<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Jean_Dominique_Cassini_1625_AGE_V08_1801.jpg/173px-Jean_Dominique_Cassini_1625_AGE_V08_1801.jpg" class="right">
You received a letter from Jean Dominique Cassini, the Royal Astronomer of King Louis XIV.
"We eventually were able to see the new reflecting telescope made by the Isaac Newton, of Cambridge. Our group here in Paris were intrigued by the idea of not using lenses to capture and focus starlight but a mirror. Newton crafted a parabolic mirror, allowing easy observations of Jupiter and its moons. This is a special focus for us, as you know."
[[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, $mirror to false, $lens to false, $cabinet to false, $support to false, $origami to false, $almanac 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, $v9 to true)}<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cc/NewtonsTelescopeReplica.jpg/276px-NewtonsTelescopeReplica.jpg" class="right">
"The compactness of Newton's design means that you could perhaps construct a reflecting telescope yourselves, in New Spain. The light is collected by the main parabolic mirror and focused on a small flat mirror. This sends the image to a simple lens on the side of the telescope. You need someone skilled with mirrors, someone skilled with lenses, and someone skilled in cabinetry. The mathematics of the parabola and the reflecting characteristics caused me to reread Apollonius, the great Greek geometer."
[[next|one]]<img src="https://upload.wikimedia.org/wikipedia/commons/0/0c/Telescope_newton_schema.png" class="right">
"It's critical to understand how the curved surface of the main mirror focuses all the light. Write back with a demonstration of how different points on the parabola, say 3, reflect the rays from a distant planet or star on to the focus point. This will let me judge your skill and if you are having any problems."
"Oh, and if you get the chance, please use our almanac for the moons of Jupiter to make a careful calculation of the hours and minutes to the west that Mexico City is from Paris."
[[next|main]]<div class="right" style="border: 2pt solid firebrick; width:420px; padding: 4px">
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/34/Herigone_Tome_V.png/372px-Herigone_Tome_V.png" style="float:center">
<br>Find 3 points on a parabola and show how vertical rays reflect and go through the focus. Perhaps calculate the hours and minutes Mexico City is west of Paris.</div>
Choose who you wish to talk to. Try not to draw too much attention to yourselves.
[[astronomer]]
[[bronzesmith]]
[[cabinet maker]]
[[Japanese samurai]]
[[lens grinder]]
[[Nahuatl scholar]]
[[Viceroy]]
Cards drawn: $log{(if: $v1 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v1 to false, $almanac to true)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/37/Gerard_Dou_-_Astronomer_-_WGA06646.jpg/269px-Gerard_Dou_-_Astronomer_-_WGA06646.jpg" class="right">
<h4>Francisco Becerra</h4>
You take a short trip to Puebla to visit the architect and astronomer Becerra.
"Well to find the time difference between two locations we need to determine local noon. Imagine Mexico City is on a line running from the North Pole to the South Pole. If we can exactly draw a segment of that line then we can use a sundial pointer to see when the Sun's shadow crosses it. We set a clock to local noon at that moment."
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Meridian_%28PSF%29.png/199px-Meridian_%28PSF%29.png" class="left">
"Then you need to know when something will happen at the distant location and watch for when it happens at your location. In the past we had good estimations of the start of lunar eclipses. But rather than wait for one to occur, we can use the tables prepared in Paris for eclipses of the moons of Jupiter. If Io is predicted to go behind Jupiter at 3 am Paris time and you observe it at 9:30 pm, then you are 5 hours, 30 minutes west of Paris."
He generously allows you to borrow his almanac of Jupiter's moons, prepared in Paris.
[[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, $mirror 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 viceroy recommended this bronzesmith from Michoacán. He appreciates his craftsmanship in various works he 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 tin alloy mirror. How many inches across?" he asks.
You reply that about 4 inches in diameter would be good. Also, it needs to be bowl-like, parabolic to be specific. Could he make anything like that?
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Parabolic_mirror_focal_point.svg/242px-Parabolic_mirror_focal_point.svg.png" class="left" style="background:white">
"I have made mirrors before that size but to make it convex like that will be more difficult." You try to explain the shape and how it focuses the rays. (if: $origami is true)[You show him the origami parabola and that helps.] "Since the viceroy sent you, I will start on it right away."
[[next|main]]
Cards drawn: $log
{(if: ($mirror is true and $lens is true) and ($cabinet is true and $supply 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, $cabinet 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>Nicolás Enríquez</h4>
More and more cabinet makers are setting up shop in Mexico City to supply the growing demand for both European and Asian style furniture.
"A telescope? Hmm, do you mean a spyglass that has nested tubes?" You explain that you are making an astronomical telescope and because it uses mirrors instead of lenses, isn't actually that large.
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/30/ColonialEraChestMayer.jpg/254px-ColonialEraChestMayer.jpg" class="left">
"Oh well, that can be arranged. I assume you have the support of the viceroy?" You assure him that everything will be arranged.
