Page:A Source Book in Mathematics.djvu/659

Multiplication:

or by placing

Yet it must be noticed that the converse is not always given by a differential equation, except with a certain caution, of which [I shall speak] elsewhere.

Next, division:

(or ${\displaystyle z}$ being placed equal to ${\displaystyle {\frac {v}{y}}}$)

Until this sign may be correctly written, whenever in the calculus its differential is simply substituted for the letter, the same sign is of course to be used, and ${\displaystyle +dx}$ [is] to be written for ${\displaystyle +x}$, and ${\displaystyle -dx}$ [is] to written for ${\displaystyle -x}$, as is apparent from the addition and subtraction done just above; but when an exact value is sought, or when the relation of the ${\displaystyle z}$ to ${\displaystyle x}$ is considered, then [it is necessary] to show whether the value of the ${\displaystyle dz}$ is a positive quantity, or less than nothing, or as I should say, negative; as will happen later, when the tangent ${\displaystyle ZE}$ is drawn from the point ${\displaystyle Z}$, not toward ${\displaystyle }$A, but in the opposite direction or below ${\displaystyle X}$, that is, when the ordinates ${\displaystyle z}$ decrease with the increasing abscissas ${\displaystyle x}$. And because the ordinates ${\displaystyle v}$ sometimes increase, sometimes decrease, ${\displaystyle dv}$ will be sometimes a positive, sometimes a negative quantity; and in the former case the tangent ${\displaystyle IVIB}$ is drawn toward ${\displaystyle A}$, in the latter ${\displaystyle 2V2B}$ is drawn in the opposite direction. Yet neither happens in the intermediate [position] at ${\displaystyle M}$, at which moment the ${\displaystyle v}$’s neither increase nor decrease, but are at rest; and therefore ${\displaystyle dy}$ becomes equal to ${\displaystyle 0}$, where nothing represents a quantity [which] may be either positive or negative, for ${\displaystyle +0}$ equals ${\displaystyle -0}$; and at that place the ${\displaystyle v}$, obviously the ordinate ${\displaystyle LM}$, is maxi- mum (or if the convexity turns toward the axis, minimum) and the tangent to the curve at ${\displaystyle M}$ is drawn neither above ${\displaystyle X}$, where it approaches the axis in the direction of ${\displaystyle A}$, nor below ${\displaystyle X}$ in the contrary direction, but is parallel to the axis. If ${\displaystyle dv}$ is infinite