To divide a given Line into any Number of Parts, which Parts shall be in the same Proportion to each other as the Parts of some other given Line, whether those Parts arc equal or unequal.

Let a b be the given line which it is required to divide in the same manner and proportion as the line c d, whether the parts are equul or unequal. On the base line C d, form an equilateral triangle in the manner already described in a former problem. Then take the distance a b, and with e as a centre, describe the arc F G, and join the points F and G, and F G shall be equal to A B. Now, if from the points h i k, which are the divisions of the line c, we draw lines to e, as h e, I e, and k e, these lines will cut F G in the points a b c which will divide the line F G into parts proportionate to the divisions of the line c D.

Fig. 35.