ON THE PROPORTION OF NUMBERS.
170. When two numbers are named in any problem, it is usually
necessary, in some way or other, to compare the two; that is, by
considering the two together, to establish some connexion between
them, which may be useful in future operations. The first method
which suggests itself, and the most simple, is to observe which is
the greater, and by how much it differs from the other. The connexion
thus established between two numbers may also hold good of two other
numbers; for example, 8 differs from 19 by 11, and 100 differs from
111 by the same number. In this point of view, 8 stands to 19 in the
same situation in which 100 stands to 111, the first of both couples
differing in the same degree from the second. The four numbers thus
noticed, viz.:
8, 19, 100, 111,
are said to be in _arithmetical[26] proportion_. When four numbers are
thus placed, the first and last are called the _extremes_, and the
second and third the _means_. It is obvious that 111 + 8 = 100 + 19,
that is, the sum of the extremes is equal to the sum of the means.
And this is not accidental, arising from the particular numbers we
have taken, but must be the case in every arithmetical proportion; for
in 111 + 8, by (35), any diminution of 111 will not affect the sum,
provided a corresponding increase be given to 8; and, by the definition
just given, one mean is as much less than 111 as the other is greater
than 8.
[26] This is a very incorrect name, since the term ‘arithmetical’
applies equally to every notion in this book. It is necessary, however,
that the pupil should use words in the sense in which they will be used
in his succeeding studies.
171. A set or series of numbers is said to be in _continued_
arithmetical proportion, or in arithmetical _progression_, when the
difference between every two succeeding terms of the series is the
same. This is the case in the following series:
1, 2, 3, 4, 5, &c.
3, 6, 9, 12, 15, &c.
(1½), 2, (2½), 3, (3½), &c.
The difference between two succeeding terms is called the common
difference. In the three series just given, the common differences are,
1, 3, and ½.
172. If a certain number of terms of any arithmetical series be taken,
the sum of the first and last terms is the same as that of any other
two terms, provided one is as distant from the beginning of the series
as the other is from the end. For example, let there be 7 terms, and
let them be,
_a_ _b_ _c_ _d_ _e_ _f_ _g_.
Then, since, by the nature of the series, _b_ is as much above _a_ as
_f_ is below _g_ (170), _a_ + _g_ = _b_ + _f_. Again, since _c_ is as
much above _b_ as _e_ is below _f_ (170), _b_ + _f_ = _c_ + _e_. But
_a_ + _g_ = _b_ + _f_; therefore _a_ + _g_ = _c_ + _e_, and so on.
Again, twice the middle term, or the term equally distant from the
beginning and the end (which exists only when the number of terms is
odd), is equal to the sum of the first and last terms; for since _c_
is as much below _d_ as _e_ is above it, we have _c_ + _e_ = _d_ + _d_
= 2_d_. But _c_ + _e_ = _a_ + _g_; therefore, _a_ + _g_ = 2_d_. This
will give a short rule for finding the sum of any number of terms of
an arithmetical series. Let there be 7, viz. those just given. Since
_a_ + _g_, _b_ + _f_, and _c_ + _e_, are the same, their sum is three
times (_a_ + _g_), which with _d_, the middle term, or half _a_ + _g_,
is three times and a half (_a_ + _g_), or the sum of the first and
last terms multiplied by (3½), or ⁷/₂, or half the number of terms. If
there had been an even number of terms, for example, six, viz. _a_,
_b_, _c_, _d_, _e_, and _f_, we know now that _a_ + _f_, _b_ + _e_, and
_c_ + _d_, are the same, whence the sum is three times (_a_ + _f_), or
the sum of the first and last terms multiplied by half the number of
terms, as before. The rule, then, is: To sum any number of terms of an
arithmetical progression, multiply the sum of the first and last terms
by half the number of terms. For example, what are 99 terms of the
series 1, 2, 3, &c.? The 99th term is 99, and the sum is
99 100 × 99
(99 + 1)---, or --------, or 4950.
2 2
The sum of 50 terms of the series
1 2 4 5 ( 1 50 ) 50
---, ---, 1, ---, ---, 2, &c. is (--- + ---)---,
3 3 3 3 ( 3 3 ) 2
or 17 × 25, or 425.
173. The first term being given, and also the common difference and
number of terms, the last term may be found by adding to the first term
the common difference multiplied by one less than the number of terms.
For it is evident that the second term differs from the first by the
common difference, the _third_ term by _twice_, the _fourth_ term by
_three_ times the common difference; and so on. Or, the passage from
the first to the _n_th term is made by _n_-1 steps, at each of which
the common difference is added.
EXERCISES.
_Given._ | _To find._
Series. |No. of terms.| Last term. | Sum.
4, (6½), 9, &c. | 33 | 84 | 1452
1, 3, 5, &c. | 28 | 55 | 784
2, 20, 38, &c. | 100,000 | 1799984 | 89999300000
174. The sum being given, the number of terms, and the first term,
we can thence find the common difference. Suppose, for example, the
first term of a series to be one, the number of terms 100, and the sum
10,000. Since 10,000 was made by multiplying the sum of the first and
last terms by ¹⁰⁰/₂, if we divide by this, we shall recover the sum
of the first and last terms. Now, ¹⁰,⁰⁰⁰/₁ divided by ¹⁰⁰/₂ is (122)
200, and the first term being 1, the last term is 199. We have then to
pass from 1 to 199, or through 198, by 99 equal steps. Each step is,
therefore, ¹⁹⁸/⁹⁹, or 2, which is the common difference; or the series
is 1, 3, 5, &c., up to 199.
