435. _Another_ ORRERY. In this Machine, which is the simplest I ever
saw, for shewing the diurnal and annual motions of the Earth, together
with the motion of the Moon and her Nodes; _A_ and _B_ are two oblong
square Plates held together by four upright pillars; of which three
appear at _f_, _g_, and _g_2. Under the Plate _A_ is an endless screw on
the Axis of the handle _b_, which works in a wheel fixed on the same
Axis with the double grooved wheel _E_; and on the top of this Axis is
fixed the toothed wheel _i_, which turns the pinion _k_, on the top of
whose Axis is the pinion _k_2 which turns another pinion _b_2, and that
other turns a third, on the Axis _a_2 of which is the Earth _U_ turning
round; this last Axis inclining 23-1/2 Degrees. The supporter _X_2, in
which the Axis of the Earth turns, is fixed to the moveable Plate _C_.
In the fixed Plate _B_, beyond _H_, is fixed the strong wire _d_, on
which hangs the Sun _T_ so as it may turn round the wire. To this Sun is
fixed the wire or solar ray _Z_, which (as the Earth _U_ turns round its
Axis) points to all the places that the Sun passes vertically over,
every day of the year. The Earth is half covered with a black cap _a_,
as in the former Orrery, for dividing the day from the night; and, as
the different places come out from below the edge of the cap, or go in
below it, they shew the times of Sun-rising and setting every day of the
year. This cap is fixed on the wire _b_, which has a forked piece _C_
turning round the wire _d_: and, as the Earth goes round the Sun, it
carries the Cap, Wire, and solar Ray round him; so that the solar Ray
constantly points towards the Earth’s Center.
On the Axis of the pinion _k_ is the pinion _m_, which turns a wheel on
the cock or supporter _n_, and on the Axis of this wheel nearest _n_ is
a pinion (hid from view) under the Plate _C_, which pinion turns a wheel
that carries the Moon _V_ round the Earth _U_; the Moon’s Axis rising
and falling in the socket _W_, which is fixed to the triangular piece
above _Z_; and this piece is fixed to the top of the Axis of the last
mentioned wheel. The socket _W_ is slit on the outermost side; and in
this slit the two pins near _Y_, fixed in the Moon’s Axis, move up and
down; one of them being above the inclined Plane _YX_, and the other
below it. By this mechanism, the Moon _V_ moves round the Earth _T_ in
the inclined Orbit _q_, parallel to the Plane of the Ring _YX_; of which
the Descending Node is at _X_, and the Ascending Node opposite to it,
but hid by the supporter _X_2.
The small wheel _E_ turns the large wheels _D_ and _F_, of equal
diameters, by cat-gut strings crossing between them: and the Axis of
these two wheels are cranked at _G_ and _H_, above the Plate _B_. The
upright stems of these cranks going through the Plate _C_, carry it over
and over the fixed Plate _B_, with a motion which carries the Earth _U_
round the Sun _T_, keeping the Earth’s Axis always parallel to itself;
or still inclining towards the left-hand of the Plate; and shewing the
vicissitudes of seasons, as described in the tenth chapter. As the Earth
goes round the Sun the pinion _k_ goes round the wheel _i_, for the Axis
of _k_ never touches the fixed Plate _B_; but turns on a wire fixed into
the Plate _C_.
On the top of the crank _G_ is an Index _L_, which goes round the Circle
_m_2 in the time that the Earth goes round the Sun; and points to the
days of the months; which, together with the names of the seasons, are
marked in this Circle.
This Index has a small grooved wheel _L_ fixed upon it, round which, and
the Plate _Z_, goes a cat-gut string crossing between them; and by this
means the Moon’s inclined Plane _YX_ with its Nodes is turned backward,
for shewing the times and returns of Eclipses § 319, 320.
The following parts of this machine must be considered as distinct from
those already described.
Towards the right hand, let _S_ be the Earth hung on the wire _e_, which
is fixed into the Plate _B_; and let _O_ be the Moon fixed on the Axis
_M_, and turning round within the cap _P_, in which, and in the Plate
_C_ the crooked wire _Q_ is fixed. On the Axis _M_ is also fixed the
Index _K_, which goes round a Circle _h_2, divided into 29-1/2 equal
parts, which are the days of the Moon’s age: but to avoid confusion in
the scheme, it is only marked with the numeral figures 1 2 3 4, for the
Quarters. As the crank _H_ carries this Moon round the Earth _S_ in the
Orbit _t_, she shews all her Phases by means of the cap _P_ for the
different days of her age, which are shewn by the Index _K_; this Index,
turning just as the Moon _O_ does, demonstrates her turning round her
Axis as she still keeps the same side towards the Earth _S_ § 262.
