Now, on to the actual mathematics!

"Understand thus..." says the first line of the treatise, giving the definitions of the earliest level; of course, without a formalized tetrational notation, he had to improvise. In modern notation, we can summarize: hn=n^^n (now we would call the same function "megafuga").

Now we have hhn=h(hn), hhhn=h(hhn), and so on; culminating in the general case mhn=h^{m}(n).

[I use the superscript here in the modern sense; a different meaning occurs in the treatise itself - though that doesn't really create a conflict, since it fortunately doesn't include this case!

Specifically, it does not include cases like h^{x}n, with only one h to the left of the superscript and one number to the right - which, for obvious reasons, includes all the cases where I'll be using it in the modern meaning. Which is kind of lucky.]

Note that hn=n^^n=n^^^2, hhn=(n^^n)^^(n^^n)>n^^(n^^n)=n^^^3, and similarly 3hn>n^^^4, ..., mhn>n^^^(m+1) (which in turn is more than n^^^m). So we're already in pentational territory.

And the next step brings us to the limits of pentation: let hmhn=h^{hm}n; this is more than n^^^(m^^m).

In turn, we can have hhmhn=h^{hhm}n, and so on, the number to the left of the last hn being the number of iterations for the h function.

This means that hkhmhn=h^{hhkm}n; we can easily iterate this further, to outrageous examples like h7h2h15h4h5h8h9=h^{hhhhhh7215458}9 (for all I know, this number, being the largest listed specifically in the treatise - further examples always involve a n somewhere - might well have been the largest number ever specifically referred to in all of mathematical history up to that year).

Now we are certainly passing into hexational territory - already nhnhn>n^^^(nhn)>n^^^(n^^^n)=n^^^^3. Similarly, nhnhnhn>n^^^(nhnhn)>n^^^^4, and so on.

At this point, Kharms focuses on a specific case - that where all the coefficients m in hmhm...hmhn are identical; it is for this case that he used the notation mh^{k}n (where the sequence "hm" repeats k-1 times). In particular, for m=n, we have hnhn...hn=nh^{k}n (where the sequence "hn" repeats k times).

It is this case - and specifically its version where k=n - that Kharms uses as the next function, nh^{n}n=tn.

We can easily see that nh^{k}n=h^{nhk-1n}n (this combines the original and modern superscript notations); then nh^{k}n>n^^^(nh^{k-1}n). Since nh^{2}n=hnhn=h^{hn}n>n^^^(hn)>n^^^n=n^^^^2, by induction we have nh^{k}n>n^^^^k, and tn>n^^^^n (though not by much, googologically speaking - it is probably less than n^^^^(n+1)).

Kharms continues, defining the first-level circle notation, which is already as powerful as the full might of (non-iterated) Knuth's up-arrows; but for that, see part 2 (if I ever decide to write it).