Stewart Chapter 7: Cosimo's Stars

Table of Contents

1. Introduction
2. Shapelore Learning 14-2: Lengths in Right Threenooks
3. A Rapoport Probability Problem
4. Probability in California Geometry Classes
5. Probability and the Coronavirus
6. Music: "Roots," "Ghost of a Chance," "One Billion is Big"
7. Calculating the Cosmos Chapter 7: Cosimo's Stars
8. Lemay Chapter 8 Part 2: "Java Applet Basics"
9. More About Applets in Java
10. Conclusion

Introduction

We just finished Chapter 5 of our Stewart reading, so we're ought to be ready for Chapter 6 now. But this is one chapter that we already covered on the blog, as a preview in my November 15th post. And so it's safe to move on to Chapter 7.

I could, of course, just reblog Chapter 6 anyway so that the chapters are in order. But I promised that I would avoid excessive reblogging, and I wish to keep that promise for one more post. And so I'll post Chapter 7 today and then reblog Chapter 6 in my next post.

Shapelore Learning 14-2: Lengths in Right Threenooks

Lesson 14-2 of the U of Chicago text is called "Lengths in Right Triangles." You already know that the official Anglish word for "triangles" is threenooks. But in our Plain English, we'll be using the word threesides instead.

Our translation begins here:

Sometimes the two means in a splitting ("proportion") are worthsame ("equal"), as in 2/10 = 10/50.

The word "proportion" definitely needs to be replaced. The Anglish website suggests the words cleaving and splitting for "divide"/"division." While I will grandfather in the word "divide" and change "division" to "dividing," the suggested words can be used for "proportion" instead.

I sort of like splitting for "proportion," since it reminds me of the Side-Splitting Theorem of Lesson 12-10 (and the sides really are split in proportion in that theorem). It might be better, though, to use our grandfathered word "divide" instead. A proportion is a division equation, and we're already using the word worthlink for equation. So a proportion would be a divideworthlink.

When this happens, the number that appears twice is called a shapeloreful mean of the other two numbers. Above, the truth of the divideworthlink shows that 10 is a shapeloreful mean of 2 and 50.

Actually, I don't like using shapeloreful for "geometric mean" in this situation, even though Geometry itself is Shapelore. Many people wonder why, for example, the sequence 2, 10, 50, 250, and so on is called a "geometric sequence" when it has no apparent connection to Geometry. Our text argues that the presence of the geometric mean in today's lesson is the reason why -- but still, now's our opportunity to use a better word than "geometric" for sequences and means -- especially since the appearance of geometric sequences in Algebra I have no apparent connection to Geometry.

Geometric sequences and means have an obvious connection to multiplication -- whatever we multiply the first term by to get the second, we multiply the second term by to get the third. So let's call it a timesish mean. An arithmetic mean can be called a plusish mean.

Oh, and by the way, notice that while the word mean exists in Anglish, the use of this word to denote "middle" or "average" is French, and derives from the same Latin word as "median." But since the word mean has only four letters, I'm willing to grandfather in its French/Latin definition.

So here is our new definition:

Let a, b, and g be forward numbers. g is a timesful mean of a and b if and only if a/g = g/b.

The meaning provides a way of calculating the timesful mean of any pair of forward real numbers.

Actually, the suggested word for "real" in the sense of numbers is true, but again, since real has only four letters, we'll grandfather it in.

Here is our first example:

Find the timesful mean of 7 and 12.

Answer: Let g be the timesful mean. From the definition of timesful mean, 7/g = g/12.
Using the Means-Outers ("Extremes") Law, g^2 = 84. So g = +/-sqrt(84).

From the definition of timesful mean, g is forward; so the timesful mean of 7 and 12 is sqrt(84).

Check: sqrt(84) is about 9.17. Is 7/9.17 = 9.17/12? Yes, 0.7634 = 0.7642.

I know that in the Means-Extremes Property, "extremes" should definitely be changed to outers, but should we change "means" to inners here? Notice that there's a clear connection between "means" (vs. extremes) and "geometric mean" here -- if the means in a proportion are equal, then that is exactly the geometric mean. So there's a reason why we might keep "means" here.

