tions. To the early cultivators of the general algebraic theory, however, we are indebted for much that has been turned to useful account in the treatment of equations with numerical coefficients; and to later labourers in the same field, for a deeper insight into the texture of algebraic expressions and the laws of algebraic combinations, than we should otherwise have possessed. In this respect the "Mathematical Researches" of Mr. G. B. Jerrard, and the numerous contributions of Mr. James Cockle to the Philosophical Magazine, the Cambridge Journal, the Mathematician, and the Mechanics' Magazine, are peculiarly valuable; and may be consulted by the advanced student of algebra with very great profit. CHAPTER X. ON PROPORTION AND PROGRESSION; AND ON SIMULTANEOUS QUADRATIC EQUATIONS. DEFINITIONS, ETC. 100. When four quantities, a, b, c, d, are so related that a-b-c-d, they are said to be in arithmetical proportion; and when their relation is such as to furnish the equation a ī с they are said to be in geometrical proportion: in the former case, a, b, and c, d, have equal differences: in the latter, they have equal quotients. The difference is sometimes called the arithmetical ratio of the two quantities compared; and the quotient the geometrical ratio; but the term arithmetical ratio is falling into disuse, difference being employed instead; so that, in modern algebraical writings, the word ratio always means quotient; the phrase "the ratio of a to b," meaning "the quotient of a by b;" and thus geometrical proportion is frequently defined as an equality of ratios. 101. Although the terms quotient and ratio, in reference to two numerical or algebraical quantities, mean the same thing, yet they are usually expressed by different forms of notation; the quotient of a by b being written, and the ratio of a to b a b с being written a b; also an equality of ratios, instead of being written is more generally expressed in the form abcd, which is read thus: a is to b as c is to d, the form of expression employed to denote that the four quantities a, b, c, d, are in proportion. The four dots, put between the two equal ratios, are however sometimes replaced by the sign of equality, the proportion being expressed thus: a b c d ; but this is not the usual form. = 102. The first term, in the expression for a ratio, is called the antecedent, the other the consequent; and the first and last terms of a proportion are called the extremes; the two intermediate ones the means: thus, in the proportion above, a and c are the antecedents of the ratios, and b and d the consequents; also a and d are the extremes of the proportion, and b and c the means. 103. If ratios are multiplied together, the antecedents by the antecedents and the consequents by the consequents, the ratio of the results is called the compound ratio, or the ratio compounded of the proposed ratios: thus, if the proposed ratios be ab, cd, ef, the ratio ace : bdf, is the compound ratio derived from the given ratios. Should the given be all equal ratios, as ab, a:b, a: b, &c. then the compound ratio derived from two of them, namely, a2: b2, is called the duplicate ratio; and the compound ratio derived from three, namely, a3: b3, the triplicate ratio; also ab is called the sub-duplicate, and a/b the subtriplicate ratio; but these terms are of but infrequent use in the more modern writings of algebraists. 104. From these definitions and remarks, the learner will perceive that the subject of ratio and proportion cannot present to him any new difficulties: every ratio may be expressed as a fraction, and every proportion as an equation, the two members of which are equal fractions; and, conversely, a fraction may be expressed as a ratio, and an equation between two fractions, as a proportion. It thus appears that, by converting any proportion into an equation, and then applying the usual axiomatic principles to the latter, derived equations may be deduced which, being converted into proportions, will furnish so many inferences, in the form of proportions, from the proportion originally proposed. A variety of such inferences is actually exhibited in the "Treatise on Algebra:" in this "Introduction" it will suffice to mention but two or three. 105. Let the proportion be ab::c:d; or, which is virtu a с ally the same, let then clearing fractions, adbe; so b = d that, in a proportion, the product of the extremes is equal to the product of the means. Again, since a с b d that bad:c; that is, four quantities in proportion are also in proportion when taken INVERSELY. Moreover that is, a:cb: d, or the quantities are in proportion α when taken ALTERNATELY. Lastly, since the equation = с authorises the equation a+b c+d a+b (page 113), the given c+d proportion supplies this other, namely: ab: ab:: cd :cd; or, alternately, a±b:c±d::ab: c=d. 106. What is here said applies, of course, exclusively to geometrical proportion: when arithmetical proportion is referred to, the distinctive prefix arithmetical is always introduced: the only peculiarity worth notice in reference to four quantities, a, b, c, d, in arithmetical proportion, and of which the relation is expressed by the equation a-b-c—d, is that a+d=b+c; that is, that the sum of the extremes is equal to the sum of the means. ARITHMETICAL PROGRESSION. 