The English academy, or, A brief introduction to the seven liberal arts grammar, arithmetick, geometrie, musick, astronomie, rhetorick & logic : to which is added the necessary arts and mysteries of navigation, dyaling, surveying, mensuration, gauging & fortification, practically laid down in all their material points and particulars, highly approved to be known by the ingenious, and as such are desirous to profit, or render themselves accomplished : chiefly intended for the instruction of young scholars, who are acquainted with no other than their native language, but may also be very useful to other persons that have made some progress in the studies of the said arts / by John Newton.

THE ENGLISH ACADEMY: The FIFTH PART. Of ASTRONOMIE. (Book 5)

CHAP. I. Of the General Subject of Astronomie.

AStronomy, is an Art, by which we are Taught the Measure and Moti∣on of the heavenly Orbs and Stars that are in them.

2. The Heavenly Orbs are either 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, without Stars, as the Primum Mobile, or 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, such as have Stars in them, as the eight inferiour Orbs.

3. The Stars are either fixed or movea∣ble: The fixed Stars are those which always keep the same distance from one another; but the moveable Stars, otherwise called

Planets, are such as do not always keep the same distance.

4. All the Stars, as well fixed as moveable have a double motion; the one occasioned by the Primum Mobile, from East to West, the other natural or proper to themselves, by which they move from West, to East.

5. According to this double motion of the Stars, this Art of Astronomy is divided in∣to two Parts; the first sheweth the motion of the Primum Mobile, and how the several Heavenly Orbs are by that carried round the World, from East to West, which is called the Diurnal motion of the Stars.

The second part of Astronomy, sheweth the Periodical motion of the Stars, in which the inferiour Orbs, according to their own proper and natural motion, do move from West to East.

6. For the better understanding of these several motions, the Primum Mobile, or tenth Orb, is usually represented by a Sphere or Globe, with such lines drawn about it as the Stars in their motions are supposed to make, or may help to discover unto us, the quantity of their motions, and shew the time of their Risings and Settings, and such like.

7. This Sphere or Globe, is a round bo∣dy, containing one Superficies, in the mid∣dle whereof there is a Point, from whence all Right Lines drawn to the Superficies are equal.

8. In the Sphere or Globe, there are ten imaginary Lines or Circles, of which six are great, and four are small.

9. The great Circles are these which di∣vide the Sphere or Globe into two equal Hemispheres, and such are the Horizon, Ae∣quinoctial, Zodiack and the two Colures; the two first of which are called external and mu∣table, the other internal and immutable.

10. The Lesser Circles, are those which divide the Sphere or Globe, into two une∣qual Hemispheres, whereof one is more, and the other less than the half of the Sphere or Globe; such are the two Tropicks of Can∣cer and Capricorn, and the Artick and An∣tartick Circles, all which are represented in Fig. 9.

11. The Horizon, which is also called the Finitor, is a Circle, which divideth the visible part of the Heavens from the not vi∣sible; that is, the lower Hemisphere from the upper, as the line AB; one of whose Poles is in the Point directly over our heads, and is called the Zenith, the other Diame∣trically opposite, called the Nadir, and no∣ted with the Letters Z. N.

12. The Horizon, is either Sensible or Ra∣tional.

13. That is called the Sensible Horizon, which bounds our sight, and seemeth to di∣vide the Heavens into two equal Hemis∣pheres.

14. And that is called the Rational or In∣telligable Horizon, which doth indeed bisect the Heavens; and this is Right, when it passeth through the Poles of the World; or Oblique, when one of the Poles is some∣what elevated, and the other depressed; or Parallel, when one Pole is in the Verti∣cal Point or Zenith, for then the Horizon is Parallel to the Aequator; it otherwise makes therewith either Right or Oblique Angles.

15. Hence there is a threefold position of Sphere. 1. A Right, where the Horizon is Right; that is, where the Aequator pas∣seth through the Zenith and Nadir, 2. Ob∣lique, when the Horizon is Oblique; that is, when one Pole is somewhat elevated and the other depressed. 3. Parallel, when one of the Poles of the world is in the Zenith.

16. In a Right Sphere, all the Stars do Rise and Set, but in an Oblique Sphere, some are hid from our sight, and some are always above the Horizon.

17. The Meridian is a great Circle pe∣culiar and proper to every place, and drawn through the Vertical point and the Poles of the World, to which when the Sun comes in his Diurnal motion, in the Day-time he mak∣eth the Mid-day, and in the Night time, he maketh Midnight. There may be as many Meridians as there are Vertical points, but upon the Globe they are usually drawn thro' every tenth or fifteenth Degree of the Ae∣quator.

CHAP. II. Of the Internal and Immutable great Circles.

HItherto of the two External and Muta∣ble Circles, the Horizon and Meridi∣an, I come now to the Internal and Immu∣table.

