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.

PROBL. II. The Latitudes of two Places being given, and Difference of Longi∣tude of both Places, To find the Rhomb and Distance.

TO the intent the Application may be the more evident, our Examples shall be of two Places before-expressed on the Plain Chart.* 1.1

Suppose the Latitude of the Island of Lundy in the Mouth of Seavern, to be at A, 51 deg. 22 min. and the Latitude of Barbadoes 13 deg. 10 min. at B, and the Diffe∣rence of Longitude 52 deg. 55 min. CD, that the Barbadoes is to the VVestward of the Island of Lundy; The Course and Distance from the one Place to the other is de∣manded.* 1.2

First you may demonstrate the Question by the Scale. Draw the Right Line AC for the Meridian; and in regard the Difference of Latitude is 38 deg. 12 min. con∣vert them into Leagues, by multiplying them by 20, the Number that goes to a De∣gree,

[illustration] geometrical diagram

and the odd Minutes divide by 3, and the Difference in Leagues will be 764; which lay from A to B,* 1.3 for the Common Difference of Latitude. Then take the Diffe∣rence of the two Latitudes inlarged, 934 5/10 Leagues, and lay from A to C; then draw the two Parallel Lines, as BE and CD. Then 52 deg. 55 min. the Difference of Lon∣gitude, converted into Leagues, as before-directed, is 1058 ••/3, which lay from C to D, and draw the Line AD, which is the true Course from A to D, and the Distance according to the True Chart inlarged: Therefore AE is the true Rhomb-line and Di∣stance found out, produced by the former Work. And as D is the true Point by Mercator's Chart of Barbadoes, so is E the true Point of the same Place of Barbadoes by the Plain Chart; and AE the true Distance, BAE the true Course. And as CD is the true Longitude by the Globe, so is BE the true Meridian-distance between Lun∣dy and Barbadoes.* 1.4 You may work afterwards by the Rules of the Plain Chart; and you need not work Mercator's way any more, without you have a Mercator's True Chart; and to work by that, you shall be directed in the following Discourse. There∣fore to work by the Plain Rules all the Voyage after, measure AE, and you will have 1157 Leagues for the Distance; and for the Course take GH, and apply it to the Points on the Scale, and you will find the true Course S. VV. a little more than a Quarter VVesterly, which is all one Course with the True Sea-Chart; but the Di∣stance inlarged is 1408 Leagues AD. Now by the Plain Sea-Chart the Course is BAF, S. W. above ¾ of a Point VVesterly; and the Distance is AF 1306 Leagues: So that the Plain Chart sheweth the Distance more than it is by 149 Leagues, and the Course more VVesterly by half a Point, and the Meridian-distance too much by 194 Leagues, which is a gross Error; and in such Distances grosly are these Men mistaken, that use a Plain Chart.

A Table of Meridional Parts to the 10 part of a League, and for every 10 Mi∣nutes of Latitude from the Aequinoctial to the Poles.