107. If there be a series of quantities, such that the first be greater or less than the second, the second than the third, the third than the fourth, and so on, by the same common difference, the series of quantities is said to form an arithmetical progression: it is an increasing progression if the common difference is continually added, and a decreasing progression if subtracted: thus, a, a+d, a + 2d, a + 3d, &c. is an increasing, and a, a-d, a-2d, a-3d, &c. a decreasing arithmetical progression. The first of these series may stand indifferently for either kind if the sign of d be regarded as arbitrary. If n be made to stand for either of the numbers, 1, 2, 3, 4, &c. we may say that the nth term of such a progression is equal to the first term plus the product of the common difference by the number of terms minus 1; that is, the expression for the nth term will be a+(n−1)d, d being either positive or negative: this is obvious from a glance at the series; for the second term consists of a plus once d, the third of a plus twice d, the fourth of a plus three times d, and so on; so that the nth must consist of a plus (n-1) times d. Hence an arithmetical progression may always be written thus:— a, a+d, a+2d......, a+(n−1)d. The sum of the first and last terms of such a series is 2a+ (n-1)d, and it is plain that the same thing is also the sum of the second term and last but one; of the third and last but two; and so on; because as we advance from the first, the terms increase by d; and as we recede from the last, the terms decrease by d, so that the sum of any two terms, equidistant from the extreme terms, must always be 2a+(n-1)d. If the series consist of an odd number of terms, there must be a middle term: the term next before this will have one d fewer, and the term next following, one d more; therefore, the sum of the next preceding and next following term must be double the middle term; for whatever be taken for a middle term in the series 1, 2, 3, 4, 5, 6, 7, &c. the sum of the two adjacent numbers is double that term. Hence, when the arithmetical progression has an odd number of terms, the expression 2a+ (n-1)d represents equally the sum of any pair of equidistant terms, or double the middle term. 108. To find an expression for the sum of an arithmetical progression: Let the progression be a+(a+d)+(a+2d)+(a+3d)+......, being put for the last term; or, writing the terms in reverse order, 1+(l—d)+(l—2d)+(l—3d)+......a; adding these two equal series together, we have (a+1)+(a+1)+(a+1)+(a+1)+......(a+1), for twice the sum; and as there are n terms in each of the three series here written, twice the sum of the proposed is n(a+1), since the terms of the last series are all equal. Hence, putting S for the sum, we have Sin(a+1), or Sn{2a+(n-1)d}, since the expression for the last term is la+(n-1)d. Consequently any two of the five quantities, a, d, n, l, S, may be determined when the other three are given. (1) Required the sum of twelve terms of the series 1, 3, 5, 7, &c. Here a=1, d=2, n=12.. l=a+(n−1)d=23; ...S=\n(a+1)=6x24=144; so that the sum of twelve terms is 122; and from the second expression for S above, we see that the property is general; for when a 1, and d=2, that expression is Sn2. And the same expression shows that whenever the first term is a square, and the common difference double that square, the sum of any number of terms must be a square; thus if a=p2, and d=2p2, S is (pn)2. (2) The first term of an arithmetical progression is 5, and the 15th term is 47, what is the common difference of the terms? Here a 5, n=15, and 7-47=5+14d..d= 47-5 14 3. (3) How many terms of the series 12, 11, 11, 101, &c. must be taken to make up the number 55? Here a=12, d=-1, and S=55; therefore, by the formula, 55 in (24-1(n-1)}.: 220=n(49-n)..n2-49n=-220. The roots of this quadratic are n= 4939 -5, or 44; con sequently the sum of the terms will make 55, whether their number be five or forty-four. That forty-four terms must give the same sum as five, is easily seen: the decreasing progression can extend only to twenty-four positive terms, the twenty-fifth term being zero; the series then proceeds with negative terms, increasing in numerical value just as it before decreased; so that, after the zero term, there are 19 negative terms, which balance the 19 positive terms before the zero, leaving, for the aggregate of the whole 44 terms, only the five leading ones. (4) Insert four arithmetical means between and . Here the first term of the series is a, the last 1, the number of terms n=6, and it is required to find the common difference d. Since l=a+(n−1)d..1=1+5d::d=-20; 1 201 Hence the four intermediate terms are -2-2-2 ; that is, they are, 20, 10.* 3 *The insertion of any number of arithmetical means between two extremes is easily accomplished without any formula; for if the last extreme be subtracted from the first, and the remainder divided by the number of terms beyond the first, the quotient, with changed sign, will obviously be the common difference. |