2. The first Internal and Immutable Cir∣cle is called the Aequator, or Equinoctial Circle, which divideth the whole Sphere òr Globe into two equal parts between the Poles, to which when the Sun cometh, which is twice in the Year, the days and nights are equal in all places but in a Parallel Sphere: this Circle is noted with the letters EF.

3. This Circle is also the measure of Time; for as oft as 15 Degrees of this Cir∣cle do ascend above the Horizon, so many hours are compleated in its going round.

4. The second Immutable Circle is called the Zodiak, which is a great Oblique broad Circle, under which the Planets do always move; the Poles of this Circle are distant, from the Poles of the world 23 Degrees, 31 Minutes, and 30 Seconds, or 23.53 Cen∣tesms.

5. This Circle doth differ from other Circles in the Heavens, in that other Circles to speak properly, have Longitude or

Length, but no Breadth, whereas this Cir∣cle is allowed to have both.

6. In respect of Longitude, this Circle is divided as other Circles commonly are into 360 Degrees, but more peculiarly into 12 parts, constituting, as it were, the 12 Parts or Months of the Year, or 12 Constellati∣ons of Stars, called Signs, each Sign being subdivided into 30 Degrees or Parts. The Names and Characters of these 12 Constel∣lations, or Signs, are as followeth. A∣ries ♈, Taurus ♉, Gemini ♊, Cancer ♋, Leo ♌, Virgo ♍, Libra ♎, Scorpio ♏, Sagit∣tarius ♐, Capricornus ♑, Aquarius ♒, Pis∣ces ♓.

7. The Zodiack, in respect of Latitude, is divided into 16 Degrees, that is, into 8 Degrees North-ward, and 8 Degrees South-ward, because all the Planets, except the Sun, do in their Motions vary from the middle Line, sometimes one way, and some∣times another; to the quantity of 8 De∣grees; and the middle Line in which the Sun moves, is the Ecliptick Line, because when the Sun and Moon are in Conjunction, the Sun is Eclipsed, but when they are in Opposition, the Moon is Eclipsed.

8. Of these 12 Signs, 4 are called Cardi∣nals, viz. Aries and Libra, in which do happen the Vernal and the Autumnal Ae∣quinoctials; Cancer and Capricorn, in which do happen the Summer and the Winter Sol∣stices.

9. Again these Signs are distinguished in∣to Northern and Southern; the Northern signs are those which decline from the Ae∣quator towards the North Pole, as ♈, ♉, ♊, ♋, ♌, ♍; And the Southers signs are those which decline from the Aequator to∣wards the South Pole, as ♎, ♏, ♐, ♑, ♒, ♓.

10. All other Constellations or fixed stars are referred to some one or other of the 12 signs, whether they be the 21 Northern con∣stellations, called Vrsa Minor, Vrsa Major, Draco, Cepheus, Arctephylax, Corona Bore alis, Engonasus, Lyra, Avis, Cassiopeia, Persius, Heniochus, Serpentarius, Serpens, Sagitta, Aqui∣la, Delphinus, Equisectio, Pegasus, Andromeda, Triangulus. Or whether they be the 15 Sou∣thern constellations, called Cetus, Orion, Eri∣danus, Lupus, Canis Major, Precyon, Argo, Hy∣dra, Crater, Corvus, Centaurus, ••••ra, Ara, Corona Austr. Pisces Austra.

11. The two other great Circles called the Colures, are the two Circles which pass through the Poles of the World, and the four Cardinal points in the Zodiack.

12. That circle which passeth thro' the Poles of the world, and the two Solstitial points in the Zodiack, which are the beginnings of ♋ and ♑, and is called the Solstitial Colure.

13. That Circle, which passeth through the poles of the world and the two Aequino∣ctial points, or first entrance into ♈ and ♎, is called the Aequinoctial Colure, and in Fig. 9. represented by the line D. C.

14. The Lesser Circles of the Sphere are the two Tropicks of ♋ and ♑ with the Ar∣tick and Antartick Circles.

15. The Tropick of ♋ is a Circle joyned to the Zodiack in the beginning of ♋, and is de∣scribed by the Suns Diurnal Motion, when he is in the Summers Solstitial point, and is di∣stant from the Aequinoctial towards the North Pole 23 deg. 31' 30" or in Decimal Numbers, 23 deg. 5.25. to which when the Sun cometh, he causeth the longest day and shortest night to all Northern; the shortest day and longest night to all Southern Inha∣bitants; and is noted with G ♋.

16. The Tropick of ♑, is a Circle joyned to the Zodiack in the beginning of ♑, and de∣scribed by the Suns Diurnal Motion, being in the winters Solstitial point, and is distant from the Aequinoctial towards the South Po•••• ••3 deg. 31' 30", or in Decimal Num∣bers, 23 deg. 5.25 parts, to which, when the Sun cometh, he maketh the longest day and shortest night, to all Southern; the shortest day and longest night to all Northern Inha∣bitants, and is noted with H ♑.

These two Circles are called of the Greeks 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, à convertendo, because when the Sun toucheth any of the Circles, he is at his greatest distance from the Aequator, and returneth thither again.

