Sect. 5. Of fuch Terms as are generally used in Geometry. Whatsoever is propofed in Geometry will either be a Problem or a Theozem. Both which Euclid includes in the general Term of Propofition. A Problem is that which proposes something to be done, and relates more immediately to practical than fpeculative Geometry; That is, it's generally of fuch a Nature, as to be perform'd by fome known or Commonly-receiv'd Rules, without any Regard had to their Inventions or Demonftrations. A Theorem is when any Commonly-receiv'd Rule, or any New Propofition is required to be demonftrated, that fo it may from thenceforward become a certain Rule, to be rely'd upon in Practice when Occafion requires it. And therefore feveral Rules are often call'd Theorems, by which Operations in Arithmetick, and Conclufions in Geometry, are perform'd. Note, By Deinonftration is underflood the highest Degree of Proof that human Reafon is capable of attaining to, by a Train of Arguments deduced or drawn from fuch plain Axioms, and other Selfevident Truths, as cannot be denied by any one that confiders them. A Collary, or Confectary, is fome Confequent Truth drawn or gain'd from any Demonftration. A Lemita is the Demonftration of fome Premises laid down or proposed as preparative to obviate and shorten the Proof of the Theorem under Confideration. A Scholium is a brief Commentary or Obfervation made upon fome precedent Difcourfe. N. B. I advise the young Geometer to be very perfect in the Definitions, viz. Not to rest fatisfied with a bare Remembrance of them; but, that he endeavour to gain a clear Idea or Underftanding of the Things defined; and for that Reafon I have been fuller in every Definition than is ufual. And, that he may know from whence most of the following Problems and Theorems contain'd in the two next Chapters are collected, I have all along cited the Propofition, and Book of Euclid's Elements where they may be found. As for Inftance; at Problem 1. there is (3. e. 1.) which shews that it is the Third Propofition in Euclid's First Book. The like must be understood in the Theorems. CHA P. II. The First Rudiments, or Leading and Preparatory Problems, in Plane Geometry, IN order to perform the following Problems, the young Geometer ought to be provided with a thin freight Ruler, made either of Brafs or Box wood, and two Pair of very good Compaffes, viz. one Pair called Three-pointed Compaffes, being very useful for drawing of Figures or Schemes, either with Black Lead or Ink; and one Pair of common Compafles with very fine Points, to meafure and fet off Diftances; alfo he foould have a very good Steel Drawing Pen: And then he may proceed to the Work with this Caution; that he ought to make himself Master of one Problem before he undertakes the next : That is, he ought to underfland the Defign, and, as far as he can, the Reafon of every Problem, as well as how to do it; and then a little Practice will render them very eafy, they being all grounded upon these following Poftulates. Poftulates 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, encreased, or made langer from either of its Ends. 3. That upon any given Point (or Center) and with any given Diflance (viz. with any Radius) a Circle may be described. PROBLEM I. Two Right lines being given, to find their Sum and Difference. (3. e. 1.). To bifect, or divide a Right-line given (as AB) into two equal Parts. (10. e. 1.) From each End of the given Line (viz. A and B) with any Radius greater than half its Length, defcribe two Arches that may cross each other in two Points, as at D and F: then join thofe Points D, Fwith a Right-line, and it will bifect the Line A B in C; viz. it will make ACCB; as was required. .D. B To bifect or divide into two equal Angles a Right-lined Angle given. (9. e. 1.) Upon the Angular Point, as at C, with any convenient Radius, defcribe an Arch AB; and from thofe Points A and B, defcribe with the fame Radius two Arches croffing each other, as at D; then join the Points C and D with a Right-line, and it will bifect the Arch AB, and confequently the Angle; as was requir'd. PROBLEM IV. A D B At a Point A, in a Right-line given AB, to make a Right-lin'd Angle equal to a Right-lin'd Angle given C. (23. e. 1.) Upon the given Angular Point C defcribe an Arch, as FD, (making C D any Radius at Pleafure) and with the fame Radius defcribe the like Arch upon the given Point A, as fd; that is, make the Arch fd equal to the Arch FD; Then join the Points A and f with a Right-Line, and it will form the Angle requir'd. D PRO To draw a Right-line, as FD, parallel to a given Right-line AB, that shall pass thro' any affign'd Point, as at x, viz. at any Difance requir'd. (31. e. 1.) F M... Take any convenient Point in the given Line, as at C, (the farther off the better;) make Cx Radius, and with it upon the Point C, defcribe a Semicircle, as HMx N; then make the Arch HM equal to the A H C -D N Arch x N; thro' the Points M and x draw the Right-line FD, and it will be parallel to the Line A C, as was requir'd. To let fall a Perpendicular, as Cx, upon a given Right-line AB, 1 A L B PROBLEM VII. To erect or raife a Perpendicular upon the End of any given Right-Line, as at B; or upon any other Point affign'd in it. (II. e. 1.) C. Upon any Point C taken out of the given Line, as a Center, defcribe fuch a Circle as will pass through the Point from whence the Perpendicular must be raised as at B, (viz. make CB Radius): And from the Point where the Circle cuts the given Line, as at A, draw the Circle's Diameter ACD; then from the Point D draw the Rightline DB, and it will be the Perpendicular requir'd. PRO. PROBLEM VIII. To divide any given Right-line, as AB, into any propofed Number of equal Parts. (10. e. 6.) At the extream Points (or Ends) A and B of the given Line, make two equal Angles (by Prob. 4.) continuing their Sides AD and BC to any fufficient Length; then upon those Sides, beginning at the Points A and B, fet off the propofed Number of equal Parts (Juppofe 'em 5.) If Right-lines be drawn (cross the given Line) from one Point to the other, as in the 3 2 B J 2 3 D annexed Figure, thofe Lines will divide the given Line A B into the Number of equal Parts required. PROBLEM IX. To deferibe a Circle that Points given, not lying hall pass (or cut) thro' any Three in a Right-line, as at the Points A, B, D. Join the Points AB and BD with Right-lines; then bisect both those Lines (per Problem 2.) the Point where the bifecting Lines meet, as at C, will be the Center of the Circle required. The Work of this Problem being well understood, 'twill be eafy to perform the two following, without any Scheme, viz. B where 1. To find the Center of any Circle given. (1. e. 3.) By the last Problem 'tis plain, that if three Points be any taken in the given Circle's Periphery, as at A, B, D, the Center of that Circle may be found as before, 2. A Segment of any Circle being given, to compleat or defcribe the whole Circle. This may be done by taking any three Points in the given Seg ment's Arch, and then proceed as before. P ́R O |