The historical discussion of the contributions below from `Quaternions in Three Dimensions' may be downloaded here: (.pdf)
Caspar Wessel (St. Andrews, Wikipedia) 
Leonhard Euler (St. Andrews, Wikipedia) 
Euler's Study of the Axis of an Infinitesimal Rotation E177 (1752) (Euler Archive)
Use of a Cross Product Formula
Euler's 1770 Study of Rotation E407 (1770) (Euler Archive)
Coordinate Plane `Euler Angle' Parametrization of Solutions,
Euler's Study of the Axis of a Rotation E478 (1775) (Euler Archive)
det(R-1I)=0 condition stated but not proved
Constructive Geometric Proof of Existence of a Fixed Axis
Euler's Study of the Matrix of a Rotation with a specified Axis and Angle E479 (1775) (Euler Archive)
Euler's Formula Involving Cross Product Matrix
Olinde Rodrigues (St. Andrews, Wikipedia) 
Rodrigues' Study of the Rotation with a specified Axis and Angle (1840), First page
Rodrigues' Rotation Matrix Formula
Rodrigues' Axis-Angle Composition Formula (SU(2) a.k.a. Unit Quaternion Multiplication)
Sir William Rowan Hamilton (St. Andrews, Wikipedia) 
Hamilton's Introduction of the Non-Commutative Quaternion Algebra (1843)
Sir Arthur Cayley (St. Andrews, Wikipedia) 
Cayley's Observation of the Correspondence Between Hamilton's Algebra and Rodrigues' Composition Formula (1845)
Cayley's Introduction of the Non-Associative Octonion Algebra (1845)
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