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PREVIOUS to commencing the various subjects of Land and Engineering Surveying, it will be necessary to give a clear view of Practical Geometry, which is especially requisite for those who are unacquainted with this branch, as well as those parts of the mathematics which are equivalent to it.



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.



4. A right or straight line lies wholly in the same direction,

as AB.


5. Parallel lines are always at the same distance, and never meet when prolonged, Cas A B and C D.

6. An angle is formed by the meeting of two lines, as A C, CB. It is called the angle A CB, 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 E angle, as DB C.

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


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

and has three angles.

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11. A right angled triangle has one right angle, as AB 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 2).

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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 the diagonal, as A B.

22. A trapezoid is a quadrilateral, having two of its opposite sides parallel, and the remaining two not, as 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 called regular polygons, when all their sides and angles are equal, otherwise irregular polygons.

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

25. An arc of a circle is a part of the circumference, as A B.



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

each of which is A chord is a right

It divides the If the radii CF,

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 it is estimated by the number of degrees contained in that arc :thus the arc A B, the centre of which is C, is the measure of the angle A CB.

If the


angle A CB contain 42 degrees, 29 minutes, and 48 seconds, it is thus written 42° 29′ 48′′.


(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.)

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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 A B in the middle point E.


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



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From any two points m and r, in the line AB, 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.


Through a given point r, to draw a straight line CD parallel to a given straight line AB,


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


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.

A m

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 P m, describe two arcs cutting each other in C; through C, draw CP, and it will B be the perpendicular required.

2. When the point P is at the end of the line. With the centre P, and any radius, describe the arc mrs;-from the point m, with the same radius, turn the compasses twice on the arc, as at r and s:again, with centres r and s, describe ares intersecting in C:-draw CP, and it will be the perpendicular required.



NOTE. This problem and the following one are usually done with an instrument called the square.


From a given point C to let fall a perpendicular to a given line. 1. When the point is nearly opposite the middle of the line.

From C, as a centre, describe an arc to cut AB in m and n;-with centres m and n, and the same or any other radius, describe arcs intersecting in o: through C and o draw Co, the perpendicular required.


2. When the point is nearly opposite the end of the line. From C draw any line C m to meet B A, in any point m;-bisect Cm in n, and with the centre n, and radius C n, or m n, describe an arc cutting BA in P. Draw CP for the perpendicular required.




(Euc. I. 22.)

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To construct a triangle with three given right lines, any two of which must be greater than the third. Let the three given lines be 5, 4 and 3 yards. From any scale of equal parts lay off the base AB = 5 yards; with the centre A and radius A Č= 4 yards, describe an arc; with centre B and radius A



CB 3 yards; describe another arc cutting the former arc in C-draw A C and CB; then A B C is the triangle required.


Given the base and perpendicular, with

the place of the latter on the base, to construct the triangle.

Let the base AB=7, the perpendicular C D 3, and the distance AD 2 chains. Make A B = 7 and AD = 2;—at D erect the


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