_The Uses of Points._--The sole use of points is to facilitate the employment of the principle of Convergence to Simplicity. By this principle some simple relations in appropriate circumstances become true, when objects are considered which are sufficiently restricted in time or in space. The introduction of points enables this principle to be carried through to its ideal limit. For example, suppose _g_ (_a_, _b_, _c_) represents some statement concerning three enclosure-objects, _a_, _b_, _c_, which may be true if the objects are sufficiently restricted in extent. Let _A_, _B_, _C_ be three given points, then we define _g_ (_A_, _B_, _C_) to mean that _whatever_ three enclosure-objects _a_, _b_, _c_ are chosen, such that _a_ is a member of _A_, _b_ of _B_, and _c_ of _C_, it is _always possible_ to find three other members of _A_, _B_, _C_, namely, _x_ a member of _A_, _y_ of _B_, and _z_ of _C_, such that _aEx_, _bEy_, _cEz_, and _g_ (_x_, _y_, _z_). So by going far enough down in the tail-ends of _A_, _B_, _C_ we can always secure three objects _x_, _y_, _z_ for which _g_ (_x_, _y_, _z_) is true. For example, let _g_ (_A_, _B_, _C_) mean "_A_, _B_, _C_ are three points in a linear row." This must be construed to mean that whatever three objects _a_, _b_, _c_ we choose, members of _A_, _B_, _C_ respectively, we can always find three objects _x_, _y_, _z_, also members of _A_, _B_, _C_ respectively, and such that _a_ encloses _x_, _b_ encloses _y_, _c_ encloses _z_, and also such that _x_, _y_, _z_ are in a linear row. Sometimes a double convergence is necessary, namely, a convergence of conditions as well as a convergence of objects. For example, consider the statement, "the points _A_ and _B_ are two feet apart." Now, the exact statement "two feet apart" does not apply to objects. For objects _x_ and _y_ we must substitute the statement, "the distance between _x_ and _y_ lies between the limits (2 ± _e_) feet." Here _e_ is some number, less than two, which we have chosen for this statement. Then the points _A_ and _B_ are two feet apart; if, _however we choose the number e_, whatever enclosure-objects _a_ and _b_, members of _A_ and _B_ respectively, we consider, we can always find enclosure-objects _x_ and _y_, members of _A_ and _B_ respectively, such that _a_ encloses _x_ and _b_ encloses _y_, and also such that the distance between _x_ and _y_ lies between the limits (2 ± _e_) feet. It is evident, since _e_ can be chosen as small as we please, that this statement exactly expresses the condition that _A_ and _B_ are two feet apart.