MO,

be projected into the Curve DFCG, in which draw FAG.

Then is the Angle B. FA,

a right one; and fince DA is perpendicular to B A,

it will be perpendicular to FA

another Line in the fame Plane FB A. Draw the Lines 1 M, NP,

parallel to the Diameters F G, BE;

Then is the Angle B FA

NIP,

and both right ones; also because NP, and P I, are parallel to BA, and FA;

the Angle BAF = NPI.

Therefore the Triangles B E A, NIP,
are Similar; and FA: IP::BA: NP.
Therefore it will be FAq, IPq:: BAq: NPq.
But it is BAq: NPq:: CAX AD: CPX PD,
(By 35 Prop. 3 Lib. Euclid.) Then FAq: IPq::
CAXAD:CPX PD.

But

!

But this is a Property of the Circle and Ellipfis ; therefore the Figure CFDG muft be one of them. But because the Circle is oblique to the Plane of the Projection, its Diameter B E is projected into a Line FG lefs than it felf, by Theorem 5. therefore fince FG is lefs than BE DC, it follows the Curve FDGC is an Ellipfis, whofe longer Diameter is DC, and the fhorter FG. 2. E. D.

THEOREM X.

An Ellipfis ftanding at Right-Angles with the Plane of the Projection, is projected into a Right Line; if it be parallel to the Plane of the Projection, 'tis projected into an Ellipfis equal to it felf; if it be oblique to the faid Plane, 'twill be projected into an Ellipfis less than it self; except, laftly, when the longer Diameter is so far short'ned by the Projection, that it be equal to the fhorter Diameter, and then 'tis projected into a Circle.

Self-evidence is the Demonstration of this Theorem in every of its Parts; as is eafy to conceive from, what has been hitherto demonftrated of Lines and Circles.

THEOREM XI.

The Quantity of any Right Line is to its Quantity when projected, as Radius to the Co-fine of the Inclination of that Right Line to the Plane of the Projection.

VOL. II...

D

Demon

Demonftration.

Let the given Right Line be DC,

its Projection (by Theorem 4.) will be FC;

D

E

F

C.

The Angle of Inclina- A tion is ACD,

whofe Co-fine is DE FC.

Make the given Line Radius DC:
Then, as the given Right Line DC
Is to its Projection FC DE

:: Radius: to the Co-fine of AC D. Q. E. D. Corollary.

Hence the Radius DC, the Arch DG, and its right Sine D E, are all evidently projected into the right Line FC

Sine.

THEOREM XII.

The Area of any Circle is to the Area of its Elliptic Projection, as the Radius to the Co-fine of its Inclination to the Plane of the Projection.

By fimiliar Triangles C D F, C df,
it will be CD: CF :: Cd(= qo): Cf.
And, fo are all the Lines go, go, go, go, &c.
to all the Lines Cf, Cf, Cf, &c.
But all the Lines qo,

conftitute the Quadrant C GB;
And all the Lines C f,

conftitute the Area of its Projection CFH: Wherefore, as the Radius CD,

: to the Co-fine of the Inclination CF; :: fo is the Area of the Quadrant C G B, : to the Area of its Projection C F H.

And fo is the Area of the whole Circle to its Projection. 2, E. D.

CHA P. III.

Problems of the Orthographical Projection of the Sphere, or Analemma.

B

PROBLEM 1.

Y the Sector to defcribe an Ellipfis, having the Tranfverfe and Conjugate Diameters gi

ven.

VOL. II.

D 2

Practice.

· Sector from go to

90, in the Line of Sines;

Then lay off the Parallel Sines of 10, 20, 30, &c.

on each Semidiameter from E to A, and E to C.

Then on each Divifion, raife the Perpendiculars 10a, 20a, 30a, &c.

And thro' the Points of Interfection a, a, a, a, &c.

Draw the Curvelineal Arch Aa, a, a, &c. C...

After the fame manner draw the other three Arches CB, BD, DA;

fo fhall the Whole Ellipfis ACBD

be compleat as required. 2 E. F.

PROBLEM II.

To divide a right Circle in the Plane of the Pro

jection into its proper Degrees.

Practice