On the other hand, this shows why we might also prefer the opposite approach -- use inners all the way through, including timesful inners and plusful inners. But I prefer means here.

Actually, this is one thing that annoys me about the Means-Extremes Property -- we never call the parts of a proportion "means" and "extremes" unless it's in the context of this theorem. I'll think I will keep it as the Means-Outers Law for another reason -- Inners-Outers Law might also confuse students with the "outer" and "inner" in FOIL.

From the definition of timesful mean, if a/g = g/b then g^2 = ab; thus g = sqrt(ab). This shows:

Timesful Mean Provedsaying:
The timesful mean of the forward numbers a and b is sqrt(ab).

The timesful mean sqrt(ab) is always a number between a and b. As you will explore in Question 21, it is always closer to the smaller of a and b.

You may recall that the average of two numbers is called their plusful mean. The plusful mean is exactly midway between the numbers. Both kinds of means have applications in arithmetic, algebra, and geometry (or scorelore, frothering, and shapelore).

This is where our text explains the origin of the term "geometric mean."

As usual, the small letters a and b are used for the lengths of the legs of a right threeside, and c for the longestside. The letter h is for the height to the longestside, which splits c into two lengths, x and y, so x + y = c.

Angle A = BCD since both equals 90 - Angle B. So Threesides I, II, and ABC are all right threesides and each inholds ("includes," which might be considered a mathematical term since the "included angle," or inheld angle, is mentioned for SAS) an angle worthsame in breadth to Angle A. Thus all three threesides are shapesame: Threeside ABC ~ CBD ~ ACD. Since the threesides are shapesame, matching sides are divideworthful ("proportional"). Now look for divideworthlinks where the same quantity (or worth) appears twice.

Consider threesides I and II. A length of a side in both is h. x/h = h/y.

For threeside I and the original (biggest) threeside, a is a side in both, and for threeside II and the original threeside, b is a side in both. x/a = a/c and y/b = b/c.

Thus the height h and the legs a and b are timesful means of other lengths. Almost everyone remembers these lengths by their positions in the original threeside, as stated in the following provedsaying:

Right Threeside Height Provedsaving:

In a right threeside,

a. the height to the longestside is the timesful mean of the liths ("segments") into which it divides the longestside; and

b. each leg is the timesful mean of the longestside and the lith of the longestside neighboring the leg.

In the shape above, h = sqrt(xy), a = sqrt(cx), and b = sqrt(cy).

Here is the second example:

CD is the height to the longestside of right threeside ABC, as shown below. If AD = 3, DB = 12, find CD, CA, and CB.

Answer: CD is the timesful mean of AD and BD. So CD = sqrt(3 * 12) = sqrt(36) = 6.

CA is the timesful mean of BA and DA. So CA = sqrt(3 * 15) = sqrt(45) = 3sqrt(5).

CB is the timesful mean of AD and BD. So CB = sqrt(15 * 12) = sqrt(180) = 6sqrt(5).

Let's try some of the questions now. Unfortunately, only the first three can be done without a diagram for you to see.

In 1 and 2, find the timesful mean of the given numbers to the nearest hundredth.

1. 2 and 50 (Answer: sqrt(100) = 10)

2. 9 and 12 (Answer: sqrt(108) = 10.39)

3. True or untrue/false? If g is the timesful mean of a and b, and a < b, then g is closer to a than b. (Answer: true)

By the way, this last statement is often all the Arithmetic-Geometric Mean Inequality. It often appears in college-level math proofs.

I think I will include one more problem here. Question #11 is a proof of the Pythagorean Theorem using the theorems of this lesson. This is, in fact, the only proof of Pythagoras that appears in the Common Core Standards:

11. Provide the missing reasons in this argument verifying the Pythagorish Provedsaying:

Findings                           Reasons

1. a is the timesful mean  1. Right Threeside Height Provedsaying

    of c and x,

    b is the timesful mean

    of c and y.

