1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the cicle. The cicle is a vey easy, natual object to picute in you head. Actually poducing a pefect cicle is much moe difficult. Ancient cultues wanted to know all they could about this gloious shape, its size, its countepats, it bounday. In ode to look at the cicle moe caefully, we must fist lean a bit about angles. Imagine you have two ays o lines with a common endpoint, as pictued below in figue 1. The angle is the amount of otation sepaating these two ays. In the figue below, as is customay, we denote the amount as θ. Angle comes fom the Latin angulus, meaning cone. Geek has its own cognate, αγκύλoς, meaning cooked. Of couse, we have not yet decided on a unit of measuement fo angles, so it does not make much sense to talk about measuing them yet. θ Figue 1 2. A bief histoy of angles. As fa back as Babylon, ancient cultues have been studying angles. We know that the Babylonians split the cicle into 60 units of measuement. We still use this numbe 60, and we know them as degees. We know that an angle of 180 ceates a flat line. O an angle of 0 means that the two ays ovelap. The degees epeat evey 60. That is 0 is the same thing as 60, and 90 is the same thing as 450, which is the same as 270, and so on. How did the Babylonians stumble acoss 60 as a unit of measuement, which we still use today? The answe is unknown, but some ecently-uncoveed ancient Babylonian tablets give us some clues. Some speculate that since an Eath yeas as about 60 days, the Babylonians thought 60 would be a fit numbe to descibe a cicle. A much moe likely speculation is obseving thei connection between the hexagon and thei numbe system. Conside figue 2 below. 1
2 The Babylonians knew that the peimete of a egula hexagon was equal to the adius of a cicumscibed cicle. We also know that the Babylonians used a base 60 numbe system. Figue 2 So it is vey likely that the Babylonians attibuted 60 units to each side of the hexagon. Its esulting peimete would then be 60! As a quick aside, if you have neve been exposed to numbe systems that use something othe than base 10, it is good to expeience them. We use the Aabic numbe system with ten diffeent symbols : 0, 1, 2,, 4, 5, 6, 7, 8, 9. To epesent numbes beyond 9, we have to stat putting symbols in the tens place o hundeds place, etc. The Babylonians had sixty diffeent symbols in thei numbe system. Fo example, suppose that they used the symbol δ was thei symbol fo ou 59. Then ou 10 would be thei 60 (1 in the sixtys place, 0 in the ones place). 11 would be thei 61. 1δ would be thei 119 (1 in the sixtys place, 59 in the ones place). It is a little stange to get used to, but that is how thei cultue counted.. Back to the Geeks While the Geeks wee extemely fascinated by cicles and acs, they used a tool o concept that made appoximating thei measuements much moe easily. They used something called a chod. Refe to figue below. Imagine we take a piece of the cicle, so some ac. This ac will have an coesponding angle θ. In figue, we can see two ays extending fom the cente of the cicle to edge. θ is the angle between them. The cod is the staight line between the two points whee the ays meet the edge of the cicle. (The chod is dawn in ed in figue.)
θ Figue Notice that the two ays and the chod make an isosceles tiangle. Also note that, if we know what θ and the adius of the cicle ae, we also know all the lengths of the tiangle as well as the thee angles. The Geeks wee much moe familia with tiangles than acs, so the chods poved to be a useful tool. 4. Radians We ae also familia with anothe unit of measuement fo angles : the adian. Refe to figue 1.8 in you text; is the adius of a cicle, θ is an angle, and s is the length of the ac detemined by θ. The adian is defined as the atio of the length of the ac to the adius, o s. So the angle in adians will be θ = s. A natual question to ask about the adian is if it depends at all on the size of the cicle we ae using. i.e. if you have θ = s = s = θ, ae the angles epesented by θ and θ actually the same angle? We would like to show that this definition does not depend on the cicle we ae using. Refe to figue 1.9 in you text. Imagine we have two cicles, one with adius and anothe with adius R. Let s assume R >. We will pick an abitay angle θ. Let s be the coesponding ac length detemined by θ on the smalle cicle. Let S be the coesponding ac length detemined by θ on the lage cicle. We would like to show that s = S. To pove this, we will use the ancient geek tool, chods. We will now patition the wedge into pieces. The numbe of pieces doesn t matte. In figue 1.10 in you book, the wedge is patition into 4 equal pieces. We then daw the chods fo each coesponding piece fo both cicles. In the figue, d 4 epesents the chod in one of the pieces on the smalle cicle, and D 4 epesents the chod in one of the pieces on the lage cicle. Remembe that chods ceate tiangles that we ae able to wok with moe easily. It is impotant to note that each of the tiangles inscibed in the smalle cicle is simila to those inscibed in the lage one. This is because the angles ae equal. Also note that d 4 = D 4.
