1797.] Analogy between the Circie: and other Curves. 289 | 
in the fifth century, and whocommented manufcripts, with a view to re-eftablith 
on the works of Ariftotle. In this para- this paffage, has, moreover, difcovered a 
phrafe we meet with a paflage relative valuable fragment of the text of Simpli- 
to Xenophon, ' contradicting all the cius, which tiad been omitted by all the . 
other remains of antiquity which treat other copyifts, and the omiffion of which 
of that great captain. Schweighaufer, had rendered defective the paflage which 
in fearching through a number of ancient related to Zenophon. 
oe NASI 
MATHEMATICAL CORRESPONDENCE 

For the Monthly Magazine. 
Or THE ANALOGY BETWEEN THE CIRCLE AND OTHER CURVES (CONTINUED). 
"THE next problem which we fhall confider, was fuggeited by the following remarkable pro- 
perty of the Circle, of which we have not mct with a demonftration : 
' Theorem I. Fig. 3. 
If, from a given point, without a circle, two tangents be drawn, and from one of the points 
of conta, a perpendicular be let fall upon the diameter pafling through the other, it will be. bi- 
feted, by a ftraight line, joining the point without the circle, and the fartheft extremity of the 
diameter. 
Let ADB be the circle, C the point without it, and. 
CB CD the two tangents: draw the diameter BA, and 
from D let fall the perpendicular DE: join AC, which 
will bife@& DE in F. 
Draw CG parallel to BA, and through A D draw AG 
to mect it in G: join DB, CO, OG, OD, and let OG 
meet AC in K; and OC meet BD in Q. 
Then, becaufe BO=OD, the angle OBC=the angle 
CDO, and CO common, the angle BOC is = the angle 
COD, and therefore = the angle BAD: confequently, 
AD is parailel to CO. Now CG being parallel to AB, , 
aud AD parallel to CO, AGCO isa parallelogram, and 
therefore CG=-AO=BO, andGK=OK. Alfo becaufe CG is equal and parallel to BO, OG 
- 38 equal and paraidel to BC: but DE is parallel to BC, wherefore OG is parallelto DE. Now 
GK being equal to KO, and GO parallel to DE, DF is equal toEE. Confequently, if. from a 
given point withouta circle, &c. Bi Bios D. 
’ This property, with little variation, takes place in the ellipfis, and may be demonftrated in the 
following manner : 

Theorem IT. Figs 3. 
If, from a given point without an ellipfis, two tangents be drawn, and from one of the points 
ef contaa, an ordinate be drawn to the diameter pafling through the other, it will be bifccted by 
a ftraight line, joining the point without the ellipfis, and the fartheft extremity of the dia~ 
WMmciek 60 
Let ADB be the ellipfis, O the centre, C the point without it, and CB, CD the two tangents : 
through one of the points of contact B, draw the di:meter BA, and Sm the other an ordinate, 
DE, be drawn to this diameter: then if AC be drawn,Jt 71 bifea DE in F. 
Draw CG parallel to AB, and through A B 4raw AG to mect it in G: 
AC in K, and produce CD to meet BA in L. 
Then, by Simfon’s Con. Lect. lib. ii. prop.17, AL is to OL,-as AE is to OB, ane by Cor, 11. 
fame prop. AL isto OL, as LE to LB: hence AK istoOB, as LEtuLB. Buta known pros 
perty of the cllipfis DE is parallel to BC, and therefore the triangics LDE, LCB are fimilar: 
confequently, LE is to LB,as DE to BC. Wherefore AE is to OB, as DE to BC, and as the 
angles DEA and CBO are equal, the triangles DEA, CBO are fimilar, and the angle DAE equal 
to the anele COB. A 
Now the angles DAE, COB being equal, the lines AD, CO are parallel, and confequently 
CH=AO=BO: alfo becaufe CH is equal and parallel to BO, OH ts parallel to BC or to DE, 
and, confequently, as OH is bifected by AC in K, DE is bife&ted by AC in F. Toerefore, ii 
from a given point, without an ellipfis, two tangents, &&c. we aay WB) 
| SCHOLIUM. 
This new and curious property fuggefis an eafy method of drawing a tangent to any point D 
of an ellipfis. For, it BA be the tranfverfe diameter, and BC a perpendicular from B ~draay DE 
perpendicular to BA, and bifeé this perpendicular in F: through A, B draw AC tu mect BC in 
C; joinCD, which willbe the tangent requred. The demonttration is obvious. . 
§. CYGNI. 
TT 6 
join OG, meeting 
Aberdeeny Jan. 1797 « [To be continued.) 





















