Machin-like Formula
Summary: §142 of Chapter 8. To compute rapidly without irrational denominators, Euler decomposes as a sum of two arctangents of small rational numbers:
Combining with the arctangent series gives
two rational geometric-rate series, “with much more ease than with the series mentioned before.”
Sources: chapter8 (§142)
Last updated: 2026-04-27
The decomposition
Suppose , so . The §128 addition formula gives
(source: chapter8, §142). Choose . Then
Hence
The two series
Substituting in :
Substituting :
Therefore
(source: chapter8, §142). Both series have only rational terms, decay geometrically at rate and respectively, and avoid the that complicates the §141 series.
Convergence rate
Per term in the series, the magnitude shrinks roughly by . In the series, by . So:
| Terms used | Approximate digits of |
|---|---|
| 5 | 4 |
| 10 | 8 |
| 15 | 12 |
| 20 | 16 |
This is a dramatic improvement over Leibniz (, no convergence in any practical sense) and a significant improvement over §141’s series (irrational terms, factor per step).
Naming and history
Euler attributes this style of identity to no-one in §142 — he simply derives it. The general technique is named after John Machin, who in 1706 used the identity
to compute to 100 digits — the first 100-digit computation of in history. Euler’s identity is a simpler cousin in the same family. Both rely on the same algebraic manipulation: choose rational with small denominator, solve for the complementary arc, and check that is also rational with small denominator.
The general “Machin-like formula” pattern with integers and rational is, after this section, the canonical method for computing to high precision. It dominated computation from the 18th into the early 20th century, when iterative methods (Gauss–Brent–Salamin, etc.) finally surpassed it.
Pattern: combining arctangents
The §142 derivation generalizes. From :
Picking small and rational so that also lands at a useful arc — e.g., 1 (giving ), , etc. — produces an unending family of identities. The §142 choice , gives , exactly the value at which .
Why Euler stops here
The chapter ends at §142, with the Machin-style formula as the final word on rapid computation of . No further arctangent decompositions are explored, and the Introductio’s treatment of pauses here. Later chapters and other works return to the topic — Euler himself derived several improved Machin-like formulas in subsequent papers — but in the Introductio the §142 identity stands as the definitive practical tool.