The mariners magazine, or, Sturmy's mathematical and practical arts containing the description and use of the scale of scales, it being a mathematical ruler, that resolves most mathematical conclusions, and likewise the making and use of the crostaff, quadrant, and the quadrat, nocturnals, and other most useful instruments for all artists and navigators : the art of navigation, resolved geometrically, instrumentally, and by calculation, and by that late excellent invention of logarithms, in the three principal kinds of sailing : with new tables of the longitude and latitude of the most eminent places ... : together with a discourse of the practick part of navigation ..., a new way of surveying land ..., the art of gauging all sorts of vessels ..., the art of dialling by a gnomical scale ... : whereunto is annexed, an abridgment of the penalties and forfeitures, by acts of parliaments appointed, relating to the customs and navigation : also a compendium of fortification, both geometrically and instrumentally / by Capt. Samuel Sturmy.

THE Mariners Magazine; OR, STURMY's Mathematical and Practical ARTS. The Second Book. (Book 2)

The ARGUMENT.

YOu're come to see a Sight, the World's the Stage; Perhaps you'l say, 'Tis a Star-gazing Age. Come out and see the Ʋse of Instrument, Can Speculation yield you such Content? That you can rest in Learning; But the Name Of flying Pegasus, or swift Charles-Wain. And would you learn to know how he doth move About his Axis, set at work by Jove? If you would learn the Practice, read, and then I need not thus intreat you by my Pen, To tread in Arts fair Steps, or gain the way; Go on, make haste, Delinquent, do not stay. Or will you scale Olympick Hills so high? Be sure take fast hold on Astronomy; Then in that fair-spread Canopy no Way From thee is hid, no not Galaxia. They that descend the Waters deep, do see Our great God's Wonders there, and what they be. They that contemplate on the Starry Sky, Do see the Works that he hath fram'd so high. Then learn the Worlds Division, and that Art Which I shall shew you in this Second Part.

IN this Book is contained both a general and particular Description, Making, and Use of all the most necessary Instruments belonging to the Art of Navigation; As the Mathematical Ruler, on which are these Scales following; viz. The Line of Chords, Points, Leagues, Longitude, Natural Sines, Tangents, Secants, at one End; at the other is Dialling Scales, viz. The Art of Dialling of all sorts, resolved by the Chords and Gnomon Line, and Scale of Six Hours; Scale of Inclination of Me∣ridians, and two Scales inlarging Hours; Lines upon any reclining, inclining, or de∣clining, Plain without a Center, called the greater and lesser Pole: On the other side is a Line of Artificial Signs, Tangents, and Numbers; A Meridian Line, according to Mercator's or Mr. Edward Wright's Projection; And Tables for the making of these Scales, with a Line of Longitude and Reduction, which are the Lines on the Mathe∣matical

Scale; also, A Portable most useful Travis-Scale, with a Table for to make it, with artificial Rhombs, Points, ½ Quarters, and Tangent-Rhombs, and the making of the Sinical Quadrant, and so ordered, that by the help of an Index, and Lines thereon, it shall answer most of the useful Questions in Astronomy and Navigation. Also the making the plain Sea-Chard, and the true Sea-Chard, and particular Chards for any Place; with the most useful and necessary Semicircle, that will protract any Angle, or run upon any Chard, without drawing Rhomb-lines to fill the Chard; that so, by help of this Instrument, the Chard may serve for many Voyages. Also the Ma∣king and Use of a Compleat Instrument, made in the manner and on the back-side of a Nocturnal, with 31 of the most useful and easiest Stars to be known in the North and South Hemisphere, of the first, second, and third Magnitude; which in a Mo∣ment, the Instrument being rectified, sheweth the Hour of the Night that any Star cometh to the Meridian, with his Declination N. and S. Also a Table of the De∣clination, Right Ascension, Latitude, and Longitude calculated from Tycho's Tables, re∣ctified for the year 1671. On the other side a Nocturnal so ordered, that it shall give you the Hour of the Night by the North-Star, and the brightest Guard, and his bearing every Point of the Compass from the Pole, whereby you may take the true De∣clination; and also being so rectified, sheweth the Suns place in each Sign and De∣gree in the Ecliptick every day in the year. The Making and the Use of the Cross-staff, Back-staff, Quadrant; The Making and Use of the small Pocket-Instrument, on which is contained the most useful Lines, Scales, and Proportions, that in an Instant will shew the Diameter of any sort of Ordnance at the Bore, and the length and weight of the Gun, and Shot, and Powder, in Brass or Iron; and the Diameter and Names of each Piece, Diameter of the Shot to each Piece, and the weight of any Iron Shot, the Diameter being given in Inches, with the breadth and length of the Ladle; And how many Paces point-blank any Piece will shoot, and of Randoms for the sixth Point of the Quadrant, which may by this Instrument be answered near enough for so short a time, to give any reasonable Man an answer to any useful Question in the Art of Gunnery. Also the Description of the Mariners Azimuth-Compass, so ordered that it shall measure all kind of Grounds whatsoever, whether Wood-land or other; and for taking of Heights and Distances, whether accessible or inaccessible: And by the help of the aforesaid Semicircle, to protract any Plot of a Field or Plantation whatso∣ever, as soon as any Instrument, as the Plain Table, the Theodolit or Circumferenter, with much delight and pleasure to the Ingenious Mariner, it agreeing so well with his Travisses at Sea. All which shall be shewn in the following Treatise in its due Place.

A DESCRIPTION OF INSTRUMENTS. CHAP. I. Of Instruments in general.

THe particular Description of the several Instruments that have from time to time been invented for Mathematical Practice, would make a Treatise of it self; and in this place is not so necessary to be insisted on every of the Inventors in their Construction. To omit therefore the Description and Super∣fluity of unnecessary Instruments, I shall immediately begin with the Description of those which are the Grounds and Foundation of all the rest, and are now the only Instruments in esteem amongst Navigators and Mariners at Sea, which are chiefly these; viz. The Mathematical Ruler, the Plain Scale, the Sinical Quadrant, the Plain Sea-Chard, and the True Sea-Chard, the particular Chard, the Semicircle or Protractor, the Nocturnal, the Cross-staff, Back-staff, and Quadrant; the Gunter's Scale, and the Mariner's Azimuth-Compass. Now as I would not confine any Man to the Use of any particular Instrument for all Imployments; so I would advise any Man not to incumber himself with Multiplicity, since these aforesaid are sufficient for all Occa∣sions. These special Instruments have been largely described already by divers; As namely, by Mr. Blundevil, Mr. Wright, Mr. Gunter, and others: but not fitted with Tables for the making of them, or demonstrated so plain to the Capacities of Sea∣men, as they are here. Therefore in this place it will be very necessary to give a parti∣cular Description of them, because that if any Man hath a desire to any particular In∣strument, he may give the better direction for the making thereof, or making of it himself.

Forasmuch as there is a continual use both of Scales and Chords, which are on the Mathematical Scale, in drawing of Schemes in the Art of Navigation, and all other sorts in this Treatise; Therefore we will demonstrate the fundamental Diagram of the Mathematical Scale, that all Mariners may understand (that have not the know∣ledge already) the making of them, which is a most commendable Vertue in an expert Mariner. I could wish that all Masters and Mates were able to make their own Instruments, that if they should be long at Sea, and by disaster break or lose their In∣struments; or if any in the Ship discovers the Practice, he may be able to make more for himself and others, without the help of the Artificer's Labour, and supply that defect by their own pains.

This Diagram plainly sheweth the making of the Scale of Degrees or Chords, and Points of the Mariner's Compass, in a Right Line B 8, being the Degrees, containing in all 90; and F 8 is the Scale for the Points of the Compass, being in all 8 Points for the ¼ part of the whole Circle.

Now for the Sines, Tangents, and Secants, you shall note, That the Semi-dia∣meter AB must be divided into a Radical Number, for the more ease in Calculation; as into 100, or 1000, 10000, 100000; and that by the Table of Natural Sines,

Tangents, Secants, Chords, and Points, which I have fitted on purpose for this Work. You may take off so many Numbers as the Table directs you, as shall be shewn.

[illustration] geometrical diagram

Here followeth a Table of 90 Degrees of the Quadrant. He that desires it larger, may make it to the Parts of a Degree. I have joyned the Chord proper to it, which is the Natural Sine of half the Arch doubled.

For Example, If you double the Natural Line of 6. 15. 25. 30 Deg. you shall produce the Chords of 12. 30. 45. 60 Degrees; thus 10453 is the Sine of 6 Degrees, being doubled, the Sum will be 20906 the Chord of 12 Degrees; and so of the rest, as in the Table following.

The Table of Degrees and Chords.

De	Chord	De	Chord	De	Chord	De	Chord	De	Chord	De	Chord
01	17	16	278	31	534	46	781	61	1015	76	1231
02	25	17	296	32	551	47	797	62	1030	77	1245
03	52	18	313	33	568	48	813	63	1045	78	1259
04	70	19	330	34	585	49	830	64	1060	79	1273
05	87	20	347	35	601	50	845	65	1074	80	1286
06	105	21	364	36	618	51	861	66	1089	81	1299
07	122	22	382	37	635	52	876	67	1104	82	1312
08	139	23	398	38	651	53	892	68	1118	83	1325
09	151	24	416	39	668	54	908	69	1133	84	1338
10	175	25	432	40	684	55	923	70	1147	85	1351
11	192	26	450	41	700	56	939	71	1161	86	1364
12	209	27	466	42	717	57	954	72	1176	87	1377
13	228	28	484	43	733	58	970	73	1190	88	1389
14	244	29	501	44	749	59	984	74	1204	89	1402
15	261	30	518	45	76	60	1000	75	1217	90	1414

This done, Proportion the Radius of a Circle to what extent you please; make AB equal thereto, which must be divided into equal Parts, as before-directed, by half thereof, and this Table, the Chord of any Arch proportionable to this Radius, may speedily be obtained. As for Example, Let there be required the Chord of Thirty De∣grees, the Number in the Table is 518; or in proportion to this Scale of 100 equal Parts, AB is 52 almost; I take therefore 52 from the Scale of equal Parts, and set them from B towards 8 to h and o, and draw the Line h o, which is the Chord desired 30 Degrees: Thus may you find the Chord of any other Arch agreeable to this Radius. Or if your Radius be of a greater or lesser extent, if you make the Base of your Right Angle AB equal thereunto, You may in like manner find the Chord of any Arch, agreeable to any Radius given. Only remember, That if the Chord of the Arch desi∣red exceed 60 Deg. AB which is divided into 100 equal parts, you must continue the Base AB in the division of such parts, as need shall require.

In this manner is made the Line of Chords in the Fundamental Diagram answerable to that Radius.

And in this manner you may find the Chord of the Rhomb, Points, halfs, and quarters, and the Sines, Tangents, and Secants of any Arch proportionable to any Radius, by help of these Tables following [which is an abbreviation of the Canon of Natural Sines, Tangents, and Secants] and proportioning the Base AB thereunto, which is the Scale of equal parts; as by Example may more plainly appear.

A Table for the Angles which every Rhomb maketh, with the Meri∣dian, and the Chords of every Quarter and Point of the Compass.

North.	South.	deg. mi. sec.	Chor.	South.	North.
		2 48 45	49
		5 37 30	98
		8 26 15	147
N. b. E.	S. b. E.	11 15 00	195	S. b. W.	N. b. W.	1
		14 3 45	244
		16 52 30	293
		19 41 15	333
N. N. E.	S. S. E.	22 30 00	390	S. S. W.	N. N. W.	2
		25 18 45	427
		28 7 30	485
		30 56 15	533
N. E. b. N.	S. E. b. S.	33 45 00	580	S. W. b. S.	N. W. b. N.	3
		36 33 45	627
		39 22 30	673
		42 11 15	720
N. E.	S. E.	45 00 00	767	S. W.	N. W.	4
		47 48 45	811
		50 37 30	855
		53 26 15	899
N. E. b. E.	S. E. b. E.	56 15 00	942	S. W.b. W.	N. W.b. W	5
		59 3 45	985
		61 52 30	1028
		64 41 15	1069
E. N. E.	E. S. E.	67 30 00	1111	W. S. W.	W. N. W.	6
		70 18 45	1151
		73 7 30	1190
		75 56 15	1230
E. b. N.	E. b. S.	78 45 00	1268	W. b. S.	W. b. N.	7
		81 33 45	1305
		84 22 30	1343
		87 11 15	1378
East.	East.	90 00 00	1414	West.	West.	8

Let there be required the Chord of the first Point of the Scale, 11 Deg. 15, in this Table, as I have fitted for every Point, Half, and Quarter, for ¼ of the Compass.

The Numbers: answering to 11 Deg. 15 Min. is 195. I take therefore with my Compasses 19, or reckon so many on the Scale of Equal parts, which is joyned with a Scale intended to be made; and so with a Square for that purpose, as shall be shewed, mark from F towards 8 the first Point 11 Deg. 15, where the Radius of the Circle is AB; and so of the rest.

The Scale of Longitude.

THis Scale is made also by the Table of Degrees and Chords, as before.

EXAMPLE.

It is required to know how many Miles make a Degree in the Parallel of 10 Deg. If you extend the Compasses from A, to the Complement of the Latitude 80 Deg. in the Line of Sines, and setting one Foot in F, turn that distance from F toward A, you will find it reach 59 Miles nearest, in the former Diagram.

Another EXAMPLE.

It is required in the Latitude of 60 Degrees to know the Miles answering to a De∣gree. In that Parallel extend the Compasses from A to the Complement of the Latitude 30, in the Line of Sines; and setting one Foot of the Compasses in F, turn that di∣stance towards A, and you will find it reach 30 Miles, that makes a Degree in that Parallel; and so of the rest.

But if it be required how to make a Scale of Longitude in Miles answerable to the Radius of the same Scheme, for the Parallel of 10 Degrees, you will find in the Ta∣ble, the Chord for 10 Degrees is 17.5 for the first Mile, and for 60 Degrees 1000, take 100 from A to B, as you was before-directed, and so do with the rest, until you have made the whole Scale. Remember, that 60 Miles must begin where the first Degree of the Chords doth on the Scale, and so diminish towards the Pole 90 Degrees of the Scale, as reason will give you,

SINES.

NOte, That a Sine falls al∣ways within the Quadrant of a Circle, as CD, which is the Sine of the Arch BC 60 Degrees; and by the Table of Natural Sines, to every Degr. of the Quadrant which I have fitted for this purpose, whose Radius is 1000, you shall find the Sine of 60 deg. to be 86.6. I take therefore with my Compasses 86 from my Scale of Equal Parts, and set them from A towards 8 in the Line of Sines for 60 Degrees, where the Radius of the Circle is AB, and CE is the Comple∣ment thereof, or Sine of 30 Degrees of the Arch C 8, the Number in the Table an∣swering 30 Degrees is 500; take therefore with your Compasses 50 equal Parts of A B, and lay it from A upon the Line of Sines for 30 towards 8; and so of the rest.

A Table of Natural Sines to the Radius of 1000.

De	Sines.	De	Sines.	De	Sines.	De	Sines.	De	Sines.	De	Sines.
1	17	16	275	31	515	46	719	61	874	76	970
2	34	17	292	32	529	47	731	62	888	77	974
3	52	18	309	33	544	48	743	63	891	78	978
4	69	19	325	34	559	49	754	64	898	79	981
5	87	20	342	35	573	50	766	65	906	80	984
6	104	21	358	36	587	51	777	66	913	81	987
7	124	22	374	37	601	52	788	67	920	82	990
8	139	23	390	38	615	53	798	68	927	83	992
9	156	24	406	39	629	54	809	69	933	84	994
10	178	25	422	40	642	55	819	70	939	85	996
11	190	26	438	41	656	56	829	71	945	86	997
12	207	27	453	42	669	57	838	72	951	87	998
13	224	28	469	43	682	58	848	73	956	88	999
14	241	29	484	44	694	59	857	74	961	89	999
15	258	30	500	45	707	60	866	75	965	90	1000

TANGENTS.

A Table of Natural Tangents to every Degree of the Quadrant.

De	Tan.	De	Tan.	De	Tan.	De	Tan.	De	Tangents.
1	17	19	344	37	753	55	1428	73	3270
2	34	20	363	38	781	56	1482	74	3487
3	52	21	383	39	809	57	1559	75	3732
4	69	22	404	40	839	58	1600	76	4010
5	87	23	424	41	869	59	1664	77	4331
6	105	24	445	42	900	60	1732	78	4704
7	122	25	456	43	932	61	1804	79	5144
8	140	26	487	44	965	62	1880	80	5671
9	158	27	509	45	1000	63	1962	81	6313
10	176	28	531	46	1035	64	2650	82	7115
11	194	29	554	47	1072	65	2144	83	8144
12	212	30	577	48	1110	66	2246	84	9514
13	230	31	600	49	1150	67	2355	85	11430
14	249	32	624	50	1191	68	2475	86	14300
15	267	33	649	51	1234	69	2601	87	19081
16	286	34	674	52	1279	70	2747	88	28336
17	305	35	700	53	1327	71	2904	89	57289
18	324	36	726	54	1376	72	3177	90	0000000
									Infinite.

A Tangent Line is always falling without the Quadrant, and is drawn at the end of a Semidiameter at Right Angles, as B 6 in the Fundamental Di∣agram, which is the Tangent of the Arch BC 60 Degrees, as in the Table of Tangents you shall find it to be 1732 equal parts, which take with your Compasses from A, when you have conti∣nued the Line beyond B, take 173 parts, and that will reach from B to G, the Tangent of 60 Degrees in the Scale, and 8 H is the Complement Tangent 30 Degrees 577 parts; therefore take 57 parts, it will reach from B to the length of 30 Degrees; and so of the rest.

A SECANT.

A Table of Secants to every Degree of the Quadrant.

De	Sec.	De	Sec.	De	Sec.	De	Sec.	De	Secants.
1	1000	19	1057	38	1269	56	1788	74	3627
2	1000	20	1064	39	1286	57	1836	75	3863
3	1001	21	1071	40	1305	58	1887	76	4133
4	1002	22	1078	41	1325	59	1941	77	4445
5	1003	23	1086	42	1345	60	2000	78	4809
6	1005	24	1094	43	1367	61	2062	79	5240
7	1007	25	1103	44	1390	62	2130	80	5758
8	1009	26	1112	45	1414	63	2202	81	6392
9	1012	27	1122	46	1439	64	2281	82	7185
10	1015	28	1132	47	1466	65	2366	83	8205
11	1018	29	1143	48	1494	66	2458	84	9566
12	1022	30	1154	49	1524	67	2559	85	11473
13	1026	31	1166	50	1555	68	2669	86	14335
14	1030	32	1179	51	1589	69	2790	87	19107
15	1035	33	1192	52	1624	70	2923	88	28653
16	1040	34	1206	53	1661	71	3071	89	57298
17	1045	35	1220	54	1701	72	3236	90	0000000
18	1051	36	1228	55	1743	73	3420		Infinite.
		37	1252

A Secant Line is drawn always from the Center of the Circle, until it cut the Tangent Line; as A G in the foregoing Diagram cuts the Tangent of the Arch BC 60 Degrees in G: so is AG the Secant of 60 Degrees, which in this Table of Se∣cants is found 2000 equal parts; therefore take off such parts as are in proportion to AB 200, it shall reach from A to G for the Secant of 60 Degrees, and AH is the Complement-Se∣cant, or Secant of the Arch 8 C, 30 Degrees, which in this Table of Secants is found to be 1154; therefore take with your Compas∣ses, or other Instruments, 115 equal parts, and it shall reach from A towards G for the Secant of 30 Degrees, as you may find by the Scale in the Diagram.

Versed Sines.

A Versed Sine is found by substracting his Complement-Sine out of the Radius. Example. For to know the Versed Sine of 60 Degrees, you must substract EC or AD, which is the Complement or Sine of 30 Degrees, viz. 500 out of the Radius 1000, or Sine of 90, AB, the remain will be DB 500, for the Versed Sine of the Arch BC 60 Degrees. In like manner E 8 will be found 134 for the Versed Sine of the Arch C 8, being 30 Degrees; and so work in like manner for any other De∣gree. The Word Versed is a sufficient Direction, to let them understand, that do not, That the Degrees of this Scale, or sort of reckoning, begins at B or F, and con∣tinues to 180 Degrees, the Diameter of the Circle, or the Line of Sines Reversed, by putting the two beginnings of Degrees together of the Quadrant or Seale, and so be∣gin to count at one End; for 80 Degrees must be placed 10, for 70 Degrees 20 Deg. and so to 180; and of the first 90 or middle of the Scale, count the Sun's greatest Declination 23 Degrees 30 Min. towards both ends, that is, 47 Degrees asunder in that distance; by the side thereof must be placed the Reversed six Northern Signes, ac∣cording to the Sun's Declination, and place in the Ecliptick at such Declination: And likewise 23 Degrees 30 Min. the space for dividing the Reversed Southern Signes to∣ward 180; and are reckoned double, as occasion requireth.

Either of the Semidiameters AB or AF, the Sides of the Quadrant, you may take the equal divisions thereof, and make a Scale of Leagues or Miles, or Equal Parts, for the demonstration of all plain Triangles, which you cannot be without, having it upon the Ruler.

CHAP. II. A Description of what Instruments of Brass, Steel, Iron, and Wood, you must be provided with before you can make Instruments for Mathe∣matical Uses.

BEfore we explain the other half of the Fundamental Diagram and Semicircle, it will be necessary for to give a Description of what Innstruments in Brass, Steel, Iron, or Wood, you must have by you in readiness, before you can make a Mathematical Instrument; That Men that are ingenuous may be provided in some measure with such, before they go to Sea, in spending their spare-time on this Practice. In brief, they are these. First, For Instruments of Wood, you must be provided with several Scales of Equal Parts,* 1.1 of several lengths, which must be exactly and carefully divided, the length you intend to make the Radius of the In∣struments by it. First, divide this Line into 10 equal parts, and each 10 into 10 more; so is your Line divided into 100; and so you may continue it into 200, 300, 400, so much as you please, as the Instrument you are making will re∣quire; which you may quick∣ly see by the Table. You must be fitted with some pieces of Box (dry,* 1.2 clean from Knots, straight, and smooth planed) or other Wood, on which you may make what Scale you please. You must have by you a true Square of Brass and Wood,* 1.3 such as you may see in this Figure,* 1.4 with a pair of Cramps made of Iron, with Screws to fasten the Scale of Equal

[illustration] geometrical diagram

Parts, and the Scale to be made together, so as they may not slip, whereby may be made no mistake in Graduating. Or for small Scales,* 1.5 you may fasten the Scale of Equal Parts, and the Scale to be made by it, on a piece of Deal Board, with the Heads of Scuper Nails, so as they will not stir; but for greater Instruments, and Cross-staves, and Gauging-staves, you must do by them as in this Figure. You must have a Gauge made of Brass, with a good Steel Pin, for the drawing of straight Lines on your Scale, for the division of the Columnes for Graduation. You must have two or three Sorts and Sets of Steel Letters and Figures, and Figures for Ornament,* 1.6 with a neat Hammer to use with them: And the Figures, and Letters, and Ornament-Figures, set in an Alphabet-Box, with written Letters and Figures before them, for the ready find∣ing of them; with Characters of the Signes, and Planets, and Stars, in like man∣ner.

