{"version":"1.0.0","segments":[{"startTime":0.18,"endTime":1.2,"body":"- So today, I want to talk to you"},{"startTime":1.2,"endTime":4.6,"body":"about the incredible"},{"startTime":1.2,"endTime":4.6,"body":"sine wave, and its uses."},{"startTime":4.6,"endTime":8.03,"body":"We've all seen a sine wave"},{"startTime":4.6,"endTime":8.03,"body":"in our lives, I'm sure,"},{"startTime":8.03,"endTime":11.34,"body":"but we also all learned at school"},{"startTime":11.34,"endTime":13.91,"body":"that sine is opposite"},{"startTime":11.34,"endTime":13.91,"body":"over hypotenuse, right?"},{"startTime":13.91,"endTime":16.94,"body":"So that is associating to some angle"},{"startTime":16.94,"endTime":19.02,"body":"in a right-angled triangle"},{"startTime":19.02,"endTime":20.24,"body":"a ratio of lengths,"},{"startTime":20.24,"endTime":23.74,"body":"a number that must be"},{"startTime":20.24,"endTime":23.74,"body":"between zero and one,"},{"startTime":23.74,"endTime":27.23,"body":"so how does that very un-curvy thing,"},{"startTime":27.23,"endTime":29.21,"body":"a number associated to a triangle,"},{"startTime":29.21,"endTime":32.17,"body":"end up being related to"},{"startTime":29.21,"endTime":32.17,"body":"this very curvy thing"},{"startTime":32.17,"endTime":34.16,"body":"that goes on forever in both directions"},{"startTime":34.16,"endTime":37.09,"body":"and takes both positive"},{"startTime":34.16,"endTime":37.09,"body":"and negative values,"},{"startTime":37.09,"endTime":38.42,"body":"how is that?"},{"startTime":38.42,"endTime":41.05,"body":"Well, I'm going to give you"},{"startTime":38.42,"endTime":41.05,"body":"today a brief history of sine"},{"startTime":41.05,"endTime":43.6,"body":"to get us from triangles to sine waves,"},{"startTime":43.6,"endTime":46.78,"body":"and look at some of"},{"startTime":43.6,"endTime":46.78,"body":"their huge range of uses."},{"startTime":46.78,"endTime":47.613,"body":"Along the way,"},{"startTime":47.613,"endTime":49.97,"body":"we'll encounter banned books, archery,"},{"startTime":49.97,"endTime":52.32,"body":"and the workings of the human ear."},{"speaker":"So let's get started","startTime":55.01,"endTime":59.093,"body":"the ancient Greeks did not"},{"speaker":"So let's get started","startTime":55.01,"endTime":59.093,"body":"have sin, cos, and tan,"},{"speaker":"So let's get started","startTime":60.18,"endTime":63.29,"body":"so how does it happen that Hipparchus,"},{"speaker":"So let's get started","startTime":63.29,"endTime":66.48,"body":"the ancient Greek astronomer"},{"speaker":"So let's get started","startTime":63.29,"endTime":66.48,"body":"and mathematician,"},{"speaker":"So let's get started","startTime":66.48,"endTime":68.87,"body":"is called the father of trigonometry?"},{"speaker":"So let's get started","startTime":68.87,"endTime":71.24,"body":"Well he didn't use sin, cos, and tan,"},{"speaker":"So let's get started","startTime":71.24,"endTime":73.45,"body":"but he did work with triangles,"},{"speaker":"So let's get started","startTime":73.45,"endTime":75.68,"body":"and I want to just briefly show you"},{"speaker":"So let's get started","startTime":76.64,"endTime":78.733,"body":"a little bit about what he did."},{"speaker":"So let's get started","startTime":80.59,"endTime":83.04,"body":"He used chords, he worked"},{"speaker":"So let's get started","startTime":80.59,"endTime":83.04,"body":"with chords of circles,"},{"speaker":"So let's get started","startTime":83.04,"endTime":88.04,"body":"so lines joining places on"},{"speaker":"So let's get started","startTime":83.04,"endTime":88.04,"body":"the circumference of circles,"},{"speaker":"So let's get started","startTime":89.05,"endTime":91.903,"body":"in astronomical calculations."},{"speaker":"So let's get started","startTime":91.903,"endTime":94.08,"body":"He did lots of amazing"},{"speaker":"So let's get started","startTime":91.903,"endTime":94.08,"body":"things in astronomy,"},{"speaker":"So let's get started","startTime":94.08,"endTime":97.4,"body":"he measured the procession of equinoxes,"},{"speaker":"So let's get started","startTime":97.4,"endTime":99.83,"body":"he calculated an estimated distance"},{"speaker":"So let's get started","startTime":99.83,"endTime":101.72,"body":"from the Earth to the Moon,"},{"speaker":"So let's get started","startTime":101.72,"endTime":105.37,"body":"he said it was between"},{"speaker":"So let's get started","startTime":101.72,"endTime":105.37,"body":"59 and 67 Earth radii,"},{"speaker":"So let's get started","startTime":105.37,"endTime":106.65,"body":"which is not bad,"},{"speaker":"So let's get started","startTime":106.65,"endTime":109.82,"body":"the true value is around"},{"speaker":"So let's get started","startTime":106.65,"endTime":109.82,"body":"about 60 Earth radii."},{"speaker":"So let's get started","startTime":109.82,"endTime":113.96,"body":"He did that using estimates"},{"speaker":"So let's get started","startTime":109.82,"endTime":113.96,"body":"for the parallax of the Sun,"},{"speaker":"So let's get started","startTime":113.96,"endTime":115.46,"body":"which I'm not going to go into,"},{"speaker":"So let's get started","startTime":115.46,"endTime":116.91,"body":"but just to give you an illustration"},{"speaker":"So let's get started","startTime":116.91,"endTime":121.11,"body":"of how chords of circles"},{"speaker":"So let's get started","startTime":116.91,"endTime":121.11,"body":"could be used in astronomy,"},{"speaker":"So let's get started","startTime":121.11,"endTime":123.87,"body":"I'll just show you how you could use them"},{"speaker":"So let's get started","startTime":123.87,"endTime":126.29,"body":"to work out the distance"},{"speaker":"So let's get started","startTime":123.87,"endTime":126.29,"body":"from the Earth to the Moon."},{"speaker":"So let's get started","startTime":126.29,"endTime":127.87,"body":"It isn't Hipparchus' method,"},{"speaker":"So let's get started","startTime":127.87,"endTime":129.92,"body":"but it's a method that can be used,"},{"speaker":"So let's get started","startTime":129.92,"endTime":131.27,"body":"and has been used."},{"speaker":"So let's get started","startTime":131.27,"endTime":134.0,"body":"So imagine there's the"},{"speaker":"So let's get started","startTime":131.27,"endTime":134.0,"body":"Earth, there's the Moon,"},{"speaker":"So let's get started","startTime":134.0,"endTime":135.577,"body":"I'm going to draw in some lines here,"},{"speaker":"So let's get started","startTime":135.577,"endTime":136.53,"body":"and what you can do is"},{"speaker":"So let's get started","startTime":136.53,"endTime":138.99,"body":"you and your friend can"},{"speaker":"So let's get started","startTime":136.53,"endTime":138.99,"body":"stand as far away as possible"},{"speaker":"So let's get started","startTime":138.99,"endTime":140.01,"body":"from each other"},{"speaker":"So let's get started","startTime":140.01,"endTime":141.82,"body":"on the surface of the Earth"},{"speaker":"So let's get started","startTime":141.82,"endTime":143.08,"body":"and look at the Moon,"},{"speaker":"So let's get started","startTime":143.08,"endTime":145.54,"body":"and look at its position in the sky,"},{"speaker":"So let's get started","startTime":145.54,"endTime":148.883,"body":"and you can compare it to"},{"speaker":"So let's get started","startTime":145.54,"endTime":148.883,"body":"something that's so far away"},{"speaker":"So let's get started","startTime":148.883,"endTime":151.99,"body":"that it looks like it's in"},{"speaker":"So let's get started","startTime":148.883,"endTime":151.99,"body":"the same place in the sky"},{"speaker":"So let's get started","startTime":151.99,"endTime":152.98,"body":"to everybody on Earth,"},{"speaker":"So let's get started","startTime":152.98,"endTime":154.41,"body":"so a distant star"},{"speaker":"So let's get started","startTime":154.41,"endTime":157.0,"body":"will look fixed in the"},{"speaker":"So let's get started","startTime":154.41,"endTime":157.0,"body":"same position for everyone,"},{"speaker":"So let's get started","startTime":157.0,"endTime":159.86,"body":"but the Moon will be at a"},{"speaker":"So let's get started","startTime":157.0,"endTime":159.86,"body":"slightly different inclination,"},{"speaker":"So let's get started","startTime":159.86,"endTime":160.88,"body":"a slightly different angle."},{"speaker":"So let's get started","startTime":160.88,"endTime":162.5,"body":"So you can measure the angles,"},{"speaker":"So let's get started","startTime":162.5,"endTime":164.74,"body":"and then compare notes with your friend,"},{"speaker":"So let's get started","startTime":164.74,"endTime":168.35,"body":"and that gives you the"},{"speaker":"So let's get started","startTime":164.74,"endTime":168.35,"body":"angles in this diagram here,"},{"speaker":"So let's get started","startTime":168.35,"endTime":170.75,"body":"and you can then work out,"},{"speaker":"So let's get started","startTime":170.75,"endTime":173.89,"body":"I guess the height of that"},{"speaker":"So let's get started","startTime":170.75,"endTime":173.89,"body":"big isosceles triangle,"},{"speaker":"So let's get started","startTime":173.89,"endTime":176.1,"body":"and that tells you the"},{"speaker":"So let's get started","startTime":173.89,"endTime":176.1,"body":"distance to the Moon."},{"speaker":"So let's get started","startTime":176.1,"endTime":178.37,"body":"You can work that out as"},{"speaker":"So let's get started","startTime":176.1,"endTime":178.37,"body":"long as you know the base,"},{"speaker":"So let's get started","startTime":178.37,"endTime":179.87,"body":"and what's the base of that triangle,"},{"speaker":"So let's get started","startTime":179.87,"endTime":181.2,"body":"that red line there?"},{"speaker":"So let's get started","startTime":181.2,"endTime":182.93,"body":"Well that's a chord of a circle."},{"speaker":"So let's get started","startTime":182.93,"endTime":184.37,"body":"If you slice through the Earth,"},{"speaker":"So let's get started","startTime":184.37,"endTime":187.03,"body":"you get a circle, ish,"},{"speaker":"So let's get started","startTime":187.03,"endTime":188.26,"body":"and that's a chord."},{"speaker":"So let's get started","startTime":188.26,"endTime":190.49,"body":"So knowing the length of that chord,"},{"speaker":"So let's get started","startTime":190.49,"endTime":192.61,"body":"being able to work that out,"},{"speaker":"So let's get started","startTime":192.61,"endTime":194.293,"body":"is a useful thing in astronomy."},{"speaker":"So let's get started","startTime":195.885,"endTime":198.91,"body":"I've just thrown away the"},{"speaker":"So let's get started","startTime":195.885,"endTime":198.91,"body":"extraneous parts of that diagram,"},{"speaker":"So let's get started","startTime":198.91,"endTime":201.39,"body":"so this is the kind of diagram"},{"speaker":"So let's get started","startTime":198.91,"endTime":201.39,"body":"that we're interested in."},{"speaker":"So let's get started","startTime":201.39,"endTime":203.25,"body":"And for astronomical applications,"},{"speaker":"So let's get started","startTime":203.25,"endTime":205.743,"body":"like the ones Hipparchus was working with,"},{"speaker":"So let's get started","startTime":206.75,"endTime":208.12,"body":"this is the kind of thing we want to know,"},{"speaker":"So let's get started","startTime":208.12,"endTime":210.73,"body":"you want to know the length"},{"speaker":"So let's get started","startTime":208.12,"endTime":210.73,"body":"of that red line, the chord,"},{"speaker":"So let's get started","startTime":210.73,"endTime":211.94,"body":"and the yellow lines there,"},{"speaker":"So let's get started","startTime":211.94,"endTime":214.18,"body":"each one of those is a"},{"speaker":"So let's get started","startTime":211.94,"endTime":214.18,"body":"radius of that circle,"},{"speaker":"So let's get started","startTime":214.18,"endTime":218.27,"body":"so this is why chord"},{"speaker":"So let's get started","startTime":214.18,"endTime":218.27,"body":"lengths were interesting."},{"speaker":"So let's get started","startTime":218.27,"endTime":220.5,"body":"And later on, actually,"},{"speaker":"So let's get started","startTime":220.5,"endTime":223.51,"body":"it turned out that"},{"speaker":"So let's get started","startTime":220.5,"endTime":223.51,"body":"maybe half chord lengths"},{"speaker":"So let's get started","startTime":223.51,"endTime":226.04,"body":"were the things you really"},{"speaker":"So let's get started","startTime":223.51,"endTime":226.04,"body":"wanted to know about,"},{"speaker":"and here's why","startTime":226.873,"endTime":228.88,"body":"if instead of being the Earth,"},{"speaker":"and here's why","startTime":228.88,"endTime":231.64,"body":"that circle is now the"},{"speaker":"and here's why","startTime":228.88,"endTime":231.64,"body":"whole celestial sphere,"},{"speaker":"and here's why","startTime":231.64,"endTime":233.66,"body":"or at least a slice through it,"},{"speaker":"and here's why","startTime":233.66,"endTime":235.92,"body":"actually, you can't see"},{"speaker":"and here's why","startTime":233.66,"endTime":235.92,"body":"all of that at night"},{"speaker":"and here's why","startTime":235.92,"endTime":237.1,"body":"when you're doing your astronomy"},{"speaker":"and here's why","startTime":237.1,"endTime":238.513,"body":"because the horizon gets in the way,"},{"speaker":"and here's why","startTime":238.513,"endTime":240.21,"body":"there's the horizon."},{"speaker":"and here's why","startTime":240.21,"endTime":241.79,"body":"So if you want to find out something"},{"speaker":"and here's why","startTime":241.79,"endTime":245.25,"body":"like the height of a"},{"speaker":"and here's why","startTime":241.79,"endTime":245.25,"body":"star above the horizon"},{"speaker":"and here's why","startTime":245.25,"endTime":246.83,"body":"in the celestial sphere,"},{"speaker":"and here's why","startTime":246.83,"endTime":249.04,"body":"it's actually the half chord length"},{"speaker":"and here's why","startTime":249.04,"endTime":250.64,"body":"that you might want to know."},{"speaker":"and here's why","startTime":250.64,"endTime":252.3,"body":"That's a more convenient thing"},{"speaker":"and here's why","startTime":252.3,"endTime":255.47,"body":"because lots of the calculations"},{"speaker":"and here's why","startTime":252.3,"endTime":255.47,"body":"want the half chord length,"},{"speaker":"and here's why","startTime":255.47,"endTime":257.73,"body":"but even if you then do"},{"speaker":"and here's why","startTime":255.47,"endTime":257.73,"body":"want the full chord length,"},{"speaker":"and here's why","startTime":257.73,"endTime":259.44,"body":"of course you just double,"},{"speaker":"and here's why","startTime":259.44,"endTime":261.13,"body":"and doubling is easier than halving,"},{"speaker":"and here's why","startTime":261.13,"endTime":263.43,"body":"so if you're going to"},{"speaker":"and here's why","startTime":261.13,"endTime":263.43,"body":"write a table of things,"},{"speaker":"and here's why","startTime":263.43,"endTime":265.58,"body":"maybe half chord lengths"},{"speaker":"and here's why","startTime":263.43,"endTime":265.58,"body":"are what you want."},{"speaker":"and here's why","startTime":266.47,"endTime":268.24,"body":"So we can see now the link"},{"speaker":"and here's why","startTime":268.24,"endTime":271.28,"body":"between chords and the sine function,"},{"speaker":"and here's why","startTime":271.28,"endTime":273.48,"body":"because if I just zoom in"},{"speaker":"and here's why","startTime":273.48,"endTime":276.11,"body":"on the important bit of that diagram,"},{"speaker":"and here's why","startTime":276.11,"endTime":278.37,"body":"you see, look, there's that horizon,"},{"speaker":"and here's why","startTime":278.37,"endTime":279.92,"body":"the horizontal diameter,"},{"speaker":"and here's why","startTime":279.92,"endTime":282.04,"body":"you've got the radii of the circle,"},{"speaker":"and here's why","startTime":282.04,"endTime":283.8,"body":"radius r, let's say,"},{"speaker":"and here's why","startTime":283.8,"endTime":285.72,"body":"the chord is there in red,"},{"speaker":"and here's why","startTime":285.72,"endTime":288.5,"body":"and that half chord length I've called L."},{"speaker":"and here's why","startTime":288.5,"endTime":289.89,"body":"We know in a right-angled triangle,"},{"speaker":"and here's why","startTime":289.89,"endTime":290.9,"body":"which we've got here,"},{"speaker":"and here's why","startTime":290.9,"endTime":293.18,"body":"the sine of the angle theta there"},{"speaker":"and here's why","startTime":293.18,"endTime":296.063,"body":"is opposite over hypotenuse,"},{"speaker":"and here's why","startTime":293.18,"endTime":296.063,"body":"so it's L over r."},{"speaker":"and here's why","startTime":297.01,"endTime":299.02,"body":"And what happened?"},{"speaker":"and here's why","startTime":299.02,"endTime":301.99,"body":"So the ancient Greeks were"},{"speaker":"and here's why","startTime":299.02,"endTime":301.99,"body":"looking at chord lengths,"},{"speaker":"and here's why","startTime":301.99,"endTime":304.59,"body":"over in India a few centuries later,"},{"speaker":"and here's why","startTime":304.59,"endTime":306.75,"body":"they started out looking"},{"speaker":"and here's why","startTime":304.59,"endTime":306.75,"body":"at half chord lengths,"},{"speaker":"and here's why","startTime":306.75,"endTime":308.93,"body":"and then they looked at the ratio"},{"speaker":"and here's why","startTime":308.93,"endTime":313.93,"body":"of the half chord length to"},{"speaker":"and here's why","startTime":308.93,"endTime":313.93,"body":"the radius of the circle,"},{"speaker":"and here's why","startTime":313.99,"endTime":316.28,"body":"and that is how we came to have sin-theta"},{"speaker":"and here's why","startTime":316.28,"endTime":317.95,"body":"So it started with chords,"},{"speaker":"and here's why","startTime":317.95,"endTime":319.62,"body":"ended up with sin-theta,"},{"speaker":"and here's why","startTime":319.62,"endTime":322.3,"body":"and cos-theta is where you"},{"speaker":"and here's why","startTime":319.62,"endTime":322.3,"body":"can take the cos of the angle,"},{"speaker":"and here's why","startTime":322.3,"endTime":324.76,"body":"that would be just the"},{"speaker":"and here's why","startTime":322.3,"endTime":324.76,"body":"horizontal, the width there,"},{"speaker":"and here's why","startTime":324.76,"endTime":326.073,"body":"divided by the radius."},{"speaker":"and here's why","startTime":327.85,"endTime":331.8,"body":"If you are trying to gather"},{"speaker":"and here's why","startTime":327.85,"endTime":331.8,"body":"together information,"},{"speaker":"and here's why","startTime":331.8,"endTime":334.84,"body":"I'm going to write down all the"},{"speaker":"and here's why","startTime":331.8,"endTime":334.84,"body":"chord lengths I need to know,"},{"speaker":"and here's why","startTime":334.84,"endTime":335.673,"body":"when you're doing that,"},{"speaker":"and here's why","startTime":335.673,"endTime":338.85,"body":"the chord length depends on the angle,"},{"speaker":"and here's why","startTime":338.85,"endTime":342.41,"body":"but also on the radius of"},{"speaker":"and here's why","startTime":338.85,"endTime":342.41,"body":"the circle that you're in,"},{"speaker":"and here's why","startTime":342.41,"endTime":345.29,"body":"whereas the sine of the angle,"},{"speaker":"and here's why","startTime":345.29,"endTime":346.36,"body":"that's already a ratio,"},{"speaker":"and here's why","startTime":346.36,"endTime":348.8,"body":"you don't need to know"},{"speaker":"and here's why","startTime":346.36,"endTime":348.8,"body":"the radius of the circle"},{"speaker":"and here's why","startTime":348.8,"endTime":350.11,"body":"ahead of time."},{"speaker":"and here's why","startTime":350.11,"endTime":352.58,"body":"If you then want to know"},{"speaker":"and here's why","startTime":350.11,"endTime":352.58,"body":"the actual chord length,"},{"speaker":"and here's why","startTime":352.58,"endTime":354.52,"body":"you multiply by the radius of the circle."},{"speaker":"and here's why","startTime":354.52,"endTime":359.24,"body":"So somehow, it's more convenient"},{"speaker":"and here's why","startTime":354.52,"endTime":359.24,"body":"to have the sine function,"},{"speaker":"and here's why","startTime":359.24,"endTime":360.073,"body":"because, then,"},{"speaker":"and here's why","startTime":360.073,"endTime":360.97,"body":"if you were trying to tabulate"},{"speaker":"and here's why","startTime":360.97,"endTime":362.373,"body":"and get a list of values,"},{"speaker":"and here's why","startTime":363.23,"endTime":364.44,"body":"if you're doing chord lengths,"},{"speaker":"and here's why","startTime":364.44,"endTime":367.29,"body":"it's implicit that somehow"},{"speaker":"and here's why","startTime":364.44,"endTime":367.29,"body":"you have to make an assumption"},{"speaker":"and here's why","startTime":367.29,"endTime":368.47,"body":"about the radius of your circle"},{"speaker":"and here's why","startTime":368.47,"endTime":369.41,"body":"when you're doing it,"},{"speaker":"and here's why","startTime":369.41,"endTime":373.91,"body":"so your list of values has"},{"speaker":"and here's why","startTime":369.41,"endTime":373.91,"body":"a fixed radius in mind."},{"speaker":"and here's why","startTime":373.91,"endTime":376.42,"body":"You can then scale it up"},{"speaker":"and here's why","startTime":373.91,"endTime":376.42,"body":"or down if you want to,"},{"speaker":"and here's why","startTime":376.42,"endTime":379.15,"body":"but the sine function doesn't"},{"speaker":"and here's why","startTime":376.42,"endTime":379.15,"body":"have that disadvantage,"},{"speaker":"and here's why","startTime":379.15,"endTime":380.35,"body":"so that's probably why"},{"speaker":"and here's why","startTime":380.35,"endTime":383.193,"body":"sine won out over chord"},{"speaker":"and here's why","startTime":380.35,"endTime":383.193,"body":"lengths in the end."},{"speaker":"and here's why","startTime":384.88,"endTime":386.02,"body":"So yeah, that's why,"},{"speaker":"and here's why","startTime":386.02,"endTime":387.94,"body":"'cause you just have to know the angle"},{"speaker":"and here's why","startTime":387.94,"endTime":391.38,"body":"to find out the sine of the angle there,"},{"speaker":"and here's why","startTime":391.38,"endTime":393.603,"body":"you don't need to know"},{"speaker":"and here's why","startTime":391.38,"endTime":393.603,"body":"the radius to begin with."},{"speaker":"and here's why","startTime":394.62,"endTime":397.02,"body":"This diagram also tells us"},{"speaker":"and here's why","startTime":397.02,"endTime":400.72,"body":"how sine got its name, because,"},{"speaker":"and here's why","startTime":400.72,"endTime":401.95,"body":"well, I'll show you this diagram"},{"speaker":"and here's why","startTime":401.95,"endTime":405.093,"body":"superimposed on a photograph right now"},{"speaker":"and here's why","startTime":405.093,"endTime":406.36,"body":"just to show you,"},{"speaker":"and here's why","startTime":406.36,"endTime":407.683,"body":"this is the archery link."},{"speaker":"and here's why","startTime":408.66,"endTime":410.75,"body":"That picture that we just saw,"},{"speaker":"and here's why","startTime":410.75,"endTime":413.96,"body":"if you imagine the arc of"},{"speaker":"and here's why","startTime":410.75,"endTime":413.96,"body":"the circle is now the bow,"},{"speaker":"and here's why","startTime":413.96,"endTime":416.61,"body":"and the bowstring would"},{"speaker":"and here's why","startTime":413.96,"endTime":416.61,"body":"hang vertically down,"},{"speaker":"and here's why","startTime":416.61,"endTime":419.12,"body":"but when you draw it, when"},{"speaker":"and here's why","startTime":416.61,"endTime":419.12,"body":"you pull it back like that,"},{"speaker":"and here's why","startTime":419.12,"endTime":421.24,"body":"I'm standing with my bow, like that,"},{"speaker":"and here's why","startTime":421.24,"endTime":424.3,"body":"you get the radii of the circle,"},{"speaker":"and here's why","startTime":424.3,"endTime":426.11,"body":"and then the horizontal part,"},{"speaker":"and here's why","startTime":426.11,"endTime":427.86,"body":"which was our diameter,"},{"speaker":"and here's why","startTime":427.86,"endTime":430.88,"body":"that's the arrow, or part of your arm."},{"speaker":"and here's why","startTime":430.88,"endTime":432.83,"body":"So the whole diagram"},{"speaker":"and here's why","startTime":432.83,"endTime":435.24,"body":"looks like a bow and arrow being drawn,"},{"speaker":"and here's why","startTime":435.24,"endTime":438.44,"body":"and the bit we're interested"},{"speaker":"and here's why","startTime":435.24,"endTime":438.44,"body":"in, the chord length,"},{"speaker":"and here's why","startTime":438.44,"endTime":440.62,"body":"is the bowstring,"},{"speaker":"and here's why","startTime":440.62,"endTime":442.98,"body":"so the Sanskrit word for bowstring"},{"speaker":"and here's why","startTime":442.98,"endTime":445.75,"body":"is what was used for sine originally."},{"speaker":"and here's why","startTime":445.75,"endTime":449.86,"body":"Now, when those texts got"},{"speaker":"and here's why","startTime":445.75,"endTime":449.86,"body":"translated into Arabic,"},{"speaker":"and here's why","startTime":449.86,"endTime":452.363,"body":"as the knowledge passed"},{"speaker":"and here's why","startTime":449.86,"endTime":452.363,"body":"through the ancient world,"},{"speaker":"and here's why","startTime":453.58,"endTime":455.61,"body":"they transliterated that word,"},{"speaker":"and here's why","startTime":455.61,"endTime":457.17,"body":"they didn't try and translate bowstring,"},{"speaker":"and here's why","startTime":457.17,"endTime":459.11,"body":"they just transliterated"},{"speaker":"and here's why","startTime":459.11,"endTime":462.25,"body":"and wrote down the letters"},{"speaker":"and here's why","startTime":459.11,"endTime":462.25,"body":"that would make the same sound,"},{"speaker":"and here's why","startTime":462.25,"endTime":465.17,"body":"and so we got, in Arabic, jiba,"},{"speaker":"and here's why","startTime":465.17,"endTime":466.003,"body":"something like this."},{"speaker":"and here's why","startTime":466.003,"endTime":467.45,"body":"That's not a word in Arabic,"},{"speaker":"and here's why","startTime":467.45,"endTime":469.6,"body":"so then as these things got recopied,"},{"speaker":"and here's why","startTime":469.6,"endTime":473.03,"body":"the scribes would sort of"},{"speaker":"and here's why","startTime":469.6,"endTime":473.03,"body":"go to the nearest word,"},{"speaker":"and here's why","startTime":473.03,"endTime":475.0,"body":"which was jayb,"},{"speaker":"and here's why","startTime":475.0,"endTime":476.39,"body":"and that means cavity."},{"speaker":"and here's why","startTime":476.39,"endTime":479.21,"body":"And so the final step"},{"speaker":"and here's why","startTime":476.39,"endTime":479.21,"body":"is that when, in turn,"},{"speaker":"and here's why","startTime":479.21,"endTime":482.72,"body":"those Arabic texts got"},{"speaker":"and here's why","startTime":479.21,"endTime":482.72,"body":"translated into Latin,"},{"speaker":"and here's why","startTime":482.72,"endTime":484.8,"body":"they translated that word cavity"},{"speaker":"and here's why","startTime":484.8,"endTime":486.5,"body":"into the Latin word for cavity,"},{"speaker":"and here's why","startTime":486.5,"endTime":487.91,"body":"which is sinus,"},{"speaker":"and here's why","startTime":487.91,"endTime":489.3,"body":"and that's how we get like,"},{"speaker":"and here's why","startTime":489.3,"endTime":490.33,"body":"we all have sinuses, right,"},{"speaker":"and here's why","startTime":490.33,"endTime":491.75,"body":"that's how we get this word sinus,"},{"speaker":"and here's why","startTime":491.75,"endTime":494.49,"body":"and then in English it becomes sine."},{"speaker":"and here's why","startTime":494.49,"endTime":497.82,"body":"So that just tiny, little vignette"},{"speaker":"and here's why","startTime":497.82,"endTime":498.73,"body":"shows us something"},{"speaker":"and here's why","startTime":498.73,"endTime":500.87,"body":"about how mathematical"},{"speaker":"and here's why","startTime":498.73,"endTime":500.87,"body":"knowledge is transferred,"},{"speaker":"and here's why","startTime":500.87,"endTime":502.27,"body":"from the etymology of that word,"},{"speaker":"and here's why","startTime":502.27,"endTime":504.07,"body":"I think it's fascinating."},{"speaker":"and here's why","startTime":504.07,"endTime":504.91,"body":"But we're not here to talk about that,"},{"speaker":"and here's why","startTime":504.91,"endTime":506.49,"body":"we're here to talk about sine,"},{"speaker":"and here's why","startTime":506.49,"endTime":508.27,"body":"so let's do so."},{"speaker":"and here's why","startTime":508.27,"endTime":512.47,"body":"We can see from the origin story of sine"},{"speaker":"and here's why","startTime":512.47,"endTime":515.06,"body":"how it is that we can get sine of angles"},{"speaker":"and here's why","startTime":515.06,"endTime":517.1,"body":"that are bigger than 90 degrees,"},{"speaker":"and here's why","startTime":517.1,"endTime":519.25,"body":"that don't occur in"},{"speaker":"and here's why","startTime":517.1,"endTime":519.25,"body":"right-angled triangles,"},{"speaker":"because we can think of it like this","startTime":522.66,"endTime":526.33,"body":"what is the sine of that"},{"speaker":"because we can think of it like this","startTime":522.66,"endTime":526.33,"body":"angle theta telling us?"