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His Work On Math (Newton's Contribution)
Philip Chiu

Unlike the Principia where Newton spent years polishing his mathematical findings, my focus will be on Newton's earlier, less refined mathematical works; more specifically, I will discuss Newton's algorithm for finding subnormals and some of his proposed applications for them. Although Newton was not the first to use subnormals to find the tangents of curves, nor was he the first to find tangents to curves, his new method was more powerful and more general. [Westfall ] Newton states the problem as follows:

Haveing y tex2html_wrap174 nature of a crooked line expressed in Algebr: termes to find its axes, to determine it & describe it geometrically &c.

If fd=x, db=y, & y being perpendicular to x describes y tex2html_wrap174 crooked line w tex2html_wrap176 one of it extreames. Then reduce y tex2html_wrap174 equation (expressing y tex2html_wrap174 nature of y tex2html_wrap174 line) to one side soe y tex2html_wrap180 it be =0. Then find y tex2html_wrap174 perpendicular bc w tex2html_wrap174 is done by finding dc=v, for vv+yy=bc tex2html_wrap183 [Math 236]

Here is my interpretation of his proposed problem: Given an axes, Newton draws a perpendicular line down from a specific point on the algebraic curve to the axes. In this case, Newton refers to line db as the perpendicular from the point on the equation to the axes. The line that db is perpendicular to is the line containing fd. Given the length of that line db and an arbitrary length on the given axes, essentially the ``y'' and ``x'' variables, Newton claims to be able to find the subnormal. Graphically, Newton's subnormal is the remaining leg of a right triangle with segment db as the other leg. Once Newton found the value of the subnormal,he can calculate the slope of the normal through the given point and from there found the tangent or the perpendicular to the normal. To find the subnormal, Newton had the following algorithm:

In finding dc=v [subnormal] observe this rule. Multiply each terme of y tex2html_wrap174 equat: by so many units as x hath dimensions in y tex2html_wrap180 terme, divide it by x & multiply it by y for a Numerator. Againe multiply each terme of y tex2html_wrap174 equation by soe many units as y hath dimensions in each term & divide it by -y for a denom: in y tex2html_wrap174 valor of v. [Math 236]
Clarifying Newton's notations, y tex2html_wrap174 stands for the equation of the curve, and y tex2html_wrap180 stands for the equation where everything is moved to one side, i.e. when we ''reduce y tex2html_wrap174 equation to one side soe y tex2html_wrap180 it be =0.'' Unfortunately, Newton does not provide a proof for his algorithm. However, given his earlier analysis on the relation of particular axes to the equation of a curve. The inspiration for his tangent algorithm appears to stem from his research into the relationship between standardized perpendicular lines and subnormals. [Westfall 107]

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