[math-fun] What is "geometric" about a geometric progression?
Why is a geometric progression called geometric? Wikipedia's answer is "A geometric progression gains its geometric character from the fact that the areas of two geometrically similar plane figures are in "duplicate" ratio to their corresponding sides; Given two squares whose sides have the ratio 2:3, then their areas will have the ratio 4:9; we can write this as 4:6:9 and notice that the ratios 4:6 and 6:9 both equal 2:3; so by using the side ratio 2:3 "in duplicate" we obtain the ratio 4:9 of the areas, and the sequence 4, 6, 9 is a geometric sequence with common ratio 3/2. Furthermore, the volumes of two similar solid figures are in "triplicate" ratio of their corresponding sides. Similar to with the squares, you can take two cubes whose side ratio is 2:5. Their volume ratio is 8:125, which can be obtained as 8:20:50:125, the original ratio 2:5 "in triplicate", yielding a geometric sequence with common ratio 5/2." but there's no citation given, and I'm not sure I believe that. The terminology is older than higher-dimensional geometry, I think, so there would be no reason to think of a progression of more than 4 terms as being "geometric". Any evidence for wikipedia's theory? Any alternate theories? Andy
Why is a geometric progression called geometric?
....
Any evidence for wikipedia's theory? Any alternate theories?
Each term is the *geometric mean* of the terms either side of it, which in turn gains its name from the fact that the geometric mean has an elegant compass-and-straight-edge construction. Sincerely, Adam P. Goucher
On Wed, Mar 2, 2011 at 5:52 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Why is a geometric progression called geometric?
....
Any evidence for wikipedia's theory? Any alternate theories?
Each term is the *geometric mean* of the terms either side of it, which in turn gains its name from the fact that the geometric mean has an elegant compass-and-straight-edge construction.
This sounds much more plausible to me than the etymology in wikipedia. I had thought of geometric progression as the primary one, with geometric mean being called that because the geometric mean of a and b is the number c such that a, c, b is a geometric progression (with positive ratio). But the theory that the geometric progression gets its name from the geometric mean, rather than the other way around, seems quite plausible. I've edited wikipedia to reflect this. Hoping this doesn't produce an edit war... Andy
Sincerely,
Adam P. Goucher
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Does the OED have anything to say on it? They're a pretty canonical arbiter of such things... On Wed, Mar 2, 2011 at 3:49 PM, Andy Latto <andy.latto@pobox.com> wrote:
On Wed, Mar 2, 2011 at 5:52 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Why is a geometric progression called geometric?
....
Any evidence for wikipedia's theory? Any alternate theories?
Each term is the *geometric mean* of the terms either side of it, which in turn gains its name from the fact that the geometric mean has an elegant compass-and-straight-edge construction.
This sounds much more plausible to me than the etymology in wikipedia. I had thought of geometric progression as the primary one, with geometric mean being called that because the geometric mean of a and b is the number c such that a, c, b is a geometric progression (with positive ratio). But the theory that the geometric progression gets its name from the geometric mean, rather than the other way around, seems quite plausible.
I've edited wikipedia to reflect this. Hoping this doesn't produce an edit war...
Andy
Sincerely,
Adam P. Goucher
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On Thursday 03 March 2011 00:10:23 Mike Stay wrote:
Does the OED have anything to say on it? They're a pretty canonical arbiter of such things...
OED regards "geometrical" as more fundamental than "geometric"; it has citations for "geometrical progression" going back to 1557, though their 1557 citation is rather obscure in meaning. (Another, from 1594, is entirely convincing. The same 1594 source is quoted under "arithmetic[al] progression".) All the OED has to say about the origins of the phrase (as opposed to the word "geometric[al]") is: <<< arithmetical progression, †proportion, †ratio, etc. (see arithmetical adj.) relate to differences instead of quotients. The term geometrical points to the fact that problems involving multiplication were originally dealt with by geometry and not by arithmetic.
OED doesn't have much to say about "geometric[al] mean", regrettably. The entry for "mean" has a citation from about 1450 concerning the "meene proporcionalle", which is clearly the same thing as the GM (the OED entry for "mean proportional" quotes the same thing and seems to think it might mean something other than the GM, but I've no idea what else they think it could be). There's nothing for "arithmetic[al] mean" or "geometric[al] mean" until much later. So it's inconclusive, but seems to me like weak evidence against the theory that "geometric mean" came *before* "geometric progression". -- g
On Wed, Mar 2, 2011 at 6:36 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
OED doesn't have much to say about "geometric[al] mean", regrettably. The entry for "mean" has a citation from about 1450 concerning the "meene proporcionalle", which is clearly the same thing as the GM (the OED entry for "mean proportional" quotes the same thing and seems to think it might mean something other than the GM, but I've no idea what else they think it could be). There's nothing for "arithmetic[al] mean" or "geometric[al] mean" until much later.
