Chapter XXI: Part 21

+-------------------+---------------------------+ | After Christ. | Hundreds of Years. | +-------------------+------+------+------+------+ | | 100 | 200 | 300 | 400 | | | 500 | 600 | 700 | 800 | | | 900 | 1000 | 1100 | 1200 | | | 1300 | 1400 | 1500 | 1600 | | | 1700 | 1800 | 1900 | 2000 | | Years less than | 2100 | 2200 | 2300 | 2400 | | an Hundred. | 2500 | 2600 | 2700 | 2800 | | | 2900 | 3000 | 3100 | 3200 | | | 3300 | 3400 | 3500 | 3600 | | | 3700 | 3800 | 3900 | 4000 | | +------+------+------+------+ | | C | E | G | B A | +----+----+----+----+------+------+------+------+ | 1 | 29 | 57 | 85 | B | D | F | G | | 2 | 30 | 58 | 86 | A | C | E | F | | 3 | 31 | 59 | 87 | G | B | D | E | | 4 | 32 | 60 | 88 | F E | A G | C B | D C | +----+----+----+----+------+------+------+------+ | 5 | 33 | 61 | 89 | D | F | A | B | | 6 | 34 | 62 | 90 | C | E | G | A | | 7 | 35 | 63 | 91 | B | D | F | G | | 8 | 36 | 64 | 92 | A G | C B | C D | F E | +----+----+----+----+------+------+------+------+ | 9 | 37 | 65 | 93 | F | A | C | D | | 10 | 38 | 66 | 94 | E | G | B | C | | 11 | 39 | 67 | 95 | D | F | A | B | | 12 | 40 | 68 | 96 | C B | E D | G F | A G | +----+----+----+----+------+------+------+------+ | 13 | 41 | 69 | 97 | A | C | E | F | | 14 | 42 | 70 | 98 | G | B | D | E | | 15 | 43 | 71 | 99 | F | A | C | D | | 16 | 44 | 72 | | E D | G F | B A | C B | +----+----+----+----+------+------+------+------+ | 17 | 45 | 73 | | C | E | G | A | | 18 | 46 | 74 | | B | D | F | G | | 19 | 47 | 75 | | A | C | E | F | | 20 | 48 | 76 | | G F | B A | D C | E D | +----+----+----+----+------+------+------+------+ | 21 | 49 | 77 | | E | G | B | C | | 22 | 50 | 78 | | D | F | A | B | | 23 | 51 | 79 | | C | E | G | A | | 24 | 52 | 80 | | B A | D C | F E | G F | +----+----+----+----+------+------+------+------+ | 25 | 53 | 81 | | G | B | D | E | | 26 | 54 | 82 | | F | A | C | D | | 27 | 55 | 83 | | E | G | B | C | | 28 | 56 | 84 | | D C | F E | A G | B A | +----+----+----+----+------+------+------+------+

TAB. VI. _Shewing the Days of the Months for both Styles by the
Dominical Letters._

