, or the plane cut the opposite
cones, the section is an hyperbola.
Cor. 4. The | major and | minor axes of the ellipse and hyperbola are
c sin. (3 c sin. /3
2 sin. (+/3)' and " ' ' Vsin - * sin ( + 0.
2 cos. sin. (a -}- /S)
Cor. 5. The lat. rect. of the parabola = 4 c sin.2 ^-.
Cor. 6. The parallel and subcontrary sections of an oblique cone are
circles.
CONGELATION. See Heat.
CONGELATION, point of perpetual. See Atmosphere.
CONIC SECTIONS, properties of.
PARABOLA. \
Latus rectum or L = 4 S A.
T N = 2 A N.
S Y2 S P. S A ; i. e. p V~o~7,
65
COT?
S P or = - , where / traced out by
1 4. cos. o
2 cos.* 4
rad. vect. S P.
Ch. curv. = 4 S P.
Diain. curv. =
Equation to the curve # 2 a x (a L).
JVofe. The general equation to a parabolic curve is a 1 ~ y x n '
If n = 3, it is called the cubical parabola.
If n = , it is called the semi-cubical parabola.
ELLIPSE!
SP-fPHr=2AC.
A S. S M = B C 2 .
S P. P H = C D 2 .
A C 2 + C B 2 C P* + C D 8 .
A C. C B = C D. PF, or if the perpendicular P F be called P..
52 _ fi s
Pv. vG X CD 2
B C a V i e , where e eccentricity =r ,
- ___
^ (I e 2 COS .S g) V 1 C 2 COS.*
p _ Af __ !_ __ _ a (1 e)
a ' 1 + e cos. 1 4- e cos. ^ '
Ch. curv. through centre = - _,
c *
66
CIi. curv. through focus
CON
2 C D' ?
A C '
C' D'- S P 3
Diameter curv. " p ^ , or L X ^y- { .
Equation to the curve, when referred to its principal diameters,
IT- + -JF = L
And when the coordinates originate at the vertex,
Or yi = - (a 8 .r 8 ) when the origin is at the centre,
HYPERBOLA.
HP SP 2 AC.
AS. S M - B C 2 .
2 BO*
L-
AC '
S P. P H = C D 2 .
A C 2 C B2 - C P2 C D*.
ab
A C. C B - C D. P F, or P= ,
C*.=; ^fT^'
B C = a ^(ea _i.)
C p= - 7 ^^-^z,ay > -^i T ,
SP = ^ 1 ^ (g-l)
' 1 _|_ e cos. fi 1 + e cos. fl *
2 C I3 2
Ch. curv. through centre = .
Ch. curv. through focus =
2 CD
2 C D* r S P
Diam. curv. = --- or = L. -~,
coo
Equation to the curve, when referred to its principal diumei<-.v>,
A 8 7/2
55 - T"
And when the coordinates originate at the vertex, y* -^ (2 a # -f-
Or #2 -- (^ 2 as), when the origin is at the centre.
Equation to the hyperbola, when referred to its asymptotes, is xy
~jj , where y is parallel to the other asymptote.
2
If the hyperbola is equilateral, xy .
The general equation to an hyperbolic curve \&yx n =. a n + l
Note. The general Equation to the Conic Sections, referred to their
axes is
y* m x -f- w .r 2 , where in is the latus rectum, and the conic
section is a parabola, ellipse, or hyperbola, according as n = o t or is nega-
tive, or positive.
CONTACT of Curves. (Higman.)
Let there be two curves, whose equations are y / (x] t and y'
, and -^ +* : (3) that besides
the preceding conditions -r | ^ 2 and so on ; then will the dis-
tance between the curves be infinitely greater in the first case, than it is
in the second ; infinitely greater in the second than it is in the third j and
so on continually.
CONTINUED Fractions. See Fractions.
COORDINATES Polar, tnfind the relation between. (Higman.)
If the relation between the rectangular coordinates A- and y in any
curve be given, that between the polar o?ies and ti may be determined j
and conversely.
For x = cos. 8, and y sin. Oj substitute these values la the given
equation, and the polar one will be found.
C U B
Let.yz
CORDS, strength of. (Gregory.)
The best mode of estimating the strength of a cord of hemp is to mul-
tiply by 200 the square of its number of inches in girth, and the product
will express in pounds the practical strain it may be safely loaded with.
For cables, multiply by 120, instead of 200. The ultimate strain is pro-
bably double this.
For the utmost strength that a cord will bear before it breaks, a good
estimate will be found by taking- -J. of the square of the girth of the cord,
to express the tons it will carry. This is about double the rule for prac-
tice just given ; and is, even for an ulterior measure, too great for tarred
cordage, which is always weaker than white.
In cables, the strength when twisted, is to the strength when tho
fibres are parallel, as about 3 to 4.
The following fa the breaking strain, l>y experiment, in the best bower ca-
bles at present employed in the British navy. (Encyc. Metrop.)
fci/,i-s, circum.
in inches.
No. oi thread?
in each.
Breaking
strain.
23
2736
TONS.
lU
21
22(>8
89
is
1(556
ftf
Ml
1080
40
From the experiments of Mr. LaMHardiere, it appears, that if we call
the strength of flux HHIO; that of the American aloe will be 50fi j of hemp
1390 ; of Now Zealand flax lift*. ; and of silk 289 i. f Young's Nat. Phil)
COSINES,/#re of. See Fin
CUBATURES of Solids. See Solids.
CUBE Roots of Numbers. -Sec Involution.
00
C Y C
CURVATURE radius. of t in any curre, irJtose equation is g/'ren.
Let x, y, and z represent the abscissa, ordinate, and curve, then
Rad. = , * (rfa-beinsr constant - f '] ( 1 -\ ~-^) '
d^vd-y (hy \ d d* J
Or Rad - = ^ bein& coustaut} =
For the Curvature of Spirals see Spiral.
CYCLE.
A circulation of time between the returns of the same event.
Cycle of the sun, a space of 28 years, in which time the days of the
month return again to the same days of the week, and the sun's place to
the same degrees of the Ecliptic on the same days, so as not to differ 1
in 100 years ; and the leap years return again in respect to the days of
the week on which the days of the months fall. To find it, add 9 to the
given year of Christ, and divide the sum by 28, and the quotient is the
number of cycles elapsed since his birth, and the remainder is the cycle
for the given year ; if nothing remain the cycle is 28.
Cycle of the moon, or golden number, a revolution of 19 years, in which
time the conjunctions, oppositions, and all other aspects of the moon,
return on the same days of the months as they did 19 years before, but
about 1| hours sooner. To find it, add I to the given year of Christ, and
divide the sum by 19, and the quotient is the number of cycles elapsed
from the birth of Christ, and the remainder is the cycle for the given
year, or the golden number ; and if nothing remain, 19 is the cycle.
Cycle of Indiction, a revolution of 15 years, but has no depeudance on
the motions of the heavenly bodies. It was used by the Romans for in-
dicating the times of certain payments made by the subjects to the re-
public, established by Constantino,, A. D. 312. To find it, subtract 312
from the given year, and divide by 15.
Julian period. From the multiplication of the Solar cycle of 28 years,
into the Lunar of 19, and Indiction of 15, arises the Julian period of
7980 years, in which time they all return again in the same order. The
Julian'period, commencing before all the known epochs, is, as it were,
a common receptacle of them all, and to which they may all be reduced
(see JEra.) To find it, add to any year of Christ, 4713, and it gives the
year of the Julian period ; or subtract for any time before Christ
70
DAY
CYCLOID, principal properties of.
1. Circ. arc E G G C.
2. Tangent at C is parallel to the
chord EG.
3. Cyduidal ar<- E C - 2 chord
4. Area of cycloid 3 times area of the generating circle.
.1. Solid generated by the revolution of the cycloid about its base A U :
it-' circumscribing cylinder :: 5 : 8.
(\ Centre of gravity of the whole cycloid = - of the axis from the
vertex.
7. Had. curv. at E = 2 D E.
8. Equations to the cycloid ; put a diameter E D ; jc and y the co-
ordinates E K, K C ; x = arc E C ; z> - arc E G j then
d x a* x~ " dx.
and ij #'-}- Va x x*.
For the oscillation of a body in a cycloid, see Pendulum,
D
DAMS. See Fluids.
DATES. See Chronology.
DAY of the week to find. See Dominical Letter*
DAYS, length and increase of, c.
TABLE,
Shewing^ ti-ith sufficient accuracy for common purposes, thf length and
increase of the days in this country, at different seasons of the year, to-
get her with the beginning and end of twilight.
JANUARY.
FEBRUARY.
Days.
Loncrth
of Day.
Day
inc.
0~(5
12
2-2
32
Ifi
1.
Day
breaks
6. 0~
5. 5H
54
49
11
38
T\vi.
ends.
Days.
~y
ii
ifi
2\
26
Length
of Day.
9. 4
:^0
10
58
io K;
38
Day
inc.
ElJO
3H
66
2. U
98
5->
Day
breaks
Twi.
end>.
1
g
11
16
-21
2g
7. 50
56
S. fi
l(^
:!0
44
fi.
6
11
If?
5. 3!
21
1H
4. 58
49
. J?9
J7
45
54
7 3
]2
DAY
MARCH.
APRIL.
Days.
i~
6
11
16
21
26
Length
of Day.
10. 48
11. 8
28
48
1-3. 8
20
Day
inc.
571
24
41
4. 4
24
42
Day
breaks
4. 44
1
'!
3. 50
Twi.
ends.
Days.
Length
of Day.
Day
inc.
5. 6
20
46
6. 6
24
\9.
Day
breaks
Twi.
ends.
7. 11
29
40
50
8. 1
13
6
11
16
21
m
12. 50
13. 10
30
50
11. 8
20
3. 33
21
8
2. 51
10
26
8. 28
40
53
9. 7
21
35
MAY.
JUNE.
Days.
Length
of Day.
Day
inc.
Day
breaks
Twi.
ends.
Days.
Length
of Day.
Day
inc.
Day
breaks
Twi.
ends.
1
6
11
16
21
26
14. 44
15. 2
18
34
48
16.
1.
18
31
50
8. 4
16
2. 1
1. 52
30
1!
0. 32
No rat
9. 55
10. 10
33
56
11. 48
I night.
1
6
11
16
tl
26
hi. 12
m
28
32
34
34
8.28
38
41
48
50
50
No real night
but constant
day or twi-
light.
JULY.
AUGUST.
Days.
Length
of Day.
Day
dec.
Day
breaks
Twi.
ends.
Days.
Length
of Day.
Day
dec.
Day
breaks
Twi.
end;;.
1
6
11
16
21
26
16, 30
24
16
6
15. 56
42
0. 4
10
18
28
38
52
No real night
0. 44 jll. 14
6
11
16
21
26
15. 24
8
14. 50
34
16
13. 56
I. 10
26
44
2.
18
38
1. 22
42
2.
18
33
48
10. 35
15
9. 57
40
25
10
SEPTEMBER.
OCTOBER.
Days.
Length
of Day.
Day
dec.
1 Day I Twi.
breaks) ends.
Days.
Length
of Day.
Day
dec.
Day
breaks
Twi.
ends.
1
6
11
16
21
26
13. 34
16
U. 56
36
16
11. 58
3.
18
38
58
1. 18
36
3. 5
19
32
43
51
4. 5
8. 54
40
27
16
5
7. 5t
1
6
11
10
21
26
11. 38 4. 56
18 5. 16
10. 58 36
38 56
20 6. 14
1 34
4. 17
23
38
48
57
5. 6
7. 41
31
z\
11
2
6. 53
NOVEMBER.
DECEMBER.
Days.
Length
of Day.
<). 38
20
|
8. IS
3ty
Day
dec.
6.56
1. 14
30
46
**. 2
11
(Day
breaks
"sTis
22
2. 44.
37
30
24
1H
12
1
6
11
16
-21
16
8. 8
7. 58
52
46
M
46
6. 6
;;
1
DEC
DEGREE, decimal parts of. See Time.
DEGREES, $c. converted into Time. See Tims.
DEGREES of Latitude and Longitude.
TABLE of the lengths of different degrees in fathoms t computed by Cot,
Lambtontfor every three degrees from the Equator to tJie Pole. ( Phil,
Trans. 1818 J
Lat.
Degrees on the
Meridian.
Degrees on the
Perpendicular.
Degrees of
Longitude.
604-59,2
60848,0
608-18,0
3
60460,8
60818,4
60765,0
6
60465,6-
60850,1
60516,8
9
60173,5
60852,8
60103,6
12
60484,5
60350,5
595^6,7
15
60498,4
60861,1
58"; 87,3
18
60515,1
6086(>,7
57887,7
21
60534,3
608^3,2
56830,0
24
60556,0
60880,5
55628,1
27
60579,8
60888,5
54?52,0
SO
60605,5
60397,1
52738,4
S3
60632,7
60606,2
51080,^
26
6066L3
60915,8
49i81,9
39
BOHH),8
60925,7
473-18,2
42
60721,3
60935,7
45: ; S1,0
45
60751,8
60940,1
43095,4
48
6078s?,3
609.06,4
40^87,8
51
60812,5
60866,5
38S67,5
54
60842,1
60976,5
35841,1
57
60870,7
60086,1
83215,4
60
0898,0
60995,2
30497,6
63
60923,7
61003,8
27695/2
66
60947,5
61011,8
24815,7
69
60969,1
61018,9
21867,2
72
60938,3
610^5,6
18857,9
75
61005,1
61031,0
157i^6,0
78
61018,9
61035,8
12680,1
81
610-9,9
61029,5
9548,7
84
61037,8
6104-2,1
6330,6
87
61042,6
61043,7
3194,8
eo
6104t,3
61044,3
DEGREE French.
The French usually divide the circumference of the circle into 401005,
each degree into 100', and each minute into 100". Hence if n number
of French degrees, &c. the corresponding number of English = n
i i. e. from the number we must subtract the same, after the decimal
point has been removed one place to the left
73 E
D I A
Exs. What number of degrees, minutes, &c, in the English seal* cor
espond to 71. 15'., and to 2C 3 . 0735, in the French scale.
26.0735
2.G07S5
3.48615
60
27.96900
58.14000 Answer 23. 27'.
DEW. See Rain.
DIALLING.
In all Dials universally, the style or gnomon is parallel to the earth's
axis, and, on account of the great distance of the sun, may be imagined
actually to coincide with it. In like manner the dial plate is parallel to,
and supposed actually to coincide with, some great circle of the earth j
and the hours may be conceived to be traced out by the shadow of the
axis of the earth (here supposed hollow) upon one of these great circles.
Hence there may be an infinite number of different kinds of dials, as
they depend upon the position of the plane (on which the shadow of the
earth's axis falls) with respect to
the meridian and horizon. Thus
if the shadow be receivni upon
the Equator E Q, the dial is cal-
led an Equatorial Dial ; if upon
H R (a great circle of the earth
in the plane of the horizon), a __ /
Horizontal one ; if upon Z N, J
which is in the plane of the
prime vertical, a North or South
Dial, &c. &c. And in these
three last cases, it is obvious
that the shadow of the earth's
axis, when the sun is on the
meridian, or at 12 o'clock, will cut these several circles in Q, R, and N.
At 1 o'clock, or when the / Q P 1 is 15, it will cut them at 1, 1', 1"; at
2 o'clock, or when the / at P is 30, in 2, 2', 2", &c. ; which are .*, the
12 o'clock, 1 o'clock, 2 o'clock, &c, marks.
74
E
D 1 A
Equatorial Dial.
In this Dial, since the sun moves uniformly 15<> per hour, the ,/s. at P,
and consequently the arcs of the circle Q E, which measure them, will
increase uniformly. Hence we have only to take from Q the area 15,
30, 45, &c., and they will be 1 o'clock, 2 o'clock, 3 o'clock, &c., marks.
This Dial, unless graduated on both sides, will only shew the hours for
the six summer months, viz. from the vernal to the autumnal equinoxes.
Horizontal Dial.
Here the arcs R 1', 1' 2' &c. are not equal, but must be calculated by
the resolution of the right /d. As. P R 1', PR 2', &c., where RP 1' =
15, R P 2' 300, & c>> then we shall have
tan. R 1' = sin. lat. X tan. 150.
tan. R 2' = sin. lat X tan. 2 X 15.
&c. &c.
This Dial shews the hour throughout the year, whenever the sun is
above the horizon. In order to fix a horizontal dial, find the time by the
sun's alt. when it is at or near the solstices, and set a well regulated
watch to that time ; then when the watch shews 12 o'clock, at that iu-
stant set the dial to 12 o'clock, and it stands right.
Vertical North and South Dial's.
Here to find the arcs N 1", N2", &c., we have in the right ^d, A
P' N 1",
tan. N 1" = cosin. lat. X tan. 15o.
tan. N 2" zr cos. lat. X tan. 2 X 15<>.
&c. &e.
If P be the North Pole, this represents a South Dial. The construc-
tion for the Vertical North Dial is nearly the same. In this Dial the
number of hours shewn in a day can never exceed twelve, which is the
case at both the equinoxes ; at any other season of the year, the num.
ber of hours shewn is less.
To find whether a wall be full south for a vertical south Dial, erect a
gnomon perpendicular to it, and hang a plumb line from it ; then when
the watch shews 12, if the shadow of the gnomon coincide with tha
plumb line, the wall is full south.
D I A
Vertical East Dial
Here the plane of the Dial is in the meri-
dian, and the gnomon a parallelogram per-
pendicular to it (as represented in the Fig.)
and the shadows upon the plane will evi-
dently be all parallel to the gnomon, and to
one another. Moreover, at 6 o'clock, the
sun, being due east, will be in the plane of
the gnomon, and .*. cast the shadow per-
pendicularly upon the Dial or on Pp. To
find the 7 o'clock mark, let S be the sun at
that hour, and S F a ray proceeding from it cutting the Dial in 7j then
in the plane right /d. A C F7, C 7, = C F X tan. ^ C F7 height of
style X tan. 15 . C 8 = height of style X tan. 2 X 15. &c. Similarly
may be constructed a vertical West Dial. The East Dial will not shew
the hour after 12 o'clock at noon, nor the West Dial before.
General Problem*.
1. Given the latitude of the place, and
the position of the plane of the Dial,
both with respect to the meridian and
horizon ; it is required to find the ele-
vation of the style, the distance of the j_jf
sub-style from the meridian, and the
arc intercepted between the meridian
and any other given hour line.
Let B O A be the plane of the dial,
given in position both with respect to
the horizon H R, and the meridian P E A C ; then in the right angled A
B N R, the /s. B N R, N B R are given, .'. B R may be found ; but P R
== latitude, .". P B is known. Now let a plane pass through O P, and
let it be turned about till it becomes perpendicular to BOA, and let it
cut the circumference of B A in M, then P M is that meridian which is
perpendicular to B O A, .'. in the right angled A PMB, PB and /
PBM are known, .'. PM = elevation of the style, and MB, the dis-
tance of the substyle from the meridian, may be found. Draw P T,
making an ^ of 15<> with P B ; then will T be the 1 o'clock mark, and to
find it we have PB, and BPT 15, and / PBT supplement of
N B R, .*. B T may be found, and so on for the other hours.
2. To determine the curve, traced out by the extremity of the shadow
#f a vertical gnomon on a horizontal plane. (Noddy.)
76
D I F
Conceive a line A B to be the gnomon, A P the shadow, A N the di-
rection of the meridian shadow. Draw P N perpendicular to A N, and
let A N = .r, P N y> A B = a, I latitude of the place, S = sun's de-
clination ; then
_ (cos. 2 / sin.* 3). A* -f- 2a sin. I cos. I x -f- (sin, a I sin.g S). at.
siiPl
Cor. If cos. = sin. , or I = 90o , the curve is a parabola, if cos. I
is greater than sin. , or I less than 90 S, an hyperbola, if cos I is less
than sin. 5, or J greater than 90 S, an ellipse.
DIFFERENTIALS.
TABLE I.
Differentiation of Algebraic and Transcendental Functions ; and of th*
higher orders of Differentials.
QUANTITY. DIFFERENTIAL.
ax ... adx.
ax+by ~+ e ... adx + bdy A
dx.
( m i to \ n 1 / m i ~"
a + x J n (a -f *
xy xdg+ydx.
m n
* y
y
Hyp. log. x ... .
Hyp. log. 1 *f * .
77 E2
D I T
S5d. 5A. 48m. 51 fit ; being less than in th
time of Hipparchus by 11,?."
Mean velocity of earth in its orbit 59'. 10,7" each day.
Velocity in its perihelion lo. 1'. 9,9."
Do. in aphelion 57.' 10,7"
Revolution about the line of the apsides, or anomalistic year, 3654. 6ft.
14m. 2*.
Tropical revolution of apsides performed in 20,931 years.
Inclination of axis to Ecliptic 23. 7'. 57"., which decreases at the rate
of 52,1". in a century, but this decrease can never exceed 2. 42'.
Nutation of axis = 19,3".
Precession of the equinoxes 50,1" annually, or 1. 23'. 30". in a century,
A complete revolution performed in 25868 years.
Length of sidereal day 237z. 5fiw. 4,1*. j and has not varied the hundredth
part of a second since the time of Hipparchus.
The interval between the vernal and autumnal equinoxes is (on ac-
count of the excentridty of ths earth's orbit and ifcs unequal velocity
therein) nearly eight days longer than the interval between the autum-
nal and vernal equinoxes. These intervals are at present nearly aa fol-
lows :
d. h. .in.
d, h. m.
185. 35. 20,
From spring equinox to snmm*>rl 09 o] 45 ^
solstice J "" ' /
From summer solstice to autuni-7 TO bk A
f SO. lo, <>Q, t
nal equinox -> j
From autumn aquinox to winter! gg j^ 47 "\
solstice > ' /
From winter solstice to spring") go 3 42 V
equinox ,., ,....,-^
equinox .
Difference 7. 16. 51.
E3
EAR
EARTH, figure of. ( Play fair > Maddy.)
1. To find the radius of curvature at any point of the terrestrial meri-
dian, supposing the earth to be an oblate spheriod.
Let a and b be the Equatorial and Polar % axes, r the rad. of curv. to
the latitude X, c = a b compression, m =. 57o. 2957795 the number
of degrees in an arc radius ; then
r a 2 c + 3 c sin. 2 X.
c 3 c
or = a cos. 2 x.
and if D = length of a degree in lat. X, r = m D
as. c 3 c \
..D = (I cos. 2 X. )
m \ 2a 2a J
Cor. 1. At the Equator m D = a 2 c ; at the Pole m D = a -f c; and
in lat. 45o. = a c. Hence if E, P, and M = the degree at the Equa-
tor, Pole, and lat. 45. ; M = | (P + E).
Cor. 2. The excess of a degree in any lat. above that at the Equator, or
D E, varies as sin. 2 X.
2. The lengths of two degrees of latitude, of which the middle points
are in given latitudes, being known by admeasurement, the Equatorial
and Polar diameters of the earth may be calculated from the following 1
formulae.
Let D and D' be the given degrees (the least, or that nearest the Equa-
tor being D) X and X' the latitudes of their middle points, then
m. CD' - D.)
3 sin. (X' + X) X sin. (X' X)'
and the compression, or ellipticity of the earth
_ e D' D
~~ ~a ~~ 3 D. sin. (X' + X) X sin. (X' X)'
from which two equations a and r, and consequently a and (, mny be
found.
EAR
The following are the five arcs, which have been measured with th
greatest care :
Latitude.
Degrees in
Fathoms.
Country.
By whom.
Oo. 0'. 0".
60480.2 ...
Peru
Condamine, &c.
11. 0. 0.
60486.6 ...
India
Major Lambton.
45. 0. 0.
60759.4 ...
France ...
Cassini, &c.
52. 2. 2.
60826.6 ...
England...
Colonel Mudge.
66. 20. 10.
60952.4 ...
Lapland...
Swanberg, &c.
By combining these in pairs, and taking the mean, we get the follow-
ing results.
a : b :: 312 : 311.
D 69.044 .3299 X cos. 2 A in miles,
or D = 60759.472 290.576 X cos. 2 A in fathoms, which ex-
presses the degrees of the meridian in any latitude.
- = .0032 =
1
312.5 *
c 12.<580 miles.
a 3962.349 miles.
b 3949.669 miles.
Hence circumference of elliptic meridian = 24855.84 miles ; do. of equa-
tor 24896.16 miles ; .'. difference 40 miles nearly.
3. The figure of the earth may also be determined, by comparing a de-
gree of the meridian with the degree of a great circle perpendicular to
the meridian in the same latitude, by the following formulae.
Let A be the degree of the curve perpendicular to the meridian, the
rest as before, then
e = (A D) X V-.
2 * cos.* A
and =
A D
i nearly.
2 A cos. A
4k To find the compression by means of a second's pendulum, consider-
ing the earth as a spheroid of equilibrium.
67
EAR
Let p and p' be the lengths of two pendulums oscillating seconds in la-
titudes A and A', c tlie compression, the equatorial radium being unity j.
then
c _ pp'
jtsin. A p'sin.2 A'
5. Comparison of the figure of the earth, deduced from actual admea-
surement of a degree in different latitudes, with that deduced from the
theory of gravity.
If a homogeneous fluid revolve on an axis, it will form itself into an
oblate spheroid, of which the Polar | axis : radius of Equator : : attrac-
tion at Equator centrifugal force at Equator : attraction at tlie Pole.
In the case of the earth, this ratio will be :: 229 : 230.
If the earth be not homogeneous, but composed of strata that increase
in density towards the centre, the spheroid will have less oblateness than
if it were homogeneous, and it is demonstrable that if the density in-
crease so that it be infinite at the centre, the ellipticity -r=o which is
oTo
the case of the least ellipticity ; -^ is the case with the greatest.
Hence as the ellipticity of the earth has been shewn to be less than
230" ^ viz ' "312") ' Jt is eviaent tnat if tne eartn is a spheroid of equili-
brium, it is denser towards the interior. This has been indisputably
proved to be the case by actual experiment. See Mountain, attraction of.
But after all, whether the eatth be a spheroid of equilibrium, whether
the N. and S. f spheres be equal and similar to each other, and whn.t is
the ratio of an arc of the meridian, measured ia a given latitude, to tna
whole meridian, are questions to which complete solutions have not yet
been given.
EAR
. TABLE of the eilipticities of the earth.
Authors.
Eilipticities.
Principles.
1
Theory of Gravity
230
1
Lambton
Sabine ...
312
1
310
1 to !
Do.
Treisnecker
312.6 314.3 ""
1
329
1
Occupation of Stars.
Precession and Nutation.
La Place
334
1
Theory of the Moon
[SOS
1
Upon the whole, the ellipticity probably lies between -^ and -^.
But Captain Sabine, from some very recent experiments on the length
of the Pendulum (see Pendulum), states the ellipticity at . For
Tables of Degrees of Latitude and Longitude, see Degree.
EARTH'S Surface, extent of.(Encyc. Britt. Supplt.)
The extent of the four great divisions of the world is as follows :
Europe, Witll itS IsleS *r*^fr**v+r*t*rr*r*s.r*f*tfr*r*>
Sq. Eng. Miles.
rxx w OTWWJ , 3 43-^^000
Africa with Madagascar ^WJX^WWXXWXJW/^.T
^^^^ 1 1 ,4^0,000
Continental Asia -M* f M ;
at 110 yards depth, 53. 6 j at 336 yards, 60. 8 ; at 472 yards, 74. G.
In Saxony, in four of the deepest mines, annual temperature at sur-
face is 460. 4 ; at 170 to 200 yards depth, 54. 5 ; at 280 yards, 58 ; at 360
yards, 6->. 0.
In the coal mine of Killing-worth, the deepest in Britain, annual tern-
perature at surface is 48 ; at 300 yards, 70 ; at 400 yards, 77. In seven
others of the deepest coal mines in Britain, a corresponding gradation
was observed.
In these British mines, the increment of temperature is about 1 for
15 yards of descent. In the Vosges it is about 1 for 20 yards, and in
Saxoay 1 for 22 yards. Taking 20 yards as a mean, if the increase fol-
lows the same arithmetical ratio to a considerable depth, we should
find the temperature of the Bath waters (116) at 1320 yards below the
surface ; and that of boiling water at 3300 yards, or nearly two miles.
EARTH, pressure of against walls. 'Gregory.)
Let D A E F be the vertical section of a
wall, behind which is placed a bank or tor-
race of earth, of which a prism, whose sec-
tion is represented by D A G, would detach
itself and fall down, were it not prevented by
th* wall. Then A G is called the line ofrup.
twre, or the natural slope, or natural decli-
vity, In sandy or loose earth, the
10
B F
A E
t A ft
seldom exceeds 30 ; in stronger earth it becomes 37 ; and in some fa *
rourable cases more than 45.
1. If h r= A D, x = A E, 6 = D A G, and S and s represent the speci-
fic gravities of the wall and earth, the state of equilibrium is expressed
by this equation,
| A-2. S - l / 6 7*2. g. tan, 8 I 8,
PX. Suppose the wall to be 30.37 feet high, of brick, specific gravity
8000, and the bank of earth specific gravity 1428, and the natural slope
63; then
#*. 2000 % X 39.S7 2 X 1428 X tan.s 2GJV
.". x 9.6 feet thickness of wall..
The following practical results may be found useful.
Values ofD G for different materials.-
Bank of vegetable earth ~~~*~~***~~*~.~~~ D G =z .618 h,
Do. of sand ~~~~~~ ~,*~~^~~~^~ D G = .677 h.
Do. of vegetable earth mixed wHh small gravel *~~ D G =. .646 h.
Do. of rubbles ~~~~ r ~~~^**~~~*~~~ ^ r ^^^ rff ^. DG = '4J4 h.
Do. of vegetable earth mixed with large gravel ~~~ D G .613 h,.
Thickness of walls, both faces vertical
1. "Wall brick, 109 Ibs. per cubic foot, bank vegetable 1
earth carefully laid course by course ^- , D F = .16 : ^.
2. Wall unhewn stones, 135 Ibs. per cubic foot, earth
as before ^-^ ,~~*^+~~*~~ ^^ DF .15-&.
3. Wall brick, earth clay well rammed ~~~~ , DF .17 h.
4. Wall unhewn stones, earth as before ^^r^^.^^^ 13 F .16 h,
5. Wall of hewn freestone, 170 Ibs. per cubic foot,
bank vegetable earth ^^^^^^^^^^^^^^ D F .13 K.
6. Do. bank clay ^^^ fff ^, w ~ ww ~ ww ,. w ~. D F -= .14 h.
7. Bank of earth mixed with large gravel, wall of
bricks ~~^ . D F .19' 7>.
Do. of unhewn stone ,v~*~^,^.~^v~~^,~~~~~ D F = .17 h.
Do. of hewn freestone *,,..*,..***** D F = .16 h.
8. Bank of sand.
Wall of bricks . ~~**~~*~*~. D F - .33 h.
Do. of unhewn stones .., Mn ,. flf ^ f ~~~ f ~**~~*+ D F .30 A.
Do. of hewn freestone *,....,., w ,.,.. D F .26 h.
When the earth of the bank is liable to be much saturated with water*
%B proportional thicknesses of the walls must at least be doubled,
9i F
E C C
2. For walls with an interior slope, or a slope towards the bank, let
the base of the slope be - of the height, then
where m .0424 for vegetable or clayey earth, mixed with large gravel ;
m = .0464 if the earth be mixed with small gravel j m = .1528 for sand ;
and m = .166 for semifluid earths.
Ex. Let the height of a wall be 20 feet, and _L of the height for the
20
base of the slope, suppose also the specific gravity of the wall and bank
to be 2600 and 1400, and the earth semifluid j then
=i 5 feet, while the thickness of the wall at the bottom
will be 6 feet.
RASTER, to find it on any year. '(Delambre.)
1. Divide the year proposed by 19 Call remainder a.
2. Divide the same number by 4 Call remainder b.
3. Divide it also by 7 Call remainder c.
4. Divide (19 a -f- M) by 30 Call remainder d.
5. Divide (2& + 4c + 6d-fN)by7 Call remainder e.
6. Then Easter day will fall either on (22 -f- d -f- e) of March j or on
(d _|_ _ 9) of April.
Values of M and N in the above calculation.
M. N.
From 1700 to 1799 23 3
1800 to 1899 23 4
^900 to 1999 24 5
Exceptions to this rule :
1. If the computation give April 26, substitute the 19th.
2. If it give April 25, substitute the 18th.
ECCENTRICITY of a Planet's orbit (Woodhouse, Playfair.)
Let e be the eccentricity of the orbit, g the greatest equation of the
centre, found by observation, and put 5?0 ^9573 = A * then
E C L
In the earth's orbit h is very small, .*. e | h nearly.
The secular diminution 18". 79, and .". if this diminution continued
uniform (which, however, we have not a right to suppose) the earth's
orbit would become a circle in about 36300 years,
ECHO.
That an echo may return one syllable as ?oon as it is pronounced, the
reflecting surface should be 80 or feet distant ; for a dissyllablic echo
170 feet, &c. This is upon the supposition that sound proceeds at the
rate of 1142 feet per second, and that the ear can distinguish the succes-
sion of two sounds or syllables, when the interval between them is th
of a second.. (Play fair.)
An echo in Woodstock Park repeats 17 syllables by day, and 20 by
night. An echo on the north side of Shipley church in Sussex, repeats
21 syllables. (Young's Nat. Phil.)
ECLIPSES. ( Woodliouse, Play fair.)
\. Eclipses of the Moon.
1. The length of the earth's shadow varies, according to the distance of
the sun and earth, between the limits of 212,896, and 220,238 semidiame-
ters of the earth j its mean length being 216,531. And in general if r be the
earth's radius, - the apparent semidiameter, and p the horizontal par-
allax of the sun, the length of the shadow, reckoned from the earth's
centre,
~ sin. (R'-* sin. JJ
\ 2 P ' 220
2. Hence half the angle subtended at the earth's centre by the section
-of the shadow, at the distance of the moon, (if P be the horizontal paral-
lax of the moon) is
From this formula the apparent diameters of the earth's shadow may
be computed for various distances of the sun and moon, as in the follow-
ing Table.
Apparent diam. of
earth's shadow.
r Moon in apogee ... .............. 1. 15'. 24".3036
Sun in perigee ......... < at mean distance ................. 1. 23. 2'31
tin perigee ....... ................ tf 1. 30. 40.3164
EC L
Nin a; mean distance
San in apogee .,,.,..,.
4
parent diam. of
firth's shadow.
15. 56.8656
. 23. 34.872
31. 12.8784
16. 28.2936
. 24. 6.3
31. 44.3064
< at mean distance
3. The distance of the centres of the moon and of the earth's shadow,
when the moon's disk just touches the shadow (if d = moon's diameter)
Cor. If P = 57'. 1", p 8", 8, and =: 16'. 1".3, we have the mean
apparent | diameter of the earth's shadow =. 41'. 8".5, which is nearly
three apparent diameters of the moon. Hence since the moon in the
space of an hour moves over a space nearly equal to its diameter, the
moon may be entirely within the shadow, or a total eclipse may endure,
about two hours.
4. The apparent | diameter of a section of the penumbra at the moon's
orbit
And the distance of the moon's centre and of the centre of the shadow,
v/heu the moon first enters the penumbra, is
5. To find the time, duration, and magnitude of a lunar eclipse.
Let m moon's motion in longitude,
n moon's motion in latitude,
s sun's (or the shadow's centre's) motion in longitude,
X moon's latitude when in opposition,
t time from opposition,
c ~ distance of moon and earth's shadow,
jind-let m ^ 8 = tan. 6.
then t = 1 5 A sin. B -j- sin. 9 V (c I* cos. ^)
5*
E C L
from which expression may be deduced values of the time, correspond-
ing to any assigned values of c, as in the following instances.
(j) To determine the time at which the moon first enters the penum-
bra, for c put P -f- P + IT + "5" > ^ nas * wo va l ues > an d the second value
will denote the time at which the moon quits the penumbra.
(jj) To determine the time at which the moon enters the umbra, put
(jjj) To determine the time when the whole disk has just entered tha
shadow, we must deduct d from the preceding value, and make c = P-f.
p - -- j and similarly for other phases.
(Jijj) To find the middle of the eclipse, we have t X Sin ' 8 g , and in
that case the distance of the centres (c) is = Pi cos. 6.
(v) The nearest approach of the centres being known, the magnitude
of the eclipse is easily ascertained. Thus on the supposition that A cos. 9
is less than the distance (P -f- p 4- -~- j at which the moon's
limb just touches the shadow, some part of the moon's disk is eclipsed -,
and the portion of the diameter of the eclipsed part is
The portion of the diameter of the non-eclipsed part is the moon's ap-
parent diameter d t minus the preceding expression, and therefore is
A cos. 8+ ^ 4.5_p~p.
If this expression should be equal nothing, the eclipse would be just a
total one. If the expression should be negative, the eclipse may be eaid
to be more than a total one, since the upper boundary of the moon's disk
would be below the upper boundary of the section of the shadow.
(vj) If in the expression
sin. 6 V (c* A* cos.* 0).
n
we substitute for <*, P 4- p 4- -5- -5- we have the time from the moon's
A A
first entering to her finally quitting the shadow or umbra. And if in th
95 F2
E C L
*aine expression we substitute for c, P 4. p -f. -^ -f- , we have thi
whole time of an eclipse, from the moon's first entering, till her finally
quitting the penumbra.
6. Ecliptic limits. When the mean opposition is 12o. 3& distant from
the node, there can be no eclipse ; and when it is less than 9. distant
from it, there must be aa eclipse. Between these limits 120. SG' and 9.
the matter is uncertain, and must be decided by the calculation of the
true place of the moon.
II. Eclipses of the Sun.
1. -Let r, R be the radii of the moon and earth, the rest as before ; then
Jhe length of the moon's shadow
__ N P^
-By means of this formula, we have
Length of Moon'*
shadow, distance.
Sun in apogee, moon in perigee ... 59.730 | 55.902
San in perigee, moon in apogee 57.760 I 63.862
Hence in the latter case, the moon's shadow never reaches the earth,
and the eclipse cannot any Avhcre be total.
The moon's mean motion about the centre of the earth is 33' in an hour ;
and the shadow of the moon .'. traverses the surface of the earth, when
it falls on the surface perpendicularly, with a velocity of about 380 miles
in a minute. W T hen the shadow falls obliquely, its velocity appears
greater in the inverse ratio of the sine of the obliquity.
The duration of a total eclipse ia any given place cannot exceed 1m.
An annular eclipse may last 12?. 24*.
.2. The apparent f diameter of the moon's shadow = f ^ 1- __ .
Hence when d D apparent J diameter =r o, or the vertex of the coni-
cal shadow just reaches the earth. When tf is less than D, the expres-
si-m is negative, ia ether words the shadow never reaches the earth.
In a similar manner may the formulas for the penumbra of the earth
be transformed and adapted to the case of the moon.
(jjj) The solar ecliptic.limits =' 17. 21'. 27". If the conjunction hap-
pens nearer to the node than this, t&ere may be an eclipse. If it be more
distant, there can be none.
.96
E C L
Solar eclipses arc more difficult of computation than lunar ones ; nor
is it possible to enter here upon the methods that have been employed,
We shall .'. conclude this article with an account of the number of
eclipses that may take place in a year.
III. Eclipses, number of.
In the space of 18 years, there are usually about 70 eclipses, 29 of the
moon, and 41 of the sun.
Seven is the greatest number of eclipses that can happen in a year, and
two the least.
If there are seven, five must be of the sun, and two of the moon. If
there are only two, they must be both of the sun j for in every year there
are at least two eclipses of the sun.
There can never be more than three eclipses of the moon in a year ;
and in some years there are none at all.
Though tlie number of solar eclipses is greater than of lunar in the ra-
tio of 3 to 2, yet more lunar than solar eclipses are visibla in any parti-
cular place, because a lunar eclipse is visible to an entire hemisphere,
and a solar is only Visible to a part.
ECLIPTIC, obliquity of.f Woodhouse, Vines.)
The mean obliquity of the Ecliptic in January 1, 1827 = 23. 27'. 43".7.
For the variations in the obliquity, see Precession. But besides these va-
riations in the obliquity, arising- from solar inequality and nutation, the
former of which passes through all its changes in the period of half a
year, and the latter in 9 years and 3 months, the obliquity of the Eclip-
tic has, as far back as observation goes, been diminishing from the action
of the planets, particularly Venus and Jupiter. This diminution, called
the secular diminution, is at present 52" in a century. There is, how-
ever, a mean to the obliquity which it cannot pass, and round which it
oscillates backwards and forwards. According to La Grange, the incli-
nation will never vary more than 5. 23' from the year 1700.
Hence if we have given the mean obliquity for any time, and wish to
find the true obliquity, we must correct the given mean obliquity by
the secular diminution, the solar inequality, and the nutation. The ana-
lytical expression for the obliquity, including these corrections, is
E ~~L 4. 0".4345 X cos. 2 sun's longitude 4. 9".63 X cos. N
E being the mean obliquity at the beginning of the year, N the supple-
ment of the node, and n the number of days from the beginning of the
year.
E L A
ELASTIC bodies, equilibrium of.(Whewell.)
This subject may be comprised under three heads. (1.) Elasticity of
Extension and Compression, as in the case of a string stretched by a
force. (2.) Elasticity of Flexure, as when wires and laminae of different
metals and other substances exert a force to unbend themselves when
forcibly bent. (3.) The Elasticity of Torsion, as when twisted threads
of metal exert a force to untwist themselves. Our view of these several
subjects must necessarily be very limited and imperfect.
1. Elasticity of Extension.
1. When an elastic string of given length is stretched by a given force,
to find its length.
The increase of length is proportional to the tension. Let i be the
measure of the extensibility of the string, whose length at first is a j t
the force or weight with which the string is stretched, which of course
measures the tension ; then the increase of length = a it, and the length
I when stretched will .'. be
a -}- a it, or a (1 + i f)
We may determine i, if we know the original length of the string, and
its length for any given value of t. It may be convenient to know it in
terms of the force which will draw out the string to double its length.
Let E be this force j hence
a (1 -f- i E) = 2 a, and t = -g.
Hence the length of the string under a tension t becomes
=(>+*)
E may be expressed by a length of the given string, whose weight
would draw the string a to double its length. E is then called the mo-
dulus of elasticity.
2. A uniform elastic string hangs vertically, stretched by its own
weight .* to find its length.
The same notation being retained,
Cor. 1. I
Cor. 2. Since I a f 1 + ~-\ , it appears that the weight of the
string stretches it half as much, as if it were ail collected at the lowe*t
point
98
E L A
2. Elasticity and resistance of solid materials.
Here we suppose that all solid bodies may be considered as made up of
elastic fibres capable of extension and compression ; and that the resia-
tance to extension is proportional to the extension in each fibre.
When a solid body is acted on by any
force, it may be partly extended and
partly compressed. Thus let a mass
A B Q P be acted upon by a force F
.compressing it in the direction EF.
The surface P N Q may be brought in.
to the direction p N q j in this case all
the fibres R R' which are on one side
of N are shortened ; all those on the
other side of N are lengthened. N N'
remains the same as in the natural
State. N is called the neutral point ;
and the line which separates the parts fT ^
of the body which are compressed from those which are elongated is call-
ed the neutral line.
1. When a rectangular prismatic mass is compressed by a force paral.
lei to the direction of the axis : to find the neutral line.
Let P M = M Q = cr, M F = h, M N n, then
Cor. 1. If h | a, n a, or the neutral point is in the surface, and
the whole beam is compressed.
2. When a rectangular prism is acted upon by any force in any direc-
tion ; to find the neutral point at any part.
Let a force/ act in the line y F on a prism A B P Q, then the same no,
tation being retained, we have as before
Cor. If the force act perpendicularly to the axis, A is infinite, n = o,
and the neutral point is in the axis.
3. When a rectangular prismatic beam is made to deviate a little from
a straight line by the action of a f*iven force perpendicular to it, to find
the deflexion,
E L A
Since the force is perpendicular to the . ^_ ^^ 13
beam, and the beam is nearly a straight '
line, we may (by Cor. last Art.) suppose
the neutral point coincident with the axis.
Let A M E represent the axis bent by a
force acting- perpendicularly to A D its original position ; and let F be a
length of the beam equivalent to the force/, I length b breadth, and
a = thickness of the beam, E the modulus of elasticity, then the whole
deflexion S
-
or if for F we put its value o^-i
a
Cor. 1. Hence for a given breadth and thickness, the deflexion is as the
force and cube of the length ; and for a given weight and length, the de-
flexion is inversely as the breadth and cube of the thickness.
Cor. 2. Let the direction of the tangent at E make an ^ 6 with the tan-
gent at A ; then 6 may be called the angular deflexion, and we have
The angular deflexion is as the force and square of the length.
4. When a rectangular prismatic beam in a horizontal position is bent
by its own weight j (its thickness being vertical) to find the deflexion.
The same notation being retained, the whole deflexion
3ft
8Ea*'
Cor. In this and the last Art. 3 being objerved, E may be found.
5. A rectangular prismatic beam is compressed by a given force acting
in a direction parallel to the axis j to find the deflexion.
Let a be | the thickness of the beam, I - f the length, h distance
of the force from the axis j then if E be very large compared with F,
we have the deflexion
7
= h (eec. - - 1).
aVE
Cor. If the force act at the extremities of the axis, h o, and there
will be no deviation except
100
L A
Hence we may find the weights which columns of given materials
will support. Thus, if in fir-wood the modulus E be 10,000,000 feet, a
bar, an inch square, and 10 feet long-, may begin to bend, when
F = .8225 X -~f- X 10,000,000 = 571 feet.
3. Elasticity of Torsion.
1. Let/and/' be the forces necessary to twist a metallic thread, from
the position in which it would naturally hang, through the ,/s. 8 and fr ;
then if 6 and 6' be very small,
/ _ *
J> ~ p-
On this principle depends the Torsion Balance of Coulomb, which has
been employed for the purpose of measuring very small repulsive and
attractive forces. In some cases the instrument was constructed with
so much delicacy, that each degree of torsion required a force of only
Height of the Modulus of Elasticity in thousands of feet ( Encyclop.
Brit. Supplem.)
Iron and steel ~~~~~ ~~~~* 10,000 Fir wood ,~~~~~ 10,000
Copper *, 5,700 Elm 8,000
Brass 5,000 Beech 8,000
Silver ww ~ w ~ w , ,, w 3,240 Oak *~~~ ~~ , * 5,060
Tin^^ w ~~ ^^^^^ ,850
The following Table is the result of experiments by Mr. Rennie, pub-
lished in the first part of the Phil. Trans, for 1818.
Mr. Rennie found a cubical inch of the following bodies crushed by
the following weights :
Ibs. av.
^ WWJ ^r WW ^ JJWWWWJ ^r W r WJ -x J u- JJJ - JW x Ww r J ^nn 1284
American Pine _ 1606
White Deal ,_ , 1928
English Oak ^ 5860
Cube* of 1% inch.
Sp. gr,
1127
Derby Grit-
Portland *.
101
E Q U
frai"leith White freestone rrr^nr^..-*^
Sp. gr.
~~~~~ 2452
12346
12856
13632
13632
14302
14918
17354
20610
20742
21254
Yorkshire Pavin" j^j^ffsj^jir^fju^^f^^t^fsv-t nrfr,s TJ
^ rjj ^ rj ^ f ^ t ?507
White Statuary Marble ^j^^^^^ru^^.,,,^
ww*, 2760
Bromley Fell Sandstone, near Leeds ~~~*~~
Cornish. Granite *^^r*jxj-^ JJJ _n,jx l *n.j^xxx^jxrjj^ J - sjswu
2506
2662
Dundee Sandstone j-s**^** -jj^sjj^j-jjj-sffjsvfjjfff^f.
rnssMM^ 2530
Compact Limestone f,********!********-***^^^***^
1JVU ^ WVMJ 2584
KSWSSWS+ ?599
Cluck Brabant Marble *jwjs**~*rrwrf****j-rr*jrj-Tj-s
n* ?697
Verv Hard Freestone ~~~~~~~+~~~~~~~*~^
2528
Cubes of different metals of ^ inch were crushed by the following
weights :
Cast Iron
Cast Copper *~~~
Fine Yellow Brass
Ibs. av.
9773
7318
10304
Wrought Copper *
Cast Tin --
Cast Lead ,
Ibs. av.
6440
483
Bars of different metals six inches long 1 , and ^ inch square, were sus-
pended by nippers, and broken by the following weights :
Ibs. av.
1166
1218
Ibs. av,
2273
2112
1192
1123
296
Cast Iron, horizontal ~~ 1166 Gun Metal
Ditto, vertical <..~..~.~ 1218 Copper hammered
Cast Steel ~~~,~~ -~~ 8391 Cast Copper
Blistered Steel hammered 8322 Fine Yellow Brass
Shear Steel do 7977 Cast Tin
Swedish Iron do. ~~ 4504 Cast Lead r*~~~~~> \\k
English Iron do. ~~* 3492
ELASTIC bodies, theory of.^-See Collision.
ELLIPSE, principal properties of. See Conic Sections.
ELLIPTICITY of the Earth. See Earth, figure of.
EMBANKMENT. See Dyke, and Earth pressure of.
EPOCH. See JEra.
EQUATIONS of condition. (flay fair, Maddy.)
Any equation expressing the relation that obtains among the coem.
cients of another equation, is called an Equation of condition. These
equations are used in determining by observation the constant coefficient*
in an assumed or given function of a variable quantity. Thus let us sup-
pose that the form of the function is known from theory, but that the
onstant quantities that enter into it, are to be determined by observe
102
tion ; required, considering that every observation is liable to error, in
what way these quantities may be most accurately determined.