17. The Artick Circle, is distant from the North Pole of the world, as much as

the Tropick of ♋ is distant from the Aequi∣noctial and is noted with KL.

The Antartick Circle is distant from the South Pole as much as the Tropick of ♑ is di∣stant from the Aequator, & is noted with OM.

18. By the Intersection of any three of the greatest Circles of a Sphere is made a Spherical Triangle.

19. A Spherical Triangle, is either Right Angled or Oblique.

20. A Right Angled Spherical Triangle, hath one Right Angle at the least.

21. An Oblique Angled Spherical Triangle, is either Acute or Obtuse.

22. An Acute Angled Spherical Triangle, hath all its Angles Acute.

23. An Obtuse Angled Spherical Triangle, hath all his Angles, either Obtuse or mixt, that is one Angle at the least Obtuse, and the other Acute.

24. In Spherical Triangles, there are 28 Varieties or Cases, 16 in Rectangular, and 12 in Oblique Angular, whereof all the Rectan∣gular and 10 of the Oblique Angular, may be resolved by one Catholick, and Universal Proposition; for the understanding where∣of, some things must be premised.

1. That in a Right Angled Spherical Triangle, the Hypotenuse and both the Acute Angles are supposed to be noted with their Complements.

2. That the Right Angle is not reckoned a∣mongst the Circular parts, and therefore one of the other five will be always a middle part, and the other four extreams Conjunct or Disjunct.

The Proposition is this:

A Rectangle made of the Sine of the middle part & Radius, is equal to the Rectangle made of the Tangents of the Extreams Conjunct, or of the Cosins of the Extreams Disjunct: Therefore,

When two things are given, and a third re∣quired, you must first find out the middle part, and where the other Terms be Extreams Con∣junct or Disjunct; if the things given and in∣quired lie together, the middle is the middle part, but if they be disjoyned, that which stan∣deth by it self is the middle part.

Note also, that when a Complement in the Proposition doth chance to concur with a Com∣plement in the Circular Parts, you must take the Sine it self, or the Tangent it self, because cs of the cs=S. and ct of the ct=t.

25. These things being understood, the Analogies to be used in every of the 16 Cases of a Right angled Triangle, will from this Proposition be as followeth.

	Bata	Q.	Analogia.
1	BCA	AB	Rad cot . A ∷ tBC . sAB.
2	AAB	BC	cot A . Rad ∷ sAB . tBC.
3	ABBC	A	tBC . sAB ∷ Rsd . ctA.

4	ABA	AC	tAB . csA ∷ Rad. cotAC.
5	ACAB	A	Rcot . AC ∷ tAB . csA.
6	ACA	AB	cot AC . Rad ∷ csA. tAB.
7	AC	AC	Rad . ctC ∷ ct A.csAC.
8	ACA	C	ctA . csAC ∷ ctC.
9	ACA	BC	Rad . sA ∷ sAC . sBC.
10	BCA	AC	SA . sBC ∷ Rad. sAC.
11	ACBC	A	SAC . sBC ∷ Rad . sA.
12	ABA	C	Rad . sA ∷ csAB . csC.
13	ABC	A	csAB . csC ∷ Rad. sA.
14	AC	AB	sA . csC ∷ Rad. csAB.

15	ABBC	AC	Rad csAB ∷ csBC . csAC.
16	ACAB	BC	csAB . csAC ∷ Rad . csBC.

26. In Oblique angled Spherical Triangles, there are, as hath been said, 12 Cases, 10 whereof may be resolved by the Catholick Proposition, if the Spherical Triangle pro∣pounded be first converted into two Right, which may be done by this General Rule.

From the end of a side given, being adjacent to an Angle given, let fall the Perpendicular.

A Type of the several Varieties here followeth.

1.	...AC ...CD D	A	...Rad . SCD ∷ sD . sBC. ...sAC . Rad ∷ sBC . sA.
2.	...CD ...A ...D	AC	sA . sD ∷ sCD . sAC.
3.	...AC ...CD ...D	AD	...cot CD . R ∷ csD . tBD. ...cosBD . csCD ∷ R . csBC. ...R . csBC ∷ csAC . csAB. BD + AB = AD. 1 Tri. BD − AB = AD. 2. Tri.

