« PreviousContinue »
If two equal ratios a:b and a : 6 be compounded, the resulting retio is that of a?: 6, which is called the duplicate ratio of a to b.
In the same manner, if three equal ratios be compounded, the resulting ratio is a' :b, and is named the triplicate ratio of a to b.
In a series of ratios, if the antecedent of a succeeding ratio be always equal to the consequent of a preceding ratio, then the ratio compounded of these ratios will be equal to the ratio of the first antecedent to the last consequent.
When two quantities of the same kind are compared by means of numbers, the numbers must express units of the same magnitude, and the ratio of the two quantities will also be the same as the ratio of two abstract numbers which denote the number of units in the two quantities.
It will often happen that one of the two quantities does not contain the other a certain number of times exactly; in that case the ratio between them is the ratio of the numbers which express how many times each quantity contains exactly some third quantity of the same kind.
And sometimes the strict meaning of the definition of a ratio is pot admissible, as in cases in which the measure of the ratio of two quantities is neither an integer nor a rational fraction.*
* If ABCD be a square whose side A B contains a units, the diagonal AC is equal to a 2 (Euc. I., 47); and the ratio of AC to AB is as 2 to 1 ; also if AB be the edge of a cube ABCDE, the diagonal AE is equal to av3; and the ratio AE to AB is as ✓3 to 1; both of these ratios are incommensurable, or such that there is. no integer or rational fraction which can exactly measure the side and diagonal of a. square, or the side and diagonal of a cube. In Euc. X., 117, is demonstrated the proposition, that the side and diagonal of a square are incommensurable.
Although the ratio of the diagonal to the side of a square is incommensurable, approximations may be made to the true ratio. Since v2 =1•4142136 ..... If two, three, four, &c., figures of this number be taken, these will be successive approximations to, but each less than 2. But if the last figures on the right of each of these numbers be increased by unity, then these numbers will be successive approximations to, but each greater than ✓2. So that 14, tót, 1884, &c., are successive approximations, but each less than, and t&, thé, tout, &c., are also approximations, but each greater than the true value of the ratio.
These approximations may be continued to any extent, and therefore the value of the ratio v2 can be expressed to any degree of accuracy required.
It may be seen that the difference of each pair of the successive approximations becomes less and less, according as the number of figures becomes greater in each successive approximation.
The ratio of the circumference of a circle to its diameter, or the semi-circumference to the radius, is an irrational number.
355 The numbers 37, and 3:14159 113
...: are approximations to the value of the ratio of the circumference of a circle to the diameter.
In the volume of the Berlin Memoirs for 1761, M. Lambert has printed a paper entitled “Mémoire sur quelques propriétés remarquables des quantités
A variable quantity is one which admits of gradual change of its magnitude, and a quantity which can be made greater than any assignable magnitude, is said to increase without limit; and a quantity which can be made less than any assignable magnitude, is said to decrease without limit.
The limit of a variable quantity is defined to be that constant quantity towards which it may be made to approach nearer than by any difference that can be assigned.*
Two variable quantities may be supposed to approach towards equality or to some constant quantity, either by the continued diminution of their difference, or by the approach of their ratio either to a ratio of equality or some constant quantity.
The limiting ratio of two variable quantities is some constant quantity towards which their ratio may be made to approach nearer than by any other ratio which can be assigned.
transcendantes Circulaires et Logarithmiques,” in which he has proved that both the numerical values of the semi-circumference of a circle and the square of it are irrational numbers. The proof he has given depends on continued fractions.
Dr. Brewster, in 1824, published a translation of the Geometry of Legendre, and in the note 4, page 239, he has given in English Lambert's demonstration.
The notion of a limit is necessary to understand what is called the sum of a converging series of quantities in geometrical progression indefinitely continued.
A convergent series is one in which each successive term becomes less and less than the preceding term, and is such as to admit of a limit.
A divergent series is one in which each successive term in general, becomes greater than the preceding, and is of such a nature as not to admit of a limit.
