Chapter II – Of the Subtraction of Compound Quantities
Summary: Gives the rule for subtracting compound expressions: change every sign in the expression being subtracted, join the resulting terms to the minuend, and then reduce like terms. (source: chapter-1.2.2)
Sources: chapter-1.2.2
Last updated: 2026-04-26
§263–266: Changing Signs in the Subtrahend
Euler first writes subtraction symbolically as
(source: chapter-1.2.2)
To carry out the subtraction, every term in the second expression changes sign before being joined to the first:
(source: chapter-1.2.2)
The reason is that subtracting a positive quantity decreases the result, while subtracting a negative quantity increases it. Euler explains this by the analogy that removing a debt is equivalent to giving something. (source: chapter-1.2.2)
§267: Reducing the Remainder
Once the signs are changed, the remainder may be abbreviated by combining like terms exactly as in addition. (source: chapter-1.2.2)
§268: Difference of the Sum and Difference
Euler shows that subtracting the difference of two quantities from their sum gives twice the lesser:
(source: chapter-1.2.2)
§269: Examples
The chapter works through examples involving powers and radicals, such as:
(source: chapter-1.2.2)
These examples reinforce that subtraction of compound quantities is a sign-change operation followed by reduction. (source: chapter-1.2.2)