350 PHILOSOPHICAL TRANSACTIONS. [aNNO I7I6. 



An Account of the Cause of the late remarkable Appearance of the Planet 

 Fenusy seen this Summer, 1 7 1 6, for many Days together, in the Day-time. 

 By Edmund Halley, R. S. Seer. N° 349, P- -166. 



It may justly be reckoned one of the principal uses of the mathematical 

 sciences, that they are in many cases able to prevent the superstition of the 

 unskilful vulgar; and by showing the genuine causes of rare appearances, to 

 deliver them from the vain apprehensions they are apt to entertain of what they 

 call prodigies; which sometimes, by the artifices of designing men, have been 

 employed to very bad purposes. 



Of this kind was the late appearance of Venus in the day-time, generally 

 taken notice of about London and elsewhere; and by some reckoned to be 

 prodigious. This put me on the inquiry, how it came to pass that at that time 

 the planet should be so plainly seen by day, whereas she rarely shows herself so, 

 unless to those who know exactly where to look for her. To resolve this, the 

 following problem arose, viz. to find the situation of the planet in respect of 

 the earth, when the area of the illuminated part of her disk is a maximum. 



To investigate this maximum, I found it requisite to assume the following 

 lemmata. 1 . That the visible areas of the disk of the same planet, at differing 

 distances, are always reciprocally as the squares of those distances, which is 

 evident from the first principles of optics. 2. That the area of the whole disk 

 of the planet is to the area of its illuminated part, as the diameter of a circle to 

 the versed sine of the exterior angle at the planet, in the triangle at whose 

 angles are the sun, earth, and planet. 3. That in all plane triangles, 4 times 

 the rectangle of the sides containing any angle, is to the excess of the square 

 of the sum of the sides above the square of the base, as the diameter is to the 

 versed sine of the complement of the contained angle to a semicircle, which I 

 call the exterior angle; this is a new theorem, of good use in trigonometry, 

 and easily proved from the 12th and 13th of the 2d Elem. Euclid, 



This premised, putting m for the distance of the sun and earth, and n for 

 that of the sun and Venus, and x for the distance of the earth and Venus, or 

 the third side of the triangle which we seek; by the third lemma, Anx will be 

 to the excess of the square of n -f- a: above the square of m, as the area of the 

 whole disk of Venus, to the area of the part illuminated; and by the first lemma, 

 the areas of her whole disk are at all times as the squares of x reciprocally; 



with the ecliptic j which affected the accuracy of his conclusions. He was mistaken also in thinking 

 the external contact might be observed to a second of timej astronomers had disagreed 20 seconds in 

 observing the internal contacts of Mercury, which is a similar phenomenon. 



Greenwich, May 26, 1803. N^ M.** 



