Grammelogia, or, The mathematicall ring extracted from the logarythmes, and projected circular : now published in th[e] inlargement thereof unto any magnitude fit for use, shewing any reasonable capacity that hath not arithmeticke, how to resolve and worke, all ordinary operations of arithmeticke : and those that are most difficult with greatest facilitie, the extract on of rootes, the valuation of leases, &c. the measuring of plaines and solids, with the resolution of plaine and sphericall triangles applied to the practicall parts of geometrie, horo[l]ogographic, geographie, fortification, navigation, astronomie, &c, and that onely by an ocular inspection, and a circular motion / invented an[d] first published, by R. Delamain, teacher, and student of the mathematicks.

Of the Examination of the Graduation of the Circles of the Ring, which may serve as an in∣ducement and furtherance to the Learner, to fit and acquaint him how with prompt∣nesse to conceive of opposite numbers in the answering of Questions following.

FIrst, to examine the Circle of Numbers, bring any number in the moveable to halfe of that number in the fixed: so any number or part in the fixed shall give his double in the moveable, and so may you trie of the thirds, fourths, &c. of numbers, vel contra.

2. Bring 2. in the moveable unto 3. in the fixed, so against 3. in the moveable is 4. and 5. tenths in the fixed, against 4. and 5. tenths in the moveable is 6. and 75. 100. in the fixed, and so may you goe on in try∣ing the divisions of the Circle of Numbers in continuall proportion to other numbers, according to the Table A.

3. The Instrument not removed from the rectification of 2. unto 3. right against 3. in the moveable is 4. and 5. tenths in the fixed, and against 4. in the moveable is 6. in the fixed, but against 4. in the fixed is 2. and 66. 100ths in the moveable; against 5. in the moveable is 7. & 5. tenths in the fixed, but against 5. in the fixed is 3. and 33. 100ths in the moveable, and so may you proceede in examining farther, ac∣cording to the Table B. in which M. at that head of the Table signifieth moveable, and F. at the head of the Table signifieth fixed, and so against the numbers under M. or F. is one anothers answer.

4. Bring 3. in the moveable unto 2. in the fixed, so right against the sine of 90. in the moveable is 41. gr. 44. m. in the fixed; ••gainst 60. gr. is 41. gr. 02. m. against 75. gr. in the moveable is 40. gr. 05. m. in the fix∣ed, and so you may examine further, as in the Table C.

5. To examine the sines amongst them selves in continuall proportion, as 6. gr. to 7. gr. bring 6. gr. in the moveable to 7. gr. in the fixed, so right against 7. gr. in the moveable is 8. gr. 10. m. in the fixed, and right against this 8. gr. 10. m. in the moveable is 9. gr. 32. m. in the fixed, and so may you goe on in examining other sines on the Instru∣ment in continuall proportion, according to the Table D.

6. The Instrument being at this stay against 10 gr. in the moveable as 11. gr. 41. in the fixed, but against 10. gr. in the fixed is 8. gr. 34. m. in the moveable; against 15. gr. in the moveable is 17. gr. 34. m. in the fixed, but against 15. gr. in the fixed is 12. gr. 50. m. in the moveable, and so may you examine other sines, according to the Table E.

7. In the examination of the graduation of the Tangents: the Instru∣ment not removed right against the Tangent of 6. gr. in the moveable is the Tangent of 7. gr. & the sine of 7. gr. 3. m. in the fixed; against the Tangent of 7. gr. in the moveable is the Tangent of 8. gr. 9 m. & the sine of 8. gr. 14. m. in the fixed, against the Tangent of 8. gr. in the movea∣ble is the Tangent of 9. gr. 18. m. & the sine of 9. gr. 26. m. in the fixed, & so may you examine further according to the Table G.

A
M	F
2	3
3	4. 5
4. 5	6. 75
6. 75	10. 10
10. 10	15. 2
15. 2	22. 8
22. 8	34. 2
34. 2	51. 2
51. 2	76. 9
76. 9	115. 4
C
90	41. 49
80	41. 02
75	40. 05
70	38. 47
65	37. 10
60	35. 16
55	33. 06
50	30. 42
45	28. 07
40	25. 22
35	22 29
30	19. 28
25	16. 22
20	13. 10
15	9. 56
10	6. 39

B
M		F
2	F	3	M
3	4. 5	3	2. 0
4	6. 0	4	2. 66
5	7. 5	5	3. 33
6	9. 0	6	4. 00
7	10. 5	7	4. 66
8	12. 0	8	5. 33
9	13. 5	9	6. 00
10	15. 0	10	6. 66
15	22. 5	15	10. 00
20	30. 0	20	13. 33
25	37. 5	25	16. 66
35	52. 5	35	23. 33
45	67. 5	45	30. 00
55	82. 5	55	36. 66
65	97. 5	65	43. 33
75	102. 5	75	50. 00
85	127. 5	85	56. 66
95	142. 5	95	63. 33
100	150. 0	10	66. 66

D
M	F
6. 0	7. 0
7. 0	8. 10
8. 10	9. 32
9. 32	11. 08
11. 08	13. 00
13. 01	15. 13
15. 13	17. 50
17. 50	20. 55
20. 55	24. 35
24. 35	29. 20
29. 01	34. 27
34. 27	41. 15
41. 15	50. 15
50. 15	63. 41

