THE IRRIGATION AGE. 



781 



THE PRIMER OF HYDRAULICS* 

 By FREDERICK A. SMITH, C. E. 



Substitute the given 



Applied Problems. 1. Find the depth of a rectangular box 

 if the width is 5 ft. the length 8 ft. and the volume equals 



82 cubic ft. Solution : In formula : d = . Substitute the 



bl 



82 82 

 given quantities : d ~ = =; 2.05 ft. 



3x8 40 



:'.. Find the breadth of a bux containing 100 cubic ft. 

 if the depth is Z]/ 2 ft. and the length 16 ft.: 



V 



Solution : In formula : b = . 



Id 



100 100 

 quantities : b = = = 2.5 ft. 



16x2.5 40 



:'.. Find the length of a box containing 1 gallon (231 

 cubic inches) if the depth is 4 in. and the breadth is 5 in. 



v 

 Solution : In formula / = substitute the given quantities, 



bd 



231 231 

 then': / = := = 11.55 inches. 



5x4 20 



This shows the usefulness of formulae to solve practical 

 problems. 



Article 3. Geometrical Principles. 

 1. Introductory. The science relating to the forms or 

 shapes of things regardless of the substances enclosed is 

 called Geometry. Elementary conceptions in geometry are 

 points, lines, surfaces and solids. A point has no extention 

 or size, but merely denotes location in space. A geometrical 

 line has no breadth or thickness, but possesses length. A 

 surface has no thickness, but has length and width, and a 

 solid has length, breadth and thickness. All geometrical 

 quantities are within space, which is everywhere and which 

 is infinite. All geometrical figures drawn in a plane surface 

 belong in the field of plane geometry, and all geometrical 

 figures lying in more than one plane are part of solid geom- 

 etry. 



2. Angles and Triangles. 



When two straight lines, AB and CD, intersect each 

 other as shown in Fig. 4, they form angles with 

 each other ; let O be the point of intersection, then 

 there are formed four angles. Angles are desig- 

 nated by naming three points like AOC, which 

 angle includes the space bounded by AO and CO. 

 B The point O is called the vertex, and AO and CO 

 Fig. 4. are called the sides or legs of the angle ; these 

 sides must be considered of infinite length, so that the size 

 of an angle is not affected by the length of its sides ; 

 sometimes angles are expressed also by writing a letter 

 into the angle area near the vertex like a, x, a, ft, etc. 

 This simplifies the work many times; thus, if we set for 

 angle AOC the letter a and for angle BOD the letter C, 

 for angle BOC the letter c, and for angle AOD the letter 

 d, then it is much easier to refer to any one of the 

 angles. Angles in the position of a arid b are called ver- 

 tical angles, the legs of the one being extensions of the 

 legs of the other ; likewise c and d are vertical angles, 

 and the first important principle in geometry is that vertical 

 angles are equal ; thus angle a = angle 6 and angle d = 

 angle r. While angle a angle b and angle c = angle d, 

 it is seen that there is much difference between angles a 

 and c and means are provided to compare different angles 

 together by measuring them with a unit angle in the follow- 

 ing manner : 



Conceive the whole plane divided into 360 equal angles, 

 as indicated in Fig. 5, all radiating from the 

 same point A like BAG, CAD, etc., then 

 each of these angles is called 1 degree (1). 

 Each degree is again divided into 60 equal 

 Fig. 5. angles, called minutes ('), an d each minute 



is divided into 60 equal angles called seconds (") ; thus 



Copyright by D. H. Anderson. 



the whole plane measures 360, or 21,600'. or 1,296,000", so 

 that by using these units any angle may be exactly expressed 

 m degrees, minutes and seconds. 



If two lines intersect each other so as to make 

 tour equal angles as in Fig. 6, in which the four 

 < angles formed are equal, a = b c d, then 



each angle equals 360 -=- 4 = 90. Such angles 

 are called right angles, and the lines forming 



B them are said to stand perpendicular to each 

 other; thus the line AB stands perpendicular to 

 the line CD in the point O. An angle smaller than 

 MS. 6. 90 j s ca ii ec i an acute angle, ijk e angle ABC in 



l;ig. ~, and an angle greater then 90 is called an obtuse angle, 

 like angle DBF in Fig. 8. If the sum of two angles equals 

 '.10 , like angles a and b, in Fig. 9, they are called com pie- 



Fig, i. 



Fig. 9. 



Fig. 10. 



Fig. 11. 



inenltu-y angles, and if the sum of two angles like c and d, in 

 Kig. 10, is equal to 180, they are called supplementary angles. 

 A geometrical figure bounded by three intersecting 

 straight lines is called a Triangle. Thus, in 

 Fig. 11, ABC is a triangle. Each triangle has 

 three sides and three angles. The sum of 

 the three angles of every triangle equals ISO" ; 

 thus, if a, b and c are the three angles in the 

 triangle ABC, then a + b + c = 180. Therefore, if two 

 angles are known in any triangle the remaining angle can be 

 found ; for instance : Angle a = 70 and angle b = 80 ; find 

 angle c. Solution : a + b + c 180 ; substitute known 

 values for a + b : 



70 + 80 -f c = 180. 

 150 + c = 180. 

 Subtract 150 from both sides : 

 c= 180 150. 

 c = 30. 



If in a triangle, two sides are equal, it is called isosceles 

 (See Fig. 12.) If AC = CB, and if the line CM is drawn so 

 that AM = EM, then the line CM divides the 

 angle ACB in two equal parts and stands per- 

 pendicular to AB at point M. 



If in a triangle two sides are equal, the 

 angles are also . equal opposite their equal 

 sides. Thus, if AC = CB, in Fig. 12, then 

 angle b = angle a. 



If in a triangle the three sides are equal to 

 each other is is called an equilateral triangle, as 

 in Fig. 13. If the side AB = BC = AC, then the 

 angles must also be qual, hence angle a = an- 

 gle b = angle c, and such a triangle is also 

 called equiangular, so that an equilateral trian- 

 gle is also equiangular.. Furthermore, since the 

 sum of the three angles of every triangle equals 

 180, then in the equiangular triangle each angle must be 180 

 -^- 3 60. So, if the three sides in a triangle are equal, it 

 Jells at once that each angle in the triangle equals 60, and if 

 ftich angle in a triangle equals 60 it shows that the three 

 sides are equal to each other. 



If in a Triangle ABC, Fig. 14. the angle ABC = 90, it is 

 called a right angled triangle, and the side AB op- 

 posite the right angle, is called the Hypothenuse ; 

 hence, both of the other two angles must be acute 

 angles, and as their sum makes 90, they are also 

 complementary angles. 



In any triangle the sum of two sides is greater 

 c then the third side. 



Fig. 14. j n a right angled triangle the square of the Hy- 



pothcnuse is equal to the sums of the squares of the other two 

 sides. Tims, if in Fig. 14 AB 4, BC = 3 and AC = 5, then 

 ."i" = 4 2 + 3 : , or 25 = 16 + 9. This is a very important prin- 

 ciple and is referred to as the forty-seventh problem of Euclid. 

 The fact that the three dimensions, 3, 4 and 5, or multiples 

 thereof form right angled triangles, is made use of to plot 

 right angles as follows : Suppose it is required to draw a 

 perpendicular at point B to the line BC; let BC = 30', then 

 AB must be 40' and AC must be 50' to make the line AB 

 stand perpendicular to BC; a closed string 120 feet long with 

 knots 30, 40 and 50 foot apart would set out a right angle 

 anywhere ; also, smaller dimensions, for instance, 15', 20' and 

 25', or 9', 12' and 15' will do the same. 



