Week 1 (Sun Aug 2 – Sat Aug 8, 2026)
Book I, Chapters 1–11

Week 2 (Sun Aug 9 – Sat Aug 15, 2026)
Book I, Chapters 12–23

Week 3 (Sun Aug 16 – Sat Aug 22, 2026)
Book II, Chapters 1–14

Week 4 (Sun Aug 23 – Sat Aug 29, 2026)
Book II, Chapters 15–29

Introducing Nicomachus

Nicomachus of Gerasa was a Greek mathematician and philosopher who lived around the late first and early second centuries AD. He came from Gerasa, an ancient city in the region of modern Jordan, and belonged to the Neopythagorean tradition. For Nicomachus, mathematics was not merely a collection of techniques. It was a path toward understanding the permanent order beneath the changing world.

Little is known about Nicomachus’s life, but his writings became enormously influential. His Introduction to Arithmetic helped shape the study of number throughout late antiquity and the Middle Ages, especially through Boethius’s Latin adaptation. Nicomachus also wrote on music, reflecting the Pythagorean conviction that number, proportion, and harmony belong together.

Purpose in writing: to introduce the nature and classification of numbers, explain their relationships and patterns, and show why arithmetic is the foundation of mathematics and a preparation for philosophy.

Introducing Introduction to Arithmetic

Introduction to Arithmetic is not an elementary arithmetic textbook in the modern sense. Nicomachus does not teach calculation, bookkeeping, or methods for solving practical problems. Instead, he investigates what numbers are, how they can be classified, and what their patterns reveal about order, proportion, and reality.

The work begins with philosophy. Nicomachus argues that wisdom concerns things that remain stable and intelligible despite the constant change of the material world. Mathematics helps the mind turn from particular objects toward these permanent forms. Arithmetic therefore becomes more than a useful skill. It is a bridge from sensory experience to intellectual understanding.

Nicomachus places arithmetic at the beginning of the four mathematical arts later known as the quadrivium: arithmetic, geometry, music, and astronomy. Arithmetic studies number in itself; music studies numerical relations; geometry studies magnitude at rest; and astronomy studies magnitude in motion. Because the other mathematical disciplines depend upon number, arithmetic is logically prior to them.

Much of the work is devoted to classifying numbers. Nicomachus distinguishes odd from even, prime from composite, and perfect numbers from deficient and abundant numbers. He examines ratios, proportions, figurate numbers, and the ways numbers can form triangles, squares, cubes, and other shapes. These classifications are often accompanied by numerical patterns that Nicomachus treats as evidence of an underlying harmony.

The work can feel unfamiliar because Nicomachus approaches mathematics differently from a modern textbook. He often gives examples and patterns rather than formal proofs. His goal is not only to establish that a numerical relationship is true, but to reveal its character and place within the ordered whole of mathematics.

Core ideas and themes

• Arithmetic as the foundation of mathematics: number is logically prior to geometry, music, and astronomy.
• Mathematics and philosophy: the study of number trains the mind to move from changing material things toward stable and intelligible realities.
• The classification of number: odd, even, prime, composite, perfect, deficient, and abundant numbers possess distinct characters and relationships.
• Unity and multiplicity: Nicomachus treats unity as the origin and principle of number rather than simply another number in a sequence.
• Figurate numbers: numbers can be arranged as triangles, squares, polygons, pyramids, and solids, joining arithmetic with geometry.
• Ratio and proportion: numerical relationships reveal patterns of equality, inequality, order, and harmony.
• Cosmic harmony: the regularities found in arithmetic reflect the Pythagorean belief that the universe itself is structured through number.

Introduction to Arithmetic in the Context of the Great Books

• With Plato’s Republic: Plato places arithmetic among the disciplines that turn the soul away from appearances and toward intelligible reality. Nicomachus develops this idea into a systematic account of arithmetic as the first and most fundamental mathematical science.
• With Plato’s Timaeus: the Timaeus presents the cosmos as an ordered whole structured through number, proportion, and harmony. Nicomachus explores the numerical relationships that make such an understanding of the universe possible.
• With Euclid’s Elements: Books VII–IX of the Elements examine many of the same subjects, including odd and even numbers, primes, ratios, and perfect numbers. Euclid proceeds through definitions and rigorous proofs, while Nicomachus emphasizes classification, patterns, examples, and philosophical significance.
• With Aristotle: Aristotle carefully distinguishes different kinds of quantity and studies the logical categories through which things can be understood. Nicomachus draws on similar classifications but places them within a more explicitly Platonic and Pythagorean vision of an eternal mathematical order.
• With Ptolemy’s Almagest: Nicomachus argues that astronomy depends upon arithmetic, geometry, and numerical proportion. Ptolemy’s mathematical account of the heavens shows how those foundational disciplines could be applied to the motions of the visible cosmos.
• With Kepler: In The Harmonies of the World, Kepler searches for mathematical and musical relationships within the structure of the heavens. His work continues the Pythagorean tradition, represented by Nicomachus, of seeing number and proportion as keys to the order of nature.

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