So it's inconclusive, but seems to me like weak evidence against the theory that "geometric mean" came *before* "geometric progression".
With regards to the geometric mean, I thought you all might be interested in a construction I just made of four means. See the PDF file here: <http://www.symbo1ics.com/files/means.pdf> Key ==================================================== Light Blue : Arbitrary Value : x Dark Blue : Arbitrary Value : y Dark Green : Arithmetic Mean : (x+y)/2 Light Green : Harmonic Mean : 2/(1/x + 1/y) Red : Geometric Mean : sqrt(x*y) Orange : Root Mean Square : sqrt[(x^2 + y^2)/2] ==================================================== The constructions should be easy to infer from the diagram. :) This does make for some nice identities, somewhat interesting methods of computation, and easy derivation of inequalities. -Robert
On Wed, Mar 2, 2011 at 7:10 PM, Mike Stay <metaweta@gmail.com> wrote:
Does the OED have anything to say on it? They're a pretty canonical arbiter of such things...
I don't think the OED is really the best source for what is essentially a question of history of mathematics. There is a definition of elliptic, but it won't tell you that elliptic functions are called that because they are the inverses of the integrals used to describe the arc length of ellipses. The OED says, in explaining "geometrical ratio": "The term geometrical points to the fact that problems involving multiplication were originally dealt with by geometry and not by arithmetic". But I think that's just wrong. The babylonians knew how to multiply, and did it arithmetically, rather than by use of geometry. Andy
On Wed, Mar 2, 2011 at 3:49 PM, Andy Latto <andy.latto@pobox.com> wrote:
On Wed, Mar 2, 2011 at 5:52 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Why is a geometric progression called geometric?
....
Any evidence for wikipedia's theory? Any alternate theories?
Each term is the *geometric mean* of the terms either side of it, which in turn gains its name from the fact that the geometric mean has an elegant compass-and-straight-edge construction.
This sounds much more plausible to me than the etymology in wikipedia. I had thought of geometric progression as the primary one, with geometric mean being called that because the geometric mean of a and b is the number c such that a, c, b is a geometric progression (with positive ratio). But the theory that the geometric progression gets its name from the geometric mean, rather than the other way around, seems quite plausible.
I've edited wikipedia to reflect this. Hoping this doesn't produce an edit war...
Andy
Sincerely,
Adam P. Goucher
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On Thursday 03 March 2011 02:11:35 Andy Latto wrote:
On Wed, Mar 2, 2011 at 7:10 PM, Mike Stay <metaweta@gmail.com> wrote:
Does the OED have anything to say on it? They're a pretty canonical arbiter of such things...
I don't think the OED is really the best source for what is essentially a question of history of mathematics. There is a definition of elliptic, but it won't tell you that elliptic functions are called that because they are the inverses of the integrals used to describe the arc length of ellipses.
It's not that far off, actually. "2. elliptic integrals: a class of integrals discovered by Legendre in 1786, so named because their discovery was the result of the investigation of elliptic arcs. elliptic functions: certain specific functions of these integrals. (Formerly the term elliptic functions was applied to what are now called elliptic integrals.)"
"The term geometrical points to the fact that problems involving multiplication were originally dealt with by geometry and not by arithmetic".
But I think that's just wrong. The babylonians knew how to multiply, and did it arithmetically, rather than by use of geometry.
Yeah, "originally" is probably wrong. If it had said, say, "formerly", I don't think we know it's wrong. (It's referring to the ancient Greeks rather than the Babylonians, right?) -- g
On Mar 2, 2011, at 9:11 PM, Andy Latto wrote:
But I think that's just wrong. The babylonians knew how to multiply, and did it arithmetically, rather than by use of geometry.
