NOTES Showing how to get the Angle OF PARALLAX OF A COMET, Or other PHAENOMENON At Two Observations, To be taken in any One Station, or Place of the Earth, and thereby the distance from the Earth.  
By. R. HOLLAND.  
OXFORD, Printed by L. LICHFIELD, Printer to the University. for RICHARD DAVIS, 1668.   

TO THE READER.  
IN this I mean to be as brief as conveniently I may, intending only the practical part, therefore I set not down the Diagrams of Demonstration, which may be found in Authors, but refer to the places where they may be seen by such as think good to look upon them, only in the 13, and 14 Notes I have set forth such Figures for Demonstration as are elsewhere wanting. Moreover I advise the Reader to be prepared before hand with trigonometry, called also the Doctrine of Triangles, for without it nothing can be calculated, which is requisite in this desired Art.   

Of the Parallax  

Definitions.  
I. PArallax, that which I intent ●o speak of, and to find, is only the Angle a, c, b, contained between Two Lines, the one a, c, drawn from the Centre of the Earth, the Other b, c. from the Superficies thereof, to the Centre 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 than indeed it is, and the higher that a Phaenomenon is from the Horizon, the lesser is the Parallax thereof, and contra.   calculation  

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 Centre of any Star.   

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 apparent place of the Star, 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 Compliment of the Altitude before mentioned, and is the Arch c, g.   depiction of instrument for calculations  

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 show 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 add it to the lowest Altitude to give the height of the Pole.  
But in takeing the Altitude of any fixed Star that is less than 20 gs. high, take the Refraction from the Altitude observed, to leave the true height, for the Refraction maketh the Star to seem higher than indeed it is, as in Tycho Brahe's Table of the Refraction of fixed Stars here following.  
Alt. Refr. Alt. Refr.  ′ ″  0 30. 00 1 21. 30 11 5. 00 2 15. 30 12 4. 30 3 12. 30 13 4. 00 4 11. 00 14 3. 30 5 10. 00 15 3. 00 6 9 00 16 2. 30 7 8. 15 17 2. 00 8 6. 45 18 1. 15 9 6. 00 19 0. 30 10 5. 30 20 0. 00   

Proportion between the Two Arches of Parallax.  
VII. Seeing that in taking Parallax there is always two observations to be made, and the sum or difference of the Two Parallaxes is expected thereby, it is therefore necessary, whether I have their sum 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 Star, 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 Author, as it is spoken.   

To Separate the Arches of Parallax, when the sum of them is given.  
VIII. If the sum of Two Arches of Parallax be given in one Arch, than they may be separated, by help of the sixth Proposition of Clavius' Triangula Rectilinea, thus; Take the natural sins of the two distances of the Star from the Zenith (for they are in the same Proportion with the sins of the Two Parallaxes, as in the last Note) Add these two sins together, and take half their sum; Also take the fine of the lesser distance from the Zenith, from the said half sum to have their difference, and then the proportion is,  
As the said half sum of the sins of the distance from the Zenith, Is to the Tangent of the half sum of Parallax So is the difference aforesaid To the Tangent of another Arch.  
Which Arch being added to the half sum of the Parallaxes, giveth the greater Parallax, and being subtracted from the half sum of the Parallaxes, leaveth the lesser Parallax.   

But if the Difference of Two Parallaxes be given.  
IX. Then seeing the sins of the Distances from the Zenith of the Start, is the terms of Proportion between the two unknown Parallaxes thereof, as by he 7th. Note, thereforee take the natural sins 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 add 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 sins from the Zenith, To the Tangent of the half difference of the Parallaxes, So is the Aggregate of the half difference of the sins aforesaid, and the lesser sine of distance from the Zenith, To the Tangent of an other Arch.  
To which Arch add 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' 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 sum of the Parallaxes, subtract the lesser Meridional distance from the Pole (which is always that which is at his greatest Altitude) from the greatest Meridional distance from the Pole (to be found at his lesser Altitude) and the remain is the sum of the Parallaxes. See the Demonstration hereof, in the Tenth Problem of Mr. Thomas Diggs his Ala, sen Scala Mathematica.  
Now having the sum of the Parallaxes, they may be separated and known by the 8th. Note before.  
XI. But if the Phaenomenon rise to the Zenith, than the Compliment of the Latitude taken from the Meridional distance of the same, from or below the Pole, is the Parallax. But if the Phaenomenon do not set at the North, and yet cometh up beyond the Zenith toward the South, than the difference of the several Meridional distances from the Pole, is the difference of Parallaxes. Corollary the Second of the same Tenth Problem of Mr. Tho. Diggs, his Ala, etc. And then the several Parallaxes may be found by the 9th. Note before going.   calculation 
As the Cousin 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, etc. showeth 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 sum subtracted from d, c, the Compliment of the side a, d, leaveth s, g, the sum or Aggregate of the Two Parallaxes. And these may be separated by the Eighth Note.   

If the Comet doth Rise and Set.   calculation 
As Radius, To the sine of Latitude, being Compliment of z, P, So Cousin of P, i, To Cousin of z, i.  
Thus having found the difference of Parallax i, o, the Parallaxes d, c, and r, oh, may be severally found as in the Ninth Note.   

If the Comet be under, or below the Equinoctial.   calculation 
As Radius, To the Cousin 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 Cousin of z, P, To the Cousin of g, P, So the Cousin of z, h, To the Cousin 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 than the Arch z, oh, 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 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 add 90 gs to the Angle of Altitude above the Horizon, to give the Angle a, b, c, at the Eye, to this add the Angle of Parallax a, c, b, and subtract the aggregtae from 180 gs, it leaveth the Angle b, a, c, at the Centre of the Earth. And then the proportionis always.  
For the distance from the Eye, 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 Centre 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.   
FINIS.