RULE. Substitute the quantities known by observation fory and x t in
the given formula (each observation being supposed to afford a value both
of # aud#), and thus, as many equations of condition will be obtained,
as there are observations. If these exceed the number of quantities to
be found, or of the equations wanted, let there be composed from the ad-
dition of them into separate sums, as many equations as are necessary,
each consisting of as many of the given equations as the question admits
of. From the equations thus obtained, the quantities sought may be de-
termined with the least probability of error.
Suppose the general formula to be
y = A sin. x 4. B sin. 2 x,
and that from observation we have eight values of x and y t viz.
Values of
140
135
130
125
120
115
110
105
Values of y.
73'.5
80.2
87.0
94.1
99.5
104.5
107.5
110.2
Hence,
.6428 A .9848 B = 73.5
.7071 A 1.0000 B - 80.2
.7660 A .9848 B = 87.0
.8191 A .9337 B 94.1
.8660 A .8660 B = 99.5
.9063 A .7660 B 104.5
.9397 A .6428 B =r 107.5
.9660 A .5000 B = 110.2
By adding the first four into one, and also the second four, we get
2.9350 A 3.9033 B = 334.8, and
3.6780 A 2.7748 B = 421.7 j
and therefore,
- 1.7 ~ 2.7748 X 334.8
3.678 X 3.9033 2.935 X 2.7748"'
or A = i.55*00.
103 F3
E Q U
In like manner, B = K2; so that the equation becomes,
y = (lo. 54'.2) sin. x -f (1'.2) sin. 2 jr.
This is nearly the equation of the centre in the earth's orbit.
In this \vay all the elements of any of the planetary orbits may be de
termined simultaneously, or corrected if they are already nearly known.
In the construction of Astronomical Tables, the number of equations
combined has amounted to many hundreds.
In the example above, no method was to be followed, but that of di-
viding the original equations into two parcels or groups, from the sums
of which the new equations were to be deduced. But when it happens
in the given equations, that the terms involving the same unknown quan-
tity have different signs, the best way is to order all the equations so that
one of the unknown quantities, as A, shall have the same sign through-
out j and then to add them together, for the first of the derivative equa-
tions. Let the same be done with B, C, &c. whatever be the number of
the quantities sought. Thus, each of the unknown quantities will occur
in one of the equations, with the greatest possible coefficient; and the
coefficients of the same unknown quantity, in the different equations, will
become by that means as unequal as they can be rendered, which con-
tributes to make the divisor by which that quantity is to be found, as
large, and itself of course, as accurate as the case will admit of.
Ex. Let the equations be
3 x+y 2z = o
5 3 .r 2 y + 5 * o
21 4.r # 4# o
14-h.r 3 # 3z = o
changing the signs of the last equation, and adding,
15 9x + y + 2z = o
similarly for #,37 5 or 1 y = o
forz, 33 # y Uz = o
From these equations x = 2.486
y = 3.517
z = 1.928
Second Method.
Let m + ax + by -f- cz + &c. = o,
m' + a' x -f b' y -f- c' z -f &c. = o,
m" -f- a"x -f b"y + c" z + &c. o,
&c. ,..,, , ,., ~ o r
104
E Q U
be the equations ; multiply the first by a, the second by a', and so on ;
then by addition,
ma 4- m> a' 4- &c. -f- (as 4- a'2 4- &c.) JP + (a 6 -f ' &' 4- &c.) y 4-
(a c 4- a' c' 4- &c.) z = o,
Similarly
(w 6 4- ' 6' 4- &c.) -I- (ab + a' b' + &c.) x -f (ft, + i'a + &c.) y -f-
(b c 4- b' c' 4- &c.) x 4- &c. = o,
(w c + ;' c x -f- &c.) + (ac + a' c' + &c.) ^ + (6 c + b' c' -f- &c.) ^ +
(c 8 + c'* H- &c.) 5r + &c. o,
&c. 4- &c .................................................................................. o.
By this means as many equations are formed as there are unknown
quantities, and from them #, y } z, &c. may be determined.
The method applied to the example in the preceding" article gives the
reduced equations
884-27^4-6^ = 0,
70 4- 6 x 4- 15 z o,
107 4- y 4- 51 x = o.
From whence x 2.470, y 3,551, z = 1.916.
The above mode of reducing the linear equations, which is called the
Me f hod of Least Squares, Avas invented by Gauss.
EQUATION of Payments.
Common rule.
Let p and p' be the sums due at the end of the times n and n' * =
equated time
p n -\- p> n'
then x = - -, - .
p-rp'
i.e. equated time is found by multiplying each sum by the time at which
it is due, and dividing by the sum of the payments.
This rule is erroneous in principle, being founded upon the supposi-
tion that the receiver gains interest upon the latter sum by receiving it
before it is due ; Avhereas in fact he ought only to gain the discount. In
most questions, however, that occur in business, the error is so trifling,
that the above rule Avill always be made use of as the most eligible me-
thod.
Correct rule.
Let r interest of 1. for one year, the rest as before, put
= ^ prnn> + p>n> + pn =
p r p
a V 4 b
105
E Q U
EQUATION of Time.
The equation of time, relatively to its causes, depends on two circum-
stances; (1) the obliquity of the ecliptic j and (2) the unequal angular
motion of the s^n in its orbit. The equation of time, as arising- from
the first cause, wouldbe " (Tie^'difference of the sun's longitude and its
right ascension converted into time. In the first and third quadrants,
apparent time would precede true ; in the second and fourth quadrants,
true time would precede apparent j and at the Tropics and Equinoxes,
true and apparent time would coincide. Also upon this supposition,
the equation would be a maximum at 4 points, viz. when the cosine of
the sun's declination is a mean proportional between radius, and the
cosine of the obliquity of the ecliptic.
The equation of time, as arising from the second cause, would be the
difference between the true and mean anomaly. Hence true and appa-
rent time would coincide at the higher and lower apsides. From the
higher to the lower apside, apparent time would precede true ; from the
lower to the higher apside, true time would precede apparent. The
equation, in this case, would be greater at two points than at any other,
viz. when the earth's distance from the sun is a mean proportional be-
tween the | axes of its orbit. To find it, when both causes are consider-
ed together, let A be the sun's time right ascension, M his mean longi-
tude, v the equation of the Equinoxes in longitude j then v X cos.
obliquity = the equation of the Equinoxes in right ascension, and
Equation of time = A - M -, X cos, obliquity
15
which is to be added to apparent time if positive, and subtracted if
negative.
As the sun's true right ascension is deduced from the true longitude
and the apparent obliquity of the ecliptic, both of which vary from one
age to another ; hence tables of the equation of time, constructed for
any one time, are not true for another. The following Table, there-
fore, taken from the Nautical Almanack for 1828, or leap year, though
inapplicable when any very nice determinations of the time are requir,
d, may yet be xiseful for regulating common clocks or watches, as the
error for the next half century will only amount to a few seconds.
106
E Q U
TABLE.
Equation of Time for every Day in the Year JS28.
^jJan. j Feb.
Mar.i Apr.
May.j June July.
Aug.!
Sept.
Oct.
Nov.
Dec."
Q Add Add
Add
Add jSub.
Sub. |Add
Add
Sub.
Sub.
Sub.
Sub.
m s \ m s
m s
m s
m s
m s
m s
m s
m s
m s
m s
m s
1 3 35 13 52
12 35
351
3 5
2 33 3 25
5 57
15
10 25
16 17
1038
24 414
12 23
3 So
3 13
2 24 3 37
5 53
31
1043
16 17
10 14
3 4 34 14 7
12 10
3 IS
3 19 2 14
3 48
5 49
53
11 2
16 17
9 50
4 4 59,14 13
11 5G' 3
3 26 2 4 3 58
5 44
1 12
11 20
16 16
926
5 5 27il4 18
11 42
2 4-.'
3 31
1 54 4 9
5 38
1 32
11 38
16 14
9 1
6 5 54 14 23
11 28 2 25
3 37
1 43 4 19
5 32
1 51
11 55
16 11
8 35
7 6 SO 14 21
11 17
2 7
3 41
1 33 4 29
525
2 11
12 12
16 8
8 9
8 6 46 14 30
10 59 1 50
3 45
1 21
4 39
5 18
2 32
12 29
16 3
7 43
9 7 11
10 7 36
14 32)10 43 1 33
14 34 10 28 1 17
3 48
3 51
1 10 4 48
58 4 57
5 10
5 1
2 52
3 12
12 45:i5 58
13 0'15 52
7 15
648
11 8 114 3510 12
12 8 25 14 35 9 55
1 1
45
3 53
3 55
046
034
554 52
5 13 4 43
3 33
3 54
13 15J15 45
13 30il5 37
6 20
5 52
13 8 48
1434
9 39
29
3 55
1 5 20
4 32
4 15
13 44 15 29
5 23
14 9 10
14 33
9 22
11
S 56
9 5 27
422
4 36
13 58
15 19
4 55
Sub.
Add
15 935
14 SO
9501
3 56
4
5 34
4 10
4 57
14 11
15 9
4 25
16 9 51
14 S?8
8 471 16
3 55
17
5 40
3 58
5 18 14 24
14 58
3 56
1710 15
14 24
8 30 30
3 54
30
5 45
3 46
5 39 14 36
14 46
3 27
1810 35
14 19
8 12 44
3 52
43
5 50 3 33
6 14 47
14 34
2 57
19 10 54
14 14
7 54 58
3 49
56
5 54 3 20
6 21
14 58
14 20
2 27
2011 12
14 9
7 36 111
3 46
1 9
5 58
3 6
6 42
15 8
14 6
1 58
21 11 30
14 2
7 18] 1 24
3 43
1 22
6 1
2 51
7 3
15 18
1351
1 28
22 11 47
13 55
7 o 1 sr>
3 39
1 35
6 4
2 37
7 24
15 27
1335
58
2312 3
13 47
6 41
1 48
3 34
1 47
6 6
2 21
745
1535
13 18
28
Add
24 12 18
1339
6 23
1 59
3 29
2
6 7
2 5
8 5
15 43
13 1
2
2512 33
13 29
6 4
2 10
3 24
2 13
6 8
1 49
8 26
15 50
12 42
32
26 12 47
13 20
5 46
2 '20
3 18
2 25
6 8
1 33
8 46
15 56
12 23
1 2
2713
13 9
527
230
3 12
2 38
6 8
1 16
9 7J16 1
12 4
1 32
28 13 12
12 59
5 8
2 40
3 5
2 50
6 7 58
9 26 16 6
11 43
2
29 13 23
12 47
4 50
2 40
2 57
3 2
6 5 41
9 46 16 10
11 22
2 30
30 13 33
431
2 57
2 50
3 14
6 3
23
10 6
16 13
11
3
31 13 43
4 13
2 41
6004
16 15
329
The above Table contains the equation of time for leap year; but the
equation ma.y be found for other years as follows. For the first year
after leap year take one-fourth of the difference between the equations
for the given and preceding days, w hich is to be added to the equation
for the given day, if at that time the equation is decreasing ; but sub-
tracted if it is increasing. In the second after leap year, take half the
difference between the equations ; and in the third, take three-fourths
of the difference, and apply this correction in the same manner as be-
fore.
107 F4
E Q U
Note. The word add in the Table denotes that the equation of time,
as there expressed, must be added to the apparent time, shewn by a
Dial or other instrument, in order to give the mean or equated time.
In those columns to which the word sub is prefixed, it implies that the
equation of time must be subtracted from the apparent time, in order to
give the true or correct time.
If it be proposed to convert mean time into apparent, this is done by
a contrary process, by applying the equation of time to the mean time
given, with its title or sign changed, viz. subtracting instead of adding,
and adding instead of subtracting.
EQUILIBRIUM of Floating Bodies. (Play fair, Eland.)
1. "When the centre of gravity of a floating body is in the same verti-
cal line with the centre of gravity of the fluid displaced, the body re-
mains in equilibrium.
2. If in a floating body, of which the transverse section is the same
from one end of the body to the other, a be the length of the water line,
c* the area of the section of the immersed part, d the distance between
the centre of gravity of the whole and the centre of gravity of the im-
mersed part, and i an indefinitely small inclination from the position of
equilibrium, the momentum of the force tending to restore the equili-
brium is
If 3 z is greater than d, the force tends to restore the body to ita
state of equilibrium, or the equilibrium is that of stability.
If -rpY d, there is no force tending either to restore or destroy
the equilibrium ; or the equilibrium is that of indifference.
If ^ 3 g be less than d, the force becomes negative, and tends to over-
set the body ; or the equilibrium is that of instability.
When W remains the same, the stability is proportional to
When the centre of gravity of the body is lower than the centre of
gravity of the immersed part, d is negative, and the quantity -^^ d
is affirmative, whatever be the magnitude of . t -*
~ 108
EVA.
If in the axis of the solid, or in the line passing through the two cen-
tres, there be taken a point distant from the centre of the immersed
part by f g , this point is called the metacentre ; and $ie stability
will be positive or negative or nothing, according as the metacentre is
above, below, or coincident with the centre of gravity of the floating
body.
3. If a rectangular parallelepiped float in a fluid, with its altitude a
perpendicular to the surface ; if its breadth be 5, and its specific gravity
n, that of the fluid being 1, its stability will be as w (1 w) a 8 .
When it has no stability, w (1 w) a* zr 0, and a = -=>
V6 n (1 w)
/I W
and n = | + V 3 .
4 6 a*
Cor. 1. When - ? is less than -7-, or when the height of the solid has
6 a 4 4
a greater proportion to the base of the section than V!T : Vlf, two va-
lues may assigned to the specific gravity of the body, which will cause
it to float in the equilibrium of indifference.
Cor. 2. If n = , as is nearly the case with fir, a = J v --r-
nearly. The truth of this conclusion may be shewn by experiment.
EQUILIBRIUM of an Elastic Body See Elastic Bodies equilibrium
of.
EQUILIBRIUM of a Point. See Forces composition of*
EQUINOXES, precession of. See Precession.
ERRORS in Time, in Astronomy. See Time*
EVAPORATION.
Mean monthly evaporation from the surface of water, from the ex-
periments of Dr Dobson, of Liverpool, in the years n72, 1773, 1774 A and
1775. (Phil. Trans.) and Manchester Memoirs.)
Inches. Inches
January ..... ............. 1,50 July .......*....*.............. 5.11
February 1.77 August 5.01
March- 2.64 September 3.18
April 3.30 October 2.51
May 4.34 November 1.51
June , mmmmfim ., mw , 4.41 December .,., 1.4&
109 G
From some very acnmito experiment^ made by Mr Dalton, the mean
annual evaporation, over the whole surface of the globe, has been esti-
mated at 35 inches ; this gives 94,450 cubic miles for the water annually
evaporated over the whole globe. tee Bain.
EVECTION.S'^ Moon.
EVOLUTES of Curves. (Higman.)
To find the equation to the evolute.
Let A N = , and N O /3, then may
the relation between a and /3 be found by
eliminating x and y from the equations
/ \
,N
JEx. Required the evolute of the parabola.
Here y* m y
Find values of ^ and .r from the two last equations, substitute them in
the first, and we shall have
.". the evolute is the semicubical parabola.
DEVOLUTION. See Involution.
EXPANSION of liquids and solids by Heat. See Heat.
EXPANSION of Water See Heat.
EYE, dimensions of, t$c,(Coddington.)
The proportions of the spaces occupied by the three humours of the
eye vary in different animals, as may be seen frm the following Table,
110
EYE
taken from M. Cuvier's Anatomie Comparee, which shews the parts of
the axis lying in the several humours.
Aqueous
Humour.
Chrystal-
line.
Vitreous
Humour.
]V[ an ,
3
4
15
T)(\ar
22
5
22
8
22
8
Ox
21
5
il
14
21
18
Sheep ... , .>...>.
$f
4
37
Jl
37
12
Ilor^e < in
17
9
17
16
17
18
Owl ,
43
8
43
11
43
8
Herring .'.,..
27
1
7~
27
5
7
27
1
7
The radii of the surfaces of the chrystalline are in
Man as 12 to 16
Dog 12 to 14
Ox 6 to 21
Rabbit 14 to 14
Owl 16 to 14
The specific gravities of the different parts are as folloxvs, that of
distilled water being 1.
In the In the
0-jf. Cod Fish.
Aqueous humour 1 1
Vitreous humour 1.016 1.013
Chrystalline lens (mean) 1.114 1.165
Outer part of ditto , 1.070 1.140
Inner 1.160 I 1.200
As to their refractive powers, they must be more considerable than
their density indicates, on account of the inflammable particles which
enter into their composition.
Dr. Wollaston makes the refracting power of the vitreous humour
equal to that of water, and that of the chry.stalliue lens of the ox jrroater
111
EYE
in the ratio of from 1.38 to 1.447 to 1. Dr. Brewoter gives the following-
Table, deduced from experiments made on a recent human eye :
'Water 1.3358
The Aqueous humour 1.3366
Vitreous humour 1.3394
.f "4 outer coat of chrystalline 1.3767
power of _ middle 13786
central parts 1.3990
whole chrystalline 1.3839
Dr. Brewster also gives the following dimensions :
Inch.
Diameter of the chrystalline , 0.378
cornea 0.400
Thickness of the chrystalline 0.172
coinea 0.012
If the humours of the eye be too convex or too flat, an imperfection
in vision is in either case the consequence : a conrave lens will remedy
the former defect, and a convex one the latter. The following problems
embrace nearly every thing connected with the theory of spectacles.
1. Given tlfb distance at which a short-sighted person can see distinct-
ly, to find the focal length of a concave glass which will enable him to
see distinctly at any other given distance.
Let A" = distance at which he can see distinctly, A a greater distance
at which he wishes to view objects, F focal length of the required
lens, then (see Refraction jjj, Art. 2.)
_---
A" ~ F A ' ~ A A"'
Cor. If A be indefinitely great, F = A".
2. Given the distance at which a long-sighted person can see distinct-
ly, to find the focal length of a convex glass which will enable him to
see distinctly at any other given distance.
Let A" distance at which he can see distinctly, A a shorter dis-
tance at which he wishes to view objects, F focal length of the lens,
then
1 1 1 . r A A"
2F = A* F ;andF== -A^A'
Cor. If A" be indefinitely great, or the eye require parallel rays,
F=A.
113
F I G
TABLE,
Of ihe focal length of the convex or magnifying glasses, commonly requir-
ed at various ages. ( Kitchiner.)
Years
of age.
Inches.
Focus.
Remarks.
40 ...
36
45 ...
50 ...
55 ...
30
24
20
Convex Spectacles are seldom want-
ed except to read by candle light,
till 45 or 50.
58 .
18
GO ...
16
Concave glasses called No. 1, are
65 ...
14
equivalent to a convex of 24
70 ...
12
inches focus ; No. 2 to a 21 inch
75 ...
10
convex ; No. 3 to an 18 inch.
80 ...
9
The following is an easy method of finding 1 which of two concave or
convex glasses magnifies most. Hold one in .each hand about one foot
from your eye, and about five feet from a window frame, and the lens,
through which the panes of glass appear least> magnifies most. This is
the readiest way of ascertaining their comparative power.
F
FIGURE of the sines, $c.
Figure of the sines, cosines, tangents, secants ; to find the area of,
4. Figure of the sines.
Let 6 arc or abscissa, then area =- r X ver. siu. 0,
When 6 is a quadrant, area - r*.
2. Figure ih in feet, and it be required to find the pres-
bure on agate, which, standing across the car. a!, would darn the water
up, we have area of trapez. | B -f- -b. d ; and depth of centre of gra-
26+~B . d . 6 -4- B ,.
vity = ^ ~ j . the whole pressure in ounces 500. ~ . d*.
3. B-f 6
M. The strongest angle of position for a pair -of gates for the lock of a
-rsuialor riv sin.s 9
g '
And in a direction perpendicular to that of its motion, is
Sfsin.0Xcos. e
2g
Ex. At what / must the rudder of a vessel be inclined to the stream,
that the effect produced may be a maximum ?
The effect varies (by the 3d Formula) as sin. 8 8 X cos. = max., .%
sin. 6 = J |.
3. If a plane figure, or a solid generated by the revolution^ a plane
figure round its axis, move in a fluid in the direction of its axis ; to deter-
mine the ratio of the resistances on the curve or surface, and on the base.
119
FLU
In a plane figure,
Res. on base I that on the curve ". y ' fl. ~-
And in a solid,
Res. on base : that on the surface :: # 2 : fi. ---35
l +7&*
Ex. 1. Let the curve be a semicircle.
Res. on base I res. on curve \\ y I y ^-^- t which, when y r> be-
comes as 3 : 2.
Ex. 2. Let the solid be a sphere,
Res. on base : res. on surface : : | y* '. | y* i 1 " ' * 2 I 1 wheny = r.
Hence resistance to a cylinder is double that of the inscribed sphere.
Cor. Hence if n density of a globe, whose radius is r, and the speci-
fic gravity of the fluid be 1,
R *r*v* lienv* 3v*
and R' =. = ; -- = -- - - = ^r - ; or if ar space fallen
w 4>g 3 IGgnr '
through to acquire the velocity
4. Let a sphere of given diameter be projected in a resisting medium,
whose specific gravity is to that of the sphere as 1 : n. Having given
the velocity of projection, to find the velocity of the sphere at any dis-
tance x y and the time of description.
Let e = No. whose hyp. log. = 1, and suppose when x o, # = a,
then
V 2ga
5. Let a spherical body descend in a fluid from rest by the action of
120
FOR
gravity (the rest as before), to find the velocity at any point of the do
scent, and the time of description.
Here V = J 8 r - n ~ l x J
\\
J -"^
71 r/ 2 r 1 + 1 P fi i-
r^rrr X hyp. log. -JL-.-L^-glJLL-
3*-l / --3.v
1 "" 1 e 8 u r
Cor. 1. If or be increased sine limite, vanishes, and V
fJIQgr. n 1 _ t j ie g rea test velocity that can be acquired by a spheri-
3
cal body descending in a fluid.
FLUID elastic. '-See Atmosphere.
FLUXIONS. See Differentials.
FORCES, the composition and resolution of. ( Wheu'ell,)
\. If any two forces act at the same point, the force, which is equiva-
lent to the two, is represented in direction and magnitude by the diago-
nal of the parallelogram, of which the sides represent the magnitude and
direction of the component forces.
Cor. If p and q be the component forces, which contain an angle 6, the
resultant will be V/> 2 -f- 2 p q cos. 8 -j- #2.
2. Forces may be represented by lines parallel to their direction, and
proportional to them in magnitude.
Cor. 1. If two sides of a A taken in order represent the magnitude and
direction of two forces, the third side will represent a force equivalent
to them both.
Cor. 2. If three forces, represented in magnitude and direction by the
three sides of a A taken in order, act on a point, they will keep it at rest ;
and conversely.
Cor. 3. If three forces keep a. body in equilibrium, and three lines be
drawn making with the directions of the forces three equal angles to-
wards the same parts, these three lines will form a A, whose sides will
represent the three forces respectively.
Cor. 4. If three forces keep a point at rest, they are each inversely as
the sine of the ^ contained by the. other two.
121 G 3
FOR
Cor. 5. If the ^ between two given forces be diminished, the resultant
is increased.
Cor. 6. If any number of forces be represented by sides of a polygon
taken in order, their resultant will be represented by the line which com-
pletes the polygon.
Cor. 7. A number of forces which are represented by all the sides of a
polygon taken in order, acting upon a point, will keep it at rest.
3. If the edges of a parallelepiped drawn from the same point, repre-
sent three component forces, the diagonal will represent the resultant.
Cor. 1. If any number of forces be represented by sides, taken in order,
of a polygon, which is not in the same plane, their resultant will be re-
presented by the line which completes the polygon.
Cor. 2. If any number of forces be represented by all the sides, taken
in order, of a polygon, they will keep a point at rest.
4. To find, by means of equations among the symbols, which the forces
and their positions introduce, the resultant of two forces acting at a
point.
If we suppose a line, as A x, to pass
through A, we may determine the posi-
tions, both of the components and resul-
tant, by the /s. which they make with
this line.
Let p and q be the forces in A P, AQ -,
a,, (B the /s. which they make with A x.
Resolve p into two forces in the direc-
tions A x t and A y perpendicular to
A x t then the resolved parts will be p -A. 3VI X.
cos. , p sin. <*. In like manner q is equivalent to q cos. /3 in the direc-
tion A or, and q sin. /3 in the direction Ay. Hence the forces are equi-
valent to
p cos. , q cos. /3 in A x.
p sin. , q sin. (Sin Ay.
And the resultant of p and q will be the resultant of these four forces.
If we put
p cos. a, + q cos. /3 = X.
p sin. -f- q sin. /3 = Y.
and take in A x, Ay, A M = X, A N = Y, and complete the rectangle
A M R N, A R will be the resultant ofp and q t and if r be this resultant,
and 8 the / which it makes with A x> we have
122
a
FOR
whence the magnitude and position of the resultant are known.
Cor. 1. By putting- the values of X and Y in the expression for r, we
shall get
r- ij $p* + 2pq cos. (* /3) -f q*
which agrees with the result obtained in Cor. Art. 1.
Cor. 2. If we call $ and ^ the /s. P A R and Q A R, we shall have
- - -- ^
y [^ a 4- 2^ ? COS. (-/3) + e? 2 j
& sin . 4 = _^ -- Ptinj.*^ __
y [^ 2 4- 2^? cos. (-/3) -f j*j
5. To find the resultant of any number of forces, p> p, p, ......... p, ia
123 n
the same plane ; their directions making with the line A x angles
, , at, ......... ce, respectively.
123 n
J5y proceeding precisely as before, we shall have, by putting
p cos. -f p cos. os. -f p cos. a. ...... + p cos. a, = X
'2233 n n
p sin. a + j sin. + j sin. ...... +p sin. * = Y
/ / a 2 3 s n n
r=v(X8 + Y)j tan.fl=~.
6. To find the resultant of forces, whose directions are not all in the
same plane.
In the preceding case, the forces were resolved in the directions of two
lines at right /s. to each other. In this case we must resolve them in
the directions of three lines each at right /s. to the other two, and meet-
ing together in a point. Let us suppose these three lines to be A A*, Ay
A z, and let p be a force, and , /3, y the ^s. which it makes with A x t
Ay, A x ; the force will then be equivalent to three forces
p cos. a. in A JT, p cos. ft in Ay t p cos. -y in Az.
Hence if we have forces p t p, p ...... p
making with A x angles , , ......
183 n
123
F R A
with A y angles ft, /3, (3 3
* 3
with A wangles 7, ^, y y
i s w
and make
p COS. -}- jtf COS. +/> COS. a, -f- p COS. a =: X
112 233 n n
p cos. ,3 -f p cos. /3 -f ;> cos. /3 + j cos. = Y.
1 1 2 2 3 3 ?J 7i
p COS. y -f-P COS. y -i- p COS. 7 -\- p COS. ^ Z
112833 n n
the forces will be equivalent to X in A .r, Y in A j/, and Z in A^.
If R be the resultant, and 9, v, the /s. which it makes with A r, Ay,
A # respectively, we shall have
R = v ' (X* + Y* + Z)
. X Y Z
cos. & , cos. j = -^, cos. ? -^
One of the three last Equations is superfluous.
7. When a point is acted upon by any forces, to find the conditions of
equilibrium.
In order that there may be an equilibrium, the resultant of all the
forces must be o. And in order that this may be the case, it is evident
we must have in Art. 5, X = o, Y = o; and in Art. 6, X = o, Y = o t
Z o. Hence we have for the conditions of equilibrium in the former
*ase
P COS. K, +p COS. a, -I. p COS. a. 4- ~ O
1 12 2 ^3 3
p sin a. 4- P sin. et. -4- p sin. ex, -4- =: o
1 12 23 3
And in the latter case
p cos. u. -\-p cos. u, -4- p cos. a, 4. = o
1 1 1 2 83 3
p cos. /3 + p cos. /9 4- p cos. 8 + ...... = o
1 12 23 3
p cos. y -{- /> cos. y 4- P cos - y +
FORCE. Sec iVo^iw//.
FORCE mating, or motive. .SVt 3 Momentum.
FORCES, centripetal and txntrifirgalrSee C&ifrat'Itorvet.
FR ACTIONS continued
Continued fractiww ar- very useful when we have a fraction or ratio
F R A
in very large numbers which are prime to one another, as by their means
we may find an approximate value in less terms.
To represent y in a continued fraction.
Divide as in the rule for finding the greatest common measure, thus
ft) a (p a I
__
d)c(r 9+ r
e)d(s T+-J&C.
&c. &c.
The first approximation is p, which is too small, the next p 4. - ,
1 9
which is too large, the next p + - p which is too small ; and thus
t + 7
\ve may form a series of fractions, each succeeding one being nearer the
true value of the proposed fraction than the one which preceded it.
This series of fractions requires some trouble in their formation after
the first two or three ; but the 3d, 4th, &c. may be expeditiously found
thus. Arrange the figures of the quotients in a line, as
P) q, r, s, t, &c. let the successive fractions be -~, y, , , , &c. then
to find any of them after the 2d, as -^, we have - = ^ ! 1^ J j =
*g+._ ; ~ = - !t-, &c. where the law of formation is evident.
s h ~\-f n tl -f- n
Ex. To approximate to J,'^ , proceeding as if finding the greatest
common measure we have for the quotients
3, C, 1, 1, 2, 1, &c.
1 1 19
Now first approximation p 3 ; 2d. = p -f = 3 -f ~ = , ,*.
vve have by the rule the following series of fractions
3, W f , I ^L, * where 3 is too small, too large, &c.
FRACTIONS vanishing.
If u , where P and Q are functions of x, which are both = o,
Q
when x a, then the value of u, in this case, is the same as the values
. d P ds P d* P .
in this case of ^-, ^, ^, &c.
125 G4
F R I
tience the value of a vanishing- fraction may bo found by differentia-
tion, as in the following examples :
r 2 _ yt
Ex. 1. Required the value of - when x .
rfP Zxtl-x
Ex. -2. Required the value of ' ' __ ' . when x 1. f
But if it so happen that on substituting a instead of x in -r^y this
fraction also becomes , we must treat it in the same manner as the
o
first, and so on, till we arrive at a value of which one term at least is
finite.
p a a' 2 4- a c 2 2acx o
Ex. Let TT = ~T -TV -- r~r- 5 > which = when x c.
O bxZ ^_ 2bcx -|- ic 2 o
But ^ = -7- which is the value of -^ in this case.
d*Q b Q
FREEZING. See Congelation.
FRICTIO N. (Play fair. )
The following must only be considered as a short abstract of the most
interesting general results on the subject of Friction, as deduced from
experiments made by Coulomb and others.
1. The retardation which friction opposes to motion is nearly uyform,
or the same for all velocities.
2. The force of friction is the greater, the greater the force with which
the surfaces, moving on one another, are pressed together, and is com-
monly equal to between | and of that force j but it is very little affect-
ed by the extent of the surfaces.
M. Friction may be distinguished into two kinds, that of sliding, and
that of rolling bodies. The force of the latter is very small compared
with that of the former.
I, Hie distance to which a given body will be moved by percussion in
F R t
opposition to friction, is as the square of the velocity communicated to
it. Thus a nail is driven by a blow of no great force, into a piece of
wood where the mere friction is sufficient to retain it against a great
force applied to draw it out.
5. When motion begins, the intensity of friction diminishes j it does
not, however, change afterwards as the velocity changes, but continues,
as already said, to retard with a uniform force. Coulomb found the
friction of wood sliding on wood to become less when the body began to
move, than it had been the instant before in the ratio nearly of 2 to 9.
6. Friction may be measured by placing the body on a plane of vari-
able inclination, and increasing that inclination till the body begin to
slide. If the weight of the body W, and the inclination of the plane
when the body begins to slide = 0, the friction W X tan. 0.
7. Time is often required for friction to attain its maximum, and in
this respect different substances differ much from one another.
8. Friction is diminished by unctuous substances ; those that are thin-
nest and least tenacious are the best ; plumbago reduced to powder, and
rubbed on the surface of wood, metal, stone, &c. serves greatly to di-
minish friction.
9. The effect of friction may be diminished by drawing a body in a line
inclined at a certain angle to the plane on which it rests. Thus if the
weight of a body be to its friction on a horizontal plane as n to 1, it will
be drawn with the greatest ease in the direction which makes with that
plane an angle, having for its tangent .
10. The friction of cylinders rolling upon an horizontal plane is in a
direct ratio of their weights, and in the inverse ratio of their diameters,
11. The momentum of friction is diminished by friction wheels in the
ratio of the radius of the axis of any one of the wheels (they are suppos-
ed equal) to the perpendicular height of the axis that rests upon them,
above the line joining their centres,
12. In wheel carriages, the plane on which they move, and the line of
draught, being both horizontal, the advantage for surmounting an im-
movoable obstacle, of a given height, is as the square root of the radius
of the wheel.
Let the whole weight to be moved be W, the radius of the wheel r,
f the force which drawing horizontally will raise the carriage over an
immoveable obstacle of the height h s then/= W X V - .
127 H
1 R I
13. The stiffness of rop
Diluted nitric acid
.. 7 parts. Fj . om + 320 tQ _ 3()0
From + 32 to 40.
Muriate of lime
Potash
..3 parts. From + 3 2to _ 51 o.
Greatest artificial cold yet measured 9
G
G AUG1NG. ( Hutton. )
Rule for finding the dimensions of a cask, in wine, ale, or imperial gal-
Jonfi.
Let B bung- diameter, H head diameter, L = length of cak, all
in inches ; then
(39 B 3 + 25 H* + 26 B H) X
114
Is the content in inches, which being divided by 231 for wine gallons ; or
by 282 for ale gallons ; or by 277.274 for imperial gallons, will be the con-
tent required.
GEOMETRICAL Prugrexfion.Stf Progression.
GEORGIUM Sidus.
This planet was discovered by Dr Herscnel, March 13, 1781. For its
elements, &c. w Planet*, detuentx of; and for it* satellites', we Satel-
lite*.
GOLDEN Kumlvr. See Cycle.
GBAVJTY, CeufrrtfSce Centre of Gravity.
G R A
GRAVITY specific. ' J'inct, Bland.)
1. Of the specific gravities of a body and fluid, having 1 given the one,
to find the other.
Case 1. When the body is heavier than the fluid.
Let w weight lost by the body when immersed in the fluid, W its
whole weight in vacuo, * spec. grav. of the fluid, S .= that of the body j
then
w : w : : : s.
whence * or S may be found.
\V
Cor. I. If different bodies be weighed in the same fluid, S is as .from
w
whence we may compare the spec. grav. of two bodies.
Cor. 2. If the same body is weighed in different fluids, s is as tc ; from
whence \ve can compare the spec. grav. of t\vo fluids.
Case 2. When the body Q is lighter than the fluid in which it is weigh-
ed.
Connect it with a heavier body P, so that together they may sink.
Find the weight lost by P -f Q, and the weight lost by P, when im-
mersed ; then the difference = the weight lost by Q ; and .*. its specific
gravity may be found by the last case.
2. If the specific gravity of air be called m, that of water being 1, and
W the weight of any body in air, and W' its weight in water ; then its
weight in vacuo is nearly
W + m (W -, W).
3. If ff be the specific gravity of a body ascertained by weighing it in
air and water, and m the specific gravity of the air at the time when the
experiment was made j the correct specific gravity, or that which would
have been found, if the body had been weighed in vacuo, instead of air,
is
ff 4. m (1 . 1-500 '00176
Nitrous iras . . 1'lUt
134
G R A
Kirwan. Lavoisier.
Barometer, 30. Thermometer 52o.
Hepatic gas . . 1-106
Oxygen gas . . M03 '00137
Atmospheric air . I -000 -00128
Nitrogen gas . . -985 -00120
Ammoniacal gas . '600
Hydrogen gas . . '084 -OOOOU6
VEGETABLE PRODUCTIONS.
Sugar, white .,..., 1606
Gum Arabic ,..,.. 1452
Honey ....... 1450
Catechu .,,,,,. 1398
Pitch ,,.,.,. 1150
Copal, opaque . , , . . . 11-10
Yellow amber ...... 1078
Malmsey,. Madeira ..... 1038
Cider 1018
Vinegar, distilled . , . . . 1009
Water at 60 . . . f . 1000
Bourdeaux wine , 994
Burgundy wine .. 991
Turpentine liquid ..... 991
Camphor .. . . . 988
Linseed oil ...... 940
Elastic gum f ^ 933
ANIMAL SUBSTANCES.
Pearl ....... 2750
Coral ....... 2680
Sheep's bone, recent ..... 2222
Oyster shell . . . . . 2092
Ivory ....... 1917
Stag's horn ...... 1875
Ox's horn ...... 1840
Isinglass . . . . . . 1111
Egg of a hen ...... J090
Human blood ....... 1053
Milk cow's ...... 1032
Wax, white ....... 968
yellow ...... 965
135
G R A
Spermaceti
Butter .
Tallow .
Fat of hogs
veal
mutton
beef
Ambergrease
Lamp oil
343
943
9-12
937
934
923
923
926
923
Pomegranate tree
Lignum vitse
Box, Dutch
Ebony .
Heart of oak, 60 years felled
Oak, English, just felled "I
the same, seasoned J
usually stated at
Bog oak, of Ireland .
Teak, of the East Indies
Mahogany
Pear tree trunk
Medlar tree
Olive wood
Logwood
Beech ...
Ash
Yew, Spanish .
Dutch
Alder .
Elm
Apple tree
Plum tree
Maple .
Cherry tree
Quince tree
Orange tree
Walnut .
Pitch pine
Red pine
^Yellow pine
1354
1333
1328
1177
1170
C1113
i 743
925
1046
from 745 to 657
from 1063 to 637
646
944
927
931
852
, from 845 to 600
807
800
from 800 to 600
793
755
755
715
705
'705
671
660
657
529
GUN
White pine ... . . . 420
Fir, of New England ..... 553
of Riga ...... 753
of Mar Forest, Scotland .... 696
Cypress ....... 644
Lime tree ...... 604
Filbert wood ...... 600
Willow ...... 585
Cedar ....... 560
Juniper ....... 556
Poplar, white Spanish .... 529
common ..... 383
Sassafras wood ..... 482
Larch, of Scotland ..... 530
Cork ....... 240
GREGORIAN Calendar. See Calendar.
GULDINUS' Property. See Solids and Surface*.
GUNNERY, leading principles of.(Hutton.)
1. To find the initial velocity of a shot.
Let JP = weight of powder, B of the ball, v the initial velocity, then
- 2000 >.
Cor. 1. The initial velocity of a shot varies from 1600 to 2000 feet per
second.
Cor. 2. B e* = (2000)*. P, i.e. the effect of a shot is nearly as the quan-
tity of gunpowder.
2. If w = weight of any ball, d its diameter.
w .5236 d 3 in pounds.
3. To find the resistance of the air to any ball or projectile.
Let d = diameter of ball, v its velocity, r = resistance in avoirdupois
pound?, then
(ft x 2 e* N
-1660(3600-*-)
Ex. Resistance to a ball, whose diameter = 2.78 inches (or weight
31bs.), when thrown with a velocity of 1800 feet per second, 176 Iba.,
more than 58 times ite own weight.
4. Supposing the air to resist according to the law just assigned, re-
quired the height to which a ball will ascend perpendicularly,
137
H A R
Let d diameter of ball, c tho velocity of projection, It height as-
cended, then
Ex. A ball of 1.05 Ibs., discharged with a velocity of 2000 feet, will
ascend to the height of 2920 feet j in vacua it would have ascended to the
height of 1 If miles.
5. If a body descending in the atmosphere has acquired such a velocity
that the resistance is equal to its weight, the accelerating and retarding
forces being equal, its motion will become uniform; to find this termi.
nal velocity,
o 7 ,s
.556
a quadratic equation, from whence v may be found.
Ex. For an iron ball of 1 Ib, the terminal velocity 244 feet ; for one
of 42 Ibs. it is 456.
6. The best charge of powder is about or of the weight of the
1
ball j for battering : a 24-pounder with 16 pounds of gunpowder at
o
an elevation of 45 ranges 20,250 feet, about ~ of the range that would
take place in a vacuum. The resistance is at first 400 poands or more,
and reduces the velocity in a second from 2000 to 1500 feet in the first
1500 feet. (Young's Nat. Phil.)
GUNPOWDER. &?e Gunnery and Steam.
GYRATION, Centre of. See Centre of Gyration.
H
HARMONICAL Progression. Sec Progression,
HARVEST Moon.(MaddyJ
To find the retardation of the Moon's rising on successive nights.
Let the moon's daily motion = m, the inclination of the moon's orbit
to the horizon , latitude of the place = /, moon's declination = o t
then the difference of the times of rising on succeeding days (D) is
_^ __ m. sin, n
Vcos.*3 sin.*;
133
H E A
Hence may be explained the phenomenon of the Harvest Moon, pre-
mising that when the 1st point of Aries rises, the ecliptic makes the least
angle with the horizon. For if the moon's orbit be supposed to coincide
with the ecliptic (which it does nearly) sin. n is least when the moon
rises in Aries ; therefore the numerator of the above expression is then
least ; and because cos.* 51, the denominator is then greatest; .*. on
both accounts D is least, and if the sun be at the same time in Libra, the
moon is then at the full ; therefore the full moon, which takes place near
the autumnal equinox rises nearly at the same time for several nights,
and as this is near the time of harvest in north latitudes, it is called the
Harvest Moon.
HEAT, various Talks relating to.
TABLE I.
Table oftJie effects of heat on different substances according to Fahrenheit'*
thermometer and Wedgwood's. ( Wedgwood.)
Fahr. Weds:
Extremity of the scale of Wedgwood ^^ 32277 240o
Greatest heat of his small air furnace ~~~ 21877 160
Chinese porcelain softened ^^ ^. - 156
Cast iron melts ~~ - - 17977 130
Greatest heat of a common smith's forge *~~ 17327 125
Derby porcelain vitrifies ~~+ *** - 1 12
Welding heat of iron greatest ~~~ v~- 13427 95
least - - 12777 90
Fine gold melts _ __ _ 5237 32
Fine silver melts - ^ _ 4717 28
Swedish copper melts ,~~~ *~~ 4587 27
Brass melts ~~> - - .~~, 3807 21
Enamel colours burnt on ~~~ *~~ 1857 6
Hed heat fully viable in day light ~~ 1077
in the dark -- 917 1
fiT*?
Mercury boils - - 600 ---- 3^
Water boils -- - *~~
Vital heat - -- -
Water fropzps ^^ - ^^
Proof spirit freezes **, ^^ * ,
139 H 2
H A R
Let d = diameter of ball, c the velocity of projection, 7i r= height as-
cended, then
Ex. A ball of 1.05 Ibs., discharged with a velocity of 2000 feet, will
ascend to the height of 2920 feet j in vacua it would have ascended to the
height of llf miles.
5. If a body descending in the atmosphere has acquired such a velocity
that the resistance is equal to its weight, the accelerating and retarding
forces being equal, its motion will become uniform ; to find this termi-
nal velocity,
a quadratic equation, from whence v may be found.
Ex. For an iron ball of 1 Ib, the terminal velocity 244 feet ; for one
of 42 Ibs. it is 456.
6. The best charge of powder is about - or of the weight of the
!
ball ; for battering : a 24-pounder with 16 pounds of gunpowder at
6 1
an elevation of 45" ranges 20,250 feet, about of the range that would
take place in a vacuum. The resistance is at first 400 pounds or more,
and reduces the velocity in a second from 2000 to 1200 feet in the first
1500 feet. ( Young's Nat. Phil.)
GUNPOWDER. See Gunnery and Steam.
GYRATION, Centre of. See Centre of Gyration.
H
HARMONICAL Progremon.See Progression.
HARVEST Moon,(MaddyJ
To find the retardation of the Moon's rising on successive nights,
Let the moon's daily motion = m, the inclination of the moon's orbit
to the horizon w, latitude of the place = I, moon's declination = J,
thru the difference of the times of rising on succeeding days (D) is
t) _ m > sin, n
"" Vco8.sin.*/
H E A
Hence may be explained the phenomenon of the Harvest Moon, pre-
mising that when the 1st point of Aries rises, the ecliptic makes the least
angle with the horizon. For if the moon's orbit be supposed to coincide
with the ecliptic (which it does nearly) sin. n is least when the moon
rises in Aries ; therefore the numerator of the above expression is then
least; and because cos.* S 1, the denominator is then greatest; .". on
both accounts D is least, and if the sun be at the same time in Libra, the
moon is then at the full ; therefore the full moon, which takes place near
the autumnal equinox rises nearly at the same time for several nights,
and as this is near the time of harvest in north latitudes, it is called the
Harvest Moon.
HEAT, various Tables relating to.
TABLE I.
Table oftJie effects of heat on different substances according to Fahrenheit'*
thermometer and Wedgwood's. ( Wedgwood.}
Fahr. Wcdg.
Extremity of the scale of Wedgwood ~~+ 32277 240o
Greatest heat of his small air furnace 21877 160
Chinese porcelain softened . ^^, 156
Cast iron melts .^ 17977 130
Greatest heat of a common smith's forge *~~ 17327 125
Derby porcelain vitrifies ~~> *~~ 1 12
Welding heat of iron greatest ,~~ 13427 95
least 12777 90
Fine gold melts ,~~ 5237 32
Fine silver melts +~~ ~~~ 4717 28
Swedish copper melts ~~~ ^~, 4587 27
Brass melts ~~* 3807 21
Enamel colours burnt. on ~~ ~~~ 1857 6
lied heat fully visible in day light *~* 1077
in the dark ,~~ 917 1
Mercury boils 600 3^
Water boils
Vital heat
Water froo/ps ^^ *~~ *^*
Proof spirit fre2CS ,*** ~*~ mw M
m H a
H E A
Fahr. IVedg.
Mercury freezes ***** ***** ***** 40 ... - %\^QQ
TABLE II.
Table of the congealing or concreting temperatures of various liquids by
Fahrenheit's scale. ( Ure.}
Sulphuric ether ***** ***** ***** 46o
Liquid ammonia ***** ***** ***** 46
Nitric acid sp. gr. 1.424 ***** ***** ***** 45.5
Sulphuric acid sp. gr. 1.6415 ***** ***** 45
Mercury ***** *~~ ***** 39
Nitric acid sp. gr. 1.3290 ***** ***** 2.4
Brandy ~*~ 7.0
Alochol 1, water 1. . 7
Alcohol 1, water 3 -f- 7
Oil of turpentine ***** ***** ***** 14
Strong wines ***** ,~ ***** ^- 20
Blood 25
Vinegar *~++ **** ^*w. ***** 23
Sea water ***** ***** ***** 28
Milk ***** ***** ***** ***** 30
Water ***** *. , . ***** 32
Olive oil ***** * ***** ***** 36
Sulphuric acid, sp. gr. 1.7il ***** ***** 42
Tallow 92
Spermaceti ***** ***** ***** 112
Yellow wax ***** ***** ***** 142
White do. 155
Tin 442
Lead . 612
Zinc ***** ***** ***** ***** 680
The concreting temperature of the bodies above tallow in this Table,
is usually called their freezing or congealing point, and of tallow and the
bodies below it the fusing or melting point.
TABLE III.
Table of the boiling points by Fahrenheit's scale of a few of the most im-
portant liquids, under a mean barometrical pressure of 30 inches.
(Ure.)
Ether sp. gr. 0.7365 at 48 9 Gay Lussac ***** 100 9
Alcohol sp. gr. 0.813 ^^ ^^ Ure *,, ,^~ 173.5
140
H E A
210
212
, 232
~, 222
r~~ 220
316
~~ 240
~~ COO
640
*~* 656
TABLE IV.
Boiling temperature of water.
Heiglit of the boiling point in Fahrenheit? s Thermometer at different
heights of the Barometer.
Nitric acid sp. gr.
1.500
* Dalton
Water
v~ ,w
Muriatic acid sp. gr. 1.094 ^
Dalton
Do.
1.047 _
Do.
Nitric acid ^^.
1.16 ^
Do.
Oil of turpentine
~v~ w*
- Ure ~
Sulphuric acid, sp
, gr. 1.30
Daltori
Do.
1.848
. Ure
Linseed oil ~^,
Mercury ^^
* *VV
Barom. Ht. of boiling point.
10.0 2J30.571
30. 5 212- 79
30- 212. 00
29. 5 211. 20
29. 210. 38
28. 5 209. 55
28. 208. 69
27. 5 207. 84
27. 206. 96.
And in general Dr Horsley's rule dedu-
ced from De Luc is, height = -^ ~
8990000
log. #. 92.804, where z height of Baro-
meter in lOths of an inch.
In an exhausted receiver water boils at
98. or looo. in Papin's digester at 412<>.
From tliis variation in the height of the boiling point, arising from the
variation of the pressure of the atmosphere, an ingenious instrument
called the Therm ometrical Barometer has been invented by Mr Wol-
laston, for ascertaining the heights of mountains ; it appearing from
General Roy's experiments, that a difference of 1. in the boiling point
corresponds to 535 feet in height. Let .*. n difference of boiling points
at the bottom and top of a mountain, then 1 \ no I*. 535 feet I n X 535
= approximate height. To correct it for the temperature of the air, let
m mean temperature of the top and bottom, ascertained by a common
thermometer, then (see Barometer) n, 535 X (1 + m ~ 38 X .00244)
correct height.- (Phil. Trans.)
TABLE V.
Linear expansion of solids by heat.
