Al-Karaji's c.1000 theorem (for
)
that the sum of the cubes equals the square of the sum, & his "inductive
style" proof with picture.
Umar Al-Khayyami's c.1070 conic solution of the cubic
,
with sketches.
Al-Baghdadi's c.1000 proof that between any two rational magnitudes (he uses
and
)
there exist infinitely many irrational magnitudes.
Dunham Chap. 6: Great Theorem and Proof: Cardano's 1545 Solution of the
Depressed Cubic
.
Be able to solve a cubic in the manner of Cardano: Start with a general cubic
,
depress it (meaning substitute
),
then use Cardano's formulas to find a solution for
,
and finally solve for
.
Bombelli's 1560 proof that the Cardano solution
to
the cubic
really
equals
(using his "wild thought").
Regiomontanus (Johannes Müller) Theorem II-1, The Law of Sines and his 1463 proof.
Galileo's 1638 proof of his theorem on projectile motion (based on experiment and geometry).
Fermat's 1636 technique for drawing a tangent to a curve using his "adequality" argument.
Fermat's 1636 derivation of the area under the "higher parabola"
,
from
to
,
using Roberval's inequality.
Two connected results:
-
Gregory of St. Vincent's 1647 Theorem and proof about a relationship between certain points under the hyperbola
and
certain areas under that curve.
-
The derivation of de Sarasa's 1649 observation about how this relates to the logarithmic product rule.
Newton's 1684 proof in "De Motu" of Theorem 1 (Kepler's 2nd Law).
Dunham Chap. 7: Great Theorem and Proof: Newton's 1670 Approximation of
.
Dunham Chap. 8: Great Theorem and Proof: Johann Bernoulli's 1689 proof that
the harmonic series
diverges.
Simpson's 1737 derivation of the rule
using Roger Cotes' 1716 proof.
Dunham Chap. 9: Great Theorem and Proof: Euler's 1734 Proof that
.