Astronomia Britannica exhibiting the doctrine of the sphere, and theory of the planets decimally by trigonometry, and by tables : fitted for the meridian of London ... / by John Newton ...

CHAP. 9. Of the Theory and Motion of the Moon.

THe Moon according to our Hypothesis is a secondary planet, mo∣ving about the earth, as the earth and other planets doe about the Sun, and so not onely the earth, but the whole Systeme of the Moone is also carryed about the Sun in a yeare, And hence according to Hypparchus there ariseth a twofold, but according to Tycho a threefold inequality in the Moons motion. The first is periodicall, and is to be obtained, after the same manner, as was the excentrique Equation of the Sun or Earth; in order whereunto her middle motions should be first stated, the which Bullialdus by the rules delivered in the fourth and fifth Chapters preceding hath for the Meridian of Uraniburge determined to be as followeth.

From the Equinoctiall to the beginning of the Christian Aera, the

Moons middle motion was	135d.	16′	27″
The Meane Anomalie	355	5	18
And the Radix of her latitude	366	29	56

These then we will take for granted, until there be a more exact, and true Geometrical way propounded to us,; onely we will convert them into Decimall numbers, and reduce them to the Meridian of London.

From the Equinoctial to the beginning of the Christian Aera, The Moons middle motion in decimal numbers at Uraniburge was 135. 27417

For the Difference of Merid. adde

. 45750

The Moons mean longitude at London

135. 73167

The Meane Anomaly,

At Uraniburge

355. 08833

Differ. Merid. adde

. 45361

Mean Anomaly at London

355. 54194

The Radix of Latitude

At Vraniburge

366. 49889

Differ. Merid. adde

. 45944

Latitude at London

366. 95833 d.

The Diurnal Motion of the Moons

Mean longitude

13. 17639

Anomaly

13. 06500

Latitude

13. 22944

The Annual Motion.

In longitude

129. 38389

Anomaly

88. 71889

Latitude

148. 71278

According to which limitations of the Moones middle motions, we have composed our Tables, by help whereof and the Semi-excentricity of the Moons Orbe, which according to Bullialdus is 4362 the Moons excentrique equation, or place first equated may be found, as before was shewed in the Sun. Save onely that here the Moons Anomaly is given without subtraction.

Example.

Anno 1587, August 17 ho. 19. 41667 in the apparent time, or ho. 19. 28973 in the middle time, the Moon being in the meridian of Vraniburge noble Tycho observed her in 26 deg. 38333 of Gemini with latitude 5 deg. 23333 S. from which middle time if you subtract 83333 for the difference of the meridians of London and Vraniburge, the time in our meridian is, ho. 18. 45640.

And the Suns true place

154. 07347.

The Suns distance from the Earth

100895.

The Log••rithme of that distance

5. 0038707

The Moons middle motions for the same time are as here you see them.

Time given	☽ Longitud	☽ Anomaly	☽ Latitude
Years 1500	072. 88194	313. 06916	017. 17805
80	174. 24805	158. 80139	281. 61167
6	069. 48028	18••. 37750	185. 50583
Iuly	27••. 39555	249. 77639	284. 62194
D••yes 16	••10. 82222	209. 0••972	211. 66944
H••••res 18	••9. 88222	9. 79861	9. 92222
Paris 4564	. 250••1	. 24848	. 25152
Mean Longitude	810. 96077	1126. 11125	990. 76067
Ded••ct	720.	1080.	720.
There rests	90. 96077	46. 11125	270. 76067

The Moones meane Anomaly 46. 11125 is the angle A M E in the preceding Ellip••is, or the summe of the angles M E H and M H E. Therefore in the triangle M E H we have given, 1. The side M E 200000. 2. The side M H 8724. 3. The angle E M H the complements of the Moones Anomaly, to find M E H, whose double, is the excentrique E∣quation M B H. I say then,

As the summe of M E and M H	208724 co. ar.	4. 6804276
Is to their difference	191276	5. 2816605
So is the tang. half summe of the opposite angles,	23. 05562	9. 6290228
To the tang halfe diff.	21. 30786	9. 5911109
Differ.	1. 74776 is the angle M E H

Differ▪ doubled 3. 49552 is the angle M B H or the equation sought which being subtracted from the Moons mean longitude, because the Anomaly is lesse then a semicircle you shall have the Moones place first equated.

