I have been interested in the following paper ("On systems of linear indeterminate equations and congruences" by
Henry J. Stephen Smith,
Philosophical Transactions of the Royal Society of London, Vol. 151 (1861), pp. 293-326) [JSTOR open link](https://www.jstor.org/stable/108738?seq=1).


On page 300, the author states "if we apply Euler's method for the resolution of indeterminate equations to the system (16.)" The latter is a system of $n$ linear equations in $n+m$ variables, with no constants. I am not following what he means by Euler's method in this context. Does someone know what it means?



What is Euler's method in linear algebra?
Euler's method is described, for example, by James Fogo in [Linear indeterminate problems](https://www.jstor.org/stable/27957030?seq=1). This applies to systems of equations where there are more unknowns than there are equations, and a solution can be found by restricting the solutions to integers. The method was published by Euler in his book [Elements of Algebra](https://en.wikipedia.org/wiki/Elements_of_Algebra) (1770). For the historical context, see [The historical background of a famous indeterminate problem and some teaching perspectives.](https://www.researchgate.net/publication/358022392_The_Historical_Background_of_a_Famous_Indeterminate_Problem_and_Some_Teaching_Perspectives)


Here is an example described by Euler, for the case of one equation with two unknowns: express the two unknowns in terms of a
single auxiliary variable and then use the integer condition to restrict that single variable.


![](https://i.stack.imgur.com/ZFbDL.png)
![](https://i.stack.imgur.com/fqMRg.png)
[page 312 of Euler's Algebra](https://books.google.nl/books?id=X8yv0sj4_1YC), describing the *Regula Caeci* ("blind man's rule") also known as the [The Rule of False Position.](https://en.wikipedia.org/wiki/Regula_falsi)  

 I'm not sure why this name is appropriate here; also note that the English translation from 1822 reproduced above is corrupted, for "Position, or The Rule of False" read "or The Rule of False Position"