E
M		F
6	F	7	M
10	11. 41	10	8. 34
15	17. 34	15	12. 50
20	23. 03	20	17. 04
25	29. 31	25	21. 15
30	35. 39	30	25. 24
35	41. 58	35	29. 28
40	48. 33	40	33. 27
45	55. 32	45	37. 20
50	63. 16	50	41. 04
55	72. 51	55	44. 38
		60	47. 58
		65	51. 00
		70	53. 42
		75	55. 56
		80	57. 38
		85	58. 42
		90	59. 04

G
M	F	F	M	F	F	M	F	F	M	F	F
	T	S		T	S		T	S		T	S
6	7. 0	7. 3	16	18. 29	19. 32	26	29. 38	34. 39	36	40. 16	57. 54
7	8. 9	8. 14	17	19. 37	20. 52	27	30. 43	36. 27	37	41. 18	61. 28
8	9. 18	9. 26	18	20. 45	22. 16	28	31. 48	38. 19	38	42. 20	65. 23
9	10. 28	10. 38	19	21. 52	23. 40	29	32. 52	40. 15	39	43. 22	70. 46
10	11. 37	11. 51	20	23. 00	25. 06	30	33. 57	42. 18	40	44. 22	78. 03
11	12. 46	13. 06	21	24. 07	26. 35	31	35. 00	44. 28
12	13. 55	14. 21	22	25. 14	28. 06	32	36. 04	46. 46
13	15. 04	15. 37	23	26. 20	29. 40	33	37. 09	49. 15
14	16. 12	16. 54	24	27. 26	31. 16	34	38. 11	51. 51
15	17. 21	18. 12	25	28. 52	32. 56	35	39. 14	54. 33

A Type of the Ringe and Scheme of this Logarithmall projection, the use followeth.

These Instruments are made in siluer or Brasse by Iohn Allen neare the Sauoy in the strand

[illustration]

The mouable Compasse

The f•••• ed Proiection on a plaine

The mouable Proiection In a Ring M. mouable F. fixed

In Astronomie.

Pro. 1. The sunnes place or distance from the Aequinoctiall points knowne, to find his Declination.

Constructio. BRing the sine of 90. in the moveable unto the sine of the Tropicall point, viz. 23. gr. and a halfe in the fixed, so right against the sine of the degree of the Suns neerest distance from ♈ or ♎ in the moveable is the sine of the Suns declination of that degree in the fixed.

Declaratio. So if the Suns place where in the beginning of ♊. ♌. ♐. or ♒. which is 60. gr. of distance from the Aequinoctiall points; right against this 60. gr. in the moveable, is the Declination in the fixed, viz 20. gr. 12. m. if the d••¦stance were 15 gr. 12. m. the Declination would be 6. gr. if 10. gr. of distanceth •• in the moveable gives 3. gr. 58. m. in the fixed, if the Sunne have 3. gr. of distance this in the moveable gives 1. gr. 12. m. in the fixed.

Otherwise you may turne the 3. gr. or such which are lesse) into minutes by al∣lowing 60. minuts to a degree which 3. gr. makes 180 m. minuts, this sought out in the moveable amongst the Numbers gives 72. minuts in the fixed, which is 1. gr. 12. m. of Declination as before: But if you make a degree to containe 100. minuts or parts, then the 3. gr. will be 300. minutes or parts, so right against this in the moveable amongst the Numbers is 1••0. in the fixed, which is 1. gr. & 20. hunderth of a gr. of Declination. The Instrument being not removed, you may have the Decli∣nation for any other parte of the Ecclipticke, as for 1. minut for right against 60. seconds in the moveable (which is answerable to 1. minut) is 24 seconds in the fixed, the declination belonging to 1. minute, &c.

Pro. 2. To finde the Sunnes amplitude, or distance of rising or setting from the East or West knowing the Sunnes place and Latitude.

Constructio. Bring the sine of the Complement of the Latitude in the moveable unto the sine of 23. gr. 30 m. in the fixed, so against the sine of the Sunnes distance from ♈ or ♎ in the moveable is the sine of the Suns Ampli∣tude

[illustration]

Amplitude.

in the fixed.

Declaratio. So the Latitude being 51. gr. 30. m. the Comple∣ment is 38. gr. 30. m. Bring this to the sine of 23. gr. 30. m. in the fixed: now if the Sunne have 90. gr. of Longitude: right gainst this 90. in the moveable is 39 gr. 50. m. in the fixed, the greatest Amplitude of the Sun in that Latitude.

If the Longitude were 70. 60. 50. 40. 30. 20. 10. or 5. right against any of these num∣bers in the moveable (or any other) is the Suns Amplitude in the fixed, viz. against the Longitude of	70	is	37. 00
60	33. 42
50	29. 23
40	24. 19
30	18. 38
20	12. 39
10	6. 23
5	3. 12

As for any Longitude which is neere the Equinoctiall point, the Am∣plitude of it may be had on the Numbers, as in the former Ex∣ample.

In Astronomie. But if the Declination of the Sunne, or a Starre be knowne; the Amplitude may be found thus.

Constructio. BRing the Sine of the Complement of the Latitude in the moveable unto the sine of 90. in the fixed, so right against the sine of the Sunne or Starres declination in the moveable, is the sine of the Sunne or Starres Ampli∣tude in the fixed.

