Of the Art of

NAVIGATION.

CHAPTER I.

Practical Geometry explained by Definitions, Problems and Proportions.

GE

EOMETRY is that Science, whereby we explain the various Properties of Magnitude; that is, of Lines, Superficies, and Solids, whofe Original is from the Motion of Points, Lines, and Surfaces.

Section I. Of Linear Geometry, or the firft Kind of Magnitude.

Definitions. A Point is faid to have no Parts and therefore can

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Plate 1. Fig. 1.

not be divided, fuch we suppose the Point A.

2. A Line hath no Breadth or Thickness, only Length; it is made by the Motion of a Point, and (confidered in itself) is either Regular or Irregular.

3. Regular, is either a Right-line or an Arc.

4. A Right-line is the fhortest Distance between two Points as the Line BC. Plate 1. Fig. 1.

5. An Arc is not the shortest Distance between two Points, but bendeth evenly, as the Arc DE.

6. Irregular, as any crooked Line that bendeth unevenly, as FG. Plate 1. Fig, 1.

7. Lines, compared, are either parallel, or inclining; from whence proceed the following Problems.

Problem I. To draw a Line parallel to a given Line.

Definition. Parallel Right-lines are thofe which produced (being in the fame

never meet, fuch are the Lines AB and CD.

Example. AB the Line
C is a Point given.

being infinitely Superficies) will Plate x. Fig. 2.

Through this Point C, and parallel to the Line AB, a Line

is required to be drawn.

1. Take

1. Take (with a pair of Compaffes) the neareft diftance between the given Point C, and the Line AB.

2. With that Distance and one Foot of the Compaffes (placed any where in the Line AB,) draw (on the fame Side with the Point C) an Arc D.

3. From C, draw a Line to touch the Arc D, and it's done; for the Line CD, is parallel to the Line AB, as was required.

Prob. II. To bifest or divide a given Right-line into two equal Parts. Example. AB is a given Line. Plate 1. Fig. 3.

To find the Middle thereof is required? 1. WITH any Diftance, (greater than half the given Line AB) and one Foot of the Compasses on A described the

Arc CD.

2. With the fame Diftance, and one Foot on B, cross the former Arc in C and D.

3. By C and D draw a Line which will cut AB in E, the middle; and if AE is equal to EB, it's done true; and E is then in the middle of the Line AB, as was required.

Prob. III. To erect a Perpendicular from a Peint in a given Line.

Definitions.

Nelining Lines are not equally Diftant, but if produced, will meet on one Side, as the Lines AB and CD. Plate 1. Fig. 4.

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2. The meeting of inclining Lines, (called an Angle) is either Direct or Oblique.

3. Direct-meeting of Lines, is when the Angles on each Side are equal, as EGF, and FGH; and this kind of meeting is called Perpendicular. Plate 1. Fig. 4.

Example AB is a Line given.

The Point A, one end of it, from whence to erect a Perpendicular is required. Plate 1. Fig. 5

I With any Diftance, and one Foot in A, draw an Arc to cut the Line AB in D.

2. With the fame Distance, and one Foot in D, draw an Arc, to cut the former Arc in C.

3 With the fame Distance, and one Foot in C, describe an Arc DE, to cut the Line AB in D.

4. By C and D, draw a Line to cut the Arc DE in E.

5. Then by A and E draw a Line and it's done; For the Line AE is perpendicular to AB as was required.

Problem IV. To let fall a Perpendicular, from a given Point to a given Line.

Example. AB is a Line given, C is a given Point, from whence to let fall a Perpendicular to the Line AB is required? Plate 1. Fig. 6.

Draw a Line (at pleasure) from C to AB, as is the Line CD. 2. By Problem 2. bifect the Line CD in E.

3. With the Distance EC, equal to ED, and one Foot in E, crofs the Line AB in A.

4. By A and C draw a Line, and it's done; For AC is a Perpendicular let fall from the Point C to the Line AB, as was required.

Problem V. To make a Plane Angle.

Definition 1. The meeting of inclining Lines is called an Angle, and the Lines fo meeting are called Sides of that Angle, as AB and AC. Plate 1. Fig. 7.

2. An Angle is either a Right-Angle, or an Oblique Angle. 3. A Right-Angle is where two Lines are perpendicular to each other, as ED and DF. Plate 1. Fig. 7.

Note, A Right-Angle is juft 90 Degrees.

4. An Oblique-Angle, is either an Acute less than 90 Degrees, as BAC, or Obtufe more than 90 Degrees, as GHI. Figure 7. Note; An Angle is written with three Letters, the middle Letter fignifieth the Angular Point as BAC fignifieth the Angle A.

An Angle is measured by an Arc, whofe Center is the Angular Point, and is drawn from one Side to the other of the Angle, as the me fure of the Angle KLM is the Arc NO. Plate 1. Fig. 7. What a Degree is you may fee in Problem 9. Definition 1. Example 1. At A in the Line AB, to make a Right-Angle, Plate 1. Fig. 7.

The Rule. Upon A (by Problem 3.) erect the Perpendicular AC, and it's done; For the Angle BAC is a Right-Angle. Example 2. At A in the Line AB to make an Acute-Angle equal to 41 Degrees. Plate 1. Fig. 8.

