<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/1754_Bellin_Map_of_Mexico_City_%5E_Environs_-_Geographicus_-_MexicoCity2-bellin-1754.jpg/401px-1754_Bellin_Map_of_Mexico_City_%5E_Environs_-_Geographicus_-_MexicoCity2-bellin-1754.jpg" class="right">
The group of you join together to discuss the latest letter sent to you in Mexico City. Monsieur Bernard Fontenelle, member of the French Academy of Sciences wrote with a question about polynomials.
[[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, $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)
(set: $top to $deck's 1st, $second to $deck's 2nd, $third to $deck's 3rd, $fourth to $deck's 4th, $fifth to $deck's 5th, $sixth to $deck's 6th, $seventh to $deck's 7th)}<span class="right" style="background:white;font-size:12px;padding:8px"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Bernard_Fontenelle.jpg/162px-Bernard_Fontenelle.jpg"><br>Bernard Le Bovier de Fontenelle</span>
"Greetings to your group in New Spain. Because of your unique connections to Asia, I write to ask you about a method to solve polynomial equations. As perhaps you know, in the last century, the Italian mathematician Cardano published equations to find solutions to cubic (third degree) polynomials. However, these can be rather unwieldy. Lately, I have heard of methods developed in China or Japan to quickly evaluate polynomial functions. This allows for quick approximations of solutions. Perhaps you can find out something about this method and then illustrate it by finding a solution to a polynomial equation like 2x<sup>3</sup> - 3x<sup>2</sup> - 11x + 6."
[[next|main]]<span class="right" style="background:white"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/8/88/Funci%C3%B3n_c%C3%BAbica.svg/180px-Funci%C3%B3n_c%C3%BAbica.svg.png"><br>2x<sup>3</sup> - 3x<sup>2</sup> - 11x + 6</span>
Choose who you wish to talk to. Try not to draw too much attention to yourselves.
[[Nahuatl scholar|Alva]]
[[Japanese samurai|Encio]]
[[Chinese barber|Gart]]
[[Jesuit natural philosopher|Kino]]
[[Mathematics professor|Carlos]]
[[Philosopher nun|Cruz]]
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/7/70/The_Florentine_Codex-_Lunar_Eclipse.tiff/lossy-page1-326px-The_Florentine_Codex-_Lunar_Eclipse.tiff.jpg" class="right">
<h4>Fernando de Alva Ixtlilxochitl</h4>
You find Ixtlilxochitl working on a bilingual manuscript. You ask about Aztec mathematics and anything he knows about Chinese that have come to Mexico (New Spain).
"Certainly we have the mathematics to, for instance, make predictions of eclipses," he says, gesturing to the manuscript he's working on. "Of course, it looks different than the algebra I think you're asking about. As to your second question, there have been some Asian immigrants arriving here via the Manila-Acapulco trade route. The Spanish call them 'chino' indiscriminately. But a few years back, the Spanish crown pronounced them all 'indios' as well, hence freed from bondage. Perhaps you could ask Gart Solomedras, a Chinese barber. I have discussed mathematics with him and he seems rather knowledgeable."
[[back|main]]
Cards drawn: $log{(if: $v2 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v2 to false)}
<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 wait for Encío to pause sharpening his sword. You ask about Japanese mathematics and if they have a method to evaluate polynomials and find their roots.
<img src="https://upload.wikimedia.org/wikipedia/commons/4/47/Counting_board.jpg" class="left">
Encio shows you a set of what look like short sticks. They have been made smooth from much use. "In Japan, I learned how to use the rod numbers to carry out many mathematical procedures. With a series of multiplications and additions, we can evaluate polynomials. You first put down the coefficients of your polynomials. Then you multiply, add, multiply, add, etc."
<img src="https://www.chilimath.com/wp-content/uploads/2017/02/p1_animated_v2.gif" class="right">
"Here's an example for evaluating x^4 - 3x^3 - 11x^2 + 5x + 17 for x = -2. The result is 3."
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Cards drawn: $log{(if: $v3 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v3 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Chinese_Barber.jpg/320px-Chinese_Barber.jpg" class="right">
<h4>Gart Solomedras</h4>
Solomedras finishes with a customer and then invites you to have a seat. You explain that you really want to ask what he knows about Chinese mathematics and in particular, a method to evaluate polynomials.
"Ah, you want the other kind of services that an //algebrista// can offer." He explains the Arabic roots of the word algebrista and how it means both bone-setter and algebraist. "Certainly. Let me show you the counting board, as described in //Jigu suanjing//, a book written about 1,000 years ago. I guess the title would translate as **The continuation of ancient mathematics**." You see a board with engraved lines and a pile of small wooden rods, some colored black and others red.
<img src="https://www.chilimath.com/wp-content/uploads/2017/02/p2_animated.gif" class="left">
"For instance, to evaluate what European mathematicians would call a fifth degree polynomial at 1, for instance x^5 - 3x^3 -4x - 1. We set up the board with the coefficients and then multiply, add, multiply, add, and so on. The result is negative 7."
[[back|main]]
Cards drawn: $log{(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/d/d3/Padre_Misionero_Francisco_Eusebio_Kino.jpg" class="right">
<h4>Father Eusebius Kino</h4>
When you ask the priest natural philosopher about Chinese mathematics, he nods in understanding.
"Yes, our order of Jesuits has done much work in China, spreading the gospel and learning the language and culture. Father Matteo Ricci, and others, have learned much of the deep mathematics developed in China. The method you describes goes something like this: put down the coefficients of the polynomial and then multiply each by the value you're evaluating and add."
<span class="left" style="background:white"><img src="https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut37ex1f.gif"></span>
He illustrates with an example. "Take a polynomial like 2x<sup>3</sup> - 3x<sup>2</sup> + 4x - 1. If we put -1 in for x, this is what the Chinese method gives us: -10."
[[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/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>
You find the professor after his lecture on algebra. He also received the letter from Fontenelle that you want to ask about.
"If we think naively about how many steps to evaluate a polynomial like 2x<sup>3</sup> - 3x<sup>2</sup> - 11x + 6, we might think we would need 3 multiplications for the third degree term, 2 for the second degree term, etc. So 6 multiplications and 3 additions. But with some factorization, we can rewrite it as ((2x - 3)x - 11)x + 6. That's just 3 multiplications and 3 additions."
[[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)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Sor_Juana_In%C3%A9s_de_la_Cruz_%281772%29.jpg/381px-Sor_Juana_In%C3%A9s_de_la_Cruz_%281772%29.jpg" class = "right">
<h4>Sor Juana Inés de la Cruz</h4>
"If we imagine a plane, like Monsieur Descartes invites us to, we can draw a line separating it into the positive and the negative. Then if we draw a curve that crosses that line, we know the curve will have positive values on one side and negative values on the other side."
You agree, this seems apparent.
"But do we know that the curve will always have a value of zero somewhere in between the positive and negative. After all, the physical line we draw is just a representation of the idea of the curve. Possibly the actual curve just 'skips over' zero."
You ponder this thought.
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Cards drawn: $logDouble-click this passage to edit it.Double-click this passage to edit it.