For Spun Silver -- Was Thomas Aquinas wrong?

[sillylionlove]

Triginomitry? Are you kidding me?? I think I last studied that sometime in 1988!!!!

Polymer might have laid it out Math but lets face it he could be right here next to me and I still would not get it.



I bow down to the Math gods

 

[PolymerBob]

So the sunrise, as seen from the ground, travels from point G to the ground below Michelle's tree in 30 seconds.

There are 3600 seconds in an hour, and 24 x 3600 or 86,400 seconds in a day. The Earth rotates 360 degrees in a day, or 360 deg / 86,400 sec = 0.00416667 degrees per second.

In 30 seconds, this is 0.00416667 x 30 = 0.125 degrees. This is the angle of the right triangle at the center of the Earth. ............... to be continued.

OK, let's advance one more step. The segment GC is the radius of the Earth; let's call that R. The segment CT is Connecticut ....... no, it's the same length with the addition of a 50 foot tree. So it's R + 50.

These are 2 sides of a right triangle. R + 50 is the hypotenuse and R is one of the other two sides. The angle between these sides is 0.125 degree.

Therefore ..... R / ( R + 50 ) = COS ( 0.125 deg )

Now who can bring us home?

 

[Mathman]

Oh no.....math word problems...the bain of my existence!!!

This is actually the whole thing in a nutshell. Everyone can do arithmetic, everyone can learn algebra, everyone can master the rudiments trignometry, calculus, etc., if such things perk his/her interest.

But NOBODY can do story problems, not even people who are a whiz at math.

How can this be?

Because math is easy and words are hard! :yes:

(Unfortunately, life is a story problem.) hwell:

 

[Joesitz (RIP)]

Wait, wait! Don't give up! Get all the B's and D's on one side of the equation:

B+3D = 32
2B+3D = 43. :agree:

Let;s see B = 32 - D, therefore

2 32 - D x6D = 43

64 -6D squared =43

Help! I think I'm winding up with an imaginary number.

Joe

 

[PolymerBob]

Let;s see B = 32 - D, therefore

2 32 - D x6D = 43

64 -6D squared =43

Help! I think I'm winding up with an imaginary number.

Joe

It's actually possible to write the problem to come up with a fractional number of birds and a negative number of dogs. :scratch:

 

[Mathman]

That is, it could have a fractional number of dogs (at worst, thirds of a dog), right? (Assuming the problem is exactly as stated, only changing the integers 32 and 43.)

 

[PolymerBob]

OK, let's advance one more step. The segment GC is the radius of the Earth; let's call that R. The segment CT is Connecticut ....... no, it's the same length with the addition of a 50 foot tree. So it's R + 50.

These are 2 sides of a right triangle. R + 50 is the hypotenuse and R is one of the other two sides. The angle between these sides is 0.125 degree.

Therefore ..... R / ( R + 50 ) = COS ( 0.125 deg )

Now who can bring us home?

So let's see what we have.

R / ( R + 50 ) = ( R + 50 - 50 ) / ( R + 50 ) =
( R + 50 ) / ( R + 50 ) - 50 ( R + 50 )

= 1 - 50 / ( R + 50 ) .......... so ......... 1 - 50 / ( R + 50 ) = cos( 0.125 deg )

Does this help????

 

[Tonichelle]

Because math is easy and words are hard! :yes:

um, no, words are EASY

math is just poopy!

 

[PolymerBob]

So let's see what we have.

R / ( R + 50 ) = ( R + 50 - 50 ) / ( R + 50 ) =
( R + 50 ) / ( R + 50 ) - 50 ( R + 50 )

= 1 - 50 / ( R + 50 ) .......... so ......... 1 - 50 / ( R + 50 ) = cos( 0.125 deg )

Does this help????

So to finish it up :

1 - 50 / ( R + 50 ) = cos( 0.125 deg ) .... so .... - 50 / ( R + 50 ) = cos( 0.125 deg ) - 1

or 50 / ( R + 50 ) = 1 - cos( 0.125 )

( R + 50 ) / 50 = 1 / ( 1 - cos( 0.125 ) )

R + 50 = 50 / ( 1 - cos( 0.125 ) )

R = 50 / ( 1 - cos( 0.125 ) ) - 50 = 50 / ( 1 - 0.99999762 ) - 50

= 50 / 0.00000238 - 50 = 21,009,919 feet

21,009,919 feet / 5280 ft/mile = 3979 miles = radius of the Earth

circumphrence = 2 x PI x radius = 2 x 3.142 x 3979 miles = 25,005 miles. QED

 

[Mathman]

See, everybody? Nothing to it!

Circumference = 2 x Pi x radius.

C = 2 pi R. Would Thomas Aquinas agree with Euclid that even God can’t draw a circle whose circumference and radius were not related in this way?

