AN

INTRODUCTION

TO THE

Mathematicks.

A

PART III,

CHAP. I.

Of Geometrical Definitions, &c.

Sect. I.

Of Lines and Angles.

POINT hath no Parts: That is, a Geometrical Point is not any Quantity, but only an affignable Place in any Quantity, denoted by a Point:

As at A. and B.

}

A.

B.

Such a Place may be conceived fo infinitely fmall, as to be void of Length, Breadth, and Thickness; and therefore a Point may be faid to have no Parts.

2. A LINE is called a Quantity of one Dimenfion, because it may have any fuppofed Length, but no Breadth nor Thickness, being made or reprefented to the Eye, by the Motion of a Point.

That is, if the Point at A, be moved (upon the fame Plane) to the Point at B, it will defcribe a Line either right or circular (viz. crooked) according to its Motion.

Therefore the Ends or Limits of a Line are Points.

3. A RIGHT LINE, is that Line which lieth even or freight betwixt those Points that limit its Length, being the fhorteft Line that can be drawn between any Two

Points. As the Line AB.

}

A

B.

Therefore, between any two Points, there can lie or be drawn but one

right Line.

4. A CIRCULAR, crooked or OBLIQUE Line, is that which lies bending between thofe Points

which limit, its Length, as the Lines CD or FG, &c.

Of thefe Kinds of Lines there are various Sorts; but thofe of the Circle, Parabola, Ellipfis, and Hyperbola

C

F

D

are of most general Ufe in Geometry; of which a particular Account fhall be given further on.

5. PARALLEL LINES, are thofe that lie equally diftant from one another in all their Parts, viz. fuch Lines as being infinitely extended (upon the fame Plane) will never meet: As the Lines A B and a b: or C D and c d.

A

a

-b

D

6. LINES not PARALLEL, but INCLINING (viz. leaning) one

towards another, whether they are Right Lines, or Circular Lines, will (if they are extended) meet, and make an Angle; the Point where they meet is called the Angular Point, as at A. And according as fuch Lines ftand, nearer or further off each other, the Angle is faid to be leffer or greater, whether the Lines that include the Angle be long or short. That is, the

d

B

d

B

C

Lines Ad, and Af include the fame Angle as A B, and AC doth; notwithstanding that AB is longer than Ad, &c.

7. All ANGLES included between Right Lines are called Rightlin'd Angles; and thofe included between Circular Lines are called Spherical Angles. But all Angles, whether Right-lin'd or Spherical, fall under one of thefe Three Denominations.

8. A RIGHT ANGLE is that which is included betwixt Two Lines, that meet one another Perpendicularly.

That

That is, when a Right Line, as
DC, meets with another Right-
Line, as AB, fo directly as that

it neither inclines nor declines to
one Side more than the other, but
makes the Angles on both Sides of
it equal, as at x, x; then are thofe
Angles called Right Angles; and
the Lines fo meeting are faid to be
Perpendicular to each other.

That is, AC, and C B, are Perpendicular to DC, as well as D, C is to either or both of them.

9. An OBTUSE ANGLE is that which is greater than a Right Angle. Such is the Angle included between the Lines AC and CB.

10. An ACUTE ANGLE is that

which is less than a Right Angle:

A

B

As the Angle included between the Lines C B and CD.

Thefe Two Angles are generally called OBLIQUE Angles.

Sect. 2. Of a Circle, &c.

Before a Circle and its Parts are defined, it will be convenient ta give a brief Account of Superficies in general.

I. A SUPERFICIES OF SURFACE is the Upper, or very Out-fide of any vifible Thing. But by Superficies in GEOMETRY, is meant only fo much of the Out-fide of any Thing as is inclofed within a Line or Lines, according to the Form or Figure of the Thing defigned; and it is produced or formed by the Motion of a Line, as a Line is defcribed by the Motion of a Point; thus:

Suppofe the Line A B were equally moved (upon the fame Plane) to C D; then will the Points at A and B defcribe the Two Lines AC and BD; and by fo doing they will form (and inclofe) the SUPERFI

A

D

Confequently the

CIES or Figure ABCD, being a Quantity of Two Dimenfions, viz: it hath Length and Breadth, but not Thickness. Bounds or Limits of a Superficies are Lines.

Note,

Note, The Superficies of any Figure, is ufually called its Area.

2. A CIRCLE is a plain regular Figure, whofe Area is bounded or limited by one continued Line, called the CIRCUMFERENCE or PERIPHERY of the Circle, which may be thus defcribed or drawn.

Suppofe a Right Line, as C B, to have one of its Extream Points, as C, fo fix'd upon any Plane, as that the other Point at B may move about it; then if the Point at B be moved round about (upon the fame Plane) it will deferibe a Line equally diftant in all its Parts from the Point C, which will be the Circumference or Periphery of that Circle; the Point C, will be its CENTER, and the contained Space

will be its Area, and the Right Line CB, by which the Circle is thus defcribed, is called RADIUS. Confectary.

From hence 'tis evident, that an infinite Number of Right Lines may be drawn from the Center of any Circle to touch its Periphery, which will be all equal to one another, because they are all Radius's.

And with a little Confideration it will be easy to conceive, that no more than two equal Right Lines can be drawn from any Point within a Circle to touch its Periphery, but from the Center only. (9. e. 3.)

3. EQUAL CIRCLES are those which have equal Radius's; for it's plain by the laft Definition, that one and the fame Radius (as CB) muft needs defcribe equal Circles, how many foever they

are.

4. The Diameter of a Circle, is twice its Radius joined into one Right Line; as AB drawn through the Center C, and ending at the Periphery on each Side. That is, the Diameter divides the Circle into Two equal Parts.

D

B

A

5. A Semicircle (viz. Half a Circle) is a Figure included between the Diameter, and Half the Periphery cut off by the Diameter; as AD B.

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 Measure of a Right Angle.

A

G

M

B

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 S 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.

A

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.

C

D

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 Circle.

Sect. 3. Of TRIANGLES.

There are two Kinds of Triangles, viz. Plain and Spherical; but I shall 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.