21, (Pl. 12, Fig. 1), let thofe legs BP, CP, of the moveable angles PBN, PCN, by the concourfe of which the trajectory was described, be made parallel one to the other; and retaining that pofition, let them revolve about their poles B, C, in that figure. In the mean while let the other legs CN, BN, of thofe angles, by their concourfe K or k, defcribe the circle BKGC. Let O be the centre of this circle; and from this centre upon the ruler MN, wherein thofe legs CN, BN did concur while the trajectory was defcribed, let fall the perpen<dicular OH meeting the circle in K and L. And when thofe other legs CK, BK meet in the point K that is nearest to the ruler, the first legs CP, BP will be parallel to the greater axis, -and perpendicular on the leffer; and the contrary will happen if thofe legs meet in the remoteft point L. Whence if the centre of the trajectory is given, the axes will be given; and those being given, the foci will be readily found.

But the fquares of the axes are one to the other as KH to LH, and thence it is easy to defcribe a trajectory: given in kind through four given points. For if two of the given points are made the poles C, B, the third will give the moveable an- gles PCK, PBK; but those being given, the circle BGKC may be defcribed. Then, becaufe the trajectory is given in kind, the ratio of OH to OK, and therefore OH itself, will be given. About the centre O, with the interval OH, defcribe another circle, and the right line that touches this circle, and ~paffes through the concourse of the legs CK, BK, when the firft legs CP, BP meet in the fourth given point, will be the ruler MN, by means of which the trajectory may be defcribed. Whence alfo on the other hand a trapezium given in kind (excepting a few cafes that are impoffible) may be infcribed in a given conic fection.

There are alfo other lemmas, by the help of which trajectories given in kind may be defcribed through given points, and touching given lines. Of fuch a fort is this, that if a right -line is drawn through any point given by pofition, that may cut a given conic fection in two points, and the distance of the interfections is bifected, the point of bifection will touch another conic fection of the fame kind with the former, and

having its axes parallel to the axes of the former. But I'haften to things of greater use.

LEMMA XXVI.

To place the three angles of a triangle, given both in kind and magnitude, in respect of as many right lines given by pofition, provided they are not all parallel among themselves, in fuch manner that the feveral angles may touch the feveral lines. (Pl. 12, Fig. 2.)

*

Three indefinite right lines AB, AC, BC, are given by pofition, and it is required fo to place the triangle DEF that its angle D may touch the line AB, its angle E the line AC, and its angle F the line BC. Upon DE, DF, and EF, defcribe three fegments of circles DRE, DGF, EMF, capable of angles equal to the angles BAC, ABC, ACB respectively. But thofe fegments are to be defcribed towards fuch fides of the lines DE, DF, EF, that the letters DRED may turn round about in the fame order with the letters BACB; the letters DGFD in the fame order with the letters ABCA; and the letters EMFE in the fame order with the letters ACBA; then, completing those fegments into entire circles, let the two former circles cut one the other in G, and fuppofe P and Q to be their centres. Then joining GP, PQ, take Ga to AB as GP is to PQ; and about the centre G, with the interval Ga, describe a circle that may cut the first circle DGE in a. Join aD cutting the fecond circle DFG in b, as well as aE cutting the third circle EMF in c. Complete the figure ABCdef fimilar and equal to the figure abcDEF: I fay, the thing is done.

J

For drawing Fc meeting aD in n, and joining aG, bG, QG, QD, PD, by conftruction the angle EaD is equal to the angle CAB, and the angle acF equal to the angle ACB; and therefore the triangle anc equiangular to the triangle ABC. Wherefore the angle anc or FnD is equal to the angle ABC, and confequently to the angle FbD; and therefore the point n falls on the point b. Moreover the angle GPQ, which is half the angle GPD at the centre, is equal to the angle GaD at the circumference; and the angle GQP, which is half the angle GQD at the centre, is equal to the comple

ment to two right angles of the angle GbD at the circumference, and therefore equal to the angle Gab. Upon which account the triangles GPQ, Gab, are fimilar, and Ga is to ab as GP to PQ; that is (by construction), as Ga to AB. Wherefore ab and AB are equal; and confequently the triangles abc, ABC, which we have now proved to be fimilar, are alfo equal. And therefore fince the angles D, E, F, of the triangle DEF do refpectively touch the fides ab, ac, bc of the triangle abc, the figure ABCdef may be completed fimilar and equal to the figure abcDEF, and by completing it the problem will be fölved. Q.E.F.

COR. Hence a right line may be drawn whofe parts given in length may be intercepted between three right lines given by pofition. Suppofe the triangle DEF, by the accefs of its point D to the fide EF, and by having the fides DE, DF placed in directum to be changed into a right line whose given part DE is to be interpofed between the right lines AB, AC given by pofition; and its given part DF is to be interposed between the right lines AB, BC, given by pofition; then, by applying the preceding construction to this case, the problem will be folved.,

PROPOSITION XXVIII. PROBLEM xx. To defcribe a trajectory given both in kind and magnitude, given parts of which shall be interpofed between three right lines given by pofition. (Pl. 12, Fig 3.)

Suppose a trajectory is to be described that may be fimilar and equal to the curve line DEF, and may be cut by three right lines AB, AC, BC, given by pofition, into parts DE and EF, fimilar and equal to the given parts of this curve line.

Draw the right lines DE, EF, DF; and place the angles D, E, F, of this triangle DEF, fo as to touch thofe right lines given by pofition (by lem. 26). Then about the triangle defcribe the trajectory, fimilar and equal to the curve DEF. Q.E.F.

LEMMA XXVII.

To defcribe a trapezium given in kind, the angles whereof may be fo placed, in respect of four right lines given by position,