XVII. If three quantities be in continued proportion, the first will have to the third the duplicate ratio of that which it has to the second. XVIII. If four quantities be in continued proportion, the first will have to the fourth the triplicate ratio of that which it has to the second. Let a, b, c, d be four quantities in continued proportion, so that a:b::b:c::c:d; then, also, a:d::a3: b3. XIX. If two proportions be multiplied together, term by term, the products will form a proportion. The compatibility of any change in the order of the terms of a proportion may be tested by forming the product of the extremes and means in both the original and changed proportion, when, if they agree, the change is correct. Thus, a:b::c:d may be written d:b::c: a, for we have ad=bc in both. EXAMPLES IN PROPORTION. (1) The mercurial barometer stands at a height of 30 inches, and the specific gravity of quicksilver is 1333. How high would a water barometer stand? (2) The weights of a lever have the same ratio as the lengths of the oppoThe ratio of the weights is 5, and the longer arm 10 inches. What is the length of the shorter arm? Ans. 2 inches. site arms. (3) The weights of a lever are 6 and 8 pounds, and the length of the shorter arm 18 inches. What is that of the longer? Ans. 24 inches. (4) At the end of an arm of a lever 5 inches long, what weight can be supported by 21 pounds acting at the end of an arm 43 inches long? (5) Triangles are to each other as the products of their bases by their altitudes. The bases of two triangles are to each other as 17 and 18, and their altitudes as 21 and 23. What is the ratio of the triangles themselves? Ans. 119:138. (6) The force of gravitation is inversely as the square of the distance. At the distance 1 from the centre of the earth this force is expressed by the number 32.16. By what is it expressed at the distance 60? Ans. 0.0089. (7) The motion of a planet about the sun for a short space is proportional to unity divided by the duplicate of the distance. If the motion be represented by v when the distance is r, by what will it be expressed when the distance is r'? r2y Ans. 72* (8) The times of revolution of the planets about the sun are in the sesquiplicate ratio of their mean distances. The mean distance of the earth from the sun being expressed by 1, that of Jupiter will be 5.202776; the time of revolution of the earth is 365.2563835 days. What is the time of revolution of Jupiter? Ans. 4332.5848212 days. EQUATIONS. PRELIMINARY REMARKS. 134. As equation, in the most general acceptation of the term, is composed of two algebraic expressions which are equal to each other, connected by the sign of equality. Thus, ax=b, cx2+dx=e, cr3+gx2=hx+k, mx2+nx3+px2+qx+r=0, are equations. The two quantities separated by the sign are called the members of the equation, the quantity to the left of the sign = is called the first member, the quantity to the right the second member. The quantities separated by the signs + and are called the terms of the equation. 135. Equations are usually composed of certain quantities which are known and given, and others which are unknown. The known quantities are in general represented either by numbers, or by the first letters in the alphabet, a, b, c, &c.; the unknown quantities by the last letters, s, t, x, y, z, &c. 136. Equations are of different kinds. 1o. An equation may be such that one of the members is a repetition of the other; as, 2x-5=2x-5. 2o. One member may be merely the result of certain operations indicated in the other member; as, 5x+16=10x-5-(5x-21), (x+y)(x—y)=x2-y3, 3o. All the quantities in each member may be known and given; as, 25=10 +15, a+b=c-d, in which, if we substitute for a, b, c, d the known quantities which they represent, the equality subsisting between the two members will be self-evident. In each of the above cases the equation is called an identical equation. 4. Finally, the equation may contain both known and unknown quantities, and be such that the equality subsisting between the two members can not be made manifest, until we substitute for the unknown quantity or quantities certain other numbers, the value of which depends upon the known numbers which enter into the equation. The discovery of these unknown numbers constitutes what is called the solution of the equation. When found and put in the place of the letters which represent them, if they make the equality of the two members evident, the equation is said to be verified, or satisfied. The word equation, when used without any qualification, is always understood to signify an equation of this last species; and these alone are the objects of our present investigations. x+4=7 is an equation properly so called, for it contains an unknown quantity x, combined with other quantities which are known and given, and the equality subsisting between the two members of the equation can not be made manifest until we find a value for 2, such that, when added to 4, the result will be equal to 7. This condition will be satisfied if we make x=3; and this value of r being determined, the equation is solved. The value of the unknown quantity thus discovered is called the root of the equation, being the radix out of which the equation is formed; the term root here has a different sense from that in which we have hitherto used it, viz., that of the base of a power. 