The Instrument that you graduate with, the Edge must be very thin and sharp,* 1.7 and you may have several of them; or the end of a Pen-knife may do for a shift. You must have a Brass pair of Compasses to go with an Arch and Screws, to fasten at any distance; and four Steel Points to take in and out; two long Points for to reach a great distance. I have a pair by me will extend 3 Foot,; on a large Scale of Artificial Sines, Tangents, and Numbers, they are to be used. The other are short Points. One is to be made round for a Center-Point, that it may not go too far into the Wood; and the other pointed like a Dutch Knife, and the Shoulder fitted square as the other Points, to be fastned, and taken in and out at pleasure.* 1.8 The Use of these Points is to draw Circles on round Instruments, as Nocturnals, and the like. You may have two pair of Dividers, the least 3 Inches and ½, and the biggest 7 ½ Inches long. I hold them best that go with a Bow at the Head, and to be set together by a Screw in the midst. Be sure they be made of good Steel. These are to divide equal Parts, and any other equal Division. You must have for great Instruments, as Bows, Quadrants, and the like, a pair of Beam-Compasses, for to sweep the Arches of them.* 1.9 You should have a Hand-Vice, so made as to screw into the edge of a Board for your use, and to take out again; with three or four sorts of small Files, for to file and make Pins, which you will have occasion for. These Brass and Iron Instruments or Scales you may now give direction to an ingenuous Smith (Thomas Moone) in Bristol (if you cannot have them before of these sorts) and he will fit you with them:* 1.10 Or you may have them ready made of Walter Hayes, at the Cross-Daggers in Moor-Fields, with many useful Instruments in Brass; Or of Andrew Wakely, Mathematician, at his Shop on Redriff-Wall, near the Cherry-Garden Stairs; Or in Bristol of Philip Staynred, Math.—And now I have shewn the Practitioner what Instruments he must be fur∣nished with, I will return to the Explanation of the other half of the Fundamental Diagram of the Mathematical Ruler. I had almost forgot a Receit for setting off the Graduation, when it is newly done on Box-Instruments, which is this. Take Charcoal,* 1.11 and beat it to a fine Powder, and temper it with Linseed-Oyl; and let it be rubb'd on the Instrument newly made, and lie so on it for a time, untill it be pretty dry; and then with some S••llet-Oyl rub the Instrument, and make it clean: So will you have the Graduation and Figures set off very neatly on Box Instruments, with Black.

[illustration] depiction of geometrical instruments

The Figure of the Foure Poynted Compasses

The Long Point

The Short Points

The Figure of y^e Gaudge

The figuer of the Diuide

CHAP. III. The Explanation of the other half of the former Semicircle; being a De∣scription of the Fundamental Diagram, of the Dialling-Scale on the Ma∣thematical Ruler.

THis annexed Diagram sheweth plainly the Description of the Dialling-Scales on the Mathematical Ruler; It being the most easie and exact Instrument used in that Art, as by the use will be manifest in the Seventh Book.

[illustration] geometrical diagram

How to make the Diagram.

FIrst, Make a Semicircle by a less Radius, as ADB, and upon the midst of the Arch at D, with the distance DA describe the Quadrant-Arch, as AEB, which must be divided into six equal Parts, for the 6 Hours in the ¼ of the Sphere; which is sufficient to resolve the whole; and from each Point draw Lines to the Center at D; So will it cut the Line AB in 1, 2, 3, 4, 5, 6, for the Hour-Lines upon the said Scale for Dialling: and thus you see it is a Tangent Line, for which use it is more certainly done by this Table of Natural Tangents for three Hours, if you do but observe where the Right Line DE cuts the Tangent Line AB, which you see in the middle or Center of the Semicircle at R; therefore you must begin to make this Scale in the midst, and lay the distance of parts answering the Hours both ways from R to∣wards

B and A: As by Example, To Graduate 2Hours and 4 Hours, you see in the Table, the Numbers answering to 2 Hours and 4 Hours in the first Column to the left hand; is in the second 60 Minutes, or in the third 15 Degrees, and in the fourth Column the Tan∣gent-parts 267; therefore if you take 267 such Parts whereof the Semidiameter RB is divided into 1000, as was shewed in the former Diagram, and put one Foot of the Compasses with that extent at R the midst or 3 Hours, and turn the other toward B, it will make the distance of 4 Hours; and turn that distance towards A, it will be 2 Hours of the Scale: And so do with the rest of the Hours, and distance of the Minutes.

A Table for the dividing of the Hours and Minutes upon the Dialling-Scale.

Ho.	M	D. M.	Tan. par.
3	10	2 30	43
	20	5 00	87
	30	7 30	131
	40	10 00	176
	50	12 30	221
2 4	60	15 00	267
	10	17 30	315
	20	20 00	363
	30	22 30	414
	40	25 00	466
	50	27 30	520
1 5	60	30 00	577
	10	32 30	637
	20	35 00	700
	30	37 30	767
	40	40 00	839
	50	42 30	916
0 6	60	43 00	1000

In like manner for the Scale of Inclination of Meri∣dians, you must take out the Tangent-parts out of the Table of Tangents to every Degree, and graduate in the same manner as before, from the Center which is the midst of the Scale 45 Degrees, as is shewn plain in the Diagram.

For the Gnomen-Line, as others call it the Line of Latitude, Let BA be the Semidiameter; so on B de∣scribe the Quadrant ABC, whose Arch AC divide into 90 Degrees, from whence you may project the Line of Sines BC.

Now from each Degree of those Sines, draw Lines toward the Center of them at A, and note where they cut the Arch of the Quadrant BD: Then from B as a Center, take the distance of each of these Intersections, and lay them on the Line BD; so shall you have the Division of the Gnomon-Line, or Line of Latitude.

For the more ready making of this Scale, here is a Table of Lati∣tudes calculated to the 90 Degrees of the Qua∣drant, and the way to calculate it your self. As for Example, To find the Latitude-parts for 30 Degrees of Lati∣tude,

A Table of Latitudes for Dialling.

Deg.	Par.	Deg.	Par.	Deg.	Par.	Deg.	Par.	Deg.	Par.	Deg.	Par.
		80	992	60	926	45	817	30	632	15	354
		78	989	59	920	44	807	29	617	14	332
		76	985	58	915	43	797	28	601	13	310
		74	980	57	909	42	787	27	585	12	288
		72	975	56	903	41	776	26	568	11	265
90	1000	70	969	55	896	40	765	25	551	10	242
89		69	965	54	890	39	753	24	533	9	219
88		68	962	53	883	38	741	23	515	8	195
87		67	958	52	875	37	729	22	496	7	171
86		66	954	51	868	36	717	21	477	6	147
85	998	65	950	50	860	35	704	20	458	5	123
84		64	945	49	852	34	690	19	438	4	98
83		63	941	48	844	33	676	18	419	3	74
82		62	936	47	835	32	662	17	397	2	49
81		61	931	46	826	31	648	16	376	1	25

First, Find the Sine thereof in the Natural Table of Sines, which will be found to be 50000; which sought for in the Table of Tangents, giveth an Arch of 26 Deg. 34 Min. Then the Pro∣portion will hold,

As the Radius	100000
To the Secant 45 Deg.	141421
So is the Sine of 26 Deg. 34 Min.	44724
Ʋnto the Latitude-parts	63249

Which answers to the Radius 100000: But in my Table the Parts 632 answer to the Radius 1000, which will be sufficient for the Graduating the Line of Gnomons or Latitude.

But observe, To make 30 Degrees of Latitude on your Scale, you must take off

632 such Parts as the Line is divided into 100, or 1000, as you have been shewn in the former Diagram.

How to make the Line of Chords, you have been fully instructed already in the for∣mer Figure; which is only by dividing the Arch of the Quadrant AD into 90 equal parts; And from A as a Center, take the distance, and lay them down in a straight Line AD: So shall you have the Line of Chords or Sublemes. Or you may do it by the Table of Chords, as before-directed.

How to make the two Lines or Scales of Inlarging Hour-Lines upon any reclining Plain, without a Center, called by me the greater and the lesser Pole.

Invexed, you have a Table ready fitted for the making thereof.

First, You must make choice of the length of this Scale, that is in Proportion to the former Lines of the Scale.

A Table of Tangents for 5 Ho. to every 5 Min. of an Hour, for inlarging the Hour-Line Scale.

Hours.	Mi.	Deg Min	Tan. pa	Hours.	Mi	Deg. Min.	Tan. pa
	5	1 15	21		5	46 15	1044
	10	2 30	43		10	47 30	1091
	15	3 45	65		15	48 45	1140
	20	5 00	87		20	50 00	1191
	25	6 15	100		25	51 15	1245
	30	7 30	131		30	52 30	1303
	35	8 45	153		35	53 45	1363
	40	10 00	176		40	55 00	1428
	45	11 15	198		45	56 15	1496
	50	12 30	221		50	57 30	1569
	55	13 45	244		55	58 45	1647
1	60	15 00	267	4	60	60 00	1732
	5	16 15	291		5	61 15	1822
	10	17 30	315		10	62 30	1920
	15	18 45	339		15	63 45	2027
	20	20 00	363		20	65 00	2044
	25	21 15	388		25	66 15	2272
	30	22 30	414		30	67 30	2414
	35	23 45	440		35	68 45	2571
	40	25 00	466		40	70 00	2747
	45	26 15	493		45	71 15	2945
	50	27 30	520		50	72 30	3171
	55	28 45	548		55	73 45	3430
2	60	30 00	577	5	60	75 00	3732
	5	31 15	606
	10	32 30	637
	15	33 45	668
	20	35 00	700
	25	36 15	733
	30	37 30	767
	35	38 45	802
	40	40 00	839
	45	41 15	876
	50	42 30	916
	55	43 45	957
3	60	45 00	1000

The first 3 Hours must be di∣vided into 10 parts, and each of them into 10 more, which stand for 100, or as you have been shewd for 1000. You must have two of these Lines of Equal parts, of two proportionable Lengths, for the greater and les∣ser Pole; And so take of the Tangent-parts answerable to eve∣ry 5 Minutes of an Hour: As you see the first and second Co∣lumns of the Table are Hours and Minutes, the third Degrees, and the fourth Tangent-parts. So the Tangent of the first 2 Hours of the Scale or 30 Degrees, is 577 Parts; take of your two Scales 57 Parts; First of the largest Radius for 2 Hours on the greater Scale, and the like num∣ber of the smaller Radius, or Line of Equal Parts for 2 Hours of the lesser Pole-Scale. And so in the same manner you must work to finish the whole Scales of what Radius you please, by these Tables, as hath been di∣rected.

The Use hereof is fully shewn in the Seventh Book, 29th and 30 Chap. of the Art of Dialling.

These Scales are sufficient to make any sort of Dials, in any Latitude (as is there shewn) with ease and exactness.

There are two Lines called by the Names of Style and Substyle-Scale; but is only for this Lati∣tude, but may be found for any. But the Scales before-explained are most useful, and do the same, as you will find in the Seventh Book and Twelfth Chapter of the Art of Dialling. And these are the Scales of one Side of the Ruler.

CHAP. IV. The Scales or Lines on the Back-side of the Mathematical Ruler, are these: A Line of Numbers, A Line of Artificial Tangents, A Line of Sines, A Meridian Line according to Mercator's or Mr. Wright's Projection; and the Scale of Equal Parts, by which the Numbers were taken off for the Graduating these Scales; and a Line of Longitude or Equinoctial, with a Scale of Reduction, as followeth.

I. HOw to divide the Line of Numbers is thus. You must prepare a Ruler of what length you please, and also a Scale of Equal Parts, divided into 100 or 1000: You must count them. But if you divide the Artifici∣al Tangents and Sines with the Line of Number, you were best to divide the Line into 2000 Parts; so will you have 100 on the Line of Numbers. This Table is taken out of the Lo∣garithms, by rejecting the In∣dex or first Figure. It is best to omit the first Number, by rea∣son they will take up so much room; and begin at 1 or 11, and take the Logarithm-part at 41 for the first 10th or Integer. But if you intend to make 100 on your Line of Numbers, first take 100, which is reckoned 1000, as you see in the fore∣going Table, Parts of the Scale of Equal Parts, for the first 10 or middle of the Scale: Then suppose you were to make the first 2 or 20, take with your Instrument or Com∣pass 301 equal Parts, and lay it from 1 to 2, and the same di∣stance will reach from 10 in the middle to 20. In the like manner do with the rest; for 3 or 30 the equal parts is 477, and for 4 or 40, the Log. parts is 602: So you may easily perceive how to do it, by what is written.

A Table for the Division of the Line of Artificial Numbers.

Num.	Log. parts.	Num.	Log. parts.	Num	Log. parts.	Num.	Log. parts.	Num.	Log. parts.
1	00	21	322	41	612	61	785	81	908
2	30	22	342	42	623	62	792	82	913
3	47	23	361	43	633	63	799	83	919
4	60	24	380	44	643	64	806	84	924
5	69	25	397	45	653	65	812	85	929
6	77	26	414	46	662	66	819	86	934
7	84	27	431	47	672	67	826	87	939
8	90	28	447	48	681	68	832	88	944
9	95	29	462	49	690	69	838	89	949
10	100	30	477	50	698	70	845	90	954
11	41	31	491	51	707	71	851	91	959
12	79	32	505	52	716	72	857	92	963
13	113	33	518	53	724	73	863	93	968
14	146	34	531	54	732	74	869	94	973
15	176	35	544	55	740	75	875	95	977
16	204	36	556	56	748	76	880	96	982
17	230	37	568	57	755	77	886	97	986
18	255	38	579	58	763	78	892	98	991
19	278	39	591	59	770	79	897	99	995
20	301	40	602	60	778	80	903	100	1000

II. How to make the Line of Artificial Tangents on the Ruler.

THe Artificial Tangents are made in the same manner as before directed, beginning upon a Right Line of Numbers, omitting the first 30 Minutes, and beginning at 40 Minutes. The Tangent-parts are 106, taken off the former Scale, and applied as before-directed upward, will make 40 Minutes on your Scale: So the first and 89 Degree, the Tangent-part answer∣ing thereunto is 241; with them do in like manner, and so of the rest, until you have fi∣nished the whole Line or Scale, as you may see in the Figure.

The following Table is so plain to be under∣stood, that I need write no more, but, That the first Column to the left hand is Minutes, The se∣cond Tangent-parts answering to the Minutes and Degrees over each Column to 30 Degrees, and af∣ter to every 20 Minutes, as you may see in the Table.

III. A Table for the Division of the Line of Artificial Tangents to 45 Deg. and the Minutes fit to be set thereon.

	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.
Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.	Tang. parts.
Minutes.		89	88	87	86	85	84	83	82	81	80	79
0	1	2	3	4	5	6	7	8	9	10	11
0	00	241	546	719	844	941	1021	1089	1147	1199	1246	1288
10	46	308	577	742	862	956	1033	1099	1156	1207	1253	1295
20	76	366	610	765	879	970	1045	1106	1165	1215	1260	1301
30	96	410	640	786	895	983	1056	1119	1174	1223	1267	1308
40	106	463	668	806	911	996	1067	1129	1183	1231	1274	1314
50	162	505	694	826	927	1009	1078	1138	1191	1238	1281	1321
Min.									70
12	13	14	15	16	17	18	19	20	21	22	23
0	1327	1363	1396	1428	1457	1485	1511	1536	1561	1584	1606	1627
10	1333	1369	1402	1433	1462	1489	1516	1541	1564	1587	1610	1631
20	1339	1374	1407	1438	1466	1494	1220	1545	1568	1591	1613	1634
30	1345	1380	1412	1442	1471	1498	1524	1549	1572	1595	1617	1638
40	1351	1385	1417	1447	1476	1503	1528	1553	1576	1599	1620	1641
50	1357	1391	1422	1452	1480	1507	1532	1557	1580	1602	1624	1645
Min							60
24	25	26	27	28	29	30	31	32	33	34	35
0	1648	1668	1688	1707	1725	174••	1761	1738	1795	1812	1828	1845
10	1651	1671	1691	1710	1728	1746
20	1655	1675	1694	1713	1731	1749	1767	1784	1801	1818	1834	1850
30	1658	1678	1697	1716	1735	1752
40	1662	1681	1700	1719	1737	1755	1773	1790	1806	1823	1839	1855
50	1665	1684	1704	1722	1742	1758
Min					50					45
36	37	38	39	40	41	42	43	44	45
0	1861	1877	1892	1908	1923	1939	1954	1969	1984	2000
20	1866	1882	1898	1913	1928	1944	1959	1974	1989
40	1871	1887	1903	1918	1934	1949	1964	1979	1994

IV. How to make the Scale or Line of Artificial Sines to 90 Deg. and Minutes fit to be set thereon.

HOw to make this Line, was shewn by making the last; only that is sufficient to 45 Degrees, and this must be to 90 Degrees. And if your Line of Equal Parts be divided into 100, or as they be reckoned 1000, you may omit the last Fi∣gure of the Number: But if you number the Scale to 2000, as the Tables are made to, if you would set off the Sine of 30 Degrees, the Parts answering thereunto is 1698; therefore take off your Scale of Equal Parts with your Compasses 169, and it will reach from the beginning, to 30 Degrees on the Line of Sines.

So I hope you understand how to do the rest, it being made so plain and easie for the meanest Capacity, by what hath been writ already.

A Table for the Division of the Artificial Sines on the Ruler.

	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.	Deg.
Sine parts.	Sine parts.	Sine parts.	Sine parts.	Sine parts.	Sine parts.	Sine parts.	Sine parts.	Sine parts.	Sine parts.	Sine parts.
Minutes.	0	1	2	3	4	5	6	7	8	9	10
0	00	241	542	718	843	940	1019	1085	1143	1194	1239
10	46	308	577	742	861	954	1031	1096	1152	1202	1246
20	76	366	609	764	878	968	1042	1105	1161	1209	1253
30	94	417	639	785	894	981	1053	1115	1169	1217	1260
40		463	667	805	910	994	1064	1125	1178	1225	1267
50	162	505	693	825	925	1007	1075	1134	1186	1232	1274
Min.	11	12	13	14	15	16	17	18	19	20	21
0	1280	1317	1352	1383	1412	1440	1465	1489	1512	1534	1554
10	1287	1323	1357	1388	1417	1444	1470	1493	1516	1537	1557
20	1293	1329	1362	1393	1422	1449	1474	1497	1519	1540	1560
30	1299	1335	1368	1398	1426	1453	1478	1501	1523	1544	1564
40	1305	1340	1373	1403	1431	1457	1482	1505	1527	1547	1567
50	1312	1346	1378	1408	1435	1461	1486	1508	1530	1551	1570
Min.	22	23	24	25	26	27	28	29	30	31	32
0	1573	1591	1609	1625	1641	1657	1671	1685	1698	1711	1724
10	1576	1594	1612	1628	1644	1659	1673	1687
20	1579	1597	1614	1631	1646	1661	1676	1690	1703	1716	1728
30	1582	1600	1617	1633	1649	1664	1678	1692
40	1585	1603	1620	1636	1652	1666	1680	1694	1707	1720	1732
50	1588	1606	1623	1639	1654	1669	1683	1696
Min.	33	34	35	36	37	38	39	40	41	42	43
0	1736	1747	1758	1769	1779	1789	1798	1808	1816	1825	1833
20	1739	1751	1762	1772	1782	1792	1801
40	1743	1754	1765	1776	1786	1795	1805
Min.	44	45	46	47	48	49	50	51	52	53	54
0	1841	1849	1856	1864	1871	1881	1884	1890	1896	1902	1907
Min.	55	56	57	58	59	60	61	62	63	64	65
0	1913	1918	1923	1928	1633	1937	1941	1945	1949	1953	1957
Min.	66	67	68	69	70	71	72	73	74	75	76
0	1960	1964	1967	1970	1972	1975	1978	1980	198••	1984	1986
Min.										86
77	78	79	80	81	82	83	84	85	87	90
0	1988	1990	1991	1993	1994	1995	1996	1997	1998	1998	2000

V. How to make a Meridian Line according to the true Sea-Chard, or Mercator and Mr. Wright's Projection.

THis Line is made out of the Table of Meridian parts, called also the Division of the Meridian Line. To every 10 Minutes of Latitude, nearer we have no Chards or Plots made, which I have as yet seen; but they may be made by Mr. Wright's Tables to every Minute, if any Person will be so curious.

For the Graduating this Line in the Scale, you must note the Number answering to the first Degree is 200; therefore divide the Degrees of the Aequinoctial into 20 equal Parts, which stand for 200 of the Numbers of your Table. As by Example,

Suppose you would make the first 10 Degrees from the Aequator, towards either of the Poles, on the Scale; the Number answering 10 Degrees is 201, omitting the last Figure 0: Therefore you may take out of the Line of Longitude (which is Equal Parts, or the same Line by which you made all the rest) 201 or 20 Parts, and lay that distance for the beginning of the first 10 Degrees; and for 20 Degrees 40, 8; and for 30 Degrees 62, 9; and so of the rest.

But if you are to make a particular Line, you must take the difference of the De∣grees and Minutes, as shall be fully shewn in the Treatise of making a general and particular Sea-Chard, according to Truth, and Mr. Wright's Projection; but what hath been done already will serve for both, if you follow direction.

There is demonstrated and shewn the making of Mercator's Scale, to measure Di∣stance in any Parallel of Latitude in any true Sea-Chart.

VI. How to Calculate a Table, and by it how to take out the Numbers, and make a Scale of Reduction, to be used in Surveying of Land.

A Table for the Division of the Scale of Reduction.

	Statute Acres.
10	272 25
11	255 00
12	189 06
13	161 10
14	138 91
15	121 00
16	106 35
16 ½	100 00
17	094 21
18	84 03
19	75 42
20	68 7
21	61 73
22	56 25
23	51 47
24	47 27
25	43 56
26	40 27
27	37 35
28	34 73
29	32 37
30	30 25
31	28 33
32	26 59
33	25 00
34	23 55
35	22 21
36	21 1
37	19 89
38	18 85
39	17 90
40	17 02

THis Scale consisteth of one part or Line of Numbers and Artificial Sines on the Ruler, for the more ready use thereof, as will be shewed: I shall first shew the Cal∣culation and Proportion used in making the Table, which is thus as followeth.

Example, For a Perch whose Measure is 21 Foot (which is the Irish Chain) this must be done by the back Rule of Three.

As 16½ squared, to 100 Acres:

So is 21 squared, to 61, 73 Acres.

So a piece of Ground being measured by the Statute-Chain of 16½ Foot to the Perch, should contain 100 Acres.

Then the same piece of Ground being measured by the Irish Chain of 21 Foot, will contain but 61 73/100 Acres, as you may see in the Table, which is near 61 Acres 2 Quarters 38.

By the Line of Artificial Numbers extend the Compasses from 16 ½ to 21, the same will reach twice repeated from 100 unto 61, 73 in the same Line of Numbers.

To make this Line on the Scale, Take the Numbers off the Line of Numbers of the same Scale you make this Line upon.

EXAMPLE.

I shall place the first 10 on the Scale, to this Number answers 272. 25″; there∣fore extend the Compasses from 100 to 272 ¼, or from 27 2/10, and lay one Foot of the Compasses at A, and the other will reach to B the distance to 16 ½. From 16 ½ you must lay all your other Numbers. As suppose you would set down 14 on the Scale, the Statute Numbers answering thereunto is 138 and 91″. Extend the Com∣passes from 100 to 138 and 91″, and that distance will reach from 16 ½ at B, to 14 of the Scale. The like if you would set off 20, the Numbers to that is 68. 7; and that distance will reach from 16 ½ at B, to 20: and so do with the rest. Thus have I done with this Scale, being sufficient to resolve all manner of Mathematical Conclusions whatsoever. The Use follows in the succeeding Treatise.

[illustration] geometrical diagram

CHAP. V. A Table for the Division of the Artificial Rhomb, or Points, Halfs, and Quarters on the Travis-Scale.

Points.	Nor. South.	Deg. Min.	Sine parts	Tang Rhomb.	Tang. qu.p
	N. b. E. 1	2 48	688		689
	S. b. E. 2	5 37	990		992
	S. b. W. 3	8 26	1166	1	1177
1	N. b. W. 4	11 15	1290	7	1298
	N. N. E.	14 3	1385		1398
	S. S. E.	16 52	1462		1481
	S. S. W.	19 41	1527	2	1553
2	N. N. W.	22 30	1582	6	1617
	N. E. b. N.	25 18	1630		1673
	S. E. b. S.	28 7	1673		1727
	S. W. b. S	30 56	1710	3	1777
3	N. W. b. N.	33 45	1744	5	1824
	N. E.	36 33	1774		1870
	S. E.	39 22	1802		1914
	N. W.	42 11	1827	4	1957
4	S. W.	45 00	1849	4	2000
	N. E. b. E.	47 48	1869
	S. E. b. E.	50 37	1890
	S. W. b. W.	53 26	1904
5	N. W. b. W.	56 15	1919
	E. N. E.	59 03	1933
	E. S. E.	61 52	1945
	W. S. W.	64 41	1956
6	W. N. W	67 30	1965
	E. b. N.	70 18	1973
	E. b. S.	73 7	1989
	W. b. S.	75 56	1986
7	W. b. N.	78 45	1991
	East & West.	81 33	1995
	84 22	1997
	87 11	1999
	90 00	2000

THe Use of this Tabe is easily understood: The first Co∣lumn is the Number of Points in one quarter of the Compass, and the second their Names in the whole; The third the Degrees answering to each quarter of a Point in the Quadrant; The fourth the Sines and Equal Parts answering thereunto; The fifth the Tangent-Rhombs; The sixth the Tangent-parts answering to each Quarter and Point to 45 Degrees ½, which is sufficient.

