Did Brook Taylor develop his formula also in many variables by himself? I was wondering whether Brook Taylor was also familiar with analysis in many variables at that time. I found no information about it online.
Greetings Eu2718

After some googling, I believe the first one who wrote down the Taylor-like expansion for two-variable functions is sure Joseph-Louis Lagrange. Also my hunch was him for his contributions in Lagrange multipliers and calculus of variations.
In his book Théorie des fonctions analytiques page 92, he considered the function $f(x+i,y+o)$, and expanded it fixing $i$ and $o$ respectively. Then in page 93 he reached the following expansion:

$$
f(x+i,y+o) =  f(x,y)+if'(x,y)+of_{\prime}(x,y) 
\\
+ \frac{i^2}{2}f''(x,y) + io f'_{\prime}(x,y)+ \frac{o^2}{2}f_{\prime\prime}(x,y) + \frac{i^3}{2\cdot 3} f'''(x,y) 
\\
+ \frac{i^2o}{2} f''_{\prime}(x,y) + \frac{io^2}{2} f'_{\prime\prime}(x,y) +  \frac{o^3}{2\cdot 3} f_{\prime\prime\prime}(x,y) + \& c. 
$$

Translating into modern mathematical notation, it is just 
$$
f(x+\delta x, y+\delta y) = f(x,y) + f_x(x,y) \delta x + f_y(x,y)\delta y
\\
+ \frac{1}{2!}f_{xx}(x,y)  (\delta x) ^2 +f_{xy}(x,y) \delta x\delta y  + \frac{1}{2!}f_{yy}(x,y)  (\delta y) ^2
+ \frac{1}{3!} f_{xxx}(x,y) (\delta x) ^3
\\
+ \frac{1}{2} f_{xxy}(x,y) (\delta x) ^2 \delta y+ \frac{1}{2} f_{xyy}(x,y)  \delta x   (\delta y)^2 +  \frac{1}{3!} f_{yyy}(x,y) (\delta y)^3 + 
\text{Error term}.$$
My French is almost amateur, but it seems Lagrange didn't mention the order of partial derivatives in his notation.
My investigation started with the quote in the entry Taylor's theorem of wikipedia:

Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1712. Yet, an explicit expression of the error was provided much later on by Joseph-Louis Lagrange. 