Declaratio. So if th•• Latitude were 51. gr. 30. m. the Comple∣ment

[illustration]

is 38. gr. 30. m. bring this in the moveable unto the sine of 90. in the fixed, and if the Declination of the Sunne or a Starre were 20. gr. right against this 20. in the moveable is the Amplitude in the fixed, viz. 33. gr. 20. m.

The Instrument not removed you may for any Decli∣nation have the Amplitude of it: for right against the Declination in the moveable is the Amplitude in the fixed, &c. and there may you see what Declination such Starres have, which never rise, or set in that Latitude.

This Proposition may be inverted and applied to practice in Navigation to finde the Latitude, by knowing the Sunnes place and Amplitude, for if you bring the degree of the Sunnes Amplitude amongst the Sines in the moveable, unto the degree of the Sunnes place in the fixed; right against 23. gr. 30. m. in the move∣able, is the degree of the Complement of the Latitude in the fixed.

Pro. 3. In any Latitude to finde what higth the Sunne most have to be due East or West knowing the Sunnes place.

Constructio. Bring the sine of the Latitude in the moveable unto the sine of 23. gr. 30. m. in the fixed; so right against the sine of the Sunnes distance from ♈ or ♎ in the moveable, is the Sine of the Sunnes he••ght in the fixed:

[illustration]

when he is due East or West.

Declaratio. So if the Latitude were 51. gr. 30. m. bring the Sine 51. gr. 30. m. in the moveable, unto the Sine of 23. gr. 30. m. in the fixed: and if the Sunnes place were from ♈ or ♎ 90. gr. 80. 70. 60. 50. 40. 30. 20. or 10. gr.

			G. M.
Right against this	90	In the moveable is	30. 38	in the fixed, the Sunnes height answerable to his place.
80	30. 07
70	28. 37
60	26. 11
50	23. 00
40	19. 08
30	14. 46
20	10. 02
10	5. 04

In Astronomy.

HEre note that if the Zenith be betweene the Tropickes, that then the Sun sometimes will not be East or West, & what degrees those are you may easily try, for any Latitude, by moving the moveable soft∣ly along, that as the sine of any Latitude in the moveable passeth by the sine of the Tropicall point in the fixed, so any degree in the move∣able that passeth by the sine of 90. in the fixed, doth shew that degree to be the greatest degree of Longitude in the Eclypticke from ♈ or ♎ that the Sun will be due East or West in that Latitude, and if the Instru∣ment stay at that Elevation, any degree of Altitude in the fixed doth shew his degree of Longitude in the moveable, or any degree of Lon∣gitude in the moveable will shew at what Altitude the Sunne is due East or West in the fixed, for the one is opposite to the other. So if the Latitude were 23. gr 30. m. every degree of Longitude in the moveable would give the same degree of Altitude in the fixed to bee due East or West; so if the Sunne be in Longitude 20. gr from ♈ or ♎ then the Sunnes A••titude would be also 20. degrees when it comes due East or West: if the Suns place be 50. from ♈ or ♎ then the Altitude would be 50. to make due East, &c. But if the Latitude were lesse than 23. degrees, 30. m. as admit 15. degrees bring the sine of this 15. in the moveable unto the sine of 23. degrees, 30. in the fixed: so right a∣gainst 90. degrees in the fixed is 40. degrees, 28. m. in the moveable: so when the Sun commeth to be in Longitude, from ♈ or ♎ above 40 degrees 28. m. that is beyond the 10. degrees, 28. m. of ♉ the Sun will not be East or West for more than 100. daies, untill hee come to the 59. degree, 32. m. of ☊, and then every day after the Sunne will crosse the Verticall Circle of East or West, untill the S••nne passe the o∣ther Equinoctiall point of ♎.

If the declination of the Sunne or the Starre bee knowne, you may finde the Altitude thereof at the point of East in any Lati∣tude by this Rule.

Bring the sine of the Latitude in the moveable unto the sine of 90. in the fixed: so right against the sine of any Declination in the moveable is the sine of the Sunnes Altitude in the fixed.

[illustration]

Hence you may conceive, that if the Zenith bee be∣tweene the Tropicks, then the Declination that is equall to the Latitude, shewes the greatest Altitude to bee East: and any greater Declination shewes you that the Sunne will not be East: but if the Declination bee lesse than the Latitude, then right against it in the moveable is the Sunnes Altitude in the fixed.

In Astronomie.

Pro. 4. Knowing the height of the Sunne at the point of East or West, in any Latitude to finde the houre of the day.

Construct. BRing the Tangent of 45. in the moveable unto the Sine of the comple∣ment of the Latitude in the fixed; so right against the Tangent of the Suns Altitude in the moveable, is the Tangent of the houre from six

[illustration]

Declaratio. Let the Latitude be 51. gr. 30. m. and the Suns Altitud•• bring the Tangent of 45. gr. in the moveable unto the Sine of 38. gr. ••¦ed, so right against the Tangent of 14. gr. 45. m. in the moveable is 9. gr. 18. m. in the fixed, which reduced into Time, by allowing to eve∣ry degree 4. minutes makes 37. minutes of time: so the Sunne was due East that day 37. minutes after 6. in the morning, or due West 37. minutes before 6. in the afternoone.

The Instrument not removed, you have the time of the Sunnes being East or West for any other Altitude: for right against the Suns Altitude in the moveable is the degree of time that the Sunne comes due East or West in the fixed.

