A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

226

226 ~A SEQTJEL TO EUCLID.

associated figures is due. See Mazthesis, ome, vi., pp. 97, 148,
196, from which this Section, except propositions 4 and 6, is
taken.
Prop. 6.-ITf AjA2 A. A. l be, a Tarry's line,
II, I2... I, the invariable points, then the angle LIA,AA
I=,A2AA3.. I,,,,, and the angle ]11A.,Al = IA3A,
Dem.-By hypothesis the triangles IAA,, I12AsA
*.. I,1A,,A,,,1 f orm an associated system. Hence they
are equiangular, and theref ore the proposition is
proved.
DEF. vi.-The lines A,I,, Aj1,.. A,jT,, being corresponding lines passing the invariable points, meet on the,
circle of similitude. In like manner, A,11, AJ12...
AnI meet on the circle of similitude. The points of concurrence are called the lirocard points of the system, and
denoted by f~, f2'.
DEF. VIL.-The base angles of the equiangular triangles
-IjAjA2, I2A2A3... are called its Brocard angles, and
denoted by w, w', viz. IAA2 by co, and IA2AI by w'.
DEF. viii —Thte perspective centre [Def. iv.] of a,
Tarry's line, being such that perpendiculars from it on the
several _parts of that line are _proportional to the _parts, is
catted the symmedian point of the line.
Prop 7.-iieing given two consecutive sides AA2,
A2A, of a Tarry's line, and its -Brocard angles, to construct it.
Sol.-UIpon AA,, AA3 construct two triangles
A11,A2, A,12A3, having their base angles equal to wt, (o',
respectively, viz., A2A11, = A3A212, = o,, and AjA2I[ =
A2A312 = w'. Then the vertices II, 1, are invariable
points. The linies AI,, A21L will meet in one of the
Brocard points t~ [IDef. vi.], and the lines A,11, Aj 12 in1
the other IBrocard point f72. Then the four points I,,
12,, L, LI' are concyclic, and the circle Z, through them,