+-------------+----+----+----+----+----+----+----+ | Week Day. | A | B | C | D | E | F | G | +-------------+----+----+----+----+----+----+----+ | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | | 8 | 9 | 10 | 11 | 12 | 13 | 14 | | January 31 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | | October 31 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | | | 29 | 30 | 31 |----|----|----|----| +-------------+----|----|----| 1 | 2 | 3 | 4 | | | 5 | 6 | 7 | 8 | 9 | 10 | 11 | | Feb. 28-29 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | | March 31 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | | Nov. 30 | 26 | 27 | 28 | 29 | 30 | 31 |----| +-------------+----+----+----+----+----+----+ 1 | | | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | | 9 | 10 | 11 | 12 | 13 | 14 | 15 | | April 30 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | | July 31 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | | | 30 | 31 |----|----|----|----|----| +-------------+----|----| 1 | 2 | 3 | 4 | 5 | | | 6 | 7 | 8 | 9 | 10 | 11 | 12 | | | 13 | 14 | 15 | 16 | 17 | 18 | 19 | | August 31 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | | | 27 | 28 | 29 | 30 | 31 |----|----| +-------------+----|----|----|----|----| 1 | 2 | | | 3 | 4 | 5 | 6 | 7 | 8 | 9 | | | 10 | 11 | 12 | 13 | 14 | 15 | 16 | | Septemb. 30 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | | Decemb. 31 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | | | 31 |----|----|----|----|----|----| +-------------+----| 1 | 2 | 3 | 4 | 5 | 6 | | | 7 | 8 | 9 | 10 | 11 | 12 | 13 | | | 14 | 15 | 16 | 17 | 18 | 19 | 20 | | May 31 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | | | 28 | 29 | 30 | 31 |----|----|----| +-------------+----|----|----|----| 1 | 2 | 3 | | | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | | 11 | 12 | 13 | 14 | 15 | 16 | 17 | | June 30 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | | | 25 | 26 | 27 | 28 | 29 | 30 | | +-------------+----+----+----+----+----+----+----+

CHAP. XXII.

_A Description of the Astronomical Machinery serving to explain and
illustrate the foregoing part of this Treatise._

[Sidenote: Fronting the Title Page.

The ORRERY.]

434. The ORRERY. This Machine shews the Motions of the Sun, Mercury, Venus, Earth, and Moon; and occasionally, the superior Planets, Mars, Jupiter, and Saturn may be put on; Jupiter’s four Satellites are moved round him in their proper times by a small Winch; and Saturn has his five Satellites, and his Ring which keeps its parallelism round the Sun; and by a Lamp put in the Sun’s place, the Ring shews all the Phases described in the 204th Article.

[Sidenote: The Sun.

The Ecliptic.]

In the Center, No 1. represents the SUN, supported by it’s Axis inclining almost 8 Degrees from the Axis of the Ecliptic; and turning round in 25-1/4 days on its Axis, of which the North Pole inclines toward the 8th Degree of Pisces in the great Ecliptic (No. 11.) whereon the Months and Days are engraven over the Signs and Degrees in which the Sun appears, as seen from the Earth, on the different days of the year.

[Sidenote: Mercury.]

The nearest Planet (No. 2) to the Sun is _Mercury_, which goes round him in 87 days 23 hours, or 87-23/24 diurnal rotations of the Earth; but has no Motion round its Axis in the Machine, because the time of its diurnal Motion in the Heavens is not known to us.

[Sidenote: Venus.]

The next Planet in order is _Venus_ (No. 3) which performs her annual Course in 224 days 17 hours; and turns round her Axis in 24 days 8 hours, or in 24-1/3 diurnal rotations of the Earth. Her Axis inclines 75 Degrees from the Axis of the Ecliptic, and her North Pole inclines towards the 20th Degree of Aquarius, according to the observations of _Bianchini_. She shews all the Phenomena described from the 30th to the 44th Article in Chap. I.

[Sidenote: The Earth.]

Next without the Orbit of Venus is the _Earth_ (No. 4) which turns round its Axis, to any fixed point at a great distance, in 23 hours 56 minutes 4 seconds of mean solar time (221 & _seq._) but from the Sun to the Sun again in 24 hours of the same time. No. 6 is a sidereal Dial-Plate under the Earth; and No. 7 a solar Dial-Plate on the cover of the Machine. The Index of the former shews sidereal, and of the latter, solar time; and hence, the former Index gains one entire revolution on the latter every year, as 365 solar or natural days contain 366 sidereal days, or apparent revolutions of the Stars. In the time that the Earth makes 365-1/4 diurnal rotations on its Axis, it goes once round the Sun in the Plane of the Ecliptic; and always keeps opposite to a moving Index (No. 10) which shews the Sun’s daily change of place, and also the days of the months.