A geometrical series is convergent when the common ratio is less, but divergent when it is greater than unity.
The words infinite and indefinite are negative terms, and the exact meanings they have in mathematical reasonings must be determined by the sense of the positive terms finite and definite, which respectively express an opposite or contrary relation. A finite number is one which can be assigned, but an infinite number is one which cannot be assigned, and which can bear no conceivable relation to a finite number.
In the reasonings on prime and ultimato ratios, the symbol o has been assumed to denote a number greater than can be assigned. If the symbol o be assumed to denote a number less than can be assigned, and not absolutely nothing (as it is employed in the numerical scale to denote absence of number), some confusion of ideas migbt be avoided by the learner.
A number or qnantity may be considered to become, by continual increase or decrease, indefinitely great or indefinitely small, so that at length it may be conceived to become greater or less than any number or quantity thit can be assigned, and in that case, it is said to be greater or less than any assignable number or quantity.
If two quantities increase or decrease without limit, their ratio does not necessarily increase or decrease without limit, but may ultimately have a finite limit which cannot be exceeded. † The following are illustrations of the definition.
x and y be two variable quantities connected by the equation y=x”, and if x receive an increment h, and a', y be the corresponding values of x and y, then x=x+h and y' - ( + h)? = x2 + 2ch +h?.
2. Prop. To ascertain the effect produced on a ratio by adding the same quantity to both its terms.
Let a: 6 be the given ratio, and let x be added to both its terms, so that it becomes the ratio of a+x:b+x. Then the ratio of a+x:6+x> or < ratio of a : 6,
ats according as <
a > or <
or the limit of y' - y
y'- Y .: y'- Y=2xh+ha, the increment of y, and
h x Now, if the increment h be supposed continually to diminish, so as at length to
ข่ become less than any assignable quantity, then the limit to which tends, as h is diminished, will be 2x, which it can never exceed ; or the ultimate ratio of the corresponding increments of y and x will be 2x,
=2x, when y=x*.
a' This result admits of a geometrical interpretation. If x denote the side or edge, and y the area of a square, the limiting ratio of the corresponding increments of the area and the side of a square is equal to twice the length of the side.
In the same manner, if y=23, and if y, a' be the corresponding values of y and x when x receives an increment h, so that a' – x=h,
y =(x +h)3 =r3 +3x? h + 3cha ths,
*' - -Y = 3x2 + 3xh+ha,
h a' and the limit of “–Y = 3.c? when the increment h is diminished indefinitely. This result also admits of a geometrical interpretation.
If x denote the edge, and y the volume of a cube, the limiting ratio of the corresponding increments of the volume and the edge of the cube whose edge is x, will be 3 times the area of one of the faces of the cube.
Bishop Berkeley, perhaps, was not mistaken in the objections he raised against the mathematicians of his time, when, in 1734, he wrote in the Analyst : "I have no controversy about your conclusions, but only about your logic; and it must be remembered that I am not concerned about the truth of your theorems, but only about the way of coming at them." The bishop, in full persuasion of the truth of the old aphorism "cx nihilo nihil fit,” could not admit that, when a finite quantity was assumed to vanish or become zero, any intelligible finite result as a consequence, could logically follow from such a change made in the original hypothesis. In the former example, if y=x2, then 44–=2x+h, is true so long as the incre
ad ment h is finite ; but if the increment h be made, or supposed to become absolutely zero, then 2-x=0 and y-y=0, and 3–Y=2x +h becomes ? =22+0.
-3 And the bishop could not admit that the quotient arising from the division of zero by zero could ever be equal to any definite number or quantity. But if it be admitted that the symbol 0 may denote a number less than any
0 number that can be assigned, the expression may mean that the quotient or the
0 ratio of two such quantities, each of which is less than any assignable quantity, may be equal to a finite quantity.
ab+bar that is as
bx > or < at,
6 > or < a. If 6 > a, then the ratio a+x:6+x is > the ratio of a: 6. If b <a, then ratio of a+x: +x< the ratio of a :b.