4.	... AC ... CD ... D	C	... ct D . Rad ∷ csCD . ct BCD ... ctCD . csBCD ∷ R . tBC. ... R. tBC ∷ ctAC . csACB. BCD + ACB = ACD. 1. Tri. BCD − ACB = ACD. 2. Tri.
5.	... A ... D ... CA	C	... ctA . R∷ csAC . ctACB. ... sACB . csCAB ∷ R . csBC. ... csBC . R ∷ csBDC . csBCD. ACB + BCD = ACD. 1. Tri. BCD − ACB = ACD. 2. Tri.
6.	... A ... D ... A C	AD	... ct . AC . R ∷ csDAC . tAB. ... ct . DAC . RAB ∷ R . tBC. ... R . tBC ∷ ctADC . sBD. AB + BD = AD. 1. Tri. RD − AB = AD. 2. Tri.
7.	... A ... A ... AC	AD	... ct . CAB . R ∷ csAC . ctACB. ACD − ACB = BCD. 1. Tri. ACD + ACB = BCD. 2. Tri. ... s ACB . csCAB ∷ R . csBC. ... R . csBC ∷ sBCD . csCDB.
8.	... A ... C ... AC	DC	... ctCAB . R ∷ csAC . ctACB. ACD − ACB = BCD. 1. Tri. ACD + ACB = BCD. 2. Tri. ... ctAC . csACB ∷ R . tBC. ... tBC . Rad. ∷ csBCD. ctDC.

9.	...AD ...AC ...A	DC	...ctAC. R ∷ csCAD. tAB. AD − AB = BD. 1. Tri. AD + AB = BD. 2. Tri. ...csAB. csAC ∷ R. csBC. ...R. csBC ∷ csBD. csDC.
10.	...AC ...AD ...A	C	...ctAD. R ∷ csCAD. tAE. AE − AC = CF in 1. Tri. AE + AC = CF 2. Tri. ...ctCAD. sAE ∷ R . tDF. ...tDF. R ∷ sCF . ctDCF.
11.	...AC ...AD ...DC	C	...sAC x sCD . Rad. square. ...s ½ z − AC x s ½ z − CD. ...Q. s 1/ C.

The Twelfth, is but the Converse of the last taking the Angles for Sides, and the Sides for Angles; so shall the Angle found, be the Side inquired.

CHAP. III. Of the Ascensions and Descensions of the Parts of the Zodiack.

HItherto we have spoken of the general Principles of Astronomy, from whence the motion of the Primum Mobile is explai∣ned; come we now to these affections which properly belong to the motion there∣of, and these are the Ascension and Descension

of the Parts of the Zodiack, or Astronomi∣cal Rising and Setting.

2. Astronomical Rising and Setting, is the Elevation of the parts of the Zodiack or E∣cliptick above the Horizon, and Depressed under it, compared to the Ascension and de∣scension of the parts of the Aequator; and this comparison is in reference to diverse Elevations of the Poles.

3. But this Astronomical Rising and Set∣ting, takes his Denomination from the parts of the Zodiack; which are above the Horizon or beneath it, and are measured with respect unto the Aequator; for Astro∣nomers do not refer the Aequator to the Zodiack, but the Zodiack to the Aequator, for it is Zodiack, and not the Aequator which stands in need of measuring.

4. And an Arch of the Ecliptick or Zodi∣ack, is to be understood two manner of ways; namely, Continued or Discreet; A Continued Arch, is when it is reckoned in the Aequator in a Continued Series, from the beginning of Aries, and so forward into the consequent Signs.

5. A Discreet Arch, is so called, because it is not reckoned from the first Degree of Aries, but from any other point; as from the fourteenth of Gemini, to the fourteenth of Taurus.

6. Any part of the Zodiack is then said to Ascend Right, when a greater part of the

Aequator riseth above the Horizon than of the Zodiack. And that is said to be a grea∣ter Arch of the Aequator, which is more than 90 Degrees.

7. Any part of the Zodiack is then said to Descend Right, when a greater part of the Aequator than of the Zodiack is beneath the Horizon.

8. Any part of the Zodiack therefore is said to Ascend Obliquely, when a less part of the Aequator than of the Zodiack doth As∣cend; as also, to Descend Obliquely, when less of the Aequator than of the Zodiack is below the Horizon.

9. Ascension, is either Right or Oblique.

10. Right Ascension or Descension, is that which is in a Right Sphere.

11. In a Right Sphere, the four Quadrants of the Zodiack beginning from the Aequino∣ctial and Solstitial Points, do equally Ascend and Descend, so that in these whole Qua∣drants, as many Degrees of the Aequator as of the Zodiack do Ascend; but the interme∣diate parts of those Quadrants in the Zodi∣ack do vary, and have not equal Ascension and Descension with the parts of the Aequa∣tor.

12. Those Signs that are equally distant from any of those Points, have also equal As∣cension, as Gemini and Cancer. And the As∣cension of a Sign is always equal to the Des∣cension of the same.

13. In a Right Sphere therefore, four Signs only do rise Right, all the rest do rise Obliquely.

14. In an Oblique Sphere, the two halves that begin at the two Aequinoctial Points, do rise together, but the parts of those halves do rise Obliquely. And those Signs that rise Rightly, do Descend Obliquely, and the contrary.

15. The Ascension of opposite Signs in an Oblique Sphere, taken together, are equal to the Ascension of the same in a Right Sphere. And those signs that are equall di∣stant from either of the Aequinoctial Points, have equal Ascensions, because they equally Decline from the Aequator.

