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.

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