**Si in planis inaequalibus, quorum eadem sit elevatio, descendat mobile, spatium quod in ima parte longioris conficitur in tempore aequali ei {10} in quo conficitur totum planum brevius, est aequale spatio quod componitur ex ipso breviori plano et ex parte ad quam idem brevius planum eam habet rationem, quam habet planum longius ad excessum quo longius brevius superat. ** | **If a body descends along two inclined planes of different lengths but of the same vertical height, the distance which it will traverse, in the lower part of the longer plane, during a time-interval equal to that of descent over the shorter plane, is equal to the length of the shorter plane plus a portion of it to which the shorter plane bears the same ratio which the longer plane bears to the excess of the longer over the shorter plane. ** |

Sit planum AC longius, AB vero brevius, quorum eadem sit elevatio AD, et ex ima parte AC sumatur CE aequale ipsi AB, et quam rationem habet totum CA ad AE, nempe ad excessum plani CA super AB, hanc {20} habeat CE ad EF: dico, spatium FC esse illud quod conficitur, post discessum ex A, tempore aequali tempori descensus per AB. Cum enim totum CA ad totum AE sit ut ablatum CE ad ablatum EF, erit reliquum EA ad reliquum AF ut totum CA ad totum AE; sunt itaque tres CA, AE, AF continue proportionales: quod si ponatur, tempus per AB esse ut AB, erit tempus per AC ut AC; tempus vero per AF erit ut AE, et per reliquum FC erit ut EC: est autem EC ipsi AB aequale: ergo fit propositum. | Let AC be the longer plane, AB, the shorter, and AD the common elevation; on the lower part of AC lay off CE equal to AB. Choose F such that CA:AE = CA:CA-AB = CE:EF. Then, I say, that FC is that distance which will, after fall from A, be traversed during a time-interval equal to that required for descent along AB. For since CA:AE = CE:EF, it follows that the remainder EA: the remainder AF = CA:AE. Therefore AE is a mean proportional between AC and AF. Accordingly if the length AB is employed to measure the time of fall along AB, (Condition 2/03-th-03-cor) then the distance AC will measure the time of descent through AC; (Condition 2/02-th-02-cor2) but the time of descent through AF is measured by the length AE, (Condition 2/11-th-11) and that through FC by EC. Now EC = AB; and hence follows the proposition. |