16. Besides the Astronomical Rising and setting of the stars, or their rising and set∣ting, in respect of the Horizon and Aequa∣tor, there are other affections of the stars to be considered, namely, those which they have in respect of the sun.

17. In respect of the Celestial Circles, that is in respect of the Zodiack, Aequator, and Horizon, there is a fourfold affection of the stars. 1. Longitude. 2. Of Altitude. 3. Of Latitude. 4. Of Declination.

18. The Longitude of a star is his distance from the first Degree or Point of Aries, ac∣counting from West to East.

19. The Altitude of a star is to be conside∣red generally or specially. Generally con∣sidered,

the Altitude of a star is the height thereof above the Circle of the Horizon.

20. Specially considered, the Elevation of the Pole star above the Horizon, is called the Altitude.

21. The Latitude of a star is his dDstance from the Ecliptick, that is from the very mid∣dle of the Zodiack towards either Pole, whe∣ther North or South.

22. The Declination of a star, is his Di∣stance from the Aequator, and as he declines from thence either Northward or South∣ward, so is his Declination nominated ei∣ther North or South.

23. Thus much of these affections of the stars, which they have in respect of the Ce∣lestial Circles; come we now to those which they have in respect of the sun; usually cal∣led the Poetical rising and setting; and this is threefold. The first of these in Latin, is called Ortus Matutinus sive Cosmicus, The Morning or Cosmical Rising. The second, Vespertinus five Achronicus, The Evening or Achronical; and the last, Heliacus vel Sola∣ris, Heliacal or Solary.

24. The Cosmical or Morning Rising of a star, is when it Riseth above the Horizon, together with the sun. And the Cosmical or Morning setting of a star is, when it-set∣teth at the opposite part of Heaven, when the sun riseth.

25. The Achronical, or Evening Rising of

a Star, is when it Riseth on the opposite part, when the Sun setteth; And the Achro∣nical Evening setting of a Star, is when it setteth at the same time with the Sun.

26. The Heliacal Rising of a Star, which you may properly call the Emersion of it, is when a Star that was hid by the Sun beams, beginneth to recover it self out, and to ap∣pear. And so likewise, the setting of such a star, which may be also called the Occultati∣on of the same, is when the Sun by his own proper motion overtaketh any star, and by the brightness of his beams doth make it invisible unto us.

And thus having briefly shewed the chief affections of the Primum Mobile; how the quantity of these affections may be compu∣ted, by the Doctrine of Spherical Triangles, shall be declared in the Problems following.

Problem I. To find the Suns Greatest Declination and the Poles Elevation.

In Fig. 9. AZ, BN represents the Meri∣dian, EF the Aequinoctial, HR the Zodiack, P the North Pole; O, the South; AB, the Horizon; Z, the Zenith; N, the Nadir; HC, a Parallel; of the Suns Diurnal Moti∣on at H, or the Suns greatest Declination from the Aequator towards the North Pole;

RQ, a Parallel of the Suns greatest Declina∣tion from the Aequator towards the South Pole. From whence it is apparent, that from A to H, is the Suns greatest Meridian Altitude, from A to Q, his least; if there∣fore you deduct AQ, the least Meridian Al∣titude, from AH, the greatest, the Differ∣ence will be HQ, the Suns greatest Declina∣tion on both sides the Aequator; and because the Angles E D H, and F D R, are equal, therefore the Suns greatest Declination to∣wards the South Pole is equal to his greatest Declination towards the North, and conse∣quently, half the Distance of the Tropicks; that is, EQ, or EH, is the quantity of the Suns greatest Declination; and then if you deduct the Suns greatest Declination, or the Arch HE, from the Suns greatest Meridian Altitude, or the Arch AH, the Difference will be AE, the height of the Aequator a∣bove the Horizon, the Complement whereof to a Quadrant, is the Arch AO equal to BP, the height of the Pole.

Example.

The Suns greatest Meredian Altitude, taken June the Ele∣venth at London. —

61.99167

The Suns least Meridian Al∣titude December the tenth —

14.94167

Their Difference is the distance of the Tropicks.—

47.05000

Half that is the Suns greatest Declination whose difference from the Suns least Meridian Al∣titude, is —

23.52500

The Elevation of the Aequator and the Complement thereof to 90, is the Elevation of the Pole—

38.46667 51.53333

Problem II. The Suns greatest Declination being given, to find his Declination in any point of the Ecliptick.

In Fig. 9. In the Right Angled Spherical Triangle GLD, we have given the suns grea∣test Declination GDL, and the suns distance from the next Aequinoctial point LD, to find the present Declination GL, for which the Proportion is Rad. sLD ∷ sD. sGL.

Problem III. The suns greatest declination and his distance from the next Equinoctial point given, to find his Right Ascension.

In Fig. 9. In the Right Spherical Triangle GDL, we have given as before the Angle, GDL, and the Hypotenuse DL, to find

the suns Right Ascension DG; the Propor∣tion is Rad. tDL ∷ csD . tDG.

