Chapter IV – Of the Division of Compound Quantities
Summary: Introduces division of compound expressions, first by dividing termwise when the divisor is simple, then by finding a quotient term by term when the divisor is itself compound. (source: chapter-1.2.4)
Sources: chapter-1.2.4
Last updated: 2026-04-26
§282–284: Representation and Simple Divisors
Euler represents division either as a fraction or with the division sign, for example . (source: chapter-1.2.4)
When the divisor is simple, each term of the dividend is divided separately:
(source: chapter-1.2.4)
If a term is not exactly divisible, that part remains fractional, so dividing by gives
(source: chapter-1.2.4)
§285–287: Searching for the Quotient
When the divisor is compound, Euler says division may fail, in which case the quotient must simply be left as a fraction. (source: chapter-1.2.4)
When actual division is possible, he finds the quotient by asking what expression, multiplied by the divisor, reproduces the dividend. (source: chapter-1.2.4)
For instance,
(source: chapter-1.2.4)
He then works recursively: find one part of the quotient, subtract the corresponding product from the dividend, and divide the remainder again. (source: chapter-1.2.4)
§288: Ordered Long Division
Euler recommends arranging the divisor so that the term with the highest power comes first, and arranging the dividend according to descending powers of the same letter. (source: chapter-1.2.4)
This produces a long-division procedure very close to modern polynomial division. (source: chapter-1.2.4)
His examples include:
and
(source: chapter-1.2.4)