[Sidenote: PL. VIII.]
At the other end of the Plate _C_, a Moon _N_ goes round an Earth _R_ in
the Orbit _p_; but this Moon’s Axis is stuck fast into the Plate _C_ at
_S_2; so that neither Moon nor Axis can turn round; and as this Moon
goes round her Earth she shews herself all round to it; which proves,
that if the Moon was seen all round from the Earth in a Lunation, she
could not turn round her Axis.
_N. B._ If there were only the two wheels _D_ and _F_, with a cat-gut
string over them, but not crossing between them, the Axis of the Earth
_U_ would keep its parallelism round the Sun _T_, and shew all the
seasons; as I sometimes make these Machines: and the Moon _O_ would go
round the Earth _S_, shewing her Phases as above; as likewise would the
Moon _N_ round the Earth _R_; but then, neither could the diurnal motion
of the Earth _U_ on its Axis be shewn, nor the motion of the Moon _V_
round that Earth.
[Sidenote: The CALCULATOR.]
436. In the year 1746 I contrived a very simple Machine, and described
it’s performance in a small treatise upon the Phenomena of the Harvest
Moon, published in the year 1747. I improved it soon after, by adding
another wheel, and called it _the Calculator_. It may be easily made by
any Gentleman who has a mechanical Genius.
[Sidenote: Fig. I.]
The great flat Ring supported by twelve pillars, and on which the twelve
Signs with their respective Degrees are laid down, is the Ecliptic;
nearly in the center of it is the Sun _S_ supported by the strong
crooked Wire _I_; and from the Sun proceeds a Wire _W_, called _the
Solar Ray_, pointing towards the center of the Earth _E_, which is
furnished with a moveable Horizon _H_, together with a brazen Meridian,
and Quadrant of Altitude. _R_ is a small Ecliptic, whose Plane
co-incides with that of the great one, and has the like Signs and
Degrees marked upon it; and is supported by two Wires _D_ and _D_, which
enter into the Plate _PP_, but may be taken off at pleasure. As the
Earth goes round the Sun, the Signs of this small Circle keep parallel
to themselves, and to those of the great Ecliptic. When it is taken off,
and the solar Ray _W_ drawn farther out, so as almost to touch the
Horizon _H_, or the Quadrant of Altitude, the Horizon being rectified to
any given Latitude, and the Earth turned round its Axis by hand, the
point of the Wire _W_ shews the Sun’s Declination in passing over the
graduated brass Meridian, and his height at any given time upon the
Quadrant of Altitude, together with his Azimuth, or point of Bearing
upon the Horizon at that time; and likewise his Amplitude, and time of
Rising and Setting by the hour Index, for any day of the year that the
annual Index _U_ points to in the Circle of Months below the Sun. _M_ is
a solar Index or Pointer supported by the Wire _L_ which is fixed into
the knob _K_: the use of this Index is to shew the Sun’s place in the
Ecliptic every day in the year; for it goes over the Signs and Degrees
as the Index _U_ goes over the months and days; or rather as they pass
under the Index _U_, in moving the cover plate with the Earth and its
Furniture round the Sun; for the Index _U_ is fixed tight on the
immoveable Axis in the Center of the Machine. _K_ is a knob or handle
for moving the Earth round the Sun, and the Moon round the Earth.