Problem IV. The Elevation of the Pole, and declination of the sun being given, to find his Amplitude.

In Fig. 9. In the Right Angled spherical Triangle DTV, we have given the Comple∣ment of the Poles Elevation or Angle VDT, and the suns declination VT, to find DT, the suns Amplitude; for which the Propor∣tion is; sVDT. Rad. ∷ sVT, sDT.

Problem V. The Poles Elevation and suns declination be∣ing given, to find the ascentional difference.

In Fig. 9. In the Right Angled spherical triangle DVT, we have given the Comple∣ment of the Poles Elevation, or Angle VDT, and suns declination VT, to find the Ascen∣sional Difference DV; the Proportion is, tVDT, Rad. ∷ tVT . sDV.

Problem VI. The right ascension, and ascensional diffe∣rence being given, to find the Oblique ascension and declination.

In Fig. 9. GV, represents the right As∣cension; DV, the Ascensional difference; GD, the Oblique Ascension, which is found by deducting the Ascensional difference DV, from the Right Ascension GV; for if the Declination be North,

Add	The Ascensional diff. to or from the right Ascen. and it will give	Obl.	Asc.
Sub.	Desc.

If the Declination be South,

Sub.	The Ascensional diff. to or from the right Ascen. and it will give	Obl.	Ascen.
Add	Desce.

Problem VII. To find the time of the Suns rising and setting with the Length of the day and Night.

First find the Ascensional difference, as hath been shewed in the fifth Problem; which, when the Sun is in the Northern

signs, is to be added to the Semi-diurnal Arch of the Right sphere, which is 90, but is to be subtracted from the same, if he be in the southern signs, and the sum or difference will be the semi-diurnal Arch, which doub∣led, is the day Arch, whose Complement to 24, is the night Arch, which Bisected, is the time of the suns rising.

Problem VIII. The Poles Elevation, and the suns Declinati∣on given, to find the time when he will be due East and West.

In Fig. 10. In the Right Angled spherical Triangle TPZ, we have given PZ, the Complement of the Poles Elevation, and TP, the Complement of the suns declination, to find the Angle TPZ, for which the Propor∣tion is, Rad. . tPZ ∷ ctTP . csTPZ. whose Complement to a Quadrant TPD, being converted into time, sheweth how much it is after six in the Morning, when the sun will be due East, and before six at night, when he will be due West.

Problem IX. The Poles Elevation, with the Suns Altitude and Declination given, to find the Suns Azimuth.

In Fig. 10. In the Oblique Angled spheri∣cal Triangle SPZ, we have given SP the Complement of the suns declination, PZ the Complement of the Poles Elevation, and SZ the Complement of the suns Altitude, to find the Angle SZP, the suns Azimuth from the North; for which by the eleventh case of Oblique Angled spherical Triangles, the Proportion is; As the Rectangle of the sines of SZ, and ZP, is to the square of Radius, so the Rectangle made of sines of the differen∣ces of those containing sides and half sum of three sides given, to the Square of the sine of the half angle inquired.

CHAP. IV. Of the Secondary or Periodical Motion of the Stars.

Having done with the first part of Astro∣nomy, the motion of the Primum Mobile, and the affections of the stars, occasioned by that motion; we are now to speak of their

own Proper or Periodical motion; in which contrary to the motion of the Primum Mo∣bile, they are carried from West to East.

2. This motion of the fixed stars is very slow; for they alter their places but little in many Years, but are not immoveable as some thought; the quantity of their annual moti∣on, according to Tycho Brahe is 50 seconds, and 37 thirds of a degree, and others since him do conceive that 50 seconds only is the quantity of their annual motion, that is most agreeable unto truth and observation.

3. This motion in the Planets is more swift, and although they never move out of the Zodiack, yet they do move sometimes in one part of Heaven, sometimes in another, sometimes towards the south Pole, sometimes towards the North, sometimes near one fix∣ed star, sometimes near another, and some∣times nearer, sometimes farther from one a∣nother also, whereas the fixed stars do al∣ways keep the same distance from one another.

4. The Planets do not move in one Orb, but every Planet hath a several Orb, where∣as the infinite number of fixed stars do all move in one only sphere or Orb.

5. The Names and Characters of the pla∣nets are these:

1. Saturn, whose mark is ♄, finisheth his re∣volution in 29 Years, 174 Days, 4 Hours.

2. Jupiter, whose mark is ♃, finisheth his Revolution in 11 Years, 317 Days, 15 Hours.

3. Mars, whose mark is ♂, finisheth his Revolution in 1 Year, 321 Days, 23 Hours.

4. The Earth or Sun, marked thus ☉, finisheth his Course in 365 Days, 5 Hours, 49 Minutes, 4 seconds, and 21 thirds.

5. The Moon, marked thus ☽, finisheth her Course in 27 Days, 7 Hours, 43 Minutes, and 6 seconds, but returneth not into Conjuncti∣on with the Sun, under 29 Days, 12 Hours, 44 minutes, and 3 seconds.

