B I. By its SIDES. 2. An EQUILATERAL TRIANGLE, is that which hath all its Three Sides equal; as the Figure ABC That is, AB=BC=AC. D 3. An Isosceles TRIANGLE, is that which bath only Two of its Sides equal, as the Figure BDG: That is, BD=DG; but the Third Side B G may be either greater or less, as Ocs casion requires. B H 4. A SCALENE TRIANGLE, K is that which hath all its Three Sides unequal; such as the Figure HKM, M 2. By its ANGLES. 5. A RIGHT-ANGLED Triangle, is that which hath one Right Angle that is, when Two of its Sides are Perpendicular to each other, as CA is supposed to be to B A. Therefore the Angle at A, is a Right Angle, per B A Defin. 8. Sc&t. I. Note, The longest Side of every Right-angled Triangle (as BC) is called the Hypotenuse, and the longest of the other Two Sides which include the Right Angle (as B A) is called the Bass : The Third Side (as C A) is called the Cathetus or Perpendicular, 6. An OBTUSE-ANGLED Triangle, is that which hath one of its Angles Obtuse, and it's called an Amblygonium Triangle. Such is the Third Triangle H KM. 7. An Acute-ANGLED TRIANGLE, is that which hath all its Angles Acute, and it's called an Oxygonium Triangle; such are the First and Second Triangles ABC, and BDG. Note, All Triangles that have not a Right Angle, whether they are Acute, or Obtuse, arc in general Terms, called Oblique Trian glesa gles, without any other Distinction, as before. And the longest Side of every Oblique Triangle is rufually called the Bale; the other two are only called Sides or Legs. 8. The Altitudc or Weight of any Plain Triangle, is the Length of a Right Line let fall perpendicular from any of its Angles, upon the Side opposite to that Angle from whence it falls; and may be either within, or without the Triangle, as Occasion requires, being denoted by the Two prick'd Lines, in the annexed Triangles. Sezt. 4. Of Four-fiocd figures. 1. A Square is a plain regular Figure, Ai В. whose Area is limited by Four equal Sides all perpendicular one to another. That is, when AB=BC-CD=DA, and the Angles A, B, C, D are all equal, then it's usually called a Geometrical Square. D C с 2. A Rhombus, or Diamond-like Figure, is that which hath Four equal Sides, but no Right-angle. That is, a Rhombus is a Square mov'd out of iis right Position, as the annexed Figure. 3. A Redangle, or a Right-angled Parallelogram (often called an Oblong, or long Square) is a Fi. B C gure that hath four Right-angles and its two opposite Sides equal, viz. BC=HD and BH=CD. 'n 4. A Rhomboides, is an Oblique-angled Parallelogram ; that is, it is a Parallelogram moved out of its right Position, like the annexed Figure. 5. The alricude or Height of any Oblique-angled Parallelogram, viz, either of the Rhombus or Rhombrides, is a Right-line let fall perpendicular from any Angle upon the side opposite to that Angle; and may either be within or without the Figure : As the prick'd Lines in the annexed Figure. PP 6. Every D 6. Every Four-fided Figure, diffetent from those before-mentioned, is called a rapezium. That is, when it has neither opposite Sides, nor opposite Angles equal; as the Figure ABCD. B A А C 7. A Right-line, drawn from any Angle in a Four-sided Figure to its opposite Angle, is called a Diaganal Line, and will divide the Area of the Figure into two Triangles, being denoted by the pricka Line AC in the last Figure. 8. All Right-lin'd Figures, that have more than four Sides, are call’d Polygons, whether they be regular or irregular. 9. A Kegular Polygon is that which hath all its Sides equal, standing at equal Angles, and is named according to the Number of its Sides (or Angles.) That is, if it have five equal Sides, it is called a Pentagoit; if fix equal Sides, it is calld a Heragon; if feven, 'tis a bheptagon ; if eight, 'tis an Ddagon, &c. Note, All Regular Polygons may be infcrib'd in a Circle; that is, their Angular Points, how many soever they have, will all just touch the Circle's Periphery. 10. An Irregular Polygon is that Figure which hath many unequal Sides standing at unequal Angles (like unto the annexed Figure, or otherwise); and of such Kind of Polygons there are infinite Varieties, but they may all be reduced to regular Figures by drawing Diagonal Lines in them; as shall be Thew'd farther on. These are the most general and useful Definitions that concern plain or superficial Geometry. As for those which relate to Solids, I thought it convenient te omit given any Account of them in this place, because they would rather puzzle and amuse the Learner, than improve him, until he has gain'd a competent Knowledge in the most useful Theorems concerning Superficies; for then those Definitions may be more easily understood, and will help them to form a clearer Idea of their respective Solids, than 'cis possible to conceive of them before; and therefore I have reserv'd those Definitions until we come to the Fifth Part. Seel. Sen. 5. Of such Terms as are generally used in Geometry. Whatsoever is proposed in Ġcometry will either be a Problem or a beorem. Both which Euclid includes in the general Term of Proposition. A Problem is that which proposes something to be done, and to lates more immediately to practical than speculative Geometry; That is, it's generally of such a Nature, as to be perforın'd by some known or Commonly receiv'd Rules, without any Regard had to their Inventions or Demonsirations. A Lheorem is when any commonly-receiv'd Rule, or ony Neris flood the highest Degree of Proof that human Reason is capable of attaining to, by a Train of li. guments deduced or drawn from such plain Axions, and other Self-evident Truths, as cannot be denied by any one that considers them. A Corollary, or Conledary, is fome Consequent Truth drawn or gain'd from any Demonstration. A Lemma is the Demonstration of fome Premises laid down or proposed as preparative to obviate and shorten the Proof of the Thes orem under Contideration. A Scholium is a brief Commentary or Obfervation made upon some precedent Discourse. N. B. I advise the young Geometer to be very perfect in the Definitions, viz. Not to run Jatisfied with a bare Remembrance of them; but, that he endeavour to gain a clear Idea or Understanding of the Things defined; and for that Reason I have been fuller in every Definicion than is usual. 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 Infance; at Problem 1. there is ( 3. e; 1.) which shews that it is the Third Proposition in Euclid's First Book. The like must be underfood in the Theorems, Рp2. CHAP CH A P. II. НА Probleins, in Plain Geometry. ought to be provided with a thin streight Ruler, made either of Brass or Box-wood, and two pair of very good compasses, viz.one Pair called Three-pointed Compafies, being very useful for drawing of Figures or Schemes, either with Black Lead or Ink; and one Pair of plain Compasses with very fine Points, to measure and set off Distances; also he should have a very good Steel Drawing.Pen: And then he may proceed to the Work with this Caution; that he ought to make himself Mafier of one Problem before he undertakes the next: That is, he ought to understand the Design, and, as far as be can, tbe Reason of every Problem, as well as how to do it; and then a little Practice will render them very easy, they being all grounded upon these following Poftulates. Počulates 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 longer from either of its Ends. 3. That upon any given Point (or Center) and with any given Distance (viz. with any Kadius) a Circle may be described. PROBLEM I. Difference. (3.6, 1.) |