Minutes.	DEGREES.	Minutes.
0	1	2	3	4	5	6	7	8	9
Le. 1/10	Le. 1/10	Le. 1/10	Le. 1/10	Le.1/10	Le. 1/10	Le. 1/10	Le. 1/10	Le. 1/10	Le. 1/10
0	00	200	400	600	801	1001	1202	1403	1605	1807	0
10	33	233	433	634	834	1035	1236	1437	1639	1841	10
20	67	267	467	667	867	1068	1269	1471	1673	1875	20
30	100	300	500	700	901	1102	1303	1504	1706	1909	30
40	133	333	534	734	934	1135	1336	1538	1740	1943	40
50	167	367	567	767	968	1109	1370	1571	1774	1976	50
M.	10	11	12	13	14	15	16	17	18	19	M.
0	2010	2214	2418	2623	2828	3035	3242	3451	3661	3827	0
10	2044	2248	2452	2657	2863	3069	3277	3486	3696	3907	10
20	2078	2282	2486	2691	2897	3104	3312	3521	3731	3942	20
30	2112	2316	2520	2725	2931	3138	3347	3556	3766	3977	30
40	2164	2350	2554	2760	2966	3173	3381	3591	3801	4013	40
50	2180	2384	2588	2794	3000	3208	3416	3626	3836	4048	50
M.	20	21	22	23	24	25	26	27	28	29	M.
0	4084	4297	4512	4729	4947	5167	5388	5612	5837	6065	0
10	4119	4333	4548	4765	4983	5203	5425	5648	5875	6103	10
20	4155	4369	4584	4801	5020	5240	5462	5686	5913	6141	20
30	4196	4405	4620	4838	5056	5277	5500	5724	5951	6179	30
40	4226	4440	4656	4874	5090	5314	5537	5761	5989	6218	40
50	4262	4476	4692	4910	5130	5351	5574	5799	6027	6256	50
M.	30	31	32	33	34	35	36	37	38	39	M.
0	6294	6527	6761	6998	7238	7481	7726	7975	8277	8483	0
10	6333	6565	6800	7038	7278	7521	7768	8017	8270	8526	10
20	6371	6604	6840	7078	7319	7562	7809	8059	8312	8569	20
30	6410	6644	6879	7118	7359	7603	7850	8101	8355	8612	30
40	6449	6683	6919	7158	7400	7644	7892	8143	8397	8655	40
50	6488	6722	6938	7198	7440	7685	7934	8185	8440	8699	50
M.	40	41	42	43	44	45	46	47	48	49	M.
0	8742	9005	9272	9543	9819	10100	10385	10675	10971	11273	0
10	8786	9049	9317	9589	9865	10147	10433	10724	11021	11324	10
20	8829	9093	9362	9635	9912	10194	10481	10773	11071	11375	20
30	8873	9138	9407	9681	9959	10241	10530	10823	11121	11426	30
40	8917	9182	9452	9727	10005	10289	10578	10872	11172	11478	40
50	8961	9227	9498	9773	10052	10337	10626	10922	11222	11529	50
M.	50	51	52	53	54	55	56	57	58	59	M.
0	11581	11896	12217	12545	12881	13226	13581	13941	14313	14696	0
10	11633	11949	12271	12600	12938	13284	13640	14004	14377	14761	10
20	11685	12002	12325	12656	12995	13342	13700	14063	14440	14826	20
30	11737	12055	12380	12712	13052	13401	13759	14121	14504	14892	30
40	11790	12109	12435	12768	13110	13460	13820	14188	14568	14958	40
50	11842	12163	12490	12825	13168	13520	13881	14251	14632	15024	50
M	60	61	62	63	64	65	66	67	68	69	M.
0	15091	15497	15917	16350	16798	17263	17745	18246	18769	19315	0
10	15158	15586	15988	16423	16874	17342	17827	18332	18858	19408	10
20	15225	15635	16059	16497	16951	17421	17920	18448	18948	19502	20
30	15293	15705	16131	16572	17028	17501	17993	18505	19035	19597	30
40	15300	15775	16204	16647	17106	17582	18077	18592	19130	19693	40
50	15428	15846	16277	16722	17184	17662	18161	18680	19222	19789	50
M.	70	71	72	73	74	75	76	77	78	79	M
0	19886	20485	21116	21781	22485	23234	24033	24890	25815	26819	0
10	19984	20588	21224	21896	22607	23368	24172	25039	25976	26995	10
20	20083	20692	21333	22011	22730	23495	24312	25190	26140	27173	20
30	20183	20796	21444	22128	22854	23628	24455	25344	26306	27354	30
40	20283	20901	21555	22246	23979	23762	24598	25499	26474	27530	40
50	20384	21008	21668	22365	23106	23897	24743	25656	26645	27727	50
M.	80	81	82	83	84	85	86	87	88	89	M
0	27917	29130	30484	32019	33789	35881	38441	41740	46387	54331	0
10	28110	29345	30726	32296	34112	36270	38929	42395	47385	56420	10
20	28307	29564	30974	32579	34445	36673	39439	43090	48477	58977	20
30	28507	29787	31226	32870	34788	37096	39973	43830	49684	62274	30
40	28711	30015	31484	33168	35141	37523	40532	44621	51034	66920	40
50	28919	30247	31748	33474	35505	37973	41120	45970	52564	74863	50

By the Tables.