2. a=sqrt(cx), b=sqrt(cy) 2. Definition of timesful mean (meaning)

3. a^2 = cx, b^2 = cy       3. Timesing Law of Worthlink

4. a^2 + b^2 = cx + cy    4. Adding Law of Worthlink

5. a^2 + b^2 = c(x + y)   5. Dealing Law

6. x + y = c                      6. Betweenness Provedsaying

7. a^2 + b^2 = c^2           7. Standing-In Law of Worthlink

The name Dealing Law for "Distributive Property" appears at the Anglish website. The full name listed there is Dealing Law of Manifolding over Eking, where manifolding means "multiplication" and eking means "adding."

Actually, the word "manifold" has another meaning in mathematics -- I might have discussed it on the blog back when we were reading about the Poincare Conjecture. Instead, I've just been using "times" for multiplication (and times really is Anglish). Meanwhile, neither "add" nor "plus" is Anglish, and so it's right to use eke instead of those other words. But I've grandfathered in both "add" and "plus," so there's no need for eke in Plain English.

Earlier, I used swapping for "substitution." But the same webpage that mentions the Dealing Law also uses swapping for "commutative" -- Swapping Law of Eking. Actually, I do like swapping for commutative, and so I'll let standing-in stand in for "substitution" instead.

A Rapoport Probability Problem

There is still no Geometry this week on the Rapoport calendar. So I'll do today's problem only:

Today on her Daily Epsilon of Math 2020, Rebecca Rapoport writes:

An urn contains x white balls (x > 2) and 9 black balls. The chance that two balls chosen without replacement are white increases by exactly 2% when an extra white ball is added (i.e. is 1.02 times greater).

This is a probability question. Recall that under the California Common Core Standards, probability should be included as part of the Geometry course. Since this is a Common Core Geometry blog and I'm a Californian, I should actually post probability questions much more often than I do.

Since our urn contains x white balls and 9 black balls, the probability that a chosen ball is white is written as x/(x + 9). The second ball is chosen without replacement, so the probability that it's white now becomes (x - 1)/(x + 8).

First probability:    x    * x - 1

                            x + 9    x + 8

Then we are asked to find the probability if we had started with one more white ball. Again, we are choosing two white balls without replacement:

Second probability: x + 1 *    x   

                                x + 10   x + 9

The second probability is 1.02 times the first -- that is, the first probability times 1.02 equals the second probability:

1.02 *    x    * x - 1 = x + 1 *    x   

           x + 9    x + 8  x + 10   x + 9

We notice that the factor x/(x + 9) appears on both sides of the equation, so let's cancel it:

102 * x - 1 = x + 1

100    x + 8  x + 10

Cross-multiplying, we get:

102(x - 1)(x + 10) = 100(x + 1)(x + 8)

102(x^2 + 9x - 10) = 100(x^2 + 9x + 8)

102x^2 + 918x - 1020 = 100x^2 + 900x + 800

2x^2 + 18x - 1820 = 0

x^2 + 9x - 910 = 0

(x + 35)(x - 26) = 0

x = -35 or x = 26

Since there can't be a negative number of balls in the urn, the number of white balls must be 26 -- and of course, today's date is the 26th.

Probability in California Geometry Classes

I'm not quite sure whether today's Rapoport problem is really the type we'd use to begin teaching probability in our Geometry classes. After all, it involves some messy algebraic manipulation, including solving a quadratic equation. (Then again, since California is including probability in the Geometry classes, the students would have completed Algebra I. Therefore anything taught in Algebra I -- including quadratic equations -- would be fair game.)

Oh, and the need for that whole explanation about "increases by 2%" (i.e. is 1.02 times greater) is a bit convoluted. It's awkward indeed when we compare two quantities by using "percent greater" when the quantities can themselves be percentages. For example, 2% more than 50% isn't 52%, but 51%, just because 50% * 1.02 = 51%. The actual percentages in this problem are closer to 55% and 56%, but you get the idea.

(This came up in an episode of Mathnet -- part of Square One TV. George Frankly learns that at a certain insurance company, revenue is up 2%, yet claims are up 3%. George thinks that the difference between the two percentages is insignificant, until the president of the company reminds him that 3% is 50% more than 2%.)