OK, this is the last one. (No, no, everybody, there's no use begging for more! )

Michelle weighs the Earth

(I mean, Michelle undertakes the task of measuring the mass of the Earth, not Michelle needs to go on a diet after her long time off the ice. )

Michelle, having successfully measured the circumference of the Earth with nothing but a pocket watch, next undertook to determine the earth’s mass. Fortunately, she had all the tools she needed -- an ordinary yardstick and a tunnel through the center of the earth. Using these resources, Michelle measured the diameter of the earth.

Now, Michelle knew from her Euclid that indeed the circumference should be related to the radius by the formula C = 2 x Pi x R. But when she actually made the measurement, the radius turned out to be 1.48 millimeters too long!

“Very interesting,” said Michelle. “The mass of the earth is 6,000,000,000,000,000,000,000 metric tons.”

 

[Mathman]

Hint: According to Einstein’s General Theory of Relativity, the presence of mass distorts the geometry of space. A formula due to Karl Schwarzchild quantifies this distortion in the case of a large spherical body like the earth. According to the Schwarzchild solution of the Einstein field equations, if a great circle is drawn around a spherical gravitating mass of uniform density, the observed radius ro of the circle is longer than the expected Euclidean value re. The exact formula is

Ro = Re sqrt[(c^2 Re / 2GM) arcsin [sqrt (2GM / c^2 Re)]​

where M is the mass of the sphere, c = 3.00 x 10^8 m/sec is the speed of light and G = 6.67 x 10^(-11) m^3 kg^(-1) s^(-2) is the universal gravitational constant.

Michelle also remembered her calculus. For very small values of the argument, the first two terms of the MacLaurin series for the acrsine function give an excellent approximation:

arcsin(t) = t + (1/6)t^3.​

(For t = sqrt(2GM / c^2 Re), G is tiny and c^2 is huge. In fact, 2GM is always much less than c^2 Re for ordinary material objects. If 2GM = c^2 Re, the object is, by definition, a black hole.)

This was enough for Michelle to calculate the mass of the earth.

 

[sillylionlove]

Oh my!! Is this written in a foreign language??? MY brain sure thinks so!!

:banging:

 

[PolymerBob]

OK, now try this one.

A room is occupied by only birds and dogs.

You look inside the room and count the heads and legs on these animals. You find that :

A) There are 32 more legs than heads.

B) There are 43 more legs than dogs.

How many birds and dogs are in the room?

By the way, any more progress on this one yet? :scratch:

 

[Mathman]

By the way, any more progress on this one yet? :scratch:

OK, let's see now. We got it down (post #34) to the system of equations

2B + 3D = 43
1B + 3D = 32

Or, in matrix notation:

A([B,D]) = [43, 32],

where A is the matrix of coefficients [[2 , 3] / [1, 3]] and where [B, D] and [43, 32] are column vectors.

The solution, then, is

[B, D] = A* [43, 32],

where A* is the inverse matrix, A* = [[1, -1] / [-1/3, 2/3]].

Thus [B, D] = [1 x 43 - 1 x 32, -1/3 x 43 + 2/3 x 32] = [11, 7].

 

[Mathman]

Oh my!! Is this written in a foreign language??? MY brain sure thinks so!!

Not many people are interested in following the details of these formulas, but the basic idea is what this thread is actually about.

In Thomas Aquinas' day, as in Euclid's, it was just taken for granted that the Euclidean geometry that we learn in high school really does describe the way things are in the real world.

The great scientific discoveries of the twentieth century overturned this. Physics is geometry. That is, physical entities distort the geometry of space away from the Euclidean model, and on the other hand, the new geometry causes physical objects to exhibit the behavior that we see.

This is a huge revolution in thought, inconceivable in Thomas Aquinas' time.

Edited to add:...proving -- if any proof is required -- that God is infinitely more subtle than all our theologians and scientists, after all.

 

[PolymerBob]

“Very interesting,” said Michelle. “The mass of the earth is 6,000,000,000,000,000,000,000 metric tons.”

By the way, you have to be very careful using the term "sextillion". People will think it's something pornographic.

 

[Tonichelle]

only those with crude minds.... let's try to stay out of the gutter, shall we?

 

[Dee4707]

only those with crude minds.... let's try to stay out of the gutter, shall we?

Why Toni,.......... oh...........you're just not any fun.

Dee

 

[PolymerBob]

Edited to add:...proving -- if any proof is required -- that God is infinitely more subtle than all our theologians and scientists, after all.

Perhaps. But there is, and always be, a huge difference between the Pythagorean theorem and God. You can prove the Pythagorean theorem to someone who doesn't believe it.

 

[Tonichelle]

I still say math is a religion and should be banned from public schools because they FORCE people to believe that the numbers do exactly what they say they do