As the Earth is carried round the Sun, its Axis constantly keeps the
same oblique direction, or parallel to itself § 48, 202, shewing thereby
the different lengths of days and nights at different times of the year,
with all the various seasons. And, in one annual revolution of the
Earth, the Moon _M_ goes 12-1/3 times round it from Change to Change,
having an occasional provision for shewing her different Phases. The
lower end of the Moon’s Axis bears by a small friction wheel upon the
inclined Plane _T_, which causes the Moon to rise above and sink below
the Ecliptic _R_ in every Lunation; crossing it in her Nodes, which
shift backward through all the Signs and Degrees of the said Ecliptic,
by the retrograde Motion of the inclined Plane _T_, in 18 years and 225
days. On this Plane the Degrees and Parts of the Moon’s North and South
Latitude are laid down from both the Nodes, one of which, _viz._ the
Descending Node appears at 0, by _DN_ above _B_; the other Node being
hid from Sight on this Plane by the plate _PP_; and from both Nodes, at
proper distances, as in the other Orrery, the limits of Eclipses are
marked, and all the solar and lunar Eclipses are shewn in the same
manner, for any given year, within the limits of 6000, either before or
after the Christian Æra. On the plate that covers the wheel-work, under
the Sun _S_, and round the knob _K_ are Astronomical Tables, by which
the Machine may be rectified to the beginning of any given year within
these limits, in three or four minutes of time; and when once set right,
may be turned backward for 300 years past, or forward for as many to
come, without requiring any new rectification. There is a method for its
adding up the 29th of _February_ every fourth year, and allowing only 28
days to that month for every other three: but all this being performed
by a particular manner of cutting the teeth of the wheels, and dividing
the month circle, too long and intricate to be described here, I shall
only shew how these motions may be performed near enough for common use,
by wheels with grooves and cat-gut strings round them, only here I must
put the Operator in mind that the grooves are to be made sharp (not
round) bottomed to keep the strings from slipping.
The Moon’s Axis moves up and down in the socket _N_ fixed into the bar
_O_ (which carries her round the Earth) as she rises above or sinks
below the Ecliptic; and immediately below the inclined Plane _T_ is a
flat circular plate (between _Y_ and _T_) on which the different
Excentricities of the Moon’s Orbit are laid down; and likewise her mean
Anomaly and elliptic Equation by which her true Place may be very nearly
found at any time. Below this Apogee-plate, which shews the Anomaly,
&_c_. is a Circle _Y_ divided into 29-1/2 equal parts which are the days
of the Moon’s age: and the forked end _A_ of the Index _AB_ (Fig II) may
be put into the Apogee-part of this plate; there being just such another
Index to put into the inclined Plane _T_ at the Ascending Node; and then
the curved points _B_ of these Indexes shew the direct motion of the
Apogee, and retrograde motion of the Nodes through the Ecliptic _R_,
with their Places in it at any given time. As the Moon _M_ goes round
the Earth _E_, she shews her Place every day in the Ecliptic _R_, and
the lower end of her Axis shews her Latitude and distance from her Node
on the inclined Plane _T_, also her distance from her Apogee and
Perigee, together with her mean Anomaly, the then Excentricity of her
Orbit, and her elliptic Equation, all on the Apogee Plate, and the day
of her age in the Circle _Y_ of 29-1/2 equal parts; for every day of the
year pointed out by the annual Index _U_ in the Circle of months.
Having rectified the Machine by the Tables for the beginning of any
year, move the Earth and Moon forward by the knob _K_, until the annual
Index comes to any given day of the month; then stop, and not only all
the above Phenomena may be shewn for that day, but also, by turning the
Earth round its Axis, the Declination, Azimuth, Amplitude, Altitude of
the Moon at any hour, and the times of her Rising and Setting, are shewn
by the Horizon, Quadrant of Altitude, and hour Index. And in moving the
Earth round the Sun, the days of all the New and Full Moons and Eclipses
in any given year are shewn. The Phenomena of the Harvest Moon, and
those of the Tides, by such a cap as that in Plate 9 Fig. 10. put upon
the Earth and Moon, together with the solution of many problems not here
related, are made conspicuous.
[Sidenote: PL. VIII.]
The easiest, though not the best way, that I can instruct any mechanical
person to make the wheel-work of such a machine, is as follows; which is
the way that I made it, before I thought of numbers exact enough to make
it worth the trouble of cutting teeth in the wheels.
[Sidenote: Fig. III.]
Fig. 3d of Plate 8 is a section of this Machine; in which _ABCD_ is a
frame of wood held together by four pillars at the corners, whereof two
appear at _AC_ and _BD_. In the lower Plate _CD_ of this Frame are three
small friction-wheels, at equal distances from each other; two of them
appearing at _e_ and _e_. As the frame is moved round, these wheels run
upon the fixed bottom Plate _EE_ which supports the whole work.
