Proposition I.22
Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary 1 that two of the straight lines taken together in any manner should be greater than the remaining one. [I.20] 2 3
Let the three given straight lines be A
, B
, C
, and of these let two taken together in any manner be greater than the remaining one, namely A
, B
greater than C
,
A
,C
greater thanB
,
nd B
, C
greater than A
; thus it is required to construct a triangle out of straight lines equal to A
, B
, C
.
Let there be set out a straight line DE
, terminated at D
but of infinite length in the direction of E
, and let DF
be made equal to A
, FG
equal to B
, and GH
equal to C
. I.3
With centre F
and distance FD
let the circle DKL
be described; again, with centre G
and distance GH
let the circle KLH
be described; and let KF
, KG
be joined;
I say that the triangle KFG
has been constructed out of three straight lines equal to A
, B
, C
.
For, since the point F
is the centre of the circle DKL
,
FD
is equal toFK
.
But FD
is equal to A
;
- therefore
KF
is also equal toA
.
Again, since the point G
is the centre of the circle LKH
,
GH
is equal toGK
.
But GH
is equal to C
;
- therefore
KG
is also equal toC
. AndFG
is also equal toB
;
therefore the three straight lines KF
, FG
, GK
are equal to the three straight lines A
, B
, C
.
Therefore out of the three straight lines KF
, FG
, GK
, which are equal to the three given straight lines A
, B
, C
, the triangle KFG
has been constructed.
- Q. E. F.
References
Footnotes
-
thus it is necessary.
This is usually translated (e.g. by Williamson and Simson)
which is however inaccurate, since the Greek is notBut it is necessary,δεῖ δέ butδεῖ δή . The words are the same as those used to introduce theδιορισμός in the other sense of thedefinition
orparticular statement
of a construction to be effected. Hence, as in the latter case we saythus it is required
(e.g. to bisect the finite straight lineAB
, I. 10), we should here translatethus it is necessary.
↩ -
enunciation note
This is the first case in theElements of aδιορισμός to a problem in the sense of a statement of the conditions or limits of the possibility of a solution. The criterion is of course supplied by the preceding proposition. ↩ -
enunciation note
To this enunciation all the MSS. and Boethius add, after theδιορισμός , the wordsbecause in any triangle two sides taken together in any manner are greater than the remaining one.
But this explanation has the appearance of a gloss, and it is omitted by Proclus and Campanus. Moreover there is no corresponding addition to theδιορισμός of VI. 28. ↩