‘ 
558 
tain this differerice jn a great diftance, as 
for inftance exceeding 20 miles; the bett 
method will be to find the fecant of the 
are by the following analogy—Tabular 
radius : 251523000 :: tabular fecant 
of the arc : fecant required, - - - ufing 
a table of natural fecants extending to 
zo or 12 places, and fubtraéting from 
the fecant thus found, the before-men- 
tioned radius 251523000, the remainder 
will be the difference of levels required in 
inches.—Iin this latter cafe, the follow- 
ing table will be found of fome ufe for 
the reduétion of chains into degrees, 
minutes, and feconds, of a great circle : 
Degrees. 
000180416 
.000 3608 33 
000541249 
000721666 
.000902082 
001082498 
.001262915 
.001443331 
001623748 
.001804164 
Chain 
1 
2 
3° 
4 
ey hs 
6 
7 
8 
Q visas 
TO esis 
‘The fame anfwered by Mr. I. H. 
Not having ever feen ‘* Waddington’s 
Land Surveyor’s Companion,’ Iam not 
able to enter into his method of drawing 
the rule given in his queftion; but it 
appears to be obtained by fuppofing the 
earth’s radius equal to 3963 miles ; for 
3968 ~2 
—_— 
WN aD Wa Wt 

80* 
then ———=64, and ZAL 
100 
= 
which is Mr. Waddington’s multiplier. 
But as the earth’s diameter is now found 
7958 
de = 126, which 

to be 7958 miles, 
will be a nearer multiplier than 124, as 
may be tried from what follows. 
LictaAn irepretent (the Bi) VC ay 
earth’s centre. Aline Re Jf 
5 - . 14 We! 
equally diftant from A, is JEN. 
called the line of true 
level; but the line of figh 
BCDE is the apparent | // 
level, and the difference |/ 
between them is evidently 4 
CF, DG, EH. By the 47th Euclid’s +f, 
s/ AB?4+BC2=AC, ‘thén AC—AF= 
€F, the difference required. Suppofe 
‘BC=2 miles, the earth’s radits==3979, 
then / AB?+BC2—AF =.00050263882, 
which multiplied by 5280=(Feet in one 
Mile) .2.65393297 feet = 2 feet, 7.847 
inches.. Whence, by the fame rule, may 
any difference of the true and apparent 
level be obtained. 
Mathematical Correfpondence. 
[Aug. 
To make a table for a number of 
diftances, being an Herculean tafk.which 
fcarcely any of your correfpondents will 
go into, more eipecially when it is done 
by that able mathematician, Doétor Hut- 
ton, in his ufeful Di€tionary; and as 
many of your readers may not have an 
opportunity of confulting that book, the 
following extract may be very ufeful. 
Your's, &c. 

i; . 
Diftance, Difference | Diftance, Difference 
or BC. of Level,jor | or BC. of Levei, 
Cr. or CF. 
Yards. Inches. | Miles, Ft. In. 
100 0.026 | epee Ht ee 
200 0.103 Paes iS 2. 
300 0.23 75 eo ae 
400 O-4rt Ts o's 
50 0.643 | 2 2 %& 
600 0.925 3 6 co 
760 7.260 4 10 7 
800 1.645 | 5 167 
990 2.081 6 230K 
1.000 2.570 7 32 6 
1300 3.110 | 8 42 6 
1200 3-702 9 Ee pe! 
1300 4.344 IO 66 4 
1400 5.038 | II SO 3 
1500 5-784 12 qs 7 
1600 6.580 | 13 Ti2* 2 
1700 7-425 14.” Fao at 

ussTioN XIV (No. V).—Anfwered by 
Mr. F. F——+. 
The length of the curve of the cy- 
cloid, which the nail defcribes in each 
revolution of the wheel, being equal to 
4 times the diameter of its generating 
circle; and the {pace paffed over by the 
coach in each revolution (the bafe of the 
cycloid) being eqnal to the circum- 
ference of the wheel ;—we fhall have 
giiar6 (sp PP 4S Sag rz bye sees eee 
in an hour, for the mean velocity of the 
nail. 
The fume axfwvered by Mr. Wm. Adam, of 
the Free School, Wooburn. 
It is evident, that the nail in the 
coach wheel defcribes a cycloid. Hence, 
as. 3:1415927:4.:: 7 miles = ages 
miles, the mean velocity of the nail re- 
quired. See the article CyCLorpD in 
Do¢tor Hutton’s Diétionary. 
Errata. In No. HI, pag. 214, inftead of 
Cor. 3 tothe problem, fubititute the following = 
Cor. 3. If the equal fides be conftant, and 
the bate vary, the locus of the point E will bea 
circle, whofe centre is C: alfo the folid under 
AE, BE, and CE, will be conftant. 
In 