},{"speaker":"because we can think of it like this","startTime":526.33,"endTime":528.08,"body":"Well let's take a unit circle,"},{"speaker":"because we can think of it like this","startTime":528.08,"endTime":529.8,"body":"a circle of radius one,"},{"speaker":"because we can think of it like this","startTime":529.8,"endTime":532.3,"body":"then sin-theta is just telling us"},{"speaker":"because we can think of it like this","startTime":532.3,"endTime":536.99,"body":"exactly the height above the horizon,"},{"speaker":"because we can think of it like this","startTime":536.99,"endTime":538.36,"body":"of the horizontal diameter,"},{"speaker":"because we can think of it like this","startTime":538.36,"endTime":541.09,"body":"the height of that point"},{"speaker":"because we can think of it like this","startTime":538.36,"endTime":541.09,"body":"on the circumference."},{"speaker":"because we can think of it like this","startTime":541.09,"endTime":543.17,"body":"So we can imagine taking a radius,"},{"speaker":"because we can think of it like this","startTime":543.17,"endTime":546.0,"body":"and just rotating it around the circle,"},{"speaker":"because we can think of it like this","startTime":546.0,"endTime":548.9,"body":"and inside a circle of radius one,"},{"speaker":"because we can think of it like this","startTime":548.9,"endTime":551.83,"body":"the height of where the"},{"speaker":"because we can think of it like this","startTime":548.9,"endTime":551.83,"body":"radius meets the circumference"},{"speaker":"because we can think of it like this","startTime":551.83,"endTime":553.44,"body":"will exactly be sin-theta."},{"speaker":"because we can think of it like this","startTime":553.44,"endTime":556.32,"body":"And so we can find sine"},{"speaker":"because we can think of it like this","startTime":553.44,"endTime":556.32,"body":"of zero very easily,"},{"speaker":"because we can think of it like this","startTime":556.32,"endTime":559.65,"body":"that's when the radius is"},{"speaker":"because we can think of it like this","startTime":556.32,"endTime":559.65,"body":"just flat on the horizontal,"},{"speaker":"because we can think of it like this","startTime":559.65,"endTime":561.17,"body":"so of course sine of zero must be zero,"},{"speaker":"because we can think of it like this","startTime":561.17,"endTime":563.42,"body":"it's at height zero above the horizontal."},{"speaker":"because we can think of it like this","startTime":563.42,"endTime":564.65,"body":"And as we go round, and round, and round,"},{"speaker":"because we can think of it like this","startTime":564.65,"endTime":566.34,"body":"we get to the vertical,"},{"speaker":"because we can think of it like this","startTime":566.34,"endTime":568.71,"body":"and when it's vertical, so 90 degrees,"},{"speaker":"because we can think of it like this","startTime":568.71,"endTime":570.5,"body":"the height is exactly one radius,"},{"speaker":"because we can think of it like this","startTime":570.5,"endTime":572.56,"body":"and this is a circle of radius one,"},{"speaker":"because we can think of it like this","startTime":572.56,"endTime":573.59,"body":"so it's exactly one,"},{"speaker":"because we can think of it like this","startTime":573.59,"endTime":575.15,"body":"so sine of 90 is 1."},{"speaker":"because we can think of it like this","startTime":575.15,"endTime":576.94,"body":"And then we can keep going and going,"},{"speaker":"because we can think of it like this","startTime":576.94,"endTime":578.35,"body":"there's nothing to stop us,"},{"speaker":"because we can think of it like this","startTime":578.35,"endTime":580.43,"body":"and the height starts decreasing again"},{"speaker":"because we can think of it like this","startTime":580.43,"endTime":582.66,"body":"as we get back down to the horizontal,"},{"speaker":"because we can think of it like this","startTime":582.66,"endTime":584.32,"body":"so at 180 degrees,"},{"speaker":"because we can think of it like this","startTime":584.32,"endTime":586.71,"body":"sine of 180 will also be 0,"},{"speaker":"because we can think of it like this","startTime":586.71,"endTime":589.94,"body":"'cause it's zero height"},{"speaker":"because we can think of it like this","startTime":586.71,"endTime":589.94,"body":"above the horizontal."},{"speaker":"because we can think of it like this","startTime":589.94,"endTime":591.12,"body":"And we can keep going,"},{"speaker":"because we can think of it like this","startTime":591.12,"endTime":592.95,"body":"my arm can't, but we can,"},{"speaker":"because we can think of it like this","startTime":592.95,"endTime":595.86,"body":"and then we go underneath the horizontal,"},{"speaker":"because we can think of it like this","startTime":595.86,"endTime":598.04,"body":"so our height above it becomes negative,"},{"speaker":"because we can think of it like this","startTime":598.04,"endTime":599.19,"body":"we have negative heights,"},{"speaker":"because we can think of it like this","startTime":599.19,"endTime":601.62,"body":"so that's when sine can"},{"speaker":"because we can think of it like this","startTime":599.19,"endTime":601.62,"body":"take negative values"},{"speaker":"because we can think of it like this","startTime":601.62,"endTime":604.267,"body":"all the way down to -1 at 270 degrees,"},{"speaker":"because we can think of it like this","startTime":604.267,"endTime":605.61,"body":"and then back upwards again,"},{"speaker":"because we can think of it like this","startTime":605.61,"endTime":607.26,"body":"ad we get back to zero."},{"speaker":"because we can think of it like this","startTime":607.26,"endTime":612.26,"body":"So this origin of how we"},{"speaker":"because we can think of it like this","startTime":607.26,"endTime":612.26,"body":"came to use the sine function"},{"speaker":"because we can think of it like this","startTime":612.93,"endTime":615.33,"body":"tells us how to get sine of angles"},{"speaker":"because we can think of it like this","startTime":615.33,"endTime":617.463,"body":"other than those strictly between 0 and 90"},{"speaker":"because we can think of it like this","startTime":617.463,"endTime":620.98,"body":"in a very natural way."},{"speaker":"because we can think of it like this","startTime":620.98,"endTime":624.21,"body":"I'm talking there in"},{"speaker":"because we can think of it like this","startTime":620.98,"endTime":624.21,"body":"degrees, you've noticed,"},{"speaker":"because we can think of it like this","startTime":624.21,"endTime":625.41,"body":"that's because, for most of us,"},{"speaker":"because we can think of it like this","startTime":625.41,"endTime":627.81,"body":"that's the first kind"},{"speaker":"because we can think of it like this","startTime":625.41,"endTime":627.81,"body":"of measurement of angles"},{"speaker":"because we can think of it like this","startTime":627.81,"endTime":628.643,"body":"that we learn,"},{"speaker":"because we can think of it like this","startTime":628.643,"endTime":631.34,"body":"and I will mostly talk in"},{"speaker":"because we can think of it like this","startTime":628.643,"endTime":631.34,"body":"degrees for that today,"},{"speaker":"because we can think of it like this","startTime":631.34,"endTime":634.03,"body":"but I've used that word mundane."},{"speaker":"because we can think of it like this","startTime":634.03,"endTime":635.3,"body":"What I really mean by that,"},{"speaker":"because we can think of it like this","startTime":635.3,"endTime":637.69,"body":"360 degrees in a circle"},{"speaker":"because we can think of it like this","startTime":637.69,"endTime":642.56,"body":"is a very Earth-centered measurement unit."},{"speaker":"because we can think of it like this","startTime":642.56,"endTime":646.17,"body":"Dividing circles into 360 equal parts"},{"speaker":"because we can think of it like this","startTime":646.17,"endTime":648.2,"body":"is done on the planet Earth"},{"speaker":"because we can think of it like this","startTime":648.2,"endTime":650.12,"body":"because early astronomers noticed"},{"speaker":"because we can think of it like this","startTime":650.12,"endTime":652.87,"body":"there were about 360 days in a year,"},{"speaker":"because we can think of it like this","startTime":652.87,"endTime":655.97,"body":"that's the reason we have"},{"speaker":"because we can think of it like this","startTime":652.87,"endTime":655.97,"body":"360 degrees in a circle."},{"speaker":"because we can think of it like this","startTime":655.97,"endTime":659.56,"body":"So that's not a very mathematical reason,"},{"speaker":"because we can think of it like this","startTime":659.56,"endTime":661.98,"body":"I'm fine with it, we love all the history,"},{"speaker":"because we can think of it like this","startTime":661.98,"endTime":664.19,"body":"but actually, mathematically speaking,"},{"speaker":"because we can think of it like this","startTime":664.19,"endTime":666.117,"body":"there's a slightly better,"},{"speaker":"because we can think of it like this","startTime":668.688,"endTime":671.56,"body":"a way that makes sense more mathematically"},{"speaker":"because we can think of it like this","startTime":671.56,"endTime":672.88,"body":"of measuring angles."},{"speaker":"because we can think of it like this","startTime":672.88,"endTime":674.63,"body":"So I just thought I'd mention that to you"},{"speaker":"because we can think of it like this","startTime":674.63,"endTime":675.89,"body":"because that allows me"},{"speaker":"because we can think of it like this","startTime":675.89,"endTime":677.837,"body":"to accidentally say"},{"speaker":"because we can think of it like this","startTime":675.89,"endTime":677.837,"body":"that instead of degrees"},{"speaker":"because we can think of it like this","startTime":677.837,"endTime":679.74,"body":"and not get into trouble."},{"speaker":"because we can think of it like this","startTime":679.74,"endTime":681.59,"body":"But this is what you will start using"},{"speaker":"because we can think of it like this","startTime":682.732,"endTime":685.01,"body":"if study mathematics at"},{"speaker":"because we can think of it like this","startTime":682.732,"endTime":685.01,"body":"university or beyond,"},{"speaker":"because we can think of it like this","startTime":685.01,"endTime":688.58,"body":"you tend to switch to this"},{"speaker":"because we can think of it like this","startTime":685.01,"endTime":688.58,"body":"other way of measuring angles."},{"speaker":"because we can think of it like this","startTime":688.58,"endTime":691.43,"body":"The other way of measuring"},{"speaker":"because we can think of it like this","startTime":688.58,"endTime":691.43,"body":"an angle is to say,"},{"speaker":"because we can think of it like this","startTime":691.43,"endTime":694.26,"body":"how far around the"},{"speaker":"because we can think of it like this","startTime":691.43,"endTime":694.26,"body":"circumference have I traveled,"},{"speaker":"because we can think of it like this","startTime":694.26,"endTime":695.66,"body":"has this point gone,"},{"speaker":"because we can think of it like this","startTime":695.66,"endTime":697.343,"body":"in this circle of radius one."},{"speaker":"because we can think of it like this","startTime":699.445,"endTime":702.22,"body":"The circumference of a"},{"speaker":"because we can think of it like this","startTime":699.445,"endTime":702.22,"body":"circle of radius r, two-pi-r,"},{"speaker":"because we can think of it like this","startTime":702.22,"endTime":704.3,"body":"so if the radius, r, equals one,"},{"speaker":"because we can think of it like this","startTime":704.3,"endTime":705.76,"body":"then the whole way"},{"speaker":"because we can think of it like this","startTime":704.3,"endTime":705.76,"body":"around the circumference"},{"speaker":"because we can think of it like this","startTime":705.76,"endTime":707.283,"body":"is a distance of two-pi."},{"speaker":"because we can think of it like this","startTime":708.63,"endTime":711.59,"body":"So we will say our unit"},{"speaker":"because we can think of it like this","startTime":708.63,"endTime":711.59,"body":"of measuring angles"},{"speaker":"because we can think of it like this","startTime":711.59,"endTime":713.3,"body":"is something called the radian,"},{"speaker":"because we can think of it like this","startTime":713.3,"endTime":716.79,"body":"it's just the distance traveled"},{"speaker":"because we can think of it like this","startTime":713.3,"endTime":716.79,"body":"around the circumference"},{"speaker":"because we can think of it like this","startTime":716.79,"endTime":717.9,"body":"in the unit circle"},{"speaker":"because we can think of it like this","startTime":717.9,"endTime":719.25,"body":"corresponding to that angle."},{"speaker":"because we can think of it like this","startTime":719.25,"endTime":721.95,"body":"So if you travel the whole 360 degrees,"},{"speaker":"because we can think of it like this","startTime":721.95,"endTime":723.89,"body":"then you've traveled a distance of two-pi,"},{"speaker":"because we can think of it like this","startTime":723.89,"endTime":727.04,"body":"therefore that angle is two-pi radians."},{"speaker":"because we can think of it like this","startTime":727.04,"endTime":730.43,"body":"If you only travel a quarter"},{"speaker":"because we can think of it like this","startTime":727.04,"endTime":730.43,"body":"of the way around, 90 degrees,"},{"speaker":"because we can think of it like this","startTime":730.43,"endTime":732.5,"body":"a quarter of two-pi traveled,"},{"speaker":"because we can think of it like this","startTime":732.5,"endTime":734.79,"body":"so 1/2 pi radians,"},{"speaker":"because we can think of it like this","startTime":734.79,"endTime":736.62,"body":"that's a right angle."},{"speaker":"because we can think of it like this","startTime":736.62,"endTime":740.18,"body":"So that gives you a bit information,"},{"speaker":"because we can think of it like this","startTime":740.18,"endTime":744.51,"body":"it's mathematically a nice"},{"speaker":"because we can think of it like this","startTime":740.18,"endTime":744.51,"body":"unit to use for angles"},{"speaker":"because we can think of it like this","startTime":744.51,"endTime":747.35,"body":"because if you have your angle in radians,"},{"speaker":"because we can think of it like this","startTime":747.35,"endTime":748.41,"body":"then the angle itself"},{"speaker":"because we can think of it like this","startTime":748.41,"endTime":751.06,"body":"tells you how far you've traveled"},{"speaker":"because we can think of it like this","startTime":748.41,"endTime":751.06,"body":"around the circumference,"},{"speaker":"because we can think of it like this","startTime":751.06,"endTime":752.02,"body":"sine of the angle"},{"speaker":"because we can think of it like this","startTime":752.02,"endTime":756.27,"body":"tells you your height"},{"speaker":"because we can think of it like this","startTime":752.02,"endTime":756.27,"body":"above the horizontal,"},{"speaker":"because we can think of it like this","startTime":756.27,"endTime":757.88,"body":"and actually, cosine of the angle"},{"speaker":"because we can think of it like this","startTime":757.88,"endTime":760.75,"body":"tells you how far to the right"},{"speaker":"because we can think of it like this","startTime":757.88,"endTime":760.75,"body":"of the vertical line you are."},{"speaker":"because we can think of it like this","startTime":760.75,"endTime":762.93,"body":"So if you're plotting"},{"speaker":"because we can think of it like this","startTime":760.75,"endTime":762.93,"body":"these as coordinates,"},{"speaker":"because we can think of it like this","startTime":762.93,"endTime":764.98,"body":"you'd have the coordinates of that point"},{"speaker":"because we can think of it like this","startTime":764.98,"endTime":766.89,"body":"and you'd have the distance traveled"},{"speaker":"because we can think of it like this","startTime":766.89,"endTime":768.12,"body":"just by knowing the angle,"},{"speaker":"because we can think of it like this","startTime":768.12,"endTime":769.133,"body":"which is quite nice."},{"speaker":"because we can think of it like this","startTime":770.09,"endTime":772.83,"body":"And this also gives us our sine curve,"},{"speaker":"because we can think of it like this","startTime":772.83,"endTime":774.17,"body":"because you can draw,"},{"speaker":"because we can think of it like this","startTime":774.17,"endTime":775.83,"body":"and we'll do it in a moment,"},{"speaker":"because we can think of it like this","startTime":775.83,"endTime":778.41,"body":"if you take your angle,"},{"speaker":"because we can think of it like this","startTime":778.41,"endTime":780.86,"body":"the picture in a minute"},{"speaker":"because we can think of it like this","startTime":778.41,"endTime":780.86,"body":"has it measured in radians,"},{"speaker":"because we can think of it like this","startTime":780.86,"endTime":784.0,"body":"so it goes from zero to"},{"speaker":"because we can think of it like this","startTime":780.86,"endTime":784.0,"body":"two-pi, a full circle,"},{"speaker":"because we can think of it like this","startTime":784.0,"endTime":786.23,"body":"and you just work up the angles,"},{"speaker":"because we can think of it like this","startTime":786.23,"endTime":789.27,"body":"and the radius travels around the circle,"},{"speaker":"because we can think of it like this","startTime":789.27,"endTime":791.32,"body":"and what we're going to plot on our graph"},{"speaker":"because we can think of it like this","startTime":791.32,"endTime":794.12,"body":"is just the height above the horizontal,"},{"speaker":"because we can think of it like this","startTime":794.12,"endTime":796.07,"body":"I guess it will now be the x axis,"},{"speaker":"because we can think of it like this","startTime":796.07,"endTime":800.76,"body":"the height of that point above"},{"speaker":"because we can think of it like this","startTime":796.07,"endTime":800.76,"body":"the x axis, the horizontal."},{"speaker":"because we can think of it like this","startTime":800.76,"endTime":802.42,"body":"And you'll see when we do that,"},{"speaker":"because we can think of it like this","startTime":802.42,"endTime":805.23,"body":"there's it going from zero"},{"speaker":"because we can think of it like this","startTime":802.42,"endTime":805.23,"body":"all the way up to two-pi,"},{"speaker":"because we can think of it like this","startTime":805.23,"endTime":806.63,"body":"that's exactly what happened."},{"speaker":"because we can think of it like this","startTime":806.63,"endTime":807.51,"body":"Just follow it around,"},{"speaker":"because we can think of it like this","startTime":807.51,"endTime":809.26,"body":"it'll happen again in a moment,"},{"speaker":"because we can think of it like this","startTime":809.26,"endTime":811.61,"body":"we just see that point"},{"speaker":"because we can think of it like this","startTime":809.26,"endTime":811.61,"body":"going around the circle,"},{"speaker":"because we can think of it like this","startTime":812.66,"endTime":816.81,"body":"it's almost like we're unrolling"},{"speaker":"because we can think of it like this","startTime":812.66,"endTime":816.81,"body":"the circle onto a graph."},{"speaker":"because we can think of it like this","startTime":816.81,"endTime":817.84,"body":"It's going to go again now,"},{"speaker":"because we can think of it like this","startTime":817.84,"endTime":820.5,"body":"we're just plotting the height,"},{"speaker":"because we can think of it like this","startTime":820.5,"endTime":821.333,"body":"and there we are,"},{"speaker":"because we can think of it like this","startTime":821.333,"endTime":822.8,"body":"so that is our sine curve."},{"speaker":"because we can think of it like this","startTime":822.8,"endTime":825.413,"body":"And actually, once you"},{"speaker":"because we can think of it like this","startTime":822.8,"endTime":825.413,"body":"start doing it like this,"},{"speaker":"because we can think of it like this","startTime":826.399,"endTime":829.31,"body":"you don't have to stop when"},{"speaker":"because we can think of it like this","startTime":826.399,"endTime":829.31,"body":"you get back to the beginning,"},{"speaker":"because we can think of it like this","startTime":829.31,"endTime":830.91,"body":"you could keep going if you like,"},{"speaker":"because we can think of it like this","startTime":830.91,"endTime":831.94,"body":"and that would mean the graph"},{"speaker":"because we can think of it like this","startTime":831.94,"endTime":833.562,"body":"can carry on going, and going, and going,"},{"speaker":"because we can think of it like this","startTime":833.562,"endTime":836.42,"body":"and you can get just these"},{"speaker":"because we can think of it like this","startTime":833.562,"endTime":836.42,"body":"nice, infinite undulations"},{"speaker":"because we can think of it like this","startTime":836.42,"endTime":837.51,"body":"of the sine curve."},{"speaker":"because we can think of it like this","startTime":837.51,"endTime":839.391,"body":"You could even change"},{"speaker":"because we can think of it like this","startTime":837.51,"endTime":839.391,"body":"direction and go backwards,"},{"speaker":"because we can think of it like this","startTime":839.391,"endTime":841.0,"body":"and that would give you negative angles,"},{"speaker":"because we can think of it like this","startTime":841.0,"endTime":844.23,"body":"so it can carry on infinitely"},{"speaker":"because we can think of it like this","startTime":841.0,"endTime":844.23,"body":"in both directions."},{"speaker":"because we can think of it like this","startTime":844.23,"endTime":847.16,"body":"So that is where we get"},{"speaker":"because we can think of it like this","startTime":844.23,"endTime":847.16,"body":"the sine curve from,"},{"speaker":"because we can think of it like this","startTime":847.16,"endTime":849.65,"body":"and it's to do with the"},{"speaker":"because we can think of it like this","startTime":847.16,"endTime":849.65,"body":"way of thinking of sine"},{"speaker":"because we can think of it like this","startTime":849.65,"endTime":852.173,"body":"as being a height in a circle."},{"speaker":"because we can think of it like this","startTime":853.67,"endTime":857.27,"body":"So we've said the beginnings of the study"},{"speaker":"because we can think of it like this","startTime":857.27,"endTime":859.34,"body":"of what we now call trigonometry"},{"speaker":"because we can think of it like this","startTime":859.34,"endTime":860.37,"body":"were really chord lengths"},{"speaker":"because we can think of it like this","startTime":860.37,"endTime":863.14,"body":"to do with astronomical calculations,"},{"speaker":"because we can think of it like this","startTime":863.14,"endTime":866.28,"body":"and there are loads and"},{"speaker":"because we can think of it like this","startTime":863.14,"endTime":866.28,"body":"loads of applications"},{"speaker":"because we can think of it like this","startTime":866.28,"endTime":867.8,"body":"throughout the centuries"},{"speaker":"because we can think of it like this","startTime":867.8,"endTime":870.83,"body":"of these trigonometrical functions."},{"speaker":"because we can think of it like this","startTime":870.83,"endTime":872.46,"body":"It took a while for things to settle down"},{"speaker":"because we can think of it like this","startTime":872.46,"endTime":875.24,"body":"for them to be fixed on sin, cos, and tan,"},{"speaker":"because we can think of it like this","startTime":875.24,"endTime":877.06,"body":"and one over sine, and all over those,"},{"speaker":"because we can think of it like this","startTime":877.06,"endTime":879.7,"body":"but throughout this time,"},{"speaker":"because we can think of it like this","startTime":879.7,"endTime":880.87,"body":"many, many applications."},{"speaker":"because we can think of it like this","startTime":880.87,"endTime":881.823,"body":"This is, for example,"},{"speaker":"because we can think of it like this","startTime":881.823,"endTime":883.89,"body":"this is a picture of astronomers"},{"speaker":"because we can think of it like this","startTime":883.89,"endTime":887.64,"body":"working at the famous Istanbul Observatory"},{"speaker":"because we can think of it like this","startTime":887.64,"endTime":889.37,"body":"at the time of the Ottoman Empire,"},{"speaker":"because we can think of it like this","startTime":889.37,"endTime":891.46,"body":"all of their mathematical instruments,"},{"speaker":"because we can think of it like this","startTime":891.46,"endTime":894.653,"body":"they were needing to know"},{"speaker":"because we can think of it like this","startTime":891.46,"endTime":894.653,"body":"about sines and coses."},{"speaker":"because we can think of it like this","startTime":896.27,"endTime":897.103,"body":"Another application,"},{"speaker":"because we can think of it like this","startTime":897.103,"endTime":899.55,"body":"another use of trigonometrical functions"},{"speaker":"because we can think of it like this","startTime":899.55,"endTime":900.79,"body":"is in navigation."},{"speaker":"because we can think of it like this","startTime":900.79,"endTime":901.623,"body":"This ship, by the way,"},{"speaker":"because we can think of it like this","startTime":901.623,"endTime":902.86,"body":"it's a lovely picture,"},{"speaker":"because we can think of it like this","startTime":902.86,"endTime":906.14,"body":"it hasn't got any particular"},{"speaker":"because we can think of it like this","startTime":902.86,"endTime":906.14,"body":"mathematical importance"},{"speaker":"because we can think of it like this","startTime":906.14,"endTime":907.033,"body":"on its own, this ship,"},{"speaker":"because we can think of it like this","startTime":907.033,"endTime":908.9,"body":"it's just a nice picture of a ship"},{"speaker":"because we can think of it like this","startTime":908.9,"endTime":910.05,"body":"whose name I like,"},{"speaker":"because we can think of it like this","startTime":910.05,"endTime":912.58,"body":"because the name of this ship"},{"speaker":"because we can think of it like this","startTime":910.05,"endTime":912.58,"body":"is the Peter Pomegranate,"},{"speaker":"because we can think of it like this","startTime":912.58,"endTime":914.219,"body":"which I think is a great name for a ship,"},{"speaker":"because we can think of it like this","startTime":914.219,"endTime":918.58,"body":"it was built in 1510"},{"speaker":"because we can think of it like this","startTime":914.219,"endTime":918.58,"body":"for Henry Tudor's navy,"},{"speaker":"because we can think of it like this","startTime":918.58,"endTime":920.69,"body":"but this is just a picture to illustrate"},{"speaker":"because we can think of it like this","startTime":920.69,"endTime":924.83,"body":"navigation requires the use of triangles,"},{"speaker":"because we can think of it like this","startTime":924.83,"endTime":927.5,"body":"and therefore trigonometry,"},{"speaker":"because we can think of it like this","startTime":924.83,"endTime":927.5,"body":"and sines, and cosines,"},{"speaker":"because we can think of it like this","startTime":927.5,"endTime":928.55,"body":"and things like this."},{"speaker":"because we can think of it like this","startTime":929.4,"endTime":930.443,"body":"I should say,"},{"speaker":"because we can think of it like this","startTime":932.1,"endTime":935.64,"body":"the sine curve and the cosine"},{"speaker":"because we can think of it like this","startTime":932.1,"endTime":935.64,"body":"curve are the same curve,"},{"speaker":"because we can think of it like this","startTime":935.64,"endTime":937.59,"body":"just one is shifted along a bit,"},{"speaker":"because we can think of it like this","startTime":937.59,"endTime":940.23,"body":"because just as you can measure"},{"speaker":"because we can think of it like this","startTime":937.59,"endTime":940.23,"body":"the height of that radius"},{"speaker":"because we can think of it like this","startTime":940.23,"endTime":942.56,"body":"above the horizontal,"},{"speaker":"because we can think of it like this","startTime":942.56,"endTime":945.53,"body":"you could equally measure the"},{"speaker":"because we can think of it like this","startTime":942.56,"endTime":945.53,"body":"width of it from the vertical,"},{"speaker":"because we can think of it like this","startTime":945.53,"endTime":948.38,"body":"and you'd get exactly the same graph,"},{"speaker":"because we can think of it like this","startTime":948.38,"endTime":951.05,"body":"but just shifted by 90 degrees,"},{"speaker":"because we can think of it like this","startTime":951.05,"endTime":951.883,"body":"if you just think about"},{"speaker":"because we can think of it like this","startTime":951.883,"endTime":953.66,"body":"rotating that whole picture 90 degrees,"},{"speaker":"because we can think of it like this","startTime":953.66,"endTime":954.493,"body":"that's what you'd get,"},{"speaker":"because we can think of it like this","startTime":954.493,"endTime":957.44,"body":"so sine curves and cosine"},{"speaker":"because we can think of it like this","startTime":954.493,"endTime":957.44,"body":"curves are the same things,"},{"speaker":"because we can think of it like this","startTime":957.44,"endTime":958.733,"body":"just shifted a bit."