t I  3 3,  I
3 ar 1


multiplying by a a - xx) will be
ddexx-2a abdex+ aabbdd.
dd-t-eee


Whence, laly, from the given  Quantities a, D,  a, andu  e
x may be found by Rules given hereafter; and at that
Interval, or Diflance x or  B C, a right Line drawn pa-
rallel to A D, will cut C D in the Point fought C.
If, inficead of Geometrical Defcriptions, we ufe Equations
to denote the Curve Lines by 5 the Computations will
thereby become as much fhorter and eafier, as the gaining
of thofe Equations can make them. Thus, fuppofe the
Interfection C, of the given Ellipfis A C E, Fig. to. with
the right Line C D given Poflition, fought : To denote
the IEllipfis, take fome known Equation proper to it,
as r X- ex-         y y, where x is indefinitely put for any
Part of the Axis A b, od AB, and y for the Perpendi-
cular b c, or B C, terminated at the Curve; and r and q
are given from  the given Species of the Ellipfis. Since
therefore C D  is given in  Pofition, A D  will be alfo
given, which call a; and B D will be a - x; alfo the
Angle A D C will be given, and thence the Ratio of B D
to BC, which call I to e i and B C (y) will beea-e a -
ex, whofe Square eeaa-2eea x+ eexx, will be
equal to r x -     v.  And thence by Redudion there


2. aa ee x ± rx - aa e e


will arnie x x  =     -     eVe- +r          ,   X
q                    E
aee+º±~e^^ar+e rr *a Add, that tho' aA
r
q                                  d
Curve be denominated by a Geometrical Defcription, or t,
by a Sedion of a Solid, yet thence an Equation may be
obtained, which Ihall' define the Nature of the Curve, and I
confequently all the Difficulties of Problems propofed  t
about it, may be reduced hither. Thus, in the former I
Example, if A B be called x, and B C, y, the third Pro- I
portional B F will be Y   whofe Square, together with
the Square   of B C,    is equal to C F q,    that is,
y +yy=aa3 or y4+xxyy=aaxx. And this
is an Equation, by which every Point C, of the Curve i
A K C, agreeing or correfponding to any Length of the
Bafe (and confequently the Curve it felf) is defined; and
from whence confequently you may obtain the Solutions
of Problems propofed concerning this Curve.
.After the fame Manner almoft, when a Curve is not given
in Specie, but propofed to be determined, you may feign
an Equation at Pleafure, that may contain its general Na-
ture; and aflume this to denote it, as if it was given,
that from its Affumption you fome Way arrive at Equa-
tions, by which the Affumptions may be determined.
What remains of the Dodtrine and Pratilce of Equa-
rions, relates to their Reduajon to the loweit and fimpleft
Terms, the better to come at the Value of the unknown
Q1uantity in the Equation ; and their Geometrical Con-
ftruflion.
For the Reduaion of EO.UATIONS. See REDUCTION
ff Equations.
Extraaion of the Roots of EQUATIONS. See Ex-
,TRACTION of the Roots of Equation.
Conjirnia on of Equations.   See CONSTRUCTION     of
.I qvatiolSs.
iqyUATION of 2ime, in Aftronomy, the Difference be-
tween mean and apparent Time; or the Reducfion of the
apparent unequal Time, or Motion of the Sun, or a
Planet, to Equable and mean Time or Motion.       See
TIME and MOTION.
Time is only meafured by Motion; and as Time, in it
!elf, flows ever equably; to meafure it, fuch a Motion
nmuff be ufed as is equable, or which always proceeds at
the fame Rate.
The Motion of the Sun, is what is commonly ufed for
this Purpofe; as the moft eafy to be obferved: Yet it
wants the great Qualification of a Chronometer. In Effelf,
the Afironomners find that the Sun's apparent Motion is
no Ways equal: That he now and then flackens his
Pace, and afterwards quickens it again.  Confequently
equal Time cannot be meafured thereby. See SUN.
Hence, the Time which the Sun's Motion lhews, call'd
the aPtarent 21me, becomes different from  the true and
equable Time, wIherein all the Celeffial Motions are to be
eflimated, and accounted.
This Inequality "of Time is thus accounted for: The
Natural, or Solar Day is meafured, not, properly, by one


