<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Leonid_Meteor.jpg/235px-Leonid_Meteor.jpg" class="right">
At an outdoor evening party, many of you witness a shooting star. It's not the first time you've seen one but how vivid this one is makes you wonder about the phenomenon.
[[next]]
{(set: $cards to (a:"A ♥","K ♥","Q ♥","J ♥","10 ♥","9 ♥","8 ♥","7 ♥","6 ♥","5 ♥","4 ♥","3 ♥","2 ♥","A ♦","K ♦","Q ♦","J ♦","10 ♦","9 ♦","8 ♦","7 ♦","6 ♦","5 ♦","4 ♦","3 ♦","2 ♦","A ♠","K ♠","Q ♠","J ♠","10 ♠","9 ♠","8 ♠","7 ♠","6 ♠","5 ♠","4 ♠","3 ♠","2 ♠","A ♣","K ♣","Q ♣","J ♣","10 ♣","9 ♣","8 ♣","7 ♣","6 ♣","5 ♣","4 ♣","3 ♣","2 ♣"))
(set: $deck to (shuffled: ...$cards))
(set: $place to 1, $ball to false, $clock to false, $plank to false, $log to (a:))
(set: $v1 to true, $v2 to true, $v3 to true, $v4 to true, $v5 to true, $v6 to true, $v7 to true, $v8 to true)}<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Night_sky_near_Flagstaff_with_meteor.jpg/320px-Night_sky_near_Flagstaff_with_meteor.jpg" class="right">
"Surely it's a weather type phenomenon, like lightning," one person says.
"Maybe it's a rock falling so fast that friction heats it up," says another person. But the rock would have to travel very fast and somehow get up many miles into the sky. The party breaks up and you head home thinking about falling objects.
[[next|one]]<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/44/Johannes_Hevelius.PNG/192px-Johannes_Hevelius.PNG" class="right">
The next day you get a letter from Jan Hevelius, writing from Poland. "Greetings, I hope all is well with you. Perhaps you can help us with a discussion that still causes controversy. Aristotle taught that a rock falls because it seeks it's natural place, earth below. When Galileo sought to probe exactly how things fall, he ingeniuosly decided to slow the falling process down. If you drop a round metal ball, it falls too fast to time it with a clock. But let that same ball roll down an inclined plane and then we can time it's progress."
[[next|two]]<span class="right" style="background:white"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Inclined-plane-experiment.svg/320px-Inclined-plane-experiment.svg.png"></span>
Hevelius continues "Galileo claimed that the ball's velocity increased in a uniform way as it rolled down the inclined plane. It sped up in a predictable way. Specifically, after one second the ball traveled a short distance (depending on the steepness of the track). And then in the next second, the ball would travel 3 times as far, then 5 times, and so on."
"Some have argued that Galileo must have made a mistake, since Aristotle held that everything seeks it's proper place with heavier objects wanting to get there faster than lighter objects. They don't see what would cause the ball to keep moving faster. Or they're not sure the pattern he observed really happened that way."
[[next|three]]<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/76/Piano_inclinato_inv_1041_IF_21341.jpg/269px-Piano_inclinato_inv_1041_IF_21341.jpg" class="right">
"Please discuss this or even try to replicate the experiment. I know that Father Riccioli some years ago carefully timed the fall of objects from a high tower in Bologna, verifying Galileo's findings. My questions to you are: if each second the ball travels in an odd numbered sequence, does that always add up to a square number? And can we calculate the instantaneous velocity of the ball at say 2 seconds or 5 seconds after release?"
[[next|four]]<img src="https://upload.wikimedia.org/wikipedia/commons/2/21/Domino_-_2.png">
"I include this little sketch to suggest how to prove that 1 + 3 + ... + 2n - 1 = n<sup>2</sup> but I need some help to clarify it. For the instantaneous speed, I know Fermat did work with infinitesimals that might also be used."
[[next|main]]<span class="right2"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/25/Free-fall.gif/120px-Free-fall.gif"><br>"... if each second the ball travels in an odd numbered sequence, does that always add up to a square number? And can we calculate the instantaneous velocity of the ball at say 2 seconds or 5 seconds after release?"</span>
Choose who you wish to talk to. Try not to draw too much attention to yourselves.
