Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

SECT. II. Of Astronomical Propositions.

PROP. I. The Distance of the Sun from the next Equinoctial point (either Aries or Libra) being known, to find his Declination.

THE Analogy or Proportion.

As Radius or S. 90°,

To S. of the Sun's distance from the next Equi∣noctial point,

So it S. of the Sun's greatest Declination,

To the S. of the Sun's present Declination sought.

PROP. II. The Sun's place given, to find his Right-Ascen∣sion.

This is the Analogy or Proportion.

As Radius or S. 90°,

To T. of the Sun's Longitude from the next Equi∣noctial point,

So is the Sc. of his greatest Declination,

To T. of his Right-Ascension from the next Equi∣noctial point.

PROP. III. To find the Sun's place or longitude from Aries, his Declination being given.

This is the Analogy or Proportion.

As S. of the Suns greatest Declination,

To Radius or S. 90° 00',

So is S. of his present Declination,

To S. of the Suns Place or Longitude from Aries* 1.1

PROP. IV. By knowing the Suns Declination, to find his Right Ascension.

This is the Analogy or Proportion.

As Radius or S. 90°,

To Tc. of the Suns greatest Declination,

So is T. of the Declination given,

To S. of the Suns right Ascension required† 1.2.

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PROP. V. By knowing the Latitude of a Place, and the Suns Declination, to find the Ascensional Difference.

This is the Analogy or Proportion.

As Radius or S. 90°,

To Tc. of the Latitude given,

So is T. of the Suns Declination given,

To the S. of the Ascensional difference required.

☞ Note that if you reduce the degrees, &c. of the Ascensional difference into hours, it will shew you how much the Sun riseth, or setteth before, or after six a Clock, in that Latitude.

PROP. VI. To find the Suns Oblique Ascension or Descension.

First find the Ascensional Difference by the 5th Proposition, and his Right-ascension by the fourth: Now if the Suns Declination be Northerly, de∣duct the Ascentional Difference out of his Right Ascension, from the beginning of ♈, (for the six Northern Signs ♈ ♉ ♊ ♋ ♌ ♍) it leaves the Oblique Ascension; and added unto the Right∣ascension, giveth the Oblique-descension.

But if the Suns Declination be Southerly, the Ascentional Difference, added to the Right-ascensi∣on, (for the six Southern Signs ♎ ♏ ♐ ♑ ♒ ♓) giveth the Right-ascension, and substracted there from leaves the Oblique-descension.

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Plate 11 Page 105

PROP. VII. By knowing the Suns Declination, and the Latitude of a Place, to find the Suns Amplitude.

This is the Analogy or Proportion.

As Sc. of the Latitude,

To the Radius or S. 90°.

So is the S. of the Suns Declination,

To the S. of the Amplitude from the East or West Points of the Horizon.

PROP. VIII. By knowing the Suns Declination and Amplitude, from the North part of the Horizon, to find the Latitude.

This is the Analogy or Proportion.

As Sc. of the Amplitude from the North,

To Radius or S. 90° 00'

So is S. of his Declination given,

To Sc. of the required Latitude.

PROP. IX. By knowing the Latitude of a place, and the Sun's Declination, to find at what time the Sun will be on the true East or West Points.

The Analogy or Proportion is.

As T. of the given Latitude,

To T. of the Sun's Declination propounded,

So is Radius or S. 90° 00',

To, Sc. of the Hour from Noon.

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PROP. X. By knowing the Sun's Declination, and Latitude of a place, to find his Altitude at six a Clock.

This is the Analogy or Proportion.

As Radius or S. 90° 00',

To S. of the Sun's Declination,

So is S. of the Latitude of the place,

To S. of the Sun's Altitude at six a Clock.

PROP. XI. By knowing the Latitude of a place, and the Sun's Declination, to find the Azimuth at six.

This is the Analogy or Proportion.

As Radius or S. 90° 00',

To the T. of the Sun's Declination,

So is Sc. of the Latitude of the place,

To the T. of the Azimuth sought.

