To begin with Simple Differential Equations of the first order. Suppose, for facility of geometrical illustration, we consider the equation $y\frac{dy}{dx}=a$, or the equation $y\frac{dx}{dy}=b$, of which the former, translated geometrically, indicates that the subnormal of a plane-curve (to be found) is constant, and the latter indicates that the subtangent is constant. The algebraical solutions are easily found: in each, there is a constant, and is not defined by it; a constant of that class described (perhaps improperly) by the term ``arbitrary,'' but which really means ``not yet determined, but enabling us by proper determination of its value so to fix the value of $x$ corresponding to a given value of $y$ that we can adjust the solution to some specific condition.''
-The reader is requested to observe that instead of the term ``arbitrary constant,'' we shall always use the term ``undetermined constant.''
+The reader is requested to observe that instead of the term ``arbitrary constant,'' we shall always use the term ``undetermined constant.''
+If we treat the geometrical translation of the differential equation by a geometrical translation of the differential equation by a geometrical process, always drawing a normal or tangent so as to make the subnormal or subtangent constant, then drawing by means of it a small portion of the curve, then repeating the process, \&c., we may produce a polygon which will approach to the strict solution, with smaller errors (by taking the sides small enough) than any small quantity that can be assigned. In each case, however, a starting point is necessary.
+In both ways of treating the problems these conditions manifestly hold:
+ \item The curve, in each case, is one definite curve.
+ \item The curve, in each case, is a continuous curve, expressed by the same equation through its whole extent.
+ \item Even if there be isolated points or curves, still the same one equation defines the whole.
+ \item It is necessary to introduce on undetermined quantity, enabling us to adjust the curve to a specific condition given by special considerations: that undetermined quantity is however a simple constant.
+Let us now consider a Simple Differential Equation of the second order: such for instance as is given by this problem, ``To find the curve in which the radius of curvature is a function, given in form, of the ordinate.'' here the algebraic solution gives a formula requiring two undetermined quantities, still simple constants: the equivalent geometrical treatment shews that we require two elements, (as for instance, the value of $x$ and the inclination of the tangent, for some one value of $y$). But all the conditions hold which are mentioned in Article 5: the only difference being that, instead of \emph{one} undetermined quantity, a simple constant, there must now be \emph{two} undetermined quantities, simple constants.
+The conditions which will be found to hold in the solutions of Partial Differential Equations differy very remarkably from these.