Ch 1.4.1 — Of the Solution of Problems in General
Summary: Euler defines algebra as the science of finding unknown quantities from known ones, introduces the notion of an equation, states the principles governing valid equation transformations, and classifies equations by the highest power of the unknown.
Sources: chapter-1.4.1
Last updated: 2026-05-03
Algebra defined (§563–565)
Euler opens Section IV by restatting the central goal of algebra:
The science which teaches how to determine unknown quantities by means of those that are known. (§563)
Every earlier chapter illustrates this: addition finds the sum of given numbers; subtraction finds the difference; the work on roots, powers, and proportions all reduce to uncovering something unknown from something known.
Setting up an equation (§566–568)
When a problem is posed, the standard strategy is:
- Assign a letter (usually from the end of the alphabet, such as ) to the unknown.
- Express all quantities in the problem in terms of that letter and the given data.
- Identify a condition the problem imposes — an equality — between two such expressions.
- Write down the resulting equation and solve it.
Worked example (§567): Twenty people, men and women, dine together. Each man pays 8 shillings, each woman 7 shillings, the total bill is 145 shillings. Let the number of men be ; then the number of women is . The total cost is
Setting this equal to 145 gives , so there are 5 men and 15 women.
A second example (§568) produces a fractional equation , anticipating the more complex techniques of the next chapter.
Number of equations required (§570)
When several unknown quantities appear, the problem is determinate only if it supplies as many independent equations as there are unknowns. Fewer equations leave the problem indeterminate; redundant equations must be consistent.
Valid transformations of equations (§571)
Two sides of an equation remain equal under any of the following operations applied to both sides simultaneously:
- Adding or subtracting the same quantity
- Multiplying or dividing by the same non-zero number
- Raising both sides to the same power
- Extracting roots of the same degree
- Taking logarithms
These are the only tools Euler uses throughout Section IV.
Classification of equations by degree (§572)
| Class | Condition | Example |
|---|---|---|
| Simple (first degree) | appears only to the first power | |
| Second degree | appears after reduction | |
| Third degree | appears after reduction |
The present section treats all three in order, starting with simple equations in ch1.4.2-resolution-simple-equations.