First Geometric Interpretation of Negative and Complex Numbers

John Wallis (1616-1703), a contemporary of I. Newton, was the first to divest the notion ofnumber from its traditional association with quantity. As quantities neither negative or complex numbers make a lot of sense. The problem was not in denying their existence but in finding a suitable model: Fiction is a form in search of an interpretation [Dantzig, p. 205,Swetz, p. 481]. As Wallis wrote in Algebra (1673) [Smith, pp. 46-54],

... But it is also Impossible, that any Quantity (though not a Supposed Square) can be Negative. Since that is not possible that any Magnitude can be Less than Nothing, or any Number Fewer than None.Subsequently, Wallis interprets negative numbers as distances in the direction opposite to the positive, to the left of a given point. His geometric interpretation of complex numbers begins with the following diagram: where the circle has AC as a diameter and A plays the role of the origin. (Wallis still does not use the second, vertical axis.) He explains: If (for instance) Forward from A, I take AB = +b; and Forward from thence, BC = +c; (makingAC = +AB+BC = +b+c, the Diameter of a Circle:) Then is the Sine, or Mean Proportional BP = +bc. But if Backward from A, I takeAB = -b; and then Forward from that B, BC = +c; (makingAC = -AB+BC = -b+c, the Diameter of the Circle;) Then is the Tangent or Mean Proportional BP = -bc.

Spherical Trigonometry

Let a spherical triangle be drawn on the surface of a sphere of radius , centered at a point , with vertices , , and . The vectors from the center of the sphere to the vertices are therefore given by , , and . Now, the angular lengths of the sides of the triangle (in radians) are then , , and , and the actual arc lengths of the side are , , and . Explicitly,

			(1)
			(2)
			(3)

Now make use of , , and  to denote both the vertices themselves and the angles of the spherical triangle at these vertices, so that the dihedral angle between planes  and  is written , the dihedral angle between planes  and  is written , and the dihedral angle between planes  and  is written . (These angles are sometimes instead denoted , , ; e.g., Gellert et al. 1989)

Consider the dihedral angle  between planes  and , which can be calculated using the dot product of the normals to the planes. Assuming , the normals are given by cross products of the vectors to the vertices, so

			(4)
			(5)

However, using a well-known vector identity gives

			(6)
			(7)
			(8)