1 00:00:05,572 --> 00:00:07,140 >> So that's the assignment. 2 00:00:07,207 --> 00:00:10,877 5,000 steps by hand... 3 00:00:10,944 --> 00:00:13,780 or write a computer program 4 00:00:13,847 --> 00:00:17,250 that will do it in a minute or two. 5 00:00:17,317 --> 00:00:20,286 It's interesting historically 6 00:00:20,353 --> 00:00:24,124 that in the years before computers, of course, 7 00:00:24,190 --> 00:00:25,925 they didn't have that choice. 8 00:00:25,992 --> 00:00:29,963 It was literally 5,000 steps by hand. 9 00:00:30,029 --> 00:00:35,235 And the introduction of Euler's Method, 10 00:00:35,301 --> 00:00:40,507 largely by Katherine Johnson and her group at NASA, 11 00:00:40,573 --> 00:00:45,478 changed what scientists were able to do 12 00:00:45,545 --> 00:00:50,784 by making these numerical computations possible. 13 00:00:50,850 --> 00:00:53,353 The clip from the movie "Hidden Figures" 14 00:00:53,420 --> 00:00:57,323 illustrates that scene very dramatically, 15 00:00:57,390 --> 00:01:02,629 with Taraji P. Henson playing Katherine Johnson. 16 00:01:03,630 --> 00:01:04,898 >> It's when the capsule moves 17 00:01:04,964 --> 00:01:08,701 from an elliptical orbit to a parabolic orbit. 18 00:01:08,768 --> 00:01:11,504 There's no mathematical formula for that. 19 00:01:11,571 --> 00:01:13,807 Yes, we can calculate launch and landing 20 00:01:13,873 --> 00:01:16,476 but without this conversion, the capsule stays in orbit. 21 00:01:16,543 --> 00:01:20,280 We can't bring it back home. 22 00:01:20,346 --> 00:01:22,515 >> Maybe we've been thinking about this all wrong. 23 00:01:22,582 --> 00:01:24,551 >> How's that? 24 00:01:28,021 --> 00:01:31,791 >> Maybe it's not new math at all. 25 00:01:31,858 --> 00:01:34,527 >> It could be old math. 26 00:01:34,594 --> 00:01:36,396 Something that looks at the problem numerically 27 00:01:36,463 --> 00:01:38,431 and not theoretically. 28 00:01:38,498 --> 00:01:41,401 Math is always dependable. 29 00:01:41,468 --> 00:01:43,536 >> For you, it is. 30 00:01:45,772 --> 00:01:49,175 (gentle music) 31 00:01:49,242 --> 00:01:51,444 >> Euler's Method. >> Euler's Method? 32 00:01:51,511 --> 00:01:53,980 >> Yes. >> But that's ancient. 33 00:01:54,047 --> 00:01:55,582 >> But it works. 34 00:01:55,648 --> 00:01:58,251 It works numerically. 35 00:01:58,318 --> 00:02:33,453 ♪ ♪ 36 00:02:33,520 --> 00:02:35,088 >> That's it. 37 00:02:35,154 --> 00:02:37,824 >> So that's Hollywood. 38 00:02:37,891 --> 00:02:39,926 Let's actually take a second 39 00:02:39,993 --> 00:02:43,363 and look at the real Katherine Johnson, 40 00:02:43,429 --> 00:02:48,835 who is gonna describe the type of calculation she made 41 00:02:48,902 --> 00:02:54,173 for the Apollo and the Mercury astronaut missions. 42 00:02:54,240 --> 00:02:55,775 ♪ ♪ 43 00:02:55,842 --> 00:02:56,910 >> (narrator) Commander Alan B. Shepard 44 00:02:56,976 --> 00:03:00,480 was to become the first man sent into suborbital flight. 45 00:03:00,547 --> 00:03:02,682 The Mercury capsule is right on course. 46 00:03:02,749 --> 00:03:04,717 >> (Katherine Johnson) Our office computed 47 00:03:04,784 --> 00:03:07,754 for every mission that went out at that time. 48 00:03:07,820 --> 00:03:10,657 The height, the speed, and so on. 49 00:03:10,723 --> 00:03:13,426 It became a geometry problem. 50 00:03:13,493 --> 00:03:15,328 >> (announcer) At ignition sequence start. 51 00:03:15,395 --> 00:03:18,031 >> (Katherine) I felt most proud of the success 52 00:03:18,097 --> 00:03:20,900 of the Apollo mission. 