PROP. XII. By knowing the Latitude of a place, and the Sun's Declination, to find the Sun's Altitude when he i•• on the true East or West points.

This is the Analogy or Proportion.

As S. of the Latitude,

To the Radius or S. 90° 00',

So is the S. of the Declination,

To the S. of the Sun's Altitude being due Ea•••• or West.

PROP. XIII. To find the Sun's Altitude at any time of the day.

The Analogy or Proportion is.

As Radius or S. 90° 00',

To Tc. of the Poles height,

So is S. of the Sun's Distance,

From the Hour of Six,

To the T. of an Arch: which being substract∣ed from the Sun's Distance from the Pole; say,

As Sc. of the Arch found,

To Sc. of the remaining Arch of the Sun's Di∣stance from the Pole,

So is S. of the Poles height,

To the S. of the Sun's Altitude at the Hour required.

PROP. XIV. By knowing the Latitude of a Place, with the Sun's Declination, and Altitude, to find the Hour of the Day.

To solve this Conclusion do thus: get the Sum of the Complements of the Latitude, Declina∣tion and Altitude given* 1.3, Then find the Difference betwixt their half Sum, and the Complement of the Altitude; then say,

As Radius or S. 90° 00',

To Sc. of the Sun's Altitude,

So is Sc. of the Latitude of the Place,

To a fourth Sine: then again say,

As the fourth S.

To the S. of ½ Z. of the Lat. Declin. and Alt.

So is the S. of X. of the Altitude to the ½ Z,

To a fifth S. unto which Sine, if you add the Radius or 90° 00', half that Sum shall be the Sine of an Arch, whose double Complement is the Hour from the Meridian.

PROP. XV. To find the Time of the Sun's Rising or Setting, and consequently the Length of the Day or Night.

To resolve this Conclusion, first by prop. the 5. find the Ascensional Difference, which redu∣ced into Hours, and Minutes of Time, by allow∣ing for every 15 Deg. one Hour, and for every Deg. less than 15°, 4', of Time, and for every 15 Min. one Minute of Time.

Secondly, If the Sun's Declination be Norther∣ly, the Ascentional Difference added unto 6 Hours, gives the Time of Sun-setting, and sub∣stracted therefrom, leaves the Time of Sun∣Rising: On the contrary, if the Sun's Declinati∣on be Southerly, the Ascentional Difference added unto 6 Hours, gives the Time of Sun-Rising, and deducted therefrom, the Time of Sun∣setting.

Thirdly, If you double the Time of Sun∣Rising, it gives you the length of the Night; and the Time of Sun-setting, the length of the Day.

PROP. XVI. The Sun's Declination, Altitude and Azimuth known, to find the Hour of the Day.

The Analogy or Proportion is.

As the Sc. of the Sun's Declination,

To the S. of the Azimuth,

So is the Sc. of the Altitude,

To the S. of the Hour from Noon: which con∣verted into Time, will shew the Hour of the Day.

PROP. XVII. By knowing the Sun's Declination, Altitude, and Hour from Noon, to find the Azimuth.

The Analogy or Proportion is.

As Sc. of the Sun's Altitude,

To S. of the Hour from Noon,

So is Sc. of the Sun's Declination,

To the S. of the Azimuth, required.

PROP. XVIII. By knowing the Latitude of a place, the Altitude of the Sun, and the Hour from Noon, to find the Angle of the Sun's Position.

This is the Analogy or Proportion.

As Sc. of the Sun's Altitude,

To S. of the Hour from Noon,

So is Sc. of the Latitude,

To S. of the V. of the Sun's Position, at the time of the Question.

PROP. XIX. By knowing the Sun's Altitude, Declination, and Azimuth; to find the Latitude.

The Analogy or Proportion is.

As S. of the Sun's Azimuth,

To S. of his Distance from the North-pole,

So is S. of V of the Sun's Position,

To Sc. of the Latitude required.