Example.

The Moones meane longitude

90. 96077

Equation subtract

3. 49552

The Moones place first equated

87. 46525

And to find her distance from the Umbilique point at H.

As the sine of M B H	3. 49552 co. ar.	1. 2148808
Is to the side M H	8724	3. 9407156
So is the Sine of B M H	46. 11125	9. 8577468
To the side B H	103120	5. 0133432

But whilest the Moone is thus making her owne periodicall revoluti∣on, her whole Systeme is by the motion of the Earth removed from the proper seats thereof, whence there ariseth another revolution which is called Synodicall, the beginning whereof is the line which passeth through the center of the earth to the Sun, and therefore the Moone in that line is void of this second inequality, which is both at the Conjunction and op∣position, but being in or about her quarters, where she is farthest removed from the said Zyzigiacal line the angle of her evection is 2 deg. 50′ as is cleerely proved by the observations of Tycho and Bullialdus, whose me∣thod we follow, in our calculation of this inequality of the Moon, ma∣king 4362 the sine of the greatest evection to be the Diameter K D in the little circle K C D H. The motion of this libration of the Moone must be measured by her double distance from the Sun, because she is void of this inequality at her conjunctions and oppositions as was said before.

Now then let the angle M H B represent the Moones equated Anoma∣ly, found by subtracting the former Equation from the simple Anomaly, which is 46. 11125

Aequation subtract

3. 49552

Aequated Anomaly

42. 61573

[illustration]

And if from the place of the moon first equated

87. 46525

You subtract the Suns true place

154. 07347

Their distance is

293. 39178

The double distance

226. 78356

is the arch K C D H and drawing the lines F H and H K there shall be made the equicrurall Triangle H F K, whose exteriour angle H F D is known, viz. the excesse of the Sun and Moons double distance above a se micircle 46. 78356. The halfe whereof is the angle D K H 23. ••9178, e∣qual to the angle A H G, which being subtracted from the Equated A∣nomaly A H B 42. 61573. The angle G H B or the Synodical Anomaly will be 19. 2239••, whose complement to a semicircle is the angle B H K 160. 77605. The side H B 103120 as before, and the side H K may be found in this manner. The arch H V K is the complement of the double distance of the Sun and Moon to a whole circle 13••. 21644 the subtense of this arch is H K, H V the halfe arch is 66. 60822, and H X is the right sine thereof.

Now then, As the Radius,
Is to the Diameter K D	4••62.	3. 6396857
So is the sine of H V or H X	66. 60822	9. 9627533
To the Subtense H K	4003	3. 6024390

Therefore in the Triangle B H K we have known,

1. The angle B H K	160. 77605.
2. The side H B	103120 To find the angle H B K.
3. The side H K	4003 To find the angle H B K.
As the summe	107123 co. ar.	4. 9701173
To the differ.	99117	4. 9961481
So tang. ½ the opposite ang.	9. 61197	9. 2287638
To the tang. ½ differ.	8. 90502	9. 1950292

Their difference 0. 70695 is the angle of the evection, H B K to be added if the Synodicall Anomaly be more then 180, and to be sub∣tracted from the place of the Moone first equated when lesse, as here it is in our example, and therefore to be subtracted from the Moons place 87 46525, and then her place secondly equated will be 86. 7••830.

And according to this Analogie may be made a table of the Moons e∣vection, when she is in quadrature or 90 degrees distant from the Sun; for then the equated and Synodicall Anomalies are both the same, and therefore in the Triangle E M H we have give the angle E H M, or the e∣quated

Anomaly which suppose 25 degrees, the side M H 8724 and the side H E ••00000, to find the angle at E.

[illustration]

As the sum of H E and M H	208724 co. ar.	4. 6804276
Is to their difference	191276	5. 2816605
So is the tang. half summe		10. 6542447
To tang. half differ.	76. 40435	10. 6163328
Their differ.	1. 09565 is the angle M E H
Whose double is	2. 19130 is the angle M B H
Then as the sine of M B H	2. 1913 co. ar.	1. 4175273
To the sine of B M H	27. 1913	9. 6598808
So is the side M H	. 8724	3. 9407156
To the side B H	10. 4161	5. 0181235
or the Moons distance from the umbilique.

