Discorsi Propositions 2/02-th-02-cor2 Discorsi Proposition2/02-th-02-cor2

 {214} COLLARIUM II. {214} COROLLARY II Colligitur, secundo, quod si a principio lationis sumantur duo spatia quaelibet, quibuslibet temporibus peracta, tempora ipsorum erunt inter se ut alterum eorum ad spatium medium proportionale inter ipsa. Secondly, it follows that, starting from any initial point, if we take any two distances, traversed in any time-intervals whatsoever, these time-intervals bear to one another the same ratio as one of the distances to the mean proportional of the two distances. Sumptis enim a principio lationis S duobus spatiis ST, SV, quorum medium sit proportionale SX, tempus casus per ST ad tempus casus per SV erit ut ST ad SX, seu dicamus, tempus per SV ad tempus per ST esse ut VS ad SX.Cum enim demonstratum sit, spatia peracta esse in duplicata ratione temporum, seu (quod idem est) esse ut temporum quadrata; ratio autem spatii VS ad spatium ST sit dupla rationis VS {10} ad SX, seu sit eadem quam habent quadrata VS, SX; patet, rationem temporum lationum per SV, ST esse ut spatiorum, seu linearum, VS, SX. For if we take two distances ST and SY measured from the initial point S, the mean proportional of which is SX, the time of fall through ST is to the time of fall through SY as ST is to SX; or one may say the time of fall through SY is to the time of fall through ST as SY is to SX.(Condition 2/02-th-02) Now since it has been shown that the spaces traversed are in the same ratio as the squares of the times; and since, moreover, the ratio of the space SY to the space ST is the square of the ratio SY to SX, it follows that the ratio of the times of fall through SY and ST is the ratio of the respective distances SY and SX.

 Discorsi Propositions 2/02-th-02-cor2 Discorsi Proposition2/02-th-02-cor2