6. Venus, marked thus ♀, finisheth her Course in 224 Days, 16 Hours, 40' and 30".

7. Mercury, marked thus ☿, finisheth his Course in 87 Days, 23 Hours, 00' and 15".

6. The Civil Year, though it doth not ex∣actly agree, yet hath it some proportion with the Motions of the sun and Moon in every Nation; Romulus the founder of Rome, ap∣pointed the year at first to consist of 10 Moons, or Months, and called the first March, 2. April, 3. May, 4. June, the rest Quin∣tilis, Sextilis, S••ptember, October, November, December, because they were 5, 6, 7, 8, 9, and 10 Months distant from March.

After whom, Numa Pompilius added two Months more, and called them January and February, and appointed each Month to con∣tain 29 and 30 Days, whereby the Year, did consist of 354 Days, in which time the Moon returneth into Conjunction with the Sun, and this is the quantity of the Year in Turky

at this day; only in every third Year, they reckon 355 days. The Persians and Aegypti∣ans do also count 12 Moons or Months to their Year, but their Months are proporti∣oned to the time of the Suns continuance in every of the 12 signs: In their Year there∣fore, which is solar, there are always 365 days, that is, 11 days more than the Lunar Year.

And the Julian Year, which is the accompt of all Christendom, doth differ from the o∣ther only in this, that by reason of the suns excess in motion above 365 days, which is about 5 Hours, 49 Minutes, it hath a day in∣tercalated once in four Years, and by reason of this Intercalation, it is more agreeable with the motion of the Sun, the former dif∣fering from the Numan Year, 11 days and 6 Hours, the which 11 days, Julius Caesar di∣stributed amongst the Months, and the month Quintilis, was by him called July, ac∣cording to his own name; and Augustus Cae∣sar called the Month Sextilis, by the name of August, and altered the Position of days in each month to that which we now use, in which there are 52 Weeks, and one odd day, and this one day supernumerary, maketh an alteration in all the rest, so that the days of the Week, which used to be assigned by the Letters of the Alphabet, fall not alike in se∣veral Years, but Sunday this Year, must fall out upon the next years Monday, & so forward, till

seven years; and because the six odd Hours do make a day in four years, every fourth year hath a day added to its accompt, and such a year doth consist of 366 days, which doth occasion the Sunday letter still to alter till four times 7, that is, 28 Years be gone about. This Revolution is called the Cycle of the Sun, taking name from ☉, Sunday, the Letter whereof it doth appoint for every year, as by the Table may be seen.

To find which of 28 the present is, add 9 to the Year of Our Lord, because this Cir∣cle was so far gone about at that the time of Christ's birth, divide the whole by 28, what remains, is the present year; if nothing re∣main, the Cycle is out, and that year you must call the last, or 28.

This Intercalation of a day, placed in Fe∣bruary, doth occasion the Letter F to be twice repeated in the latter end of that Month, viz. upon the 24 and 25 days, and in such a year St. Matthias day is to be observed upon the 25 of that month, and the very next Sunday doth change and alter this letter, from which Leaping or Changing, such a year is called the Leap Year, and the Number of days in each Month is well expressed in these Distichs.

Thirty Days hath September,

April, June, and November;

February hath Twenty Eight alone,

All the rest hath Thirty and One:

—But when of Leap Year cometh the time,

Then Days hath February Twenty & Nine.

That this Accompt is somewhat too long, is acknowledged and confess••d by the most skilful Astronomers, as for the Number of days in a year, the Emperours Mathemati∣cians were in the right, for it is certain, no Year can consist of more than 365 days, but for the odd Hours, it is a certain that they cannot be fewer than five, nor so many as six, so that the doubt is upon the minutes, sixty whereof goeth to the making of an Hour; a small matter one would think, and how great in the recess and consequence we shall see.

Julius Caesar alotted 365 days, 6 hours, to his Revolution; but the Sun goeth about in less time, that is, (according to the most exact accompt,) in 365 days, 5 hours, 49 Minutes, and a little more; so that the Em∣perours year must of necessity breed a diffe∣rence in so many Minutes every year, be∣twixt the year which the Sun it self describes in the Zodiack, and that which is reckoned upon in the Calender, which though for a year or two may pass insensibly, yet in the space of 134 years it will rise to a whole day

that is, the beginning of the year in the Calender must be set one day back.

As for Example.

Let the Year begin in the Vernal Aequi∣nox, or Spring, in the Emperours time, that fell out to be the 24th of March, but now this year it fell out the 10th of March, 13 days backward and somewhat more; and so if it be let alone, will go back to the first of March, and first of February, till Easter come to be on Christmas day, and so infi∣nitely.