The Description of the Travis-Scale.

The making of this Scale is all one in a man∣ner as you made the former; only the Line of Sines is there but once made, and here the Parts answering each Quarter are twice put down, or in two Lines marked with N. S. which stands for to shew the Line to be Northing, Southing; and E. W. signifies Easting and Westing. The first is the Sine, the second is the Complement that any Point or Quar∣ter maketh an Angle with the Meridian. The Line marked with T. is the Tangent-Rhomb and Quar∣ters, and the first Line is a Line of Numbers, which you have been already shewn to make.

[illustration] geometrical diagram

The Traverse Scale.

One Example I will give the Learner, notwith∣standing it is so easie; for some there are that will not understand, though they see it often done; yet (to my knowledge) are Mates to good Ships.

EXAMPLE.

Suppose you was to set the first and seventh Rhomb or Artificial Point on the Scale, which is 11 Degrees 15 Minutes, the Equal Parts answering thereunto is 1290; therefore take 129 of your Scale of Equal Parts, and lay it from the beginning upwards, and you have by that distance the first and seventh Rhomb of your Scale.

In like manner do for any other of the Points and Quarters by these Numbers, until you have fi∣nished the Scale; and when you have done, you have an Instrument the most easie, ready, and necessary that I know of, for the working of Travises, and correcting your dead Reckoning, which shall be shewn in the Part of Sailing by the Plain Chard, in the Fourth Book. On the back side of this Scale you may set a Line of Chords and Equal Parts, and Points, for the ready protraction of Angles.

CHAP. VI. How to make a Quadrant which will resolve many Questions in Astrono∣my, by the help of an Index; and also very useful in Navigation.

AFter you have made choice of the Radius of your Quadrant CD, draw Pa∣rallel Circles thereunto, to hold the Degrees of the Quadrant and Columns, for the Figures, Points, and Quarters, as P 8. Then divide the Semi∣diameter or Side of the Quadrant CD and CM into 60 equal parts, and draw Pa∣rallels

to each of the Divisions, as Sides, CD and CM. First divide them into 6, and then each of them into 10 more, as you see in the Figure; at every 5 Parts make a Point, for the ready numbring of the Divisions.

[illustration] geometrical diagram

fol.65.

Make an Index answerable to the Radius CD, with a Line of Sines on the first Side, and a Center-Ear to put over the Center-Brass-Pin of the Quadrant C, as oc∣casion shall require: And on the other Edge make a Line of Equal Parts, equal to the 60 Divisions of the Side CD, with an Ear in like manner to remove at pleasure.

Make on the Edge a Tangent-Line; from it you must take the Sun's Declination, as you shall be fully shewn in the Use thereof.

This Quadrant will serve excellent well for a Protractor, with a long Index divided into 100 or 200 Equal Parts, with an Ear as the former, and a Needle put into a Stick, to put through the Center of the Index and Quadrant on any Point, in a Plain or Mercator's Chard, by which you may Protract any Rhomb without drawing Lines upon the said Chard; as likewise the Protractor or Semicircle which follows may do the same, being made in the same manner. On the back side of the Quadrant you may put Mr. Gunter's or Mr. Samuel Foster's Quadrant, or any other as you shall think fit.

The Use of the Quadrant in Astronomy.

SECT. I. Having the Latitude of the Place of the Sun's Declination, It is required to find the Time of the Sun-Rising and Setting.

The Latitude 51 Deg. 30 Min. Northward, and the Declination 20 Degrees, the difference of Ascension will be found thus.

First, Lay the Center Ear at E of the Index, over the Brass-Pin in the Center at A of the Quadrant, and lay the Edge of the Index to EL, to the Latitude of the Place on the Arch DM; and take of the Tangent-Line on the Edge of the Quadrant 20 Degrees the Sun's Declination; and lay that distance from the Center at A towards D, at that distance run with your Eye along the Parallel-Lines, and mark where it toucheth the Edge of the Index; there follow that Parallel-Line to the Arch, and reckon the Degrees from B to that Parallel-Line will be 27 Deg. 14 Min. the diffe∣rence of Ascension between the Sun's Rising and Setting, and hour of 6, according to the time of the Year.

The Degrees resolved into Hours and Minutes, is 1 Hour 49 Min. which is 4 of the Clock and about 11 Min, for the Sun Rising in the Morning, and 7 of the Clock 49 Min. his Setting in the Evening. In the same manner you must work for all La∣titudes.

SECT. II. Having the Latitude of the Place, and the Distance of the Sun from the next Aequinoctial Point, To find the Amplitude.

So the Latitude being 51 Deg. 30 Min. and the place of the Sun in one Degree of Aquarius, that is 59 Degrees from the next Aequinoctial Point; therefore set the Ear at S of the Line of Sines of the Index on the Pin at A, and the Edge thereof to the Latitude, and reckon 59 Degrees the Sun's distance from the first Aequinocti∣al Point, from the Center to C along the Line of Sines of the Index; there note the Line that cuts the 59 Degrees following with your Eye, to the Degrees in the Arch, and reckon the Minutes of Degrees from M to the Edge of the Index, and you will find it about 33 Deg. 20 Min. the Amplitude required.

SECT. III. Having the distance of the Sun from the next Aequinoctial Point, To find his Declination.

The Sun being either in 29 Degrees of Taurus, or 1 Deg. of Aquarius, or 1 Deg. of Leo, or 29 Deg. of Scorpio, that is 59 Degrees from the next Aequinoctial Point, To find his Declination do thus: Put the Ear of the Line of Sines on the Pin and Edge of the Index; put to 23 Deg. 30 Min. in the Sun's greatest Declination, reckoned from M on the Arch; then count the Sun's distance 59 Deg. on the Deg. of Sines of the Index: From the Center put one Foot of your Compasses by the Degree, with the other take the nearest distance to the Line or Side CM; apply that distance in the Line of Sines of the Index, from S along, and the other Foot will reach to 20 Degrees, the Declination required when the Sun is in the aforesaid Degrees and Sines. In like manner you must do for any other Degrees of the Sun's Place.

SECT. IV. Having the Latitude of the Place, and the Declination of the Sun, To find the Sun's true Amplitude from the true East and West.

This is a most excellent ready way by this Quadrant, and as near the Truth as any Man can make any rational use of this Problem at Sea: It is thus. Suppose the Latitude to be 13 Degrees, and the Sun's Declination 20 Degrees Northward, the Sun's true Amplitude of Rising and Setting is required, from the true East and West.

Set the Ear of the Side of the Index on which is the Line of Sines on the Center, and Edge to the Latitude 13; then count from M 20 Degrees of Declination, and carry your Eye upon the Parallel-Line from that Degree of the Arch, and mark what Degree it cuts of the Index and Line of Sines, as in this Question it doth 20 De∣grees 23 Minutes, and that is the true Amplitude required.

Secondly, Suppose you was about the Cape of Virginia, in Latitude 37 Degrees and 30 Min. and Declination 10 Deg. If you work as before-directed, you may find the true Amplitude to be 12 Deg. and about 36 Min. You may estimate the Min. but you cannot Steer by a whole Deg. when you have rectified your Compass by this; therefore this is sufficient for that Use, to shew you the difference between the true Compass and the Steering Compass, if you observe his Rising and Setting by it.

Note, The Amplitude is the distance of Rising or Setting of the Sun or Stars from the true East and West Points upon the Horizon.

As for the foregoing Work,—In the Latitude of 13 Deg. the Sun or Star having North-Declination 20 Deg. therefore they will rise 20 Deg. 33 Min. to the Northward of the East, and set 20 Deg. 33 Min. to the Northward of the West. But if the De∣clination had been 20 Deg. South, then they would have risen 20 Deg. 33 Min. Southward of the East, and set 20 Deg. 33 Min. to the Southward of the West.

And so if you bring these Degrees and Minutes into Points and Quarters, and use the Variation-Compass upon the Instrument of the Moon in the First Book, you may readily rectifie the Compass you Steer by.

SECT. V. The Ʋse of the Quadrant and Variation-Compass in the First Book, on the Instrument of the Moon for shifting of Tydes.

This Instrument contains two Parts or Rundles, which are the two uppermost in the aforesaid Instrument made of Wood or Brass, moving one upon the other, as there you may see. The biggest of the two uppermost Rundles represents the Compass you Steer the Ship by, which is subject to Variation: but the upper Compass doth repre∣sent the true Compass that never varieth, whereby you have a most necessary Instru∣ment to rectifie the Compass, as Mr. Wakely hath made Tables to be used with it; but this will serve for use as near by the Quadrant.

Admit I am in the Latitude of 27 Deg. and Declination 20 Deg. Northward, and I observe the Sun's Rising and Setting to be due East and by North, and West by South Point of my Steering or Variation-Compass; the Variation in that Latitude is required.

The Sun having North-Declination, and in that Latitude of 27 Deg. if there be no Variation the Sun will rise (as you may presently find his Amplitude by the Qua∣drant and Index, 22 Deg. 34 Min. which is but 4 Min. not to be taken notice of,* 1.12 above) E. N. E. and sets W. N. W. But according to the foregoing Propositions, the Sun did rise at E. b. N. and set at W. b. N. Therefore it plainly appeareth that there is a full Point Variation.

Therefore on the Variation-Compass on the Instrument of the Moon, you must al∣ways bring the true Point of Rising and Setting on the upper Compass, to touch the false Point or Rising and Setting, found by Observation and Steering-Compass, on the

middle Rundle, being set in this Position. You will find the E. b. N. to be the true E. N. E. and the W. and by N. to be the true W. N. W. and the N. b. E. to be the true N. and the S. b. E. to be the true S. and the S. E. ½ Point Southerly, to be the true S. E. b. S. ½ Southerly; and the South 3.1/4 East, to be the South ¼ West: And so you may do with ease in all other Observations, in like manner as you have been shewn, by Points, Halfs, and Quarters, which is on the two Trundles; and be sure nearer than ¼ of a Point I never did see any man Steer or sheape a Course.

SECT. VI. To know the Variation by the Quadrant.

You may do the same thing by the Quadrant, without the help of the Rectifier before spoken of, if you will remember, That this Quadrant hath eight Points, or ¼ of the whole Compass, by which you may orderly reckon the whole, and set the Index to the greatest difference either from the East, Southward, or Northward, or West. In like manner as in the foregoing Proposition, the true Amplitude of Rising and Setting was 2 Points, or 22 Deg. 34 Min. E. N. E. Set the Index to the Degrees and Points, reckoning the Deg. from D on the Arch of the Quadrant, toward E the Scale of Leagues of the Index; then reckon the Point and Degree taken by Observa∣tion, which is 11 Deg. and 15 Min. a just Point of the Compass: therefore it being but E. by N. short a Point of the true Amplitude, therefore the E. b. N. of the Steer∣ing-Compass, respecteth the true E. N. E. and the N. b. E. respecteth the true N. and so account all round the Compass a Point more than the Steering-Compass sheweth: And if you would know which way the Variation is, you see it is a Point more from the E. than your Compass sheweth Northerly.

But if the Steering or Azimuth-Compass, had shewn a Point more than the true Amplitude found by the Quadrant and true Point, the Variation had been Westerly.

But suppose the Amplitude found had been a Point Southerly, E. b. S. and the Sun's Rising and Setting had been a Point Northerly; by Observation of the Ship-Com∣pass, you see there is two Points difference: therefore set the Index to two Points, from M the East or West side of the Quadrant, as in this Proposition you must reckon it, and you may see plainly that the East Point by the Steering-Compass is the true East-South-East Point; and the South Point is the true South-South-West Point; and the North is the true North-North-West Point; and so of all the rest: And the Va∣riation is Southerly. So that you see how readily this Quadrant doth these things, when the Points of the Compass is imprinted in a Mans mind, which must be, and is in all Masters and Mates.

Suppose I would know by the Quadrant the true Point of the Compass, when Bootes Arcturus riseth and setteth: In the Latitude of 40 Deg. Bootes Arcturus De∣clination is 20 Deg. 58 Min.

Set the Side of the Index and Sine to the Latitude of 40, and count the Declina∣tion 21 Degrees almost, from M on the Arch; and run your Eye up the Parallel, and it will cut the Index about 27 Deg. 50 Min. which is reduced into Points and Quar∣ters by allowing Gr. 15 Min. to a Point, his Rising will be almost E. N. E. ½ a Point Northerly, his Setting W. N. W. ½ Westerly. But if the Declination of a Star of the South side the Aequinoctial, the Rising had been E. S. E. ½ Southerly, and his Setting W. S. W. ½ Southerly.

In the like manner you may know the Rising and Setting of any Star in an instant, by this Quadrant and Index, which I hold to be as necessary an Instrument as Seamen can use, in respect of its plainness, and brevity, and portability, so made as you see the Figure, the larger the better: And on it you may work all manner of Travisses to the distance of 60 Leagues or Miles which is on the side of the Index. It being so plain and easie, I need not write any thing thereof; but for the Learner's sake, take these few Rules following.

SECT. VII. To finde the Number of Miles answering to one Degree of Longitude, in each several Degree of Latitude.

In Sailing by the Compass, the Course sometimes holds upon a great Circle, some∣time upon a Parallel to the Aequator, but most commonly upon a crooked Line, winding towards one of the Poles, which Lines are well known by the Name of Rhombs.

If the Course hold upon a great Circle, it is either North or South under some Meridian; or East or West under the Aequator.

Deg.	Min.	Miles.
00	00	60
18	12	57
25	15	54
31	48	51
36	52	48
41	25	45
45	34	42
49	28	39
53	08	36
56	38	33
60	00	30
63	01	27
66	25	24
69	30	21
72	32	18
75	31	15
78	28	12
81	23	09
84	15	06
87	08	03

In these Cases every Degree requires an allowance of 60 Miles, or 20 Leagues; every 60 Miles or 20 Leagues will make a Degree difference in the Sailing; therefore as was shewn in the first Di∣agram, and use of the Line of Sines, may be sufficient here, which is the Rule of Proportion.

But if the Course hold East or West, on any of the Parallels to the Aequator,

As the Radius is to 60 Miles, or 20 Leagues, the Measure of one Degree of the Aequator:

So is the Sine-Compl. of the Latitude, to the Measure of Miles or Leagues to one Degree in that Latitude.

But if you would know by the foregoing Quadrant the Miles answering to a Degree in each Parallel of Latitude, it is thus.

Set the Ear E on the Center-Pin, and reckon the Degrees of Latitude from D: to which set the Edge of the Index, and note the Parallel-Line that is at the Degree; carry your Eye on it to the Side CD, and from the Center to that Line you have the Number of Miles answering a Degree in that Latitude.

EXAMPLE.

In the Latitude of 18 degrees 12 min. set the Index 18 gr. 12 min. from D, and the Parallel-Line rising with that Degree, with your Eye or a Pin follow to the Edge, and you will find it to be 57 Miles, the Miles answering one Degree of Longitude and 51 Miles, in the Latitude of 31 gr. 48 min. as in the foregoing Table; and so work for any other Latitude in like manner.

But if the Course hold upon any of the Rhombs between the Parallel of the Aequator and the Meridian, we are to consider besides the Aequator of the World to which we Land, which must be always known.

• First, The difference of Longitude, at least in general.
• 2. The difference of Latitude, and that in particular.
• 3. The Rhomb whereon the Course holds.
• 4. The distance upon the Rhomb, which is the distance we are here to consider, and is always somewhat greater than the like distance upon a great Circle. The first follows in the next Proposition.

I. To find how many Leagues do answer to one Degree of Latitude, in every several Rhomb.

In this Table is the Degrees of every quarter Point, ½, and whole Point in the Quadrant; as the first quarter is 2 gr. 49 m. so the half Rhomb is 5 gr. 37 m. the third is 8 gr. 26 m. and the first Point from the Meridian is 11 gr. 15 m. and so you may plainly see the rest.

Rhombs.	Inclinati∣on to the Meridian.	Number of Leagues.
	Gr. Min.	Leag par.
	2 49	20 2
	5 37	20 10
	8 26	22 22
1	11 15	20 39
	14 4	20 62
	16 52	20 90
	19 41	21 24
2	22 30	21 65
	25 19	22 12
	28 7	22 68
	30 56	23 32
3	33 45	24 05
	36 34	24 90
	39 22	25 87
	42 11	26 99
4	45 00	28 08
	47 49	29 78
	50 37	31 52
	53 26	33 57
5	56 15	36 00
	52 4	38 90
	61 5	42 43
	64 41	36 76
6	67 30	52 26
	70 19	59 37
	73 7	68 90
	75 56	82 31
7	78 45	102 52
	81 34	136 30
	84 22	205 14
	87 11	407 60
8	90 00	ad infinit.

As the Sine-Complement of the Rhomb from the Meri∣dian, is to 20 Leagues or 60 Miles, the Measure of 1 Degree at the Meridian:

So is the Radius or Sine of 90, to the Leagues or Miles answering to one Degree upon the Rhomb.

Suppose by the Quadrant it were required to answer this Question,

Sailing N. N. E. from 40 Degrees of North-Latitude, How many Leagues shall the Ship run before it can come to 41? By reason this is the second Rhomb from the Meridian, and the Inclination thereof is 22 deg. 30 m.

Therefore set the side of the Index EL to the second Point from the Meridian N. N. E. 22 d. 30 m. and reckon from C 20 Leagues towards D, and with your Eye or a Pin fol∣low the Parallel-Line to the Index, and you will find it cut 21 Leagues 65 parts (or better than ½ more) the number of Leagues you must Sail before you can reach 1 Degree.

You may do the same by the Travis-Scale thus. Extend the Compasses from 2 Points nearest the end of the Scale, and greatest Number of the Line of Numbers that is N. N. E. 2, and E. N. E. 6 Points, unto 20 Leagues on the Line of Numbers; remove the Compasses to 100 in the Line of Num∣bers, and the other Point of the Compass will reach to 21 Leagues 6/10 ½ or 65 parts, as before in the Line of Numbers.

This may be found also by a Line of Chords and Equal Parts, if you draw a Right Line, and take with your Com∣passes 20 parts, and lay it from one end on the Line; then take 60 deg. and sweep an Arch, and take 2 Points with your Compasses, and lay from the Meridian on that Arch from N. N. E. and draw the Secant or Rhomb-Line, at 20 Leagues draw a Perpendicular or Line at Right Angles there to the former, and measure the distance from the Center, to the Intersection of the Line drawn from 20, with the Rhomb-Line on the Scale of Leagues or Equal Parts, and you will find it the same as before. And so the Qua∣drant shews you all at one sight, if you understand without more words. By the Artificial Sine and Number, Extend your Compasses from the Sine of the Rhomb 67 deg. 30 to 20 in the Line of Numbers, the same Extent will reach from 8 Points, or 90 deg. or 100 in the Line of Numbers, to 21 Leagues 65 parts, as before.— This consider in general; I shall shew you more particularly in 12 Proofs (how of these four, any two being given, the other two may be found, both by Mercator's Chart, and all other ways, as is usual) when I come to treat more particularly of Na∣vigation.

II. By one Latitude, Rhomb, and Distance, To find the difference of Latitudes.

Let the place given be C in the Latitude of 40 Degrees, that is in the Center of the Quadrant, the second Latitude unknown; The distance upon the Rhomb 21 Leagues 65 parts of a League; the Rhomb N. N. E. the second from the Meridian: Therefore set the Index to the Point, and count 21 Leagues 10 parts, and run your Eye up the Parallel-Line you there meet with, and reckon the Leagues from the Center C to that Line, and you will find it 20 Leagues; and such is the difference of Latitude required.

It is easie to be understood how to lay it down by the Plain Scale; therefore I shall forbear to write any more of that Way.

As the Radius, to the Co-sine of the Rhomb from the Meridian:

So the distance upon the Rhomb, to the difference of the Latitudes.

Extend the Compasses from the Sine of 90, to the Co-sine of the Rhomb 67 deg. 30 m. the same distance will reach from 21-65 Leagues in the Line of Numbers, to the difference of Latitude 20 Leagues. In like manner you must work for all such Propositions, let the Number be greater or less, by either Instrument.

The Travis-Scale is the same manner of Work, as the Artificial Sines, Tangents, and Numbers; For extend the Compasses from 8 Points, to 2 Points, the same distance will reach from 21: 65 in the Line of Numbers, to 20 the difference required.

III. By the Rhomb and both Latitudes, To find the Distance upon the Rhomb.

As suppose the one place given were C the Center of the Quadrant, in the Latitude 40 deg. the second place in the Latitude 41 deg. and the Course the second from the Meridian.

Set the Index to the Rhomb, and account 20 Leagues, which is 41 deg. the second Latitude, and carry your Eye on that Parallel that leads to the Index; and there it will cut the distance upon the Rhomb, which in this Question is 21 Leagues 65 parts.

Extend the Compasses from the Co-sine of the Rhomb from the Meridian, to the Radius or Sine of 90—

The same Extent will reach from 20 Leagues, the difference of Latitude, to 21: 65 in the Line of Numbers, the distance upon the Course required.

IV. By the distance and both Latitudes, To find the Rhomb.

Suppose the Place given was at C, in Latitude 40 deg. and the second Place a Degree or 20 Leagues further Northward, and the distance was 21-65 Leagues upon the Course.

From the Center C reckon 20 Leagues towards D, follow that Parallel, and set the Index, and count the distance until it touch the Parallel, and look in Arch of the Quadrant, and you will find the Rhomb 22 gr. 30 m. or N. N. E. 2 Points from the Meridian.— Or, Extend the Compasses from the distance upon the Rhomb 21: 65, to the distance of Latitudes 20 Leagues; The same Extent will reach from the Radius or Sine of 90, to the Sine-Complement of the Rhomb 67 deg. 30, which was required.

V. By the difference of Meridians, and Latitude of both Places, To find the Rhomb.

As if the Place given was C the Center of the Quadrant, 40 deg. and 20 Leagues was the difference of Latitude Northward, that is 41 deg. and the difference of Longi∣tude 8 Leagues 45 parts of a League.

First, count from C the difference of Latitude 20 Leagues, on that Parallel count 8 Leagues 45 parts; to that put the Index; and in the Arch you will find the Course 22 gr. 30. from the Meridian.

Extend the Compasses from 20 Leagues to 8: 45, the same Extent will reach from 90 to the Tangent of the Rhomb 22 gr. 30 min. as before.

VI. By the Rhomb and both Latitudes, To find the difference of Longi∣tude, or departure from the Meridian.

Let the Rhomb be 2 Points from the Meridian, the one Latitude given 40 deg. the other Latitude 41 deg. the difference 20 Leagues.

Set the Index to the Point and Rhomb 20 gr. from the Meridian, and count 20 Leagues the difference; on that Parallel reckon the Leagues between the Side and the Index, and you will find it in this Question 8 Leagues 45 parts, the Meridians-distance required.— Or, Extend the Compasses from the Tangent of the Rhomb 22 gr. 30, to Radius 90, the same Extent will reach from the difference of Latitude 20 Leagues, to the departure from the Meridian 8 Leagues 45 parts.

These six last Propositions depend one upon the other, as you may plainly see; which may be sufficient for the Explanation of the Quadrant, by which may be un∣derstood much more.

CHAP. VII. How to make a most Ʋseful Protractor.

THis Instrument is always to be made in Brass or Copper, but best in Brass. On the Center C draw the Semicircle BO, and divide it into two 90, or 180 Degrees, as you may see the Figure; and let the sixteen Points of the Compass PP be set in the inward Circle, with the Quarter-Points. And let the

[illustration] diagram of a protractor

[illustration]

[illustration] diagram of a compass

A: Nocturnal:

Index AE be two Diameters and ½ long, and so fitted as the Semidiameter of the Cir∣cle may be the distance from the Center, for the ready setting to any number of De∣grees, or Points and Quarters the Edge thereof, and divide from the Center to the end of the Index into 100 equal Parts, which are accounted sometime Leagues, and rec∣koned by the Surveyors of Land Perches, or any other Denomination of Numbers: You may call it for protraction according as you have occasion to use it.

Let the Index be fastned to the Center with a Brass Rivet, and through the midst of the Rivet there must be a Hole drill'd; you must put the Pin or Needle spoken of in the last Chapter, upon any Point assigned, in any Chart or Protraction whatsoever. You may divide the Edges into equal parts, by which you may make a Meridian-Line on the blank Charts, according to Mercator's or Wright's Projection.