},{"speaker":"because we can think of it like this","startTime":960.395,"endTime":963.98,"body":"Why might you use"},{"speaker":"because we can think of it like this","startTime":960.395,"endTime":963.98,"body":"trigonometry in navigation?"},{"speaker":"because we can think of it like this","startTime":963.98,"endTime":966.4,"body":"I'll just give you a"},{"speaker":"because we can think of it like this","startTime":963.98,"endTime":966.4,"body":"silly, little example."},{"speaker":"because we can think of it like this","startTime":966.4,"endTime":968.75,"body":"Suppose you set off from your home port,"},{"speaker":"because we can think of it like this","startTime":968.75,"endTime":969.89,"body":"and you have to decide on a direction."},{"speaker":"because we can think of it like this","startTime":969.89,"endTime":970.93,"body":"You've got some maps,"},{"speaker":"because we can think of it like this","startTime":970.93,"endTime":972.79,"body":"so you're heading off somewhere,"},{"speaker":"because we can think of it like this","startTime":972.79,"endTime":974.22,"body":"so you set your bearing,"},{"speaker":"because we can think of it like this","startTime":974.22,"endTime":975.053,"body":"you've got an angle,"},{"speaker":"because we can think of it like this","startTime":975.053,"endTime":976.99,"body":"and off you go in that direction,"},{"speaker":"because we can think of it like this","startTime":976.99,"endTime":978.56,"body":"and then at some point,"},{"speaker":"because we can think of it like this","startTime":978.56,"endTime":979.76,"body":"there's a storm or something,"},{"speaker":"because we can think of it like this","startTime":979.76,"endTime":980.84,"body":"your ship gets damaged,"},{"speaker":"because we can think of it like this","startTime":980.84,"endTime":981.75,"body":"and you have to say,"},{"speaker":"because we can think of it like this","startTime":981.75,"endTime":983.93,"body":"we're going to have to pull"},{"speaker":"because we can think of it like this","startTime":981.75,"endTime":983.93,"body":"in at the nearest port,"},{"speaker":"because we can think of it like this","startTime":983.93,"endTime":984.83,"body":"so where is that?"},{"speaker":"because we can think of it like this","startTime":984.83,"endTime":986.64,"body":"So you look at your charts, and you say,"},{"speaker":"because we can think of it like this","startTime":986.64,"endTime":988.97,"body":"there's an island nearby,"},{"speaker":"because we can think of it like this","startTime":988.97,"endTime":990.5,"body":"let's head there,"},{"speaker":"because we can think of it like this","startTime":990.5,"endTime":994.44,"body":"what direction, what bearing"},{"speaker":"because we can think of it like this","startTime":990.5,"endTime":994.44,"body":"are you going to go in?"},{"speaker":"because we can think of it like this","startTime":994.44,"endTime":996.41,"body":"And to do that calculation,"},{"speaker":"because we can think of it like this","startTime":996.41,"endTime":1000.97,"body":"you know the initial"},{"speaker":"because we can think of it like this","startTime":996.41,"endTime":1000.97,"body":"direction you set out in,"},{"speaker":"because we can think of it like this","startTime":1000.97,"endTime":1004.76,"body":"and you know what direction you"},{"speaker":"because we can think of it like this","startTime":1000.97,"endTime":1004.76,"body":"would have set out from home"},{"speaker":"because we can think of it like this","startTime":1004.76,"endTime":1007.55,"body":"in order to reach this"},{"speaker":"because we can think of it like this","startTime":1004.76,"endTime":1007.55,"body":"island going directly,"},{"speaker":"because we can think of it like this","startTime":1007.55,"endTime":1010.267,"body":"so you've got two sides of this triangle,"},{"speaker":"because we can think of it like this","startTime":1010.267,"endTime":1013.439,"body":"and the angle there,"},{"speaker":"because we can think of it like this","startTime":1013.439,"endTime":1014.767,"body":"so you've got two sides of a triangle"},{"speaker":"because we can think of it like this","startTime":1014.767,"endTime":1016.4,"body":"and the angle between them,"},{"speaker":"because we can think of it like this","startTime":1016.4,"endTime":1018.54,"body":"and there's a law of trigonometry"},{"speaker":"because we can think of it like this","startTime":1018.54,"endTime":1019.83,"body":"called the cosine rule"},{"speaker":"because we can think of it like this","startTime":1019.83,"endTime":1020.9,"body":"that tells you how, then,"},{"speaker":"because we can think of it like this","startTime":1020.9,"endTime":1023.01,"body":"to calculate the rest of the triangle."},{"speaker":"because we can think of it like this","startTime":1023.01,"endTime":1025.11,"body":"So you can work out the"},{"speaker":"because we can think of it like this","startTime":1023.01,"endTime":1025.11,"body":"other angles, the other side,"},{"speaker":"because we can think of it like this","startTime":1025.11,"endTime":1026.76,"body":"and then you've got your bearing."},{"speaker":"because we can think of it like this","startTime":1026.76,"endTime":1030.94,"body":"So that's one use of"},{"speaker":"because we can think of it like this","startTime":1026.76,"endTime":1030.94,"body":"trigonometry in navigation,"},{"speaker":"because we can think of it like this","startTime":1030.94,"endTime":1033.35,"body":"we've got uses in astronomy,"},{"speaker":"because we can think of it like this","startTime":1033.35,"endTime":1036.693,"body":"another really important"},{"speaker":"because we can think of it like this","startTime":1033.35,"endTime":1036.693,"body":"application is in surveying,"},{"speaker":"because we can think of it like this","startTime":1037.78,"endTime":1039.78,"body":"the technique called triangulation."},{"speaker":"because we can think of it like this","startTime":1039.78,"endTime":1041.82,"body":"It sounds like it will involve triangles,"},{"speaker":"because we can think of it like this","startTime":1041.82,"endTime":1042.99,"body":"and it does."},{"speaker":"because we can think of it like this","startTime":1042.99,"endTime":1044.86,"body":"In that situation,"},{"speaker":"because we can think of it like this","startTime":1044.86,"endTime":1047.21,"body":"suppose you want to make"},{"speaker":"because we can think of it like this","startTime":1044.86,"endTime":1047.21,"body":"a map of your country,"},{"speaker":"because we can think of it like this","startTime":1048.15,"endTime":1050.31,"body":"we haven't got a nice, flat,"},{"speaker":"because we can think of it like this","startTime":1050.31,"endTime":1053.0,"body":"I could make a map of"},{"speaker":"because we can think of it like this","startTime":1050.31,"endTime":1053.0,"body":"this stage very easily"},{"speaker":"because we can think of it like this","startTime":1053.0,"endTime":1055.45,"body":"because I could just pace"},{"speaker":"because we can think of it like this","startTime":1053.0,"endTime":1055.45,"body":"it out, or measure it,"},{"speaker":"because we can think of it like this","startTime":1055.45,"endTime":1059.18,"body":"lay flat things out and"},{"speaker":"because we can think of it like this","startTime":1055.45,"endTime":1059.18,"body":"measure my distances, all easy,"},{"speaker":"because we can think of it like this","startTime":1059.18,"endTime":1061.99,"body":"but countries tend to have annoying things"},{"speaker":"because we can think of it like this","startTime":1061.99,"endTime":1064.02,"body":"like mountain ranges, and forests,"},{"speaker":"because we can think of it like this","startTime":1064.02,"endTime":1065.59,"body":"and rivers, and lakes, and swamps,"},{"speaker":"because we can think of it like this","startTime":1065.59,"endTime":1067.62,"body":"and all sorts of things"},{"speaker":"because we can think of it like this","startTime":1065.59,"endTime":1067.62,"body":"that get in the way"},{"speaker":"because we can think of it like this","startTime":1067.62,"endTime":1070.46,"body":"of you just pacing out"},{"speaker":"because we can think of it like this","startTime":1067.62,"endTime":1070.46,"body":"nicely your distances,"},{"speaker":"because we can think of it like this","startTime":1070.46,"endTime":1074.75,"body":"so what you do in that case"},{"speaker":"because we can think of it like this","startTime":1070.46,"endTime":1074.75,"body":"is you use triangulation."},{"speaker":"because we can think of it like this","startTime":1074.75,"endTime":1077.82,"body":"You measure out very, very"},{"speaker":"because we can think of it like this","startTime":1074.75,"endTime":1077.82,"body":"accurately an initial line,"},{"speaker":"because we can think of it like this","startTime":1077.82,"endTime":1079.51,"body":"which you do have to use with chains,"},{"speaker":"because we can think of it like this","startTime":1079.51,"endTime":1080.97,"body":"or something like this,"},{"speaker":"because we can think of it like this","startTime":1080.97,"endTime":1082.483,"body":"measure it out very carefully."},{"speaker":"because we can think of it like this","startTime":1083.98,"endTime":1085.31,"body":"So you've got this initial line,"},{"speaker":"because we can think of it like this","startTime":1085.31,"endTime":1086.88,"body":"which is going to be"},{"speaker":"because we can think of it like this","startTime":1085.31,"endTime":1086.88,"body":"the base of a triangle,"},{"speaker":"because we can think of it like this","startTime":1086.88,"endTime":1090.6,"body":"and you know that length"},{"speaker":"because we can think of it like this","startTime":1086.88,"endTime":1090.6,"body":"as exactly as you can."},{"speaker":"because we can think of it like this","startTime":1090.6,"endTime":1093.63,"body":"Then you send your"},{"speaker":"because we can think of it like this","startTime":1090.6,"endTime":1093.63,"body":"associate a few miles away,"},{"speaker":"because we can think of it like this","startTime":1093.63,"endTime":1095.53,"body":"maybe they go and stand on"},{"speaker":"because we can think of it like this","startTime":1093.63,"endTime":1095.53,"body":"the top of a nearby hill,"},{"speaker":"because we can think of it like this","startTime":1095.53,"endTime":1096.383,"body":"or something,"},{"speaker":"because we can think of it like this","startTime":1097.46,"endTime":1101.28,"body":"so then they are standing"},{"speaker":"because we can think of it like this","startTime":1097.46,"endTime":1101.28,"body":"at the third apex,"},{"speaker":"because we can think of it like this","startTime":1101.28,"endTime":1103.31,"body":"the third vertex of a triangle,"},{"speaker":"because we can think of it like this","startTime":1103.31,"endTime":1106.28,"body":"and you measure the"},{"speaker":"because we can think of it like this","startTime":1103.31,"endTime":1106.28,"body":"angles then very carefully"},{"speaker":"because we can think of it like this","startTime":1106.28,"endTime":1107.69,"body":"in that triangle,"},{"speaker":"because we can think of it like this","startTime":1107.69,"endTime":1110.151,"body":"so you don't have to be trudging over"},{"speaker":"because we can think of it like this","startTime":1110.151,"endTime":1111.52,"body":"your great, big theodolite"},{"speaker":"because we can think of it like this","startTime":1111.52,"endTime":1113.47,"body":"up and down hills and through marshes,"},{"speaker":"because we can think of it like this","startTime":1113.47,"endTime":1115.34,"body":"you're just looking and"},{"speaker":"because we can think of it like this","startTime":1113.47,"endTime":1115.34,"body":"measuring the angles."},{"speaker":"because we can think of it like this","startTime":1115.34,"endTime":1117.01,"body":"So now, in this triangle,"},{"speaker":"because we can think of it like this","startTime":1117.01,"endTime":1119.0,"body":"you've got your base line you know,"},{"speaker":"because we can think of it like this","startTime":1119.0,"endTime":1120.68,"body":"and you've got two angles,"},{"speaker":"because we can think of it like this","startTime":1120.68,"endTime":1122.38,"body":"well you can work out the third for free"},{"speaker":"because we can think of it like this","startTime":1122.38,"endTime":1125.36,"body":"'cause angles in a triangle"},{"speaker":"because we can think of it like this","startTime":1122.38,"endTime":1125.36,"body":"add up to 180 degrees,"},{"speaker":"because we can think of it like this","startTime":1125.36,"endTime":1126.543,"body":"or pi radians,"},{"speaker":"because we can think of it like this","startTime":1127.53,"endTime":1130.83,"body":"and then you use a magic"},{"speaker":"because we can think of it like this","startTime":1127.53,"endTime":1130.83,"body":"bit of trigonometry,"},{"speaker":"because we can think of it like this","startTime":1130.83,"endTime":1131.95,"body":"the sine rule,"},{"speaker":"because we can think of it like this","startTime":1131.95,"endTime":1132.8,"body":"and this tells you"},{"speaker":"because we can think of it like this","startTime":1132.8,"endTime":1134.51,"body":"there's a lovely relationship"},{"speaker":"because we can think of it like this","startTime":1134.51,"endTime":1136.88,"body":"between these ratios of side length"},{"speaker":"because we can think of it like this","startTime":1136.88,"endTime":1139.56,"body":"to sine of opposite angle."},{"speaker":"because we can think of it like this","startTime":1139.56,"endTime":1141.12,"body":"And in our situation,"},{"speaker":"because we can think of it like this","startTime":1141.12,"endTime":1142.52,"body":"we know all the angles,"},{"speaker":"because we can think of it like this","startTime":1142.52,"endTime":1145.32,"body":"so we can look up the"},{"speaker":"because we can think of it like this","startTime":1142.52,"endTime":1145.32,"body":"sines of those angles,"},{"speaker":"because we can think of it like this","startTime":1145.32,"endTime":1146.153,"body":"it's pre-calculators, or anything,"},{"speaker":"because we can think of it like this","startTime":1146.153,"endTime":1147.62,"body":"you have to have a little table,"},{"speaker":"because we can think of it like this","startTime":1147.62,"endTime":1148.98,"body":"and you'd look them up,"},{"speaker":"because we can think of it like this","startTime":1148.98,"endTime":1152.34,"body":"and then you can use that"},{"speaker":"because we can think of it like this","startTime":1148.98,"endTime":1152.34,"body":"relationship, the sine rule,"},{"speaker":"because we can think of it like this","startTime":1152.34,"endTime":1154.32,"body":"to work out the missing lengths."},{"speaker":"because we can think of it like this","startTime":1154.32,"endTime":1156.84,"body":"So now you've got all the"},{"speaker":"because we can think of it like this","startTime":1154.32,"endTime":1156.84,"body":"lengths of this triangle,"},{"speaker":"because we can think of it like this","startTime":1156.84,"endTime":1158.78,"body":"that's great, you've got those distances,"},{"speaker":"because we can think of it like this","startTime":1158.78,"endTime":1161.08,"body":"those two distances that you now know"},{"speaker":"because we can think of it like this","startTime":1161.08,"endTime":1164.07,"body":"can now in turn be the"},{"speaker":"because we can think of it like this","startTime":1161.08,"endTime":1164.07,"body":"bases of new triangles,"},{"speaker":"because we can think of it like this","startTime":1164.07,"endTime":1166.44,"body":"and so you can put little"},{"speaker":"because we can think of it like this","startTime":1164.07,"endTime":1166.44,"body":"triangles all over your country"},{"speaker":"because we can think of it like this","startTime":1166.44,"endTime":1167.85,"body":"and you can measure out all the lengths,"},{"speaker":"because we can think of it like this","startTime":1167.85,"endTime":1170.45,"body":"and that's how you do your"},{"speaker":"because we can think of it like this","startTime":1167.85,"endTime":1170.45,"body":"surveying and map making."},{"speaker":"because we can think of it like this","startTime":1172.421,"endTime":1173.38,"body":"You may have noticed I said"},{"speaker":"because we can think of it like this","startTime":1173.38,"endTime":1176.05,"body":"angles in a triangle"},{"speaker":"because we can think of it like this","startTime":1173.38,"endTime":1176.05,"body":"add up to 180 degrees,"},{"speaker":"because we can think of it like this","startTime":1176.05,"endTime":1179.07,"body":"well they do, not disagreeing with that,"},{"speaker":"because we can think of it like this","startTime":1179.07,"endTime":1181.863,"body":"but they only do on a"},{"speaker":"because we can think of it like this","startTime":1179.07,"endTime":1181.863,"body":"flat surface, on a plane."},{"speaker":"because we can think of it like this","startTime":1182.805,"endTime":1184.24,"body":"When your triangles are small enough"},{"speaker":"because we can think of it like this","startTime":1184.24,"endTime":1189.24,"body":"compared to the vast"},{"speaker":"because we can think of it like this","startTime":1184.24,"endTime":1189.24,"body":"size of the planet Earth,"},{"speaker":"because we can think of it like this","startTime":1189.41,"endTime":1191.37,"body":"then that's absolutely fine,"},{"speaker":"because we can think of it like this","startTime":1191.37,"endTime":1194.93,"body":"but when your triangles get very big,"},{"speaker":"because we can think of it like this","startTime":1194.93,"endTime":1197.188,"body":"there starts to be an issue."},{"speaker":"because we can think of it like this","startTime":1197.188,"endTime":1198.83,"body":"This picture here, at the bottom of it,"},{"speaker":"because we can think of it like this","startTime":1198.83,"endTime":1199.663,"body":"if you can read the writing,"},{"speaker":"because we can think of it like this","startTime":1199.663,"endTime":1201.45,"body":"it says \"Calcutta base line.\""},{"speaker":"because we can think of it like this","startTime":1201.45,"endTime":1203.5,"body":"This is a drawing"},{"speaker":"because we can think of it like this","startTime":1203.5,"endTime":1208.04,"body":"representing the Great"},{"speaker":"because we can think of it like this","startTime":1203.5,"endTime":1208.04,"body":"Trigonometric Survey of India,"},{"speaker":"because we can think of it like this","startTime":1208.04,"endTime":1209.94,"body":"which was carried out in the 19th century."},{"speaker":"because we can think of it like this","startTime":1209.94,"endTime":1212.27,"body":"It took over 70 years, I think,"},{"speaker":"because we can think of it like this","startTime":1212.27,"endTime":1213.963,"body":"it finished in about 1878,"},{"speaker":"because we can think of it like this","startTime":1214.9,"endTime":1218.223,"body":"surveying and mapping the"},{"speaker":"because we can think of it like this","startTime":1214.9,"endTime":1218.223,"body":"whole of India very accurately."},{"speaker":"because we can think of it like this","startTime":1219.496,"endTime":1221.78,"body":"The triangles there were so big"},{"speaker":"because we can think of it like this","startTime":1221.78,"endTime":1222.95,"body":"that the curvature of the Earth"},{"speaker":"because we can think of it like this","startTime":1222.95,"endTime":1224.29,"body":"had to be taken into account,"},{"speaker":"because we can think of it like this","startTime":1224.29,"endTime":1226.01,"body":"the angles in a spherical triangle"},{"speaker":"because we can think of it like this","startTime":1226.01,"endTime":1228.57,"body":"add up to more than 180 degrees."},{"speaker":"because we can think of it like this","startTime":1228.57,"endTime":1230.29,"body":"Luckily, that's not an issue,"},{"speaker":"because we can think of it like this","startTime":1230.29,"endTime":1232.01,"body":"there's a modified"},{"speaker":"because we can think of it like this","startTime":1230.29,"endTime":1232.01,"body":"version of the sine rule,"},{"speaker":"because we can think of it like this","startTime":1232.01,"endTime":1236.847,"body":"we know by how much the"},{"speaker":"because we can think of it like this","startTime":1232.01,"endTime":1236.847,"body":"angle will exceed 180 degrees"},{"speaker":"because we can think of it like this","startTime":1236.847,"endTime":1238.26,"body":"the sum of the angles,"},{"speaker":"because we can think of it like this","startTime":1238.26,"endTime":1240.33,"body":"it's proportional to the"},{"speaker":"because we can think of it like this","startTime":1238.26,"endTime":1240.33,"body":"area of the triangle,"},{"speaker":"because we can think of it like this","startTime":1240.33,"endTime":1242.19,"body":"so that's not a problem,"},{"speaker":"because we can think of it like this","startTime":1242.19,"endTime":1244.62,"body":"but you just have to"},{"speaker":"because we can think of it like this","startTime":1242.19,"endTime":1244.62,"body":"take that into account."},{"speaker":"because we can think of it like this","startTime":1244.62,"endTime":1247.77,"body":"Spherical trigonometry is"},{"speaker":"because we can think of it like this","startTime":1244.62,"endTime":1247.77,"body":"another kind of trigonometry"},{"speaker":"because we can think of it like this","startTime":1247.77,"endTime":1250.72,"body":"made with triangles on"},{"speaker":"because we can think of it like this","startTime":1247.77,"endTime":1250.72,"body":"the surface of spheres."},{"speaker":"because we can think of it like this","startTime":1250.72,"endTime":1253.1,"body":"So these many, many applications,"},{"speaker":"because we can think of it like this","startTime":1253.1,"endTime":1254.5,"body":"and others I haven't mentioned,"},{"speaker":"because we can think of it like this","startTime":1254.5,"endTime":1258.93,"body":"all mean that we have"},{"speaker":"because we can think of it like this","startTime":1254.5,"endTime":1258.93,"body":"needed the technology"},{"speaker":"because we can think of it like this","startTime":1258.93,"endTime":1263.7,"body":"to be able to work with these triangles"},{"speaker":"because we can think of it like this","startTime":1263.7,"endTime":1266.54,"body":"that we encounter in"},{"speaker":"because we can think of it like this","startTime":1263.7,"endTime":1266.54,"body":"all these applications,"},{"speaker":"because we can think of it like this","startTime":1266.54,"endTime":1271.54,"body":"and that means it became"},{"speaker":"because we can think of it like this","startTime":1266.54,"endTime":1271.54,"body":"very, very important"},{"speaker":"because we can think of it like this","startTime":1272.0,"endTime":1277.0,"body":"to get good lists, or tables, of,"},{"speaker":"because we can think of it like this","startTime":1277.02,"endTime":1277.853,"body":"given an angle,"},{"speaker":"because we can think of it like this","startTime":1277.853,"endTime":1279.54,"body":"what's the sine of it, cosine of it,"},{"speaker":"because we can think of it like this","startTime":1279.54,"endTime":1280.373,"body":"and vice versa,"},{"speaker":"because we can think of it like this","startTime":1280.373,"endTime":1281.3,"body":"given a sine or cosine,"},{"speaker":"because we can think of it like this","startTime":1281.3,"endTime":1283.603,"body":"can we go backwards and"},{"speaker":"because we can think of it like this","startTime":1281.3,"endTime":1283.603,"body":"say what the angle was."},{"speaker":"because we can think of it like this","startTime":1284.66,"endTime":1285.93,"body":"This is absolutely pressing,"},{"speaker":"because we can think of it like this","startTime":1285.93,"endTime":1287.9,"body":"it's vitally important, life-saving work"},{"speaker":"because we can think of it like this","startTime":1287.9,"endTime":1290.67,"body":"to be able to get these"},{"speaker":"because we can think of it like this","startTime":1287.9,"endTime":1290.67,"body":"things correctly right,"},{"speaker":"because we can think of it like this","startTime":1290.67,"endTime":1292.08,"body":"because if you're navigating,"},{"speaker":"because we can think of it like this","startTime":1292.08,"endTime":1293.02,"body":"and you look at your table,"},{"speaker":"because we can think of it like this","startTime":1293.02,"endTime":1295.26,"body":"and you get the wrong answer,"},{"speaker":"because we can think of it like this","startTime":1295.26,"endTime":1296.74,"body":"lives could be lost,"},{"speaker":"because we can think of it like this","startTime":1296.74,"endTime":1298.17,"body":"so it's important."},{"speaker":"because we can think of it like this","startTime":1298.17,"endTime":1300.62,"body":"I just want to mention a"},{"speaker":"because we can think of it like this","startTime":1298.17,"endTime":1300.62,"body":"couple of people in this story."},{"speaker":"because we can think of it like this","startTime":1300.62,"endTime":1302.433,"body":"Regiomontanus is one,"},{"speaker":"because we can think of it like this","startTime":1302.433,"endTime":1304.32,"body":"there's a picture of him looking at,"},{"speaker":"because we can think of it like this","startTime":1304.32,"endTime":1305.25,"body":"that's an astrolabe, isn't it,"},{"speaker":"because we can think of it like this","startTime":1305.25,"endTime":1307.11,"body":"he's looking very"},{"speaker":"because we can think of it like this","startTime":1305.25,"endTime":1307.11,"body":"cleverly at an astrolabe,"},{"speaker":"because we can think of it like this","startTime":1307.11,"endTime":1311.597,"body":"he wrote one of the first"},{"speaker":"because we can think of it like this","startTime":1307.11,"endTime":1311.597,"body":"trigonometry textbooks in Europe,"},{"speaker":"because we can think of it like this","startTime":1311.597,"endTime":1313.087,"body":"\"De Triangulis Omnimodis,\""},{"speaker":"because we can think of it like this","startTime":1314.587,"endTime":1315.8,"body":"\"Every Kind of Triangle,\""},{"speaker":"because we can think of it like this","startTime":1315.8,"endTime":1317.53,"body":"triangles for dummies."},{"speaker":"because we can think of it like this","startTime":1317.53,"endTime":1318.7,"body":"He says, not quite,"},{"speaker":"because we can think of it like this","startTime":1318.7,"endTime":1320.81,"body":"I'm sure it was very clever"},{"speaker":"because we can think of it like this","startTime":1318.7,"endTime":1320.81,"body":"and wonderful people,"},{"speaker":"because we can think of it like this","startTime":1320.81,"endTime":1323.787,"body":"he says, \"You who wish to study"},{"speaker":"because we can think of it like this","startTime":1320.81,"endTime":1323.787,"body":"great and wonderful things,"},{"speaker":"because we can think of it like this","startTime":1323.787,"endTime":1325.877,"body":"\"who wonder about the"},{"speaker":"because we can think of it like this","startTime":1323.787,"endTime":1325.877,"body":"movement of the stars,"},{"speaker":"because we can think of it like this","startTime":1325.877,"endTime":1328.02,"body":"\"must read these theorems"},{"speaker":"because we can think of it like this","startTime":1325.877,"endTime":1328.02,"body":"about triangles,\""},{"speaker":"because we can think of it like this","startTime":1328.02,"endTime":1329.11,"body":"it's absolutely vital,"},{"speaker":"because we can think of it like this","startTime":1329.11,"endTime":1331.02,"body":"you have to understand about triangles"},{"speaker":"because we can think of it like this","startTime":1331.02,"endTime":1333.52,"body":"if you're going to be"},{"speaker":"because we can think of it like this","startTime":1331.02,"endTime":1333.52,"body":"able to do all this stuff."},{"speaker":"because we can think of it like this","startTime":1333.52,"endTime":1336.26,"body":"He produced an early book"},{"speaker":"because we can think of it like this","startTime":1333.52,"endTime":1336.26,"body":"of trigonometric tables,"},{"speaker":"because we can think of it like this","startTime":1336.26,"endTime":1338.253,"body":"lists of values of these functions."},{"speaker":"because we can think of it like this","startTime":1339.34,"endTime":1341.11,"body":"In case you're worrying"},{"speaker":"because we can think of it like this","startTime":1339.34,"endTime":1341.11,"body":"about the date there,"},{"speaker":"because we can think of it like this","startTime":1341.11,"endTime":1342.38,"body":"it was posthumously printed,"},{"speaker":"because we can think of it like this","startTime":1342.38,"endTime":1345.697,"body":"so it's not a typo, it came out in 1490."},{"speaker":"because we can think of it like this","startTime":1345.697,"endTime":1349.63,"body":"And others also produced tables like this,"},{"speaker":"because we can think of it like this","startTime":1349.63,"endTime":1352.053,"body":"I want to mention Georg Rheticus."},{"speaker":"because we can think of it like this","startTime":1353.16,"endTime":1358.16,"body":"His tables were extremely precise,"},{"speaker":"because we can think of it like this","startTime":1359.36,"endTime":1360.42,"body":"but even better than that,"},{"speaker":"because we can think of it like this","startTime":1360.42,"endTime":1361.87,"body":"they were extremely accurate."