Hence a ratio of less inequality is increased, and of greater inequality is diminished, by adding the same quantity to both terms of the ratio.*
In a similar manner, it may be shown that a ratio of greater inequality is increased, and of less inequality is diminished, by subtracting the same quantity from both its terms.
3. Prop. If the terms of a ratio a:b be multiplied by the same quantity, the ratio remains unaltered. The ratio of a: b is represented by the fraction ; and
Õmb' any fraction is unaltered by multiplying the numerator and denominator by the same quantity.
Hence the ratio of ma : na is the same as the ratio of a:b.
Conversely. If the terms of a ratio ma: mb be divided by the same quantity, the ratio remains unaltered. The fraction for any fraction is unaltered in value when
mbo the numerator and denominator are divided by the same quantity. Hence the ratio a:b is the same as the ratio of ma : mb.
Hence, the two numbers which express the ratio or relative magnitude of two quantities, do not always express the actual magnitude of the quantities compared, as it appears that different pairs of numbers can denote the same ratio.
4. “Four numbers are proportionals when the first is the same multiple of the second, or the same part or parts of it, as the third is of the fourth.” (Euc. VII., Def. 20.)
Or. A proportion may be defined to be the equality of two ratios, or when the first of four quantities divided by the second gives the same quotient as the third divided by the fourth. As, if a, b, c, d be four quantities such that then a, b, c, d are
* It appears that the successive addition of the same quantity to both terms of a ratio, tends to make it to approach continually to a ratio of equality.
If the terms of the ratio 5 to 3 be successively increased by unity, the values of the successive ratios are $, *, }, %, 4, , &c., or 13, 14, 14, 13, 14, 15, &c., and the successive differences are 1, to, ts, ito , &c., each of which obviously becomes less and less, and therefore the successive values of the ratios tend to equality.
This tendency to equality is illustrated by the fact, that the ages of two persons, one older than the other, in successive years, continually tend to become relatively equal, while the difference between their ages remains constant.
proportionals; and if a, b, c, d be proportionals, then
37 A proportion is sometimes written a : 6::c:d, and is expressed by saying that a is to b as c is to d; the first and fourth terms are called the extremos, and the second and third the means, of the proportion.
If the third term be equal to the second, the proportion consists of three terms, a, b, d, and the second term b is called a mean proportional between a and d, the first and the third.
If the successive terms of four or more quantities, as a, b, c, d, have the same ratios, a to b, as b to c, as c to d; they are said to be in continued proportion.
There is a distinction between direct and inverse proportion. Four quantities a, b, c, d are said to be in direct proportion when the first and second terms are directly proportional to the third and fourth terms, as 6 - ,; and it is obvious from the definition that if the first of four proportionals has the same ratio to the second, as the third has to the fourth ; then if the first term be greater than the second, the third is greater than the fourth ; if equal, equal; and if less, less.
Four quantities a, b, c, d are said to be in inverse proportion when the first and second terms are directly proportional to the reciprocals of
1 1 the third and fourth terms, as a : 6::-:
or as a : 6::d: 0.
d' And in this case, if the first term be greater than the second, the reciprocal of the third is greater than the reciprocal of the fourth ; if equal, equal; and if less, less.*
* The distinction between a direct and inverse proportion has a real existence in the forces of nature, as shewn by the following examples :
(1) If a agents produce an effect e in time t; and a' agents produce the effect e' in time † ; find the relation between the effects, times, and agents, the units being the same in each case.
Since a agents produce effect e in time t ; 1 agent produces effect in time t, and
1 agent produces effect in an unit of time. at
é Similarly, in the second case, 1 agent produces effect in an unit of time. But
é as the units of agent and time are equal, .. and ea'ť = d'at, dividing
at a't' at each by a'e't, ..
or e : é :: at : a't'; that is, the effects produced are ‘irectly proportional to the product of the numbers which denote the agents and times in each case.
(2) If the times be the same in ach case, then t' =ť and ea't' =eat, becomes ta' = e'a, and or € :6::a: a'.