And now you have a necessary Instrum••nt, which will protract any Travis or piece of Land upon Paper, with as much speed as any Instrument I ever yet knew, and readier by much, the use whereof shall be shewn in this Treatise.

CHAP. VIII. The Projection of the Nocturnal.

IT consists, as you see, of three Parts. The greatest or handle-part hath on it two Circles divided: On the first or outmost is the Ecliptick, divided into 12 equal Parts, in which is put the 12 Signes; and each of those 12 Parts is divided into 30 equal Parts or Degrees, in each Signe, numbered 10, 20, 30, as you see the Figure plainly sheweth. The inward Circle is the 12 Months of the Year, set in by a Table of the Sun's place every day of the Year, accounting the Degrees of the outward Cir∣cle, and the number of Days in each Month equally divided and set down 10, 20, 30.

Note, Where to begin to divide the Months and Days is thus. Observe the bright∣est Guarde, or by Calculation or the Globes find when he comes to the Meridian just at 12 a Clock at Night. In the following Tables the Right Ascension of the brightest Guard is 223 Degrees 31 Minutes; from it substract 12 Hours, or 180 Degrees, the Remainder is 43 deg. 31 min. the Right Ascension of the Sun the 26th day of April, in 16 deg. of Taurus, which must be uppermost next the Zenith in the middle.

The other Part or middle Rundle equally divideth the outward Circle into twice 12 Hours; and within that is a Circle equally divided into 32 Parts, or Points of the Mariner's Compass projected thereon. The upper part is an Index, the length is from the Center to the Foot of the Instrument; all three being fastned with a piece of Brass, so Rivetted that the Center is an Hole through which you may see the North-Star. You may make it in Brass, or good dry Box.

The Ʋse of the Nocturnal.

THe Use of the Nocturnal is easie and ready. Let the Tooth or Index of the mid∣dle Circle be set to the Day of the Month, and it will cut in the outward Circle the Sun's Place in the Ecliptique. Then hold the Instrument on high, a pretty distance from you, with the Foot AB right with the Horizon level: Then look through the Hole of the Center, and see the North-Star, turning the long Index or Pointer up∣wards or downwards, untill you see the brightest of the Guards over or under the Edge that comes straight to the Center. Then look on the Hour-Circle by the Edge of the Pointer, and it shews the Hour of the Night, and likewise the Point of the Com∣pass the Guard beareth from the Pole; by the which you may have his Declination by the following Tables exactly.

The Hour of the Night may be also found by the Right Ascension of the Sun and Stars. Thus, When that you see any Stars in the South, whose Right Ascension is known, and also the Right Ascension of the Sun for that day, you shall substract the Sun's Ascension from the Star's; that which remaineth divide by 15, to bring it into

Hours;* 1.13 for 15 Degrees makes an Hour, and 4 Minutes make a Degree; thereby you have the Hour of the Night. If the Sun's Ascension be more than the Star's, in such case you shall add 360 Degrees to the Ascension of the Stars, and then substract the Sun's Ascension, as before directed, you have the time also.

CHAP. IX. How to use the Pole-Star's Declination and Table, and thereby to get the Latitude.

THe Pole-Star, being so very well known to all Sea-men, is therefore made the most use of by them: Therefore know, That this Table is made for the year 1660. the true Declination being 2 deg. 30 min. but will serve for many years after. This Table is made contrary to the two former Tables; for whereas the North Point of the Nocturnal is the first Point you reckon from, and was on the for∣mer Nocturnal reckoned from South: so of this Nocturnal you must take the Point at the Tooth for North, and so reckon forward North and by East, and so on to East and South-East, South and West, to North again.

And likewise in the Table, you must begin in like manner at that part of the Ta∣ble that lies directly under the Pole; which, as before-said, is properly called the North, and so proceed about the Pole, ascending from this lowest or North Point of the Me∣ridian, as was said before, to the North-East, East, and South-East, so to the South or highest Point of ascending, being directly over the Pole: From the South they descend again by the West, and so return to the North again.

Observe this, That the brightest of the Guards is the first of the little Bear, which is the Star you are to observe, and is almost in opposition to the Pole-Star.

Note, That when the Guard-Star is under the Pole, then the Pole-Star is above the Pole; and when the Guard-Star is above the Pole, then the Pole-Star is under the Pole so many Degrees and Minutes as the Table shews you.

The Use of this Table and Instrument is this: Look with the Nocturnal, and see what Point the Guards bears from the Pole, as before-directed; and if you find the Guard is not on a full Point, stay a little longer until he is just, and then observe the height of the Pole-Star exactly as you can; then knowing by your Dead Reckoning within a Degree or two what Latitude you are in, look for the nearest to that Lati∣tude in the top of the Table; and if you find the Point of the Compass which the Guard-Star is upon, in the first Column of the Table, and in that Line under the Co∣lumn of your Latitude, you shall find the number of Degrees and Minutes the Pole-Star is either above or below the Pole, according to the direction of the last Column of the Table, which you must thus make use of; If the Star be any thing above the Pole, substract the Number in the Table from the height of the Star observed: but if the Star be under the Pole, then add the Number found in the Table to the height observed, by which you shall have the height of the Pole.

For Example. Estimate the Latitude to be near 40 Degrees, and observe the Pole-Star. Suppose you find the Altitude 40 Degrees, and the Guard-Star bears N. N. E. from the Pole; therefore look for N. N. E. in the first Column, and right under your estimated Latitude 40, in the same Line with N. N. E. you will find the Declination to be 1 Degree 30 Minutes; substract that from 40 Deg. the Altitude observed leaves the true Latitude 38 Degrees 30 Minutes.

d.	m.
40	00
01	30
38	30

But if the Guard-Star had been S. S. W. then the Pole-Star had been 1 Degree 33 Minutes under the Pole; which being added to the Altitude observed 40 Deg. the Latitude would have been exactly 41 Degrees 30 Minutes by the Star. So the Star's Altitude by observation being 55 Deg. the Guard bears from the Pole-Star S. E. b. S. the Declination against that Point is 2 Degrees 30 Minutes, added to 55. had been 57 Degrees 30 Minutes for the Latitude: but if your estimated Latitude had been near 50, and the Guard bear from the Pole North-West

by North, the Declination against that Point is 2 deg. 30 min. substracted from 55 Degrees, the Altitude observed, there remains 52 deg. 30 min. the Latitude of the place by the Star.

A Table of the North-Star's Declination in these several Latitudes.

	The True Point of the Compass.	0	20	30	40	50	60	70
D. M.	D. M.	D. M.	D. M.	D. M.	D. M.	D. M.
If the former of the Guards be ascending from the North or lower part of the Meridian.	North.	2 10	2 10	2 10	2 09	2 09	2 08	2 07	Above the Pole.
N. b. E.	1 53	1 53	1 53	1 52	1 52	1 51	1 49
N. N. E.	1 31	1 31	1 30	1 30	1 29	1 28	1 25
N. E. b. N.	1 06	1 05	1 04	1 03	1 02	1 01	0 58
N. E.	0 39	0 38	0 37	0 36	0 35	0 33	0 30
N. E. b. E.	0 10	0 09	0 08	0 07	0 06	0 04	0 01
E. N. E.	0 18	0 19	0 20	0 21	0 22	0 23	0 26	Ʋnder the Pole.
E. b. N.	0 49	0 50	0 50	0 51	0 52	0 53	0 56
East.	1 15	1 15	1 16	1 17	1 18	1 19	1 21
E. b. S.	1 38	1 39	1 39	1 40	1 41	1 42	1 44
E. S. E.	2 00	2 00	2 00	2 00	2 00	2 01	2 02
S. E. b. E.	2 15	2 15	2 15	2 15	2 16	2 16	2 16
S. E.	2 25	2 25	2 25	2 25	2 25	2 25	2 25
S. E. b. S.	2 30	2 30	2 30	2 30	2 30	2 30	2 30
S. S. E.	2 29	2 29	2 29	2 29	2 29	2 29	2 29
S. b. E.	2 22	2 22	2 22	2 22	2 22	2 22	2 22
If the former of the Guards be descending from the South or upper part of the Meridian.	South.	2 10	2 10	2 10	2 11	2 11	2 11	2 12
S. b. W.	1 58	1 53	1 54	1 53	1 55	1 55	1 57
S. S. W.	1 31	1 32	1 32	1 33	1 34	1 35	1 38
S. W. b. S.	1 07	1 07	1 08	1 10	1 11	1 13	1 13
S. W.	0 39	0 40	0 41	0 40	0 43	0 44	0 47
S. W. b. W.	0 10	0 11	0 12	0 13	0 14	0 16	0 19
W. S. W.	0 19	0 19	0 17	0 16	0 15	0 13	0 10	Above the Pole.
W. b. S.	0 48	0 47	0 46	0 45	0 44	0 43	0 42
West.	1 15	1 14	1 13	1 12	1 11	1 10	1 08
W. b. N.	1 39	1 39	1 38	1 37	1 36	1 35	1 33
W. N. W.	2 00	1 59	1 59	1 58	1 58	1 57	1 56
N. W. b. W.	2 15	2 15	2 14	2 14	2 14	2 13	2 12
N. W.	2 25	2 25	2 25	2 25	2 24	2 24	2 24
N. W. b. N.	2 30	2 30	2 30	2 30	2 30	2 30	2 30
N. N. W.	2 29	2 29	2 29	2 29	2 29	2 29	2 29
N. b. W.	2 22	2 22	2 22	2 22	2 22	2 21	2 21

I hope the young Seamen are pleased for Examples, it being made so plain to their Capacity, and as profitable for their Use as any Rule whatsoever.

CHAP. X. How to make a most Ʋseful Instrument of the Stars, and by it to know most readily when any of 31 of the most notable Stars will come to the Meridian, what Hour of the Night, at any time of the Year, at the first sight.

THis Instrument consisteth of two parts, which is two Rundles; on the back side of the foregoing Nocturnal it may be very fitly made: On the matter or greater Rundle are three Circles divided; the outermost is the 12 Months of the Year, begun the 10th of March, the day the Sun enters into Aries; and the Days equally divided to the Number of Days in each Month. The second Circle re∣presenteth the 24 Hour Circle, divided equally into 24 Hours, ½ and ¼, beginning the 10th of March at 12 a Clock at Noon, the time the Sun comes to the Meridian, and the first Degree of Aries. The third and inward Circle is the Aequinoctial, divided into 360 Degrees; by them is accounted the Right Ascension of these 31 Stars in the Table following.

A Table of the Longitude, Latitude, Right Ascensions, and Declinations, of 31 of the most Notable Fixed Stars: Calculated from Tycho his Tables, rectified for the Year of our Lord, 1671.

	Longitude.	Latitude.	Ascensi∣on.	Decli∣nation.	Nor. Sou.
The Whale's Tail	27 56 ♓	20 47 S	06 45	19 48	S	2
The Bright Star in the South Foot of Androm.	09 39 ♉	27 46 N	25 57	40 44	N	2
The Bright Star in the Right Side of Perseus.	27 17 ♉	30 05 N	44 16	48 36	N	2
The Bright Star of the 7 Stars or Pleiades.	24 24 ♉	04 0 N	52 00	23 03	N	3
The South Eye of the Bull Aldebaran.	05 12 ♊	05 31 S	64 17	15 48	N	1
The Bright Star in the left Foot of Orion Riges.	12 17 ♊	31 11 S	74 44	8 37	N	1
Orion's right Shoulder towards the East.	24 12 ♊	16 6 S	84 23	07 18	N	1
The glittering Star in the Mouth of the great Dog.	09 35½ ♋	39 30 S	97 42	16 14	S	1
The Little Dog's Thigh Procyon.	21 18½ ♋	15 57 S	110 34	06 03	N	2
In the South Arm of the Crab.	09 03½ ♌	05 08 S	125 22	20 48	N	3
The Bright Star called the Heart of the Hydra.	22 45 ♌	22 24 N	137 54	07 15	S	1
The Lion's Heart Regulus.	25 17 ♌	00 26 N	147 43	13 33	N	1
The lower of the Pointers.	14 43 ♌	45 03 N	160 18	58 08	N	2
The White or North Pointer.	10 34 ♌	49 40 N	160 48	63 32	N	2
The Lion's Tail.	17 03 ♍	12 18 N	173 04	16 25	N	1
The First between the Tail and the Body.	04 10 ♍	54 18 N	189 03	57 47	N	2
The second of the Tail of the Horses.	10 56 ♍	56 14 N	197 37	56 41	N	2
The Fore-Horse, or last in the Tail.	22 12 ♍	54 25 N	203 37	51 00	N	2
In the Skirt of his Garment Arcturus.	19 39 ♎	31 2½ N	210 13	20 58	N	1
The South Balance of Libra.	10 31 ♏	00 26 N	218 13	14 37	S	2
The Brightest of the Guards.	08 16 ♌	72 51 N	223 37	75 38	N	2
The Scorpion Heart Antares.	08 13 ♐	04 27 S	242 23	25 37	S	1
Engonasis Head Hercules.	11 31 ♐	37 23 N	254 12	14 50	N	3
The Bright Star of the Harp Lyra.	10 43 ♑	61 47 N	276 27	38 30	N	1
The Swan's Bill.	26 44 ♑	49 02 N	289 23	27 18	N	3
The Bright Star the Eagle's Heart.	27 09 ♑	29 21½ N	293 41	08 03	N	2
The Dolphin's Tail.	09 32 ♒	29 08 N	304 24	10 14	N	3
The Mouth of Pegasus the winged Horse.	27 22 ♒	22 7½ N	322 03	08 24	N	3
The Bright Star of Pegasus Neck.	1 39½ ♓	17 41 N	336 21	09 08	N	3
The Southern Star in the Wing of Pegasus Macrobe	18 56½ ♓	19 26 N	342 07	13 28	N	2
Andromeda her Head.	09 47 ♈	25 42 N	357 54	27 18	N	2

[illustration] diagram of a celestial chart instrument

On the other Rundle or upper part, is placed all these aforesaid Stars; and any other you may set thereon, if you follow this Rule.

For Example, First set the Index to the 10th day of March of the upper Circle; on the under Circle, which will be at 12 a Clock at Noon, or 360 Degrees, stop it fast there, that it may not move, until you have placed the Stars on it as you intend to set thereon. As suppose you would set the Whale's Tail on the Rundle in his place; look in the foregoing Table, and you will find his Right Ascension 6 deg. 45 min. account that from the 10th of March, and on the Aequinoctial Circle, and lay a Ruler from the Center over the 6 deg. 45 min in the Matter and Aequator or inward Circle, and draw the Line from the Center to the outward Edge of the upper Circle, and thereon set the Name of the Star, next to that the Declination of the Star, and the Letter S or N. representing South or North Declination: on the inward Circle, set before each Star the Magnitude of the Star, whereby you may know the better, as the Figure following shews you all plain.

Take this Example more. Suppose you would set the Lion's Heart in his place; In the Tables I find his Right Ascension is 147 deg. 43 min. reckon that Number on the Aequinoctial Circle next the Rundle, and draw the Line as before-directed, 1 signify∣ing the First Magnitude, secondly his Name, thirdly 13 deg. 33 N, for his North Declination.

The Instrument in this posture, you will find the Lion's Heart will come to the Meridian at almost 10 a Clock in the Evening the 10th day of March in any Lati∣tude.

How to know the Hour of the Night any Star comes to the Meridian in any Latitude.

YOu have been in a manner shewed it before in the last Example. Set the Index or Hand of the upper Rundle to the Day of the Month, and right against the Star is the Hour of the Night, in the Matter the Star will be on the Meridian.

For Example, Suppose you would know the Hour of the Night the Bull's Eye comes to the Meridian the 20th Day of October; Set the Index to the Day, and right against the Bull's Eye is ¾ of an Hour past 1, the time in the Morning that Star will be on the Meridian South. And in the same manner you may see the Stars, and Hours they come to the Meridian that Night and Day. For note the upper half of the Circle, and 12 Hours is the Day-hours, and the lower and Handle-half is the Night-hours. You begin to reckon the Day-hours on the left side of the Instrument, and the Night-hours on the right side; so round with the Sun.

How to know what Stars are in Course at any time or Day of the Year.

THe Course and seasonable coming to the Meridian of the Stars, and what are fit to be observed, is shewn you at once, the Instrument in the former posture, if you look against each Star, you have the Hour of the Night and Day, being the whole 24 Hours. This is so plain, you need no further Precept.

How to know the Hour of the Night, by the Stars being on the Meridian.

SUppose it were required to know the Hour of the Night the 10th of December, the brightest of the 7 Stars being on the Meridian South: Set the Index to the day of the Month, and right against the brightest of the 7 Stars, is half an Hour past 9 at Night, which is the Hour required.

CHAP. XI. Of the Crosiers.

WHen the Mariners pass the Aequinoctial Line towards the South, so that they cannot see the North-Star, they make use of another Star, which is the Constellation called the Centaur; which Star, with three other no∣table Stars which are in the same Constellation, maketh the Figure of a Cross (be∣twixt his Legs) for which cause they call it the Crosiers. And it is holden for certain, That when the Star A (which of all four cometh nearest the South-Pole) is

[illustration] celestial diagram

North and South by the Star B, then it is rightly scituated to take the Height by: And because this Star A, which is called the Cocks-foot, is 30 Degrees from the South-Pole, it cometh to pass, it being scituated as before-said, we take the Height thereof (which is then the greatest that it can have) this Height will truly shew how far we are distant from the Aequinoctial: For if the said Height be 30 Degrees, then we are under the very Aequinoctial: But if it be more than 30 deg. then are we by so much past the Aequinoctial, toward the South: And if it be less than 30 deg. so much as it wanteth, we are to the North of the Aequinoctial. And here it is to be noted, That when the Guards are to the North-East, then is the Star in the Crosiers fitly scituated for observation, because then they are in the Meridian.

CHAP. XII. How to make the Cross-Staff.

THe Mariner's Cross-Staff is that which by the Astronomers is called Radius Astronomicus, by which we observe the Celestial Lights above the Horizon. The Mathematicians have invented many kinds of Instruments, whereof the Cross-Staff and Quadrant are the most useful above all the rest. At Sea it is not every Mans Work to make and mark a Cross-Staff, and other Instruments, for want of Practice needful thereunto; yet notwithstanding it is fit and necessary that a Master, his Mates, and Pilot, who are to have the Use of it, should at least know when it is well ••ade.

For to mark well a Cross-Staff, you shall make a plain flat Board of good dry

Wood, fifteen or sixteen Inches broad, and about four Foot or three Foot long: Paste it well with good Paper; draw along the one Side a Right Line, as in the next fol∣lowing Figure CAD; out of the Line C draw a Square Line upon AC, as CB, and upon the Center C draw the Arch AEB, being a Quadrant or fourth part of a Circle; divide that into two parts; the one half thereof, as AE, divide into 90 Equal Parts or Degrees, thus; first into three Parts, and each of the same again into three; these Parts each into two, and each of the last Parts into five: so the Arch AE shall then be divided into 90 Parts. Then take a right Ruler, lay the one end on the Point or Center C, the other upon each Point of the foresaid several Divisions, and draw small Lines out of C, through each of the foresaid Points or Degrees of the Quadrant, so long as they can stand upon the Board, as you may see it plain in the Figure. Then take with a pair of Compasses, just the half length of the Cross that you would mark the Staff after; prick it from the Point C towards B; as by Exam∣ple, from C to F, and from D to G; joyn these two Points with a Line to one ano∣ther; and even into such Parts as that Line is cut, and divided by the aforesaid Lines coming out of the Center of the Quadrant, must your Staff be marked, whether the Cross be long or short, as appeareth by the Lines HI and KL, which are drawn for Crosses: the half thereof is so long as CH, or CK, or CF. If the aforesaid Quadrant, for want of good Skill or Practice, be not well divided, or Lines not well drawn, the Staves being marked thereafter will also be faulty. Therefore they may be marked more exactly by Points equally divided, in manner as followeth.

[illustration] geometrical diagram

A Table for the Division of the Cross-Staff.

D.	Parts	D.	Parts.	D.	Parts.
1	176	31	7675	61	28667
2	355	32	8040	62	30108
3	538	33	8418	63	31653
4	724	34	8807	64	33315
5	913	35	9210	65	35107
6	1106	36	9626	66	37046
7	1303	37	10057	67	39152
8	1504	38	10503	68	41445
9	1708	39	10965	69	43955
10	1918	40	11444	70	46713
11	2131	41	11943	71	49758
12	2349	42	12460	72	53197
13	2572	43	12998	73	56912
14	2799	44	13558	74	61154
15	3032	45	14142	75	65958
16	3270	46	14751	76	71445
17	3514	47	15386	77	77769
18	3764	48	16051	78	85144
19	4019	49	16746	79	95854
20	4281	50	17475	80	104301
21	4550	51	18239	81	117062
22	4826	52	19042	82	133007
23	5108	53	19887	83	153499
24	5399	54	20777	84	180811
25	5697	55	21716	85	219038
26	6003	56	22708	86	276332
27	6318	57	23759	87	371885
28	6643	58	24874	88	561810
29	6976	59	26059	89	1135891
30	7320	60	27321	90	Endless.

Prepare you a Staff, draw thereon a Right Line so long as your Staff, and take with a sharp pair of Compasses the half Length of the Cross after which you desire to mark your Staff: prick it so often along the aforesaid Line, as it can stand upon the same. Divide each of the Lengths of the half Cross into 1000 Equal Parts. Then prick upon the Staff you will mark from the Center-end, just half the Length of the Cross; and mark there a small thwart Stroke. Off from thence prick for each Degree so many of the same Parts as the Cross is divided in his half Length, like as is marked in the Table here annexed for every Degree. For the first Degree you shall mark off from 90 the aforesaid thwart Stroke 176, for the fourth Degree 724, for the 10 Degree 1918 of those Parts, and so of the rest. If you cannot divide the half Cross, by reason he is so little, into 1000, divide him into 100, and leave out the two last Figures, and that shall satisfie your desire: For 30 De∣grees take 73, and for 40 Degrees 114, and for 10 Degrees 19 Parts, and so of the rest.

CHAP. XIII. How to use the Cross-Staff.

SEt the end of the Cross-Staff to the outside of the Eye, so that the end of the Staff come to stand right with the Center of the Motion of the Sights. Then move the Cross so long off from you or towards you, holding it right up and down, and winking with your other Eye, till that the upper end come upon the middle or Center of the Sun or Star, and the lower end right with the Horizon, the Cross then shall shew upon the side of the Staff belonging thereunto, the Degrees of the Altitude of the Sun or Star. Note, The Staff is marked with two Lines of Numbers, with 90 Degrees next the Eye, and diminishing from 90 to 80 and 70, 60 towards the outmost end: The Complement-Sine beginneth towards the Eye-end, and encreaseth contrarywise towards the outmost end, as from 10, to 20, 30. The first Number sheweth you the Altitude, the second Number is the Sun's distance from the Zenith.

The Sun or Stars being high elevated from the Horizon, the Cross cometh nearer the Eye than when they are but a little elevated, and do stand neer the Horizon; thereby the eye mak••th (seeing now to the lower, and then again to the upper end of the Cross) greater

motion in looking up and down, than when the Sun or Star doth stand low. And inso∣much the Center of the Sight, by such looking up and down together with the end of the Staff, a Man seeth then smaller Angles then if it did remain stedfast, in regard where∣of the Cross cometh nearer to the Eye than it should, and there is found too much Altitude. This being found by many, besides my self, by experience, they were there∣fore wont to cut off a piece of the end of their Staff, or set the Crosses a Degree and ½ or two Degrees nearer the Eye; but it is not the right means for to amend the afore∣said Errors. The best means of all in my opinion is this; That upon each several Height which men will observe, they do try with two Crosses set upon the like De∣grees, how the Staff must be set, that they may see the end of the same two Crosses right one with the other.

[illustration] geometrical diagram

Having found that, and then taking off one of the Crosses, and setting the Staff again, in the same manner as before, all Errours will be so prevented, which by the lifting up or casting down of the Eye, might any manner of way happen.

EXAMPLE.