},{"speaker":"because we can think of it like this","startTime":1362.94,"endTime":1363.883,"body":"The first version,"},{"speaker":"because we can think of it like this","startTime":1364.892,"endTime":1367.7,"body":"a shorter version came out in 1551,"},{"speaker":"because we can think of it like this","startTime":1367.7,"endTime":1370.6,"body":"it didn't quite use the"},{"speaker":"because we can think of it like this","startTime":1367.7,"endTime":1370.6,"body":"standard sin, cos, and tan"},{"speaker":"because we can think of it like this","startTime":1370.6,"endTime":1371.61,"body":"that we use today,"},{"speaker":"because we can think of it like this","startTime":1371.61,"endTime":1374.03,"body":"but what it used was equivalent,"},{"speaker":"because we can think of it like this","startTime":1374.03,"endTime":1376.45,"body":"you could deduce one from the other."},{"speaker":"because we can think of it like this","startTime":1376.45,"endTime":1378.15,"body":"And I just wanted to show you,"},{"speaker":"because we can think of it like this","startTime":1378.15,"endTime":1379.77,"body":"I don't know if you can see here,"},{"speaker":"because we can think of it like this","startTime":1379.77,"endTime":1382.25,"body":"I won't go into detail"},{"speaker":"because we can think of it like this","startTime":1379.77,"endTime":1382.25,"body":"about what it all means,"},{"speaker":"because we can think of it like this","startTime":1382.25,"endTime":1386.74,"body":"but the length of the"},{"speaker":"because we can think of it like this","startTime":1382.25,"endTime":1386.74,"body":"numbers that are shown there,"},{"speaker":"because we can think of it like this","startTime":1386.74,"endTime":1389.85,"body":"these are subdivisions of the circle"},{"speaker":"because we can think of it like this","startTime":1389.85,"endTime":1392.41,"body":"into very, very tiny, little increments,"},{"speaker":"because we can think of it like this","startTime":1392.41,"endTime":1394.46,"body":"and the number of places,"},{"speaker":"because we can think of it like this","startTime":1394.46,"endTime":1397.86,"body":"the number of significant"},{"speaker":"because we can think of it like this","startTime":1394.46,"endTime":1397.86,"body":"figures given is very high,"},{"speaker":"because we can think of it like this","startTime":1397.86,"endTime":1400.15,"body":"and this is back in 1551,"},{"speaker":"because we can think of it like this","startTime":1400.15,"endTime":1403.433,"body":"so this is very, very,"},{"speaker":"because we can think of it like this","startTime":1400.15,"endTime":1403.433,"body":"very accurate tables."},{"speaker":"because we can think of it like this","startTime":1404.72,"endTime":1405.863,"body":"You can see by looking at this"},{"speaker":"because we can think of it like this","startTime":1405.863,"endTime":1408.16,"body":"that it's extremely salacious,"},{"speaker":"because we can think of it like this","startTime":1408.16,"endTime":1410.483,"body":"and it was banned by the Catholic Church."},{"speaker":"I jest","startTime":1413.21,"endTime":1414.4,"body":"it was banned,"},{"speaker":"I jest","startTime":1414.4,"endTime":1416.07,"body":"but it probably wasn't because"},{"speaker":"I jest","startTime":1416.07,"endTime":1420.82,"body":"trigonometry was so sinful, or anything,"},{"speaker":"I jest","startTime":1420.82,"endTime":1422.65,"body":"but actually, more likely,"},{"speaker":"I jest","startTime":1422.65,"endTime":1425.23,"body":"it's because Rheticus is"},{"speaker":"I jest","startTime":1422.65,"endTime":1425.23,"body":"famous for something else"},{"speaker":"I jest","startTime":1425.23,"endTime":1427.64,"body":"other than these amazing"},{"speaker":"I jest","startTime":1425.23,"endTime":1427.64,"body":"tables he produced,"},{"speaker":"I jest","startTime":1427.64,"endTime":1430.49,"body":"and that is that he was the only pupil,"},{"speaker":"I jest","startTime":1430.49,"endTime":1434.24,"body":"and the really important person"},{"speaker":"I jest","startTime":1434.24,"endTime":1437.67,"body":"in the popularization of"},{"speaker":"I jest","startTime":1434.24,"endTime":1437.67,"body":"the work of Copernicus,"},{"speaker":"I jest","startTime":1437.67,"endTime":1441.74,"body":"and Copernicus was not in"},{"speaker":"I jest","startTime":1437.67,"endTime":1441.74,"body":"great favor at that time"},{"speaker":"I jest","startTime":1441.74,"endTime":1442.91,"body":"with the Catholic Church,"},{"speaker":"I jest","startTime":1442.91,"endTime":1445.829,"body":"and so probably by association,"},{"speaker":"I jest","startTime":1445.829,"endTime":1448.223,"body":"Rheticus, oh he's that Copernicus guy,"},{"speaker":"I jest","startTime":1449.59,"endTime":1450.81,"body":"he's the pupil of Copernicus,"},{"speaker":"I jest","startTime":1450.81,"endTime":1452.25,"body":"he can't be up to any good."},{"speaker":"I jest","startTime":1452.25,"endTime":1456.8,"body":"So this \"Canon Doctrinae"},{"speaker":"I jest","startTime":1452.25,"endTime":1456.8,"body":"Triangulorum\" was banned,"},{"speaker":"I jest","startTime":1456.8,"endTime":1458.73,"body":"but then a much longer version came out,"},{"speaker":"I jest","startTime":1458.73,"endTime":1460.22,"body":"hundreds of pages,"},{"speaker":"I jest","startTime":1460.22,"endTime":1462.37,"body":"a bit later on,"},{"speaker":"I jest","startTime":1462.37,"endTime":1463.375,"body":"and that wasn't banned,"},{"speaker":"I jest","startTime":1463.375,"endTime":1466.14,"body":"so it's all right now,"},{"speaker":"I jest","startTime":1466.14,"endTime":1467.22,"body":"we're allowed to look at this"},{"speaker":"I jest","startTime":1467.22,"endTime":1471.9,"body":"without feeling like our"},{"speaker":"I jest","startTime":1467.22,"endTime":1471.9,"body":"souls might be in danger."},{"speaker":"I jest","startTime":1471.9,"endTime":1473.68,"body":"And these tables are so, so accurate"},{"speaker":"I jest","startTime":1473.68,"endTime":1475.92,"body":"that they were used for hundreds of years."},{"speaker":"I jest","startTime":1475.92,"endTime":1477.728,"body":"So that's all great,"},{"speaker":"I jest","startTime":1477.728,"endTime":1479.47,"body":"we know we need these things"},{"speaker":"I jest","startTime":1479.47,"endTime":1482.84,"body":"to do our navigation, our"},{"speaker":"I jest","startTime":1479.47,"endTime":1482.84,"body":"astronomy, our surveying,"},{"speaker":"I jest","startTime":1482.84,"endTime":1484.96,"body":"how do you actually make"},{"speaker":"I jest","startTime":1482.84,"endTime":1484.96,"body":"one of these tables?"},{"speaker":"I jest","startTime":1484.96,"endTime":1486.337,"body":"We can't get our calculator out"},{"speaker":"I jest","startTime":1486.337,"endTime":1489.62,"body":"and just go, what's sine of 36.725,"},{"speaker":"I jest","startTime":1489.62,"endTime":1494.62,"body":"how on earth does Rheticus"},{"speaker":"I jest","startTime":1489.62,"endTime":1494.62,"body":"produce a table like this?"},{"speaker":"I jest","startTime":1496.51,"endTime":1497.99,"body":"In making this new technology,"},{"speaker":"I jest","startTime":1497.99,"endTime":1499.7,"body":"new mathematics was required,"},{"speaker":"I jest","startTime":1499.7,"endTime":1502.65,"body":"and tables could be produced"},{"speaker":"I jest","startTime":1502.65,"endTime":1506.413,"body":"using some rules that were found,"},{"speaker":"I jest","startTime":1507.6,"endTime":1508.748,"body":"new lamps for old,"},{"speaker":"I jest","startTime":1508.748,"endTime":1513.748,"body":"there are ways of finding out"},{"speaker":"I jest","startTime":1508.748,"endTime":1513.748,"body":"the sines and coses of angles"},{"speaker":"I jest","startTime":1515.01,"endTime":1516.83,"body":"based on things you already know."},{"speaker":"I jest","startTime":1516.83,"endTime":1518.31,"body":"So I've just put a couple up there,"},{"speaker":"I jest","startTime":1518.31,"endTime":1521.59,"body":"so if you know sin and cos of an angle x,"},{"speaker":"I jest","startTime":1521.59,"endTime":1522.43,"body":"then there are formulae"},{"speaker":"I jest","startTime":1522.43,"endTime":1526.64,"body":"that tell you how to find"},{"speaker":"I jest","startTime":1522.43,"endTime":1526.64,"body":"those values for 1/2 x."},{"speaker":"I jest","startTime":1526.64,"endTime":1530.11,"body":"If you know the sin and 'cause"},{"speaker":"I jest","startTime":1526.64,"endTime":1530.11,"body":"of angle a and an angle b,"},{"speaker":"I jest","startTime":1530.11,"endTime":1531.0,"body":"then there are formulae"},{"speaker":"I jest","startTime":1531.0,"endTime":1532.84,"body":"that tell you how to find the sin and cos"},{"speaker":"I jest","startTime":1532.84,"endTime":1535.08,"body":"of a plus b and a minus b,"},{"speaker":"I jest","startTime":1535.08,"endTime":1537.03,"body":"so you can do sums and differences,"},{"speaker":"I jest","startTime":1537.03,"endTime":1538.66,"body":"and there are other things as well."},{"speaker":"I jest","startTime":1538.66,"endTime":1539.51,"body":"So that's all very well,"},{"speaker":"I jest","startTime":1539.51,"endTime":1541.38,"body":"but you need somewhere to start,"},{"speaker":"I jest","startTime":1541.38,"endTime":1544.67,"body":"and we can start with triangles we know."},{"speaker":"I jest","startTime":1544.67,"endTime":1547.97,"body":"Here's a couple of triangles"},{"speaker":"I jest","startTime":1544.67,"endTime":1547.97,"body":"we know, collectively."},{"speaker":"I jest","startTime":1547.97,"endTime":1549.49,"body":"There's an equilateral triangle there,"},{"speaker":"I jest","startTime":1549.49,"endTime":1550.84,"body":"you can use Pythagoras' theorem"},{"speaker":"I jest","startTime":1550.84,"endTime":1552.56,"body":"to find the height of that triangle,"},{"speaker":"I jest","startTime":1552.56,"endTime":1554.57,"body":"and that gives you a right-angled triangle"},{"speaker":"I jest","startTime":1554.57,"endTime":1556.413,"body":"with angles 60 and 30,"},{"speaker":"I jest","startTime":1557.994,"endTime":1560.17,"body":"we know all the lengths there,"},{"speaker":"I jest","startTime":1560.17,"endTime":1562.783,"body":"so we can find sin and"},{"speaker":"I jest","startTime":1560.17,"endTime":1562.783,"body":"cos of those angles."},{"speaker":"I jest","startTime":1563.81,"endTime":1565.29,"body":"Regular pentagon,"},{"speaker":"I jest","startTime":1565.29,"endTime":1568.29,"body":"we've known ever since ancient Greek times"},{"speaker":"I jest","startTime":1568.29,"endTime":1569.94,"body":"the diagonal of a regular pentagon,"},{"speaker":"I jest","startTime":1569.94,"endTime":1571.19,"body":"if it has side one,"},{"speaker":"I jest","startTime":1571.19,"endTime":1574.1,"body":"the diagonal is precisely"},{"speaker":"I jest","startTime":1571.19,"endTime":1574.1,"body":"the golden number, phi,"},{"speaker":"I jest","startTime":1574.1,"endTime":1575.92,"body":"one plus root-five, over two,"},{"speaker":"I jest","startTime":1575.92,"endTime":1577.18,"body":"so we know the diagonal,"},{"speaker":"I jest","startTime":1577.18,"endTime":1579.59,"body":"we can then use Pythagoras"},{"speaker":"I jest","startTime":1577.18,"endTime":1579.59,"body":"again to find the height,"},{"speaker":"I jest","startTime":1579.59,"endTime":1581.34,"body":"and that's another right-angled triangle,"},{"speaker":"I jest","startTime":1581.34,"endTime":1583.2,"body":"all of whose sides we know,"},{"speaker":"I jest","startTime":1583.2,"endTime":1587.32,"body":"and we therefore can find"},{"speaker":"I jest","startTime":1583.2,"endTime":1587.32,"body":"the functions for 72,"},{"speaker":"I jest","startTime":1587.32,"endTime":1589.92,"body":"and I guess 36 is on top;"},{"speaker":"I jest","startTime":1589.92,"endTime":1593.52,"body":"maybe not 36, got to add them up to 180."},{"speaker":"I jest","startTime":1593.52,"endTime":1596.04,"body":"So we've got 60 and 72,"},{"speaker":"I jest","startTime":1596.04,"endTime":1598.61,"body":"and we've got a difference formula,"},{"speaker":"I jest","startTime":1598.61,"endTime":1600.53,"body":"if we know sine of a and sine of b,"},{"speaker":"I jest","startTime":1600.53,"endTime":1602.51,"body":"we can find sine of a minus b,"},{"speaker":"I jest","startTime":1602.51,"endTime":1605.03,"body":"so we've got 60 and 72,"},{"speaker":"I jest","startTime":1605.03,"endTime":1606.72,"body":"that enables us to find the sine of 12,"},{"speaker":"I jest","startTime":1606.72,"endTime":1609.34,"body":"and then we have half angle formulae,"},{"speaker":"I jest","startTime":1609.34,"endTime":1610.58,"body":"so if we know the sine of 12,"},{"speaker":"I jest","startTime":1610.58,"endTime":1612.11,"body":"we can find the sine of six,"},{"speaker":"I jest","startTime":1612.11,"endTime":1614.23,"body":"and we can do that again"},{"speaker":"I jest","startTime":1612.11,"endTime":1614.23,"body":"and find the sine of three."},{"speaker":"I jest","startTime":1614.23,"endTime":1616.63,"body":"So this kind of consideration"},{"speaker":"I jest","startTime":1616.63,"endTime":1618.95,"body":"allows you to get a long way"},{"speaker":"I jest","startTime":1618.95,"endTime":1621.66,"body":"and find a lot of these functions"},{"speaker":"I jest","startTime":1621.66,"endTime":1623.693,"body":"for smaller and smaller angles."},{"speaker":"I jest","startTime":1625.11,"endTime":1628.8,"body":"So that's a hint of how"},{"speaker":"I jest","startTime":1625.11,"endTime":1628.8,"body":"these tables were produced,"},{"speaker":"I jest","startTime":1628.8,"endTime":1630.06,"body":"it's still an amazing feat"},{"speaker":"I jest","startTime":1630.06,"endTime":1634.243,"body":"of calculation and"},{"speaker":"I jest","startTime":1630.06,"endTime":1634.243,"body":"mathematical achievement."},{"speaker":"I jest","startTime":1635.52,"endTime":1638.537,"body":"We're going to get onto uses"},{"speaker":"I jest","startTime":1635.52,"endTime":1638.537,"body":"of sine waves in a moment,"},{"speaker":"I jest","startTime":1638.537,"endTime":1641.22,"body":"but before we leave talking about tables,"},{"speaker":"I jest","startTime":1641.22,"endTime":1642.053,"body":"I just want to mention"},{"speaker":"I jest","startTime":1642.053,"endTime":1645.12,"body":"that there was a really"},{"speaker":"I jest","startTime":1642.053,"endTime":1645.12,"body":"clever use of this technology"},{"speaker":"I jest","startTime":1645.12,"endTime":1646.32,"body":"that had been developed."},{"speaker":"I jest","startTime":1646.32,"endTime":1648.05,"body":"I think of these tables,"},{"speaker":"I jest","startTime":1648.05,"endTime":1649.97,"body":"this is state of the art technology,"},{"speaker":"I jest","startTime":1649.97,"endTime":1651.67,"body":"this is like having your supercomputer,"},{"speaker":"I jest","startTime":1651.67,"endTime":1653.353,"body":"here is your book that tells"},{"speaker":"I jest","startTime":1651.67,"endTime":1653.353,"body":"you all of these things,"},{"speaker":"I jest","startTime":1653.353,"endTime":1655.073,"body":"and you can calculate with them."},{"speaker":"I jest","startTime":1656.14,"endTime":1659.43,"body":"And this technology that was already there"},{"speaker":"I jest","startTime":1659.43,"endTime":1662.29,"body":"was given an alternative use"},{"speaker":"I jest","startTime":1662.29,"endTime":1664.34,"body":"at the end of the 16th century"},{"speaker":"I jest","startTime":1665.25,"endTime":1668.14,"body":"to solve a completely different problem,"},{"speaker":"I jest","startTime":1668.14,"endTime":1671.95,"body":"and it's the problem of"},{"speaker":"I jest","startTime":1668.14,"endTime":1671.95,"body":"multiplying long numbers."},{"speaker":"I jest","startTime":1671.95,"endTime":1673.29,"body":"This is hard to do."},{"speaker":"I jest","startTime":1673.29,"endTime":1675.8,"body":"Adding is easy, subtracting is easy,"},{"speaker":"I jest","startTime":1675.8,"endTime":1677.173,"body":"multiplying is harder,"},{"speaker":"I jest","startTime":1678.66,"endTime":1680.003,"body":"and it takes more time."},{"speaker":"I jest","startTime":1681.77,"endTime":1682.81,"body":"At some point,"},{"speaker":"I jest","startTime":1682.81,"endTime":1685.91,"body":"logarithms were discovered/invented,"},{"speaker":"I jest","startTime":1685.91,"endTime":1688.99,"body":"and people started using"},{"speaker":"I jest","startTime":1685.91,"endTime":1688.99,"body":"log tables, and slide rules,"},{"speaker":"I jest","startTime":1688.99,"endTime":1690.04,"body":"and things like that,"},{"speaker":"I jest","startTime":1691.03,"endTime":1692.95,"body":"and then obviously, nowadays,"},{"speaker":"I jest","startTime":1692.95,"endTime":1694.03,"body":"we don't tend to use that,"},{"speaker":"I jest","startTime":1694.03,"endTime":1696.21,"body":"we would use calculators or computers."},{"speaker":"I jest","startTime":1696.21,"endTime":1698.76,"body":"But before logs came along, logarithms,"},{"speaker":"I jest","startTime":1698.76,"endTime":1700.6,"body":"which turn multiplication into addition"},{"speaker":"I jest","startTime":1700.6,"endTime":1703.193,"body":"in a way that I won't go into right now,"},{"speaker":"I jest","startTime":1704.13,"endTime":1705.83,"body":"a different technology was invented,"},{"speaker":"I jest","startTime":1705.83,"endTime":1707.98,"body":"and it made use of things"},{"speaker":"I jest","startTime":1705.83,"endTime":1707.98,"body":"that already existed,"},{"speaker":"I jest","startTime":1707.98,"endTime":1710.12,"body":"these tables that we already had."},{"speaker":"I jest","startTime":1710.12,"endTime":1712.68,"body":"And it's got a brilliant"},{"speaker":"I jest","startTime":1710.12,"endTime":1712.68,"body":"name, prosthaphaeresis,"},{"speaker":"I jest","startTime":1712.68,"endTime":1715.01,"body":"which kind of means"},{"speaker":"I jest","startTime":1712.68,"endTime":1715.01,"body":"adding and taking away."},{"speaker":"I jest","startTime":1715.01,"endTime":1719.06,"body":"And it uses this nice, little fact,"},{"speaker":"I jest","startTime":1719.06,"endTime":1722.61,"body":"which is that there's this formula"},{"speaker":"I jest","startTime":1722.61,"endTime":1726.23,"body":"that turns a product of two cosines"},{"speaker":"I jest","startTime":1726.23,"endTime":1727.98,"body":"into a sum,"},{"speaker":"I jest","startTime":1727.98,"endTime":1729.3,"body":"or I guess, strictly speaking,"},{"speaker":"I jest","startTime":1729.3,"endTime":1730.133,"body":"it's an average, isn't it,"},{"speaker":"I jest","startTime":1730.133,"endTime":1731.84,"body":"add them up and divide by two."},{"speaker":"I jest","startTime":1731.84,"endTime":1734.19,"body":"And we know addition is easier,"},{"speaker":"I jest","startTime":1734.19,"endTime":1735.04,"body":"we can halve things,"},{"speaker":"I jest","startTime":1735.04,"endTime":1735.96,"body":"that's not a problem,"},{"speaker":"I jest","startTime":1735.96,"endTime":1738.05,"body":"adding is easier than multiplication,"},{"speaker":"I jest","startTime":1738.05,"endTime":1740.24,"body":"so this is, before logarithms,"},{"speaker":"I jest","startTime":1740.24,"endTime":1743.203,"body":"a way to turn multiplication"},{"speaker":"I jest","startTime":1740.24,"endTime":1743.203,"body":"into addition."},{"speaker":"I jest","startTime":1744.43,"endTime":1746.53,"body":"It's a little bit more involved"},{"speaker":"I jest","startTime":1746.53,"endTime":1748.04,"body":"than what we might do later on,"},{"speaker":"I jest","startTime":1748.04,"endTime":1750.08,"body":"but I just want to show you"},{"speaker":"I jest","startTime":1748.04,"endTime":1750.08,"body":"a tiny, little toy example,"},{"speaker":"I jest","startTime":1750.08,"endTime":1751.99,"body":"because I think this is"},{"speaker":"I jest","startTime":1750.08,"endTime":1751.99,"body":"a really great way of,"},{"speaker":"I jest","startTime":1751.99,"endTime":1755.25,"body":"somehow it's using the"},{"speaker":"I jest","startTime":1751.99,"endTime":1755.25,"body":"existing infrastructure"},{"speaker":"I jest","startTime":1755.25,"endTime":1756.393,"body":"to solve a problem."},{"speaker":"I jest","startTime":1757.56,"endTime":1760.78,"body":"So let me just give you a"},{"speaker":"I jest","startTime":1757.56,"endTime":1760.78,"body":"very tiny, little toy example,"},{"speaker":"I jest","startTime":1760.78,"endTime":1763.88,"body":"let's multiply 123 by 456."},{"speaker":"I jest","startTime":1763.88,"endTime":1766.43,"body":"So what you do is you scale,"},{"speaker":"I jest","startTime":1766.43,"endTime":1768.57,"body":"you divide by a suitable power of 10"},{"speaker":"I jest","startTime":1768.57,"endTime":1771.16,"body":"to create two numbers that"},{"speaker":"I jest","startTime":1768.57,"endTime":1771.16,"body":"are between zero and one,"},{"speaker":"I jest","startTime":1771.16,"endTime":1774.903,"body":"so here, A is 0.123, B is 0.456."},{"speaker":"I jest","startTime":1777.03,"endTime":1778.8,"body":"Because they're between zero and one,"},{"speaker":"I jest","startTime":1778.8,"endTime":1780.7,"body":"both of those are cos of something,"},{"speaker":"I jest","startTime":1780.7,"endTime":1782.84,"body":"so you get out your trig tables,"},{"speaker":"I jest","startTime":1782.84,"endTime":1784.83,"body":"and you find x and y"},{"speaker":"I jest","startTime":1784.83,"endTime":1788.62,"body":"such that cos of x is A and cos of y is B."},{"speaker":"I jest","startTime":1788.62,"endTime":1790.72,"body":"So now, that formula at the top"},{"speaker":"I jest","startTime":1790.72,"endTime":1792.51,"body":"is going to tell you what's A times B"},{"speaker":"I jest","startTime":1792.51,"endTime":1794.52,"body":"in terms of adding some things together."},{"speaker":"I jest","startTime":1794.52,"endTime":1796.01,"body":"You've got your x and y,"},{"speaker":"I jest","startTime":1796.01,"endTime":1799.163,"body":"so you find x plus y, you find x minus y."},{"speaker":"I jest","startTime":1800.47,"endTime":1802.96,"body":"Done, addition is easy,"},{"speaker":"I jest","startTime":1802.96,"endTime":1804.93,"body":"and then all you need to do"},{"speaker":"I jest","startTime":1804.93,"endTime":1806.55,"body":"is take the average of those things,"},{"speaker":"I jest","startTime":1806.55,"endTime":1807.95,"body":"so add them up, divide by two,"},{"speaker":"I jest","startTime":1807.95,"endTime":1810.07,"body":"again, we're just adding, very easy,"},{"speaker":"I jest","startTime":1810.07,"endTime":1814.2,"body":"and you get the answer 0.056088,"},{"speaker":"I jest","startTime":1814.2,"endTime":1815.9,"body":"and that is your A times B."},{"speaker":"All we have to do now is scale back up","startTime":1819.03,"endTime":1820.46,"body":"we divided by 1,000 twice,"},{"speaker":"All we have to do now is scale back up","startTime":1820.46,"endTime":1822.38,"body":"we've got to multiply by 1,000 twice,"},{"speaker":"All we have to do now is scale back up","startTime":1822.38,"endTime":1826.577,"body":"and we find that 123 times 456 is 56,088,"},{"speaker":"All we have to do now is scale back up","startTime":1827.82,"endTime":1829.91,"body":"and we've used our trig tables"},{"speaker":"All we have to do now is scale back up","startTime":1829.91,"endTime":1833.95,"body":"to get away from having to do"},{"speaker":"All we have to do now is scale back up","startTime":1829.91,"endTime":1833.95,"body":"that difficult multiplication,"},{"speaker":"All we have to do now is scale back up","startTime":1833.95,"endTime":1836.77,"body":"and that can be a really useful thing"},{"speaker":"All we have to do now is scale back up","startTime":1836.77,"endTime":1839.98,"body":"if you're multiplying very,"},{"speaker":"All we have to do now is scale back up","startTime":1836.77,"endTime":1839.98,"body":"very long numbers together."},{"speaker":"All we have to do now is scale back up","startTime":1839.98,"endTime":1843.01,"body":"It may be that you don't"},{"speaker":"All we have to do now is scale back up","startTime":1839.98,"endTime":1843.01,"body":"get the exact answer"},{"speaker":"All we have to do now is scale back up","startTime":1843.01,"endTime":1844.44,"body":"because you might have"},{"speaker":"All we have to do now is scale back up","startTime":1843.01,"endTime":1844.44,"body":"to approximate slightly,"},{"speaker":"All we have to do now is scale back up","startTime":1844.44,"endTime":1847.08,"body":"but you get a good"},{"speaker":"All we have to do now is scale back up","startTime":1844.44,"endTime":1847.08,"body":"enough approximate answer"},{"speaker":"All we have to do now is scale back up","startTime":1847.08,"endTime":1848.56,"body":"for your calculations."},{"speaker":"All we have to do now is scale back up","startTime":1848.56,"endTime":1851.87,"body":"So it's a very clever use of these tables"},{"speaker":"All we have to do now is scale back up","startTime":1851.87,"endTime":1853.6,"body":"before logarithms came along,"},{"speaker":"All we have to do now is scale back up","startTime":1853.6,"endTime":1854.643,"body":"which I quite like."},{"speaker":"All we have to do now is scale back up","startTime":1855.79,"endTime":1857.41,"body":"So we've seen that sine,"},{"speaker":"All we have to do now is scale back up","startTime":1857.41,"endTime":1860.92,"body":"and its friends cosine and also tan,"},{"speaker":"All we have to do now is scale back up","startTime":1860.92,"endTime":1862.05,"body":"are useful,"},{"speaker":"All we have to do now is scale back up","startTime":1862.05,"endTime":1864.39,"body":"there are really, really useful, important"},{"speaker":"All we have to do now is scale back up","startTime":1864.39,"endTime":1865.35,"body":"practical applications,"},{"speaker":"All we have to do now is scale back up","startTime":1865.35,"endTime":1867.89,"body":"and that is why all this"},{"speaker":"All we have to do now is scale back up","startTime":1865.35,"endTime":1867.89,"body":"technology developed,"},{"speaker":"All we have to do now is scale back up","startTime":1867.89,"endTime":1870.98,"body":"and people got very, very good"},{"speaker":"All we have to do now is scale back up","startTime":1867.89,"endTime":1870.98,"body":"at working with these things."},{"speaker":"All we have to do now is scale back up","startTime":1870.98,"endTime":1874.71,"body":"But we haven't yet used, or seen"},{"speaker":"All we have to do now is scale back up","startTime":1874.71,"endTime":1876.87,"body":"this nice curve that can go on forever,"},{"speaker":"All we have to do now is scale back up","startTime":1876.87,"endTime":1879.0,"body":"all of these angles that we're using,"},{"speaker":"All we have to do now is scale back up","startTime":1879.0,"endTime":1881.03,"body":"they're angles within the circle."},{"speaker":"All we have to do now is scale back up","startTime":1881.03,"endTime":1883.883,"body":"But a bit after this point,"},{"speaker":"All we have to do now is scale back up","startTime":1885.09,"endTime":1885.95,"body":"we got to the stage"},{"speaker":"All we have to do now is scale back up","startTime":1885.95,"endTime":1887.36,"body":"where mathematicians and scientists"},{"speaker":"All we have to do now is scale back up","startTime":1887.36,"endTime":1892.36,"body":"could start understanding and"},{"speaker":"All we have to do now is scale back up","startTime":1887.36,"endTime":1892.36,"body":"solving some physical problems"},{"speaker":"All we have to do now is scale back up","startTime":1892.38,"endTime":1896.32,"body":"that involve oscillation"},{"speaker":"All we have to do now is scale back up","startTime":1892.38,"endTime":1896.32,"body":"and periodic motion,"},{"speaker":"All we have to do now is scale back up","startTime":1896.32,"endTime":1897.49,"body":"where you get some motion,"},{"speaker":"All we have to do now is scale back up","startTime":1897.49,"endTime":1899.3,"body":"and I'll show you actually, straightaway,"},{"speaker":"All we have to do now is scale back up","startTime":1899.3,"endTime":1902.24,"body":"you get some motion that repeats."},{"speaker":"All we have to do now is scale back up","startTime":1902.24,"endTime":1903.88,"body":"This is very soothing,"},{"speaker":"All we have to do now is scale back up","startTime":1903.88,"endTime":1907.07,"body":"a very soothing, little animation there"},{"speaker":"All we have to do now is scale back up","startTime":1907.07,"endTime":1909.29,"body":"of a pendulum swinging"},{"speaker":"All we have to do now is scale back up","startTime":1907.07,"endTime":1909.29,"body":"backwards and forwards,"},{"speaker":"All we have to do now is scale back up","startTime":1909.29,"endTime":1912.72,"body":"and its path is being traced below,"},{"speaker":"All we have to do now is scale back up","startTime":1912.72,"endTime":1917.06,"body":"and we're moving the graph"},{"speaker":"All we have to do now is scale back up","startTime":1912.72,"endTime":1917.06,"body":"paper along underneath"},{"speaker":"All we have to do now is scale back up","startTime":1917.06,"endTime":1921.1,"body":"so you can see that the"},{"speaker":"All we have to do now is scale back up","startTime":1917.06,"endTime":1921.1,"body":"shape that's being produced"},{"speaker":"All we have to do now is scale back up","startTime":1921.1,"endTime":1922.65,"body":"is a sine wave."