PROP. XX. To find the length of the Crepusculum, or Twilight

The Crepusculum or Twilight, is nothing else but the Refraction of the Sun's Beams in the Density of the Air. Which the Learned Pet. Nonnius found the length of the Crepusculum (by his many strict obser∣vations * 1.4) to continue from the time of the S〈…〉〈…〉 passing below the Hori∣zon of a place, untill the Sun had run below the said Horizon 18° 00', and then followed the shut∣ting in of the Twilight, and untill the Sun was

departed so low the Twi∣light continued. — To find which observe this Ana∣logy or Proportion.

As Radius or S. 90°,

To Sc. of the Sun's Declination,

So is Sc. of the Poles-height,

To a fourth Sine: which keep.

Then out of the Sun's Distance from the South-Pole, subduct the Complement of the Pole; and of that remains and the degrees 62, being added to it, their Sum and Difference found, say again.

As the fourth Sine found,

To S. ½ Z of the remainder and 62° 00',

So is S. ½ X. of the remainder and 62° 00',

To a Number, which being multiplyed by the Radius is equal unto the Quadrat of the Sine of the ½ Angle of the Sun's Distance at the Ending of the Twilight, from Noon next ensuing.

Then from the Sun of the whole Angle con∣verted into Hours, substract the Hour of the Sun's setting * 1.5, it gives you the length of the Crepusculum, or Twilight.

But the Sun being in the Winter Tropick, makes the Twilight longest of a∣ny* 1.6 Twilight, the whole Winter half year: Now in a certain Parallel, be∣twixt that Tropick, and the Equinoctial is the shortest Crepusculum: the Declination of which Parallel, is thus found.

As the Tc. of the Pole,

To the S of the Pole,

So is the T. of 99° 00',

To S. of the Declination of the Parallel, in which the Sun maketh the shortest Crepusculum of the Year.

But before the Crepusculum come to be short∣est, there is another Parallel, in which the Crepusculum is equal to that of the Equinoctial: the Declination of which is found thus.

As the Radius or S. 90° 00',

To S. of the Poles Elevation or Altitude,

So i•• S. of 18° 00',

To S. of the Declination of the Parallel, in which the Sun maketh the Crepusculum equal to that in the Equinoctial.

PROP. XXI. To find the Quantity of the Angles, which the Cir∣cles of the 12 Houses make with the Meri∣dian.

This is the Analogy or Proportion.

As the Radius or S. 90°,

To T. of 60°: for the 11th, 9th, 5th and 3d House, Or to the T. 30° for the 12th, 8th, 6th, and 2d House,

So is the Sc. of the Pole,

To the Tc. of any House with the Meridian.

PROP. XXII. To find the Right Ascension of the Point in the Equinoctial: and also the Point in the Ecliptick; called Medium Coeli or Cor Coeli.

First, To find the Right Ascension of the Point of the Equinoctial; called Medium Coeli, vel Cor Coeli, find out the Sun's Right Ascension, by prop. 2. Then reduce the whole Time from Noon last past into degrees, which add unto the right Ascension of the Sun, so shall their A∣gragat, be the right Ascension of the point, which in the Equinoctial, is called Medium Coeli, vel Cor Caeli, required to be found.

Secondly, By the 2 propositions aforegoing, you may find the right Ascension of the point in the Ecliptick Culminant in the Meridian, cal∣led Medium Coeli vel Cor Coeli, which is the Cuspis of the tenth House: and his Declination by prop. the first.

PROP. XXIII. To find the Angle of the Ecliptick with the Meri∣dian.

The Analogy or Proportion is.

As the Radius or S. 90°,

To S. of the Sun's Greatest Declination,

So is Sc. of the Sun's right Ascension, from the next Equinoctial point,

To Sc. of the V. of the Ecliptick, with the Me∣ridian.

PROP. XXIV. To find the Angle of the Ecliptick with the Ho∣rizon.

The Analogy or Proportion is.

As Radius or S. 90°,

To Sc. of the Altitude of Cor Coeli,

So is S. of the V. Ecliptick with the Meridian,

To Sc. of the V. of the Ecliptick and Horizon sought.

PROP. XXV. To find the Amplitude Ortive of the Ascendent, or Horoscopus.

