Chapter IX: Section VIII: 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.