},{"speaker":"All we have to do now is scale back up","startTime":1922.65,"endTime":1924.57,"body":"And those kinds of situations"},{"speaker":"All we have to do now is scale back up","startTime":1926.43,"endTime":1928.923,"body":"sometimes give you a"},{"speaker":"All we have to do now is scale back up","startTime":1926.43,"endTime":1928.923,"body":"sine wave as a solution,"},{"speaker":"All we have to do now is scale back up","startTime":1928.923,"endTime":1931.21,"body":"or a sine function as a solution."},{"speaker":"All we have to do now is scale back up","startTime":1931.21,"endTime":1932.41,"body":"In particular,"},{"speaker":"All we have to do now is scale back up","startTime":1932.41,"endTime":1934.84,"body":"the kind of situation"},{"speaker":"All we have to do now is scale back up","startTime":1932.41,"endTime":1934.84,"body":"where you get a sine wave"},{"speaker":"All we have to do now is scale back up","startTime":1936.05,"endTime":1938.483,"body":"is where you've got a force,"},{"speaker":"All we have to do now is scale back up","startTime":1939.74,"endTime":1940.87,"body":"like a resulting force,"},{"speaker":"All we have to do now is scale back up","startTime":1940.87,"endTime":1941.93,"body":"you cancel everything out,"},{"speaker":"All we have to do now is scale back up","startTime":1941.93,"endTime":1943.53,"body":"and the net effect is a force"},{"speaker":"All we have to do now is scale back up","startTime":1943.53,"endTime":1945.94,"body":"that's pulling you towards"},{"speaker":"All we have to do now is scale back up","startTime":1943.53,"endTime":1945.94,"body":"an equilibrium position."},{"speaker":"All we have to do now is scale back up","startTime":1945.94,"endTime":1946.773,"body":"So what I've got here"},{"speaker":"All we have to do now is scale back up","startTime":1946.773,"endTime":1949.32,"body":"is a pendulum swinging"},{"speaker":"All we have to do now is scale back up","startTime":1946.773,"endTime":1949.32,"body":"backwards and forwards,"},{"speaker":"All we have to do now is scale back up","startTime":1949.32,"endTime":1951.02,"body":"so it's got rigid rod"},{"speaker":"All we have to do now is scale back up","startTime":1951.02,"endTime":1953.37,"body":"and then a bob, or weight at the end,"},{"speaker":"All we have to do now is scale back up","startTime":1953.37,"endTime":1955.67,"body":"and it's just swinging"},{"speaker":"All we have to do now is scale back up","startTime":1953.37,"endTime":1955.67,"body":"backwards and forwards"},{"speaker":"All we have to do now is scale back up","startTime":1955.67,"endTime":1956.67,"body":"in arcs of circles"},{"speaker":"All we have to do now is scale back up","startTime":1956.67,"endTime":1958.57,"body":"in a very soothing way,"},{"speaker":"All we have to do now is scale back up","startTime":1958.57,"endTime":1960.58,"body":"and what happens there"},{"speaker":"All we have to do now is scale back up","startTime":1960.58,"endTime":1963.44,"body":"is there's a force"},{"speaker":"All we have to do now is scale back up","startTime":1960.58,"endTime":1963.44,"body":"downwards due to gravity,"},{"speaker":"All we have to do now is scale back up","startTime":1963.44,"endTime":1964.48,"body":"but the part of it"},{"speaker":"All we have to do now is scale back up","startTime":1964.48,"endTime":1966.34,"body":"that's trying to pull that bob away"},{"speaker":"All we have to do now is scale back up","startTime":1966.34,"endTime":1968.34,"body":"from the center of the circle"},{"speaker":"All we have to do now is scale back up","startTime":1968.34,"endTime":1972.23,"body":"is exactly counteracted"},{"speaker":"All we have to do now is scale back up","startTime":1968.34,"endTime":1972.23,"body":"by the tension in the rod."},{"speaker":"All we have to do now is scale back up","startTime":1972.23,"endTime":1976.81,"body":"So the distance from the center"},{"speaker":"All we have to do now is scale back up","startTime":1972.23,"endTime":1976.81,"body":"of the circle can't change,"},{"speaker":"All we have to do now is scale back up","startTime":1976.81,"endTime":1979.62,"body":"so that bit of the force due to gravity"},{"speaker":"All we have to do now is scale back up","startTime":1979.62,"endTime":1981.22,"body":"is canceled out,"},{"speaker":"All we have to do now is scale back up","startTime":1981.22,"endTime":1983.22,"body":"and the remaining bit of the force,"},{"speaker":"All we have to do now is scale back up","startTime":1983.22,"endTime":1985.51,"body":"all it can do is pull downwards,"},{"speaker":"All we have to do now is scale back up","startTime":1985.51,"endTime":1989.46,"body":"but along the circumference of the circle."},{"speaker":"All we have to do now is scale back up","startTime":1989.46,"endTime":1992.2,"body":"And the steeper the angle you get,"},{"speaker":"All we have to do now is scale back up","startTime":1992.2,"endTime":1995.5,"body":"the more proportion of"},{"speaker":"All we have to do now is scale back up","startTime":1992.2,"endTime":1995.5,"body":"that vertical gravity force"},{"speaker":"All we have to do now is scale back up","startTime":1995.5,"endTime":1997.83,"body":"is along that circumference."},{"speaker":"All we have to do now is scale back up","startTime":1997.83,"endTime":1999.13,"body":"The other thing that you require"},{"speaker":"All we have to do now is scale back up","startTime":1999.13,"endTime":2001.4,"body":"in order to get a sine"},{"speaker":"All we have to do now is scale back up","startTime":1999.13,"endTime":2001.4,"body":"function out of this"},{"speaker":"All we have to do now is scale back up","startTime":2001.4,"endTime":2006.4,"body":"is that the force that's"},{"speaker":"All we have to do now is scale back up","startTime":2001.4,"endTime":2006.4,"body":"being pulled downwards"},{"speaker":"All we have to do now is scale back up","startTime":2008.93,"endTime":2010.15,"body":"towards the equilibrium position,"},{"speaker":"All we have to do now is scale back up","startTime":2010.15,"endTime":2010.983,"body":"which in this case"},{"speaker":"All we have to do now is scale back up","startTime":2010.983,"endTime":2014.32,"body":"is just the pendulum"},{"speaker":"All we have to do now is scale back up","startTime":2010.983,"endTime":2014.32,"body":"vertically downwards at rest,"},{"speaker":"All we have to do now is scale back up","startTime":2014.32,"endTime":2015.89,"body":"that force should be proportional"},{"speaker":"All we have to do now is scale back up","startTime":2015.89,"endTime":2018.88,"body":"to the distance from the"},{"speaker":"All we have to do now is scale back up","startTime":2015.89,"endTime":2018.88,"body":"equilibrium position."},{"speaker":"All we have to do now is scale back up","startTime":2018.88,"endTime":2022.63,"body":"And if you look at the equations"},{"speaker":"All we have to do now is scale back up","startTime":2018.88,"endTime":2022.63,"body":"you get from a pendulum,"},{"speaker":"All we have to do now is scale back up","startTime":2022.63,"endTime":2025.15,"body":"then for small oscillations,"},{"speaker":"All we have to do now is scale back up","startTime":2025.15,"endTime":2027.16,"body":"for small angles of oscillation,"},{"speaker":"All we have to do now is scale back up","startTime":2027.16,"endTime":2028.28,"body":"that's basically true,"},{"speaker":"All we have to do now is scale back up","startTime":2028.28,"endTime":2030.31,"body":"you have the make the odd"},{"speaker":"All we have to do now is scale back up","startTime":2028.28,"endTime":2030.31,"body":"approximation here and there,"},{"speaker":"All we have to do now is scale back up","startTime":2030.31,"endTime":2032.4,"body":"but that's basically what happens,"},{"speaker":"All we have to do now is scale back up","startTime":2032.4,"endTime":2037.4,"body":"and so you get this sine"},{"speaker":"All we have to do now is scale back up","startTime":2032.4,"endTime":2037.4,"body":"wave coming out of it,"},{"speaker":"All we have to do now is scale back up","startTime":2037.52,"endTime":2040.01,"body":"it's all squashed into a vertical,"},{"speaker":"All we have to do now is scale back up","startTime":2040.01,"endTime":2041.52,"body":"but that's what you get."},{"speaker":"All we have to do now is scale back up","startTime":2041.52,"endTime":2042.61,"body":"And in particular,"},{"speaker":"All we have to do now is scale back up","startTime":2042.61,"endTime":2046.56,"body":"if you set this thing going"},{"speaker":"All we have to do now is scale back up","startTime":2042.61,"endTime":2046.56,"body":"at some time, t equals zero,"},{"speaker":"All we have to do now is scale back up","startTime":2046.56,"endTime":2049.42,"body":"then the displacement after some time, t,"},{"speaker":"All we have to do now is scale back up","startTime":2049.42,"endTime":2051.853,"body":"will depend on the sine of t."},{"speaker":"All we have to do now is scale back up","startTime":2053.599,"endTime":2054.56,"body":"There are other situations"},{"speaker":"All we have to do now is scale back up","startTime":2054.56,"endTime":2056.49,"body":"where you get precisely"},{"speaker":"All we have to do now is scale back up","startTime":2054.56,"endTime":2056.49,"body":"this same thing happening,"},{"speaker":"All we have to do now is scale back up","startTime":2056.49,"endTime":2058.42,"body":"and I want to show you"},{"speaker":"All we have to do now is scale back up","startTime":2056.49,"endTime":2058.42,"body":"just one other of them,"},{"speaker":"All we have to do now is scale back up","startTime":2058.42,"endTime":2063.0,"body":"because it was something"},{"speaker":"All we have to do now is scale back up","startTime":2058.42,"endTime":2063.0,"body":"due to a predecessor of mine"},{"speaker":"All we have to do now is scale back up","startTime":2063.0,"endTime":2065.09,"body":"as Gresham professor of geometry,"},{"speaker":"All we have to do now is scale back up","startTime":2065.09,"endTime":2065.923,"body":"Robert Hooke,"},{"speaker":"All we have to do now is scale back up","startTime":2065.923,"endTime":2066.756,"body":"and this is the picture of him"},{"speaker":"All we have to do now is scale back up","startTime":2066.756,"endTime":2068.5,"body":"that's hanging in Gresham College now."},{"speaker":"All we have to do now is scale back up","startTime":2068.5,"endTime":2071.25,"body":"It wasn't painted at the time of his life,"},{"speaker":"All we have to do now is scale back up","startTime":2071.25,"endTime":2073.11,"body":"I'm not able to see that there are any"},{"speaker":"All we have to do now is scale back up","startTime":2073.11,"endTime":2077.04,"body":"reliably, provably"},{"speaker":"All we have to do now is scale back up","startTime":2073.11,"endTime":2077.04,"body":"accurate pictures of Hooke"},{"speaker":"All we have to do now is scale back up","startTime":2077.04,"endTime":2078.51,"body":"painted during his lifetime,"},{"speaker":"All we have to do now is scale back up","startTime":2078.51,"endTime":2081.65,"body":"but this is the one that's"},{"speaker":"All we have to do now is scale back up","startTime":2078.51,"endTime":2081.65,"body":"hanging up at Gresham now,"},{"speaker":"All we have to do now is scale back up","startTime":2081.65,"endTime":2083.7,"body":"and he's holding in his hand"},{"speaker":"All we have to do now is scale back up","startTime":2083.7,"endTime":2085.84,"body":"the thing that I want to talk about,"},{"speaker":"All we have to do now is scale back up","startTime":2085.84,"endTime":2087.5,"body":"which is a spring."},{"speaker":"All we have to do now is scale back up","startTime":2087.5,"endTime":2090.49,"body":"And we may all remember"},{"speaker":"All we have to do now is scale back up","startTime":2087.5,"endTime":2090.49,"body":"having learned at school"},{"speaker":"All we have to do now is scale back up","startTime":2090.49,"endTime":2091.323,"body":"Hooke's Law,"},{"speaker":"All we have to do now is scale back up","startTime":2091.323,"endTime":2092.156,"body":"and here it is,"},{"speaker":"All we have to do now is scale back up","startTime":2092.156,"endTime":2094.643,"body":"I'm sure you recognize it, 1676."},{"speaker":"All we have to do now is scale back up","startTime":2097.16,"endTime":2098.603,"body":"This is a jumble of letters."},{"speaker":"All we have to do now is scale back up","startTime":2100.07,"endTime":2101.47,"body":"This is Hooke's Law,"},{"speaker":"All we have to do now is scale back up","startTime":2101.47,"endTime":2102.973,"body":"but at the time,"},{"speaker":"All we have to do now is scale back up","startTime":2104.64,"endTime":2106.7,"body":"people often did things like this,"},{"speaker":"All we have to do now is scale back up","startTime":2106.7,"endTime":2110.7,"body":"he actually published"},{"speaker":"All we have to do now is scale back up","startTime":2106.7,"endTime":2110.7,"body":"an anagram of his law,"},{"speaker":"All we have to do now is scale back up","startTime":2110.7,"endTime":2113.55,"body":"I'm sure you're quickly"},{"speaker":"All we have to do now is scale back up","startTime":2110.7,"endTime":2113.55,"body":"solving this right now,"},{"speaker":"All we have to do now is scale back up","startTime":2113.55,"endTime":2115.697,"body":"and getting the Latin phrase,"},{"speaker":"All we have to do now is scale back up","startTime":2115.697,"endTime":2118.41,"body":"\"Ut tensio, sic vis,\""},{"speaker":"All we have to do now is scale back up","startTime":2118.41,"endTime":2120.343,"body":"as the extension, so the force."},{"speaker":"All we have to do now is scale back up","startTime":2121.52,"endTime":2124.02,"body":"This was a thing done"},{"speaker":"All we have to do now is scale back up","startTime":2124.02,"endTime":2129.02,"body":"to prevent, or try and"},{"speaker":"All we have to do now is scale back up","startTime":2124.02,"endTime":2129.02,"body":"prevent priority disputes."},{"speaker":"All we have to do now is scale back up","startTime":2129.27,"endTime":2131.19,"body":"He's thought of this great, new law,"},{"speaker":"All we have to do now is scale back up","startTime":2131.19,"endTime":2132.71,"body":"if he just publishes it,"},{"speaker":"All we have to do now is scale back up","startTime":2132.71,"endTime":2135.02,"body":"what's to stop someone"},{"speaker":"All we have to do now is scale back up","startTime":2132.71,"endTime":2135.02,"body":"else coming up and saying,"},{"speaker":"All we have to do now is scale back up","startTime":2135.02,"endTime":2136.4,"body":"I thought of that last week,"},{"speaker":"All we have to do now is scale back up","startTime":2136.4,"endTime":2137.72,"body":"look, here's my bit of parchment"},{"speaker":"All we have to do now is scale back up","startTime":2137.72,"endTime":2139.5,"body":"with the date written on, last Tuesday,"},{"speaker":"All we have to do now is scale back up","startTime":2139.5,"endTime":2140.783,"body":"I thought of it first."},{"speaker":"All we have to do now is scale back up","startTime":2142.16,"endTime":2143.8,"body":"If you don't want that kind of nuisance,"},{"speaker":"All we have to do now is scale back up","startTime":2143.8,"endTime":2145.06,"body":"and maybe they did think of it first,"},{"speaker":"All we have to do now is scale back up","startTime":2145.06,"endTime":2148.4,"body":"it's very hard to then"},{"speaker":"All we have to do now is scale back up","startTime":2145.06,"endTime":2148.4,"body":"prove who got there first,"},{"speaker":"All we have to do now is scale back up","startTime":2148.4,"endTime":2149.91,"body":"so what you would do"},{"speaker":"All we have to do now is scale back up","startTime":2149.91,"endTime":2151.88,"body":"would be you'd publish something,"},{"speaker":"All we have to do now is scale back up","startTime":2151.88,"endTime":2153.44,"body":"or in a letter to a friend,"},{"speaker":"All we have to do now is scale back up","startTime":2151.88,"endTime":2153.44,"body":"or something like this,"},{"speaker":"All we have to do now is scale back up","startTime":2153.44,"endTime":2157.54,"body":"you'd put some encrypted version"},{"speaker":"All we have to do now is scale back up","startTime":2153.44,"endTime":2157.54,"body":"of what you wanted to say,"},{"speaker":"All we have to do now is scale back up","startTime":2157.54,"endTime":2160.07,"body":"so Hooke's just put the"},{"speaker":"All we have to do now is scale back up","startTime":2157.54,"endTime":2160.07,"body":"letters in alphabetical order,"},{"speaker":"All we have to do now is scale back up","startTime":2160.07,"endTime":2161.26,"body":"hasn't he, there,"},{"speaker":"All we have to do now is scale back up","startTime":2161.26,"endTime":2163.91,"body":"and then later on, you wait a bit,"},{"speaker":"All we have to do now is scale back up","startTime":2163.91,"endTime":2165.07,"body":"if no one comes up and says,"},{"speaker":"All we have to do now is scale back up","startTime":2165.07,"endTime":2168.947,"body":"ah, he must mean, \"Ut tensio, sic vis,\""},{"speaker":"All we have to do now is scale back up","startTime":2169.88,"endTime":2173.0,"body":"then you're safe to reveal"},{"speaker":"All we have to do now is scale back up","startTime":2169.88,"endTime":2173.0,"body":"your amazing discovery."},{"speaker":"All we have to do now is scale back up","startTime":2173.0,"endTime":2174.63,"body":"So that's Hooke's Law,"},{"speaker":"All we have to do now is scale back up","startTime":2174.63,"endTime":2177.34,"body":"what it actually means is"},{"speaker":"All we have to do now is scale back up","startTime":2174.63,"endTime":2177.34,"body":"we've got a spring now,"},{"speaker":"All we have to do now is scale back up","startTime":2177.34,"endTime":2179.4,"body":"it's going to bounce up"},{"speaker":"All we have to do now is scale back up","startTime":2177.34,"endTime":2179.4,"body":"and down quite quickly,"},{"speaker":"All we have to do now is scale back up","startTime":2179.4,"endTime":2181.26,"body":"it's very energetic,"},{"speaker":"All we have to do now is scale back up","startTime":2181.26,"endTime":2182.79,"body":"there it is oscillating away."},{"speaker":"All we have to do now is scale back up","startTime":2182.79,"endTime":2184.317,"body":"So what happens here,"},{"speaker":"All we have to do now is scale back up","startTime":2184.317,"endTime":2185.9,"body":"and what Hooke discovered,"},{"speaker":"All we have to do now is scale back up","startTime":2185.9,"endTime":2187.2,"body":"is that, take a spring,"},{"speaker":"All we have to do now is scale back up","startTime":2187.2,"endTime":2188.56,"body":"you put a weight on it,"},{"speaker":"All we have to do now is scale back up","startTime":2188.56,"endTime":2190.22,"body":"if you just let it go,"},{"speaker":"All we have to do now is scale back up","startTime":2190.22,"endTime":2192.49,"body":"then the tension in the spring"},{"speaker":"All we have to do now is scale back up","startTime":2192.49,"endTime":2194.27,"body":"increases with the extension,"},{"speaker":"All we have to do now is scale back up","startTime":2194.27,"endTime":2196.123,"body":"with how stretched it is,"},{"speaker":"All we have to do now is scale back up","startTime":2196.966,"endTime":2200.582,"body":"the force of gravity is pulling downwards,"},{"speaker":"All we have to do now is scale back up","startTime":2200.582,"endTime":2203.89,"body":"the tension in the spring"},{"speaker":"All we have to do now is scale back up","startTime":2200.582,"endTime":2203.89,"body":"is pulling upwards,"},{"speaker":"All we have to do now is scale back up","startTime":2203.89,"endTime":2205.43,"body":"there'll be an equilibrium position"},{"speaker":"All we have to do now is scale back up","startTime":2205.43,"endTime":2207.62,"body":"where they exactly balance out."},{"speaker":"All we have to do now is scale back up","startTime":2207.62,"endTime":2209.83,"body":"If you disturb it from"},{"speaker":"All we have to do now is scale back up","startTime":2207.62,"endTime":2209.83,"body":"that equilibrium position"},{"speaker":"All we have to do now is scale back up","startTime":2209.83,"endTime":2211.36,"body":"at some time, t,"},{"speaker":"All we have to do now is scale back up","startTime":2211.36,"endTime":2215.12,"body":"then if you pull it down below"},{"speaker":"All we have to do now is scale back up","startTime":2211.36,"endTime":2215.12,"body":"the equilibrium position,"},{"speaker":"All we have to do now is scale back up","startTime":2215.12,"endTime":2216.14,"body":"then the tension in the spring"},{"speaker":"All we have to do now is scale back up","startTime":2216.14,"endTime":2219.0,"body":"is higher than the force of"},{"speaker":"All we have to do now is scale back up","startTime":2216.14,"endTime":2219.0,"body":"gravity that's acting downwards,"},{"speaker":"All we have to do now is scale back up","startTime":2219.0,"endTime":2220.91,"body":"so it gets pulled upwards,"},{"speaker":"All we have to do now is scale back up","startTime":2220.91,"endTime":2223.23,"body":"and the amount of force is proportional"},{"speaker":"All we have to do now is scale back up","startTime":2223.23,"endTime":2225.98,"body":"to how far you are from"},{"speaker":"All we have to do now is scale back up","startTime":2223.23,"endTime":2225.98,"body":"the equilibrium position."},{"speaker":"All we have to do now is scale back up","startTime":2225.98,"endTime":2226.813,"body":"On the other hand,"},{"speaker":"All we have to do now is scale back up","startTime":2226.813,"endTime":2228.9,"body":"if you lift it above that"},{"speaker":"All we have to do now is scale back up","startTime":2226.813,"endTime":2228.9,"body":"equilibrium position,"},{"speaker":"All we have to do now is scale back up","startTime":2228.9,"endTime":2231.0,"body":"this time, the force of gravity"},{"speaker":"All we have to do now is scale back up","startTime":2231.0,"endTime":2233.78,"body":"outweighs the tension in the"},{"speaker":"All we have to do now is scale back up","startTime":2231.0,"endTime":2233.78,"body":"spring that's pulling you up,"},{"speaker":"All we have to do now is scale back up","startTime":2233.78,"endTime":2234.99,"body":"and so you get pulled down,"},{"speaker":"All we have to do now is scale back up","startTime":2234.99,"endTime":2235.823,"body":"and again,"},{"speaker":"All we have to do now is scale back up","startTime":2235.823,"endTime":2239.41,"body":"proportionally to your distance"},{"speaker":"All we have to do now is scale back up","startTime":2235.823,"endTime":2239.41,"body":"from the equilibrium point."},{"speaker":"All we have to do now is scale back up","startTime":2239.41,"endTime":2240.943,"body":"And so because of that,"},{"speaker":"All we have to do now is scale back up","startTime":2243.0,"endTime":2245.1,"body":"when you solve that equation that you get,"},{"speaker":"All we have to do now is scale back up","startTime":2245.1,"endTime":2246.75,"body":"the answer is"},{"speaker":"All we have to do now is scale back up","startTime":2246.75,"endTime":2249.67,"body":"that the displacement from"},{"speaker":"All we have to do now is scale back up","startTime":2246.75,"endTime":2249.67,"body":"the equilibrium position"},{"speaker":"All we have to do now is scale back up","startTime":2249.67,"endTime":2251.25,"body":"as this boings up and down"},{"speaker":"All we have to do now is scale back up","startTime":2251.25,"endTime":2253.65,"body":"is going to depend on, not the time,"},{"speaker":"All we have to do now is scale back up","startTime":2253.65,"endTime":2256.64,"body":"but sine of time, t."},{"speaker":"All we have to do now is scale back up","startTime":2256.64,"endTime":2257.68,"body":"So these are two places"},{"speaker":"All we have to do now is scale back up","startTime":2257.68,"endTime":2259.993,"body":"where you genuinely do"},{"speaker":"All we have to do now is scale back up","startTime":2257.68,"endTime":2259.993,"body":"get a sine function."},{"speaker":"All we have to do now is scale back up","startTime":2260.93,"endTime":2264.263,"body":"There are lots of other sine"},{"speaker":"All we have to do now is scale back up","startTime":2260.93,"endTime":2264.263,"body":"waves in physical phenomena."},{"speaker":"All we have to do now is scale back up","startTime":2265.17,"endTime":2266.77,"body":"Light is one,"},{"speaker":"All we have to do now is scale back up","startTime":2266.77,"endTime":2268.18,"body":"light waves are sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2268.18,"endTime":2269.89,"body":"so that's anything from light,"},{"speaker":"All we have to do now is scale back up","startTime":2269.89,"endTime":2271.65,"body":"X-rays, radio waves,"},{"speaker":"All we have to do now is scale back up","startTime":2271.65,"endTime":2272.95,"body":"all of these things are sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2272.95,"endTime":2276.0,"body":"and it's important to"},{"speaker":"All we have to do now is scale back up","startTime":2272.95,"endTime":2276.0,"body":"understand the sine function"},{"speaker":"All we have to do now is scale back up","startTime":2276.0,"endTime":2277.06,"body":"and the sine wave"},{"speaker":"All we have to do now is scale back up","startTime":2277.06,"endTime":2281.57,"body":"in order to be able to understand"},{"speaker":"All we have to do now is scale back up","startTime":2277.06,"endTime":2281.57,"body":"phenomena like refraction,"},{"speaker":"All we have to do now is scale back up","startTime":2281.57,"endTime":2283.05,"body":"interference patterns,"},{"speaker":"All we have to do now is scale back up","startTime":2283.05,"endTime":2286.58,"body":"those all rely on the"},{"speaker":"All we have to do now is scale back up","startTime":2283.05,"endTime":2286.58,"body":"properties of sine waves."},{"speaker":"All we have to do now is scale back up","startTime":2286.58,"endTime":2288.67,"body":"Other kinds of waves, though,"},{"speaker":"All we have to do now is scale back up","startTime":2288.67,"endTime":2290.35,"body":"may or may not be sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2290.35,"endTime":2292.86,"body":"so you can get periodic things,"},{"speaker":"All we have to do now is scale back up","startTime":2292.86,"endTime":2295.39,"body":"repeating patterns,"},{"speaker":"All we have to do now is scale back up","startTime":2295.39,"endTime":2298.22,"body":"which might or might not"},{"speaker":"All we have to do now is scale back up","startTime":2295.39,"endTime":2298.22,"body":"themselves be sine waves."},{"speaker":"All we have to do now is scale back up","startTime":2298.22,"endTime":2300.223,"body":"An example is water waves."},{"speaker":"All we have to do now is scale back up","startTime":2300.223,"endTime":2303.36,"body":"When the wind blows water in the ocean"},{"speaker":"All we have to do now is scale back up","startTime":2303.36,"endTime":2304.67,"body":"and creates waves,"},{"speaker":"All we have to do now is scale back up","startTime":2304.67,"endTime":2308.15,"body":"there is a lot going on in that situation,"},{"speaker":"All we have to do now is scale back up","startTime":2308.15,"endTime":2309.93,"body":"a lot of confounding factors,"},{"speaker":"All we have to do now is scale back up","startTime":2309.93,"endTime":2312.47,"body":"the temperature of the"},{"speaker":"All we have to do now is scale back up","startTime":2309.93,"endTime":2312.47,"body":"water can have an effect,"},{"speaker":"All we have to do now is scale back up","startTime":2312.47,"endTime":2313.64,"body":"the depth of the seabed,"},{"speaker":"All we have to do now is scale back up","startTime":2313.64,"endTime":2315.01,"body":"which of course is constantly changing,"},{"speaker":"All we have to do now is scale back up","startTime":2315.01,"endTime":2316.12,"body":"can have an effect,"},{"speaker":"All we have to do now is scale back up","startTime":2316.12,"endTime":2317.83,"body":"the strength of the wind,"},{"speaker":"All we have to do now is scale back up","startTime":2317.83,"endTime":2320.12,"body":"I don't know, passing whales swimming by,"},{"speaker":"All we have to do now is scale back up","startTime":2320.12,"endTime":2323.15,"body":"all sorts of things can"},{"speaker":"All we have to do now is scale back up","startTime":2320.12,"endTime":2323.15,"body":"affect what's going on here"},{"speaker":"All we have to do now is scale back up","startTime":2323.15,"endTime":2325.17,"body":"in these water waves or ripples."