EQU
[ 335 I
EQU
fquaring its Parts, and multiplying by a a ■ reduced to this Form.
■ xx) will be
. bd e :
,-bbdd
+ aaee
.aaabdetf+aabbdd.
dd-|- ee Whence, laftly, from the given Quantities a, b, d, and c; X may be found by Rules given hereafter; and at that Interval, or Diftancc 101BC, a right Line drawn pa- rallel to A D, will cut C D in the Point fought C.
If inifcad of Geometrical Defcriptions, we ufe Equations to denote the Curve Lines by; the Computations will thereby become as much (hotter and eafier, as the gaining of thofe Equations can make them. Thus, fuppofe the Interfecf ion C, of the given Ellipfis ACE, Fig. to. with the right Line C D given Pofition, fought : To denote the Ellipfis, take forne known Equation proper to it,
as r x — — ■ x x = y y, where x is indefinitely put for any Part of the Axis A b, or AB, and y for the Perpendi- cular be, or B C, terminated at the Curve; and r and q are given from the given Species of the Ellipfi therefore C D is given in Pofition, A D given, which call a; and B D will be a . Angle ADC will be given, and thence the Ratio of BD toBC, which call i to e; and BC (y) will be = ea — e x, whole Square ceaa->eea.i-j- et«, will be
And thence by Reduction there
entire Revolution of the Equinocfial, or 24 Equinocfial Hours; but by the Time which pafles while the Plane of a Meridian palling thro' the Centre of the Sun, does, by the Earth's Converfion round its Axis, return again to the Sun's Centre : Which is the Time between one Mid-day and the next. See Day and Meridian.
Now, had the Earth no other Motion but that round its Axis; all the Days would be precisely equal to each other, and to the Time of the Revolution of the Equi- nocfial : But the Cafe is otherwife; for while the Earth is turning round its Axis, it is likewife proceeding forward in irs Orbit. So that when a Meridian has compkated a whole Revolution from the Sun's Centre, its Plane has not yet arrived at the Sun's Centre again : As will appear from the Figure.
Let the Sun be S, {7ab. Aftronom. Fig. 50.) ana let A B be a Portion of the Ecliptic : Let the Line M D, tepre- fent any Meridian, whole Plane produced, pafles thro' the Sun when the Earth is in A. Let the Earth proceed in its Orbit, and in making one Revolution .round its Axis, let it arrive at B; then, will the Meridian M D be in the Pofition m d parallel to the former M D; and con-
equal to xx xx.
will arife x x =
aaeeaj-f- r x — ee-|-_r
- e \/ a r -j
Since
alfo the fcquwtly has not yet pafled rhro' the Sun, nor have the Inhabitants under that Meridian, yet had their Mid-day. But the Meridian d m, muft ftill proceed with its angular Motion, and defcribe the Angle d B f e're its Plane can pafs thro' the Sun. See Earth.
Hence it appears, that the Solar Days are all longer than the Time of one Revolution of the Earth round its Axis. However, were the Planes of all the Meridians perpen- dicular to the Plane of the Earth's Orbit; and did the Earth proceed with an equal Motion in its Orbit; the Add that tho' a Angle d B f would be equal to the Angle B S A and the Arches A f and A B be fimilar : Conlequently, the Times would be always equal; the Arch A B, and the Angle d B f, of the fame Quantity; all the Solar Days equal to 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-
■._ A_L„K — J^.-lkor. o lute ft t-.-h mrl in its
Curve be denominated by a Geometrical Defcription, or by a Section of a Solid, yet thence an Equation may be
obtained, which ftiall define the Nature of the Curve, and ^
confequently all the Difficulties of Problems propofed tion, but in its Aphelion, defcribes a lefs Arch, and in its
about it, may be reduced hither. Thus, in the former Perihelion a greater, in the fame Time; befide, that the
Example, if AB be called x, and B C, y, the third Pro- Planes of the Meridians, are not perpendicular to the
• 1 t,t."-h 1. yy uz-c 1. • 1. Ecliptic, but to the Equator. Confequentl 5 , the Time ot
portional B F will be " , whofe Square, Together with jhe ^ r g u i ar Motion d B f, which is to be added to the
the Square of B C, is equal to C F q, that is, entire Revolution in order to make a whole Day, is not
v* , « 1 « j a.- always of the fame Quantity.
