The seaman's companion being a plain guide to the understanding of arithmetick, geometry, trigonometry, navigation, and astronomy. Applied chiefly to navigation: and furnished with a table of meridional parts, to every third minute: with excellent and easie ways of keeping a reckoning at sea, never in print before. Also, a catalogue of the longitude and latitude of the principal places in the world with other useful things. The third edition corrected and amended. By Matthew Norwood, mariner.

[illustration] depiction of geometrical figure

The Projection of the Sphere in Plain Lat. 50 deg.

There be many that do not attain to the Doctrine of Spheri∣cal Triangles, and are loth (indeed or cannot) spend time to study them: I have thought good to shew the way to measure any thing that is useful for Seamen by the plain Scale (before made:) and though this cannot be so exact as to come to a minute of a degree, yet it is exact enough for our use at Sea: For in taking an Azimuth, or Amplitude, or such like, half a degree of the Compass will not make your Course the worse (after you have allowed for Variation) it being not full the 22^t•. part of a point, which is no sensible error in steering: or for finding the length of the day (for ordinary uses) it will breed but little error, 30 minutes being but 2 minutes of time; but for other things that require exactness to a minute or less, of a degree, this way is of no considerabie use; the way by the Tables being exact, and so better. (I shall do both) The reason I handle this here, is in order to Navigation, that you may understand the Circles of the Sphere. This by way of Advertisement.

BEfore I shew how to project the Sphere, it will be necessary to understand the Circles of the Sphere.

A Sphere is a Scheme or Figure which represents the Heavens, and therefore is round exactly, though upon a Plain it doth not seem so; from whence it is evident, that the lines here drawn in it cannot be streight lines, but Circles: Now there are two sorts of Circles, namely, the Greater Circles, and the Lesser; the Greater Circles are such as go round the very Body of the Globe, and so cut through two opposite points, dividing it into two Hemispheres, and are six in number, which are these:

• 1. The Axis of the World. (From South to North Pole.)
• 2. The Horizon.
• 3. The Aequinoctial.
• 4. The Ecliptick.
• 5. The East and West Azimuth.
• 6. The Meridian.

The smaller Circles of the Sphere, are all such Circles as do not divide the Sphere of Heaven into two Hemispheres, and so are less than those that do; and they be four, namely, The two Tropicks, and the two Polar Circles, which to distinguish

the Southermost from the Northermost, we call the Souther∣most Tropick, Capricorn; the Northermost, The Tropick of Cancer; the Southermost Polar Circle, The Antartick; and the Northermost, The Artick Circle.

All Circles that cut through the two Poles of the World, are called Meridians; and are also Great Circles, because they di∣vide the Sphere into two Hemispheres, cutting in two opposite points: it is certain, if a Circle goeth through two opposite points, it is as great a Circle as that globous Body can bear, and must divide that Body into two Hemispheres, as doth the Meridian A B C D.

The Poles are two opposite places in the Heavens, and are the ends (as it where) of that Line called the Axletree, which the Heavens may be imagined to turn upon: here that Line is called the Axis of the World.

The Aequinoctial is a Great Circle of the Sphere, which lieth between the Pole so equally, that its distance from either Pole is 90 degrees (the half of the distance between the Poles) and because it is so equal between them, there begins Latitude; so that whatsoever Latitude you are in, so many degrees and mi∣nutes the Aequinoctial is below that part of the Heavens which is right over your head, and is called the Zenith; or from being right under you, which is called the Nadir. All Meridians cut this Aequinoctial at right Angles.

The East and West Azimuth is a Great Circle of the Sphere that cuts through two opposite points of the Heavens, namely, the Zenith and Nadir: it also cuts the Horizon at right Angles in the points of East and West; and therefore is called the East and West Azimuth: all Azimuths are great Circles and cut the Horizon at right Angles.

The Horizon is a Great Circle of the Sphere, and divides that part of the Heavens which we do not see, from that part which we do see: or it is the furthest part of the Heavens which we can see for the Body of the Sea; it is only to be seen at Sea, or upon the shore where there is no land between you and it, and so you have divers Horizons according to your motion, for as you raise one part of the World, you lay the other.

