4

Fig. 9.

Fig. 10.

Fig. 10.

Fig. 11, 12.

Fig. 13.

Fig. 13.

Fig. 12.

Fig. 10, 11.

Fig. 14, 15.

Fig. 15.

Fig. 16.

Fig. 17.

20. The Diameter is a right Line (B A) drawn thro' the Center, and on both Sides terminated at the Circumference; and confequently it divides the Circle into two equal Parts, (as is abundantly manifest from the exa&t Agreement of two Semicircles when laid one upon another.) 21. The Semi-diameter or Radius is the right Line AF drawn from the Center to the Circumference.

22. A Semi circle is a Figure (BLC) which is contain'd by the Diameter BC, and half the Circumference (BLC.)

Mathematicians are wont to divide the Circumference into 360 equal Parts (which they call Degrees) the Semi circumference into 180, the Quadrant or Quarter in

to 90.

23. A Right-lin'd Figure is a plain Surface bounded on every Side with right Lines.

24. A Triangle is a plain Surface contained by Three right Lines,

This is the first and most fimple of all Right-lin’d Figures, and that into which they are all refolv'd.

25. An Equilateral Triangle is that which hath all the Sides equal.

26. An Ifofceles or equicrural Triangle is that which hath only two Sides equal.

27. A Scalenum is that which hath Three unequal Sides.

28. A right-angled Triangle is that which hath one Angle right.

29. An obtufe-angled Triangle is that which hath one obtufe Angle.

30. An acute-angled Triangle is that which hath three acute Angles.

31. Amongst quadrilateral Figures, the Rectangle is that which hath Four right, and confequently equal Angles; whether the Sides be equal or not.

32. A Square is that which hath equal Sides, and is Right-angled, and confequently Equi-angled.

Every Square is a Rectangle; but every Rectangle is not a Square.

33. A Rhombus is a quadrilateral or four-fided Figure, which is equilateral, but not equiangled.

34. A Rhomboides is that which hath the oppofite Sides and Angles equal, but is neither Equilateral, nor Equiangled.

16, 17.

35. A Parallelogram is a quadrilateral Figure, which Fig. 14, 15 hath each Two of its oppofite Sides (A B, FC, and BF, AC) parallel to each other. Now what parallel Lines are, will be fhewed in the following Definition.

Every Rectangle and Square is a Parallelogram; but every Parallelogram is not a Rectangle or a Square.

36. Right Lines are Parallel or Equi-diftant, which Fig. 18. being in the fame Plane, and drawn out on both Sides infinitely, are diftant from one another by equal Intervals.

The Intervals are faid to be equal, in refpect of the Perpendiculars. Wherefore if all the Perpendiculars (QL) unto one of the two Parallels (A B) fhall be equal, the right Lines (AB, CF) are faid to be Parallel.

Parallels are produc'd, if the right Line (LQ) which is perpendicular to the right Line (A B) be moved along (A B) always perpendicularly; for then its Extremity L defcribes the Parallel CF.

37. The Diameter or Diagonal of a Parallelogram, Fig. 17. and every Quadrilateral, is a right Line (AF) drawn thro' the oppofite Angles.

38. Plain Figures contain'd by more Sides than Four, are called Many-fided or Many-angled, and by a Greek, Word Polygones.

39. The external Angle of a right-lin'd Figure, is Fig. 19. that which arifeth without the Figure when the Side is produc'd. Such are FBC, GCA, HAB. Every Figure therefore hath fo many external Angles as it hathSides, and internal Angles.

Poftulates.

A
Poftulate is that which is manifeft in it felf, that it'
may eafily be done, or conceiv'd to be done. It is
required therefore to be granted that we may,

1. From any Point given draw a right Line unto any other Point given.

2. Draw forth a finite right Line in Length ftill farther.

3. From any Center at any Interval describe a Circle.

Fig. 21.

Axioms.

N Axiom is a Truth manifeft of it felf.

AN

1. Thofe things which are equal to the fame thing, are equal alfo amongst themfelves. And that which is greater or leffer than one of the Equals, is alfo greater or lefs than the other of them.

