Chapter 1 Supplemental Notes and Hints
General instructions:
Whenever it asks for 10 repeated measurements, do 5.
Section I.
The "nomagraph" is on page 10. Let's say you get a measurement of 15
centimeters on your crossstaff with the medium sight. In order to
calculate the angle, take a straightedge and connect 15 centimeters
with the medium sight width, follow the straightedge to the angle
line. The number that hits the angle line is what your measured angle
is. (In the case of 15 centimeters and a medium sight, you can see by
the dashed line that angle is 20 degrees.)
Section II.
A radian is a unit for measuring an angle, just like a degree is. To
visual a radian, first imagine a circle. Then imagine a line going
1/360th the way around the circle. That is a degree.
Now imagine a piece of string connecting the center of the circle with
the edge of the circle. The length of that piece of string is the
radius of the circle. Wrap that piece of string around the circle.
It goes a certain distance around. That angle is a radian.
Now, from basic geometry, you know that the circumference of a circle
is 2 Pi times the radius. This means that you can wrap your imaginary
piece of string, 2 Pi times the circle. In other words, there are 2
Pi radians in a full circle. Since you know that there are a 360 degrees
in a circle, you should be able to set
360 degrees = 2 Pi radians = 1 full circle
and calculate the number of degrees in a radian as well as the number of
radians in a degree.
Question 1. This question asks you to calculate the angular size of the
moon in units of minutes, knowing that the size of the moon is half a
degree. You do not have to do any measurements.
Section IV.
The formula here uses what is called "sigma notation." The best way of
explaining sigma notation is to give an example of it......
means add up all of the numbers from one to five. In other words
means take each number between one and four, square it, and then add
them together.
Question: What is the difference between
and
Answer: The first means, take each number from zero to nine, square it
and then add them all together. The second means take each number
from zero to nine, add them all together, and then square it.
In order to explain the standard deviation formula, it is necessary to
explain one more piece of notation. Suppose you have fifty numbers,
and you want to express the idea of the sum of these random numbers.
You could do it the hard way and give each of the numbers it's own
variable, but there is a better way, you could call the first number
x , the second number x , the third number x , and so forth. That
way you can express the sum of your fifty numbers as:
which using our summation rules is the same thing as:
Now to "interpret" the standard deviation formula.
The heart of the formula is the expression.
Now this is really shorthand for this:
where x_i is your i-th measurement. What this translates into is.
In other words, take each of your measurments, subtract the average,
square the difference, and then add them all together. Once you
have calculated
You can calculate this
by dividing by the number of measurements minus one, and square
rooting the whole thing.
Section VI.
Section IX. Using your hand as a measuring tool.
Your hand can serve as a tool for measuring angles when there is
nothing else available. Hold your arm out. The different parts of
your hand cover a different number of degrees.
The problem is that these numbers are good only for the average
person. No one is really average, and you need to find out how big
YOUR hands are in order to use this method.
To do this use the following procedure. Take a meterstick and put it
a fixed distance away from you. Put out your hand, make a fist, and
see how many centimeters your fist appears to cover.
For example, I did this experiment and found that my fist seems to cover
centimeters of a meterstick that's ten meters away. What this means is
that my extended fist of an angular size such that an object which is
Now from the formula on page 7, we know that
So if my fist covers an object which is cm long and ten meters away, then
it's angular size is