In all these meanings the notion of straightness is involved, and could we say in lieu of straightness first directness and then direction and holding fast in thought this sense of the word, make a noun of it, so that a direction would intend the same as a straightness and no more, it might obtain a useful geometric term and notion. To define it we might first define a line thus: A line is a space boundary that is indefinitely long but not otherwise of any extent. Then, a direction is a line such that between the points that bound any assigned parcel of it no copy of said parcel is possible. But direction purports to our author the second of the meanings above set forth, namely, the indefinite straight sense of the procession of a motion. Definite parcels of a direction thus understood are identical with vectors. Now the notion of straightness is after the notions of point and line the most fundamental one of geometry and the one which is altogether the most prominent and useful. It is the necessary means for any definition of a vector or of the notion which our author deems so important. As straightness is attributable only to lines and long things which a line may represent it makes no difference whether we define straightness or a straight line, but a masterful performance of this definition is absolutely necessary before the foundations of geometry can be abidingly certified. Our author defines a straight line thus: “A straight line is a continuous series of points extending from each of them in the same two directions.” What kind of a thing a continuous series of points may be we are not told but as a point is defined to be “a portion of matter so small that for the purpose in hand variations of positions within it may be neglected” we take it that a straight line is a continuous series of particles of matter. The “purpose in hand” in this case must of course be the purpose of geometry. In defining an angle our author first lays down that “The difference between two directions is called their _inclination_ to one another” and then “The measure of an inclination is called an _angle_.” Considering that it is the doctrine of the author that every straight line has two contrary directions the measure of whose inclination is an angle of one hundred and eighty degrees, we imagine a northeast southwest line cutting an east west line and wonder if the right hand upper angle is really two angles according to whether or not the directions both pass to the left or both pass to the right or pass one to the left and the other to the right.