167 a с с a HARMONICAL PROGRESSION AND PROPORTION. When the reciprocals of a series of numbers form an arithmetical progression, the numbers themselves constitute an harmonical progression. Thus, į, , 1, ... }, }, s, it, ..., or oran ... constitute an harmonical a + B'a + 2b'a + 3b series, or an harmonical progression. If, therefore, we are required to form an harmonical progression, or to find the law from a sufficient number of terms given in any part of it, we have only to form the reciprocals, and thence the arithmetical progression, and finally to take the reciprocals of these. The resulting series is the one sought. Thus, to find a fourth harmonical to the given terms }, }, }, we have only to take 2, 3, 4, and find the fourth arithmetical proportional to 2, 3, 4, viz. 5. Then the reciprocal of this, viz. }, is the fourth term sought. It is, however, sufficient to give two of the terms immediately preceding that sought, as the arithmetical progression of reciprocals follows from these two, without employing any of the more remotely precedent ones of the series *. It will be obvious, that as an arithmetical series does not admit of indefinite extension in one direction, without employing negative numbers ; so also the harmonical series does not admit of indefinite extension in the other direction, without also employing negative numbers. Employing, however, as is always done in Algebra, both positive and negative numbers, all three series admit of indefinite extension both ways. It is with this explanation that the remark made by writers on these classes of series is to be understood, when they say that the arithmetical series admits of indefinite extension only upwards, the harmonical only downwards, and the geometrical, in both directions, both increasing and decreasing. The following properties furnish a specimen of those which belong to numbers in barmonical progression. 1. If there be three terms in harmonical progression, then, the first is to the third as the difference of the first and second is to the difference of the second and third. For, by the definition, if a, b, c be three numbers in arithmetical progression, 1 1 1 b-C - b will be in harmonical progression. Also, in virtue of the a b c abc atithmetical progression, and by proportion we obtain from this ..:: 4 ab b 1 1 1 bc The converse of this is obviously true: that if three given numbers fulfil this proportion they are harmonicals. 2. The harmonical mean between two numbers is equal to twice the product of those numbers divided by their sum. For in the preceding proportion we 2 bave from which ab a abc 1 1 a-6 a с 1 : b An extension of the definition, so as to render the fourth term dependent on the three immediately precedent ones, has been given by several authors, though that definition has no reference whatever to musical intervals, nor do the terms of it form the reciprocals of an arithmetical series. 1 с a 1 where a, b, c are the arithmetical reciprocals of the three harmonical terms. 3. A third harmonical to two given terms is equal to the product of those terms divided by the difference between twice the first and the second terms. For from the same we have 1 b 4. In an harmonical series, any three terms, the extremes of which are equi. distant from the extremes, are in harmonical progression. For their reciprocals are in arithinetical progression *. 5. Let a, b, c, h, k, be an harmonical series : then The product of any two adjacent terms is to the product of any other two adjacent terms, as the difference of the first pair is to the difference of the second pair. 1 1 1 For by definition. a k k h hk k - h h - K 6. 6. When the first two terms a and b are given, the nth may be thus expressed: ab b and b ab 6 (n − 1) a. - (n ub 1) a (n b (12 h a or a a 2) 6 a ز a 1 a 2) b * As the doctrine of geometrical proportion, which essentially involves four terms, has been applied to three only, by taking the second and third as identical (thereby constituting continued proportionals), so in the harmonical progression continued (each step of which essentially contains three terms), the second term has been supposed to be replaced by a different one, which stands as the third, whilst that which originally stood as the third, thereby becomes the fourth. In this case the four numbers have been said to be in harmonical proportion, when the first is to the fourth as the difference of the first and second is to the difference of the third and fourth : thus, if a, b, c, d fulfil the condition a:d::ambicnd, the four quantities a, b,c,d , are harmonicals. In this case, if c be made = h, and d = c, the harmonicals previously defined will result; but the definition of harmonicals there given does not apply to any of the other numbers which fulfil this condition. Some writers also speak of contra-hurmonicals. Into the study of each of these kinds of proportion, the