},{"speaker":"All we have to do now is scale back up","startTime":2325.17,"endTime":2328.51,"body":"And experiments have been done that say"},{"speaker":"All we have to do now is scale back up","startTime":2328.51,"endTime":2330.7,"body":"when the amplitude, the height of the wave"},{"speaker":"All we have to do now is scale back up","startTime":2330.7,"endTime":2333.64,"body":"is small in comparison"},{"speaker":"All we have to do now is scale back up","startTime":2330.7,"endTime":2333.64,"body":"with the wave length,"},{"speaker":"All we have to do now is scale back up","startTime":2333.64,"endTime":2336.8,"body":"then, actually, they are"},{"speaker":"All we have to do now is scale back up","startTime":2333.64,"endTime":2336.8,"body":"quite like sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2336.8,"endTime":2338.56,"body":"nice periodic things,"},{"speaker":"All we have to do now is scale back up","startTime":2338.56,"endTime":2341.31,"body":"but as the amplitude of"},{"speaker":"All we have to do now is scale back up","startTime":2338.56,"endTime":2341.31,"body":"the wave gets larger,"},{"speaker":"All we have to do now is scale back up","startTime":2341.31,"endTime":2344.23,"body":"and becomes big compared"},{"speaker":"All we have to do now is scale back up","startTime":2341.31,"endTime":2344.23,"body":"to the wave length,"},{"speaker":"All we have to do now is scale back up","startTime":2344.23,"endTime":2347.3,"body":"then you don't get things that"},{"speaker":"All we have to do now is scale back up","startTime":2344.23,"endTime":2347.3,"body":"quite look like sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2347.3,"endTime":2350.44,"body":"they have more crests and peaks like that,"},{"speaker":"All we have to do now is scale back up","startTime":2350.44,"endTime":2352.42,"body":"and they get closer to a curve"},{"speaker":"All we have to do now is scale back up","startTime":2352.42,"endTime":2353.35,"body":"that we've briefly mentioned"},{"speaker":"All we have to do now is scale back up","startTime":2353.35,"endTime":2354.42,"body":"in a previous lecture, actually,"},{"speaker":"All we have to do now is scale back up","startTime":2354.42,"endTime":2355.253,"body":"a trochoid,"},{"speaker":"All we have to do now is scale back up","startTime":2356.92,"endTime":2360.91,"body":"it isn't, but it looks a bit"},{"speaker":"All we have to do now is scale back up","startTime":2356.92,"endTime":2360.91,"body":"like semicircles upside-down."},{"speaker":"All we have to do now is scale back up","startTime":2360.91,"endTime":2362.01,"body":"Those things, then,"},{"speaker":"All we have to do now is scale back up","startTime":2362.01,"endTime":2363.67,"body":"they aren't sine waves."},{"speaker":"All we have to do now is scale back up","startTime":2363.67,"endTime":2366.39,"body":"So is all lost, is that"},{"speaker":"All we have to do now is scale back up","startTime":2363.67,"endTime":2366.39,"body":"the end of sine waves?"},{"speaker":"All we have to do now is scale back up","startTime":2366.39,"endTime":2367.66,"body":"No, no,"},{"speaker":"All we have to do now is scale back up","startTime":2367.66,"endTime":2370.72,"body":"because these things are"},{"speaker":"All we have to do now is scale back up","startTime":2367.66,"endTime":2370.72,"body":"periodic, they repeat,"},{"speaker":"All we have to do now is scale back up","startTime":2370.72,"endTime":2373.82,"body":"and because of the next guy,"},{"speaker":"All we have to do now is scale back up","startTime":2373.82,"endTime":2376.493,"body":"that means there's a link to sine waves."},{"speaker":"All we have to do now is scale back up","startTime":2376.493,"endTime":2378.27,"body":"The next guy is Joseph Fourier,"},{"speaker":"All we have to do now is scale back up","startTime":2378.27,"endTime":2380.8,"body":"and it's exactly 200 years this year"},{"speaker":"All we have to do now is scale back up","startTime":2380.8,"endTime":2385.8,"body":"since his world-changing"},{"speaker":"All we have to do now is scale back up","startTime":2380.8,"endTime":2385.8,"body":"book on heat flow,"},{"speaker":"All we have to do now is scale back up","startTime":2385.917,"endTime":2387.71,"body":"\"The Analytic Theory of Heat,\""},{"speaker":"All we have to do now is scale back up","startTime":2387.71,"endTime":2390.073,"body":"it came out in 1822, 200 years ago."},{"speaker":"All we have to do now is scale back up","startTime":2391.02,"endTime":2393.21,"body":"He was working on the heat equation"},{"speaker":"All we have to do now is scale back up","startTime":2393.21,"endTime":2398.21,"body":"which governs how waves of"},{"speaker":"All we have to do now is scale back up","startTime":2393.21,"endTime":2398.21,"body":"heat pass through objects."},{"speaker":"All we have to do now is scale back up","startTime":2398.79,"endTime":2399.623,"body":"To do this,"},{"speaker":"All we have to do now is scale back up","startTime":2399.623,"endTime":2401.81,"body":"he did something very"},{"speaker":"All we have to do now is scale back up","startTime":2399.623,"endTime":2401.81,"body":"clever with sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2401.81,"endTime":2403.13,"body":"and before I tell you what that is,"},{"speaker":"All we have to do now is scale back up","startTime":2403.13,"endTime":2406.59,"body":"I just want to give you a"},{"speaker":"All we have to do now is scale back up","startTime":2403.13,"endTime":2406.59,"body":"few sine waves to look at,"},{"speaker":"All we have to do now is scale back up","startTime":2406.59,"endTime":2409.32,"body":"because we can take our"},{"speaker":"All we have to do now is scale back up","startTime":2406.59,"endTime":2409.32,"body":"standard sine wave, there it is,"},{"speaker":"All we have to do now is scale back up","startTime":2409.32,"endTime":2410.81,"body":"y equals sin-x,"},{"speaker":"All we have to do now is scale back up","startTime":2410.81,"endTime":2414.02,"body":"from 0 to 2-pi, aka 360 degrees,"},{"speaker":"All we have to do now is scale back up","startTime":2414.02,"endTime":2415.86,"body":"it goes from 1 to -1,"},{"speaker":"All we have to do now is scale back up","startTime":2415.86,"endTime":2417.41,"body":"that's your sine wave,"},{"speaker":"All we have to do now is scale back up","startTime":2417.41,"endTime":2419.98,"body":"but you can tweak those a little bit,"},{"speaker":"All we have to do now is scale back up","startTime":2419.98,"endTime":2421.033,"body":"you can scale them."},{"speaker":"All we have to do now is scale back up","startTime":2422.28,"endTime":2423.69,"body":"Here's one way of scaling it,"},{"speaker":"All we have to do now is scale back up","startTime":2423.69,"endTime":2424.73,"body":"you can just double the height,"},{"speaker":"All we have to do now is scale back up","startTime":2424.73,"endTime":2427.29,"body":"you've stretched it vertically."},{"speaker":"All we have to do now is scale back up","startTime":2427.29,"endTime":2430.31,"body":"That's the same as me starting"},{"speaker":"All we have to do now is scale back up","startTime":2427.29,"endTime":2430.31,"body":"with a circle of radius,"},{"speaker":"All we have to do now is scale back up","startTime":2430.31,"endTime":2431.92,"body":"not one, but two."},{"speaker":"All we have to do now is scale back up","startTime":2431.92,"endTime":2434.296,"body":"So you can make one of double the height,"},{"speaker":"All we have to do now is scale back up","startTime":2434.296,"endTime":2435.129,"body":"it's a sine wave,"},{"speaker":"All we have to do now is scale back up","startTime":2435.129,"endTime":2436.62,"body":"we say it's sinusoidal,"},{"speaker":"All we have to do now is scale back up","startTime":2436.62,"endTime":2439.42,"body":"it's like a sine wave, just tweaked a bit."},{"speaker":"All we have to do now is scale back up","startTime":2439.42,"endTime":2440.43,"body":"Another thing you could do is"},{"speaker":"All we have to do now is scale back up","startTime":2440.43,"endTime":2441.77,"body":"you can make your radius,"},{"speaker":"All we have to do now is scale back up","startTime":2441.77,"endTime":2443.12,"body":"you can stick with a unit circle"},{"speaker":"All we have to do now is scale back up","startTime":2443.12,"endTime":2446.53,"body":"and just make the radius"},{"speaker":"All we have to do now is scale back up","startTime":2443.12,"endTime":2446.53,"body":"rotate at a different speed,"},{"speaker":"All we have to do now is scale back up","startTime":2446.53,"endTime":2448.47,"body":"so then you might get"},{"speaker":"All we have to do now is scale back up","startTime":2446.53,"endTime":2448.47,"body":"something like this one,"},{"speaker":"All we have to do now is scale back up","startTime":2448.47,"endTime":2450.05,"body":"sine of two-x,"},{"speaker":"All we have to do now is scale back up","startTime":2450.05,"endTime":2451.44,"body":"so you're doubling the angle,"},{"speaker":"All we have to do now is scale back up","startTime":2451.44,"endTime":2453.36,"body":"that means you go around twice as fast,"},{"speaker":"All we have to do now is scale back up","startTime":2453.36,"endTime":2455.19,"body":"and that means you fit twice as many waves"},{"speaker":"All we have to do now is scale back up","startTime":2455.19,"endTime":2456.56,"body":"into the same space,"},{"speaker":"All we have to do now is scale back up","startTime":2456.56,"endTime":2457.73,"body":"so it gets squashed,"},{"speaker":"All we have to do now is scale back up","startTime":2457.73,"endTime":2460.6,"body":"concertinaed in the horizontal direction."},{"speaker":"All we have to do now is scale back up","startTime":2460.6,"endTime":2463.993,"body":"So those are both fine,"},{"speaker":"All we have to do now is scale back up","startTime":2460.6,"endTime":2463.993,"body":"they're sine waves."},{"speaker":"All we have to do now is scale back up","startTime":2464.92,"endTime":2466.287,"body":"You can then combine them if you like,"},{"speaker":"All we have to do now is scale back up","startTime":2466.287,"endTime":2468.95,"body":"you can combine sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2468.95,"endTime":2470.91,"body":"so you can add them together."},{"speaker":"All we have to do now is scale back up","startTime":2470.91,"endTime":2471.743,"body":"Here, for example,"},{"speaker":"All we have to do now is scale back up","startTime":2471.743,"endTime":2473.16,"body":"is the red one and the blue one,"},{"speaker":"All we have to do now is scale back up","startTime":2473.16,"endTime":2476.03,"body":"I've just now made a graph"},{"speaker":"All we have to do now is scale back up","startTime":2473.16,"endTime":2476.03,"body":"of the sum of those two,"},{"speaker":"All we have to do now is scale back up","startTime":2476.03,"endTime":2477.267,"body":"so I've just added those points together,"},{"speaker":"All we have to do now is scale back up","startTime":2477.267,"endTime":2480.13,"body":"sin-2x plus 2-sin-x,"},{"speaker":"All we have to do now is scale back up","startTime":2480.13,"endTime":2480.963,"body":"and I get this thing,"},{"speaker":"All we have to do now is scale back up","startTime":2480.963,"endTime":2482.93,"body":"which is definitely not a sine wave,"},{"speaker":"All we have to do now is scale back up","startTime":2482.93,"endTime":2485.65,"body":"it's got a bit of a kink"},{"speaker":"All we have to do now is scale back up","startTime":2482.93,"endTime":2485.65,"body":"in the middle of it,"},{"speaker":"All we have to do now is scale back up","startTime":2485.65,"endTime":2487.02,"body":"but it is periodic,"},{"speaker":"All we have to do now is scale back up","startTime":2487.02,"endTime":2489.9,"body":"it does repeat every two-pi,"},{"speaker":"All we have to do now is scale back up","startTime":2489.9,"endTime":2493.05,"body":"because those things that"},{"speaker":"All we have to do now is scale back up","startTime":2489.9,"endTime":2493.05,"body":"made it up repeat as well."},{"speaker":"All we have to do now is scale back up","startTime":2493.05,"endTime":2495.44,"body":"So this is a periodic wave"},{"speaker":"All we have to do now is scale back up","startTime":2495.44,"endTime":2496.53,"body":"that's not a sine wave,"},{"speaker":"All we have to do now is scale back up","startTime":2496.53,"endTime":2499.36,"body":"but is somehow made of sine waves."},{"speaker":"All we have to do now is scale back up","startTime":2499.36,"endTime":2500.79,"body":"What Fourier did,"},{"speaker":"All we have to do now is scale back up","startTime":2500.79,"endTime":2502.083,"body":"it's pretty amazing,"},{"speaker":"All we have to do now is scale back up","startTime":2503.06,"endTime":2505.03,"body":"there's an asterisk, which I"},{"speaker":"All we have to do now is scale back up","startTime":2503.06,"endTime":2505.03,"body":"will come back to a second,"},{"speaker":"All we have to do now is scale back up","startTime":2505.03,"endTime":2505.95,"body":"in this statement,"},{"speaker":"All we have to do now is scale back up","startTime":2505.95,"endTime":2508.06,"body":"for the mathematicians in the audience,"},{"speaker":"All we have to do now is scale back up","startTime":2508.06,"endTime":2511.21,"body":"Fourier showed that any periodic function"},{"speaker":"All we have to do now is scale back up","startTime":2511.21,"endTime":2513.05,"body":"is made up sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2513.05,"endTime":2515.8,"body":"you can find sine waves to add together"},{"speaker":"All we have to do now is scale back up","startTime":2515.8,"endTime":2517.79,"body":"to make any periodic function,"},{"speaker":"All we have to do now is scale back up","startTime":2517.79,"endTime":2519.35,"body":"and we know how to do it,"},{"speaker":"All we have to do now is scale back up","startTime":2519.35,"endTime":2520.61,"body":"that's the crucial thing."},{"speaker":"All we have to do now is scale back up","startTime":2520.61,"endTime":2522.97,"body":"It's all very well having a"},{"speaker":"All we have to do now is scale back up","startTime":2520.61,"endTime":2522.97,"body":"theoretical thing that says,"},{"speaker":"All we have to do now is scale back up","startTime":2522.97,"endTime":2525.74,"body":"but there is a standard technique,"},{"speaker":"All we have to do now is scale back up","startTime":2525.74,"endTime":2526.94,"body":"it involves a bit of calculus,"},{"speaker":"All we have to do now is scale back up","startTime":2526.94,"endTime":2531.12,"body":"that allows you to pull"},{"speaker":"All we have to do now is scale back up","startTime":2526.94,"endTime":2531.12,"body":"out the exact sine waves"},{"speaker":"All we have to do now is scale back up","startTime":2531.12,"endTime":2532.92,"body":"that make up your periodic function."},{"speaker":"All we have to do now is scale back up","startTime":2533.83,"endTime":2535.96,"body":"I'll show you some examples in a second."},{"speaker":"All we have to do now is scale back up","startTime":2535.96,"endTime":2539.07,"body":"This asterisk is for the"},{"speaker":"All we have to do now is scale back up","startTime":2535.96,"endTime":2539.07,"body":"benefit of any mathematicians"},{"speaker":"All we have to do now is scale back up","startTime":2539.07,"endTime":2539.903,"body":"who are watching,"},{"speaker":"All we have to do now is scale back up","startTime":2539.903,"endTime":2540.736,"body":"who are saying,"},{"speaker":"All we have to do now is scale back up","startTime":2540.736,"endTime":2545.2,"body":"Sarah, do not lie to the"},{"speaker":"All we have to do now is scale back up","startTime":2540.736,"endTime":2545.2,"body":"good folk in your audience."},{"speaker":"All we have to do now is scale back up","startTime":2545.2,"endTime":2549.31,"body":"There are some really"},{"speaker":"All we have to do now is scale back up","startTime":2545.2,"endTime":2549.31,"body":"weird periodic functions"},{"speaker":"All we have to do now is scale back up","startTime":2549.31,"endTime":2551.25,"body":"where this kind of doesn't work."},{"speaker":"All we have to do now is scale back up","startTime":2551.25,"endTime":2553.04,"body":"There's something called"},{"speaker":"All we have to do now is scale back up","startTime":2551.25,"endTime":2553.04,"body":"the Dirichlet conditions,"},{"speaker":"All we have to do now is scale back up","startTime":2553.04,"endTime":2554.92,"body":"which you can go and look up if you like,"},{"speaker":"All we have to do now is scale back up","startTime":2554.92,"endTime":2557.23,"body":"that specify exactly when it's possible."},{"speaker":"All we have to do now is scale back up","startTime":2557.23,"endTime":2558.64,"body":"It's almost always possible"},{"speaker":"All we have to do now is scale back up","startTime":2558.64,"endTime":2560.7,"body":"for any kind of function"},{"speaker":"All we have to do now is scale back up","startTime":2558.64,"endTime":2560.7,"body":"you would encounter"},{"speaker":"All we have to do now is scale back up","startTime":2560.7,"endTime":2563.837,"body":"in your everyday scientific experiments,"},{"speaker":"All we have to do now is scale back up","startTime":2563.837,"endTime":2565.89,"body":"but there are some really weird,"},{"speaker":"All we have to do now is scale back up","startTime":2565.89,"endTime":2569.55,"body":"we call them pathological"},{"speaker":"All we have to do now is scale back up","startTime":2565.89,"endTime":2569.55,"body":"functions in mathematics,"},{"speaker":"All we have to do now is scale back up","startTime":2569.55,"endTime":2573.46,"body":"you can define things on"},{"speaker":"All we have to do now is scale back up","startTime":2569.55,"endTime":2573.46,"body":"purpose, just to be mean,"},{"speaker":"All we have to do now is scale back up","startTime":2573.46,"endTime":2575.42,"body":"that can't be treated in this way."},{"speaker":"All we have to do now is scale back up","startTime":2575.42,"endTime":2576.91,"body":"I'm thinking of the kind of function"},{"speaker":"All we have to do now is scale back up","startTime":2576.91,"endTime":2580.9,"body":"where it's valued at one"},{"speaker":"All we have to do now is scale back up","startTime":2576.91,"endTime":2580.9,"body":"on all rational numbers"},{"speaker":"All we have to do now is scale back up","startTime":2580.9,"endTime":2583.75,"body":"and zero on all irrational numbers,"},{"speaker":"All we have to do now is scale back up","startTime":2583.75,"endTime":2584.66,"body":"something like that."},{"speaker":"All we have to do now is scale back up","startTime":2584.66,"endTime":2587.52,"body":"So okay, small asterisk"},{"speaker":"All we have to do now is scale back up","startTime":2587.52,"endTime":2589.723,"body":"which I now propose to forget all about."},{"speaker":"All we have to do now is scale back up","startTime":2591.11,"endTime":2592.85,"body":"Let's see an example."},{"speaker":"All we have to do now is scale back up","startTime":2592.85,"endTime":2593.83,"body":"This thing on the top here,"},{"speaker":"All we have to do now is scale back up","startTime":2593.83,"endTime":2595.83,"body":"this is what's called a Fourier series,"},{"speaker":"All we have to do now is scale back up","startTime":2595.83,"endTime":2599.3,"body":"so it's a sum of lots"},{"speaker":"All we have to do now is scale back up","startTime":2595.83,"endTime":2599.3,"body":"of different sine waves."},{"speaker":"All we have to do now is scale back up","startTime":2599.3,"endTime":2601.25,"body":"In theory, there's"},{"speaker":"All we have to do now is scale back up","startTime":2599.3,"endTime":2601.25,"body":"infinitely many of them,"},{"speaker":"All we have to do now is scale back up","startTime":2601.25,"endTime":2602.46,"body":"but we'll do a few terms"},{"speaker":"All we have to do now is scale back up","startTime":2602.46,"endTime":2604.39,"body":"and we'll see what we think is happening."},{"speaker":"All we have to do now is scale back up","startTime":2604.39,"endTime":2605.77,"body":"So we're just going to start adding,"},{"speaker":"All we have to do now is scale back up","startTime":2605.77,"endTime":2607.15,"body":"and you can see the pattern here,"},{"speaker":"All we have to do now is scale back up","startTime":2607.15,"endTime":2608.49,"body":"you start with sin-x,"},{"speaker":"All we have to do now is scale back up","startTime":2608.49,"endTime":2612.16,"body":"and then a 1/3-sin-3x, 1/5-sin-5x,"},{"speaker":"All we have to do now is scale back up","startTime":2612.16,"endTime":2614.02,"body":"one, three, five, seven,"},{"speaker":"All we have to do now is scale back up","startTime":2612.16,"endTime":2614.02,"body":"nine, dot, dot, dot,"},{"speaker":"All we have to do now is scale back up","startTime":2614.02,"endTime":2615.083,"body":"all the odd numbers."},{"speaker":"All we have to do now is scale back up","startTime":2616.5,"endTime":2618.34,"body":"The first term of the"},{"speaker":"All we have to do now is scale back up","startTime":2616.5,"endTime":2618.34,"body":"series is just sin-x,"},{"speaker":"All we have to do now is scale back up","startTime":2618.34,"endTime":2619.63,"body":"there's my sine curve,"},{"speaker":"All we have to do now is scale back up","startTime":2619.63,"endTime":2622.38,"body":"then I'm going to start"},{"speaker":"All we have to do now is scale back up","startTime":2619.63,"endTime":2622.38,"body":"adding in the other terms"},{"speaker":"All we have to do now is scale back up","startTime":2622.38,"endTime":2624.37,"body":"until we think we know what's going on."},{"speaker":"All we have to do now is scale back up","startTime":2624.37,"endTime":2627.82,"body":"So I'll add in next 1/3-sin-3x,"},{"speaker":"All we have to do now is scale back up","startTime":2627.82,"endTime":2630.153,"body":"so I've scaled it and I've squashed it."},{"speaker":"All we have to do now is scale back up","startTime":2631.26,"endTime":2632.16,"body":"So I've added that in,"},{"speaker":"All we have to do now is scale back up","startTime":2632.16,"endTime":2634.89,"body":"and now we get this"},{"speaker":"All we have to do now is scale back up","startTime":2632.16,"endTime":2634.89,"body":"weird, jelly-like shape."},{"speaker":"All we have to do now is scale back up","startTime":2634.89,"endTime":2637.794,"body":"Now I'm going to add in the 1/5-sin-5x,"},{"speaker":"All we have to do now is scale back up","startTime":2637.794,"endTime":2640.22,"body":"so I've now got three"},{"speaker":"All we have to do now is scale back up","startTime":2637.794,"endTime":2640.22,"body":"terms of that series."},{"speaker":"All we have to do now is scale back up","startTime":2640.22,"endTime":2642.333,"body":"Then I'm going to go up to five terms,"},{"speaker":"All we have to do now is scale back up","startTime":2643.49,"endTime":2645.76,"body":"and you'll be guessing what"},{"speaker":"All we have to do now is scale back up","startTime":2643.49,"endTime":2645.76,"body":"you think is going to happen."},{"speaker":"All we have to do now is scale back up","startTime":2645.76,"endTime":2647.75,"body":"10 terms,"},{"speaker":"All we have to do now is scale back up","startTime":2647.75,"endTime":2649.203,"body":"50 terms,"},{"speaker":"All we have to do now is scale back up","startTime":2650.64,"endTime":2651.6,"body":"500 terms,"},{"speaker":"All we have to do now is scale back up","startTime":2651.6,"endTime":2655.07,"body":"we'll just do one more, 5,000 terms."},{"speaker":"All we have to do now is scale back up","startTime":2655.07,"endTime":2656.65,"body":"And look at that,"},{"speaker":"All we have to do now is scale back up","startTime":2656.65,"endTime":2659.33,"body":"out of these curvy, curvy waves,"},{"speaker":"All we have to do now is scale back up","startTime":2659.33,"endTime":2662.173,"body":"we've made something that"},{"speaker":"All we have to do now is scale back up","startTime":2659.33,"endTime":2662.173,"body":"just is straight lines."},{"speaker":"All we have to do now is scale back up","startTime":2663.21,"endTime":2664.83,"body":"People tend to call this a square wave,"},{"speaker":"All we have to do now is scale back up","startTime":2664.83,"endTime":2666.87,"body":"but really, it's rectangular,"},{"speaker":"All we have to do now is scale back up","startTime":2666.87,"endTime":2667.75,"body":"but the point is,"},{"speaker":"All we have to do now is scale back up","startTime":2667.75,"endTime":2669.04,"body":"we've taken something,"},{"speaker":"All we have to do now is scale back up","startTime":2669.04,"endTime":2673.37,"body":"this is about as far from a"},{"speaker":"All we have to do now is scale back up","startTime":2669.04,"endTime":2673.37,"body":"curve as you can think of,"},{"speaker":"All we have to do now is scale back up","startTime":2673.37,"endTime":2674.95,"body":"I mean it's got to be periodic,"},{"speaker":"All we have to do now is scale back up","startTime":2674.95,"endTime":2677.02,"body":"so it's just a step,"},{"speaker":"All we have to do now is scale back up","startTime":2677.02,"endTime":2679.51,"body":"or crenelations in a castle,"},{"speaker":"All we have to do now is scale back up","startTime":2677.02,"endTime":2679.51,"body":"or something like that,"},{"speaker":"All we have to do now is scale back up","startTime":2679.51,"endTime":2681.96,"body":"but these are straight"},{"speaker":"All we have to do now is scale back up","startTime":2679.51,"endTime":2681.96,"body":"lines, right angles,"},{"speaker":"All we have to do now is scale back up","startTime":2681.96,"endTime":2683.6,"body":"this is far from a sine wave."},{"speaker":"All we have to do now is scale back up","startTime":2683.6,"endTime":2684.94,"body":"So if you can make that,"},{"speaker":"All we have to do now is scale back up","startTime":2684.94,"endTime":2686.267,"body":"it seems like you ought"},{"speaker":"All we have to do now is scale back up","startTime":2684.94,"endTime":2686.267,"body":"to be able to do anything,"},{"speaker":"All we have to do now is scale back up","startTime":2686.267,"endTime":2688.65,"body":"and you, asterisk, can."},{"speaker":"All we have to do now is scale back up","startTime":2688.65,"endTime":2691.01,"body":"We'll just do one more."},{"speaker":"All we have to do now is scale back up","startTime":2691.01,"endTime":2692.02,"body":"I had all the odd numbers there,"},{"speaker":"All we have to do now is scale back up","startTime":2692.02,"endTime":2694.09,"body":"let's just have all the numbers,"},{"speaker":"All we have to do now is scale back up","startTime":2694.09,"endTime":2695.71,"body":"one, two, three, four, and so on,"},{"speaker":"All we have to do now is scale back up","startTime":2695.71,"endTime":2698.835,"body":"so I'm adding sin-x to 1/2-sin-2x,"},{"speaker":"All we have to do now is scale back up","startTime":2698.835,"endTime":2700.8,"body":"to 1/3-sin-3x, and so on."},{"speaker":"All we have to do now is scale back up","startTime":2700.8,"endTime":2702.5,"body":"So we'll start with one term,"},{"speaker":"All we have to do now is scale back up","startTime":2702.5,"endTime":2704.21,"body":"two terms,"},{"speaker":"All we have to do now is scale back up","startTime":2704.21,"endTime":2705.99,"body":"three terms,"},{"speaker":"All we have to do now is scale back up","startTime":2705.99,"endTime":2709.25,"body":"five, and it's getting"},{"speaker":"All we have to do now is scale back up","startTime":2705.99,"endTime":2709.25,"body":"wigglier and wigglier,"},{"speaker":"All we have to do now is scale back up","startTime":2709.25,"endTime":2710.083,"body":"10,"},{"speaker":"All we have to do now is scale back up","startTime":2711.15,"endTime":2711.983,"body":"100,"},{"speaker":"All we have to do now is scale back up","startTime":2711.983,"endTime":2714.3,"body":"and let's go all the way"},{"speaker":"All we have to do now is scale back up","startTime":2711.983,"endTime":2714.3,"body":"to 5,000 straightaway,"},{"speaker":"All we have to do now is scale back up","startTime":2714.3,"endTime":2716.83,"body":"5,000 is essentially infinity"},{"speaker":"All we have to do now is scale back up","startTime":2717.9,"endTime":2719.8,"body":"if you're doing these things."},{"speaker":"All we have to do now is scale back up","startTime":2719.8,"endTime":2720.79,"body":"So look at that,"},{"speaker":"All we have to do now is scale back up","startTime":2720.79,"endTime":2722.77,"body":"this is called a sawtooth wave,"},{"speaker":"All we have to do now is scale back up","startTime":2722.77,"endTime":2724.09,"body":"again, straight lines."},{"speaker":"All we have to do now is scale back up","startTime":2724.09,"endTime":2725.159,"body":"And I like this one"},{"speaker":"All we have to do now is scale back up","startTime":2725.159,"endTime":2728.38,"body":"'cause it's like we've come full circle,"},{"speaker":"All we have to do now is scale back up","startTime":2728.38,"endTime":2731.18,"body":"in all the ways you can"},{"speaker":"All we have to do now is scale back up","startTime":2728.38,"endTime":2731.18,"body":"interpret that phrase:"},{"speaker":"All we have to do now is scale back up","startTime":2731.18,"endTime":2733.87,"body":"we started off with"},{"speaker":"All we have to do now is scale back up","startTime":2731.18,"endTime":2733.87,"body":"right-angled triangles,"},{"speaker":"All we have to do now is scale back up","startTime":2733.87,"endTime":2735.46,"body":"and turned them into sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2735.46,"endTime":2737.42,"body":"and now we've taken some sine waves"},{"speaker":"All we have to do now is scale back up","startTime":2737.42,"endTime":2739.23,"body":"and turned them into"},{"speaker":"All we have to do now is scale back up","startTime":2737.42,"endTime":2739.23,"body":"right-angled triangles,"},{"speaker":"All we have to do now is scale back up","startTime":2739.23,"endTime":2741.52,"body":"so that's all very nice."