Hence in the first Diagram of this Chapter, in the Triangle B H K we have given A H B the equated Anomaly, 25 degrees, the Moons di∣stance H B 104161, with the side H K, or rather D K, the Diameter of the little circle 4362, to find the angle H B K.

As the summe of H B and H K	108523 co. ar.	4. 9644782
To their difference	99799	4. 9991262
So is tang. halfe summe	12. 50	9. 3447552
To tang. halfe differ.	11. 52314	9. 3093596

whose difference . 97686 is the evection sought and by Bullial∣dus . 97805 which is so little differing from what we have found, that I have taken his Table and converted it into Deci∣mall numbers.

And for the finding the part proportial between the quadrature and the Zyzygia, Bullialdus whom we follow, hath annexed scruples of pro∣portion in this manner.

As Radius to 60 minutes or one degree; so is the fine of halfe the de∣grees of the equated Anomaly to the scruples of proportion required.

But this proportion in the Sexagenary Canon will not give the scruples either so easily or so exactly as the Decimal Canon will, because the seconds must still be found in that Canon by the part proportional, which in the beginning of the Canon cannot be true, but working by a Decimal Canon the natural sines of halfe the degrees, are the Decimall parts required, as the naturall sine of 4 degrees; 069756, are the Decimall parts for 8 degrees of equated anomaly, and so of the rest.

Having done with the first and second inequalities of the Moon, we come to the third which Tycho calls the variation, Bullialdus the Reflecti∣on for as the Moons Systeme is carryed about by the earth, the place of her Apogaeon is changed, or doth reflect contrary to the succession of the Signes, by reason of which reflection the angle of her Evection is some∣times more sometimes lesse then it will be found by the former directions, but the quantity of this variation according to Tycho doth never exceed 40′ 30″ or in Decimal numbers 67500, to be added to or subtracted from the place of the Moone secondly equated, and the proportion by which he finds it is thus,

As the Radius

To the sine of the complement of the double distance of the Sun and Moon if lesse then a Semicircle: To the excesse if more:

So is the sine of the greatest variation, or Reflection.

To the variation required, which is to be added to the Moons place, secondly, equated if the double distance be lesse then 180 deg. to be sub∣tracted, when it is more.

Therefore in our Example,

As the Radius
To the sine of H F D	46. 78356	9. 8625917
So is the sine of	67500	8. 0711591
To the variation	49189	7. 9337508

The Moones place secondly equated

86. 75830

Variation subtract

49189

The Moones place in her Orbe

86. 26641

Lastly, to find the Moones latitude and place in the Ecliptique, take the middle motion of her latitude for the time propounded, the which rejecting the whole circles is 270. 76067 and in which according to Tycho there is a twofold variation, The first is occasioned by the various intersection of the Moones orbe with the Zodiack, and the o∣ther by the reciprocal progression and retrogradation of the Nodes. In the New and Full Moones the limits of her greatest latitude, are 4. 97500, but in her quarters 5. 29167, as Tycho hath experimented by many dili∣gent and accurate observations, whose method of calculation is as fol∣loweth.

From the meane motion of the Moones latitude

270. 76067

Subtract the Moones absolute Equation

4. 694 6

The Equated latitude of the Moon

266. 06631

Then to find the Equation of the Nodes, let the line A D or the angle A T D 5. 13333 represent the meane inclination of the Moones Orbe with the Ecliptique, let the least inclination be represented by A B 4. 97500, and the greatest by A C 5. 29166. And from the distance of the Sun and Moon before found,

 

293. 39178

Subtract the evection and variation

••••1. ••9884

True distance of the Sun and Moon

292. 19••94

The double thereof is

224. 385••8

which being numbred from B by C to F in the Triangle F D A we have known A D 5. 13333 the meane inclination of the Moones Orbe. 2. D F 15833 the halfe difference of the least and greatest inclination. 3. With the angle F D A 135. 61412, the complement of the double di∣stance of the Sun and Moon to a whole circle: whence to find the angle F A D the Aequation of the Nodes, by the Doctrine of spherical Triangles say,

