Let AB be the Axis of an Ellipfis, equal to

4P

✓ bit bq2
let G be the Centre of the Ellipfe,
and the Axis to the Parameter as p to b. At G,
raife a Perpendicular to the Axis, and on it take
bg, and on the other Side in the Perpen-

GD=

bq

2p

pab. Let DE

dicular continued take GKqx Pb.

P

and KC be parallel to the Axis: take KC=b-p, and from C as a Center,with the Radius DCq+br defcribe a Circle meeting the Ellipfe in P, and the Ordinate PM, on the Line DE, fhall be one of the Roots of the propofed Equation.

Let PM (=y) produced meet AB in R, and KC in N; and calling DM=x, then CPq= NPq+NCq, that is, q+pb+ip2+br = b — ÷ p—xl2 + y + 9; and therefore

-b

x—br=o, the Equation to

the Circle, which was to be constructed.
And fince PRq: GBq-GRq::b:p, there

bq2

: br+ -x2::b:p; and con

4P

fore, y+

fequently,

(

the Equation that was to be conftructed.

Now that their Interfections will give the Roots required, appears thus..

For in the first Equation fubftitute the Va

lue you deduce for it from the fecond, viz.

that is, "— =x, and x2= ; which fubftituted

for

4

62

and x in the firft Equation, gives,

"/ 2 + y2+p—bx ~ ~ +gy—bɩ=0; that is,

y^*+bpy2+b2 qy—b3r=o.

And if you fubftitute them in the 2d Equation,

b

there will arifey+y2+ bq

that is, y+*+bpy2+b2qy—b3r=o, the very fame as before; and thus it appears that the Roots of the Equation y**-bpy2+b2qy— bro are the Ordinates that are common to the Circle and Ellipfe, or that are drawn from their Interfection.

End of the Third Part.

APPEN

DE

Linearum Geometricarum Proprietatibus generalibus

TRACTATUS.

AUCTORE

COLINO MAC-LAURIN.

LONDINI:

M.DCC.XLVIII.

1

DE

LINEARUM GEOMETRICARUM

Proprietatibus generalibus.

D

E Lineis fecundi Ordinis, five fectionibus conicis, fcripferunt uberrime Geometræ veteres

& recentiores; de figuris quæ ad fuperiores Linearum Ordines referuntur pauca & exilia tantum ante NEWTONUM tradiderunt. Vir illuftriffimus, in Tra&tatu de Enumeratione Linearum tertii Ordinis, doctrinam hanc, cum diu jacuiffet, excitavit, dignamque effe in qua elaborarent Geometris oftendit. Expofitis enim harum Linearum proprietatibus generalibus, quæ vulgatis fectionum conicarum affectionibus funt adeo affines ut velut ad eandem normam compofitæ videantur, alios fuo exemplo impulit ut Analogiam hanc five fimilitudinem quæ tam diverfis intercedit figurarum generibus bene cognitam & fatis firme animo conceptam atque comprehenfam habere ftuderent. In qua illuftranda & ulterius indaganda curam operamque merito pofuerunt; cum nihil fit omnium quæ in difciplinis purè mathematicis tractantur quod pulchrius dicatur, aut ad Animum Veri inveftigandi cupidum oblectandum aptius, quam rerum tam diverfarum confenfus five harmonia, ipfiufque do&trinæ compofitio & nexus admirabilis, quo pofterius priori convenit, quod fequitur fuperiori refpondet,