Taylor expansion was introduced in its currently well known form by Brook Taylor. Though the concept as this page says, has been formulated by James Gregory. Among his other works, Gregory established $$\theta = \tan \theta - \frac13 \tan^3 \theta + \frac15 \tan^5 \theta - \ldots$$ which can easily be established when $|\tan \theta|<1$.
I was now thinking, how did Gregory thought of this problem? what did he know back then? I will be happy with any historical note/book. Many thanks.