7. A plane surface, (planus, level, flat,) is that in which any two points being taken, the straight line joining them lies wholly in that surface. -HERON. “A plane surface is that which lies evenly, or equally, with the straight lines in it." EUCLID. “ A plane surface is one whose extremities hide all the intermediate parts, the eye being placed in its continuation.”—PLATO. A plane surface is measured by the number of square units of surface, contained within its boundaries ;-as, 4 square inches ; 9 square feet; 13 square yards, &c. 8. A plane angle (angulus, a corner), is the inclination of two lines to each other in a plane, which meet together in the same point, but are not in the same straight line. A plane angle is the opening of two lines from their point of meeting, or of intersection, that point being the vertex. 9. A plane rectilineal angle (rectus, straight, and linea, a thread), is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. The magnitude of the angle is altogether independent of the length of the lines; angle B is greater than A FDG, GDE. grees in the arc of a circle, of which the centre is at F G 10. When a straight line standing on another. straight line makes the adjacent angles equal to each other, each of these angles is called a right angle ;D and the straight line which stands on the other is called a perpendicular to it. The Angle CDA being equal to angle CDB, each of them is a right angle, and CD is perpendicular с arc of 90 degrees, i.e., to the fourth part of the circumference of a circle. 11. An obtuse angle (obtusus, blunt), is an angle greater than a right angle; as angle EDB. The measure of an obtuse angle is always greater than A D B 90 degrees. 12. An acute angle (acutus, sharp), is an angle less than a right angle; as angle FDB. The measure of an acute angle is always less than 90 degrees. A boundary is the limit within which anything is contained. 14. A figure (figura, shape, form), is a surface enclosed by one or more boundaries. “When all the points in a figure are also points in the same plane, the figure is called a plane figure. 15. A circle (circulus, a ring, or hoop), is a plane figure contained by one line, which is called the circumference (circumferre, to carry round), and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. Any portion of a circumference is named an arc, D H Α. B 16. And the point (from which the C equal lines are drawn) is called the centre F (kentron, a goad, a point), of the circle; as, C the centre of the circle ADBE, having the lines CA, CD, CB, &c., all equal. The straight lines from the centre to the circum ference are radii. 17. A diameter of a circle (diametros, a measure through), is any straight line drawn through the centre, and terminated both ways by the circumference; as AB, or DE, in the circle ADBE. A diameter is double of a radius, (radius, the spoke of a wheel). 18. 100 A semicircle (semi, half), is the figure contained by the diameter and the part of the circumference cut off by the diameter; as ADBCA. The Semicircle divided into 180 D 90 с B the angle. Or, if it be required to draw an angle of a fixed number of degrees the semicircle thus divided will enable us to do it; for draw a line, and determine where the angular point in it is to be, then apply to that point the centre C, and lay the edge CB, or CA, along the line already drawn; note by a mark the degrees required, and join that mark to the angular point C by a st. line, the two strait lines will form the angle required. The Semicircle may be employed with a plummet suspended from D, or 90 degrees, to ascertain both the horizontal and vertical lines. An instrument of this kind is made use of by builders and others, and for purposes of levelling, &c. In surveying, for the measurement of angles from any point to distant objects, several instruments have been constructed : as the Sextant, an arc of 60 degrees; the Quadrant, an arc of 90 degrees ; and the Theodolite, a whole circle of 360 degrees, divided into two semicircles, of 180 degrees each. The Theodolite (a word of uncertain derivation), E B G S А D covered by the intersection of the wires of the object-glass of the telescope : the number of degrees traversed by the telescope from S, to T, or from F, to G, is noted, and that number is the measure of the angle ACB. 19. A segment of a circle, (segmentum, a cutting, a slice), is the figure contained by a straight line and the part of the circumference which it cuts off; as in Def. 15, the Figure FGEF. In a segment the straight line, as FG, is the chord, (chordee, a harp-string), the part of the circumference cut off, as FEG, the arc. A sector (sector, a cutter), is any portion of a circle bounded by two radii and the arc which those radii intercept, as BCDB, BCIB. 20. Rectilineal figures are those which are bounded by right or straight lines. 21. Trilateral figures, (trilaterus, three sided), or Triangles (triangulus, three cornered), are bounded by three straight lines. All other rectilineal figures may be resolved into triangles, by joining some one angular point and the other angular points. 