Dimensions which, a bar takes at 212 whose length at 32 if 1.000000.-^.
(Urej
Glass tube ^ Smeaton 1.00083333
Do. ~~ .^^ ^^^ Roy ~~ *~** 1.00077615
Deal *** .^, r^, Roy, as glass
H E A
Tlatina
***** Troughton
1.00099190
Cast iron prism
***** Roy ***** *****
1.00110940
Steel rod '
***** Rov , **^.
1.00114470
Iron ~~* -
, , Smeaton *****
1.00125800
Iron wire *****
^ Troughton ^^
J.00144010
Gold *****
Ellicot
1.00150000
'Copper *****
~~* Trough ton *~~
1.00191880
Brass *****
~ . Laplace *~~
1.00186G7I
Brass wire ~ .
*** Smeaton *~~*
1.00193000
Silver *****
*~~ Troughton ^^~
1.0020826
Tin ** M
~~+ Laplace -*v^
1.00217298
Lead
~~~ Smeaton ,,
1.00-^86700
TABLE VI.
Expansion of liquids.
Dtlatattoti of the volume of liquids by being heated from 32 to 212. ( Ure.)
.Mercury ***** Lord C. Cavendish *****
0.018870
1
53
Do. ***** Roy *****. ***** *****
0.017000
1
59
Do. ***** Shuckburgh ***** *****
0.018510
1
54
Do. ***** Du Long and Petit *****
0.0180180
1
5575
Do. , Do. from 212 to 392
0.1301843
1
54i;
Do. ***** Do. from 392o to 5720
0.0188700
1
53
Water Kirvvan from 39o. its max. dens.
0.04332
Muriatic acid sp, gr. 1.137 Dalton *****
0.0600
1
17
Nitric acid, sp. gr. 1.40 ***** Do. *****
0.1100
~
Sulphuric acid sp. gr. 1.85 Do. *****
O.OtJOO
1
17
Alcohol **+ *,***. Do. *****
0.1100
1
Water saturated \vith salt Do, *~*.
0.0500
i
142
H R
Sulphuric ether
~~ Do. v
^ e.0700
14
Fixed oils ,**
~*~ Do: *
~* 0.0800
1
1-2.5
TABLE VII.
Expantiorfpf water. ( UreJ
The maximum density of water is at 39., and it is a singular fact that
the expansion of water is the same for any number of degrees above or
below the maximum of density ; thus the density of water at 32 and at
4fio is precisely the same. The following Table, the result of experi-
ments by Sir Charles Blagden and Mr Gilpin, shews this in a clear light.
Sp. Gr.
Bulk of
water.
Temperat.
Bulk of
water.
Sp. Gr.
1.00000
390
1.00000
1.00000
1.00000
38
40
1.00000
1.00000
0.99999
1.00001
37
41
1.00001
0.99999
0.99998
1.00002
36
42
1.00002
0.99998
0.99990
1.00004
35
43
1.00004
0.99996
0.99994
1.00006
34
41
1.00006
0.99994
099991
1.00008
33
45
1.00008
0.99991
0.99988
1.00012
32
46
1.00012
0.99988
This law of maximum density does not prevail in the case of sea water ;
on the contrary, Dr Marcet found that sea water gradually increases in
weight down to the freezing point.
HORIZON, Dip or depression of.
In observing an altitude at sea with the sextant or reflecting circle,
the image of the object is made to coincide with the visible horizon, but
as the eye is elevated above the surface of the sea by the height of the
ship's deck, the visible horizon will be below the true horizontal plane.
The following Table gives the dip or apparent depression of the hori-
zon for different elevations of the eye, allowing y-r for terrestrial refrac-
tion. The dip must be always subtracted from the observed altitude
when taken, by the fore obscrvatioDj but added to it in the back observa-
tion.
145
H O R
TABLE I. Of the dip of the horizon.
H. of
Eye.
Dip of i
Horiz.
H. ot I
Eye.
Dip. of
Horiz.
H. of
Eye.
Dip of
Horiz.
H. of
Eye.
Dip of
Horiz.
Feet.
1
U
1*
If
2
7
13
19
24
Feet,
6
6*
2 26
2 32
2 38
2 43
2 48
Feet.
16
1?
B*
3 58
4 2
4 5
4 9
4 12
Feet.
32
33
34
35
36
5 37
542
5 47
5 53
5 58
Ol GM
2-43254
2-898278
4-103933
5-791816
8-147252.
37
2-49335
2-985227
4-268090
6-081407
8-636087
38
2-55568
3-074783
4-438813
6-385457
9'15k'52
39
2-61957
3-167027
4-616366
6-704751
9-703507
40
2-68506
3-262038
4-801031
7-039989
10-285718
41
2-75219
3-359899
4-993061
7-391988
10-902861
42
2-82100
3-460696
5-192784
7-761588
11-557033
43
2-89152
3564517
5-400495
8-149667
12-250455
44
2-96381
3-671452
5616515
8-557150
12-985482
45
3-03JJ90
3-781596
5-841176
8-985008
13-764611
46
3-1 1385
3-895044
6-074823
9-434258
14-590487
47
3 : 19170
4-011895
6-317816
9-905971
15-465917
48
3"i7l49
4-132252
6-5705S8
10-401270
16-393872
49
3-35328
4-256219
6-833349
10-921333
17-377504
50
3-43711
4-383906
7-106683
11-467400
18-420154
TABLE II.
iig the present rain? of one pound to be received at the end of any
number of years not exceeding 50.
Years. 2% per Cent 3 per Cent, j 4 per Cent. 5. per Cent. 6 per Cent.
1
975610
970874
961538
952381
943396
2
951814
942596
924556
907029
889996
3
928599
915142
888996
883838
839619
i
905951
888487
854804
822702
792094
5
888864
862609
821927
783526
747258
6
862297
R37-484
790315
-746215
704961
7
841265
813092
759918
710681
665057
8
820747
789409
730690
676839
627412
9
8C0728
766417
702587
6-1461B -
591898
10
781193
744094
675564
613913
558395
11
762145
722421
619581
584679
526788
12
743556
^01380
624597
556837
496969
13
725420
680951
600574
530321
468839
14
707727
661118
577475
505068
442301
15
690466
641862
555265
481017
417265
16
673625
623167
533908
458112
393646
17
657195
605016
513373
436297
371364
18
641166
587395
493628
415521
350344
19
6255->8
570286
474642
395734
330513
20
610271
553676
456387
376889
311805
21
595386
537549
438834
358942
294155
22
580865
521893
421955
341850
277505
23
566697
506692
405726
325571
261797
24
552875
491934
390121
310068
246979
25
539391
477606
375117
295303
232999
26
526235
463695
360689
281241
219810
27
513400
' '450189
346817
267848
207368
28
500878
437077
333477
255094
195630
29
488661
424346
320651
242946
184557
30
476743
411987
308319
231377
174110
31
465115
399987
296460
220359
164255
32
453771
388337
285058
209866
154957
33
44>703
377026
274094
199873
146186
34
431905
366045
263552
190355
137912
35
421371
355383
253415
181290
130105
36
411094
345032
243669
172657
122741
37
401067
334983
234297
164436
115793
38
391285
325226
225285
156605
109239
39
381741
315754
216621
149148
103056
40
372431
306557
208289
142046
097222.
41
363347
297628
200278
135282
091719
42
354485
288959
192575
128840
086527
43
345839
280543
185168
122704
081630
44
337404
272372
178046 .
U6861
077009
-15
329174
204139
171198
111297
072650
46
321146
256737
164614
105997
068538
47
313313
240259
158283
100949
061658
48
305671
2-11999
152195
096142
050998-
49
298216
23-1950
146341
091561
057546
.50
29094-2
228107
140713
087204
054288
INT
INTERPOLATIONS.!' Woodhouse, Vines.)
If a, a', a", &c. are successive values of a quantity a, differing by a ooa-
stant interval 1, and if the 1st, 2d, 3d, &c. differences be d' t d", d"' t &c. i
then any intermediate value (#), distant from a by the interval x, is equal
. ..
A r ofe In taking- the differences, the preceding quantity must always
be subtracted from the succeeding} they will .". be positive or negative
according as the series of quantities is increasing or decreasing.
If the law of the quantities be such that their last differences always
become o, we shall get at any intermediate time the accurate value of
that quantity ; but if the differences do not at last become accurately ~
o t we shall then get only an approximate value.
In general the quantities d', d", &c. diminish very fast, and it will not
often be necessary to proceed farther than d'".
Ex, 1. Given the squares of 2, 3, 4, and 5, to find the square of 2|.
4, 9, 16, 25 ...... quantities
5, 7, 9 ......... 1st order of differences.
2, 2 ......... *. 2ddo.
.............. 3d do.
Here a = 4, d 1 5, d" =. 2, d'" = 0, x the required interval ; .".
y - 4 + i X 5 1 X 2 = 6, 25.
Ex. 2. Given the log. of 110 2.04139, of 111 = 2.04532, of 112 ~
2.04922, and of 113 = 2.05308] required the log. of 110.5.
2.01139, 2.04532, 2,01922, 2,05308
.00393, .00390, .00386
. 00003, . 00004.
Here a - 2.04139. d' = .00393, d" - .00003, and x = $, /.
?/ - 2.01139 + .1 X .00393 J- X -00003 = 2.043359.
Ex. 3. Given five places of a comet as follows ; on Nov. 5th at 8h.
llm. in Cancer 2. 30' = 150' ; on the Gth at 87*. \~*m. in 4. 1' = 247' ; on
the 7th at Q/t. Urn. in 60. 20' = 380' ; on the 8th at 87*. 17w. in 9. 10'
550' ; on the 9th at 87*. 170*. in 12. 40' = 760'. To find its place on the
7th at 14ft. 17z.
First subtract 5d. 87*. Mm. from Id. 14ft. I7w., and there remains 5d. 6ft,
^ 2,25 for the interval of time between the first observation and the
given time at which the place is required ; this .'. is the value of A; t^
which we want to find the corresponding value of y ; hence
150, 247, 380,. 550, 760
97, 133, 170, 210
3G,' 37, 40
1, 3
153
1 N V
Here a = 150, d' = 97, d" = 36, d"> = 1, d"" 2 ; hence y 150 -f
1 2
97 X 2,25 + ~ X 2,25 X 1,25 -{- X 2,25 X 1,25 X ,25 -f * -f-
2,25 X 1,25 X ,25 X ,75 = 418', 96 = 6. 53'. 57", the place required.
But besides the use of the above equation, to find the value of any
term of a series from its position being- given, the converse is often re-
quired, i. e. having given any term, to find its position or distance from
the first term.
Ex. On March, 1783, the sun's declination at noon at Greenwich was
as follows : On the 19th, N. 28'. 41" =. 1721" ; on the 20th, N. 5' = 300" ;
on the 21st, S. 18'. 41" = 1121" ; to find the tune of the equinox.
1721, 300, 1121
1421, 1421
Here a - 1721, d' 1421, hence y 1721 1421 X x ; now when
the sun comes to the Equator, y the declination becomes o ; .'. 1721
1721
1421 x = o, and x = j^ = Id. 5. 3m. 53*., the time from the 19th j
hence 20d. bh. 3m. 53*. is the time required.
We have here supposed that the quantities to be interpolated were
taken at equal intervals of time ; for a formula when the intervals are
unequal, see fence's Astronomy, vol. 2.
INVOLUTION and Evolution.
TABLE of the first nine powers of numbers.
1st
1
2
3
2d
_
4
9
3d
1
~8~
27
4th
1
16
81
5th | 6th
7th 1
8th
9th
1
1
1
1
1
32
G4
128
256
512
213
729
2187
6561
196S3
4
5
6
16
64.
~i25"
216
256
625
1024
4096
16384
65536
262144
25
36
3125
7776
15625
46656
78125
390625
1953125
1296
279i)36
1679616 j 10077696
7
~8~
~Q~
49
64,
81
343
2401
16807
117649
823543
5764801
40353607
512
729
4096
6561
32768
262144 | 2097152
167-.7216
13421772S
59049
531441
4782969
43046721
387420489
I N V
TABLE of squares, cubes, squarr. roots, r>tbe ro-jts, rind reciprocals^ of
of all numbers from 1 to 100. (Barlow.)
Num. Squares.
Cubes. ! Sq. Roots.
Cu. Roots.
Reciprocals.
I
1
1
1
1
1
2
4
8
1-4142136
1-2599210
500000000
3
9
27
1 "7320508
1 -4422496
333333333
4
16
64
2'OOOOOGO
1-5874011
250000000
5
25
1:25
2-2360680
1-7099759
200000000
6
36
216
2-4494897
1-8171206
166666667
7
49
343
2-6457513
1-9129312
142857143
8
6t
512
2-8284271
2-0000000
125000000
9
81
729
30000000
2-0800837
111111111
10
100
1000
3-1622777
2-1544347
100000000
11
121
1331
3-3166248
2-2239801
090909091
12
144
1728
3-1641016
2-2894286
083333333
13
169
2197
3-6055513
2-3513347
076923077
14
196
2744
3-7416574
2-4101422
071428571
15
225
3375
3-8729833
24662121
066666667
16
256
4096
4-0000000
2-5198421
062500000
17
289
4913
4-1231056
2-5712816
058823529
18
324
5832
4-2426407
2'6>07414
055555556
19
361
6859
4-3588989
2-6684016
052631579
20
400
8000
4-4721360
2-7144177
050000000
21
441
9261
4-5825757
2-7589243
047619048
22
484
10648
4-6904158
2-8020393
045454545
23
5>9
12167
4-7958315
2-8438670
043478261
24
576
13824
4-8989795
2-8844991
041666667
25
625
15625
5-0000000
29240177
040000000
26
676
17576
5-0990195
2-9624960
038461538
27
729
19683
5-1961524
3-0000000
037037037
38
784
21952
5-2915026
3-0365889
035714286
29
841
24389
5-3851648
3-0723168
034482759
30
900
27000
5-4772256
3-1072325
033333333
31
961
29791
5-5677644
3-1413806
032258-65
32
1024
32768
5-6568542
3-1748021
031250000
33.
1089
35937
5-7445626
3-2075343
030303030
34
1156
39304
5-8309519
3-2396118
029411765
35
1225
42875
5-9160798
3-2710663
028571429
36
1296
46656
60000000
3-3019272
027777778
37
1369
50653
6-0827625
3-3322218
027027027
38
1444
51872
6-1644140
33619754
026315789
39
1521
59319
6-2449980
3-3912114
025641026
40
1600
64000
6-3245553
3-4199519
025000000
41
1681
68921
6-4031242
3-4482172
024390244
42
1764
74088
6-4807407
3-4760266
023809524.
43
1849
79507
6-5574385
3-5033981
023255814
44
1936
85184
6-6332496
3-5303483
022727273
45
2025
91125
6-7082039
3-5568933
022222222
46
2116
97336
6-7823300
3-5830479
021739130
47
2209
103823
68556546
3-6088261
021276600
48
2304
110592
6-9282032
3-6342411
020833333
49
2401
117649
7-0000000
3-6593057
020408163
50
2500
125000
7-0710678
3-6840314
020000000
I N V
Num.
Squares.
Cubes.
Sq. Roots.
Co.. Roots, i Reciprocals.
51
2601
132651
7-1414284
37084298
019607843
52
2704
140603
7-2111026
3-7385111
0192S0769
53
2809
148377
7-2801099
3-7368858
018867925
54
2916
157461
7-3484692
3-7797631
018518519
55
3025
166375
7-4161985
38029525
018181818
56
3136
175616
7-4833148
3-8?5S624
017857143
57
3249
185193
7-54PS344
3-8485011
0173=13860
58
3364
195112
7-6157731
38708766
'017241379
59
3481
205379
7-6811457
3-8929965
'0169-19153
60
3600
216000
7-7459667
3-9148676
016666667
61
3721
226981
7-8102497
3-9364972
018393413
62
3844
238328
7-8740079
3-9578915
016129032
63
3969
250047
7-9372539
3-9790571
SI 5873016
64
4096
262144
8-0000000
4-1 OOOCC3
015625000
65
4225
274625
8-0622577
4-0207256
015384615
66
4356
287496
81240384
4-0412401
015151515
67
4489
300763
8-1853528
4-0615480
014925373
68
4624
314432
8-2162113
4-0316551
014705882
69
4761
32S509
8-3066239
4-1015661
014462754
70
4900
343000
8-3666003
41212853
0142S5714
71
5041
357911
8-4-261498
4-1408178
0141384507
72
5184
373248
8-4852814
4-1601676
013888839
73
5329
389017
8-5440037
4'179a390
013098830
74
5476
405224
8-6023253
4-1983364
013513514
75
5625
421875
8.6602540
4-2171633
OiaSS3383
76
5776
438976
8-71 77979
4-2358236
0131 57895
77
5929
456533
8-7719644
4-2543210
012987013
78
6034
474552
88317609
4-2726586
0128.0513
79
6241
4S3039
88881944
4-2908404
% '012658228
80
6400
512000
8 9442719
4-3088695
012500003
81
6561
531441
9-0000000
4-3267487
012345679
82
6724
551368
9-0553851
4-3444815
012195122
83
6889
571787
9-1104336
4-3820707
012048193
84
7056
592704
91651514
4-3795191
011904762
85
7225
614125
9-21954-15
4-3968296
01176470G
86
7396
63(3056
9-2736185
4-4140049
0116-17907
87
7569
658503
9-3273791
4-4310476
011494253
88
7744
6S1472
9-SS08315
4-4479602
011363636
89
7921
704969
9-4339811
4-4647451
011235955
90
8100
729000
'9-4868330
4-4814047
011111111
91
8281
753571
9-5393920
4-4979414
010GS9011
92
8464
778688
9-5916630
4-5143574
010889565
93
8T49
804357
9-6436508
4-5306549
010752688
94
8S36
8305S4
9-6953597
45468359
010638298
95
90?5
857375
9-7467943
4-5629026
010526316
96
9216
884736
97979590
4-5788570
010416667
97
9409
912673
98488578
4-5947009
01030927S
93
9604
941192
9-8994949
4-6101363
010204082
Q;>
9801
970299
99498744.
4-6260650
010101010
156
L A T
The use of the first five columns is obvious : the column of reciprocal*
is useful for converting- a vulgar into a decimal fraction, as in the fol-
lowing example.
Express ~ as a decimal.
By Table -^ is .035714286
JULIAN Period. See Cyrle.
JUNO.
This planet was discovered by Mr Harding', at Lilienthal, September
1st, 180-1. For its elements, &c. see Planets, elements of.
JUPITER. See Planets, element* of.
JUPITER'S Satellites.!^'*? Satellites.
L
LAND Surveying. See Surveying 1 .
LATITUDE Geographical. (Woodhouse. )
1st Method, by the Altitudes of circumpolar stars.
Co-latitude half the sum of the greatest and least zenith distance*
corrected for refraction.
Or the latitude may be found by Captain Kater's method, from an ob-
served altitude of the pole star when out of the meridian thus (Gal*
Iraith.)
To the constant log. 5.3144-25, add the log-, tangent of the star's polar
distance p, and the log. cos. of the meridian distance t in degrees, the
sum of these will be log. of an arc u in seconds. Now to the log. secant
p add the log. cosine u t and cosine of the zenith distance z ; the sum will
be the cosine of ($ i w) an arc which being increased or diminished by
the arc w, will be the co-latitude $/.
To find t, calculate the time of the star's meridian passage (see Time},
f hf difference between which and the time of observation gives t.
In the application of u attention must be paid to the sign of the arc t,
according to its situation in the circle which the star describes round tH
157 13
LEA
This method is commonly used at sea, but as the sun must be on the
meridian, clouds may prevent its being used. A subsidiary method there-
fore is provided, in which the latitude may be computed from two ob-
served altitudes of the sun, and the interval of time between the obser-
vations.
Let Z be the zenith, P the pole, S, j? two
positions of the sun ; then the following
are the steps in this process.
(1.) Find Ssj let t interval of time,
p - P S then
S s f
2 sin. 8 -y ^in.2p. 2. sin.2 ;
/. log. sin. ^- log. sin. p + log. sin.
. 10.
tan. SsP-
cos. p '
,*. log. tan. S s P 10 -f. log. cot. ^- log. cos. p.
(3.) Find / Z s S ;
Let a and a' be the observed altitudes, then sin.s | Z * S
- cos, f (S * + a' -f- *) sin. * (S s -|- a' - a)_ . 2 lo* sin 4 Z S =
sin. S s X tos. a
20 + log. cos. | (S s + a' -f a] 4. log. sin. | (S s + a> a] log. sin.
S * log. cos. a.
Hence Z*P S*P ZsSis known.
(4.) Find Z P ;
Assume 9 such that
. cos. a', ein. p. ver. sin. Z s P
tan " *= ver. sin. (900- a'- J9) ;
ZP , 90 a' p
then sin. ^- = sm. ^ x sec. & ;
:. log. sin. ^j~ 10 + log. sin. % (90 a' p} log. cos. 0.
},KAP Vear.See Calendar.
LEMNISCATA, equation to.
160
L E V
a y x v 2 _. #x
a of James Bernouilli.
or considered as a spiral
= a Vcos~2~0.
LENGTHS ofctt.rve9.-See Rectification.
LENS. See Refraction.
LEVELLING.
T\vo or more places are on a true level, when they are equally distant
from the centre of the earth ; and a line equally distant from that centre
in all its points, is called the line of true level. This line is nearly an
arc of a circle, and will evidently pass below the line of apparent level,
which, as determined by the instrument, will be a tangent or a parallel
to a tangent at the earth's surface at the point of observation. Hence
the depression of the true below the apparent level is always equal to the
excess of the secant of the arc of distance above the radius of the earth.
To find this depression, let L be the arc of distance in English miles, D
the depression in feet ; then
TABLE shewing the height of the apparent above the true level for every
100 yards of distance on the one hand, and for every mile on the other.
Distance
of base.
Viff. of
lend.
Distance
of base.
Difference
of level.
Yards.
Inches.
Miles.
Feet. In.
- 100
0.026
i
0. 0|
200
0.103
I
0. 2
300
0.231
0. 4
-100
0.411
r
0. 8
500
0.013
g
2. 8
600
0.025
9
6.
700
1.260
4
10. 7
800
1.645
5
16. 7
900
2.08!
23. 11
.
2.570
7
32. 6
lion
3.110
R
42. 6
1200
3.701
<
53. f>
1300
4.344
10
66. 4
1400
5.038
11
80. .'!
1600
A. 781
12
95. 7
1600
a580
l:j
112. 2
1700
7.125
14
130. J
LEV
Example. Suppose a spring to be on one side of a hill, and a house on
;,n opposite hill, with a valley between them ; and that the spring seen
from the house appears by a levelling instrument to be on a level with the
foundation of the house, which suppose is at a mile distance from it ;
then (by Table) the spring is eight inches above the true level of the
house ; and this difference would be barely sufficient for the water to be
brought in pipes from the spring to the house, the pipes being laid all the
way in the ground.
In the above Table, the effects of refraction have not been considered,
which, however, should not be neglected, if the distances are consider-
able. In that case, the correct formula is
which expression includes the effects both of curvature and refraction.
See Refraction terrestrial.
LEVER.
Levers may be divided into three kinds. In levers of the first kind,
the fulcrum is between the power and the Aveight, as in the balance,
steelyard, scissors, poker, &c. In levers of the second kind, the weight
is between the fulcrum and the power, as in oars, doors, cutting knives
fixed at one end, &c. In levers of the third kind, the power acts be-
tween the fulcrum and the weight, as in tongs, sheers for sheep, mus-
cles of animals, &c.
1. Two weights or forces, acting perpendicularly upon a straight lever,
will balance each other, when they are reciprocally proportional to their
distances from the fulcrum.
Cor. 1. When the power and weight act on the same side of the ful-
crum, and keep each other in equilibrio, the weight sustained by the
fulcrum is equal to the difference between the power and the weight.
Cor. 2. If the same body be weighed at the two ends of a false balance
(one arm of which is longer than the other), its true weight is a mean
proportional between the apparent weights,
Cor. 3. If a weight be placed upon a lever supported upon two props,
the pressures upon the props are Inversely proportional to their distances
from the weight.
2. If two forces, acting upon the arms of ant/ lever, keep it at rest, they
are to each other inversely as the perpendiculars drawn from the centre
of motion to the directions in which the forces act} or inversely as th*
arms, multiplied into the bines of the angle*, which, the direction of the
forces make with them.
J, I ir
if ;i man, balanced in a common pair of *c;ii*'s, pi :.-.> upw.'K
n>e;ms of a rod, against any point of tlio beam, except that from which
the scale is suspended, he will preponderate.
;{. In a compound lever, where one is made to turn another, there is
an equilibrium, when W I P II the product of all the arms taken alter-
nately, beginning with that to which the power is applied : the product
of all the other arms,
4. Any weights will keep each other in equilibrio on the arms of a
straight lever, when the products, which arise from multiplying each
xveight by its distance from the fulcrum, are equal on each side of the
fulcrum.
Cor. 1. If in the above Propositions we would allow for the weight of
the lever itself, we must suppose its weight to be united iu the centre of
gravity, and to act there as a third force added to the power or the
weight, according to the side of the fulcrum on which it is placed.
Cor. 2. If the weights do not act perpendicularly to the arms of Ihc
lever, we must for the distances substitute the perpendiculars, (see Art.
2.)
Cor. 3. Let A D be the common "F C "P O D
steelyard, whose fulcrum is C, and i *~~T\ *~ ~~* j
let the moveable weight P, when
placed at E, keep the lever at rest ; O W OP
then when W and P are suspended upon the lever, a^id the whole r<.
mains at rest, WXAC PXDC-|- PXEC^PXDE; .*. W
varies as E D ; the graduation must .'. begin from E, and if P when
placed at F support a weight of one pound at A, take FG, GD, &c,
equal to one another and to E F ; and when P is placed at G it will sup-
port two pounds ; and when at D it will 'support three pounds, &c,
LIFE Annuities. See Annuities Life.
LIFE Assurances. See Annuities.
LIGHT, Phenomena of.
Light, propagation of.
1. In a free medium the force and intensity of light, which propagates
itself in rays emanating from the same point, are inversely as the squares
of the distances from that point.
Prob. Having given the position of two lights of known intensities,
to determine the nature and equation of the surface, of which every
point shall be equally illuminated by the two lights.
Let A and B be the two point" at which the lights are placed, m and n
1G3 . K
L I G
their intensities at, any assumed unit of distance, and let n = A B ; them
it may bo shewn that the required surface, is a sphere of which the ra-
dius ~ / M , and whose centre has for an abscissa - .
n m v m ' m
Cor. If m n the radius is infinite, as also the abscissa from the cen-
tre ; in this case the surface is a plane perpendicular to the middle of the
line A B.
LigJity velocity of.
2. Light takes up about 16 minutes in passing- over a space the dia-
meter of the earth's orbit, which is nearly 190 millions of miles; .'. it
travels at the rate of almost -200,000 miles per secnod.
iminution of y under various circumstances.
3. If the spaces through which light passes through a uniformly dense
diaphonous medium increase in arithmetical progression, the quantity
will decrease in geometrical progression.
Let the space be divided into equal portions or laminae, and suppose
th part of the whole light to be lost or absorbed in its passage thro' the 1st
lamina ; then = quantity of light entering the 2d lamina ; 5
rr do. entering the 3d ; ^ rr do. entering the 4th, &c.
TABLE from Buttguer, shewing the intensity of the sun's light at differ.
ent altitudes y and the thickness of air it has to penetrate at each angle.
Sun's
altitude.
Thickness of
air in toises.
Intensity oj light
the whole being
10,000.
90
3911
8123
80
3971
8098
70
4162
8016
60
4516
7866
50
5104
7624,
40
6086
7237
30
7784
6613
20
11341
5474
15
14880
4535
10
21745
3140
5
39893
1 1201
3
58182
454
1
100930
47
138823
6
1G4
L I G
4. According to Leslie, in passing through sea water, light is diminish-
ed four times for every five fathoms of vertical descent; and Bouguer
asserts, that the whole effect of the sun's light would be lost by passing
through b'79 feet of sea water, and that the same effect would take place
by its passing through 3,110,310 feet of air.
5. Bouguer computes that of 300,000 rays which the moon receives ;
172,000, or perhaps 204,100 are absorbed ; and that the light of the sun :
ditto of the full moon :: 300,000 : 1.
6. Euler makes the light of the sun equal to that of 6560 candles at one
foot distance ; that of the moon to a candle at 7| feet ; of Venus to a can-
dle at 421 feet ; and of Jupiter to a candle at 1620 feet, partly from Bou-
guer's experiments. Hence the sun would appear like Jupiter, if re-
moved to 131,000 times his present distance (Young's Nat. Phil.}
Light, refrangibility of.
7. The sun's light consists of rays which differ in refrangibility and
colour.
The 1 primary colours are red, orange, yellow, green, blue, indigo, and
violet, of which the red rays are the least refrangible, and the violet ones
the most ; while green and blue are the colours which have a mean de-
gree of refrangibility. Sir Isaac Newton found their degrees of re-
frangibility in passing out of glass into air to be as the numbers 77,
77JL > 77J_ } 77-L, 77__ 9 77 77 ' and 78, those being the values of the
8 o 3 2 3 "
sines of refraction to the common sine of incidence 50. Some substances,
however, separate the different coloured rays more widely than others,
and the dispersive power of media does not appear to depend at all upon
their mean refracting power.
To find a measure of the dispersing power, take a constant small / 6
for the / of refraction, the / of incidence will then be m and will dif-
fer according to the value of m. The difference between these two or
(m- 1) 6 is the refraction; and if m and m be values of m for led and
r v
violet rays, the difference of refraction will be (m 1 ) 8 -~ (7/1 1} 5 or
v r
m m
(m . m . 0. Its ratio to the refraction will consequently be
v r m '
taking the mean value of m '. this is the usual measure of the dispers*
iug power.
In flint glass its value is about 0.05; in crown glass 0,033.
t Having- giveu th refracting- power* of two mediums, to find the
ratio rif the focal lpiijrth-> of two louses formed of these substances, which,
when united, will produce images nearly free from colour.
Let | and $' be the focal lengths of the lenses, 1 + r and 1 -f v the ra-
tio of refraction belonging to the red and violet rays respectively in the
1st lens, and 1 4. /' and 1 -f v = ditto of the other ; then
Hence it appears that p' and must be of dilfererit signs, or one lens
concave and the other convex ; and that they are as the respective dis-
persive powers of the substances of which the lenses are made.
The common practice of opticians, is to use flint glass and crown glass,
the dispersive powers of which are in the ratio of 50 to 33 ; and .". a
compound lens, in which the separate focal lengths for the same kind of
homogeneous light, are as 50 I 33 will make the red and violet rays, con-
verge accurately to one point.
9. Having given the aperture of any lens, and the foci to which rays
of different colours, belonging to the same pencil, converge ; to find the
least circle of aberration through which these rays pass.
Let D diameter of the least circle of aberration, , aperture of the
lens, the rest as before ; then
D =
r + r
Suppose, for instance, the lens be of crown glass, v = .56, r = .54 ;
.'. r^r ; D .". is of the aperture.
0*4. r 5a ' 5 )
Light, aberration ofsec Aberration.
For a concise account of other physical properties of light, such as the
phaenomena of coloured rings, double refraction, polarization of light,
&c. see Coddington's Optics ; these subjects, as requiring diffuse expla*
nations, cannot here be entered upon.
LINE right.
Equations and Problems relating to, the co-ordinates being supposed
rectangular. (Hamilton. )
1. The equation to a straight line is y
ax + b, where a is the tangent of the
angle which the line makes with the axis
X A x t and b is the distance from A at
which it intersects the axis Ay,
L 1 N
2. Required the equation to a straight line passing through a given
point, whose co-ordinates are a", y'.
Any point of which the co-ordinates are x, y being assumed in the
line, we have y a x + b ; also y 1 - a x' -f b ; /. equation required is
y y - a (# #')
For the sake of brevity it is usual to designate the point, whose co-
ordinates are #', y' y as the point (#', y') ; and the straight line, whose
equation is y ax -4- b, as the straight line y = a x -f- b.
2. Required the equation to the line which passes through two given
points (,f, y'} and (-1"', #").
3. Required the angle formed by the intersection of two given lines.
Let y a x -\- b and y = a'x 4. b' be the given lines, and the given
angle ; then
a'
tan. 6 =
1 +aa'
a a 1
1 +'
the positive sign being used when the / is acute, the negative when it
is obtuse.
4. Required the equation to a straight line drawn through a given
- i
point (#', y'}, and making an angle tan. m with the line y = a x + b.
TT am,
Here y y' = - (* ^')-
l _|_ am
Hence (1 ) when the lines are perpendicular,
(2) When they are parallel,
yy> = a(x A").
5. Required the distance (r) between two points (x> y] and (.r', /').
" v} + (./' - ?/}* 1
When a," arid //' o, r ^ (.r* -|- yi] ; which therefore expre*se the
distance of a point from the origin.
167 K 2
LOG
6. "If <,p} be the perpendicular dropped from a given point (A", v%) o
the straight line y a x 4- b ; then
- . --
va-f 8 ; *
LITUUS. See Spiral.
LOGARITHMS.
1. Properties of Logarithms.
Log. aXb Log. a -f. Log. fc.
Log. y = Log. a Log. 6.
Log. a l =. m Log. a.
|
> 1 x
Log. a = - Log. .
Log. a = Log. -.
2 20 X 3^ X 2 031
Ex. I. Log. - 17 x 935Q J - = 20 lo 8T- 2 + 7 log. 3 + log. 2.013
(log. 17 + log. 9350).
8 x 2. Log. 5V 1<2 X ^ 51 3 ~ - = -jr (2 log. 317 -f * log. 3 + *
log. 5 log. 251).
2. Given a number, to find its Logarithm.
Let 1 + x be the number, -m the modulus,
then log. 1 + x = m ( x - ~ + |* - ? + & c .)
and log. 1 -JT = m X ( - A- - ^ - ^ - f. 4 - & c .)
^^ = 2 W x( * + f + + & c .)
Or since N = ^^y we may for x substitute ^[ p and we
hall have
L O G
both of which last series converge very fast.
Ex. If N = 2, Lll - I ; .'. log. 2 - .3010300.
In hyp. logarithms m = 1, in the common system z 9 - - ^ ~-
,43424948. And since different systems of logs, are as their moduli, if any
common log. be divided by this modulus, it gives the corresponding hyp.
log. ; or if any hyp. log. be multiplied by it, it gives the corresponding"
common logarithm.
3. Given a logarithm, to find its number.
Let 1 + x =. No., y its log. m the modulus.
then 1 -f x = 1 + -S- + J. + ^L. + & c .
r m T 2 m T 2. 3. m* T
If m = 1 ( l+o;-l+3/ + ^- + ^T 4- &c - = No - whose hyp. log.
4. Modular ratio is the ratio of which the modulus is the measure, or
the number of which the modulus is the logarithm, and = 1 + 1 +
+ - + &c. : 1 ; or 2.7182818 : 1 ; which is therefore the same for
every system, being independent of m and y.
Hence in Napier's or hyp. logs., where the modulus is 1, the log. of
2.7182818 is 1 i in Brigg's or the common system, log. 2.7182818 ia
,43424948.
Hence also since in every system the log. of the base is 1 ; 2.7182818 is
the base of Napier's logs. ; in Brigg's the base is 10.
In general if a = base of any system, whose modulus is m, m = h ^ ^.
The following Table of Logarithmic series will be found useful on various
occasions.
1. Log. a = ~ x (-l) -i (-!)* + 1 (a- 1) 3 - &c.
2
Log. a = X
T
a ^ b\ 3 r fa -*~ b\ 5 >
5T*) + -f I^M) + &c ' 5
7. Log. . =tog. (._!) + ^ X i+^ + jL 4-4^
8. Log. a =,o ff . (_,) + x -L.__
___
10. Log. a = ^X J(a a" 1 ) S(o ! " S ) + J(a3 ""') &c.f
11.
.3. Log. , .) =,og.l x ^ + 1 a -^V f
< (wv/a 1} J (w^/a I) 2 -f i( Wx /a ])'-&c. >
LOGARITHMIC Cwrpe, Equation to, 8>c,~( Uigman. )
x
y a .
The curve con-sists of one branch infinite on each side of the origin, to
which the axis of abscissas is an asymptote.
J70
L O N
If x o, y 1 ; and if x 1, ^ a.
If the abscissas increase in arithmetic progression, the ordinates in-
crease in geometric.
The subtangent is a constant quantity, and = modulus of the system
of logarithms, whoso base a.
Area between any two ordinates y and b in (y b), where m is the
modulus or subtangent.
Content -^ (yt 62).
Arc = ^/( m* -f #2) ^( m 2 4. 52)
b (^(
+ m log. -
y(^/(m* + &) w
Surface ~- T ( ?/ ^(w/2 4- ?/ 8 ) 6 ^ (wa 4. &g)
LOGARITHMIC Spiral See Spiral
LONGITUDE Geographical (Woodhouse, Fince.J
\sl Method, by a chronometer.
Suppose a chronometer to be adjusted to mean solar time at Green-
wich, then if its motion were equable, and of the proper rate, we should
always know, whatever the place, the time at Greenwich. Compute .*.
the apparent, and by means of the equation of time, the mean time, at the
place of observation. The difference between this latter time, and that
shewn by the chronometer, would be the longitude, east or west of
Greenwich.
2d Method, by an eclipse of the moon or of Jupiter's satellites.
Having the times calculated when the eclipse begins and ends at
Greenwich, observe the times when it begins and ends at any other
place ; the difference of these times, converted into degrees, gives the
difference of longitudes.
3d. Method, by the moon's distance from the sun or a fixed star.
The steps by which we find the longitude by this method are these :
From the observed altitudes of the moon and the sun or a fixed star,
and their observed distance, compute the moon's true distance from the
sun or star.
171
L O N
From the Nautical Almanack find the lime at Greenwich when the
moon AVHS at that distance.
From the altitude of the sun or star, find the time at the place of ob-
servation.
The difference of the times thus found, gives the difference of the
longitudes.
Formula for deducing the true from the observed distance.
Conceive S, M to be the true places of the star and moon in two ver-
tical circles S Z, M Z forming at the zenith Z the /_ M Z S ; then since
both parallax and refraction take place entirely in the direction of ver-
tical circles, some point s above S, in the circle Z S, will be the apparent
place of the star, and m below M (since in the case of the moon the de-
pression by parallax is greater than the elevation by refraction) Avill be
the apparent place of the moon : let
D (S M) be the true, d (s m} the apparent distance,
A, a (900 Z M, 90 Z S) the true altitudes,
H, A (90 Z m) 990 _ z s) the apparent altitudes,
cos. A. cos. a
then if F =
cos. H. cos. k *
or if we make the fraction, on the right hand side of the equation =
sin. 8 0, we shall have
sin.* = cos.2 i ( A + a) , cos.- 6.
and sin. = cos. (A -f- ) . cos. 8.
The true distance of the moon from the sun or star being thus found,
we are next to find the time at Greenwich corresponding to this true
distance. To do this, we must observe that the true distance is com-
puted in the Naut. Almanack for every three hours for the meridian of
Greenwich. Hence considering that distance as varying uniformly, the
time corresponding to any other distance may be thus computed. Look
into the Naut. Almanack, and take out two distances, one next greater,
and the other next less, than the true distance deduced from observa-
tion, and the difference D of these distances gives the access of the moou
to, or recess from, the sun or star in three hours ; then take the differ-
ence d between the moon's distance at the beginning of that interval,
and the true distance deduced from observation, and then say, D '. d '. '
3 hours : the time the moon is acceding 1 to or receding- from the sun or
*hu* by flip quantify d, which added to the time at the beginning* of thff
interval, gives the apparent time at Greenwich corresponding to the
Driven true distanrfe of the moon from the sun or star.
Having- thus found the time at Greenwich, compute the time at the
place of observation from the corrected altitude of the sun or star, the
sun's or star's north polar distance (furnished by Tables) and the lati-
tude.
The difference between this latter time and the time at Greenwich, is
the longitude.
The other methods of finding- the longitude are, by an occupation of
a fixed star by the moon ; by a Solar eclipse j and by the passage of the
*noon over the meridian.
LOOKING Glass, method of judging of. (" Coddington. )
To mid the thickness of a looking glass, bring a pin or other slender
object into contact with the fore surface of the glass, and observe its
image, as shown by reflection ; then the thickness of the glass will be
q
equal to -ths of the apparent distance between the objects and its image.
In a looking glass it is not only necessary that each plane should b
perfect, but they must be also parallel to each other. If the images of a
candle seen very obliquely, and under different degrees of obliquity, and
from all parts of the glass, do not always keep pretty nearly at equal dis-
tances from one another, it is a proof that the sides of the glass are neither
plane nor parallel.
Another method of trying the goodness of a glass is as- follows : Stick
a pin or slender wire in the bar of a window sash, so that the pin may
be nearly horizontal, and in the plane of the window. Then hold the
looking-glass, and turn it about so as to see the image of the pin very ob-
liquely and from all parts of the glass. In this case two images will be
visible ; and if these images keep always straight, parallel, and at regu-
lar distances one from another, the glass may be considered as being well
figured. These phenomena will be more conspicuous if two pins be
stuck parallel to one another, and at a small distance asunder.
With respect to the polish of a glass ; we may observe, caeteris pari-
bus, that the darker the colour of the glass of the speculum is, the better
generally is the polish.
For the theory of plane mirrorswe Reflection.
LUNAR inequalities.' See Moon,
ITS
M
MACLAURIN'S Theorem. See Taylor's 7V
MAGNETIC Needle, variation and dip of.Sf-r Va>
MARS. See Planets, elements of.
MARS phases of. See Venus.
MAXIMA and Minima of quantities.
1. To determine in what cases any quantity y, depending- upon .r, may
become a maximum or minimum, we must find the differential of the
equation which expresses the relation that they bear to each other, and
make the quantity -~ = o. The resulting equation, combined with the
original one, will give the values of x and y in which y is a maximum
or minimum.
2. To determine when y is a maximum and when a minimum ; find
rf 2 ?/
the value of -y- ^, and if it be negative, y is a maximum ; if it bf posi-
tive, a minimum.
3. If -jt- and -j-^ both vanish, but -~ remain, then y will be neither
a maximum or minimum at that place, but will pass through a point of
contrary flexure parallel to the abscissa. In like manner, if dy, d*y,
d s y vanish, but d* y remain, the ordinate y will be a maximum or mini-
num ; and if dy, d?y, canic matter resting on in . Krt
Antisana (highest volcano, Andes) f primitive rockS) such M >
Cotopaxi (voJLgano, Andes) . . J gneiss, mica slate, &c. 18,875
C Mount St Elie 18,090
Popocatepetl (voj^ano of Puebla Mexico) .... 17,720
Cotocatche (Andes) . 16,450
Tonguragua (volcano, Quito) 16,270
Mouna Roa (Owhyhee), volcanic from top to bottom ; all the
rocks of the island are igneous 15,871
Mont Blanc (Alps, highest in Europe) granite, syenite, horn-
blende slate, in vertical layers 15,W5
188
M O U
Mont Rosa (Alps), talc slate and serpentine .... 15,527
Ortler Spiltze (Tyrol) alpine or Jura limestone, with organic
remains 15,430
Mount Cervin (Switzerland) primitive slaty rocks . . 14,780
Mount Ophir (Sumatra) 13,842
Peak of Jungfraa (Switzerland) alpine limestone . . . 13,735
Pambamarca (Andes) 13,500
Brat-horn (Switzerland) granite and gneiss .... 12,815
Sochonda (China) primitive, probably granite . . . 12,800
Finisteraaharn (Alps) granite and gneiss .... 12,210
Lake of Toluca (Mexico) 12,195
Peak of Teneriffe, volcanic from top to bottom . . . 12,176
Town of Micupampa (Peru) 11,670
Mulahacen (Spain) . 11,670
Peak of Venlatta (Spain) 11,390
Mont Perdu (Pyrenees) calcareous, with organic remains . 11,265
Le Viguemal (do.) summit granite, flanks calcareous . . 11,010
Mount JEtna (Sicily) volcanic ashes, scoria, lava, &c. . . 10,955
Italitzkoi (Altaic chain, Asia) 10,735
Pic Blanc (Alps) 10,205
Ouito 9,630
Awatsha (volcano, Kamtchatka) 9,600
Mount Libanus (Turkey) . 9,535
Real del Monte (mine, N. Spain) poryhyry slate . . . 9,125
Imbabura (Quito) a great volcanic dome resting on primitive
rocks 8,960
Mont St Gothard (Alps) granite and gneiss . . . 8,930
Peak of Lomnitz (Carpathian Mountains) primitive rocks,
details unknown 8,640
Mount Valina (highest of the Apennines) .... 8,300
Sneebutten (Norway) gneiss, mica slate, and other primitive
slates . .......... 8,295
Blue Mountains (Jamaica) ....... 8,180
Volcano, Isle of Bourbon , " t ?,6SO
789
M O U
Mexico . 7,525
Mount Cenis (Alps) transition slate, &c 6,780
Mount Olympus (Turkey) primitive limestone, with serpen-
tine, syenite, porphyry . . , . . . . . 6,500
: Stony Mountains (N. America) 6,250 '
Mont d'Or (France) . 6,130
Roettruck (Sweden) gneiss and mica slate .... 6,000
Mount Reculet (Switzerland) . 5,590
Puy de Dome (France) ancient volcanic rocks (trachyte) . 5,225
La Souffriere (Guadaloupe) volcanic 5,110
Hecla (Iceland) volcanic from top to bottom, scoria, lava,
tuff, porphyry, slate, &c 5,010
Mount Ida (Turkey) 4,960
Ben Nevis (highest in Britain) feldspathic slate, greenstone
slate, &c 4,370
Jorulla (volcano, Mexico) volcanic scoria .... 4,265
Ben Lawers (Scotland) 4,015
Mount Vesuvius (Italy) volcanic ashes, scoria, lava . . 3,935
Ben Wyvis (Scotland) 3,720
Snowden (highest in Wales) transition slate, with organic re-
mains, greenstone slate, &c. 3,571
Town of Caracas 3,490
Ben Lomond (Scotland) feldspathic slate, greenstone slate,
&c 3,250
Sea Fell (Cumberland) chloritic slate, greenstone slate, com-
pact felspar, &c 3,166
Helvellyn do. . .,...* 3,055
Skiddaw, clay slate, with crystals of chiastolite . . . 3,022
Cadir Idris, compact felspar, greenstone slate, &c. . . 2,914
Cross Fell, mountain limestone, and millstone grit . . 2,901
Cheviot, porphyry 2,653
Plyulimmon 2,463
Whernside (Ingleton Fells) mountain limestone, gritstone,
&e. . 2,38i
190
M O U
Ingleborough, mountain limestone, gritstone, c. . . 2,361
Madrid 2,276
Pennigent do. .... 2,270
Whernside (Kettlewell) do. .... 2,263
Fountains Fell do. .... 2,180
Snea Fell (Isle of Man) clay slate 2,004
Pendle Hill, mountain limestone, millstone, grit. . . . 1,803
Rye Loaf Hill do If.SS
Malvern Hills, Syenitic granite 1,441
^Cataract of Tequendama (S. America) 600
St Peter's, summit of the Cross 535
Pyramids (Egypt) 4"j 500
Natural Bridge of Iconozo (S. America) .... 300
Caspian sea below the ocean T***^ 306
Gay Lussac highest altitude ever attained by a balloon,
Sept. 16, 1804 22,900
Highest flight of the Condor 21,000
Height attained by Humboldt up the Andes, June 23, 1802 19,400
Highest limit of lichen plants . 18,225
Lower limit of perpetual snow on the Equator . . . '15,720
Farm House of Antisana 13,435
Highest limit of trees 11,125
Superior limit of oaks in the torrid zone .... 10,500
Convent on Mont St Bernard (Switzerland) . . . 8,040
Do. of St Gothard (Alps) . 6,810
MOUNTAINS, visibility of. See Refraction.
MOUNTAINS, attraction of.
Dr Maskelyne was the first who satisfactorily proved the attraction of
mountains by their effect in drawing the plumb line from its vertical
direction. The mountain selected was Schehallien in Scotland, the mean
height of which above the surrounding valley is 2000 feet, and above the
level of the sea 3550. The attraction of this mountain was found = 5",8 :
from which Dr Hutton calculated the mean density of the earth to be
near 5 times that of water, or as 99 to 20, and almost double the density
of rocks near the earth's surface, Mr Cavendish, upon totally different
191
N E 15
principles, found the density of the earth to be to that of water as 5.48
: 1. The internal parts of the earth are .*. much denser than those at
the surface ; though in what manner the dense parts are disposed of must
he uncertain.
MOUNTAIN, correction for height of. See Refraction.
MOUNTAINS, visibility of See Refraction.
N
NEBULA, and Clusters of Stars. (Herschel.)
On Herschel's Catalogue of new Nebuke, and clusters of stars.
The telescope used was a Newtonian reflector of 20 feet focal length,
and 18 inches aperture. The sweeping power was 157. The field of
view 15'. 4".
The Nebulae are divided into classes like the double stars. (See Stars
double. ) Thus in the 1st class, the degree of brightness of the Nebula
has been the leading feature, as most likely to point out those which or-
dinary instruments may be expected to reach. The 1 st class .'. contains
the brightest of them j the 2d those which shine but with a feeble light ;
and in the 3d are placed all the very faint ones. It should be observed,
that what Herschel calls bright, or very bright among those of the first
class, are commonly less distinguishable than what Messier, in his Cata-
logue des Nebules (given in Wollaston) calls faint ; on account of the su-
periority in the instruments of the former observer.
