Division of Compound Quantities
Summary: Euler’s division of compound quantities combines simple termwise division with a long-division method for cases where a compound divisor exactly divides the dividend. (source: chapter-1.2.4)
Sources: chapter-1.2.4
Last updated: 2026-04-26
Simple Divisors
If the divisor is a single quantity, each term of the dividend is divided separately. (source: chapter-1.2.4)
Examples include
and
(source: chapter-1.2.4)
Compound Divisors
If the divisor is itself compound, exact division is not guaranteed; when it fails, Euler leaves the quotient in fractional form. (source: chapter-1.2.4)
When division succeeds, the quotient is built term by term by matching the highest-order part of the dividend and repeatedly subtracting products of the divisor with the partial quotient. (source: chapter-1.2.4)
Ordering Principle
Euler recommends ordering terms by descending powers of the leading letter before starting the division. (source: chapter-1.2.4)
That arrangement makes the successive quotient terms easier to detect and aligns closely with modern polynomial long division. (source: chapter-1.2.4)