.. Welcome in the year 1643 for time-passengers ..

Isaac Newton (1643 - 1727);

Euclid

Isaac Newton was an English mathematician, physicist, alchemist, astronomer, and theologian. Along with Gottfried Leibniz (1646 - 1716), he is considered a co-inventor of infinitesimal calculus. It is important to mention that the French mathematician Pierre de Fermat (1601 - 1665) can be considered a pioneer of infinitesimal calculus. Newton's differential calculus is based on Fermat's work from 1636, titled Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum. Newton derived Johannes Kepler's (1571 - 1630) laws of planetary motion using his own mathematical formulation of gravity, conclusively establishing that the Sun is at the center of the solar system. He formulated the laws of motion, which were later "refined" by Albert Einstein's (1879 - 1955) theory of relativity. Newton's conclusion that the Earth is an oblate spheroid was subsequently proven correct. He is also regarded as a pioneer in the development of regression analysis, vector calculus, and the calculus of variations. In 1668, Newton built the first practical reflecting telescope, now known as the Newtonian telescope. This telescope used a primary concave mirror to gather light and a smaller mirror to reflect it (modern telescopes use lenses). His telescope could achieve a magnification of 30-40, was 15.2 cm long, and featured mirrors with diameters between 2.5 and 3.8 cm. Using this telescope, Newton observed the four Galilean moons - Io, Europa, Ganymede, and Callisto - which had originally been discovered by Galileo Galilei (1564-1642) in 1610 with his homemade telescope. Newton also observed the crescent phases of Venus and noted the flattening of Jupiter. The book "Philosophiae Naturalis Principia Mathematica", written in Latin by Newton, where he presents the laws of motion and his law of universal gravitation, can be described as one of the most important works in the history of science. It was published in 1687. This work was honored by renowned scientists such as Joseph-Louis Lagrange, Pierre-Simon Laplace, Alexis Clairaut, and others. His law of universal gravitation states that every particle attracts every other particle in the universe with a force proportional to the product of their masses (m₁ and m₂) and inversely proportional to the square of the distance R between their centers of mass.

In other words: F (gravitational force) = G × m₁ × m₂ / R² and where G is the gravitational constant (G ~ 6.6743 × 10^-11m³kg^-1s^-1).

Just to avoid confusion, Newton considered G to be merely a constant; he did not calculate its value. It was the experimental and theoretical physicist and chemist Henry Cavendish (1731-1810) who calculated G in 1798 using a torsion balance. Newton also authored two other books: "De Motu Corporum" (Dealing with motion in a resisting medium) and "De Mundi Systemate" (On the system of the world), which discusses the implications of gravitational laws in astronomy.

example 1: Newton's interpolation formulas ..

Let's look at the equation: f(x) = 2x⁷ - 15x⁶ + 3x³ - 100 = 0, the root is x₁ ~ 7.49909552207

Assuming f(x+h) is the exact root we approximate h by -f/f' (f is just a short form for f(x) and f' for f'(x))
So let's look at the following approximations for the value of h:

1^st approximation -f/f'
2^nd approximation -f/[f' - ff''/2f']
3^rd approximation -f/[f' - ff''/2f' + f'''f²/6f'²]

/derivative
dx:{-1_x*-1+c-til c:count x}
p1:dx p:2 -15 0 0 1 0 0 -100f
p2:2 dx/ p
p3:3 dx/ p

g1:{x-(x sv p)%x sv p1}
g2:{x-c0%c1-(x sv p2)*(c0:x sv p)%2*c1:x sv p1}
g3:{c0:x sv p;c1:x sv p1;x-c0%c1+(neg c0*(x sv p2)%2*c1)+c0*c0*(x sv p3)%6*c1*c1}

The more specialized our approximation function is, the more "erratic" the process can become if we rely on formulas blindly.

So lets take as start value x = 6
g1 arrives at x₁ after 46 iterations
g2 arrives at x₁ after 55 iterations
g3 arrives at x₁ after 4 iterations only (!!)

Next start value x = 1
g1 arrives at x₁ after 14 iterations
g2 arrives at x₁ after 14 iterations
g3 no convergence to x₁

example 2:

With the total differential df of function which is (in the 2-dim case) f_xdx + f_ydy we can also here define our linear correction value:

f_x(x - x_k) + f_y(y - y_k) + f(x_k,y_k) = 0 and where x_k and y_k are the initial values!

Let be:

f(a,b) = 2a³ - 6ab² - 3a²b + b³ + 4a + 3b + 1714 = 0
g(a,b) = 6a²b - 2b³ + a³ - 3ab² + 4b - 3a + 642 = 0

Btw: g_a = -f_b and g_b = f_a

V:-1f*(f[xk;yk],g[xk;yk]);
xyd:V mmu inv flip 0f+2 2#(fx[xk;yk];fy[xk;yk];gx[xk;yk];gy[xk;yk]);
xk:xk+d0:xyd 0;
yk:yk+d1:xyd 1;

For startvalue (2,2) we obtain 5 8, for (-3,0) we get -9.3104 0.2582 and for (4.-12) we get 4.3104 -8.2582 (decimals are rounded)

We solved the complex cubic equation: (2 + i)z³ + (4 - 3i)z + 1714 + 642i = 0

One other example of the Newton method is displayed separately --> see year 1880 Bairstow method

Go back to startpage