Achard of St. Victor (d. 1171) and the Eclipse of the Arithmetic Model of the Trinity

In the first book of De doctrina christiana, Augustine of Hippo famously teaches that only the Trinity is to be enjoyed; all other things and even people are to be used toward this singular end. The brevity of Augustine’s passing remarks on the Trinity gives no hint that he will later devote many pages to the topic. He writes:

These three have the same eternal nature, the same unchangeableness, the same majesty, the same power. In the Father there is unity, in the Son equality, and in the Holy Spirit a harmony of unity and equality. And the three are all one because of the Father, all equal because of the Son, and all in harmony because of the Holy Spirit.¹

These few lines comprise the lengthiest discussion of the Trinity in De doctrina christiana. We can now trace Augustine’s triad of unitas, aequalitas, and concordia to a saying of the neo-Pythagorean Moderatus of Gades (Cádiz) (fl. ca. 50 CE), as reported by Porphyry.² Marius Victorinus had already adopted a portion of the same passage on Moderatus (along with others from Porphyry) when formulating his own Pythagorean analogy of the Trinity.³ But Augustine, immediately after introducing his triad in De doctrina christiana, repudiates the high-minded philosophical analogy, calling it an example of the painful failure of language in the face of God’s ineffability.⁴ Indeed, the bishop of Hippo disowned the idea altogether, never repeating it again. There is no trace of the triad in Augustine’s earlier works, despite his avowedly Pythagorean views of number in De ordine, De musica, and De libero arbitrio, and despite the arithmological interests of his several commentaries on Genesis.⁵ Nor does Augustine so much as mention the triad in the fifteen books of De trinitate that he wrote two decades later.

Despite such inauspicious origins, Augustine’s triad of unitas, aequalitas, and concordia in hindsight now belongs on any short list of classical analogies of the Trinity. Medieval readers of Augustine found it a fruitful analogy of Trinitarian relations and considered its meaning alongside other notable triads: Hilary of Poitiers’s aeternitas, species, usus, much discussed by Augustine and then by early scholastics; Augustine’s own more noteworthy memoria, notitia, amor from De trinitate; variations on the Plotinian triad of One, Mind, and World Soul, such as survive in Macrobius; later Neoplatonic triads conveyed from Ps.-Dionysius’s Divine Names via John Scotus Eriugena’s translations (mansio, processio, reditus; or less commonly, esse, vivere, intelligere); and finally, the controversial potentia, sapientia, benignitas from Hugh of St. Victor and Peter Abelard.⁶ Peter Lombard carried Augustine’s triad of unity, equality, and harmony into the Sentences, and Aquinas found a home for it in the Summa.⁷ But where these authorities cited the analogy as a confirmation of standard accounts of intradivine relations, its most noteworthy defenders, Thierry of Chartres and Nicholas of Cusa, relished the deeper mathematical mystery that the three words seemed to conceal.

In his boldly naturalistic hexaemeral commentary from the 1130s, Thierry of Chartres (d. 1157) breathed new life into the Augustinian triad by illuminating its connections with contemporary medieval science.⁸ Thierry sought to interpret the six days of creation secundum physicam, and likewise the Creator, so to speak, secundum quadrivium. Unitas and aequalitas were the grounding principles, respectively, of arithmetic and harmonics (or music), the two pillars of the mathematical disciplines in Boethius’s account.⁹ Thierry went on to pursue this arithmetical reconstruction of Augustine’s triad throughout his commentaries on Boethius’s De trinitate in the 1140s and 1150s.¹⁰ He adjusted Augustine’s third term from concordia to conexio, a minor amendment but one that drew attention to the implicit arithmetical link between unity and equality. Numerical oneness, explained Thierry, can be multiplied by itself and remain oneness. This perfect self-equality of unity is an analogy of the Son’s generation by the Father. But the oneness that results after the self-generation stems both from unity and from unity...