An 11th-century mathematical treatise in which Adelbold, Bishop of Utrecht, presents his method for calculating the volume of a sphere to Pope Sylvester II, the renowned "Mathematician Pope." This remarkable text demonstrates how medieval scholars reconstructed Archimedean mathematics through sources like Macrobius and Boethius, offering a rare glimpse into the scientific culture of the cathedral schools during what is too often dismissed as the "Dark Ages."

1. Historical Context: The 10th-Century Scientific Renaissance

The Libellus de Ratione Inveniendi Crassitudinem Sphaerae (“A Short Work on the Method of Finding the Volume of a Sphere”) emerges from one of the most intellectually vibrant periods in early medieval Europe. The late 10th and early 11th centuries witnessed a remarkable revival of mathematical learning, centered in the cathedral schools of places like Reims, Liège, and Chartres.

This was not the “Dark Ages” of popular imagination. Instead, it was an era when scholars like Gerbert of Aurillac (later Pope Sylvester II) reintroduced the abacus to Western Europe, brought Hindu-Arabic numerals to European attention through contact with Islamic Spain, and constructed sophisticated astronomical instruments including armillary spheres.

The text represents a continuation of scholarly correspondence between Adelbold and Gerbert—a relationship that exemplifies how mathematical knowledge circulated among the educated elite of medieval Christendom. As the Encyclopedia of Pope Sylvester II notes, Gerbert’s circle at Reims began to revive Greek geometry (often via Boethius and Arabic sources) under a Christian framework, and this treatise is a direct product of that milieu—evidence of the “vivid interest in mathematics” among the clergy. Earlier, Adelbold had posed a problem to Gerbert about calculating the area of an equilateral triangle, demonstrating that mathematical puzzles were a form of intellectual currency in this learned network.

2. The Author: Adelbold of Utrecht (c. 975–1026)

Adelbold (also spelled Adalbold) was a medieval scholar, mathematician, and churchman who became Bishop of Utrecht from 1010 until his death on November 27, 1026.

Education and Formation

Born around 975, probably in the Low Countries, Adelbold received an exceptional education for his time:

  • Lobbes Abbey: Began studies at this Benedictine monastery in Hainaut under the learned Hériger
  • Liège Cathedral School: Continued under Bishop Notker (Notger), one of the foremost scholars of the era. The Liège school was exceptional—students flocked there “from all Christendom”
  • Reims: Became one of the privileged students of Gerbert of Aurillac, then making Reims perhaps the leading center of mathematical learning in Europe

Episcopal Career

  • Became a canon and teacher at Laubach
  • Served as chaplain and notary to Emperor Henry II, who held him in high regard
  • Appointed Bishop of Utrecht in 1010
  • Completed the great Romanesque Cathedral of Saint Martin at Utrecht
  • Expanded the territorial possessions of the diocese, acquiring Drente (1024) and Teisterbant (1026)

Military Challenges

As bishop, Adelbold faced frequent Norman raids and conflicts with neighboring nobles. He was defeated at the Battle of Vlaardingen (1018) against Dirk III, Count of Holland, in a dispute over the Merwede district between the mouths of the Maas and the Waal rivers.

Other Writings

Beyond this mathematical treatise, Adelbold also wrote:

  • A philosophical exposition of a passage of Boethius
  • Adelboldi Fragmentum de Rebus Gestis S. Henrici (also in PL 140)
  • Possibly a Vita Heinrici II imperatoris (attribution debated)

3. The Recipient: Pope Sylvester II (c. 946–1003)

Gerbert of Aurillac was arguably the most brilliant mind of 10th-century Europe and one of the most remarkable figures of the medieval period.

The “Mathematician Pope”

Born around 946 in Aurillac, Auvergne, Gerbert’s intellectual journey took him far beyond the typical monastic education:

  • Aurillac: Entered the Benedictine monastery of Saint-Gerald in 963
  • Catalonia: In 967, Count Borrell II of Barcelona took him to study mathematics in Spain
  • Islamic Learning: Studied under Bishop Atto of Vich and received instruction in Arabic mathematical and astronomical learning at Seville and Córdoba

Mathematical Achievements

Gerbert’s contributions to medieval science were transformative:

  • First Christian to teach using Hindu-Arabic numerals (though he called them “Hindu numerals” following his Arabic sources)
  • Reintroduced the abacus to Europe: His abacus had 27 divisions with 9 number symbols and 1,000 characters crafted from animal horn
  • Revived the quadrivium (arithmetic, geometry, music, astronomy) as master of the Reims cathedral school from 972 onward
  • Created astronomical instruments including armillary spheres and celestial globes

Career and Papacy

  • Master at the Cathedral School of Reims (made it the leading European center of learning)
  • Abbot of Bobbio
  • Archbishop of Reims (991–997)
  • Archbishop of Ravenna (998)
  • Pope Sylvester II (999–1003)

He was the first French pope. The papal name “Sylvester” recalled Sylvester I, who worked alongside Emperor Constantine—Gerbert similarly worked alongside Emperor Otto III in what they envisioned as a renewed Christian Roman Empire.

