By John Casey

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Additional info for A Sequel to the First Six Books of the Elements of Euclid, Fifth Edition

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Then, since AB is bisected in G, we have (16) AL + EL = 2GL ; and since GC is divided into (1 + 2) equal parts in H, so that HG contains one part and HC two parts ; then 2GL + CL = 3 HL ; AL + EL + CL = 3HL, & c &c. Hence the Proposition is :prov e d � = - ' = • e. " 13 BOOK 1. L s is zero. Hence some of the 1- s must be nega.. L s on the positive side equal to the sum of those on the negative side. Prop . - We may extend the foregoing Definition as follows :-Let there be any system ofJJoints A, B, C, D, 9'-0.

43 Prop. -If in the line Joining the centres of two ()1:rcles a poz'nt D he found, G Buch that the tangents DE, DE' from it to the c�'rcles are equal, and if through D a hOn e DG he drawn perpendt"cular to the hOn e Join�Ong the centres, then r-_-,;C_:rl -t-_�-I--+-�--I the tangents from any other pot"nt G �On DG to the c�orcles w�oll he equal. -Let GF, GF' be the tangents. Now, by hyp o the sis , DE2 = DE/2. To each add DG2, and we h av e GD2 + D E'2, GF2 = GF'2 ; . . GF = GF' . -The line GD is called the railical axis of the two circles ,.

And PF2 = DF2. Hence, by addition, making use of 11. , and 11. iii. EP 2 = DF . DH = twice rectangle contained by the radii. Prop. -If a m;rcle PGD touoh a oirole AB C in D and a chord AB in P, and tt/ EF be perpendioular to AB , at its middle point, and at the 8tt'ae opposite to that of D the o£rcle PGD, the rectangle contatt'ned by EF and the dtt'ameter ofthe circle PGDis equal At--��--+E-----J- B to the rectangle AP . PB. -Let PG be at right L s to AB, then PG is the diameter of the 0 H F PGD .

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A Sequel to the First Six Books of the Elements of Euclid, by John Casey
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