22. Quadrilateral figures (quadrilaterus, four sided), are bounded by four straight lines. The Diagonals of a quadrilateral are the lines joining the opposite angles. 23. Multilateral figures, (multilaterus, many sided), or Polygons, (polugonos of many angles), are bounded by more than four right lines. They are named from the number of angles : as, pentagon, a figure of five angles, or sides ; hexagon, of six ; heptagon, of seven ; octagon, of eight, &e. 24. An Equilateral Triangle (equilaterus, equal sided), is that which has three equal sides; as, ABC. 25. An Isosceles Triangle с (isoskelees, having equal legs), is that which has two equal sides, or legs; namely, DF and EF in AFDE. 26. A Scalene Triangle (skaleenos, of unequal sides), is that which has three unequal А H sides; as A GHI. 27. A Right-angled Triangle is that which has a right angle; as, ABC. F I BAAA AAA AL E G H 28. An Obtuse-angled Triangle is that which has an obtuse angle; as DEF. 29. An Acute-angled Triangle is that which has three acute angles; as, GHI. The right and the obtuse-angled triangles, have also two acute angles. 30. Of quadrilaterals, a Square has all its sides equal, and all its angles right angles; as fig. A. P P C D D 31. An oblong (oblongus, rather long), is a figure which has all its angles right angles, but not all its sides equal; as B. The oblong is the same as the rectangle of book ii. 32. A rhombus (rhombos, a turbot, a lozenge), has all its sides equal, but its angles are not right angles; as C, 33. A rhomboid (rhomboeidees lozenge-like), has its opposite sides equal to each other, but all its sides are not equal, nor are its angles right angles; as D. The term parallelogram may supersede that of rhomboid. 34. All other four-sided figures, as E, are called T'rapeziums, the master sat at the broad end, that he might be seen by all his E 35. Parallel straight lines (paralleelos placed "A -B along opposite to each other), are such as are in the same plane, and which being produced ever so far both ways, do not meet; as AB, CD. С -D The least objectionable definition is that which, as Potts says (p. 50),“ simply expresses the conception of equidistance ;” thus,“ Parallel lines are such as lie in the same plane, and which neither recede from, nor approach to, each other. In the first Six Books of Euclid, all the lines are supposed to be in the same plane; the test of parallelism is, that two lines, being in the same plane, never meet, though indefinitely produced. A. A parallelogram (paralleelogrammon a drawing with parallel sides), is a four-sided figure, А. B of which the opposite sides are parallel, as AB to K CD, and AC to BD. The Diameter, or diagonal, E F AD, is the straight line joining two opposite angles: the parallelograms about the diagonals AEKG and KADF, are those through which the diagonal passes; and the Complements to fill up the figure, с H are the two parallelograms, ECHK and GKFB, through which the diagonal does not pass. Any figure with an equal number of equal sides, as four, six, eight, &c., will have its opposite sides parallel ; but in the Elements of Euclid'the name parallelogram is restricted to four-sided figures. OBSERVATION_" It is necessary to consider a solid,- that is, a magnitude which has length, breadth, and thickness,-in order to understand aright the definitions of a point, a line, and a superficies. A solid, or volume, considered apart from its physical properties, suggests the idea of the surfaces by which it is bounded ; a surface, the idea of the line or lines which form its boundaries ; and a finite line, the points which form its extremities. A solid is therefore bounded by surfaces; a surface is bounded by lines; and a line is terminated by two points. A point marks position only; a line has one dimension, length only, and defines distance ; a superficies bas two dimensions, length, and breadth, and defines extension ; and a solid has three dimensions, length, breadth and thickness, and defines some definite portion of space.' " It may also be remarked, that two points are sufficient to determine the position of a straight line ; and three points not in the same straight line are necessary to fix the position of a plane.”—Potts Euclid, p. 44. POSTULATES. 1. Let it be granted, that a straight line may be drawn from any one point to any other point: 2. That a terminated straight line may be produced to any length in a straight line: 3. And that a circle may be described from any centre at any distance from that centre. The first and second Postulates concede the use of a ruler, but not of a scale the third, that of the compasses,—but not that a circle can be described round a given centre with a radius, or distance in the compasses, of a given length. Euclid himself gave three other postulates, which modern Editors place as the tenth, eleventh, and twelfth Axioms. AXIOMS. I. The Seven Axioms, which apply to number and quantity as well as to magnitude, were called by Euclid COMMON NOTIONS. They are 1. Things which are equal to the same thing, are equal to one another. |