Explore spherical and elliptic geometries in this comprehensive mathematics lecture. Begin with an introduction to longitude and latitude on a sphere, and discover the historical connections between spherical geometry and astronomy dating back to ancient Greek, Indian, and Arab civilizations. Examine the relationship between spherical geometry and Euclid's 5 postulates, and learn how elliptic geometry emerges from identifying antipodal points on a sphere. Delve into measurement techniques on spherical surfaces using angles and additional functions to define distances. Investigate Ptolemy's tables of chords and subsequent Indian and Arab work on sine tables. Conclude with an exploration of Menelaus' theorem, which originated in the spherical context using chords.