Wilson (1901)

Wilson's "Vector analysis" [58] was, historically, one of the most influential textbooks on vector analysis in the early 20th century. It was based on the lectures of Gibbs, and it helped to popularize all aspects of what we now understand as vector algebra — including the notion of a right-handed coordinate system. He begins simply:

By the $X$-, $Y$-, or $Z$-axis the positive half of that axis is meant. The $X\,Y$-plane means the plane which contains the $X$- and $Y$-axis, i.e., the plane $z=0$.

Now, he gets to the heart of it:

In one case (Fig. 8, first part) the $Z$-axis lies upon that side of the $X Y$-plane on which rotation through a right angle from the $X$-axis to the $Y$-axis appears counterclockwise or positive according to the convention adopted in Trigonometry. This relation may be stated in another form. If the $X$-axis be directed to the right and the $Y$-axis vertically, the $Z$-axis will be directed toward the observer. Or if the $X$-axis point toward the observer and the $Y$-axis to the right, the $Z$-axis will point upward. Still another method of statement is common in mathematical physics and engineering. If a right-handed screw be turned from the $X$-axis to the $Y$-axis it will advance along the (positive) $Z$-axis. Such a system of axes is called right-handed, positive, or counterclockwise.

This is followed by a footnote giving another description that is more common now:

A convenient right-handed system and one which is always available consists of the thumb, first finger, and second finger of the right hand. If the thumb and first finger be stretched out from the palm perpendicular to each other, and if the second finger be bent over toward the palm at right angles to first finger, a right-handed system is formed by the fingers taken in the order thumb, first finger, second finger.

I've left out some more discussion; in the above, he is just drawing the distinction between left- and right-handed systems. But eventually, he gets to the final statement of his choice:

In this book the right-handed or counterclockwise system will be invariably employed.

Finally, he makes the important connection between these vectors and Cartesian coordinates:

Definition: The three letters $𝐢$, $𝐣$, $𝐤$ will be reserved to denote three vectors of unit length drawn respectively in the directions of the $X$-, $Y$-, and $Z$-axes of a right-handed rectangular system.

In terms of these vectors, any vector may be expressed as

\[𝐫 = x 𝐢 + y 𝐣 + z 𝐤.\]

The coefficients $x$, $y$, $z$ are the ordinary Cartesian coördinates of the terminus of $𝐫$ if its origin be situated at the origin of coördinates. The components of $𝐫$ parallel to the $X$-, $Y$-, and $Z$-axes are respectively

\[x 𝐢, y 𝐣, z 𝐤.\]

The rotations about $𝐢$ from $𝐣$ to $𝐤$, about $𝐣$ from $𝐤$ to $𝐢$, and about $𝐤$ from $𝐢$ to $𝐣$ are all positive.

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