THE NEW PRACTICAL BUILDER, &c. CHAPTER I. GEOMETRY THE ELEMENTS OF GEOMETRY. EOMETRY is a science which considers the properties of lines and angles, as formed according to some certain law; as, also, the construction of all manner of figures, according to given data. It is divided into two branches; one of which considers the relations, positions, and properties, of lines, so as to render a proposition clear to the understanding without the aid of compasses or other instruments; being demonstrated, by a continued chain of reasoning, from certain principles previously established and laid down as axioms; so that the conclusions from one truth become part of the data for the proof of a succeeding proposition. This, which is called the Theory of Geometry, is fully explained by EUCLID, in his celebrated "ELEMENTS," which have served as the basis of all succeeding treatises on the subject: and so much of those Elements as may be required in the practice of Architecture will be found included in the present work. The other branch of geometry is entirely practical, and may be acquired without the theory, according to the directions hereafter given; although with a knowledge of the reasons of the rules it will be more satisfactory. It is this practical branch that enables the architect to regulate his designs, and the artizan to construct his lines, so as to enable him to execute the work. Without the aid of this branch of knowledge, the workman will be unfit for any undertaking whatever; and, so long as he is ignorant of the methods of geometrical construction, he must remain under the control and direction of a superior in his own class. The definitions and problems, which follow, are calculated to instruct the uninformed mechanic, and will qualify him for proceeding to the remaining parts of this treatise, wherein it will be found that the application of this branch of science is absolutely necessary. The uses of Geometry are not confined to Carpentry and Architecture: Astronomy, Navigation, Perspective, and numerous other branches, are entirely dependent upon it. "It conducts the soldier in the field, and the seaman on the ocean; it gives strength to the fortress, and elegance to the palace." In short, there is no mechanical profession that does not derive considerable advantage from it. One workman is superior to another, in proportion to his knowledge of the subject we are now commenting upon, and which we are about to explain. The Terms are here as clearly defined as the nature of the subject will admit, and the Problems are put in a regular succession; so that nothing is introduced, in any problem, as taken for granted, but what has been explained in some problem previously given. This selection, though not very numerous, is sufficient to enable the student to proceed with the remaining parts of the work, to which it is specially adapted: and every attention has been paid to divest the diagrams of superfluous lines, without rendering them less intelligible. GEOMETRIC DEFINITIONS. 1. A POINT is considered as that which has position without magnitude. Practically, a Point is the smallest visible mark upon a surface, as at figure 1, plate I. 2. A LINE is considered as length, without breadth or thickness; having extension only in one direction, as figures 2 and 3, (plate I,) which may be conceived to be made by the trace of a point, pen, or pencil. 3. Á RIGHT or STRAIGHT LINE is that which lies evenly between its extremes or ends. If two straight lines coincide in two points, all the intermediate points will coincide also. Thus, fig. 3, (pl. I,) represents a straight line, and fig. 2, a curve, or crooked line; the latter may be formed either by regular inflexions, or portions of straight lines, or both. 4. A SUPERFICIES, or SURFACE, is that which is considered as having length and breadth without depth. Thus the outward parts of any body, which are exhibited to the eye, are termed the superficies of that body. 5. A PLANE SUPERFICIES OF PLANE SURFACE, is that on which a straight line, drawn through any given point, in any position, will coincide. 6. A PLANE FIGURE, or DIAGRAM, or SCHEME, is the representation of any thing on a plane surface, by means of lines. When the lines are straight, the figure is said to be rectilineal. 7. An ANGLE is a space between two lines meeting in a point. A PLANE RECTILINEAL ANGLE is the space between two straight lines so meeting. Thus, fig. 4, (pl. I,) is a plane rectilineal angle. 