Discorsi Propositions 2/02-th-02-cor1 Discorsi Proposition2/02-th-02-cor1

 COLLARIUM I. COROLLARY I Hinc manifestum est, quod si fuerint quotcunque tempora aequalia consequenter sumpta a primo instanti seu principio lationis, utputa AD, DE, EF, FG, quibus conficiantur spatia HL, LM, MN, NI, ipsa spatia erunt inter se ut numeri impares ab unitate, scilicet ut 1, 3, 5, 7: (Condition 2/02-th-02) Hence it is clear that if we take any equal intervals of time whatever, counting from the beginning of the motion, such as AD, DE, EF, FG, in which the spaces HL, LM, MN, NI are traversed, these spaces will bear to one another the same ratio as the series of odd numbers, 1, 3, 5, 7; haec enim est ratio excessuum quadratorum linearum sese aequaliter excedentium et quarum excessus est aequalis minimae ipsarum, seu dicamus quadratorum sese ab unitate consequentium.Dum igitur gradus velocitatis augentur iuxta seriem simplicem numerorum in temporibus aequalibus, spatia peracta iisdem temporibus incrementa suscipiunt iuxta seriem numerorum imparium ab unitate. {20} for this is the ratio of the differences of the squares of the lines (which represent time), differences which exceed one another by equal amounts, this excess being equal to the smallest line (viz. the one representing a single time-interval): or we may say (that this is the ratio) of the differences of the squares of the natural numbers beginning with unity.While, therefore, during equal intervals of time the velocities increase as the natural numbers, the increments in the distances traversed during these equal time-intervals are to one another as the odd numbers beginning with unity.

 Discorsi Propositions 2/02-th-02-cor1 Discorsi Proposition2/02-th-02-cor1