the Figure projected, as the Co-fine of the Inclination Lecture of the Planes is to the Radius.

FOR any Figure can always be refolved into Parallelograms, or Triangles, whofe Bafes are parallel to the common Section of the Planes; and therefore, they will be parallel to the Plane on which they are projected: Wherefore, the Bafes of these Parallelograms, or Triangles, and their Projections on the Plane, are always equal to each other, and parallel, as we have thewed in Lecture XIII. But the Perpendiculars let fall from the Summits, or Tops of the Triangles, and Parallelograms upon the Bafes, are also perpendicular to the common Section of the Planes, by the 29th of the firft El. And therefore, the Inclination of the Perpendiculars to the Plane is equal to the Inclination of the Planes to each other. And confequently, the Projections of thefe Perpendiculars will be to the Perpendiculars themselves, as the Co-fine of the Inclination of the Plane is to the Radius. Wherefore, every Parallelogram, or Triangle, is projected into another, whofe Bafe is equal to the Bafe of the Triangle, or Parallelogram projected; and its Height is to the Height of the Figure projected, as the Co-fine of the Inclination of the Planes is to the Radius. But Triangles, and Parallelograms, whofe Bafes are equal, are as the Perpendiculars let fall from the Tops upon the Bafes. The Projection therefore, of each Triangle, is to the Triangle projected, in a conftant and given Proportion; confequently, all the Projections of all the Triangles, or Parallelograms, are to the Figures projected in the fame Proportion; that is, the Projection is to the Figure projected, as the Co-fine of the Inclination is to the Radius.

If the Orbit of the Earth be orthographically projected on the Plane of the Æquator, by letting fall from each of its Points Perpendiculars, the Projection will be an Ellipfe, in whofe Perimeter the Extremity of a right Line, let fall from the Earth perpendicular to the Plane of the Equator, will conftantly

Y 2

XXV.

Lecture conftantly move: And this Point, by its Motion, XXV. will mark out the right Afcenfion of the Earth, or

Fig. 5.

its Motion, according to the Equator, as it is to be

feen from the Sun; to which the right Afcenfion of the Sun, feen from the Earth, is always equal. Pla. XXIV. Let VAC be the Ellipfe in which the Orbit of the Earth is projected, S the Point of Projection of the Sun's Center, S the common Interfection of the Equator and the Ecliptick, A any Point, where a Perpendicular from the Earth meets with the Projection: The Angle YSA will measure the right Afcenfion of the Sun. Now, I fay, that this Point A, which marks out the Motion of right Afcenfion, will fo proceed in the Ellipfe T AC, that it will defcribe about the Point S, elliptick Area's proportional to the Times. For, in a given Time, let A move through the elliptick Arch AB; draw the Lines AS, and BS; and the trilineal Figure ASB will be the Projection of the correfpondent Area, which the Earth defcribes in the Plane of the Ecliptick, in the fame Time, round the Sun: And therefore, the Projection ASB, will be to the correfpondent Area in the Earth's elliptick Orbit, as the Co-fine of the Inclination of the Æquator, and the Ecliptick, is to the Radius. But in the fame Proportion is the whole elliptick Area Y AC, to the whole Area of the Earth's Orbit: Therefore, by Permutation of Proportion, the trilineal Figure ASB will be to the whole elliptick Area AC, as the Area, described in the Earth's Orbit round the Sun, is to the whole Area of the Earth's Orbit ; that is, as the Time in which that Area in the Orbit of the Earth, or the Area ASB in the Projection, as defcribed, is to the whole periodical Time. Therefore, the Point A moves in the Perimeter of the Ellipfe at fuch a Rate, that it describes, about S, Areas that are continually proportional to the Times.

THE fame Things being laid down; at the Center S, and Distance SA, which is a mean Proportional between half the greater, and half the leffer Axis of the Eilipfe, defcribe a Circle; this Circle will be equal

XXV.

