Page 7 - Foundations Of Differential Calculus

vi Preface

of time the shot is in the air. Unless the same cannon is used throughout
the experiment, we must also bring into our calculations the length of the
barrel and the weight of the shot. Here, we will not consider variations in the
cannon or the shot, lest we become entailed in very complicated questions.
Hence, if we always keep the same quantity of powder, the elevation of
the barrel will vary continuously with the distance traveled and the shot’s
duration of time in the air. In this case, the amount of powder, or the
force of the explosion, will be the constant quantity. The elevation of the
barrel, the distance traveled, and the time in the air should be the variable
quantities. If for each degree of elevation we were to define these things,
so that they may be noted for future reference, the changes in distance
and duration of the flight arise from all of the different elevations. There
is another question: Suppose the elevation of the barrel is kept the same,
but the quantity of powder is continuously changed. Then the changes that
occur in the flight need to be defined. In this case, the elevation will be the
constant, while the quantity of powder, the distance, and duration are the
variable quantities. Hence, it is clear that when the question is changed, the
quantities that are constant and those that are variables need to be noted.
At the same time, it must be understood from this that in this business
the thing that requires the most attention is how the variable quantities
depend on each other. When one variable changes, the others necessarily
are changed. For example, in the former case considered, the quantity of
powder remains the same, and the elevation is changed; then the distance
and duration of the flight are changed. Hence, the distance and duration
are variables that depend on the elevation; if this changes, then the others
also change at the same time. In the latter case, the distance and duration
depend on the quantity of charge of powder, so that a change in the charge
must result in certain changes in the other variables.
Those quantities that depend on others in this way, namely, those that
undergo a change when others change, are called functions of these quanti-
ties. This definition applies rather widely and includes all ways in which one
quantity can be determined by others. Hence, if x designates the variable
quantity, all other quantities that in any way depend on x or are determined
2
by it are called its functions. Examples are x , the square of x, or any other
powers of x, and indeed, even quantities that are composed with these pow-
ers in any way, even transcendentals, in general, whatever depends on x in
such a way that when x increases or decreases, the function changes. From
this fact there arises a question; namely, if the quantity x is increased or
decreased, by how much is the function changed, whether it increases or
decreases? For the more simple cases, this question is easily answered. If
2
the quantity x is increased by the quantity ω, its square x receives an
2
2
increase of 2xω + ω . Hence, the increase in x is to the increase of x as ω
2
is to 2xω + ω , that is, as 1 is to 2x + ω. In a similar way, we consider the
ratio of the increase of x to the increase or decrease that any function of x