46S THE MATHEMATICAL PRINCIPLES [BooK III. After the same method, from five observed latitudes, we may find the latitude to a given time. If the differences of the observed longitudes are small, suppose of 4 or 5 degrees, three or four observations will be sufficient to find a new longitude and latitude : but if the differences are greater, as of 10 or 20 degrees, five observations ought to be used. LEMMA VII. Through a given point P to draw a right line BC, rvhose parts PB, PC, cut off by two right lines AB, AC, given in position, may be one to the other in. a given proportion. P i\ From the given point P suppose any right line PD to be drawn to either of the right lines given, as AB; and produce the same towards AC, the other given right line, as far as E; so as PE may be to PD in the given proportion. Let EC be parallel to A D. Draw CPB, and PC will be to PB as PE to PD. Q.E.F. LEMMA VIII. Let ABC be a parabola, having its focus in S. By the chord AC bi sected in I cut off the segment ABCI, ivhose diameter is Ip and vertex I'. In I/i produced take pO equal to one half of I//. Join OS, and produce it to £ so as S£ may be equal to 2SO. Now, supposing a comet to revolve in the arc CBA, draw £B, cutting AC in E ; I say, the point E will cut off from the chord AC the segment AE, nearly proportional to the time. For if we join EO, cutting the parabolic arc ABC in Y, and draw //X touching the same arc in the vertex //, and meeting EO in X, the curvi linear area AEXjuA will be to the curvilinear area ACY//A as AE to AC ; and. therefore, since the triangle ASE is to the triangle ASC in the same proportion, the whole area ASEXju A will be to the whole area ASC Y/^A as