I desire to observe the Sun or any Star in the South: I make my Estimation, as neer as I can, how high that shall stand, or take the Height of them a little before they shall come to the South, which I take to be 50 Degrees. I set therefore the two

Crosses each upon their 50 Degrees, and the end of the Staff in the hollow of the Eye-bone, on the outside of the Eye, and bow the Head forward or backward, or over the one side or the other, till I see the utmost end of both the Crosses right one with the other, according as is shewn by these Lines AB and CD, as is apparent enough by the foregoing Figure; That the Sight-beams over the ends of the Crosses shall then agree with the Lines which might be drawn over the end of the Crosses, to the Point or Center at the end of the Staff, which doth agree with the Center of the Quadrant, or the beginning of the Equal Points upon which the Staff is marked. Keeping in memory such standing of the Staff, I take off the one Cross, and set the Staff again in the aforesaid manner to the Eye, and observe without any errour of the Eye.

In taking the Height of the Sun with the Cross-Staff, Men do use red or blew Glasses, for the saving and preserving the Eyes; yet it is notwithstanding a great let, and very troublesom for the Sight, especially if it be high: therefore the Quadrant and Back-Staff is much better, as will be shewed in the next Chapter.

Thus I have shewed you how to take an Observation by the Fore-Staff. The next thing that followeth in course will be to shew you how to work your Observation; which to do, take notice of these following Rules.

To Work your Observation.

IF the Sun hath North Declination, and be on the Meridian to the Southwards of you, then you must substract the Sun's Declination from your Meridian Altitude, and that Remainder is the Height of the Aequinoctial, or the Complement of the La∣titude North. But if the Sun hath South Declination, you must add the Sun's De∣clination to your Meridian Altitude, and the Sum is the Height of the Aequator, or the Complement of the Latitude North. If the Sun hath North Declination, and be on the Meridian to the Northwards, then add the Sun's Declination to his Meridian Al∣titude, and the Sum is the Height of the Aequator, or the Complement of the Latitude South, if the said Sum doth not exceed 90 deg. but if it doth exceed 90 deg. you must substract 90 deg. from the said Sum, and the Remainder is your Latitude North.

If the Sun hath South Declination, and be to the Northwards at Noon, you must then substract the Sun's Declination from his Meridian Altitude, and the Remainder is the Complement of your Latitude South. When the Sun hath no Declination, then the Meridian-Altitude is the Complement of the Latitude. If the Sun be in the Ze∣nith, and if at the same time the Sun hath no Declination, then you are under the Aequinoctial.

But if the Sun hath North Declination, and in the Zenith, then look how many Degrees and Minutes the Declination is, and that is the Latitude you be in North.

But if your Declination be South, then you are in South Latitude. If you observe the Sun or Star upon the Meridian beneath the Pole, then add your Meridian Alti∣tude to the Complement of the Sun or Stars Declination, and the Sum is the Height of the Pole.

Rules for Observation in North Latitude.

SUppose I am at Sea, and I observe the Sun's Meridian Altitude to be 39 deg. 32 min. and the same time the Sun's Declination is 15 deg. 20 min. North; I de∣mand the Latitude I am in.

	deg.	min.
The Meridian Altitude	39	32
The Declination North, subst.	15	20
The Complement of the Lat.	24	12
	90	00
The Latitude I am in	65	48 North.

Suppose I were in a Ship at Sea the 18th of April, Anno 1667. and by Observati∣on I find the Sun's Meridian Altitude to be 62 deg. 15 min. The Latitude is required.

	deg.	min.
The Meridian Altitude	62	15
The Declination North, subst.	14	18
The Complement of the Latitude	47	57
	90	00
The Latitude I am in, required	42	03

Admit you were in a Ship at Sea the 5th of November, Anno 1679. and I find the Sun's Meridian Altitude to be 24 deg. 56 min. The Latitude is required I am in.

	deg.	min.
The Meridian Altitude	24	56
The Declination South, add	18	37
The Complement of the Latitude	43	33
	90	00
The Latitude I am in	46	27

Suppose a Ship at Sea the 28th of May, Anno 1666. and I find the Sun's Me∣ridian Altitude by Observation 56 deg. 45 min. The Latitude is required I am in.

	deg.	min.
The Meridian Altitude	56	45
The Declination North, Subst.	22	46
The Complement of the Latitude	33	59
	90	00
The Latitude required I am in	56	01

Admit a Ship at Sea the 11th of June 1668. and find the Sun's Meridian Al∣titude by Observation 79 deg. 30 min. North, It is required the Latitude I am in.

	deg.	min.
The Meridian Altitude	79	30
The Declination North	23	31 add.
	103	01
	90	00
The Latitude I am in	13	01 required.

Suppose I were at Sea the 14th of May 1693. and the Meridian Altitude of the Sun was 69 deg. 07 min. North, I demand the Latitude the Ship is in at that time.

	deg.	min.
The Meridian Altitude	69	07
The Declination North	20	53 add.
	90	00
	90	00
The Ship is under	00	00 the Aequinoctial.

Rules for Observation in South Latitude.

SUppose I at Sea in a Ship the second of June, Anno 1666. and I find the Sun's Meridian Altitude by Observation to be 64 deg. 45 min. The Latitude the Ship is in, is required.

	deg.	min.
The Meridian Altitude North	64	45
The Declination North, add	23	15
The Complement of the Latitude	88	00
	90	00
The Latitude the Ship is in-	02	00 South.

Suppose a Ship at Sea the 28th of December, Anno 1695. and in Longitude 169 deg. East, and I find the Meridian Altitude by Observation to be 59 deg. 52 min. The Latitude the Ship is in, is required. The Declination in the Meridian of Bristol for the 28th of December, is 22 deg. 25 min. and the daily difference of Declination is at this time 8 min. Therefore if you look in the Table of Proportion following, you will find the Proportional Minutes to be about 4, which you must add to the Declination of the Meridian of Bristol, and the Sum will be the true Declination for the Longitude 169 deg. East, which is 22 deg. 29 min.

	deg.	min.
The Meridian Altitude North	59	52
The Declination South, substr.	22	29
The Complement of the Latitude	37	23
	90	00
The Latitude the Ship is in, which was	52	37 required.

Suppose I were at Sea in a Ship the 29th of June, 1679. and I find the Sun's Me∣ridian Altitude to be 62 deg. 30 min. North, The Latitude is required.

	deg.	min.
The Meridian Altitude North	62	23
The Declination North, add	22	26
The Complement of the Latitude	84	49
	90	00
The Latitude the Ship is in	05	11 South.

Admit I am in a Ship at Sea the 20th of January 1667. the Sun's Declination 20 deg. 4 min. and the Sun's Meridian-Altitude 79 deg. 36 min. South, I require the Latitude the Ship is in.

Answer, 9 deg. 30 min. South.

Admit a Ship were at Sea, the Sun's Declination 13 deg. 53 min. South, and the Sun's Meridian Altitude 80 deg. 43 min. South, The Latitude is required.

	deg.	min.
The Declination South	13	53
The Meridian Altitude	80	43 add.
	94	36 the Sum.
Substr.	90	00
The Latitude the Ship is in	04	36 South.

If you observe the upper part of the Sun, you must substract 16 min. But to the contrary, if you observe the lower part of the Sun, you must add 16 min. for the Sun's Diameter, and the Sum will be the true Altitude of the Sun's Center.

Rules for Observing the Stars.

SUppose I am at Sea, and observe the Brightest of the 7 Stars upon the Meridi∣an, and find his Meridian Altitude to be 47 deg. 20 min. and the Latitude were required.

	deg.	min.
The Declination of this Star is	23	03 North.
The Meridian Altitude	47	20
Substract the North Declination	23	03
The Complement of the Latitude	24	17
	90	00
The Latitude I am in	65	43

Admit I were at Sea, and observe Hydra's Heart on the Meridian, his Altitude is 36 deg. 15 min. and his Declination is 7 deg. 15 min. South, The Latitude of the Place is demanded.

	deg.	min.
The Meridian Altitude is	36	15
The Declination is South	07	15 add.
The Height of the Aequinoctial above	43	30 the Horizon.
	90	00
The Latitude the Ship is in	46	30 required.

This you see is plain, and needs no further Precept but what is already said.

CHAP. XIV. A Description of the Back-Staff or Quadrant.

THe Back-Staff or Quadrant is a double Triangle, as this Figure following sheweth; whereof the Triangle ABC the Arch is equally divided into 60 deg. and the other Triangle is divided equally into 30 deg. ADF the Vanes are fitted neat. In proportion to him, the Use followeth.

The Ʋse of the Back-Staff or Quadrant.

SEt the Vane G to a certain number of Degrees, as the Altitude of the Sun re∣quireth; and looking through the Vane F, to the upper Edge of the Slit of the Sight of the Horizon, if you see all Skie and no Water, then draw your Sight-Vane a little lower towards E: but if you see all Water and no Skie, then put your Eye-Vane up higher towards F; and when you have done so, observe again; and then if you see the Shade lie upon the upper part of the Slit, on the Horizon-Vane, and you at the same time do see the Horizon through the Sight-Vane, then that is all you can do untill the Sun be risen higher; and tending the Sun until he be upon the Meridi∣an, you will perceive he is descending, or as we commonly say he is fallen, you will see nothing but Water; your Vanes fast in this posture, you have done observing the Sun upon the Meridian that day: Therefore reckon the Degrees from B to the up∣per side of the Vane G; to it add the number of Degrees from E to the Eye-Sight, and their Sum is the Distance of the Sun from the Zenith to the upper Edge of the Sun; to which Sum if you add 16 minutes, which is the Sun's Semidiameter, you will have the true distance of the Sun's Center from the Zenith or Complement of the Meridian Altitude. Note this, If you observe the upper Limb of the Sun by the up∣per

part of the Shade, then it is the upper Limb that gives the Shade; but if you observe the lower part of your Shade, then it is the lower side of the Sun that gives the Shade: Therefore you must substract 16 min. from what your Back-Staff gives you, and the Sum or Difference gives you the right Distance of the Sun from the Zenith. You may have the Altitude of the Sun from your Quadrant, if you work thus; from C to G is 40 deg. for the Vane stands at 20 deg. from D to F is 16 deg. being added together makes 56, the Altitude or Height of the Sun above the Hori∣zon, which you may use as you were shewn by the Fore-Staff: But in regard the English Navigators work their Observation by the Complement of the Sun's Altitude, when he is upon the Meridian, being so ready to be counted by their Quadrant; Therefore we will direct you in general, and after in particular Rules.

[illustration] geometrical diagram

The Figure of the Quadrant

First, If the Sun hath North Declination, and you in North Latitude, and the Sun upon the Meridian, South of you; then if you add the Sun's Declination to his Ze∣nith-Distance, that is the Complement of the Sun's Meridian Altitude, the Sum will be the Latitude you are in.

But if the Sun hath South Declination, you must substract the Complement of the Meridian Altitude, and the Remainder will be the Latitude the Ship is in.

If you be to the Southward of the Aequinoctial, and the Sun to the Northwards of the Aequinoctial, in such case you must add the Sum of the Declination to the Zenith-distance, and the Sum will be your Latitude South.

But if the Sun be to the Northwards of the Aequinoctial (that is, have North De∣clination)

you must substract the Declination from the Zenith-distance, and the Re∣mainder will be the Latitude South.

If you understand the fore-going Rules given of the Use of the Fore-Staff, you cannot mistake the Use of the Quadrant or Back Staff.

We will now come to Examples what are needful.

Observe these Rules for North Latitude.

ADmit we were in a Ship at Sea the fifth of May, Anno 1694. and by Observation I find the upper side of the Sun to be distant from the Zenith 37 deg. 36 min. the Sun being upon the Meridian, I require the Latitude the Ship is in.

	deg.	min.
The Sun's Distance from the Zenith	37	36 his upper Edge.
The Sun's Semidiameter, add	00	16
The Center of the Sun from the Zenith	37	52
North Declination	19	02
The Latitude required, the Ship is in	56	54

Suppose a Ship at Sea the 29th of July, Anno 1682. and I find the Complement of the Sun's Meridian Altitude by observation to be 32 deg. 54 min. The Latitude of that Place the Ship is in, is required.

	deg.	min.
The Complement of the Altitude is	32	54
The Sun's Semidiameter add to it	00	16
The Distance of the Sun's Center from Zenith	33	10
North Declination, add	16	07
The Latitude the Ship is in	49	17

Suppose a Ship were at Sea the 13th of Sept. 1683. and I find the Complement of the Sun's Meridian Altitude, or Distance from the Zenith, 45 deg. 42 min. I demand what Latitude the Ship is in.

	deg.	min.
The Complement of the Altitude is	45	42
The Sun's Semidiameter add to it	00	16
The Distance of the Center of the Sun	45	58
The Declination South, substract	00	07
The Latitude the Ship is in	45	51

Admit a Ship were at Sea the fourth of December, Anno 1690. and the Comple∣ment of the Sun's Meridian Altitude that day were 49 deg. The Latitude the Ship is in, is required.

	deg.	min.
The Complement of the Meridian Altitude	49	07
The Sun's Semidiameter—Add	00	16
The Center of the Sun distant from the Zenith	49	23
The Sun's Declination South, substract	23	30
The Latitude the Ship is in North	25	53

Suppose I were in a Ship at Sea the 23d of May, Anno 1695. and I am also in Longitude to the East of the Meridian of London 135 deg. and I find the Comple∣ment of the Meridian Altitude by Observation to be 13 deg. 12 min. The Latitude is required.

	deg.	min.
Declination in the Meridian of London	22	17
The Proportional Minutes, substract	00	03
The Sun's Declination in the Meridian given	22	14
The Complement of the Sun's Altitude	13	12
The Sun's Semidiameter, added	00	16
The True Zenith Distance of the Sun	13	28
The True Latitude the Ship is in North	08	46

That is, by reason the Sun is to the Northward of my Zenith, and the Declination more than the true Distance from the Zenith or Complement of the Meridian Altitude; therefore substract 13 deg. 28 min. from 22 deg. 20 min. and the Remainder is the true Latitude 8 deg. 52 min. North.

EXAMPLE.

Let the Complement of the Sun's Altitude be ZS, the Altitude 76 deg. 32 min. in the North BS, the Declination North ES 22 deg. 14 min. if you add the Altitude SB 76 deg. 32 min. to the Declination SE 22 deg. 14 min. the Sum is BE 98 deg. 46 min. the Distance of the Aequinoctial from the Horizon in the North BZ 90 deg. being substracted from it, remaineth for ZE, the Distance of the Aequinoctial from the Zenith towards the South, 8 deg. 46 min. just BP the Latitude of the Place, and Altitude of the Pole above the Horizon.

	d.	m.
BS	76	32
SE	22	14
BE	98	46
	90	00
B	08	46

[illustration] geometrical diagram

Let the Complement of the Sun's Altitude be ZS 13 deg. 28 min. the North Decli∣nation ES 22 deg. 14 min. being more than the Distance of the Sun from the Zenith, substract ZS 13 deg. 28 min. the Complement of the Altitude from SE the Declinati∣on, there remaineth 8 deg. 46 min. the Distance of the Aequinoctial from the Zenith ZE or Latitude of the Place BP, as before. I have been the more large on this, by reason I would have Learners perfect in it, it being most useful Questions.

When that you Sail far Northward or Southward, that the Sun goeth not down, as they find that Sail about the North Cape, and to Spitsberghen, or Greenland; and that you would observe the Altitude by the Sun, also when he is in the North at the lowest,

First, There must be added to the Altitude of the Sun taken above the Horizon, the Complement of the Sun's Declination; that is, the Distance betwixt the Sun and the Pole, that Number sheweth the Altitude of the Pole.

Secondly, Or else the observed Altitude must be substracted from the Declination, that which remaineth is the depression or depth of the Aequinoctial under the Hori∣zon in the North, just to the Altitude of the same in the South, the Complement thereof is the Altitude or Height of the Pole.

Thirdly, If you take the Complement of the Sun's Altitude, and substract from it the Complement of the S••n's Declination, there remaineth the Distance of the Pole from the Zenith, or the Altitude of the Aequinoctial in the South; the Complement thereof is the Altitude or Height of the Pole.

Fourthly, Or else if you add the Declination to the Complement of the Altitude, and you substract 90 Degrees out of that Number, there remaineth the Depth of the Aequinoctial in the North under the Horizon; that being substracted out of 90, there remaineth the Altitude of the Pole.

Directions for Observation in South Latitude.

ADmit a Ship at Sea the 7th of July, Anno 1695. and I am in Longitude 135 deg. East, and the Sun being upon the Meridian, I find the Complement of his Meridian Altitude by Observation to be 42 deg. 34 min. North; The Latitude is demanded, the Ship is in.

	deg.	min.
The Complement of the Meridian Altitude	42	34
The Sun's Semidiameter, add	00	16
The Sun's Center distant from the Zenith	42	50
The Declination North, substract	21	16
The Latitude the Ship is in	21	34

Admit I were in a Ship the fifth of November, Anno 1687. and in Longitude 120 deg. West, and the Complement of the Sun's Meridian Altitude by Observation is 31 deg. 37 min. North; The Latitude is required, the Ship is in.

	deg.	min.
The Complement of the Meridian Altitude	31	37
The Sun's Semidiameter, add	00	16
The Sun's Center distant from the Zenith	31	53
The Declination South	18	37
The Proportional Minutes	00	05 Added.
The Sun's Declination in the Meridian given	18	42
Which add to the Zenith-distance	31	53
The Latitude the Ship is in	50	35

Suppose I were in a Ship at Sea to the Southwards of the Aequinoctial, the third of January, Anno 1683. and I find the Sun upon the South part of the Meridian, and by Observation his Meridian Altitude is 75 deg. 38 min. The Latitude the Ship is in, is required.

	deg.	min.
The Complement of the Meridian Altitude	14	22
The Sun's Semidiameter, add	00	16
The Sun's Center from the Zenith	14	38
The Declination South	21	28
The Latitude the Ship is in South	06	50

EXAMPLE.

In this Figure let C be the South, and P the North Pole, DE the Aequinoctial, AB the Horizon, Z the Ze∣nith. Let AF be the Altitude of the Sun a∣bove the Horizon, in the North 58 deg. DF South Declination 8 deg. If you substract the Declination DF 8 deg. from FA the Altitude, there remains 50 deg. the Height of the Ae∣quinoctial above the Horizon in the North; that being deducted out of 90 deg. there remaineth AP 40 deg. for the Depth of the North Pole under the Horizon, just to BC the Elevation or Alti∣tude of the South-Pole above the Horizon in the South.

[illustration] geometrical diagram

	deg.	min.
Sun's Altitude	58	00
South Declination	08	00
Height of the Aequator	50	00
	90	00
The Latitude is	40	00

The Ʋse of the Fore-Staff in South Latitude for the Sun and Stars.

SUppose I were at Sea in a Ship the second of June, Anno 1694, and I find the Sun's Meridian Altitude by Observation to be 59 deg. I demand the Latitude the Ship is in.

	deg.	min.
The Meridian Altitude North	59	00
The Declination North, add	23	15
The Complement of the Latitude	82	15
	90	00
The Latitude required	07	45

Admit I were at Sea in a Ship to the Southward of the Aequinoctial, the 12th of January, Anno 1682. and in Longitude 135 East; and I find by Observation the Me∣ridian Altitude 63 deg. 34 min. North: There is required the Latitude the Ship is in.

The Declination for this Meridian, the Lands-end of England, is about 19 deg. 33 min. the daily difference in Declination at this time is 14 min. Therefore if you look in the Table of Proportion, you will find the Proportional Minutes to be 5, which you must add to the Declination of the former Meridian, and the Sum will be the true Declination for the Longitude of 135 deg. East, which is 19 deg. 38 min.

	deg.	min.
The Meridian Altitude	63	34
The Declination South	19	38 substract.
The Complement of the Latitude	43	56
	90	00
The Latitude required	46	04

Admit a Ship were at Sea the third of August, Anno 1675. and I find the Sun's Meridian Altitude to be 59 deg. 36 min. North, The Latitude is required.

	deg.	min.
The Meridian Altitude North	59	36
The Declination North, add	14	42
The Complement of the Latitude	74	18
	90	00
The Latitude the Ship is in	15	42

Suppose a Ship at Sea, the Sun's Declination being 21 deg. 42 min. South, and the Sun's Meridian Altitude 74 deg. 23 min. South, The Latitude is required the Ship is in.

	deg.	min.
The Complement of the Sun's Meridian Altitude	15	37
Substracted from the Sun's Declination	21	42
The Latitude the Ship is in	06	05

This being made so plain and easie to be understood, need no more Precedent: But observe this, If you observe the upper Edge or part of the Sun, you must sub∣stract 16 minutes; if the lower part, add 16 minutes for the Semidiameter of the Sun, and the Sum sheweth the true Altitude of the Center of the Sun.

CHAP. XV. Directions for Observing the Stars.

SUppose I am at Sea in a Ship, and I observe the bright Star in the left Foot of Orion Rigel, upon the Meridian, and find his Altitude 44 deg. 32 min. his Declination is 8 deg. 37 min. North; The Latitude is required, the Ship is in.

	deg.	min.
The Meridian Altitude	44	32
The Declination North	08	37 substract.
The Complement of the Latitude	35	55
	90	00
The Latitude the Ship is in	54	05

Suppose I am at Sea, and I observe the South Balance of Libra, and by Observati∣on of the Star upon the Meridian, I find his Altitude 39 deg. 27 min. and his Declination South 14 deg. 27 min. I require the Latitude I am in.

	deg.	min.
The Meridian Altitude	39	27
The Declination of the Star	14	37 South.
The Height of the Aequinoctial	54	04
	90	00
The Latitude the Ship is in	35	56

I have furnished the Practitioner with all useful and needful Examples, which I thought necessary for direction, which explains the following Tables, and shews the most easie and perfect way of Observation, and how to work them on either side the Aequator. Others I confess have been larger, but none more plain: for he that can∣not understand these Rules and Directions, is not fit to be a Mate of any Ship or Vessel, nor fit to be ranked among the Ingenious Mariners.

CHAP. XVI. The Description and Ʋse of the most Ʋseful Quadrant for the taking Alti∣tudes on Land or Sea, of the Sun or Stars, backwards or forwards, or any other Altitude of Hills, Trees, Steeples, or Castles, or any thing what∣ever.

THis Quadrant is made of well-seasoned and smooth dry Box Wood or Pear-tree. The Sides or Semidiameter of the Circle is about 19 or 20 Inches. C V and CH the Arch of the Quadrant is divided into 90 Degrees first, and each Degree into 6 Equal Parts, each Part being 10 Minutes, which is near enough for Sea or Land Observations, and numbred as you see from 10 to 90 deg. The two Sides next the Center, EF and GF, are divided each of them into 100 Equal Parts:

[illustration] geometrical diagram

That which is next the Horizon GFH, are called the Parts of Right Shadow: the other Side EFV, is the Parts of contrary Shadow. In the Center at C there is a Brass-Pin, and on it hang the Thred and Plummet; and on the Side there is a Sight made of Brass at E. There is also an Horizon-Vane, let in upon the Center C, with two Laggs that the Brass-Pin comes upon in the middle of the Slit; and a Shade-Vane and Sight-Vane, for Back-Observation. The Use of the Quadrant is,

EXAMPLE.

Admit I am ashore upon any Land or Island, and would know the Sun's Me∣ridian Altitude, and true Latitude of the Place. Take the Altitude thus; The String and Plummet being hanged on the Center C, turn the Brass-Pin to the Sun, and hold up the Center until the Shade of the Brass-Pin strikes on the Sight and Line of E, the Thred and Plummet playing easily by the Side: mark where it cuts the Arch of the Quadrant, as at F, that is the Sun's Altitude, and reckoned from H; and the Latitude is found by the same Rules as you have been given in the Use of the Fore-Staff. The best way to hold the Quadrant steady, is to skrew it with a Brass-Pin through at K, to a Staff set perpendicular, and then you may raise it by degrees, as the Sun rises.

PROPOSITION I. For Back-Observation at Sea.