In the Center of this last mentioned Plate is fixed the upright Axis _f_
_FFG_, and on the same Axis is fixed the wheel _HHH_ in which are four
grooves _I_, _X_, _k_, _L_ of different Diameters. In these grooves are
cat-gut strings going also round the separate wheels _M_, _N_, _O_ and
_P_.
The wheel _M_ is fixed on a solid Spindle or Axis, the lower pivot of
which turns at _R_ in the under Plate of the moveable frame _ABCD_; and
on the upper end of this Axis is fixed the Plate _o o_ (which is _PP_,
under the Earth, in Fig. I.) and to this Plate is fixed, at an Angle of
23-1/2 Degrees inclination, the Dial-plate below the Earth _T_; on the
Axis of which, the Index _q_ is turned round by the Earth. This Axis,
together with the Wheel _M_, and Plate _o o_, keep their parallelism in
going round the Sun _S_.
On the Axis of the wheel _M_ is a moveable socket on which the small
wheel _N_ is fixed, and on the upper end of this socket is put on tight
(but so as it may be occasionally turned by hand) the bar _ZZ_ (_viz._
the bar _O_ in Fig. I.) which carries the Moon _m_ round the Earth _T_,
by the Socket _n_, fixed into the bar. As the Moon goes round the Earth
her Axis rises and falls in the Socket _n_; because, on the lower end of
her Axis, which is turned inward, there is a small friction Wheel _s_
running on the inclined Plane _X_ (which is _T_ in Fig. I.) and so
causes the Moon alternately to rise above and sink below the little
Ecliptic _VV_ (_R_ in Fig. I.) in every Lunation.
On the Socket or hollow Axis of the Wheel _N_, there is another Socket
on which the Wheel _O_ is fixed; and the Moon’s inclined Plane _X_ is
put tightly on the upper end of this Socket, not on a square, but on a
round, that it may be occasionally set by hand without wrenching the
Wheel or Axle.
Lastly, on the hollow Axis of the Wheel _O_ is another Socket on which
is fixed the Wheel _P_, and on the upper end of this Socket is put on
tightly the Apogee-plate _Y_, (that immediately below _T_ in Fig. I.)
all these Axles turn in the upper Plate of the moveable frame at _Q_
which Plate is covered with the thin Plate _cc_ (screwed to it) whereon
are the fore-mentioned Tables and month Circle in Fig. I.
The middle part of the thick fixed Wheel _HHH_ is much broader than the
rest of it, and comes out between the Wheels _M_ and _O_ almost to the
Wheel _N_. To adjust the diameters of the grooves of this fixed wheel to
the grooves of the separate Wheels _M_, _N_, _O_ and _P_, so as they may
perform their motions in the proper times, the following method must be
observed.
The Groove of the Wheel _M_, which keeps the parallelism of the Earth’s
Axis, must be precisely of the same Diameter as the lower Groove _I_ of
the fixed Wheel _HHH_; but, when this Groove is so well adjusted as to
shew, that in ever so many annual revolutions of the Earth, its Axis
keeps its parallelism, as may be observed by the solar Ray _W_ (Fig. I.)
always coming precisely to the same Degree of the small Ecliptic _R_ at
the end of every annual revolution, when the Index _M_ points to the
like Degree in the great Ecliptic; then, with the edge of a thin File
give the Groove of the Wheel _M_ a small rub all round; and by that
means, lessening the Diameter of the Groove, perhaps about the 20th part
of a hair’s breadth, it will cause the Earth to shew the precession of
the Equinoxes; which, in many annual revolutions will begin to be
sensible as the Earth’s Axis slowly deviates from its parallelism § 246,
towards the antecedent Signs of the Ecliptic.
The Diameter of the Groove of the Wheel _N_, which carries the Moon
round the Earth, must be to the Diameter of the Groove _X_ as a Lunation
is to a year; that is, as 29-1/2 to 365-1/4.
The Diameter of the Groove of the Wheel _O_, which turns the inclined
Plane _X_ with the Moon’s Nodes backward, must be to the Diameter of the
Groove _k_ as 20 to 18-225/365. And,
Lastly, the Diameter of the Groove of the Wheel _P_, which carries the
Moon’s Apogee forward, must be to the Diameter of the Groove _L_ as 70
to 62.