To reform this difference in the accompt, some of the latter Roman Bishops earnestly endeavoured, and the thing was brought to that perfection it now standeth, (so much as it is,) by Gregory the Thirteenth, in the year 1582. his Mathematicians, (whereof Lylius was the chief) advised him thus: That considering there had been in agitation in the Council of Nice, somewhat concerned in this matter upon the motion of the Que∣stion about the Celebration of Easter; and that the Fathers of that Assembly, after due deliberation with the Astronomers of that time, had fixed the Vernal Aequinox; at the 21th of March, and considering also, that since that time a difference of 10 whole days had been past over in the Calender, that is, that the Vernal Equinox or Spring, which began upon the 21th of March, had prevented so much as to begin in Gregories

days, at the 10th of the same, 10 days dif∣ference, or thereabout; they advised, that 10 days should be cut off from the Calender, which was done, and the 10 days taken out of October, in the Year 1582, as being the Month of that Year in which the Pope was born, so that when they came to the 5th of the Month, they reckoned the 15th, and so the Aequinox was come up to its place again, and happened upon the 21th of March, as at the Council of Nice. But that Lylius should bring back the beginning of the year to the time of the Nicene Council, and no farther is to be mar∣velled at; he should have brought it back to the Emperours own time, where the mi∣stake was first entred; and instead of 10, cut off 13 days; however this is the reason why these two Calenders differ the space of 10 days one from another.

And thus I have given you an accompt of the year as it now stands with us in England, and with the rest of the Christian World, in respect of the Sun; some other particulars there are with us and them, that do depend upon the motion of the moon, for the better understanding of them, I will give you a brief accompt of her Revolution.

The Solar year consisting, as hath been said, of 11 days more than the Lunar year; those 11 days called the Epact, are there∣fore added to the Lunar Year, to made it e∣qual

with the Solar, by the addition of which access, in every three years there is gotten a number more than 30, but because the Moon between change and change doth never pass 30 days, the Epact cannot exceed that num∣ber, and the time in which the Moon is sup∣posed to make her several Motions and so return to the place where she first began, is a circle or Revolution of 19 years, first found out by Meton, and Athenian, who lived about 439 years before Christ, this Cycle is therefore called Cyclus Decenno ••ennalis, and from the Autor A••nus Metonicus from whose Atheni∣ans, the Aegyptians may seem to have receiv'd it, as the Romans from them, in letters of Gold, from whence (if not from the more precious use of it) it attained to be called, as yet still it is, the Numerus Aureus, or Golden Number: It is made Christian, by the Fathers of the Ni∣cene Council, as being altogether necessary to the finding out of the Neomenea Paschalis, up∣on which the Feast of Easter and all the rest of the moveable Feasts depend: It self is found by adding a Unite to the year of our Lord, and dividing the whole by 19, the Remainder shall be the Cycle of the Moon, or if nothing remain, the Cycle is out, that is, 19.

And the Epact is found by Multiplying the golden Number by 11, and dividing the Pro∣duct by 30, what remains is the Epact; but to save this trouble of Calculation, you have it set down to your hand in the table before the Ca∣lender

the use whereof as of the Golden Number is to find the Change of the Moon, for the Ancient Philosophers supposing the Moon to make a perfect Revolution in 19 years, did Calculate the several Changings of the Moon that happened in each Month for that time, and placed the golden number for each year, right against the day of the Month on which the Moon changed, that so having found the golden number, they might thereby presently know on what day of the month the Moon did change, in any Month of the year for ever, as also the time, when the Feast of Easter was to be observed, according to the Canon made at a General Council held at Nice, in the year of our Lord, 322, in which it is commanded that Easter should be Celebra∣ted upon the next Sunday following the first Full Moon after the Vernal Aequinox, which then was upon the 21th day of March, and accor∣ding to this rule is this Feast observed with us at this day, and not according to the true Mo∣tion of the Moon, or precise time of the Ver∣nal Aequinox, which now is about the 10th of March; This use of the Golden Number is well expressed in these Distichs.

In March after the first C, Look the Prime where e're it be; The third Sunday after Easter day shall be, And if the Prime on the Sunday be, Then reckon that for one of the three.

To find the New Moons by the Epact, do thus, To the Epact for the year given add the number of the months from March including both Months, and the days of that Month past, the Sum of these three Numbers shall give you the age of the Moon, if they exceed not 30, if more than 30, cast away 30, and the Remainder shall be the Age as before.

Example.

I would know the Age of the Moon on the Fifteenth day of August, 1672. The Epact is 11, and the Months from March to August are 6, and 15 the day of the Month, all which put together, do make 32, from whence take 30, and there rests 2, the Age of the Moon that day. And to know the day of the Change, do thus: To the Epact add all the months from March, and if they joyned together, come not to 30, look what they lack of 30, and at so many days of the month the moon chan∣geth: If they be above 30, and the month you desire have 31 days, then Subtract 30; but if 30 days, then Subtract 29, and that rest take from 30, then look what remains, and at so many days of the month the moon changeth, by either of these ways the time of the new moons may indeed be guest at, but not exactly found: How that may be done, is shewed in the larger Treatises of this Subject; this we deem sufficient for our present purpose.