[[next|main]]
Cards drawn: $log
{(if: ($mirror is true and $lens is true) and ($cabinet is true and $supply 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, $origami to true)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Hasekura_in_Rome.jpg/163px-Hasekura_in_Rome.jpg" class="right">
<h4>Luis de Encío (Soemon Fukuchi)</h4>
You find Encio folding a colored piece of paper but he puts it aside and asks how he can help. You tell him about the letter from Cassini.
"Let me show you how the parabola can be shaped from the tangent lines." He picks up a new piece of paper and puts a small dot about two inches from the edge.
<img src="https://www.cutoutfoldup.com/images/0507-i.gif" class="left" width="350">
"If we fold so that the edge meets the dot, the focus, then repeat that in different spots, all the fold creases start to make the parabola."
[[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, $lens to true)}
<img src="https://ethw.org/w/images/thumb/2/26/17th_century_lathe.jpg/300px-17th_century_lathe.jpg" class="right">
<h4>Jaime Spinoza</h4>
You take a short trip to Puebla, knowing of the glassworks there. You find the lens grinder.
"Yes, I have the lens you need, let's see ..." Eventually, he finds an appropriate lens.
[[next|main]]
Cards drawn: $log
{(if: ($mirror is true and $lens is true) and ($cabinet is true and $supply is true))[<br>
[[Don Gongora|prof]] ]}{(if: $v6 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v6 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/37/Gerard_Dou_-_Astronomer_-_WGA06646.jpg/269px-Gerard_Dou_-_Astronomer_-_WGA06646.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: $v7 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v7 to false, $mirror to true)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Ave_espejo_ante_Moctezuma_malos_presagios_C%C3%B3dice_Florentino_1_XII_f.3r.JPG/312px-Ave_espejo_ante_Moctezuma_malos_presagios_C%C3%B3dice_Florentino_1_XII_f.3r.JPG" class="right">
<h4>Fernando de Alva Ixtlilxochitl</h4>
Ixtlilxochitl has a book in front of him, with text in Nahuatl and Spanish. You came to him to ask about Aztec mirrors.
"Yes, we made fine mirrors for members of the nobility. They were often made of polished pyrite, jade, or obsisdian" he explains.
You ask if he has any examples, perhaps.
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f6/John_Dee%27s_Aztec_Scrying_Mirror.jpg/320px-John_Dee%27s_Aztec_Scrying_Mirror.jpg" class="left">
"Here, I need to get rid of this actually. Some people think mirrors are used in dark arts, like scrying the future. It wouldn't do for me to be denounced with this." He hands you an obsidian mirror and you thank him.
[[next|main]]
Cards drawn: $log
{(if: ($mirror is true and $lens is true) and ($cabinet is true and $supply is true))[<br>
[[Don Gongora|prof]] ]}{(if: $v8 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v8 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/37/Gerard_Dou_-_Astronomer_-_WGA06646.jpg/269px-Gerard_Dou_-_Astronomer_-_WGA06646.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: $v9 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v9 to false, $supply to true)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/c5/CondeDeParedes.jpg/369px-CondeDeParedes.jpg" class="right">
<h4>Tomás de la Cerda y Aragón, 3rd Marquess of la Laguna de Camero Viejo</h4>
The viceroy agrees with your plan to build a telescope.
"Please proceed to make the arrangements and I will contact Don Sigüenza y Góngora. He could help you with your mathematics questions."
[[next|main]]
Cards drawn: $log
{(if: ($mirror is true and $lens is true) and ($cabinet is true and $supply is true))[<br>
[[Don Gongora|prof]] ]}<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>
Sigüenza helps you finish the assembly of the telescope. You test it and adjust it until Sigüenza thinks it's ready to go. The viceroy looks on with clear interest.
Sigüenza looks through the telescope at Jupiter. "Such an amazing view. We should be able to clearly see the eclipses of the moons" explains the math professor.
You ask about how the parabolic shape of the mirror focuses the light and mention the letter from Cassini. "Three points? Hmm ..."
"Let's look at the points where the tangent lines have slopes of 0, 1, and -1" says Sigüenza. "A slope of zero would be a horizontal line so a vertical ray of light would reflect straight back. And for slopes of 1 and -1, those would be at a 45° angle to the vertical so reflecting horizontally." He pauses writing and you nod in understanding.
You ask about the time difference between Paris and Mexico City. Earlier, you observed local noon and set the viceroy's astronomical clock.
"If we have an almanac from Paris then we should be able to make that calculation tonight."
(if: $almanac is true)[You show Sigüenza the almanac you borrowed in Puebla. "Well it looks like we have a few hours until Io goes behind Jupiter so let's keep observing" he explains.<br>At 9:37 pm, you see Io eclipsed by Jupiter. The almanac says that occurred at 4:23 am Paris time.]
[[next|main]]