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3. Divide 234.0 by .
OF PROPORTION IN GENERAL,
NUMBERS are compared together to discover the relations they have to each other.
There must be two numbers to form a comparison : the number which is compared, being written first, is called the antecedent; and that to which it is compared, che consequent. Thus of these numbers 2:4::3:6, 2 and 3 are called the antecedents; and 4 and 6, the consequents.
Numbers are compared to each other two different ways: one comparison considers the difference of the two numbers, and is called arithmetical relation, the difference being sometimes named the arithmetical ratio ; and the other considers their quotient, and is termed geometrical relation, and the quotient the geometrical ratio. So of these numbers 6 and 3, the difference or arithmetical ratio is 63 or 3; and the geometrical ratio is çor 2.
If two or more couplets of numbers have equal ratios, or differences, the equality is named proportion ; and their terms similarly posited, that is, either all the greater, or all the less, taken as antecedents, and the rest as consequents, are called proportionals. So the two couplets 2, 4, and 6, 8, taken thus, 2, 4, 6, 8, or thus, 4, 2, 8, 6, are arithmetical proportionals ; and the two couplets 2, 4, and
8, 16, taken thus, 2, 4, 8, 16, or thus, 4, 2, 16, 8, are geometrical proportionals.*
Proportion is distinguished into continued and discontinued.
If, of several couplets of proportionals written down in a series, the difference or ratio of each consequent and the antecedent of the next following couplet be the same as
the common difference or ratio of the couplets, the pro6
portion is said to be continued, and the numbers themselves a series of continued urithmetical or geometrical proportionals. So 2, 4, 6, 8, form an arithmetical progression ; for 4-2 = 6-4=8-6= 2; and 2, 4, 8, 16, a geometrical progression ; for 1 = = = 2.
But if the difference or ratio of the consequent of one couplet and the antecedent of the next couplet be not the same as the common difference or ratio of the couplets, the proportion is said to be discontinued. So 4, 2, 3, 6, are in discontinued arithmetical proportion"; for 4-2=8-6=2, but 8-2=6; also 4, 2, 16, 8, are in discontinued geometrical proportion ; for == 2, but = 8.
Four numbers are directly proportional, when the ratio of the first to the second is the same as that of the third to the fourth. As 2:4:: 3:6.
Four numbers are said to be recipr ally or inversely proportional, when the first is to the second as the fourth is to the third, and vice versa. Thus, 2, 6, 9 and 3, are reciprocal proportionals į 2:6:: 3:9.
Three or four numbers are said to be in harmonical proportion, when, in the former case, the difference of the first
* In geometrical proportionals a colon is placed between the terms of each couplet, and a double colon between the couplets ; in arithmetical proportionals a colon may be turned horizontally between the terms of each couplet, and two colons - written between the couplets. Thus the above geometrical proportionals are written thus, 2:4::8:16, and 4 : 2 :: 16:8; the arithmetical, 2. 4 :: 6. 8, and 4
and second is to the difference of the second and third, as the first is to the third ; and, in the latter, when the difference of the first and second is to the difference of the third and fourth as the first is to the fourth. Thus, 2, 3 and 6 ; and 3, 4, 6 and 9, are harmonical proportionals ; for 3--2=1:6--3=3:: 2 :6; and 4--3=1:9-6=3 ::3:9
Of four arithmetical proportionals the sum of the extremes' is equal to the sum of the means. Thus of 2 4::6 .. 8 the sum of the extremes (2+8)= the sum of the means (4+6)=10. Therefore, of three arithmetical proportionals, the sum of the extremes is double the
Of four geometrical proportionals the product of the extremes is equal to the product of the means.f Thus, of 2 : 4::8:16, the product of the extremes (2X 16) is equal to the product of the means (4X8)=32. Therefore of three geometrical proportionals, the product of the extremes is equal to the square of the mean.
Hence it is easily seen, that either extreme of four geometrical proportionals is equal to the product of the means
* DEMONSTRATION. Let the four arithmetical proportionals be A, B, C, D, viz. A. B :: C.. D; then, 1-B=C-D and B+D being added to both sides of the equation, A-B+B+D =C-D+B+D; that is, A+D the sum of the extremes =C+B the sum of the means.--And three A, B, C, may be thus expressed, A .. B :: B. C ; therefore A+C=B+B=2B.
Q. E. D.
+ DEMONSTRATION. Let the proportion be A:B::C:D, and let ==r; then A=Br, and C=Dr ; multiply the former of these equations by D, and the latter by B ; then ADS BrD, and CB=DrB, and consequently A D the product of the extremes is equal to BC the product of the means. And three may be thus expressed, A:B::B:C, therefore A C-BX B=B2. Q. E. D.
divided by the other extreme ; and that either mean is equal to the product of the extremes divided by the other mean.
SIMPLE PROPORTION, O'R RULE OF
The Rule of Three is that, by which a number is found, having to a given number the same ratio, which is between two other given numbers. For this reason it is sometimes named the Rule of Proportion.
It is called the Rule of Three, because in each of its questions there are given three numbers at least. And because of its excellent and extensive use, it is often named the Golden Rule:
RULE." W down the number, which is of the same kind the diswer or number required.
* DEMONSTRATION. The following observations, taken collectively, form a demonstration of the rule, and of the reductions mentioned in the notes subsequent to it.
i. There can be comparison or ratio between two numbers, only when they are considered either abstractly, or as applied to things of the same kind, so that one can, in a proper sense, be contained in the other. Thus there can be no comparison between 2, men and 4 days ; but there may be between 2 and 4, and between 2 days and 4 days, or 2 men and 4 men. Therefore, the 2 of the 3 given numbers, that are of the same kind, that is, the first and third, when they are stated according to the rule, are to be compared together, and their ratio is equal to that, required between the remaining or second number and the fourth or answer.
2. Consider whether the answer ought to be greater or less than this number : if greater, write the greater of the two remaining numbers on the right of it for the third,
2. Though numbers of the same kind, being either of the same or of different denominations, have a real ratio, yet this ratio is the same as that of the two numbers taken abstractly, only when they are of the same denomination. Thus the ratio of 11. is the same as that of 1- to 2 =1; is. has a real ratio to 21. but it is not the ratio of 1 to 2 ; it is the ratio of is. to 40s. that is, of 1 to 40 =16. Therefore, as the first and third numbers have the ratio, that is required between the second and answer, they must, if not of the same denomination, be reduced to it ; and then their ratio is that of the abstract numbers.
3. The product of the extremes of four geometrical proportionals is equal to the product of the means ; hence, if the product of two numbers be equal to the product of two other numbers, the four numbers are proportionals ; and if the product of two numbers be divided by a third, the quotient will be a fourth proportional to those three numbers. Now as the question is r. solvable into this, viz. to find a number of the same kind as the * second in the statement, and having the same ratio to it, that the greater
of the other two has to the less, or the less has to the greater; and as these two, being of the same denomination, may sidered as abstract numbers ; it plainly follows, that the fourth number or answer is truly found by multiplying the second by one of the other two, and dividing the product by that, which remains.
4. It is very evident, that, if the answer must be greater than the second number, the greater of the other two numbers must be the multiplier, and may occupy the third place ; but, if less, the less number must be the multiplier.
5. The reduction of the second number is only performed for convenience in the subsequent multiplication and division, and not to produce an abstract number. The reason of the reduction of