Now, we can go back to the original problem--the thing we've been trying to solve this whole unit.
As you remember we have the sun's rays striking perpendicular to the earth here at Syene.
We've seen that these rays all came in parallel like what's around here and here in Alexandria
which is north of Syene, the rays no longer strike the earth perpendicular to the surface of the earth.
They strike at this angle α.
We showed before that must be equal to this angle α as well.
And by comparing the full angular measure of the circle 360 degrees to this portion α, we will
define that if we knew d and we knew α, we can calculate the circumference of the earth.
We got d by timing a camels distance from Syene to Alexandria. Now we just need to find α.
To determine this angle α, let's imagine what it would be like to be standing here in Alexandria.
Well, if I were actually standing there, I can now say the earth is flat
because I don't notice the curvature of the earth on a daily basis
and I can imagine sticking some sort of pole into the ground.
And maybe I know the length of this pole.
Now I know the sun's rays are coming in at a certain angle.
I'm going to draw the rays in red just to make it a little easier to see.
So here comes the sun's rays, you can tell they're being blocked by the pole in some places.
So when the pole blocks the sun's rays, we get a shadow--here's the pole shadow.
And now you can see we're almost there. You can see the right triangle that's emerged.
We have the sun's ray that just barely missed the edge of the pole,
we have the shadow of the pole, and we have the pole itself forming a right triangle.
Now from this drawing, you can actually see that the angle we called alpha.
If I imagine making a perpendicular line here perpendicular to the earth,
this is the angle we called alpha.
There's no problem. That's also equal to this angle α and now we're almost there.
So here's our triangle with three sides--one, two, three, opposite, adjacent to α, and hypotenuse
and we just need to know what this angle is.
Now that we have a right triangle, we just need to consult the trigonometric table
and we should be able to figure out what alpha is and what's the circumference of the earth is.
So we set up the experiment.
We have our vertical bar with its shadow and what Eratosthenes data may have looked like
with the length of the bar would have been something around 1 meter
and the length of the shadow about 0.126 meters which is 12.6 cm.
He, of course, has access to his trigonometric tables and here's a portion of one such table.
Now, can you put yourselves in Eratosthenes' shoes and tell me what is the value of alpha?
Ahora podemos regresar al problema original, lo que hemos estado tratando de resolver toda esta unidad.
Como recordarás tenemos los rayos solares golpeando pendicularmente a la tierra aquí en Siena.
Hemos visto que esos rayos llegan en paralelo, como por aquí. Y Aquí en Alejandría
que esta al norte de Siena, los rayos ya no golpean a la tierra perpendicular a su superficie.
La golpean en un ángulo α.
Mostramos antes que esto es igual a este ángulo α también.
Y al comparar el total de la medida angular del circulo, 360°, con esta porción α, definiremos
que si conocemos d y conocemos α, podemos calcular la circunferencia de la tierra.
Obtuvimos d cronometrando un camello de Siena a Alejandría. Ahora solo necesitamos encontrar α.
Para determinar éste ángulo α, imaginemos lo que sería estar parado aquí en Alejandría.
Bueno, Si estuviera en realidad parado aquí, yo podría decir que la tierra es plana
porque no noto la curvatura de la tierra diariamente
y me puedo imaginar clavando algún poste en el suelo.
Y tal vez conozco la longitud del poste.
Ahora sé que los rayos del sol llegan con cierto ángulo.
Voy a dibujar los rayos en rojo solo para hacerlo un poco más fácil de ver.
Aquí llegan los rayos solares, pueden notar que son bloqueados por el poste en algunas partes.
Así que cuando el poste bloquea los rayos solares, obtenemos una sombre, aquí esta la sombra del poste.
y ahora puedes ver ya casi llegamos. Puedes en triángulo recto que se ha formado.
Tenemos los rayos solares que apenas pasaron por el borde del poste,
tenemos la sombra del poste, y tenemos el poste formando un triángulo recto.
