1795+} 
Having found the depreffion of the vi- 
fible horizon, we have the height of the 
mountain from one analogy, viz. Co-fine 
angle depreflion : radius ::,earth’s femi- 
diameter : earth’s femidiameter + se 
height of the mountain required. 
We may alfo, to avoid the neceflity of 
ufing fuch large numbers as will occur in 
the preceding analogy, take the follow- 
ing for a near approximation : Reduce 
an arc of the earth’s circumference, 
whofe quantity is equal to that of the 
angle of depreifion before found (and 
which will confequently be the femidia- 
meter of the vifible horizon from the 
mountain) into yards or feet, and we 
fhall have radius : tangent angle depref- 
fion :: the diftance fo found : twice the 
height required, very nearly. 
In the cafe before us, the difference of 
the times of fu rifing given, is fuppofed 
to be the difference of the times of the 
true rifing of his centre above the rational 
horizon of the place, and the vifible ho- 
rizon from the fummit of the mountain, 
both properly ‘corrected for refraction 
and parallax. 

The fame anfwered hy F. H. 
Having the latitude and declination 
given, perfpherical trigonometry, as ra- 
dius is to the co-tangent of the comple- 
ment of latitude, fo is the tangent of de- 
clination to a third number, which is the 
time of fun rifing before fix o’clock (if 
latitude and declination are both north). 
To this add the given difference of fun 
rifing on the top of the hill; and that 
will be the included angle of a fpherical 
triangle, the two fides of which are giv- 
en, viz. the fun’s polar diftance and the 
co-latitude, whence the third fide, or ze- 
nith diftance, will be found, and confe- 
quently the fun’s depreifion from the 
true horizon, or the diftance from the 
bottom of the hill on the arc of a great 
circle, where a tangent drawn from the 
top of the hill to the fun, will touch the 
furface of the earth. If-that point of 
contact, the top of the hill, and the centre 
of the earth, be joined by three lines, a 
right-angled triangle will be formed, in 
which are all the angles, and the earth’s 
femidiameter given; whence, as the co- 
fine of the angle of the fun’s depreifion 
(above found) is tothe earth’s femidia- 

— Mathematical Queftions...To Correfpondents. 
88r 
meter, fo is radius to the hypothenufe ; 
from which fubtraét the femidiameter, 
and that gives the height of the hill re- 
quired. 
2bis Quefion was allo anfwered by Mr. 
Foon Dawes, and Mr. Fobu Haycock. 

QUESTION XIX (No. VI).—Anfwvered 
by Plitalethes. 
Of feven numbers in continued geo- 
metrical progreffions, having given the 
fum of the two leaft = 90, and the fum 
of the two greateft = 2812503, make 
4 =the firft number, and x =the com- 
mon ratio; then will v, rz, rz, 723, rx4, 
vz5, #28, reprefent the feven numbers. 
Therefore, by the queftion +twre=g0, 
and #25--1%°==281250; dividing the lat- 
ter of thefe by the former, gives z3= 
281250 — 90 = 3125; therefore x = 
/ 312 baie Confequently, ‘=90-+1-- 
%—==15; and, therefore, the numbers - 
fought are, 15, 75, 375: 1875, 9375, 
46375, 234375. 
. £——r, after his folution of this 
queftion, adds this remark, viz. If there 
be 7 number in geometrical progreffion, 
w being the firft term, and r the common 
ratio, the fum of the two firft being =a, 
and that of the two laft=4; then, 
Brn 6 
generally, it will be y=2—2,/-, and r== 
2 a 
Anfwers 10 this Queffion were alfo giver 
by Mejis. W. Adam, W. Clavey, Fobn Cole 
lins, A. Cox, L.W.D. F. H. Fobn Haycoch, 
Laycey, B.W., X. and Hermes of Bath. 

New MatueMaTiIcaAL QUESTIONS. 
QuEsTION XXIITI.—By Mr. B. W. 
Which is the greateft, an arithmetical 
or a geometrical mean, between any twe 
quantities, aand 0 ? 
QUESTION XKIV.—By ihe fame. 
If a pendulum, 39 inches long, fwing 
feconds, in what time will it fwing, when 
carried into a latitude where its weight 
is diminifhed by the 3ooth part of an 
inch. and its length increafed by the heat 
the roth part of an inch ? 
Erratum, p. 721. In the folution to 
Queftion XVI, |.5, for 4%, read 43. 
TO CORRESPONDENTS. 
The Notice to fome of our Poetical Priends 
to the pieces figned E. 8. F. : 
Our obligtig. correfoondent will obferve 
Work from Cambridge. 
ri, 
» in our laft Number, will particularly apply 
that notice has been taken of the Mufical 
Eis future correfpondence will be acceptable. 
Biographical Notices of remarkable and diftinguifhed Characters are folicited. 
Ro SSS 
