T TI K I R R I G A T I X A G E . 



741 



THE PRIMER OF HYDRAULICS* 



By FREDERICK A. SMITH, C. E. 



Art. 1. General Properties of Matter Continued. 



Inertia. This means that all bodies have the tendency 

 to remain in their state of rest or state of motion, and that it 

 takes some external force to change such state. Thus, a 

 wagon standing still will remain standing still until some out- 

 side force makes it move; this principle is easily seen, but 

 the other part of the claim that bodies in motion will remain 

 in motion until stopped by some external force requires 

 some study to prove in all cases; for instance, roll a ball 

 along the ground and it is observed that its motion grows 

 slower and slower and finally comes to rest. This looks at 

 first to be a violation of the principle of Inertia, yet it is 

 in full accord therewith, for the stoppage of the motion of 

 the ball is caused by two external causes or forces, namely, 

 the impenetrability of the air which offers a certain resist- 

 ance to the rolling ball, and the friction upon the surface on 

 which the ball rolls bring it gradually to a stop. 



The phenomena of motion and the causes producing them 

 will be further treated and in more detail in the article on 

 Mechanical Principles. 



Art. 2. Algebraic Principles. 



A knowledge of mathematical principles is necessary for 

 the successful application of scientific principles to practical 

 problems. This applies also to hydraulics and in order to con- 

 vey this knowledge to those who have not had it before, or 

 to refresh it in those who have forgotten it articles 2 and 3 

 have here been introduced. 



Algebra is a general arithmetic teaching the principles 

 of arithmetic by the use of symbols. It is presumed that the 

 four elementary operations, addition, subtractKm, multiplica- 

 tion and division are known so far as common numbers are 

 concerned and the following explanations illustrate their 

 adaptability to algebraic symbols : 



1. Addition. The act of adding two or more numbers to- 

 gether means to find a new number which contains as many 

 units as the two or more given numbers; thus: 7+3=10 

 {read: 7 plus 3 equal 10) represents that operation arith- 

 metically ; the algebraic representation is just like it : A+B=C 

 (read: A plus B equal C) and it means that the number of 

 units in A are added to the number of units in B and that 

 the number C contains as many units as there are in <i and b 

 together. The numbers to be added together are called 

 summands and the result of the addition is called the sum. 

 Thus a and b are summands and c is the sum. 



The expression A+B=C is also called an equation, be- 

 cause the value of A+B is equal to C ; A+B is the left side 

 and C the right side of the equation and as the principle of 

 equation is very important in this work the following axioms 

 should be mastered right now : If two things are equal 

 they remain equal if the same quantity is added to each, or 

 if the same quantity is subtracted from each, or if each is 

 multiplied by the same number, or if each is divided by the 

 same number. That these principles are true is easily seen. 

 Thus, if 



7+3=10 then 



7+3+2 = 10+2 (adding 2 to both sides), 



7+3 5=10 5 (subtracting 5 from both sides), 



7+3X4=10X4 (multiply both sides by 4) and 



7+3 10 

 = dividing both sides by 2; for in the first oper- 



ation we have 12=12; in the second operation 5=5; in the 

 third operation 40=40, and in the fifth operation we have 

 r> = 3 - The application of these principles applies with equal 

 force to algebraic equations as will be seen hereafter. 



2. Subtraction. Unknown quantities are usually expressed 

 by the symbol X, Y, Z, etc., though any letter may be used 

 for that purpose. If in the expression : 



*Copyright by D. H. Anderson, December. 1910. 



one of the summands X is unknown it is found by subtract- 

 ing the other summand 7 from both sides, thus: 



7+X 7=10 7 



The equality of the two sides has not been disturbed by sub- 

 tracting 7 units from both sides ; on the left side this leaves 

 X and on the right side it leaves 10 7, hence: 

 X=10 7 



X=3 



This is a very important principle and the expression 10 7=3 

 shows how a summand is found by subtracting the given 

 summand from the sum. It also illustrates the principle of 

 subtraction. The number 10 from which is subtracted is 

 Called the minuend, the number 7 which is subtracted is called 

 the subtrahend, and the number :i which is the result of the 

 subtraction is called the difference. 



As a general thing when in an equation all but one 

 quantity are given then that one unknown quantity can be 

 found. Thus, if in the above equation the number 10 was 

 pot known we give it the symbol X and then write X 7=3. 

 Now, in order to find X, add 7 to both sides of the equation : 



X 7+7=3+7 



On the left side 7+7 cancel, hence: 

 X=3+7 



This also shows the principle that the minuend is equal to 

 the subtrahend plus the difference. 



3. Negative Numbers. In common arithmetic all numbers 

 are greater than zero ; in algebra, however, we say there are 

 just as many numbers smaller than zero than greater than 

 zero or with other words there is an ascending series of 

 numbers greater than zero and a descending series of num- 

 bers smaller than zero ; the former are called positive num- 

 bers the latter negative numbers. The positive numbers are 

 indicated by the + (plus) sign though position numbers need 

 not have the sign affixed, thus the numbers 7 or A indicate 

 position numbers which 7 or A would mean negative num- 

 ber* A negative number bears the same relation to a posi- 

 tion number as debts bear to property, thus adding a nega- 

 tive number to a positive number diminishes the positive 

 number as many units as there are in the negative num- 

 ber. Thus: adding 4 dollars debts to 7 dollars property 

 would leave 3 dollars property. This would be indicated 

 thus: 



+7+(-3)=+4 



On the other hand to subtract a negative number from 

 a positive number increases the latter as many units as the 

 negative number has, thus : 



+7 (3) =+10 



Also if a negative number is added to a negative number 

 it decreases the value of that negative number, thus : 



7+ (3) =10 



And if a negative number is subtracted from a negative 

 number it increases the value of that negative number : 

 -7-(-3)=-4 



