A history of Greek mathematics - Wilbourhall.org

.
HYPSICLES 217
II. If Oj, a 2
... a n
... a 2 «-i
is a descending arithmetical progression
of 2n—l terms with 8 as common difference and a n
is the middle term, then
a 1
+ « 2
+...+a 2n _ 1
= (27i-l)a n
.
III. If a 1}
a 2
...a n
, a n+1 ...a 2n is a descending arithmetical
progression of 2n terms, then
a x
+ a 2
+ .
. . +
a 2 n
= w (a x
+ a 2 n ) = n (a 2
+ a 2 n _ 2 ) = .
.
Now let J.,
= ^(a w + a n+1 ).
5, G be the descending series the sum of which
is 105, and D, E, F the next three terms in the same series
the sum of which is 75, the common difference being 8; we
then have, by (I),
and 8= 3|.
A + B + G-(D + E+F) = 98, or 30 = 98,
Next, by (II), A + B + C =3B, or 3B = 105, and B = 35
;
therefore ^4, i?, (7, 2), i?, .Fare equal to 38§, 35, 31 §, 28-J, 25,
21§ time-degrees respectively, which the author of the tract
expresses in time-degrees and minutes as 38*20', 35*, 31*40',
28* 20', 25*, 21* 40'. We have now to carry through the same
procedure for each degree in each sign. If the difference
between the times taken to rise by one sign and the next
is 3* 20', what is the difference for each of the 30 degrees in
the sign 1 We have here 30 terms followed by 30 other terms
of the same descending arithmetical progression, and the
formula (I) gives 3* .
20' 2
= (30)