Regula Caeci

Summary: The Regula Caeci (Rule of the Blind, also called Position or the Rule of False) is Euler’s method for solving two linear equations in three or more unknowns over the positive integers, by eliminating one variable and reducing to a single indeterminate equation.

Sources: chapter-2.0.2

Last updated: 2026-05-04

---

Name and Context

The name Regula Caeci (Latin: “rule of the blind”) appears in common arithmetic books for problems where a trial-and-error guess is used; Euler explains its algebraic foundation. The same rule is also called Position or The Rule of False. It applies when two equations contain three or more unknowns, making the system underdetermined. (source: chapter-2.0.2, §24)

Standard Setup

The canonical form is:

$x+y+z=afx+gy+hz=b$

with $f,g,h$ given coefficients (not necessarily ordered). Eliminate $z=a-x-y$ from the second equation to obtain a single equation in $x$ and $y$, which is then solved by the indeterminate method of Chapter I. (source: chapter-2.0.2, §25)

Feasibility Condition

Assume $f$ is the largest and $h$ the smallest among $f,g,h$. Then the system has positive integer solutions only if

$h\cdot a<b<f\cdot a$

If $b\ge fa$ or $b\le ha$, the system is impossible. If $b$ is very close to either bound, the number of solutions may be small or zero. (source: chapter-2.0.2, §27)

Proof sketch: If all quantities were at the maximum rate $f$, total expenditure would be $fa$; if all at the minimum $h$, it would be $ha$. Any feasible mixture must lie strictly between these bounds.

Applications

Mixtures and alloys: A goldsmith blending metals of different purities sets up $x+y+z=\text{total weight}$ and $fx+gy+hz=\text{total fine content}$; the Regula Caeci determines all valid blends. (source: chapter-2.0.2, §28)

Purchase problems: Buying several categories of goods (hogs, goats, sheep; oxen, cows, calves, sheep) at different prices for a fixed total — the classic “100 animals for 100 pounds” type — is the paradigmatic application. (source: chapter-2.0.2, §25–26, §29)

Extension to Four or More Unknowns

With four unknowns and two equations, eliminate two variables in succession; the remaining free parameters each range over a finite integer interval. The total number of solutions is the product of the sizes of those intervals (minus excluded boundary cases). (source: chapter-2.0.2, §29)

Generalised Coefficients

When the first equation is $\alpha x+\beta y+\gamma z=a$ rather than $x+y+z=a$, the same elimination technique applies; feasibility bounds become $h\cdot a/{\alpha }_{\min }$ and $f\cdot a/{\alpha }_{\min }$ appropriately adjusted. (source: chapter-2.0.2, §30)

Related pages

• indeterminate-analysis
• linear-diophantine-equations
• systems-of-linear-equations
• ch2.0.2-regula-caeci