EQU


ire Revolution of the Equinocfial, or  24 Equinodial
ours; but by the Time which pafes while the Plane of
MIeridian pang thro' the Centre of the Sun, does, by
Earth's Converfion round its Axis, return again to the
n's Centre: Which 's the Time between one Mid-day
d the next. See DAY and MERIDIAN.
Now, had the Earth no other Motion but tfiat round
Axis; all the Days would be precisely cq-yi  to each
ler, and to the Time of the Revolution of the Equi-
Etial: But the Cafe is otherwife; for while tile Eirth
turning round its Axis, it is likewife proceeding  forward.
its Orbit. So that when a Meridian has compited a
iole Revolution from the Sun's Centre, its Plane has not
t arrived at the Sun's Centre again: As will appear
Om the figure.
Let the Sun be S, (Iab. Ayfronom. Fig. 50.) and let A B
a Portion of the Ecliptic: Let the Line M D, repre-
at any Meridian, whofe Plane  roduced, pafes thro' the
in when the Earth is in A. LIet the Earth proceed in
Orbit, and in making one Revolution  round its Axis,
it arrive at B.  then, will the Meridian M D be in
e Pofition m d parallel to the former M D; and con-
quently has not yet paffed thro' the Sun, nor have the
habitants under that Meridian; yet had their Mid-day.
ut the Meridian d m, mufif Pill proceed with its angular
Lotion, and defcribe the Angle a B. f e're its Plane can
afs thro' the Sun, See EARTH.
Hence it appears, that the Solar Days are all longer than
Le Time of one Revolution of the Earth round its Axis.
However, were the Planes of all the Meridians perpen-
icular to the Plane of the Earth's Orbit ; and did the
arth proceed with an equal Motion in its Orbit; the
Lngle d B f would be equal to the Angle B S A, and the
Lrches A f and A B be fimilar: Confequently, the Times
,ould be always equal; the Arch A B, and the Angle
I B f, of the fame Quantity; all the Solar Days equal
t each other; and the apparent and real Time agree.
But, as it is, neither of thofe is the Cafe: For the
Earth does not proceed in its Orbit with an equable Mo-
ion, but in its Aphelion, defcribes a lefs Arch, and in its
Perihelion a greater, in the fame Time X beide, that the
)lanes of the Meridians, are not perpendicular to the
Ecliptic, but to the Equator. Confequenrit, the Time of
.he Angular Motion d B f, which is to be added to the
entire Revolution in order to make a whoie Day, is not
always of the fame Quantity.
The fame will be found, if, fetting afide the Confidera-
tion of the real Motion of the Earth, we confider the
apparent Motion of the Sun in lieu thereof i as being
what we meafure Time by.
On this Principle we obferve, that the Day not only
includes the 'lime of one Converfion of the Globe on its
Axis, but is increas'd by fo much as anfwers to that Part
of the Sun's Motion, performed in that Time. For when
that Part of the Equinodial, which, with the Sun, was at
the Meridian yefterday at Noon, is come thither again to
Day ; it is not yet Noon; the Sun not being now at the
Place where he yeflerday was, but gone forward near a
Degree more or lefs. And this Additament above the 14
Equinocfial Hours is upon a double Account unequal.
IL. In that the Sun, by Reafon of his Apogee and Pe-
rigee, does not at all Times of the Year difpatch an
equal Arch of the Ecliptic in one Day; but greater
Arches near the Perigaum, which is about the middle
of December; and letfer nearer the Apogwum, which is
about the middle of 7vne.
a0. In that tho' the Sun fhould always move equably in
the Ecliptic, yet equal Arches of the Ecliptic do not in
all Parts of the Zodiac, anfwer to equal Arches of the
Equator, by which we are to eflimate Time; by Reafon
fome Parts thereof, as the two Solflitial Points, lie nearer
to a parallel Pofition to the Equinodial than others, eigr.
thofe about the Equinoctial Points, where the Ecliptic
and Equinodial interfe~l.  Whereupon an Arch of the
Ecliptic, near the Solflitial Points, anfwers to a greater
Arch of the Equinodial, than an Arch equal thereto
near the Equinodial Points.
- The apparent Motion of the Sun to the Eafi, then,
being unequal; the natural and apparent Days are no
Ways proper to he applied to meafure the Cceleflial Mo-
tions, which have no Dependance on that of the Sun.
And hence Aftronomers have been obliged to invent
other Days for the Ufe of their Calculations: Thofe others
are equal; and a mean between the Ihorted and longeft of
the unequal ones.
Thefi are had by confidering the Number of Hours in
the whole Revolution of the Sun in the Ecliptic, and di.'
viding the whole Time into as many Equal Parts ac
there are Hours, 14 of which conflitute the Day. And
this Redudtion of the Days conflitutes the Eq4zaion Of
natural Days.                                  Confh


EQU




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