_Given._ | _To find._
Sum. |No. of terms.|First term.|Last term.|Common diff.
1809025 | 1345 | 1 | 2689 | 2
44 | 10 | 3 | ²⁹/₅ | ¹⁴/₄₅
7075600 | 1330 | 4 | 10636 | 8
175. We now return to (170), in which we compared two numbers together
by their difference. This, however, is not the method of comparison
which we employ in common life, as any single familiar instance will
shew. For example, we say of A, who has 10 thousand pounds, that he is
much richer than B, who has only 3 thousand; but we do not say that
C, who has 107 thousand pounds, is much richer than D, who has 100
thousand, though the difference of fortune is the same in both cases,
viz. 7 thousand pounds. In comparing numbers we take into our reckoning
not only the differences, but the numbers themselves. Thus, if B and D
both received 7 thousand pounds, B would receive 233 pounds and a third
for every 100 pounds which he had before, while D for every 100 pounds
would receive only 7 pounds. And though, in the view taken in (170), 3
is as near to 10 as 100 is to 107, yet, in the light in which we now
regard them, 3 is not so near to 10 as 100 is to 107, for 3 differs
from 10 by more than twice itself, while 100 does not differ from 107
by so much as one-fifth of itself. This is expressed in mathematical
language by saying, that the _ratio_ or _proportion_ of 10 to 3 is
greater than the _ratio_ or _proportion_ of 107 to 100. We proceed to
define these terms more accurately.
176. When we use the term _part_ of a number or fraction in the
remainder of this section, we mean, one of the various sets of _equal_
parts into which it may be divided, either the half, the third, the
fourth, &c.: the term multiple has been already explained (102). By
the term _multiple-part_ of a number we mean, the abbreviation of the
words _multiple of a part_. Thus, 1, 2, 3, 4, and 6, are parts of 12;
½ is also a part of 12, being contained in it 24 times; 12, 24, 36,
&c., are multiples of 12; and 8, 9, ⁵/₂, &c. are multiple parts of 12,
being multiples of some of its parts. And when multiple parts generally
are spoken of, the parts themselves are supposed to be included, on
the same principle that 12 is counted among the multiples of 12, the
multiplier being 1. The multiples themselves are also included in this
term; for 24 is also 48 halves, and is therefore among the multiple
parts of 12. Each part is also in various ways a multiple-part; for
one-fourth is two-eighths, and three-twelfths, &c.
177. Every number or fraction is a multiple-part of every other number
or fraction. If, for example, we ask what part 12 is of 7, we see
that on dividing 7 into 7 parts, and repeating one of these parts 12
times, we obtain 12; or, on dividing 7 into 14 parts, each of which
is one-half, and repeating one of these parts 24 times, we obtain 24
halves, or 12. Hence, 12 is ¹²/₇, or ²⁴/₁₄, or ³⁶/₂₁ of 7; and so on.
Generally, when _a_ and _b_ are two whole numbers, _a_/_b_ expresses
the multiple-part which _a_ is of _b_, and _b_/_a_ that which _b_ is
of _a_. Again, suppose it required to determine what multiple-part
(2⅐) is of (3⅕), or ¹⁵/₇ of ¹⁶/₅. These fractions, reduced to a common
denominator, are ⁷⁵/₃₅ and ¹¹²/₃₅, of which the second, divided into
112 parts, gives ¹/₃₅, which repeated 75 times gives ⁷⁵/₃₅, the first.
Hence, the multiple-part which the first is of the second is ⁷⁵/₁₁₂,
which being obtained by the rule given in (121), shews that _a_/_b_, or
_a_ divided by _b_, according to the notion of division there given,
expresses the multiple-part which _a_ is of _b_ in every case.
178. When the first of four numbers is the same multiple-part of the
second which the third is of the fourth, the four are said to be
_geometrically[27] proportional_, or simply _proportional_. This is
a word in common use; and it remains to shew that our mathematical
definition of it, just given, is, in fact, the common notion attached
to it. For example, suppose a picture is copied on a smaller scale,
so that a line of two inches long in the original is represented by a
line of one inch and a half in the copy; we say that the copy is not
correct unless all the parts of the original are reduced in the same
proportion, namely, that of 2 to (1½). Since, on dividing two inches
into 4 parts, and taking 3 of them, we get (1½), the same must be done
with all the lines in the original, that is, the length of any line in
the copy must be three parts out of four of its length in the original.
Again, interest being at 5 per cent, that is, £5 being given for the
use of £100, a similar proportion of every other sum would be given;
the interest of £70, for example, would be just such a part of £70 as
£5 is of £100.
[27] The same remark may be made here as was made in the note on the
term ‘arithmetical proportion,’ page 101. The word ‘geometrical’ is,
generally speaking, dropped, except when we wish to distinguish between
this kind of proportion and that which has been called arithmetical.
