# Proposition 29

## Proposition 29 Problem

a trajectory given in kind, that in given by position, into parts given describe lines may 21

be cut by order, kind, four right and proportion. that may be Suppose a trajectory is to be described similar to the curve line FGHI, and whose parts, similar and proportional to the parts FG, GH, HI of be intercepted between the right lines the other, AB may AD, and BD, and AD, BD and CE given by po pair of those lines, sition, viz., the second between the second, and the third between the the third. first between the Draw the XXVII) and (by Lem. first right lines FG, GH, HI, FI; describe a trapezium fghi that be similar to the trapezium FGHI, and whose an touch the right lines given by posi gles/, g, h, i, may And then about tion AB, AD, BD, CE, severally according to their order. will be similar to the that a describe trajectory bins trajectory, may trapezium curve line FGHI. SCHOLIUM. This problem may be likewise constructed in the following manner. to Y, and join FH, IG, and make Joining FG, GH, HI, FI, produce GFOF NATURAL PHILOSOPHY. SEC. VI 153 DAL the angles CAK. equal to Let the angles PGH, VFH. meet the right line AK, AL BD K in draw and L, and thence LN, of which let KM, KM make the angle A KM equal CHI, and be to the angle to AK HI as is to GH ; itself and let LN make the angle ALN equal to the angle FHI, and be itself But AK, KM. AL, LN are to be drawn to AL as HI to FH. El AD, AK, AL, that the letters may be carried round in the same towards those sides of the lines OA.KMC, ALKA, DALND order as the letters line CE and let PE FGHIF Make L in MN and draw ; meeting the right v iEP equal to the angle IGF, GI and through P draw PQ/ the angle be to Ei as FG to that ; ADE may PQE contain an angle with the right line equal to the angle FIG, and may meet the right line AB in /, and join fi. But PE and PQ arc- to be drawn towards those sides of the lines CE, PE, that the circular PEtP and PEQP be the same as of the letters same order of letters, and similar to the trapezium FGHI, a trapezium /^//.i is constructed, and a trajectory given in kind is circumscribed about it, the Problem will be solved. order of the letters FGHIF and ; if upon the line/i, may in the So far concerning the finding of the mine the motions of bodies in the orbits orbits. Section How the motions are to be It remains that we deter so found.