John Wallis and his method with ratios

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burt

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I was asked the following question:

>Using Wallis's method, calculate the values \(n=0,\frac12,1,\frac32,2,\frac52\) in row \(p=\frac32\) of his ratio table.

The Wallis rereferred to here is John Wallis of England (1616-1703). The ratio table looks like this:
\[\begin{matrix}p/n&0&1&2&3&4&5&6&7&...\\0&1&1&1&1&1&1&1&1&...\\1&1&2&3&4&5&6&7&8&...\\2&1&3&6&10&15&21&28&36&...\\3&1&4&10&20&35&56&84&120&...\\…&...&...&...&...&...&...&...&...&...&\\\end{matrix}\]

I don't understand how to read this table or how to actually go about solving the question. The table's goal is to find the ratio of the area of the unit square to the area enclosed in the first quadrant by the curve \(y=(1-x^{\frac1p})^n\).
 
I think you'll have to show us what you were taught of his method, not just his table. It is not at all clear yet what "row p=3/2" means (or meant to him). Of course, the table appears to be a version of Pascal's triangle. But though I happen to have a math history book sitting right in front of me, I don't even know what to look up in it, other than all the references to Wallis.
 
Dr.Peterson said:
I think you'll have to show us what you were taught of his method, not just his table. It is not at all clear yet what "row p=3/2" means (or meant to him). Of course, the table appears to be a version of Pascal's triangle. But though I happen to have a math history book sitting right in front of me, I don't even know what to look up in it, other than all the references to Wallis.
Click to expand...
I hear you - and that's my problem. I do not understand his method at all.
 
burt said:
I hear you - and that's my problem. I do not understand his method at all.
Click to expand...
Well, if you have not been taught anything about his method, and therefore have nothing to show us and no link to provide, then neither you nor I can be expected to answer the question, can we?

On the other hand, if you have been taught something but don't understand it, you'll have to show us what you don't understand.
 
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