Since, then, the part which _a_ is of _b_ is expressed by the fraction
_a_/_b_, or any other fraction which is equivalent to it, and that
which _c_ is of _d_ by _c_/_d_, it follows, that when _a_, _b_, _c_,
and _d_, are proportional, _a_/_b_ = _c_/_d_. This equation will be
the foundation of all our reasoning on proportional quantities; and
in considering proportionals, it is necessary to observe not only the
quantities themselves, but also the order in which they come. Thus,
_a_, _b_, _c_, and _d_, being proportionals, that is, _a_ being the
same multiple-part of _b_ which _c_ is of _d_, it does not follow that
_a_, _d_, _b_, and _c_ are proportionals, that is, that _a_ is the
same multiple-part of _d_ which _b_ is of _c_. It is plain that _a_ is
greater than, equal to, or less than _b_, according as _c_ is greater
than, equal to, or less than _d_.
179. Four numbers, _a_, _b_, _c_, and _d_, being proportional in the
order written, _a_ and _d_ are called the _extremes_, and _b_ and _c_
the _means_, of the proportion. For convenience, we will call the two
extremes, or the two means, _similar_ terms, and an extreme and a mean,
_dissimilar_ terms. Thus, _a_ and _d_ are similar, and so are _b_ and
_c_; while _a_ and _b_, _a_ and _c_, _d_ and _b_, _d_ and _c_, are
dissimilar. It is customary to express the proportion by placing dots
between the numbers, thus:
_a_ : _b_ ∷ _c_ : _d_
180. Equal numbers will still remain equal when they have been
increased, diminished, multiplied, or divided, by equal quantities.
This amounts to saying that if
_a_ = _b_ and _p_ = _q_,
_a_ + _p_ = _b_ + _q_,
_a_ - _p_ = _b_ - _q_,
_ap_ = _bq_,
_a_ _b_
and --- = ---.
_p_ _q_
It is also evident, that _a_ + _p_-_p_, _a_ -_p_ + _p_, _ap_/_p_, and
_a_/_p_ × _p_, are all equal to _a_.
181. The product of the extremes is equal to the product of the means.
Let _a_/_b_ = _c_/_d_, and multiply these equal numbers by the product
_bd_. Then,
_a_ _abd_
--- × _bd_ = ----- (116) = _ad_,
_b_ _b_
_c_ _cbd_
and --- × _bd_ = ----- = _cb_: hence (180), _ad_ = _bc_.
_d_ _d_
Thus, 6, 8, 21, and 28, are proportional, since
6 3 3 × 7 21
--- = --- = ------ = --- (180);
8 4 4 × 7 28
and it appears that 6 × 28 = 8 × 21, since both products are 168.
182. If the product of two numbers be equal to the product of two
others, these numbers are proportional in any order whatever, provided
the numbers in the same product are so placed as to be similar terms;
that is, if _ab_ = _pq_, we have the following proportions:--
_a_ : _p_ ∷ _q_ : _b_
_a_ : _q_ ∷ _p_ : _b_
_b_ : _p_ ∷ _q_ : _a_
_b_ : _q_ ∷ _p_ : _a_
_p_ : _a_ ∷ _b_ : _q_
_p_ : _b_ ∷ _a_ : _q_
_q_ : _a_ ∷ _b_ : _p_
_q_ : _b_ ∷ _a_ : _p_
To prove any one of these, divide both _ab_ and _pq_ by the product of
its second and fourth terms; for example, to shew the truth of _a_: _q_
∷ _p_: _b_, divide both _ab_ and _pq_ by _bq_. Then,
_ab_ _a_ _pq_ _p_
---- = ---, and ---- = ---; hence (180),
_bq_ _q_ _bq_ _b_
_a_ _p_
--- = ---, or _a_ : _q_ ∷ _p_ : _b_.
_q_ _b_
The pupil should not fail to prove every one of the eight cases, and to
verify them by some simple examples, such as 1 × 6 = 2 × 3, which gives
1: 2 ∷ 3: 6, 3: 1 ∷ 6: 2, &c.
183. Hence, if four numbers be proportional, they are also proportional
in any other order, provided it be such that similar terms still remain
similar. For since, when
_a_ _c_
--- = ---,
_b_ _d_
it follows (181) that _ad_ = _bc_, all the proportions which follow
from _ad_ = _bc_, by the last article, follow also from
_a_ _c_
--- = ---,
_b_ _d_
184. From (114) it follows that
_a_ _b_ + _a_
1 + --- = ---------,
_b_ _b_
_a_
and if --- be less than 1,
_b_
_a_ _b_ - _a_
1 - --- = ---------,
_b_ _b_
_a_
while if --- be greater than 1,
_b_
_a_ _a_ - _b_
--- - 1 = ---------.
_b_ _b_
_a_ + _b_ _a_ - _b_
Also (122), if --------- be divided by ---------
_b_ _b_
_a_ + _b_
the result is ---------.
_a_ - _b_
Hence, _a_, _b_, _c_, and _d_, being proportionals, we may obtain other
proportions, thus:
_a_ _c_
Let --- = ---
_b_ _d_
_a_ _c_
Then (114) 1 + --- = 1 + ---
_b_ _d_
_a_ + _b_ _c_ + _d_
or --------- = ---------
_b_ _d_
or _a_ + _b_: _b_ ∷ _c_ + _d_: _d_
That is, the sum of the first and second is to the second as the sum of
the third and fourth is to the fourth. For brevity, we shall not state
in words any more of these proportions, since the pupil will easily
supply what is wanting.
Resuming the proportion _a_: _b_ ∷ _c_: _d_
_a_ _c_
or --- = ---
_b_ _d_
_a_ _c_ _a_
1 - --- = 1 - ---, if --- be less than 1,
_b_ _d_ _b_
_b_ - _a_ _d_ - _c_
or --------- = ---------
_b_ _d_
that is, _b_-_a_: _b_ ∷ _d_-_c_: _d_ or, _a_-_b_: _b_ ∷ _c_-_d_: _d_,
_a_
if --- be greater than 1.