Anyway, our first probability problems in Geometry class should be simpler than this one. Let's look more closely at the probability standards listed in the Common Core:

Understand independence and conditional probability and use them to interpret data

CCSS.MATH.CONTENT.HSS.CP.A.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

CCSS.MATH.CONTENT.HSS.CP.A.2
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

CCSS.MATH.CONTENT.HSS.CP.A.3
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

CCSS.MATH.CONTENT.HSS.CP.A.4
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

CCSS.MATH.CONTENT.HSS.CP.A.5
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

Use the rules of probability to compute probabilities of compound events.

CCSS.MATH.CONTENT.HSS.CP.B.6
Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.

CCSS.MATH.CONTENT.HSS.CP.B.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

CCSS.MATH.CONTENT.HSS.CP.B.8
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

CCSS.MATH.CONTENT.HSS.CP.B.9
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Use probability to evaluate outcomes of decisions

CCSS.MATH.CONTENT.HSS.MD.B.6
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

CCSS.MATH.CONTENT.HSS.MD.B.7
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

That's clearly a lot of standards. There are more standards listed under the MD strand, but I've only listed the standards specifically required for California Geometry.

So far, I've posted Geometry over the lifetime of this blog and ignored probability. But if I were to become a Geometry teacher, I'd have to cover those standards. So when exactly during the year would be a good time to teach probability?

A few years ago, I purchased an old Geometry text from a library book sale, published by HRW (Holt, Rinehart, and Winston). Chapter 5 of that text is called "Perimeter and Area" (corresponding roughly to Chapter 8 of the U of Chicago text), and Lesson 5.7 is called "Geometric Probability."

So here is a clear link between Geometry and probability. The classic example is the probability that a dart will hit the target -- a circle inscribed in a square dartboard. If we assume that this is the unit circle, then it has area pi -- and the square has a side length equal to the diameter of circle or 2, and hence its area is 4. Therefore the desired probability is the ratio of the area of the circle to that of the square, namely pi/4.

Anyway, the idea suggested here is to include probability with the area chapter. A California teacher using the HRW text might cover the first five chapters of the Geometry text, then continue with probability between Chapters 5 and 6, and then finish Chapters 6-12 of Geometry.

And if we were to follow this with the U of Chicago text, then we should cover probability right after Chapter 8 on area. Indeed, I could maintain the digit pattern by teaching probability in lieu of Chapter 9, which is currently a dead zone. It introduces 3D figures but doesn't give their surface areas or volumes, which don't appear until Chapter 10. (Indeed, on the blog I currently fill this dead zone with some old 3D proofs from Euclid.)

After Chapter 15 of the text might work as well -- instead of SBAC review, use that remaining time to teach probability. (We might even start after Lesson 15-3 on the Inscribed Angle Theorem -- the last lesson that's relevant to the SBAC.)

Notice that approximately one-fifth of the Geometry standards are on probability. Thus we ought to spend about one-fifth of the school year teaching those standards -- and a Geometry text with 15 chapters (like our U of Chicago text) would need to devote three chapters to probability.

We can also allot the necessary time by using the new Third Edition of the U of Chicago text, with only 14 chapters. Once we reach the Inscribed Angle Theorem midway through Chapter 14, and following the digit pattern, we should have about one-fifth of the year left for probability.

Again, if I were a Geometry teacher, before making a decision I would actually count the number of students who are juniors (i.e., SBAC test takers). If there are many of them, then I'd make sure that they get as much material on tested standards (including probability and Inscribed Angle Theorem) before the start of the SBAC testing window. This likely means that probability would be taught right after the area chapter, unless there is somehow extra time left after the Inscribed Angle Theorem. If on the other hand most of the Geometry students are in Grades 9-10, then I can safely wait until the end of the year for probability.

Of course, this plan ignores the coronavirus and its impact on the upcoming school year. As far as we know, there might not even be an SBAC this year, and thus no reason to use the test to determine what to teach.

And that takes us to:

Probability and the Coronavirus

The number of cases of and deaths due to the coronavirus are on the rise, even as many places are starting to reopen. It's starting to get me worried that schools may resort to full distance learning in the fall, thereby curtailing my chance to earn any money as a sub. So there's surely a virus-related probability that I'm concerned with -- the probability that schools will reopen face-to-face.

Of course, there's another probability that matters here -- the probability of catching the virus. This is the probability that determines whether schools will re-open in the fall.