2. If to Equals you add Equals, the Wholes will be equal.

3. If from Equals you take away Equals, the Remainders will be equal.

4. If to Unequals you add Equals, the Wholes will be unequal.

5. If from Unequals you take away Equals, the Remainders will be unequal.

6. What things are each of them half of the fame Quantity, are equal amongst themfelves; and what things are double, or treble, or quadruple of the fame, are equal amongst themselves.

7. What things do mutually agree with one another, are equal.

8. If right Lines be equal, they will mutually agree with one another; and the fame thing is true of Angles.

9. The whole is greater than its part.

10. All right Angles are equal amongst themselves. 11. Parallel Lines have a common Perpendicular: That is, the right Line which is perpendicular to one of them, is perpendicular alfo to the other.

12. The two perpendicular Lines (LO, QI) intercept equal Parts of the Parallels.

13. Two right Lines do not comprehend a Space; for unto this there are required three at the leaft.

14. Two right Lines cannot have one common Seg ment; for that they cut one another only in a Point.

Of Propofitions fome propofe fomething to be done, and are called Problems; in others we proceed no further than bare Contemplation, which therefore are na

med Theorems.

PRO

PROPOSITIONS.

THE requifite Citations are found in the Margin. When Propofitions are cited, the firft Number defigns the Propofition; the Letter 1. with the Number following, fignifies the Book. As when you meet with (per 5.1. 3.) you must read it thus, (by the 5th Propofifition of the 3d Book.) The Figure is always to be fought amongst the Figures of that Book in which we are then converfant. The reft of the Citations are easy to be understood.

The primary Affections of Triangles and Parallelograms are deliver'd in this Book. The more famous Propofitions are, 32, 35, 37, 41, 44, 45, 47.

u

PROPOSITION I. Problem.

Pon a given Right Line (A B) to make an Equi-
lateral Triangle.

Fig. 23.

From the Centre A, with the Interval (AB) (a) de-(a) Per Pofcribe the Circle FCB: and from the Centre B with the jul. 3. fame Interval B A defcribe the Circle ACL, cutting the former in the Point C, from which Point draw the right Lines CA, CB.

I fay, that the Triangle A CB now made, is Equilate

ral. For the right Line AC is equal to the right (b)(b) Per Line A B, feeing they are Semi-diameters of the fame Def. 18. Circle FCB. And again, the right Line B C is equal to the fame right Line BA, feeing they are both Semidiameters of the Circle LCA. Therefore A C, BC are (c) equal betwixt themselves. And therefore all the (c) Per Sides of the Triangle are equal. Therefore the Triangle Axiom 1. (d) ACB is both Equilateral, and made upon the (d) Per given Line A B; which was the thing to be done. Def. 25. 2EF.

Corollary. Hence we may measure an inacceffible Fig. 77. Line, as AB. For fuppofe any Equilateral Triangle whatfoever BDE applied to the Point B along the Line B A. Looking from the Point B along the Line B E, mark as many Points as you conveniently can in

B 4

the

Fig. 24.

Fig. 24.

Fig. 25.

the Line BC. Then remove the Triangle BDE along the Line BC, from one place to another of that Line, until by taking aim along the fide of the Triangle ED or CF, you fee the inacceffible Point A in a Continuation of that Line. Thus the Triangle B AC is as well Equilateral as BDE. If therefore you shall now meafure the acceffible Line BC, you have the Measure of the inacceffible AB. QE. F.

PRO P. II. Problem.

Rom a given Point A to draw a right Line equal to one given EF,

FR

Take with a Pair of Compaffes the Interval E F, and transfer it from A to D, the right Line AD will be equal to the given E F.

T

PRO P. III. Problem.

WO unequal right Lines being given, from the greater of them GH to cut off GI equal to the lefs EF.

Take with a Pair of Compaffes the Interval of the leffer given Line EF, and transfer it unto the greater from G to I.

Ive

PROP. IV. Theorem.

F in two Triangles (X, Z) one fide of the one (BA) be equal to one fide FL of the other, and another Jide (CA) of the one equal to another fide (IL) of the other, and the Angles (A and L) made by thofe fides be alfo equal; then the Bafes (BC, FI) are likewife equal, as alfo the Angles at the Bafes (B, F, and C, I) which are oppofite to equal fides, and confequently the whole Triangles are equal..

For if we fuppofe the Triangle Z to be laid upon the Triangle X, the Sides LF, LI will perfectly agree and

fall