Right Angles, at the Center (Def. 11.) the Arks AD, DB, being each a fourth part of the whole Circumference, or half the Semi-circumference: hence, a Right Angle is faid to be of 90 Degrees. 3. If the Ark AD be bifećted in E, and EC be drawn, the Angles ACE, ECD, will be each of 45 deg. half ACD a Right Angle, or 90 Degrees. And, if the Ark DB be trifected, at F & G, (i. e. divided into three equal parts), and FC be drawn, the Ark DF containing 30, and FGB 60 deg. the Angle DCF is faid to be an Angle of 30, and FCB an Angle of 60 deg. By which means, an Angle of any quantity may be obtained, or measured. 4. On the fame Center, C, with any other Radius, as Ca, let the Arch aed fb be drawn, which is alfo a Semicircle. It is very obvious, that it is alfo divided into the fame number of parts, and in the fame proportion, as the Arch AEDFB; for it is bifected in d, and ad is again bisected in e, and db is alfo trifected at f and g; wherefore, AD, ad are each a fourth; ED, ed an eighth; BF, bf, a fixth; and FD, fd, a twelfth part of their respective Circles; and the Angles ACD, aCd; ECD, eCd, &c. are the fame in both. From all which, it is clear, that, Angles may be formed or measured by an Ark or Circle of any Radius. And alfo, that equal Arks of the fame, or of equal Circles, or an equal number of degrees in a Circle of any Radius, will form equal Angles at the Center. If you would have an Angle of 60 degrees at the point C, of the line BC. With With any Radius, at pleasure, defcribe the Ark BD, cutting the Line BC in B; with the fame Radius, on the Center B, cut the Ark BD at F, and draw CF. It is very clear, that the Angle BCF would be the fame, if a lefs Radius had been taken, as C b. For, draw the Chord Lines F B and fb, each will be equal to the Radius of its refpective Circle; and, the Triangles CFB, Cfb are equilateral; whofe Angles are of 60 degrees each (Cor. 1. 9. 1.); for the Arks BF and bf are, each a fixth part of the whole Circumference of their respective Circles, of which C is the common Center, (fee Prop. 11. 4.); and confequently, each contains 60 degrees on the circumference of that Circle, of which it is a Part; whofe Radius is CB or C b. A defcription of the Inftrument called a PROTRACtor, with the application of it, in measuring and making Angles, of any known quantity or measure, may not be improper in this place. It is of fpecial ufe in Surveying, in drawing Plans of any piece of Ground for building on, &c. or of Buildings, already erected, being readier and more exact than a Line of Chords. The Protractor is a Semicircle, divided on its Limb or Semi-circumference, AEDFB, into 180 equal parts; having a fmall Notch at C, the Center. See the laft Figure. Some Protractors have a Scale, added to the Semicircle, which are the beft and readieft in ufe. In measuring an Angle, apply the Diameter, i. e. the Edge or Right Line AB, to either Side of the Angle ACE, with the Vertex, C, of the Angle, at the center of the Protractor; and, where-ever the Side CE, cuts the Limb or circular edge of the Inftrument, obferve how many De grees there are from A to E, the Ark intercepted between the Sides AC and CE, of the Angle ACE. If If it contains 45 or 50, or whatever number of Degrees it happens to be, (as 45 by the Figure) the Angle ACE is of fo many Degrees. If it had cut the Arch at D, as ACD, it is a Right Angle; and if beyond D, as ACF, it is obtufe ; the Complement FCB, i. e. the Ark FB, being fubtracted from the Arch of the Semicircle, ADB, or 180 degrees, gives the quantity of the Angle ACF. 2. If it is required to lie down or make an Angle of some known Quantity (as 45 deg.) at the Point C, of the Line AB. Apply the edge of the Protractor, as above, with the Center, C, at the Point given; make a Mark or Point at E; take away the Inftrument, and draw E C. Thus, may any Angle whatever be laid down on Paper. A Scale of equal Parts is nothing more than a Right Line divided into any number of equal Parts, at pleasure. Each Part may reprefent any measure you please, as an Inch, a Foot, a Yard, &c.; for, being equal, whatever measure any Object or Figure contains, in length and breadth, a fimilar Figure may be conftructed, on a Plane, having or containing the fame number of Divifions, each way, on the Scale, as the real Object contains of Feet, Yards, &c. One of thofe Parts is generally fubdivided into parts of the next inferior denomination, or into tenths and hundredths, denoting the Decimal Parts. Obferve, that the Divifions on the Scale, (whatever measure is reprefented by them,) muft always be adapted to the Proportion you would delineate any Object, or form a Defign. See the Appendix (Page 15 and 16) for the conftruction of Scales. N. B. A pair of Compaffes or Dividers, a Drawing Pen, and a ftreight Ruler, are all the Utenfils that are requifite, in Plane Geometry. The Board or Paper, on which we draw any geometrical Figure, is fuppofed to be a Plane. ABBREVIATIONS &c. EXPLAINED. When, what is fup And all the reft. posed to follow may be readily understood. Id eft. That is. When, what has been faid requires to be further explained. &c. Et cætera. i. e. viz. e. g. Videlicet. To wit; or, that is to say. When any thing advanced is given in Grofs, which is more particularly specified, as follows after. Exempli gratia. For instance; or, for the fake of example. When an Example is to be given of what is advanced. N. B. Nota Bene. Mark well. That is, take particular notice of that Paragraph. QE. F. Quod erat faciendum. Which was required to be done. Diag. Diagonal. Par. AB. or P.AB. Rect. AB. }ParallelogramAB. Rectangle AB, or ABCD, &c. R. Ang. ABC. Right Angle ABC, &c. Trap. Trapezium. Pent. Pentagon, &c. Note. When it is required to join any two Points, it is meant, that a Right Line be drawn between them, i. e. from one Point to the other. AB ABBREVIATIONS, by way of Reference. Poft. 1. or 2. Refers to the firft or second Poftulate; where it is requested that fuch or fuch things may Def. 6. or 7. Or, Def.6. 3. or 6. 5. &c. Ax. 3. be granted. Refers to the fixth or feventh Definition in the general Introduction. But, when there are two numbers, it refers to the 6th Definitions of the 3d or 5th Book. Refers to the third Axiom for Demonftration, arifing from felf-evident properties of things. 2. Th. P. A. Refers to the fecond Article in the Theory of Plane Angles, for illuftration or Proof. Pr. 1. or 2. &c. Refers to the firft or fecond Problem, for the conftructing of fome Figure, &c. P. 1. or 2. &c. Refers to the first or fecond Proposition of that Book, for proof of the Affertion. To the fecond Propofition of the third Book. To the 20 Corollary, of the 4th Propofition of the first Book. P. 2. 3. &c. C. 2. 4. I. Hyp. Sup. Con. That the thing is fo by the Hypothefis, or is given in the Premifes. That it is fo by Suppofition, only. That it is fo by Construction; i. e. the thing was formed or made fo. Conf. confequently. Th. therefore. Wh. wherefore, it is fo. Note. When there is but one Number within Parenthesis or otherwife, as (15.) &c. it refers to the fifteenth Problem or Theorem of that fame Book. But if there be two Numbers, as (10. 1.) or (12. 3.) &c. it refers to the 10th Prop. of the first Book, or to the 12th of the third, &c. INSTRUC |