THE COLLECTION. BOOK YII 
425 
For EA 2 + AF 2 = ED 2 + DA 2 + AB 2 + B.P 2 
= ED 2 + BC 2 + CD 2 + BF 2 . 
Also £A 2 + AF 2 = EF 2 + 2 EA. AF. 
Therefore 
2 EA.AF= EA 2 + AF 2 - EF 2 
= A7J 2 + i?(7 2 + CD 2 + BF 2 - EF 2 
= {ED 2 + CD 2 ) + (.EG 2 + BF 72 ) - EF 2 
= EC 2 + 2 ED. DC + GF 2 + 2 GB. BF- ATP 2 
= 2ED.DC+2GB.BF-, 
i.e. EA .AF= ED. DC + GB. BF. 
This is equivalent to sec 6 cosec 6 = 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 + 2 b) a + (b + x) (b — x) = {a + h + x) (a + b — x). 
Prop. 195. 4tt 2 = 2{{a — x) (a + x) + (a — y) {a + y) + x 2 + y 2 ]. 
Prop. 196. 
(a + h — x) 2 + {a + b + x) 2 — {x — h} 2 + (x + b) 2 + 2 {a + 2 6) a. 
Props. 197, 199, 198. ' 
If {x + y + a) a + x 2 = {a + x) 2 , \ 
or if {x + y + a) a + x 2 = {a + y) 2 , l then x — y. 
or if {x + y — a) a + (x — a) 2, = y 2 , j 
r-l TC/ 7\ JO II 2 6 -(- tt h -J- X . 
Props. 200, 201. It (a + b)x = 6“, then —-— = ^—- and 
(2 b + a)a = (a + b) (a + h—x). 
Prop. 207. If (a + h)h — 2a 2 , then a = h. 
(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