AN INTRODUCTION TO THE Mathematicks. PART III, A С НА Р. І. Sect. 1. Of Lines and Angles. Quantity, denoted by a Point : } A. } B. Such a Place may be conceived so infinitely small, as to be void of Length, Breadth, and Thickness ; and therefore a Point may be said to have no Parts, 2. A Line is called a Quantity of one Dimension, because it may have any supposed Length, but no Breadth nor Thickness, being made or represented to the Eye, by the Motion of a Point. That is, if the Point at A, be moved ( upon the same Plane) to the Point at B, it will describe a Line either right or circular (viz. crooked) according to its Motion. Therefore the Ends or Limits of a Line are Points. 3. A Right Line, is that Line which liech even or freight betwixt those Points that limit its Length, being the shortest Line that can be drawn between any Two AB. Points. As the Line A B. } Therefore, between any two Points, there can lie or be drawn but one right Line, Oo2 4. A 4. A CIRCULAR, crooked or OBLIQUE Line, is that which lies bending between those Points which limit, its Length, as the Lines CD or FG, &c. Ď Of these kinds of Lines there are various Sorts; but those of the Circle, F Parabola, Ellipsis, and Hyperbola are of most general Use in Geometry; of which a particular Account shall be given further on. C с 6. Lines not PARALLEL, but INCLINING (viz. leaning) one towards another, whether they are B Right Lines, or Circular Lines, will 4 (if they are extended) meet, and make an Angle; the point where they meet is called the Angular Point, as at A. d And according as such Lines stand, nearer or further off each other, the Angle is said to be lesser or greater, whether the Lines that include the Angle be long or short. That is, the с Lines Ad, and Af include the fame Angle as A B, and AC doth; notwithstanding that A B is longer than Ad, &c. B 7. All ANGLEs included between Right Lines are called Rightlin'd Angles; and those included between Circular Lines are called Spherical Angles. But all Angles, whether Right-lin'd or Spherical, fall under one of these Three Denominations. 2 A Bight Angle. An Acute angle. That That is, when a Right Line, as D XX B Angles called Right Angles ; and с the Lines so meeting are said to be Perpendicular to each other. That is, AC, and C B, are Perpendicular to DC, as well as D,C is to either or both of them. 9. An Obtuse Angle is that which is greater than a Right Angle. Such is the Angle included between the Lines AC and B СВ. . D 10. An Acute Angle is that A which is lefs than a Right Angle: с As the Angle included between the Lines CB and CD. These Two Angles are generally called OBLIQUE Angles. Set. 2. Of a Circle, &c. Before a Circle and its Parts are defined, it will be convenient to give a brief Account of Superficies in general. 1. A SUPERFICIES or Surface is the Upper, or very Out-fide of any visible Thing. But by Superficies in GEOMETRY, is meant only so much of the Out-side of any Thing as is inclosed within a Line or Lines, according to the Form or Figure of the Thing designed; and it is produced or formed by the Motion of a Line, as a Line is described by the Motion of a Point ; thus : Suppose the Line A B were equally moved (upon the same Plane) to C B D; then will the Points at A and B describe the Two Lines AC and BD; and by so doing they will с D form (and inclose) the SUPERFI. CIES or Figure A B CD, being a Quantity of Two Dimensions, viz: it hath Length and Breadth, but not Thickness. Consequently the Bounds or Limits of a Superficies are Lines. Note, A Note, The Superficies of any Figure, is usually called its AREA. 2. A CIRCLE is a plain regular Figure, whose Arca is bounded or limited by one continued Line, called the CIRCUMFERENCE or PERIPHERY of the Circle, which may be thus described or drawn. Suppose a Right Line, as CB, to have one of its Extream Points, as C, fo fix'd upon any Plane, as that the other Point at B may move about B В it ; then if the Point at B be moved round aþout (upon the fame Plane) it will describe a Line equally difiant in all its Parts from the C Point C, which will be the Circumference or Periphery of that Circle; the Point C, will be its CENTER, and the contained Space will be its Area, and the Right Line CB, by which the Circle is thus described, is called RADIUS. Confectary. From hence 'tis evident, that an infinite Number of Right Lines may be drawn from the Center of any Circle to touch its Periphery, which will be all equal to one another, because they are all Radius's. And with a little Confideration it will be easy to conceive, that ng more than two equal Right Lines can be drawn from any Point within a Circle to touch its Periphery, but from the Center only. (9.4. 3.) 3. EQUAL CIRCLES are those which have equal Radius's; for it's plain by the last Definition, that one and the fame Radius (as CB) must needs describe equal Circles, how many foever they are. D 4. The Diameter of a Circle, is twice its Radius joined into one Right Line; as AB drawn through the Center C, and ending at the Periphery on each side. That is, the Diameter divides the Circle into Two equal Parts, A B 5. A Semicircle (viz. Half a Circle) is a Figure included between the Diameter, and Half the Periphery cut off by the Diameter; as ADB 6. A 6. A QUADRANT is Half a Semicircle, viz. one Quarter of a Circle; and tis made by the Radius (as DC) standing Perpendicular upon D the Diameter at the Center C, cutting the Periphery of the Semicircle in the Middle, as at D. Therefore a Quadrant, or half the Semicircle, is the Measure A B of a Right Angle. 7. A CHORD Line, or the Subtense G of an Arch, is any Right Line that cuts the Circle into Two unequal Parts, as the Line SG; and is always lefs than the Diameter. 8. A SEGMENT of a Circle, is a Figure included betwixt the Chord and that Arch of the Periphery which is cut off by the Chord: And it may either be greater or less than a Semicircle; as the Figure SDG, or S MG. 9. A Sector is a Figure included between Two Radius's of the Circle, and that Arch of its Periphery B where they touch, as the Figure ACB: And the Arch AB is the Measure of the Angle at C, included betwixt the Radius's AC, and B C. C D Note, All Angles of SeEtors are called Angles at the Center of a Circle. 10. An Angle in the Segment of a Circle is that which is included between Two Chords that flow from one and the fame Point in the Periphery, as at D, and meet with the Ends of another Chord Line, as at F and G. That is, the Angles at D, at F, and at G, are called Angles at the Periphery, or Angles standing on the Segment of a Circle. Sect. 3.' Of TRIANGLES. There are two kinds of Triangles, viz. Plain and Spherical; but I shall not give any Definition of the Spherical, because they more immediately relate to Astronomy. 1. A PLAIN TRIANGLE is a Figure whose Area is contained within the Limits of Three Right Lines called Sides, including Three Angles : And it may be divided, and takes its Name either according to its Sides or Angles, |