Copernicus, De Revolutionibus orbius caelestium 1543
Kepler, Astronomia nova, Heidelberg, 1609
LAW 1: The orbit of a planet/comet about the Sun is an ellipse with the Sun's center of mass at one focus
LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time
LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor axes:
$T_a^2 / T_b^2 = R_a^3 / R_b^3$
1615, Nova stereometria doliorum vinariorum.
Cavalieri, 1634
Galileo, Dialogue Concerning the Two Chief Systems of the World 1630
Discorsi e dimostrazioni matematiche, intono a due nuove scienze Leiden 1638
Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth.
Descartes, Discours de la méthode; La Geometrie Leiden 1637
Newton De analysi per aequationes numero terminorum infinitas 1669
Rule 1: Area under $y=ax^{m/n}$ is $an/(n+m) x^{1+(m/n)}$
Rule 2: If $y$ is given by a sum of more terms (possibly infinite) then the area under $y$ is given by the sum of the areas of all the terms.
Rule 3: To calculate the area under a curve $f(x,y)=0$, expand $y$ as a sum of terms of the form $x^{m/n}$ and apply rule 1 and rule 2.
De methodis serierum et fluxionum 1670
1. Given the length continually (that is at every time), to find the speed of motion an any time.
2. Given the speed of motion continually, to find the length of the space described at
any time proposed.
Principia 1687
Leibniz, Nova Methodus pro Maximis et Minimis, itemque Tangentibus,.. 1684
Aspects of self-view:
1) Relation to authority.
2) Relation to experiment.
3) Mathematics and nature.
Algebra vs. Geometry
Level of generality: (Descartes) algebraic geometry in all degrees;
(Newton) all differentiable functions.
Foundations:
limits (prime and ultimate ratios);
fluxions / derivatives ;
geometrical atoms.
Euclid V. vs. the new ratios. Inhomogeneity. Functions.