There are two ways to think about this. The pro-reopening side often cites certain conditional probabilities (which are mentioned in the Common Core standards), such as:

P(dying of coronavirus)
P(dying of coronavirus | catching coronavirus)
P(dying of coronavirus | being under 80 years of age)

They argue that this last probability, rounded to the nearest percent, is 0%. Thus, the argument goes, businesses should not have closed at all (especially not due to local or state mandates). They argue that the economy is more important -- that the probability of dying of starvation due to not having a job is greater than that of dying of the coronavirus.

On the other hand, there are those who look at other probabilities, such as:

P(dying of coronavirus | having preexisting conditions)
P(dying of coronavirus | having a relative with preexisting conditions)

They have reason to think that their probability of dying doesn't round to zero. These people are on the side of keeping places closed.

Some people believe that reopening too soon is akin to murder. But others believe the opposite -- that keeping places closed is akin to murder. And never the twain shall meet -- there is no single plan that will satisfy both groups.

I've noticed that those who wish to reopen are often business owners -- either large businesses (such as Disney), or small businesses (such as barbers and so on). They might genuinely believe that their probability of dying of the virus rounds to 0. Those who wish to keep things closed are more likely to be employees. They fear that rushing them to work when conditions aren't safe might kill them -- raise their death probability away from 0.

Of course, this discussion can quickly turn political. I will keep today's post apolitical by emphasizing my biases here. And I'm definitely biased -- even though I'm an employee, it's a sub, a job that depends on face-to-face instruction being available. I'm biased on the side of reopening schools at least, since that's how I earn any money at all.

Along this same line of thinking, I don't mind keeping things closed until it's close to when the first day of school should be. If keeping places closed will lower the cases and deaths enough for leaders to feel safe reopening schools -- and for parents to feel safe sending their kids there -- then let's keep places closed.

On one hand, my "summer" will be twice as long as a normal summer -- instead of nearly three months, I'll have been away from the classroom for over five months. But on the other, it won't feel like much of a summer if places remain closed. As a Californian, I often go to the beach in the summer -- and if beaches must close due to the virus surge, then it won't feel as if I've truly enjoyed this summer at all.

But again, as much as I enjoy going to the beach, I enjoy earning money even more. If closing beaches lowers cases enough to raise the probability that schools reopen face-to-face so that I can earn some money, then I say, close the beaches!

So these are the probabilities that we're looking at more and more as it gets closer to the first day of school, by which tough decisions need to be made. I'll be discussing the virus and its impact on the upcoming school year more as the summer proceeds.

As we leave probability, know there's a good Anglish word for "probability" -- likelihood. Another word that works is "chance" -- it's short, but it's French. Meanwhile, the word "odds" is Norse -- not quite Anglish, but still Germanic. Therefore a Shapelore class that wishes to use Plain English can replace "probability" with either true Anglish likelihood or Plain English "chance." (I see that "odds" aren't quite the same as "probability," so maybe we should avoid "odds.")

Music: "Roots," "Ghost of a Chance," "One Billion is Big"

Here are the songs that I sang at the old charter school during Weeks 11-12 of that school year.

The first song is "Roots":

ROOTS

Square root of 1 is 1,
Hey, this is so much fun.
Square root of 4 is 2,
So here's what we should do.
Square root of 9 is 3,
So let's all come and see.
Square root of 16 is 4,
So please show us some more.
Square root of 25 is 5,
And now I feel so alive,
Square root of 36 is 6,
So now there ain't no more tricks.

Cube root of 1 is 1,
Hey, this is so much fun.
Cube root of 8 is 2,
So here's what we should do.
Cube root of 27 is 3,
So let's all come and see.
Cube root of 64 is 4,
So please show us some more.
Cube root of 125 is 5,
And now I feel so alive,
Cube root of 216 is 6,
So now there ain't no more tricks.

This is the only song that needs a new tune. My last song debuted on October 20th of that year, and so I suggested writing it in 20EDL. Well, this song debuted on October 25th. Odd-numbered EDL's are awkward since they'll lack the octave. I could use 24EDL (with the justification that I likely wrote this song the night before, on the 24th).