},{"speaker":"All we have to do now is scale back up","startTime":2741.52,"endTime":2743.415,"body":"So you can do this,"},{"speaker":"All we have to do now is scale back up","startTime":2743.415,"endTime":2745.03,"body":"you can make these Fourier series,"},{"speaker":"All we have to do now is scale back up","startTime":2745.03,"endTime":2746.62,"body":"anything you encounter,"},{"speaker":"All we have to do now is scale back up","startTime":2746.62,"endTime":2748.21,"body":"even if it wasn't a sine wave,"},{"speaker":"All we have to do now is scale back up","startTime":2748.21,"endTime":2750.42,"body":"you can do this, and break down"},{"speaker":"All we have to do now is scale back up","startTime":2750.42,"endTime":2753.56,"body":"using a well understood process,"},{"speaker":"All we have to do now is scale back up","startTime":2753.56,"endTime":2756.01,"body":"to decompose any periodic function"},{"speaker":"All we have to do now is scale back up","startTime":2756.01,"endTime":2758.12,"body":"into a combination of sine waves."},{"speaker":"All we have to do now is scale back up","startTime":2758.12,"endTime":2761.29,"body":"So at the heart of all periodic things"},{"speaker":"All we have to do now is scale back up","startTime":2761.29,"endTime":2762.18,"body":"are these sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2762.18,"endTime":2763.013,"body":"and that's why the sine wave"},{"speaker":"All we have to do now is scale back up","startTime":2763.013,"endTime":2764.96,"body":"has so many uses and applications,"},{"speaker":"All we have to do now is scale back up","startTime":2764.96,"endTime":2765.793,"body":"it's because of this."},{"speaker":"All we have to do now is scale back up","startTime":2765.793,"endTime":2767.84,"body":"Even if you don't start"},{"speaker":"All we have to do now is scale back up","startTime":2765.793,"endTime":2767.84,"body":"off with the sine wave,"},{"speaker":"All we have to do now is scale back up","startTime":2767.84,"endTime":2771.12,"body":"still, what you have can"},{"speaker":"All we have to do now is scale back up","startTime":2767.84,"endTime":2771.12,"body":"be broken into sine waves,"},{"speaker":"All we have to do now is scale back up","startTime":2771.12,"endTime":2774.49,"body":"so they're the absolute basic kind of wave"},{"speaker":"All we have to do now is scale back up","startTime":2774.49,"endTime":2776.57,"body":"that underlies everything."},{"speaker":"All we have to do now is scale back up","startTime":2776.57,"endTime":2777.72,"body":"I've got a few minutes left,"},{"speaker":"All we have to do now is scale back up","startTime":2777.72,"endTime":2779.8,"body":"and I want to talk about just one context"},{"speaker":"All we have to do now is scale back up","startTime":2779.8,"endTime":2781.81,"body":"in which we see this kind of thing,"},{"speaker":"All we have to do now is scale back up","startTime":2781.81,"endTime":2783.463,"body":"and that is music."},{"speaker":"All we have to do now is scale back up","startTime":2784.944,"endTime":2787.61,"body":"I did give a couple of lectures previously"},{"speaker":"All we have to do now is scale back up","startTime":2787.61,"endTime":2788.88,"body":"about mathematics and music,"},{"speaker":"All we have to do now is scale back up","startTime":2788.88,"endTime":2790.82,"body":"so if you want all the details"},{"speaker":"All we have to do now is scale back up","startTime":2790.82,"endTime":2792.95,"body":"that I can't give you"},{"speaker":"All we have to do now is scale back up","startTime":2790.82,"endTime":2792.95,"body":"in a couple of minutes,"},{"speaker":"All we have to do now is scale back up","startTime":2792.95,"endTime":2794.6,"body":"go and find those, and watch those,"},{"speaker":"All we have to do now is scale back up","startTime":2794.6,"endTime":2799.6,"body":"but just a brief, brief"},{"speaker":"All we have to do now is scale back up","startTime":2794.6,"endTime":2799.6,"body":"moment on the context here,"},{"speaker":"All we have to do now is scale back up","startTime":2801.7,"endTime":2804.573,"body":"going back thousands of years,"},{"speaker":"All we have to do now is scale back up","startTime":2805.6,"endTime":2808.82,"body":"people like Pythagoras did experiments"},{"speaker":"All we have to do now is scale back up","startTime":2808.82,"endTime":2811.26,"body":"with plucking strings,"},{"speaker":"All we have to do now is scale back up","startTime":2811.26,"endTime":2813.52,"body":"with basic musical instruments,"},{"speaker":"All we have to do now is scale back up","startTime":2813.52,"endTime":2815.92,"body":"where you would pluck a"},{"speaker":"All we have to do now is scale back up","startTime":2813.52,"endTime":2815.92,"body":"string, you would hear a sound,"},{"speaker":"All we have to do now is scale back up","startTime":2815.92,"endTime":2818.12,"body":"and then you could change"},{"speaker":"All we have to do now is scale back up","startTime":2815.92,"endTime":2818.12,"body":"the length of the string"},{"speaker":"All we have to do now is scale back up","startTime":2818.12,"endTime":2822.44,"body":"and that would produce a"},{"speaker":"All we have to do now is scale back up","startTime":2818.12,"endTime":2822.44,"body":"sound of a different pitch."},{"speaker":"All we have to do now is scale back up","startTime":2822.44,"endTime":2824.49,"body":"If you made it shorter,"},{"speaker":"All we have to do now is scale back up","startTime":2822.44,"endTime":2824.49,"body":"the pitch would be higher,"},{"speaker":"All we have to do now is scale back up","startTime":2824.49,"endTime":2826.67,"body":"if you made it longer,"},{"speaker":"All we have to do now is scale back up","startTime":2824.49,"endTime":2826.67,"body":"the pitch would be lower."},{"speaker":"All we have to do now is scale back up","startTime":2826.67,"endTime":2829.08,"body":"We now know there is"},{"speaker":"All we have to do now is scale back up","startTime":2826.67,"endTime":2829.08,"body":"an inverse relationship"},{"speaker":"All we have to do now is scale back up","startTime":2829.08,"endTime":2830.61,"body":"between the length of your string"},{"speaker":"All we have to do now is scale back up","startTime":2830.61,"endTime":2833.35,"body":"and the pitch, or the"},{"speaker":"All we have to do now is scale back up","startTime":2830.61,"endTime":2833.35,"body":"frequency that you hear."},{"speaker":"All we have to do now is scale back up","startTime":2833.35,"endTime":2835.69,"body":"If you halve the length of the string,"},{"speaker":"All we have to do now is scale back up","startTime":2835.69,"endTime":2837.81,"body":"the frequency doubles, for example,"},{"speaker":"All we have to do now is scale back up","startTime":2837.81,"endTime":2839.7,"body":"so there's this inverse relationship."},{"speaker":"All we have to do now is scale back up","startTime":2839.7,"endTime":2842.01,"body":"What Pythagoras discovered"},{"speaker":"All we have to do now is scale back up","startTime":2842.01,"endTime":2846.66,"body":"was that there are some sounds"},{"speaker":"All we have to do now is scale back up","startTime":2842.01,"endTime":2846.66,"body":"that sound pleasing together,"},{"speaker":"All we have to do now is scale back up","startTime":2846.66,"endTime":2848.28,"body":"and others don't,"},{"speaker":"All we have to do now is scale back up","startTime":2848.28,"endTime":2850.98,"body":"and he found that the ones"},{"speaker":"All we have to do now is scale back up","startTime":2848.28,"endTime":2850.98,"body":"that sound pleasing together"},{"speaker":"All we have to do now is scale back up","startTime":2850.98,"endTime":2854.56,"body":"tend to be where the"},{"speaker":"All we have to do now is scale back up","startTime":2850.98,"endTime":2854.56,"body":"length of the strings,"},{"speaker":"All we have to do now is scale back up","startTime":2854.56,"endTime":2857.39,"body":"and so the frequencies too,"},{"speaker":"All we have to do now is scale back up","startTime":2857.39,"endTime":2861.66,"body":"are in very nice, small"},{"speaker":"All we have to do now is scale back up","startTime":2857.39,"endTime":2861.66,"body":"whole number ratios."},{"speaker":"All we have to do now is scale back up","startTime":2861.66,"endTime":2863.81,"body":"So he talked about halving"},{"speaker":"All we have to do now is scale back up","startTime":2861.66,"endTime":2863.81,"body":"the length of a string,"},{"speaker":"All we have to do now is scale back up","startTime":2863.81,"endTime":2865.01,"body":"doubling the frequency,"},{"speaker":"All we have to do now is scale back up","startTime":2865.01,"endTime":2866.69,"body":"those two sounds, if"},{"speaker":"All we have to do now is scale back up","startTime":2865.01,"endTime":2866.69,"body":"you play them together,"},{"speaker":"All we have to do now is scale back up","startTime":2866.69,"endTime":2869.71,"body":"they sound very pleasing,"},{"speaker":"All we have to do now is scale back up","startTime":2866.69,"endTime":2869.71,"body":"very harmonious together."},{"speaker":"All we have to do now is scale back up","startTime":2869.71,"endTime":2873.16,"body":"We would now call that"},{"speaker":"All we have to do now is scale back up","startTime":2869.71,"endTime":2873.16,"body":"gap in pitch an octave,"},{"speaker":"All we have to do now is scale back up","startTime":2873.16,"endTime":2876.46,"body":"and actually, they sound so"},{"speaker":"All we have to do now is scale back up","startTime":2873.16,"endTime":2876.46,"body":"similar and pleasing together"},{"speaker":"All we have to do now is scale back up","startTime":2876.46,"endTime":2878.3,"body":"almost we think of them as the same note."},{"speaker":"All we have to do now is scale back up","startTime":2878.3,"endTime":2880.56,"body":"If you've got a mixed choir singing,"},{"speaker":"All we have to do now is scale back up","startTime":2880.56,"endTime":2882.6,"body":"the ladies would be usually singing"},{"speaker":"All we have to do now is scale back up","startTime":2882.6,"endTime":2884.92,"body":"an octave higher than the gentlemen,"},{"speaker":"All we have to do now is scale back up","startTime":2884.92,"endTime":2888.01,"body":"and still, they sound like"},{"speaker":"All we have to do now is scale back up","startTime":2884.92,"endTime":2888.01,"body":"we're singing in unison."},{"speaker":"All we have to do now is scale back up","startTime":2888.01,"endTime":2890.95,"body":"So that's the first thing,"},{"speaker":"All we have to do now is scale back up","startTime":2890.95,"endTime":2893.0,"body":"doubling the frequency,"},{"speaker":"All we have to do now is scale back up","startTime":2893.0,"endTime":2894.8,"body":"those two sound very pleasing together,"},{"speaker":"All we have to do now is scale back up","startTime":2894.8,"endTime":2897.45,"body":"but also, other ratios like three to two"},{"speaker":"All we have to do now is scale back up","startTime":2897.45,"endTime":2899.44,"body":"in the string length and the frequencies,"},{"speaker":"All we have to do now is scale back up","startTime":2899.44,"endTime":2902.21,"body":"that we now call a pitch"},{"speaker":"All we have to do now is scale back up","startTime":2899.44,"endTime":2902.21,"body":"difference of a perfect fifth,"},{"speaker":"All we have to do now is scale back up","startTime":2902.21,"endTime":2903.34,"body":"those sound nice together,"},{"speaker":"All we have to do now is scale back up","startTime":2903.34,"endTime":2904.92,"body":"three to four, four to three,"},{"speaker":"All we have to do now is scale back up","startTime":2904.92,"endTime":2908.93,"body":"these very simple ratios"},{"speaker":"All we have to do now is scale back up","startTime":2904.92,"endTime":2908.93,"body":"make pleasing sounds,"},{"speaker":"All we have to do now is scale back up","startTime":2908.93,"endTime":2910.48,"body":"pleasing harmonies."},{"speaker":"All we have to do now is scale back up","startTime":2910.48,"endTime":2912.47,"body":"But why? We didn't know why."},{"speaker":"All we have to do now is scale back up","startTime":2912.47,"endTime":2915.12,"body":"And many people worked on"},{"speaker":"All we have to do now is scale back up","startTime":2912.47,"endTime":2915.12,"body":"this over the centuries,"},{"speaker":"All we have to do now is scale back up","startTime":2915.12,"endTime":2917.76,"body":"people like Galileo, Mersenne,"},{"speaker":"All we have to do now is scale back up","startTime":2917.76,"endTime":2919.45,"body":"I want to mention Joseph Sauveur"},{"speaker":"All we have to do now is scale back up","startTime":2919.45,"endTime":2921.69,"body":"because he did"},{"speaker":"All we have to do now is scale back up","startTime":2921.69,"endTime":2924.48,"body":"a lot of really good and"},{"speaker":"All we have to do now is scale back up","startTime":2921.69,"endTime":2924.48,"body":"accurate experiments."},{"speaker":"All we have to do now is scale back up","startTime":2924.48,"endTime":2926.853,"body":"A lot of the other people, Galileo,"},{"speaker":"All we have to do now is scale back up","startTime":2927.88,"endTime":2930.763,"body":"had mostly worked theoretically,"},{"speaker":"All we have to do now is scale back up","startTime":2931.871,"endTime":2934.73,"body":"Sauveur did detailed,"},{"speaker":"All we have to do now is scale back up","startTime":2931.871,"endTime":2934.73,"body":"accurate experiments,"},{"speaker":"All we have to do now is scale back up","startTime":2934.73,"endTime":2935.62,"body":"he studied acoustics,"},{"speaker":"All we have to do now is scale back up","startTime":2935.62,"endTime":2937.77,"body":"in fact he coined the term acoustics,"},{"speaker":"All we have to do now is scale back up","startTime":2937.77,"endTime":2940.29,"body":"that was his invention, that word,"},{"speaker":"All we have to do now is scale back up","startTime":2940.29,"endTime":2943.27,"body":"he did experiments with"},{"speaker":"All we have to do now is scale back up","startTime":2940.29,"endTime":2943.27,"body":"undulating strings,"},{"speaker":"All we have to do now is scale back up","startTime":2943.27,"endTime":2944.78,"body":"vibrating strings,"},{"speaker":"All we have to do now is scale back up","startTime":2944.78,"endTime":2946.46,"body":"looking at the frequencies,"},{"speaker":"All we have to do now is scale back up","startTime":2946.46,"endTime":2948.44,"body":"and on also what that phrase says,"},{"speaker":"All we have to do now is scale back up","startTime":2948.44,"endTime":2950.4,"body":"the nodes of undulating strings."},{"speaker":"All we have to do now is scale back up","startTime":2950.4,"endTime":2955.08,"body":"And I think one of the reasons, perhaps,"},{"speaker":"All we have to do now is scale back up","startTime":2955.08,"endTime":2956.75,"body":"that he did the experiment he did,"},{"speaker":"All we have to do now is scale back up","startTime":2956.75,"endTime":2958.17,"body":"in the way he did it,"},{"speaker":"All we have to do now is scale back up","startTime":2958.17,"endTime":2960.81,"body":"was because, actually,"},{"speaker":"All we have to do now is scale back up","startTime":2960.81,"endTime":2962.41,"body":"in his study of acoustics,"},{"speaker":"All we have to do now is scale back up","startTime":2962.41,"endTime":2964.42,"body":"he did it in spite of, or"},{"speaker":"All we have to do now is scale back up","startTime":2962.41,"endTime":2964.42,"body":"maybe because of the fact"},{"speaker":"All we have to do now is scale back up","startTime":2964.42,"endTime":2969.063,"body":"that he himself suffered"},{"speaker":"All we have to do now is scale back up","startTime":2964.42,"endTime":2969.063,"body":"from hearing difficulties,"},{"speaker":"All we have to do now is scale back up","startTime":2970.1,"endTime":2970.933,"body":"he wasn't quite deaf,"},{"speaker":"All we have to do now is scale back up","startTime":2970.933,"endTime":2973.77,"body":"but he had hearing troubles"},{"speaker":"All we have to do now is scale back up","startTime":2973.77,"endTime":2976.3,"body":"that meant he had to use"},{"speaker":"All we have to do now is scale back up","startTime":2973.77,"endTime":2976.3,"body":"different techniques, sometimes,"},{"speaker":"All we have to do now is scale back up","startTime":2976.3,"endTime":2978.723,"body":"to study this science."},{"speaker":"All we have to do now is scale back up","startTime":2979.7,"endTime":2981.9,"body":"The nodes of the undulating strings,"},{"speaker":"All we have to do now is scale back up","startTime":2981.9,"endTime":2983.9,"body":"he found that when you pluck a string,"},{"speaker":"All we have to do now is scale back up","startTime":2983.9,"endTime":2985.763,"body":"or you displace it in some way,"},{"speaker":"All we have to do now is scale back up","startTime":2986.81,"endTime":2989.64,"body":"he'd put little pieces"},{"speaker":"All we have to do now is scale back up","startTime":2986.81,"endTime":2989.64,"body":"of paper on the strings,"},{"speaker":"All we have to do now is scale back up","startTime":2989.64,"endTime":2991.11,"body":"and he found that there were,"},{"speaker":"All we have to do now is scale back up","startTime":2991.11,"endTime":2993.96,"body":"depending on how you set"},{"speaker":"All we have to do now is scale back up","startTime":2991.11,"endTime":2993.96,"body":"the string in motion,"},{"speaker":"All we have to do now is scale back up","startTime":2993.96,"endTime":2996.07,"body":"there were sometimes places on the string"},{"speaker":"All we have to do now is scale back up","startTime":2996.07,"endTime":2997.7,"body":"that didn't appear to move,"},{"speaker":"All we have to do now is scale back up","startTime":2997.7,"endTime":2999.67,"body":"and the pieces of paper did not fall off,"},{"speaker":"All we have to do now is scale back up","startTime":2999.67,"endTime":3002.47,"body":"so they're essentially still, those nodes."},{"speaker":"All we have to do now is scale back up","startTime":3002.47,"endTime":3004.74,"body":"And depending on how you"},{"speaker":"All we have to do now is scale back up","startTime":3002.47,"endTime":3004.74,"body":"set the string moving,"},{"speaker":"All we have to do now is scale back up","startTime":3004.74,"endTime":3006.403,"body":"you might get a node in"},{"speaker":"All we have to do now is scale back up","startTime":3004.74,"endTime":3006.403,"body":"the middle of the string"},{"speaker":"All we have to do now is scale back up","startTime":3006.403,"endTime":3008.06,"body":"that was stationary,"},{"speaker":"All we have to do now is scale back up","startTime":3008.06,"endTime":3011.05,"body":"or you might get two nodes,"},{"speaker":"All we have to do now is scale back up","startTime":3011.05,"endTime":3012.99,"body":"each at the thirds of"},{"speaker":"All we have to do now is scale back up","startTime":3011.05,"endTime":3012.99,"body":"the way along the string,"},{"speaker":"All we have to do now is scale back up","startTime":3012.99,"endTime":3014.44,"body":"1/3 and 2/3,"},{"speaker":"All we have to do now is scale back up","startTime":3014.44,"endTime":3017.21,"body":"or quarters, or fifths,"},{"speaker":"All we have to do now is scale back up","startTime":3017.21,"endTime":3019.94,"body":"these small whole number fractions."},{"speaker":"All we have to do now is scale back up","startTime":3019.94,"endTime":3020.87,"body":"And he found that out,"},{"speaker":"All we have to do now is scale back up","startTime":3020.87,"endTime":3023.31,"body":"and that's a visible"},{"speaker":"All we have to do now is scale back up","startTime":3020.87,"endTime":3023.31,"body":"experiment that you can see,"},{"speaker":"All we have to do now is scale back up","startTime":3023.31,"endTime":3025.92,"body":"does the piece of paper fall off or not."},{"speaker":"So again, this is another puzzle","startTime":3028.81,"endTime":3030.34,"body":"we know that frequencies"},{"speaker":"So again, this is another puzzle","startTime":3030.34,"endTime":3033.14,"body":"that are related by"},{"speaker":"So again, this is another puzzle","startTime":3030.34,"endTime":3033.14,"body":"nice whole number ratios"},{"speaker":"So again, this is another puzzle","startTime":3033.14,"endTime":3034.6,"body":"seem to sound nice together,"},{"speaker":"So again, this is another puzzle","startTime":3034.6,"endTime":3037.42,"body":"we now have vibrating, undulating strings"},{"speaker":"So again, this is another puzzle","startTime":3037.42,"endTime":3040.73,"body":"that somehow have sometimes fixed points"},{"speaker":"So again, this is another puzzle","startTime":3040.73,"endTime":3044.93,"body":"at nice, small fractions of"},{"speaker":"So again, this is another puzzle","startTime":3040.73,"endTime":3044.93,"body":"the way along their length,"},{"speaker":"So again, this is another puzzle","startTime":3044.93,"endTime":3046.113,"body":"what's going on?"},{"speaker":"So again, this is another puzzle","startTime":3047.71,"endTime":3049.67,"body":"The wave equation is what's going on,"},{"speaker":"So again, this is another puzzle","startTime":3049.67,"endTime":3051.55,"body":"and it's a rather complicated equation,"},{"speaker":"So again, this is another puzzle","startTime":3051.55,"endTime":3052.7,"body":"so I might just show it to you,"},{"speaker":"So again, this is another puzzle","startTime":3052.7,"endTime":3055.73,"body":"but there's no expectation"},{"speaker":"So again, this is another puzzle","startTime":3055.73,"endTime":3058.27,"body":"that you'll instantly understand"},{"speaker":"So again, this is another puzzle","startTime":3055.73,"endTime":3058.27,"body":"all the symbols in it,"},{"speaker":"So again, this is another puzzle","startTime":3058.27,"endTime":3060.5,"body":"I just want to show you it"},{"speaker":"So again, this is another puzzle","startTime":3058.27,"endTime":3060.5,"body":"and talk you through it,"},{"speaker":"So again, this is another puzzle","startTime":3060.5,"endTime":3062.3,"body":"and if you recognize it, and you're happy,"},{"speaker":"So again, this is another puzzle","startTime":3062.3,"endTime":3063.26,"body":"that's all great,"},{"speaker":"So again, this is another puzzle","startTime":3063.26,"endTime":3065.06,"body":"if you're not, don't worry about it."},{"speaker":"So again, this is another puzzle","startTime":3066.09,"endTime":3068.02,"body":"Imagine a string that's"},{"speaker":"So again, this is another puzzle","startTime":3066.09,"endTime":3068.02,"body":"fixed at both ends,"},{"speaker":"So again, this is another puzzle","startTime":3068.02,"endTime":3069.91,"body":"like a violin string,"},{"speaker":"So again, this is another puzzle","startTime":3069.91,"endTime":3070.95,"body":"and then you're going to do something,"},{"speaker":"So again, this is another puzzle","startTime":3070.95,"endTime":3072.76,"body":"you'll pluck it, or bow it,"},{"speaker":"So again, this is another puzzle","startTime":3070.95,"endTime":3072.76,"body":"or move it, or something,"},{"speaker":"So again, this is another puzzle","startTime":3072.76,"endTime":3073.86,"body":"disturb it,"},{"speaker":"So again, this is another puzzle","startTime":3073.86,"endTime":3076.03,"body":"at an initial time, zero."},{"speaker":"So again, this is another puzzle","startTime":3076.03,"endTime":3077.19,"body":"And then you're interested in"},{"speaker":"So again, this is another puzzle","startTime":3077.19,"endTime":3079.37,"body":"the vertical displacement of this,"},{"speaker":"So again, this is another puzzle","startTime":3079.37,"endTime":3081.77,"body":"what shape is that string going to make,"},{"speaker":"So again, this is another puzzle","startTime":3081.77,"endTime":3083.59,"body":"so we'll call that displacement y,"},{"speaker":"So again, this is another puzzle","startTime":3083.59,"endTime":3086.43,"body":"and that's going to depend"},{"speaker":"So again, this is another puzzle","startTime":3083.59,"endTime":3086.43,"body":"on how much time has passed,"},{"speaker":"So again, this is another puzzle","startTime":3086.43,"endTime":3087.82,"body":"so the value of t,"},{"speaker":"So again, this is another puzzle","startTime":3087.82,"endTime":3090.43,"body":"and also how far along the string you are,"},{"speaker":"So again, this is another puzzle","startTime":3090.43,"endTime":3091.89,"body":"we'll call that x."},{"speaker":"So again, this is another puzzle","startTime":3091.89,"endTime":3093.423,"body":"So I'll give you the equation,"},{"speaker":"So again, this is another puzzle","startTime":3094.48,"endTime":3096.06,"body":"don't worry about it at all,"},{"speaker":"So again, this is another puzzle","startTime":3096.06,"endTime":3097.68,"body":"but just to somehow say"},{"speaker":"So again, this is another puzzle","startTime":3097.68,"endTime":3099.93,"body":"there's some stuff in"},{"speaker":"So again, this is another puzzle","startTime":3097.68,"endTime":3099.93,"body":"there, there's some symbols"},{"speaker":"So again, this is another puzzle","startTime":3099.93,"endTime":3101.59,"body":"that are constants,"},{"speaker":"So again, this is another puzzle","startTime":3101.59,"endTime":3102.65,"body":"they just depend on the string,"},{"speaker":"So again, this is another puzzle","startTime":3102.65,"endTime":3105.6,"body":"so that big T is the"},{"speaker":"So again, this is another puzzle","startTime":3102.65,"endTime":3105.6,"body":"tension in the string,"},{"speaker":"So again, this is another puzzle","startTime":3105.6,"endTime":3107.14,"body":"the mu is the linear density,"},{"speaker":"So again, this is another puzzle","startTime":3107.14,"endTime":3109.663,"body":"fine, those are just"},{"speaker":"So again, this is another puzzle","startTime":3107.14,"endTime":3109.663,"body":"qualities of the string."},{"speaker":"So again, this is another puzzle","startTime":3110.56,"endTime":3113.44,"body":"The thing on the left is an expression,"},{"speaker":"So again, this is another puzzle","startTime":3113.44,"endTime":3114.88,"body":"it's called a partial derivative,"},{"speaker":"So again, this is another puzzle","startTime":3114.88,"endTime":3118.91,"body":"it tells you how the"},{"speaker":"So again, this is another puzzle","startTime":3114.88,"endTime":3118.91,"body":"vertical displacement changes"},{"speaker":"So again, this is another puzzle","startTime":3118.91,"endTime":3120.9,"body":"in terms of the time,"},{"speaker":"So again, this is another puzzle","startTime":3120.9,"endTime":3122.37,"body":"and the thing on the right"},{"speaker":"So again, this is another puzzle","startTime":3122.37,"endTime":3124.28,"body":"is telling you how that"},{"speaker":"So again, this is another puzzle","startTime":3122.37,"endTime":3124.28,"body":"displacement changes"},{"speaker":"So again, this is another puzzle","startTime":3124.28,"endTime":3126.71,"body":"in terms of the distance along the string."},{"speaker":"So again, this is another puzzle","startTime":3126.71,"endTime":3128.48,"body":"So what the equation says"},{"speaker":"So again, this is another puzzle","startTime":3128.48,"endTime":3131.27,"body":"is that those are closely"},{"speaker":"So again, this is another puzzle","startTime":3128.48,"endTime":3131.27,"body":"related to each other,"},{"speaker":"So again, this is another puzzle","startTime":3131.27,"endTime":3133.33,"body":"one depends on the other."},{"speaker":"So again, this is another puzzle","startTime":3133.33,"endTime":3136.67,"body":"And the French mathematician d'Alembert"},{"speaker":"So again, this is another puzzle","startTime":3136.67,"endTime":3138.09,"body":"found a method to solve this."},{"speaker":"So again, this is another puzzle","startTime":3138.09,"endTime":3139.63,"body":"Note, this is before Fourier,"},{"speaker":"So again, this is another puzzle","startTime":3139.63,"endTime":3142.19,"body":"so he found a method to solve this,"},{"speaker":"So again, this is another puzzle","startTime":3142.19,"endTime":3143.21,"body":"and what he said was,"},{"speaker":"So again, this is another puzzle","startTime":3143.21,"endTime":3145.33,"body":"any solution to this equation"},{"speaker":"So again, this is another puzzle","startTime":3145.33,"endTime":3146.7,"body":"is the sum of two waves,"},{"speaker":"So again, this is another puzzle","startTime":3146.7,"endTime":3148.1,"body":"so there's a wave, A,"},{"speaker":"So again, this is another puzzle","startTime":3148.1,"endTime":3149.807,"body":"it moves along the string somehow,"},{"speaker":"So again, this is another puzzle","startTime":3149.807,"endTime":3152.28,"body":"and when it gets to the end of the string,"},{"speaker":"So again, this is another puzzle","startTime":3152.28,"endTime":3155.01,"body":"that string is fixed,"},{"speaker":"So again, this is another puzzle","startTime":3152.28,"endTime":3155.01,"body":"it's fixed at both ends,"},{"speaker":"So again, this is another puzzle","startTime":3155.01,"endTime":3156.793,"body":"so that energy that's in that wave,"},{"speaker":"So again, this is another puzzle","startTime":3157.65,"endTime":3159.38,"body":"something has to happen to it."},{"speaker":"So again, this is another puzzle","startTime":3159.38,"endTime":3161.02,"body":"Basically, there has to be another wave,"},{"speaker":"So again, this is another puzzle","startTime":3161.02,"endTime":3163.01,"body":"that's wave A inverted,"},{"speaker":"So again, this is another puzzle","startTime":3163.01,"endTime":3164.58,"body":"and traveling in the other direction"},{"speaker":"So again, this is another puzzle","startTime":3164.58,"endTime":3166.7,"body":"so that it cancels out,"},{"speaker":"So again, this is another puzzle","startTime":3166.7,"endTime":3169.45,"body":"any vertical displacement in the wave A"},{"speaker":"So again, this is another puzzle","startTime":3169.45,"endTime":3171.49,"body":"has to be canceled out by that wave B,"},{"speaker":"So again, this is another puzzle","startTime":3171.49,"endTime":3174.07,"body":"because that end of the string can't move."