To Survey with the Chain only-Four Problems and numerous Ex- To find the differences of Levels of several points on the Earth's surface 120 To draw a Sectional Line of several points in the Earth's surface, the THE METHOD OF LAYING OUT RAILWAY CURVES ON THE GROUND. PROB. I.-To lay out a Railway Curve by the common Method-Cases PROB. II.-To lay out a Railway Curve by Offsets from its Tangent.— PROB. I.-To find the Contents of Railway Cuttings from their Depths 132 PROB. II. To find the Contents of Railway Cuttings from Sectional Areas 133 General Rule for finding the Contents of Solids, with Examples . INTRODUCTION. Ir will at once be seen that condensation of the materials produced by previous authors, and the introduction of a judicious selection of matter, adapted to the expanded intellect of the present age, are the proper requisites for a work on Mensuration. To this plan, the author trusts, from his long experience in engineering pursuits, that he has strictly adhered. In the first part, on PRACTICAL GEOMETRY, numerous examples are introduced, wherein the dimensions of certain parts are given to find the dimension of their corresponding parts, which has been rarely or never done by previous authors. This part is succeeded by a second part, on the MENSURATION OF LINES; which is not added for the sake of novelty only, but because it seemed to be the natural order of a work of this kind. The third and fourth parts treat of the MENSURATION OF SUPERFICES AND OF SOLIDS; while in all the three last-named parts the rules are not only given in words at length, in the usual way, but the same rules are expressed by FORMULA, together with other formulæ depending thereon, by which the rules receive considerable extension. Some of the rules and examples are taken verbatim from Dr. Hutton's Mensuration; for the author conceives that it would be disreputable to attempt, by verbal alterations in such rules, to give an air of originality to his work, as all other authors have done since Dr. H.'s time: the originality of this work consists in the new matter, every where added, to adapt it to the wants of modern times. Timber measure and Artificer's work, the latter with considerable modern improvements, are next introduced, with concise and practical methods of finding the surfaces and solidities of vaulted roofs, arches, domes, &c. Concise, and the author trusts, clear systems of Mensuration applied to land, i.e. surveying, levelling, laying out railway curves and finding the contents of railway cuttings, complete the work, and may serve as an introduction to other more extended treatises on Land and Engineering Surveying, adapted to modern practice. The demonstrations of all the rules and formulæ, in the four leading parts of the work, will be found in Dr. Hutton's Large Mensuration and in the Rudimentary Geometry of this series. Conic Sections and their solids are very briefly treated of in this work, and chiefly in as far as they may be useful to those who intend to become excise officers, whose actual practice is best learnt from an experienced officer. An extended article is not generally useful to practical men. The weights and dimensions of balls and shells may be found by Prob. VIII., Part IV., in conjunction with the Table and Rules for finding the specific gravities of bodies. The method of piling balls and shells, finding their number in a given pile, and the quantity of powder contained in a given shell or box, form no essential part of a work on mensuration, being only useful in an arsenal, and are also omitted The author has thus secured space for the ample discussion of subjects really useful to the great majority of students and practical men, in the compass of a volume less than half the size and one-fifth of the price of the works of his predecessors on Mensuration. The plan being thus briefly detailed, it will now be proper, previous to studying the following work, to give the following SUGGESTIONS TO STUDENTS. Mensuration treats of the various methods of measuring and estimating the dimensions and magnitudes of figures and bodies. It is divided into four parts, viz., Practical Geometry, and Mensuration of Lines, of Superfices, and of Solids, with their several applications to practical purposes. The beginner, for a first course, may omit the problems beyond the thirty-second in Practical Geometry, and Problems III., VIII., IX., XI., and XII., in the Mensuration of Lines, with the formulæ and examples depending on them. He may also omit all the formulæ in the Mensuration of Superfices and Solids, with the examples depending on them, as well as the problems beyond the tenth in the Mensuration of Solids, except it is required he should learn the method of gauging casks, in which case omit only the two last problems. But if he require an extensive knowledge of some or all the subjects here treated of, he will do well to learn the use of such of the formulæ and the other parts, omitted in the first course, according to what he may require as a practical man. MENSURATION. PART I. PRACTICAL GEOMETRY. DEFINITIONS. 1. A point has no dimensions, neither length, breadth, nor thickness. 2. A line has length only, as A. 3. A surface or plane has length and breadth, as B. A B 4. A right or straight line lies wholly in the same direction, as A B. 5. Parallel lines are always at the same distance, and never meet when prolonged, as A B and C D. 6. An angle is formed by the meeting of two lines, as A C, C B. It is called the angle ACB, the letter at the angular point C being read in the middle. 7. A right angle is formed by one right line standing erect or perpendicular to another; thus, A B C is a right angle, as is also A BE. 8. An acute angle is less than a right angle, as DB C. A A E B -B 9. An obtuse angle is greater than a right angle, as DB E. 10. A plane triangle is a space included by three right lines, and has three angles. G D E A B C 11. A right angled triangle has one right angle, as A B C. The side A C, opposite the right angle, is called the hypothenuse; the sides A B and B C are respectively called the base and perpendicular. 12. An obtuse angled triangle has one obtuse angle, as the angle at B. 13. An acute angled triangle has all its three angles acute, as D. 14. An equilateral triangle has three equal sides, and three equal angles, as E. 15. An isosceles triangle has two equal sides, and the third side greater or less than each of the equal sides as F. 16. A quadrilateral figure is a space bounded by four right lines, and has four angles; when its opposite sides are equal, it is called a parallelogram. ·17. A square has all its sides equal, and all its angles right angles, as G. 18. A rectangle is a right angled parallelogram, whose length exceeds its breadth, as B, (see figure to definition 3). K B 19. A rhombus is a parallelogram having. all its sides and each pair of its opposite angles equal, as I. 20. A rhomboid is a parallelogram having its opposite sides and angles equal, as K. 21. A trapezium is bounded by four straight lines, no two of which are parallel to each other, as L. A line connecting any two of its angles is called a diagonal, as A B. In some works this is also called a scalene triangle. |