The Triangle Correspondence

Before this treatise on spheres, Adelbold had posed a problem to Gerbert: In an equilateral triangle with base 30 feet and height 26 feet, two methods of finding the area give different answers. Gerbert correctly explained that half the base times the height (giving 390 square feet) is the correct method. This earlier exchange established the pattern of mathematical correspondence between student and teacher that continues in the present text.

4. The Mathematical Content

Figure concept: A sphere inscribed in a cylinder – Archimedes' classical model illustrating sphere volume. Archimedes proved that a sphere's volume equals exactly ⅔ of its circumscribing cylinder's volume.

The Problem from Macrobius

Adelbold explicitly addresses a passage from Macrobius’s Commentary on the Dream of Scipio (Commentarii in Somnium Scipionis), one of the most influential textbooks of the medieval period.

Macrobius (fl. c. 400 CE) stated that when one sphere’s diameter is double another’s, its “thickness” (crassitudo, meaning volume) is eight times greater. Adelbold wants to understand and verify the mathematical reasoning behind this claim. Notably, the Latin term crassitudo sphaerae is adopted from Cicero to mean “volume”—a reminder that medieval scholars lacked our modern vocabulary and had to adapt classical terminology.

This connection is significant because it shows:

  1. The continuity of classical learning through late antique sources
  2. How medieval scholars engaged critically with inherited texts
  3. The role of Macrobius as an intermediary for mathematical knowledge

Adelbold’s Method: A Worked Example

The treatise demonstrates a sophisticated understanding of spherical geometry through a concrete numerical example. Adelbold works through the calculation step by step:

Step 1: Establish the Circle

Take a sphere with diameter 7 (length units). Using the Archimedean approximation π ≈ 22/7:

  • Circumference = 22
  • Radius = 3½

Step 2: Calculate the Circle’s Area

The circle’s area is calculated as radius × half-circumference:

\[3.5 \times 11 = 38.5\]

Step 3: Inscribe the Sphere in a Cube

Adelbold inscribes the sphere in a cube of side 7. The cube’s volume is:

\[7^3 = 343\]

Step 4: Subtract the Excess

To find the sphere’s volume, Adelbold “cuts away” the parts of the cube that exceed the sphere. By removing 1/21 of the cube repeatedly:

  • Removing 10 times gives: 343 − (10 × 343/21) = 343 − 163⅓ ≈ 179⅔
  • Removing 11 times gives: 343 − (11 × 343/21) = 343 − 179⅔ ≈ 163⅓

He concludes the sphere’s volume is 179⅔ cubic units.

Step 5: Verification

This result exactly matches the classical formula \(\frac{4}{3}\pi r^3\) when using π ≈ 22/7:

\[\frac{4}{3} \times \frac{22}{7} \times (3.5)^3 = \frac{4}{3} \times \frac{22}{7} \times 42.875 = 179\frac{2}{3}\]

The agreement is striking—Adelbold has successfully reconstructed the Archimedean result!

The 11/21 Ratio

Adelbold arrives at the sphere being approximately 11/21 of the cube’s volume—meaning 10/21 must be “cut away” from the cube to leave the sphere.

Mathematical Significance

The actual ratio of sphere to cube volume (where the sphere is inscribed in the cube) is π/6 ≈ 0.5236. Using π ≈ 22/7, this gives approximately 11/21 ≈ 0.5238—remarkably close! This demonstrates sophisticated understanding of the relationship between these geometric forms.

Modern historian Curtze (in his Abhandlungen zur Geschichte der Mathematik) remarks that Adalbold’s arithmetic indeed reproduces Archimedes’ result, yielding the familiar π = 22/7 approximation.

The Archimedes Connection

Figure concept: Archimedes' sphere and cylinder – the sphere's volume is ⅔ that of its circumscribing cylinder. This relationship was so beloved by Archimedes that he requested it be inscribed on his tombstone.

While Adelbold works toward sphere volume calculations, the ultimate source for this mathematics is Archimedes’ treatise On the Sphere and Cylinder (c. 225 BCE). However, Archimedes’ original work was largely unknown in Western Europe in Adelbold’s time.

The Byzantine manuscripts preserving Archimedes were being copied in Constantinople (Codex A was copied in 888 CE), but this knowledge had not yet reached the Latin West. This makes Adelbold’s mathematical reasoning all the more remarkable—he was essentially reconstructing Archimedean results through medieval intermediaries like Macrobius and Boethius.