},{"speaker":"So again, this is another puzzle","startTime":3174.07,"endTime":3175.53,"body":"So that means there"},{"speaker":"So again, this is another puzzle","startTime":3174.07,"endTime":3175.53,"body":"must be this other wave"},{"speaker":"So again, this is another puzzle","startTime":3175.53,"endTime":3177.3,"body":"going backwards in the other direction,"},{"speaker":"So again, this is another puzzle","startTime":3177.3,"endTime":3178.33,"body":"and off it goes,"},{"speaker":"So again, this is another puzzle","startTime":3178.33,"endTime":3179.78,"body":"and then when it gets to the other end,"},{"speaker":"So again, this is another puzzle","startTime":3179.78,"endTime":3180.78,"body":"the same thing has to happen,"},{"speaker":"So again, this is another puzzle","startTime":3180.78,"endTime":3182.46,"body":"it's going to flip, and reverse,"},{"speaker":"So again, this is another puzzle","startTime":3182.46,"endTime":3184.37,"body":"and produce wave A again."},{"speaker":"So again, this is another puzzle","startTime":3184.37,"endTime":3185.96,"body":"So what this means is"},{"speaker":"So again, this is another puzzle","startTime":3185.96,"endTime":3190.96,"body":"that the solution to wave"},{"speaker":"So again, this is another puzzle","startTime":3185.96,"endTime":3190.96,"body":"equation has to be periodic,"},{"speaker":"So again, this is another puzzle","startTime":3190.96,"endTime":3192.38,"body":"you go all the way along one length,"},{"speaker":"So again, this is another puzzle","startTime":3192.38,"endTime":3194.2,"body":"say the string has length l,"},{"speaker":"So again, this is another puzzle","startTime":3194.2,"endTime":3196.54,"body":"you go along that whole length l,"},{"speaker":"So again, this is another puzzle","startTime":3196.54,"endTime":3197.373,"body":"you flip, and invert,"},{"speaker":"So again, this is another puzzle","startTime":3197.373,"endTime":3199.17,"body":"and go back again length l,"},{"speaker":"So again, this is another puzzle","startTime":3199.17,"endTime":3201.03,"body":"and then you get back to"},{"speaker":"So again, this is another puzzle","startTime":3199.17,"endTime":3201.03,"body":"where you started from,"},{"speaker":"So again, this is another puzzle","startTime":3201.03,"endTime":3202.46,"body":"you get that wave A again."},{"speaker":"So again, this is another puzzle","startTime":3202.46,"endTime":3205.18,"body":"So any solution has to be periodic,"},{"speaker":"So again, this is another puzzle","startTime":3205.18,"endTime":3206.32,"body":"i.e. repeating,"},{"speaker":"So again, this is another puzzle","startTime":3206.32,"endTime":3208.23,"body":"and the period is two-l,"},{"speaker":"So again, this is another puzzle","startTime":3208.23,"endTime":3211.13,"body":"every two-l, you get the same thing again."},{"speaker":"So again, this is another puzzle","startTime":3211.13,"endTime":3214.928,"body":"So our eyes light up at this point,"},{"speaker":"So again, this is another puzzle","startTime":3214.928,"endTime":3216.83,"body":"'cause we know the future,"},{"speaker":"So again, this is another puzzle","startTime":3216.83,"endTime":3219.98,"body":"we know that thanks to Fourier,"},{"speaker":"So again, this is another puzzle","startTime":3219.98,"endTime":3222.29,"body":"every solution, then,"},{"speaker":"So again, this is another puzzle","startTime":3219.98,"endTime":3222.29,"body":"of this wave equation,"},{"speaker":"So again, this is another puzzle","startTime":3222.29,"endTime":3224.07,"body":"has to be a sum of sine waves,"},{"speaker":"So again, this is another puzzle","startTime":3224.07,"endTime":3227.65,"body":"and in particular, ones"},{"speaker":"So again, this is another puzzle","startTime":3224.07,"endTime":3227.65,"body":"of that period, two-l."},{"speaker":"So again, this is another puzzle","startTime":3227.65,"endTime":3229.41,"body":"So here are the first few,"},{"speaker":"So again, this is another puzzle","startTime":3229.41,"endTime":3230.28,"body":"and we see exactly,"},{"speaker":"So again, this is another puzzle","startTime":3230.28,"endTime":3233.14,"body":"look, there are those points"},{"speaker":"So again, this is another puzzle","startTime":3230.28,"endTime":3233.14,"body":"there that Sauveur saw,"},{"speaker":"So again, this is another puzzle","startTime":3233.14,"endTime":3234.86,"body":"his little bits of paper,"},{"speaker":"So again, this is another puzzle","startTime":3234.86,"endTime":3237.95,"body":"sometimes, depending on"},{"speaker":"So again, this is another puzzle","startTime":3234.86,"endTime":3237.95,"body":"what you do to begin with,"},{"speaker":"So again, this is another puzzle","startTime":3237.95,"endTime":3239.57,"body":"you could get fixed points, or nodes,"},{"speaker":"So again, this is another puzzle","startTime":3239.57,"endTime":3240.403,"body":"that are still"},{"speaker":"So again, this is another puzzle","startTime":3240.403,"endTime":3242.56,"body":"halfway along, or thirds, or quarters,"},{"speaker":"So again, this is another puzzle","startTime":3242.56,"endTime":3243.393,"body":"so that's really cool,"},{"speaker":"So again, this is another puzzle","startTime":3243.393,"endTime":3244.803,"body":"and that's why that happens."},{"speaker":"So again, this is another puzzle","startTime":3245.69,"endTime":3247.94,"body":"Those waves correspond to the frequencies,"},{"speaker":"So again, this is another puzzle","startTime":3247.94,"endTime":3250.64,"body":"an initial fundamental"},{"speaker":"So again, this is another puzzle","startTime":3247.94,"endTime":3250.64,"body":"frequency for the longest wave,"},{"speaker":"So again, this is another puzzle","startTime":3250.64,"endTime":3251.94,"body":"let's call it f,"},{"speaker":"So again, this is another puzzle","startTime":3251.94,"endTime":3255.09,"body":"and then if you have two"},{"speaker":"So again, this is another puzzle","startTime":3251.94,"endTime":3255.09,"body":"sine waves that fit in there,"},{"speaker":"So again, this is another puzzle","startTime":3255.09,"endTime":3257.84,"body":"you double the frequency,"},{"speaker":"So again, this is another puzzle","startTime":3255.09,"endTime":3257.84,"body":"or treble, or quadruple."},{"speaker":"So again, this is another puzzle","startTime":3257.84,"endTime":3259.25,"body":"Different instruments"},{"speaker":"So again, this is another puzzle","startTime":3259.25,"endTime":3262.123,"body":"will have different combinations"},{"speaker":"So again, this is another puzzle","startTime":3259.25,"endTime":3262.123,"body":"of those sine waves,"},{"speaker":"So again, this is another puzzle","startTime":3263.12,"endTime":3266.14,"body":"and you can analyze exactly what."},{"speaker":"So again, this is another puzzle","startTime":3266.14,"endTime":3267.73,"body":"You can draw yourselves little pictures,"},{"speaker":"So again, this is another puzzle","startTime":3267.73,"endTime":3271.52,"body":"you can analyze these sounds"},{"speaker":"So again, this is another puzzle","startTime":3267.73,"endTime":3271.52,"body":"with modern equipment,"},{"speaker":"So again, this is another puzzle","startTime":3271.52,"endTime":3274.76,"body":"and you can break them"},{"speaker":"So again, this is another puzzle","startTime":3271.52,"endTime":3274.76,"body":"down using Fourier analysis"},{"speaker":"So again, this is another puzzle","startTime":3274.76,"endTime":3278.06,"body":"to work out exactly what"},{"speaker":"So again, this is another puzzle","startTime":3274.76,"endTime":3278.06,"body":"sine waves you have."},{"speaker":"So again, this is another puzzle","startTime":3278.06,"endTime":3279.74,"body":"There's an initial transient sound as well"},{"speaker":"So again, this is another puzzle","startTime":3279.74,"endTime":3281.23,"body":"which I won't talk about."},{"speaker":"So again, this is another puzzle","startTime":3281.23,"endTime":3284.27,"body":"So you can break down and do analysis,"},{"speaker":"So again, this is another puzzle","startTime":3284.27,"endTime":3286.41,"body":"break these sounds into"},{"speaker":"So again, this is another puzzle","startTime":3284.27,"endTime":3286.41,"body":"their individual sine waves,"},{"speaker":"So again, this is another puzzle","startTime":3286.41,"endTime":3288.69,"body":"and then if you want to"},{"speaker":"So again, this is another puzzle","startTime":3286.41,"endTime":3288.69,"body":"synthesize them later,"},{"speaker":"So again, this is another puzzle","startTime":3288.69,"endTime":3289.82,"body":"you precisely can"},{"speaker":"So again, this is another puzzle","startTime":3289.82,"endTime":3293.253,"body":"by reproducing those exact"},{"speaker":"So again, this is another puzzle","startTime":3289.82,"endTime":3293.253,"body":"combinations of sine waves."},{"speaker":"So again, this is another puzzle","startTime":3294.47,"endTime":3295.74,"body":"It also answers the question"},{"speaker":"So again, this is another puzzle","startTime":3295.74,"endTime":3298.77,"body":"of why some frequencies"},{"speaker":"So again, this is another puzzle","startTime":3295.74,"endTime":3298.77,"body":"sound nice together,"},{"speaker":"So again, this is another puzzle","startTime":3298.77,"endTime":3300.85,"body":"because every sound"},{"speaker":"So again, this is another puzzle","startTime":3300.85,"endTime":3302.87,"body":"will have this basic,"},{"speaker":"So again, this is another puzzle","startTime":3300.85,"endTime":3302.87,"body":"fundamental frequency,"},{"speaker":"So again, this is another puzzle","startTime":3302.87,"endTime":3303.9,"body":"let's say f,"},{"speaker":"So again, this is another puzzle","startTime":3303.9,"endTime":3306.71,"body":"and then various amounts"},{"speaker":"So again, this is another puzzle","startTime":3303.9,"endTime":3306.71,"body":"of multiples of f,"},{"speaker":"So again, this is another puzzle","startTime":3306.71,"endTime":3308.33,"body":"two-f, three-f, and so on."},{"speaker":"So again, this is another puzzle","startTime":3308.33,"endTime":3310.5,"body":"So if we take, say,"},{"speaker":"So again, this is another puzzle","startTime":3310.5,"endTime":3312.48,"body":"one string of a particular length,"},{"speaker":"So again, this is another puzzle","startTime":3312.48,"endTime":3315.293,"body":"and another one with,"},{"speaker":"So again, this is another puzzle","startTime":3312.48,"endTime":3315.293,"body":"say, half the length,"},{"speaker":"So again, this is another puzzle","startTime":3316.14,"endTime":3318.01,"body":"the one of half the length,"},{"speaker":"So again, this is another puzzle","startTime":3318.01,"endTime":3320.04,"body":"it will have that fundamental frequency"},{"speaker":"So again, this is another puzzle","startTime":3320.04,"endTime":3322.02,"body":"that's double the frequency,"},{"speaker":"So again, this is another puzzle","startTime":3322.02,"endTime":3325.03,"body":"but it will also have those"},{"speaker":"So again, this is another puzzle","startTime":3322.02,"endTime":3325.03,"body":"overtones, those harmonics"},{"speaker":"So again, this is another puzzle","startTime":3325.03,"endTime":3327.9,"body":"that are multiples of that number."},{"speaker":"So again, this is another puzzle","startTime":3327.9,"endTime":3331.13,"body":"And so those string"},{"speaker":"So again, this is another puzzle","startTime":3327.9,"endTime":3331.13,"body":"lengths and frequencies"},{"speaker":"So again, this is another puzzle","startTime":3331.13,"endTime":3335.3,"body":"that are small whole"},{"speaker":"So again, this is another puzzle","startTime":3331.13,"endTime":3335.3,"body":"number ratios of each other"},{"speaker":"So again, this is another puzzle","startTime":3335.3,"endTime":3338.42,"body":"will share infinitely many"},{"speaker":"So again, this is another puzzle","startTime":3338.42,"endTime":3340.95,"body":"of the same overtones and harmonics,"},{"speaker":"So again, this is another puzzle","startTime":3340.95,"endTime":3343.35,"body":"and that's why they sound nice together,"},{"speaker":"So again, this is another puzzle","startTime":3343.35,"endTime":3346.04,"body":"because they do have this"},{"speaker":"So again, this is another puzzle","startTime":3343.35,"endTime":3346.04,"body":"resonance between them,"},{"speaker":"So again, this is another puzzle","startTime":3346.04,"endTime":3347.27,"body":"so that's the answer to that question,"},{"speaker":"So again, this is another puzzle","startTime":3347.27,"endTime":3350.173,"body":"and the answer to Sauveur's"},{"speaker":"So again, this is another puzzle","startTime":3347.27,"endTime":3350.173,"body":"little pieces of paper."},{"speaker":"So again, this is another puzzle","startTime":3351.14,"endTime":3353.29,"body":"And the amazing thing is"},{"speaker":"So again, this is another puzzle","startTime":3353.29,"endTime":3354.29,"body":"that as you're sitting here,"},{"speaker":"So again, this is another puzzle","startTime":3354.29,"endTime":3356.61,"body":"listening to me now,"},{"speaker":"So again, this is another puzzle","startTime":3356.61,"endTime":3360.233,"body":"your ear is doing this kind of analysis."},{"speaker":"So again, this is another puzzle","startTime":3360.233,"endTime":3362.59,"body":"This is a picture of your inner ear,"},{"speaker":"So again, this is another puzzle","startTime":3362.59,"endTime":3363.71,"body":"well not yours personally,"},{"speaker":"So again, this is another puzzle","startTime":3363.71,"endTime":3365.07,"body":"an inner ear,"},{"speaker":"So again, this is another puzzle","startTime":3365.07,"endTime":3366.55,"body":"and the sound comes in,"},{"speaker":"So again, this is another puzzle","startTime":3366.55,"endTime":3367.92,"body":"and it travels to the cochlea,"},{"speaker":"So again, this is another puzzle","startTime":3367.92,"endTime":3371.58,"body":"which is this lovely, spirally"},{"speaker":"So again, this is another puzzle","startTime":3367.92,"endTime":3371.58,"body":"part of your ear there"},{"speaker":"So again, this is another puzzle","startTime":3371.58,"endTime":3373.17,"body":"that's filled with fluid,"},{"speaker":"So again, this is another puzzle","startTime":3373.17,"endTime":3377.27,"body":"and the sound comes in"},{"speaker":"So again, this is another puzzle","startTime":3373.17,"endTime":3377.27,"body":"and vibrates that fluid,"},{"speaker":"So again, this is another puzzle","startTime":3377.27,"endTime":3379.22,"body":"and the individual component waves,"},{"speaker":"So again, this is another puzzle","startTime":3379.22,"endTime":3380.38,"body":"this is something that was suggested"},{"speaker":"So again, this is another puzzle","startTime":3380.38,"endTime":3383.55,"body":"by the scientist Georg Ohm,"},{"speaker":"So again, this is another puzzle","startTime":3383.55,"endTime":3387.28,"body":"knowing that sound could be"},{"speaker":"So again, this is another puzzle","startTime":3383.55,"endTime":3387.28,"body":"broken up into sine waves,"},{"speaker":"So again, this is another puzzle","startTime":3387.28,"endTime":3388.177,"body":"he suggested it,"},{"speaker":"So again, this is another puzzle","startTime":3388.177,"endTime":3390.83,"body":"and then it was confirmed by"},{"speaker":"So again, this is another puzzle","startTime":3388.177,"endTime":3390.83,"body":"experiment many years later,"},{"speaker":"So again, this is another puzzle","startTime":3390.83,"endTime":3393.13,"body":"the sound comes into the cochlea,"},{"speaker":"So again, this is another puzzle","startTime":3393.13,"endTime":3395.1,"body":"and the individual components,"},{"speaker":"So again, this is another puzzle","startTime":3395.1,"endTime":3396.24,"body":"the individual sine waves"},{"speaker":"So again, this is another puzzle","startTime":3396.24,"endTime":3398.03,"body":"that are all joined together,"},{"speaker":"So again, this is another puzzle","startTime":3398.03,"endTime":3400.57,"body":"each one of those is part of the sound,"},{"speaker":"So again, this is another puzzle","startTime":3400.57,"endTime":3403.397,"body":"and the cochlea has this"},{"speaker":"So again, this is another puzzle","startTime":3400.57,"endTime":3403.397,"body":"thing in the middle of it,"},{"speaker":"So again, this is another puzzle","startTime":3403.397,"endTime":3405.38,"body":"the basilar membrane,"},{"speaker":"So again, this is another puzzle","startTime":3405.38,"endTime":3407.43,"body":"it gets narrower, and"},{"speaker":"So again, this is another puzzle","startTime":3405.38,"endTime":3407.43,"body":"narrower, and narrower,"},{"speaker":"So again, this is another puzzle","startTime":3407.43,"endTime":3408.33,"body":"and at some point,"},{"speaker":"So again, this is another puzzle","startTime":3408.33,"endTime":3411.0,"body":"each individual sine wave"},{"speaker":"So again, this is another puzzle","startTime":3408.33,"endTime":3411.0,"body":"component of the sound"},{"speaker":"So again, this is another puzzle","startTime":3411.0,"endTime":3411.96,"body":"reaches a stage"},{"speaker":"So again, this is another puzzle","startTime":3411.96,"endTime":3415.25,"body":"where its wavelength is"},{"speaker":"So again, this is another puzzle","startTime":3411.96,"endTime":3415.25,"body":"too big to go any further,"},{"speaker":"So again, this is another puzzle","startTime":3415.25,"endTime":3417.47,"body":"it gets to that point where"},{"speaker":"So again, this is another puzzle","startTime":3415.25,"endTime":3417.47,"body":"it can't travel any further,"},{"speaker":"So again, this is another puzzle","startTime":3417.47,"endTime":3420.26,"body":"So it will collapse at that stage."},{"speaker":"So again, this is another puzzle","startTime":3420.26,"endTime":3422.35,"body":"but the energy from it"},{"speaker":"So again, this is another puzzle","startTime":3420.26,"endTime":3422.35,"body":"has to go somewhere,"},{"speaker":"So again, this is another puzzle","startTime":3422.35,"endTime":3424.95,"body":"and the energy goes into that membrane,"},{"speaker":"So again, this is another puzzle","startTime":3424.95,"endTime":3427.87,"body":"which has lots of tiny,"},{"speaker":"So again, this is another puzzle","startTime":3424.95,"endTime":3427.87,"body":"little hairs on it,"},{"speaker":"So again, this is another puzzle","startTime":3427.87,"endTime":3432.26,"body":"and they are set vibrating by"},{"speaker":"So again, this is another puzzle","startTime":3427.87,"endTime":3432.26,"body":"that collapsing wavelength."},{"speaker":"So again, this is another puzzle","startTime":3432.26,"endTime":3435.94,"body":"And so your brain then receives"},{"speaker":"So again, this is another puzzle","startTime":3432.26,"endTime":3435.94,"body":"signals from the sound,"},{"speaker":"So again, this is another puzzle","startTime":3435.94,"endTime":3437.04,"body":"and it gets signals"},{"speaker":"So again, this is another puzzle","startTime":3437.04,"endTime":3440.26,"body":"from each place along"},{"speaker":"So again, this is another puzzle","startTime":3437.04,"endTime":3440.26,"body":"the path of the cochlea"},{"speaker":"So again, this is another puzzle","startTime":3440.26,"endTime":3442.7,"body":"where a wavelength collapsed,"},{"speaker":"So again, this is another puzzle","startTime":3442.7,"endTime":3445.46,"body":"and then it can, from that,"},{"speaker":"So again, this is another puzzle","startTime":3445.46,"endTime":3448.89,"body":"work out what sine waves"},{"speaker":"So again, this is another puzzle","startTime":3445.46,"endTime":3448.89,"body":"exactly were going together"},{"speaker":"So again, this is another puzzle","startTime":3448.89,"endTime":3450.18,"body":"to produce that sound,"},{"speaker":"So again, this is another puzzle","startTime":3450.18,"endTime":3452.77,"body":"and interpret that sound accordingly."},{"speaker":"So again, this is another puzzle","startTime":3452.77,"endTime":3454.57,"body":"So your brain, right now,"},{"speaker":"So again, this is another puzzle","startTime":3454.57,"endTime":3456.74,"body":"and all the way through"},{"speaker":"So again, this is another puzzle","startTime":3454.57,"endTime":3456.74,"body":"this talk about sine waves,"},{"speaker":"So again, this is another puzzle","startTime":3456.74,"endTime":3458.67,"body":"has been doing Fourier analysis,"},{"speaker":"So again, this is another puzzle","startTime":3458.67,"endTime":3460.86,"body":"and breaking down the"},{"speaker":"So again, this is another puzzle","startTime":3458.67,"endTime":3460.86,"body":"sound into sine waves,"},{"speaker":"So again, this is another puzzle","startTime":3460.86,"endTime":3462.64,"body":"which I think is lovely."},{"speaker":"So again, this is another puzzle","startTime":3462.64,"endTime":3464.22,"body":"So I will stop there"},{"speaker":"So again, this is another puzzle","startTime":3464.22,"endTime":3466.2,"body":"with a little advert for,"},{"speaker":"So again, this is another puzzle","startTime":3466.2,"endTime":3468.34,"body":"only 48 hours from now,"},{"speaker":"So again, this is another puzzle","startTime":3468.34,"endTime":3469.68,"body":"very, very close,"},{"speaker":"So again, this is another puzzle","startTime":3469.68,"endTime":3471.36,"body":"6 o'clock, 25th of May,"},{"speaker":"So again, this is another puzzle","startTime":3471.36,"endTime":3474.18,"body":"that's two days from now at"},{"speaker":"So again, this is another puzzle","startTime":3471.36,"endTime":3474.18,"body":"the time that I'm speaking,"},{"speaker":"So again, this is another puzzle","startTime":3474.18,"endTime":3475.61,"body":"you only have to wait two more days"},{"speaker":"So again, this is another puzzle","startTime":3475.61,"endTime":3478.01,"body":"for another mathematics lecture."},{"speaker":"So again, this is another puzzle","startTime":3478.01,"endTime":3479.6,"body":"This one is joint from Gresham"},{"speaker":"So again, this is another puzzle","startTime":3479.6,"endTime":3481.17,"body":"and the London Mathematical Society,"},{"speaker":"So again, this is another puzzle","startTime":3481.17,"endTime":3482.61,"body":"it's something we do every year,"},{"speaker":"So again, this is another puzzle","startTime":3482.61,"endTime":3484.24,"body":"it'll be given by Hugh Hunt,"},{"speaker":"So again, this is another puzzle","startTime":3484.24,"endTime":3485.7,"body":"who is a fantastic speaker,"},{"speaker":"So again, this is another puzzle","startTime":3485.7,"endTime":3488.31,"body":"on the mathematics of"},{"speaker":"So again, this is another puzzle","startTime":3485.7,"endTime":3488.31,"body":"gyroscopes and boomerangs,"},{"speaker":"So again, this is another puzzle","startTime":3488.31,"endTime":3491.62,"body":"so come back then to"},{"speaker":"So again, this is another puzzle","startTime":3488.31,"endTime":3491.62,"body":"hear about boomerangs."},{"speaker":"So again, this is another puzzle","startTime":3491.62,"endTime":3493.293,"body":"All right, thank you."},{"speaker":"So again, this is another puzzle","startTime":3493.293,"endTime":3496.293,"body":"(audience applauds)"},{"speaker":"So again, this is another puzzle","startTime":3501.151,"endTime":3502.48,"body":"- Thank you so much, Professor Hart,"},{"speaker":"So again, this is another puzzle","startTime":3502.48,"endTime":3503.46,"body":"for the fascinating lecture."},{"speaker":"So again, this is another puzzle","startTime":3503.46,"endTime":3504.9,"body":"I've got some questions from online,"},{"speaker":"So again, this is another puzzle","startTime":3504.9,"endTime":3506.75,"body":"which I think we should take two of first"},{"speaker":"So again, this is another puzzle","startTime":3506.75,"endTime":3508.45,"body":"before going to the room."},{"speaker":"So again, this is another puzzle","startTime":3508.45,"endTime":3510.333,"body":"The first question is,"},{"speaker":"So again, this is another puzzle","startTime":3510.333,"endTime":3511.166,"body":"\"On a calculator,"},{"speaker":"So again, this is another puzzle","startTime":3511.166,"endTime":3511.999,"body":"\"you often get settings"},{"speaker":"So again, this is another puzzle","startTime":3511.999,"endTime":3514.757,"body":"\"to work in degrees,"},{"speaker":"So again, this is another puzzle","startTime":3511.999,"endTime":3514.757,"body":"radians, or gradians.\""},{"speaker":"So again, this is another puzzle","startTime":3514.757,"endTime":3517.337,"body":"\"Where did gradians come"},{"speaker":"So again, this is another puzzle","startTime":3514.757,"endTime":3517.337,"body":"from, and who uses them?\""},{"speaker":"So again, this is another puzzle","startTime":3518.81,"endTime":3520.55,"body":"- I believe,"},{"speaker":"So again, this is another puzzle","startTime":3520.55,"endTime":3521.7,"body":"I don't want to say anything wrong,"},{"speaker":"So again, this is another puzzle","startTime":3521.7,"endTime":3524.113,"body":"I believe this was a"},{"speaker":"So again, this is another puzzle","startTime":3521.7,"endTime":3524.113,"body":"French Revolution thing."},{"speaker":"So again, this is another puzzle","startTime":3525.607,"endTime":3528.69,"body":"There are 400 gradians in a circle,"},{"speaker":"So again, this is another puzzle","startTime":3528.69,"endTime":3531.993,"body":"i.e. 100, nice decimal"},{"speaker":"So again, this is another puzzle","startTime":3528.69,"endTime":3531.993,"body":"100 in every right angle,"},{"speaker":"So again, this is another puzzle","startTime":3534.26,"endTime":3536.7,"body":"that's where I think that originated."},{"speaker":"So again, this is another puzzle","startTime":3536.7,"endTime":3539.7,"body":"I believe they still have"},{"speaker":"So again, this is another puzzle","startTime":3536.7,"endTime":3539.7,"body":"some uses in engineering,"},{"speaker":"So again, this is another puzzle","startTime":3539.7,"endTime":3541.41,"body":"but if you're a mathematician,"},{"speaker":"So again, this is another puzzle","startTime":3541.41,"endTime":3542.53,"body":"you tend to use radians,"},{"speaker":"So again, this is another puzzle","startTime":3542.53,"endTime":3544.05,"body":"although, if you're like me,"},{"speaker":"So again, this is another puzzle","startTime":3544.05,"endTime":3546.45,"body":"you still have degrees"},{"speaker":"So again, this is another puzzle","startTime":3544.05,"endTime":3546.45,"body":"in the back of your head."},{"speaker":"So again, this is another puzzle","startTime":3547.64,"endTime":3548.473,"body":"- Thank you, Professor Hart."},{"speaker":"So again, this is another puzzle","startTime":3548.473,"endTime":3549.72,"body":"I've got another question here,"},{"speaker":"So again, this is another puzzle","startTime":3549.72,"endTime":3552.987,"body":"again possibly more about the sine wave,"},{"speaker":"So again, this is another puzzle","startTime":3552.987,"endTime":3555.88,"body":"\"Why is mains electricity a sine wave?\""},{"speaker":"So again, this is another puzzle","startTime":3555.88,"endTime":3557.35,"body":"- Oh?"},{"speaker":"So again, this is another puzzle","startTime":3557.35,"endTime":3558.359,"body":"Well everything's a sine wave,"},{"speaker":"So again, this is another puzzle","startTime":3558.359,"endTime":3560.496,"body":"there you go, next question."},{"speaker":"So again, this is another puzzle","startTime":3560.496,"endTime":3563.55,"body":"(audience laughs)"},{"speaker":"So again, this is another puzzle","startTime":3563.55,"endTime":3566.35,"body":"Anything that's"},{"speaker":"So again, this is another puzzle","startTime":3563.55,"endTime":3566.35,"body":"electromagnetic is a sine wave,"},{"speaker":"So again, this is another puzzle","startTime":3566.35,"endTime":3569.054,"body":"light waves, and all of those things are."},{"speaker":"So again, this is another puzzle","startTime":3569.054,"endTime":3570.38,"body":"- Thank you so much, Professor Hart."},{"speaker":"So again, this is another puzzle","startTime":3570.38,"endTime":3571.8,"body":"Please do come back here"},{"speaker":"So again, this is another puzzle","startTime":3571.8,"endTime":3574.18,"body":"for Professor Hart's series next year,"},{"speaker":"So again, this is another puzzle","startTime":3574.18,"endTime":3575.87,"body":"she's still with us next year, hooray,"},{"speaker":"So again, this is another puzzle","startTime":3575.87,"endTime":3578.09,"body":"so please come back for"},{"speaker":"So again, this is another puzzle","startTime":3575.87,"endTime":3578.09,"body":"her series next year."},{"speaker":"So again, this is another puzzle","startTime":3578.09,"endTime":3579.99,"body":"If you get on our mailing list,"},{"speaker":"So again, this is another puzzle","startTime":3579.99,"endTime":3582.62,"body":"you'll get notifications about that series"},{"speaker":"So again, this is another puzzle","startTime":3582.62,"endTime":3585.4,"body":"when it launches at the end of July."},{"speaker":"So again, this is another puzzle","startTime":3585.4,"endTime":3586.845,"body":"Thanks so much."},{"speaker":"So again, this is another puzzle","startTime":3586.845,"endTime":3589.845,"body":"(audience applauds)"}]}