A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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
i.e. EA .
= ED 2 + BC 2 + CD 2 + BF 2 - EF 2
= (.ED2 + CD 2 ) + (.BO 2 + BF 2 )
- EF 2
= EC 2 + 2ED.DC+CF 2 + 2CB.BF-EF 2
= 2ED.DC+2CB.BF;
4F = 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
or if (a; + 2/
— a) a + (# — a)
2 = 2/
2
,
Props. 200, 201. If (a + b)x = b 2 , then
(2b + a)a = (a + 6) (a + & — a?).
Prop. 207. If (a + b)b — 2a 2 , then a = b.
, L then x = y.
2b + a
a
b + x
b — x
and
(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
426 PAPPUS OF ALEXANDRIA
is fond of giving, to cover a class of propositions. The
enunciation may be translated as follows :
'
If AB be a straight
line, and CD a straight line parallel to a straight line given in
position, and if the ratio AD . DB : DC 2 be given, the point C
lies on a conic section. If now AB be no longer given in
position, and the points A, B are no longer given but lie
(respectively) on straight lines AE, EB given in position, the
point G raised above (the plane containing AE, EB) lies on
a surface given in position. And this was proved.' Tannery
was the first to explain this intelligibly
in length.
and his interpretation only requires the
very slight change in the text of substituting
evOeiais for evOeia in the phrase
yivrjTai St 777)0? decrei evdeta reus AE, EB.
It is not clear whether, when AB ceases
to be given in 'position, it is still given
If it is given in length and A, B move on the lines
AE, EB respectively, the surface which is the locus of G is
a complicated one such as Euclid would hardly have been
in a position to investigate. But two possible cases are
indicated which he may have discussed, (1) that in which AB
moves always parallel to itself and varies in length accordingly,
(2) that in which the two lines on which A, B move are
parallel instead of meeting at a point. The loci in these two
cases would of course be a cone and a cylinder respectively.
The second Lemma is still more important, since it is the
lirst statement on record of the focus-directrix property of
the three conic sections. The proof, after Pappus, has been
set out above (pp. 119-21).
(1) An unallocated Lemma.
Book VII ends (pp. 1016-18) with a lemma which is not
given under any particular treatise belonging to the Treasury
of Analysis, but is simply called 'Lemma to the 'Ai/a\v6/jLeuos\
If ABC be a triangle right-angled at B, and AB, BG be
divided at F, G so that AF : FB = BG : GC = AB: BC, and
if AEG, CEF be joined and BE joined and produced to D,
then shall BD be perpendicular to AC.
The text is unsatisfactory, for there is a long interpolation
containing an attempt at a proof by reductio ad absurdum
;