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.
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.
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.