CONGRUENCE 109

By the hypothesis g, where 0 < & < =+ 1, is the exact or
lower approximate quotient obtaind by dividing 7;_, by 7;_;.

For example, ¢, is the exact or lower approximate quotient
obtaind by dividing 7_; by 7, that is, a by b.

Hence, if ¢ <5, ¢ = 0; if a=0b ¢qg=1;if a > b and
a > b, ¢ is positiv, since the exact quotient of two positiv
numbers is positiv; if @ > b and a —>% b, ¢1 is positiv by § 83.

Thus g; is positiv when £ = 1, except when ¢ < b, in which
case it is zero.

There remains to be discussed the case when 1 <k<n++1.
Now, since, if 0 < 2 < n + 1, 7 is the remainder obtaind

by d1V1d1ng () by Yl = k. § 80.
Putting 2 — 1 for 2, 7., >, f0<k—1<n+41
that is, it <k<n-t?2

Hence the statement 7;,_» > 7, is tru if 1 <k<mn+4+1,
which is the case remaining to be discussed.

Then, since ¢ is the exact or lower approximate quotient
obtaind by dividing 74— by 7., g is positiv if 7,y >> 7,4,
since the exact quotient of two positiv numbers is positiv, and
gr 1s positiv if 7,_s —>> 74 by § 83. -

Therfor ¢ is positivif 1 < B < + 1.

Hence ¢ is positiv in all cases except when 2 =1 and
a < b, in which case it is zero.

Now consider s;.

We have Sa=1
So = 0
S1 =Sa4+q@gso=14+0=1
52=So+9281=0+42=92

Hence, since g, is positiv, s; and s, are positiv.
Therfor the first two numbers in the series 81539, Sty -, 8
are positiv,