Take the Handle of the Quadrant at H in your Hand, after the Vanes are set on, and fix the Shade-Vane; then hold your Quadrant as upright as you can; then bring your Sight-Vane to your Eye, and look through your Sight upon the Horizon-Vane. You must be sure to hold your Quadrant, so that the upper part of your Shad•• Vane, may be upon the upper part of the Slit on your Horizon-Vane, and look through the Slit for the Horizon: But if you cannot see the Horizon, but all Skie and no Water, you must draw your Sight-Vane a little lower down towards H; but if, on the contrary, you do see all Water and no Skie, then slide your Sight-Vane a little higher towards V, and then make Observation again; and then if the upper part of the Shade do lie upon the upper part of the Slit, and you see the Horizon at the same time, then it is well, and you must wait a little longer as your Judgment thinks fit, till the Sun is upon the Meridian, and so do as you did before; and if the Sun be to the Westward of the Meridian, and falling, you will see all Water and no Skie, the Work is done for that time and day. Then look what Degrees the Shade-Vane is put at, which in the Figure is at 70 deg. which note. Look also what Degrees and Minutes do stand against your Sight, which substract from the former Degrees by the Shade-Vane, and the Remainder is the Sun's Meridian Altitude. As in the Figure, The Sight-Vane is at 25 deg. 30 min. which taken out of 70 deg. the Remainder is 44 deg. 30 min. the Sun's Altitude, or the distance of the upper part of the Sun from the Horizon; from which if you substract 16 min. which is the Sun's Semidi∣ameter, the Remainder will be the Distance of the Sun's Center from the Horizon, or the true Meridian Altitude. And the way of working your Observation, is the very same as you have been given in the Use of the Fore-Staff.

PROP. II. Any Point being given, To find whether it be level with the Eye, or not.

Take the Quadrant and look through the Sight at E and Center-pin C, unto the Point given, or the Place you would know whether it be level, or not. If the Thred fall on CH the Horizontal Line, then is the Place level with the Eye: But if it should fall within, upon any of the Divisions, then it is higher; if without the Quadrant, then it is lower than the level of the Eye.

PROP. III. To find the Height of an House, Steeple, Tower, or Tree, from the Ground, at one Observation; and the length of the Ladder which will Scale it.

If you can approach the bottom or foot of the Thing whose Height you desire, the thing is easily performed by this Quadrant or Cross-Staff, holding up your Qua∣drant to the Place whose Height you would know, and looking through the Sight on the Side EC, going nearer or further from it, till the Thred cut 45 deg. or fall upon 100 Parts in the Quadrat: So shall the Height of the Thing above the level of your Eye, be equal to the Distance between the Place and your Eye.

If the Thred fall on 50 parts of a right Shadow, or 26 deg. ½ or Vanes on the Cross-Staff, set to the Number of Deg. the Height is but half the Distance.

If the Thred cut 25 Parts in the Quadrat, or 13 deg. 55 min. in the Arch of the Quadrant, it is but a quarter of the Distance: But if it fall on 75 Parts, or 36 deg. 53, it is three quarters of the Distance. The Rule is,

As 100, to the Parts on which the Thred falleth:

So is the Distance, to the Height required.

And on the contrary,

As the Parts cut by the Thred, are to 100:

So is the Height, to the Distance.

[illustration] geometrical diagram

But when the Thred shall fall on the parts of the contrary Shadow, if it fall on 50 Parts, or 63 deg. 30 min. as it doth at C, the Height is double unto the Distance CD. If on 25, it is four times the Distance. If the Thred fall upon the contrary Shadow, this is the Rule,

As the Parts cut by the Thred, are unto 100:

So is the Distance, unto the Height.

On the contrary,

As 100, are unto the Parts cut by the Thred:

So is the Height, unto the Distance.

These are the Rules Mr. Gunter shews by the Quadrat, And what hath been said

of Height and Distance, the same may be understood of Height and Shadow; but here follows more useful Rules than these before-going.

PROP. IV. The Distance being given, To find the Altitude.

Suppose EFGD were a Tower, or Steeple, or Tree, or House, whose Altitude you would know, and you cannot come so neer as to measure between your Station of 45 deg. and the Base of the Thing, by reason of some Wall or Moat; yet by the Pro∣portion of the Line of Quadrature, you may help your self by going backwards.

Thus if you could not measure the Distance from B to D, then go backward from B to A, until the Thred cut the 26 deg. 30 min. of your Quadrant; and measure the Distance between B and A, as suppose it to be 32 Foot or Yards, equal to the Height DE 32; The whole Line DA being 64 Feet or Yards, which is double to the Height. By the Tables,

Suppose then the Angle made by the Thred on the Quadrant ADB, be equal to the Angle EAD 26: 30 min. and the Distance AD be 64 Yards, or 192 Foot, to find the Height DE, I say,

As the Radius 90 Degrees	100000
To the Tangent of the Angle EAD 26 deg. 30 min.	969773
So is AD 64 Yards	180618
To the Height required, DE 32 Yards	X, 50391

PROP. V. The Distance being given, To find the Distance from the Eye to the top of the Tower.

Let the Distance AD be 192 Feet, the Angle at the Eye A 26 deg. 30 min. and the Distance from the Eye or Hypotenusa AE is required.

As the Sine of the Angle AED 63 deg. 30 min.	995179
Is to the Radius 90 deg.	10
So is AD 192 Feet	1228330
To AE 214 5/10 Feet	233151

PROP. VI. Some part of the Distance being given, To find the Distance from the Eye or Hypotenuse.

Let the Part of the Distance given be AB 96 Feet, and it is required to find the Distance from the Eye or Hypothenuse EB, which is the Length or Hypothenuse to the Triangle DBE. First find the other Angles thus:

AE is 214 5/10 Feet	The Sum is 310 5/10	349206
AB is 96 5/10 Feet	The Difference 118 5/10	307371

The Angle at B is 153 deg. 30.	To Tang. ½ Sum of opposite Angle 76 deg. 45.	1062806
The Half-Tangent is 76 deg. 45.	1370177
1020971

	deg.	min.
The Half-Tang. difference is	58	20.
The Half-Tangent Sum	76	45
Sum	135	05
Taken out of	180	00
Whose Complement remains	44	55 Angle EBD.

Then to find the Length of BE,

As the Sine of the Angle ABE 135 deg. 5 min. or 44 deg. 55 min.	984885
To AE 214 5/10 Feet	333142
So is the Sine of the Angle EAB 26 deg. 30 min.	964952
To the Distance from the Eye BE 135 5/10 Feet or Hypotenuse	1298094
	313209

PROP. VII. Some part of the Distance being given, To find the Altitude.

Keep the Angle and Distance from the Eye found by the former Proposition.

As the Radius 90 deg.	10
Is to the Sine of the Angle EBD 44 deg. 55 min.	984885
So is EB 135 6/10	313225
To ED the Height required, 95 8/10 Feet	1298100

which is the same as before found, without sensible difference. By the same Rule you may find the nearest Observation in the Figure to the Tower.

PROP. VIII. To do the same thing by the Quadrant, and Scale of Equal Parts, another way.

Without Calculation, by your Quadrant or Scale of Equal Parts, you may be re∣solved of all the foresaid Propositions, by the help of a Line of Chords; you may lay it down and demonstrate it, as you see the Figure, by the same Scale of Equal Parts as you measured, the first Distance, will answer all the rest. This is so plain, it needs no other Precept.

Here is another way to find the Length of the Scaling-Ladder without Calculation, which in many Cases is the chief thing looked after; which cannot be so well done by the Quadrat, as by observing the Angles of the Quadrant; and this is the best way I know.

Let your Station be any where at random, or as neer as you can come to the Foot of the Tower or Wall, for the Ditch, or Moat, or Cannon shot: As suppose at B, and observe there the Angle of the Height of the Thing, which let be any Degrees what∣soever, as here is 45 deg. I say, If you go so far backwards from this Station of B, toward H, till you make the thing appear just at half the aforesaid Angle, which is here 22 deg. 30 min. the half of 45 deg. That then this Distance from B to H is the true Length of the Sloap-side BE, without farther trouble; and a Ladder of that Length will Scale the said Moat or Wall, allowing only the Height of your Eye from the Ground.

PROP. IX. Part of the Distance being given, To find the Remainder of the Distance.

Let part of the Distance given be AB 96 Feet, and the Remainder of the Distance cannot be measured, by reason of danger of Shot, or Moat of Water, or some other Impediments; Therefore by the 7 Rule I found the Angle at B to be 44 deg. 55 min. So that the Angle BED 45 deg. 05 min. is the Complement thereof: Which known, I say,

As the Radius	10
To the Sine of the Angle BED 45 deg. 5 min.	985011
So is the Distance from the Eye BE 135 6/10 Feet	313225
To the Remainder of the Distance BD 96 Foot	298236

PROP. X. By the Height of the Sun, and the Length of the Shadow, To find the Height of any Tree, Tower, or Steeple.

This Conclusion may be tried by a little Quadrant or Pocket-Instrument, by which you may take the Sun's Altitude to a Degree, ½ or ¼, which is near enough for these Conclusions.

[illustration] geometrical diagram

Suppose DE to be a Turret, Tree, or Steeple, whose Height is required to be found by the Shadow it makes on Level Ground, the Rule is thus, viz. Let the Height of the Sun be 37 deg. 00 min. and the Length of the Shadow 40 Foot, the Rule is,

As the Radius 90 deg.	10
To the Length of the Shadow 40 Foot	160206
So is the Tangent of the Sun's Height 37 deg.	987711
To the Height of the Thing desired	147917

which is found to be 30-14 Parts, which shews the Height to be a little above 30 Foot.

Here is another way to do the same without the help of a Quadrant and Sun's Altitude, viz. Set up a Staff of any Length, suppose 3 Feet in Length, as CB, and

the Shadow which it makes from B to A is 4 Feet; Because the Shadow of the Tower from the Base thereof to B is 40 Feet, I say,

As the Shadow of the Staff, is to the Height of the Staff:

So is the Shadow of the Steeple, to the Height of the Steeple.

The Operation may be performed by Natural Numbers, or by Logarithms, thus, viz.

As 4 Feet, to 3 Feet: So 40 Feet, to 30 Feet.

〈 math 〉〈 math 〉

As the Shadow of the Staff 4 Feet AB, Log.	4.0,60206
To the Length of the Staff 3 Feet BC	3.0, 47712
So is the Shadow of the Steeple 40 Feet DB	40, 1, 60206
	2, 47918
To the Height of the Tower 30 Feet DE	1, 47712

A Constant Kalendar; OR AN ALMANACK For Three Hundred Years. But more exactly serving for Nineteen Years, BEING THE CIRCLE of the MOON, OR THE GOLDEN NUMBER. With New Exact TABLES OF THE Suns Declination, Rectified by the best Hypothesis, until the LEAP-YEARS 1695. BY Capt. SAMUEL STURMY.

London, Printed Anno Domini 1668.

AN ALMANACK For XXXII. Years. According to the English and Foreign Accounts.

Anno Dom.	Prime.	••pact	Sund. Letter	Shrove Sunday.	Easter Sunday.	White Sunday.	Diff.
1664	12	12	CB	Feb. 21	10	May 29	1
1665	13	23	A	5	March 26	14	0
1666	14	04	G	25	April 15	June 03	0
1667	15	15	F	17	7	May 26	1
1668	16	26	ED	2	March 22	10	0
1669	17	7	C	21	April 11	30	0
1670	18	18	B	13	3	22	1
1671	19	29	A	March 05	23	June 11	5
1672	01	11	GF	Feb. 18	7	May 26	0
1673	2	22	E	9	March 30	18	1
1674	3	3	D	March 1	April 19	June 07	5
1675	4	14	C	Feb. 14	4	May 23	0
1676	5	25	BA	6	March 26	14	0
1677	6	6	G	25	April 15	June 03	1
1678	7	17	F	10	March 31	May 19	0
1679	8	28	E	March 2	April 20	June 08	5
1680	9	9	DC	Feb. 22	11	May 30	0
1681	10	20	B	13	3	22
1682	11	1	A	26	16	June 04
1683	12	12	G	18	8	May 27
1684	13	23	FE	10	March 30	18
1685	14	04	D	March 1	April 19	June 07
1686	15	15	C	Feb. 14	4	May 23
1687	16	26	B	6	March 27	15
1688	17	7	AG	26	April 15	June 03
1689	18	18	F	10	March 31	May 15
1690	19	20	E	March 2	April 20	June 08
1691	1	11	D	Feb. 22	April 12	May 31
1692	2	22	CB	7	March 27	May 15
1693	3	3	A	Feb. 26	April 16	June 4
1694	4	14	G	18	8	May 27
1695	5	25	F	Feb. 3	March 24	12

The Ʋse of the Alma∣nack for 32 Years.

THis Table shew∣eth first the Date of the Years; secondly, the Prime, or Golden Num∣ber; thirdly, the Epact; fourthly, the Dominical Letter for these Years; and then in their Or∣der the Chief Movable Feasts (viz.) Shrove-Sunday, Easter-day, and Whitsunday, upon which all the rest depend. The Foreign Account is com∣monly ten days before us; but their Mova∣ble Feasts fall sometimes at the same time with ours, sometimes 1, 2, 3, 4, or 5 Weeks before ours, as you see in the last Column of the Ta∣ble.

Of the Terms.

THere are four times of the Year ap∣pointed for the Deter∣mining of Causes; these are called Terms. Two of these Terms (viz.) Hillary Term, and Michaelmas Term, are at a constant time of the Year: but Easter Term and Trinity Term are sooner or later, as those Feasts happen. Each of these Terms hath several Returns, and each Return hath four Days belonging to it. The first is the Day of Return or Essoin, for the Defendant in a Personal Action, or the Tenant in a Real Action. The second is the Day of Exception, for the Plaintiff or Defendant to lay an Exception. The third is the Day whereon the Sheriff must re∣turn the Writ. The fourth is the Day of Appearance in the Court. These four Days follow each other in order, except a Sunday or Holyday take up any of them, and then the Day following serves for both Occasions.

The Beginning and End of Hillary Term and Michaelmas Term, with all their Returns, you shall find in this following Kalendar, which are constant if no Sunday hinder them.

Easter Term begins Wednesday Fortnight or 17 Days after Easter, and ends the Munday after Holy-Thursday, or the Munday before VVhitsunday: It hath these five Returns.

• Quind. Pasc. A Fornight after Easter.
• Tres. Pasc. Three Weeks after Easter.
• Mens. Pasc. A Month after Easter.
• Quinq. Pasc. Five Weeks after Easter.
• Crast. Ascen. The Day after Holy-Thursday.

Trinity Term begins the Friday after Trinity-Sunday, which is next Sunday to Whit-sunday, and hath these four Returns.

• Crast. Trin. The Munday after Trinity-Sunday.
• Oct. Trin. A Week after Trinity-Sunday.
• Quind. Trin. A Fortnight after Trinity-Sunday.
• Tres. Trin. Three Weeks after Trinity-Sunday.

The Exchequer opens four Days before Trinity-Term; but eight Days before the other Terms.

Lo! here a Trade surpasseth all the rest; No Change annoys the Lawyer's Interest. His Tongue buyes Lands, builds Houses, without Toil; The Pen's his Plough, the Parchment is his Soil; Him Storms disturb not, nor Militia Bands. The Tree roots best, that in the Weather stands.

How to Rectifie the Tables of the Sun's Declination at any Time by Prostaphaereses.

Prostaphaereses of the Suns Declination.

Day Month.	Jan.	Feb.	Mar.	Apr.	May.	June.	July.	Aug.	Sept.	Octob.	Nov.	Dec.
Sec.	Sec.	Sec.	Sec.	Sec.	Sec.	Sec.	Sec.	Sec.	Sec.	Sec.	Sec.
1	17	36	42	40	28	8	15	33	42	41	30	9
2	18	37	43	39	28	7	15	34	43	42	29	8
3	19	37	43	40	27	6	16	35	43	42	29	7
4	20	38	44	40	27	5	17	35	43	42	28	6
5	20	37	44	39	26	4	18	34	43	41	28	5
6	21	37	43	39	25	4	19	35	43	41	28	4
7	21	38	43	38	25	3	19	35	44	41	27	3
8	22	39	43	38	24	2	19	35	43	40	26	2
9	24	38	44	38	23	2	20	36	44	40	25	1
10	24	39	45	37	23	1	21	36	43	40	24	1
11	25	39	44	37	22	0	21	37	43	40	24	0
12	25	40	44	37	22	1	22	38	43	39	23	0
13	26	41	43	37	21	1	22	38	43	39	22	1
14	26	40	44	37	20	2	23	38	44	38	21	2
15	27	41	44	35	20	3	23	38	43	38	21	3
16	28	42	43	35	19	4	24	39	44	37	20	4
17	28	41	42	35	19	4	25	39	43	38	20	5
18	29	41	43	35	18	5	26	39	43	37	19	6
19	30	41	43	34	17	6	26	39	43	37	18	7
20	31	42	42	34	17	7	27	40	43	36	17	8
21	31	42	43	33	16	8	27	40	43	36	16	9
22	32	42	43	33	15	8	28	41	44	35	16	10
23	32	43	43	32	14	9	29	40	43	35	15	11
24	32	43	42	32	14	10	29	41	42	34	14	12
25	33	44	42	31	12	11	30	41	43	34	13	12
26	33	44	41	3••	12	12	30	41	42	33	13	13
27	34	43	41	30	11	13	30	41	42	34	12	14
28	35	43	41	29	9	14	31	41	43	32	11	16
29	35	43	41	29	9	14	31	41	43	32	11	16
30	35		41	29	9	14	32	42	42	32	10	16
31	36		41		8		33			31		17

The Ʋse of the Table.

* 2.1IN this Kalendar, Printed for the Year 1665, 66, 67, and 1668. the Sun's Declination to be trusted sufficient; but for any Year after 1668. the Rule is thus.

For Example. I would know the Sun's Declination for the Year 1689. you must always substract 1668. from the Year gi∣ven, which is here 1689. the Remainder is 21 Years; which being divided by 4, the Quotient is 5 Leap-Years, and 1 remains, which sheweth it is the first Year.

Now I desire to rectifie the Table for the first day of April, which in the Kalendar you have 8 deg. 36 min. and in this Table you have 40 Seconds;* 2.2 which multiplied by 5 Leap-years, give 200 Seconds, that is 3 m. 20 sec. to be added to 8 deg. 36 min. So have you 8 deg. 39 min. for the Sun's Decli∣nation in 1689.

A Table of the Sun's Declination.

JANUARY XXXI.
South Declination.					Leap-ye.		First.		Second.		Third.		South Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Day Mo.	Da. Week	☽ Epact Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	A	29	Circumcis.	21 50		21 43		21 45		21 47
2	B	28	4 1 8	21 40	10	21 33	10	21 35	10	21 38	9
3	C	26 12	4 2 8	21 30	10	21 22	11	21 25	10	21 28	10
4	D	25 08	4 3 8	21 20	10	21 11	11	21 14	11	21 17	11
5	E	23 21	4 4 8	21 09	11	21 00	11	21 03	11	21 6	11
6	F	✶	Twelf day.	20 57	12	20 48	12	20 51	12	20 54	12
7	G	22 10	4 7 8	20 45	12	20 36	12	20 39	12	20 42	12
8	A	21	4 8 8	20 33	12	20 24	12	20 27	12	20 30	12
9	B	20 06	☉ in ♒	20 21	12	20 11	13	20 14	13	20 17	13
10	C	18 18	4 11 8	20 8	12	19 58	13	20 1	13	20 4	13
11	D	✶	4 12 8	19 55	13	19 44	14	19 49	14	19 51	13
12	E	17 15	4 14 8	19 41	14	19 30	14	19 33	14	19 37	14
13	F	16	Hilary.	19 27	14	19 16	14	19 19	14	19 23	14
14	G	15 03	4 11 8	19 12	15	19 01	15	19 04	15	19 8	15
15	A	✶	4 18 8	18 57	15	18 46	15	18 50	14	18 54	14
16	B	14 0	4 20 8	18 42	15	18 31	15	18 35	15	18 39	15
17	C	12 12	4 21 8	18 27	15	18 15	16	18 19	16	18 23	16
18	D	✶	4 23 8	18 11	16	17 59	16	18 3	16	18 7	16
19	E	11 1	☉ to ♒	17 55	16	17 42	17	17 47	16	17 51	16
20	F	9 0	Oct. Hilar.	17 38	17	17 26	16	17 30	17	17 34	17
21	G	7 10	Except.	17 22	16	17 9	17	17 13	17	17 17	17
22	A	✶	Ret. Brev.	17 5	17	16 51	18	16 56	17	17 00	17
23	B	6 6	Term beg.	16 47	18	16 34	17	16 38	18	16 42	18
24	C	4 18	4 33 8	16 30	17	16 16	18	16 20	18	16 25	17
25	D	✶	Conv. Paul	16 12	18	15 58	18	16 2	18	16 7	18
26	E	3 7	4 37 8	15 54	18	15 40	18	15 44	18	15 49	18
27	F	2 0	Qu. Hilar.	15 35	19	15 21	19	15 25	19	15 30	19
28	G	1 3	Except.	15 16	19	15 2	19	15 7	18	15 11	19
29	A	✶	Ret. Brev.	14 57	19	14 43	19	15 48	19	14 52	19
30	B	29 16	Appear.	14 38	19	14 22	21	14 28	20	14 33	19
31	C	28 12	4 46 8	14 18	20	14 4	28	14 8	20	14 13	20

A Table of the Sun's Declination.

FEBRUARY XXVIII.
South Declination.					Leap-ye.		First.		Second.		Third.		South Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Da. Mo.	D. Week.	☽ Epact. Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	D	27	4 18 8	13 59		13 44		13 48		13 53
2	E	26 1	Purif. Mar.	13 39	20	13 24	20	13 28	20	13 33	20
3	F	25	Crast. Pur.	13 10	20	13 3	21	13 8	20	13 13	20
4	G	23 10	Except.	12 58	22	12 43	20	12 48	20	12 52	2••
5	A	22 22	Ret. Brev.	12 38	20	12 22	21	12 27	21	12 32	20
6	B	21	Appear.	12 17	21	12 1	21	12 6	21	12 11	21
7	C	20 18	4 59 8	11 56	21	11 40	21	11 45	21	11 50	21
8	D	19	☉ in ♓	11 35	21	11 19	21	11 24	21	11 29	21
9	E	18 8	Oct. Pur.	11 14	21	10 57	22	11 2	22	11 8	21
10	F	18 18	Except.	10 52	22	10 36	21	10 40	22	10 47	21
11	G	17 3	Ret. Brev.	10 30	22	10 14	22	10 19	21	10 24	23
12	A	15 16	Term ends.	10 8	22	9 52	22	9 57	22	10 2	22
13	B	✶	5 11 7	9 46	22	9 30	22	9 35	22	9 40	22
14	C	14 12	Valentine.	9 24	22	9 7	23	9 13	22	9 18	22
15	D	13	5 14 7	9 2	22	8 45	23	8 50	23	8 55	23
16	E	12 1	5 16 7	8 39	23	8 22	22	8 27	23	8 33	22
17	F	11 14	5 18 7	8 17	22	8 0	23	8 5	22	8 11	22
18	G	9 10	☉ 10 ♓	7 54	23	7 37	23	7 42	23	7 44	22
19	A	8	5 22 7	7 31	23	7 14	23	7 20	22	7 26	20
20	B	7 0	5 24 7	7 8	23	6 51	23	6 57	23	7 3	25
21	C	6 19	5 26 7	6 45	23	6 28	24	6 34	23	6 39	24
22	D	5	5 28 7	6 22	23	6 4	23	6 10	24	6 16	23
23	E	4 7	5 30 7	5 59	23	5 41	23	5 47	23	5 53	23
24	F	3 20	Matthew.	5 36	23	5 18	23	5 24	23	5 30	23
25	G	2	5 34 7	5 12	24	4 55	24	5 0	24	5 5	25
26	A	1 16	5 36 7	4 49	23	4 31	23	4 37	23	4 43	22
27	B	✶	5 38 7	4 25	24	4 8	24	4 13	24	4 19	24
28	C	29 5	☉ 20 ♓	4 2	23	3 44		3 50	23	3 56	23
				3 39	23

A Table of the Sun's Declination.