I don't agree, unless you can prove to me the Babylonians were manipulating algebraic expressions. The term "geometric" probably derives from the context in which the progression arises, such as subdividing plane geometry figures into infinite sequences of similar figures. A famous one involves dividing up the golden rectangle into a square and a smaller golden rectangle. Could the Babylonians have dealt with this example arithmetically? Veit
I think that there's evidence that the Babylonians were pretty good at arithmetic, and if you practice it, as I have, you'll amaze yourself at how much you can do in your head using base 60. Here's an example of how they probably had ideas about limiting processes. YBC7289 gives strong evidence that they (some of them, anyway) knew the sometimes-called Heronian algorithm for square roots. If you want the sq root of 2, make a guess, say 1. Divide into 2 and get 2, so 1's too small and 2's too big. Take the average, 3/2. Divide into 2 and get 4/3. Average 17/12. Next stage 577/408. It's clear that the Babylonians got that far, but I reckon that they also noticed that, after the first guess, all the answers were too big, and that they differed from one another by surprisingly smaller amounts, AND that the process could be continued indefinitely. Let's see what happens if you translate 577/408 into Babylonian. Correct to 3 sexagesimal places, it's 1, 24 51 11. But that's not what they inscribed: 1, 24 51 10 --- because they KNEW that the former was too big, and they probably had a good idea of how much too big it was. As I said, if you work in base 60, calculations are easier, but as most of us are used to base 10, we'll have to do the harder comparison: 577/408 = 1.414215686274509803921568627 1+24/60+51/60^2+10/60^3 = 1.414212962962962962962962963 sqrt(2) = 1.414213562373095048801688724 (577^2+2*408^2)/(2*408*577) = 1.414213562374689910626295579 I'm sure the Babylonians, some of them, were good at geometry, and the above, and the famous Plimpton tablet, were obviously inspired by geometry (actually, astronomy), but the Plimpton tablet is a monumental piece of ARITHMETIC. R. On Thu, 3 Mar 2011, Veit Elser wrote:
On Mar 2, 2011, at 9:11 PM, Andy Latto wrote:
But I think that's just wrong. The babylonians knew how to multiply, and did it arithmetically, rather than by use of geometry.
I don't agree, unless you can prove to me the Babylonians were manipulating algebraic expressions. The term "geometric" probably derives from the context in which the progression arises, such as subdividing plane geometry figures into infinite sequences of similar figures. A famous one involves dividing up the golden rectangle into a square and a smaller golden rectangle. Could the Babylonians have dealt with this example arithmetically?
Veit
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How pleasant it was to read your comments. Hope things are going well for you. We're having a cold winter -- Mark and I get together about once a week to work on the second edition of Wohascum, but he's terribly busy with teaching and writing letters of recommendation, and playing bridge and recorder and ping-pong and .... Loren ----- Original Message ----- From: "Richard Guy" <rkg@cpsc.ucalgary.ca> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, March 03, 2011 11:07 AM Subject: Re: [math-fun] What is "geometric" about a geometric progression?
I think that there's evidence that the Babylonians were pretty good at arithmetic, and if you practice it, as I have, you'll amaze yourself at how much you can do in your head using base 60.
Here's an example of how they probably had ideas about limiting processes. YBC7289 gives strong evidence that they (some of them, anyway) knew the sometimes-called Heronian algorithm for square roots. If you want the sq root of 2, make a guess, say 1. Divide into 2 and get 2, so 1's too small and 2's too big. Take the average, 3/2. Divide into 2 and get 4/3. Average 17/12. Next stage 577/408. It's clear that the Babylonians got that far, but I reckon that they also noticed that, after the first guess, all the answers were too big, and that they differed from one another by surprisingly smaller amounts, AND that the process could be continued indefinitely.
Let's see what happens if you translate 577/408 into Babylonian. Correct to 3 sexagesimal places, it's 1, 24 51 11. But that's not what they inscribed: 1, 24 51 10 --- because they KNEW that the former was too big, and they probably had a good idea of how much too big it was.
As I said, if you work in base 60, calculations are easier, but as most of us are used to base 10, we'll have to do the harder comparison:
577/408 = 1.414215686274509803921568627 1+24/60+51/60^2+10/60^3 = 1.414212962962962962962962963 sqrt(2) = 1.414213562373095048801688724 (577^2+2*408^2)/(2*408*577) = 1.414213562374689910626295579
I'm sure the Babylonians, some of them, were good at geometry, and the above, and the famous Plimpton tablet, were obviously inspired by geometry (actually, astronomy), but the Plimpton tablet is a monumental piece of ARITHMETIC. R.
On Thu, 3 Mar 2011, Veit Elser wrote:
On Mar 2, 2011, at 9:11 PM, Andy Latto wrote:
But I think that's just wrong. The babylonians knew how to multiply, and did it arithmetically, rather than by use of geometry.
I don't agree, unless you can prove to me the Babylonians were manipulating algebraic expressions. The term "geometric" probably derives from the context in which the progression arises, such as subdividing plane geometry figures into infinite sequences of similar figures. A famous one involves dividing up the golden rectangle into a square and a smaller golden rectangle. Could the Babylonians have dealt with this example arithmetically?
Veit
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participants (8)
-
Adam P. Goucher -
Andy Latto -
Gareth McCaughan -
Loren and Liz Larson -
Mike Stay -
quad -
Richard Guy -
Veit Elser