The reasoning is not geometric, it's numerical. The simplest reasoning is that there is no way to evaluate all values of transcendental functions exactly. We may know some important information about them (e.g. that they are differentiable, what their derivatives are, specific values of this function), but we don't know ALL the important information...

Really the only calculations of numbers that can be done are addition, subtraction, multiplication, division and exponentiation. So for transcendental functions, the only way to evaluate them at any point is to find the correct combination of the five operations. Thus, a polynomial, which is a general combination of the five, is required.

So supposing that we have a function and we want to evaluate it at some point . The only way we can do this is if we know the correct combination of the five operations. So we will write down a general polynomial and hope to find the correct combination from there...

.

Notice that if we were to substitute into the equation, it would eliminate everything except . So

.

Now here is the problem - how do we find the other constants? We need an operation that will reduce exponents and remove the known constants. This operation is differentiation. So if is (infinitely) differentiable in a close enough neighbourhood to , we can take the derivative of both sides

.

Now if we let we can eliminate everything but ...

.

Differentiating both sides again gives

.

Letting we find

.

Differentiating both sides again gives

.

Letting we find

.

Are you starting to see a pattern here? If we continue on this way to evaluate all the constants, we find

and now if we wanted to evaluate the function at , we can substitute into the Taylor polynomial and simplify to whatever degree of accuracy we like. This is also the method that calculators use - they are programmed with Taylor Polynomials...