137. Equations are divided into degrees according to the highest power of the unknown quantity which they contain. Those which involve the simple or first power only of the unknown quantity are called simple equations, or equations of the first degree; those into which the square of the unknown quantity enters are called quadratic equations, or equations of the second degree: so we have cubic equations, or equations of the third degree; biquadratic equations, or equations of the fourth degree; equations of the fifth, sixth, nth degree. Thus, ax+b=cx+d x23+px2=2q is a simple equation. is a quadratic equation. x"+px11+qx12+, &c., =r, is an equation of the nth degree. 138. Numerical equations are those which contain numbers only, in addition to the unknown quantities. Thus, x3+5x=3x+17 and 4x=7y are numerical equations. 139. Literal equations are those in which the known quantities are represented by letters only, or by both letters and numbers. Thus, 23+px2+qx=r, x-3px3y+5qxy+rry3=5 are literal equations. 140. Let us now pass on to consider the solution of equations, it being understood that to solve an equation is to find the value of the unknown quantity, or to find a number which, when substituted for the unknown quantity in the equation, renders the first member identical with the second. The difficulty of solving equations depends upon the degree of the equations and the number of unknown quantities. We first consider the most simple case. ON THE SOLUTION OF SIMPLE EQUATIONS CONTAINING ONE UNKNOWN QUANTITY. 141. The various operations which we perform upon equations, in order to arrive at the value of the unknown quantities, are founded upon the following axioms: If to two equal quantities the same quantity be added, the sums will be equal. If from two equal quantities the same quantity be subtracted, the remainders will be equal. If two equal quantities be multiplied by the same quantity, the products will be equal. If two equal quantities be divided by the same quantity, the quotients will be equal. These axioms, when applied to the two equal quantities which constitute the two members of every equation, will enable us to deduce from them new equations, which are all satisfied by the same value of the unknown quantity, and which will lead us to discover that value. 142. The unknown quantity may be combined with the known quantities in the given equation by the operations of addition, subtraction, multiplication, and division. We shall consider these different cases in succession. I. Let it be required to solve the equation x+a=b. If, from the two equal quantities r+a and b, we subtract the same quantity a, the remainders will be equal, and we shall have or x+a-a-b-a, x=b-a, the value of r required. So, also, in the equation x+6=24. Subtracting 6 from each of the equal quantities x+6 and 24, the result is x=24-6 II. Let the equation be 18, the value of x required. x-a-b. If, to the two equal quantities x-a and b, the same quantity a be added, the sums will be equal; then we have x=a+a=b+a, or x=b+a, the value of r required. So, also, in the equation x-6=24. Adding 6 to each of these equal quantities, the result is x=24+6 =30, the value of x required. It follows from (I.) and (II.) that We may transpose any term of an equation from one member to the other by changing the sign of that term.* We may change the signs of every term in each member of the equation with out altering the value of the expression.† If the same quantity appear in each member of the equation affected with th same sign, it may be suppressed. Dividing each of these equals by 6, the result is r=4, the value of r required. From this it follows that, When one member of an equation contains the unknown quantity alone, affected with a coefficient, and the other member contains known quantities only, the value of the unknown quantity is found by dividing each member of the equation by the coefficient of the unknown quantity Multiplying each of these equals by a, the result is x=ab, the value of x required. When one member of the equation contains the unknown quantity alone, divided by a known quantity, and the other member contains known quantities only, the value of the unknown quantity is found by multiplying each member of the equation by the quantity which is the divisor of the unknown quantity. V. Let the equation be In order to solve this equation, we must clear it of fractions; to effect this, reduce the fractions to equivalent ones, having a common denominator (Art. 41), the equation becomes Multiply these equal quantities by the same quantity ben, or, which is evi * If we transpose a plus term, it subtracts this term from both members; and if we transpose a minus term, it adds this term to both. † This is, in fact, the same thing as transposing every term in each member of the equa tion, or multiplying throughout by -1. |