On Wed, 24 May 2006, Michael Dyck wrote:
Michael Hart wrote:
On Wed, 24 May 2006, Michael Dyck wrote:
However, if it makes things easier, I'll gladly refuse to acknowlege that statement *now*. A linear growth curve *is* a growth curve, and a pretty ordinary one at that. It certainly doesn't make it impossible to create an ordinary growth curve. But that's just disagreeing with what you said, rather than what you meant.
No, a line is not a curve.
Well, it is to a mathematician.
OK, back to basics, I have consulted with some mathematicians, not that I think you didn't know this, but you are pressuring me to make the point, so I will, as gently as possible mostly with the presentation aids provided by the mathematicians. Presumed givens: A 9 year period in which 1 title was added each year to the index of what would later become known as Project Gutenberg. 1971-1979 A ~15 year period of increasing growth as represented by the graph previously presented. 1991-2006 Results: First for: A 9 year period in which 1 title was added each year to the index of what would later become known as Project Gutenberg. 1971-1979 The equation that would describe this is known as a "Linear Equation" of the variety y = mx + b When plotted on the normal "x,y" plane this would be a straight line, no matter what numeric variables you plugged into the equation. When describing such results the term "line" is used to represent a straight line of this nature, while various terms describing curves are used in higher order equations. Very often the term "exponential" is used in common parlance to describe what is really just a "multiplicative" or "geometric" growth pattern, but there is no need to go into that here. When describing such a linear equation in opposition to curved equations, the usual terms are "line" and "curve." We would normally say that a line "intersects" with a curve, in such a case where the equations have a common solution. Trying to fit a straight line into an equation for a curve has been one of the mathematical problems of the ages. Look up "squaring the circle" for some history on this. However, generally speaking, a curve could not contain a portion of a graph that was identical to the portion of a straight line. In this case it has been speculated that the growth of Project Gutenberg listings could be approximated via a curve, and in particular that the end result should be in some total comparison to the curve known as Moore's Law Which specifies that x should double every 18 months. Obviously it would be very rare indeed for real world curves to exactly match mathematical equations in the sense of human endeavors, but a quick look at what we have seen as the report of the dates of each 500 book level passed by Project Gutenberg, would indicate the approximate match to several well known curves. I'll leave it to you to choose which is the best fit.
But if you want to use that definition of "curve", fine, I agree, linear growth is straight, not bendy. I'm *quite* positive that I never refused to acknowlege such a thing.
You certainly seemed to be yesterday, apparently demanding the above explanations of the difference between lines and curves when describing graphs, intersections, etc.
(If you look close enough, PG's ebook count is actually a step function, piecewise constant: after a book is posted, the number of books is constant until the next one is posted. Dunno if that satisfies your definition of "curve".)
I think that is why our mathematical friends above said "approximates" a curve. . .since we are only using the "counting numbers." A true curve would include many other kinds of numbers. However, in this case, using counting numbers as input, and 1/4 year increments on the graph, you do get graphs that would normally be described as curves. Growth curves is the term normally used. In this case the growth "line" does not fit the growth "curve."
However, the case is even more drastic than that, as there was a period of over a decade when 0 eBooks were added, due to the hassles of the US Copyright Act of 1976, which took us forever to find out about, and the truth is that we would probably NEVER have figured them out without the help of one of the top dozen copyright lawyers in the US.
Thus, if you INSIST on talking about curves, the curve was downward.
Uh, if no books are added, the number is constant, which I would consider "flat" rather than "downward". But if you want to define it as "downward", fine.
The number used in the equation to create a graph in approximation to the performance would decrease, hence the term "downward" might be applicable; a line with no growth lies "downward" of lines that represent growth statistics.
Just one more obvious reason why you can't talk about growth curves for this period of Project Gutenberg's history.
A downward curve is still a curve. Though if you want to say it isn't a "growth curve", fine. (Mind you, it's not hard to find talk of "negative growth".)
In this case it would literally be a "negative growth" of the slope of the line. Technically the second order derivative, which is where these terms probably go beyond what is appropriate here.
You also mentioned that you can't fit real world items into such growth curves, but you never mentioned that the graph I included is a remarkably good overall fit,
(Well, now you're blurring a distinction I made between two meanings of "fit": real world data doesn't "fit" (= rigidly conform to) exponential curves, but you can "fit" (=approximate) it to an exponential curve. But anyway.)
I think all this was anticipated in the help I received above. If not, ask for more detailed explanations.
And in fact, I HAVE mentioned how closely the PG numbers are approximated by an exponential curve. See, e.g., http://lists.pglaf.org/private.cgi/gutvol-d/2005-January/001263.html and http://lists.pglaf.org/private.cgi/gutvol-d/2005-January/001456.html
This was also apparently anticipated by our mathematical friends, when they mentioned "approximate match to several well known curves." and "I'll leave it to you to choose which is the best fit." Obviously there are a number of equations that make approximate fits, obvious even to someone who hadn't seen your example in the URLs.
But now we (well, you, really) have strayed from the topic that brought me in, the comparison between Google's progress and PG's (and dang, I wish I'd changed the subject line at that point), so my interest in this discussion is probably fading.
Ah, it would appear that you already knew you were painting us into a corner. Then I hope that the great effort spent in replying to your messages was not a total waste for either yourself or the rest of us. It is only a waste to me if no one gains an apprecation of how seriously I take your messages, and of my willingness to provide the best answers. However, as I stated in my opening paragraph, I presumed you already knew all of this and thus presumed you were only asking the question for other reasons. May I ask what those reasons were?
-Michael
Thanks!!! Give the world eBooks in 2006!!! Michael S. Hart Founder Project Gutenberg Blog at http://hart.pglaf.org