MARCH XXXI.
South Declination.					Leap-ye.		First.		Second.		Third.		South Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Day Mo.	Da. Week	☽ Epact Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	D	28 1	David.	3 15		3 20		3 26		3 32
2	E	26 14	5 44 7	2 51	24	2 57	23	3 3	23	3 9	23
3	F	✶	5 46 7	2 28	23	2 33	24	2 29	24	2 45	24
4	G	25 10	5 48 7	2 4	24	2 9	24	2 16	13	2 21	24
5	A	24	5 50 7	1 40	24	1 46	23	1 52	24	1 58	23
6	B	23 0	5 52 7	1 16	24	1 22	24	1 28	24	1 34	24
7	C	21 11	5 54 7	0 53	23	0 59	23	1 5	23	1 11	23
8	D	21	5 56 7	0 29	24	0 35	24	0 41	24	0 46	24
9	E	20 7	5 58 7	0 5	24	0 11	24	0 17	24	0 23	23
-	10	F	18 20	☉ in ♈	Nor. 18	24	Nor. 13	24	North 6	11	North 1	22	Aequinoctial.
North Declination.	11	G	✶	6 2 6	0 42	24	0 37	23	0 30	24	0 25	24	North Declination.
12	A	17 16	6 4 6	1 6	23	0 00	24	0 54	24	0 49	24
13	B	16	6 6 6	1 29	24	1 24	23	1 18	24	1 12	23
14	C	15 5	6 8 6	1 53	23	1 47	24	1 42	24	1 36	24
15	D	14 1	6 10 6	2 16	24	2 11	24	2 5	23	2 00	32
16	E	12 14	6 12 6	2 40	24	2 35	23	2 29	24	2 23	23
17	F	✶	6 14 6	3 4	23	2 58	23	2 52	23	2 46	23
18	G	11 2	6 18 6	3 27	23	3 21	24	3 15	23	3 10	24
19	A	10	6 16 6	3 50	24	3 45	23	3 39	24	3 33	23
20	B	9 0	☉ 10 ♈	4 14	23	4 8	23	4 2	23	3 56	23
21	C	7 11	6 22 6	4 37	23	4 31	23	4 25	23	4 19	23
22	D	✶	6 24 6	5 00	1	4 54	23	4 49	24	4 43	24
23	E	6 7	6 26 6	5 23	23	5 17	23	5 12	23	5 6	23
24	F	4 20	6 28 6	5 46	23	5 40	23	5 35	23	5 29	23
25	G	✶	An. Mar.	6 9	23	6 3	22	5 57	22	5 52	23
26	A	3 8	6 32 6	6 31	22	6 25	22	6 20	23	6 15	23
27	B	2	6 34 6	6 54	23	6 48	23	6 42	22	6 37	22
28	C	1 7	6 36 6	7 16	22	7 11	22	7 5	23	7 0	23
29	D	29 17	6 38 6	7 38	22	7 33	22	7 28	23	7 22	22
30	E	28 14	☉ 20 ♈	8 1	23	7 55	22	7 50	22	7 45	23
31	F	27	6 42 6	8 23	22	8 17	22	8 12	22	8 7	22

A Table of the Sun's Declination.

APRIL XXX.
North Declination.					Leap-ye.		First.		Second.		Third.		North Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Da. Mo.	D. Week.	☽ Epact. Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	G	26 2	6 44 6	8 45		8 39		8 34		8 29
2	A	25 22	6 46 6	9 7	22	9 1	22	8 56	22	8 51	22
3	B	24 0	6 48 6	9 29	22	9 23	22	9 18	22	9 13	22
4	C	23 11	6 50 6	9 50	21	9 44	21	9 39	21	9 34	21
5	D	✶	6 52 6	10 11	21	10 6	22	10 1	22	9 56	22
6	E	22 0	6 54 6	10 32	21	10 27	21	10 22	21	10 17	21
7	F	20 20	6 56 6	10 53	21	10 48	21	10 43	21	10 38	21
8	G	19	6 57 6	11 14	21	11 9	21	11 4	21	10 59	21
9	A	18 9	6 59 6	11 35	21	11 30	20	11 25	21	11 20	21
10	B	✶	☉ in ♉	11 55	20	11 50	20	11 46	21	11 40	20
11	C	17 5	7 2 5	12 15	20	12 10	20	12 6	20	12 1	21
12	D	15 6	7 4 5	12 35	20	12 30	20	12 26	20	12 21	20
13	E	14 14	7 6 5	12 55	20	12 50	20	12 46	20	12 41	20
14	F	13	7 8 5	13 25	20	13 10	19	13 6	20	13 1	20
15	G	12 2	7 10 5	13 34	19	13 29	19	13 25	19	13 20	19
16	A	11 15	7 11 5	13 53	19	13 48	19	13 44	19	13 40	20
17	B	10	7 13 5	14 12	19	14 7	19	14 3	19	13 59	19
18	C	9 11	7 15 5	14 31	19	14 26	19	14 22	19	14 18	19
19	D	8	7 17 5	14 50	19	14 45	18	14 41	19	14 36	18
20	E	7 0	☉ 10 ♉	15 8	18	15 3	18	14 59	18	14 55	19
21	F	6 20	7 29 5	15 26	18	15 21	18	15 17	18	15 13	18
22	G	5	7 22 5	15 44	17	15 39	17	15 35	18	15 31	18
23	A	4 9	George.	16 1	17	15 56	18	15 53	18	15 48	17
24	B	3 21	7 25 5	16 18	17	16 14	17	16 10	17	16 6	18
25	C	2	Mark.	16 35	17	16 31	16	16 27	17	16 23	17
26	D	1 18	7 49 5	16 52	17	16 47	18	16 44	17	16 40	17
27	E	✶	7 31 5	17 59	16	17 5	16	17 1	17	16 57	17
28	F	29 6	7 32 5	17 25	16	17 21	16	17 17	16	17 13	16
29	G	28 2	7 34 5	17 41	15	17 37	16	17 33	16	17 29	16
30	A	27	☉ 20 ♉	17 56	15	17 52	15	17 49	16	17 44	15

A Table of the Sun's Declination.

MAY XXXI.
North Declination.					Leap-ye.		First.		Second.		Third.		North Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Day Mo.	Da. Week	☽ Epact. Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	B	✶	Ph. & Ja.	18 11		18 8		18 4		18 00
2	C	25 11	7 39 5	18 26	15	18 23	15	18 19	15	18 15	15
3	D	24	7 40 5	18 41	15	18 37	14	18 34	15	18 30	15
4	E	23	7 42 5	18 55	14	18 52	15	18 48	14	18 45	15
5	F	22 13	7 44 5	19 9	14	19 6	14	19 2	14	18 59	14
6	G	21	7 45 5	19 25	14	19 19	13	19 16	14	19 13	14
7	A	20 9	7 46 5	19 36	14	19 33	14	19 30	14	19 27	14
8	B	18 21	7 47 5	19 49	13	19 46	13	19 43	13	19 40	13
9	C	✶	7 49 5	20 2	13	19 59	13	19 56	12	19 53	13
10	D	17 18	☉ in ♊	20 14	12	20 11	12	20 8	12	20 5	12
11	E	16	7 50 5	20 26	12	20 24	13	20 20	12	20 18	13
12	F	15 6	7 53 5	20 38	12	20 36	12	20 32	12	20 30	12
13	G	14 2	7 54 5	20 49	11	20 47	11	20 43	12	20 41	11
14	A	12 15	7 55 5	21 00	11	20 57	12	20 55	11	20 52	11
15	B	✶	7 56 5	21 11	10	21 9	10	21 6	10	21 3	11
16	C	11 4	7 57 5	21 21	10	21 19	10	21 16	10	21 14	11
17	D	10	7 58 5	21 31	9	21 29	9	21 26	10	21 24	10
18	E	9 0	7 59 5	21 40	10	21 38	9	21 36	9	21 34	10
19	F	7 13	8 0 4	21 50	9	21 47	9	21 45	9	21 43	9
20	G	✶	8 1 4	21 59	8	21 56	9	21 54	8	21 52	9
21	A	6 9	☉ 10 ♊	22 7	8	22 5	8	22 2	9	12 1	9
22	B	4 1	8 3 4	22 15	7	22 13	8	22 11	9	22 9	8
23	C	✶	8 4 4	22 22	7	22 21	7	22 19	7	22 17	8
24	D	3 10	8 5 4	22 29	7	22 28	7	22 24	7	22 31	7
25	E	2	8 6 4	22 36	7	22 35	6	22 33	7	22 38	7
26	F	1 6	8 7 4	22 43	6	22 41	6	22 40	6	22 45	7
27	G	29 19	8 8 4	22 49	6	22 47	6	22 46	6	22 51	6
28	A	28 15	8 8 4	22 55	5	22 53	5	22 52	5	22 56	5
29	B	27	8 9 4	23 00	5	22 58	5	22 57	5	23 1	5
30	C	26 4	8 10 4	23 5	4	23 3	5	23 2	5	23 6	5
31	D	✶	8 10 4	23 9	4	23 8	5	23 7	5

A Table of the Sun's Declination.

JUNE XXX.
North Declination.					Leap-ye.		First.		Second.		Third.		North Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Da. Mo.	D. Week.	☽ Epact. Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	E	25 0	☉ 20 ♊	23 13		23 12		23 11		23 10
2	F	23 13	8 11 4	23 17	4	23 16	4	23 15	4	23 14	4
3	G	✶	8 11 4	23 20	3	23 19	3	23 19	4	23 18	4
4	A	22 1	8 12 4	23 23	2	23 22	3	23 22	3	23 21	3
5	B	20 21	8 12 4	23 25	2	23 25	3	23 24	2	23 24	3
6	C	19	8 12 4	23 27	2	23 27	2	23 26	2	23 26	2
7	D	18 10	8 13 4	23 29	1	23 28	1	23 28	2	23 28	2
8	E	✶	8 13 4	23 30	1	23 29	1	23 29	1	23 29	1
9	F	17 6	Days at a stand.	23 31	0	23 30	1	23 30	1	23 30	1
10	G	15 19	23 31	0	23 31	1	23 31	1	23 31	1
11	A	14 15	☉ in ♋	23 31	1	23 31	0	23 31	0	23 31	0
12	B	13	Days shorten.	23 30	1	23 31	0	23 31	0	23 31	0
13	C	12 4	23 29	0	23 30	0	23 30	1	23 30	1
14	D	11 16	8 13 4	23 29	2	23 29	1	23 29	1	23 29	1
15	E	10	8 13 4	23 27	2	23 27	2	23 28	1	23 28	1
16	F	9 13	8 12 4	23 25	3	23 25	2	23 26	2	23 26	2
17	G	8	8 12 4	23 22	3	23 23	2	23 23	3	23 24	2
18	A	7 1	8 12 4	23 19	3	23 20	3	23 20	3	23 21	3
19	B	6 21	8 11 4	23 16	4	23 17	4	23 17	3	23 19	2
20	C	5	8 11 4	23 12	4	23 13	4	23 13	4	23 15	4
21	D	4 10	8 11 4	23 8	5	23 9	5	23 9	4	23 11	4
22	E	✶	☉ 10 ♋	23 3	5	23 4	4	23 5	4	23 6	5
23	F	3 0	8 10 4	22 58	5	23 0	5	23 1	4	23 2	4
24	G	1 19	John Bapt.	22 53	6	22 55	6	22 56	5	22 57	5
25	A	✶	8 8 4	22 47	6	22 49	4	22 51	5	22 51	6
26	B	29 6	8 8 4	22 41	7	22 43	7	22 45	6	22 45	6
27	C	28 4	8 7 4	22 34	7	22 36	7	22 38	7	22 39	6
28	D	26 17	8 6 4	22 27	7	22 24	7	22 31	7	22 33	6
29	E	✶	Peter Ap.	22 20	7	22 22	8	22 34	7	22 26	7
30	F	25 13	8 4 4	22 12	8	22 14	8	22 16	8	22 18	8

A Table of the Sun's Declination.

JULY XXXI.
North Declination.					Leap-ye.		First.		Second.		Third.		North Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Day Mo.	Da. Week	☽ Epact Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	G	24	8 3 4	22 4		22 6		22 8		22 10
2	A	23 1	☉ 20 ♋	21 56	8	21 58	8	22 00	8	22 2	8
3	B	22 14	8 2 4	21 47	9	21 49	9	21 51	9	21 54	9
4	C	21	8 1 4	21 38	9	21 40	9	21 42	9	21 45	9
5	D	20 10	8 0 4	21 28	10	21 31	10	21 33	9	21 35	10
6	E	19	7 58 5	21 18	10	21 21	10	21 23	10	21 26	9
7	F	18 0	7 57 5	21 8	10	21 11	11	21 13	10	21 16	••0
8	G	17 19	7 56 5	20 57	11	21 00	11	21 3	10	21 5	11
9	A	16	7 55 5	20 46	11	20 49	12	20 52	11	20 54	11
10	B	15 8	7 54 5	20 34	12	20 37	11	20 40	12	20 43	11
11	C	14 4	7 53 5	20 23	11	20 26	12	20 29	11	20 31	12
12	D	12 17	7 52 5	20 11	12	20 14	12	20 17	12	20 20	11
13	E	✶	☉ in ♌	19 59	12	20 2	13	20 5	12	20 8	12
14	F	11 5	7 49 5	19 46	13	19 49	13	19 52	13	19 55	13
15	G	10	Swithin.	19 33	13	19 36	13	19 39	13	19 42	13
16	A	9 1	7 46 5	19 20	13	19 23	13	19 26	13	19 24	13
17	B	7 14	7 45 5	19 6	14	19 10	14	19 13	13	19 16	13
18	C	✶	7 43 5	18 52	14	18 56	14	18 59	14	19 2	14
19	D	6 10	Dog d. beg.	18 30	14	18 42	15	18 45	14	18 48	14
20	E	5	7 40 5	18 23	15	18 27	15	18 30	15	18 34	14
21	F	4 0	7 39 5	18 8	15	18 12	15	18 15	15	18 19	15
22	G	3 11	Magdalen.	17 53	15	17 57	16	18 00	15	18 4	15
23	A	2	☉ 10 ♌	17 37	16	17 41	16	17 45	15	17 49	15
24	B	1 8	7 34 4	17 21	16	17 25	16	17 29	16	17 23	16
25	C	29 20	S. Jam. Ap.	17 5	16	17 9	16	17 13	16	17 17	16
26	D	28 14	7 0 5	16 49	16	16 53	16	16 57	16	17 1	16
27	E	27	7 29 5	16 33	16	16 37	17	16 41	16	16 45	16
28	F	26 5	7 27 5	16 16	17	16 20	17	16 24	17	16 28	17
29	G	✶	7 25 5	15 59	17	16 3	18	16 7	17	16 11	17
30	A	25 1	7 24 5	15 41	18	15 45	18	15 50	17	15 54	17
31	B	23 14	7 22 5	15 23	18	15 27	18	15 32	18	15 36	18

A Table of the Sun's Declination.

AUGUST XXXI.
North Declination.					Leap-ye.		First.		Second.		Third.		North Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Da. Mo.	D. Week.	☽ Epact. Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	C	*	Lamas.	15 5		15 10		15 14		15 19
2	D	22 3	7 19 5	14 47	18	14 52	18	14 56	18	15 1	18
3	E	21 0	☉ 20 ♌	14 29	18	14 34	18	14 38	18	14 42	19
4	F	20 0	7 16 5	14 10	19	14 15	19	14 19	19	14 24	18
5	G	18 12	7 14 5	13 51	19	13 56	19	14 00	19	14 5	19
6	A	*	7 12 5	13 32	19	13 37	19	13 41	19	13 46	19
7	B	17 8	7 10 5	13 13	19	13 18	19	13 52	19	13 27	19
8	C	15 20	7 8 5	12 54	19	12 59	19	13 3	19	13 08	19
9	D	14 17	7 7 5	12 34	20	12 39	20	12 43	20	12 48	20
10	E	13	Laurence.	12 14	20	12 19	20	12 23	20	12 28	20
11	F	12 5	7 3 5	11 54	20	11 59	20	12 3	20	12 8	20
12	G	11 18	7 1 5	11 33	21	11 38	21	11 43	20	11 48	20
13	A	10	☉ in ♍	11 13	20	11 18	21	11 23	20	11 28	20
14	B	9 14	6 58 6	10 52	21	10 57	21	11 3	20	11 7	21
15	C	8	6 56 6	10 32	20	10 36	21	11 42	21	10 47	20
16	D	7 3	6 54 6	10 11	21	10 15	21	10 21	21	10 26	21
17	E	*	6 52 6	9 50	21	9 54	21	10 00	21	10 5	21
18	F	6 0	6 50 6	9 28	22	9 33	21	9 38	22	9 44	21
19	G	4 12	6 48 6	9 6	22	9 12	22	9 17	21	9 22	22
20	A	*	6 46 6	8 44	21	8 50	21	9 55	22	9 1	21
21	B	3 0	6 44 6	8 23	22	8 29	22	8 33	22	8 39	22
22	C	1 20	6 42 6	8 1	22	8 7	22	8 12	21	8 17	22
23	D	*	☉ 10 ♍	7 39	22	7 45	22	7 50	22	7 55	22
24	E	29 9	Barthol.	7 17	22	7 23	23	7 28	22	7 33	22
25	F	28 5	6 36 6	6 56	23	7 00	23	7 5	23	7 11	22
26	G	26 18	6 34 6	6 32	22	6 38	22	6 43	22	6 45	23
27	A	*	6 32 6	6 10	23	6 15	23	6 21	22	6 26	22
28	B	25 14	Dog d. end.	5 47	22	5 53	22	6 58	23	6 4	22
29	C	24 6	6 29 6	5 25	23	5 30	23	5 35	23	5 41	23
30	D	23 3	6 27 6	5 2		5 8	22	5 13	22	5 18	23
31	E	22 15	6 25 6	4 39	27	4 44	24	4 50	23	4 55	23

A Table of the Sun's Declination.

SEPTEMBER XXX.
North Declination.					Leap-ye.		First.		Second.		Third.		North Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Day Mo.	Da. Week	☽ Epact Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	F	21	Giles.	4 16		4 22		4 27		4 32
2	G	20 12	6 21 6	3 53	23	3 59	23	4 4	23	4 9	23
3	A	19	☉ 20 ♍	3 30	23	3 35	24	3 41	23	3 46	23
4	B	18 0	6 17 6	3 7	23	3 12	23	3 17	24	3 23	23
5	C	17 20	6 15 6	2 43	24	2 49	23	2 54	23	3 00	23
6	D	16	6 13 6	2 20	23	2 26	23	2 31	23	2 37	23
7	E	15 9	6 12 6	1 57	23	2 3	23	2 8	23	2 13	24
8	F	14 6	Lady Fair.	1 33	24	1 39	24	1 45	23	1 50	23
9	G	12 18	6 7 6	1 10	23	1 16	23	1 21	24	1 27	23
10	A	*	6 5 6	0 46	24	0 52	24	0 58	23	1 3	24
11	B	11 7	6 3 6	0 23	23	0 29	23	0 30	24	0 40	24
12	C	10	6 1 6	0 1	22	0 5	24	0 10	24	0 16	24
-	13	D	9 3	☉ in ♎	South 24	23	South 18	24	South 13	7	South 7	9	Aequinoctial.
South Declination.	14	E	7 16	5 57 7	0 48	24	0 42	23	0 36	23	0 31	24	South Declination.
15	F	*	5 55 7	1 11	23	1 5	24	1 00	26	0 54	23
16	G	6 12	5 53 7	1 35	24	1 29	24	1 23	23	1 18	24
17	A	5	5 51 7	1 58	23	1 52	23	1 47	24	1 41	23
18	B	4	5 49 7	2 22	24	2 16	24	2 10	23	2 5	24
19	C	3	5 47 7	2 45	23	2 40	24	2 34	24	2 28	23
20	D	2	5 45 7	3 9	24	3 3	23	2 57	23	2 52	24
21	E	1 9	Matth. Ap.	3 32	23	3 27	24	3 21	24	3 15	23
22	F	29 22	5 42 7	3 56	24	3 50	23	3 44	23	3 38	23
23	G	28 18	☉ 10 ♎	4 19	23	4 13	23	4 7	23	4 2	24
24	A	27	5 38 7	4 42	24	4 36	23	4 30	23	4 25	23
25	B	26 17	5 36 7	5 06	23	4 59	23	4 54	24	4 48	23
26	C	*	5 34 7	5 29	23	5 23	24	5 17	23	5 12	24
27	D	25 3	5 32 7	5 52	23	5 36	23	5 41	24	5 35	23
28	E	23 16	5 30 7	6 15	23	6 09	23	6 4	23	5 58	23
29	F	*	Michael.	6 38	23	6 32	23	6 16	22	6 21	23
30	G	22 4	5 26 7	7 01	23	6 55		6 49	23	6 44	23

A Table of the Sun's Declination.

OCTOBER XXXI.
South Declination.					Leap-ye.		First.		Second.		Third.		South Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Da. Mo.	D. Week.	☽ Epact. Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	A	21	5 24 7	7 24		7 17		7 12		7 06
2	B	20 0	5 22 7	7 46	22	7 40	23	7 35	22	7 29	23
3	C	18 13	☉ 20 ♎	8 9	23	8 2	22	7 58	23	7 52	23
4	D	*	5 18 7	8 31	22	8 25	23	8 20	22	8 15	23
5	E	17 9	5 16 7	8 53	22	8 47	22	8 42	22	8 37	22
6	F	22 22	5 14 7	9 15	22	9 9	22	9 4	22	8 59	22
7	G	14 18	5 12 7	9 37	22	9 32	23	9 27	23	9 21	22
8	A	13	5 10 7	9 59	22	9 54	22	9 49	22	9 43	22
9	B	12 7	5 8 7	10 21	22	10 16	22	10 10	21	10 5	22
10	C	11 19	5 6 7	10 43	22	10 37	21	10 32	22	10 27	22
11	D	10	5 4 7	11 04	21	10 59	22	10 53	21	10 48	21
12	E	9 16	5 2 7	11 25	21	11 20	21	11 15	22	11 10	22
13	F	8	☉ in ♏	11 46	21	11 41	21	11 36	21	11 31	21
14	G	7 4	4 59 8	12 7	21	12 2	21	11 57	21	11 52	21
15	A	*	4 57 8	12 28	21	12 23	21	12 18	21	12 13	21
16	B	6 0	4 55 8	12 48	20	12 44	21	12 39	21	12 34	21
17	C	4 13	4 53 8	13 9	21	13 4	20	12 59	20	12 54	20
18	D	*	Luke.	13 29	20	13 24	20	13 20	21	13 14	20
19	E	3 1	4 49 8	13 49	20	13 44	20	13 40	20	13 34	20
20	F	2	Tres. Mic.	14 9	20	14 4	20	13 59	19	13 54	20
21	G	1 0	Except.	14 28	19	14 24	20	14 19	20	14 14	20
22	A	29 11	Ret. Brev.	14 48	20	14 43	19	14 38	19	14 34	20
23	B	28 7	Term beg.	15 7	19	15 2	19	14 57	19	14 53	19
24	C	26 19	4 40 8	15 26	19	15 21	19	15 16	19	15 12	19
25	D	*	Crispin.	15 44	18	15 40	19	15 35	19	15 31	19
26	E	25 16	☉ 13 ♏	16 2	18	15 58	18	15 53	18	15 49	18
27	F	24	Mens. Mi.	16 20	18	16 16	18	16 11	18	16 7	18
28	G	23 4	Sim. Jude.	16 38	17	16 33	17	16 29	18	16 25	18
29	A	22 17	Ret. Brev.	16 55	17	16 51	18	16 47	18	16 32	17
30	B	21	Appear.	17 12	17	17 8	17	17 4	17	17 00	18
31	C	20 13	4 2 8	17 29	17	17 25	17	17 21	17	17 17	17

A Table of the Sun's Declination.

