I have seen the formulation "ordinary differential equations in more than two variables" in two books.
Is this terminology really correct?
The books:
The first book is Elements of Partial Differential Equations by Ian N. Sneddon. Chapter 1 is called Ordinary differential equations in more than two variables (the chapter at Google Books) :
In this chapter we shall discuss the propertires of ordinary differential equations in more than two variables.
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... if in the rectangular Cartesian coordinates $(x,y,z)$ of a point in three-dimensional space are connected by a single relation of the type $$ f(x,y,z)=0 \tag{1}$$ the point lies on a surface.
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To demonstrate this generally we suppose a point $(x,y,z)$ satisfying equation $(1)$. Then any increments $(\delta x, \delta y, \delta z)$ in $(x,y,z)$ are related by the equation $$\frac{\partial f}{\partial x}\delta x + \frac{\partial f}{\partial y}\delta y +\frac{\partial f}{\partial z}\delta z=0$$ so that two of them can be chose arbitrarily.
The second book is A Treatise on Differential Equations by George Boole and chapter XII is called Ordinary differential equations in more than two variables (the chapter at Google Books):
The class of equations which we shall first consider in this Chapter, is represented by the typical form, $$ P\, dx + Q\, dy + R\, dz = 0, $$ $P$, $Q$ and $R$ being functions of the variables $x$, $y$, $z$; and it is usually termed a total differential equation of the first order with three variables.
I'm confused over the terminology of using "ordinary" and "many variables" in the same sentence. Isn't a ordinary differential equation always one variable and if more than one variable it's instead a partial differential equation?