Or if the declination of the Sunne bee knowne. Bring the Tangent of 45. in the moveable unto the Tangent of the Latitude in the fixed (if it bee greater than 45. gr.) so right against the Tangent of any Declination in the move∣able

[illustration]

is the degree of time amongst the Sines in the fixed, that the Sunne or Starre will bee due East or West according to that declination; but if the Latitude be lesse than 45. gr. then bring the Tangent of the Latitude in the moveable unto the Tangent of 45. gr. in the fixed, so right against the Tangent of the Declina∣t••on in the moveable is the degree of time in the fixed amongst the Sines. Thus if the Latitude were 70. bring the Tangent of 45 unto the Tangent of 70. in the fixed, so right against the Tangent of the Suns Decli∣nat••on in the moveable admit 23. gr. 30. m. is the sine of 9. gr. 6. m. the Instrument not removed, you have the time of the Sunnes comming East or West, for any other Declination, for right against the Declination in the moveable are the degrees of time in ••he fixed.

But if you move the moveable softly along, as the Tangent of 45. in the moveable passeth by the Tangent of any Latitude in the fixed, so the degrees of Declination in the moveable passeth by their degrees of time in the fixed, amongst the Sines at what houre any such degrees are due East or West, untill the Tangent of 45. in the moveable bee opposite unto 45. gr. in the fixed: for then as the Tangent of any Latitude lesse than 45. in the moveable passeth by the Tangent of 45. in the fixed, so any degree of Declination in the moveable will shew his de∣gree of time amongst the Sines in the fixed, that the Sunne is due East or West.

This Proposition of finding the houre of the Sunnes being East or West, may serve to great use both on Sea and Land, in rectifying of Glasses, Watches, or such like, to keepe and Regulate the account of Time.

In Astronomie.

Pro. 5. To finde the time of the Sunne Rising or Setting in any Latitude, the Sunnes declination knowne.

Constructio. BRing the Tangent of 45. gr. in the moveable unto the Tangent of the Latitude in the fixed (if it be under 45. but if the Latitude be 〈◊〉〈◊〉 above 45. then bring the Tangent of the Complement of Latitude in the move∣able unto the sine of 90. in the fixed.) So against the degree of Declination amongst the Tangents in the moveable, is the degree of the Ascentionall difference among the Sines on the fixed.

Declaratio. So if the Latitude were 20. gr. and the Declination were 10. gr. Bring the Tangent of 45. gr. in the moveable unto the Tangent of 20. gr. in the fixed, so against the Tangent of 10. gr. in the moveable is 3. gr. 41. m. amongst the Sines in the fixed, which being reduced into time, makes neere 15. m. and this added unto 6. (if the Sun have South declination, gives the Sunne Ri∣sing to be a quarter past 6. and that 15. minutes taken from 6.

[illustration]

gives the Sunnes setting: But if the Sunne have North Declina∣tion; in stead of Addition, use Substraction, &c. The Instrument not removed, you may at one instant have the difference of Ascention for any Declination, for right against the Declination in the moveable is the difference of Ascention in the fixed. Or contrarily, the difference of Ascention knowne, you have the Declination answerable unto it: In like manner may you worke for any Starre. Or you may speedily compare the longest day of any two, three, or more places by knowing their Latitudes, for having rectified the Instrument ac∣cording to the former directions, right against the Tropicall point in the movea∣ble, viz. the Tangent of 23. gr. 30. m. is the Sine of the greatest ascentionall diffe∣rence in the fixed for that Latitude; by which according to the former directions you may have the Sunne setting, this doubled as before gives the lengest or shor∣test day for that place. Hence for Practice, if you move the moveable softly along, as the Tangent of 45. gr. in the moveable, passeth by the Tangent of any degree of Latitude in the fixed, so the Tropicall point in the moveable, will passe by the Sine of the greatest difference of Ascention in that Latitude.

	G.		G. M.
So as 45. passeth by the Lati∣tude of	25	The Tropicall point, viz. 23. gr. 30. m. will passe by	11. 41	The greatest diffe∣rence of Ascensi∣on for those La∣titudes.
30	14. 32
35	17. 43
40	21. 24
45	25. 27
50	31. 12
55	38. 23
60	48. 51
65	68. 49

In Dyalling.

Pro. 1. To finde the distance of the houres, in horizontall plaines for an oblique Sphaere.

Constructio. BRing the Tangent of 45 gr. in the moveable unto the sine of the la∣titude in the fixed, so right against the degrees of the houres, in a right Sphaere amongst the Tangent in the moveable under 45. gr. are the degrees of the distances of the houres, from 12. amongst the Tangents on the fixed for an oblique Sphaere, but if the degrees of the houres bee more than 45. then right against the Tangent of those degrees in the fixed, are the degrees of the houre distances in the moveable.

[illustration]

☞ Note, that the houres in a right Sphaere are equall, the one unto another, viz. 15. gr. for one houre, 30. gr. for two houres, 45. gr. for three houres, &c. hence here after they are called equall houres, &c.

Declaratio. So if the Latitude were 51. gr. 30. m bring the Tangent of 45 in the moveable unto the sine of 51. gr. 30. m. in the fixed, so right against 15. gr. in the moveable a∣mongst the Tangents is 11. gr. 50. m. in the fixed amongst the Tangent, and so much is the distance of the houre of 1. or 11. from 12. in the Lati∣tude of 51. gr. 30. m. The Instrument not removed you may have any other hou•••• distance, halfe houres, quarters, &c.