Ahora de este dibujo, puedes ver en realidad que el ángulo que llamamos alpha.
Si me imagino una línea perpendicular aquí, perpendicular a la tierra,
este es el ángulo al que llamamos alpha.
No hay problema. También es igual a este ángulo α, y ahora ya casi lo tenemos.
Aquí está nuestro triángulo con tres lados, uno, dos, tres, opuesto, adyacente a α e hipotenusa
y solo necesitamos saber la medida de este ángulo.
Ahora que tenemos un triángulo recto, solo tenemos que consultar la tabla trigonométrica
y deberíamos ser capaces de encontrar el valor de alpha, y cual es la circunferencia de la tierra.
Así que preparamos el experimento.
Tenemos nuestra barra vertical con su sombre y lo que podrían haber sido los datos de Eratóstones
con el largo de la barra, podría haber sido alrededor de 1 metro
y el largo de la sombra alrededor de 0.126 metros que es 12.6 cm.
Él, por supuesto, tenía acceso a sus tablas trigonométricas y aquí esta una parte de esta tabla.
Ahora, ¿puedes ponerte en los zapatos de Eratóstenes y decirme cual es el valor de alpha?
이제 우리는 원래 문제로 돌아가볼 수 있습니다. 이 전체 단원에서 풀려고 했던 내용입니다.
기억하겠지만, 우리는 여기 시에네에서 지구까지 직각으로 내리쬐는 태양 광선을 봅니다.
우리는 이 광선이 모두 시에네의 북쪽에 있는 여기 알렉산드리아와 이 주위를
평행으로 온다는 사실을 보았습니다. 광선은 더이상 지구의 표면에 직각인 지구를 내리쬐지 앟습니다.
그들은 이 각도 알파로 내리쬡니다.
우리는 마찬가지고 저것이 이 각도 알파와 같아야 함을 이전에 보여주었습니다.
그리고 이 비인 알파까지 360도를 측정하는 모든 각도를 비교해봄으로써 우리는
만약 d와 알파를 안다면 지구의 원주를 계산할 수 있다고 정의할 것입니다.
우리는 시에네에서 알렉산드리아까지의 거기를 낙타를 통해서 시간으로 측정하여 d를 얻습니다. 이제 우리는 알파를 찾을 필요가 있습니다.
이 각 알파를 정하기 위해서, 여기 알렉산드리아에서 여기까지 이것과 무엇이 같을지 상상해 봅시다.
만약 내가 실제로 저기 서있었다면, 나는 이제 지구가 평평하다고 말할 수 있습니다.
왜냐하면 나는 매일 지구의 곡률을 알지 못하기 때문입니다.
그리고 나는 땅에 어떤 막대기로 태양이 내리쬘지 상상할 수 있습니다.
그리고 아마도 나는 이 막대기의 길이를 압니다.
이제 나는 태양의 광선이 특정한 각도에서 오고 있다는 사실을 압니다.
나는 조금 더 보기 쉽게 하기 위해서 빨간 색으로 광선을 그리겠습니다.
그러므로 여기에 태양 광선이 있고, 어떤 장소에서 막대기 때문에 그림자가 생긴다고 말할 수 있습니다.
그러므로 막대기로 태양 광선에 그림자가 생길 때, 여기에 막대기와 그림자가 있습니다.
그리고 이제 우리가 거기에 거의 다다르고 있다는 사실을 볼 수 있습니다. 여기서 발생한 직각 삼각형을 볼 수 있습니다.
우리는 막대기의 가장자리를 거의 스쳐지나가는 태양 광선을 봅니다.
우리는 막대기의 그림자를 보고, 우리는 직각 삼각형을 형성하는 막대기를 구합니다.
이제 이 그림으로부터, 우리가 알파라고 부르는 저 각도를 실제로 볼 수 있습니다.
내가 여기 지구에 직각인 선과 직각을 만든다고 상상한다면
이것이우리가 알파라고 불렀던 각입니다.
