METU Department of Mathematics Math 304 History of Math Concepts Spring 2013

Midterm 2

May 7 2013, 17:40 – 18:50 120 points in 16 questions

Name and the student number:

Q 1. (4 pts) Which discoveries caused a priority conflict between Legendre and Gauss ? When these discoveries were made ? Why did they lead to a conflict ?

Q 2. (8 pts) When did Ruffini, Abel and Galois sent their manuscripts concerning non-existence of a solution to the equations of gegree ≥ 5 ? To whom did they send and what was the result ? What is the difference between the results achieved by these mathematicians ?

Q 3. (7 pts) Which mathematicians are related to the discovery of non-Euclidean Geometry ? Who made a decisive breakthrough, and when these discovery was published ? Why this even is considered as a revolutionary event in Geometry ?

Q 4. (6 pts) Who worked in the fields of Analytic, Synthetic, and Differential geometry in the end of 18th and beginning of 19th century (give names for each of these three fields separately) ? How did mathematicians distinguish the analytic and the synthetic geometry in 19th century, what was the difference ?

Q 5. (5 pts) Who first proposed the following mathematical terms:

determinant group matrix Jacobian Toplogy

Q 6. (12 pts)Who developed the following theories and concepts.

Matrix multiplication and the inverse matrices. Theory of convergence of power series. Geometric meaning of the complex numbers. Theory of congruences modulo n. The modern definition of a function Homogeneous coordinates for lines. Theory of Heat. Theory of gravitational collapse and black holes. Stability of Solar system. Cubic residues. Vector algebra.

Q 7. (8 pts) Give the names of mathematicians working in the following fields in the end of 18the and beginning of 19th centuries.

Theory of elliptic functions. Celestial mechanics. Complex Analysis. Mathematical Physics, namely, theory of electricity and magnetism.

Q 8. (12 pts) Give a formula or explain the meaning of the following the number of points which determines a conic curve the heat equation Laplace equation. Pfaffian Möbius transformation Riemann surface Q 9. (30pts) What were the mathematical interests of the following scientists ? Describe their famous results, give the names of their works and years of publication.

Lagrange Laplace Legendre Cauchy Gauss Abel Jacobi Dirichlet Riemann Cayley Q 10. (4 pts) Who was the author of the best in 19th century exposition of Euclid’s Geometry ? When it was written and how it was called ?

Q 11. (4 pts) What was the historical importance of the Habilitation Lecture of Riemann ? When was it delivered, what was its content ?

Q 12. (6 pts) Find the quadratic residues modulo 11. Apply the quadratic reciprocity Law to determine if 11 is a quadratic residue modulo 37.

Q 13. (6 pts) Formulate the Fifth Postulate of Euclid. Give three other claims equivalent to it, in particular, the Playfair axiom.

Q 14. (3 pts) Give formulas for the area of a spherical and a hyperbolic triangle.

Q 16. (5 pts) Give a formula for the zeta-function. What did Riemann prove concerning its convergence and its zeroes ? Formulate the Riemann Conjecture.