8. Two straight lines are said to converge, when they meet each other, if produced or continued; as in fig. 5. 9. When one straight line stands upon another, and makes the angles on each side equal to each other, each of the equal angles is called a RIGHT ANGLE, and the line which stands upon the other is called a perpendicular to that other line. Thus, in fig. 6, (pl. I,) if the line CD stand upon AB, and make the angles on both sides of CD equal; each of these angles is a right angle. In fig. 7, the line CD does not make the angles on each side of it equal to each other: in this case, CD is said to stand at oblique angles to AB; and in the former case, fig. 6, CD is said to stand at right angles to AB. 10. An ACUTE ANGLE is that which is less than a right angle. In fig. 7, as CD makes the angles on each side of it unequal, one of them must be greater than the other: the greater must, therefore, be an obtuse angle, and the less an acute angle. And, as the space around the point C is the same, whatever be the position of the line CD, with respect to AB, what the one angle has in excess above the right angle, the other will have as much in defect. Figure 8, (pl. I,) is an acute angle; fig. 9, a right angle; and fig. 10, an obtuse angle. 12. A PLANE TRIANGLE is a space inclosed by three straight lines. Thus, figures 11, 12, 13, and 14, are triangles. 13. A RIGHT-ANGLED TRIANGLE is that which has one right angle. Thus, fig. 11 is a right-angled triangle. 14. An ACUTE-ANGLED TRIANGLE is that which has all its angles acute; as figures 12 and 13. 15. An OBTUSE-ANGLED TRIANGLE is that which has one obtuse angle; as fig. 14. 16. AN EQUILATERAL TRIANGLE is that which has all its sides equal; as fig. 12. 17. An ISOSCELES TRIANGLE is that which has two equal sides; as fig. 13. 18. A SCALENE TRIANGLE is that which has no two of its sides equal; as fig. 14. 19. PARALLEL LINES are lines on the same plane, which cannot meet, though produced or continued ever so far from each extremity (fig. 15.) 20. A PARALLELOGRAM is a figure whose opposite sides are parallel. Thus, figures 16, 17, 18, and 19, are parallelograms. 1 21. When the parallelogram has one of its angles a right-angle, it is called a RECTANGLE. Thus, figures 16 and 17 are rectangles. 22. When the sides of the rectangle are equal, it is called a SQUARE. Thus, fig. 16 is a square. 23. When the two adjacent sides are unequal, the rectangle is called an OBLONG; as fig. 17. 24. When only two opposite angles of a parallelogram are equal, it is called a RHOMBUS; as figures 18 and 19. 25. When two adjacent sides of a rhombus are equal, it is called a кHOмBOID (pron. rhom-bo-id); as fig. 19. 26. Every figure, inclosed by four straight lines, is called a QUADRANGLE or QUADRILATERAL. Thus, figures 16, 17, 18, 19, 20, and 21, are quadrilaterals. 27. When all the sides of a quadrilateral are unequal, it is called a TRA PEZIUM. 28. When two sides of the trapezium are parallel, it is called a TRAPEZOID; as fig. 21. 29. Equilateral and equiangular figures, contained by more than four straight lines, are called REGULAR POLYGONS. 30. A regular polygon of five sides, is called a PENTAGON; as fig. 22. 31. A regular polygon of six sides, is called a HEXAGON; as fig. 23. 32. A regular polygon of seven sides, is called a HEPTAGON; as fig. 24. 33. A regular polygon of eight sides, is called an OCTAGON; as fig. 25, and so on. The words enea, deca, undeca, dodeca, having the termination gon subjoined, signify regular polygons of nine, ten, eleven, and twelve, sides. Other polygons are commonly expressed as such, with the number of sides. 34. A CIRCLE is a plain figure, contained under one line only, which is called its circumference. From the circumference, straight lines, called radii, being drawn to a certain point within the figure, are equal. 35. The point to which the equal lines from the circumference are drawn, is called the CENTRE of the circle. Thus, in fig. 26, c is the centre, and c d the radius of the circle a bd. 36. The DIAMETER of a circle is a straight line, drawn through the centre, and terminated by the circumference; as the line a b, fig. 27. 37. A CHORD of a circle is a straight line, drawn through the circle, and terminated by the circumference. Thus the line ab, fig. 28, is a chord; and a b, fig. 27, is a chord passing through the centre. |