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be constant, then A ∞ BC; let B be constant, then C ∞ AD; let C be constant, then B ∞ AD. Next, let A and B be both constant, then D∞ C; let A and C be constant, then D∞ B; let D and B be constant, then A ∞ C; let D and C be constant, then A ∞ B; let A and D be constant, then B and C will be both constant, or
vary inversely as each other, that is, B
(Art. 111.) in like manner, if B and C be constant, then A and D will both be constant, or vary inversely as each other, namely,
and D x
Lastly, if three of the quantities be con
stant, the fourth will evidently be constant.
119. To shew the use and great convenience of the conclusions derived in the preceding articles, the following examples are subjoined.
EXAMPLES.-1. Let Pany principal or sum of money lent out at interest, R=the ratio of the rate per cent. T=the time it has been lent at interest, and I=the interest; to determine the relative value of each.
First, supposing all the quantities variable,
(Art. 114.) Let I be given, then P∞ R∞
(Art. 104.) let P be given, then I ∞ RT, R ∞
(Art. 111.) let T be given, then I ∞ PR, P ∞
and T cc
; let I
let P and T be given, then I ∞ R. Lastly, let R and T be given, then I ∞ P; and if any three of the quantities be given, the fourth will be given.
2. Suppose the quantities of motion in two moving bodies to be in the ratio compounded of the quantities of matter, and the velocities, to determine the other circumstances.
First, let M the quantity of motion, Q=quantity of matter,
V=velocity; then M ∞ QV by hypothesis, wherefore Q ∞ i
and if M be given, Q c
; if Q be given, then M ∞ V; and if V be given, M oc Q.
Secondly, suppose the quantity of matter Q to be in the compound ratio of the magnitude m, and density D, or Q œ mD; by substituting mD for Q in the above expressions where Q is
found, we shall have M ∞ mDV, mD mD & T M being
given; V x or V∞ mD'
DM being given; from these it is plain that a great variety of other expressions may be obtained, and still more, by considering one or more of the quantities invariable.
Lastly, since the magnitudes of bodies are as the cubes of their homologous lines, (or d3,) that is, d3 ∞ m; if d3 be substituted for m, by proceeding as before, we shall obtain at length all the possible relations of the above quantities: but the prosecution of this is left as an exercise for the learner.
120. To investigate the rules and theorems of Geometrical Progression.
Let a the least term,
z=the greatest term,} called also the extremes.
r=the common ratio,
s=the sum of all the terms.
Then will a+ar+ar2+ar3, &c. to ar"-1, be an increasing geometrical progression.
i A progression, consisting of three or four terms only, is usually called geometrical proportion, or simply proportion. One important property of a geometrical progression is this, namely, the product of the two extreme terms is equal to that of any two terms equally distant from the extremes: hence, in
From the former of these we have ar11the last term of the series, but z=the last term by the notation, wherefore ar"-1=z;
from this equation we obtain a= (THEOR. 1.) z=ar”—1
(THEOR. 2.) r= (THEOR. 3.) and since 1:r::a+ar+ ar2: ar+ar2+ar3, (Art. 72.) that is, 1:r::s-z: s-a, therefore s-a=r.s-z, whence r= 3—4 (THEOR. 4.) a=s—r.s—Z (THEOR. 5.)
(THEOR. 7.) but since .1 by th. 2. substitute this value for z in th. 7, and s=
(THEOR. 6.) and s=
(THEOR. 8.) whence a=
(THEOR. 9.) and since r==
(th. 3.) and s=- (th. 7.) if for r in the latter, its
(THEOR. 10.) and because (th. 4.) s—a=sr—zr, and (th. 1.)
therefore (s—a=) s—
-=sr-zr, or sr—s= (zr—
(THEOR. 11.) con
The above theorems are all that can be deduced in a general manner, without the aid of logarithms in some cases, and of equations of several dimensions in others. The theorems wanting are four for finding n, two for r, one for a, and one for z: the four theorems for finding the value of n, may be expressed
four proportionals, the product of the two extremes is equal to the product of the two means; and in three proportionals, the product of the extremes is equal to the square of the mean.
logarithmically; the remaining four cannot be given in a general manner, but their relation to the other quantities may be expressed in an equation, by means of which any particular value will be readily known.
121. We proceed then, first, to deduce the equations from whence the remaining values of r, a, and z, may be found in any particular case; next, we shew how the theorems found are to be turned into their equivalent logarithmic expressions; and lastly, we shall deduce logarithmic theorems for the four expressions of the value of n.
First, because z=ar11 (th. 2.) and z=
a-s, or rn-.
whence ar"=sr-s+a, or ar¤—sr=
===3 (THEOR. 13.) which is as near as we can
get to the value of r, and which (supposing a, s, and n, given in numbers) if n be greater than 2, will require the solution of a high equation to find its value.
Secondly, because s—a=sr—zr, (th. 4.) and (th. 1.) a=
, therefore (s—a=) s— =sr-zr, and zr"-z=sr"
(THEOR. 14.) this equation being solved, the value of r will be
Thirdly, since s—a—sr-zr, (th. 4.) and r=
therefore s-a=s.— -2. ', or a.s—al"—1=z.s—2—1,
(THEOR. 15.) by the solution of which equation (s, n, and z, being given) a will be found.
Fourthly, by the same equation, viz. a.s
(THEOR. 16.) s, n, and a, being given, z will likewise be known.
122. It remains now to put the above theorems into a logarithmical form, to place the whole in one point of view, and to deduce the four theorems for finding the value of n: observing that to multiply two factors together, we add their logarithms together; to divide, we subtract the logarithm of the divisor from that of the dividend; to involve or evolve, we multiply
or divide respectively the logarithm of the root or power by its index, as directed in Vol. I. Part 2.
And L the logarithm of the quantity to which it is prefixed; then will the following synopsis exhibit the whole doctrine of geometrical progression, as investigated in the preceding articles.
* Some of the following logarithmic expressions are extremely inconvenient, particularly theor. 10. The best method of computing the value of s in that theorem, will be, first to find the log. of z, subtract the log. of a from it, add this remainder to the log. of z, and divide the sum by n-1; find the natural number corresponding to the quotient, from which subtract a, and find the log. of the remainder. Secondly, from the log. of z, subtract the log. of a, divide the remainder by n-1, find the natural number corresponding to the quotient, subtract 1 from it, and subtract the log. of this remainder from that of the former; and the like in other cases.