Besides this general division, there are added a 4th and 5th class, which
contain Nebulae deserving a more particular description. The 4th class
contains Planetary Nebulae, i.e. stars with burs, with milky chevelure,
with short rays, remarkable shapes, &c. The 5th class very large Ne-
bulae.
The 6th, 7th, and 8th classes contain clusters of stars sorted according
to their apparent compression, like the Catalogue of double stars, so that
the closest and richest clusters take up the first or 6th class ; the bright-
est, largest, and pretty much compressed ones, the second or 7th class ;
and those which consist only of scattered and less collected large stars,
are put into the last.
Note. When a superior power and telescope increase the brightness
of a nebula, but at the same time only make the tinge of it more nU
192
NEB
formly united, and of a milky appearance, it may be concluded to be
purely nebulous ; but when by using a superior instrument, its appear-
ance is a mixture of nebulosity and extremely fine points, so that we can
almost see stars, the nebula is said to be easily resolvable, and may be
concluded to be a cluster of stars.
Conjecture on the nature of nebula, not resolvable.
In the Philosophical Transactions for 1811, Herschel has started a new
conjecture respecting the nature of nebulae. He no longer considers
them as clusters of stars, which assume a nebulous appearance by reason
of their immense distance, but that they consist of a luminous and ex-
tremely rare substance. That this substance, at its first formation, is
pretty equally diffused through the nebula; but that in the course of
ages, this matter, by the preponderance of some part of it, formstme or
more centres, to which all the other matter gravitates ; that in conse-
quence of this, the nebula gradually decreases in size, and increases in
density, till at last a nucleus is formed ; and the nebula becomes plane-
tary surrounded by nebulous matter ; which last again is finally absorb-
ed by the central body ; and the whole then is, or has all the appearance
of, a fixed star. This connexion between nebulous matter and a fixed
star, and the conversion of the one into the other, he endeavours to
establish, by arranging the nebulae into classes, according to their sup-
posed age and degree of condensation, beginning with extensive and uni-
formly diffused nebulosity, and establishing the connexion between this
and a fixed star by such nearly allied intermediate steps, as makes it not
improbable that every succeeding state of the nebulous matter is the re-
sult of the action of gravitation upon it while in aforegoing one, and by
such steps the successive condensation of it has been brought up to the
planetary condition. From this the transit to the stellar form requires
but a very small additional compression of the nebulous matter; and in
Horschel's observations of many of these it became doubtful whether
they were not stars already.
The steps by which he arrives at this conclusion are nearly as fol-
lows :
1. Extensively diffused nebulosity.
2. Nebulosities joined to nebulae.
3. Nebulae of various shapes, but nearly uniform brightness,
4. Nebulae that are gradually a little brighter in the middle.
5. Nebulae which are gradually brighter in the middle.
G. Nebulae which are gradually much brighter in the middle.
193 L3
NEB
7. Nebulae that have a cometic appearance.
8. Nebulae that are suddenly much brighter in the middle.
9. Round nebulae increasing gradually in brightness up to a nucleus
in the middle.
10. Nebulae that have a nucleus.
11. Round nebulae that are of an almost uniform light.
12. Nebulae that draw progressively towards a period of final conden-
sation.
13. Planetary nebulae.
14. Stellar nebulae.
15. Stellar nebulae nearly approaching to the appearance of stars.
Clusters rf stars.
We have seen according to Herschel's doctrine, that extensive ne-
bulosities are in process of time broken up into separate and distinct
nebulae ; and that these last again, after becoming gradually more and
more condensed, form stars. Upon the same principle he accounts for
the formation of clusters of stars. He conceives that in rich portions of
the heavens, as for instance the milky way, various centres of attraction
are formed, to which the neighbouring stars gravitate ; that thus the
whole is broken up into separate systems or clusters of stars. That these
clusters at first are of various irregular figures, and consist of stars coarse-
ly and unequally scattered over the mass ; that by the progress of con-
densation they become more insulated and detached from the neighbour-
ing stars, their figures are more regular and spherical, and the stars more
rich and closely connected ; till they at length form those minute and
beautiful phaenomena which are undoubtedly the most interesting ob-
jects for our finest telescopes. He arranges them as follows, according
to their degree of condensation.
1. Aggregation of stars, or patches of stars, which seem beginning to
form clusters.
2. Irregular clusters of various unascertained sizes.
3. Clusters variously extended and compressed.
4. Considerably compressed clusters of stars.
5. Gradual concentration and insulation of clusters of stars,
6. Globular clusters of stars requiring a very fine telescope
7. More distant globular clusters of stars.
8. Still more distant globular clusters.
194
O B S
NEUTRAL Point. See Elastic bodies., equilibrium of.
NIGHT-GLASS, or Siveeper.
These are Telescopes of t\vo, or two and a half feet in length, with
large apertures, the object glass either a single lens of 3 or 3| inches dia-
meter, or an achromatic of 2% ; their magnifying power, 6, 8, or 10
times ; field of view 5 or 6 degrees : they are occasionally furnished with
a system of cross wires, and a diagonal eye piece. Their use is for a ra-
pid survey of any part of the heavens, and for fixing upon such objects
as may be proper for examination with finer telescopes. They are also
useful, provided the observations are recorded, in detecting minute
changes in the heavens upon a subsequent review ; or in searching for
any object supposed to be movdable, as an asteroid. For this purpose
delineations of the telescopic constellations, near the place where it is
suspected to be, should be drawn upon paper ; and after some days in-
terval, the moving star will be discovered. This can only be done with
a night glass of very low magnifying power. Herschel's small sweeper
was a Newtonian reflector of 2 feet focal length, aperture 4,2 inches,
magnifying power 24, and field of view 2 12'. (Phil. Trans, )
NONIUS. See Vernier.
NORMAL. See Subnormal.
NORMAL, equation to See Tangent.
NUTATION of the Earth's axis. See Precession.
O
OBSERVATORY.
TABLE,
Of the Latitudes and Longitudes of the principal Observatories of Europe,
from the most recent and accurate determinations. (Lax.)
Amsterdam ^^ ^^,
LATITUDE N.
5^o 22' 11"
LONGITUDE.
Qh 19m 33* E
Armagh *,~~ *w* ****
54 21 15
26 SOW
Berlin *~~ *~~ ^^
52 31 45
53 29 E
Berne ~*~ *~>~ +~~
46 56 55
29 45
Bologna *~~ ~~~
44 30 12
45; |26
Bremen *** w* w>~v
53 4 38
35 12
195
O B S
Brunswick ***** , .
LATITUDE N.
52o 16' 29"
Lo.\(JlTl. ; E.
07i42m&?
Buda
47 29 44
1 16 10
Cadiz
36 32
25 9W
Cambridge *****
52 12 36
17 E
Cassel
51 19 20
38 21
Coimbra ***** ***** *****
40 12 30
33 39 W
Copenhagen ***** *****
55 41 4
31 38 E
Cracow ***** ***** ~~ *
50 3 38
1 19 49
Dantzic ***** ***** *****
54 20 48
1 14 32
Dorpat ***** ***** *****
58 22 47
1 46 48
Dresden ***** ***** *****
51 2 50
54 52
Dublin *****
53 23 13
25 22 W
Edinburgh *****
55 57 21
12 41
Florence ***** ***** *****
43 46 41
45 3 E
Geneva ***** ***** *****
46 12
24 38
Genoa
44 25
35 52
Glasgow ***** ***** *****
55 51 32
17 4 W
Gotha
50 56 8
42 56 E
Gottingen *****
51 31 50
39 46
Greenwich ***** *****
51 28 39
000
Koenigsburg ***** *,***
54 42 12
1 21 57
Leipsic ***** ** *****
51 20 16 ......
49 27
Lilienthal
53 8 30
35 37
Lisbon ***** ***** *****
38 42 24
36 3! W
London .
1
Madrid
51 30 49
40 24 57
23
14 49
Marseilles ***** *****
43 17 49
21 29 E
Milan ***** ***** *****
15 28 2
36 46
Moscow ***** *
55 45 45
2 30 12
Munich ***** ***** *****
18 8 20
46 18
Naples ***** *****
to 50 15
57 3
Nuremberg ,**** *****
49 26 55
41 17
!9tf
P T
Oxford ~~*
LvrrruoK N. J,o,N(iiTi;nii.
51 45 39 ; ()/i 5/ l,v W
Padua ~~* . f ~~.
J5 21 2
47 20 E
Palermo ~*~ ~~~
38 (5 4.4
53 28
Paris ,~~ ~~~ ,v~~
48 50 It
9 21
Pavia ~~~ -~wi ff *~
45 10 17
36 39
Petersburg- ,, ^
59 56 23
2 1 15
Pisa ,~~ T~~ *+**,
43 43 11 ......
41 36
Portsmouth ~*~ ,
50 48 3
4 24 W
Rome ~~* ,.
41 53 54
49 59 E
SlOUgh ^r^, ^r^.
51 30 20
2 24 W
Stockholm ^^, ^,
59 20 31
1 12 14 E
Strasburgh , ,
48 34 56
30 59
Toulouse +~~ ~~, *~~
43 35 46
5 46
Turin ,r,~.~ .w~ -*-~*
45 4
30 41
Upsal *~~ ,~~v, ~~,
59 51 50
1 10 36
Utrecht ~~~
52 5 31
20 29
Venice ~~v ^^, ^^,
45 25 32
49 24
Verona *~~ *~~ ^^,
45 26 7
44 5
Vienna , *~~ *~~
48 12 40
1 5 31
Wilna
54 41 2
1 41 12
OPERA-GLASS. KitcJiiner.
An Opera-g-lass should not magnify more than three, or at the most
four times ; this also makes a pleasant prospect glass. If it have besides
a power magnifying twice, it will be an excellent assistant in giving- a
general view of the constellations, and will be a good finder for sweep-
ing the sky for a comet. The best Opera-glasses at present made by op-
ticians, have an achromatic object glass of one and a half inch in diame-
ter, magnifying four times ; the price in a plain mounting, about two
guineas and a half.
OPTICS, laws of.
The theory of Optics reposes on three laws, which depend for their
proof upon observation and induction,
1, The rays of light are straight lines.
J97 L4
P A R
2. The angles oi' incidence and reflexion are in the same plane ano
>qual.
The angles of inrideuee and refraction are in the same plane, and their
sines bear an invariable ratio to one another for the .same medium.
For the various subjects connected with this branch of science, see the
respective heads.
OSCILLATION, Centre of See Centre.
PALLAS.
This planet was discovered by Dr Olbers, of Bremen, March 28, ISCtt.
For its elements see Planets, elements of.
PARABOLA, principal properties of. See Conic Section*.
PARACENTRIC velocity. See Central Forces.
PARALLAX ( Woodkoitse, Play fair.)
1. If P be the horizontal parallax of a heavenly body, p the parallax
at a zenith distance z.
p P X sin. z.
Cor. If R be the radius of the earth j r the tubular radius, d the dis-
tance of the body, then
d = X R.
To adapt this to computation, r must be expressed in degrees, minutes,
&c. then
2. If two observers under the same meridian, but at a great distance
from one another, observe the zenith distances of the same planet, when
it passes the meridian on the same day, they can from thence determine
the horizontal parallax.
Let L and L' be the two latitudes, z and z' the observed zenith dis-
tances, then
sin. # 4- sin, '
198
PAR
This formula was employed by Laeaille, at the Cape of Good Hope,
and Wargentin, at Stockholm, for finding the parallax of Mars. It can-
not be successfully applied to the Sun, or to Jupiter, Saturn, or tlu>
Georgian ; for where the parallax does not exceed 10 or 12 seconds, tl.e
probable errors of observation will bear so large a proportion to it, as
materially to affect the certainty of the result.
The moon, however, whose parallaxes are considerable, is a proper
instance for the method, though in that case it will require some modi
fication ; as we must take into consideration the spheroidical figure of
the earth ; thus
Let R be the radius of the equator, r and r' the radii of the earth at
the two places of observation, z and x 1 the zenith distances found as be-
fore, but corrected for the /'s between the vertical and the radius, then
the horizontal parallax at the equator is
x + *' "" ( L L/ )
r sin. z -J- r sin. K'
3. The parallax of a planet in R. A. being found by observation, to find
its horizontal parallax.
Let s be the R. A. in time, taken out of the meridian, then
.p _^ 15 s x cos - ttec.
cos. lat. x sin. hour angle
If the R. A. be taken both before and after the meridian, and h and 7t' be
the two hour angles, and ,v the sum of the parallaxes in R. A. on the east
and west of the meridian,
15 * X cos. dec. .
P =
cos. lat. x (sin. h -f- sin. /*')
15 .v x cos, dec.
. h + h' h h r
2 cos. lat. x sin. -~ x cos. - -
4-. The greatest horizontal parallax of the sun and planets.
Sun 8",75 Venus 29",1G Jupiter 2",08
Mercury 14",58 Mars 17",50 Saturn. 1",027
Georgian ... 0,"415,
For Sun's parallax in altitude nee Sun,
5. Parallax of the fixed stars.
If t\\e anmial parallax does not exceed 1", the distances of the fixed
stars cannot be less than 206365 times the radius of the earth's erbit 16
1! M
f> K N
w probable, however, that the parallax of a star of the second magnitude
is not more than of a second ; and of a star of the sixth magnitude,
iiot more than or of that quantity.
PENDULUMS, oscillation of, %c.(Woof], Whev-cU, &>:
1. Let T ~ time of vibration of a simple pendulum iu a cycloidal arc,
L length, F = accelerating force, g = force of gravity 3aV 6 feet,
x =. 3.14159, &c., n number of vibrations in a given time T', then
71 '= ij
" , or in case of gravity T = \- --^~-
I jp 'JVg / o- 1'/2
and n V , T , or in case of gravity n V ft r .
* L IT* Li
Cor. Hence if x = space fallen through by gravity in 1" in any lati-
tude, and L ~ length of a seconds pendulum, then if .v be given, L
; and if L b given, x = -^- .
5T2
By help of this last formula x is found more exactly than can be done
by direct experiment. In the latitude of London L 39.126 inches,
hence x = 16.09 feet.
2. To find the vibration of a pendulum in a circular arc, let a ver.
sin. of % arc of vibration, r radius of the circle ; then
t- x , + _/ 2 -f- ifi -f- n* ............ -f n 1 ; and .'.
nn _ ]
* nt^T X *
Thus the whole number of permutations and combinations that can be
made of the 1 letters a, b, c, d, when they are taken by two's, three's,
and four's
4. Supposing there are m sets of things, one set containing n things,
another p t another 7, &c., then the total number of combinations that
can be formed by taking one from each 6et
n X P X q ......... to m factors.
Thus suppose there are 4 companies, in one of which there are tS men,
in another 8, in each of the other two .0, then the number of changes
that can be made by taking one out of each company = 68. n. t> = L>8SS.
204
P L A
PIPES, leaden and iron,, weight of. (Gregory.)
Let I be the length in feet, d the interior diameter, and t the thickness
both in inches and parts of an inch, W the weight in hundred weights j
then,
In a leaden pipe, W = ,1382 It (d + f.)
In a cast iron pipe, W = ,0876 lt(d+t.)
PIPES for conveying Water. See Fluids t pressure of.
PLANE inclined. See Inclined Plane.
PLANETS, Elements, 8fc. of.
What are usually called the elements of a planet's orbit are in number
seven.
1. The longitude of the ascending node of the orbit.
2. The inclination of the orbit to the plane of the Ecliptic.
3. The mean motion of the planet round the sun.
4. The mean distance of the planet from the sun.
5. The eccentricity of the orbit.
6. The longitude of the aphelion.
7. The epoch at which the planet is in the aphelion.
Elements and general view of the Planetary System. ( Laplace t
Maskelyne, Fince, Play fair.}
Names
of the
Planets.
Sidereal Pe-
riods of the
Planets.
Mean dis-
tances or %
axes of the
orbits.
Ratio of the
eccentrici-
ties to the y%
axes at the
beginning
0/1801.
Mean longitudes
reckoned from
the mean Equi-
nox at the epoch
of the mean noon
of Jan. 1, 1801,
Greenwich.
d.
o. ' "
Mercury
87.969258
0.387098
0.205514
166. 0. 48.2
Venus ...
224.700824
0.723332
0.006853
11. 33. 16.1
Earth ....
365.256381
1.000000
0.016853
100. 39. 10
Mars
686.979619
1.523694
0.093134
64. 22. 57.5
Vesta....
1335.205 ...
2,373000
0.093220
267. 31. 49
Juno
1590.998 ...
2-667163
0.254944
290. 37. 16
Ceres ....
1681.539 ...
2.767406
0.078319
261. 51. 34
Pallas ....
1681.709 ...
2.767592
0.245384
252. -43. 32
Jupiter..
4332.596308
5.202791
0.048178
112. 15. 7
Saturn...
10758.969810
9.538770
0.056168
135. 21. 32
Georgian
30688.7 i-i'6S7
19.183305
0.016670
177. 47. 38
205
Name*
of the
Planch:
Mean longi-
tudes of the Pe-
rihelion for the
same epoch as
before.
Inclinations of
orbits to the
Ecliptic, for
Jan. 1, 1801.
Longitudes of
the ascending
nodes on the
Ecliptic,
Jan. 1, 1801.
Mercury 74. 21. 46
o / n
7. 0. 1
/ //
45. 57. 31
Venus ...
1-28. 37. 0.8
3. 23. 3-2
74. 5-2. 38.6
Earth ....
99. 30. 5
0. 0.
0. .0
Mars
332. 24. 24
1. 51. 3.6
48. 14. 38
Vesta
249. 43.
7. 8. 46
171. 6. 37
Juno 53. 18. 41
13. 3. 28
103. 0. 6
Ceres
14fx 39. 39
10. 37. 34
80. 55. 2
Pallas .... 1 121. 14. 1
34. 37. 7.6
17-2. 3-2. 35
Jupiter .. | 11. 8. 35
1. 18. 51
98. 25. 31
Saturn ...
89. 8. 58
2. 29. 31.8
111. 55. 46
Georgian
167. 21. 1-2
0. 46. 26
72. 51. 14
Names
of the
Planet*.
Mean
diame-
ter in
English
miles.
Mean
dist
from
the
sun in
mil. of
miles.
Mean
appar.
diam.
as seen
fromthe
earth.
Mean
diam-
eter as
seen
from
the
sun.
Appa-
rent
diam.
of sun
as-
seen
from
each.
Diurnal ro-
tation on
their own
axes.
Inclina-
tion of
axes
to orbits.
The Sua
883246
32' 1".5
25m8wO,.
8241'0"
Mercury
3224
37
10
16"
80'
1 5 28
.t. ....,,
Venus .
7687
68
58
30
45,7
23 20 54
Earth ...
7912
95
17.2
32
1000
66 32
Mars ....
4189
144
27
10
21,33
24 39 22
59 22
P
163
263
1
Pallas
80
265
0.5
1425
252
3
Vesta
238
225
0.5
Jupiter...
89170
490
30
37
6,15
9 55 37
M3 near.
Saturn ...
79042
900
18
16
3,37
10 16 -2
(50 prob.
Georgian
3511-2
1800
3.5
4
],W
'206
P L A
Names
of the
Planets.
Place of
ApheHit in
Jan. 1800.
Secular
motions
ofih*
a p he! in, :
jrogres-
sive.
Secular
motions
of
nodes.
Greatest
Equation
of the
Centre.
Elon.
gation
when
sta-
tion-
ary.
Arc
of
retro-
gra-
da~
tion.
Time
of re-
tro-
gra-
da-
tion.
Sun
Mercury
Venus ...
Earth ....
Mars
Ceres ....
Pallas ....
Juno
Vesta-'....
Jupiter ..
Saturn ...
Georgian
8,yl420'50"
10 7 59 1
9 8 40 12
5 2 24 4
4 25 57 15
10 1 7
7 29 49 33
2 9 42 53
6 11 8 20
8 29 4 11
11 16 30 31
lo 33' 15"
1 21
19 35
1 51 40
10 12' 10"
51 40
23 40' 0"
17 20
1 5530.9
18 00'
28 48
13o 30'
16 12
23 d.
42
40 10
10 40 40
9 20 8
28 25
136 48
16 12
73
121
139
151
1 34 33
1 50 7
1 29 2
59 30
55 30
1 44 35
5 30 38
6 26 42
5 27 16
115 12
108 51
103 30
9 54
6 18
3 36
Names
of the
Planets.
Synodic
revolu-
tion.
Densities.
Quantities
of
matter.
Gravity
on
surface.
Intensities
of light
and heat.
0,25226
333928
27,7
Mercury
MSd.
2,5833
0,16536
1,0333
6,25
Venus ...
584
1,024
0,88993
0,9771
2,04
Earth....
1
1
1
1
Mars
^80
0,6563
0,08752
0,3355
0,44375
Jupiter ..
399
0,20093
312,101
2,3287
0,036875
Saturn ...
378
0,10349
97,762
1,0154
0,01106
Georgian
370
0,21805
16,837
0,9285
0,00276
For the telescopic appearance of the Planets see Telescope.
PLANETS, disturbances occasioned by their mutual action upon each
other. -( Pla if fair. )
The orbit of every planet, by the action of the other planets, is chan-
ged in all its elements bat t\vo ; the mean motion, and the mean distance
from the sun. Thus in Mercury and Venus the line of the nodes, thy
20?
POL
inclination to the Ecliptic, the line of the apsides, the eccentricity, and
consequently the greatest equation of the centre all vary. In Mars the
eccentricity, the lines of the apsides and nodes, vary by the action of
Venus, the Earth, and Jupiter; as also his place in his orbit, which is
not the case with Mercury and Venus, in consequence of their nearness
to the Sun. In Jupiter and Saturn, the place in their orbit, the -motion
of the apsides, and the change of eccentricity, are chiefly produced by
their action on each other ; but in the disturbance of the inclination the
other planets have a sensible effect. Uranus, from his great distance,
Buffers no disturbance in his motion but from Jupiter and Saturn.
Two interesting results are obtained from the investigation of the
planetary disturbances. 1. That both in the system of primary and se-
condary planets, two elements of every orbit remain secure against all
disturbances, the mean distance, and the mean motion, i. e. the trans-
verse axis of the orbit, and the periodic time. 2. That all the inequali-
ties in the planetary motions are periodical, and after a certain time run
through the same series of changes. This accurate compensation of the
planetary inequalities arises from three conditions ; 1st. that the eccen-
tricities of the orbits are small j 2d. that the planets all move in the same
direction, ?'. e. from west to east ', 3d. that the planes of their orbits are
but little inclined to one another.
PLANET, time of its passing over the meridian. See Time.
POLAR Sean.Sce S^aa, Polar.
POLYGONS regular, to find tlie area of.
Let ^ represent the length of one of the equal sides, n the number of
them j then
'n . / 90 H 1900 N
Area = s 2 x -r * an - f )
4V n J
.Hence the following Tablo :
Trigon - .v2 X 0.4330127
Tetragon = ** X 1.0000000
Pentagon sv X 1.7204774
Hexagon r= ** X 2.5980762
Heptagon = ** X 3.6339124
Octagon = **, X 4.8284271
Nonagon = g X 6.1818242
Decagon = ** X 7.6942088
Undecagon = s* X 9.3656399
Dodecagon * 2 X/
2GB
POPULATION, increase of. ( Bridge. )
Of the method of finding the increase of population in any country,,
under given circumstances of births and mortality.
Let P represent the population of a country at any given period j
the fractional part of the population which die in a year (or ratio
of mortality j) -j- the proportion of births in a year ; then, if A repre-
sent the state of the population at the end of n years,
Or Log. A = Log. P -f n X log. 1 +
Of the quantities A, P, m, b, n, any four being given, the fifth may be
found.
Ex. I. Suppose the population of Great Britain, in the year 1800, to have
been ten millions ; that th part die annually ; and the number of births
are ^ , and that no emigration takes place during the present century j
What will be the state of its population in the year 1900 ?
Here A = 22930000.
Ex. 2. Suppose the population of France in the year 1792 to have been
27000000 ; the ratio of mortality, during the 18th century, to have been
th, and the number of births th j What was the state of its popula-
tion in the year 1700 ?
Here P = 16864396.
Ex. 3. Suppose the population of North America to have, been five
millions in the year 1800 j in how many years will it amount to 16 mil-
lions, taking the ratio of mortality at -Uh, and the annual proportion of
births at ^th ?
Here n = 60.3 years.
Ex. 4. The population of a province, in the year 1760, was estimated
at 500000 persons j in the year 1800 it amounted to 720000 persons ; from
the bills of mortality it appeared, that, upon an average, p^th part of
209
P R K
the population had died annually ; no rector \va> kopt of the birth? ;
What was the annual proportion of them during that period *i
Here I = '.',\.\.
The annual proportion of births was about th.
Supposed Population of the World. (Enc. Brit. Suppl.)
ElirODe ^r^ff t^^r^r^f^rf^ff^f^f^^r^rr^f 185 000 000
Asia (with Australia and Polynesia) ~~, 270,000,000
Africa ~~ 55,000,000
America v~~~~~~~,~~~~~~ ,~~ ,.~~ : ~ 40,000,000
350,000,000
POWERS of numbers. See Involution.
POWERS Mechanical See Mechanical Powers.
PRECESSION of the Equinoxes. (Woodliouse* Vince, Play fair.)
I. The mean annual precession = 50",34, which gives nearly 1 for
the precession in 71| years, or about 25745 years for the entire revolu-
tion of the pole of the Equator round that of the Ecliptic. The part of
the precession arising from the action of the sun 15",3, that from the
moon = 35". If the effect of the sun bo reduced to 12",5 ; that of the
moon will be triple of it, whicli is agreeable to the latest results deduced
from the theory of the tides.
The precession affects the situation of stars in Declinatiou or North
Polar distance, and in right ascension ; hence the following Formulae.
Annual Precession in Declination.
This = 50",34 X sin. obliquity X cos. star's R. A.
Cor. When the right ascension (R. A.) is between 90* and 270, the
declination is diminished by the effect of precession. And when the R. A.
is between and 90, or between 270 and 360, the declination is
increased.
Annual Precession in R. A.
This = 50",34 X (cos. I + sin. I X sin. star's R. A. X tan. star's decli-
nation) where I obliquity of ecliptic. In this expression the first part,
50" ,34 cos. I is common to all stars.
Cor. The precession in R. A. is nothing when the angle of position is
aright^; it is also positive when that / is acute, ai^d negative when
obtuse.
II. Precession Solar t inequality of.
The mean annual precession has been stated at 50",34 ; but it cannot
have been equably produced, For the sun is sometimes in the Equator,
210
when its force causing precession is nothing- ; at other times more than
230 distant, when its force is greatest. Hence the sun's action in produ-
cing precession must continually vary from the Equinox in March to the
solstice in June. The correction due to this solar inequality is called
the semi-annual Solar Equation. In consequence of this solar inequali-
ty, the pole of the earth describes, half yearly according to the order of
the signs, round the place of the mean pole a circle whose radius =
0".434o.
The solar inequality affects the precession of the stars in longitude,
declination, and right ascension, also the obliquity of the ecliptic ; hence
the following formulae.
Equation of precession in longitude.
This = 1", 1 X sin. 2 sun's longitude.
Substitute this expression for 50",31 in the above formulae for preces-
sion, and we shall have the equations of precession in declination and
R. A.
Correction of the obliquity.
This = 0",4345 X cos. 2 sun's longitude.
The variation in the obliquity of the ecliptic arising from the sun, is
called the correction of the obliquity ; that from the moon is called the
equation of the obliquity.
III. Precession, lunar inequality of.
The lunar inequality of precession is called Nutation, to distinguish it
from the solar inequality. In consequence of the lunar action, the true
pole of the earth describes about the place of the mean pole, in 18 years
7 months, contrary to the order of the signs, an ellipse of which the ma-
jor axis 19",2, and minor axis 15".
The nutation affects the precession of the stars in longitude, declina-
nation, and R. A. and also the obliquity of the ecliptic ; hence the follow-
ing formulae.
Equation oftJie Equinoxes in R. A.
The variation in the precession, or in the equinoctial points, usually
called the Equation of the Equinoxes in R. A. is
17",2 sin. longitude moon's node.
This affects the longitude of all the stars equally.
Nutation in declination.
Let A be the R. A. of a star, D its declination, N the longitude of the
moon's node j then nutation in declination ,
1",1 sin. (A + N) + 8",5 sin. (A N).
211 M3
PRO
Nutation in Right Ascension.
This 8",5 tan. dec, cos. (N A) + 1",1 tan. dec. cos. (N -f A) and
if to this be added the equation of the equinoxes, the whole effect of
nutation will
4. 8,"5 tan. dec. cos. (N A) + 1",1 tan. dec. cos. (N -f. A) 4.
17", 2 sin. N.
Equation of the obliquity.
This = 9",63 cos. N.
PRESSURE of Earth against walls. 'See Earth, pressure of.
PRESSURE of Fluids. See Fluids.
PRESSURE, centre of See Centre.
PRISM. See Refraction.
PROGRESSION, Arithmetical ', Geometrical, and Harmonica!.
I. Arithmetical Progression.
All the cases of Arithmetical progression may be solved by the follow-
ing formulae :
1. Let a first term, I last, b common difference, n number of
terms, s = sum of the series ; then
Of the quantities , /, b t , any three being given, the other may be
found by the equation
I a + n l. b.
2. Of the quantities a, 6, , .v, any three being given, the other may be
found by the equation
3. Of the quantities , I, n, s t any three being given, the other may be
found by the equation
II. Geometrical Progression.
1. All the cases of Geometrical Progression may be solved by the fo?
lowing formulae :
Let a = first term, I last, r = common ratio, n = number of terrn^,
s = sum of the series ; then
Of the quantities a, l t r t n t any three being given, the other may be
found by the equation
PRO
2. Of the quantities , r, w, s, any three being given, the other may be
.found by the equation
a r n a
S =-T=
3. Of the quantities a, /, r, s t any three being given, the other may be
found by the equation
Ir a
TTCT-
4. When n, or the number of terms is infinite, then of the quantities
a, /, s, any t\vo being given, the other may be found by the equation
III. Harmonical Progression.
1. Let a, b t c be in Harmonical Progression; then a I c I", a 6 I
*. c.
2. Let a, b, c, &c. be as before, then
, y, , &c. are in arithmetical progression.
3. Let a and b be the two first terms of an Harmonical Progression, to
^continue the series.
ab ab
4. To find an harmonic mea n (x) between two quantities a and b.
x= 2 ab
a+b
5. If between two quantities a and b t an harmonic mean .r, and aa
.arithmetical mean y, be inserted,
a : x ;: y : b.
G. If between two quantities a and b an arithmetic mean x t a geome-
tric mean y> and an harmonical % } be inserted
x \ y \\ y \ z.
7. If a fourth proportional be found to three quantities in Arithmetical
progression, the three last terms are in Harmonical progression.
PROJECTILES in a vacuum. ( Whewell.)
Formulae for finding the range, altitude, and time of flight, of bodies
projected along planes inclined to the horizon.
J. Let r range, A = greatest altitude, t time of flight, v yelc*
213
PRO
city of projection, h height due to this velocity, a, = /.of projectioa
above the horizontal plane, i = elevation of the plane above the hori-
zon, g ~ 32 % feet ; then \ve have the following equations.
A v * sin.8( <) _ , sin.* ( )
~ 2g ' cos. 3 ; cos. 2 >
_ 2p sin. (-<) _ /2 A 2 sin. ( <)
5- ' cosTT " g- ' cos. <
Greatest range =
2. When t = o; r will be the horizontal range, and the above equa-
tions will become
r = . sin. 2 2 h sin. 2 .
A = -- . sin.2 = h sin. .
t = . sin. = V ^- . 2 sin. .
#
Greatest range = 2 A.
3. The curve described by a projectile is a parabola, the principal pa-
rameter of which = 4 h cos. 2 , and the velocity at any point is that ac-
quired by falling from the directrix.
4. To find an .equation to the curve, re-
ferred to horizontal and vertical co-ordi- s -n-
nates.
Let AB = jr, B C = #, t = any time ;
then
x v t cos. a.
y = v t sin. <* and eliminating t.
y x tan. ---
2 r* cos.* . '
the equation to the curve.
Cor. 1. If, as before, h = -~,
PRO
xz
?/ = A- tan. r 5.
4 h cos. 2
Cor. 2. To find where the curve meets the horizontal piano, we must
put y = o, .*. x tan. , T-T j = o, .'. # = 4 . x .
2 h cos. 8 oe,
Ex. 1. Let a body be projected from the top of a tower horizontally with
a velocity acquired in falling down its height j at what distance from the
base will it strike the horizon ?
Here if a ==. altitude of tower, y a, z= o, and vz =. 2 g a, .'.
. a -^ , and # = 2 a.
J?A\ 2. A body is projected at an / of 45, with a velocity of 50 feet per
second ; find its horizontal range.
Here 45, v 50, .". when y o
250
_ _.
Ex. 3. A projectile is thrown across a plain 120 ject wide, to strike ;t
mark .'30 feet high, the velocity of projection being that acquired down
SO feet ; required the / of projection.
215
PRO
Here y ;30, x = 1-20, v? = 1GO g, .'.
a
1 = 4 tan.
2 (cos.) 2
.*. (tan.) 2 - tan. , -77-,
o o
and tan. a = 1 or ~r, and . = 45.
Ex. 4. A body projected from the top of a tower at an / of 45o above
the horizontal direction, fell in 5" at a distance from the bottom equal to
its altitude ; required the altitude.
Let a = height, then a = 45, t = 5, and y = a t
.;a = a tan. 45 -J. 25,
/. a = 200.
PROJECTILES, resistance of air to. See Gunnery.
PROJECTION, principles of.( Vince.)
I. Orthographic Projection.
1. The figure of a straight line is a straight line in the projection.
2. The figure of 'the projection of a circle is an ellipse, of which the
minor axis is the cosine of inclination of the circle to the plane of pro-
jection. Hence if the circle be parallel to the plane of projection, the
projection will be a circle equal to it. If the circle be perpendicular to
the plane of projection, the circle is projected into its diameter ; any arc,
reckoned from its intersection with the plane, into its versed sine ; and
the remainder of the quadrant into the sine of that remainder, or into
the cosine of the first mentioned arc.
3. In this projection the area of the circle I the area of the ellipse into
which it is projected II radius : cosine of inclination of the plane of the
body to the plane of projection ; hence the area of the circle will be di-
minished in the ratio of radius I the cos. of this inclination. And this is
true whatever be the form of the projected body. Also the projection
is not similar to the body. Hence equal parts upon the surface of a
sphere will not be projected into parts cither equal or similar.
This projection is not convenient for maps, but is used in the con-
struction of solar eclipse*.
PRO
II. StereograpJiic Projection.
J. The projection of an arc, measured from the pole, is equal to the
tangent of half that arc.
2. The projection of every circle is a circle.
3. The projection of all circles parallel to the plane of projection will
be concentric circles, the radii of which are the tangents of % the dis-
tances of the circles from the pole.
4. The projection of every great circle passing through the pole is a
straight line.
5. The radius of projection of any other great circle is the secant of the
angle between the plane of the circle and the plane of projection.
From these Arts, it appears, that the projection of the parts of the
sphere will not properly represent, in magnitude and situation, the parts
themselves.
6. If the place of the eye be the pole of the earth, the meridians will
be projected into straight lines (Art. 4) ; and the parallels to the equator
will be projected into circles (Art. 3). This is called the Polar Projec-
tion.
7. If the eye be placed in the equator 90 distant from the point from
which the longitude is reckoned, the projection of the radius of any me-
ridian will be the secant of its longitude (Art. 5). And the radius of
projection of the parallels of latitude is the cotangent of their latitude.
This is called an Equatorial Projection.
The stereographic projection is chiefly used in delineating maps of the
world.
III. Mercator's Projection.
1. In this projection the meridians are parallel linos, the degrees of
longitude are all equal j the parallels of latitude are also parallel lines,
but unequal, a degree of latitude being to a degree of longitude '.'. rad.
I cos. latitude, and .*. the length of a degree of longitude being constant,
the length of a degree of latitude will be inversely as the cosine of lati-
tude, and will .", increase in going towards the pole.
2. To find the length of the meridian on this projection for any num-
ber of degrees of latitude.
Let x "= required length, r earth's radius then
cot. comp. latitude
% = r x n. 1. .
r
If .'. we take the latitude = 1, 20, ,3 , 90 we can construct a
217 N
P U L
Table shewing 1 the length of the meridian on the projection for every de-
gree of latitude ; in like manner it may be constructed for every minute,
Such a table is called a table of Meridional Parts.
This projection is of great use in navigation, on account of its being
constructed by right lines only j the rhumb lines or lines of azimuth be-
ing also straight lines.
Suppose for example a ship wants to go from any place A to B laid
down upon Mercator's map, and it is required to find the rhumb or point
of the compass it must sail upon j we have only to join A B, and that is
the rhumb. Now to determine what rhumb this is, there is always in
these maps one or more points, from which are drawn 32 straight lines,
representing the 32 points of the compass. Apply .*. one edge of a par-
allel ruler to the line A B, and bring the other edge over the point from
which the lines of the compass are drawn, and it immediately gives the
direction in which the ship must sail.
PULLEY.
1. In the single fixed pulley, there is an equilibrium when the power
and weight are equal.
2. In the single moveable pulley whose strings are parallel, P '. W ::
1 :2.
3. In a system where the same string passes round any number of pul-
lies, 1 and the parts of it between the pullies are parallel, P .* W : : \ '. n y
n being the number of strings at the lower block.
* Cor. If we consider the weight of the pullies, it is only requisite to add
the weight of the lower block ; hence if a be this block,
W n P a.
4. In general in the single moveable pulley, P : W :.' rad. '. 2 cosine of
the angle which either string makes with the direction in which the
weight acts ; or : : sin. angle which the two strings make with each
other : sin. of the whole angle.
5. In a system where each pulley hangs by a separate string, and the
strings are parallel, P : W ; : 1 : 2 n where n is the number of moveable
pullies.
Cor. 1. Hence W =2 n P. If the weight of the pullies be taken into
the account, and a = weight of each, W =2 n P a (2 n 1) j hence the
weight W is less as a is greater.
Cor. 2. When the strings are not parallel, P : W ;; (rad.) w : 2 X
218
cos. * X cos. /3 X cos. y &c. where , /3, y, &c. are the angles which the
string's make with the direction in which the weight acts in each case.
6. In a system of n pullies each hanging by a separate string, where the
strings are attached to the weight, P : W : : 1 .' 2 n 1.
Cor. Supposing the weight of each pulley a, then the part of the
weight sustained by the pullies a X (2 n n 1) ; and .*. W
(2 n 1) P + (2 n n 1) a.
PULLEY, on the ascent or descent of bodies over.
1. If two bodies P and Q are connected by a string and hung over a
fixed pulley, the accelerating force, supposing P the heaviest, is g X
P-Q
-. Substitute this for F in the formulae for the rectilinear descent
Xg*.
of bodies ( see Motion) and we get
2. If two bodies P and Q are connected to-
gether by a cord going over a fixed pulley, and
one of them Q descends down an inclined
plane, we have the moving force of Q = Q X
TT
=r- ; hence the moving force of Q when con-
nected with P = -^ P =
QH PL
O H P T
~~ j and accelerating force
irx-
LX
If P draws up Q, accelerating force = g X , which may
L X " "
be substituted for F as in the last Art.
Let both bodies P and Q move upon inclined
planes, whose lengths are L and I respectively,
and having a common altitude H, and let Q be
.the descending body ; then moving force of >
219
P U M
the moving force of the system =
L Q T ""/ l P X H, and accelerating force of Q = g X L Q ~ /PX JL
L * LJX P + Q
which may be substituted as before.
PUMP. Air Pump.
1. If b represent the capacity of one of the barrels, and r that of the
receiver together with the pipes and gages connected with it j then the
quantity of air extracted after every turn : the quantity before that turn
C : b I 2 b + r. And the quantity left in : the quantity before : : b + r
Cor. Hence if P represent the quantity of air in the machine before the
first turn, the quantity left in after n turns is
And the quantity exhausted is P P. ( ^^) -
P.
2. The density of the air in the receiver at first : the density after t
turns :: (2i + r)* : (&+*)*
3. When the density of the air is diminished in the ratio of n : I, the
number of turns t = WLJ2. .
log. 2b -f- r log. b -f. r
4. As the air is exhausted, the mercury will rise in the gage, and the
defects of the mercury in the gage from the standard altitude, after each
successive turn, form a geometric series, the ratio of whose terms is
2 b + r I b + r. And the ascents of the mercury at each successive turn
form a geometric series, the ratio of whose terms is 2 b + r I b 4. r.
PUMP condensing, or condenser.
If b represent the capacity of the barrel of the syringe, and r that of
the receiver, then after t descents of the sucker, the density of the air
in the receiver, will be to the density at first in the ratio ofr+tblr.
PUMP, common or sucking.
1. In the common pump the force necessary to overcome the resistance
experienced by the piston, in ascending, is equal to the weight of a co-
P Y R
\itmn of water, having the same base as the piston, and an altitude equal
to that of the surface of the water in the body of the pump above that in
the reservoir.
2. In a sucking pump, if the height of the lower or fixed valve above
the surface of the water #, the length of the stroke of the piston b,
and the height of a column of water in equilibrium with the pressure of
the atmosphere = h, the height to which the water is raised by the
first stroke is
a 4. b + h V( + b -f. h}* kbh
2
3. The same notation being retained, and c being put for a -f b or tire
greatest height to which the piston ascends, b must be greater than
j-r- otherwise the water will not rise above the piston.
4. Height to which water will rise in a vacuum in different states of
the barometer.
Barom. in inches. Height of water in feet.
28 ,,,~.,~wyv w ^ w ~,~ w 31.66
28| w 32.23
29 32.79
29| 33.36
30 33.92
30| 34.49
31 ,~~~~ ~~ 35.05
Hence the valve of the piston in the common pump must be nearer to
the surface of the water in the reservoir than 33 feet, otherwise the wa-
ter can never rise above it.
PYROMETER, Wedgwood's, for measuring very high temperatures.
The scale of this Pyrometer, or the point marked commences at red
heat fully visible in day light, and is equivalent to 1077|o of Fahrenheit's
scale, and one degree of the former is = 130 of the latter. The extre-
mity of Wedgwood's scale is 240, but the highest heat he measured with
it is 160. It appears .'. that this pyrometer includes an extent of about
32000 of Fahrenheit's degrees, or about 54 times as much as that between
the boiling and freezing points of mercury, by which mercurial ones are
naturally limited ; that if the scale be produced downward in the same
manner as Fahrenheit's has been supposed to be produced upward for an
ideal standard, the freezing point of water would fall nearly 8 below
221 N2
R A I
of Wedgwood's and the freezing point of mercury a little below 8|, and
that there are 80 from the freezing of water to full ignition.
Q
QU ADRATRIX of Dinostrates, Equation to.
y V 7-2 $2 s ( r _ A ^ } where r = radius, and s the sine of the circ.
arc, by the help of which the curve is generated.
The radius of the generating circle is a mean proportional between the
quadrantal arc and the base of the quadratrix.
If with the base of the quadratrix as radius, there be described a qua-
drantal arc, this will be equal in length to the radius of the generating
circle.
QUADRATURE of Curves. See Area.
R
RADIUS vector of a planet's orbit. See Anomaly.
RAIN, quantity of at different places.
Mean annual quantity of rain for 30 years, as observed at the apartment*
of the Royal Society. The gage is placed 75 feet 6 incJies above the
ground.
YEARS.
1790
1791
1792
1793
1794
1795
1796
1797
179S
f '99 vw^rw^yvy^^
INCHES.
16.052
15.310
19.489
17.128
18.466
16.864
14.779
- 22.697
. 19.411
. 19.662
YEARS.
1801
INCHES.
18.925
. 19.197
1802
1803 ~~~~~
13.946
1804
20.973
1805
20.396
1806 ~~
1807 ~~~~~v
.^~~~ 20.427
14.206
1808 ~~~~v>
18.475
ff ^ fffff ^ 20.711
YEARS.
1810 and 1811*^
1813
R
INCHES.
18.348
15953
A I
YEARS.
1817 **+**,***,
INCHES.
MATMW 15299
1819
11.636
13 727
1815 ******.
^ 16.367
r*~ 12.968
1820
18381
1821
****** 23567
1816 ****************,
15.174
17.548
23.567
11.636
me period of '30
INCHES,
1.979
1.489
1.564
1.712
**, 1.985
1.520
Average of
Greatest me
Least do, .
Mean quantity of r<
January *********
February *******
March ************
April wwjfjfjffw
:he 30 years
an quantity d
uring this period
tin for each in
yet
INCHES.
1.253
1.004
***** 0.884
1 269
onth during the ab
irs.
July *************
August **********
September *****
October ********
November *****
December *****
May **************.
1.476
June **************,
1.411
It appears from observation, that the quantity of rain, as shewn by
two gages, is not materially influenced by the height of the places above
the level of the sea, provided the heights of the gages above the ground
are equal ; but it is a singular fact, which has not been satisfactorily ac-
counted for, that it is very considerably affected by the height of the
gages above the surface of the earth, though all other circumstances are
the same. This will appear by a comparison of the following results,
given in the Philosophical Transactions.
Quantity of rain observed by Mr Daines Barrington, for upwards of
four months in 1770, as shewn by two gages, the one placed upon Mount
Rennig, in Wales, the other on the plain below at about half a mile dis-
tant ; the perpendicular height of the mountain being 1350 feet, and each
gage being at the same height above the surface of the ground.
INCHES.
Bottom of mountain ********************** 8.766
Top of mountain *************************** 8.165
Quantity Of rain observed by Dr Heberden, from July 7, 1766, to July
7, 1767, as shewn by three gages, one placed below the top of a house, a
second upon, the top of a house, and the third upofl Westminster Abbey,
223
H A I
INCHES.
Lowest gage f ^^ f ^^^^ fJ ^ ff ^ ff ^ f 22.608
Middle gage 18.139
Highest ~~~^ 12.099
The same result was obtained from the two gages belonging to the
Koyal Society, the one placed 75 feet 6 inches above the ground, the other
a few feet distant from the other and 11 feet 6 inches lower.
Mean annual quantity of rain, as shewn by the two gages.
YEAR. LOWER GAGE ix INCHES. HIGHER.
1812 ^^^^^ 22.03 18.348
1813 ,~~ 18.296 15.953
1814 ^^ 20.723 ~~~~ 16.367
These facts should be attended to, in order to prevent any inaccurate
conclusions from a comparison of different gages.
Estimate by Homboldt of the quantity of rain in different latitudes.
Latitude. Eng. inches. Latitude. Eng. inches.
GO swvswww 96 45 *~~~w~~~ 29
19 ro^vwwr** 80 60 ~~~v~w~~ 17
Professor Leslie has given the following empirical rule for the annual
deposit of rain and dew in any latitude.
Quantity = 75 (1 sin. lat.) -f 8 =r depth in inches.
Annual fall of rain at different places y according to Dalton and others.
( Young's Nat. Phil) INCHES.
Granada, Antilles 126
Cape Frangois, St Domingo ....... 120
Calcutta 81
Bombay 64
Charlestown 50.9
Pisa 43.2
Rome 39.0
Venice 36.1
Tadua 34-5
Zurich 33.1
Madeira 31.0
Loyrten 30.2
R A I
INCHES.
Hague 28.4
Algiers 27.0
Utrecht 24.7
Lisle 24.0
Dublin 22.2
Edinburgh 22
Berlin 20.6
Petersburgh . . . 17.3
Upsal 16.7
Keswick, Cumberland, 7 years 67.0
Kendal, Westmorland, 25 years 63.9
Garsdale, Westmorland, 3 years .... 52.3
[Lancaster, 20 years 39.7
Townley, Lancashire, 15 years . . . . . 41.5
Dover, 5 years . . , . . . . . 37.5
Liverpool, 18 years ...,.,. 34.4
Manchester, 33 years 36.1
Bristol, 3 years ........ 29.2
Chatsworth, Derbyshire, 15 years .... 27.8
Barrowby, near Leeds, 6 years ..... 27.5
Fyfield, Hampshire, 7 years 25.9
Norwich, 13 years . . 25.5
Lyndon, Rutlandshire, 21 years 24.3
Near Oundle, Northamptonshire, 14 years . . 23.0
South Lambeth, 9 years 22.7
Dalton's mean for all England ..... 31.3
Dalton's mean, for rain and dew together, for all
England 36.0
M. Cotte's mean annual quantity of rain falling at 147
places from N. lat. IK to 60 34.7
The superficies of the globe consists of 170,981,012 square miles; sup-
posing therefore that the mean annual quantity of rain for the wholo
globe to be 34 inches, the quantity of rain falling annually will amount
225
R A I
to somewhat more than 91,751 cubic miles of water, which must lie sup-
plied by evaporation from the surface of the earth and sea. See Eva-
poration. The dry land amounting to 52,745,253 square miles, the quan-
tity of rain falling on it will amount to 30,960 cubic miles. The quantity
of water running annually into the sea is estimated at 13,140 cubic miles,
a quantity of water equal to which must be supplied by evaporation from
the sea, otherwise the land would be soon completely drained of its
moisture.
The area of England and Wales = 46,450 square miles, taking there-
fore Dalton's mean at 36 inches, we shall have the annual quantity of
rain and dew falling in England and Wales = 28 cubic miles of water.
