Definitions.

I. PArallax, that which I intend [...]o speak of, and to find, is only the Angle a, c, b, contained between Two Lines, the one a, c, drawn from the Center of the Earth, the O­ther b, c. from the Superficies thereof, to the Center of a Comet or other Phaenomenon, as in the Figure follow­ing.

But in an other sense, it is taken for an Arch of the Eight Sphereh, compre­hended between the Lines a, c, and b, c, continued forth, and is called the Di­versity of Aspects, such is the Arch d, e, in the Figure.

Note that the Parallax maketh the Phaenomenon appear lower then indeed it is, and the higher that a Phaenome­non is from the Horizon, the lesser is the Parallax thereof, and contra.

Azimuth.

II. I need not here define the A­zimuth to any, who knoweth the use of the Globe, yet I will Note how it is accounted in this case. Azimuth is an Arch of the Horizon contained be­tween the North or South points of the Horizon, where the Meridian cutteth it, and a Vertical Circle, which falleth on the Horizon from the Zenith through the Center of any Starr.

Altitude.

III. The Altitude is the Arch of a Vertical Circle, accounted in the Sphere on the Phaenomenon, contain­ed between the Visible Horizon, and the apparen [...] place of the Starr, such is the Arch, b, e, in the Figure before; But the Parallax or Difference of As­pect, is the Arch e, d, in the Eighth Sphere, and measureth the Angle d, c, e, and the Vertical thereof, a, c, b.

IV. The Distance from the Ze­nith is the Complement of the Alti­tude before mentioned, and is the Arch c, g.

V. In the next place it will be convenient to make an Instrument, wherewith to observe, that may take the Altitude and the Azimuth both to­gether, wherein every man is left to his own Devise; yet I will pass my opinion thus; First let a round Stone polished (be it White Marble or the like) be setled Horizontally, this may be 2½ Foot Diameter, and on the Cen­ter [Page 4]

draw a Circle, and draw thereon al­so a Meridian Line, and cross it in the Center at right Angles with a Line of [Page 5]East and West, commonly called a Prime Vertical, divide each Quadrant thereof into 90 equall parts, and fi­gure them from each end of the Meri­dian line, with 10, 20, 30, &c. to 90; And let it have a round Pin or Cylin­der of Iron fixed Perpendicular in the Center; Also let a bearer for the Qua­drant be made of convenient hight, with a hole in the lover end, to slip upon the Pin or Cylinder fastened in the Center of the Stone, and let the bearer also have a cross foot to stand up­on the Stone to keep it Perpendicular thereto, one barre of the cross foot is to be continued out to the devided limb of the Stone, in form of a Diopter, running through the Center of the Stone, to shew the Azimuth. Upon the bearer fasten a Quadrant in the Center of Gravity of the Quadrant, and this Quadrant may be about 2½ foot Semi­diameter (but the larger the better) with a broad limb to be devided into Sexagenarie or centesimal minutes with Diagonal lines, the Quadrant may have 2 Pin in the Center, a, whereon to hang a Thred & a Plummet, the Thred may be a small Steel Cittern string, and it will bear a round Plummet or Bullet of about two pound weight.

To find the Altitude of the Pole by Observation.

VI. Because in this case the Alti­tude of the Pole is necessary to be known to the nearest, and that I sup­pose it cannot be found near enough by help of the Sun's Meridian Altitude, and Tables of his Declination, there­fore I thought meet to shew how it may be done to the nearest at the place where the Instrument is to be placed, thus; Take two Meridian Altitudes (viz. the highest and the lowest) of some one of the Northern Stars, which setteth not, neither riseth to the Zenith (such may be some in the Tail of the Great Bear, or one of the Stars in Cas­siopea, for these may be observed in December, near six in the Morning; and six at Night) Subtract the lowest Altitude from the highest, and take half the difference, and adde it to the lowest Altitude to give the height of the Pole.

But in takeing the Altitude of any fixed Starr that is less then 20 gs. high, take the Refraction from the Altitude observed, to leave the true height, for the Refraction maketh the Starr to seem higher then indeed it is, as in Ticho Brahe's Table of the Refraction of fix­ed Starrs here following.

Proportion between the Two Arches of Parallax.

VII. Seeing that in taking Pa­rallax there is alwaies two observations to be made, and the summe or differ­rence [Page 8]of the Two Parallaxes is expected thereby, it is therefore necessary, whe­ther I have their summe or difference, to know what is the Proportion be­tween them, that thereby they may be found severally; Concerning this, read Dr. John Dee's Third Theorem of his Parallaticus Nucleus which is.

In whatsoever two divers Parallaxes of the same Starr, or the like Phaeno­menon (so that in the mean time it be conceived to be carried only with the Diurnal motion of the whole) there will be the same reason, or proportion of the right sine of the greater Paral­lax, to the right sine of the lesser, that is, distance from the Vertex, to the right sine of the lesser apparent distance, from the Vertex.