The Zodiack is a great Circle, it is the bounds of the twelve Signs: now in the middle of it is the Ecliptick Line, in which Line the Center of the Sun goeth, and passeth every day its mo∣tion of Declination, till it comes to its utmost bounds, which is to the Tropicks: it cuts the Aequinoctial in two opposite points, and makes from these points an Angle of 23 deg. 30 min. which is the Angle S R t, so the Sun is never out of it. The twelve Signs divide this Ecliptick into twelve equal parts, and to every Sign theee is a Name, and a Character for that Name, and a Month to that Sign; which for your better knowledge I have here following inserted.

The Names and Characters of the twelve Signs, with the Months they belong to.

March	Aries	♈	These be the Northern Signs
April	Taurus	♉
May	Gemini	♊
June	Cancer	♋
July	Leo	♌
August	Virgo	♍
September	Libra	♎	These be the Southern Signs
October	Scorpio	♏
November	Sagittarius	♐
December	Capricornus	♑
January	Aquarius	♒
February	Pisces	♓

Six of these be Northern Signs, and are in the North part of the Zodiack; and six of them are Southern Signs, because they be in the South part of the Zodiack. The twelve Signs are twelve Constellations, the Months answering to them are agree∣able. Thus much for the great Circles of the Sphere.

The smaller Circles are the Polar Circles and the Tropicks.

The Polar Circles are distant 23 deg. 30 min. from the Poles, and between them and the Poles is counted the Frozen Zones.

The Tropicks are the bounds of the Suns Declination, and they go parallel to the Aequinoctial (and 23 deg. 30 min. from it) the Tropick of Cancer being 23 deg. 30 min. to the North∣wards, the Tropick of Capricorn being as far to the South∣wards, which is the Suns furthest distance from the Aequator at any time.

How to Project a Sphere.

THis Sphere or Scheme is projected on this wise: First, take an Arch of 60 deg. from your Scale of Chords, and describe the Meridian M A E R. Now the reason you take just 60 deg. is because that number of degrees taken from the Scale is the Semidiameter of the Circle the Scale was made by: so then whatsoever you take from that Circle, and apply to the Scale, it will be the quantity of it in degrees and minutes; or from the Scale if you take any distance, and apply it to the Circle, it will be the same quantity upon the Circle, that it was upon the Scale.

The Meridian being thus described, draw the line A R through the Center, which represents the Horizon: Cross that at right Angles, and draw the line E M, which is the East and West Azimuth.

Then because the Sphere is made for the Latitude of 50 deg. take 50 deg. from your Scale of Chords, and fix one foot in the Horizon at A, and set the other foot upwards to B, (make a prick or mark) which is to signifie that the North Pole is ele∣vated or raised above the Horizon so many degrees as the La∣titude is.

Keep the same distance in your Compasses, and go to E with one foot, and let fall the other upon the Meridian on the left hand, which will be at G, and will represent that point of the Aeqinoctial that cuts the Meridian of that place which is the nearest point of the Aequinoctial to that Zenith, and is as much below the Zenith, as the Pole is above the Horizon.

[illustration] depiction of geometrical figure

Latitude 50 deg. 00 min. Declination. Northerly 13 deg. 15 min.

Keep the same distance, and carry your Compasses to the Ho∣rizon at R, and let fall the other point to K in the Meridian, which is the South Pole; for as much as the North Pole is ele∣vated above the North part of the Horizon (at A) so much is the South Pole depressed below the South part of the Horizon at K.

Continue the same distance between your Compasses, and fix one foot in the Nadir at M, and set the other foot upon the Meridian upwards to O, which is the other point of the Aequi∣noctials intersection in the Meridian opposite to the former; and it stands to good reason, that as much as the Aequinoctial is below the Zenith, so much it ought to be above the Nadir: These marks being set off, draw the Axis of the World B K, and the Aequinoctial G O. And because (as I said before) the outmost bounds of the Suns Declination is 23 deg. 30 min. (either to the Northwards or Southwards of the Aequinoctial, which bounds are expressed by the Tropicks) take 23 deg. 30 min. from your Scale of Chords, and fix your Compasses at O in the Aequinoctial, and set it off on both sides to P and N, carry the same distance to G, and set it off in like manner from G to F and H, then draw F P and H N, which are the Tropicks: also draw the Ecliptick from F to N, which will fall through the Center. Always set the same distance from both Poles on both sides upon the Meridian, and draw D C the Artick, and I L the Antartick Circles. And thus you may project a Sphere by a Plain Scale: The like for any other.

Before I proceed to the Questions, it will be necessary to give you the meaning of things that will be spoke to.