substituting this value in the expression for x, we have x=a+ b bb'+1' for the third approximation to x. We may find a fourth by 1 making y=b'+; for if b' designate the entire number immediately below y', we shall have We have already seen (218), that the smallest of the positive roots of this equation is found between and, that is, between 1 and 2; we make therefore x=1+ y ; we shall then have y3 — 4y2 + 3y+1=0. The limit to the positive roots of this last equation is 5, and by substituting successively 0, 1, 2, 3, 4, in the place of y, we immediately discover, that this equation has two roots greater than unity, one between 1 and 2, and the other between 2 and 3. Hence that is x=1+ and x=1+1, x=2 and x=3. These two values correspond to those, which were found above between § and §, and between § and, and which differ from 흉 each other by a quantity less than unity. In order to obtain the first, which answers to the supposition of y = 1, to a greater degree of exactness, we make We find in this equation only one root greater than unity, and that is between 2 and 3, which gives 1 Again, if we suppose y' = 2+, we shall be furnished with the equation 3 y's — 3 y′′2 — 4y"—1=0; we find the value of y" to be between 4 and 5 ; taking the smallest of these numbers 4, we have y' = 2+1, y=1+3=1, x=1+ //}=}{} • It would be easy to pursue this process, by making y" =4+ and so on. I return now to the second value of x, which, by the first approximation, was found equal to, and which answers to the 1 supposition of y = 2. Making y=2+, and substituting this expression in the equation involving y, we have, after changing the signs in order to render the first term positive, 2 This equation, like the corresponding one in the above operation, has only one root greater than unity, which is found between 1 and 2; taking y' = 1, we have we are furnished with the equation y's — 3y's — 4 y" — 1 = 0, in which y" is found to be between 4 and 5, whence y' = 4, y = 14, 19 We may continue the process by making y" = 4+ The equation x3-7x+7=0 has also one negative root In order to approach it more nearly, between 3 and we make x=- 3 whence y3 — 20 y2 — 9 y — 1 = 0, y ▷ 20 and ◄ 21, To proceed further, we may suppose y = 20 + shall then obtain successively values more and more exact. The several equations transformed into equations in y, y', y", &c. will have only one root greater than unity, unless two or more roots of the proposed equation are comprehended within the same limits a and a + 1; when this is the case, as in the above example, we shall find in some one of the equations in y, y', &c. several values greater than unity. These values will introduce the different series of equations, which show the several roots of the proposed equation, that exist within the limits a and a + 1. The learner may exercise himself upon the following equation x3-2x-5=0, the real root of which is between 2 and 3; we find for the entire values of y, y', &c. 10, 1, 1, 2, 1, 3, 1, 1, 12, &c. and for the approximate values of x, 21 23 111 155 576 † 1, 11, 11, W13, 14, 178, 381, 1307, 16415 Of proportion and progression. 7837 223. ARITHMETIC introduces us to a knowledge of the definition and fundamental properties of proportion and equidifference, or of what is termed geometrical and arithmetical proportion. I now proceed to treat of the application of algebra to the principles there developed; this will lead to several results, of which frequent use is made in geometry. I shall begin by observing, that equidifference and proportion may be expressed by equations. Let A, B, C, D, be the four terms of the former, and a, b, c, d, the four terms of the latter; we have then equations, which are to be regarded as equivalent to the expres Hence it follows, that in equidifference the sum of the extreme terms is equal to that of the means, and in proportion the product of the extremes is equal to the product of the means, as has been shown in Arithmetic (127, 113), by reasonings, of which the above equations are only a translation into algebraic expressions. The reciprocal of each of the preceding propositions may be easily demonstrated; for from the equations and consequently, when four quantities are such, that two among them give the same sum, or the same product, as the other two, the first are the means and the second the extremes (or the converse) of an equidifference or proportion. When BC, the equidifference is said to be continued; the same is said of proportion, when b = c. We have in this case A+D=2B, ad=b2: that is, in continued equidifference the sum of the extremes is equal to double the mean; and in proportion, the product of the extremes is equal to the square of the mean. From this we deduce the quantity B is the middle or mean arithmetical proportional between A and D, and the quantity b the mean geometrical proportional between a and d. from which it is evident, that we may change the relative places of the means in the expressions A. B: C. D, a:b::c:d, and in this way obtain A. C: B. D, a:c::b: d. In general, we may make any transposition of the terms, which is consistent with the equations A+D=B+C and ad=bc (Arith. 114.) I have now done with equidifference, and shall proceed to consider proportion simply. b a 224. It is evident, that to the two members of the equation d = we may add the same quantity m, or subtract it from c reducing the terms of each member to the same denominator, we obtain These two proportions may be enunciated thus; The first consequent plus or minus its antecedent taken a given number of times, is to the second consequent plus or minus its antecedent taken the same number of times, as the first term is to the third, or as the second is to the fourth. Taking the sums separately and comparing them together and also the differences, we obtain or rather, by changing the relative places of the means b+ma:b-ma::d+mc:d-mc; and if we make m=1, we have simply baba::d+c:d-c, which may be enunciated thus ; The sum of the two first terms is to their difference as the sum of the two last is to their difference. |