D

6. A QUADRANT is Half a Semicircle, viz. one Quarter of a Circle; and tis made by the Radius (as DC) ftanding Perpendicular upon the Diameter at the Center C, cutting the Periphery of the Semicircle in the Middle, as at D. Therefore a Quadrant, or half the Semicircle, is the Meafure of a Right Angle.

A

B

C

G

M

7. A CHORD LINE, or the Subtenfe of an Arch, is any Right Line that cuts the Circle into Two unequal Parts, as the Line 8 G; and is always less than the Diameter.

8. A SEGMENT of a Circle, is a Figure included betwixt the Chord and that Arch of the Periphery which is cut off by the Chord And it may either be greater or less than a Semicircle; as the Figure SDG, or S MG.

B

9. A SECTOR is a Figure included between Two Radius's of the Circle, and that Arch of its Periphery where they touch, as the Figure ACB: And the Arch AB is the Measure of the Angle at C, included betwixt the Radius's AC, and B C.

Note, All Angles of Sectors are called Angles at the Center of a Circle.

D

C

10. An ANGLE in the Segment of a Circle is that which is included between Two Chords that flow from one and the fame Point in the Periphery, as at D, and meet with the Ends of another Chord Line, as at F and G.

That is, the Angles at D, at F, and at G, are called Angles at the Periphery, or Angles ftanding on the Segment of a Circle.

Sect. 3. Of TRIANGLES.

There are two Kinds of Triangles, viz. Plain and Spherical; but I fhall not give any Definition of the Spherical, because they more immediately relate to Aftronomy.

I. A PLAIN TRIANGLE is a Figure whofe Area is contained within the Limits of Three Right Lines called Sides, including Three Angles: And it may be divided, and takes its Name either according to its Sides or Angles.

B

I. By its SIDES.

2. An EQUILATERAL TRIANGLE, is that which hath all its Three Sides equal; as the Figure ABC

That is, AB=BC=AC.

3.

A

D

An ISOSCELES TRIANGLE, is that which hath only Two of its Sides equal, as the Figure BDG: That is, BD DG; but the Third Side B G may be either greater or less, as Occafion requires.

4. A SCALENE TRIANGLE,

is that which hath all its Three Sides unequal;

fuch as the Figure HK M.

2. By its ANGLES.

B

K

M

H

C

5. A RIGHT-ANGLED Triangle, is that which hath one Right Angle; that is, when Two of its Sides are Perpendicular to each other, as CA is fuppofed to be to B A. Therefore the Angle at A, is a Right Angle, per Defin. 8. Sect. 1.

Note, The longest Side of every Right-angled Triangle (as BC) is called the Hypotenufe, and the longest of the other Two Sides which include the Right Angle (as B A) is called the Bafs: The Third Side (as C A) is called the Cathetus or Perpendicular.

6. An OBTUSE-ANGLED Triangle, is that which hath one of its Angles Obtufe, and it's called an Amblygonium Triangle. Such is the Third Triangle HKM.

7. An ACUTE-ANGLED TRIANGLE, is that which hath all its Angles Acute, and it's called an Oxygonium Triangle; fuch are the First and Second Triangles ABC, and BDG.

Note, All Triangles that have not a Right Angle, whether they are Acute, or Obtufe, are in general Terms, called Oblique Trian

gles

gles, without any other Diftinction, as before. And the longest Side of every Oblique Triangle is ufually called the Bafe; the other two are only called Sides or Legs.

8. The altitude or eight of any Plain Triangle, is the Length of a Right Line let fall perpendicular from any of its Angles, upon the Side oppofite to that Angle from whence it falls; and may be either within, or without the Triangle, as Occafion requires, being denoted by the Two prick'd Lines, in the annexed Triangles.

Sett. 4.

Of Four lioed Figures.

1. A Square is a plain regular Figure, whofe Area is limited by Four equal Sides all perpendicular one to another.

That is, when AB-BC-CD=DA, and the Angles A, B, C, D are all equal, then it's usually called a Geometrical Square.

2. A Rhombus, or Diamond-like Figure, is that which hath Four equal Sides, but no Right-angle. That is, a Rhombus is a Square mov'd out of its right Pofition, as the annexed Figure.

B

A

B

D

σ

3. A Rectangle, or a Right-angled Parallelogram (often called an Oblong, or long Square) is a Figure that hath four Right-angles and its two oppofite Sides equal, viz. BC=HD= and BH CD.