J__ Xvn ^.90. nr v* -U v. X v v = a a X X. And this '"llJ"
y 4 -|-a?a?yy— 2.3. x x.
«« ■'" = aa
is an Equation, by which every Point C, of the Curve 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
The fame will be found, 'if, fetting afide the Confidera- tion of the real Motion of the Earth, we conficler toe apparent Motion of the Sun in lieu thereof; as being what we mcafure Time by.
On this Principle we obferve, that the Day not only includes the Time of one Converfion of the Globe on its Axis but is increas'd by fo much as anfwers to that Part
in Specie, but propofed to be determined, you may feign of the Sun's Motion, performed in rhat Time. For when
that Part of the Equinoctial, which, with the Sun, was at
the Meridian ycfterday 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
Dearee more or lefs. And this Additament above the 24 _ & . „. . ,, . i . 1 .1 .. 1
lates to their ReiuBion to the lowcft and (impleft Equinocfial Hours is upon a double Account unequal.
an Equation at Pleafure, that may contain its general Na- tute; and affume this to denote it, as if it was given that from its Affumption you fome Way arrive ar Equa- tions, by which the AlTumptions may be determined.
What remains of the Doctrine and Practice of Equa tions, relates to their Re. '
Terms, the better to come at the Value of the unknown Quantity in the Equation; and their Geometrical Con- RruBion. ■
For the ReiuBion of Equations. See Reduction of Equations.
ExtraBion of the Roots of Equations, traction of the Roots of Equation.
See Ex-
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 Perigxum, which is about the middle of ^December; and letTcr nearer the Apogaium, which is about the middle of June.
2 . In that tho' the Sun ihould always move equably in ConftniBion of Equations. See Construction of the Ecliptic, yet equal Arches of rhe Ecliptic do not in Earations. all Parts of the Zodiac, anfwer to equal Arches of the
Equation of T'illle, in Aftronomy, the Difference be- Equator, by which we are to eftimate Time; by Reafon tween mean and apparent Time; or the Reduflion of the fomc Parts thereof, as the two Solifitial Points, lie nearer apparent unequal Time, or Motion of the Sun, or a to a parallel Pofition to the Equinoctial than others, e.gr. Planet, to Equable and mean Time or Motion. See thofe about the Equinoctial Points, where the Ecliptic Time' and Motion. and . Equinocfial interfect. Whereupon an Arch of the
Time is only meafured by Motion; and as Time, in it Ecliptic, near the Solftitial Points, anfwers ro a greater felf flows ever equably; to meafure it, fuch a Motion Arch of the Equinoctial, than an Arch equal thereto muft be ufcd as is equable, or which always proceeds at near the Equinocfial Points.
the fame Rate. The apparent Motion of the Sun to the Eaif, then,
The Motion of the Sun, is what is commonly ufed for being unequal; the natural and apparent Days are no this Purpofe; as the moil eafy to be obferved : Yet it Ways proper to be applied to mcafure the Cceleftial Mo- wants the great Qualification of a Chronometer. In Erfcft, tions, which have no Dependance on that of the Su» the Aftronomers find that the Sun's apparent Motion is And hence A-ftronomers have been obliged . " no Ways equal : That he now and then ifackens his other Days for the Ufe of their Calculations : Th Pace, and afterwards quickens it again. Confequently are equal; and a mean between the /horteil and longelt c
invent Thofe others
equal Time cannot be meafured thereby. See Sun.
the unequal ones.
Hence the Time which the Sun's Motion ihews, call'd Thefe are had by confidering the Number of Hours in
•' e whole Revolution of the Sun in the Ecliptic, and di-
thc apparent c Eime, becomes different from the true and the whole Revo equable Time, wherein all the Celeftial Motions are to be viding the whole cltimatcd, and accounted.
This Inequality of Time is thus accounted for : The Natural, or Solar Day is meafured, not, properly, by one
Time into as many Equal Parts as
there"arc Hours, 24 of which conftitute the Day : And
this Reduclion of the Days conftitutes the Equation of
natural 2)ays.
Come-