This is the Analogy or Proportion.

As Radius or S. 90°,

To S. of Altitude of Med. Coeli,

So is T. of V. Ecliptick with the Meridian,

To Tc. of the Amplitude Ortive of the Ascen∣dent, or the distance of the Azimuth from the Meridian.

PROP. XXVI. To find the Ascendent degree of the Ecliptick, or the Cuspis of the first House.

The Amplitude Ortive of the Ascendent, is equal to the Distance of the Azimuth of 90°, from the Meridian, wherefore the Cuspis of the

first House, or Ascendent Degree of the Ecliptick, may be found thus.

As Radius or S. 90°,

To Sc. of the V. Ecliptick with the Meridian,

So is Tc. of the Altitude of Med. Coeli,

To T. of the Distance of Med. Coeli, from the Ascendent Degrees.

PROP. XXVII. To find the Distance of the Cuspis of any House, from Med. Coeli.

This is the Analogy or Proportion.

As Sc. of the remaining part of V. of the E∣cliptick with the Meridian, (found by prop. 28.)

To Sc of the former part of the V,

So is T. of the Altitude of Med. Coeli,

To T. of the Distance of the Cuspis of that House sought, from Med. Coeli.

PROP. XXVIII. To find the parts of the Angle of the Ecliptick with the Meridian, cut with an Arch perpendicular to the Circle of any of the Houses.

The Analogy or Proportion is:

As Radius or S. 90°,

To Sc. Altitude of Med. Coeli,

So is T. of the Circle of any House with the Meridian,

To Tc. of that part of that Angle which is next the Meridian:

Then substract that part found, out of the whole Angle, for the remaining or latter part

PROP. XXIX. To find the Pole's Altitude, above any of the Circles of the Houses.

The Analogy or Proportion is.

As the Radius or S 90°,

To S. of V. of the Circle of the House with the Meridian: (found by the 21 prop.)

So is the S of the Poles Elevation, above the Horizon of the Place,

To S. of the Altitude of the Pole, above the Circle of Position.

PROP. XXX. By knowing the Latitude and Longitude of any fixed Star, to find his Right Ascension and De∣clination.

The Analogy or Proportion is.

1. As Radius or S. 90°,

To S. of the Stars Longitude from the next Equinoctial point,

So is Tc. of the Stars Latitude,

To T. of a fourth Arch.

Which compared with the Arch of Distance betwixt the Poles of the World and the Ecliptick 23°, 30'; And if the Latitude and Longitude of the Star be both of one Dignity, i e. when the Star hath North Latitude in the six Northern Sines, ♈, ♉, ♊, ♋, ♌, ♍, or South Latitude in the six Southern Sines, ♎, ♏, ♐, ♑, ♒, ♓: Then shall the difference between this found Arch, and the

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Page 216 Plate III

Distance of the Poles be your fifth Arch: But if the Latitude and Longitude of the Star be of contrary qualities, i. e the one Northern, and the other Southern, then add this fourth Arch to the Distance of the Poles 23° 30', and the Sum thereof shall be your fifth Arch; with which,

AGAIN, say.

2. As S. of the fourth Arch,

To S. of the fifth Arch,

So is T. of the Stars Longitude,

To T. of the Stars Right-ascension from the next Equinoctial point.

3. As Sc of the fourth Arch,

To Sc. of the fifth Arch,

So is S of the Stars Latitude,

To S. of the Stars Declination.

I might also shew how by having the La∣titude and Longitude of any two fixed Stars, to find their Distance: but because 'tis the very same with finding the Distance of any two Places on Earth, I refer you to the Directions of Prop 1, 2 and 3. of Chap. 7, ensuing, where you will see the plain Demonstration thereof.

PROP. XXXI. By knowing the Pole's Altitude, to find when any fixed Star shall be due East or West.

This is the Analogy or Proportion.

As Radius or S. 90°,

To T. of the Stars Declination,

So is Tc. of the Pole,

To Sc. of the Stars Horary Distance from the Meridian.

