ON PRACTICAL GEOMETRY, MENSURATION, Conic Sections, Gauging, and Land-Surveying, WITH AN ESSAY ON THE SPECIFIC GRAVITY OF BODIES, THE TONNAGE OF SHIPS, FOR THE USE OF THE IRISH NATIONAL SCHOOLS. DUBLIN: PRINTED AT THE HIBERNIA PRESS OFFICE, 1834. 1874. f.8. A TREATISE ON MENSURATION IN THEORY AND PRACTICE. SECTION I. PRACTICAL GEOMETRY, DEFINITIONS, 1. GEOMETRY teaches and demonstrates the properties of all kinds of magnitudes, or extension; as solids, surfaces, lines, and angles. 2. Geometry is divided into two parts, theoretical and practical. Theoretical Geometry treats of the various properties of extension abstractedly; and Practical Geometry applies these theoretical properties to the various purposes of life. When length and breadth only are considered, the science which treats of them is called Plane Geometry; but when length, breadth and thickness, are considered, the science which treats of them is called Solid Geometry. 3. A Solid is a figure, or body, having three dimensions, viz. length, breadth, and thickness, as A. A The boundaries of a solid are surfaces, or superficies. B 4. A Superficies, or surface, has length and breadth only; as B. The boundaries of a superficies are lines. 5. A Line is length without breadth, and is formed by the motion of a point; as CBC. The extremities of a line are points. B -B Note. It is likewise necessary to conceive that a line is composed of an infinite number of points, each less than any assignable quantity. 6. A Straight Line is the shortest distance between two points, and lies evenly between these two points. 7. A Point is that which has no parts or magnitude; it is indivisible; it has not length, breadth, or thickness. If it had length, it would then be a line; were it possessed of length and breadth, it would be a superficies; and had it length, breadth, and thickness, it would be a solid. Hence a point is void of length, breadth, and thickness, and is only the creature of imagination, 8. A Plane rectilineal Angle is the inclination of two right lines which meet in a point, but are not in the same direction; as S. S then be equal; or by conceiving the D point F to recede from D, till x n becomes equal to m n, then the angles at B and E would be equal. Hence it appears that the nearer the extremities of the lines forming an angle approach each other, while the point at which they meet remains fixed, the less the angle; and the farther the extreme points recede from each other, the vertical point remaining fixed, as before, the greater the angle. 10. A Circle is a plane figure contained by one line called the circumference, which A is every where equally distant from a point within it, called its centre; as ò: and an arc of a circle is any part of its circumference ; as A B. 11. The magnitude of an angle does not consist in the length of the lines which form it: the angle CBG is less than the angle ABE, though the lines CB, GB are longer than AB, EB. A B F G E B 12. When an angle is expressed by three letters, as ABE, the middle letter always stands at the angular point, and the other two any where along the sides; thus the angle ABF is formed by A B and BE. The angle ABG by AB and G B. 13. In equal circles, angles have the same ratio to each other as the arcs on which they stand, (33. vi). Hence also, in the same, or equal circles, the angles vary as the arcs on which they stand; and therefore the arcs may be assumed as proper measures of angles. Every angle then is measured by an arc of a circle, described about the angular point as a centre; thus the angle ABE is measured by the arc AE; the angle ABG by the arc AF. 14. The circumference of every circle is generally divided into 360 equal parts, called degrees; and every degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. The angles are measured by the number of degrees contained in the arcs which subtend them; thus, if the arc AE contain 40 degrees, or the ninth part of the circumference, the angle ABE is said to measure 40 degrees. |