문제가 없습니다. 저것이 각 알파와 같고, 이제 우리는 거의 왔습니다.
그러므로 세 변, 1, 2, 3, 반댓변, 알파의 인접변, 그리고 빗변이 있는 삼각형이 여기 있습니다.
그리고 우리는 이 각이 무엇인지 알 필요가 있습니다.
이제 우리는 직각 삼각형을 구하고, 삼각표를 만들어볼 필요가 있습니다.
그리고 우리는 무엇이 알파인지, 지구의 둘레가 무엇인지 찾을 수 있어야 합니다.
그러므로 우리는 실험을 하겠습니다.
우리는 수직인 막대와 그림자를 구했고 에라스토테네스 데이터가 어떻게 보이는지 구했습니다.
그리고 막대기의 길이는 대략 1미터가 될 것입니다.
그리고 12.6cm인 0.126미터정도 되는 그림자의 길이입니다.
물론 그는 삼각표에 접근했고 여기에 이런 한 표의 일부가 있습니다.
이제, 에라스토테네스의 입장이 되어서 알파의 값이 무엇인지 말해줄 수 있습니까?
Podemos agora voltar ao problema original, o que tentamos resolver desde o início.
Como se lembram, temos os raios de sol a atingir a terra perpendicularmente aqui em Syene.
Já vimos que estes raios chegam paralelos, como aqui e aqui em Alexandria
que fica a norte de Syene, os raios já não atingem a terra na perpendicular.
Atingem-na com este ângulo α.
Já mostramos que esse tem que ser igual a este ângulo α.
E ao comparar a circunferência total de 360º com esta porção α, vamos definir que
se conhecermos d e conhecermos α, podemos calcular a circunferência da Terra.
Obtemos d cronometrando um camelo que vá de Syene a Alexandria. Precisamos de α.
Para determinar este ângulo α, vamos imaginar que estamos aqui em Alexandria.
Bem, se eu estivesse mesmo aqui, poderia dizer que a Terra é plana,
Porque não noto a curvatura da Terra no dia-a-dia
e posso imaginar espetar um bastão no chão.
E talvez eu conheça o comprimento do bastão.
Sei que os raios de sol chegam com um determinado ângulo.
Vou desenhar estes raios a vermelho para ser mais fácil ver.
Portanto os raios chegam aqui, podem ver que alguns são bloqueados pelo bastão.
Quando o bastão bloqueia os raios de sol, faz uma sombra - está aqui a sombra.
E podem ver que nos falta pouco. Podem ver que apareceu um triângulo retângulo.
Temos este raio de sol que passou mesmo ao lado da ponta do bastão,
temos a sombra do bastão, e o próprio bastão, que formam um triângulo retângulo.
Com este desenho, podemos ver o ângulo a que chamamos alfa.
Se eu imaginar uma linha perpendicular à Terra aqui,
este é o ângulo a que chamamos alfa.
Não há problema. É também igual a este ângulo α e estamos quase lá.
Aqui está o nosso triângulo - um, dois, três, oposto, adjacente e hipotenusa
e só nos falta saber este ângulo.
Agora que temos um triângulo retângulo, só temos que consultar a tabela
e devemos ser capazes de descobrir o nosso alfa e a circunferência da Terra.
Portanto montamos a experiência.
Temos o nosso bastão com a sombra e os dados que Eratóstenes teria
com o comprimento do bastão, que seria algo como um metro,
E o comprimento da sombra algo como 0.126 metros, ou seja 12.6cm.
Ele, claro, tinha acesso às tabelas trigonométricas, temos aqui parte de uma.
Podem colocar-se no lugar de Esratóstenes e dizer-me o valor de alfa?
