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Literal resolution of affected equations by Isaac Newton. (English) Zbl 1445.01005

In De analysis (1669) and De methodis (1671) Isaac Newton attempted the solution of an affected equation, meaning an algebraic equation \(f(x,y)=0\) which is not binomial. The solution is given by an infinite series in which two cases, whether \(x\) is close to \(0\) or whether \(x\) is sufficiently large, have to be considered. When \(x\) is close to \(0\) the series is now known as Puiseux series. The authors analyse Newton’s solution algorithm and prove that
1.
when \(f(0,c)=0\) and \(f_y(0,c) \neq 0\) Newton’s algorithm given in De analysi is well defined and converges asymptotically to the implicit function \(f(x,y)=0\),
2.
when \(x\) is sufficiently large an infinite series can be constructed by the algorithm in De analysi which converges asymptotically,
3.
when \((0,c)\) is a singular point of \(f(x,y)=0\) and an infinite series can be constructed by Newton’s diagram method from De methodis, then the series converges asymptotically to one of the branches of \(f(x,y)=0\).

MSC:

01A45 History of mathematics in the 17th century
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
58C15 Implicit function theorems; global Newton methods on manifolds

Biographic References:

Newton, Isaac
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