ibn Ezra to Giqatilla, and from Giqatilla to later theosophical Kabbalistic texts, offers a glimpse into the workings of a dynamic and creative force of Jewish theological speculation, producing conceptual shifts within a multifaceted intellectual history.
Tzvi Schoenberg, PhD
Arts and Letters Provost Postdoctoral Fellow
Medieval Institute
University of Notre Dame
Sources:
The Bavarian State Library, MS Cod. hebr. 54, fol. 175r
The Bavarian State Library, MS Cod. hebr. 76 fol. 154v
The National Library of France, MS héb. 811, 30v
The British Library, MS Add. 11416, fol. 147r
The British Library, MS Or.13261, fol. 55r
The Austrian National Library, MS Cod. hebr. 194, fol. 90v
The Russian State Library, MS Guenzburg 1446, fol. 182r
Further reading:
Yosef Avivi, Kabbalat ha-Ari, vol. 1 (Jerusalem: Yad Ben-Zvi Institute, 2008), 445 (Hebrew)
Avishai Bar-Asher and Jeremy Phillip Brown, Light is Sown: The Cultivation of Kabbalah in Medieval Castile (New York: Oxford University Press, 2025), 73-115 (esp. 99-106)
J.H. Chajes, “Spheres, Sefirot, and the Imaginal Astronomical Discourse of Classical Kabbalah,” Harvard Theological Review, 113: 2 (2020) 230–262
J. H. Chajes, The Kabbalistic Tree (Pennsylvania State University Press, 2022)
Elke Morlok, Rabbi Joseph Gikatilla’s Hermeneutics (Tübingen: Mohr Siebeck, 2011)
Yakir Paz and Tzahi Weiss, “From Encoding to Decoding: The AṬBḤ of R. Hiyya in Light of a Syriac, Greek, and Coptic Cipher,” Journal of Near Eastern Studies 74: 1 (2015): 95–114
James T. Robinson, “The ‘Secret of the Heavens’ and the ‘Secret of Number’: Immanuel of Rome’s Mathematical Supercommentaries on Abraham Ibn Ezra in His Commentary on Qohelet 5:7 and 7:27”, Aleph, 21:. 2 (2021): 279-308
Shlomo Sela, Abraham Ibn Ezra and the Rise of Medieval Hebrew Science (Leiden: Brill, 2003)
Judith Weiss, “Spherical Sefirot in Early Kabbalah.” Harvard Theological Review 117: 4 (2024): 770-792
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