RAINBOW. (Wood.)
1. If a ray of light refracted into a sphere, emerge from it after any
given number of reflections, to find the angle contained between the di-
rections in which it is incident and emergent.
Let $ and $ =. angles of incidence and refraction, p = number of re-
flections, then the deviation, or inclination of the emergent to the inci-
dent ray is
20 2(p + l)?' or 2 (p + l)0' 2$.
Cor. In the primary rainbow p = 1, /_. deviation = 4 $' 2 . 10'.
4. Construction of the primary and secondary Rainbow.
The red rays we have seen
are efficacious when the /
between the incident and
emergent rays = 42. 2', and
the violet rays when the
same /_ 40. 16' ; hence if
H Q be the horizon, S, S',
S" rays proceeding from the
sun, O the eye of the spec-
tator, and the / P O R ( =i
/ S" R O) be taken = 42. 2'
the drop R will transmit the
red rays to the eye ; and if
P O V (= S' V O) be taken = 40o. 16' the drop V will transmit the violet
rays. The drops betwixt R and V will transmit to the eye the other
eolours in their proper order.
If O R and O V revolve about the axis O P, every drop of water in
the surface of the cones thus described \vill respectively transmit to the
eye a small parallel pencil of red and violet rays ; and thus a red and
violet arc, whose radii (measured by the angles which they subtend at
the eye) are 42. 2', and 40. 16' respectively, will appear in the falling-
rain opposite to the sun ; and the same may be said of the other colours.
The parallel pencils of red &c. rays
which emerge from other drops fall
above or below the eye.
The secondary rainbow is formed by
two refractions and two reflections.
In this case, as we have seen, the vio-
let rays are efficacious when the ^
contained by the incident and emer-
gent rays 54. 10', and the red
rays when the same / = 50. 58'.
Hence as in the primary bow, if ^ ** "" \ x
P O V = 54o. 10', the drop V will trans-
mit the violet rays to the eye ; and if PO R = 50<>. 68' the drop R will
transmit the red rays.
227
EEC
Hence the colours in the two bows lie in a contrary order, the reti
forming- the exterior ring of the primary, and the interior ring of tlte
secondary bow.
5. To find the altitude and breadth of the rainbow.
In the primary, the altitude of the highest point cf the red arc = 42.
2' sun's altitude j and of the violet 40. 16' sun's altitude. Hence
the breadth of the bow, supposing the sun a point = 1. 46' ; this breadth
must, however, be increased by 30' the sun's apparent diameter, and .*.
the true breadth = 2<>. 1&.
In the same manner it may be shewn that the altitude of the highest
point of the secondary = 54. 10' sun's altitude; and breadth = 3. 42'.
Cor. When the sun is in the horizon, the altitude of the bow is equal
to its radius j if the sun's altitude equal or exceed 42. 2', there can be no
primary bow j and if it equal or exceed MO. 10', there can be no secon-
dary.
6. Given the radius of an arc of any colour in the primary rainbow, to
find the ratio of the sine of incidence to the sine of refraction, when rays
of that colour pass out of air into water.
The ra dius of the arc" 4 0' 2 q> ; let the tangent of 2 $' <*, half
tlu's angle, be a, the tangent of $' ; then
2 z 3 3 a z* a o.
The value of z being thus obtained, the angles $' and and conse-
quently their sines may be found from the tables.
RECIPROCALS of numbers. See Involution.
RECIPROCAL Spiral See Spiral
RECTIFICATION of Curves.
Let z = curve, x and y the abscissa and ordinate ; then
Ex. 1. In the semicubical parabola, where a x* = y* t
- ( 9 ff-M)4- Sa
27 | " 27 '
2. In the common parabola, = =- x (y* + b* y*) 9 + I *
824
R E F
3. In a circular arc, whose tangent is t,
For rectification of Spirals see Spiral.
REFLEXION in Optics. ( Coddington. )
1. Reflexion at plane surfaces.
1. To find the direction in which a ray of light, emanating from a
given point, takes after reflexion at a plane mirror.
Let the ray proceed from a point Q, and a perpendicular Q C be drawn
to the surface of the reflector, and let the ray after reflexion cut Q C
produced in q ; then will Q and q be on opposite sides of C, and Q C will
-Cq.
2. To find the same when the ray is reflected alternately by two plane
mirrors inclined to each other at any given angle.
Let if be the / of incidence at the first reflexion
I. second
i the inclination of the two mirrors j then we shall have this series
of equations,
(n 1) t
* n
or Again < "
O O'" ~2c + 2a( JOO=2e
2. Suppose now that the mirrors, instead of being parallel, are inclined
to each other, in this case the number of images will be limited, and will
evidently lie in the circumference of a circle, whose centre is the inter-
section of the two planes, and radius the distance of the object from that
intersection.
Now let H I and K I be the mirrors, O the
object, then as before there will be two series
of images
O', O", O'" &c. and O, O, O &c.
to determine the distances O O', O O" &c. mea-
sured along the circumference of the circle, put
e I O or II O - 6, O I K or O K = fi', H I K
t or H K = ;, then
231
REF
00' -20 X /-OO-20'
O O" = 2 < f J O O = 2 *
. Again ^
O O" = 2
and the number of images in the first series is the least whole number
greater than *""" ; and the number of images in the other series is
the least whole number greater than **'"" .
If i be a measure of ar, the whole number of images is ; and in this
case two images of the different series coincide.
3. If the object placed before a spherical reflector be a circular arc con-
centric with it, the image will also be a circular arc concentric with and
similar to the object, and its position and magnitude may be determined
by the proportion
F 9 : FE :: FE : FQ.
4. If the object placedbefore a spherical reflector be a straight line, the
image is a conic section ; and is a parabola, ellipse, or hyperbola, ac-
cording as the distance of the object from the centre of the mirror is
equal to, greater, or less than, half its radius.
REFRACTION in Optics. (Coddington.)
I. Refraction at plane surfaces.
1. Given the direction in which a ray falls on a plane surface bounding
a refracting medium ; to find the direction of the refracted ray.
Let the ray proceed from the point Q, and a perpendicular Q C be
drawn to the surface of the refracting medium, and let the ray after re-
fraction cut Q C or Q C produced in q ; put A and A' = C Q and Cq ; 6
and 6' ^'s. of incidence and refraction ; in the ratio of the sine of inci-
dence : sine of refraction, usually called the ratio of refraction , then
t>r when 6 is small, as it is usually supposed to be,
A' m A nearly.
2. Let a ray pass through a refracting substance bounded by two
parallel plane surfaces ; to determine its direction after emergence.
R E F
Let D = distance of the foci of incident and emergent rays, T th
thickness of the medium, then
3. To determine the refraction which a ray experiences in passing 1
through a medium bounded by planes not parallel ; for example a tri-
angular prism_of glass,
Let i =. vertical angle of the prism.
$ / of incidence.
^ {_ of emergence.
of deviation of the incident and emergent rays j then
S =
~ r *" "A
N4
R E F
In order to find the principal focal distance, which we call / (see Itc-
fexion II) we have of course only to make A infinite in the equations just
given ; we have then
1 _ m 1 _ w/__
I -rn i -. m
Case 2. , or/ zz -
/ ni r m 1
Case 3. -f ^ , or/ r ?
It is important to observe that in all cases the distance of the principal
focus from the surface, is to its distance from the centre, as the sine of
incidence to the sine of refraction.
If we introduce the distance /into the formulae, we shall have in
Cases 1 & 2,1=}+^,
3& ^ = 7 + Z-
The following are corresponding values of A and A', in different posi-
tions of the conjugate foci Q and q.
00
Co*e 2.
( A = X , ^-r, 0, r. 00
i **f-*r^
| A m r m r
A> = - srrr 00 - - -"==!
Ta^e 3. -
234
R E F
Case 4.
2. To find the direction of a ray after refraction through a lens.
The method here is to consider a ray refracted at the first surface, as
incident on the second, and there again refracted.
Let A" be the distance of the focus after the second refraction, r' the
radius of the second surface, t the thickness of the lens, the other sym-
bols as above ; then
To find the principal focal distance F, put A infinite in the above ex-
pression, i.e. suppose the rays parallel, and we have
a,nd then = ^- + ~.
Ill -*-
If we put for ~ in the above equation,
Hence arise different values of rr according" as is positive or
negative.
In the concavo-convex lens, either r is less than f and F positive j or
when the lens is turned the contrary way and r greater than r> they are
both negative, we have then
In the moniecut, either r ia greater than f, both beicg positive, and.
then
or : 4 if leso than r' and both are negative, 88
00014
* Hydro- ea
1.00014
o oooi
1 00029
00012
1 00045
00018
239 O2
n E F
REFRACTION, terrestrial ( Vince, Playfair, $cj
1. To determine it, let E apparent elevation of a mountain from a
point in the plain below ; D = apparent depression of that point from
the top of the mountain observed at the same moment; A = ^ subtend-
ed at the earth's centre by the distance between them ; then
A 4. E D
Refraction = ^ .
The terrestrial refraction found by this theorem, when the elevation
is not very great, varies from - to pj of the ^ A, but in the mean
state of the atmosphere - _L of A, which, in taking the elevation of
any object, must be subtracted from the observed E to give the cor-
rect elevation. Also the radius of curvature of the ray varies from
twice to 12 times the earth's radius, but in the mean state of the atmos-
phere 7 times earth's radius. When the ray is not horizontal it =
7 times earth's radius
Bin. appar. zen. disk
2. But in determining the height of a mountain, a correction may be
made at once both for the curvature of the earth and for refraction thug.
Let L = horizontal distance of the object in English miles, then the cor-
2 La
rection for curvature in feet is , (see Levelling} and for refrac-
21
2 1 2 9 L* 4 LS
-- -^^ = - = feet which must be added to
3 21 7
computed height, and it will give correct height both for curvature and
refraction.
3. To determine the most distant point on the earth's surface that can
be seen from the top of a given height with and without refraction.
Let A = given height in miles, r = earth's radius, thpn in the mean
state of the atmosphere, the distance of the farthest visible point =
tj j and distance, if there was no refraction, = \'2r h ; .'. dis-
tance which the eye can reach with refraction : do. without : : ^ ;
Ve :: 14 : 13 nearly.
Cor. A/-^- = 96.1 miles, .'. the distance of the farthest visible
point in miles, allowing for refraction, = 90. 1 ^ 7T. Or by the last Art,
4. I 9 \/7 fat
if 7i> -height in feet, -- : = h', .'. L = -^.
J . fa * , / R E B ,
n ** , />iv ^^, ^ ^ ^
By the last formula the following Table was computed
TABLE,
Shewing the distance of the farthest risible point in miles
from the top of a given height, taking into account the
tion.
Height in feet.
500
700
1COO
1500
2000
2500
3000
3500
4COQ
5000
6030
7000
8000
9000
10000
15000
that can le teen
effect of refrac.
Dist. in miles,
, 29.5
. S5.0
. 41.8
,51
.79
. 83
.94
, 102
. 110
. 118
. 125
. 132
. 162
, 187
Ex. 1. The topmast of a ship 50 feet high was just visible to a specta-
tor situated 20 feet above the level of the sea ; required the distance of
the ship,
Feet. Miles.
By table 50 ..., give 9.35
20 do 5.91
Required distance 15.26
Ex. 2. The summit of Mouna Roa (whose height is supposed 15,000
feet) was observed at 180 miles distance ; required the height of the ob.
server. Miles.
Observed distance 180
Distance due to 15000 feet 162
Difference 13
which answers to a little lees than 200 feet altitude.
241
ft E F
TABLE of distances at which mountains are said to "hav?. been
observed.
AUTHORITIES. MILES.
Himalaya mountains ^^^^^^^v^^v,^^ Sir W. Jones 244
Mount Ararat *v~^**~.^~^^~^,^^^^,^,-^. Bruce 240
Mouna Roa, Sandwich Isles (53 leagues) 180
Chimborazo (47 leagues) ~ ^, 1GO
Peak of Teneriffe from Cape of Lauzerota 135
Do. from ship's deck ^^ . 115
Peak of Azores ~ r~~~~ Humboldt 1G
Temaheud ~^~r^~^~^~^r~^r*^r~~~~* Blorier 100
Mount A thus ~-~ ~, Dr Clarke 1GO
Adam's Peak +~~,~~ ~~~ C5
Ghaut at the back of Tellii-hery ~ 4
Golden Mount from ship's deck ~~~~~~ *,, 93
Pulo Pera from the top of Penang >~~^~~ 75
Ghaut-at Cape Comorin ~~~~~, ~~ * 73
Pulo Penang from ship's deck ~~*~~~~ 53
Ths l:-.st six observations, and that of the Peak of Teneriffe, were made
by a writer in the Calcutta Monthly Journal.
REFRACTION of the heavenly bodies*- ( Vince, Maddy.)
1. The refraction of a star in the zenith is nothing, is greatest, in the
horizon, and at considerable altitudes is nearly as the tangent of the
zenith distance. Or more nearly as tan. (z Sr}, \lz zenith distance,
and r the refraction found by the common rule.
Cor. Refraction ~ 57" X tan. (g 3 r.)
2. To determine the refraction of a star by observation.
Observe the altitude and azimuth of a star of a known declination at
the same moment : from the azimuth, the polar distance, and the com-
plement of latitude, compute the altitude; the difference between this
and the observed altitude is the refraction.
3. To determine how much the apparent time of rising 1 nnd setting 1 of
a star is affected by refraction,
T . _ I*** refract. *^ j***v ,
ia3 ~~ 15" X cos. lat. X sin. star's azim."
Hence the time is lejst, when the star is in the equator. Or if / = la-
titude, I star's declination, r few refraction. a>b A0yv* *^
242 J
R E F
Time =r _
15o v(cos. (I + 3). cos. (I $))
4. Twilight is occasioned by the refraction and reflexion of the sun'a
rays passing through the atmosphere, and continues till the sun descends
about 18 degrees below the horizon.
To find the duration of twilight.
Let h and h' be the hour angles corresponding to the beginning and
end of twilight, I the latitude, and the sun's declination j then
cos. li tan. 1. tan.
cos. h> sin. 18. sec. 1. sec. 8 tan. I. tan. 3
hence li' h may be deduced.
Cor. Twilightwillcontinue all night, if I + S be greatarthan 72^_
To find the time of year when twilight is shortest.
sin. =. tan. 9. sin. I
and sin. h = sin. 9. sec. I
The first equation gives the sun's declination, or the time when the twi-
light is shortest; and the second gives the duration of it.
Ex. In latitude 52, the time of shortest twilight will fall about March
2, and October 11 j and the duration will be about 1A. 58m.
5. The refraction varies with the state of the barometer and thermo-
meter.
Dr Maskelyne's Formula.
Let a =- height of barometer in inches, h height of Fahrenheit's
thermometer, x = zenith distance, r = 57" tan. z j then
Refraction = ^-^ X tan. (* 3 r) X 57" X
Dr Young's Formula.
.0002825 .= v. y + (2.47 + .5 v 2 ) -}- 3600 v -~ 4- 3600 (1.235 -J-
.25 c s ) g- r being the refraction, v the sine of altitude, and s the
From this last formula, the following Table, taken from the Nautical
Almanack for 1827, is computed,
243
R E F
TABLE OF REFRACTIONS.
j, & %f D f i-
Aliuu. -ju^il A it}+\B,
JL^.
Eefr. \ Dip
2?iO ! fyr
r/i.50o 1' Alt
z>iff.
for
4-15.
%?\
-loFa
D.M.
0.0
5
10
15
20
85
30
35
40
45
50
55
i M.S. , S.
i S.
74
71
69
67
65
C3
61
59
58
56
55
53
s.
8,1
7,6
7,3
7,0
6,7
6,4
D.M.
M.S.
11.54
11.30
11.10
10.50
10.82
10.15
9.58
9.42
9.^7
911
8.5S
8.45
1 S _
2,2
2,1
2,0
1,9
1,8
1,7
1,6
1,5
1,5
1,4
1,3
'1,3
S.
S.
S3. 51
32.53
31.53
31. 5
013
9.24
1 78.37
27.51
7. 6
6.24
25.43
5. 3
11,7
11,3
10,9
10,5
10,1
! 9,7
4.0
10
30
40
50
24,1
23,4
2?,7
22,0
21,3
0,7
"g
50
49
48
4(5
45
4,7
4,6
4,5
4,4
4,2
4,0
6.0
10
SO
40
50
8.32
8.20
8. 9
7.38
7.-V7
7.37
1 y
M
\\\
1,0
1,0
J7,2 1,15
16,8 : 1,11
10,4 i 1,09
16,0 1,06
15,7 i 1,03
15,3 1,00
21. 7
SA98
0.10
19.43
19.17
18.52
~Ia29
18. 5
17.43
17.21
17.
10.40
44
43
42
40
39
39
~33~
37
36
3?5
35
34
33
33
Si
32
31
30
3,9
3,8
3,6
3,5
3,4
3,3
3,2
3,1
3,0
2,9
2,8
2,8
2,7
2,7
2,6
2,5
2,4
2,3
7.0
10
20
30
40
50
8
10
20
SO
40
50
7.27
7.-I7
7. 8
0.59
6.51
6.43
0.35"
6.28
0.21
6.14
6. 7
6.
1,0
,9
,9
;!
,8
',7
,7
,7
,7
G
15,0
14,6
14,3
14,1
13,8
13,5
l3,3~
13,1
12,S
1:,6
12,3
12,1
,93
,95
,93
,91
;!?
,85
,83
,82
,80
,79
,77
16.21
1(5. 2
15.43
15.25
15. 8
14.5!
14 35
14.19
14. 4
13.50
13.S5
1321
9.0
10
^0
30
40
50
554
5.47
5.41
5.36
5.30
5.25
5,20
5.15
5.10
5. 5
5.
4.56
,(J
jG
,6
,6
s
,5
i
&
,5
,5
,4
11,9
11,7
11,5
11,3
11,1
11,0
10,8
10,6
10,4
10,2
10,1
9,9
,76
,74
'Z 3
^1
,70
3.
5
10
15
50
25
30
29
29
8
28
27
2,3
2,2
2,2-
2,1
?i
2,0
10.0
10
30
40
50
,6L>
,67
,65
,64
,(53
,62
30
35
40
45
50
55
13. 7
12.53
12.41
12.28
12.1(5
12. 3
2,7
2,0
s$
2,4
2,4
2,3
27
26
26
25
25
25
2,0
2,0
1,9
1,9
1,9
1,8
11.0
10
30
40
50
4.51
4.47
443
4,39
435
4.31
,4
i
,4
4
,4
9,8
9,6
9,5
9,4
9,2
9,1
,60
,59
*
'%
REF
TABLE OF REFRACTIONS.
n#>~ ij
[Amu. n
t
ttefr. |
Difi
for i
Diff.
for
3f
- i" Fa
App. 1
Refr.
B ao
for
1' Alt.
S
Wff~
t*r
-\"F
iD.M
M.S. j
S.
~s7"~
~S. ,
"~D~
M.S.
S.
S,
.
f"~12
4.28,1"
~S8~ f
9 J
,550
42
T17
~OS8~
*,\ i i
~7i0^
10
4.24,4'
1&7 ]
$,SQ
,548
43
1. 2,4
,0^9
' ,< 9
,l*-5
80
4. -0.8
,':0
8,74
,511
44
1. 0,3
,034
2,02
,120
SO
40
4.17,3
4.13,9
|S3 i
8,03
8,51
,524
45
46
53,1
50,1
,034
,033
1,94
1,83
,117
,112
50
4.10,7
,32
8,41 1
,517
47
54,2
,032
1,81
,108
13.6"
47" 7,5"
,31
8~30
~509
48
5 ; ,3
,031
75
,104
10
4. 4,4;
,503
49_
50,5
,030
,69
,101
20
30
40
50
4. 1,4
353,4'
3.55,5
3. 5. ,6
'so :
,30 ;
8,' 10
8,00
7,89
7,19
,4S6
',410
50
51
52
53
48,8
47,1
45,4
43,3
lo. ; 8
,0.-7
,0i6
'53
>5 7
,097
,09 1
loss
14. "
3 49 9
^8 "" i
"7,70
,469~
5r4
42,2
,(.i6
*41
,^83
10
3'.4-,t
,28
7,01
,404
55
40,8
,0';5
,36
,03>
20
3.43,4.1
,27
7,52
,458
56
9,3
,31
,019
30
3.41,8
,;0
7,43
,453
57
37,8
,'6:5
,-,6
,07fT
40
5.0;
3.39,2
3.28,7
,',6
7,34
7,-, 6
,418
,444
53
59
25lo
lok
,22
,073
,cr?o
L5.1),
3.34,3
"","24~
7,78
~4S9~~
~00~
33,0
,6.3
~,12
,(K)7
30
01
3:^,3
,08
,005
10.
so;
17.0
3 20,6
3.14,4
3. 8,5|
,21
,20
,19
0',73
0,51
6,31
,41 1
62
63
64
31,0
29,7
28,4
^022
,0>1
,0^1
,04-
,99
,95
,002
,000
,057
so
18.0
19.0
3. 2,9
2.57,6
2.41 ,7
,18
,17
,10
0,12
136-2
,340
60
67
27,2
25,9
24,7
,0>0
,0-0
,0-0
,91
*83
,0f)5
,052
,050
20
21
22
23
24
2.38,7
2.30,5'
2.- 3,2,
2.10,5
2.10,1
,15
,13
|ll
,10
5,31
5,04
4,79
4,57
4,35
,322
,305
l'<16
,264
(58
C9
71
72
23,5
22,4
27,2
19,9
18,8
,0.0
,0-0
',019
',75
",71"
,07
,047
,045
,04.'}
,0-10
,038
f5
2. 4,2
,09
4,16
,252
73
17,7
,018
'ftO
,036
26
27
28
29
1.5% 8
1 53,8
1.49,1
1.44,7
,C<)
,03
,03
,07
3,97
3,81
3,< : 5
3,50
,241
,2SO
1209
74
75
16
17
10,6
15,5
14,4
13,4
,018
: ,018
,018
,017
156
,52
,48
,45
,033
,031
,0.-9
,027
~30~
1 .40,5
,07
~3.S6
,'<0l
18
12,3
,017
,41
,0>5
31
l.0 }
,06
3,"- 3
! 93
19
u3
: ,017
,3
,023
32
S3
34
1.23,0
1.29,5
1.26,1
,06
,06
,05
3,11
,99
2,83
,186
Ins
80
81
8>
10,2
9,2
8,->
,017"
,017
,017
",34"
,02f
,018
,016
35
1.23,0
,05
2,18
,167
3
7,1
,017
1?4
,014
36
1.20,0
,05
2,fi8
,101
84
6,1
,017
,20
,012
37
S3
39
1.17,1
1.14,4
1.11,8
,05
,05
2,58
2,49
2,40
,155
,149
,144
85
86
87
5,1
4,1
3 \
,017
,017
j017
,17
,14
10
,010
,C03
,CC6
40
1. 9,3
1. 6,9
,04
,04
2^24
,1S9
,134
; 83
* 89
2,(
1,C
) ,017
>ljm7
,07
,03
',004
,002
R I V
Explanation of the Table of Refractiont.
The apparent altitude being found in the first column, the second shor/s
the refraction when the barometer stands at 30 inches, which is its mean
height oil the level of the sea, and the thermometer at 50 of Fahrenheit.
The third column contains the difference to be subtracted or added for
every minute of altitude, reckoned from the nearest number in the first
column. The fourth shows the number of seconds to be added for every
inch that the height of the barometer exceeds 30, or to be subtracted for
each inch that it wants of SO ; and the last contains the number of se-
conds to be subtracted for each degree that the thermometer stands above
50, or to be added for each degree that its height wants of 50.
JEjr. At 7. 18'. 13". Bar. 29.87. Ther. 6Go. required refraction.
Alt. 7. 20'. R. 7'. 8" Diff. Alt. ",9 B. 14", 3 Th. ",63
4- 1.62 K 47" =: 1'. 8 .13 1C
4. 1,62 1,86 14,88
1,86
Ref. = 6. 52,83
REFRANGIBIL1TY of lightSee Light.
RESISTANCE of air to Projectiles. See Gunnery.
RESISTANCE of Fluids. See Fluid*.
RIVER. ( Du Buat, Rolison.)
1. Let V = velocity of the stream por second in inches, R the quotient
arising from the division of the section of the stream, expressed in
square inches, by its perimeter minu the superficial breadth of the
stream in linear inches, S the slope the numerator being- unity, i.e. the
quotient arising from dividing the length of the stream, supposing it ex-
tended in a straight line, by the difference of level of its two extremities,
or l^t it be the cotangent of the inclination or slope ; then the section
and velocity being both supposed uniform,
v =
V, S I h. 1. /S -{
-__ -- 5, _
10 V, Si I h. 1. /S -{- i|\ 10
When R and S are very great
V=S E*(_2 -- *) nearly.
V si - \ h. L 9 ^
The elope remaining the mine, the velocities are a* v' R~ TT; or M
R", when R u ver
216
R I V
The velocity will become nothing by making the declivity so small that
-[^ur lo = Oi but if is lesa " wBno or than
l^th of an inch to an English mile, the water will have sensible motion.
In the above formula R is called the radius of tJie section.
2. In a river the greatest velocity is at the surface and in the middle
of the stream, from which it diminishes towards the bottom and sides,
where it is least ; and it has been found by experiment, that if v = velo-
city of the stream in the middle in inches, then the velocity at the bot-
tom is
_2 W+l.
3. The mean velocity, or that with which (were the whole stream to
move) the discharge would be the same with the real discharge, is equal
to half the sum of the greatest and least velocities, as computed in the
last Prop. Hence the mean velocity v V v _j_ .
4. Suppose that a liver having a rectangular bed is increased by the
junction of another river equal to itself, the declivity remaining the
same ; required the increase of depth.
Let the breadth of the river = b t the depth before the junction .= d t
and after it .r ; then
- ^- - .. ~ t) - , a cubic, equation which can always
be resolved by Cardan's rule.
5. To find the fall of water under bridges, let the breadth of the river,
in feet b j the breadth between the piers c ; the velocity iu a se-
cond v ; ' ^.S^ig feet; then the fail of the liver will be
Tims at London bridge b - 920, c = 2S6, reduced by the piks.to IDC^^
v =. 3*4, hence, the fall is 4.739; by observation A75, ( Toa.^V Nat.
Phil)
6. When the sections of a river vary, tha quantity of water remaining-
lh same, the mcaa velocities are inversely as the ai'eas of the sections,,
7. The following Table abridged from Dr Robison serves at once to
compare the surface, bottom, and mean velocities in rivers according; to
the principles of Arts, 2, 3, (Gregory.)
\
247 O 3
R I V
Velocity in Inches.
Velocity in Inches.
Sur-
face.
Bot'.om.
Mean.
Stir-
face.
Bottom.
Mean.
4
1
2.5
56
42.016
49.003
8
3.342
5(57
60
45.509
5i>.754
12
6.071
9036
64
49.0
56.5
16
9.0
12.5
68
52.505
60.252
go
12.055
16.027
72
56.0*5
6L012
24
15.194
19.597
76
59.568
67.784
18
18.4vl
i3.->10
80
63.107
71.553
32
v 1.6-78
6.859
81
66.651
' 75.325
36
25.0
30.5
88
70.224
79.J12
40
5:8.245
31.172
92
73.788
8 -.894
44
SI. 742
37.871
C6
77.370
8S.6S5
48
35.151
41.570,
100
81.0
90.5
b2
38.564-
45.^82
8. Eytelwein, a German mathematician, gives the following- formula
for the mean velocity of the stream of a canal. Let v be the mean velo-
city of the current in English feet, a the area of the vertical section of
the stream, p the perimeter of the section, or sum of the bottom and two
sides, I the length of the bed of the canal corresponding to the fall h t all
in feet ; then
i - 0.109 + J
9582 .- 4. 0.0111
9. To find experimentally the velocity of the water in a river, and the
quantity which flows down in a given time, observe a place where the
banks of the river are steep and nearly parallel, and by taking the depth
at various places in crossing make a true section of the river. Stretch a
string at right /'s. over it, and at a small distance another parallel to
the first. Then take an apple, orange, or a pint or quart bottle partly
filled with water so as just to swim in it, and throw it into the water
above the strings. Observe when it comes under the first string by
means of a quarter second pendulum or a stop watch, and observe also
when it arrives at the second string. By this means the velocity of the
tipper surface, which in practice may frequently be taken for that of the
whole, will be obtained. The section of the river at the second string 1
must then be ascertained by taking various depths as before, and the
mean of the two will be obtained by adding both together and taking
half the sum for the mean section. Then the area of the mean section
in square feet being multiplied by the distance between the stringif in
feet will give the contents of the water in solid feet which passed from
R I V
one string to the other during- the time of observation ; and this by the
rule of three may be adapted to any other portion of .time. This opera-
tion may often be greatly abridged by noticing the arrival of the float-
ing- body opposite to two stations on the shore, especially when it is not
convenient to stretch a string across. Where a time piece is not at hand
the observer may easily construct a quarter second or other pendulum.
RIVERS, proportional lengths of, and supposed quantity of ivater dis-
charged per annum. ( Ency. Brit. Suppl.)
EUROPE
RIVERS.
LENGTH.
QV. OF WATEB.
fThames
1
1
Rhine
4i
13
Loire
4 .
10
Po
. 2* .
6
Elbe
. 4* .
8
Vistula
4$ -
12
Danube
. 0} .
65
Dntiper . . . .
. 7f .
36
Don
. 7| .
38
Wolga
. 14 .
80
Euphrates
9| .
60
Indus
. H| .
133
10
143
Kang-tse or Great river of China
21| .
258
Amour, Chinese Tartary
16 .
166
Lena, Asiatic Russia ,
. 13* .
125
LOby do.
15 .
179
Nile ......
18$ .
250
"St Lawrence including Lakes
Mississippi
Plata . . . . .
Amazon, not including Araguay
22-} .
19
13| .
22f .
. 113
338
. 490
. 1280
ASIA
AFRICA
To deduce the approximate lengths of the rivers in miles from the pro.
portiorial lengths we may multiply the latter by 180. To convert the
proportional discharge into known measures we may multiply by 1800
to pbtain the number of cubic feet per second, or by .4 or j^to find the an.
nual discharge in cubic miles.
249
ROD
Proportional lengths according to Major Remiell.
r Thames ....... 1
1 Rhine 51
EUROPE J ^
J Danube ....... 7
I Volga ....... 9
'Indus ....... C|
Euphrates ....... 8}
Ganges ....... 9|
Burrampooter ...... 8| t f
.Ava River ...... 9
ASIA... < Jiicei ....... 10
Obi ........ 10|
Amour ........ 11
Lena . . ...... . 11|
Hoang-Ho ...... . . 13J
^ [KianKu ....... 15
AFRICA Nile ....... 12J '
581 ' 38 ^ 1 ....... 8
C Amazon ....... 15|
S, equilibrium of.-(WhewelL)
1. A roof A C A', consisting of beams forming an isosceles triangle
with its base horizontal, supports a given weight at its vertex C : the
weights of the beams being also given; it is required to find the hoii-
zouttffKrce at A and A'.
,JLet B be the weight of the beam A C, C the weight at C, a. the angle
which A C makes with the horizon, H the horizontal pressure at A j
then
H =
2 tan. a.
If there is no beam joining A A', this horizontal pressure H must be
counteracted by the supports on which the ends A, A' are placed.
If te roof A C A' support a covering of uniform thickness, the formu-
la will still be true including in the weight B, the weight of that portioa
of the covering 1 which rests upon the beam.
The weight C at the point C may arise from a longitudinal team per.
pendicular to the plane A A' C.
250
ROT
2. Any number of given beams, arranged as sides of a polygon, in a
vertical plane, support each other, and support also given weights at
the ^/.s ; it is required to find the horizontal pressure at the points of
support.
Let B and B be the weights of two contiguous beams, a and the an-
gles they make with the horizon, and C the given weight at the ^, or
point of junction j then
% (B + B) + C
H rr
tan. ot tan.
This horizontal pressure is the same at all the angles.
Cor. If we suppose the weights of the beams o, H -= -
If we suppose no weights, except the beams,
H _ % (B 4. B)
tan a. tau '
3. To find the position of the beams, having given their weights
B, B, B &c. the weights C, C, C &c. and the position of two of them.
1* 2 3 123
By the last Prop, we have the following equations, , a, <* being the
^.s which the beams make with the horizon.
H (tan. tan. ) = ^(84. B) 4. C
H (tan. * tan. ) = }i (B 4. B) 4. C
&c. &c.
If there be n beams there will be n 1 weights C, C &c, and^t- I
1 9
equations. The'number of unknown quantities is n 4- 1 viz. the n tan.
gents, tan. , tan. &c. and the pressure H. Hence if we know two of
the /.s , . we can find the rest,
i a
RO OTS of numbers. See Involution.
ROPES, rigidity of. See Friction,
ROTATION of bodies about a fixed or moveable axis.
The following Proposition is of the greatest use in Mechanics, and is
general under the circumstances there mentioned, whether bodies move
in right lines or have a rotatory motion. It applies with peculiar facili-
ty to the investigation of the motion of revolving bodies, and by the help
of it the most difficult problems admit of a simple and easy eolation.
51 O i
ROT
Prttp. If a system of bodies be connected together and supported at
tiny point which is not the centre of gravity, r.nd then left to descend by
that part of their weight which is not supported ; 2 g multiplied into the
sum of all the products of each body into the space it has perpendicular,
ly descended will be equal to the sum of all the products of each body
into the square of its velocity, g being = &2% feet. (Mr J)au-son t Serf,
lergh.)
A demonstration of this Prop, may be seen in Leybourn's Mathemati-
cal Repository.
Ex. 1. Let a cylinder whose weight = W, moveable about a horizon-
tal axis passing through the centre, be put in motion by a weight P at-
tached to a string wound round it ; required the force accelerating the
body P, and the space descended in t seconds.
Let s space perpendicularly descended by P, = velocity acquired
in the time t t r radius of the cylinder, a- = distance of the centre of
gyration from t'.ie centre of the cylinder ; then by the Prop.
2gXPs = Pi*+ WXt = Pa*-f W X^
v |p T^ accelerating force.
4.v2
To find s, put c* =. r^- in the leading equation, and we shall have
Er. 2. A given cylinder with a thread wound round it is suffered to
unwrap itself and descend j required the time of its descent through a
given space.
The same notation being retained
2g X W* - Wt; 8 +Wc* X = WX
KJC. 3. P and \V are hung over a fixed pviley, to find how /ar F will
descend in t",
ROT
Let r radius of pulley, w its weight, x distance of the centra
of gyration from its centre ; then
2g X (P W) * - (P 4. W) v* 4. w v* X ^
~ (P 4. W) uz -{- iv. j, but * = -il,
2P4-2W4- w'
2?.r. 4. Let A andB represent a single fixed
and moveable pulley as represented in the an-
nexed figure ; required the space which the de-
scending- weight P describes in a given time.
Let w = weight of each pulley, v =
velocity of P, then ~ velocity of W ; also
: velocity of the centre of gyration of
A, and - g velocity of the same centre
in C ; then.
fo
PO
,-W.^-) =P V + W X^ + wX^
Ex. 5. A sphere D, whose
radius is <> and weight W, is
put in motion by a weight P
acting by means of a string
going over awheel whose ra-
dius is r: required the velo.
city acquired in the time t.
Let ?> = velocity of P, s the
space descended by P in t",
A' distance of the centre of gyration of the sphere from ita centre
then
* X ~* ; but x - ? X
P
253
ROT
,*. 2g X Ps Pra 4. Wrs X 1
t v - -
-~ " ~
2*
Or by substituting ~ for v ; s or may be found.
Ex. 6. Let a weight P, fastened to a string going over a wheel, by its
descent cause two weights W, W' to be wound up on two axles. Re-'
quired the velocity of P after it has descended t" ; the radii of the wheel
and of the two axles being r, , $'.
Here
( Ps W X H W X -} = P r* + W* x i-frW* c 8 X ?!?
\ v/X T
IX -- - (Pra -f W f 4- W ^
P V 600 miles to the eastward.
T Mackenzie's River, and from
113 to 149. 38. W. Long.
1 81. 5|. and on the ice to
'<* 82. 45*. 20. E. Long.
, 71. 10. Long. 101 to 110 W.
. 74. 15. Long. 34. 16. 45. W.
No human beings are found in the Southern Ocean below the 55th
parallel of latitude, and none beyond the 50th, except on Patagonia and
Terra del Fuego.
It is impossible to enter here into any of those points of scientific re-
search which these expeditions have been the means of communicating-.
It may not, however, be uninteresting to subjoin the result of Captain
Parry's observations on the temperature of Melville Island, iu 1819 and
,i820, as indicating a very extraordinary degree of cold.
SHI
1819 September.
October ....
* November
Greatest
Temperat.
.. 4- 37
.. 4- 17.5
Least.
. 1
. 28
47
Mean,
3.46
20.60
21.79
30.09
32.19
18.10
8.37
4. 16.66
4. 36.24
+ 42.41
+ 32.68
December .
1820 January
.-4-6
, 13
47
February .,
,. 17
.. _u 6
, 50
40
April
f May
+ 32
.. -4- 47 ..
. 32
4
, 4, 23
July
August .....
.-. 4. 60
,. 4- 45
4. 32
4- 22
Annual temperature
4- 1.33
According to Leslie's Table (see Atmosphere ) the temperature of Mel-
ville Island should have been nearly 36, whereas it is only 1<>-
o
Inches.
Greatest height of barometer was ......... 30.86
Least do. 29.W
SEA, extent of. See Earth.
SEASONS, length of. See Earth, elements of.
SECANTS, figure of. See Figure.
SEMIDIURNAL arcs.See^rcs Semidiurnal
SHIPS, tonnage of.
To find the tonnage of Ships.
RULE 1 Multiply the length of the keel, taken within the vessel, or
as much as the ship treads upon the ground, by the length of the midship
beam, taken also within, from plank to plank, and that product by half
the breadth, taken as the depth; then divide the last product by 94, and
the quotient will give the tonnage.
If the length of a ship's keel be 80 feet, and the midship-beam SO;
required the tonnage. Ans. 382.9787 4. tons.
RULE 2. Shipwrights take the dimensions on the outside of the light
mark, as the ship swims, being unladen, to find the content of the empty-
ship. But if the measure of the ship be taken from the light mark to her
263
$ H I
full draught of water, when laden, it will give the burden of the ship ;
and then the length, breadth, and depth multiplied together, and the pro-
duct divided by 100 for men of war (which gives an allowance for guns,
anchors, &c. that are all burden but no tonnage) and by 95 for merchant
ships, will give the tonnage.
N.B. A hundred solid feet make a ton.
Required the tonnage of a ship, whose length is 300 feet, breadth 50,
and depth 30. 16
Ans. 4736j^tons. '
RULE 3. At London, shipwrights multiply the length of the keel by
the extreme breadth of the ship, taken from outside to outside, and that
product by half the breadth ; and this they divide by 9-1 for merchant
ships, and by 100 for men of war j the quotients are the tonnage of the
vessels of their respective classes.
Required the tonnage of an eighty gun ship, the length of whose keel
is 149 feet 4 inches, and her extreme breadth 49 feet 8 inches.
Ans. 184 186 + tons.
The following ir ethod is used in the Royal Navy :
RULE 4. Let fall a perpendicular from the foreside of the stern at the
height of the hawse holes, and another from the back of the main port at
the height of the wing transom j from the distance between these per-
o
pendiculars deduct of the extreme breadth, and as many times 2%
inches as there are feet in the height of the wing transom above the up-
per edge of the keel, the remainder is the length of the keel for tonnage.
Then multiply the length of the keel by the extreme breadth, and that
product by half the breadth; divide this product by 94 for the tonnage.
Given the length of the keel 68 feet, and the extreme breadth 22;
required the tonnage. 6
Ans. I^\TT tons.
Ship-building.
A man-of-war of 74; guns requires about 3000 loads of timber, of 50
cubic feet each ; worth, at f>. a load, 15,000. A tree contains about
two loads, and 3000 loads would cover fourteen acres. The value of ship-
ping in general is estimated at 8. or 10. a ton.
It is said that 180,000 pounds of hemp are required for the rigging of
first-rate man-of-war./' Young's Nat. Phil.)
Note." -The above calculation of fourteen acres to a 74 gun ship is pro-
bably much too low- It will be nearer the truth to suppose each tree to
264
S I P
contain only a load and a half of timber, and that every acre contains 35
trees fit for naval purposes j this gives 57 acres of land for a 7-1 gun ship.
See Report of the Board of Commissioners of Woods and Forests, 1812.
SHOT, pile of.
Shot or shells are usually piled up in a pyramidal form, the base being
an equilateral triangle, square, or rectangle.
The following formulae give the total number of balls in any of these
piles :
Triangular pile = J- ( + 'J ( +JJ.
Rectangular pile = aJ^LLg-+ D
Where n in the two first formulae denotes the number of balls in the
side of the base ; and in the last n is the number of balls in the length of
the base, and m the number of those in the breadth.
SHOT, weight of.(Hutton.)
Let W be the weight in pounds, 5 diameter in inches, then
In iron balls, W = ^ X &.
Dfc
In leaden, W = ~ X S 3 .
In iron shells, if D and & be the external and internal diameters, W
= ~ X (1)3 - S3.)
SHOT. See Gunnery.
SIDEREAL time. See Time.
SINES, figure of. See Figure.
SINES, arithmetic of. See Trigonometry.
SIPHON, oscillatory motion of water in. (Play fair.)
1. Let an inverted siphon, partly filled with water, be composed of
three rectilinear tubes of equal diameters, of which the intermediate one
IB horizontal, and the two others inclined to the horizon at any angles
, 0> ; and let an oscillatory motion be communicated to the water; re*
265 P3
sot
quired the time of the water's oscillating in either of the legs from tire
lowest to the highest points.
Let L = length of the whole canal, g = 32% feet ; thea
W-
gX (sin. 0+ sin. 6')'
"When the two ascending tubes are vertical,
Cor. Hence if the legs are vertical, the time of one oscillation = th
time in which a pendulum would vibrate, whose length is L.
2. The vibratory motion of water in the form of waves may be com-
pared to the above reciprocation in a siphon or bent tube. And hence if
a be the altitude of a wave, and b half the breadth, the time of one undu-
lation, i.e. the time, from the wave being highest at any point, to its
being highest at that point again, is
and the space which the wave appears to pass over in a second is
b
Cor. 1. If a be neglected, the velocity of the wave becomes * g ,
which is the velocity as determined by Newton, Princip. lib. 2. Prop. 46.
Cor. 2. Hence a pendulum whose length = its distance between any
two consecutive highest and lowest points will make two vibration*
during the time of one complete undulation ; or if the pendulum is four
times the preceding, i.e. equal to the distance of any two consecutive
waves, the time of one undulation equals the time in which this latter
pendulum would perform one vibration.
SLUICES. See fluids.
SOLAR inequality. See Precestion.
SOLAR mean time. See Time.
206
SOL
SOLIDS the five regular, surface and solidity of.
Names.
Surface.
Solidity.
Tetraedron
* X 1.7320508
s X 0.1 178513
Hexaedron
2 X 6.0000000
*a X 1.0000000
Octaedron
** X 3.4641016
4-3 X 0.4714045
Dodecaedron
* X 20.6457288
^3X7.6631189
Icosaedron
* X 8.6602540
4- 3 X 2.1816950
SOLIDS, contents of.
Let x and y be the abscissa and ordinate of any curve j then if * =s
3.14159 &c.
Solid content
Ex. 1; Content of cylinder = yy*x.
2. Content of cone = l / 3 x y* x y s of circumscribing cylinder.
3. Content of paraboloid = % r */ 2 .r ^ circumscribing cylinder.
4. Content of sphere % of circumscribing cylinder.
5. Content of spheroid round ax. maj.
4 ar 5* a . 4 a 8 6 '
__ . Do. round ax. mm. .
o >
6. Content of pyramid = % content of prism of the same base and alti-
tude.
Guldinus* property.
I^et M D E K be any plane figure revolving
about an axis xy in its own plane, then the
solid generated is equal to the circumference
described by the centre of gravity multiplied
into the area of the figure.
Ex. Let D M E K be a circle, then the solid
will represent the ring of an anchor j in this
case if r radius of circle, and a A O, the
*olid = 2 * a X 9 r* = 2 ^ ar*.
sou
SOUND, velocity of. (Phil Trans. 1823. ;
The velocity with which vibrations are propagated through the air, is
the same that a heavy body would acquire by falling through half the
height of the homogeneous atmosphere, or that which the atmosphere
would be reduced to, if it were everywhere of the same density, and the
same temperature with the air at the surface of the earth.
The height of this homogeneous atmosphere has been computed at
4343 fathoms, when the temperature is that of freezing. If this height f
be called H, then r, the velocity of the aerial vibrations, = ^ 2 g H.
Hence v 1057, which is too small, see infra.
The velocity of sound has been variously given by different philoso-
phers, as appears from the following Table :
Feet.
Newton -- . 963 per second.
Roberts ~ 1300
Boyle 1200
Walker 1338
Flamstead, Halley, and Derham 1142
Florentine Academy ~~~ ^^^v^^* 1143
French Academy ~~~ f ~~, w ~~.~^~ 1172
More modern determinations.
Millington -~~ ,~~~ ~^~ 1130 Chili.
Bengenberg ~~* , 1005 Busseldorf.
LaCaille , 110G Moutaiartre.
Lacaille 1130
Flamstead's and Halley's measure, or 1142, is the one generally as-
turned by English writers.
m
S P H
JResult of Mr Goldingham's elaborate series of experiments at Madrat.
Velocity
Months,
Barometer
in
Inches.
Thermome-
ter, Fah.
Hygrome-
ter , dry.
of Sound
in a Se-
cond in
Feet.
January,
30.124
79o.05
6.2
1101
February.
3ai26
78 .84
14 .70
1117
March,
30.073
82.30
15 .22
1134
April,
30,031
85 .79
17.23
1145
May,
29,892
88.11
19.92
1151
June,
29.907
87.10
24.77
1157
July,
29,914
86.65
27.85
11(54
August,
29,931
85.02
21 .54
11 03
September,
29,963
84 .49
18 97
1153
October,
30.058
81. .33
18 .23
11*8
November,
30.1^5
81 .35
8J8
1101
December,
30.087
79 .37
1 .43
1099
Mr Goldingham concludes, that for each degree of the thermometer
1.2 feet may be allowed in the velocity of sound for a second ; for each
degree of the hygrometer 1.4 feet j and for th of an inch of the barome-
ter 9.2 feet. He concludes that 10 feet per second is the difference of the
velocity of sound between a calm and in a moderate breeze, and 21 feet
in a second, or 1275 in a minute, is the difference, when the wind is in the
direction of the motion of sound, or opposed to it. See Phil. Trans.
1823.
SPECIFIC Gravity. See Gravity specific.
SPECTACLES.?e Eye.
SPHERE, doctrine of.
In what is usually called the doctrine of the sphere is merely included
the solution of the following problem :
In a spherical triangle, whose sides are the co-declination D, the co-
latitude of the place L, the zenith distance Z, and two of whose angles
are the hour angle from noon H, and azimuth , ; if any three of these
quantities be given, the other two may be found by the rules and formu-
lae of Trigonometry.
For the solution of the several cases see Trigonometry spherical.
SPHERE, Equations to, when the axes are rectangular. (Hamilton.)
Let r radius, and suppose x', y', ' to be the coordinates of the cen-
2C9 P4,
S P I
tre, and .r, y, % those of any point on the surface j then the general equa-
tion is
(.r-.r') a + (y-y')* + (*-*') =r.
If the origin be at the centre, x', y' t and % each o, and the equation
becomes
SPHERICAL excess.
Spherical excess in Trigonometry is the excessfof the sum of the three
angles of any spherical A above two right angles. Now in surveying a
country where the sides of the A's are usually 14 or 15 miles each, the
spherical excess, with a fine instrument, is plainly discernable ; and in
strict accuracy the sides of the A's ought to be calculated by the rules of
spherical Trigonometry, which would be a most tedious process, where
many hundreds of such operations are to be performed. Legendre has
therefore furnished us with the following rule, which combines sufficient
exactness, with all the conciseness that can be expected, viz. :
A spherical A being proposed, of which the sides are very small with
regard to the radius of the sphere, if from each of its angles one-third of
the excess of the sum of its three /'s above two right /'s be subtracted,
the angles so diminished may be taken for the /'s of a rectilineal A, the
sides of which are equal in length to those of the proposed spherical tri.
angle.
SPIRALS. (Higman, VinceJ
1. Spirals y Equations to.
In the spiral of Archimedes, let r =r rad. vect. 9 = /. traced out by r ;
then
r = - . 6, or r a 8 : if a = - .
2 * ' 2*
In the reciprocal or hyperbolic spiral,
_ a
In the logarithmic spiral,
9
r a.
In the lituus,
The spiral of Archimedes, the reciprocal spiral, and the lituus are par.
ticular cases of the equation r a ffl 1 .
270
s p i
If ft be -f-, the spirals begin at the pole, and rreue to r.r.
tance j but if n be , the spirals begin at an infinite distance, and r
the pole after an infinite number of revolutions.
2. Spirals to draw tangents to,
Subtangent ~ ~~! .
Ex. I. In the spiral of Archimedes r-~aS t
:. Subtangent = , and hence p ~ ,
V a 4-r*
Ex. 2. In the reciprocal spiral,
Snb tangent tt, and p ~ - a
x. 3, In the logarithmic spiral,
Subtangent = ~ and p ~ a r,
3, Spirals to find the areas of,
Area = fL - .
