"'(8). This passage from the one system to the other may be said to consist mainly in the consideration of the variable plane of an angle. If, after tracing the equilateral triangle ABC on a card, which at first rests on a horizontal table, we then lift up that card, with the figure traced thereon, and lay it on a sloping desk, the triangle in its new position takes also a new aspect; it faces a different region of space, and may be conceived to look at, or be looked at by, a new point of the heavens, which is not now the vertical point (or ), as before. This new aspect of the figure, or of the plane (or desk) on which it is now situated, is the new circumstance introduced, in the transition from Double to Quadruple Algebra. And in fact it is easy to see that this new circumstance, of the varied position of the figure, namely, of the triangle, or simply (if we choose) of the ANGLE ABC, requires the consideration of two new numerical elements. For we have now two new questions to answer, or two new things to determine: namely, 1st, the slope of the desk (or inclination of the plane), suppose forty-five degrees, conducting to a first new number, 45 ; and 2nd, the direction of the edge (or, technically speaking, the line of the nodes), where that slope meets the table, and which may deviate from the line of north and south by any other number of degrees, suppose seventy, giving thus a second new number, in this case 70.'"