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"> and hence the angle FEG is equal to EGH, but FEG is a right one by Con- <br /> ſtruction. Let now HI be drawn from H perpendicular to AB: then the two <br /> triangles EHI and EHG having two angles in one HEI and EIH reſpectively <br /> equal to two angles in the other HEG and EGH, and alſo the ſide EH com- <br /> mon, by Euc. I. 26. HI will be equal to HG, and therefore the circle will <br /> touch alſo the other line AB: and HG or HI equals the given line Z, becauſe <br /> EF was made equal to Z, and HG and EF are oppoſite ſides of a paral- <br /> lelogram. </p> <h2 xmlns="http://www.w3.org/1999/xhtml"> <span class="headingNumber">9.</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> two circles given whoſe centers are A and B, it is required to draw <br /> another, whoſe Radius ſhall be equal to a given line Z, which ſhall alſo touch <br /> the two given ones. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">This</span> Problem has various Caſes, according to the various poſition of the <br /> given circles, and the various manner of deſcribing the circle required: but there <br /> are ſix principal ones, and to the conditions of theſe all the reſt are ſubject. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Case</span> 1ſt. Let the circle to be deſcribed be required to be touched outwardly <br /> by the given circles. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Limitation</span> . Then it is neceſſary that 2Z, or the given Diameter, ſhould <br /> not be leſs than the ſegment of the line joining the centers of the given circles <br /> which is intercepted between their convex circumferences, viz. not leſs than CD <br /> in the Figure belonging to Caſe 1ſt. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Case</span> 2d. Let the circle to be deſcribed be required to be touched inwardly by <br /> the given circles. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Limitation</span> . Then it’s Diameter muſt not be given leſs than the right line, <br /> which drawn through the centers of the given circles, is contained between their <br /> concave circumferences; viz. not leſs than CD. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Case</span> 3d. Let the circle to be deſcribed be required to be touched outwardly <br /> by one of the given circles, and inwardly by the other. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Limitation</span> . Then it’s Diameter muſt not be given leſs than the ſegment <br /> of the right line, joining the centers of the given circles, which is intercepted <br /> between the convex circumference of one and the concave circumference of the <br /> other; viz. not leſs than CD. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Case</span> 4th. Let one of the given circles include the other, and let it be re- <br /> quired that the circle to be deſcribed be touched outwardly by them both. </p> <p xmlns="http://www.w3.org/1999/xhtml"> <span class="caps" style="font-size: 75%">Limitation</span> . Then it’s Diameter muſt not be given greater than the greater <br /> ſegment of the right line, joining the centers of the given circles, which is in- <br /> tercepted between the concave circumference of one and the convex circumference <br /> of the other; nor leſs than the leſſer ſegment; viz. not greater than CD, nor <br /> leſs than MN. </p> </div>