reader who desires to enter will find ample information in Malcolm's Arithmetic, pp. 297–313, 1730. Further notice of them here would be incompatible with the plan and objects of this Course of Mathematics. from which the arithmetical progression, and thence the harmonical, is readily found. EXAMPLES. = 2, 1. Find the fifth term of an harmonical series whose first and second terms are 3 and 4, and likewise of that whose first and second terms are 4 and 3. In the first case, į and are the first and second terms of an arithmetical progression descending, since the barmonical progression ascends. Hence, is the greatest term ; and we have - = = common difference = d; and z= j, and n= = 5. Therefore the term a, or least term, is j - = } – š= 0, and the fifth term is therefore ; = infinite. In the second case the harmonical series descends, and hence the arithmetical one ascends; and, therefore, as before, d=1, and n = 5, and the least term is te: the fifth term is, therefore, št i = h = The fifth term of the harmonical series, 4, 3, &c. is therefore 1. 2. Find an harmonical mean between 3 and 4, and six harmonical means between 1 and 2. 3. An harmonical series consists of fifteen terms, and the greatest and least terms are x and y: what is the middle term ? 4. A line, whose length is 10 inches, is divided harmonically, so that the first section (from the origin) is 3 inches : how far distant is the second point of section from the first? 5. Four terms are in harmonical proportion ; the first and last are 6 and 10: what is the relation between the second and third ? Ans. 10x + 6y = 120; where x is the second, and y the third term. 6. The first and second terms of an harmonical proportion are 4,5 : and the first, second, and third terms of an harmonic progression are also 4, 5, 6. Find the fourth term of the proportion, and the fourth term of the progression. 7. Ten termes are in harmonical progression, and the last two are fo and 11: what are the terms, and what is their sum? EQUATIONS. An equation is the algebraic expression of the equality of two assemblages of quantities to one another; and consists in writing =, the sign of equality, between them *. Thus 10 — 4 = 6 is an equation expressing the equality of 10 – 4 to 6; and 4x + b = c d is an equation expressing that 4x + b is equal to c d. Equations are designated by different names, according to the manner of their composition, and the highest power of the unknown quantity which enters into them. When the highest power is the first, the equation is called a simple equation, or an equation of the first degree : when it is of the second, the equation is The mark here used was introduced into algebra by the first English author on the subject, Robert Recorde, in his “Whetstone of Witte,” (sig. Ff. 1b,) 1557. He gives his reason in his own quaint manner in the following words : “ And to avoide the tediouse repetition of these woordes : is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicause noe 2 thynges can be more equalle.” For a long period afterwards, the Continental mathematicians employed the symboles, which was, doubtless, a rapid formation of the diphthong e, the initial of the phrase æquale est. a called a quadratic equation, or an equation of the second degree : when it is of the third, the equation is called a cubic equation, or an equation of the third degree : and so on. The known, or given quantities, are represented by the earlier letters of the alphabet, a, b, c, and the unknown, or quantities whose values are sought, by the later ones, 2, Y, Z, W, It often happens that equations arise which are composed of only two terms, in which the power of the unknown is of a higher degree than the first. These are called binomial equations, as 206 = - 100; and otherwise pure equations, to distinguish them from adfected (or affected) equations. These are, for the purposes of solution, considered as simple equations, the operations that are requisite for completing the solution being purely of the arithmetical kind. Examples in which these occur are therefore classed amongst those of simple equations. When there are several equations given, the unknown of which in one is capable of such multiplications or divisions by that in some of the others, or when they admit of a ready combination with one another, so as to form results that are known to be powers of some binomial or trinomial expression, they are frequently classed also amongst the exercises on simple equations. Such a method of classification is, evidently, very arbitrary; and hence there are several questions in the following series which are by some authors distributed under a different denomination : though in general this classification accords with the most common practice of algebraical writers. The quantities which precede the mark of equality, are often called together the first member or the first