For right against those equal houres, viz.	15	Are the un¦equal houre distances, viz.	11. gr. 50. m.	For the houre of	1	or	1••
30	24. 20	2	10
45	38. 03	3	9
60	53. 35	4	8
75	71. 05	5	7
90	90. 00	6	6

By inverting this, you may trie for what Latitude ordinary pocket dialls are made, knowing the distance betweene the houre of 11. and 12. for if you bring the Tangent of 15. gr. in the moveable unto the Tangents of the distance in the fixed, right against the Tangent of 45. gr. in the moveable is the sine of the Latitude in the fixed.

Pro. 2. To finde the distance of the houres for a direct south Dyall in an oblique Sphaere.

Constructio. This differeth 〈◊〉〈◊〉 little from the former, only, here bring the Tangent of 45. gr. in the moveable unto the fine Complement of the Latitude in the fixed, so right against the Tangent of the degrees of the equall houres, are the Tangents of the degrees of the houre distances in the oblique sphaere.

Declaratio. So if the Latitude were as before 51. gr. 30. m. bring the Tangent of 45. gr. in the moveable, unto the sine of 38. gr. 30. m. in the fixed; so right against the Tangent.

Of those de∣grees, viz.	15	Are the degrees of the houre distances, viz.	9. gr. 28. m.
30	19. 45
45	31. 55
60	47. 10
75	66. 42
90	90. 00

Now if you move the moveable softly along as the Tangent of 45. gr. passeth by the sine Complement of any Latitude, so the degrees of the equall houres, will shew the degrees of the unequal houre distances, in comparison of one Latitude to another. This may be otherwise represented in this fundamentall Diagram, if EZ, and EH be divided out of the Table of naturall Tangents, so that each Radius represent 90. gr. and so to move upon E, then may you place H to P, and as you move H from P to increase in Latitude, so the meridians passing by E H will shew the houre distances in a Horizontall plaine, and the houre distances in E Z for a verticall plaine, and this kinde of projection and motion may serve to other ex∣cellent purposes, &c.

In Dyaling.

Pro. 3. Knowing the Declination of a verticall plaine, and the latitude of the place, to finde what Angle the Axis makes with the plaine, common∣ly called the height of the stile.

Constructio. BRing the Sine of 90. in the moveable unto the sine Complement of the latitude in the fixed, so right against the sine Complement of the De∣clination, in the moveable, is the degree of the stiles height amongst the sines in the fixed.

The Instrument not removed, you may at one Instant see

[illustration]

the stiles Altitude for all Declinations in that Latitude: for right against the Complement of the Declination in the moveable, is the stiles height in the fixed.

Declaratio. So if the Latitude were 51. gr. 30. m. the comple∣ment of it is 38. gr. 30. m. which seeke amongst the sines in the fixed, and bring 90. gr. to it, now if the Declination of the plaine were 10. 20. 30 40. 50. 60. 70. 80. &c.

The Complement of those are.	80	So right against these in the moveable are	37. 49	in the fixed the stiles he••ght answerable to these Declinations.
70	35. 48
60	32. 37
50	28. 22
40	23. 35
30	18. 08
20	12. 18
10	6. 12

Pro. 4. Knowing the Latitude of the place, and the Declination of a verticall plaine to finde the number of degrees betw••ene the Meridian of the place, and the Meridian of the plaine, which may be called the diffe∣rence of Meridians.

Constructio. Bring the Tangent of 45. gr. in the moveable unto the sine of the La∣titude in the fixed (which admit 51. gr. 30. m.) so against this 45. 〈…〉〈…〉 is the Tangent of 38. gr. 3. m. in the fixed: So if the Declination of the Plaine be 38. gr. 3. m. then the difference of meridians is 45. gr. But if the Declina∣tion be lesse then this 38. gr. 3. m. right against the Tangent of any degree for such a Declination in the fixed is the degree of the difference of meridians in the moveable.

So if the Decli∣nation were	35	Right against these in the fixed are	41. 49	In the moveable, the dif∣ference of meridians an∣swerable to those Decli∣nations.
30	36. 25
25	30. 44
20	24. 56
15	18. 54
10	12. 42
5	6. 23

If the Declination be above 38. gr. 3. m. you may move the Tangent of 45. softly alonge by the Tangentiall degrees of Declination in the fixed, untill 45. gr. in the moveable be opposite to 45. gr. in the fixed, and as it passeth by any Declination, the sine of the Latitude in the fixed, will give the difference of Meridians amongst the Tangents in the moveable. Lastly, if the Declination be above 45. gr. then the Declination most be accounted amongst ••he degrees of the moveable, and if you move the moveable softly along: As the degree of any Declination, passeth by the Tangent of 45 gr. in the fixed: so the Tangent of the difference of Meridians in the moveable passeth by the sine of the Latitude in the fixed, &c.

Pro. 5. To finde the houre distances in Dyalling in a Declining plaine by knowing the former, viz. the stiles height, and the difference of Meridians.

Declaratio. LET the Declination of a Plaine be S. W. 50. gr. according to the for∣mer Instruction, the stiles height would be 23. gr. 35. m. and the diffe∣rence of Meridians 56 gr. 43. m. Now before the houre distances can be knowne which are alwayes unequall, there may be made a Table of equall houres thus.