53 00:03:20,967 --> 00:03:23,803 They were going to the moon! 54 00:03:23,870 --> 00:03:27,607 I computed the path that would get you there. 55 00:03:27,674 --> 00:03:31,411 You determined where you were on Earth when you started out, 56 00:03:31,477 --> 00:03:34,814 and where the moon would be at a given time. 57 00:03:34,881 --> 00:03:37,250 We told them how fast they would be going, 58 00:03:37,317 --> 00:03:40,053 and the moon will be there by the time you got there. 59 00:03:40,119 --> 00:03:42,055 >> (man) Beautiful, just beautiful. 60 00:03:42,121 --> 00:03:46,125 >> So what Katherine Johnson was doing 61 00:03:46,192 --> 00:03:51,197 was solving a differential equation 62 00:03:51,264 --> 00:03:54,000 by use of Euler's Method. 63 00:03:54,067 --> 00:03:57,570 And in this case, the differential equation 64 00:03:57,637 --> 00:04:03,843 is here is the Earth, and here is the moon, 65 00:04:03,910 --> 00:04:09,816 and what you want to do is you want to compute a trajectory. 66 00:04:09,882 --> 00:04:12,685 Here's your little takeoff point from the Earth. 67 00:04:12,752 --> 00:04:15,488 And you wanna compute a trajectory 68 00:04:15,555 --> 00:04:18,791 that goes like that 69 00:04:18,858 --> 00:04:23,429 and lands you exactly on the moon. 70 00:04:23,496 --> 00:04:24,697 Then, when you come home, 71 00:04:24,764 --> 00:04:27,967 you have to have another complicated trajectory 72 00:04:28,034 --> 00:04:29,435 calculated. 73 00:04:29,502 --> 00:04:37,043 And the Euler's Method here is just lots and lots and lots 74 00:04:37,110 --> 00:04:40,546 of tiny little finite steps 75 00:04:40,613 --> 00:04:45,985 approximating this complicated orbit. 76 00:04:46,052 --> 00:04:51,591 And she had been doing that by hand. 77 00:04:51,658 --> 00:04:54,293 And then, there's a scene, which is accurate, 78 00:04:54,360 --> 00:05:01,467 in the movie where they first wheel in these giant machines, 79 00:05:01,534 --> 00:05:08,107 room-size machines, that they call "computing machines." 80 00:05:08,174 --> 00:05:11,711 And this was a computing machine. 81 00:05:11,778 --> 00:05:17,817 Previously, the word "computer" referred to a person. 82 00:05:17,884 --> 00:05:19,886 It was an occupation. 83 00:05:19,952 --> 00:05:21,287 "What do you do?" 84 00:05:21,354 --> 00:05:26,926 "Oh, I'm a computer for NASA... I work at Langley." 85 00:05:26,993 --> 00:05:29,362 And the interesting thing-- there's a very good book 86 00:05:29,429 --> 00:05:34,300 which I recommend called, "When Computers Were Human" 87 00:05:34,367 --> 00:05:36,736 by David Alan Grier. 88 00:05:36,803 --> 00:05:39,572 And he tells the whole story 89 00:05:39,639 --> 00:05:44,010 of the army of people called "computers." 90 00:05:44,077 --> 00:05:49,415 And, by the way, 98% of them were women 91 00:05:49,482 --> 00:05:54,721 who actually carried out hand calculations 92 00:05:54,787 --> 00:06:02,462 of artillery trajectories for the Army, 93 00:06:02,528 --> 00:06:06,199 ballistic trajectories for the Navy, 94 00:06:06,265 --> 00:06:09,469 pursuit problems for the Navy. 95 00:06:09,535 --> 00:06:14,173 Rooms full of women with hand calculators, 96 00:06:14,240 --> 00:06:18,745 calculating complicated trajectories. 97 00:06:18,811 --> 00:06:20,179 And in this case, of course, 98 00:06:20,246 --> 00:06:24,117 the differential equation that she's solving 99 00:06:24,183 --> 00:06:28,121 has to do with the positions and velocities, 100 00:06:28,187 --> 00:06:31,224 and there's an X-position of the satellite, 101 00:06:31,290 --> 00:06:35,561 of the rocket, and a Y-position and a Z-position. 