( 4 *>7 ) 
the Complement of the contained Angle to a Semicircle, which 
I call the exteriour Angle : This is a new Theorem of good ufe 
in Trigonometry, and eafily proved from the nth and i %th of 
the II. Elem , Euclid . 
This premifed, putting m for the Diftance of rhe Sun and 
Earth, and n for that of the Sun and Venus , and a: for the 
Didance of the Earth and Venus , or the third Side of the 
Triangle which we feek ; by the third Lemma, 4 n x, will be 
to the excefs of the Square of n -\-x above the Square of m, as 
the Area of the whole Disk of Venus to the Area of the part 
illuminated ; and by the firft Lemma , the Area's of her whole 
Disk are at all times as the Squares of x reciprocally ; whence 
1 ~ . nn -\-m x -\- x x — mm 
the Quantity will in all Cafes be 
4 nx l . 
proportional to the Area of the illuminated part. 
Now that this Ihould be a Maximum, it is required that the 
Fluxion thereof be equal to o, or that the N egative parts the re- 
of be equal to the Affirma tive, that is, that 1 n x -j- 1 x x x 
4 n x 3 — ii nx z x x 'nn - inx -f- * x — m m ; and 
dividing all by 4 nx^x, the Equation becomes i n x -\~i x x 
=:^nn-\-6 nx-\-^xx— 3 mm. Confequen tly 3 n n 
4 n x -}- x x ='3 mm, and therefore x ■=. vf m m -j~ » n 
— 1 n. 
From hence a ready and not inele- 
gant Geometrical Condruilion (if 1 
may be allowed to fay fo) becomes 
obvious; for with the Center S and 
Radius S T m, defcribe the Se- 
micircle T D A; and with the fame 
Center and Radius S E== n, the Semi- 
circle E y B ; which two Semicircles 
fhall reprefenc the Orbs of the Earth 
and V enus. Make che chord A D e- 
qual to the Radius ST, and from D 
towards^, lay off D F — SE; draw 
T F, and thereon place FG — BE — 
2 n } and with the Center T and Radius 
