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DEFINITIONS.

DEF. 1. A POINT is, rather to be conceived than understood to be, without dimenfions, therefore indivifible.

In Plane Geometry, it is a Mark, fuppofed to be made with a fharp-pointed Needle, or fine drawing Pen. As A.

DEF. 2. A LINE has length only, without other dimenfions, of breadth or thickness.

DEF. 3. A RIGHT LINE is the fhorteft that can be drawn between two given Points, (A and B) ufually called a freight Line.

DEF. 4. A CURVE, or CURVED LINE, is any other than a Right Line; either regular, as ACB; or irregular, as ADB.

Curve Lines are of various kinds, as circular, ellip tic, parabolic, &c. each of which has particular properties peculiar to them, and fome in common.

B DEF. 5. A SURFACE, or SUPERFICIES, is the outfide or external parts of Bodies, having no regard to thickness or fubftance; and therefore, has but two dimenfions, viz. length and breadth. As A B and CD.

Surfaces are various, as Convex, Concave, and Plane.

Irregular Surfaces are fuch as are not, uniformly, any of the three, but may be compounded of them all.

A convex Surface is one that is externally round or protuberant. Alfo, the cutlide of the curve of a Circle, &c. is convex.

A concave Surface is one that is round internally, or hollow, fuch as the infide of a round Veffel; the outfde is convex. The curve of a Circle towards the Center is concave,

DEF.

A

D

DEF. 6. A PLANE is a perfectly even and regular furface, which is neither convex nor concave, in any part; to which, if a Right Line or freight Ruler be any how applied, it will touch in every Point.

Or, if any two Points, in a Plane, be joined by a Right Line, the whole line is in that Plane.

A Plane may be conceived to be generated by the direct motion of a Right Line, laterally; or whirled around on any Point in it.

If the Right Line A B be moved, directly to CD, there will be generated the Plane ABCD.

N. B. A Plane may be of any Shape or Figure, in refpect of its bounds or limits.

A

D

DEF. 7. PARALLEL. Right Lines are Parallel A to each other, which, if produced infinitely, either way, and being in the fame Plane, would never meet. As AB and CD.

N. B. Parallel Lines are equidiftant in every part; between which all Right Lines, as a a, bb, and cc, being alfo parallel, are equal.

But, the Distance between two parallel Lines is meafured by a Perpendicular, AD or BC. See Perpendicular (Def. 10.)

The fame holds true in parallel Planes.

DEF. 8. A PLANE-ANGLE is formed by the meeting, touching, or cutting, of two Lines in the fame Plane, fo as not to fall into or conftitute one Line.

Angles are either right-lined, as A or B; curved, as C, D and E; or mixed, as F and G; which are compounded of a Right Line and a curved or crooked one.

N.B. Angles are neither increased nor diminished by the length of the Lines which form them; for, the Angle A is greater than the Angle B, notwithstanding the Lines which form the Angle B are longer than thofe of the Angle A; but, the Lines forming the Angle B are more inclined to each other.

B

a b c B

Dabc

C

A

B

F

2. Ima

D

E

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2. Imagine the dotted Lines to represent the Angle A laid upon, or applied to the Angle B; it is evident, that the Angle B is lefs than the Angle A, having a greater inclination of its Sides, or the aperture between them lefs; therefore, a lefs Angle may always be contained in a greater.

3. If the Sides of any Angle (as B) were lengthened infinitely, the Angle is not varied, if the inclination of the Lines remains the fame; but, if you fuppofe them moveable, on the angular point, like a folding Rule, and are parted farther afunder, approaching towards the dotted Lines, their inclination will then be lefs; and confequently, the aperture or opening between them greater; and therefore, the Angle is faid to be greater.

DEF. 9. VERTEX, is the angular Point in which two Lines, forming a Plane Angle, meet and touch each other.