[[Mexican attorney|attorney]]
[[Michoacán bronzesmith|bronze]]
[[German clockmaker|clock]]
[[Pai Gow player|pai gow]]
[[Italian printer|printer]]
[[Mayan sawyer|sawyer]]
[[Vicereine]]
Cards drawn: $log{(if: $v1 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v1 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/36/Pedro_Malo_De_Villavicencio.jpg/326px-Pedro_Malo_De_Villavicencio.jpg" class="right">
<h4>Rafael Ponze Borrego</h4>
"Have I heard of Pierre de Fermat, the French judge and mathematician? As it turns out, I have, although I dare say that very few other attorneys in this city could say the same."
"I also have an interest in mathematics and appreciate the fine distinctions in logic required to understand the more advanced kind. Fermat wrote of his concept of //adequality//, that is two numbers infinitely close to each other. In this way he could calculate instantaneous rates of change."
<img src="https://upload.wikimedia.org/wikipedia/commons/f/fa/Difference_quotient-chart.png" class="left">
"You find the difference quotient for the function you want to calculate the tangent for, do some algebra perhaps, then find the value infinitely close to that. I try my hand at these when I need some diversion between cases."
You don't want to take any more of the attorney's time and you thank him again.
[[next|main]]
Cards drawn: $log{(if: $v2 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v2 to false, $ball to true)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Iraq%2C_blacksmith_and_forge_LOC_matpc.13290.jpg/310px-Iraq%2C_blacksmith_and_forge_LOC_matpc.13290.jpg" class="right">
<h4>Tenoch Huerta</h4>
The vicereine recommended this bronzesmith from Michoacán. She appreciates his creftsmanship in various works she commissioned from him. You also know that he is Purépecha, known for their work in copper alloys, among other things.
"Ah, you ask about a bronze sphere. How many inches across?" he asks.
You reply that about 2 inches in diameter would be good. Does he have anything like that?
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Shiny_steel_ball.png/220px-Shiny_steel_ball.png" class="left">
"I have made smaller bearings but one that size would be possible." You explain that it needs to be very round and highly polished so that it rolls with as little friction as possible. "Since Dona Luisa sent you, I will start on it right away."
[[next|main]]
Cards drawn: $log
{(if: ($ball is true and $clock is true) and $plank is true)[<br>
[[Don Gongora|prof]] ]}{(if: $v3 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v3 to false, $clock to true)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Clocks%3B_composite_pendulum%2C_side_and_front_view_with_details_Wellcome_V0024418.jpg/289px-Clocks%3B_composite_pendulum%2C_side_and_front_view_with_details_Wellcome_V0024418.jpg" class="right">
<h4>Martin Zoller</h4>
You have been wanting to meet this clockmaker from Augsburg, in Germany. You heard that he has clocks that track hours, minutes, and seconds even.
"So you need a clock for timing experiments? Well, every time we move a clock, it requires careful calibration. So it would only make sense to do this where you could afford a long-term installation." Zoller explains.
You know that the expense would have to be covered by the viceroy or vicereine and suggest that they would be happy to have one of his fine clocks.
"Very well, I will get a pendulum clock ready."
[[next|main]]
Cards drawn: $log
{(if: ($ball is true and $clock is true) and $plank is true)[<br>
[[Don Gongora|prof]] ]}{(if: $v4 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v4 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/32/Drawing%2C_Three_Chinese_Men_Playing_Dominoes_in_a_Baxter_Street_Club-House%2C_New_York_City%2C_ca._1874_%28CH_18175141%29.jpg/320px-Drawing%2C_Three_Chinese_Men_Playing_Dominoes_in_a_Baxter_Street_Club-House%2C_New_York_City%2C_ca._1874_%28CH_18175141%29.jpg" class="right">
<h4>Gaspar Fernandes</h4>
The sketch by Hevelius jogged your memory of game pieces used in a Chinese game. It's a long shot, but maybe you can get some help making sense of the sketch.
"Hmm, it seems he's saying something about using these tiles not to play a game but standing them up in a row and then toppling them. Interesting."