This is as plainly demonstrated by the Authour, as it is spoken.

To Separate the Arches of Parallax, when the summe of them is given.

VIII. If the summe of Two Ar­ches of Parallax be given in one Arch, then they may be separated, by help of the Sixt Proposition of Clavius's Tri­angula Rectilinea, thus; Take the na­tural sines of the two distances of the Starr from the Zenith (for they are in the same Proportion with the sines of the Two Parallaxes, as in the last Note) Adde these two sines together, and take half their summe; Also take the fine of the lesser distance from the Ze­nith, from the said half summe to have their difference, and then the proporti­on is,

As the said half summe of the sines of the distance from the Zenith, Is to the Tangent of the half summe of Parallax So is the difference aforesaid To the Tangent of another Arch.

Which Arch being added to the half summe of the Parallaxes, giveth the greater Parallax, and being subtracted from the half summe of the Parallaxes, leaveth the lesser Parallax.

But if the Difference of Two Parallaxes be given.

IX. Then seeing the sines of the Distances from the Zenith of the Start, is the termes of Proportion between the two unknown Parallaxes thereof, as by he 7th. Note, thereforee take the natural sines of the said distances, and subtract the lesser sine from the great­er, and take half the difference for the First in the Rule of Three; also adde the same half difference to the lesser sine of distance from the Zenith, for the Third in the Rule of Three, and then the Proportion is,

As the half difference of the Two sines from the Zenith, To the Tangent of the half difference of the Parallaxes, So is the Aggregate of the half differ­rence [Page 11]of the sines aforesaid, and the lesser sine of distance from the Zenith, To the Tangent of an other Arch.

To which Arch adde half the differ­ence of Parallaxes, it giveth the great­er Arch of Parallax; But the half dif­ference of Parallax subtracted from the same Arch leaveth the lesser Arch of Parallax. See Clavius's 7th. Prop. of his Triangula Rectilinea.

Situation of a Comet.

X. Concerning the Situation of a Comet, it may be so near the Pole, that it shall not set at the North, nor rise so high as the Zenith; In this case to get the summe of the Parallaxes, subtract the lesser Meridional distance from the Pole (which is alwaies that which is at his greatest Altitude) from the great­est Meridional distance from the Pole (to be found at his lesser Altitude) and the remaine is the summe of the Paral­laxes. See the Demonstration hereof, in the Tenth Problem of Mr. Thomas Diggs his Ala, sen Scala Mathematica.

Now having the summe of the Parallaxes, they may be separated and known by the 8th. Note before.

XI. But if the Phaenomenon rise to the Zenith, then the Complement of the Latitude taken from the Meri­dional distance of the same, from or be­low the Pole, is the Parallax. But if the Phaenomenon doe not set at the North, and yet cometh up beyond the Zenith toward the South, then the difference of the several Meridional distances from the Pole, is the difference of Parallaxes. Corollarie the Second of the same Tenth Problem of Mr. Tho. Diggs, his Ala, &c. And then the several Parallaxes may be found by the 9th. Note before going.

XII. Moreover, seeing a Starr that setteth not at the North, nor riseth to the Zenith, Two Meridian Altitudes thereof (the higher and the lower) cannot be observed in one night except the right Ascension thereof, and the right Ascension of the sunne doe dif­fer near about 90 Degrees, therefore it [Page 13]

will be necessary to take two Altitudes thereof, viz. the highest and lowest, in one and the same Azimuth Circle, as in the Figure above, wherein a, b, t, is the Merid. r, c, t, the Horizon, a, the Zenith b, the Pole of the World, a, c, the Azim. given, in which h, g, do shew the ap­parent places of the Starr, now the two parallels i, o, of true motion, and k, n, being drawn, draw also from the Pole b, a [Page 14]great Circle b, d, r, to cut the Azi­muth a, c, at right Angles in d, and there is a right Angled-Triangle a, b, d, to be resolved, wherein the Angle of Azimuth b, a, d, is given, and the Angle a, d, b, right, together with the side a, b, the complement of Lati­tude, to find the side a, d; the Pro­portion is,

As the Cosine of the Angle d, a, b, To Radius, So the Tangent of Latitude, Compt. of a, b, To the Tangent of the side a, d.

And for as much as, Mr. Thomas Diggs, in the Eleaventh Problem of his Ala, &c. sheweth that the Arches e, h, and s, f, are equal, and that the Arch h, d, is equal to the Arch d, s, Therefore the practice is this, Take a, h, the distance from the Zenith at the highest Observation, from a, d, the side of the Triangle found, then h, d, = to d, s, added to c, g, the lowest Alti­tude, and the summe subtracted from d, c, the Complement of the side a, d, leaveth s, g, the summe or Aggregate of the Two Parallaxes. And these may be separated by the Eighth Note.

If the Comet doth Rise and Set.