Ex. 1. lu the spiral of Archimedes,
* A _ *" **
SA: 2, In the reciprocal spiral.
Suppose the area to vanish when r b % then will the area, intercepted
ietweea two radii 6 and r, (6 r).
Ex. a In the logarithmic spiral,
Area between two radii b and r =. ~ (r* *- &*), m being tht modulai.
271
Ex. 4. Inthelitutw,
Area = c 8 log. .
4 Spirals to find the lengths of.
d z* - d r* 4- r 2 d 6*.
or dz - T T . (p perpendicular on the tangent)
Vr*j*
Ex. I. In the spiral of Archimedes,
Are:=z fl. dr ^ a* + -t* t and .'. a parabolic arc, whose
latus rectum is 2 a, and whose ordinate is r (see Rectification.)
Ex. 2. In the reciprocal spiral,
Arc arc of a logarithmic curve contained between the ordi-
nates b and ;; the subtangent of the curve being equal to the subtau-
gent of the spiral.
Ex. a In the logarithmic spiral,
Arc = VC 1 4- m *) ( *)
Ex. 4. In the involute of a circle,
Arc = (a radius of the circle).
5. Spirals t curvature of.
Had. of curv. = ^ .
Zpdr
Ch. Cttrv . = _-.
Ex. 1. In the logarithmic spiral,
Rad. curv. = , and ch. curv. =2r.
in
Ex. 2. In the spiral of Archimedes,
Ha,eu r ,=^^
Ex. 3. In the recif rocal spiral,
2r (a +r*}
Ch. curv. J ~ - ; .
a
S72
S T A
rdr
6. Spirahy point of contrary flexure in.
Here the rad. of curvature is either infinite or nothing ; .'. --'-=.
or infinity, and dp is infinite or nothing-.
Ex. Let r a n , then when dp o, r a ( n. n -f. l) i. Hence
in the case of the lituus, where n |, r a ^/ 3".
SPRINGS 7io#, temperature of a few of the principal ( Ure.)
Matlock .
Bristol ,
Buxton
B;ith ,
Berege
66
~~ 74
82
. 114
Borset ,
Aix
Carlsbad
, 132
, 143
, 165
The Geyzers (Iceland) 212
SPRINGS, temperature of. See Atmosphere.
SQUARES minimum, method of.- See Equations of Condition.
SQUARE roots of numbers. See Involution.
STANDARD measures. See Weights and Measures.
STARS, Catalogue of ( Naut. Aim.)
A Catalogue of 61) principal FLved Stars for Jan. 1, 1823.
No.
Names of Stars.
A.R.
An.Var.
N.P.D.
An.Var.
1
2
3
y Pegasi
a. Cassiopeiae
II. M. ,S.
0. 4. 8,1
0.30.3] ,3
57 46 5
s.
+ 3,08
3,31
1501
75.48. 2
34.SJ6. 6
1 38 8
19,9
19,7
19 4
4
5
Arh'th
a Ceti
1.57. 13,1
y 53. i>,3
sue
3^2
67.2s?.44
86 3(\S7
17,2
14,4
6
7
g
Persei
Aldeharan
Cii'iell'i
3.11.44,3
4.i J 5.4(i,6
5 S 3*7 8
4,iO
3,43
4 4-1
40.46.39
73.51.18
44 11 37
13,3
7,7
4 3
9
M
n
12
13
14
Rigel
|8 Tauri
y Orionis
..................!.....
5. 6. 2,2
5.15. (>,8
5.15.38,7
5.->^.58,3
5.27.14.3
531 50,1
2,88
3,^8
3.^0
3J06
3,03
3,01
98 24.48
61.33. 7
83.49. 8
90.56.18
91.19.23
9^ 238
- 4,5
-3,7
4,0
3,1
2 5 6
2' 4
Id
5.45.35,6
3,1 ; 5
8238 4
] 1
16
5 46 32,9
4 39
45 4 56
1 2
17
6 37 20,9
2,64
106 28 49
4-48
18
3 85
57 43 59
170
373
S T A
No.
Names of Stars.
A.R.
An.Var.
N.P.D.
An. Var.
B
21
22
Procyon
Pollux
a. Hydrse
Regal us
H. M. S.
7.30. 2,2
7.34.23,5
9.18.53,5
9,58.53,3
8.
3,17
3,69
2,95
3^21
/ it i
84.19.43
61.33.17
SJ7.53.44
77.1016
t'i:?
ti$
3
4
a Ursse Majoris ...
/3 Leonis
10.52.43,5
11.40. 1,7
3,83
3,07
27.17.44
74.26.18
19,2
-j- 0,1
25
6
y Ursae Majoris ...
11.44.28,6
12. 6.37,2
3,^0
3,00
35.19.15
31,59.
4-19,9
4-202
iff
28
29
Spica Virffinis ...
1 Ursae Majoris ...
13.15.52,9
13.16.46,8
13.40.a3,5
3,14
2,41
2,SS
100.14.
34. 8.51
39.48.
+ 18,9
+ 18,9
+ vK
SO
s
32
u, Draconis
Arcturus
Bootis
13.59.35,9
14. 7.a%6
14.37.15,6
1,62
2,73
2,61
24.46.31
69.53.29
62.10.k8
+ 17,3
+ 19 3
+ 15,5
S3
14.41. 6,4
4-3,30
105 17.56
+ 154
34
33
36
/8 Ursse Minoris ...
ct, Cor. Bur
1451.19,6
15.27.12,0
15.35.33,5
0,32
+ 2,54
2,95
15. 7.16
62.41. 1
83 0.37
+ 14,7
+ 12,5
+ 11,7
37
g Ophiuchi
16. 5. 5,0
3,13
U3.13.49
-4- 99
3S
39
40
Antares
a, Hercuiis
16.18.34,2
17. 635,0
17 6 25,9
3,66
2,73
1,34
116. 1.43
75.24.
37 33 49
+ 8|7
+ 4,6
+ 29
41
42
43
44
Ophiuchi
y Draconis
5 Ursee Minoris ....
17.i6.43,5
17.52.30,1
18.29.22,3
1830.57,0
2,78
+ 1,38
- 19,12
4-2,03
77.18.11
39.^9.11
3.25.11
51.22.31
+ 3,2
+ 0,7
2,4
30
45
18.4:133,0
2,^0
565012
3,8
46
47
43
Aquilae
1 Draconis
5 Aquilae
18.57.16,8
19.1229,7
19.16.34,6
2,75
0,(>2
3,00
76.23.31
22.38.59
87.13.47
- 4,8
- 6,2
6,6
40
50
51
^ !!"""!"!"!
1937.50,8
19.42. 8,9
19 46 37,3
2,85
2,93
2,95
79.48 39
81.35.30
gi 1 40
8,5
- 8,9
86
5-2
53
2 K, Capricorni
20. 8.13,7
20.35.24,2
3,34
2,05
103.' 5^ 7
45 20 52
10,3
I- 5
54,
1st 61 Cygni
20.58.58,6
2 77
ff> 655
17 6
65
Cephei
21 1421,0
1,42
8 9 43
is'o
56
21.22.14,2
3,15
962038
153
57
;3 Cephei
^1 ?6 ^0 4
081
20 jr> 54
15 7
58
21 5641,5
3 09
91 1031
170
59
22 55.57 2
2 98
754442
IP
60
a, Andromedae
23.59.15,6
3,08
61.53.12
19,8
STARS double.
On KerscheVs Catalogue of double Stars,
The first Catalogue of double stars was made with a Newtonian telf-
f eope of not quite seven feet focus, and with only 4J4 inches aperture,
r! with a power of 222, The second Catalogue with an .aperture
274
S T A
. of six inches and a quarter, with a power of 227, and 460 ; when the stars
were detected, he used a gradual variety of powers from 460 to 6000.
These double stars are divided into several different classes. In the
first are placed all those which require a very superior telescope, the ut-
most clearness of air, and every other favourable circumstance to be seen
at all, or well enough to judge of them. Their distance is so extremely
small (seldom exceeding two diameters oi the largest) that it cannot be
accurately measured by the 'micrometer, but may be more correctly es-
timated by the eye in measures of their own apparent diameters. It
should be observed, that since it will require no common stretch of power
and distinctness to see these double stars, it will .*. not be amiss to go
gradually through a few preparatory steps of vision, such as the follow-
ing : for instance, when y Coron. Borealis (one of the most minute
double stars) is proposed to be viewed, let the telescope be some time be-
fore directed to Geminorum, or if not in view to either of the follow-
ing stars, Aquarii, p Draconis, g Herculis, a, Piscium, or the curious
double-double star t Lyrae. These should be kept in view for a consider-
able time, that the eye may acquire the habit of seeing such objects well
and distinctly. The observer may next proceed to the | Ursae Majoris,
and the beautiful treble star in Monoceros' right foot ; after these to t
Bootis, which is a fine miniature of a, Geminorum, to the star preceding"
K Orionis, and to n Orionis. By this time both the eye and the telescope
will be prepared for a still finer picture, which is >j- Coronae Boreaiis. It
will be in vain to attempt this latter, if all the former, at least * Bootis,
cannot be distinctly perceived to be fairly separated ; because it is almost
as fine a miniature of * Bootis as that is of , Geminorum. To try stars
of unequal magnitude, it Avill be expedient to take them in some such
order as the following : . Herculis, o> Aurigse, Geminorum, k Cygni,
t Persei, and b Draconis j from these the observer may proceed to a most
beautiful object t Bootis. As the foregoing remarks have suggested the
method of seeing how far the power and distinctness of our instruments
will reach, we may next add the way of finding how much light we have
The observer may begin with the pole star, and Lyrae, then go to the
star south of s Ajquilse, the treble star near k Aquilse, and last of all to
the star following o Aquilae. Now if his telescope has not a great deal
of good light, he will not he able to see some of the small stars that ac-
company them,
In the second class of double stars ere put all those that are proper for
estimations by the eye, or very delicate measures of the micrometer.
To compare the distances with the apparent diameters, the power of the
telescope should not be much less than 200, as they will otherwise be too
275 Q2
S T A
"ic2 for the purpose. It \\-ill be necessary here to notice that the esti-
mation made with one telescope cannot be applied to those made with
another, nor can the estimations made with different powers, though with
the same telescope, be applied to each other ; therefore if we would wish
to compare any such observations together with a view to see whether
a change in the distance has taken place, it should be done with the very
same telescope and power, even with the very same eye-glass or glasses,
In the third class are placed all those double stars that are more than
5 but less than 15" asunder. In the same manner that the stars in the
1st and 2d classes will serve to try the goodness of the most capital in-
struments, these will afford objects for telescopes of inferior power, such
as magnify from 40 to 100 times. The observer may take them in this
or the like order j Ursae Maj., y Delphini, y Arietis, a- Bootis, y Virgi-
nia, / Cassiopeae, p. Cygni. And if he can see all these he may pass over
into the second class, and direct his instrument to some of those that are
pointed out as objects for the very best teleseopes, where he will soon
and the want of superior power.
The 4th, 5th, and 6th classes contain double stars that are from 15 to
SO" ; from 30" to 1', and from 1' to 2' or more asunder. PhiL Tram, vol,
72, ':-5.
For a list of a few of the most remarkable double stars Leonis, ~v~~~~~
X/eo 4"0 Mayer j-.**.*
6 55
933
~ 9 37
16 c Virgo, vuw *.w w >.wx*rj
-^wx^^ 12 10
10 Vir^o ~~~~~~~~~~~~
^^r^u Y>- ?fl
Virgo MM IMMWM ~~
~ 13 04
u Hydra ,~~,,~,,~~~
97 Vlrro, f ~~~~*~.
13 19
14 02
JBootea, ' +*~~~ *~~~~.
^ 1404
STE
Right Ascension. Declinattoa
in 1800.
26 19 S.
2 53 N
28 47 N.
S3 57 N.
14S8N,
554S.
33 CSN.
26 32 S.
S3 16N.
52 57 N,
OSON.
53 C8 S.
37 25 N.
5753
1623
Of this catalogue of Variable Stars, Nos. 2, 3, 7, 11, 18, IP, 0, 22, 4,
and 26, belong to the list of fifteen as given by Mr Pigott. Some of the
others belong to the list of those which he suspected to be variable.
STARS, clusters of. See Nebulas.
STEAM, elasticity and dentntif of.^.(Encyc. Brit. SttpJ
Let E be the No. of atmospheres expressing the elasticity, / the tem-
perature reckoned from 21 % 2 j then
E - (1 4. .00 i/) 5
From hence is obtained the following Table of the elasticities and den.
cities :
Names of stars
1 Libra A w.wvww
in 1800.
..~~~, 14 05
15 Virgo ^WSSWMMSWJJSS-^
^* 14 37
50 Northern Crown, *~~.
31 Hercules, ^..^v^wwjwjv.
~~~*~ 15 40
16 24
u Hercules, ~-,~~~~~.
17 06
59 Sobiesld's Shield ^
20 Lyra, ,
is 3 1 :
v-v. 18 43
34 er Sagittarius, ^
18 43
V SWan *r*~~*~r~~^+~* V.M
, 19 S9
,~~. 19 41
Swan No. 295. P. ~~~~~*
n Antinous, ~~~~~ f ~~~*
25 Southern Fish, ^~^~-~
34 Swan, near y +~~
8 Cepheus, ~~~.~
Aquarius, ~~*
^ 19 42'
19 43
10
22 22
23 24
Atmos-
pheres.
Tempe-
rature.
Compar.
Density.
Atmos- } Tcmpe- \ Coinpar.
pheres. ''ra'ure. j /.'
1
2120
1.000
30
456 I SI. 834
2
?49
1.896
IP
210
3
73
8.742
50
509 81.383
4
fe92
S.5H5 '
fiO
,^-:9 I 40.40t
5
307
4 3S6
70
^17 4/:.i?S5
6
300
5150
60
M:i f' J .OS3
7
SSI
5.917
90
5/7 j WG6
a
S41
6.678
ICO
500 i 6 -.571
9
859
7.433
1000
W < '
10
S.I 70
000
1105 j
15
389
11.820
8000
M02 ; 11!
SO
7
13J82
SUN
77o&>, Bernonilli makes the expansive force of gunpowder equal to
10,000 atmospheres ; Rumford, from the bursting of a barrel of iron,
50,000, from some more direct experiments from 20,000 to 40,000. The
utmost that can be justly inferred from the bursting of the barrel is in
reality about 30,000, since the tension could by no means be equal througlQ
every part of its substance. ( Young's Nat. Philj
STEELYARD. See Lever.
STILE new. See Calendar.
STRENGTH animal See Animal strength.
SUBNORMAL, formula for.
Let x and y abscissa and ordinate of any curve j then
Subnormal = ~~.
dx
Vd-iS-f
43 11
h ra s
2 34 27
h m s f
4 S? 12
2
18 48 SO
21 41
22 53 24
48 40
2 S3 16
4 41 18
3
IS 52 55
21 4 45
22 57 7
50 i7
2 42 6
4 45 ?4
4
18 57 19
21 8 47
23 51
54 6
2 45 56
4 49 31
5
19 1 43
21 12 50
3 4 33
57 45
2 49 47
4 53 S3
G
19 6 7
21 16 51
23 8 16
1 24
2 53 S8
4 57 45
7
19 ]0 30
21 20 51
23 11 57
5 3
2 57 30
5 1 52
6
19 14 52
21 24 51
23 15 39
8 42
3 1 W
560
9
19 19 14
21 28 50
23 19 20
12 ?2
3 5 16
5 10 8
10
19 23 38
21 32 48
23 23 1
16 2
3 9 10
5 14 17
11
19 27 57
21 36 45
23 26 41
19 42
3 13 4
5 IS 55
12
19 32 17
21 40 42
23 30 22
23 23
3 16 53
5 2 34
13
19 36 37
21 44 38
23 34 1
27 3
3 20 55
5 26 43
14
19 40 57
21 48 33
23 87 41
30 45
3 24 51
5 30 52
15
19 45 15
21 52 27
23 41 21
34 26
3 28 48
5 35 2
18
19 49 33
21 56 21
23 45
38 8
3 32 45
5 39 11
17
19 53 50
22 14
23 48 39
41 50
3 S6 ^13
5 43 20
IS
19 58 7
22 4 6
23 52 1$
45 33
3 40 42
5 47 30
19
20 2 23
22 7 57
23 55 56
49 16
3 44 40
5 51 39
20 6 S8
22 11 48
2S 59 35
52 59
3 48 40
5 55 49
21
20 JO 52
22 15 38
3 13
56 43
3 52 40
5 59 59
S3
20 15 6
22 19 27
6 51
2 27
3 5f> 41
648
23
20 19 19
22 23 16
10 9
S 4 12
4 44
6 8 18
24
20 23 31
22 27 4
14 7
2 7 57
4 4 43
C 52 27
25
20 7 42
22 30 52
17 45
2 11 43
4 8 45
6 16 36
26
20 31 52
22 3* 88
SI 23
2 15 i9
4 12 48
6 20 45
27
20 36 1
22 33 25
25 1
2 19 15
4 16 51
6 24 54
28
?0 40 10
22 42 10
28 39
2 1<3 2
4 20 54
6 9 3
29
20 44 18
22 45 55
32 17
2 26 50
4 U 53
6 33 11
SO
20 48 25
35 55
2 30 33
4 29 i?
6 S7
31
52 31
3933
4 S3 7
-
SUN
Days.
Jttfe
August,
September.
October.
November.
December.
1
h in s
6 41 S8
h in s
8 46 14
h in s
10 42 14
h m s
12 30 19
h m s
14 26 39
h m s
16 30 36
2
fi 45 36
8 50 6
10 45 52
12 33 57
14 30 35
16 34- 56
3
6 49 44
8 53 5S
10 49 29
12 37 35
14 34 32
16 39 16
4
6 53 51
8 57 50
10 53 6
12 41 13
14 38 30
16 43 37
5
6 57 58
9 1 11
10 56 43
12 44 52
14 42 28
16 47 59
6
7 2 5
9 531
11 20
12 48 31
H 4T 27
16 2 21
7
7611
9 9 21
11 3 56
12 52 1 1
14 50 27
16 56 44
8
7 10 18
9 13 10
11 7 33
12 55 51
14 54 28
17 I 7
9
7 14 23
9 16 59
11 11 9
12 59 31
14 58 30
17 5 31
10
7 18 29
9 20 47
11 14 45
13 3 12
15 2 33
17 9 55
11
7 22 34
9 24 34
11 18 21
13 6 53
15 6 36
17 14 20
12
7 6 33
9 28 21
11 21 56
13 10 35
15 10 4!
17 18 44
13
7 30 4-2
9 32 7
1 1 25 32
13 14 17
15 14 46
17 23 10
14
7 34 46
9 35 53
11 29 8
13 18
15 18 52
17 27 35
15
7 38 49
9 Si) 38
11 32 43
13 21 44
15 22 59
17 32 1
10
7 42 51
9 43 23
11 36 19
13 25 i8
15 27 6
17 36 27
17
7 46 53
9 47 7
1 1 39 54
13 29 12
15 31 15
17 40 53
13
7 50 55
9 50 51
1 1 43 29
13 32 57
15 35 24
17 45 19
19
7 54 55
9 54 34
11 47 5
13 36 43
15 39 34
17 49 45
7 5S 56 9 5S 16
11 50 40
13 40 29
15 43 45
17 51 12
21
8 2 56
10 1 58
11 54 16
13 44 16
15 47 57
17 53 33
22
8 6 55
IK 5 40
11 57 51
13 48 3
15 52 9
18 3 5
23
8 10 53 i' 10 9 21
12 1 27
13 51 52
15 56 22
18 7 31
.54
8 14 51 10 13 2
12 5 3
13 55 41
16 36
18 11 53
55
8 18 49 10 16 42
12 8 29
13 59 30
16 4 51
13 16 25
56
8 22 45 1 10 20 22
12 12 15
14 3 21
16 9 7
18 20 51
S7
8 s?6 42
10 24 2
12 15 51
14 7 12
16 13 ?3
13 25 17
58
8 30 37
10 27 11
12 19 *8
14 11 4
16 17 40
18 29 44
59
8 34 3-2 10 31 20
12 23 4 ! H 14 56
16 21 58 I 18 34 10
30
8 38 26 10 34 58
12 26 42
14 18 50
16 26 17
13 33 35
31
8 42 20
10 38 36
14 22 44
18 43 1
This Table is alaptai ta L?ap Year, particularly the year 1328, and 13
only intended to answer the purposes of information when no great de-
gree of accuracy is required, and the Nautical Almanack not at hand.
la order to ad-ipt it to common years, one-fourth of the difference be-
tween the given and preceding days is to be subtracted from the right
ascension in the table for the first after Leap Year, one-half for the se-
cond after Leap Year, and three-fourths for the third ; and in the months
of January and February, the right ascension is to be taken for the day
following that given.
This Table may be employed in finding the apparent time by the altl.
tads of a star, for finding tho tims of a star's transit when that is requir-
ed, for obtaining the latitude by a meridian altitude, c.
880
SUN
TABLE II.
Sun's Declination for every Day in the Year 1823.
January. February.
March.
April.
May.
June.
South.
South.
South.
North.
North.
North.
1
o / //
S3 4 22
/ '/
17 17 44
/ //
7 28 10
/ n
438 49
i n
15 9 11
22 5 43
2
22 59 29
17 43 7 5 18
5 1 52 ! 15 27 8
22 13 37
3
22 54 8
16 43 23 6 42 21
5 24 50 ! 15 44 50
22 21 7
4
22 48 20
16 i5 46 i 6 19 18
5 47 43
16 2 17
22 i8 14
5
22 42 5
16 7 53 ' 5 56 9
6 10 29
16 19 27
22 34 57
6
22 S5 5:3
15 49 42 i 5 32 56
6 33 9
16 36 22
22 41 17
7
22 28 14
15 31 15
5 9 38
6 55 43
16 53
22 47 12
8
22 iO :,8
15 12 32
4 46 15
7 18 10
17 9 22
22 bi 44
9
22 12 36
14 53 34
4 22 50
7 40 30
17 25 26
22 57 52
10
22 4 8
14 34 VI
3 59 z\
8 2 42
17 41 13
23 2 36
11
21 55 14
14 14 53
3 35 48
8 24 46
17 ?6 43
23 6 55
12
21 45 54
13 55 10 3 12 13
8 46 42
18 11 54
23 10 50
13
21 36 9
13 35 14 2 48 '36
9 8 29
18 26 48
23 14 20
14
21 25 59
13 15 5 2 24 57
9 30 6
18 41 22
23 17 26
15
21 15 24
12 54 13
2 1 16
9 51 35
18 55 33
23 20 7
16
21 4 24
12 34 8
1 37 35
10 12 51
19 9 35
23 22 24
17
20 53
12 13 21 1 13 52
10 34 2
19 23 12
23 24 16
18
20 41 13
11 52 22
50 10
10 55 1
19 36 29
23 25 43
19
20 :9 2
11 31 13
t) 26 27
11 15 48
19 4!) 27
23 26 45
SO
20 16 27
11 952
2 45 S 11 36 24
20 2 4
23 27 22
21
20 3 30
10 48 22
20 56N 11 56 49
20 14 20
23 27 35
22
19 50 11
10 26 41
44 36
12 J7 I
20 26 16
23 27 22
23
19 36 29
10 4 52
1 8 14
12 37 2
20 37 51
23 26 45
24
19 22 26
9 42 52
1 31 50
12 56 50
20 49 5
23 25 44
25
19 8 1
9 ift 41
1 55 24
13 16 25
20 59 57
23 24 17
26
18 53 16
8 58 28
2 18 56
13 35 48
21 10 28
23 21 26
5t7
18 S8 10
8 36 5 ! 2 42 24
13 54 56
21 20 36
23 20 10
28
18 22 43
8 13 33 3 5 49
14 13 51
21 SO 23
23 17 SO
29
18 6 57
7 50 55
3 29 10
14 32 32
21 39 47
23 14 25
30
17 50 52
3 5> 27
14 50 59
21 48 49
23 10 56
31
17 34 27
4 15, 40
21 57 28
s u N-
Days.
July.
j
August. {September
October. ^November
,.,.
December,
North.
North.
North.
South.
South.
South.
j
23 7 2
17 59 23
8 13 9
3 1633
14 31 39
21 52 8
2
23 2 44
17 44 9
7 51 16
3 39 51
14 50 44
22 1 9
3
22 58 2
17 28 33
7 29 15
437
15 9 35
22 9 44
4
22 52 56
17 12 39
776
4 26 20
1528 11
22 17 53
5
22 47 26
16 56 29
6 4451
4 49
15 46 32
22 25 37
6
22 41 32
16 40 2
fi 22 28
5 12 36
16 437
22 32 54
7
22 35 14
16 23 19
5 5959
5 35 39
16 22 26
22 39 45
8
22 28 33
16 6 21
5 37 25
5 53 37
16 39 59
22 46 9
9
22 21 29
15 49 6
5 14 44
6 21 31
16 57 14
22 52 6
10
22 14 1
15 31 36
4 51 59
6 44. 19
17 14 12
22 57 8
11
22 6 11
15 13 2
4 29 8
772
17 30 52
23 2S8
12
21 57 58
14 55 53
4 6 12
72940
17 47 14
23 7 13
13
21 49 22
14 37 39
3 43 13
7 5"2 1 1
18 3 18
23 11 21
14
21 40 23
14 19 11
3 20 9
8 14 35
18 19 2
23 15
15
21 31 3
14 OSO
2 57 2
8 ?G 52
18 34 27
23 18 12
16
21 21 20
13 41 5
2 23 51
8 59 2
18 49 32
23 20 56
17
21 11 16
13 22 tf
2 10 38
9 x 1 ! 4
19 4 17
23 23 12
13
21 50
13 3 7
1 47 22
9 42 58
19 18 41
23 2459
id
2051 3
12 43 35
1 24 4
10 4 43
19 32 44
23 26 19
20 33 55
12 23 50
1 43
10 26 19
19 46 27
23 27 10
21
20 27 26
12 3 54
37 22
10 47 46
19 59 47
23 27 33
22
20 15 36
1 1 43 45
13 59N
11 9 3
20 1-2 46
S3 27 27
23
20 3 27
11 23 28
9 25 S
11 SO 10
25 22
23 26 53
24
19 50 57
11 2 58
32 50
11 51 7
20 37 35
23 25 51
25
19 38 7
JO 42 18
53 15
12 11 53
20 49 26
23 24 21
6
19 24 8
10 21 23
1 19 39
12 3:2 28
21 053
23 22 22
27
19 11 30
10 28
1 43 4
12 52 51
21 11 57
23 1955
8
18 57 42
9 39 IS
2 628
13 13 2
21 22 36
23 17
29
18 43 36
9 17 59
2 29pl
1333 1
21 32 52
23 13 37
SO
1829 11
8 56 31
2 53 13
13 52 47
21 43 43
23 946
SI
18 14 28
834 54
14 12 20
23 527
This Table, tike the last, is for the year 1828, or Leap Year. The cor*
reetion for any other year must be made as before.
8 U NF
SUN'S Semitliameter, #c. (Naztt. Aim.)
TABLE,
Of Sun's Semidiameter, and of the time of his semidiame tcr passing the
meridian.
Time of
Sttn's 'w
passing
Meridian.
; Semi,
diameter.
T/we o/
9w'* | rf/a?
passing
Meridian.
Semi-
diameter.
Jan.
M. 3.
~ M. S.
July.
i\r. s.
M. 8.
1
7
13
19
S5
1. 10,8
1. 10,5
1. 10.1
1. P,5
1. 8,9
16. 17,8
19. 17,7
16. 17,4
16. 169
16. 16,3
1
7
13
J9
S6
1. 8,5
1. 8,3
1. 8,0
1. 7,5
1. 7,0
15. 45,5
15. 45,5
15. 45,8
15. 46,1
15. 46,(>
Feb.
Aug.
\
7
13
19
25
J. 8,1
1. 7,4
1. 6,7
1. 6,1
1. 5,5
16. 15,3
16. 14,4
16. 13,2
16. 13,0
15. 10,7
I
7
13
19
8&
1. 6,5 "
1. 6,0
1. 5,5
I. 5,0
1. 4,6
15. 47,4
15. 48,3
15. 49,3
15. 50,4
15. 51,6
Mai-.
Sept.
1
7
13
19
25
1. 5,2
1. 4,8
1. 4,5
1. 4,3
1. 4k
16. 0,7
J6 8,2
16. 6,6
16. 4,9
16. 3,3
1
7
13
19
5
I. 4/2
1. 3,9
1. 3,8
1. 3,8
1. 3,9
15. 58,1
15. 51,6
15. 56,1
15. 57.7
15. 59J3
April.
Oct.
1
7
13
19
25
1. 4,2
1. 4,1
1. 4,6
1. 4 3 9
1. 5,4
16. 1,3
15. 59,7
15. 59,1
15. 56,5
15. 54,9
1
7
13
19
5
1. 4,1
1. 4,4
1. 4,8
1, 5,3
1. 5,9
16. 1,0
16. 2,6
16. 4,3
16. 5,9
16. 7,5
May.
Nov.
\
7
13
19
5
1. 5,8
1. 6,3
1. 6,8
1. 7,2
1. 7,7
15. 53,5
15. 52,1
15. 50,9
15. 49,7
15. 48,7
1
7
13
19
S5
1. 6,7
I. 74
1. #1
I. 8,7
1. 9,4
13. 9,3
16. 10,8
if., is?, i
16. 13,3
16. 14,5
**une.
Dfc.
" 1 '
7
13
19
25
1. 8,1
1. 8,3
1. 8,5
1. S,6
1. 8,6
*" 15. 47,6
15. 46,9
15. 4,3
15. 45,9
15. 45,6
1
7
13
19
25
"" 1. 10,0
1. 10,5
1. 10,3
1. 10,9
1. 11,0
16; 15,4
16. 16,2
16. 16,9
16. 17,4,
16. 17,7
S U R
SUN'S parallax m altitude.
Altitude.
Parallax.
Altitude.
Paraliax.
9"
COo
4"
10
9
(5
4
to
8
70
3
SO
8
75
2
40
7
80
2
50
6
85
1
55
5
90
SURFACES of Solids.
Let# ordinate of any curve, z length ; then
Surface fl. 2 v ydz.
Ex. 1. Surface of cone = 2 s- 6 X , where b = base, and * = slant
side, circumference of base X | slant side.
2. Surface of sphere 4 a- r* = four times the area of one of its great
circles.
S. Sorface of paraboloid =
6 '
4. Surface of cycloid = 5 (a diameter of generating circle.)
Guldinus' property.
Let M D E K (see Fig. Art. Solid} be any plane figure, revolving about
an axis xy in its own plane j then the area of the suriace generated by
the perimeter of this figure, is equal to the circumference described by
the centre of gravity of the perimeter multiplied into the perimeter.
Ex. Let D M E K be a circle, then the solid will represent the ring of
an anchor, arid if r radius of circle, and a = A O, the surface = 2 * a
X2rr = 4- 8 ar.
SURVEYING.
I. Surveying Land.
1. The area of a triangle =: base X | perppndirular altitude : or -=; th*
product of any two sides X natnr.il sine of their included / ; or when
three aides A B, AC, B C arc given, tlieir half sum being S are* =
J s X (S A B) X (S A C) X (S BC)]
84
SUE
13
2. The area of a trapezium rr base X | sum of the perpendiculars.
And the area of a trapezoid j= | sum of the parallel sides X perpendicular
distance between them.
3. To find the area of any irregular polygon, divide it into trapeziums,
or trapczoids, or triangles, and find their areas separately ; and their sum
is the area of the polygon.
4. To find the area of a long irregular figure E q D p bounded on one
side by a curve.
Divide ED into any number of equal parts, and measure the perpen-
diculars, no,pg, rs, tv c. then the area is found nearly by adding to-
gether all the perpendiculars, dividing the sum by the number of per-
pendiculars increased by unity, and multiplying by the chord of the
curve.
5. Tofinrl, by the forego-
ing rules, i he, content of
theirregular field ABCDE,
which will include most of
the cases likely to occur in
practice.
Find the area of the tra-
pezium A B D E by Art. 3,
the A B D C by Art. 1 ; and
the curvilinear areas E q D, -A. B
E b A by Art. 4 ; add the three first areas together and subtract the last,
for the content of the field. ,
Land is measured by a chain 22 yards long, and divided into 100 equal
parts or links, each link being 7.92 inches : 10 square chains, or 100,000
square links, is one acre, viz. :
625 square links is 1 perch.
25,000 square links or 40 perches, 1 rood.
100,000 square links or 4 roods, 1 acre.
The perch (which in statute measure is 16| feet) varies by custom ia
different parts of England ; and with it, consequently, varies the acre ia
proportion.
In Devonshire and part of Somersetshire, 15; in Cornwall, 18 ; in Lan-
cashire atid Yorkshire, 21 j and in Cheshire and Staffordshire, 24, feet are
accounted a perch.
285
S U R
Hence the follow ing Table \\ ill give the number of square feet in a
tquaie perch, iu the above-mentioned counties.
Statute perch 13,5 X 1(5.5 272,25 square feet.
Devonshire perch 15 X 15 225 do.
Cornwall perch 18 X 13 = ?24 , do.
Lane, and Yorks, perch 21 X 21 4-11 do.
Cheshire and Staff, perch 24 X 24 = 576 do.
Il'Mesfor reducing Statute Measure to Customary ', and the contra ty.
}. To reduce statute measure to customary, multiply the number
of perches statute measure, by the square feet in a square perch statute
measure ; divide the product by the square feet in a square perch cus-
tomary measure, and the quotient will be the answer in square perches ;
which reduce to roods and acres, by dividing by 40 and 4.
2. To reduce customary measure to statute, multiply the number of
perches, customary measure, by the square feet in a square perch custo-
mary measure ; divide the product by the square feet in a square perch
statute measure, and the quotient will be ths answer in square perches;
which red ace as before.
By these rules tables may be calculated to save the trouble of com-
puting for particular cases ; thus,
TABLE I.
To reduce Statute Measure to Customary of 21 feel to a perch.
Stat.
Acre.
Customary.
A. B. P.
Stat.
Rood.
Customary.
R. P.
1
2
3
2 1S,7
1 37,5
1 3 16,3
1
2
3
4,7
1 9,4
1 34,1
1
5
g
'2 1 35,0
3 18,8
3 '^ 3' 6
Stat.
Perch
Customary,
p.
8
9
10
30
40
50
4 i n,4
4 3 30,1
5 H 8,9
fi ?7,7
12 1 15,4
18 2 3,1
24 2
30 3 JL-.:
\
5
10
15
20
25
30
35
0,6
3,0
6,1
9,2
12,3
15,4
18,5
*M
sec
SUE
TABLE IT.
To reduce Customary Measure of 21 feet to a perch, to Statute.
Cmt.
Acre.
Statute.
A. R. P.
Cunt.
Rood
Statute.
A. R. p.
2
3
4
5
6
7
8
9
10
20
30
40
50
1 2 19,17
3 33,-M,
4 3 17,44
6 1 33,61
8 15,81-
9 2 34,83
11 1 14,8
13 3 33,28
14 2 Ij?,48
16 31,68
32 1 23,36
48 2 15, t
04 3 3(5,83
80 3 38,5(5
1
2
3
'Oust.
Perch
1 24,70
3 <),58
1 34,39
Statute.
R. P.
1
5
10
15
20
25
30
35
1,019
8,099
16,193
24/>97
3i,396
1 0,495
1 8,591
1 16,693
Er. 1. In 3fiA. lR. 10p. statute how many acres, &c. customary measure
of 21 feet to a per eh?
Reduce to perches, which will be found 5810, .'. 5310 X 272,25 =
18ol772,50; this divided by 441 gives 3535,7 perches ; divide by 40 and 4
and the result is 22A.-1R. 26,7p. The same answer may be had from
Table I.
E.r, 2. Reduce 22A. IR. 27p. customary, to statute measure.
Here the number of perches is 3537, which, multiplied by 141, and di-
vided by 272.25, gives 3(>A. IR. 10p. The same result may be obtained
from Table II.
II. Surveying TrigonometricaUy.
1. These large surveys have been undertaken principally for the ac.
complishment of one or other of tbese three objects, viz. (1) For finding-
the difference of longitude between two moderately distant and noted
meridians, as the meridians of the observatories at Greenwich and Paris.
(2) For the exact determination of the principal place: in a country, with
a view to give greater accuracy to maps. (3) For the measurement of
a degree in various situations, in order to determine from thence the
figure and magnitude of the earth.
These important objects can only be attained, by the greatest possible
degree of accuracy in the instruments employed, the operations perform-
ed, and the compatfttionjs required.
287 Q4
S U R
The following must only be considered a mere outline of the method
pursued in surveying a country : the niceties necessary to be attended
to, in crder to render such survey available for scientific purposes, can-
not be here described,
2. To carry on a measurement by a 5erii?s of triangles.
I^eta base line ABbe measured,*
and having fixed upon two objects C
and D, observe the /'s B A C, B A D,
ABC, A B D ; then in the A A B C,
the 's B A C, ABC, being known,
their supplement A C B is known, .'.
A C and C B may be found by Case 1.
Plane Trigonometry. The relative A B
bearings and distances therefore of A, B, C arc thus deterirmied. Again
in the A A B D, the ^'s D A B, D B A being known, A D B is known,
and /. D B may be found. Lastly, in the A D B C, the sides B C, B D
and the included ^ C B D are known, .'. the remaining /'s B C D, B D C
may be found, and consequently also the side CD (see Case 2. Plane
Trigonometry). The bearings .'.and distances of B, C, D are also known.
In the same way, by considering either A C, C D or D B as a new base,
and fixing upon two other points ; the measurement may be continued
at pleasure.
In conducting geodetical operations, the following rules by Hutton
should be observed, to diminish the probability of error.
(1) When one side only of a triangle is to be determined, the measured
base should be nearly equal to the required side.
(2) W T hen two sides of a triangle are to be determined, the triangle
should, if possible, be equilateral.
(3) When the base cannot be equal to one or both the required sides,
it should be as long as possible, and the two angles attbe base equal, and
not less than 20 or 30 degrees.
In the late survey of England the base first measured was upon Houn-
slow Heath. By continuing the measurement to Salisbury Plains, the
* To reduce a base on an plevatpd level to that at the surface of the
sea, let r =. rad. of earth at the surface of the sea, r -f k the rad. referred
to the level of the base measured, the altitude h being determined by
the barometer, B the length of the measured base at the altitude h, then
the correction is nearly, \vbu-h must bp subtracted from the mea-
*urpd base, to give the true bafe at tho level of the eea.
i?88
S U R
distance of two objects was there found by calculation to be 6574,4 feet,
find, by actual admeasurement, the distance was found to be 30574,3 feet,
differing very little more than an inch from the computed distance.
We shall close this necessarily imperfect article with the methods of
finding the difference of longitudes and difference of latitudes of places
upon the earth's surface, as practiced in the late government survey of
this country.
Difference of Longitude.
Let P be the pole, L G the circle described by the
G
pole star, A and B the two places. Take by the in-
strument the /_ GAB / contained by B and the
pole star when at its greatest azimuth ; then knowing
P A, P G, we may find PAG the greatest azimuth,
which added to GAB gives P A B ; hence in the
spherical A P A B, we have PA, P B and / P A B to
find /_ A P B the difference of longitude. Hence if A
be Greenwich, the longitude of B is known.
N.B. For four minutes either before or after the pole star's greatest
elongation, it moves only about a second in azimuth; h.nce a good
pocket watch gives the time of greatest a/iniuth with sufficient accuracy
Difference of Latitude.
In finding the difference of longitude above, the latitude was i
known j this was found by geometrical admeasurement thus,
Let A, B, and C be three places, A F the
meridian of A. Find the / B A D zz supple-
ment of / P A B in the former figure (and
which may be found, if A 'a latitude ia known,
as was shewn ia the last Article) ; and find by
observation A B. B C, and / A B C; then ia
the A A B D we can find A D in feet, and '
(knowing the dimensions of the earth) in second,, which gives the dif-
ference of latitudes of A and * Again,!, the A C B E, w! hT.e C B Q
the supplement of ABC and GBE (.BAD) and .'. CBE; hence
BE may be found, and /. A F, = difference of latitude of A and C
s , h r r!^ ! h r u any ** * ** A be G^:
tan^alpt^^ ff
The tetitude. thus determined are more accurate than thoso dtfuoe*
S Y N'
from astronomical observation ; since the best instruments could not
have given the zenith distances nearer than about one second, answer-
ing in these parts to lul feet on the, surface of the earth.
TABLE
Qftho fli-f.n-jit I -> norths of a degree, as measured in various parts of the
earth, (,\ fane of its measurement, the latitude of Us middle point, fyc.
(Bar
Date.
Latitude.
Extent in
En?, miles
and dec.
Measurers.
Countries.
1523
1620
49" iOi' N.
52 4~ X.
68-7(tt
Go- 91
M. Fernel
France.
Holland.
1635'
53 15 A*.
09545
England.
1641
75-06(J
Rici-ioli
Italy.
1669 7
C63945
Pioard 7
1713 j
1535
1740
6f> 20 N.
49 22 N.
45 00 N
169-119
69-403
69-121 7
69-092 \
Cassini j
Maupertuis, &c.
Cassini and La
Caille
France.
Lapland.
France.
1744
1752
33 18} S.
r 68 -751
363-732
668713
69-076
Juan and Ulloa T
Bouguer >
Condamine J
La Caille
Peru.
C. of GoodHope
1755
1764
43 N.
41 44 N.
68-998
69 -061
Boscovich 7
Italy.
1766
1768
1802
1803
47 40 N.
39 12 N.
51 2954-iN.
66 204 N.
12 32 N.
69-142
68 893
69.146
69-292
as -743
Leisganig
Mason & Dixon
Lt.-Col. Mudge
S\vanberg, &c. ..
Lambton
Germany.
America.
England.
Lapland.
Misore.
1803
44 5?f N.
68-769
Biot, Arago, &c.
France.
SYNODIC A L revolution of the planets.
1. If two planets revolve in circular orbits, to find the time from con-
junction to conjunction.
Let P = periodic time of the earth, p that of the planet, suppose an
inferior, t tlie time required j then
-
V-p
For a superior planef,
t = -
290
T A N
This will also give the time between two oppositions, or between any
two similar situations.
= -It
Cor. Since = -, p =
Therefore, from the earth's period (P) known, and the synodic (*) ob-
served, we can determine the periodic time (/>) of the planet.
For the synodical periods of the planets see Planets elements of.
2. To find the same for three bodies.
Let T = time between the conjunctions of the 1st and 2d found as
above j t = do. between the 2d and 3d.
Then \fm = least common multiple of T and t, m time between two
.conjunctions of the three bodies.
TANGENTS, method of drawing.
1. Method of drawing a tangent to any curve, whose equation is gir
Let x and y abscissa and ordinate ; then
Sub tangent .
Ex. 1. In parabola, subtangent 2 or.
y a x .r
a -
2 a .v -f A-
3. In hyperbola, subtaji. .
2. To find the equations to the tangent and normal.
Lett/' a x' -f- & be the equation to the tung-cnt; then a -/T,
also since the curve and line have a point in common, // a ,r -\- I,
and y' " y = -,-' (?' *') whifh is the requherl o<|uatiwu.
Again let y" and x" be the coordinates to the normal, tiicu since it
passes through a point whose coordinates are ..r and //, and is perpoudL-
T A Y
cular to a line whose equation is y f y -? *- (x f x), the equation
to the normal will be y" y = -- -, (JF" JT). fS / ^ u \ / fh u \
1T2 + U^a) u7i + &c * whcre (w) ' ("ST> C^)' &c - denot9
the values of u, -^-^, ^-jg &c. when .r = 0.
This theorem is only a particular case of Taylor's, for take jc = ia
Taylor's series, and we have
which is the same as the theorem above, if for h we write jr.
T A Y
Ex. 1. To expand (.r 4. h) m .
Let u = a? m and u' = (x 4- h] . By differentiation we have -3
m-i rf w v m- 2 efe M , w-*
w .1- j *-- = TO.. (?-!)* ; -T- w . (OT - 1) . (m - 2 x &e.
' d j-* ^'3
Hence, by Taylor's theorem, v,' a- + mir ^4- w ~^~~ * v " ^ z
+ &c.
i\r. 2. To expand ( ^- *}"* by Ma^-laurin's theorem.
Let u - ( + *) W ; .'. -^ = m. (a + jf^ 1 ; ^ ^ m. (m - l.J
(a 4- .r) w - 2 c. &c. Now let * = 0; then (a) = a m ^ ( ^i)
= m a m " 1 j ( ^) =c w, (m - 1.) a m ' 9 &c. ft
= (a + z) m - a + W " ! * + ~ ! m " 2 4- &e.
jr. 3. To expand a' in a series.
Let u = a T , then - = ft M (ft
' 0, then u 1 ; .*., by Madaurin's theorem,
Let u = a T , then - = ft M (ft = h. 1. a), j-~ ft* w, &c, Now let
Ex. 4. To expand log. (x + ft).
Lot u I x, and W = / (* + A) ; .*.
du _ m dzu _ w <^3 M _ 2w _
~dx = d^2 P'JI^"" 1^~
.*. by Taylor's theorem,
x 7i A2 , ^3 A*
'= w + in ( 7 - ^ + ^ - 4^7 + A
Cor. If x = 1 , we have
7?or. 5. Expand sin. x in a series.
I>et w = sin. .T. Take the successive differentials of sin. r, and find
293 R2
TEL
their value tvherx * ~ 0, and we shall have by Maclaurin'a theorem
In like manner,
TELESCOPE, theory of.(Coddington t Wood.)
1. Astronomical Telescope.
Let F, F' represent the focal lengths of the object and eyo glass j then
F
the magnifying power is ,.
Cor. The linear magnitude of the greatest visible area is measured by
the /, which the diameter of the eye glass subtends at the centre of the
object glass, increased by the difference between the /'s which the dia-
meter of the object glass subtends at the image, and at the eye glass.
2. Galileo's telescope.
The magnifying power as before .= .
Cor. The linear magnitude of the field of view, when the eye is placed
close to the concave lens, is measured by the angle which the diameter
of the pupil subtends at the centre of the eye glass, increased by the dif-
ference between the /'s which the diameter of the object glass subtends
at the pupil, and at the image.
3. HersclteVs and Newton's telescope.
L3t/and F' be the focal lengths of the speculum and eye glass ; then
the magnifying power = ^
Cor. The field of view is nearly equal to the apparent magnitude of
the eye glass seen from the speculum.
4. The Gregorian and Catsfgr&in't telescope.
Let/,/', F, be the focal lengths of the great and small mirror, and the
lens respectively, I the distance of the mirrors; then the magnifying
power of the Gregorian is nearly
(*-/')
fV '
and of Cassegrain's la
294
TEL
5. In refracting telescopes, if A be the linear aperture of the object
glass, the density of rays in the picture upon the retina varies as
A2 F'g
pa
And in the Newtonian telescope, as
A 2 F'a
~7*~"
6. To place a telescope in the meridian by tlie pole star.-( Wollaston. )
Calculate the time of the meridian passage of the star correctly, and
apply that to your chronometer. Then having the str.r in 1 ho field of
your telescope (the instrument being first truly adjusted, and the adjust-
ing screw for azimuth between your finger and thumb) and keeping it
bisected, or covered by your meridian wire till the exact instant calcu-
lated, clamp the instrument there in azimuth, and you will find it very
neaily in the meridian indeed.
Having thus placed the telescope very nearly in the meridian ; we may
adjust it accurately so, by either of the following formulae :
Formula for correcting the error of a Meridian Telescope by the observa-
tion of any circuinpolar star above and beloiv the po!c.
If the western interval be greater than the eastern one, Ihe telescoro
points to the east of that end of the true meridian which lies under the
the elevated pole (be that N. or S.) and v. v.
The angle of this deviation may be investigated thus :
To the log-, of half the difference between the intervals in seconds (or
the difference between either interval and lh. sid. time.)
Add the log. tangent of the star's PD.
And the log. secant of the lat. of the station.
The sum (abating from the Index) will give the log. of a number of
seconds of sid. time; which converted into degrees, c. will express tho
angular deviation of tho instrument from the true meridian, to be ap-
plied as above.
This method depends not at alt upon knowing truly the R A of tho
star; nor its PD with any very great accuracy: the ZD or alt. read
oS' with the instrument, as it passes the meridian, will give the latter
with fully sufficient precision.
Formula, from which the above rule is deduced. (Maddy.)
Deviation. - l^JT-^^ 1 ? > where t and t' are the two intervals,
cos.J. tan.
$ the star's declination, and I the latitude of the place.
295 \
T E L
Formula for the correction of a Meridian Telescope by Hie observation of
two stars differing considerably in polar distance.
If the southern of two stars passes a meridian telescope too soon for the
calculated difference of apparent R A between them (whether its passage
be before or after the northern one, is immaterial) the telescope when.
turned down towards the south horizon will point to the east of the true
meridian, and v. v. This holds universally, whether the latitude of the
station be N. or S.
The angle of this deviation from the meridian may be found thus:
The quantity of sidereal time, by which the observed difference of R A
varies from the calculated difference between the stars, being- reduced to
seconds of time ;
To the log. of that number of seconds ; add
the log. cosines of the declination of each star ;
the log. cosecant of the difference between them in declination ;
and the log. secant of the lat. of the station :
The sum (abating 40 from the Index) will give the log. of a number of
seconds of sidereal time ; which reduced to degrees, &c. will express the
angle made by the instrument and the true meridian.
