The commutative law in arithmetic, for instance, states that adb and b+a, or ab and ba, are equal.
If the ray suffers one internal reflection at D, then it is readily seen that, if DB be the path of the reflected ray, the angle ADB equals 2r, i.e.
7 angled at C, ADB is the semicircle described on AB as diameter, AEB the circular arc described with centre C and radius CA= CB.
The arm CD turns on the axis C, and -.~ .- is jointed at D to the middle of the bar ADB, whose length is double I of that of CD, and one of whose - ends B is jointed to a slider, sliding in straight guides along the line FIG 113.
Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis of the bar ADB; and the direction of motion of A is at every instant perpendicular to EAthat is, along the straight line ACa.