The hyperbolic law of cosines is nearly a century old result that has sprung from the soil of Einstein velocity addition law that Einstein introduced in his 1905 paper [5, 23] that founded the special theory of relativity. It was established by Sommerfeld (1868-1951) in 1909 [3] in terms of hyperbolic trigonometric functions as a consequence of Einstein’s velocity addition of relativistically admissible velocities. Soon after, Vari?ak(1865-1942) established in 1912 [68] the interpretation of Sommerfeld’s consequence in the hyperbolic geometry of Bolyai and Lobachevski. Vari?ak’s interpretation marks the first uncovered link between Einstein’s velocity addition law and hyperbolic geometry. This thesis is based on the works of two mathematicians, Helmut Karzel and Abraham Ungar which open a new perspective and new way in the relation between hyperbolic geometry and special relativity. This approach had profound similarities with the conventional approach to vector space in Euclidean geometry. These analogies allow us to utilize their knowledge of Euclidean geometry and Newtonian physics to gain of hyperbolic geometry and special relativity. The gyrovectors that give rise to gyrovector spaces are hyperbolic vectors that allow Einstein’s velocity addition to be presented as gyrovector addition. In 1924 Vari?ak had to admit to his chagrin that the adaption of vector algebra for use in hyperbolic space was just not feasible [69], as noted by Walter in [73]. Accordingly, the introduction of vector algebra into hyperbolic geometry offered in [49] was noted by Walter in [74]. In fact, hyperbolic vectors (that is, gyrovectors) are presented in [48] as equivalence Let c be the vacuum speed of light, and let e the c-ball of all relativistically admissible velocities of material particles. Einstein addition in the c-ball of is given by the equation.