ADmit a Ship is at A, in Latitude 51 deg. 22 min. North, as is Lundy, and sails, or is to sail to E, in Latitude 13 deg. 10 min. according to the Plain Chart corrected, which is Barbadoes; or by Mercator's Chart, Barbadoes is in the Point D, and the Difference of Longitude is 1058 Leagues, which is 52 deg. 55 min.. First find the Difference of Latitude inlarged, as is before-directed in the first Problem, and found to be 934 5/10 Leagues.

Now you have given AB the Difference of Latitude 38 deg. 12 min. inlarged from B to C, and CD the Difference of Longitude 52 deg. 55 min. whereby the Angles and Hypothenusal shall be found by the Fourth and Fifth Case of Plain Triangles.

But because in this kind of Projection, the Degrees of Longitude and Latitude are not equal (except in Places near the Aequinoctial) the Degrees of Latitude at every Parallel, exceeding the Degrees of Longitude, in such proportion as the Aequinoctial exceeds that Parallel; therefore these Differences of Longitude and Latitude must be expressed by some one common measure; and for that purpose serves the foregoing Table, which sheweth how many Equal Parts are from the Aequinoctial, in every Degree of Latitude, to the Poles; namely, of such Equal Parts as a Degree of Lon∣gitude contains 20 Leagues.

Wherefore, as before-directed, multiplying 52 deg. by 20, and dividing the odd Minutes, being 55, by 3, it will be 18 ⅓ Leagues; added to the former Sum, makes 1058 ⅓ Leagues, for the Meridional parts contained in the Difference of Longitude. Also by the last Problem, I find the Meridional parts contained in the Difference of Latitude to be 934 5/10 Leagues: So that AC is 934 5/10 Parts, and CD 1058 ⅓ of such Parts.

Therefore, By the Second Case of Plain Triangles.

As the Difference of Latitude inlarged AC is 934 5/10 parts	297057
Is in proportion to the Radius 90 deg.	10
So is the Difference of Longitude in such Parts CD 1058 ⅓	1302448
To the Tangent of the Rhomb at A 48 deg. 33 min.	1005391

Extend the Compasses from 934 5/10 Leagues the inlarged Latitude, to 1058 ⅓ Leagues; the same Distance will reach from the Radius to the Tangent of the Course 48 deg. 33 min. which is the Course from Lundy to Barbadoes, S. W. a little above a quarter of a Point Westerly.

By the Fifth Case of Plain Triangles.

As the Sine-Complement of the Rhomb at D 41 deg. 27 min.	982083
To the Difference of Latitude AB 764 Leagues	288309
So is the Radius o 90 deg.	10
To the Distance AE 1154 2/10 Leagues	306226

Extend the Compasses from the Complement-Sine of the Rhomb 41 deg. 27 min. to the Sine of 90 deg. the same Extent will reach from the true Difference of Latitude 764 Leagues, to the Distance AE 1157 Leagues, which is required.

Notes

• * 1.1

  Longitude.
  	d.	m.
  Lundy	25	52
  	360	00
  	385	52
  Barbad.	332	57
  Differ.	52	55
  	20
  	1040
  	18 ⅓
  	1058
• * 1.2

  51 d.	22 m.
  13	10
  38	12 (4
  20	3
  760
  4
  764
• * 1.3

  As was shewed in the last Ex∣ample.
• * 1.4

  Then you must put down the Course and Di∣stance and Me∣ridian distance in the Head of your Journal.