First, place downe 56. gr. 43. m. the difference of Meridians against 12. as in the Table, then by adding 15. gr. unto this difference of Meridians, viz. to this 56. gr. 43. m. it makes 71. gr. 43. m. for the distance betweene the houre of 11. and the Meridian of the plane: unto this 71. gr. 43. m. adde another 15. gr. makes 86. gr. 43. m. for the houre distance betweene 10. and the Meridian of the Plaine: But for the houre of 1. 2. 3. that 15. gr. must be taken from the said 56. gr. 43. m. so the houre distance of 1. is 41. gr. 43. m. the houre distance of 2. is 26. gr. 43. m. the

10	86. 43	81. 50
11	71. 43	50. 20
12	56 43	31. 20
1	41. 43	19. 35
2	26. 43	11. 22
3	11. 43	4. 45
4	3. 17	1. 15
5	18. 17	7. 32
6	33. 17	14. 35
7	48. 17	24. 10
8	63. 17	38. 25

houre distance of 3. is 11. gr. 43 m. now because 15. gr. cannot be taken out of 11. gr. 43. m. take this out of that, so the houre distance of 4. will be 3. gr. 17. m. now unto this adde 15. gr. makes 18 gr. 17. m. for the houre of 5. then adde 15. gr. to that, and so prosecute the Table which may be called the Table of equall houres from the Meridian of the plaine.

Now to finde the true houre distances, bring the Tangent of 45. gr. in the moveable unto the sine of the stiles height in the fixed, viz. 23. gr. 35. so right against the Tangent of any equall houre in the moveable under 45. gr. is the Tangent of the true houre distance in the fixed A. But if the equall houre distance be above 68. gr. 12. m. (which is right against 45. gr. in the moveable) then right against the equall houre distance in the fixed, is the true houre distance in the moveable B. Lastly, if the equall houre di∣stances be betweene 45. gr. & 68. gr. 12. m. then move the moveable softly along, and as the Tangent of any equall houre in the moveable passeth by the sine of the stiles ••e••ght in the fixed, so right against the Tangent of 45. gr. in the moveable is the Tangent of the true houre distance in fixed.

So the equall houre distan∣ces being	A	3. 17	The true houre di∣stances would be	1. 15 [illustration]
18. 17	7. 32
33. 17	14. 35
B	86. 43	81. 50
71. 43	50. 20
C	48. 17	24. 10
56. 43	31. 20
63. 17	38. 25

Having gotten the true houre distances from the Meridian of the Plaine, they may be placed against the houres as in the Table, and protra∣cted thus, draw the houre of 12. C. M. and on C. describe a semi-Circle: now seeing in the Table that the Meridian of the Plaine is from the houre of 12. in its true distance 31. gr. 20. m. protract it in the Circular Arke from D. to S. (because of West declination) otherwise contrary, and draw C. S for the Sub∣stiler; from this S. protract all the houres distances, as S. R. 4. gr. 45. m. for the houre of 3. S. Q. 11. gr. 22. m. for the houre of 2. &c. then may we draw the houre lines CN. CO. CP. CQ. CR. CT. CV. CW. CX. CΩ. and upon the said CS. place the stile A. B. C. perpendicular to the plaine, so it shall bee fitted for the casting of shadowes upon the said houre lines,

Of Plaine Triangles. Praecognita.

Theoreme. 1. IF a right line fall upon a right line, or if the side of an Angle be aug∣mented to make another Angle, those two Angles put together are equall to two right angles, that is twice 90. or 180. gr. by the thirtenth of the first of Eue. therefore knowing one of those Angles, the other is also knowne▪ So in the Triangle A B C augmenting the angular side B A to R. by the line R B and C A there is made two Angles, viz. C A B. and C A R. which together make 180. gr. and seeing the Angle C A B. is assumed to be 30. gr. necessarily by the said thirtenth Proposition, the other Angle C A R shall be 150. gr.

Theoreme 2. The three Angles of any plaine Triangle either Right, or Oblique by the 32. of the first of Euc. are equall to two right Angles, therefore one acute An∣gle knowne in a right angled Triangle, and two Angles knowne in an Oblique An∣gled Triangle, the third Angle is likewise knowne: hence in the right angled Tri∣angle A B C knowing the Angle at A. 30. gr. the Angle at C most be 60. gr. And so in the Oblique Triangle A H I. the Angle at A and I put together being 135. the Angle at H must be 45. gr. &c.

Propositio. I.
In any	Right Angled	Triangle, knowing	One Acute Angle, and a side opposite to either of the Angles, or the Hypo∣tenusae, and one side	to find the rest. [illustration]
Oblique Angled	Two Angles, and a side, or two sides and one Angle opposite to either of those sides

Axiome 1. The Axiome for the resolution of this proposi∣tion is thus.

The sides in all such Triangles, beare proportion the one to the other, as the sines of their opposite Angles doe. Or the sines of the Angles are directly proportionall to their opposite sides, by the seven & twentieth of the first Booke of Regiomontanus, the thirtenth Chap. of the 1. of Copernicus, and by the second Ax∣iome of the third booke of Pitiscus.

