$80 
former equation has only one root, which 
is unity, 3a muft be lefs than g, or 27. 
To folve equations in this form,  fhould 
be made equal to a—é, from which will 
43 
refult an equation ara, an equa- 
tion which admits of only one root ; and 
from the refolution of this equation, a 
may be found, and thence 4, which 1s 
equal to by 
3a 
Upon equations of this form, I have 
(Algebra, p. 213) inadvertently faid : 
“- When w is of fuch a magnitude that 
3ad is not equal to g, it is evident that 
whe rule faiis.’’ This is-true generally, 
and is the reafon why, in feveral cales, 
Cardan’s rule fails; but it does not ap- 
ply tothe cafe in queftion, a3--gx=r 5 
in which 3a)-may be always equal to 9. 
I go on: “* Thus. let #2--*=68, in 
which cafe a—é=4, with g=1, and, con- 
fequently, 3a muft be greater than one.” 
Now this is net true; for a—é may be 
equal to 4, and at the fame time 3ad-=1; 
for a may be a whole number, with a de- 
cimal, and 4a decimal ; fo that 3ad may 
be not only equal to unity, but to any 
afMfignable number lefs than unity. As 
a familiar inftance, let aq—d=1, and 
a=2,01,. and 6=3,0%.: 3al—3. 44,01 .% 
3O1==,1203 = number lefs than unity. 
Exotericus properly afks, what advan- 
tages will be gained, by giving up the 
mode of working by negative numbers ? 
I anfwer: the {cholar is not taught a 
falfe principle; he is not taught to take a 
number away from another lefs than it- 
felf, that is to perform an impoifibility. 
Confequently, when he comes to any 
thing leading to fuch an operation, he 
paufes; renews his work ; and adinits 
nothing which is not confiftent with 
plain fenfe. Jf it is faid, that fir Ifaac 
Newton followed this mode ; I anfwer, 
Alexander alfo cut the Gordian knot, and 
great names are no excufe for unjuiti- 
fable aétions. 
You will permit me, fir, to add my 
thanks to feveral namelefs correfpond- 
ents, and my hopes that they will conti- 
nue to favour me with their communica- 
tions. As my Algebra may not fall into 
the way of feveral of your readers, I 
have enclofed the refolution * of an equa- 
tion of the third order, true to fix 
places of decimals, which, with a little 
more trouble, might be carried on to 
twice that number. Your’s, &c. 
Inner Temple, W. FREND. 
Dec. 18, 1796. 
* Deterred till onr next, 
Mr. Frend on Cardan’s Rute. 
[ Dec. 
QuesTION XVIII (No. VI).—Axnfwered 
by Mr. F. F re 
This problem may be folved by feve- 
ral eafy methods : one, which Is perhaps 
the moft proper for our purpofe, on ac- 
count of the extenfive ufe of the theo- 
rem from which it is derived, is the fel- 
lowing : 
Fergufon, in his ** Seleét Le&ures,” 
page 362, fhows, that if we put A= fine 
ef fun’s altitude, L and / = fine and-co- 
fine latitude, D and d = fine and co-fine 
fun’s declination, and H =fine of the 
fun’s hour-angle fom V1, then theyre- 
lation of H to A will have three va- 
rieties, VIZ. 
1. When the declination is towards 
the elevated pole, and the hour nearer 
noon than VI is, A=LD-+H/d and H—= 
A—LD 
ld .4i 
2. When the declination is towards 
the elevated pole, and the hour nearer 
midnight than VI is; then AS>LD— 
Hild, and as 
3. When the declination is towards the 
depreffed pole) A=H/d—LD, and H= 
A--LD ; 
ld 
When A comes out #egative ia any oF 
tke above formule, it indicates that the 
fun is below the horizon, and is thea the 
fine of its depre(ffion. 
From the data, we eafily get the hour 
of the fun’s rifing above the w/ble bort- 
zon from tbe mountain; from which, by 
the foregoing method, we get A, which 
is, in this cafe, the fine of his then de- 
preffion below the rasional horizon of the 
piace. 
This angle of depreifion may alfo be 
obtained (as indeed all the furegoing ex- 
preffions are) by the folution of one cb- 
lique-angled {pherical triangle, two fides 
whereof are the polar diftance of the 
fun and co-latitude of the place, the 
contained angle the hour-angle from 
noon; and the third fide to be found is 
the fun’s zenith diftance. But the for- 
mula themfelves are fo extremely conve- 
nient in a great variety of other cafes, 
and fo eafily applicable even by perfons 
who are not converfant in {pherics, that 
they feemed worth infertion. 
A proper application of the column 
marked “ log-rifing,’’ in Tab. XVI of 
the “ Requifite Tables,’ publifhed by 
the: Board of Longitude, will alfo give 
the depreffion required with as much 
eafe as either of the foregoing methods. 
1 Having 

