: THE ELEMENT OF THE NEW-YOR OF GEOMETRY. BOOK I. DEFINITIONS AND REMARKS. I. THE word Geometry is from the Greek, it means the measurement of the earth, and is the name of the science which treats of space, as defined by shape or extent. II. Two portions of space the same in shape, may be called similar. The same in extent, equal. III. IV. The same in shape and extent, equivalent. V. A portion of space extending in length, breadth, and thickness, may be called a solid. VI. In length and breadth, without reference to thickness, a surface. VII. In length without reference to breadth or thickness, a line. VIII. Place, without reference to extent, may be called a point. IX. A point is considered incapable of division, and not to occupy space. X. A solid is bounded by a surface, or surfaces. XI. A surface is bounded by a line, or lines. XII. A line is bounded by points. XIII. The intersection of two lines is a point. XIV. If a line is extended, its parts will approach more nearly to the same direction, and if it is extended till its extremities are at the greatest distance which the length of the line will permit, every part will then be in the same direction, and it may be called a stretched line, or straight line. XV. A straight line may be supposed to be drawn from any point, in any direction, and to any distance. XVI. If a line that is not straight between two points be straightened, part of the line will be drawn beyond one of the points, and the remainder will extend from one to the other; therefore, a straight line is the shortest line that can be drawn between two points. XVII. To continue a straight line in the original direction, may be called producing the line. XVIII. To draw a straight line from one point to another, may be called joining the points. XIX. Two magnitudes, which being compared exactly fill the same space, may be said to coincide. XX. If one straight line is applied to another, they will coincide, except so far as one extends beyond the other, and if they are produced they will still coincide. XXI. If a straight line be drawn between two points, any other straight line between the same points will coincide with the first line. XXII. Two straight lines, in different directions from the same point, may be said to make an angle. XXIII. An angle may be named by a letter at the angular point, as E ; or by letters distinguishing the lines which make the angle, as A B C, A B D, or C B D, the letter at the angular point being between the other two. XXIV. The angle is determined by the difference in direction between the lines, and is not varied by their length, or by producing them. XXV. If two straight lines which make an angle are produced, the distance between their extremities will be increased, and the more they are produced, the more will it be increased. XXVI. If two straight lines meet and are produced, they will either coincide, or not coincide. If they coincide, and are produced, they will still coincide, and cannot enclose a space. If they do not coincide, they will make an angle, and cannot enclose a space. Therefore, two straight lines cannot enclose a space. XXVII. If a straight line standing on another straight line, makes the angles on each side equal, each of them may be called a right angle, and the lines may be said to be perpendicular to each other. XXVIII. An angle greater than a right angle, may be called an obtuse angle. XXIX. An angle less than a right angle, may be called an acute angle. XXX. An acute or obtuse angle, may be called an oblique angle. XXXI. A portion of space enclosed by one or more boundaries, may be called a figure. XXXII. A figure enclosed by straight lines, may be called a rectilineal 'figure. XXXIII. In describing a rectilineal figure, the two first lines make an angle, every succeeding line makes an angle with the preceding, and the last line makes an angle with the preceding, and with the first line. Therefore, in every rectilineal figure, the number of angles is equal to the number of sides. XXXIV. A figure enclosed by three straight lines may be called a triangle. XXXV. A triangle containing a right angle may be called a right angled triangle. XXXVI. The side opposite the right angle may be called the hypothenuse, and the other two sides the legs. One of the legs may be called the base, and the other the perpendicular. XXXVII. A triangle containing an obtuse angle may be called an obtuse angled triangle. XXXVIII. A triangle containing three acute angles may be called an acute angled triangle. XXXIX. An obtuse, or acute angled triangle, may be called an oblique angled triangle. XL. A triangle having three equal sides may be called an equilateral triangle. |