XII.

Plate IX.
Fig. 6.

Lecture towards or from the Sun, will be different from that which he has in her Orbit; and its Inclinat on to the Ecliptick will be greater, than the Inclination of the Orbit to it. Let SA be a Portion of the Moon's Orbit produced to the Ecliptick And fuppofe the Sun and Moon in Conjunction in the Node: Now, then, while the Moon in her Orbit defcribes the Space L, the Sun by his apparent Motion will describe the Space & S in the Ecliptick, and SL will be the Way of the Moon from the Sun. Now if two Bodies be both moved the fame Way, but one faster than the other, their relative Motion, whereby the one recedes from the other, is the fame as if the flowest Body ftood ftill, and the other moved on with the Difference of Velocities, as we have demonstrated in our Physical Lectures. Thro' the Place of the Moon L, draw BL parallel to the Ecliptick, to which let B be perpendicular. Now while the Moon in her Orbit defcribes the Space SL, its Motion according to the Ecliptick is equal to the Space BL; take L equal to S, and draw, it will be parallel to SL; and the Motion of the Moon from the Sun will be the fame as if the Sun had remained in the Node, and the Moon, according to the Ecliptick, had been carried with the Velocity B, which is the Difference of the Velocities of the Sun and Moon, according to the Ecliptick. Because the Angles BLS and BIS are but fmall, the Angle BL will be to the Angle BI, as Bl is to BL; that is, as the Difference of the Motions of the Sun and Moon, according to the Ecliptick, is to the Motion of the Moon, in the Ecliptick, fo will the Angle which the Orbit of the Moon makes with the Ecliptick, be to the Angle B/S, which is equal to the Angle LSE, or the Inclination of the Way of the Moon from the Sun to the Ecliptick: And by this means we can find out the Angle which a Circle of Latitude, drawn through any Point of the Eclip

tick, makes with the Way of the Moon from the Lecture Sun. For in the rectangular fpherical Triangle, XIII. which the Ecliptick, the Circle of Latitude, and the Way of the Moon from the Sun do form, we have one Angle, which is the Inclination of the Way of the Moon from the Sun to the Ecliptick, and its Bafe, which is the Distance of the CircleLatitude from the Node; and therefore we can find the other acute Angle.

Of the Projection of the Moon's Shadow on the Disk of the Earth.

I

me

Fa Right Line be projected on a Plane that is parallel to it, by letting fall from all its Points Perpendiculars on the Plane, the Projection, or the Place where all Perpendiculars meet with the Plane, will be a Right Line, parallel and equal to the former Line which was projected. For the Perpendiculars that fall from the Extremities of the Right Line, are parallel and equal; and therefore the Lines which join them will be parallel and equal.

HENCE, if two Right Lines touching one another, be parallel to any Plane, the Projections of these two Lines upon that Plane, will contain an Angle, equal to the Angle the Lines themfelves make together; this is plain by Prop. 10. Book XI. of Euclid. Hence all Plane Figures projected on a Plane parallel to themselves have

for

Lecture for their Projections, Figures exactly fimilar and equal to themselves.

XIII.

Plate X.
Fig. 1.

BUT if a Line be inclined to any Plane, its Projection upon that Plane, made by letting fall from it Perpendiculars to the Plane, will be to the Line itfelf, as the Cofine of the Inclination of the Line is to the Radius. For let AB be a Line inclined to the Plane, and let DE reprefent the Plane: Letting fall from the Points A and B, the Perpendiculars A a, Bb; ab will be the Projection of the Line AB; to which if we draw through B the parallel Line BC, meeting with the Perpendicular Aa in C, this Line BC is equal to ab: But BC is to A B, as the Sine of the Angle CAB, or the Cofine of the Angle ABC to the Radius; that is, as the Cofine of the Angle of Inclination is to the Radius, so is ab to AB. Hence it follows, that every Figure, whofe Plane is perpendicular to the Plane of the Projection, is projected in a right Line. For the Perpendiculars from every Point of the Figure, will all fall upon the common Interfection of the Plane of the Figure, An Ortho-with the Plane of the Projection. Such a Prographical jection of Lines and Figures is called an OrthograProjection phical Projection.

zubat ?

If we imagine a Plane to pafs through the Center of the Earth, fo that the Line which joins the Centers of the Sun and Earth, may be perpendicular to this Plane, it will make on the Surface of the Earth a Circle, which will feparate the illuminated Hemisphere of the Earth from the dark. This Circle we before called the Circle bounding Light and Darkness, but The Disk of We will now call it the illuminated Disk; which the Earth. Disk is directly feen by a Spectator placed at the Distance of the Moon, in the Right Line

which joins the Centers of the Sun and Earth. Upon this Circle the Earth's Equator, its PaThe Ortho- rallels, Poles, and all the other Circles which graphical we imagined, are to be fuppofed, projected OrProjection on the Disk, thographically. For all Lines drawn from the

Center