Katz 7.2.4: 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. (p.255)
Katz 7.2.5: Umar Al-Khayyami's c.1070 conic solution of the cubic
,
with sketches. (pp.260-261)
Katz 7.4.3: Al-Baghdadi's c.1000 proof that between any two rational
magnitudes (he uses
and
)
there exist infinitely many irrational magnitudes. (pp.272-273)
Dunham Chap. 6: Great Theorem and Proof: Cardano's 1545 Solution of the
Depressed Cubic
.
[Corresponds roughly to Katz 9.3.1, pp.362-363.]
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
.
Katz 9.3.2: Bombelli's 1560 proof that the Cardano solution
to
the cubic
really
equals
(using his "wild thought"). (p.367)
Katz 10.3.1: Regiomontanus (Johannes Müller) Theorem II-1, The Law of Sines and his 1463 proof (p.400)
Katz 10.5.2: Galileo's 1638 proof of his theorem on projectile motion (based on experiment and geometry). (p.424)
Katz 11.1.1: Fermat's 1637 straight line Theorem and proof. (p.435)
Katz 11.3.2: Pascal's 1654 proof of his solution to the de Mere problem on The Division of Stakes, Theorem (pp.455-456), using induction, the arithmetical triangle, and his two "principles" (from p.451, bottom), as on handout.
Katz 12.1.1: Fermat's 1636 technique for drawing a tangent to a curve using his "adequality" argument. (p.471)
Katz 12.1.2: Descartes' 1637 Circle Method for finding a normal line to a
curve, as illustrated for the curve
(p.473).
Katz 12.2.3: Fermat's 1636 derivation of the area under the "higher parabola"
,
from
to
,
using Roberval's inequality (p.481).
Katz 12.2.5: Roberval's 1637 proof that the area under half an arch of the
cycloid is
times
the area of the generating circle (pp 489-490), as presented in class.
Katz 12.2.6:
-
Gregory of St. Vincent's 1647 Theorem and proof about a relationship between certain points under the hyperbola
and
certain areas under that curve. (p.491)
-
The derivation of de Sarasa's 1649 observation about how this relates to the logarithmic product rule. (pp.491-492)
Katz 12.5.2: Newton's 1671 derivation of the infinite series for
.
(pp.508-509)
Katz 12.5.6: Newton's 1684 proof in "De Motu" of Theorem 1 (Kepler's 2nd Law) (pp.516-517)
Dunham Chap. 7: Great Theorem and Proof: Newton's 1670 Approximation of
.
Katz 12.6.2: Leibniz' 1686 proof of the Transmutation Theorem (pp.525-526)
Dunham Chap. 8: Great Theorem and Proof: Johann Bernoulli's 1689 proof that
the harmonic series
diverges.