12^ Mr, Ivory on a new Method of deducing 
consequently to the segments AB, BC, CD: then the lines AP, 
BE, CF, DQ, or the directions in which the comet is seen from 
the earth, will be given by position on account of the observed 
longitudes. Draw DM parallel to AP, and PM parallel to AD, 
and join QM : now if we assume any point whatever in the 
line AP, as p ; and draw pm parallel to PM, and mq parallel 
to MQ ; then, the \me pq being drawn, it will be cut by the 
lines BE and CF into segments having the same proportions 
as the segments of the lines AD and PQ. To demonstrate 
this : draw BH and CG parallel to AP and DM, and let those 
lines meet PM in H, G, and pm in h, g : join HE, GF ; and 
because the segments of PM are equal to the segments of AD, 
they will be proportional to the segments of PQ (hyp.); 
therefore (E. 2. VI.) HE and GF will be parallel to MQ and 
mq: draw he and gf parallel to HE and GF ; and consequently 
(E. pq will be cut in e and/in the same proportion as 
AD is cut in B and C, and PQ in H and G: and I say that BE 
will coitpq in and CF will cut it in/. For on account of the 
parallel lines, and because PM =pm, VG=. pg^ PH z=. ph, 
therefore 
QM : HE : : qm : he 
QM : GF : : qm : gf 
and, by alternation, 
QM : qm : : HE : he 
QM : qm : : GF : gf 
but, on account of the similar triangles QDM, qdm, 
QM : qm : : MD : wD 
therefore, because MD = GC = BH, and wD = C^ = BA, 
BH : BA : : HE : he 
CG:%::GF:^/ 