102 00:06:35,628 --> 00:06:38,164 And there's an X-prime, Y-prime, Z-prime. 103 00:06:38,231 --> 00:06:42,635 And Newton's Laws give you the forces of gravity that apply. 104 00:06:42,702 --> 00:06:47,907 And you just have to apply Euler's Method, 105 00:06:47,974 --> 00:06:50,042 ehh, 50,000 times. 106 00:06:50,109 --> 00:06:53,279 And because you want a short time step, 107 00:06:53,346 --> 00:06:56,449 because the longer time step saves you time, 108 00:06:56,516 --> 00:07:00,553 but you could very well, with a bigger time step, 109 00:07:00,620 --> 00:07:02,855 end up doing that, 110 00:07:02,922 --> 00:07:05,925 which you really don't wanna do. 111 00:07:05,992 --> 00:07:09,629 So you need a short time step to stay accurate 112 00:07:09,695 --> 00:07:14,000 but that means you have a lot of time steps. 113 00:07:14,066 --> 00:07:17,036 And so, the problem was, these are massive computations 114 00:07:17,103 --> 00:07:20,273 We have rooms full of women carrying them out, 115 00:07:20,339 --> 00:07:23,509 but it takes a long time. 116 00:07:23,576 --> 00:07:26,112 And then-- and this is historically accurate-- 117 00:07:26,179 --> 00:07:30,183 there's a scene in the movie "Hidden Figures" 118 00:07:30,249 --> 00:07:32,919 where they wheel in this big computer, 119 00:07:32,985 --> 00:07:37,056 and the Octavia Spencer character 120 00:07:37,123 --> 00:07:41,527 looks at the computer, and they tell her what it's for, 121 00:07:41,594 --> 00:07:45,097 and she goes back to her room full of women, 122 00:07:45,164 --> 00:07:51,170 and she says, "Ladies, we are either computer programmers 123 00:07:51,237 --> 00:07:54,207 "or we are unemployed." 124 00:07:54,273 --> 00:07:58,177 And she goes-- you remember, she steals a book on computing 125 00:07:58,244 --> 00:08:00,413 from the library because they wouldn't lend it to her 126 00:08:00,479 --> 00:08:02,815 'cause she's African American. 127 00:08:02,882 --> 00:08:08,087 She got that book on computing languages, 128 00:08:08,154 --> 00:08:10,623 and they learned computing. 129 00:08:10,690 --> 00:08:13,526 And these things, calculations, 130 00:08:13,593 --> 00:08:20,766 started to be done with these giant new IBM computers. 131 00:08:20,833 --> 00:08:24,637 But as a tiny little footnote, 132 00:08:24,704 --> 00:08:25,805 there's a scene in the movie, 133 00:08:25,872 --> 00:08:29,842 which is, I checked, it's historically accurate, 134 00:08:29,909 --> 00:08:33,512 where John Glenn, the great John Glenn, 135 00:08:33,579 --> 00:08:36,849 is going up into an orbital flight, 136 00:08:36,916 --> 00:08:41,354 and they tell him that they computed his trajectories 137 00:08:41,420 --> 00:08:44,423 with these new computers, 138 00:08:44,490 --> 00:08:46,959 and he was accustomed to working with Katherine Johnson 139 00:08:47,026 --> 00:08:49,028 and trusted her, 140 00:08:49,095 --> 00:08:55,234 and John Glenn says to the head of the program at Houston, 141 00:08:55,301 --> 00:09:00,239 he said, "Did the girl check the calculations? 142 00:09:00,306 --> 00:09:04,810 "I'm not flying unless she checked the calculations." 143 00:09:04,877 --> 00:09:06,512 And they asked her to do it. 144 00:09:06,579 --> 00:09:10,082 And she did some steps by hand to test the method. 145 00:09:10,149 --> 00:09:12,919 And she verified that it was doing it correctly. 146 00:09:12,985 --> 00:09:16,055 And Glenn said, "Okay, I will fly."