PROP. XXXII. By knowing the Poles Altitude, to find the Eleva∣tion of any fixed Star above the Horizon, being due East or West.

This is the Analogy or Proportion.

As S. of the Poles Altitude,

To Radius or S. 90°,

So is S. of the Stars Declination,

To S. of the Stars Elevation, above the Hori∣zon, at due East or West.

PROP. XXXIII. To find out the Horizontal Parallax of the Moon.

The Analogy or Proportion.

As the Moons Distance from the Center of the Earth,

To the Earth's Semidiameter,

So is Radius or S. 90°,

So S. of the Moon's Horizontal Parallax in that Distance.

PROP. XXXIV. The Horizontal Parallax of the Moon being known, to find her Parallax in any apparent Latitude.

This is the Analogy or Proportion.

As Radius or S. 90°,

To S. of the Moon's Altitude,

So is S. of the Moon's Horizontal Parallax,

To S. of the Parallax in that Altitude.

PROP. XXXV. By knowing the Moon's Place in the Ecliptick, (having little or no Latitude) and her Paral∣lax of Altitude, to find the Parallaxes of her Longitude and Latitude.

First, If the Moon be in the 90° of the Eclip∣tick, she hath then no Parallax of Longitude, and the Parallax of the Latitude, is the very Parallax in that Altitude.

Secondly, But if the Moon be not in the 90th. Degree of the Ecliptick, to find the Parallaxes of the Latitude and Longitude, the Analogy or Proportion is,

1. As Radius or S. 90°,

To T. of the V. of the Ecliptick and Horizon,

So is Sc. of the Moon's Distance from the As∣cendent, or Descendent deg. of the Ecliptick,

To Tc. of the Ecliptick's V, with the Azimuth of the Moon.

AGAIN say,

2. As the Radius or S. 90°,

To S. of that V. found,

So is the Parallax of the Moon's Altitude,

To the Parallax of her Latitude sought.

LASTLY say,

3. As the Radius or S. 90°, 00'

To Sc. of the former V. found,

So is the Parallax of the Moon's Altitude,

To the Parallax of his Longitude sought, which being added to the true Motion of the Moon, if she be on the East part of the 90° of the Eclip∣tick. Or from it to be deducted if she be on the West part of the 90° of the Ecliptick.

PROP. XXXVI. How by knowing the Refraction of a Star, to find his true Altitude.

For the speedy performance of which I have annexed this Table of Refractions of the Stars observed by Tycho Brabe a Nobleman of Den∣mark, and a most famous Astronomer.

A Table of the Refraction of the Stars observed by Tycho Brabe.

Altitude.	Refraction.
o°	30'	00"
1	21	30
2	15	30
3	12	30
4	11	00
5	10	00
6	9	00
7	8	15
8	6	45
9	6	00
10	5	30
11	5	00
12	4	30
13	4	00
14	3	30
15	3	00
16	2	30
17	2	00
18	1	15
19	0	30
20	0	00

The USE of which Table is thus.

EXAMPLE.

Suppose the Altitude of a Star were found by Observation to be 13°; the correspondent Re∣fraction is 4' 00", which substracted from 13° leaves 12°, 56', which is the true Altitude

Notes

• * 1.1

  If the Sun's Declina∣tion be North, and in∣creasing, this Proportion finds the Sun's distance from ♈; but if decreasing from ♎ in the northern Sines. But if the Sun's Declinati∣on be South, and increasing from ♎; if decreasing from ♈, among the Southern Sines.
• † 1.2

  From the next Equi∣noctial point either ♈, or ♎.
• * 1.3

  As in Case 11 of Oblique Spherical Tri∣angles.
• * 1.4

  Watched the Time after Sun-setting when the Twilight in the West was shut in, so that no more Twilight than in any other part of the Skie near the Horizon appea∣red there: then by one of the known fixed Stars, having found the true Hour of the Night, he found the length of the Twilight, to be as in the rule is mentioned.
• * 1.5

  Or ½ Diurnal Arch.
• * 1.6

  To find the length of the least Crepusculum or Twilight.