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 formula 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 CD.

6. An angle is formed by the meeting of two lines, as A C, CB. 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, ABC is a right angle, as is also A BE.

8. An acute angle is less than a right angle, as DB C.

A

C

C

E

B

-B

-D

9. An obtuse angle is greater than a right angle, as DBE

1

10. A plane triangle is a space included by three right lines, and has three angles.

G

D

E

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).

A

L

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 are called the diagonal, as AB.

* In some works this is also called a scalene triangle.

22. A trapezoid is a quadrilateral, having two of its opposite sides parallel, and the remaining two not, as M.

M

23. Polygons have more than four sides, and receive particular names, according to the number of their sides. Thus, a pentagon has five sides; a hexagon, six; a heptagon, seven; an octagon, eight; &c. They are regular polygons, when all their sides and angles are equal, otherwise irregular polygons.

24. A circle is a plain figure, bounded by a A curve line, called the circumference, which is every where equidistant from a point C within, called the centre.

25. An arc of a circle is part of the circumference, as A B. 26. The diameter of a circle is a straight

·F

B

B

line AB, passing through the centre C, and dividing the circle into two equal parts, each of which is called a semicircle. Half A the diameter AC or CB is called the radius. If a radius CD be drawn at right angles to A B, it will divide the semicircle into two equal parts, each of which is called a quadrant, or one fourth of a circle. A chord is a right line joining the extremities of an arc, as FE. It divides the circle into two unequal parts called segments. If the radii CF, CE be drawn, the space, bounded by these radii and the arc FE, will be the sector of a circle.

27. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees, and each degree into 60 minutes, each minute into 60 seconds, &c. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees.

28. The measure of an angle is an arc of any circle, contained between the two lines which form the angle, the angular point being the centre; and is estimated by the number of degrees contained:-thus the arc AB, the centre of which is C, is the measure of the angle ACB. If the angle ACB contain 42 degrees, 29 minutes, and 48 seconds, it is thus written 42° 29′ 48′′.

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.

B

G

PROBLEMS IN PRACTICAL GEOMETRY.

(In solving the five following problems only a pair of common compasses and a straight edge are required; the problems beyond the fifth require a scale of equal parts; and the two last a line of chords: all of which will be found in a common case of instruments.)

PROBLEM I.

To divide a given straight line AB into two equal parts.

From the centres A and B, with any radius, or opening of the compasses, greater than half A B, describe two arcs, cutting each other in C and D; draw CD, and it will cut AB in the middle point E.

PROBLEM II.

At a given distance E, to draw a straight line CD, parallel to a given straight line A B.

C

n

m

E

D

B

From any two points m and r, in the line A B, with a distance equal to E, describe the arcs n and s::-draw CD to touch these arcs, without cutting them, and it will be the parallel required.

NOTE. This problem, as well as the following one, is usually performed by an instrument called the parallel ruler.

PROBLEM III.

Through a given point r, to draw a straight line CD parallel to a given straight line A B.

Am

From any point n in the line A B, with the distance nr, describe the arc rm -from centre r, with the same radius, describe the arc ns-take the arc mr in the compasses, and apply it from n to s:-through r and s draw C D, which is the parallel required.

PROBLEM IV.

From a given point P in a straight line AB to erect a perpendicular. 1 When the point is in or near the middle of the line.

On each side of the point P take any two equal distances, Pm, Pn; from the points m and n, as centres, with any radius greater than Pm, describe two arcs cutting each other in C; through C, draw CP, and it will be the perpendicular required.