17960] 
Mr. Frend fays, that,.as g and 4 are both 
efs than unity, 3a4 cannot be equal to 
27. If this is really the cafe, and I fee 
no means of contradicting it, the adop- 
tion of Cardan’s Rule wat lead every 
one who depends upon it, into continual 
error, unleis there is, fome method 
pointed out by Algebraifis, which tells 
‘him, when he may apply the rule toa 
Particular cafe, or when it fails. I have 
heard, indeed, that there mutt be two 
impoffible roots in an equation, to bring 
it under Cardan’s method: but the pro- 
cefs of finding them out, muft make the 
rule very tedious and difficult of ap- 
pheation. : 
Again, Mr. Frend objeéts to the equa- 
tion ufed in explaining Cardan’s Rule, 
a3-53+..=0, and calls w abfurd: for, 
‘days he, three numbers added together, 
cannot be equal to nothing, Doubtlets, 
according to his pofition, which does net 
admit of negative numbers, the expref- 
fion4is ablard: but I fhiculd be much 
obliged to fome one of your correfpon- 
dents to inform me, what is the real ufe 
of thefe negative numbers; and whether, 
if equations can be folved without them, 
the {uppofition fhould be admitted into 
a work’ of icience> In Mr. Frend’s 
book, various equations are folved, with- 
gut admitting them: the true folution is 
brought out by one root, when, accord- 
ing to the common mode, two roots ap- 
pear; and the learner is to try which of 
them is the true one. If this method 
may be purfued throughout the whole 
ef the f{cience, there feems to me to be 
fomething gained by fimplifying the 
principles ; but, before I give up. en- 
tirely the old mode, I fhould like to be 
well informed, what lofe will be fuf- 
tained in the higher parts of algebra, 
by rejecting the negative quantity > for, 
to fay the truth, it frequently puzzled 
me fo much, that, though I can get 
through a quadratic equation, all be- 
yond feems to me to be enveloped in im- 
penctrable darknefs and myfterv. 
I remain, fir, your’s, 
Fly 20, 1790. EXOTERICUS. 
eee a s 
QuesTion XIII (No. IV).—Anfwered 
: by F. F—r. 
‘The difference between the true and 
apparent level, is the difference between 
the earth’s femi-diameter and the fecant 
of an arc of its circumference, whofe 
length is the given diftance. The verfed 
fines of circular arcs are as the {yuares of 
their chords; the verfed fines of intall 
Mathematical Correfpondence. 
557 
arcs are alfo nearly -as the above-mens, 
tioned differences, and the arcs themfelves 
nearly as the chords: therefore, the 
abovementioned differences, when the 
arcs are {mall, are as the {quares of the 
arcs, GUAML PYONLTLEC » 
A mean cf the principal meafures of 
a degree of latitude, taken fince 1736, 
by Maupertius, Caffini, . Bofcowich, 
Mafon, and Dixon; Bouguer and de la 
Condamine, de la Caille, &c. in diffe. 
rent parallels, gives 69.076947 Englith 
miles, or 5§526.15376 chains; which 
multiplied by 1.903, being about half 
the ratio of the equatorial diameter to 
the axis, gives 5542.7 chains, for a mean 
degree of a great circit 3 whofe radius 
will, contequently, be 347579 chains. 
From hence, we have 1 chain = 6490499 
of a fecond, the difference between 
whole natural fecant and radius is = 
(11)49571563 and this’ multiplied by 
317579 gives .00000157429.0f a chain, 
Or .00124683 of an inch; from whence 
the derivation of the rule is eafy. 
The conftruétion of a table from thefe 
data, is too obvious for explanation. dt 
might be calculated for every 100 chains 
as tar as neceffary ; but, as the frft difs 
ferences of the terms would not be equal, 
it would be neceflary, if confiderable 
accuracy was required, to be prepared 
with a table of equations of fecond dif. 
ference, confruéted upon the commoa 
theorcin for its interpolation; fo that, 
upon the whole, it feems better to cal- 
culate it for any particular cafe, frony 
Mr. Waddington’s rule, which will be 
fomewhat nearer the truth if we put 
1247 for 124, and cut off fix places in- 
ftead of five. Or it will be the eafich 
way of any, by ufing the number 125, 
which is nearer than Mr. Waddington’sy 
and being = one-eighth of 1000, there- 
fore only cut off two figures, and divide 
by 8, or take the 8c0oth part. 
It we ufe logarithms, we fhall get 2 
rule which, 1 think, may be touid 
fomewhat fhorter in its application, viz. 
from double the logarithm of the diftance 
in chains, fubtradé? 2 -904193, and ihe 
remainder willbe the logarithm of the diffe- 
rence im inches between the apparent and 
_ true levels. 
Hither of the above rules, the laft of 
which 1s neareft the truth, will do til 
the are becomes fo large as to render the 
error of the firft hypothefis confiderable, 
which will not be the cafe within the 
limits of any ordinary operation ‘of this 
Kind, Should it be neceffary to afcer. 
tap 