Most likely, I'll end up using 12EDL, but perhaps inserting extra notes that sound good (for example, if I use 12EDL but insert a minor sixth between Degrees 8 and 7, or a leading tone between Degrees 7 and 6, then I'm actually using Degrees 15 and 13 of the 24EDL scale.)

The other two songs I sang during this time were both Square One TV songs. The second song is "Ghost of a Chance" -- how fitting, considering that I wrote about probability earlier in this post:

Here are the lyrics, courtesy Barry Carter:

http://wordpress.barrycarter.org/index.php/2011/06/07/square-one-tv-more-lyrics/#.WBLLFforKUk

Ghost Of A Chance

Lead vocals by Cris Franco and Reg E. Cathey

One night on a Pizza Shack delivery
I walked into this spooky house
And just as I was yelling “Two with anchovies!”
The door slammed and the lights went out

Started shouting, “Someone owes me fourteen ten!”
And then I heard a creepy voice
Boy, you’ll never see the Pizza Shack again
Unless you make the proper choice

Probability
Don’t you mess with me
Help me make the most of a chance
Might be win or lose
Still I’ve got to choose
Long as there’s a ghost of a chance

There are four dusty bookcases right over there
One of them’s a secret door
Go ahead and try one of them, if you dare
Your chance is only one in four

Did a little eenie, meenie, miney, moe
Pulled the third with all my might
Probability of one-fourth is low
But lucky thing I got it right

Probability
Don’t you mess with me
Help me make the most of a chance
Might be win or lose
Still I’ve got to choose
Long as there’s a ghost of a chance

Walked in to a hallway full of rattlesnakes
Only five are real ones; forty-five are fakes
Chance is five in fifty that I’m gonna croak
Chances are you thought this was a lark, a joke!

Probability
Don’t you mess with me
Help me make the most of a chance
Might be win or lose
Still I’ve got to choose
Long as there’s a ghost of a chance

Found myself inside an old Egyptian tomb
Open up the mummy case
Behold the seven keys he clutches in the gloom
Three will let you blow this place

Three in seven chance to pick a key that fits
I picked one of the four that don’t
But now the probability becomes three-sixths;
Three will work and three still won’t

(laughs)

Son, you’ve earned your freedom; here are twelve ways out
Eleven lead you to your truck
But what about the one in twelve? My boy, don’t pout;
Good luck; here’s your fourteen bucks

Probability of one-twelfth is slim
The guy was finally being nice
But that’s the one he chose, and I’m so pleased for him
The pizza’s for my poltergeist

Probability
Don’t you mess with me
Help me make the most of a chance
Might be win or lose
Still I’ve got to choose
Long as there’s a ghost of a chance

Probability
Don’t you mess with me
Help me make the most of a chance
Might be win or lose
Still I’ve got to choose
Long as there’s a ghost of a chance

(fade out)

And here is the third song, "One Billion Is Big." For some reason, I sang this song on Tuesday, Thursday, and Friday of Week 14 (rather than one song on Tuesday and another Thursday-Friday, as I usually did):

ONE BILLION IS BIG -- by the Fat Boys

1st Verse:
Have you seen the headline? We did OK,
We sold a million records in just one day.
That's a thousand times a thousand sold,
That's plenty of vinyl, a million whole.
A million dollar bills reach for the sky,
Stack 'em about three hundred feet high.
A billion dollars is a thousand times more,
A lot more money than we bargained for.

Refrain:
One million is big,
One billion is bigger.
One thousand times one million,
That's one billion.

2nd Verse:
We're getting kinda hungry for our favorite food,
Hey, what do ya say? Are you in the mood?
Let's satisfy our special taste,
And get some lunch at the burger place.
See that sign? "One billion served!"
Beat box, that's a lot of hamburgers.
One thousand times, when ya order fries,
A million times one thousand apple pies.
(Repeat Refrain)

3rd Verse:
If we multiply one million by ten,
How close are we to one million then?
If we take a look, we will see,
We got a way to go, my friend Markie Dee.
If we multiply by one hundred this time,
Let's take a look, and we will find,
That we're not even halfway there,
We need a lot more to be a billionaire.
If we order one billion cheeseburgers,
And eat one million cheeseburgers,
It would be enough to knock us off our feet,
'Cause we'd still have almost one billion burgers to eat.
One million's not even one percent of one billion. Wow!
(Repeat Refrain)

Calculating the Cosmos Chapter 7: Cosimo's Stars

Chapter 7 of Ian Stewart's Calculating the Cosmos is called "Cosimi's Stars." As usual, it begins with a quote:

"Since it is up to me, the first discoverer, to name these new planets, I wish, in imitation of the great sages who placed the most excellent heroes of that age among the stars, to inscrive these with the name of the Most Serene Grand Duke [Cosimo II de' Medici, Grand Duke of Tuscany].

