Excerpt from Plane Trigonometry, Vol. 1: With the Use of Logarithms
3. Here, however, we have 'unconsciously introduced the idea of a line having a positive or negative value. Fora, we know, means + a; and, if in our reasonings wc should happen to arrive at some negative result as the value of a line, (if, for instance, it came out, at the end of a problem, that the value of a certain line was -b), "What would this (-) sign mean ?
"We shall show that it indicates contrariety of direction to some line which we have before represented by a positive algebraical symbol.
For let AB be a line, containing a units, which we y A c express algebraically by a or+a. Then, if from the end B we measure off in the direction BA a line BC containing b units, the length of the line A C will be a - b units, and, therefore, its algebraical expression will be a-b. Now, if b be less than a, this expression for A C will be positive, and C will also lie to the right of A, that is, AC will be measured in the same direction as AB. But, if b be greater than a, then C will fall to the left of A as at Cy and A C will be measured in the contrary direction to AB, its value in units being b-a ; while, at the same time, the expression a-b will become negative, and may be written - (b - a).
"We infer, then, that, when the algebraical expression for a line is negative, the (-) sign indicates contrariety of direction to a line which has been already assumed to be positive; and, conversely, that if a line drawn in any one direction from a given point, when represented algebraically, be assumed to be positive, then a line drawn from the same point in the opposite direction must be considered negative.
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