The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus Apollonius Pergaeus John Lawson 868387657
<?xml version="1.0" encoding="UTF-8"?> <div> <p xmlns="http://www.w3.org/1999/xhtml" id="false"> to find the center of a circle which will paſs through the two points, and like- <br /> wiſe touch the right line, which is the VIIth of the preceeding Problems. </p> <h2 xmlns="http://www.w3.org/1999/xhtml"> <span class="headingNumber">39.</span> <span class="head">PROBLEM III.</span> </h2> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Having</span> three points N, O, M, given, as likewiſe a ſphere IG, to de- <br /> ſcribe a ſphere which will paſs through the three given points, and likewiſe <br /> touch the given ſphere. </p> <p xmlns="http://www.w3.org/1999/xhtml"> The circle NOM in the ſurface of the ſphere ſought is given, and a per- <br /> pendicular to its plane from it’s center FA being drawn, the center of the <br /> ſphere required will be in this line. From I the center of the given ſphere <br /> let IB be drawn perpendicular to FA, and through F, ED parallel to IB, <br /> which, from what has been before proved, will be in the plane of the circle <br /> NOM, and the points E and D will be given. </p> <p xmlns="http://www.w3.org/1999/xhtml"> Suppoſe now the thing done, and that the center of the ſphere required is <br /> C. Then the lines CI, CE, CD, will be in the ſame plane, which is given, as <br /> the points I, E, and D are given. But the point of contact of two ſpheres is <br /> in the line joining their centers; therefore the ſphere ſought will touch the <br /> ſphere given in the point G, and the line IC will exceed the lines EC, ED, by <br /> IG the radius of the given ſphere: with center I therefore and this diſtance <br /> IG let a circle be deſcribed in the plane of the lines CI, CE, CD, and it <br /> will paſs through the point G and be given in magnitude and poſition; but <br /> the points D and E are alſo in the ſame plane; and therefore the queſtion is <br /> reduced to this, Having two points E and D given, as likewiſe a circle <br /> IGH, to find the center of a circle which will paſs through the two points <br /> and likewiſe touch the circle, which is the XIIth of the preceeding Problems. </p> <h2 xmlns="http://www.w3.org/1999/xhtml"> <span class="headingNumber">40.</span> <span class="head">PROBLEM IV.</span> </h2> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Having</span> four planes AH, AB, BC, HG, given; it is required to de- <br /> ſcribe a ſphere which ſhall touch them all four. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">If</span> two planes touch a ſphere, the center of that ſphere will be in a plane <br /> beſecting the inclination of the other two. And if the planes be parallel, it <br /> will be in a parallel plane beſecting their interval. This being allowed, <br /> which is too evident to need further proof; the center of the ſphere ſought <br /> will be in a plane biſecting the inclination of two planes CB and BA; it will <br /> likewiſe be in another plane biſecting the inclination of the two planes BA and <br /> AH; and therefore in a right line, which is the common ſection of theſe two </p> </div>