V.

Lines not parallel, but inclining towards each other, whether they are right Lines or circular, will, if they are extended, meet, and make an AAngle; the Point where they meet is called the Angular Point, as at A; and according as fuch Lines ftand nearer or farther off each other, the Angle is faid to be greater or lefs, whether the Lines that include the Angle be long or short.

All Angles included between Right Lines, are called Right-lined Angles, and fall under these three Denominations; viz. A Right Angle, an Obtufe Angle, and an Acute Angle.

VI.

A Right Angle is that which is included between two Lines that meet each other perpendicular, as DC meets A B.

D

VII.

An Obtufe Angle is that which is greater than a Right Angle, fuch as the Angle included between the Lines A C and C B.

A

B

VIIJ..

An Acute Angle is that which is less than a Right Angle, fuch as the Angle included between the Lines C B and CD; these Angles are called Oblique Angles.

D

B

NOTE. The Radius, generally ufed in defcribing Circles, is • 60°, (taken from the Line of Chords) the Circumference of which will contain 360°, the Number of equal Parts or Degrees that all Circles are fuppofed to be divided into; and each of these Degrees are divided into 60 equal Parts, called Minutes, &c.

All Angles are measured by an Arch of this Circle, defcribed upon the Angular Point as a Centre, with the Chord of 60°, as above, and the Angle is faid to be greater or lefs, according to the Number of Degrees contained between the Legs; but the Sides or. Legs are measured by a Scale of equal Parts.'

The Diameter is twice its Radius, joined into one Right Line drawn through the Centre, which divides the Circle into two equal Parts called Semi-circles.

A Quadrant is half a Semi-circle.

A Chord Line is a Right Line that cuts the Circle in two unequal Parts called Segments.

A Sector is a Figure included between two Radius's, and is lefs than a Quadrant.

All Plan Triangles are Figures comprehended under three Right Lines, and are diftinguished into three Sorts; viz. A Right-angled Triangle, an Obtufe-angled Triangle, and an Acute-angled Triangle.

X.

A Right-angled Triangle is that which hath one of its Legs perpendicular to the other, and is equal to a Quadrant or 90°; as BAC; the longest Side of which is called the Hypothenufe, as BC, and the other two Sides are called Legs.

XI.

An Obtufe-angled Triangle is that which hath one of its Angles greater than a Right Angle, as ED F.

D

XII.

An Acute-angled Triangle is that which hath all its Angles lefs than a Right Angle, G as G H I.

H

XIII.

The Three Angles of every Triangle are equal to a Semi-circle, or 180°; the two Acute Angles of every Right-angled Triangle are equal to 90o.

XIV.

Sides or Angles given, are marked with a Dash; but, when required, with a Cypher.

XV.

Degrees are always marked with a Cypher over them, (°) and Minutes with a Dash. (')

XVI.

Three Letters denote an Angle; the Middle one fhews the Angular Point, and the other two the Sides that inclose it.

GEOMETRICAL PROBLEMS.

PROBLEM I. .

To draw a Line parallel to a given Line AB, at any given Distance, as C.

TAKE with a pair of Compaffes the nearest Distance between the given

Point C and the Line A B.

D

A

B

With that Distance and one Foot of the Compaffes, any where in the Line A B, draw (on that Side where the Point Clieth) the Arch D, from the Point C draw a Line to touch the Arch D, and it is done; for the Line CD is parallel to the Line A B, as was required.

PROBLEM II.

To bifect or divide a given Line into two equal-Parts.

With any Distance (greater than Half the given Line A B) and one Foot of the Compaffes on A, defcrible the Arch C D; with the fame Distance and one Foot on B, cross the former A Arch in C and D; by C and D draw a Line, and that will cut A B in E the Middle, as was required.

E

B

To erect a Perpendicular at the end of a given Line, as AG.

With any Distance, as from A to C in your Compaffes, and one Foot in C defcribe a Circle, fo that it may just touch the End of the given Line in A, from where that cuts the Line at G, and through C draw a Line to cut the Circle in B, from B draw the Line A B, which will be the Perpendicular required.

B..

Α

Or, with any convenient Distance in your Compaffes defcribe an Arch, as F B, fet off the fame Distance from B to E, with one Foot in E defcribe an Arch, and with the F fame Distance, and one Foot in F, defcribe an Arch to cut the former Arch in C, from C to A draw a Line, and it is done.

PRO OBLEM

E

A B

IV.

From a Point as at C, to let fall a Perpendicular on the Line A B.

With one Foot in C describe an Arch to cut the given Line in A B, with one Foot in B defcribe an Arch; and with the fame Radius, and one Foot in A, defcribe an Arch to cut the former in D, and from D to C draw a Line, and it is done; for Ce is Perpendicular to A B, as was required.

C

PROBLEM

To make Plane-Angles.

..

At A in the Line A B, to make a Right C Angle, erect the Perpendicular A C, and it is done; for the Angle B A C is a Right Angle containing 90 Degrees. Or, defcribe an Arch from B to C, with the Chord of 60°, and set off 90° from B to C, and it is done.

V.

B

-B

To make an Angle equal to any given Number of Degrees.

Draw the Line A B, and take always a Chord of 60° from the Scale in your Compaffes, and with one Foot in A defcribe the Arch DE to cut the Line A B in D; take any Number of Degrees, fuppofe 42° 30', from the Line of Chords, and lay it upon the Arch from D to A

42.30

E

E; by A and E draw the Line A E C, and it is done; for the Angle BAC is an Acute Angle containing 42° 30′.