<img src="https://upload.wikimedia.org/wikipedia/commons/7/7f/ChineseDominoes.jpg" class="left" width="200">
"Maybe he means that proving something can be done in two parts. First show that a beginning case works. That's like tipping over the first tile. The second part is showing that if one tile falls, the next tile falls. Therefore all the tiles fall."
The other players call Fernandes back to the game so you take your leave.
[[next|main]]
Cards drawn: $log{(if: $v5 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v5 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Printer_in_1568-ce.png/372px-Printer_in_1568-ce.png" class="right">
<h4>Juan Pablos (Giovanni Paoli)</h4>
Pablos takes a short break to talk with you. "Galileo? No, the censor won't allow us to import or print anything by Galileo. We could possibly get something from the Jesuit astronomer Riccioli you mentioned, but you would have to wait many months."
You ask about Fermat. He shakes his head. "Sorry, but maybe you could check with an attorney who privately has read him."
[[next|main]]
Cards drawn: $log{(if: $v6 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v6 to false, $plank to true)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Vienna_-_Carpenter_workshop_tools_-_0027.jpg/320px-Vienna_-_Carpenter_workshop_tools_-_0027.jpg" class="right">
<h4>Francisco Camal</h4>
Camal takes a short break from his woodworking to talk with you, brushing sawdust out of his hair. You explain that you need two long planks joined together to form a channel. You want the planks to be very smooth, polished even, so that there's as little friction as possible.
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/a/af/Swietenia_macrophylla_%2830680883066%29.jpg/192px-Swietenia_macrophylla_%2830680883066%29.jpg" class="left">
"Well then, what you want are planks from the mahogany tree. The hardwood has a close grain and can be made very smooth. Give me a few days to do it."
You thank him and head to your next stop.
[[next|main]]
Cards drawn: $log
{(if: ($ball is true and $clock is true) and $plank is true)[<br>
[[Don Gongora|prof]] ]}{(if: $v7 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v7 to false)}
<img src="https://miro.medium.com/max/1400/1*5YYKb99Aij55AKV77yC-vQ.jpeg" class="right" width="360">
<h4>María Luisa Manrique de Lara y Gonzaga</h4>
You find the vicereine talking with Sister Juana Ines de la Cruz.
"What an interesting letter you received," the vicereine says. "I propose that we try to recreate the experiment here. If you can collect the appropriate materials, then I will contact Don Góngora. He could help us analyze our findings and explain the mathematics."
You agree to look for a metal ball, clock, and material for a ramp.
[[next|main]]
Cards drawn: $log<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Don_Carlos_de_Sig%C3%BCenza_y_G%C3%B3ngora.jpg/201px-Don_Carlos_de_Sig%C3%BCenza_y_G%C3%B3ngora.jpg" class="right">
<h4>Carlos de Sigüenza y Góngora</h4>
Góngora helps you setup the apparatus for the experiment. You test it and adjust it until Góngora thinks it's ready to go.
"We should make multiple trials and record the data. Then we can analyze that and try to answer the questions of Hevelius" explains the math professor.
You carefully run the trials and record the data. You see some room for error but, in general, your data agrees well with Galileo and Riccioli. If the ball rolls some distance in one second, the next second it rolls 3 times as far and then 5 times as far, etc.
You ask "How can we show that 1 + 3 + 5 + ... + 2n - 1 = n<sup>2</sup>?"
"Consider the case n = 1" says Góngora. "2 times 1 minus 1 is equal to 1 squared." He pauses writing and you nod in understanding.
"Now let's assume that our statement is true for some value k. So 1 + 3 + 5 + ... 2k - 1 = k<sup>2</sup>. Let's add 2(k +1) - 1 to both sides. We get 1 + 3 + 5 + ... + 2(k + 1) - 1 = k<sup>2</sup> + 2k + 1. Once we factor the right hand side we get (k + 1)<sup>2</sup>." He pauses again and you nod after a little thought.
"So, since we showed our statement is true for the first case and that if it's true for one case it's true for the next case. Therefore, it's true in all cases."
Now you understand Hevelius' sketch.
[[next|main]]