VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 701 



Let AC and b6 (fig. 1, pi. 7,) be 2 indefinite lines, intersecting each other at 

 right angles in c. Let lsm be the beam, or a rigid right line, in which assume 

 1 fixed points l and s at pleasure. If the fixed point l be kept always sliding on 

 the line Bci, and the other point s always sliding on the line ACa ; then, any 

 point M in the line ls, or that line produced, will describe an ellipse. 



Bisect LS in e, and through c and e draw the indefinite right line ckh. On 

 LS as a diameter, with the centre e, describe a semi-circle, and because lcs is a 

 right angle, it will pass through c, and ec = el. Through m draw mph per- 

 pendicular to AC, meeting ce produced in h ; and because mh is parallel to cl, 

 the triangles meh and cel are similar, and he = me, and he -j- ec = me -f el, 

 or ch =: LM. The point h therefore always falls in the circumference of the 

 circle hadg described with the centre c and radius ch = lm. Now the similar 

 triangles chp and smp give ch : sm :: ph : pm. But when l arrives at c, then 

 LM (= ch) coincides with ca ; and when s arrives at c, then sm coincides with 

 CB ; therefore ca : cb :: ph : pm, and ca' : cb^ :: ph* : pm% or ca" : CB^ :: ap 

 X Pfl : PM^, which is the property of an ellipse, whose first semi-axe is ca or lm, 

 and 2d semi-axe is cb = sm. 



Produce pm till it meets the circle in n, and draw the radius cn ; then 

 PH = pn and ca : cb y. pn : pm. Again, because pch ^ pcn, therefore nod 

 = ECL = elc and cn is parallel to lm, and cl = nm. Draw up perpendicular 

 toB^, cutting CN in n, and for the like reason cw = sm = cb, and cs := mn. 

 While the point m describes an oval, the point e describes a circle whose centre 

 is c and radius ce = iSL. 



To the ruler mel (fig. 3 and 4) fix another ruler or right line mEK passing 

 through E, so that the ruler jwek may be carried about by the ruler mel, keep- 

 ing the angle meto between the 2 rulers invariable. On wjek take ev = ek, 

 and each = es or el ; then the point v will describe a right line nvcx passing 

 through c, and making an angle ocs with ca, equal to half meot the angle 

 made by the 2 rulers ; the point k will also describe a right line Akc(3 passing 

 through c, and making an angle ;^cl, with cl, also equal to half metk. 



On the centre e (fig. 3 and 4) and with the radius ec, describe a circle, and 

 it will pass through the points s, v, c, l, k ; draw the lines vc and kc, and the 

 angles sev, and scv, both stand on the same arch sv ; the former at the centre 

 E, the latter at the circumference c ; therefore the former is double the latter. 

 In like manner the angles kel and kcl both stand on the same arch kl, the 

 former at the centre, the latter at the circumference ; therefore the former is 

 double the latter. Now as this holds in every position of the rulers during their 

 joint motion, it is manifest that the points v and k will each describe right lines, 

 namely, acaand ^c|3, passing through c, and making the angles aCA and Z'CL 

 (= BC(3) each equal to half mew. 