That is, the side A B (in the first Triangle) is in proportion to B C, as the sine of the Angle at C is to the sine of the Angle at A, or as A B. to A C. so the sine of the Angle at C. to the sine of the Angle at B. which are the opposite Angles of those sides: Againe in the oblique angled Triangle A D E, as the side A E is to E D, so is the sine of the Angle at D. to the sine of the Angle at A, or as the sine of the Angle at H, (In the fourth Triangle, is to the sine of the Angle at A, so is the side A I, to the si•••• I H, &c.

☞ And here note generally, that in the resolution of such Trian∣gles there is three things given (as afore said) by which a fourth is found, according to the method of the Golden Rule. But the disposing of the said three termes, is primarily, and princi∣paly to be considered.

How to dispose the three Termes knowne in any of the former Triangles fit for operation.

FIrst, if a side be required in a right, or oblique Triangle, & a side be knowne, and the Angles opposite to those sides bee also knowne: the Angle opposite to the side knowne shall have the first place of the Golden Rule, and the Angle opposite to the side required may have the second place of the Golden Rule, and the side knowne the third place.

Secondly, if an Angle be required, and an Angle knowne with the sides opposite to those Angles, then the side opposite to the Angle knowne shall have the first place of the Golden Rule, the side oppo∣site to the Angle required may have the second place of the Golden Rule, and the Angle knowne shall have the third place: the disposing of the Termes thus considered, the Angles have reference to the Cir∣cle of Sines upon the Ring, and the sides to the Circle of Numbers.

Example. Let the side BC. in the Rect-Angle Tri-Angle be required, knowing the side AB. 25. and the Angles at A and C. 30. gr. and 60. gr. therefore by the former. As the Sine of the Angle at C. 60. gr. to the Sine of the Angle at A. 30. gr. so the side AB. 25. to the side BC. 14. and 43. 100. and so of the rest.

[illustration]

The construction by the Ring is thus.

Bring 60. gr. amongst the Sines in the moveable to the Sine Complement of it in the fixed, viz. 30. gr. so right against AB. 25. in the Circle of Numbers in the moveable is 14. and 43. 100. in the fixed. But if the side AB. had beene any other number, you might instantly have the said BC. according to the same proportion.

☞ Here note generally, that if the two first Termes be upon the Circle of Sines, the other two termes are then upon the Circle of Numbers, vel contra. *

☞ Note further, that in all Trigonometrie, the termes or parts given either Angles or sides, are noted with a small stroke of the Pen thus — and the Termes or parts required either in the Angles or in the sides, thus O. so the sides AB. AD. AE. ••••. with the Angles at A. are given the sides AC. CB. A••. and ••••. and the Angles at E. and G. are demanded; now the mutuall proportion of these parts one unto another, is according to the former Axiome; by which infinite propositions may bee resolved of admirable consequence, lying under the habit of some one of those whose excellent use in ordinary Practicall things, I will illustrate in se∣verall kinds by sundry propositions; first, In Dementions; secondly, in Fortification; thirdly, in Navigation; fourthly, in Dyalling, as follow∣eth.

Vpon Plaine Triangles in Dementions. How to measure an Inaccessible height scituated upon a Hill, the Practice being not upon a Plaine.

THe example upon this I will take from mine owne observations which I made upon one of the goodliest Hills in this Kingdome, which belongs to Sir Ri∣chard Newport in Shropshire, called the Wreaken, not farre from Shrewsbury, which Hill I found to be neere 6. miles and a halfe in Circuit, and in the Perpendicular height 995. foot, as followeth.

Declaratio. Let the figure P. Q. S. N. represent the body of the Hill, and A B E C D a part of the side of it, now admit in the side of the Hill as at C, be a spring of water: and let a Well be sunke from the Top of the Hill at D to be le∣vill with the spring C, viz. at R: here it may be demanded how deepe this Well may be sunke, viz. D R, how farre it is from the spring C, to the botume of the Well R, and how farre it is from the said spring C, to the Top of the Hill D.

Constructio. First, with conveniencie I made choise of a Station at A, and there rectified my Instrument upon his Stay, or Rest, from thence I caused to be measu∣red to B 20. Chaines, and at B I caused another Stay or Rest to be erected of the same height that the former at A was: Then loking from A to B. by the Sights of the Instrument, I found the Angle of Ascent B A X. 4. gr. 44. m. and so I had the rectangled triangle A X B. in which A B was knowne 20. Chaines, or 1320 feet with the Angle at A 4. gr. 44. m. and therefore the Angle at B by the second The∣orem Pagr. 64. is 85. gr. 16. m. and the Angle at X by the same is 90. gr. and this is opposite to the side knowne, viz. A B. 20.

Now according to the Rule of operation by the Ring Page. 65. if the Sine of 85. gr. 16. m. in the moveable be brought to the sine of 4. gr. 44. m. in the fixed, right against 20. in the Circle of Numbers in the moveable is 1. Chaine, and 65. 100. of a Chaine in the fixed, which according to Page the ninth is 108 Foote, and 9. tenth. this shewes that the station at B was so many Foote higher, than that at A. By the same Rule A X. would be found to be 19. Chaines, and 93. hundreths of a Chaine, or 1315. Foote, and 5. tenths, those two dementions I first sought out.

Secondly, before I removed from A. I observed the Angle C A H. which I found to be 12. gr. 8. m. and also the Angle D A H. which was 15. gr. 55. m. those Angles I noted downe, and then going from A to B. there I rectified my Instru∣ment as I did formerly at A. and there I observed the Angle C B Y. which I found to be 16. gr. 45. m. and the Angle D B O. to be 22. gr. 12. m. thus for the ob∣servation.