5. Rhetorical Style and Scholarly Culture

The Humility Topos

The preface is notable for its elaborate humility topos—Adelbold apologizes profusely for daring to trouble the Pope with such questions:

“It is a great fault to disturb one engaged in public affairs with private concerns. But I have such confidence in your intellect that it can both suffice for the commonwealth and satisfy me in this matter I seek.”

This was conventional in medieval scholarly correspondence but also reflects the genuine reverence scholars held for Gerbert’s mathematical authority. The playful tone—asking for “forgiveness” for his “sin” of bothering the Pope with geometry—reveals the informal scholarly friendship underlying the formal dedication.

Mathematical Problem-Solving as Intellectual Exchange

The text reveals how mathematical problems circulated among scholars as a form of intellectual currency. Posing and solving problems demonstrated one’s learning and maintained scholarly networks across great distances. Adelbold frames his inquiry modestly:

“What I now wish to propose, I seem to see by what reasoning it can be examined and brought to understanding; but I await your diligence for the determination, so that the authority of so great a man may become either a correction or a completion of my perception.”

This shows the collaborative nature of medieval scholarship—even a bishop sought validation from his former teacher.

6. Historical Significance

Mathematics in the “Dark Ages”

This text directly counters the myth that medieval Europe was intellectually stagnant. It demonstrates:

  • Active mathematical inquiry and problem-solving
  • Critical engagement with inherited classical texts
  • Sophisticated numerical reasoning
  • International networks of scholarly correspondence

Islamic-European Knowledge Transfer

Gerbert’s education in Al-Andalus (Islamic Spain) shows the crucial role of cross-cultural scientific exchange. The mathematical knowledge Gerbert brought back—including the Hindu-Arabic numerals and computational techniques—was then transmitted through his students like Adelbold to a new generation of European scholars.

The Cathedral School Network

The text exemplifies how cathedral schools (Reims, Liège, Bamberg, Cologne) served as nodes in a network of mathematical learning:

  • Reims (Gerbert) → Liège (where Adelbold studied) → Utrecht (where Adelbold became bishop)

Students became teachers, and the cycle continued, gradually building the intellectual infrastructure that would eventually lead to the medieval universities.

Manuscript Tradition and Publication History

The text survived in manuscripts from:

  • Tegernsee Monastery
  • St. Peter’s Monastery, Salzburg

As Migne notes in Patrologia Latina, this Libellus “now appears for the first time” from old manuscripts—meaning it had not circulated widely before its 19th-century publication. It was first published by Alphonsus Hueber and included in Bernard Pez’s Thesaurus Anecdotorum Novissimus (1721), making it available to modern scholars.

In the corpus of Patrologia Latina (vol. 140), this text illustrates the Church’s custodianship of learning—even a bishop’s mathematical paper is preserved alongside sermons and theology. The Latin text appears at cols. 1103–1106 of that volume.

7. Noteworthy Excerpts (Translated)

The Opening Dedication

Latin: Domino SILVESTRO summo et pontifici et philosopho ADELBOLDUS scholasticus vitae et felicitatis perpetuitatem.

English: “To Lord Sylvester, supreme pontiff and philosopher, Adelbold the scholar wishes perpetual life and happiness.”

On the Problem from Macrobius

Latin: Macrobius super Somnium Scipionis… compertum esse ait apud geometras peritissimos, ut in duobus circulis, si diametrum unius duplum sit diametro alterius, ejus circuli crassitudo, cujus diametrum duplum sit, octupla sit crassitudini illius circuli cujus subduplum est diametrum.

English: “Macrobius, in his commentary on the Dream of Scipio… says that it has been established among the most skilled geometers that if, in two circles, the diameter of one is double the diameter of the other, the volume of the circle whose diameter is double will be eight times the volume of the circle whose diameter is half.”

The Methodological Approach

Latin: Quid autem jam inde mihi percepissem, aperiam, non, ut aiunt, Minervam litteras, sed ut monstrem quid sentiam; quatenus, si erro, ad viam a sagacitate vestra reducar.

English: “What I have so far understood from this, I shall reveal—not, as they say, to teach Minerva her letters, but to show what I think; so that, if I err, I may be led back to the path by your sagacity.”

Sources

Primary Sources

  • J.-P. Migne, Patrologia Latina vol. 140, cols. 1103–1106: Archive.org

Secondary Literature

Reference

Notes on Figures

The conceptual figures described in this article illustrate Archimedes’ classical sphere-volume theorem. No direct medieval diagram survives for Adelbold’s treatise, but the geometry described underlies his calculations.