I am writing a book called φ, π, e and i. It’s about those constants and their connections.
I’d like to include an argument by John Wallis that culminates in a beautiful representation of π. But I am reading his original and am stumped by a pattern he claims to see. Here is the relevant section. Start at COMMENT:
It is on the second page that I am befuddled, at the sentence, “And therefore (by analogy…”. His A stands for the number that is to be inserted into the given sequence like so: 1, A, 6, 30, 140, 630, …
I see of course where he gets 1 x (6/1) x (10/2) x etc. but where does he come up with A x (16/3) x (24/5) x etc. by analogy?
If you help me, I will thank you in my preface! And Tom will give you a math-related user tag, and get you pregnant.
Ugh! My iPad crashed while I was typing out a post that explains the answer. I figured it out about ten seconds after posting the above.
Briefly, Wallis’s second version of the infinite fraction gives him room to put whole numbers between those in the denominator. I’d written this all out in detail, but can’t bear to do it again. Stupid generation 1 iPad.
I can elaborate, if this puzzle piqued anyone’s interest. Please feel free to post math questions of your own in this thread now!
So hard to comment out of context, but isn’t it just talking about how if you create a mathematical series of numbers approaching zero and a series of numbers between the numbers of the original series, as the first series stretches to infinity and it’s values approach zero, then the values of the two series approach equality. Basically as series A and series B approach infinity they both approach a value of zero. Which is kind of amazing since series A is greater than series B at every element…
I remember in calc (or as my teacher would call it THE Calculus) that one of the big things at some point was that if you wanted to prove that a weirdo series approached zero the easiest way was to show that it was smaller than an easier to understand series for all elements and this seems to be touching on that.
I totally kind of understood at least 20% of DFW’s Everything and More: A Compact History of Infinity so I’m sure my answer is right.
I’m not enough of a math nerd to understand, yet alone explain it, but the pattern is 16 is halfway between 12 and 20 in the numerator, as the 3 is halfway between the 2 and the 4 in the denominator. 12(16)20(24)28 over 2(3)4(5)6. After 32/7 would be 40/9.
After reading it again, I’m not even sure if this what your asking, as you probably saw the pattern and what you really want is the reason for the analogy?
Definitely not enough help to warrant impregnation. Only a real mathematician is deserving of such a fate.
Edit:
Now I see you had it figured out well before my flailing attempts.
You nailed it, belouski. That’s the answer, and explains why Wallis doubled every number in his fraction. Thanks!
Lantz, thanks too. Wallis played around with the infinite a bit cuz he lived before Newton and Leibniz. And wow, what better way to lose a full point of cool in the eyes of students by saying THE calculus. It’s also improper to say “math” for “mathematics”. But good lord, why not just pin a KICK ME sign straight on your nuts.
