476 
A, N—Y etm l. me 2 1. TT. A 2. mn 
=—— 
BCE ere ae eer See Ie 





Eee 
nA. Tel, 2-2. 
ae eet orteetetts 


mmo oe tte Be 



ee is a) 3. 4e 5 
Wg Ame Pewets, ME 3. od. mc will 
Tet Ze. 3s) Ae Se 6 
feverally exprefs the number of combinations of 
z things by onés, by twos, by threes, by oe 
by fives, and by fixes. In the prefent cafe, 
we have »==6, and the numbers refulting from 
the above theorem, are 
6 Signals BY exhibiting 1 board in each. 

mS Oe SSR ie Sealey 8, 
20> UDO ioe SMe a My pg re ae Je 
Og a igh Wire. fasten ait 
@ DY 26) Stet Sy by Bde ee 
PDe 2 hen by ree De 
63 the total number, agreeable to the re- 
marks of J. C. at page 2.96) 
eeu Greece 
The fame anfwered by Mr. tes 
In anfwer to Queftion AI, in No. IV, of 
your Monthly Magazine, a variety of authors 
have given general rules for the doctrine of 
combinations, permutations, &e. notwithftand- 
ing your ingenious corre{pondent, J. C.has faid, 
“ thefe combinations are not to be afcertained ey 
“any known rule, but by experiment only.” 
The moft concife. and plain method of treating 
that of combinations, is that of Dr. Hutton’s, in 
his valuable Newent Mies Pee oe which he 
comprifes under two heads: First, 
Having given any number of things, 
the number im each combination. 
number of “combinations. 
with 
to find the 
This comes under 
the changes m the Telegraph; and the gene- 
ral rule is,. (ifn be the number of fhaitters) 
tH] \—7——i for any Rumber whatever. But 
as the pofitions .c of e ach fingle {hutte; ac 19) be 
added, the rule w will tend 2°—1. For ingtance, if 
re then 2° — 1==6 3, the whole number of 6 
things; if 7==g, kage 39—-T==502,, the whole 
humber of 9 things, &c. for any number what 
ever. .Second, Yo find the number of changes 
or alterations which any number of quantites 
can undergo, when; combined. in all poflible 
ways, with fhemiclece and each other, both as 
to things themfelves, and the order or pofition 
of them, which Oue authors call the Compo- 
fiction of things; the general rule, then, is, 
Bn en, 
“WI 
é°—1 
~———- X 6==55986—=the number of compofi- 
Sica 
tions that can be made out of 6 
In this cafe, if 2 be==5, it will be 
c 
different things. 
And fo may the whole different ways of placing - 
DT ae | 
24——i 

the 24 letters of the alphabet be rey: 
amountin 
g to 34 figures, 
Mathematical Correfpondence. 

(July. 
Question XII (No.IV)—Arfwered by 
lur. J. F—r. 
It is evident that 
the diftances of 
the point perpendi- 
cularly under the 
balloon from the 
ftations, will be as 
the refpeétive co- 
tangents of the an- 
gles of elevation tak- 
en at each of them. 
Put, therefore, a, 4, and he cotangents of 
the angles of elevation. obferved at A, B, and, 
refpectively,—Find the line AD: AB:: cz by 
and the line CD: BC :: a: 4.—About A and 
C with the radii AD and CD, deferibe ares 
interfecting in D, and diaw BD.—Find AE 
AB.t>€D:: BD, andiCE :CRe. a en 
and with thefe as radii, defcribe arcs about A: 
and C, whofe interfection E will be the point 
perpendicularly under the balloon. 
For becaufe AE and AB are as CD and BD, 
and the angle ABE==DBC, the triangles ABE 
and CBD are fimilar; as is alfo in Tike manner 
proved of the triangles CBE and ABD.—But 
aib::CD:iCB.3: AE: BE, andes bse AD 
:AB::CE: BE. -Therefore, AE, BE, CE are 
as.a, 6, and c, refpectively ; and, confequently, 
the point E is rightly found. From which 
the height is obtained <ither by confiruction, or 
by one analogy, viz. Cotangent angle or 
tion : diftance of perpendicular from fiation : 
rad:us : height required. 
Cor. If a “circle be defcribed about ADC, and 
DB be produced to cut its circumference on the 
fide Eof AC, the point of interfection will alfe 
be the point E required; which gives another 
method of conftruction, as eafy as "ie former. 
In the cafe given, we fhall have for caleula- 
. 
w. 
nern tenn 


TASS J SS Ge 
ee Dae Goes © 
> 2732°129-=—BEy 
: 2438°99g==CE. 
tion the following analogies; 6:¢:: AB: 
$92°708E=AD; and 4 14:3, BO F3828‘gar=— 
CD. 
Then by plain trigonometry, we get the an- 
sles B AD 239 ~ d BED 0 >t eee 
Gres DAU 33° 7 3, and DEVS 33 F535 
and hence BD==549°077.  Aifo, 
S py. Pipemns ee 
‘ee CD :: AB 23353:0%—=— ee 
As ED: 5 
and from any one of th rele, 
Com nev ABy 2:'radius= wee 22s ete yards, 
Cot, the height of the balleoa re= 
po BB ¢ 
Cot. 20° = CED 
fed, 
quire ed. 
Sete: see 
The fame anfwered by Mr, T. Hickman, 
P| 
In the an nnexed fig gure, let "4 
O repiefent the  bailoon; APN 
A, B,‘and. C, the places of J fpry\ 
x z f 7 ea 
Pais ation, being m the Pad pe 
Yi ignt line ALC: OD a per- Tae fan aN 
pendicular gom the balloon =‘ 
to the horizontal place. Draw AD, BD, CD 
ia the fame plane; aud DF perp. te "AC C; ‘then 
pat 
