(8) | [Jan 
MATHEMATICAL CORRESPONDENCE. 

Question XX (No. IX). —Arfwered by Mr. F. F——r. 
ET AB be the arc of a great circle, DCE a fpherical right angle E 
touching it in C, and AD and BE perpendiculars to AB. I fay, D 
the rectangle of the tangents of the perpendiculars AD and BE will 
be equal to the reftangle of the fines of the fegments AC and BC. 
A C B 
For, s. AC : radius ::#. AD: +. ACD; 
And, s) BC > xadis - 27: BES 7 BES 
Therefore, s. AC xX s. BC: radius?.::4.AD x #. BE: t. ACD x #. BCE. 
But, t. BCE = cot. ACD, becaufe ACD +-BCE = go°. 
Therefore, s. AC Xs. BC: radius2. :: 4. AD x t. BE: «. ACD x cot. ACD. 
But, t. ACD x cot. ACD =radius?. 
Therefore, s. AC x s. BC =+. AD x ¢. BE. Q. E. BD. 
Scholium.—The application of this theorem not unfrequently cecurs in the folution of aftrono- 
nical problems. Let us, for inftance, fuppofe, that DC and CE reprefent ares of the prime 
vertical and horizon. AB of the equinodtial, and AD and BE of meridians, or circles of might 
> afcenfion. Let AD, BE, and AB be given, and the fegments AC and BC required. We have 
nothing more to do in this cafe, than to add together the logarithmic tangents of AD and BE, 
and to find from the tables two arcs, whofe quantities are together equal to AB, and the fum of 
whofe logarithmic fines is equal to the before-mentioned fum of the logarithmic tangents of AD 
and BE; which a trial or two will give with great eafe, and we have the arcs AC and BC 
required. ake 
- This Quefiion was alfo anfwered by Mr, Fokn Hayco:k. 

Question KXI (No. IX).—Anfvered oy Mr. H. Cox. 
The fquare numbers 16, 25, 36, 49, 64, and 87, being the only fquare whole numbers ex= - 
preflible by two digits, it is plain, by infpe4tion, that the firft condition of the problem can apply 
only to the laft of them, namely 81, the root of which (9) is equal to $1, the fum of its digits. 
To the fame number the fecond condition will equally apply. For, 9X7, or 63, the produc 
of the fum and difference of its digits, being fubtraéted from 81, will leave 18, the fame digits 
mverted, - : : 
So that 81 is the number, and 1$ the age of the propofer. 

The fame anfwered by Mr. Fof. Yeung, of Norwich. : 
Let x and y reprefent the two digits. Then will rex-Ly exprefs the number fought, and 

zoy-[-x the fame digits inverted By the queftion, “ 
4/ 10x-L. =x-+y, and 
: 10x-py—«-Ly xX s—y=Soy-Lx, or 
tox-Ly—x?--y*=1o0y-+-x. 
Hence, x? — yx — 97 5 
xy? 
Bye fie ae 
Therefore, sy ae =93 
ERE i ae 
Confequently, 4/ 10x y=9 
And 1ox-++-y==81, the number fought. 
This Quefiion was alfo anfwered by Mr. F. Bonner, Mr. Fihn Haycock, and by Mr. William Saint, 
of Norwich. 

Question XXII (No. X).—Anfwered by Mr. 0. G. Gregory. 
In fUlving this Queftion, it muf firft be premifed, that the planetary ce 
orbits, though not dtriétly circular, may, in many cafes, as well as the S fe 
prefen', be fuppofed fuch, in order to facilitate the calculations. This 
being admitted, let BDFH, in the adjoining diagram, reprefent the :; 
orbit of the Earth, ACEG that of Venus, and LIK an arch of the ° 
apparent heavens. Then, fuppofing the Earth to be at F, the Sun at 
S will appear as though in the heavens at I, and Venus’s greateft elon- 
gat.on is, when that planet appears as though at L or K, viz. when fhe 
is in thefe parts of her orbit where lines, drawn from the Earth to her, 
a FG or FC, form tangents to her orbit at C or G, 