_b_
_a_ + _b_ _c_ + _d_
Again, since --------- = ---------
_b_ _d_
_a_ - _b_ _c_ - _d_ _a_
and --------- = --------- (--- being greater than 1)
_b_ _d_ _b_
_a_ + _b_ _c_ + _d_
dividing the first by the second we have --------- = ----------,
_a_ - _b_ _c_ - _d_
or _a_ + _b_ : _a_ - _b_ ∷ _c_ + _d_ : _c_ - _d_
and also _a_ + _b_ : _b_ - _a_ ∷ _c_ + _d_ : _d_ - _c_,
_a_
if --- be less than 1.
_b_
185. Many other proportions might be obtained in the same manner. We
will, however, content ourselves with writing down a few which can be
obtained by combining the preceding articles.
_a_ + _b_ : _a_ ∷ _c_ + _d_ : _c_
_a_ : _a_ - _b_ ∷ _c_ : _c_ - _d_
_a_ + _c_ : _a_ - _c_ ∷ _b_ + _d_ : _b_ - _d_.
In these and all others it must be observed, that when such expressions
as _a_-_b_ and _c_-_d_ occur, it is supposed that _a_ is greater than
_b_, and _c_ greater than _d_.
186. If four numbers be proportional, and any two dissimilar terms be
both multiplied, or both divided by the same quantity, the results are
proportional. Thus, if _a_: _b_ ∷ _c_: _d_, and _m_ and _n_ be any two
numbers, we have also the following:
_ma_ : _b_ ∷ _mc_ : _d_
_a_ : _mb_ ∷ _c_ : _md_
_a_ _c_
--- : _mb_ ∷ --- : _md_
_n_ _n_
_ma_ : _nb_ ∷ _mc_ : _nd_
_a_ _b_ _c_ _d_
--- : --- ∷ --- : ---
_m_ _m_ _m_ _m_
_a_ _b_ _c_ _d_
--- : --- ∷ --- : ---
_m_ _m_ _n_ _n_
and various others. To prove any one of these, recollect that nothing
more is necessary to make four numbers proportional except that the
product of the extremes should be equal to that of the means. Take the
third of those just given; the product of its extremes is
_a_ _mad_
--- × _md_, or -----,
_n_ _n_
_c_ _mbc_
while that of the means is _mb_ × ---, or -----.
_n_ _n_
But since _a_ : _b_ ∷ _c_ : _d_, by (181) _ad_ = _bc_,
_mad_ _mbc_
whence, by (180), _mad_ = _mbc_, and ----- = -----.
_n_ _n_
_a_ _c_
Hence, ---, _mb_, ---, and _md_, are proportionals.
_n_ _n_
187. If the terms of one proportion be multiplied by the terms of a
second, the products are proportional; that is, if _a_: _b_ ∷ _c_:
_d_, and _p_: _q_ ∷ _r_: _s_, it follows that _ap_: _bq_ ∷ _cr_: _ds_.
For, since _ad_ = _bc_, and _ps_ = _qr_, by (180) _adps_ = _bcqr_, or
_ap_ × _ds_ = _bq_ × _cr_, whence (182) _ap_: _bq_ ∷ _cr_: _ds_.
188. If four numbers be proportional, any similar powers of these
numbers are also proportional; that is, if
_a_ : _b_ ∷ _c_ : _d_
Then _aa_ : _bb_ ∷ _cc_ : _dd_
_aaa_ : _bbb_ ∷ _ccc_ : _ddd_
&c. &c.
For, if we write the proportion twice, thus,
_a_ : _b_ ∷ _c_ : _d_
_a_ : _b_ ∷ _c_ : _d_
by (187) _aa_ : _bb_ ∷ _cc_ : _dd_
But _a_ : _b_ ∷ _c_ : _d_
Whence (187) _aaa_ : _bbb_ ∷ _ccc_ : _ddd_; and so on.
189. An expression is said to be homogeneous with respect to any two or
more letters, for instance, _a_, _b_, and _c_, when every term of it
contains the same number of letters, counting _a_, _b_, and _c_ only.
Thus, _maab_ + _nabc_ + _rccc_ is homogeneous with respect to _a_, _b_,
and _c_; and of the third degree, since in each term there is either
_a_, _b_, and _c_, or one of these repeated alone, or with another, so
as to make three in all. Thus, 8_aaabc_, 12_abccc_, _maaaaa_, _naabbc_,
are all homogeneous, and of the fifth degree, with respect to _a_, _b_,
and _c_ only; and any expression made by adding or subtracting these
from one another, will be homogeneous and of the fifth degree. Again
_ma_ + _mnb_ is homogeneous with respect to _a_ and _b_, and of the
first degree; but it is not homogeneous with respect to _m_ and _n_,
though it is so with respect to _a_ and _n_. This being premised, we
proceed to a theorem,[28] which will contain all the results of (184),
(185), and (188).
[28] A theorem is a general mathematical fact: thus, that every number
is divisible by four when its last two figures are divisible by four,
is a theorem; that in every proportion the product of the extremes is
equal to the product of the means, is another.
