8 MEASUREMENT:

c) A side of a triangle. Length  d) A cube. Volume 
e) The boundary  the faces  of a cube. Area
f) The edge of a cube. Length
g) A triangle. Area  h) A pyramid. Volume  
i) A sphere. Volume  j) The surface of a sphere. Area 
k) The equator of a sphere. Length
Measurement
There is one number, clearly, that must be associated with these strokes:
But is there one number that must be associated with this length, as
its measure?
No. It will depend on the unit of measure. For if we measure in inches, we will get one number, while if we measure in meters, we will get another. Unlike counting, measurements are not absolute.
How do we "measure" AB? We take a unit of measure  1 inch, 1 meter, 1 mile  and then name the ratio of AB to that unit.
Every measurement implies a ratio to the unit of measure. Every measurement implies a proportion.
A magnitude is to the unit magnitude (of the same kind) as
A number is to 1.
For if we say that AB is 3 meters, that means
AB : 1 meter = 3 : 1.
Problem 3. What proportion is implied by each of the following?
a) The length L is 5 miles.
a) L : 1 mile = 5 : 1
b) The length L is 7.62 cm.
b) L : 1 cm = 7.62 : 1
c) The weight W is 5½ pounds.
c) W : 1 pound = 5½ : 1
d) The area A is 2.71 square meters.
d) A : 1 square meter = 2.71 : 1
e) The volume V is .035 cubic centimeters.
e) V : 1 cc = .035 : 1
Problem 4. In the previous problem, each measurement is a rational number of units. Therefore, express each ratio as a ratio of natural numbers (Lesson 7.).
a) The length L is 5 miles. 5 : 1
b) The length L is 7.62 cm.
b) 7.62 : 1 = 762 : 100
c) The weight W is 5½ pounds.
c) 5½ : 1 = 11/2 : 1 = 11 : 2
d) The area A is 2.7 square meters.
d) 2.7 : 1 = 27 : 10
e) The volume V is .035 cc.
e) .035 : 1 = 35 : 1000
Problem 5.
a)  What is "the ratio of two natural numbers;" that is, what is the relationship that natural numbers have to one another? 
One number is either a multiple of another, a part of it, or parts of it. 
b)  Do you expect that magnitudes (of the same kind) will have the same ratio as two natural numbers? 
Do you?  
c)  In particular, if 1 centimeter is the unit of length, do you expect that every length will be a rational number of centimeters? 
Do you? 
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