N. B. A fingle Angle is ufually defcribed by one Letter only, or other Character, as A or B ; but, if three or more Lines meet in a Point, then three Letters are ufed, to fpecify the Angle spoken of; as ABC, CBD or ABD; of which the middle Letter always denotes the Vertex; and means, the Angle made by the lines AB and BC, or CB and BD, &c. wherefore, the middle Letter, denoting the Angle, is underfood to be twice named.

DEF. 10. PERPENDICULAR. A Right Line is perpendicular to another, when it does not incline to the other on either Side; but makes the Angle, on each Side, equal to the other.

Thus; if the Angles ACD, DC B, are equal to one another, then CD is perpendicular to AB.

N. B. No more than one Perpendicular can be drawn from the fame Point in a Right Line, on the fame Side, and in the fame Piane.

For, if any other Line, as CE, be drawn from the fame Point, C, it cannot be a Perpendicular, but is faid to incline to AB; and, the inclination is on the fame Side of the Perpendicular CD; i. e. the Angle ACE is the Angle of Inclination of the two Lines AB and C E.

N. B. 2.

N. B. 2. If another Right Line, as FG, be drawn in the fame Plane, from any Point F, perpendicular to AB, it will be parallel to CD.

DEF. II. A RIGHT ANGLE is that which is
formed by the meeting of two Right Lines which A
do not incline to each other, but either of them
is perpendicular to the other.

N. B. If either Side of a Right Angle be produced, or drawn out beyond the Vertex, there is neceffarily generated another Right Angle. And, confequently, if both Sides are produced, there will be generated four Right Angles.

Thus, ABC is a Right Angle; if, when AB or CB is produced, towards D or E, there is made another, CBD or ABE; and if both are produced, EBD is a fourth Right Angle.

DEF. 12. An ACUTE ANGLE is lefs than a

Right Angle.

If the Line CB, meeting AB in the Point B, falls on this Side of a Perpendicular, BD, at that Point; the Angle ABC, being less than the Right Angle ABD, is therefore called Acute.

DEF. 13. An OBTUSE ANGLE is greater than a Right one.

If the Line BE falls on the other Side of the Perpendicular, BD; the Angle A B E is Obtufe.

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A

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N. B. The difference, CBD, between an Acute Angle, ABC, and a Right one, ABD, is called the A COMPLEMENT of the Angle ABC.

And, if either Side of an Obtufe Angle, as AB, be produced, the Angle EBF is the Complement of the Obtufe Angle; or its deficiency to two Right Angles, ABD, DBF.

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D

B F

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DEF. 14. EQUAL ANGLES. Angles are equal, when the Lines, which form them, have the fame Inclination each to the other, respectively.

In the Angles abc, ABC; if the Vertex, b, of the one, be applied to the Vertex, B, of the other, in fuch wife, that the Side ab falling on AB, cb alfo falls on CB; then, there is the fame inclination of cb to a b as of CB to AB, and the Angles abc, ABC are equal.

N. B. The length of the Lines, or Sides, is not confidered or regarded in the equality of the Angles, but only their inclination to each other.

Angles have other Denominations, which are derived to them only from their Situation, in respect of each other; yet ftill retain the general appellation of Right, Acute, or Obtufe Angles. Such are the following.

DEF. 15. VERTICAL and CONTIGUOUS
ANGLES.

If two Lines, AB and CD, cut and crofs each other, there are made four Angles, at the Point, E, of their mutual Interfection; either two of which, AED, CEB, or AEC and DEB, touching at their Vertices, only, are called Vertical Angles.

Any other two, as AEC, AED, or AEC and CEB, &c. having one Side, CE or AE, common to both Angles, are called Contiguous or Adjoining Angles.

DEF. 16. ALTERNATE ANGLES, and others.

If a Line croffes or interfects two Lines, there are made eight Angles, A, B, C, D, &c.; of which C and F, alfo E and D, between the two Lines, one on each Side of the cutting Line, are called Alternate Angles.

C and E, alfo D and F, are called INTERNAL ANGLES on the fame Side.

E and A, F and B, C and G, or D and H are called INTERNAL and OPPOSITE ANGLES, on the fame Side.

DEF.

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