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It seems that they author of those words in red is saying that, or something very close to that. Whether that is his meaning or not is one question. The question of whether Newton experimented with and understood the ideas underlying the Taylor series, is certainly answered as "yes". Newton was one of those very rare genius' on the level of Gauss, Hilbert, Einstein and Bohr. People with those gifts see way beyond what they formally prove and publish. Newton intuitively knew many many things that were later proved formally by mathematician's decades, and even centuries later. For example, Newton developed the basic principles of Calculus of Variations long before is was formalized by Euler and Lagrange, about 50-70 years later.
Could someone please comment on the following part? Thanks.
Taylor series is a kind of power series where coefficients are derivatives. A function f(x) is approximated using Taylor series around a fixed point, 'a', and as the function is evaluated at points away from 'a', error would increase. I'm assuming finite number of terms. In other words, Taylor series approximates a function locally around a fixed point. On the other hand, Fourier series or transform approximates a function over its defined domain and not around a fixed point. In other words, Fourier series or transform approximation is more of global. I understand comparing Taylor series and Fourier series is more like comparing oranges and apples but I'm trying to make a very general comparison between the two.