D

4. A Rhomboides, is an Oblique-angled Parallelogram; that is, it is a Parallelogram moved out

of its right Pofition, like the annex

ed Figure.

5. The altitude or Height of any Oblique-angled Parallelogram, viz. either of the Rhombus or Rhomboides, is a Right-line let fall perpendicular from any Angle upon the Side oppofite to that Angle; and may either be within or without the Figure: As the prick'd Lines in the annexed Figure.

6. Every

PP

6. Every Four-fided Figure, different from thofe before- mentioned, is called a Trapezium.

That is, when it has neither oppofite Sides, nor oppofite Angles equal; as the Figure ABCD.

A

C

7. A Right-line, drawn from any Angle in a Four-fided Figure to its oppofite Angle, is called a Diagonal Line, and will divide the Area of the Figure into two Triangles, being denoted by the prick'd Line AC in the laft Figure.

8. All Right-lin'd Figures, that have more than four Sides, are call'd Polygons, whether they be regular or irregular.

9. A Regular Polygon is that which hath all its Sides equal, fanding at equal Angles, and is named according to the Number of its Sides (or Angles.) That is, if it have five equal Sides, it is called a Pentagon; if fix equal Sides, it is call'd a Heragon; if seven, 'tis a Heptagon; if eight, 'tis an Daagon, &c.

Note, All Regular Polygons may be infcrib'd in a Circle; that is, their Angular Points, how many foever they have, will all just touch the Circle's Periphery.

10. An Irregular Polygon is that Figure which hath many unequal Sides ftanding at unequal Angles (like unto the annexed Figure, or otherwife); and of fuch Kind of Polygons there are infinite Varieties, but they may all be reduced to regular Figures by drawing Diagonal Lines in them; as fhall be fhew'd farther

on.

These are the most general and ufeful Definitions that concern plain or fuperficial Geometry.

As for those which relate to Solids, I thought it convenient to omit given any Account of them in this Place, because they would rather puzzle and amuse the Learner, than improve him, until he has gain'd a competent Knowledge in the most useful Theorems concerning Superficies; for then thofe Definitions may be more eafily understood, and will help them to form a clearer Idea of their refpective Solids, than 'tis poffible to conceive of them before; and therefore I have referv'd those Definitions until we come to the Fifth Part.

Sect. 5. Of fuch Terms as are generally used in Geometry.

Whatsoever is propofed in Geometry will either be a Problem or a Theorem.

Both which Euclid includes in the general Term of Propofition. A Problem is that which proposes fomething to be done, and relates more immediately to practical than fpeculative Geometry; That is, it's generally of fuch a Nature, as to be perform'd by fome known or Commonly-receiv'd Rules, without any Regard had to their Inventions or Demonftrations.

A Theorem is when any Commonly-receiv'd Rule, or ony New Propofition is required to be demonftrated, that fo it may from thence forward become a certain Rule, to be rely'd upon in Practice when Occafion requires it. And therefore feveral Rules are often call'd Theorems, by which Operations in Arithmetick, and Conclufions in Geometry, are perform'd.

Note, By Demonstration is understood the highest Degree of Proof that human Reason is capable of attaining to, by a Train of Arguments deduced or drawn from fuch plain Axioms, and other Self-evident Truths, as cannot be denied by any one that confiders them.

A Corollary, or Confectary, is fome Confequent Truth drawn or gain'd from any Demonftration.

A Lemma is the Demonftration of fome Premifes laid down or propofed as preparative to obviate and fhorten the Proof of the Theorem under Confideration.

A Scholium is a brief Commentary or Obfervation made upon fome precedent Difcourfe.

N. B. I advise the young Geometer to be very perfect in the Definitions, viz. Not to reft fàtisfied with a bare Remembrance of them; but, that he endeavour to gain a clear Idea or Understanding of the Things defined; and for that Reafon I have been fuller in every Definition than is ufual.

And, that he may know from whence most of the following Problems and Theorems contain' in the Two next Chapters are collected, I have all along cited the Propofition, and Bock of Euclid's Elements where they may be found.

As for Inflance; at Problem 1. there is (3. e. 1.) which shews that it is the Third Propofition in Euclid's Firft Book. The like must be underflood in the Theorems.