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.

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.