VOL. XXIX,] PHILOSOPHICAL TRANSACTIONS. 251 



whence the quantity 2!L±i!!l±i^ZL^ will in all cases be proportional to the 



area of the illuminated part. 



Now that this should be a maximum, it is required that the fluxion thereof 

 be equal to O, or that the negative parts t hereof be equal to the affirmative, 

 that is, that 2ni + 2a7i X 4nx^ = \2nx\v X nn -\- 2nx -{- jcx — mm ; and divid- 

 ing all by 4nx\v, the equation becomes Inx + 2xx = 3Jin + Qt zj? + 3xx — 3mm. 

 Consequently 3nn -f- 4nx -\- xx = 3mm, and therefore x = ^3mm -\- nn — 2n. 



From hence a ready and not inelegant geometrical construction (if I may be 

 allowed to say so) becomes obvious : for with the centre s, and radius st = m, 

 describe the semicircle tda, fig. 4, pi. 5; and with the same centre and radius 

 SE = n, the semicircle evb ; which two semicircles will represent the orbs of 

 the earth and Venus. Make the chord ad equal to the radius st, and from d 

 towards a, lay off dp = se; draw tp, on which place pg = re = 2n, and with 

 the centre t and radius tg describe the arch gv, cutting the semicircle eve in 

 v; and draw the lines sv, tv; I say the triangle stv is similar to that at whose 

 angles are the sun, earth, and Venus, at the time when the area of the 

 enlightened part of that planet's disk, as seen from the earth, is greatest. 

 How this geometrical effect follows from the equation is too evident to need 

 repetition. 



In consequence of this solution, I find this maximum always to happen when 

 the planet is about 40° distant from the sun; and the times of it about the mid- 

 dle between her greatest elongations on both sides from him, and her retrograde 

 conjunctions with him; when little more than a quarter of her visible disk is 

 luminous, and resembling the moon of about 5 days old ; and though her dia- 

 meter is at that time only 50 seconds, yet she shines with so strong a beam, as 

 to surpass the united light of all the fixed stars that appear with her, and casts 

 a very strong shade on the horizontal plane they all shine on; an irrefragable 

 argument to prove that the disks of the fixed stars are inconceivably small, and 

 next to nothing, since shining with a native light, so many of them do not 

 equal the reflex light of one quarter of a disk of less than a minute diameter. 



In this situation Venus was found in July last, on the 10th day, about which 

 time, when the sun grew low, she was very plainly seen in the day-time, for 

 many days together, as she might have been in the mornings, about the latter 

 end of September. But this, arising from the causes we have now shown, is 

 nothing uncommon ; for every 8th year it returns again, so that the planet may 

 be seen on the same day of the month and hour, very nearly in the same place, 

 as all acquainted with the heavenly motions must know. 



Lastly, it may not be amiss to note that the equation x = V 3mm -\- nn — In 



K K 2 



