Fig. 9. 20. The Diameter is a right Line (BA) drawn thro' the Center, and on both Sides terminated at the Circumference; and consequently it divides the Circle into two equal Parts, (as is abundantly manifest from the exact Agreement of two Semicircles when laid one upon another.) 21. The Semi-diameter or Radius is the right Line A F drawn from the Center to the Circumference. Fig. 10. Fig. 10. Fig. 11, 12. Fig. 13. Fig. 13. Fig. 12. 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 simple 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 Isofceles 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. Fig. 10, 11. 30. An acute-angled Triangle is that which hath three acute Angles. Fig. 14, 15. 31. Amongst quadrilateral Figures, the Rectangle is that which hath Four right, and confequently equal Angles; whether the Sides be equal or not. Fig. 15. 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. Fig. 16. 33. A Rhombus is a quadrilateral or four-fided Figure, which is equilateral, but not equiangled. Fig. 17. 34. A Rhomboides is that which hath the oppofite Sides and Angles equal, but is neither Equilateral, nor Equiangled. 35. A Parallelogram is a quadrilateral Figure, which Fig. 14, 152 hath each Two of its oppofite Sides (AB, FC, and BF, 16, 17. AC) parallel to each other. Now what parallel Lines are, will be shewed 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-distant, 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 respect of the Perpendiculars. Wherefore if all the Perpendiculars (QL) unto one of the two Parallels (AB) shall be equal, the right Lines (AB, CF) are said to be Parallel. Parallels are produc'd, if the right Line (LQ) which is perpendicular to the right Line (AB) be moved along (AB) always perpendicularly; for then its Extremity L describes 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 so many external Angles as it hath Sides, and internal Angles. Postulates. A Poftulate is that which is manifest in it felf, that it may easily 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 still farther. 3. From any Center at any Interval describe a Circle. : Fig. 21. A Axioms. N Axiom is a Truth manifest of it felf. 1. Those things which are equal to the same thing, are equal also amongst themselves. And that which is greater or leffer than one of the Equals, is also greater or less 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 themselves; 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 same 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 also 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 some propose something to be done, and are called Problems; in others we proceed no further than bare Contemplation, which therefore are named Theorems. : し PRO PROPOSITIONS. THE requifite Citations are found in the Margin. The primary Affections of Triangles and Parallelo- PROPOSITION I. Problem. Pon a given Right Line (AB) to make an Equi U lateral Triangle 7 Fig. 23. From the Centre A, with the Interval (AB) (a) de-(a) Per Poscribe the Circle FCB: and from the Centre B with the stul. 3. same Interval B A describe the Circle ACL, cutting the former in the Point C, from which Point draw the right Lines CA, СВ. I fay, that the Triangle ACB now made, is Equilateral. For the right Line AC is equal to the right (b) (b) Per Line A B, feeing they are Semi-diameters of the same Def. 18. Circle FCB. And again, the right Line BC is equal to the fame right Line BA, seeing they are both Semidiameters of the Circle LCA. Therefore AC, BC are (c) equal betwixt themselves. Sides of the Triangle are equal. (d) ACB is both Equilateral, and made upon the (d) Per given Line AB; which was the thing to be done. Def. 25. And therefore all the (c) Per REF. : Corollary. Hence we may measure an inaccessible Fig. 77. the 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 side of the Triangle ED or CF, you see the inaccessible 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 accessible Line BC, you have the Measure of the inaccessible A B. Q. E. F. Fig. 24. Fig. 25. PROP. II. Problem. Rom a given Point A to draw a right Line Take with a Pair of Compasses the Interval E F, and transfer it from A to D, the right Line AD will be equal to the given E F. PROP. III. Problem. WO unequal right Lines being given, from the greater of them GH to cut off GI equal to the less EF. Take with a Pair of Compaffes the Interval of the lesser given Line EF, and transfer it unto the greater from G to I. I 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 those fides be also equal, then the Bases (BC, FI) are likewise equal, as also the Angles at the Bases (B, F, and C, I) which are opposite to equal fides, and consequently the whole Triangles are equal. For if we suppose the Triangle Z to be laid upon the Triangle X, the Sides LF, LI will perfectly agree and fall |