-- Galileo Galilei, Sidereus Nunchus

And the chapter proper begins:

"When Galileo first observed Jupiter through his new telescope, he noticed four tiny specks of light in orbit around the planet, so Jupiter had moons of its own."

This chapter is all about the so-called "stars," namely, the moons of other planets. After our planet, Mars is the next planet with moons -- namely Phobos and Deimos:

"Both bodies are irregular, and are probably captured asteroids -- or, possibly, a captured asteroids with a duck shape like comet 67P, recently shown to be two bodies that came gently together and stuck."

Here Stewart includes his first picture of the chapter. On the left are Galileo's records of the moons. On the right are the positions of Jupiter's moons as seens from Earth, forming sine curves.
As for the many moons of Saturn, we read:

"Planets with ring systems slowly accrete moons at the edge of the rings, and then 'spit them out' one by one, like water dripping from a tap."

Even an asteroid can have its own moon. The author's next picture is of asteroid Ida on the left and its tiny moon Dactyl on the right.

We read that Galileo once tried to measure longitude at sea by examining the motions of the planets and their moons in the sky:

"The main practical issue was to make observations from the deck of a ship as it tossed on the waves, and he worked on two devices for stabilizing a telescope."

The longitude problem wouldn't be solved for another century.

Returning to Jupiter's moons, three of them -- Io, Europa, and Ganymede -- are in resonance with each other, yet they never seemed to line up at the same time:

"Their orbital periods are roughly in the ratio 1:2:4, and in 1743 Pehr Wargentin, Director of the Stockholm Observatory, showed that this relationship becomes astonishingly accurate if it's reinterpreted correctly."

Wargentin showed that a possible triple conjunction of the three moons won't happen for at least a million years:

"The equation also implies a specific pattern to conjunction of these moons, which occur in a repeating pattern:

• Europa with Ganymede
• Io with Ganymede
• Io with Europa
• Io with Ganymede
• Io with Europa
• Io with Ganymede

"Laplace decided that Wargentin's formula couldn't be coincidence, so there must be a dynamical reason."

Stewart now shows a diagram of successive conjunctions of the three innermost moons of Jupiter: Io, Europa, and Ganymede (reading outwards), in the pattern listed above.

And Laplace showed that the resonance formula ultimate goes back to Newton's laws. In fact, the moons of the dwarf planet Pluto follow a pattern similar to the Jovian moons:

"There are five Styx/Hydra conjunctions and 3 Nix/Hydra conjunctions for every two Styx/Nix conjunctions. Europa, Ganymede, and Callisto all have icy surfaces."

Returning to Jupiter's moons, Stewart tells us that we suspect there to be liquid water under the icy surfaces, though it would be dozens of miles deep:

"Callisto's ocean probably lies the same distance under the ice, with an ocean 50-200 km deep."
The author shows us a picture of where one of these oceans might be -- Conamara Chaos on Europa, where the ice is irregular and jumbled.

And it's likely that Saturn's moon Enceladus has a similar ocean:

"The ice above is probably 30-40 km thick, and the ocean is 10 km deep -- more than the average for Earth's oceans. Seven of Saturn's moons orbit just beyond the edge of the planet's outer main ring, the A ring."

It's possible that these smaller moons, such as Janus are in resonance with that A ring, forming at the same time as the planet itself. In describing this theory, Stewart concludes the chapter as follows:


"However, a moonlet such as Janus should take no more than a hundred million years to drift outwards for the A ring to its current orbit, suggesting an alternative theory: both the rings and these moonlets appeared together when a larger moon passed inside the Roche limit and broke up, some tens of millions of years ago. The simulations reduce this period to between 1 and 10 million years: the authors say, 'Saturn's rings, like a mini protoplanetary disk, may be the last place where accretion was recently active in the solar system, some 10^6-10^7 years ago."

