Section 1.1: Discovering the Generalized Binomial Expansion

Newton's mathematical contributions were extensive, but his journey began with the discovery of the generalized binomial expansion. This concept first emerged in a letter he penned to Gottfried Leibniz in 1676, referred to as epistola prior.

In Newton's original notation, the expansion is represented as follows:

Equation 1: Newton's binomial expansion using his notation.

Where A, B, C, D, etc., denote the preceding terms of the series.

The example Newton presented in his letter is:

Equation 2: The example given by Newton in his epistola.

Following this, the first two terms of the expansion can be derived, and the third term, influenced by the second term B, can be deduced analogously. This process can be continued indefinitely, resulting in:

Equation 3: Generalized result of the binomial expansion.

Transitioning to Modern Notation

We can express the binomial expansion using contemporary notation. By factoring out the P term from the left-hand side of Equation 1 and substituting A, B, C, D, etc., with their respective previous terms, we arrive at:

Equation 4: Modern representation of the binomial expansion.

How Newton Developed the Binomial Expansion

Newton built upon the prior work of mathematicians like John Wallis to extend the binomial expansion for integer exponents:

Equation 5: Known binomial expansions for integer exponents.

An insightful visualization of these coefficients is illustrated in the following figure.

Newton's ambition was to broaden these expansions to encompass non-integer exponents. Following the work of Bressoud, he organized the coefficients in a table, leaving blanks for the non-integer expansions.

Using observable patterns, Newton deduced that the values in the columns increase polynomially:

• The first column remains constant (degree zero polynomial).
• The second column grows linearly (degree one polynomial).
• The third column escalates quadratically (degree two polynomial).

Based on this observed pattern, Newton hypothesized that the fourth column would increase as a cubic polynomial. His findings allowed him to fill in the empty cells of the table.

The overall expression can be summarized as:

Equation 6: General expression for the binomial expansion.

As noted by Whiteside, although Newton's interpolation methods were seemingly valid, they lacked rigorous proof, which was a notable flaw in his mathematical approach.

Section 1.2: Deriving the Power Series of the Sine Function

Newton's derivation of the sine function's power series represented a pinnacle in his mathematical prowess. This accomplishment was detailed in his 1669 manuscript titled "On Analysis by Equations with an Infinite Number of Terms."

The necessary elements to grasp Newton's derivation are captured in the figure depicting the equation for a quadrant of a circle with unit radius:

Equation 7: Function describing a quadrant of a unit circle.

To find x (the sine), we begin with similar triangles and deduce relationships based on the properties of the unit circle.

Now, as Newton applied his binomial expansion to the relevant equation, the inversion process yielded the power series for the sine function:

Equation 8: Power series expansion for the sine function.

This approach, while different from modern methods such as Taylor's series, showcases the unique evolution of mathematical thought.