side of the equation ; those which follow it, the second member or the second side. The resolution of equations, is the finding the value of the unknown quantity, or in disengaging that quantity from the known ones; and this consists in so transforming the equation, that the unknown letter or quantity may stand alone on one side of the equation, without a coefficient; and all the rest, or the known quantities, on the other side. SIMPLE EQUATIONS, WITH ONE UNKNOWN. In these, the unknown quantity, when properly transformed, is of the first degree, as dx = :b, and its solution in this state is obvious: but as they seldom so occur, we must lay down the principles of transformation so as to disengage x from all other quantities on one side of the equation. In general, the unknown quantity is disengaged from the known ones, by performing always the reverse operations. That is, if the known quantities are connected with it by t, or addition, they must be subtracted; if by minus, -, or subtraction, they must be added ; if by multiplication, we must divide by them; if by division, we must multiply; when it is in any power, we must extract the corresponding root; and when in any radical, we must raise it to the corresponding power. The following special rules are founded on this general principle, viz.: that when equivalent operations are performed on equal quantities, the results must still be equal; whether hy equal additions, subtractions, multiplications, divisions, extractions, or involutions. I. When known quantities are connected with the unknown by + or transpose them to the other side or member of the equation, and change their signs. Which is only adding or subtracting the same quantities on both sides, -; in order to get all the unknown terms on one side of the equation, and all the known ones on the other *. The same rule applies whether the known quantities be given in numbers or in symbols. Thus, if x + 5 = 8; then transposing 5, gives x = 8 - 5 = 3. II. When the unknown term is multiplied by any quantity; divide all the terms of the equation by it. * Here it is earnestly recommended that the pupil be accustomed, at every line or step in the reduction of the equations, to name the particular operation to be performed on the preceding equation, in order to produce the next form or state of the equation, in applying each of these rules, according as the particular form of the equation may require : applying them according to the order in which they are here placed, and always allotting a single line for each operation and its description, and ranging the equations under each other, in the several lines, as they are successively produced. The master, indeed, never ought to receive a solution from his pupil in writing in which this rule is not complied with, and as much attention given to the proper concatenation of the verbal descriptions as to the mere work set down in the algebra. Due regard being had to this point would prevent algebra from becoming a mere piece of ingenious mechanism, as it now too often does become. The procedure here enforced differs in no respect from that employed by the earlier algebraical writers, as may be seen by reference to Wallis, Ronayne, Kersey, Ward, and others. It was also a useful custom, and one which has been recently revived, to number the several successive steps of the process, and to quote the equation by means of the number attached to it. The older writers ruled a column down the middle of the page in which to put the ordinal numbers, and kept the written description of the process on the left, and the work itself on the right of this column. However, in the extended equations, to which modern physical science gives rise, the great inequality in the length of the lines renders it more convenient to write the ordinal numbers, (1), (2), (3) ... at the margin of the page. The mode of taking the ordinal column down the middle is better, however, for the learner, as his work is thereby kept in one vertical column to the right of it, and is therefore much more easily inspected by himself as well as by the master. On this account its adoption is advised in the earlier stages of study, even though it may ultimately be laid aside when good and regular habits are formed. Thus, if the equations a? - y2 =a?, and x +y=b, had been given, we should have had 1 — yo = a? Given 2 x + y = 6 Divide (1) by (2) then 3 6 Add (2), (3) together, 4 2x = +0 or reducing to common den. 5 a? + 12 b In this notation a figure enclosed in a parenthesis, as (2) or (3) indicates the words “the equation marked two," whilst in the case of no parenthesis, it signifies the number 2. This subject is illustrated and enforced very elegantly in Butler's Course of Mathematics, vol. ii. p. 17. The author correctly traces the first proposal of the practice to Dr. Pell, an eminent analyst of the early English school. |