190. If any four numbers be proportional, and if from the first two,
_a_ and _b_, any two homogeneous expressions of the same degree be
formed; and if from the last two, two other expressions be formed, in
precisely the same manner, the four results will be proportional. For
example, if _a_: _b_ ∷ _c_: _d_, and if 2_aaa_ + 3_aab_ and _bbb_ +
_abb_ be chosen, which are both homogeneous with respect to _a_ and
_b_, and both of the third degree; and if the corresponding expressions
2_ccc_ + 3_ccd_ and _ddd_ + _cdd_ be formed, which are made from _c_
and _d_ precisely in the same manner as the two former ones from _a_
and _b_, then will
2_aaa_ + 3_aab_ : _bbb_ + _abb_ ∷ 2_ccc_ + 3_ccd_ : _ddd_ + _cdd_
_a_
To prove this, let --- be called _x_.
_b_
_a_ _a_ _c_
Then, since --- = _x_, and --- = ---,
_b_ _b_ _d_
_c_
it follows that --- = _x_.
_d_
But since _a_ divided by _b_ gives _x_, _x_ multiplied by _b_ will give
_a_, or _a_ = _bx_. For a similar reason, _c_ = _dx_. Put _bx_ and _dx_
instead of _a_ and _c_ in the four expressions just given, recollecting
that when quantities are multiplied together, the result is the same
in whatever order the multiplications are made; that, for example,
_bxbxbx_ is the same as _bbbxxx_.
Hence, 2_aaa_ + 3_aab_ = 2_bxbxbx_ + 3_bxbxb_
= 2_bbbxxx_ + 3_bbbxx_
which is _bbb_ multiplied by 2_xxx_ + 3_xx_
or _bbb_ (2_xxx_ + 3_xx_)[29]
Similarly, 2_ccc_ + 3_ccd_ = _ddd_ (2_xxx_ + 3_xx_)
Also, _bbb_ + _abb_ = _bbb_ + _bxbb_
= _bbb_ multiplied by 1 + _x_
or _bbb_(1 + _x_)
Similarly, _ddd_ + _cdd_ = _ddd_ (1 + _x_)
Now, _bbb_ : _bbb_ ∷ _ddd_ : _ddd_
[29] If _bx_ be substituted for _a_ in any expression which is
homogeneous with respect to _a_ and _b_, the pupil may easily see
that _b_ must occur in every term as often as there are units in the
degree of the expression: thus, _aa_ + _ab_ becomes _bxbx_ + _bxb_ or
_bb_(_xx_ + _x_); _aaa_ + _bbb_ becomes _bxbxbx_ + _bbb_ or _bbb_(_xxx_
+ 1); and so on.
Whence (186), _bbb_(2_xxx_ + 3_xx_): _bbb_(1 + _x_) ∷ _ddd_(2_xxx_ +
3_xx_): _ddd_(1 + _x_), which, when instead of these expressions their
equals just found are substituted, becomes 2_aaa_ + 3_aab_: _bbb_ +
_abb_ ∷ 2_ccc_ + 3_ccd_: _ddd_ + _cdd_.
The same reasoning may be applied to any other case, and the pupil may
in this way prove the following theorems:
If _a_ : _b_ ∷ _c_ : _d_
2_a_ + 3_b_ : _b_ ∷ 2_c_ + 3_d_ : _d_
_aa_ + _bb_ : _aa_ - _bb_ ∷ _cc_ + _dd_ : _cc_ - _dd_
_mab_ : 2_aa_ + _bb_ ∷ _mcd_ : 2_cc_ + _dd_
191. If the two means of a proportion be the same, that is, if _a_ :
_b_ ∷ _b_: _c_, the three numbers, _a_, _b_, and _c_, are said to be in
_continued_ proportion, or in _geometrical progression_. The same terms
are applied to a series of numbers, of which any three that follow one
another are in continued proportion, such as
1 2 4 8 16 32 64 &c.
2 2 2 2 2 2
2 --- --- --- --- ---- ---- &c.
3 9 27 81 243 729
Which are in continued proportion, since
2 2 2
1 : 2 ∷ 2 : 4 2 : --- ∷ --- : ---
3 3 9
2 2 2 2
2 : 4 ∷ 4 : 8 --- : --- ∷ --- : ---
3 9 9 27
&c. &c.
192. Let _a_, _b_, _c_, _d_, _e_ be in continued proportion; we have
then
_a_ _b_
_a_ : _b_ ∷ _b_ : _c_ or --- = --- or _ac_ = _bb_
_b_ _c_
_b_ _c_
_b_ : _c_ ∷ _c_ : _d_ --- = --- _bd_ = _cc_
_c_ _d_
_c_ _d_
_c_ : _d_ ∷ _d_ : _e_ --- = --- _ce_ = _dd_
_d_ _e_
Each term is formed from the preceding, by multiplying it by the same
number. Thus,
_b_ _c_
_b_ = --- × _a_ (180); _c_ = ---× _b_;
_a_ _b_
_a_ _b_ _b_ _c_ _b_
and since --- = ---, --- = --- or _c_ = --- × _b_.
_b_ _c_ _a_ _b_ _a_
_d_ _d_ _c_ _b_
Again, _d_ = --- × _c_, but --- = ---, which is = ---;
_c_ _c_ _b_ _a_
_b_
therefore, _d_ = --- × _c_, and so on.