XIII. Hitherto of a Comet that Setteth not, but such as doe Rise and Set, come not to the North part of the Meridian, but only to the South; In this case, Take the Meridian Altitude thereof, and adde the distance of it from the Zenith, to the Complement of Latitude to give the apparent di­stance thereof from the Pole; Also take the Altitude thereof when it is in the prime Vertical, or East and West Cir­cle, (if it move between the Equinocti­al and the Zenith) and then if it have the same distance from the Pole that it had at the Meridian, it hath no Par­allax, as in the Figure following, where­in let the Circle ae, S, q, N, represent the Horizon, S, N, the Meridian, ae, q, the prime Vertical, ae, k, q, the Equi­nostiall, a, c, e, the Circle or Parallel of apparent motion, it being at the Me­ridian, b, d, f, the Circle or parallel of true motion, z, the Zenith, P, the Pole of the world, c, d, the Arch of Paral­lax at the Meridian, r, o, the Arch of [Page 16]

Parallax at the prime Vertical; And to find the distance P, o, from the Pole, the Comet being at the prime Verti­cal, resolve the right Angled Spherical Triangle z, P, o, wherein these three parts are given, First z, o, the distance from the Zenith, Secondly the right Angle at z, Thirdly the Arch z, P, the Complement of Latitude, to find the Arch P, o, if this be equal to P, c, (for [Page 15] [...] [Page 16] [...] [Page 17]lesser it cannot be) then there is no Parallax, but if it be greater, then there is a Parallax; which to find, Resolve the right Angled Spherical Triangle P, z, i, wherein these Three parts are given, First the right Angle at z. Se­condly the side z, P, Complement of Latitude, Thirdly the Arch P, i, = to P, c. the distance from the Pole at the Meridian, to find the side z, i, which if it be less then the observed Arch z, o, the distance from the Zenith, in the prime Vertical, then the difference is also the difference of Parallaxes. To Resolve this Triangle, the Proport. is,

As Radius, To the sine of Latitude, being Comple­ment of z, P, So Cosine of P, i, To Cosine of z, i.

Thus having found the difference of Parallax i, o, the Parallaxes d, c, and r, o, may be severally found as in the Ninth Note.

If the Comet be under, or be­low the Equinoctial.

XIV. But if the Comet be on the South side of the Equinoctial, then as in the Figure following, Draw the Circle representing the Horizon, S, N, the Meridian, a, q, the prime Vertical, a, k, q, the Equinoctial, a, c, e, the

parallel of apparent motion (being at the Moridian) b, d, f, the parallel of true motion, z, the Zenith, P, the Pole of the world, c, d, the Arch of Paral­lax [Page 17] [...] [Page 18] [...] [Page 19] [...] the Meridian, r, o, the Arch of [...]llax to be taken on some Azimuth; And then Resolve the Oblique Spheri­ [...]al Triangle z, P, i; in doing of which, [...]ontinue forth the sides z, i, and P, i, to [...]emi-circles, and the Perpendicular P, [...] falleth without the Triangle given, [...]herefore resolve, the Triangle z, g, P, [...]e [...]ein first to get the side z, h, of the Triangle z, P, h, The Proportion is,

As Radius, To the Cosine of the Angle P, z, h, the [...] [...]u [...]h, So the Tangent of the hypotenusal z, P, the Tangent of the side z, h,

Which being had, get also the side g, [...] the Triangle g, h, P, the Propor. is,

As the Cosine of z, P, To the Cosine of g, P, So the Cosine of z, h, To the Cosine of g, h.

Which Two Arches g, h, and h, z, be­ [...]g added together, and taken from a [...]micircle, leaveth the Arch z, i, which [...] be lesser then the Arch z, o, then is [...], the difference of Parallaxes, & then [...] the 9th. each may be found.

XV. This being understood, the [...]gle of Parallax may be found, at any [Page 20]one place of the Earth, and in any situ­ation of the Comet; And the desired distances from the Earth may be calcu­lated, as in [...]e Triangle a, b, c, in the first Figure; wherein adde 90 gs to the Angle of Altitude above the Horizon, to give the Angle a, b, c, at the Eie, to this adde the Angle of Parallax a, c, b, and subtract the aggregtae from 180 gs, it leaveth the Angle b, a, c, at the Cen­ter of the Earth. And then the propor­tionis alwaies.

For the distance from the Eie, As the sine of the Angle of Parallax a, c, b, To one Semidiameter a, b, So is the sine of the Angle at the Co [...]r b, a, c, To the distance b, c, in Semidiamiters of the Earth.

For the distance from the Cen­ter of the Earth.

As the sine of the Angle of Parallax a, c, [...] To one Semidiameter a, b, So is the sine of the Angle a, b, c, To the distance a, c, in Semidiameters of the Earth.

These Semidiameters of distance, ma [...] be turned into Miles if you multipl [...] them by 34364/11.