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. II. The Projection of the Sphere in Plano, represented by the Analemma, and the Points and Circles before described.

THe Sphere may be Projected in Plano in straight Lines, as in the Ana∣lemma, if the Semi-diameters of the Circles given, be Divided in such sort as the Line of Signs in the Fundamental Diagram of the Scale.

This Scheme is fitted for the Latitude of Bristol 51 degr. 28 min. and represents the Points and Circles of the Sphere before described.

Take with your Compasses the Chord of 60 degr. and upon the Centre C describe the Circle HZON (2.) Draw the Diameter HCO which represents the Horizon; and at Right-Angles thereunto, cross it with another Diameter ZCN.

Then with the Latitude of the place, prick off 51 degr. 28 min. from O to N, and from H to S; and of the same Line of Chords, take the Complement of the Latitude 38 degr. 32 min. and prick off from HAE, and from O to Q, and draw NSC and AECQ.

Then take the Suns Declination 23 degr. 31 min. and prick off from AE to G and T, and with the like Chord do the same from N to Y and g, for the Polar Circle; and the like do from Q to D and P, and from S to X and ♄; and through these Points draw Parallels to the Equator Y g, and TSD, and G h: P, and X ♄.

And through the Centre draw the Ecliptick-Line TGP; and draw RS Parallel to the Horizon HCO, which is the Parallel of Altitude of the Hour of Six; and at any other distance, draw Parallels of Altitude E I f.

(1.) Thus are the Points before defined, represented in this Diagram; N is the North-Pole-Arctick, S the South-Pole-Antarctick; g the North, X the South-Pole of the Ecliptick; C both the Equinoctial-Points of Aries and Libra.

T The Point of the Summer-Solstice— P the Point of the Winter-Solstice, Z the Zenith over our Heads, N the Nadir-points under our Feet.

(2.) The greater Circles are HCO the Horizon, ZCN the Axis thereof, or Prime Vertical Azimuth of East and West; HZON the great Meridian, and also the Colure of the Summer and Winter-Solstice,— AECQ the Equinoctial; T C P the Ecliptick; SCN the Axis of the World, the Hour-Circle of 6; and lso it repre∣sents the Colure of the Equinoxes.

(3.) The lesser Circles are there represented, T D the Tropick of Cancer; GP the Tropick of Capricorn; Y g the Arctick Circle, about the Pole North; X ♄ the Antarctick-Circle, or South-Pole.

(4.) Other Circles not described upon Globes, are there represented; E f repre∣sents a Parallel of Altitude called an Almicanter; the Prickt Arches Z ☉, and S I be∣ing Ellipses represent the Azimuths, or Vertical-Circles.

[illustration] geometrical diagram

The Projection of the Sphere in Plano, by straight Lines or Signs.

The Prickt Arches from the Poles, represent the Meridian or Hour-Circles, which are also Ellipses; the Drawing thereof will be troublesom, and for that reason is not mentioned; and how to shun them in the resolution of any Proposition of the Sphere, by Chords shall be shewn in the several Questions following; But the Sphere may be Projected in Plano by Circular-Lines, as in the general Astrolobe of Gemma Trisius, by help of the Tangent, and ½ Tangents in the Fundamental Diagram of the Scale, and by the Directions in the 4 Book 12 Chapter beforegoing, and will resolve the same things; the directions shall be one and the same, in both, in Letters, and represents the same.

[illustration] geometrical diagram

The Convex Sphere, by Circular Lines.

Any Line drawn Parallel to the Equinoctial AEQ, as pq, TD. Yg: doth repre∣sent Parallels of Declination.

And any Line drawn Parallel to the Ecliptick TP, represents a Parallel of Latitude of the Stars and Planets in the Heavens.

(5.) Divers Arches relating to the motion of the Sun, and seen upon the Globes, and found by Calculation, are in the Convex-Sphere, represented in Right-Lines, and in the Concave-Sphere by Circular-Lines.