1. Take (always) a Chord of 60 Degrees from your Scale, and with one Foot on A draw an Arc DE, to cut the Line AB in D. 2. Make the Arc DE, equal to the Chord of 41d. that is, take 41d. from the fame Scale of Chords, and lay it on the Arc from D to E.

3. By A and E draw the Line AEC, and it's done; for the Angle BAC is an Acute Angle, containing 41 Degrees.

Example 3. At B in the Line BC, to make an Obtufe-Angle equal to 102 Degrees. Plate 1. Fig. 8.

1. As before (with one Foot on B) draw the Arc EF with a Chord of 60 Degrees, to cut BC in E.

2. On that Arc make EG equal to GF, and each equal to 51d. the half of 1024. that is, take 51d. from the fame Scale of Chords, and lay it on the Arc from E to G, and from G to F.

3. By B and F draw the Line BFD, and it's done; for the Angle CBD is an Obtuse Angle, containing 102 degrees. Section II. Of Superficial Geometry, or the fecond Kind of Magnitude.

Definition 1. A Superficies hath no Thickness, only Length and Breadth; 'tis made by the Motion of a Line, and is either plane, convex, or concave.

2. A plane Superficies, is a Figure flat, fmooth, even, and made by the Motion of a Right-line; it's either fimple or various. Simple Figure or Superficies is bounded by{

3. A{ various

one Line Lines.

4. Figure bounded by one Line, is either a Circle, or an Ellipfis.

5. A Figure bounded by Lines, is either a Triangle, a Quadrangle, or a Multangle.

6. In every Superficies there are three Things to be noted. 1. The Term, which is that Line or Lines bounding it, as BCDEB. Plate 1. Fig. 9.

2. The Center, which is a Point in the middle of it; as A. 3. The Area, which is all the Space contained within the Term, as ABCDEBA. Plate 1. Fig. 9.

4. The kinds of Plane Figures are feven, a Circle, a Triangle, a Quadrangle, and a Multangle the moft cafy to make; the more difficult are the Ellipfis, Parabola, and Hyperbola; each affords divers Problems, of which we begin with the Circle.

Problem VI. To defcribe a Circle, having its Diameter given. Definitions. 1, A Circle is a plane Figure bounded by one Line, called the Periphery; as ABCDEBA. Plate 1. Fig. 9.

2. The Periphery of a Circle is a Line encompaffing it, fo that, it's equally diftant from the Center, as BCDEB.

3. The Center of a Circle is a Point in the middle of it, from whence all Right-lines drawn to the Periphery are equal, and called Radii, each, or any one fuch Line being called Radius; as A.

4. The Diameter of a Circle, is any Right Line drawn thro' the Center to the Periphery; it bifecteth the Circle, and is the longeft Right-line that can be drawn in it, as BAD is a Diameter. Example. BD equal to 60 Inches is given; to make a Circle on or about it, is required? Plate 1. Fig. 10.

1. Bifect (by Froblem 2.) the Line BD in A.

2. With the Distance AB equal to AD, and one Foot on A, describe the Periphery BCDEB, and it's done,

Problem VII. To draw the Periphery of a Circle through any three
Points, not in a Right-line.

Example. B, C, and D, are three Points given, thro' them to draw the Periphery of a Circle is required? Plate 1. Fig. 11. 1. With any Diftance (greater than half CB, or CD) and one Foot in C, draw the Arcs FG, HI.

2. With the fame Diftance, and one Foot in B, cross the first Arc in F and G.

3. Likewife (with the fame Distance and) one Foot in D, cut the other Arcs in H and I.

4. By F and G, and H and I, draw Lines to cut each other in A which will be the Center.

5. With the Distance AB, equal to AC, equal to AD, and one Foot on A, draw the Periphery BCDB, and it's done. Problem VIII. To quarter a Circle, or in a Circle to draw two Diameters at Right-angles.

Example. BCDEB the Periphery, and A the Center of the given Circle; in which to draw two Diameters at Right-angles is required. Plate 1. Fig. 12.

1. Through the Center A, draw the Diameter BAD.

2. Bifect (by Problem 2.) BAD, by drawing the Line CAE, and it's done; and if BC, CD, DE, and EB are equal to each other, it's done true, otherwife not.

Problem IX. To find the Chord, Sine, Tangent, and Secant of an
Arc of a Circle.

Definitions. 1. If the Periphery of a Circle be divided into 360 equal Parts, they are Degrees; a Degree divided into 60 equal Parts, are Minutes; a Minute into 60 equal Parts, are Seconds, &c. Plate 1. Fig. 13.

2. The Arc of a Circle is any Part of the Periphery; as EB, or EBF; and is counted in Degrees, which are greater or lefs in Proportion to the Radius of the Circle.

3. Radius of a Circle is half its Diameter, or any Right-line drawn from the Center to the Periphery, as AB.

4. A Chord-line is drawn from one End of an Arc to the other; as EDF, or EB, are Chord-lines.

5. A Sine-line is half a Chord-line of double the Arc; as ED (being half the Chord EDF) is a Sine-line of the Arc BE, and DF is a Sine-line of the Arc BF.

6. Verfed-Sine-line lies between the Sine-linc and the Periphery, as DB is a Verfed-Sine of the Arc BE equal to BF.

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7. Tan