A history of Greek mathematics - Wilbourhall.org

,
THE COLLECTION. BOOK VII 425
For EA 2 + AF 2 = ED 2 + DA* + AB 2 + BF 2
= ED 2 + BC 2 + C£ 2 + 5i^2 .
Also EA 2 + AF 2 = EF 2 + 2EA. AF.
Therefore
2EA.AF= EA 2 + AF 2 - EF 2
= ED 2 + BC 2 + CD 2 + BF 2 - EF 2
= (.ED2 + CD 2 ) + (.BO 2 + BF 2 )
- EF 2
i.e. EA .
4F
= EC 2 + 2ED.DC+CF 2 + 2CB.BF-EF 2
= 2ED.DC+2CB.BF;
= ED . DC+CB
.
Itf 7 .
This is equivalent to sec cosec = tan 6 + cot 6.
The algebraical equivalents of some of the results obtained
by the usual geometrical algebra may be added.
Props. 178, 179, 192-4.
(a + 2b)a + {b + x) (b-x) = (a + b + x)(a + b — x).
Prop. 195. 4a 2 = 2{(a-x) (a + x) + (a-y) (a + y) + x 2 + y
2
\.
Prop. 196.
{a + b-x) 2 + (a + b + x) 2 = (x-b) 2 + (x + b) 2 +2(a + 2b)a.
Props. 197, 199, 198.
If (x + y + a)a + x 2 = (a + x) 2 , \
or if (x + y + a) a + # 2 = (a + 2/)
2
, L then x = y.
or if (a; + 2/
— a) a + (# — a)
2 = 2/
2
,
Props. 200, 201. If (a + b)x = b 2 , then
2b + a
a
b + x
b — x
and
(2b + a)a = (a + 6) (a + & — a?).
Prop. 207. If (a + b)b — 2a 2 , then a = b.
(6) The two Lemmas to the Surface-Loci of Euclid have
already been mentioned as significant. The first has the
appearance of being a general enunciation, such as Pappus