7. Tangents touching the Periphery, and its Perpendicular to a Diameter in the touching Point, as BC is a Tangent Line of the Arc BE.

8. Secant cutteth the Periphery, being drawn from the Center 'till it meet the Tangent, as AC is a Secant of the Arc BE. Example. The Chord, Sine, Tangent, and Secant of an Arc containing 45 deg. are required? Plate 1. Fig. 13.

1. With any Distance (and one Foot on A) defcribe the Periphery HIBG.

2. Quarter the Periphery (by Problem 8.) by drawing the two Diameters HAB and IAG; then is BG equal to GH, equal to HI, equal to IB, equal to 90 Degrees.

3. Divide (by Problem 2.) BI into two equal Parts in E, and then BE equal to EI, is equal to 45 Degrees.

4. Make BF equal to BÉ, and draw the Lines BE, and EDF, to cut the Diameter HAB in D.

5. At B (by Problem 3.) erect BC perpendicular to the Diaméter HAB; or draw (by Problem 1.) BC parallel to the Diameter IAG.

. 6. By A and E draw a Line to cut BC in C, and it's done. Then the Line BE is the Chord of 45 deg. the Line of EDF the Chord of 90 deg. the Line ED equal to DF, the Sine of 45 deg. DB the Verfed Sine, BC the Tangent Line, and AC the Secant Line of the fame Arc BE, equal to 45 Degrees.

This Problem is the Ground Work of the Line of Rumbs, Chords, Sines, Tangents, Secants, &c. Set on Rulers called Scales or Plane Scales; as alfo of the Degrees on the Quadrant, Crofs-Staff, and other mathematic Inftruments.

But its chief Ufe is in a Right-Angled-Triangle, as fhall be thewed in Trigonometry, when I come to explain the first Axiom in making any Side Radius, fce Chapter 2. Section 2. in Page 35.

Of Triangles, the 2d Kind of Plane Figures.

Definition 1. A Triangle is any three corner'd Figure, having three Sides and three Angles, as ABC; and in respect of its Angles, is either Rectangular or Obliquangular. Plate 1. Fig. 14.

2. A Right-Angle-Triangle hath one Right-Angle; in its making hath a Perpendicular erected, or let fall, as ABC.

3. Thofe Sides containing the Right-Angle are called Legs; as Leg AB, and Leg BC. Plate 1. Fig. 14.

4. The Side oppofite to the Right-Angle is called the Hypothenufe; as AC. Plate 1. Fig. 14.

Prob. X. To make a Right Angle Triangle, the Hypothenuse, and one Angle given.

Note 1. The three Angles of every Plane Triangle (together) are equal to 180 Degrees.

2. To make a Right Angle Triangle, two Things, (befides the Right Angle) and one a Side, must be given. Example. The Hypothenufe AC 137 Feet Angle BAC 34d. 30m.

given

With them to make a Right Angle Triangle is required. Plate 1. Fig. 15.

1. Make the Angle BAC equal to 34d. 30m. (by Problem 5. Example 2.) that is, take a Chord of 60d. and with one Foot on A, draw an Arc, and on that Arc, lay the Chord of 34d. 30m. drawing AB and AC, which includes the Angle BAC equal to 34d. 30m.

2. Make AC equal to 137 Feet; that is, from any Scale of equal Parts take 137, and lay it from A to C.

3. From C (by Prob. 4.) let fall the Perpendicular BC, to cut the Line AB in B, and it's done.

Prob. XI. To make a Right Angle Triangle; the Hypothenufe and one Leg being given.

Example. The

Hypothenufe AB 411 Perches given
Leg AB -

342

With them to make a Right Angle Triangle is required. Plate 1. Fig. 16.

1. Make AB equal to 342 Perches; that is, from any Scale of equal Parts, take 342, and lay it from A to B.

2. At B (by Prob. 3.) erect a Perpendicular BC.

3. With 411 Perches (that is, take 411 from the fame Scale of equal Parts, the 342 were taken) and with one Foot in A, cut the Perpendicular in C.

4. From A to C draw a Line, and it's done.

Prob. XII. To make a Right Angle Triangle; having one Leg and one Angle given.

Example. The Leg AB 415 Miles

Angle BAC 40d. 25m.
given.

With them to make a Right Angle Triangle is required. Plate 1. Fig. 17.

1. Make AB be equal to 415 Miles, that is, from any Scale

B

of

of equal Parts, take 415, and lay it from A to B ; and at B, (by Prob. 3.) in Page 12, erect a Perpendicular BC.

2. At A (by Prob. 5. Example 2.) in Page 13, make the Angle DAC equal to 40d. 25m. (which is thus, take a Chord of God. and with one Foot on A, draw an Arc, on which Arc Jay the Chord of 40d. 25m.) by drawing the Line AC, and it's

done.

Prob. XIII. To make a Right Angle Triangle, the Legs being given. AB

Example. The Leg{B8404} Leagues given.

With them to make a Right Angle Triangle, is required. Plate 1. Fig. 18.