_c_
_b_
If, then, ---
_a_
(which is called the _common ratio_ of the series) be denoted by _r_,
we have
_b_ = _ar_ _c_ = _br_ = _arr_ _d_ = _cr_ = _arrr_
and so on; whence the series
_a_ _b_ _c_ _d_ &c.
is _a_ _ar_ _arr_ _arrr_ &c.
Hence _a_ : _c_ ∷ _a_ : _arr_
(186) ∷ _aa_ : _aarr_
∷ _aa_ : _bb_
because, _b_ being _ar_, _bb_ is _arar_ or _aarr_. Again,
_a_ : _d_ ∷ _a_ : _arrr_
(186) ∷ _aaa_ : _aaarrr_
∷ _aaa_ : _bbb_
Also _a_ : _e_ ∷ _aaaa_ : _bbbb_, and so on;
that is, the first bears to the _n_ᵗʰ term from the first the same
proportion as the _n_ᵗʰ power of the first to the _n_ᵗʰ power of the
second.
193. A short rule may be found for adding together any number of terms
of a continued proportion. Let it be first required to add together the
terms 1, _r_, _rr_, &c. where _r_ is greater than unity. It is evident
that we do not alter any expression by adding or subtracting any
numbers, provided we afterwards subtract or add the same. For example,
_p_ = _p_-_q_ + _q_-_r_ + _r_- _s_ + _s_
Let us take four terms of the series, 1, _r_, _rr_, &c. or,
1 + _r_ + _rr_ + _rrr_
It is plain that
_rrrr_-1 = _rrrr_-_rrr_ + _rrr_-_rr_ + _rr_-_r_ + _r_-1
Now (54), _rr_-_r_ = _r_(_r_-1), _rrr_ -_rr_ = _rr_(_r_-1),
_rrrr_-_rrr_ = _rrr_(_r_-1), and the above equation becomes _rrrr_ -1 =
_rrr_(_r_-1) + _rr_ (_r_-1) + _r_ (_r_-1) + _r_-1; which is (54) _rrr_
+ _rr_ + _r_ + 1 taken _r_-1 times. Hence, _rrrr_-1 divided by _r_-1
will give 1 + _r_ + _rr_ + _rrr_, the sum of the terms required. In
this way may be proved the following series of equations:
_rr_ - 1
1 + _r_ = --------
_r_ - 1
_rrr_ - 1
1 + _r_ + _rr_ = ---------
_r_ - 1
_rrrr_ - 1
1 + _r_ + _rr_ + _rrr_ = ----------
_r_ - 1
_rrrrr_ - 1
1 + _r_ + _rr_ + _rrr_ + _rrrr_ = -----------
_r_ - 1
If _r_ be less than unity, in order to find 1 + _r_ + _rr_ + _rrr_,
observe that
1 - _rrrr_ = 1 - _r_ + _r_ - _rr_ + _rr_ - _rrr_ + _rrr_ - _rrrr_
= 1 - _r_ + _r_(1 - _r_) + _rr_(1 - _r_) + _rrr_(1 - _r_);
whence, by similar reasoning, 1 + _r_ + _rr_ + _rrr_ is found by
dividing 1-_rrrr_ by 1-_r_; and equations similar to these just given
may be found, which are,
1 - _rr_
1 + _r_ = --------
1 - _r_
1 - _rrr_
1 + _r_ + _rr_ = ---------
1 - _r_
1 - _rrrr_
1 + _r_ + _rr_ + _rrr_ = ----------
1 - _r_
1 - _rrrrr_
1 + _r_ + _rr_ + _rrr_ + _rrrr_ = -----------
1 - _r_
The rule is: To find the sum of n terms of the series, 1 + _r_ + _rr_
+ &c., divide the difference between 1 and the (_n_ + 1)ᵗʰ term by the
difference between 1 and _r_.
194. This may be applied to finding the sum of any number of terms of
a continued proportion. Let _a_, _b_, _c_, &c. be the terms of which
it is required to sum four, that is, to find _a_ + _b_ + _c_ + _d_, or
(192) _a_ + _ar_ + _arr_ + _arrr_, or (54) a(1 + _r_ + _rr_ + _rrr_),
which (193) is
_rrrr_ - 1 1 - _rrrr_
---------- × _a_, or ---------- × _a_,
_r_ - 1 1 - _r_
according as _r_ is greater or less than unity. The first fraction is
_arrrr_ - _a_ _e_ - _a_
-------------, or (192) ---------.
_r_ - 1 _r_ - 1
_a_ - _e_
Similarly, the second is ---------.
1 - _r_
The rule, therefore, is: To sum _n_ terms of a continued proportion,
divide the difference of the (_n_ + 1)ᵗʰ and first terms by the
difference between unity and the common measure. For example, the
sum of 10 terms of the series 1 + 3 + 9 + 27 + &c. is required. The
eleventh term is 59049, and ⁽⁵⁹⁰⁴⁹ ⁻ ¹⁾/₍₃₋₁₎ is 29524. Again, the sum
of 18 terms of the series 2 + 1 + ½ + ½ + &c. of which the nineteenth
term is ¹/₁₃₁₀₇₂, is
1
2 - ------
131072 131070
----------- = 3 ------.
1 - ½ 131072
EXAMPLES.