[Sidenote: PLATE IV.]
But, after all this nice adjustment of the Grooves to the proportional
times of their respective Wheels turning round, and which seems to
promise very well in Theory, there will still be found a necessity of a
farther adjustment by hand; because proper allowance must be made for
the Diameters of the cat-gut strings: and the Grooves must be so
adjusted by hand, as, that in the time the Earth is moved once round the
Sun, the Moon must perform 12 synodical revolutions round the Earth, and
be almost 11 days old in her 13th revolution. The inclined Plane with
its Nodes must go once round backward through all the Signs and Degrees
of the small Ecliptic in 18 annual revolutions of the Earth and 225 days
over. And the Apogee-plate must go once round forward, so as its Index
may go over all the Signs and Degrees of the small Ecliptic in eight
years (or so many annual revolutions of the Earth) and 312 days over.
_N. B._ The string which goes round the Grooves _X_ and _N_ for the
Moon’s Motion must cross between these Wheels; but all the rest of the
strings go in their respective Grooves _IM_, _kO_, and _LP_ without
crossing.
[Sidenote: The COMETARIUM.]
437. The COMETARIUM. This curious Machine shews the Motion of a Comet or
excentric Body moving round the Sun, describing equal Areas in equal
times § 152, and may be so contrived as to shew such a Motion for any
Degree of Excentricity. It was invented by the late Dr. _Desaguliers_.
[Sidenote: Fig. IV.]
The dark elliptical Groove round the letters _abcdefghiklm_ is the Orbit
of the Comet _Y_: this Comet is carried round in the Groove according to
the order of letters, by the Wire _W_, fixed in the Sun _S_, and slides
on the Wire as it approaches nearer to or recedes farther from the Sun,
being nearest of all in the Perihelion _a_, and farthest in the Aphelion
_g_. The Areas _aSb_, _bSc_, _cSd_ &c. or contents of these several
Triangles are all equal; and in every turn of the Winch _N_ the Comet
_Y_ is carried over one of these Areas; consequently in as much time as
it moves, from _f_ to _g_, or from _g_ to _h_, it moves from _m_ to _a_,
or from _a_ to _b_; and so of the rest, being quickest of all at _a_,
and slowest at _g_. Thus, the Comet’s velocity in its Orbit continually
decreases from the Perihelion _a_ to the Aphelion _g_; and increases in
the same proportion from _g_ to _a_.
[Sidenote: PLATE IV.]
The elliptic Orbit is divided into 12 equal Parts or Signs with their
respective Degrees, and so is the Circle _n o p q r s t n_ which
represents a great Circle in the Heavens, and to which all the fixed
Stars in the Comet’s way are referred. Whilst the Comet moves from _f_
to _g_ in its Orbit it appears to move only about 5 Degrees in this
Circle, as is shewn by the small knob on the end of the Wire _W_; but in
as short time as the Comet moves from _m_ to _a_, or from _a_ to _b_,
and it appears to describe the large space _tn_ or _no_ in the Heavens,
either of which spaces contains 120 Degrees or four Signs. Were the
Excentricity of its Orbit greater, the greater still would be the
difference of its Motion, and _vice versâ_.
_ABCDEFGHIKLMA_ is a circular Orbit for shewing the equable Motion of a
Body round the Sun _S_, describing equal Areas _ASB_, _BSC_, &c. in
equal times with those of the Body _Y_ in its elliptical Orbit above
mentioned; but with this difference, that the circular Motion describes
the equal Arcs _AB_, _BC_, &c. in the same equal times that the
elliptical Motion describes the unequal Arcs _ab_, _bc_, &c.
Now, suppose the two Bodies _Y_ and I to start from the Points _a_ and
_A_ at the same moment of time, and each having gone round its
respective Orbit, to arrive at these Points again at the same instant,
the Body _Y_ will be forwarder in its Orbit than the Body I all the way
from _a_ to _g_, and from _A_ to _G_; but I will be forwarder than _Y_
through all the other half of the Orbit; and the difference is equal to
the Equation of the Body _Y_ in its Orbit. At the Points _a_, _A_, and
_g_, _G_, that is, in the Perihelion and Aphelion, they will be equal;
and then the Equation vanishes. This shews why the Equation of a Body
moving in an elliptic Orbit, is added to the mean or supposed circular
Motion from the Perihelion to the Aphelion, and subtracted from the
Aphelion to the Perihelion, in Bodies moving round the Sun, or from the
Perigee to the Apogee, and from the Apogee to the Perigee in the Moon’s
Motion round the Earth, according to the Precepts in the 355th Article;
only we are to consider, that when Motion is turned into Time, it
reverses the titles in the Table of _The Moon’s elliptic Equation_.