[illustration]

First, As the Radius
To the cosine of F D C	44. 38588	9. 8540905
So is the tang. of D F	15833	7. 4413175
To the tang of D C	11321	7. 2954080
Adde the arch A D	5. 13333
Summe is A C	5. 24654
2 As sine D C	11321 c••. ar.	2. 7046002
To the sine A C	5. 24654	8. 9611430
So cotang. F D C	44. 38588	10. 0093107
To cotang. F A D	1. 21062	11. 6750539
From the Moones latitude equated		266. 06787
Equation Nodes subtract		1. 21062
True motion of the Moones latitude.		264. 85725
whose complement to a whole circle		95. 14275

And to find A F or the angle of 〈◊〉〈◊〉 A T ••, the 〈…〉〈…〉 of the Sun and Moone being more than 180 degrees, and lesse th•••• ••••••▪ I deduct the Moones double distance 224 d. 3858•• being numbred •••• the little circle, from B by C to F, from the Arch B C F G 270, there re∣maines F G 45. 61412, and then the A••alogie is,

As the Radius D C		10. 0000000
To the sine of F G or D F	45. 61412	9. 8540305
So is the sine of the arch, D C	. 15833	7. 4413159
To the sine of the arch D E	. 11319	7. 2953464

The aggregate is B E . 27152 which being added to the least angle of inclination A T B, or the arch A B 4. 97500 the present inclina∣tion is A F or the angle A T F 5. 24652.

Hence to find the Moones true latitude, I say,

As Radius		10. 0000000
To the sine of A T F	5. 24652	8. 9611413
So is the sine of A T	84. 85725	9. 9982481
To the sine of A F	5. 22533	8. 9593894
or the angle A S F.

And by these Analogies may be made the Table of the Moons latitude wc•• we have borrowed of Tycho, converting it onely into Decimall numbers.) For supposing the Moon to be in her Syzygial points, the angle of Incli∣nation is alwayes A T B 4. 975, and then her latitude for every degree of her true motion of latitude may be found by the last Analogie; As Radius

Example.

To the sine of A T B	4. 975	8. 9381242
So is the sine of A T	45.	9. 8494850
To the sine of A B	3. 51564	8. 7876092

And her latitude when she is in Quadratu••e or 90 degrees distant from the Sun may be found by the same analogie, if you make the angle of In∣clination A T C 5. 29166.

Example.

As Radius
To the sine of A T C	5. 29166	8. 9648517
So is the sine of A T	45.	9. 8494850
To the sine of A F	3. 73910	8. 8143367

Subtract A B 3. 51564 there rests the Excesse to be placed in the Table 0. 22346.

The proportipnal part of which excesse to be added to the Moones la∣itude

must be found by scruples of proportion, and the Scruples themselves for every degree of the Moones distance from the Sunne may thus be had.

[illustration]

As Radius▪
To the Co••ine of the Moones double distance D H	40	9. 8842539
So is the sine of D B	0. 15833	7. 4413575
To the sine of D H	0. 12142	7. 3256114
Their differ▪ is B H	0. 03691
Then as the Diameter B C	31666	5. 4994068
Is to the Diameter B C	100. 000	5. 0000000
So is B H	0. 03691	3. 5671440
To B H	0, 11656	4. 0665508

Or more readily thus D H 76604 is the sine of 50 or the Cosine of 40 the Moones double distance from the Sun, which being deducted from Radius, the remainder is the versed sine B H 23396 the halfe 11698, are the scruples of proportion answering to 20 deg. of the Moones single di∣stance from the Sun,

From the Moones place in her Orbe

86. 26641

Subtract the Moones true latitude

264. 85725

The Moones Node ascending

181. 40916

Lastly, for her Reduction▪

As Radius
To the Cosine of A T F	5. 24775	9. 9981757
So tang. of A T	84. 85725	11. 0458587
To tang. of T F	84. 83689	11. 0440344

Difference 02036 is the Reduction sought

From the place in her Orbe

86. 26641

Subtract her Reduction

. 02036

The Moones place in the Ecliptique

86. 24605