Ch2.0.2 — Of the Rule called Regula Caeci: Two Equations with Three or more Unknown Quantities
Summary: Develops the Regula Caeci (Rule of False / Position) for systems of two equations in three or more unknowns, reducing them to a single indeterminate equation and determining all integer solutions.
Sources: chapter-2.0.2
Last updated: 2026-05-04
The Setting
A single linear equation with two unknowns was treated in Chapter I. When we have two equations but three or more unknowns, the system is still indeterminate. Such problems appear in common arithmetic under the name Regula Caeci (Rule of the Blind), also called Position or The Rule of False. (source: chapter-2.0.2, §24)
Standard Method: Eliminate One Variable
Given the canonical form
(with ), eliminate by substituting into the second equation. This produces a single two-unknown equation that is solved by the method of Chapter I. (source: chapter-2.0.2, §25–26)
Example (§25): 30 persons (men, women, children) spend 50 crowns; man pays 3, woman 2, child 1 crown. Setting and gives , so ranges over , yielding nine answers (eleven including ).
Feasibility Condition
For the system , to have positive solutions, must satisfy
i.e., must lie strictly between and . If or the system is impossible; if is too close to either limit only a handful of solutions may exist. (source: chapter-2.0.2, §27)
Alloy / Mixture Problems
A standard application is alloy mixing: given metals of different purities (ounces per marc), blend them to produce a mixture of weight and purity . The two equations (total weight and total fine silver) are set up and reduced as above. (source: chapter-2.0.2, §28)
Example (§28): Silver of purities 7, , oz per marc; blend 30 marcs at 6 oz per marc. The reduction (with ) gives five solutions for .
Four Unknown Quantities
The method extends directly to three equations in four unknowns (or two equations in four unknowns by iterating elimination). After eliminating two variables, one or two free parameters remain, each ranging over a finite set of integers. (source: chapter-2.0.2, §29)
Example (§29): Buy 100 head (oxen at £10, cows at £5, calves at £2, sheep at 10s) for £100. After two eliminations, 13 solutions emerge (10 in case , three in case ), reduced to 10 if zero values are excluded.
Generalised Coefficient Case
When the first equation has the form (rather than ), the elimination proceeds identically after multiplying out; the feasibility bounds are replaced by the appropriate weighted limits. (source: chapter-2.0.2, §30)
Example (§30): and . Subtracting three times the first from the second gives , and the problem reduces to a two-variable case with exactly two solutions.