(1.) The Suns Amplitude, or Coast of Rising and Setting, from the East and West in the Analemma, is represented by CL in North Signs, and by CF in South Signs.

(2.) His Ascentional-difference, or time of Rising and Setting from Six in Summer by SL, in Winter by FH.

(3.) His Altitude at Six in Summer by RC, and his Depression at Six in Winter, by Cb.

(4.) His Azimuth at the hour of Six in Summer, by RS, or CI, equal to hb in Winter.

(5.) His Vertical-Altitude, or Altitude of East and West, by MC in Summer, and his Depression therein in Winter by CN.

(6.) His hour from Six being East and West, in Summer by MS, in Winter by Nh.

(7.) His Azimuth from the East and West upon any Altitude, is represented in the Parallel of Altitude by the Convex-Sphere, where it Intersects the Parallel of Declina∣tion by I ☉; but in the Concave-Sphere may be measured on the Horizon HO, as CV, or CI, measured on the Line of half Tangents.

(8.) The Hour of the Day from Six, to any Altitude, is always represented in the said Point of Intersection, in the Parallel of Declination, hereby q ☉, or in the Concave-Sphere by S ☉; and all these Arches thus represented in Right-Lines, are the Signs of those Arches to the Radius of that Parallel in which they happen, being ac∣counted in the midst of that Parallel.

How to measure the Quantities of those respective Arches by a Line of Chords and Signs, and by Half-Tangents; and consequently thereby to resolve the most useful Cases of Spherical Triangles; as also by Calculation, is what I intend shall be the sub∣ject of the Pages, viz. and the Art of Dialling by a Gnomical Scale.

The former Sphere or Scheme doth represent the Triangles commonly used in Cal∣culation.

Thus the Right-Angled Triangle CK ☉, Right-Angled at K; supposing the Sun at ☉, is made of CK, his Right Ascention, ☉ K his Declination; — KC ☉ the Angle of the Ecliptick, and the Equinoctial being the Suns greatest Declination 23 deg. 31′ C ☉ K, the Angle of the Suns Meridian and Ecliptick.

In the Right-Angled-Triangle LON, Right-Angled at O; supposing the Sun at LON is the Elevation of the Pole, NL the Complement of the Suns Declination, LO the Suns Azimuth from the North.

LNO the Hour from Midnight, or Complement of the Ascentional-Difference, NLO the Angle of Position, that is, of the Suns Meridian with the Horizon; and of the like parts, or their Complements, is made the Triangle CML.

In the Right-Angled-Triangle CIS, Right-Angled at I; supposing the Sun at S, there is given CS his Declination, IS his Altitude at the hour of Six, CI the Suns Azimuth from the East and West at the hour of Six, ICS the Angle of the Poles Elevation, CSI the Angle of the Suns Position.

In the Right-Angled-Triangle COM, suppose the Sun at M; dM the Suns De∣clination, Cd his hour from Six, CM the Altitude being East or West, dCM the Latitude, dMC the Angle of the Suns Position.

In the Oblique-Angled Triangle Z ☉ N, if the Sun be at ☉. ZN is the Com∣plement of the Latitude, and N ☉ the Complement of the Suns Declination, or di∣stance from the Pole. Z ☉ the Complement of the Suns Altitude, or height; ZN ☉ the Angle of the Hour from Noon; NZ ☉ the Suns Azimuth from the North-part of the Meridian; Z ☉ N the Angle of the Suns Position.

And thus we have shewed how the former Diagram or Analemma represents the Spherical Triangles used in Calculation; whereby, of the Six parts in each Triangle, if any three are given, the rest may be found by Calculation from the Proportions, and that either by Addition and Substraction, by the Artificial Signs and Tangents; and what is resolved by either of these sorts of Tables, we will resolve with the first Tables, and with Scale and Compasses, that you may see the near agreement betwixt them.