The Earth is half covered with a black cap for dividing the apparently enlightened half next the Sun, from the other half, which when turned away from him is in the dark. The edge of the cap represents _the Circle bounding Light and Darkness_, and shews at what time the Sun rises and sets to all places throughout the year. The Earth’s Axis inclines 23-1/2 Degrees from the Axis of the Ecliptic, the North Pole inclines toward the beginning of Cancer; and keeps its parallelism throughout its annual Course § 48, 202; so that in Summer the northern parts of the Earth incline towards the Sun, and in the Winter from him: by which means, the different lengths of days and nights, and the cause of the various seasons, are demonstrated to sight.

There is a broad Horizon, to the upper side of which is fixed a Meridian Semi-circle in the North and South Points, graduated on both sides from the Horizon to 90° in the Zenith, or vertical Point. The edge of the Horizon is graduated from the East and West to the South and North Points, and within these Divisions are the Points of the Compass. On the lower side of this thin Horizon Plate stand out four small Wires, to which is fixed a Twilight Circle 18 Degrees from the graduated side of the Horizon all round. This Horizon may be put upon the Earth (when the cap is taken away) and rectified to the Latitude of any place: and then, by a small Wire called _the Solar Ray_, which may be put on so as to proceed directly from the Sun’s Center towards the Earth’s, but to come no farther than almost to touch the Horizon, the beginning of Twilight, time of Sun-rising, with his Amplitude, Meridian Altitude, time of Setting, Amplitude, and end of Twilight, are shewn for every day of the year, at _that_ place to which the Horizon is rectified.

[Sidenote: The Moon.]

The Moon (No. 5) goes round the Earth, from between it and any fixed point at a great distance, in 27 days 7 hours 43 minutes, or through all the Signs and Degrees of her Orbit; which is called _her Periodical Revolution_; but she goes round from the Sun to the Sun again, or from Change to Change, in 29 days 12 hours 45 minutes, which is _her Synodical Revolution_; and in that time she exhibits all the Phases already described § 255.

When the above-mentioned Horizon is rectified to the Latitude of any given place, the times of the Moon’s rising and setting, together with her Amplitude, are shewn to that place as well as the Sun’s; and all the various Phenomena of the Harvest Moon § 273 & _seq._ made obvious to sight.

[Sidenote: The Nodes.]

The Moon’s Orbit (No. 9.) is inclined to the Ecliptic, (No. 11.) one half being above, and the other below it. The Nodes, or Points at 0 and 0 lie in the Plane of the Ecliptic, as described § 317, 318, and shift backward through all it’s Signs and Degrees in 18-2/3 years. The Degrees of the Moon’s Latitude, to the highest at _NL_ (North Latitude) and lowest at _SL_ (South Latitude) are engraven both ways from her Nodes at 0 and 0; and, as the Moon rises and falls in her Orbit according to its inclination, her Latitude and Distance from her Nodes are shewn for every day; having first rectified her Orbit so as to set the Nodes to their proper places in the Ecliptic: and then, as they come about at different, and almost opposite times of the year § 319, and then point towards the Sun, all the Eclipses may be shewn for hundreds of years (without any new rectification) by turning the Machinery backward for time past, or forward for time to come. At 17 Degrees distance from each Node, on both Sides, is engraved a small Sun; and at 12 Degrees distance, a small Moon; which shew the limits of solar and lunar Eclipses § 317: and when, at any change, the Moon falls between either of these Suns and the Node, the Sun will be eclipsed on the day pointed to by the annual Index (No. 10,) and as the Moon has then North or South Latitude, one may easily judge whether that Eclipse will be visible in the Northern or Southern Hemisphere; especially as the Earth’s Axis inclines towards the Sun or from him at that time. And when, at any Full, the Moon falls between either of the little Moon’s and Node, she will be eclipsed, and the annual Index shews the day of that Eclipse. There is a Circle of 29-1/2 equal parts (No. 8.) on the cover of the Machine, on which an Index shews the days of the Moon’s age.

[Sidenote: PLATE IX. Fig. X.]

