In the later years of his life, the Italian scientist Galileo Galilei (1564 – 1642), wrote about his experiments with motion in a treatise called Dialogues Concerning Two New Sciences. Here he described his wonderful discovery that distances covered in consecutive equal time intervals by balls rolling down inclined planes are proportional to the successive odd positive integers. Thus, if  denotes the distance covered during the  time interval, then
Where  is a suitable constant? Galileo determined that the constant  depends only on the incline and not on the mass of the ball or the material of which it is composed. He reasoned that the same result should hold for freely falling bodies if air resistance is neglected.
            Galileo’s discovery of a mathematical model for uniformly accelerated motion is considered to have been of the beginning of the science of dynamics. A mathematical model is an equation or a set of equations in which letter represent real world quantities for instance, distance and time intervals in the equation. A mathematical model is often an idealization of the real world situation if supposedly describes. For instance, Galileo’s mathematical model assumes a perfectly smooth inclined plane, no friction, and no air resistance, and it neglects the rotational movement of inertia of the balls as they rolled down the inclined plane.

            We study mathematical models related to the concept of variation.
Thus, the letters  and  in the following definition could represent variable quantities in a mathematical model.

Direct Variation:
            If there is a constant  such that
                                                           
holds for all values of , we say that  is directly proportional to  or that  varies directly as  (or with). The constant is called the constant of proportionality or the constant of variation.

For instance, if an automobile is moving at a constant speed of miles per hour, then the distance  it travels in  hours is given by the formula
                                                           
Here  and  are variables in the mathematical model  describing the motion of the automobile; the distance  is directly proportional to the time, and the constant of proportionality is miles per hour.
            Suppose that  and  are variables and that
                                                            ,
So that  is directly proportional to  with  as the constant of proportionality.
Note that  when . Also, if , we have
                                                            ,
So the fraction, or ratio, maintains the constant value as  varies through nonzero values. Thus, if a nonzero value of  and the corresponding value of  are known, the value of can be determined.

Example:
            Suppose that the rate at which impulses are transmitted along a nerve fiber is directly proportional to the diameter  of the fiber. Given that  meters per second when  micrometers:
            (a) Write a formula that express in terms of.
            (b) Find   if  micrometers.
Solution:
            Since  is directly proportional to , we have
                                                            ,
Where  is the constant of proportionality.
(a) For, we have
                                                             ,
Substituting meters per second and  micrometers into this equation, we find that . Therefore, for all values of , we have
                                                            .
(b) When  micrometers, we have
                                                meter per second.
If we say that  varies directly as the square of, we mean that  and  are related by an equation of the form
                                                             
for some constant of variation . For instance, the formula  for the area of a circle states that  is directly proportional to the square of the radius, and that is the constant of variation. More Generally:

Variation with the nth Power:
            If there are constants  and  such that
                                                           
Holds for all values of, we say that  is directly proportional to the nth power of  or that   varies directly as (or with)  the nth power of .

Example:
            The formula for the volume of a sphere states that is directly proportional to the cube of the radius and the constant of variation is . Find the volume of a sphere of radius  centimeters and round off your answer appropriately.

Solution:
             cubic centimeters
Assuming that the radius  centimeters is correct to three significant digits; we have rounded off our answer to three significant digits.

            The situation in which  is proportional to the reciprocal of  arises so frequently that if is given a special name, inverse variation.

Inverse Variation:
            If there is a constant  such that
                                                           
For all nonzero values of , then we say that  is inversely proportional to  or that  varies inversely as  (or with).

            Naturally, if  for some constant  and some positive rational number , we say that  is inversely proportional to the nth power of .

Example:
            A company is informed by its shopkeeper that the number  of units of certain item sold per month seems to be inversely proportional to the quantity, where  is the selling price per unit in dollars. Suppose that units per month are sold when the price is per unit. How many units per month would be sold if its price were dropped to  per unit?
Solution:
            Since  is inversely proportional to , there is a constant  such that
                                                           
We know that  when , so we have
                                               
Or
                                               
Therefore, the formula
                                                           
gives  in terms of . For  dollars, we have
                                                ,
So units per month would be sold at a price of per unit.