A General View of the Contents of this Work. THE System of Geometry is divided into two parts. The first contains Geometrical Definitions respecting Lines, Angles, Superficies, &c. The second part contains a number of Geometrical Problems necessary for Trigonometry and Surveying. The System of Trigonometry is also divided into two parts : and teaches the solution of Questions in Right and Oblique angled Trigonometry, by. Logarithms and also by Natural Sines. The Treatise on Surveying is divided into three parts. Part first treats of measuring Land, and is divided into three Sections. The first contains several Problems respecting Mensuration, and for finding the Area of various Right-lined Figures and Circles. The second Section teaches different methods of taking the Survey of fields ; also to protract them, and find their Area in the manner commonly practised, and likewise by Arithmetical and Trigonometrical calculations, without measuring Diagonals and Perpendiculars with a Scale and Dividers; interspersed with sundry useful rules and directions. The third Section is a particular explanation and demonstration of Rectangular Surveying, or the method of computing the Area of Fields from the Field Notes, by Mathematical Tables, without the necessity of plotting the Field. To this Section is added a useful Problem for ascertaining the true Area of a Field which has been measured by a Chain too long or too short a Part second treats of laying out Land in various shapes. Part third contains sundry Problems and Rules for dividing Land and determining the true Course and Distance of dividing Lines, or from one part of a Field to another. To this is added an uppendix conceming the Variation of the Compass and Attraction of the Needle; also, a Rule to find the difference between the present Variation, and that at a time when a Tract was formerly surveyed, in order to trace or run out the original ines T'ie Wathematical Tables, are Traverse Table, ar Table of Difference of Latitude and Departure, calculated brevery Degree and quarter of Degree, and for any distance up to 50; a Table of Natural Sines calculated for every Minute; 1 Table of Logarithms comprised in four pages, yet sufficiently extensive for common use; ad 1 Table of Logarithmic or Artificial Sines, Tangents and Secants, calculated for every 5 Minutes of a Degree. To these Tables are prefixed particular explanations of the manner of using them. GEOMETRY. G EOMETRY is a Science which treats of the properties of Magnitude. PART I. Geometrical Definitions. 1. A Point is a small Dot; or, Mathematically considered, is that which has no parts, being of itself indivisible. 2. A Line has length but no breadth. 3. A Superficies or Surface, called also Area, has length and breadth, but no thickness. 4. A Solid has length, breadth and thickness. 5. A Right Line is the shortest that can be drawn between two Points. 6. The inclination of two Lines meeting one another, or the opening between them, is called an Angle. Thus at B. PLATE I. Figure 1. is an Angle, formed by the meeting of the Lines AB and BC. 7. If a Right Line CD. Fig. 2. fall upon another Right Line AB, so as to incline to neither side, but make the Angles on each side equal, then those Angles are called Right Angles ; and the Line CD is said to be Perpendicular to the other Line. 8. An Obtuse Angle is greater than a Right Angle; as ADE. Fig. 3. B a Fig. 5. 9. An Acute Angle is less than a Right Angle ; as EDB. Fig. 3. Note. When three letters are used to express an Angle, the middle letter denotes the angular Point. 10. A Circle is a round Figure, bounded by a Line equally distant from some Point, which is called the Centre. Fig. 4.. 11. The Circumference or Periphery of a Circle is the bounding Line ; as ADEB. Fig. 4.. 12. T'he Radius of a Circle is a Line drawn from the Centre to the Circumference ; as CB. Fig. 4. Therefore all Radii of the same Circle are equal. 13. The Diameter of a Circle is a Right Line drawn a from one side of the Circumference to the other, passing through the Centre ; and it divides the Circle into two equal parts, called Semicircles; as AB or DE. 14. The Circumference of every Circle is supposdil to be divided into 360 equal parts called Degrees; and each Degree into 60 equal parts, called Minutes ; and each Minute into 60 equal parts, called Seconds ;and these into Thirds, &c. Note. Since all Circles are divided into the same number of Degrees, a Degree is not to be accounted a quantity of any determinate length, as so many inches or Feet, &c. but is always to be reckoned as being the 360th part of the Circumference of any Circle, without regarding the big: ness of the Circle. 15. An Arch or Arc of a Circle is any part of the Circumference; as BF or FD, Fig. 5; and is said to be an Arch of so many Degrees as it contains parts of 360 into which the whole Circle is divided. 16. A Chord is a Right Line drawn from one end of an Arch to the other, and is the measure of the Arch; as HG is the Chord of the Arch HIG. Fig. 6. Note. The Chord of an Arch of 60 degrees is equal in length to the Radius of the Circle of which the Arde is a part. 17. The Segment of a Circle is a part of a Circle, cut off by a Chord ; thus the space comprehended between the Arch HIG and the Chord HG is called a Segment. Fig. 6. 18. A Quadrant is one quarter of a Circle ; as ACB. Fig. 6. 19. A Sector of a Circle is a space contained between two Radii and an Arch less than a Semicircle; as BCD or ACD. Fig. 6. 20. The Sine of an Arch is a Line drawn from one end of the Arch, perpendicular to the Radius or Diameter drawn through the other end : Or, it is half the Chord of double the Arch; thus HL is the Sine of the Arch HB. Fig. 7. 21. The Sines on the same Diameter increase in length till they come to the Centre, and so become the Radius. Hence it is plain that the Radius CD Fig. 7. is the greatest possible Sine, or Sine of 90 Degrees. 22. The Versed Sine of an Arch is that part of the Diameter or Radius which is between the Sine and the Circumference ; thus LB is the Versed Sine of the Arch HB. Fig 7. 23. The Tangent of an Arch is a Right Line touching the Circumference, and drawn perpendicular to the Diameter; and is terminated by a Line drawn from the Centre through the other end of the Arch; thus BK is ; the Tangent of the Arch BH. Fig. 7. Note. The Tangent of an Arch of 45 Degrees is equal in length to the Radius of the Circle of which the Arch is a part. 24. The Secant of an Arch is a Line drawn from the Centre through one end of the Arch till it meets the Tangent; thus CK is the Secant of the Arch BH. Fig. 7. 25. The Complement of an Arch is what the Arch wants of 90 Degrees, or a Quadrant ; thus HD is the Complement of the Arch BH. Fig. 7. 26. The Supplement of an Arch is what the Arch wants of 180 Degrees, or a Semicircle ; thus ADH is the Supplement of the Arch BH. Fig. 7. |