equal to the whole elliptick Area; as it is easy to de- Lecture monftrate from the Doctrine of the Conick Sections: This Circle will cut the Ellipfis in four Points E, F,G,H: The Points of Interfection fhew the Pla. XXIV. Fig. 6. right Afcenfions of the Sun, where the Equations are greateft. Imagine a Point M, to move uniformly in the Periphery of the Circle; its Motion will then reprefent the Motion of our imaginary Star m, and will describe about the Point S, circular Sectors that are proportional to the Time: And because the Area of the whole Circle, and the Area of the Ellipfe, are equal, the Area's of the elliptick Sectors, and of the circular Sectors, defcribed in the fame Time, will be conftantly equal. Let us now fuppose, that the Point M, in the Periphery of the Circle, and the Point that marks out the Sun's right Afcenfion in the Ellipfe, be placed at the fame Time both in the right Line SLM. Let these Points afterwards be in m and A, then the elliptick Area LSA will be equal to the circular Sector MSm: And because the Arch Mm is without the Ellipfe, the Angle MSm will be less than the Angle MSA, and the Difference of the Angles measured by the Arch m A, which is the Equation of Time. When the Point, which marks out the right Afcenfion of the Sun, comes to the Interfection F of the Circle and Ellipfe; there its angular Motion round the Sun will be equal to the angular Motion of the Point m; for the Area's m Sn, and ASF, are equal, they being both defcribed in the fame Moment of Time, and at the fame Distance from S; and confequently, the Arch q F, is equal to the Arch mn. In the Point therefore, F, the Motion of right Afcenfion, is equal to the Motion of the imaginary Star, or equal to the mean Motion. The fame Thing may be fhewed at G, H, and E: But it was fhewed before, that where the Motion of right Afcenfion was equal to the Motion of the Point m, that there the Equations are greatest. Wherefore, in the Points F, G, H, and E, are the Equations greatest.

Lecture \. IF you inquire in what Points the Days are XXV. longest or fhorteft, the fame eminent Dr. Halley halfo given us a geometrical Solution of this Which are Problem Let be the Ellipfe into which the Points of he Earth's Orbit is projected, and S the Point of the Ecliptick where the he Sun's Center, K the Center of the Ellipfe. ProDays are duce KS on each Hand, fo that KG and SH, longeft, and may be to KS (which is the Projection of the Ecwhere Shai teft. centricity) as the Square of the Radius is to the Plat.XXIV. Square of the Sine of the Obliquity of the Ecliptick: Fig. 7. Thro K draw parallel to the common Section of the Aquator and Ecliptick, and cut it at right Angles with the Line Kw: Thro' G draw G F, and thro H, draw FH, parallel to the Lines v andy, and thro' S and K defcribe the Hyperbola AB, whofe Afymtots are FG, FH. This Hyperbola, and its oppofite CD, will cut the Ellipfe in the Points that are required; that is, when the Sun is in the Points of the Ecliptick which correfpond with B and D, then the Days are the longeft; and in B the Days are longer than in D: But the Points of the Ecliptick, which answer to A and C, give us the Places where the Days are fhorteft. The Demonftration of this depends on the Motion of the Point that marks the right Afcenfion of the Sun round S; for it defcribes about it Area's that are proportional to the Times. Therefore, the angular Velocity is every where reciprocally as the Square of the Distance from S; confequently, the Velocities must be greateft where these Distances are leaft; that is, where the leaft Lines that can be drawn from S fall upon the Ellipfe; and the Velocities are the leaft, where the Lines drawn from S to the Ellipfe are the greatest: But by the Construction, and the 62d Prop. Lib. V. of Apollonius's Conicks, it is evident, that the Hyperbola's will cut the Ellipfis in the Points A and C; where the right Lines SA and SC are the greateft; and in the Points B and D, where the right Lines SB and SD are the leaft: For in the e Points the Lines SA, SB, SC, SD, are perpendicular to the Curve. Hence the Mo

tion of the Sun, according to his right Afcenfion, Lecture will be quickest in B and D ; and therefore the XXVI. Days will be then the longeft; and the Motion being floweft in C and A, the Days will be there the fhorteft.

LECTURE XXVI.

Of the Theories of the other Planets.

A

FTER having explained the Theory of the Earth's annual Motion, and fhewed the Methods by which the Form of its Orbit, and the Pofition of the Apfides are determined, wẹ may then, by the Help of Aftronomical Tables, compute for any Time, the Place of the Earth in the Ecliptick feen from the Sun, and its oppofite Point in which the Sun appears to be, as he is obferved by us. We will now come to explain the Theories of the other Planets; the Knowledge of which cannot be attained without the Earth's Motion being perfectly known.

BUT the Periods of the Planets, or the Times they take to complete their Circulations, are to be The beliofound out in the first Place. For which Purpose centrick and we must observe, that when any fuperior Planer geocentrick Place of a comes to an Oppofition with the Sun, they then Panet that appear in the fame Point of the Ecliptick, feen is in OppoY 4 from ition to the Sun coincide