1. Make AB equal to 404 Leagues; and at B (by Prob. 3.) in Page 12, erect a Perpendicular BC.

2. Upon the Perpendicular make BC equal to 328 Leagues,, and from A to C draw a Line, and it's done.

Of Oblique-angle Triangles.

Definition. 1. An Oblique Triangle hath all its Angles, Oblique and is either an Obtufe or an Acute Triangle.

2. An Obtufe Triangle hath one Obtufe Angle, that is, one Angle greater than god. as BCD. Plate 1. Fig. 19.

3. An Acute Triangle hath all its three Angles Acute; that is, each Angle is lefs than god. as EFG. Plate 1. Fig. 19. Note; To make an Oblique Triangle, three Things, (and one of them a Side) must be given.

Prob. XIV. To make an Oblique Triangle, two Angles and a Side oppofite one of them being given.

Example. The

Angle BDC 26d. 40m. given.
BCD 114d. ---
Side BC 352 Miles -

With them to make an Oblique Triangle is required. Plate 1. Fig. 19.

1. Make BC equal to 352 Miles; and at C (by Prob. 5. Example 3.) in Pages 13 and 14, make the Angle BCD equal to 114d. by drawing the Line CD; that is, take a Chord of 60d. and with one Foot on C; draw an Arc, on that Arc lay the Chord of 114d. by taking it's half 57d. and laying it twice on the Arc, by which draw the Line CD, thereby you'll conclude the Angle BCD 114 Degrees.

2. Add 114d. to 26d. 40m. it makes 140d. 40m. the Sum of the given Angles; which fubtract from 180d. oom. the Sum of the Three Angles.

39d. 20m.

3. At B (by Prob. 5. Example 2.) in Page 13 make the Angle CBD equal to 39d. 20m. by drawing the Line BD; that is with a Chord of 6od. and one Foot on B, draw an Arc, on that Arc lay the Chord of 39d. 20m. by which draw the Line BD,

and it's done.

Prob. XV. To make an Oblique Triangle, having two Sides, and an Angle oppofite to one of them given.

БС

BC 274

Side { BD

BD 426

Angle BCD 108d. 3cm.

With them to make an Oblique Triangle is required. Plate 1. Fig. 20.

1. Make BC equal to 274 Miles, and at C (by Problem 5. Example 3.) in Pages 13 and 14, make, the Angle BCD equal to 108d. 30m. by drawing the Line CD.

2. With 426 Miles, and one Foot in B, cut the Line CD in D, and by B and D, draw a Line and it's done.

Prob. XVI. To make an Oblique Triangle, two Sides, and the included Angle being given.

Example. The

BC-327

SideBD-274/Feet

Angle BCD 101d. 30m.

given.

With them to make an Oblique Triangle is required. Plate t. Fig. 21.

1. Make the Side BC equal to 327 Feet; and at B (by Problem 5. Example 3.) in Pages 13 and 14 make the Angle CBD equal to 101d. 30m. by drawing the Line BD.

2. Make BD equal to 274 Feet; and by C and D draw a Line, and it's done.

Prob. XVII. To make an Oblique Triangle, having the three Sides given.

Note. Any two Sides together, must be greater than the Third, to make a Triangle.

Example. The Side

BD 525

BC 425 Leagues given.
CD 250

With them to make an Oblique Triangle is required. Plate 1. Fig. 22..

1. Make BD equal to 525 Leagues; then with 425 Leagues, and one Foot in B draw an Arc C.

2. With 250 Leagues, and one Foot in D, cut the Arc in C, and from C draw Lines to B and D, and it's done.

Sect. III. Of Proportions for the ready measuring of Superficies and Solids.

HAving fhewed how to make the most common and useful Geometrick Figures, I now proceed to their (and others) Menfuration, by exact and eafy Rules.

To prevent the Book's fwelling too great, I will not trouble. you with Arithmetic Calculation, but prefent you with an Inftrumental way of Operation, that's quick and true, eafy and ready on Gunter's-Scale, an Inftrument fo very well known, it needs the less Description; only to avoid Repetitions take this excellent Rule.

A General Rule for Gunter's-Scale.

Extend the Compaffes from the firft Term (in Geometric Proportion) to the fecond: That Extent (if right laid) from the third, will reach to the fourth Term, or Thing required.

This Rule well minded, will make what follows appear eafy, and fave many Words and much Time; and therefore to explain it, take this Proportion.

As 2 is to 6; fo is 5 to 15; which I shorten thus ;

As 265. 15.

The Extent (on the Line of Numbers) from 2 unto 6, ac-. cording to the faid General Rule) will reach from 5 unto 15, the fourth Term or Thing required.

Thefe Words may be fpar'd by the short way of writing the Proportion and minding the Order of the Terms, as in the Problems following: But, for a full Explanation of the Gunter, and how to work any Proportion by it, fee my Additions to Mr. WAKELEY's MARINER's COMPASS RECTIFIED, Page 181, &c.

Prob. XVIII. The Diameter of a Circle given; to find its Periphery.

3.14159

:: Diameter. Periphery.

Example. The Diameter 34 Inches: What's the Periphery? Anfw. Inches 106.8 Tenths of an Inch. For,

As {7: } - {22.14159}} :: 34 Inches 106.8 Tenths.