Formula from which the above rule is deduced. (Maddy.)
Let T T' be the difference of right ascensions of the two stars from
the Tables.
t t' the difference of right ascension as observed by the tele-
scope, 5 and & the declinations, I the latitude, then
Deviation
= $T - T' -(-*<)?.
3
--
cos. Asm. ($3
7. To find the field of view of a telescope.
Direct the telescope to a star in the equator, or very near it, which
will answer quite well enough for all usual purposes, and observe the
number of seconds occupied in its passage across the field of view, and
multiply this number by 4, to obtain in degrees a measure of the field.
It would evidently be inconsistent with the limits of this small work
to enter into any explanation of the nature, use, and adjustment, of
mathematical instruments j nevertheless as a telescope is in the hands
of almost every one at all conversant in scientific pursuits, the follow-
ing practical observations on this instrument, selected from the works
of eminent practical astronomers, may not be unacceptable to the inex.
perienced observer.
296
TEL
Proper size of telescopes.
The smallest achromatic that can be used with success for astronomi-
cal purposes is the 3J feet, aperture 2% inches. (Kitchiner.)
Magnifying powers of telescopes.
For day purposes, a power of CO or 100 is the maximum that can bo
generally used in this country, except on very fine days, and on objects
uncommonly well lighted up. In telescopes of different apertures, the
maximum power for day purposes is had by multiplying 1 the diameter of
the object speculum or glass in inches by 30. For astronomical pur-
poses thb rotatory motion of the earth prevents the application of a much
higher power than 300 being used with any advantage : when a higher
power than 300 is used, it requires uncommon dexterity both to find the
object and manage the instrument. The following powers are proper
for a fine achromatic. (1) A comet eye piece, made with t\vo piano con-
vexes not magnifying more than 12 or 15 times, which is also a delight-
ful eye piece for viewing nebula? and the milky way. (2) For a series of
powers for planetary observations, multiply the diameter of the object
glass in inches by SO, 30, 40, 50 and CO ; this last is the maximum that can
be used for the planets, and requires a very perfect telescope, and every
circumstance to be favourable, to admit of its application with good ef-
fect. (3) A positive eye piece magnifying SOO times for close double
stars; yet unless the telescope be an uncommonly fine one, a higher
power than 200 only renders the object less distinct. (1) A circle of six-
single double convex lenses magnifying 50, 100, 150, 00, 300, "and 4^0
times, but when the highest power is used, the distinct field of view ia
reduced to a very small diameter. (Kitchiner.)
Eye glasses for telescopes.
In very delicate observations Herschel observes, no double eye glass
should be used, as that occasions a too great waste of light. With the
double eyeglass he could not see the belts of Saturn, which he very plain-
ly saw with the single one. Of single glasses he decidedly prefers con-
cave to convex glasses, as they give a much more distinct ircnge. Their
very small field of view is a considerable imperfection, but in objects
sucli as double stars, or the satellites of Saturn, and the Georgian, this
inconvenience is not so material. (Phil. Trans.)
Best criterion of a good telescope.
The most difficult object to define in the day time, and the best test of
the distinctness and correctness of our instruments, is the dial plate of a
watch, when the sun shines upon it, placed about 100 feet from the glass.
la the night time a fixed star of the first magnitude is the best test, aa
801
TEL
the least defect in the figure or adjustment of the object g'lass is imme-
diately seen by the star not appearing round, but surrounded by false
lights and luminous accompaniments. For a test of the perfection of a
telescope as to its light and distinctness, the pole star is as proper as any,
as the small accompanying star is not visible except in a very perfect
instrument. The examination of a bright object on a dark ground, as a
card by daylight or Jupiter by night, with high magnifying powers, af-
fords the severest test of the perfect achromaticity of a telescope, by the
production of green and purple borders about their edges in the con-
trary case. ( Kitchiner. Mem. Astr. Soc.J
On the evenings and situations favourable or otherwise to astronomical
observations.
The rule upon which almost all the rest are founded is that an uniform
temperature is necessary for the proper performance of a telescope.
Upon this principle the following facts, the results of long experience,
may be satisfactorily explained.
(1) A frost after mild weather, and a thaw after frost, will derange the
telescope, till either the frost or mild weather are sufficiently settled.
(2) No telescope just brought out of a warm room can act properly. (3)
No delicate observation with high powers can be made when looking
through a door, window, or slit, in the roof of on observatory ; even a
confined place in the open air is detrimental. (4) Windy weather is un-
favourable. (5) Stars seen over the roof of a house, when very near, are
not distinct, being disturbed probably by warm exhalations from the
roof. (6) Dry air is unfavourable; but those evenings wherein the air
is saturated with moisture, so as to drop down the tube of the telescope,
are particularly favourable to distinct vision.
Upon the whole Dr Herschel observes that to use the highest magni-
fying powers to the greatest advantage, the air must be very clear, the
moon absent, no tAvilight, no haziness, no violent wind, no sudden
change of temperature ; under all these circumstances a year that will
afford 100 hours must be called a very productive one. (Herschel t Phil.
Trans.)
Rules necessary to be observed for examining delicate objects with suc-
cess.
(1) If the telescope has been kept in a warm room, the cap of the ob-
ject end should be taken off, the eye piece taken out, and the air suffer-
ed to pass through the tube for ten minutes, that it may acquire the tem-
perature of the open air. (Kitchiner.)
(2) The observer should in like manner be exposed in the open air for
15 oi % 20 minutes, and the eye carefully kept from all stvnulatin^ aud
TEL
bright objects, so that the pupil may be in its most expanded state ; it
requii'ing at least 20 minutes before the eye can admit a view of very de-
licate objects (such as faint nebulae) ; and the observation of a star,
though only of the 2d or 3d magnitude, disorders the eye again, so as to
require nearly the same time for the re-establishment of its tranquillity.
(Herschel, Phil. Tran.)
(3) We should never use a greater magnifying power than we abso-
lutely want ; the lower the power, the more beautiful and brilliant the
object appears. In objects however that require great nicety to discern,
such as the spheroidical shape of the planets, &c. it is proper in the first
instance to use a considerable power, till the eye is accustomed to the
phenomenon, after which the power may be gradually lowered. (Hers-
chely Phil. Trans.)
(I) It may be proper to observe, in order to prevent disappointment,
that in the prints usually given of Jupiter, Saturn, &c., the outlines and
all the other features of the engraving are far more distinct than we
can ever see them in the telescope in one view , it being the very inten-
tion of a copper-plate to collect together in one view all that has been
successfully discovered by repeated and occasional perfect glimpses, and
to represent it united to our conceptions. And this is the case with
all drawings in books of Astronomy. (Hersch. Phil. Trans.}
(5) In attempting to determine the apparent shape or magnitude of
any planetary body or satellite, it is useful to compare it with some
other known object of a similar kind. Thus to form an idea of the pe
Culiar shape of Saturn, compare it with Jupiter several times in succes-
sion. To form some notion of the apparent magnitudes of Juno, Pallas,
Ceres, and Vesta, compare them with each other, or with Jupiter's sa-
tellites. (Hersch. Phil. Trans.)
(6) When we wish to discover very delicate and minute objects, which,
with the finest instruments, are only to be seen under the most favour-
able circumstances, it is indispensable that we should be in a position of
the greatest ease ; no cramped or painful posture must distort the body
or irritate the mind, the whole powers of which must be concentrated
in the eye. (Kitchiner.)
(7) In adjusting the telescope to close double stars, Dr. Herschel ad-
vises the observer previously to adjust the focus of his glass with the
utmost delicacy on a star known to be single, of as nearly as possible
the same altitude, magnitude, and colour, as the star which is to be ex-
amined, carefully observing whether it be round and well defined, or
surrounded by little Hitting appendages, as is the case when the object
glass is not quite perfect. (Phil Trans.)
299
TEL
expected that we see em a er graes sance. ave nown
take up two or three months before the eye was sufficiently acquainted
with the object to judge with the requisita precision. ( Hersch. Phil.
Trans.)
(9) It is a singular fact, that a double star, where one of them is of Iho
last degree of faintness, may be best seen by directing the eye to another
part of th-j field. In this way a faint star ia the neighbourhood of a
large one will offr'n become very conspicuous, though it will totally dis-
.appear, as if suddenly blotted out, when the eye is turned full upon it.
The small companion of 23 (h] Urss Maj., is a remarkable instance of
this; also P Aurigse. 11 Monocerotis.
& Geminorum. x. Bootis.
k Cygni. /$ Scorpii.
t Persei. Polaris.
Lyra. y Herciilis.
301 R3
THE
c 40 SerpentU. 44 Bootis.
3/ Leonis. S Serpentis.
Librae Prae. t Eootis.
70 Opliiuchi. 05 Herculis.
Herculis. ft Serpentis.
Corona?. /* Herculis.
| Bootis. 2> Herculis.
Performance of different telescopes.
Dr Kitchiner has seen the small star accompanying Polaris Avith a
feet achromatic, aperture If inches; and the small star accompanying-
Rigel; but the telescope was exquisitely perfect.
t Bootis, Herculis, y Andromeda, Cygni, Aquarii, Pole Star,
Castor, Rigd, maybe seen with a fine 44 inch achromatic, ofvf aper-
ture ; but not one instrument in a hundred will shew them without a
falsa light round the larger star.
With an exquisite achromatic of 46 inches focus and a treble object
glass of SB inches aperture, Dr Kite-Inner has seen the Pole star with the
fallowing powers, 40, 80, 150, 250, S50, 4s 0, TOO, and even with 1 1>3 times
the small star was still visible. This shews only how far magnifying
power could be carried with this instrument, as it was with evident de-
triment to vision when higher than 80.
With a most perfect achromatic of 44 inches focus, aperture f indies
made by Dolload, Mr Walker made the following observations. With a
negative power of ISO, he-saw t Bootis double j t Bootis ; ?j Coronse Bo-
realis. Three satellites of Saturn ; the shadow of his ring on the planet ;
and a beltj d Serpentis j Herculis; the Pole star; / Bootis, and/i Dra-
conis ; powers 453 single eye glass, and ISO, and 133 negative powers,
Rigd with 133, and the star in Monoceros' right foot treble with powers
0,180, and 4*3.
Tfce ordinary powers used by Messrs South and Herschel, (tee Phil.
Trans.) in forming their catalogue of double stars, was 179; though oc-
casionally a lower power of 105, and a higher one of 273 were also used.
TEMPERATURE of Atmosphere. See Atmosphere.
THERMOMETER. Freezing point. Soiling point.
Fahrenheit's Thermometer 32 212
Reaumur's do. 80
Ccntrigrsde do ..,. . 100
503
T I D
To convert the degrees of Reaumur into those of Fahrenheit, and the
ontrary.
r = * + 320-and R t= (Zl^OJl*
To convert the centrigrade to Fahrenheit and the contrary.
F = ^V 32 o _ and C ^ (F-^?X5
convert the Centrigrado to Reaumur and the contrarj'-.
K = S->L* .ndc=*.
5 4
To
THERMOMETRICAL barometer. See Heat.
TIDE3. ( Fhice and Racism from Bernouilli.)
1. If a fluid sphere at rest be attracted by a distant body S also at rest,
it will put on the form of a spheroid ; and if P and Q represent respec-
tively the attraction of the spheroid at the extremities of the minor and
major axes,z be the addititious force of S upon P, and n that upon the
point E
Major axis : Minor I : P 4. m : F, 2 n.
Cor. If the sphere were the earth, and S the sun or moon ; then, upon
the above supposition, the difference of the diameters or height of the
tide, as caused by the sun, would = 2,033 feet ; and the height, as caused
by the moon, r= 5,412 feet; .'. in syzygy the height would be 7,445 feet.
2. Tha altitude of the high tidj above the level of the water, if thero
had been no tidi?, is double of the. depression of the low tida below.
3. Find (1) The elevation of the water at any point above the natural
level of the undisturbed ocean. (>) The depression below the natural
level at any point. (3) The falling 1 of the water from the highest point,
and (4) The rising of the water from the lowest point.
Put angular distance of the point from the place of high water, or
the hour / from the time of high tide ; m perpendicular height of high,
above low water ; then the equations will stand thus :
(1) Elevation = 2!^1=J x .
o
(>) Depression = ^^ -^- X m.
J
(3) Fall ......... m X sin. 3 0.
(I) Rise ......... = m X cos.s 0.
203
T I D
Cor. To find the distance of high tide from the point where the water
is at the same height at which it would have been if there had been no
tide, put 3 cos.2 8 1 = ; .*. cos. 6 = = cos. 54. 44'.
V3
4. To find the elevation and depression as before, produced by the joint
action of the sun and moon.
Let m perpendicular height of high above low water, as caused by
the sun, n ditto arising from the moon, 6 = hour angle from the time of
high tide for the sun, 6' = ditto for the moon; then the elevation above'
the natural level is
3 cos.2 _ 1 3 CCS. 2 & 1
3 X H X n ;
. 3 sin.2 2 3 shi.2 p 2 ,_
aud depression is X m -] X .
Cor. 1. If the sun and moon be in syzygy, 9 = 6' ;
.*. elevation = (m -f ?/) cos.* 8 ~T H ;
' O
and depression =: (m -f- n) sin.2 6 % (m + w).
Hence at high water, elevation = % (jn -;- w), and at low water, de-
pression = y 3 (m 4- n}.
Cor. 2. If the moon is in quadrature, elevation at S = % m % ,
and depression at M y s m % n ; also the elevation at S above the
inscribed sphere m n, and the elevation at M above the same
n m. Hence since n is greater than m in the ratio (according to Ber-
nouilli) of 2^ : 1, it is plain that when the moon is in quadrature, it is
high water under the moon, and low water under the sun.
Cor. 3. Supposing the sun and moon to he in any other position, and it
were required to find an intermediate point between them where there
is high tide ; in this case we must take the expression *"" X m
3 cos 2 6' 1
H -,- X n, arid make the differential = 0, and we shall get
m : n :: sin. 2 8 : &in. 2 0'. Hence we have only to divide an arc
2 (0 + &) into two part?, so that the ratio of the sines may be given; and
the half of each part will give 6 and 6', and thus we get the point where
the tide is highest.
Cor. 4. By computing by the last Cor. the ^'s 9 and 0' for every day
/rom the new or full moon, we might get the time of the high tide when
301
T I B
rompared with the passage of the sun and moon over the meridian ; and
thus from these we might construct a table, shewing the theoretical
times of high tide during the month.
Hitherto \ve hare supposed the luminary to be in the equator : we
come in the next place to consider the effect arising from the declination
of the moon.
5. Let D moon's declination, L = latitude of the place, 8 = hour
/ from high water j then the height of tha water from the lowest point
is
(cos. D X cos. L X cos. -f sin. D X sin. L)s X m.
Hence we may consider the following cases :
I. To find the interval from high to low tide, put cos. D X cos. L X
sin D X sin. L
cos. 4. sin. D X sin. L ; .'. cos. 6 = i=i - x ^-.
cos. D X cos. L
II. When the latitude of the place comp. of moon's declination, cos.
1 ; .*. 180, i.e. the interval between high and low tide -=i 12
hours, i.e there is only one high and one low tide in 24 hours.
III. When the distance of the place from the pole is less than the
moon's declination, the expression in Art. 5 never can become with-
in the limits of cos. I); .*. there is only om; high jind one low tide in 24
lunar hours. And if \ve make cos. & I , and cos. = 1 , we have the
difference of the altitudes of the two tides 4 cos. D X cos. L X sin. D
X sin. L X m.
IV. When D = L, make cos. 6=1, r.nd we have the greatest altitude
=. m ; also cos. - : = interval from high to low water.
V. When the moon is in the equator, the altitude of the tide = cos.*
LX m.
VI. The height of the tide, when the moon passes the meridian, =
(cos. D X cos. L 4- sin. D X sin. L) 2 X m ; and when the moon is at the
opposite meridian, the height is ( cos. D X cos. L 4. sin. D X sin. L)
X m. Hence when the moon is in the equator, sin. D o, and the
height of both tides is equal. To a place on the north of the equator,
when the moon has south declination, sin. D becomes negative, and the
latter tides are the greatest ; but when the moon has north declination,
sin. D is positive, and the former is the greatest. Hence, to us in this
case, the high tide is greater when the moon is above the horizon than.
when below. The difference of the two tides is always what is given in
Case III.
305 R i
T I D
VII. The height of the two tides, when the moon passes the meridian,
being (cos. D X cos. L -}- sin. D X sin. L) 2 X m> and ( cos. D X cos.
L -f sin. D X sin. L) 2 X m, the mean height is (cos.* D X cos. 2 L 4.
sin. 2 D X sin. 2 L) X m. Hence the same north and south declination of
the moon give the same mean altitude.
VIII. Under the equator the mean height = cos. 2 D X m.
The general phenomena of the tides agree very well with the conclu-
sions deduced from the theory of gravity, indeed much more accurately
than could have been expected, when we consider that the theory sup-
poses the whole surface of the earth to he covered with deep waters j
that there is no inertia of the waters ; that the major axis of the sphe-
roid is constantly directed to the moon ; and that there is an equilibrium
of aH the parts j none of which suppositions are strictly founded in fact.
As a sequel to this Article we will subjoin a few of the principal phse.
nomena of the tides, as deduced from actual observation. ( Play fair. )
The time from one high water to the next, is, at a mean, 12 k. 2dm. 2k.
The instant of low w r ater is not exactly in the middle of this interval ;
the tide in general taking 9 or 10 minutes more in ebbing than inflowing.
At new and full moon, or at the spring tides, the interval between the
consecutive tides is the least, viz. I2h. Win. 28*. At the quadratures, or
neap tides, the interval is greatest, viz. \2h. 30?/z. 7*.
The gradual subsidence of the waters is such, that the diminution of
heights are nearly as the squares of the times from high water.
The time of high water in the open sea is from 2 to 3 hours after the
moon has been on the meridian, either above or under the horizon ; but
on the shores of large continents, and where there are shallows and ob-
structions, there are great irregularities in this respect j but for any
given place the hour of high water is always nearly at the same distance
from that of the moon's passage over the meridian.
The highest of the spring tides is not the tide that immediately follows
the syzygy, but is in general the third, and in some cases the fourth.
At Brest, the spring tides rise to 19,317 feet ; and those of .the neap to
9,151. In the Pacific Ocean, the rise, in the first case, is 5 feet ; in the
second, 2 or 2,5. Indeed it may happen, that although the greatest ele-
vation produced by the joint action of the sun and moon, in the open
sea, does not exceed 8 or 9 feet, the tide in some singular situations may
amount considerably higher. For instance, in the harbour of Annapo-
lis-Royal, it sometimes rises 120 feet; the water accumulating to this
astonishing height in consequence of its being stopped in the Bay of
Fundy as in a hook.
306
T 1 D
The greater the rise of high water above the level of a fixed point, the
greater is the depression of the corresponding low water relatively to the
same point.
The height of the tide is affected by the vicinity of the moon to the
earth, and increases, caeteria paribus, when the parallax and apparent
diameter of the moon increase, but in a higher ratio.
The rise of the tide is affected by the declination of the luminaries j it
is greatest, caeteris paribus, at the equinoxes, and least at the solstices.
When the moon is in the northern signs, the tide of the day, in all
northern latitudes, is somewhat greater than the tide of the night : and
the contrary when the moon is in the southern signs.
If the tides he considered relatively to the whole earth, and to the
open eea, there is a meridian about SO*' eastward of the moon, where it
is always high water; on the west side of this circle, the tide is flow-
ing; on the east, it is ebbing ; and on the meridian, at right /'a to the
game, it is every where low water.
In high latitudes, whether south or north, the rise and fall of the tid
ire inconsiderable. It is probable that at the poles there are no tidea.
The tides, in narrow seas, and on shores far from the main body of
the ocean, are not produced in those seas by the direct action of the lu-
minaries, but are waves propagated from the great diurnal undulation,
and moving with much less velocity. For instance, the high water
transmitted from the tide in the Atlantic, reaches Uahant between three
nnd four hours after the moon has parsed the meridian. This wave then
divides itself into three; one passing up the British Channel, another
ranging along the west side of Ireland and Scotland, and the third en-
tering the Irish Channel. The first of these flows thrmigh the channel
at about 50 miles as hour, and reaches the Nore about 12 at night The
second moves more rapidly, so as to reach the North of Ireland by six,
and the Orkneys by nine, and the Naze of Norway by 12 ; and in 12
hours more it reaches the Nore, where it meets the morning tide, that
left the mouth of the channel only eight hours before. Thus these two
tides travel round Britain in about 18 hours, in which time the primi-
tive tide has gone round the whoio circumference of the earth and nearly
45 degree* more.
909
T I D
TABLE
(Jfthe time of High Water at rite full and change i-f the Moon, at theprin*
ripalportt chid phics on the coasts of Great Lfit^in and Ireland.
Flare?.
Situation.
Tiaie
PIaces. ,
Situation.
Time
II. M.
U 15
11 45
2 15
G 10
11 15
12 45
3 80
11 16
11 15
9 30
6
2 15
10 45
2 15
11 15
5 15
6 25
5 30
4 30
11
11 15
Jl 6
530
12 45
4 30
11 15
1 30
1 30
12 6
11 30
1036
11 45
10
6
5 15
4 30
330
11 30
5
Scotland
Wales
Wales
Ireland
Scilly Isles
Isle of Man
England
England
-
Scotland
England
Shetland
Ireland
d.-otl-.iud
Ireland
WaTei
England
Wales
England
England
England
England
England
Shetland
North Sea
England
England
England
England
England
Scotland
Wales
Orkney
St Geo. Cha
Scotland
Wales
Ireland
-Wales
England
England
England
Ireland
England
Ireland
a. M.
1-2 45
7 3(1
7 30
I; q
4 40
10 30
10 45
>. 45
10 30
11 15
9 *'t]
3
3 45
11 30
3 45
s b
5 30
U 45
11
10 15
2 15
6
2 45
5 55
7 15
10
3 30
6 45
430
6 45
10
7
2 30
9
9
1030
9
7
9
!S
1030
11 30
4 30
4 25
4 r^-
CO\VPS
T of WMit
\berdovy
Cromartie . ...
Scotluid
Ri. Thames
England
England
'k-otland
Ireland
England
England
Ireland
North Sea
Scotland
Ireland
Scotland
Ireland
England
Enff. Chan.
England
England
England
'England
England
England
Ri, Thames
Ireland
Scotland
Downs
England
Ri. Thames
England
England
England
Wales
England
England
England
Ireland
Ri. Thames
Ireland
Aberistwith
Achill Head
Agnes (St)
Cuckold's Point
Dartmouth
Deal
jAldborongh
A hie River
Amhyick Point
Dee (River)
Dingle Bay
Dover Pier .....".
;)<}\V!'S ...
Arundcl
BaHa
Dublin
u Lights
Duiibar
Baltimore"..
Bam if ...
Dundalk Bay ...
Dundee
Dungarvon
Dungenoj-'s
Eddystone
Exinouth Bar ...
Fal mouth
i'uutry Bay
Barmouth
Barnstaple Bar
Beach y, on Shore
Beach y Offing ...
Beaumaris
Flamboro' Head
Flats (Kentish)
Foreland (N) ...
Foreland (S)
Blyth
Bolt Head . .
Boston
Brassa Sound ...
Bree Bank
Bridgewater
Bridliugton
Galloper
Gal way Bay
Galloway (Mull)
Goodwyn
Britrhton ......
Bristol
Gravesend ......
Gunfleet
Burnt Island ...
Caernarvon Bar
Cairston
Harwich
Hastings
Calf of Man
Cantire (Mull) ~
Cardigan Bar ..
Carlingford
Carmarthen
Chatham
Chester-Bar
Chichester Harb
3lear Cape
Cornwall Cape ...
Cork Harb. Ent.
Helen's (St)
Holyhead Bay ...
H U H ...
H umber R. Ent.
Ives (St)
Kenmare River
Kentish Knock
Kinsale
T 1 D
Places.
Situation.
Time
Places.
Situation.
Time
Land's End
Leith Pier
England
Scotland
H. M.
4 30
2 20
Selsea Harbour
Shannon R Ent'
England
Ireland
H. M.
11 15
3 45
Lewis Islands .
Scotland
G
Sheerness
England
]v>
Liverpool
England
11 8
Shields
England
3
England
2 45
Skerries .
Ireland
4 45 1
Lyine Regis ......
England
6 45
Sligo
Ireland
6 45
England
10 W
Margate Roads
Milford Haven
England
England
11 45
6
Southampton ....
Spithead .....
England
England
11 45
9 30
Moutrose
Mount's Bay
Scotland
England
1 30
4 55
Sunderland
England
Wales
3 O 1
6 :
Ri. Thames
12
Needles
England
England
9 45
4
Scotland
2 ()'
N T ore Light
Ri. Thames
12 30
Tees River ........
England
Eng'land
3 30;
3
Orfordness
Orkney Isles
^entland Frith
England
Scotland
Scotland
England
11
10 30
10 30
4 30
Torbay
TralleeBay
Waterford Harb.
Wexford Harb
England
Ireland
Ireland
Ireland
6 10
3 45
5 30
7 30
^lymouth Sound
} oi tliuid Race
J ortland Road ...
Portsmouth Har
England
England
England
England
5 15
9 15
6 15
11 30
Weymouth .....*
Whitbv
Wicklow
Wisbeach
England
England
Ireland
England
6 30'
3 15"
9
7 30'
lamsgato
Rye Harbour ...
England
England
Ireland
11 20
10 36
5 40
Yarmouth Roads
Yarmouth Sands
Yorkshire Coast
England
England
England
Ireland
8 45
10 30
6
5
Seaford
England
10 1(1
To find the time of high water on a given day at any place where the time
of 7Kgh water at full and change is known.
Let the time of the moon's passing the meridian of the given place be
found in the Nautical Almanack, and to this time apply the correction,
from the following Table, corresponding to her meridian passage and
semidiameter, and to the result add the time of high water at full and
change at the given place, as given in the preceding Table, and the sum
will be the time of high water on the given day. If this sum exceed \2h
24w, or 24 49/, subtract those times from it, and the remainder will b
finding the tints of high. v:ater.
m | Moon's Seraidiameter.
21*1 Moon's Semidiameter.
OD
* s
5m
Sfijgfi
e u , ,,
, ,
i it \ i n
C h
S g 14 NO
15 30 16 30
S g j^S I 14 30
15 SO 1 16 SO
s
f
h m h m
h m h m h m> h ml h m
h m' h m; h ra
4
C4-0 5
!2 O: 6 00 55
1 21 12 S8
30 10
_ ft ) 5
\i 30, 6 30 46
51 58 18 SO
1 17
16 15
13 0; 7 32
03103719
1 30 24
_ 25 25
13 SO 7 30 17
16 1419 30
2 0031
SI 3(5
14 o! 8 00 1
4-0 34-0 9 iO
2 30 38
41 46
14 30 : 8 SO 4-0 8
4-0 154-0 2420 30
3 0044
49 55
15 0. 9 04-0 14
4-0 21(4-0 32
21
3 30 50
400 55
58 1 4 15 30 9 30 4- 16 4- 24 + 36
2l 1216 010 04-0154-0234-034
21 30
22
4300 58
616
!6 3010 304-0 12
4. 19 4- 29
22 SO
5 1
18 19
17 11 04-0 7
4-0 144-0 23
23
5 30 59
7 18
17 SO 1 1 30 4- 2
4-0 7|4-0 1ft
>330
f (5 56
2 12 IS 12 4
04-0 524
Ex. Required the time of high water at London, Sept. 2, 1823, the
time of the moon's transit being 22A. 39w., and her | diameter 16'. 26",
by the Naut. Aim.
h. m.
Moon's transit ~ ~*~~~~~~~~.^* H *~*~~ 2 39
Correction from the above Table ,~~~,~~..,~, 4- 29
High water at full and change by 1st Table ~
Subtract ~~~~~
Time required,*
23 8
2 45
25 53
24 49
TIMBER measuring.
The customary rule for the measurement of timber is erroneous ; for,
according to the common rule, a tree frequently contains one-fourth
more timber than it is estimated at. The following formulae give both
the customary and true content.
Let L = the length of the tree in feet and decimals, and 6 the mean
girth taken in inches ; then
- -
4OIHI
= cubic feet customary.
- = cubic feet tree eontent
310
T I M
If G as well as L be in feet,
.03 L G 2 cubic feet true content.
Sometimes a certain allowance is made in girting- a tree for the thick-
ness of the bark, which is generally one inch to every foot in girt, or
j of the whole girt ; in that case,
L G 2
-r^Tj- = cubic feet customary.
L G 2
= cubic feot true content.
If the tree tapers regularly from one end to the other, take half ths
sum of the girts at the two ends for the mean girt. If the tree do not
taper regularly, but is unequal, being thick in some places and small in
others, it is usual to take several different dimensions, the sum of which
divided by the number of them is accounted the mean girt. But when
the tree is very irregular, it is best to divide it into several lengths, and
to find the content of each separately. That part of a tree, or of the
branches, whose % girt is less than J a foot, is not accounted timber.
TIMBER, on the strength and stress of. See Elastic bodies, equilibrium,
of.
TIME, equation of. See Equation ofTimo.
TIME, various tables relating to.( Vince.)
TABLE L
For ccnverting degrees, minutes, and seconds into sidereal time.
Deg.
Min.
Hou. Min.
Min. Sec.
Deg.
Min.
Hou. Min.
Min. Sec.
Sec.
Dsc. of
Sec.
1
0. 4
30
2.
I
,067
8
0. 8
40
2.40
2
,133
3
0. 12
50
S.
3
,2
i
0. 16
60
4.
4
,S
,01066
1"
,000-28
2
,03333
2
,00056
3
,05
3
,00083
4
,06666
4
,00111
5
,08333
5
,00139
6
,t
6
,00167
7
,11666
7
,00194
8
,13333
8
,00222
9
,15
9
,00-250
10
,16666
10
,00277
20
. ,33333
20
,00556
30
,5
30
,00833
40
,66666
40
,01111
50
,83333
50
,01383
T 1 M
TABLE IV.
Decimal parts of a Degree.
Min.
Dec.
Min.
Dec.
Sec.
Dec.
Sec.
Dec.
!
,01667
31
,51667
1
,00028
31
,00861
2
,03333
32
,53333
2
,00056
32
,00889
3
,05000
33
,55000
3
,00083
33
,00917
4
,06667
34
,56667
4
,00111
34
,00944
5
,08333
35
,58333
5
,00138
35
,00972
6
,10000
36
,60000
6
,00167
36
,01000
7
,11667
37
,61667
7
,00194
37
,010*8
8
,13333
38
,63333
8
,00222
38
,01056
9
,15000
39
,65000
9
,00250
39
,01083
10
,16667
40
,66667
10
,00278
40
,01111
11
,18333
41
,68333
11
,00306
41
,01139
12
,20000
42
,70000
12
,00333
42
,01167
13
,21667
43
,71667
13
,00361
43
,01194
14
,23333
44
,73333
14
,00389
44
jOI22f
15
,25000
45
,75000
15
,00417
45
,01250
16
17
,26667
,28333
46
47
,76667
,78333
1(5
,00444
,00472
46
47
,01278
,01306
18
,30000
48
,80000
18
,00500
48
,01333
W
,31667
49
,81667
19
,005*8
4!>
,01861
20
,33333
50
,83333
20
,00556
50
,01389
21
,35000
51
,85000
21
,00583
51
,01417
22
,36667
52
,86(167
22
,00611
52
,01444
23
,38333
53
,88333
23
,00639
53
,01172
24
,40000
54
,90000
24
,00667
54
,01500
25
,41667
55
,91667
25
,00694
55
,01528
26
,43333
56
,93333
26
,00722
56
,01556
27
,45000
57
,95000
27
,00750
57
,01583
28
,46667
58
,96667
28
,00778
58
,01611
29
,48333
59
,98333
29
,00806
59
,01639
30
,50000
60
1,00000
30
,00833
60
,01667
313
T I M
TABLE V.
Decimal Numbers for each Day in the Year*
D Jan. Feb. jMar.l Apr. May June July Au Sept. jhct. INc
~1 0000 '.0.2C4 0.4-19 0.531 0.016 0.701 0.782 0.867 950
|15 0.039 O.U3 O^OO.CUSS O.S67 0452,0.534 0.019 0.703 0.785 ft870 ( '0.93i
116 0.041 O.lv? O.i03 O.i88, 0.370 0455 0.537 0.6^2 0.706 O/TSS ' .873,0,855
JI7 0.044 0. 1 v9 O.iOC O.i91 0.373 0.458 O..HO 0.6v5 0.709 0.791 876 0.9r,8
!l8 0.046 0. 131 O'i'08 0.^93 0.375 0.400 54j? 0.6>7 0.71 1 0.793 0.878 O.J61 j
19 0.019-0. 134 O.vJ | 2!)6 0.378 0.463 0.545 0.630 0.714 0.7t 6 0.882 0.161
0_052 OJ37 0I4 O.vQtf O.S81 0466 0.548 0.633 0.717 0.799 0.884 OCC6
21 0.056 0. f40 0.2 17 0.302 0.: S3 0.468 0.551 OCK6 7iO 0.802 0.887 O.L(59
,25? 0.057 142 0.210 0.304 386 471 0.553 0.638 722 0.804 0.890 0.971
: ?3 0.060 0. 145 0.22'2 (U07 0.389 0.473 0.556 0.64 1 0.7*5 0.807 0.8P3 0.974
24 O.Oftt 0. 148 0.225 0.509 0.392 476 0.559 0.644 0-7*8 0.810 8l'5 ' !>77
25 0.066 OJ5 02^7 0_312 ( 0.395 0.479 0.562 0.647 0_73I 0.813 O.WJ8 0.98Q:
i(5 O.OflS 153 0.2SO 0.315 0.397 (X482 564 0.649 OTSS 0.815 0.900 0.983
27 071 0.156 0.>33 0.318 0.400 0.485 .67 0.652 0.736 0818 0.903 0.1-85
:8 0.074 0. 159 0.226 0.3-rO 0.403 0.487 0.570 0.655 739 0.821 0.906 88
29 0.077 0.162 0.iS9 0.3-3 0.406 0.4<:0 0.573 657 0.742 0.821 0.909 0.991
.30 0.079 0.241 0.326 M. 408 0.493 575 0.660 0.744 0.826 0.912 0.994
'31 ,0.032, _ 0.244J _ 0.41 ^ _ 10.578 0.663 _ 829, _ 0.997!
: v
T I M
TABLE VI.
For reducing Sidereal to Mean Solar Tims.
Hou.
Mln. Sec.
Mia.
Sec. !
Sec.
Sec.
1
2
3
0. 9, S3
0. 19, 06
0. 29, 49
I
2
3
0,16
0,33
0,49
2
3
0,00
0,01
0,1
4
5
6
0. 39, 32
0. 49, 15
0. 58, 98
4
5
6
0,66
0,82
0,98
4
5
6
"0,01
0,01
0,02
7
8
9
. 8, 81
. 18, 64
. 28, 47
7
8
9
1,15
1,31
1,47
7
8
9
0,02
6,02
0,02
10
11
12
)3~
14
15
. 38, 30
. 48, 13
1. 57, 96
2. 7, 78
2. 17, 61
2. i7, 44
10
11
12
Is"
14
15
1,64
1,80
1,97
2,13
2,?9
2,46
10
11
12
"13
14
15
0,03
0,03
0,03
0,04
0,04
0,04
16
17
18
2. 37, .27
2. 47, 10
2. 56, 93
16
17
18
2,62
2,78
2,95
16
17
18
0,04
0,05
0,05
19
20
21
3. 0, 76
3. 16, 59
3. 26, 42
19
20
30
3,11
3,28
4,91
19
20
30
0,05
0,05
0,08
22
23
24
3. 36, 5
3. 40, 08
3. 55, 91
40
50
CO
6,55
8,19
9,83
40
50
60
0,11
0,14
0,16
RULE. Subtract the numbers found in the table corresponding to the
given sidereal time from that time, and it reduces it to mean solar time.
Ex. Reduce 17A. ]Qm. 23*. sidereal time into mean solar tima.
Ylh 2m 4 ',10*
?9>ff J - JVWJ . JJJ ^- J - JirJ .-.- JJ . J .. J . JJ - J , JJJ1Jt , JJ . rifJ . 3,11
20* ,~~~ 0,05
3* ^^^^ M. 0,01
Mean solar time .
2 50',27
17A 19 23
17 18 32,78
S15
T I M
TABLE VII. For converting Mean Solar into Sidereal Time.
Hou.
Min. Sec.
Min.
Sec.
Sec.
Sec.
2
3
0. 9, 86
0. 19, 71
0, 29, 57
1
2
3
0,16
0,33
0,49
g
3
0,00
0,01
0,01
4
5
6
0. 39, 43
0. 49, 28
0. 59, 14
4
5
6
0,66
0,82
0,99
4
5
6
0,01
0,01
0,02
7
8
9
1. 8, 99
1. 18, 85
1. 28, 71
8
9
1,15
1,31
1,48
7
8
9
0,02
0,02
0,02
10
11
12
1. 38, 56
1. 48, 42
1. 58, 28
10
11
12
1,64
1,82
1,97
10
11
12
0,03
0,03
0,03
13
14
15
2. 8, 13
2. 17, 99
2. 27, 85
13
14
15
2,14
2,30
2,46
13
14
15
0,04
0,04
0,04
16
17
18
2. 37, 70
2. 47, 56
2. 57, 42
16
17
18
2,63
2,79
2,96
16
17
18
0,04,
0,03
0,05-
19
20
21
3. 7, 27
3. 17, 13
3. 26, 98
19
20
30
3,12
3,28
4,93
~19
20
30
0,05
0,05
0,08
22
23
24
3. 36, 84
3. 46, 70
3. 56, 55
40
50
60
6,57
8,21
9,86
40
50
60
0,11
0,14
0,16
RULE. Add the acceleration or the numbers found in the table cor.
responding to the given mean solar time, to that time, and it reduces it
to sidereal time.
The application of this rule is evident, from the last example.
Time, sidereal and mean solar.
Given the hour in mean solar time, to find the sidereal time.
RULE. To the given mean solar time apply the equation of time at the
preceding noon from the Naut. Aim., but with a contrary sign, which
gives the time since the sun's centre was on the meridian ; reduce this
time so corrected to sidereal time, by adding the acceleration from Tab.
VII. ; to which add the sun's R. A. at preceding noon from the Naut. Aim.
Or thus at short
Sid. time mean solar time 4- equation of time at prec. noon 4. ac-
celeration for that hour 4. sun's R. A. at prec. noon.
Hence conversely,
Mean sol. time sid. time sun'i R. A. at prec. noon accalera-
316
T I M
lion for the hour so deduced by Tab. VI. -~ equation of time at prec. noon.
This last rule also gives the time of a star's passage over the meridian
in mean solar time, the star's R. A. being substituted for sid. time.
Ex. Given mean solar time 57*. Wm. 17,4s., Nov. 8, 1827 j to find the
corresponding sidereal time,
d. m. x.
5 19 17,4
Equation of time -f. 16 6,9
5 35 24,3
55,1
By Tab. VII.
5 36 19,4
5 36 19,4
R. A. Sun. N. Aim. 14 51 25,8
55,09
Sid. Time ~~~~~~.~ 20 27 45,2
When the longitude is different from that of Greenwich, a propor-
tional correction must be made for the difference.
If the Naut Aim. is not at hand, sidereal time may be found very
nearly by the following Table, merely adding the sun's mean R. A. in
the table to the time of day where you are. ( Woodhouse.)
Sun's mean R. A.
Hours.
Days.; M. S.
Jan. 6
19
1
3 56
21
20
2
7 53
Feb. 5
21
3
11 49
20
22
4
15 46
Mar. 7
23
5
]9 43
22
6
23 39
Apr. 7
1
' 7
27 '36-
22
2
8
31 32
May 7
3
9
35 29
22
4
10
39 28
June 7
5
11
!;J -22
22
6
1>
47 18
July 7
7
13
51 15
22
8
14
55 12
Aug. 7
15
59 8
22
10
Sept. 6
11
21
12
Oct. 6
13
21
14
Nov. 6
15 *
21
16
D c. 6
17
21
18
Ex. Given as before j to find the
sidereal time.
h. m. .r.
Nov. 6th 15
2d 7 53
Given time .
Sid. time 20 2-7 10,4
Or more accurately by adding the
acceleration 55,1 s, as found in the
last example, to the given time, we
should have sid. time 20. 28. 5,5.
317
T I M
From this same Table and the Table of R. A. of the principal star*
(tee Start, catalogue of), may also be found the time of a star's transit
over the meridian in mean solar time nearly without the aid of the
Naut. Aim. ; the rule being-,
Star's R. A. sun's mean R. A. acceleration = mean solar time at
the time of the star's transit.
To find the time of the moon or a planet's pasting the meridian,.
(Woodhouse.)
Let the increment of sun's R. A. in 24&. be a. ; do. of a planet or the
moon be A : let also the difference between the R. A. of the heavenly
body and that of the sun at the preceding noon, expressed in sidereal
time, be t; then time of a planet's transit =
a- A , /a-A y
'TST^tLlJCv
Or when the planet is retrograde, time =
In the caie of the moon, the time
A-a xA
24 i + V, *
And in the case of a star, the time ==
Time error in, corretponding to any small giren error or variation in
the declination, latitude, or altitude. ( Woodhouse,)
(1) Declination*
Let t be the exact time from noon, 3 = change of declination, i =
variation in the time, then
= & (tan. declination X cot. t tan. lat X cosec.
This formula is used in finding the time from equal altitudes of the
un, when there is a change of declination, in the interval between the
two observations, which there is always, except at the solstice*
(2) Given the error in latitude to find the error in time.
Let A = error in latitude, i = do. in time, then
* .= \ (tan. dec. X cosec. t tan. lat. X cot. t.)
Thia formula is useful at sea; for between the observation which de-
termines the latitude from the sun's meridian altitude, and the observa-
tion of the altitude, the observer, if on board ti ship, may hare changed
Us place, UB if rosy have probably changed kit latitude,
T R A
(3) Given the error in altitude to find the error in time,
Lat a, be the error in altitude, then
sin. azini. X cos. lat. .
Hence for a given error in altitude & is the least when the body is o
the prime vertical, the altitude .*. should be taken near the east or west
points. '
Time of sun's passing the meridian, or the horizontal or vertical wire
of a telescope. ( Vince.)
(1) Let d" = diameter of the sun estimated in seconds of a great cir-
cle 3 then the time of passing the meridian ia
d v ' X sec. declin.
15" , '
The same will do for the moon if d" its diameter.
(2). The time of passing an horizontal wire is
d" , rad.
The same expression must also give the time which the sun takes ia
rising.
If d" 1980" the horizontal refraction, we have the time that refrac-
tion accelerates the rising of the sun
jo.,,, y rad.a
cos. lat. X i-iu. azim.
(3) The time in which the sun would pass the vertical wire of a tele-
scope is
- X .
2490, 2498, 2733, 2741, and 298k In 1639 a transit happened at the ascend,
ing node in November, and the next transits at the same node will be in,
1874, 1882, 2117, 2125, 2360, 2368, 2603, 2611, 2846, and 2851 These tran-
sits are found to happen, by continually adding the periods, and finding
the years when they may be expected, and then computing, for each
time, the shortest geocentric distance of Venus from the sun's centre at
the time of conjunction ; and if it be less than the semidiameter of the
an, there will be a transit.
TRANSIT of a star and planet over the Meridian. Sec Time.
TRANSIT instrument, to bring it into the Meridian. See Tdetoop*
TRAPEZIUM, area of.See Surveying.
TRIANGLE, plant and tpherioal area of.See Surveying and Trifo*
nometry.
T R I
NGMETay. C Woodhouse, Barlow.)
I. PLANE TRIGONOMETRY.
Solution of the cases of right angled triangles.
"Lst a be the base, b the perpendicular, c the hypothenusa, and A, B,
C the angles opposite.
Given. Sought
Solution.
Given.
Sought.
Solution.
c, B
a
a e. cos. B
a, c
b
= ,/.-
b
A
b c. sin. B
A ~ ~ B
B
cos. B .=
c
a, B
b
c
b a. tan. B
c a. sec. B
a,b
c
B
tan.B=i
b t B
a
a - b. cot. B
c
c = 6. cosec. B
Solution of the cases of oblique angled triangles.
Let a, b t c be the sides of the A ; A, B, C the '* opposite to them.
Cote I.
Given two sides and an / opposite to oue of them j or two /'a and a
side ; to find the rest.
Solution. The sides are proportional to the sines of the opposite /'s.
Note. When two sides and an / opposite to one of them art given, the
case is sometimes ambiguous, viz. when the side adjacent is greater than,
the side opposite to the given ^, and that ^ is acute. But in practical
cases there will be found some circumstance or other to remove tlie
.ambiguity.
Case 2.
Given two sides a and b, and included ^ C.
Solution 1st. Tan. -~ - ^LrA Tan. A ^"--' Hence A + B
and A B are known, and consequently A and B.
321
T R I
Solution 2d. Let a be greater than b. Find in the tables an /_ 9 such
The latter method is the most concise in those cases in which the logs.
of a and b are given.
Case 3d.
Given a, b, c to find A, B, C.
Soli
Soli
ution 1st. Let 8 = t+J-., the,, (.to. )' = ,< X S^L^L
ution2d. (Cos. ^y = < X R '^7"' ) .
Scions, (Tan.f)' =HX^^.'
If the ^ sought be less than SO use 1st method.
If __ ____ __ . greater than 90 use 2d method.
The third method may be used in all cases, except when the /_ sought
is nearly 18O>. When the ^ sought i# very small, and great accuracy
is required, a peculiar computation is necessary.
T R I
II. SPHERICAL TaiaoNOMSTXY.
Solution of the six czses of right angled spherical triangfag.
Given. Soug.
axes where the lg^
terms required
are le^s than}
COo.
If c and b nrr> of ^
same affection .A.
If b ;utd A sire of
same affection
B Cos. B ~ cos. b X sin. A If b be less !)(R
ft ''fan. a - sin. b X tan. A If A be less Oflo.
c, A
A,B
|Taii. b - tan. c X cos. A If c and A I>H of
I same affectio
Sin. a sin. c X sin. A If A be acute
P Ci ^iJ^ jlfcand A be of
co^s.'c same affection
Cos. c cos. X cos. b If tf and b are of
Tan. A
tan. a
same, aff'ection
If a be loss than
90".
Cos. c cot. A X cot. B If A and B are
cos. A .of same affection
*' a ~ sin~B" If A be acute
(
If the A, instead of being right angled, is a quadrantal A, the suresfc,
and perhaps the most expeditious method is to take the supplemental or
polar A, and solve it by the above table, taking the supplement? of tha
323 S4
T R I
given sides for the /'s of the polar A, and the supplements of the giren
/'s for the sides.
Solution of the six cases of oblique d spherical A*.
Case 1.
Given the three sides a, b t c to find A.
Section 3d. (Ta, * )' = , X -5^-M^d .
Sometimes one of these methods may be more convenient than ano-
ther, see corresponding case in Plane Trigonometry.
Case 2.
Given A, B, C to find a, &c.
Solution 1st. Let S' = A "*" "*" C , then
So,uon 3^,=, X-T.
S' is greater than 90 and less than 270o, .*. cos. S' is positive, and
whole quantity is positive.
Case 3.
Given a, ft and included / C. Required A and B.
a b
cos. -
Solution 1st. Tan. ^--5 = - - Cot. ~
^ _^_. Cot
sm ^
from whence A -f- B and A B, and consequently A and B may be found,
as also c.
324
T R 1
Solution 2d. But if c be required alone, then it may be thus determined
independently of A and B.
T. sin, (a b.} (sec. fl)_
Case 4.
Given A, B, and included side c. Required a, b and C,
A B
cos. -
Solution 1st. Tan. ^-~ = T+.~K X tan ' T
2 cos. A "T" "
and Tan. *=
From whence a 4. 5 and a 6 and .'. a and b may be found,
Solution 2d. Or C may be determined independently of a and b thus-
Assume (tan. > = "er. sin. e. sin^A^ B , C > , =
/ 2 2
(cos.
(cos. |^) S (sec.
Given ar 6 and A opposite to a j to find the rest.
To find B, sin. B = -^ A sin - b
sin. a
To find c, sin. c = sin. a. ^-X..
sin. A
Case 6.
Given A, B and a opposite to A,
Sin, b is. iin. a. -r-fr ; then C and c as in the last case.
T R I
To find the area of a spherical A.
Let A, B, C be the three angles, then
Area - A + B + C 180. or, if r = radius of the sphere, area =
r X (A-f B 4-C 180).
III. TRIGONOMETRICAL FORMULAE.
1. If s sin. and c cos. of an arc A j the arcs, of which t is the sine*
are comprehended within the two formulae.
2 n T + A, and (2 n + 1) x ~ A, where x =: ISO.
Do., of which s is the sine, are
(2 n -f- 1) a- + A, and (2 n + 2) - A.
Do., of which c is the cosine, are
2 n tr + A and (2 n -f 2) 3- A.
Do., of which c is cosine, are
(2 n + 1) jr A and (2 n -f 1) * + A
in all which cases n may be 0, 1, 2, 3, &c.