9 terms of 1 + 4 + 16 + &c. are 87381
6 12 847422675
10 ...... 3 + --- + ---- + &c. ... ---------
7 49 201768035
1 1 1 1048575
20 ...... --- + --- + --- + &c. ... -------
2 4 8 1048576
195. The powers of a number or fraction greater than unity increase;
for since 2½ is greater than 1, 2½ × 2½ is 2½ taken more than once,
that is, is greater than 2½, and so on. This increase goes on without
limit; that is, there is no quantity so great but that some power of
2½ is greater. To prove this, observe that every power of 2½ is made
by multiplying the preceding power by 2½, or by 1 + 1½, that is, by
adding to the former power that power itself and its half. There will,
therefore, be more added to the 10th power to form the 11th, than was
added to the 9th power to form the 10th. But it is evident that if any
given quantity, however small, be continually added to 2½, the result
will come in time to exceed any other quantity that was also given,
however great; much more, then, will it do so if the quantity added to
2½ be increased at each step, which is the case when the successive
powers of 2½ are formed. It is evident, also, that the powers of 1
never increase, being always 1; thus, 1 × 1 = 1, &c. Also, if _a_ be
greater than _m_ times _b_, the square of _a_ is greater than _mm_
times the square of _b_. Thus, if _a_ = 2_b_ + _c_, where _a_ is
greater than 2_b_, the square of _a_, or _aa_, which is (68) 4_bb_ +
4_bc_ + _cc_ is greater than 4_bb_, and so on.
196. The powers of a fraction less than unity continually decrease;
thus, the square of ⅖, or ⅖ × ⅖, is less than ⅖, being only two-fifths
of it. This decrease continues without limit; that is, there is no
quantity so small but that some power of ⅖ is less. For if
5 2 1 1 1
--- = _x_, --- = ---, and the powers of ⅖ are ----, -----,
2 5 _x_ _xx_ _xxx_
and so on. Since _x_ is greater than 1 (195), some power of _x_ may be
found which shall be greater than a given quantity. Let this be called
_m_; then 1/_m_ is the corresponding power of ⅖; and a fraction whose
denominator can be made as great as we please, can itself be made as
small as we please (112).
197. We have, then, in the series
1 _r_ _rr_ _rrr_ _rrrr_ &c.
I. A series of increasing terms, if _r_ be greater than 1. II. Of
terms having the same value, if _r_ be equal to 1. III. A series of
decreasing terms, if _r_ be less than 1. In the first two cases, the sum
1 + _r_ + _rr_ + _rrr_ + &c.
may evidently be made as great as we please, by sufficiently increasing
the number of terms. But in the third this may or may not be the case;
for though something is added at each step, yet, as that augmentation
diminishes at every step, we may not certainly say that we can, by any
number of such augmentations, make the result as great as we please. To
shew the contrary in a simple instance, consider the series,
1 + ½ + ¼ + ⅛ + ¹/₁₆ + &c.
Carry this series to what extent we may, it will always be necessary to
add the last term in order to make as much as 2. Thus,
(1 + ½ + ¼) + ¼ = 1 + ½ + ½ = 1 + 1 = 2
(1 + ½ + ¼ + ⅛) + ⅛ = 2.
(1 + ½ + ¼ + ⅛ + ¹/₁₆) + ¹/₁₆ = 2, &c.
But in the series, every term is only the half of the preceding;
consequently no number of terms, however great, can be made as great as
2 by adding one more. The sum, therefore, of 1, ½, ¼, ⅛ &c. continually
approaches to 2, diminishing its distance from 2 at every step, but
never reaching it. Hence, 2 is celled the _limit_ of 1 + ½ + ¼ + &c. We
are not, therefore, to conclude that _every_ series of decreasing terms
has a limit. The contrary may be shewn in the very simple series, 1 + ½
+ ⅓ + ¼ + &c. which may be written thus:
1 + ½ + (⅓ + ¼) + (⅕ + ... up to ⅛) + (⅑ + ... up to ¹/₁₆)
+ (¹/₁₇ + ... up to ¹/₃₂) + &c.
We have thus divided all the series, except the first two terms, into
lots, each containing half as many terms as there are units in the
denominator of its last term. Thus, the fourth lot contains 16 or ³²/₂2
terms. Each of these lots may be shewn to be greater than ½. Take the
third, for example, consisting of ⅑, ¹/₁₀, ¹/₁₁, ¹/₁₂, ¹/₁₃, ¹/₁₄,
¹/₁₅, and ¹/₁₆. All except ¹/₁₆, the last, are greater than ¹/₁₆;
consequently, by substituting ¹/₁₆ for each of them, the amount of the
whole lot would be lessened; and as it would then become ⁸/₁₆, or ½,
the lot itself is greater than ½. Now, if to 1 + ½, ½ be continually
added, the result will in time exceed any given number. Still more will
this be the case if, instead of ½, the several lots written above be
added one after the other. But it is thus that the series 1 + ½ + ⅓,
&c. is composed, which proves what was said, that this series has no
limit.
198. The series 1 + _r_ + _rr_ + _rrr_ + &c. always has a limit when
_r_ is less than 1. To prove this, let the term succeeding that at
which we stop be _a_, whence (194) the sum is
1 - _a_ 1 _a_
-------, or (112) ------- - ------.
1 - _r_ 1 - _r_ 1 - _r_
The terms decrease without limit (196), whence we may take a term so
far distant from the beginning, that _a_, and therefore
_a_
-------,
1 - _r_
shall be as small as we please. But it is evident that in this case
1 _a_
------- - ------- though always less than
1 - _r_ 1 - _r_
1 1
-------- may be brought as near to -------
1 - _r_ 1 - _r_
as we please; that is, the series 1 + _r_ + _rr_ + &c. continually
approaches to the limit
1
--------.