There are two Semi-circles fixed to an elliptical Ring, which being put like a cap upon the Earth, and the forked part _F_ upon the Moon, shews the Tides as the Earth turns round within them, and they are led round it by the Moon. When the different Places come to the Semi-circle _AaEbB_, they have Tides of Flood; and when they come to the Semicircle _CED_ they have Tides of Ebb § 304, 305; the Index on the hour Circle (No. 7.) shewing the times of these Phenomena.

There is a jointed Wire, of which one end being put into a hole in the upright stem that holds the Earth’s cap, and the Wire laid into a small forked piece which may be occasionally put upon Venus or Mercury, shews the direct and retrograde Motions of these two Planets, with their stationary Times and Places as seen from the Earth.

The whole Machinery is turned by a winch or handle (No. 12,) and is so easily moved that a clock might turn it without any danger of stopping.

To give a Plate of the wheel-work of this Machine, would answer no purpose, because many of the wheels lie so behind others as to hide them from sight in any view whatsoever.

[Sidenote: Another ORRERY.

PLATE VI. Fig. I.]

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.

PLATE III. Fig. III.]

438. The _Improved Celestial Globe_. On the North Pole of the Axis, above the Hour Circle, is fixed an Arch _MKH_ of 23-1/2 Degrees; and at the end _H_ is fixed an upright pin _HG_, which stands directly over the North Pole of the Ecliptic, and perpendicular to that part of the surface of the Globe. On this pin are two moveable Collets at _D_ and _H_, to which are fixed the quadrantal Wires _N_ and _O_, having two little Balls on their ends for the Sun and Moon, as in the Figure. The Collet _D_ is fixed to the circular Plate _F_ whereon the 29-1/2 days of the Moon’s age are engraven, beginning just under the Sun’s Wire _N_; and as this Wire is moved round the Globe, the Plate _F_ turns round with it. These Wires are easily turned if the Screw _G_ be slackened; and when they are set to their proper places, the Screw serves to fix them there so, as in turning the Ball of the Globe, the Wires with the Sun and Moon go round with it; and these two little Balls rise and set at the same times, and on the same points of the Horizon, for the day to which they are rectified, as the Sun and Moon do in the Heavens.

Because the Moon keeps not her course in the Ecliptic (as the Sun appears to do) but has a Declination of 5-1/3 Degrees on each side from it in every Lunation § 317, her Ball may be screwed as many Degrees to either side of the Ecliptic as her Latitude or Declination from the Ecliptic amounts to at any given time; and for this purpose _S_ is a small piece of pasteboard, of which the curved edge _S_ is to be set upon the Globe at right Angles to the Ecliptic, and the dark line over _S_ to stand upright upon it. From this line, on the convex edge, are drawn the 5-1/3 Degrees of the Moon’s Latitude on both sides of the Ecliptic; and when this piece is set upright on the Globe, it’s graduated edge reaches to the Moon on the Wire _O_, by which means she is easily adjusted to her Latitude found by an Ephemeris.

The Horizon is supported by two semicircular Arches, because Pillars would stop the progress of the Balls when they go below the Horizon in an oblique sphere.

[Sidenote: To rectify it.]

_To rectify the Globe._ Elevate the Pole to the Latitude of the Place; then bring the Sun’s place in the Ecliptic for the given day to the brasen Meridian, and set the Hour Index to XII at noon, that is, to the upper XII on the Hour Circle; keeping the Globe in that situation, slacken the Screw _G_, and set the Sun directly over his place on the Meridian; which done, set the Moon’s Wire under the number that expresses her age for that day on the Plate _F_, and she will then stand over her place in the Ecliptic, and shew what Constellation she is in. Lastly, fasten the Screw _G_, and laying the curved edge of the pasteboard _S_ over the Ecliptic below the Moon, adjust the Moon to her Latitude over the graduated edge of the pasteboard; and the Globe will be rectified.

[Sidenote: It’s use.]