2. Sin. (| + A) ^ sin. (| -A).
3. Cos. A. = sin. ^ |- A ) = sin. f ~ 4. A ) .
4. Sin. A. = cos.
_ /I cos. 2 A 2 tan. | A
^ " "~ 2i
' cot. ^ A+ tan.| A cot. A-j- tan. 4 A'
T R I
-' cos.2 Yz A sin.2 ^
Vl 4 cot A
=12 sin. 2 y z A = 2 cos. 8 l / 2 A 1
__ /I 4 cos. 2 A ' : 1 tan.g|A
2 "14- tan.s I A
_ cot. I A tan. | A 1
_ _ ^ __
cot. $ A + tan. A ~ 1 4. tan. Alan A"
sin. A 1
- rp
0. fan. A = - - - -
cos. A cot. A
Vl s in.A
_ 2 tan. | A
" 1 tan 2 1 A'
2 cot. | A
cot* A 1 " cot. J A tan | A
1 cos. 2 A
XT cot. A 2 cot. 2 A ^ -
sin. 2 A
sin, g A __ / 1 cos. 2 A
~~ 1~+ cos. 2 A ~ 1 +~cos. 2 A'
Formulae relating to two arcs.
1. Sin. (A 4. B) sin. A . cos. B + cos. A . sin. B.
2. Sin. (A B) =. sin. A . cos. B cos. A . sin. B.
3. Cos. (A -f B) = cos. A . cos. B sin. A . sin. B.
4. Cos. (A B) cos. A . cos. B -f sin. A . sin. B.
- B '=.r^.A ^B-
- S* 111 - ( A + s ) _ tan. A 4. tan. B _ cot. B -f cot. A
Sin. (A B) tan. A fcm. B ~ "cot. B cot. A*
327
T R I
Cos. (A 4- B ) _ cot. B tan. A
' Cos. (A B) ~ cot. B 4- tau. A ~
Sin. A 4- sin. B _ tan. |_(A 4-B)
* ' Sin. A sin. B ~ tan. | ( A B)'
Cos. B + cos. A __ cot. $ (A -f- B)
~* *
cot. A tan. B
cot. A -f tan. B*
Cos. B cos. A
11. Sin. A . cos. B
12. Cos. A . sin. B
13. Sin. A . sin. B
14. Cos. A . cos. B
15. Sin. A + sin. B
16. Cos. A 4- cos. B
~* tan. (A B)*
- siu. (A + B) + sin. (A ~- B).
= | sin. (A + B) sin, (A B).
= | cos. (A -* B) *- 1 cos. (A -f B).
- % cos. (A + B) + cos. (A * B).
2 sin. & (A -f B) cos. % (A ** B).
= 2 cos. J4 (A -f B) cos. J$ (A ~ B).
19. Sin. A sin. B = 2 sin. ^ (A B) . cos. & (A + B).
20. Cos. B cos. A = 2 sin. & (A B) . sin. K (A 4. B)
82.
.
cos. A. cos.B
- = .
sin. A . sin. B
Sin.t A sin.* 6
7
Cos. 2 B - cos. Aa- J sin " ( A ~ B) ' "^ ( A + B) '
24. Cos. 8 A sin.a B cos. (A & B) . cos. (A + B).
27. Sin. B = sin. (A + B) . cos. A . sin. A . cos. ( A 4. B).
28. Cos. B = sin. (A 4. B) sin. A 4- cos. A . sin. (A 4- B).
Note. To express the formulae to rad. r, multiply each term by that
power of r that will make each term of the same dimensions as that term
>vhich has the highest dimensions.
Expressions for the sines and cosines of multiple area.
1. Cos. (n + 1) A rr 2 cos. n A. cos. A cos/(w 1) A.
2. 2 Cos. mA = (2 cos. A) w ?w (2 cos. A) w " 2 -f- .*" ~
(3 cos. A) _Tll> (sin . A) a + "' ( ""- 3 '-| ( f -"-'
(sin. A) 5 &c. (?n, odd.)
5. Sin. m A cos. A frn. sin. A " ^*^~ ^ (si- A)* -f
m. (wa 4) (ma 16) . . \ .
2. 3 4. 5 (sm ' A) &c - J ( w even -)
fl. Let 2 cos. A x -\ then 2 cos. n A x n + ^ (n any No.)
7. (Cos. A + v'"^l sin. A) m = cos. m A + V"^T sin. m A.
and (cos. A V 1 sin. A) m = cos. A V 1 sin. m A.
whence we have in another form
8. Cos. m A = (cos. A) m S^fesU ( CO s. A) m " 2 * (sin. A) +
g . (sin> 4
and sin. in A m (cos. A) OT " x sin. A -- -(cos. A) m ~
(sin. A)3 &c.
9. Also if e No. whose hyp. log. = 1 we have in terms of the impos-
feible quantity V"H1
c nA V^^-.A vCT e nA>J-\ e -n
Cos. n A -,&sin. n A
2 94/17
E.rpres* ions for the powers of the sinv and cosine of an arc.
\. '/*- 1 (cos. A " co, n A 4. . cos. ( ) A 4-n.
/ 4) A -f &5.
329 T 2
T R I
1 3 5 7 ... n 1 2 *
JVote. If n be sren the last term must alway be ' '-- '-- -
1.2.3...
2
Ext. 2 (cos. A)g = cos. 2 A 4. 1.
2 (cos. A)s cos. 3 A 4- 3 cos. A.
23 (cos. A)4 = cos. 4 A 4- 4 cos. 2 A -J- 3.
24 (cos. A)* = cos. 5 A 4- 5 cos. 3 A 4. 10 cos. A.
25 (cos. A) 6 = cos. 6 A -f 6 cos. 4 A 4. 15 cos. 2 A 4. 10.
&c. &c.
2. 2 n ~ l (sin. A) n 4- cos. n A qp cos. ( 2) . A -f n. n ~ l cos.
(n 4) A &c. where the upper sign must be used when n is 4, 8, 12, &c.
and the lower when n is 2, 6, 10, &c., and in both cases the last term is
as before.
3. 2 71 " (sin. A) n = 4. sin. n A ip sin. ( 2) A + n.
(n 4) A, &c., where the upper sign must be used when n is 1, 5, 9, &c.,
and the lower when n is 3, 7, 11, &c.
Exa. 2 (sin. A) cos. 2 A + 1.
2g (sin. A) 3 = sin. 3 A 4- 3 sin. A.
2s (sin. A)4 = cos. 4 A 4 cos. 2 A 4. 3.
24 (sin. A)5 - sin. 5 A 5 sin. 3 A 4- 10 sin. A,
25 (sin. A)6 - cos. C A 4. 6 cos. 4 A 15 cos. 2 A 4. 10.
Series for the sine and cosine in terms of the arc,
2.
Value of the sine in some of the most simple catet.
Sin. - 0.
Sin - 9 =
sin. iso =
Sin. 180-1. (V"5 l
S;n. 27o - 1
Sin. 300 - 1 .
SSO
V A R
Sin. 36orr
Sin. 450^:
Sin. 54o=
Sin. 60 =
Sin. 720 - 1 ^10 -f 2 V~5-
Sin. Sio = I
Sin. 900 - i.
TWILIGHT. See Refraction.
U, V.
VARIATION and dip of the Magnetic Needle.
TABLE I.
Shewing the variation of the Needle in various parts of the earth, from
Professor Hansteen, of Christiania.
Authority.
Date.
Variation.
Latitude.
Longitude
Place.
Luchtemacher
Bartholin
Lous, sen
Lous, jun
Do
1649
1672
1730
1765
177U
10 30' E
3 35 W
10 37
15 5
17 5
55 41'
12 35 E
Copenhagen.
Bu^sre ...
17S4
18
Wleugel
1817
1718
18 5
5 7 W
59 ''O
18 4 E
Wilcke ..
1703
11 48
toe no ra.
Do
1771
13 4
Do
1786
15 34
Cronstrand ....
Holm
1817
1761
15 36
13 50 W
63 26
10 9 2 E
Dronthtiro.
Berlin
1779
18
Vibe
1786
' 19
331
V A R
Authority.
Date.
Variation
Latitude.
Longitude
Place.
Holm
Hansteen
Billings
Schubert
Mayer
Krafit
1761
1817
1735
1805
17-26
1774
1805
15 15 W
20 3
1 5W
032E
3 15 W
4 50
11
59 55
52 17
5956
10 42 E
104 11 E
SO 19 E
Christiania.
Irkutsk.
Petersburg!!.
Do. ..
Cook
1812
1779
7 16
6 19 E
53 1
158 48
Kamtschatka
Krusenstern ...
Kirch
1805
1717
5 20
10 42 W
52 32
13 21 E
Berlin
Do
1751
14 16
Bernoulli!
Schulzft
1770
1785
16 9
18 3
Bode
1805
18 2
V. Swindea ...
Bellarmatus ...
Pieard
1797
1804
1541
1666
19 40 W
21 13
7 OE
46 12
48 50
6 9E
2 20 E
Genera.
Paris.
Cassini
La Hire
1687
1707
5 12 W
10 10
Maraldi
1720
13
Do
Do
Le Monnier ...
Cotte
1740
1760
1780
1800
1814
15 30
18 30
20 35
22 12
22 34
Kendrick
Harding'
1745
1791
18 OW
27 23
5321
353 41 E
DubUn.
Burrows
Guntcr
Gellibrand
Bond
1580
1622
1634
1 657
11 15 E
5 56|
4 6
W
51 31
0000
London.
Gellibrand
Halley
1H65
1672
1 22^
2 30
Do.
1692
6
17-^3
14 17
Do
1745
17
Do
Do
Do
Do
Do
1745
1746
Ma. 21
Ap.22
May 4
17
17 10
17 10
17 15
17 18
i
Do
14
17 20
Do
16
17 15
DO. :.:::::::::::
De 1R
17 25
Do
1747
17 30
Do
1747
17 40
Do
1748
17 JO
Heberden
Cavendish ......
1773
177*
21 9
21 16
r
V A R
Authority.
Date.
Variation.
Latitude.
Longitude
Place.
Cavendish
Ciilpin
1775
1786
'
21 43
23 17
'
51 31
00 00
London.
Do
1787
23 19
Do
1788
23 32
Do
1789
23 19
Do
1790
23 39
Do
1791
23 36
Do
1792
23 36
Do
1793
23 49
Do
1794
23 56
Do
1795
23 57
24
Do
1797
24 1
Do
1798
24 Q'6
Do
1799
24 1-8
Do
1800
24 3-6
Do
1801
24 4-2
Do
1802
24 67
Do
1803
24 8-g
Do
1804
24 8-4
Do
1805
24 8-8
Do
1809
24 11-0
Do
1814
24 167
Do
Jul.
24 17-9
Do
Au.
24 21-2
Do
Sep.
24 20-5
Martinius
Do
1638
1668
7 39 E
50 W
38 42
350 51
Lisbon.
Ross
1762
17 32
Lowenorn
Ail/out
Mathews
1782
1670
1788
1721
18)7
19 51
2 15W
17 12
5 12 W
41 54
19
1228
71 45
Rome.
Bombay.
Fontenay
Yeates
1690
1817
2 25 W
22 13
113 35
Canton.
Mathews ,
Yeates
WalLis
1722
1817
1766
2 52 W
1 OE
14 10 W
13 15
32 36
79 57
342 57
Madras.
Madeira.
Mudge
Fleurieu
Bli-h
1820
1769
1788
19 59
15 43 W
20 1
28 27
343 45
Teneriffe.
Krusenstern ...
Keeling
Yeates
1803
1609
1817
16 1
21 OW
15
20 10
57 28
Isle of France.
Daunton .....
1614
1 45 W
33 55
18 24
Table Bay.
Caille
1752
19
1804
25 4
Davis
1610
7 13 E
15 55
354 12
St Helena.
Hallev
1677
40
Wallis
1768
12 47 W
Krusenstern...
1806
17 18
V A R
Authority.
Date.
Variation.
Latitude.
Longitude
Place.
Mathews
Yeates
Mathews ........
Yeates ..
1723
1817
17-26
1917
12 20 W
6
431 E
6
12 47
17 6
56
283 UA
Socotra Isl.
P. R., Jamaica
Vancouver
Basil Hall
1795
1821
14 49 E
14 43
4 30 E
33
27 6
287 46
250 14
Valparaiso.
La Per o use ....
Cook
1786
1779
3 10
8 6 E
19 28
204
Broughton
Taxman .........
1796
1643
8 15
7 15 E
21 9
184 55
Xon^atab oo
Cook
Oxley
1777
1817
944
7 47 E
3340
148 21
New Holland.
TABLE II.
Shewing the dip of the Needle in various parts of the earth. (Hansteen.)
Authority.
Date.
Dip.
Latitude.
Longitude
Place.
Lous ..
1773
'
71 45 N
Copenhagen.
Wleugal
1813
71 26
Schubert
1805
67 ON
J.J.STSJWM
Jfjfjffrrf
Irkutsk.
Euler
Kraft
1755
1802
73 SON
76 42
Petersburg]!.
1755
71 45 N
*~~**~
*~~,*.~~
Berlin.
Euler
1769
72 45
Humboldt
1S05
69 53
Richer ..
1071
75 N
Paris.
La Caille
1754
72 15
~~~
~"
Cassini
1791
70 52
Humboldt
1806
69 12
Conn, de terns.
1814
68 36
Norman
1576
71 50 N
Gilbert
1600
72 ON
Ridley
1013
72 30 N
Bond
1676
73 30 N
Whiston
1720
C73 45
{.75 10
Graham
Nairne
1723
1772
C74 42
i74 42
72 19
JJJWJW ,
u-j-JJ-JJ-Jirxu.
London.
Cavendish
1775
72 31
Gilpin .
1786
72 8'1
Do
1787
72 2-5
Do
1788
72 40
Do
1789
71 54-8
Do
1790
71 53 7
Do
1791
71 23-7
V A R
Authority.
Date.
Dip.
Latitude.
Longitude
Place.
Gilpin
Do.
1795
1797
o /
71 11-4
70 59-4
70 55 '4
^
Do
1799
70 52-2
Do
1801
70 35 '6
vwww
*,******+*
London.
Do
1803
70 3-->'0
Do
1805
1821
7021-0
70 3'2
1640
65 40 N
Rome
Humboldt
Abercrombie ..
1806
1775
18 ;
63 48
5 15 N
63 47 N
~~~
Madras.
Madeira
La Caille
1751
1775
43 OS
45 19
Good Hope.
La Caille
d.ok
1754
1775
9 OS
11 25
St. Helena.
Panton
Vancouver
Basil Hall
Cook
1776
1795
1821
H77
4 37 N
44 15 S
38 46
40 51 S
--~-~~
IZZ
Socotra.
Valparaiso.
O \vhyee
Vancouver
Cook
1793
1777
41 24
39 1 S
Tongataboo
Cook
1776
61 52 N
Teneritt'e
1820
58 22
Basil Hall
Humboldt
Do
1821
1805
1799
12 11|N
61 35 N
13 *2 N
2 13 S
40 50 N
13 S
280 15
14 16
281 15
Guayaquil.
Naples.
Quito
Do
1805
64 45 N
44 25 N
8 58
Genoa.
Do
1805
67 ION
Lucern.
The following recent observations on the dip and variation were se-
lected by Mr. Barlow, as being 1 entitled to the greatest credit :
Place.
1
Q
Lat.
Long.
Dip.
Variation
Authority.
Tristan da Acunha
Trinidad
1821
18>1
18,
Do.
Do.
17J*
1811
1816
(80f
18U
\8'1(
Do.
Do.
Do.
Do.
37
20 30
14 51
28 28
32 38
10 25
18 50
51 31
;V2 Si
;5 4)
;t (
72 45
73 (
73 31
74 4*.
12 10
29
23 32
16 16
17 51
3 29
2 20
13 21
12 35
61 50
89 41
61 SO
77 2*
110 48
37 53 S
10 27 S
48 N
58 22 N
03 47 N
67 41 N
C8 36 N
70 34- N
69 53 N
71 20 N
83 43 N
88 26 N
81 30 N
86 4N
38 43 N
12 W
SOW
15 55 W
20 47 W
23 7 W
19 59 W
22 31 W
24 30 W
18 2 W
18 22 W
60 W
113 16 W
82 '2 W
103 46 W
W 46 E
Marry at.
Do.
j Mean of Do.
and Mudge's
3 observations
Humboldt.
Bouvard.
Kater, the dip.
Humboldt
Wleugel.
Parry.
Do.
Do.
Do.
Do.
Tenerifte
Madrid
Paris
London
Berlin
Copenhagen ...,
Davis' Strait..
Regent's Inlet
Baffin's Bay
Possession Bay ....
MelviiU' Island
335
V E L
VARIATION diurnal.
The horizontal needle, besides its annual change in direction, is also
subject to a daily change, amounting at certain seasons of the year to
about 14' or 15'. According to the most recent observations, it appears
that the needle attains its maximum direction eastward about 7 o'clock,
or y 2 past ? in the morning, that it continues moving westward till two
o'clock in the afternoon ; it then returns to the eastward till the even-
ing; it has then again a slight westerly motion, and in the course of the
night, or early in the morning 1 , attains the bearing it had 24 hours before,
or very nearly. It has also been admitted by all observers, that the daily
motion during the summer months is the greatest, and during the win-
ter months the least ; but the particular month in the summer when the
daily change is the greatest, is a little uncertain. Canton and Wargen-
tin make it about July j but Col. Beaufoy found it greater in June and
August than in July.
Table of the mean monthly diurnal variation of the compass from April
1817 to March 1819. By Colonel Beaufoy, at Stanmore Heath.
From Apri
181 ' to
March
Difference
morning,
noon,
Mean
differ.
From April
1817 to
March
Difference
morning,
noon,
Mean
differ.
1819.
evening.
euce.
1819.
evening.
ence.
n. m. r
11 48
n. m. r
846
April
n. e. ^
8 30
October
n. - e. ]
e. m. C
3 18
e. m. C
n. m. r
953
n. m. r
7 10
May
n. - e. i
7 32
Kovem.
e. m. c
2 21
June
n. m. r
n. - e. 3
e. in. c.
11 15
7 50
3 25
December
n. m. r
4 7
n. m. r
1043
n. m. r
5 3
July
n. - e. 5
e. m.6
634
4 9
January
I
August
n. m. r
n. e. )
e. m. C
11 26
834
2 52
February
n. m. r
6 3
n. m. r
9 44
n. m.r
8 22
Septem.
n. e. 4
e. ra. C
7 26
2 18
March
n. e. ^
e. m. C
7 7
1 15 j
VELOCITY angular. See Central Forcet.
VELOCITY paracentric, See Central Force*.
33ft
V E S
VENUS. See Planets, elements of.
VENUS, transit of. See Transit.
VENUS, phases of.( Vince. )
In the case of Mercury, Venus, and Mars, if = exterior / of elonga*
tion, i. e. supplement of the /, which the earth and sun subtend at the
planet, the visible enlightened part '. the whole disc I '. ver. sin. 6 : di-
^ameter.
Hence Mercury and Venus will have the same phases, from their in-
ferior to their superior conjunction, as the moon has from the new to
the full ; and the same from the superior to the inferior conjunction, as
the moon has from the full to the new. Mars will appear gibbous in
quadratures, as the / 9 will then differ considerably from two right B j
and consequently the versed sine from the diameter. For Jupiter, Sa-
turn, and the Georgian, the / never differs enough from two right /s
to make them appear gibbous, so that they always appear to shine with
a full face. In the case of the moon, the / 9 very nearly equals the /
of elongation ; .". the visible enlightened part of the moon varies very
nearly as the ver. sin. of its elongation.
Venus is brightest between its inferior conjunction and its greatest
elongation; and its elongation at that time from the sun = 39. 44'.
Also at that time the visible enlightened part '. whole disc :: 0,53 : 2.
Venus therefore appears a little more than one-fourth illuminated, and
answers to the appearance of the moon when five days old. This situ-
ation happens about 36 days before and after its inferior conjunction.
Mercury is brightest between its greatest elongation and superior
conjunction ; the elongation of Mercury at this time = 22. 18%'.
VEJINIER.
As instruments are now usually constructed, the following is a gene-
ral rule for finding the value of each division on any vernier.
Find the value of each of the divisions or sub-divisions of the limb to
which the vernier is applied. Divide the number of minutes or seconds
thus found by the number of divisions on the vernier, and the quotient
will give the value of the vernier division. Thus suppose each sub-di.
vision of the limb to be 5' or 300", and that the vernier has 20 divisions,
300
then -- - =. 15" = value of the vernier.
VESTA. This planet was discovered by Dr. Olbers, of Bremen,
March 29, 1807. For its elements, &c., see Planets, element* of,
ST7 T 3
WED
VOLCANOES.
Thfltotiilniirober- of Volcanoes knowa is about 205, of which Europe
contains 13 or 14. Of the whole number, it is computed that 107 are in
islands, and 98 on the great continents. The most remarkable are JEtna,
Vesuvius, the Lipari islands, Iceland, Kamschatka, Japan, and so along 1
the eastern coast of Asia and the Indian islands ; Cape Verd, Canary,
and other African islands ; an immense range of them, at least 60 ia
number, running from north to south on the Continent of America, and
Occupying the summits of many of the Andes, as well as the ^Jexican
and Cjilifornjan_ridges ; for a few of the principal of which, see Moun-
tains, height of.
URANUS, or Georgium Sidus.See Planets, elements of.
W.
WATER boiling, temperature of. See Heat,
WATER, expansion of. See Heat.
WATER MILL. Ser~ 1 FOOt.
109 36
~~. 3 ~~. 1
Yard.
594 193
16& , 5}
4 1 Pole.
23760 ~~ 7920
660 220
40 ~~* 1 Furlong.
190080 63360
5230 1760
320 ^^ 8 ^.
~ I Mile.
Als
o,
4 Inches ,
^ 1 Hand.
1 1 4 Feet ~~
~~~-,~~~~-.v~~,~.
1 Cubit.
6 Feet ~
^^^^^.^^
, 1 Fathom.
3 Miles .
^^^^^^, < w W ^-~ ,
w 1 League.
60 Geographical Miles ~~~.
^^< 1 Degree. 9934
Sir G. Shuckburgh's scale 35.99998
General Roy's scale 36.00088
Royal Society's standard S6.001H5
Ramsden's bar 36.00240
SQUARE OR LAND MEASURE.
1 Yard.
30i 1 Pole.
1210 40 1 Rood.
43560 4S40 16!) 4 1 Acre.
For further observations on this measure see Surveying,
WINE MEASURE.
1 Quart.
1 Gallon.
42 1 Tierce.
63 1| 1 Hogshead.
^2 li 1 Punch.
~3 ~~ 2 ~~~ l~w. 1 Pipe.
~ 6 *~~ 4 ~~* 3 *~~ 2 ~~~ I Tun.
This measure is used for wines, brandies, rum, honey, oil, vinegar, Sec.
A cask of rum, which contains from 95 to 1 10 gallons, is usuilly called a
puncheon ; a foreign pipe of wine varies from 110 to 140 gallons.
ALE AND SEER MEASURE.
Quarts.
4 1 Gallon.
1 Firkin.
2 1 Kilderkin.
4 ~~ 2 1 Barrel.
6 3 1| 1 Hogshead.
. 12 ~~ 6 3 2 1 Butt.
By the late Act the old Wine and Ale Gallons are abolished, and the
Imperial standard gallon substituted in their place. This is declared to
contain ten pounds avoirdupoise weight of distilled water weighed in
air at the temperature of 62o of Fahrenheit, the barometer being at SO
inches. From this standard gallon all other measures of capacity, a
well for wine, ale, ber, spirits, &c., as for dry goods not measured by
343
WEI
heap measure, shall be derived and computed. Two of these gallons
make a peck, and 8 such gallons make a bushel, and 8 such bushels a
quarter of corn, or other dry goods not measured by heaped measure.
The above bushel of 8 Imperial gallons is also to be used forcoals,
culm, fish, potatoes, fruit, and all other goods commonly sold by heaped"
measure, which goods are to be heaped up in the form of a cone of at
least six inches in height, the base of tlie"cone being lS l / 2 inches diam-
eter.
J- he Iin P er l al g a jfo" contains 277.274 cubic inches.
The olcf wfneljallon 231 do.
The old corn 268.8 do.
The old ale 282 do.
TABLE OF FACTORS,
For converting old measures into new, and the contrary.
By decimals.
By vulgar fractions
nearly.
Cora
Mea-
Wine
Mea-
Ale
Mea-
Corn.
Mea-
Wine
Mea-
Ale
Mea-
sure.
sure.
sure.
sure.
sure.
sure.
To convert old
measures to new.
.96943
.83311
1.01704
31
32
5
6
CO
59
To convert new
measures to old.
1.03153
1.20032
.98324
32
31
6
5
59
CO
N.B. For reducing- the prices, these numbers must all be reversed.
Ex. Reduce 63 gallons wine meaaure to the equivalent number in Im-
perial measure.
63 X .83311 or 63 X ~ = 52*4 Imperial gallons nearly,
o
DRY OR CORN MEASURE,
Pints.
8 1 Gallon.
16 2 1 Peck.
64 8 4 1 Bushel.
S66 32 16 4 1 Coomb.
512 ~
~. 64~
~ 32^,
^ 8.
2^,
^ 1 Quarter,
t560~
~320~
^160^,
^40
10^,
~ 5^
- 1 Wey.
S120~
W 640~,
~320~
^80.
20 ^
^ 10 w
^2^ 1 Lsir,
343
U
Also,
2 bushels make 1 boll.
3 bushels - 1 sack.
- ~ 1 chaldron of conls at Lor, don.
68 bushels ~~~.~~ i u >. Xewcnstle.
The . .I-.lron u el.hs 28J cwt. ; and the Nowca>;l c!
53 cwt. *
A bushel, water measure, is 5 pecks. 8 chaldrons a keel.
MEASURE ITINERARY.
Mile of .Russia . .
tffe. !
1100 1
1467
1760
2200
2200
2933
3667
4103
Mile of Poland
Yds.
'. 4400
. 5028
of Italy
of England
of Scotland Ireland
Old .league of France
. 567
. 7333
. 7333
of Hungary .
. 8800
Great league do.
TABLE OF MISCELLANEOUS
?4 Shppf of Paper make ^^r/j.^ JWJ ^^^^r^
ARTICLES.
1 quire.
1 ream.
1 printer's ream.
1 bundle.
1 bale.
1 roll of parchment,
1 gross.
1 great gross.
1 great hundred.
1 load.
1 cubic yard.
1 load.
1 load.
1 hide of land
1 sack.
1 firkin.
1 firkin.
20 Quires ff xxx^xxxxxx,rx,.,.,.r^nv-x *
211 Oiiiroa - . rrrrsj ^ ^ WJ ^
2 Reams ^^^^^^^^^
10 ReEmS M-J-jsrfs^jwnrwrr^jwrJfjfMrwwjw*
60 Skins 0^.^-^.,,^,,, j^-j^^j-juj^^jw^,
13 Dozen of any thing w-ww j^rw^x^^
) 9 Gro<=9 r^y^^*^.^ - ^^M.^. - ~ M - i.
6 SCre w/vr-^/irwr^^rxxw^xr^M^j-nrjj-jx-ju-j-J-JVi
500 Bricks ~~
8R4 Pr^Va ^^^^-^^^^^-r-v^^^.
40 Solid feet of hewn timber *~~
50 Solid feet of unhewn timber ^^v^^^
ion Acres xx ^ r ^ trrnfJ - fJa - Jf ^^ fJ - ff **
20 Stone of flour , fff ^ f
56 Pounds of butter ~,~,^~-,^,~,^,-~v~-~
Gt Pounds of soap rfJ . r ^ J .^ J . fU , J . J . J ^. JJ . J . fJX>jr) . rJJ . J .
314
19| Cwt. of lead ~~,_ ~~, 1 fother.
8-i Pounds of tea ,~~,~, ff ^. 1 chest.
168 Pounds of rice ~^~v,~. \ bag.
112 Pounds of raisins r ~~~ f ~. *~~**~* 1 barrel.
FRENCH WEIGHTS % MEASURES.
A. few of the principal old French Measures.
A point ,, ,01-18025 English inches.
A line ~~~ , . ,083315
An inch ^^^^^-.r^,,^ ,~,,-,*,.^ 1,06578
A foot 12,78936
A toise ^ ~ , 6,394665 English feet.
According to Gen. Roy, an English fathom .' a French toise '.' 1000
: 1065,75.
New, or Metre, System.
In the now system the iii^tre is the ten millionth part of the quadrant
of the meridian S.'cSl En^ij^h feet. The Arc, is the square decametre,
tmd the litre the cxibic dedmetre.
Lineal Measure.
Millimetre ^^ ~~
0.03937 English Juchea.
Centimetre ~*~~ f ^
0.39371
Decimetre ~r~~.,*~ f ,~.
r~. S. 937 10
Uletre ^.^^^^^ ,
39.37100
Decametre , *~~*
St3.71000
Hecatometre < ^-^,,^.,,
~~ Sii'37. 10000
Chiliometre , ^-,,.
w SM97K
Myrioinetre w ^,,, ~
JiOOUOfl
Superficial A/t-,'/ r :ti >v.
Are ~, ^ ^.^ .,,~~ lil'.('0!G PJujr. square yard*.
Decare .
Hecutare , ^-^. ..^,, ! ,
Measu re of cnpacit.;-.
Millilitre ~ ~ *~..+~ f . .fClOS Eng. cubic inches
Centilitre - , .610-J8
Decilitre , ^ 6.10280
Litre w ,,. ^,<^, , Cl.Oi-Sivj
W I X
some limitations. Mr Smeaton was led from experiment to conclude
that overshot wheels do most work, when their circumferences move at
the rate of 3 feet in a second, but this determination is also to be under-
stood with some latitude.
3. In an overshot wheel, the machine will be in its greatest perfection ;
when the diameter of the wheel is % of the height of the water above
the lowest point of the wheel.
4. The power of the overshot wheel is greater, caeteris paribus, than
that of the undershot, nearly in the ratio of 13 to 5.
WIND.
Winds may be divided into constant, or those which always blow in the
same direction ; periodical, or those which blow half a year in one direc-
tion, and half a year in the contrary direction, which last are called mon-
soons ; and variable, which are subject to no rules;
I. Constant or Trade Winds.
The trade wind at the Equator blows constantly from the east : from
the Equator to the northern tropic, or cren as far as the parallel 25 or
30, it declines towards the N.E., and ^ c further you recede
from the Equator: and from the Equ; , -ithern tropic, or to
the parallel 25 or 30, it has a S.E. direction. The line however that se-
parates the opposite trade winds is not precisely the Equator, but the
second or third parallel north. To a certain extent also they follow the
course of the sun, reaching a little further into the southern | sphere,
and contracting their limits in the north, when the sun is on the south
side of the Equator ; and making a reverse change when he declines to
the north. In a zone of variable breadth iu the middle of this tract, ralms
and rains prevail, caused probably by the mingling and ascending of the
opposite aerial currents. The phenomenon of the trade winds may be
thus explained. The air towards the poles being denser than that at
the Equator, will continually rus'i towards the Equator ; but as the ve-
locity of the different parts of the earth's surface, from its rotation, in-
creases as you approach the Equator; the air which is rushing from the
north will loot continue upon the same meridian, but it will be left be-
hind ; that is, in respect to the earth's surface, it will have a motion
from the east; and these two motions combined produce a N.E. wind on
the north side of the Equator. And in like manner there must be a S.E.
wind on the south side. The mr which is thus continually moving from
the Poles to the Equator, being rarified when it comes there, ascends to
the top f Hit? atmo-puc-re, tuid thf:i returns bark to the Poles,
W I N
II. Periodical Windt, or Monsoons.
Such would probably be the regular course of the trade wiuda suppos-
ing- the parts between and near the tropics were open sea. But high
lands change or interrupt their regular course. For instance, in the
Indian Ocean the trade wind is curiously modified by the lands which
surround it on the north, east, and west. There, the southern trade wind
blows regularly as it ought to do from the E. and S.E., from 100 S. lati-
tude to the tropic ; but in the space from 10 S. latitude to the Equator,
N.W. winds blow during our winter (from October to April) ; and S.E.
in the other six months, while in the whole space north of the Equator
S.W. winds blow during summer, and N.E, during winter. These
winds are called monsoons. It was observed above, that the regular
trade wind blows in the Indian Ocean from 10 S. latitude to the tropic,
but there is an exception to this in all that part of the Indian Ocean
which lies between Madagascar and Cape Comorin ; for there, between
the months of April and October, the wind blows from the S. W., and in
the contrary direction from October to April. But of both the constant
and periodical winds it may be observed, that they blow only at sea ; at
land the wind is always variable.
Particulars of the Trade Winds, from Robertson. ( Young's Xaitiral
Philosophy.)
1. For 303 on each side of the Equator, there is almost constantly an
easterly wind in the Atlantic and Pacific Oceans : it is called the trade
wind : near the Equator it is due east, further off it WOAVS towards the
Equator, and is N.E. or S.E.
2. Beyond 30<> latitude, the wind is more .uncertain.
3. The monsoons are, perhaps erroneously, deduced from a superior
current in a contrary direction.
4. In the Atlantic, between I0o and 28 N. latitude, about 300 miles
from the coast of Africa, there is a constant N.E. wind.
5. On the American side of the Caribbae Islands the N.E. wind be
comes nearly E.
6. The trade winds extend 3 or 4 further M. and S. on the W. than
on the E. side of the Atlantic.
7. Within -t of the Equator, the wind is always S.E. : it is more E.
towards America, and more S. towards Africa On the coast of BraziJ,
when the sun is far north \vards, the S.E. becomes more S., and the N.E,
snore E., and the reverse when the sun is far southwards.
t>. On the coast cf Guinea, for 1500 miles, from Sierra Lcono to '*'-.
340
\V I N
Thomas, the wind is always S. or S. W. probably from an inclination of
the trade wind towards the land.
9. Between lat. 4 and 10, and between the longitudes of Cape Yerd
and the Cape Verd Islands, there is a track of sea very liable to storms
of thunder and lightning-. It is called the rains. Probably there are op-
posite winds that meet here.
10. In the Indian Ocean, between 10 and 20 S. hititude, the wind is
regularly S.E. From Jane to November, these winds reach to within
2 of the Equator : but from December to May the wind is N. W. between
lat. 3 and 10 near Madagascar, and from 2 to 12 near Sumatra.
11. Between Sumatra and Africa, from 3 S. latitude to the coa=t* on
the N. the monsoons blow N.E. from September to April, and S. W. from
March to October: the wind is steadier, aud the weather fairer, in the
former half year.
12. Between Madagascar and Africa, and thence northwards to the
Equator, from April to October there is a S.S.W. wind, which further
N. becomes W.S.W.
13. East of Sumatra, and as far as Japan, the monsoons are N. aud S.
but not quite so certain as in the Arabian gulf.
14. From New Guinea to Sumatra and Java, the monsoons are more
N.W. and S.E. being on the south of the Equator ; they begin a month or
six weeks later than in the Chinese seas.
15. The changes of tliese winds are attended by calms and storms.
III. Winds variable.
In the temperate zones the direction of the winds is by no means so
regular as between the tropics. In the north temperate zone, however,
they blow most frequently from the S.W., in the south temperate zone,
from the N.W. ; but changing* frequently to all points of the compass,
and in the north temperate zones blowing, particularly during the
spring, from the north-east.
From an average of 10 years of the register kept by order of the Royal
Society, it appears that at London the winds blow in the following
order :
Winds, Days. Winds. Days.
South-west ~~*~~, 112 South-east 32
North-east ,~~~, 58 East ~~+~~* ~ , 26
North-west ~~ 50 South ~~^,,,~ 18
West ; , 53 North , 16
It appears from the same register, that the S.W. wind blows at an
average more frequently than any other wind during every mouth of th 2 a* sin. 2
140
2. The sails of windmills are so constructed as to have different incli-
nations to the plane of their motions at different distances from the axis ;
greatest nearer the centre, and least at their extremities. This is done
in order to make the momentum of the wind nearly the same as all dif-
ferent distances from the centre of motion.
3. Supposing the sail of a windmill to be a plane, inclined to the axis
at an angle 8, the effect of the wind to turn the sail in a plane, at right
angles to its axis, will be the greatest when cos. 6 X sin. 2 is a maxi-
mum, or when cos. 6 = %.
This gives = 54 a 44', and therefore the inclination of the sail to the
plane of its motion, or what is called the angle of u-eaiher, is 35 10'.
This is true only when the sail is at rest or just beginning to niom
352
YEA
When the sail is in motion, and of course near the extremities of the
sail, when it moves faster, tha angle of weather must be less.
Maclaurin makes the weather to vary from 260 34', at the point of the
sail nearest the centre, to 9 at its extremity. Mr Smeaton, however,
by experiment has found the following angles to answer as well as any-
The radius is supposed to be divided into six parts, and ^th reckoning
from the centre is called 1, the extremity being denoted 6.
Angle with Angle with the
No. the axis. plane of motion.
1 72 18
2 71 19
3 72 18 middle.
4 74 16
5 77| 12|
6 83 7 extremity.
4. From Smeaton 's experiments it appears, that a windmill works to
the greatest advantage, when it is so constructed that the velocity of
the sails is to their velocity when they go round without any load, as a
number between 6 and 7 is to 10 ; and also that the load, when the mill
works in this manner, is to the load that would just keep it from mov-
ing, nearly as 8,5 to 10.
5. With different velocities of wind the load that gives the maximum
effect varies nearly as the square of the velocity, and the effect itself as
the cube.
WIRE, time of sun's passing. See Time.
Y
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TABLE II.
LOGARITHMIC
SINES, COSINES, TANGENTS, AND
COTANGENTS,
TO EVERY EVEN MINUTE OF THE QUADRANT.
Note. From this Table may also be found the Logarithmic Secants
and Cosecants; the logarithm of the socunt of any arc being 20 '
log", cosine ; and the logarithmic cosecant ~ VO log-, tine.
1 E: E: 33 jSsxfjjB at sassi S 1
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9.3426139
French metre = in English yards ~+~~
French foot in English feet *~~~~*
French are =* in English acres +~~*~
French gramme imperial Ibs. Troy ,
French gramme in imperial Ibs. a-
VOirdUpOiS .wwrj-jwjwwjju-JLrj-rfrj-jjjmjjjxr-rj
French litre in imperial gallons ,^ ww ,
387
Conttantt.
Log.
AT. cmnp
Log.
Centes. degree =. in sex. degrees ~~~,
.9
9.9542425
0.0457575
minute^ minutes ~~~~
.54
9.7323938
0.2676062
second - seconds ~~ ~
.324
9.5105450
0.4894550
Mean circumference of the earth in
milpa _ r ,.,, , rr rr r . r r . r _.
24856
4 .39343 1 "
5 604."-688
Diameter ^j^jxj^jj^^xjxr^r^^^^j^^jxxx^jj^j
7912
3 8S8''86S
R iniTllTf
RBdiUS Of EqUatOr ^J-J^-J^rfJ-J^^rf^^j-rrs^^jjju-rt
3962.349
3 5979528
O. IU1 t IJi
6.4020472
Semipolar axis ~ ~~,~~~.~ ~
3949.669
3.5965608
6.4034392
Difference
12.680
1.1031193
8.S968807
Dircuraference of the Equator ~ ~~~
24896
4.3961 2C6
5 6038704
Geographical mile, in feet ~ *~~ f ~~~~~
6075.6
3.7835892
0.2164108
24 hours expressed in seconds ~*~~~~+
86400
1.936513",
5.0634863
Biurnal acceleration of stars in mean
solar seconds J w./ 1 K J ^/-/^r^r..u-*'
235.9093
2.3727451
7.6272549
Sidereal day (2'3h. 56>. 4.09*.) in mean
BOlar dayS Wf**t+sr*f*rff*wj,fsf*swrrfffrj
.99726967
9.9C8312C
0.0011874
Solar mean day (-24A. 3m. 56.5551*) in si-
dereal days ^^j-j^fj-j^^j ^^^ ji ff j ^ir, Jff i*
1.00273791
0.0011874
9.9988126
Sidereal revolution of earth in mean so-
lar dayS J-rfrJsr*-r*fr*r*fffr*sfM*JffJftff*j,
365.25636
2.5625978
7.4374022
Tropical revolution of earth in mean
solar days MM* ^, **su * **& ^r
365.24224
2,5625810
7.4374190
Cuhic inch of distilled water in grains
(Bar. 30 in. Fah. Therm 620)
252.458
2.4021891
7 5978109
An ounce of water in cubic inches ~~.
1.73298
0.2387924
9.7612076
Cubic inches in the Imperial gallon ^^,
277.276 J2.4429I24
7.5570876
Length of seconds pendulum at London
39.1393
1.592613C
3.4073870
Force of gravity at London in fret ~~
32.19081 |1.5077225
3.4922778
EXPLANATION AND USE OF THE TABLES,
TABLE I.
1. To find the log. of any given number.
If the given number be under 100, its log. is found in tlip first page of
the Table, immediately opposite to it. Thus log. 66 is 1.819544.
If the No. consist of three figures, find the given number in the column
under N, and opposite to it in the next column, marked at the top, is
the decimal part of the logarithm required, before which put an index,
which is always less by unity than the number of integral figures in the
natural number. Thus log. 448 is 2.651 2"8. If the number should con-
tsist wholly of decimals, the index of the log. is then negative, and it is
indicated by the place occupied by the first figure in the decimal. Thus
the index of the Jog. of .04 is 2 ; of .006 is - 3. But to avoid the con.
fusion that might arise by the addition and subtraction of negative in-
dices, it is customary to take the arithmetical complement or the nega-
tive indices, and to consider these complements as positive; thus 8 is put
as the index of .04 ; 7 as the index of .006.
If the No. consist of four figures, the three first are to be found as be-
fore in the side column under N ; and under the 4th at the top will be
found the logarithm required, to which prefix the index as befoie. Thus
log. 7-218 is 3.858417. If the No. be odd, and /. not contained in the
Table, take the difference of the logs, of the Nos. next greater and less
than the given one; and add % this difference to the less log. Ihus if
log. 7217 were required, we have by Table
Log. 7-218 3.858417
Log. 7216 3.858297
120
the y 2 of which, or CO, added to 3.8582-7 gives 3.858357, the log required.
If the No. consist of 5 figures or more, find the difference between the
logs, answering to the first four figures of the given No., and the next
immediately following ; multiply this difference by the remaining figures
in the given number, strike oft' as many figures from the right band as
there are in the multiplier ; and the remainder added to the log., answer-
ing to, the first 4 figures, will be the log. required nearly. Thus if log.
100176 were required, we have by last case,
Log. 1001 000434
1002 OOOSfiS
434
.". 434 X 7"6 is 32984. From this cut off two figures, and it becomes 329.84
or 3SO nearly. Whence to 000434 add 330 and supply the index, and we
have the required leg. = 5.000764.
2. To find the natural No. corresponding to any given logarithm.
Look in the different columns for the decimal part of the given log. ;
but if you cannot find it exactly, take the next less tabular log., and in a
line with the log. found in the col. on the left marked N, you have three
figures of the number sought, and at the top of the column in which the
log. is, you have one figure more, Avhich annex to the other three. As,
however, the Table contains only the logs, of the even Nos., it should be
observed that if the given log. falls between any two of the tabular logs,
and differs considerably from both j in that case we must find the log. of
the intermediate odd No. as directed above, and compare it with the given
one ; by which means the 4th figure of the No. sought (whether it be
even or odd) may be correctly ascertained. The number of integers is
always one more than the number expressed by the index. Thus the
ZXPLANATIOiT AND USE OTF THE TABLES.
No. answering to 2.993789 is 985.8. If the number be required to a great-
er No. of places than four, find the difference between the given and the
next less log. To this annex on the right hand as many ciphers as there
are figures required above four. Divide the whole by the difference be-
tween the next less and next greater log., and the quotient annexed to
the four figures formerly found will be the natural number required.
Thus required the No. to 6 places answering to the log. 4.C87956. The
nearest less log. than this is 687886 corresponding to which is the No. 4874.
The difference between 687956 and 68788(5 is 70, to this annex 2 ciphers
and it becomes 7000, which being divided by 89, the difference between
the next less and next greater log. gives 79, .'. the number required ia
48747.9.
TABLE II.
1. To find the logarithmic sine, cosine, $c. answering to any given de-
gree or minute.
Find the given degrees at the top of the page, if less than 45, and the
minutes in the left hand column ; opposite to which, and under the word
sine, cosine, &c. is the number required. But if the given degrees be
greater than 45 and less than 90, find them at tli(* bottom, and the re-
quired sine, cosine, &c. will be found above the word sine, cosine, &c.
opposite to the given number of minutes in the right hand column. If
the given arc exceed 90, find the sine, cosine, &c. of its supplement.
Thus the log. sine of 23. 28' is 9.600118; and the cotangent of 55". 57' is
9.829805. If the No. of minutes be odd, and .'. not contained in the
Table, proceed as directed for the odd numbers, Table I.
To find the logarithmic sine, tangent, $c. of an arc expressed in de-
grees, minutes, and seconds.
Find the sine, tangent, &c. corresponding to the given degree and
minute, and also that answering to .the next greater minute; multiply
the difference between them by the given number of seconds, and divide
the product by 60 ; then the quotient added to the sine, tangent, &c. of
the given degree and minute, or subtracted from the cosine, cotangent,
&c. will give the quantity required nearly.
Ex. Required the log. sine of 23. 27' 40".
Log. sin. 23<> 27' 9.599827
23 28 9.600118
Difference 291
which multiplied by 40, and divided by 60, gives 194, and this added to
9.599827 gives the required logarithm 9.600021.
2. To find the degrees and minutes answering to any given logarithmic
tine, tangent, Sfc.
Find the nearest log. to that given in the proper column : if the title
be at the top of the column, you have the number of degrees at the top
of the page, and the minutes in the column on the left hand; but should
the title be at the bottom of the column, you have the degrees at the bot-
tom of the page, and the minutes in the column on the right hand. If
the given log. seems to belong to the odd minutes, proceed as directed
Art. 2. Table I. Thus log. sin. 9.457584 answers to 16. 40'. Log. tan.
10.535401 answers t 73. 45'. But if the seconds in the arc are also re-
quired, we seek in the proper column for the logarithm which is next
less than the given one, when the logs, in the column are increasing;
but next greater, when they are decreasing, and take the degrees and
minutes corresponding to that logarithm for the degrees and minutes in
the required arc. Then to the difference between the logarithm so found
EXPLANATION AND USE OF THE TABLES.
and the given log. we annex two ciphers, and divide the result by
o
of the difference between the next less and next greater log. ; and the
quotient is the seconds to be added to the degrees and minutes before
taken out.
Ex. Required the degrees, minutes, and seconds corresponding to the
log. sin. 9.6 41 357.
The sin. 25". 58'. is 9.641324 which is the log. next less than the given
one. The difference of these two logs, is 33, which by adding two ciphers
becomes 3300, and this divided by of 260, or by 433, gives 8 nearly for
the number of seconds ; .*. required arc is 25. 58'. 8".
When the arc is small, a particular process is necessary as follows :
To find the log. sine of a small arc less than 3.
Add 4.685575 to the common log. of the arc reduced to seconds ; from
the sum subtract one-third of the log. secant less radius of the arc, and
the remainder will be the required log. sine.
To find the log. tangent of a small arc.
Add together the common log. of the arc, reduced to seconds, % of the
log. secant less radius of the arc, and 4. 68.") 575 ; and the sum will be the
required tangent. We have hence the following rules for performing
the reverse operations :
To find a small arc whose log. sine is given.
To % of the log. secant of the arc in the Table, whose log. sine most
nearly corresponds with the given log. sine, add the given log. sine, and
5.314125, and the sum will be the common log. of the seconds in the re-
quired arc.
To find a small arc when its log. tangent is given.
To the log. tangent add 5.314425, and from the sum subtract % of the
log. secant of the arc in the Table, whose tangent most nearly agrees
with the given tangent ; arid the remainder will be the log. of the se-
conds in the required arc.
Ex. 1. Required the log. sine of 1. 28'. 13". or the log. cosine of 880,
lo. 28'. 13" = 5293" log. 3.723702
Constant No 4.685575
8.409277
% log. secant lo. 28' sub. .000047
lo. 28'. 13". log. sine 8.409230
Ex. 2. Required the arc to the log. sine 7.963214.
^ log. sec. 00. 32' 000006
7.963214
Constant No 5.314425
1895" log. 3.277645
Whence the required arc is 31'. 35"
Hence the arc to log. cosine 7.963214 is 89. 28'. 25".
FINIS.
ERRATA.
Page 16. line 3. for Young's read Young.
P. 21. 1. 26. This series is the same as the last, the higher powers of a
beinsr neglected.
P. 22. 1. 7. for ?/ read ?/ ( equation of the centre.)
P. 55. I. 5. for A 4- S As -4- B -f S B* H- C 4- S C -}- c. read A X S A*
4- B X S Ba + C X S Ca -j- c.
P. S'l. 1. 3. for spheriod read spheroid.
P. 88. 1. 1(>. for with read of.
P. 107. March 7, re-id J 1. 1 1 ; April 1 1, read 0. 14 j June 13, read 0. 2L
P. 147. In some copies the Ficrure has been inverted by mistake.
P. 163. 1. 20. for .43424948 read .434^9443.
P. 175. I. 21. for mix. read rnin.
P. 25:?. 1. 26. for 2 g x W * W r ; read 2 g X W s = W c.
P. 273. 1. 11. for Berege read Barege.
P. 302. Art. Thermometer, for Centrigrade read Centigrade.
Durham ; Printed by Francis HwnMe.
JV.27506O
THE UNIVERSITY OF CALIFORNIA LIBRARY