ทีนี้, เราสามารถกลับไปยังปัญหาเดิมได้แล้ว -- สิ่งที่เราพยายามแต่มาตลอดหน่วยนี้
คุณคงจำได้ว่า เรามีรังสีดวงอาทิตย์ตรงตั้งฉากกับผิวโลกที่ไซเน
เราเห้นว่ารังสีพวกนี้มาขนานกันรอบๆ ตรงนี้และที่นี่ในอเลกซานเดรีย
ซึ่งอยู่ทางเหนือของไซเน, รังสีไม่ได้กระทบโลก แบบตั้งฉากกับผิวแล้ว
มันตกกระทบเป็นมุม α
เราได้แสดงก่อนหน้านี้แล้วว่ามันเท่ากับมุม α นี่เช่นกัน
และด้วยการเปรียบเทียบมุมเต็มวงกลม 260 องศากับส่วน α นี้, เรา
กำหนดว่า หาก เรารู้ d และเรารู้ α, เราก็หาเส้นรอบวงของโลกได้
เราหา d ได้จากการจับเวลาอูฐเดินทางจากไซเนถึงอเลกซานเดรีย ตอนนี้เราแค่ต้องหา α
ในการหามุม α นี้, ลองจินตนาการว่าเรายืนตรงนี้ที่อเลกซานเดรีย
ทีนี้, หากเรายืนอยู่ตรงนี้, ผมบอกได้ว่าโลกแบน
เพราะผมไม่สังเกตนี้ความโค้งของโลกในชีวิตประจำวัน
ผมก็จินตนการว่าผมปักแท่งอะไรสักอย่างลงกับพื้น
บางทีผมรู้ความยาวของแท่งนี้
ผมรู้ว่าแสงอาทิตย์ลงมาทำมุมอยู่
ผมจะวาดเส้นลำแสงด้วยสีแดงจะได้เห็นง่ายหน่อย
ตรงนี้ แสงอาทิตย์เข้ามา, คุณบอกได้ว่ามันถูกบังด้วยแท่งนี้บางส่วน
แล้วเมื่อแท่งบังแสงอาทิตย์, เราจะได้เงา, นี่คือเงาของแท่ง
และตอนนี้คุณบอกได้ว่าเราใกล้ถึงแล้ว คุณเห็นสามเหลี่ยมมุมฉากโผล่ออกมาแล้ว
เราได้แสงอาทิตย์ที่เกือบชนขอบแท่ง
เรามีเงาของแท่ง, และเรามีตัวแท่งเองประกอบเป็นสามเหลี่ยมมุมฉาก
และจากภาพวาดนี้, คุณเห็นได้มุมที่เราเรียกว่า α นั่น
หากผมจินตนาการว่าลากเส้นตั้งฉากตรงนี้ ตั้งฉากกับโลก
นี่ก็คือมุมที่เรียกว่าอัลฟา
ไม่มีปัญหา นั่นเท่ากับมุม α นี่ด้วย แล้วเราก็ใกล้ได้แล้ว
ตรงนี้คือสามเหลี่ยมมีสามด้าน -- หนึ่ง, สอง, สาม, ข้าม, ชิด α และฉาก
เราแค่ต้องหามุม
ทีนี้ เรามีสามเหลี่ยมมุมฉาก, เราต้องใช้ตารางตรีโกณมิติ
และเราควรหาได้ว่าอัลฟาเป็นเท่าไหร่ และเส้นรอบวงของโลกเป็นเท่าไหร่
ลองตั้งการทดลองขึ้นมา
เรามีแท่งดิ่งที่มีเงา และข้อมูลของอีราโทสธีนิสเป็นแบบนี้
ความยาวของแท่งยาวประมาณ 1 เมตร
และความยาวของเงาประมาณ 0.126 เมตร เท่ากับ 12.6 cm
เขา, แน่นอน, ใช้ตารางตรีโกณมิติ และนี่คือส่วนหนึ่งของตาราง
ทีนี้, คุณลองสวมบทอีราโทสธีนิส แล้วบอกผมได้ไหมว่า ค่าของอัลฟาเป็นเท่าไหร่?