Having thus rectified the Globe, turn it round, and observe on what points of the Horizon the Sun and Moon Balls rise and set, for these agree with the points of the Compass on which the Sun and Moon rise and set in the Heavens on the given day; and the Hour Index shews the times of their rising and setting; and likewise the time of the Moon’s passing over the Meridian.

This simple Apparatus shews all the varieties that can happen in the rising and setting of the Sun and Moon; and makes the forementioned Phenomena of the Harvest Moon (Chap. xvi.) plain to the Eye. It is also very useful in reading Lectures on the Globes, because a large company can see this Sun and Moon going round, rising above and setting below the Horizon at different times, according to the seasons of the year; and making their appulses to different fixed Stars. But, in the usual way, where there is only the places of the Sun and Moon in the Ecliptic to keep the Eye upon, they are easily lost sight of, unless covered with Patches.

[Sidenote: The PLANETARY GLOBE.

PL. VIII. Fig. IV.]

439. The _Planetary Globe_. In this Machine, _T_ is a terrestrial Globe fixed on its Axis standing upright on the Pedestal _CDE_, on which is an Hour Circle, having its Index fixed on the Axis, which turns somewhat tightly in the Pedestal, so that the Globe may not be liable to shake; to prevent which, the Pedestal is about two Inches thick, and the Axis goes quite through it, bearing on a shoulder. The Globe is hung in a graduated brasen Meridian, much in the usual way; and the thin Plate _N_, _NE_, _E_, is a moveable Horizon, graduated round the outer edge, for shewing the Bearings and Amplitudes of the Sun, Moon, and Planets. The brasen Meridian is grooved round the outer edge; and in this Groove is a slender Semi-circle of brass, the ends of which are fixed to the Horizon in its North and South Points: this Semi-circle slides in the Groove as the Horizon is moved in rectifying it for different Latitudes. To the middle of the Semi-circle is fixed a Pin which always keeps in the Zenith of the Horizon, and on this Pin the Quadrant of Altitude _q_ turns; the lower end of which, in all Positions, touches the Horizon as it is moved round the same. This Quadrant is divided into 90 Degrees from the Horizon to the zenithal Pin on which it is turned, at 90. The great flat Circle or Plate _AB_ is the Ecliptic, on the outer edge of which, the Signs and Degrees are laid down; and every fifth Degree is drawn through the rest of the surface of this Plate towards its Center. On this Plate are seven Grooves, to which seven little Balls are adjusted by sliding Wires, so that they are easily moved in the Grooves, without danger of starting out of them. The Ball next the terrestrial Globe is the Moon, the next without it is Mercury, the next Venus, the next the Sun, then Mars, then Jupiter, and lastly Saturn; and in order to know them, they are separately stampt with the following Characters; ☽, ☿, ♀, ☉, ♂, ♃, ♄. This Plate or Ecliptic is supported by four strong Wires, having their lower ends fixed into the Pedestal, at _C_, _D_, and _E_, the fourth being hid by the Globe. The Ecliptic is inclined 23-1/2 Degrees to the Pedestal, and is therefore properly inclined to the Axis of the Globe which stands upright on the Pedestal.

[Sidenote: To rectify it.]

_To rectify this Machine._ Set all the planetary Balls to their geocentric places in the Ecliptic for any given time by an Ephemeris: then, set the North Point of the Horizon to the Latitude of your place on the brasen Meridian, and the Quadrant of Altitude to the South Point of the Horizon; which done, turn the Globe with its Furniture till the Quadrant of Altitude comes right against the Sun, _viz._ to his place in the Ecliptic; and keeping it there, set the Hour Index to the XII next the letter _C_; and the Machine will be rectified, not only for the following Problems, but for several others, which the Artist may easily find out.

PROBLEM I.

_To find the Amplitudes, Meridian Altitudes, and times of Rising,
Culminating, and Setting, of the Sun, Moon, and Planets._

[Sidenote: It’s use.]