1 - _r_
Thus 1 + ½ + ¼ + ⅛ + &c. where _r_ = ½, continually approaches to
1
----- or 2, as was shewn in the last article.
1 - ½
EXERCISES.
2 2
The limit of 2 + --- + --- + &c.
3 9
1 1
or 2(1 + --- + --- + &c.) is 3
3 9
9 81
... 1 + --- + ---- + &c. ... 10
10 100
15 45
... 5 + ---- + ---- + &c. ... 8¾
7 49
199. When the fraction _a_/_b_ is not equal to _c_/_d_, but greater,
_a_ is said to have to _b_ a greater ratio than _c_ has to _d_; and
when _a_/_b_ is less than _c_/_d_, _a_ is said to have to _b_ a less
ratio than _c_ has to _d_. We propose the following questions as
exercises, since they follow very simply from this definition.
I. If _a_ be greater than _b_, and _c_ less than or equal to _d_, _a_
will have a greater ratio to _b_ than _c_ has to _d_.
II. If _a_ be less than _b_, and _c_ greater than or equal to _d_, _a_
has a less ratio to _b_ than _c_ has to _d_.
III. If _a_ be to _b_ as _c_ is to _d_, and if _a_ have a greater ratio
to _b_ than _c_ has to _x_, _d_ is less than _x_; and if _a_ have a
less ratio to _b_ than _c_ to _x_, _d_ is greater than _x_.
IV. _a_ has to _b_ a greater ratio than _ax_ to _bx_ + _y_, and a less
ratio than _ax_ to _bx_- _y_.
200. If _a_ have to _b_ a greater ratio than _c_ has to _d_, _a_ + _c_
has to _b_ + _d_ a less ratio than _a_ has to _b_, but a greater ratio
than _c_ has to _d_; or, in other words, if _a_/_b_ be the greater of
the two fractions _a_/_b_ and _c_/_d_,
_a_ + _c_
---------
_b_ + _d_
will be greater than _c_/_d_, but less than _a_/_b_. To shew this,
observe that (_mx_ + _ny_)/(_m_ + _n_) must lie between _x_ and _y_,
if _x_ and _y_ be unequal: for if _x_ be the less of the two, it is
certainly greater than
_mx_ + _nx_
----------- or than _x_;
_m_ + _n_
and if _y_ be the greater of the two, it is certainly less than
_my_ + _ny_
-----------, or than _y_.
_m_ + _n_
It therefore lies between _x_ and _y_. Now let _a_/_b_ be _x_, and let
_c_/_d_ be _y_: then _a_ = _bx_, _c_ = _dy_. Now
_bx_ + _dy_
-----------
_b_ + _d_
is something between _x_ and _y_, as was just proved; therefore
_a_ + _c_
---------
_b_ + _d_
is something between _a_/_b_ and _c_/_d_. Again, since _a_/_b_ and
_c_/_d_ are respectively equal to _ap_/_bp_ and _cq_/_dq_, and since,
as has just been proved,
_ap_ + _cq_
-----------
_bp_ + _dq_
lies between the two last, it also lies between the two first; that is,
if _p_ and _q_ be any numbers or fractions whatsoever,
_ap_ + _cq_
-----------
_bp_ + _dq_
lies between _a_/_b_ and _c_/_d_.
201. By the last article we may often form some notion of the value of
an expression too complicated to be easily calculated. Thus,
1 + _x_ 1 _x_ 1
-------- lies between --- and ----, or 1 and ---;
1 + _xx_ 1 _xx_ _x_
_ax_ + _by_ _ax_ _by_
-------------- lies between ----- and ------,
_axx_ + _bbyy_ _axx_ _bbyy_
that is, between 1/_x_ and 1/_by_. And it has been shewn that (_a_ +
_b_)/2 lies between _a_ and _b_, the denominator being considered as 1
+ 1.
202. It may also be proved that a fraction such as
_a_ + _b_ + _c_ + _d_
---------------------
_p_ + _q_ + _r_ + _s_
_a_ _b_ _c_ _d_
always lies among ---, ---, ---, and ---,
_p_ _q_ _r_ _s_
that is, is less than the greatest of them, and greater than the
least. Let these fractions be arranged in order of magnitude; that is,
let _a_/_p_ be greater than _b_/_q_, _b_/_q_ be greater than _c_/_r_,
and _c_/_r_ greater than _d_/_s_. Then by (200)
is and
less greater
_a_ + _b_ _a_ than than _b_ _c_
--------- --- --- and ---
_p_ + _q_ _p_ _q_ _r_
_a_ + _b_ + _c_ _a_ + _b_ _a_ _c_ _d_
--------------- --------- and --- --- and ---
_p_ + _q_ + _r_ _p_ + _q_ _p_ _r_ _s_
_a_ + _b_ + _c_ + _d_ _a_ + _b_ + _c_ _a_ _d_
--------------------- --------------- and --- ---
_p_ + _q_ + _r_ + _s_ _p_ + _q_ + _r_ _p_ _s_
whence the proposition is evident.
203. It is usual to signify “_a_ is greater than _b_” by _a_ > _b_ and
“_a_ is less than _b_” by _a_ < _b_; the opening of V being turned
towards the greater quantity. The pupil is recommended to make himself
familiar with these signs.