Vignettes of Ancient Mathematics
(diagram 1) For the remaining circular-arc with forward motion, Apollonius provides a little lemma of this sort, that if, in triangle ABG with BG larger than AG, and GD no smaller than AG is marked off, (diagram 2) then GD to BD has a larger ratio than angle ABG to angle BGA.
He proves it in this way. (diagram 3) For let there be filled out, he says, a parallelogram ADGE, and (diagram 4) let BA and GE be extended and meet at point Z. (diagram 5) Since AE is not smaller than AG, therefore the circle inscribed with center A and distance AE will either come through G or beyond G. (diagram 6 = general diagram) Let HEG be inscribed through G. (diagram 7) And since triangle AEZ is larger than section AEH, (diagram 8) but triangle AEG is smaller than section AEG, (diagram 9) triangle AEZ has a larger ratio to triangle AEG than section AEH has to section AEG. (a > c and b < d => a : b > c : d) (diagram 10) But as section AEH is to AEG, so is angle EAZ to angle EAG, (diagram 11) but as triangle AEZ is to AEG, so is base ZE to EG. Therefore, ZE to EG has a ratio larger than angle ZAE to angle EAG. But as ZE is to EG, so is GD t DB, (diagram 12: since GD = EA and EG = AD, while triangle ZAE is similar to ABD, because AG is parallel to AD and EA to GB, so that all respective angles are equal) (diagram 11) while angle ZAE is equal to angle ABG, and angle EAG to angle BGA. Therefore, GD to DB also has a ratio larger than angle ABG to AGB. (diagram 5) But it is clear that the ratio will also be much larger when GD, i.e. AE, is supposed not equal to AG, but larger than it.
The theorem is basically: GB
AG => AG : GB-AB > ABG