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1. The author here desires to say to the reader, that though the present chapter contains nothing new it is yet of unquestioned importance for a clear understanding of the chapters following.

2. If the lines $$A B, N O$$, for example, of a geometric figure are in different directions, and if not only their absolute lengths are considered, but their respective directions as well, it is evident that, though the lengths of these lines are equal, it cannot be said $$AB=NO$$.

3. By the expression $$A B = N O$$, in Linear Algebra and in the science of Quaternions also, it is understood that the length of $$A B$$ is equal to that of $$N O$$, and also that the direction of the line $$A B$$ is the same as that of $$N O$$, that is to say they are either on the same straight line or are parallel to each other in the same direction. But in Numerical Algebra it is the absolute equality of the lengths only of these lines which is understood.

4. In describing a line $$A B$$ for example, if we say the line $$A B$$ or simply $$A B$$ we mean the special line AB which has a determined direction and length. If we write the line $$\underline{A B}$$, or simply $$\underline{A B}$$, we mean that the lenghtlength [sic] alone is considered.

Sometimes I shall write $$N (A B)$$ for $$\underline{A B}$$ and $$N (\alpha)$$, for $$\underline{\alpha}$$. I shall also write $$N^2 (A B), N^2 (\alpha)$$ for $$N(A B) \times N(A B), N(\alpha) \times N(\alpha)$$, or for $$\underline{AB} \times \underline{AB}, \underline{\alpha} \times \underline{\alpha}$$. In these cases by the letter $$N$$ prefixed to a line will be meant the number of the abstract length of that line.

5. To represent the different lines of a figure with regard to their directions as well as lengths, Greek letters are often employed. For example, if $$\rho$$ is put for the line $$A B$$, so long as the problem is not changed, by this $$\rho$$ is understood the line $$A B$$ which by supposition has a determined length and direction.

6. It is obvious that the lines $$A B, B D$$, having a determined direction and length, the line $$A D$$ will also, necessarily have a determined direction and length; and if in departing from the point $$A$$ after having traced the line $$A B$$ in giving to it its direction and length, we trace $$B D$$ commencing at $$B$$, giving to it also its length and direction, the distance from $$A$$ to $$D$$ will represent the line $$A D$$ with its special length and direction.