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. 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.