Lemay Chapter 8 Part 2: "Java Applet Basics"


Here is the link to today's lesson:

http://101.lv/learn/Java/ch8.htm

Lesson 8 of Laura Lemay's Teach Yourself Java in 21 Days! is called "Java Applet Basics." Here's where we left off:

Passing Parameters to Applets

With Java applications, you pass parameters to your main() routine by using arguments on the command line, or, for Macintoshes, in the Java Runner's dialog box. You can then parse those arguments inside the body of your class, and the application acts accordingly, based on the arguments it is given. Applets, however, don't have a command line. How do you pass in different arguments to an applet? Applets can get different input from the HTML file that contains the <APPLET> tag through the use of applet parameters.

And as I mentioned in my last post, this will be a problem. I found out that to my surprise, I have an applet viewer on my computer. But I don't see how to pass parameters to that viewer.

Well, let me try the examples anyway. The third listing of this chapter is like the first, but renamed:

Listing 8.3. The More Hello Again applet.

 1:import java.awt.Graphics;
 2:import java.awt.Font;
 3:import java.awt.Color;
 4:
 5:public class MoreHelloApplet extends java.applet.Applet {
 6:
 7:    Font f = new Font("TimesRoman", Font.BOLD, 36);
 8:
 9:    public void paint(Graphics g) {
10:        g.setFont(f);
11:        g.setColor(Color.red);
12:        g.drawString("Hello Again!", 5, 40);
13:    }
14:}

OK, so let me try out the next listing, which requires a parameter:

Listing 8.4. The MoreHelloApplet class.

 1:  import java.awt.Graphics;
 2:  import java.awt.Font;
 3:  import java.awt.Color;
 4:
 5:  public class MoreHelloApplet extends java.applet.Applet {
 6:
 7:     Font f = new Font("TimesRoman", Font.BOLD, 36);
 8:     String name;
 9:
10:     public void init() {
11:         name = getParameter("name");
12:         if (name == null)
13:             name = "Laura";
14:
15:         name = "Hello " + name + "!";
16:     }
17:
18:     public void paint(Graphics g) {
19:         g.setFont(f);
20:         g.setColor(Color.red);
21:         g.drawString(name, 5, 40);
22:     }
23: }

I tried out this one, and -- just as expected -- there's no way for me to input a parameter. And so my applet displays the default name, which is -- Laura. Yes, I see you, Laura Lemay -- always inserting your name into these programs as the default! (Of course, there's nothing stopping me, as the coder, from changing the default to my own name.)

The last two listings are the HTML code for this applet -- one of which supplies a value for this parameter, and the other of which doesn't. There's no reason for me to cut-and-paste these, since I don't have access to HTML code. That's why I'm stuck with "Hello Laura!" on my applet viewer.

There's not much else for me to say in today's Java section, since the only topic for today is this parameter feature that I'm unable to use.

More About Applets in Java

I do wish to write another applet, but once again, what I can do with applets is quite limited until I can get to the next Lemay chapter.

So here's what I did -- I combined Lemay's Date program from an earlier lesson with our simple applets to create an applet that shows the date:

import java.util.Date;
import java.awt.Graphics;
import java.awt.Font;
import java.awt.Color;

public class DateApplet extends java.applet.Applet {

Font f = new Font("TimesRoman", Font.BOLD, 36);

    public void paint(Graphics g) {
    Date d1, d2, d3;
        g.setFont(f);
        g.setColor(Color.red);
        
        d1 = new Date();
        g.drawString("Date 1: " + d1, 5, 40);

        d2 = new Date(71, 7, 1, 7, 30);
        g.drawString("Date 2: " + d2, 5, 80);

        d3 = new Date("April 3 1993 3:24 PM");
        g.drawString("Date 3: " + d3, 5, 120);
    }
}

I hope to come up with a more interesting applet next time.

Conclusion

The traditionalists have been quiet lately, so there's no need for me to squeeze in traditionalism at the bottom of the post.

And so the progress with my summer projects continues unabated. I'm hoping that my next post will be fairly soon.