But according to the first Theoreme Page. 64. if particularly those Angles be ta∣ken from 180. gr. leaves the angle G B C. 163. gr. 15. m. and the angle G B D. 157. gr. 48. m. unto these severall Angles, Adde the Angle B A X. 4. gr. 44. m. so have you the obtuse angle A B C. 167. gr. 59. m. And the obtuse Angle A B D: 162. gr. 32. m. And seeing formerly that the Angle C A H was 12. gr. 8. m. and the Angle D A H was 15. gr. 55. m. From either of those Angles I subtract the An∣gle B A X, viz. 4. gr. 44. m. leaves the Angle C A B. 7. gr. 24, and the Angle D A B 11. gr. 11. m. By which in the oblique Triangle A B D, and A B C are knowne in each of them two Angles, and so consequently by the a••oresaid second Theoreme, Pag. 64. The other Angles are also knowne (viz.) the Angle A C B, 4. gr. 37. m. and the Angle A D B 6 gr. 16. m.

NOw by the afore going first Axiome seing that in the Triangles A B D, all the Angles are knowne, and one side to wit A B. 20. the side A D may be also knowne.

Constructio. Bring the sine of the Angle A D B 6. gr. 16. m. in the moveable un∣to the sine of the Angle A B D. 1••2. gr. 31. m. (that is to the Sine of the Comple∣ment of that Angle, viz. 17. gr. 29. m.) in the fixed, so right against 20. in the move∣able amongst the Numbers is 54. Chaines and 93. 100. the. in the fixed, which in feete is 3625. and 6. tenths. and so farre it was from the eye at A to the top of the hill at D.

Thirdly, in the Rectangled Triangle A H D knowing as afore said the Angle D A H 15. gr. 55. m. by the second Theoreme Page 64. all the other Angles are knowne, viz. A D H 74. gr. 4. m. A H D 90. and seeing by the last worke that A D was found to be 54. Chaines, and 93 100ths. the other part•• of the Triangle by the afore said Axiome will be likewise knowne. For the Instrument not removed the Sine of 90. is against the Sine of 15. gr. 55. m. and right against 54. and 93. 100ths. is 15. Chaines, and 07. 100ths. of a Chaine, which by Page the ninth makes 995. foote the higth of the hill. In like manner may you finde the other side of the Triangle viz. A H, that is from the eye under the top of the hill, viz. 52. Chaines, and 82 hunderths, or 3486. foote, and 4. tenths.

Fourthly, in the Oblique Triangle A B C. all the Angles are knowne (as afore) with the side A B, and therefore the side B C would be found to bee 32. Chaines, or 2113. foote.

Fifthly, in the rect angled Triangle B Y C the Angle C B Y was formerly known•• to be 16. gr. 45. m. the Complement of which is the Angle B C Y. 73. gr. 15. m. and the side C B was formerly found to be 32. Chaines. therefore by afore said first Axiome of plaine Triangles, the other sides of the Triangle, viz. B Y. will be found to be 30. Chaines, and 65. hunderths. or. 2023. foote, and 3 tenths, and the side C Y. to be 9. Chaines, and 23 hunderths, or 608. foote, and 9. tenths.

Sixthly, seeing B X. is equall unto Y Q. unto the said C Y. 608. foote, and 9. tinths, add. B X. formerly found, viz. 108. foote, and 9 tinths makes Q C. 717. foote, and 8. tenths. but this Q C. is equall to H R. which taken from the higth of the whole hill D H, viz. 995. foote leaves D R. 277. foote, and 2. tenths, the depth of the well.

Seventhly, for as much as B Y was found to be 2023. foote, and 3. tenths. which is equall to X Q and A X being as before 1315. foote, 5. tenths, those two put to¦••ether makes A Q 3338. foote, and 8. tenths, which taken from A H, which was ••ormerly found to bee 3486. foote, 4. tenths leaves Q H. 147. foote. 6. tenths, which is equall to C R. the distance betweene the spring C, and the botome of the well R.

Eighthly, and lastly knowing D R. 277. foote, and 2. tenths, and C R 147. foote, and 6. tenths C D is found to bee 313. and 8. tenths, according to the first Propo∣sition, Page 19ths. the example being the same with the second Proposition: thus for the Instrumentall way: such as desir•• to examine the worke by numbers may proceede by the Notes here under specified.

In Trianguli.	AXB	Data	BAX 4. gr. 44. m.	Quaeritur	A X. 1315. & 5. 10ths.
A B. 1320.	B X. 108. and 9. 10ths.
ABD	A B. 1320.	A D. 3625. & 6. 10ths.
B A D. 11. gr. 11. m. 43. s.
ABD. 162. gr. 31. m. 45. s.
AHD	D A H. 15 gr. 55. m 43. s.	D H. 995.
A D. 3625. and 6. 10ths.	A H. 3486. & 4. 10ths.
ABC	C A B. 7. gr. 24. m.	B C. 2113.
A B C. 167. gr. 59.
A B. 1320.
C B Y. 16. gr. 45. m.	B Y. 2023. & 3. 10ths.
B C. 2113.	C Y. 608. & 9. 10ths.
CRD	R D. 227. and 2. tenths.	C D. 383. & 8. 10th
C R. 147. and 6. tenths.