Plate I.

41. Acute and obtufe angled Triangles are in general called oblique angled Triangles, in all which any Side may be called the Bafe, and the other two the Sides.

42. The perpendicular Height of a Triangle is a Line drawn from the Vertex to the Base, perpendicularly: Thus if the Triangle ABC be propofed, and BC be made its Bafe, then if from the Vertex A the Perpendicular AD be drawn to BC, the Line AD will be the Height of the Triangle ABC, standing on BC as its Bafe. Fig. 16.

Hence all Triangles between the fame Parallels have the fame Height, fince all the Perpendiculars are equal from the Nature of Parallels,

43. Any Figure of four Sides is called a quadrilateral Figure.

44. Quadrilateral Figures whofe oppofite Sides áre parallel, are called Parallelograms: Thus ABCD is a Parallelogram. Fig. 3. 17. and A. B. Fig. 18. and 19.

45. A Parallelogram whofe Sides are all equal and Angles right, is called a Square, as ABCD. Fig. 17.

46. A Parallelogram whofe oppofite Sides are equal and Angles right, is called a Rectangle or an Oblong, as ABCD. Fig. 3.

47. A Rhombus is a Parallelogram of equal Sides, and has its Angles oblique, as A. Fig. 18. and is an inclined Square.

48. A

Plate I.

48. A Rhomboides is a Parallelogram whofe oppofite Sides are equal and Angles oblique ; as B. Fig. 19. and may be conceived as an inclined Rectangle.

49. Any quadrilateral Figure that is not a Parallelogram is called a Trapezium. Plate 7. Fig. 3.

50. Figures which confift of more than four Sides are called Polygons; if the Sides are all equal to each other they are called regular Polygons. They fometimes are named from the Number of their Sides, as a five-fided Figure is called a Pentagon, one of fix Sides a Hexagon, &c. but if their Sides are not equal to each other, then they are called irregular Polygons, as an irregular Pentagon, Hexagon, &c.

51. Four Quantities are faid to be in Proportion when the Product of the Extremes is equal to that of the Means: Thus if A multiplied by D, be equal to B multiplied by C, then A is faid to be to B as C is to D.

POSTULATES or PETITIONS.

1. That a right Line may be drawn from any one given Point to another.

2. That a right Line may be produced or continued at Pleasure.

3. That from any Center and with any Radius the Circumference of a Circle may be described.

4. It is alfo required, that the Equality of Lines and Angles to others given, be granted as poffible:

That

e;

That it is poffible for one right Line to be perpendicular to another, at a given Point or Diftance and that every Magnitude has its half, third, fourth, &c. Part.

Note, Though these Poftulates are not always quoted, the Reader will eafily perceive where and in what Sense they are to be understood.

AXIOMS or Self-evident TRUTHS.

1. Things that are equal to one and the fame Thing, are equal to each other.

"

2. Every Whole is greater than its Part.

3. Every Whole is equal to all its Parts taken together.

4. If to equal Things, equal Things be added, the Wholes will be equal.

5. If from equal Things, equal Things be deducted, the Remainders will be equal.

6. If to or from unequal Things, equal Things be added or taken, the Sums or Remainders will be unequal.

7. All right Angles are equal to one another.

8. If two right Lines not parallel, be produced towards their nearest Distance, they will interfect each other.

9. Things which mutually agree with each other are equal.

NOTES.

A Theorem is a Propofition, wherein fomething is proposed to be demonstrated.

A Problem is a Propofition, wherein fomething is to be done, or effected.

A Lemma is fome Demonftration, previous and neceffary, to render what follows the more eafy.

A Corollary is a confequent Truth, deduced from a foregoing Demonftration.

A Scholium, is when Remarks or Obfervations are made upon fomething going before.

The Signification of SIGNS.

The Sign, denotes the Quantities between which it ftands to be equal.

The Sign+, denotes the Quantity it precedes to be added.

The Sign, denotes that Quantity which it precedes to be substracted.

The Sign X, denotes the Quantities, between them to be multiplied into each other.

To denote that four Quantities, A, B, C, D, are proportional, they are ufually wrote thus, A: B:: C: D; and read thus A is to B, fo is C to D, but when three Quantities A, B, C, are proportional the middle Quantity is repeated, and they are wrote A: B:: B: C.

I

Plate I.

THEOREM I

F a right Line falls on another, as AB, or EB, does on CD, (Fig. 20.) it either makes with it two right Angles, or two Angles equal to two right Angles.

1. If AB be perpendicular to CD, then (by Def. 11.) the Angles CBA, and ABD, will be each a right Angle.

2. But if EB fall flantwife on CD, then are the Angles DBE+EBC= DBE+EBA (=DBA) + ABC, or to two right Angles. Q. E. D.

Corollary I. Whence if any Number of right Lines were drawn from one Point, on the fame Side of a right Line; all the Angles made by these Lines will be equal to two right Angles.

2. And all the Angles which can be made about a Point, will be equal to four right Angles. TH E 0