4 Thee was nothing special about 4, we could have chosen any othe numbe n. In geneal, we would get that dn = Dn. Now ty to think about what happens as that numbe n get vey lage, say n = 1, 000, 000. This means we would patition the wedge into 1, 000, 000 pieces. Moe impotantly, the sum of all the d n get close to s as n gets lage. So fo vey lage n, nd n s. (Also, nd n S.) This means that ndn fo vey lage n. We wite nd n lim = s nd n and lim n n R = S R. Hee, the lim means that we ae pushing n abitaily close to, o athe n making n vey lage. Finally, since dn = Dn fo any n, we get s = lim nd n nd n = lim n n R = S R. So we have poved s = S. Thus, the notion of adian does not depend on the size of the cicle and is, in fact, a good unit fo measuement of angles. s 5. The length of a cicle Now let s conside a cicle of adius 1. If we let θ be 180, the coesponding the cicumfeence of the semi-cicle wedge will be a semi-cicle. In adians, ou angle is. We 1 give a special name to this atio, and we call it π. Because the notion of adian does not depend on the cicle, we can say that fo any cicle, π is the atio of the cicumfeence of the cicle to its diamete. (This is the same as befoe, but we multiplied the atio on top and bottom by 2.) So fo the cicle of adius 1, the cicle has cicumfeence 2π. Fo any cicle, the cicumfeence is then c = 2π. One final note is that 180 = π, and thus the sum of the angles of a tiangle is π adians. Refe to table 1.1 in you text fo othe values. 6. Estimating π Although the Geeks did not use adians, they had noticed this univesal constant of the atio between a cicle s cicumfeence and its diamete. Even ealie cultues had happened upon it. Thee ae efeences in the Bible to estimations of π. King Solomon gives plans fo a pond with diamete 10 cubits and cicumfeence 0 cubits. Obviously he, as well as many othe ancient cultues appoximated π by. The Geeks wee able to use moe advanced techniques to estimate π, and wee actually able to estimate it to a few decimal places. Below we give a poof that < π <.47.
Refe to figue 1.1 in you text fo this poof. Conside a semi-cicle with adius 1, label the cente as C. We know a semicicle has cicumfeence π. We wil split this semi-cicle into thee equal pats as shown. Each pat then has angle π/. Let A and B be the points whee ou cuts intesect the semi-cicle as shown in the figue. Now let AB be the chod between the points A and B. Since all thee of these angles ae π/ (why?), this tiangle fomed by the chod is equilateal. Split this wedge (not the tiangle) in half by dawing a line fom C to the midpoint of the ac, call it P. This is again shown in the figue. Daw a line tangent to the cicle at this point P and join this with extensions of the lines though A and B foming a slightly lage tiangle simila to ou ACB tiangle. Call this meeting points A and B. Once again, this is shown in the figue. Again this tiangle A CB is equilateal (why?), so its angles ae all π/ as well. Let x be the length A P and P B. Then the ight tangle CP A have sides 1, x, and 2x. This is because of the Pythagoean theoem, 1 + x 2 = (2x) 2, o x 2 = 1, o x = 1. So the length A B has length 2x = 2. What does this mean? Imagine dawing a hexagon inside the cicle, much like in figue 2. We know this has peimete 6, and we can see that the cicumfeence of a cicle of adius 1 is lage than 6. So π >. Now imagine we daw a hexagon containing a cicle. What we have essentially just shown is that its peimiete will be 12. We can also see that this is lage than the cicumfeence of the cicle. So we have π < 6 = 2.47. So we have estimated π between and.47. This technique is simila to the one Achimedes used in his estimations of π. He used egula polygons with many moe sides. He used 96-sided polygon! Notice that as egula polygons incease in numbe of sides, they appea to become moe and moe like cicles. Achimedes noticed this phenomenon and used 96 sides. He then dew a 96-gon inside a cicle o adius 1 and one outside. Then came the tedious task of detemining both of thei peimetes. He knew that π would be in between these two numbes. So he was able to show that 10 < π < 1, o 71 7.1408 < π <.1429. This is petty dan good fo ove 2000 yeas ago! One final thought. π is a ticky numbe to calculate. The Geeks stived to find it as they believed it should be a numbe. Unfotunately, it tuns out that it is not a numbe (in the Geek sense), because it is iational. It has no pedictable decimal expansion and cetainly cannot be witten as a faction of integes. One of the most impotant numbes in existence tuns out to be something elatively foeign. The mysteies of the Univese! 5