[Sidenote: Fig. V.]
This curious Motion is performed in the following manner. _ABC_ is a
wooden bar (in the box containing the wheel-work) above which are the
wheels _D_ and _E_; and below it the elliptic Plates _FF_ and _GG_; each
Plate being fixed on an Axis in one of its Focuses, at _E_ and _K_; and
the Wheel _E_ is fixed on the same Axis with the Plate _FF_. These
Plates have Grooves round their edges precisely of equal Diameters to
one another, and in these Grooves is the cat-gut string _gg_, _gg_
crossing between the Plates at _h_. On _H_, the Axis of the handle or
winch _N_ in Fig. 4th, is an endless screw in Fig. 5, working in the
Wheels _D_ and _E_, whose numbers of teeth being equal, and should be
equal to the number of lines _aS_, _bS_, _cS_, &c. in Fig. 4, they turn
round their Axes in equal times to one another, and to the Motion of the
elliptic Plates. For, the Wheels _D_ and _E_ having equal numbers of
teeth, the Plate _FF_ being fixed on the same Axis with the Wheel _E_,
and the Plate _FF_ turning the equally big Plate _GG_ by a cat-gut
string round them both, they must all go round their Axes in as many
turns of the handle _N_ as either of the Wheels has teeth.
’Tis easy to see, that the end _h_ of the elliptical Plate _FF_ being
farther from its Axis _E_ than the opposite end _i_ is, must describe a
Circle so much the larger in proportion; and therefore move through so
much more space in the same time; and for that reason the end _h_ moves
so much faster than the end _i_, although it goes no sooner round the
Center _E_. But then, the quick-moving end _h_ of the Plate _FF_ leads
about the short end _hK_ of the Plate _GG_ with the same velocity; and
the slow moving end _i_ of the Plate _FF_ coming half round as to _B_,
must then lead the long end _k_ of the Plate _GG_ as slowly about: So
that the elliptical Plate _FF_ and it’s Axis _E_ move uniformly and
equally quick in every part of its revolution; but the elliptical Plate
_GG_, together with its Axis _K_ must move very unequally in different
parts of its revolution; the difference being always inversely as the
distance of any point of the Circumference of _GG_ from its Axis at _K_:
or in other words, to instance in two points, if the distance _Kk_ be
four, five, or six times as great as the distance _Kh_, the Point _h_
will move in that position four, five, or six times as fast as the Point
_k_ does, when the Plate _GG_ has gone half round: and so on for any
other Excentricity or difference of the Distances _Kk_ and _Kh_. The
tooth _i_ on the Plate _FF_ falls in between the two teeth at _k_ on the
Plate _GG_, by which means the revolution of the latter is so adjusted
to that of the former, that they can never vary from one another.
On the top of the Axis of the equally moving Wheel _D_, in Fig. 5th, is
the Sun _S_ in Fig. 4th; which Sun, by the Wire _Z_ fixed to it, carries
the Ball I round the Circle _ABCD_, &c. with an equable Motion according
to the order of the letters: and on the top of the Axis _K_ of the
unequally moving Ellipsis _GG_, in Fig. 5th, is the Sun _S_ in Fig. 4th,
carrying the Ball _Y_ unequably round in the elliptical Groove _a b c
d_, &c. _N.B._ This elliptical Groove must be precisely equal and
similar to the verge of the Plate _GG_, which is also equal to that of
_FF_.
In this manner, Machines may be made to shew the true Motion of the Moon
about the Earth, or of any Planet about the Sun; by making the
elliptical Plates of the same Excentricities, in proportion to the
Radius, as the Orbits of the Planets are whose Motions they represent:
and so, their different Equations in different parts of their Orbits may
be made plain to sight; and clearer Ideas of these Motions and Equations
acquired in half an hour, than could be gained from reading half a day
about such Motions and Equations.
[Sidenote: The improved CELESTIAL GLOBE.