NOVEMBER XXX.
South Declination.					Leap-ye.		First.		Second.		Third.		South Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Da. Mo.	D. We••k.	☽ Epact. Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	D	19	All Saints.	17 46		17 41		17 38		17 32
2	E	18 2	All Souls.	18 2	16	17 58	17	17 54	16	17 50	17
3	F	*	Crast. An.	18 18	16	18 14	16	18 10	16	18 6	16
4	G	17 0	Except.	18 33	15	18 30	16	18 26	16	18 22	16
5	A	15 11	Powder Tr.	18 48	15	18 45	15	18 41	15	18 37	15
6	B	14 7	Appear.	19 3	15	19 00	15	18 56	15	18 53	16
7	C	12 19	4 16 8	19 18	15	19 14	15	19 11	15	19 8	15
8	D	*	☉ 25 ♏	19 32	14	19 29	14	19 25	14	19 22	14
9	E	11 8	4 15 8	19 46	14	19 43	14	19 39	14	19 36	14
10	F	8	4 13 8	19 59	13	19 56	13	19 53	14	19 50	14
11	G	9 4	Martin.	20 13	14	20 10	14	20 6	13	20 3	13
12	A	7 17	Crast. Ma.	20 26	13	20 23	13	20 19	13	20 16	••••
13	B	*	Except.	20 38	12	20 35	12	20 31	12	20 29	13
14	C	6 13	Ret. Brev.	20 50	12	20 47	12	20 43	12	20 41	12
15	D	5	Appear.	20 2	10	20 59	12	20 55	12	20 53	12
16	E	4 2	☉ 4 ♐	21 13	11	21 10	11	21 7	12	21 5	12
17	F	3 14	4 3 8	21 23	10	21 21	11	21 18	11	21 16	11
18	G	2	Oct. Mar.	21 34	11	21 31	10	21 29	11	21 26	10
19	A	1 11	Except.	21 44	10	21 41	10	21 39	10	21 36	10
20	B	29 23	Ret. Brev.	21 54	10	21 51	10	21 49	10	21 46	10
21	C	28 20	Appear.	22 3	9	22 00	9	21 58	9	21 56	10
22	D	27	☉ 10 ♐	22 11	8	22 09	9	21 7	9	22 5	9
23	E	26 8	3 56 9	22 20	9	22 18	9	22 16	9	22 13	8
24	F	*	3 55 9	22 28	8	22 26	8	22 24	8	22 22	9
25	G	25 4	Qu. Mar.	22 35	7	22 34	8	22 31	7	22 30	8
26	A	23 17	Except.	22 42	7	22 41	7	22 39	8	22 37	7
27	B	✶	Ret. Brev.	22 48	6	22 48	7	22 45	6	22 44	7
28	C	22 6	Term ends.	22 54	6	22 54	6	22 52	7	22 50	6
29	D	21	3 51 9	23 00	6	22 59	5	22 58	6	22 56	6
30	E	20 2	Andrew.	23 05	5	23 4	5	23 03	5	23 2	6

A Table of the Sun's Declination.

DECEMBER XXXI.
South Declination.					Leap ye.		First.		Second.		Third.		South Declination.
				1664		1665		1666		1667
				1668		1669		1670		1671
				1672		1673		1674		1675
				1676		1677		1678		1679
				1680		1681		1682		1683
				1684		1685		1686		1687
				1688		1689		1690		1691
				1692		1693		1694		1695
Day Mo.	Da. Week	☽ Epact Hour.	☉ Ris. & Setting.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.	Deg. Min.	Differ.
1	F	18 15	☉ 20 ♐	23 10		23 09		23 08		23 06
2	G	✶	3 50 9	23 14	6	23 13	4	23 12	4	23 11	5
3	A	17 11	3 49 9	23 18	4	23 17	4	23 16	4	23 15	4
4	B	15 23	3 48 9	23 21	3	23 20	3	23 20	4	23 19	4
5	C	14 20	3 48 9	23 24	3	23 23	3	23 23	3	23 22	3
6	D	13	3 48 9	23 26	2	23 26	3	23 25	3	23 25	3
7	E	12 8	3 47 9	23 28	2	23 28	2	23 27	2	23 27	2
8	F	11 21	Days are at a stand.	23 30	2	23 29	1	22 29	2	23 28	1
9	G	10	23 31	1	23 30	1	23 30	1	23 30	2
10	A	9 17	23 31	0	23 31	1	23 31	1	23 31	1
11	B	8	☉ in ♑	23 31	0	23 31	0	23 31	0	23 31	0
12	C	7 6	Days in∣crease.	23 30	1	23 31	0	23 31	0	23 31	0
13	D	✶	23 29	1	23 30	1	23 30	1	23 30	1
14	E	6 2	3 47 9	23 28	1	23 29	2	23 29	1	23 29	1
15	F	4 15	3 47 9	23 26	2	23 27	2	23 27	2	23 27	2
16	G	✶	3 48 9	23 24	2	23 25	2	23 25	2	23 25	2
17	A	3 3	3 48 9	23 21	3	23 22	3	23 22	3	23 23	2
18	B	2	3 49 9	23 18	3	23 19	3	23 19	3	23 20	3
19	C	1 0	3 49 9	23 14	4	23 15	4	23 16	4	23 16	4
20	D	29 12	☉ 9 ♑	23 9	5	23 10	5	23 12	4	23 12	4
21	E	28 8	Thomas.	23 5	4	23 6	6	23 7	5	23 8	5
22	F	27	3 50 9	23 00	5	23 1	6	23 2	5	23 3	5
23	G	26	3 51 9	22 54	6	22 55	6	23 57	5	22 54	5
24	A	25 17	3 52 9	22 48	6	22 49	7	23 51	6	22 52	6
25	B	24	Christmas.	22 41	7	22 43	8	22 45	6	22 46	6
26	C	23 6	Stephen.	22 34	7	22 36	8	22 38	7	22 39	7
27	D	22 18	John.	22 27	7	22 21	8	22 31	7	22 32	7
28	E	21	Innocents.	22 19	8	22 20	8	22 23	8	22 25	7
29	F	20 15	3 56 9	22 11	8	22 12	8	22 15	8	22 17	8
30	G	19	3 57 9	22 9	8	22 4	9	22 6	9	22 8	9
31	A	18 3	3 58 9	21 53	6	21 55	9	21 57	9	21 59	9

To find the Sun's Declination upon every Day of the Year.

THe Sun's Year (that is, the time that the Sun goeth out of a certain Point of the Ecliptick, and returneth again to the same) is not of 365 days just; but about 5 Hours and 49 Minutes more (that is, little less than 6 Hours;) Wherefore after three Years, there is always added to the fourth four times 6 Hours, that is, a Day more in February, for to count the Year or the Revolution of the Sun in even Days; therefore that fourth Year is called Leap-year: Therefore when we de∣scribe the Sun's Declination in Tables, we always use to make four several Tables, for four such Years following one the other; and yet by reason of the foresaid difference, that four Revolutions of the Sun do not justly make up one Day, but wants about 48 min. bringeth in process of time so great a difference in the Declination, that it is needful every twenty Years to renew such Tables.

How to find the Leap-years, it is thus: Divide the Year of our Lord above 1600. by 4; If the Division doth fall out even, without any over-plus, that Year then is a Leap-year of 366 Days: But if out of the Division there remain any Number, that Remainder sheweth how many Years that Year propounded, is after the Leap-year.

For EXAMPLE.

I desire to know what Year the Year 1666. is. Leaving 1600, I divide 66 by 4, and find there remains 2; for 16 times 4, or 4 times 16, is 64; that taken from 66, there remains 2; whereby I find the Year 1666. to be the second Year after the Leap-year. In the like manner you must work for any other Years: Only note this, If nothing remaineth upon the Division out of the Quotient, then it is a Leap-year if it be even.

As for EXAMPLE.

It is required to know what Year 1692 is. Leaving the 1600, divide the 92 by 4, and nothing remains upon the Division, but is even 23 in the Quotient; where∣by I find that Year 1692. is a Leap-year.

For to know the same by the foregoing Tables, it is thus. Each Month hath 12 Co∣lumns; The first thereof shews the Days of the Month; The second Column, having the Dominical Letters, shews the Days of the Week; The third Column having two Rows of Figures, the first of them shews the Epact of the Moon, and the other the Hour of the Day, reckoning the said Hours always from Noon; the fourth Column shews the Chief Days of the Year, and the Terms and their Returns which are fixed and certain; and in the void places it shews the Rising and Setting of the Sun in this Latitude, and the Place of the Sun every 10 Day or Degree. These four Columns of themselves are fit for Mens ordinary use, and may be made with a little Art and Pains to perform all the Conclusions which the yearly Almanacks shew and teach, as you shall see by the following Rules and Observations.

The fifth Column of the foregoing Tables shews the Sun's Declination for every Day of the Year, for all these Years in the first Column under-written, which are all Leap-years. The sixth Column shews the Daily Difference of the Sun's Declination. The seventh Column shews the first Year from the Leap-year: The eighth, the Daily Difference of the Sun's Declination in that Year. The ninth shews the second Year from the Leap-year; The tenth, the Difference; The eleventh, the third Year from the Leap-year; The twelfth, the Difference every Day of the Sun's Declination, as you see in the Tables. This Table following shews the Leap-years, First, Second, and Third Years, as they are plainly expressed in the Head of each Table.

Leap-years.	First.	Second.	Third.
1664	1665	1666	1667
1668	1669	1670	1671
1672	1673	1674	1675
1676	1677	1678	1679
1680	1681	1682	1683
1684	1685	1686	1687
1688	1689	1690	1691
1692	1693	1694	1695

For to find the Sun's Declination, Look for the Day of the Month in the left hand of the Table, and in the common Angle of meeting you will find the Declination which you seek after.

I. EXAMPLE.

I desire to know the Sun's Declination for the 22 of May, in the Year 1693. being the first Year after the Leap-year. In the Head of the Table I find the Month and Year; on the left hand of the Table I find the Day; and in the Common Angle or Line of Meeting, I find the Declination I look for to be North 22 deg. 13 min.

II. EXAMPLE.

Upon the 5th of November in the Leap-year 1692. I desire to know the Declina∣tion of the Sun. In the Head of the Table I find the Month and Day, and in the first Column to the left hand I find the Day of the Month, and in the Common Line of Meeting, under the Year, I find the Sun's Declination required, to be 18 deg. 37 m. South Declination; and his Difference in 24 Hours, 15 min.

The foregoing Tables of the Sun's Declination is rectified properly for the Meridian of the most famous and Metropolitan City of London. The Constant Kalendar I bor∣rowed out of Ingenious Mr. Philips's Purchaser's Pattern, at the end of page 247. With some addition it is very useful with the foregoing Tables.

Of the Difference and Aequa∣tion of Declination in di∣vers Places of the Earth.

A Table by which you may proportion the Sun's Declination to any other Meridian.

The Difference in Declination daily.	M	M	M	M	M	M	M	M	M
00	03	06	09	12	15	18	21	24
	M	M	M	M	M	M	M	M	M
Degrees of Difference of Longitude either East or West.	Deg. 15	0	0	0	0	0	0	1	1	1
30	0	0	0	0	1	1	1	2	2
45	0	0	0	1	1	2	2	3	3
60	0	0	1	1	2	2	3	3	4
75	0	0	1	2	2	3	4	4	5
90	0	0	1	2	3	4	4	5	6
105	0	1	2	2	3	4	5	6	7
120	0	1	2	3	4	5	6	7	8
135	0	1	2	3	4	5	7	8	9
150	0	1	2	3	5	6	8	9	10
165	0	1	2	4	5	6	8	10	11
180	0	1	3	4	6	7	9	11	12

NOte this, They that are more Easterly from the Meri∣dian of London, have the Decli∣nation less when the Sun de∣clineth from the Line, and in∣creaseth in Declination either Northward or Southward, as well between the 10th of March and the 12th of June, as between the 13th of September and the 12th of December; and more when the Sun returneth again towards the Line, whether it be North or South of the Line, as well between the 12th of December and the 10th of March, as between the 12th of June and the 13th of September.

On the contrary, They that are more Westerly from the Meridian of London, when the Declination increaseth North or South, have more Declination, and less when the Declination decreaseth; that is, when the Sun is going towards the Aequinoctial, ei∣ther on the North or South side of the Line; the reason is, because the Sun cometh to the Meridian Eastward, to them that live there, always before it doth to us; and them that live more Westerly, have him later to their Meridian.

EXAMPLE I. Of those that are more Easterly, which increase in Declination.

On the 26th of March, the first Year after the Leap-year, I desire to know the Declination of the Sun at Noon at Bantam in the East-Indies. 〈 math 〉〈 math 〉 I find by Globes, or the Plat of Mercator, that Bantam is to the Eastward of the Meridian of London about 110 Degrees; we do not esteem of a Degree or two, because it amounteth to nothing in this Practice. The Sun for his Course round the Heavens and Earth, which is 360 Degrees, hath need of 24 Hours; What time will 110 Degrees have? Facit 7 Hours, and something more not worth the noting; whereby the Sun comes to the Meridian 7 Hours sooner at Bantam, than it doth at London; That it is 12 a Clock at Noon at Bantam, when it is 4 of the Clock in the Morning with us at London. The Sun's Declination for the 26th of March, is 6 deg. 25 min.: The Difference of the Declination of the Day following, you find is 23 min. which it is increased; Therefore I say, If in 24 Hours the Declination increaseth 23 Minutes, How much then in 7 Hours? Facit almost 7 Minutes, that the Declination is less than it is at London. So that the Declination at Bantam that Day, is but 6 deg. 18 min. North: And on the contrary, when the De∣clination decreaseth, work, and you will have the Declination South, Eastward, or Westward.

EXAMPLE II. The Ʋse of the Table.

On the 17th of September in the same Year, I desire to know the Declination that day at Noon at Bantam. The Declination for the Meridian of London is that Day 1 deg. 52 min. and the Difference of the Declination of the Day following is 24 min. decreased; and, as was said before in the last E••••mple, the difference of Longitude is 110 deg. Therefore I look in the Head of the foregoing Table, for the nearest Num∣ber to the Difference 24, and find it to fall just even on the Head of the last Column; then look on the left hand of the Table for the Difference of Longitude, and I find 105 deg. nearest, and in the Common Angle of Meeting I find 7, which is to be substracted from the Declination in the Meridian of London abovesaid, 1 deg. 52 min. and the Remainder will be the Declination for the Meridian or Longitude I am in, which is 1 deg. 45 min. South: But if the Declination decreaseth, as it doth here in∣crease, then you must have added.

	deg.	min.
In the Meridian of London the Declination	01	52
The Minutes Proportional substracted	00	07
The Declination for 110 deg. Longitude of Bantam, East	01	45
The Declination of 110 deg. West of the Meridian of London	00	07
West	01	52

EXAMPLE III.

A Ship coming on the seventh of November, in the third Year after the Leap year, into the great South-Sea, thwart of the Coast of Peru, in Longitude 76 deg. The Pi∣lot desireth the Declination there at Noon in that Meridian.

	deg.	min.
In the Meridian of London the Declination is	19	08 South.
The Minutes Proportional added	00	03
In the Longitude 76 deg. the Declination	19	11 West.
In the Longitude of 76 deg. East, the Declination is	19	05

Two Ships being in Company, they parted at the Lands-end of England: The one Sails Eastwards, and cometh upon his Reckoning upon the 28th of September 180 Degrees on the other side the Globe of the Earth (being the first Year after the Leap-year) and by the foregoing Tables finds the Sun's Declination 5 deg. 57 min. The other Ship Sails Westwards, and meeteth the first Ship at the aforesaid place, by his Reckoning not the 28th, but on the 27th of September, and findeth the Decli∣nation in these Tables for that Day; so that they differ in the Time one Day, and in Declination 24 min. the which proceedeth from this cause: The first having Sailed against the Rising of the Sun 180 Degrees, hath shortned his time 12 Hours; the other hath Sailed with the Sun 180 Degrees, hath lengthned his time 12 Hours, and thereby hath one Night less than the first. Seeing then in 24 Hours increaseth 24 Minutes, he that Sailed Eastward must reckon 12 Minutes Declination less, and he that Sailed Westward 12 Minutes more than the Table doth shew; and so both of them shall keep one manner of Declination, to wit, 6 deg. 9 min.

A Table of the Refractions of the Sun, Moon, and Stars, according to the Observation of thrice Noble Tycho Brahe.

Alti∣tudes.	Sun.	Moon	Stars		Alti∣tudes.	Sun.	Moon
min.	min.	min.		min.	min.
0	34	33	30	18	06	06
1	26	25	21	19	05	06
2	20	20	15	20	04	05
3	17	17	12	21	04	05
4	15	15	11	22	03	04
5	14	14	10	23	03	04
6	13	13	00	24	03	04
7	12	13	08	25	02	03
8	11	12	07	26	02	03
9	10	11	06	27	02	03
10	10	11	05	28	02	02
11	09	10	05	29	02	02
12	09	09	04	30	01	02
13	08	00	04	31	01	02
14	08	08	03	32	01	01
15	07	08	03	33	01	01
16	07	07	02	34	01	01
17	06	07	02	35	01	01

THe Refraction of the Sun, Moon and Stars, causeth them to appear higher above the Horizon than they are: Therefore the Refraction is alway•• to be sub∣stracted from the A ••••ude ob∣served, that the tr•• ••ltitude may be had.

As for Examp

The Sun's Meridian Altitude by Observation being 9 Degrees, I require the true Altitude.

	deg.	mi.
Altitude by Observation	9	00
Refraction substract	0	10
The true Meridian Alti∣tude	8	50

Of the Refraction of the Sun, A Dutch Ship being upon the Discovery of a North-East Passage to the East-India, was forced to Winter in Nova Zembla: the Mariners beheld the Sun 14 days sooner than he should by his De∣clination, and by Computation 5 Degrees under the Horizon; which is caused by the gross Vapours, and thickness of the Air neer the Horizon.

The USE of the CONSTANT KALENDAR.

I. To know the Day of the Month.

THis is the Chief and most useful Observation of any Almanack, and may as well be performed by this, as by any other. To this purpose, you must by the general Kalendar at the beginning hereof, know the Dominical or Sun∣day Letter for the Year; then considering with your self, whether it be the beginning, midst, or end of the Month (as you must do in any Almanack) find this Letter in the beginning, midst, or end of the Month, and reckoning from it to the Day of the Week, either Munday, or Tuesday, or whatsoever other Day it is, right against the Day of the Week you shall find what Day of the Month it is. Here is no difficulty in this; only when it is Leap-year you see there is two Sunday Letters, the first of these you may use only to the 24th of February, and the other all the Year after.

For Example. In the 1668. the Dominical Letter ED the first Sunday in January, is at the first E, which is at the fifth Day of the Month; the first Sunday in February is at the second Day of the Month; but the first Sunday in March is at the first D, which is at the first Day of the Month, and so all the Year after.

II. To know what Day of the Week any Notable Day will fall upon, in any Year.

First find the Dominical Letter in the former Table; then find your Letter in your Month next before the Day you desire, and so from thence count the Days of the Week, till you come to the Day desired. Thus if you would know what Day of the Week Lady-day, or the Annunciation of the Lady Mary falls upon this Year 1668. the Dominical Letter is D; this is three Days before the said Day, therefore that falls upon a Wednesday.

But now in the Year 1669. when the Dominical Letter is C, Lady-day will be upon the Thursday. This will be in a short time as ready to you, as if these Letters were painted out for you in Vermilion.

III. To find the Time of Sun Rising and Setting.

This is set down for most of the Days in the whole Year, for London; and may serve for all the East, South, and West Parts of England: And this is done after somewhat a briefer manner than is usual, making the Minutes which are placed in the midst, to serve both the Hours of Setting and Rising; which you must understand thus: The 7th of February you shall find these Figures, 4. 59. 8. that is, the Sun that Day sets at 4 h. 59 m. that is 59 m. after 4. and riseth at 59 m. 8 h. that is 59 m. before 8. or almost 7 a Clock. And so you must account them always, remembring, That as the Minutes follow the first Figure, so they must be reckoned in Time after: as they stand before the last Figure, so they must be reckoned in Time before it.

And think it not preposterous that the time of the Sun's Setting is set down before the Rising; for the Sun's Setting is of most use, and the other serves in a manner for the filling up of the Column.

	ho.	min.
If you double this time of Sun Setting	04	59
You have the Length of the Day	09	58
If you substract it from	12	00
You have the time of Rising, differing in shew from the Kalendar	07	01
But all one in effect; and this doubled, shews the Length of the Night	14	02

IV. To find the Place of the Sun.

THis is set down in the Kalendar, about every tenth Day, to every tenth Degree; so that reckoning a Degree for each Day between, you shall have the Place of the Sun exact enough for most ordinary Uses. Thus the 10th of March the Sun enters into Aries; therefore the 15th Day, or five Days after, the Sun is in five De∣grees of Aries.

V. To find the Day and Hour of the Change or New Moon, and thereby the Full and Quarters.

FIrst you must find the Moon's Epact for the present Year you are in: This Num∣ber is found out in the First Book, Page 12. and also in the Table before at the beginning of the Kalendar. The Change also may be found out by the Golden Num∣ber; yet that would stand so scattering and without form, that it is much hand∣somer and readier to find out by this Epact, which runs for the most part in a Constant Order, only here and there skipping a Day or a Number, which is marked with this ✶.

Having found out the Epact for this present Year, turn to the Month you desire, and there find out the said Number of the Epact in the third Column of the Months, and mark what Day of the Month it stands against; for that is the Day of the Change or New Moon. Likewise if you have respect unto the Dominical Letter, which is by it, you shall see what Day of the Week it is.

Now here in this Column there are two Rows of Figures; The first shews the Epact-Number, and the next the Time of the Day reckoned by the Hours from Noon, which are plain to understand till you come to 12 Hours after Noon, which is Midnight; but then the Numbers above 12, you must reckon to the Morning of the next Day.

So that these Hours after Noon,
13,	14,	15,	16,	17,	18,	19,	20,	21,	22,	23,	24,
are all one with these,
1,	2,	3,	4,	5,	6,	7,	8,	9,	10,	11,	12,
the next Day in the Morning.

Thus in the Year 1666. the Epact being 4, and the Dominical Letter G, you shall find this Epact-Number 4 against the 21 of July being Saturday; and the Figure o standing by it, shews that the New Moon is just at Noon.

Again, You shall find the Epact-Number against the 16th of November, being Friday; and the Figure of 2 standing by it, shews that it is about 2 Hours after Noon the Moon changeth. Now this is the true time of the New Moon, according to the Moon's mean Motion; which though it may differ half a day from the true Change, yet it seldom differs so much, and is better for the following Conclusion than the true time.

Having first found out the time of the New Moon, you may from thence reckon the Age of the Moon, and find the Quarters and Full Moon.

Thus the Moon's Age is	Days	Hours	Min.
At the First Quarter	7	09	11
At the Full Moon	14	18	22
At the Last Quarter	22	03	33
An Whole Moon	29	12	44

Or else observe the Dominical Letter that is against ••••e Epact, or Day of the New Moon; and where you find that Letter again, that is the First Quarter; for the Full Moon take two Weeks and one Day, which will fall upon the Letter next to it; for the Last Quarter take one Week more, which will fall upon the Letter of the Full Moon.

Thus if the New Moon fall upon A, the First Quarter falls upon the next A, and the Full Moon on the next B a week after, and the Last Quarter on the next B. And thus you have this brief Kalendar or Constant Almanack for many Years; only for the more exactness in the Hour of the Moon's Change and Age, it is restrained to 19 Years: For though the Change of the Moon (for the most part) hapneth again upon the same Days, for several Revolutions of the Prime or Golden Number XIX; yet not upon the same Hour of the Day, but alters every Revolution 7 Hours, 27 Minutes, 30 Seconds, proceeding forward for the most part; but the Leap-years coming in with a Day more than ordinary, keeps this Motion so much backward, that in 300 Years it neither gains nor loseth a Day, only differeth in the Hour of the Day; yet for the more exactness, it will be better to renew this every 19 Years. All these things this brief Kalendar shews plainly, with little or no trouble more than in an yearly Almanack. I shall now proceed to some other Conclusions. I have been very large already, in the First Book, of Things concerning the Use of the Moon in other Conclusions; to which I refer you for any thing of the Tides, or the Southing of the Moon, or the Rising or Setting of the Moon, or what else is necessary in Navigation.

I thought to have entred my Figure of the Sea-Compass, for the Surveying of Land, which was promised in the Argument; As likewise the Gunner's Scale and Gauging Rod: But I refer you to the several Books in the following Treatise, where the Figure and the Use of it, is together for your satisfaction.

The End of the Second Book.

Notes

• * 1.1

  Scales of equal parts
• * 1.2

  Scales of Box or o∣ther Wood.
• * 1.3

  2 Squares.
• * 1.4

  2 Cramps.
• * 1.5

  Scales fastened with Nails, on Deal-Boards. Brass a Gauge.
• * 1.6

  Steel Let∣ters and Figures.
• * 1.7

  Gauging Instru∣ments. Brass Com∣passes, to∣gether with an Arch & 4 Points,
• * 1.8

  To be taken in and out The use. A pair of Dividers.
• * 1.9

  Beam-com∣passes. Hand-vice. Files.
• * 1.10

  Where they may be made or bought.
• * 1.11

  To set off the Instru∣ment.
• * 1.12

  11 d. 15. to each Point.
• * 1.13

  Here place the North Nocturnal for the Pole Star and Guards
• * 2.1

  When the Declination increaseth, then add; and for de∣creasing, substract.
• * 2.2

  8 d. 39 m. Declination 1 April 1689.