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XIII. PROPORTION.

EXPLANATIONS.

1. The idea, annexed to the term, proportion, is easily conceived.

The rule itself is founded on this obvious principle. "The magnitude or quantity of any effect varies constantly in proportion to the varying part of the cause."

Proportion is distinguished into continual and discontinual. 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 proportion is said to be continual, and the numbers themselves, a series of continual proportionals, either arithmetical or geometrical.

2. Four numbers are said to be in direct proportion, when more requires more, or less requires less.

More requires more when the third term is greater than the first, and requires the fourth term to be greater than the second.

Less requires less, when the third term is less than the first, and requires the fourth term to be less than the second.

3. Four numbers are said to be reciprocally or inversely proportional, when the fourth is less than the second, by as many times as the third is greater than the first, or when the first is to the third, as the fourth to the second, and vice versa. Hence the phrase-if more requires less or less requires more, the proportion is Inverse.

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4. Harmonical Proportion is that, which is between those numbers which assign the lengths of musical intervals, or the lengths of strings sounding musical notes; and of three numbers it is, when the first is to the third, as the difference between the first and second is to the difference between the second and third, as the numbers, 3, 4, 6.

Thus, if the length of strings be as these numbers, the sound will be an octave 3 to 6, a fifth 2 to 3, and a fourth 3 to 4. Again, between four numbers, when the first is to the fourth, as the difference between the first and second is to the difference between the third and fourth, as in the numbers 5, 6, 8, 10 : For the strings of such lengths will sound an octave 5 to 10; a sixth greater 6 to 10; a third greater 8 to 10; a third less 5 to 6; a sixth less 5 to 8; and fourth 6 to 8.

*Vice versa is a Latin word signifying, the terms being exchanged.

Lastly-A series of numbers in harmonical proportion is, reciprocally, as another series in arithmetical proportion.

XIV. ALLIGATION.

DEMONSTRATION.

1. There is nothing in the different cases and operations in Alligation, the reason of which does not appear plain, except what relates to finding the quantities, at several different prices, to be mixed together in such proportion that one pound, bushel, &c. of the mixture, may be of a certain value, less than the highest and greater than the lowest price. The rule for doing this is called Alligation Alternate, from alligo to bind or connect together, and alterno, to change by turns; because the prices of different simples are linked together, and their differences are made to change places with one another.

The operation by this rule gives the true answer by connecting a greater and less, than the mean price, and placing their differences alternately for the quantities themselves, by which there is precisely as much gained by one quantity, as is lost by the other.

We may variously alligate the values of the ingredients, and thus obtain various results, all of which will be correct; and the results will be correct answers.

DEMONSTRATION.-Let two different qualities of grain be mixed together, one kind worth $2 a bushel, and the other $4, in such proportion that a bushel of the mixture shall be worth

$3.

Operation.

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EXPLANATION. Here it is found by the rule that there must be 1 bushel of each sort, and the price of both bushels, one at $2 and the other at $4, is $6, which divided by 2, the number of bushels mixed, gives $3, the price of a bushel of the mixture. It is also plain that the bushel which was worth $4 before it was mixed, has lost $1 by the mixture, and the bushel that was worth $2 has gained $1; therefore, there is as much gained by one, as is lost by the other, which was to be proved. The same principle will hold true in all cases, let the number of simples be what it may.

2. The difference between the greater and the less of two prices connected, is the common denominator of fractions, of which the differences between each extreme and the mean

price, are the numerators, which fractions always express the proportional quantities of the different things to be mixed, and by removing the common denominator, the numerators become whole numbers: hence the reason of the differences of the mean and extreme prices being the same as the number of things required to be mixed.

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DEMONSTRATION.

Example. Here is an example in which the mean or 3 price is 5, and the extremes are 3 and 8; the 8. or 2 quantities to be mixed are 3 of the lesser price, and 2 of the greater. The difference between 3 and 8 is 5, which is the common denominator, and 5—3—2, the numerator of the least fraction, against the greatest price, and 8-5-3, the greatest numerator of the fraction or proportion at the least price. The proportions are now found in fractions; but it is plain that if the common denominator 5, be taken away from both, it is the same as multiplying the numerators by it, and then dividing the products by the denominator; therefore, 3 and 2 are whole numbers, and they are the least whole numbers that can have the same relation to each other as to .

When several less numbers are connected with one greater, or the contrary, the principle is the same.

MISCELLANY.

1. Part 1500 acres of land between Saul, Seth and Silas; and give Seth 72 more than Saul, and Silas 112 more than Seth.

First, Seth's 72, +Silas' 72 and 112 more than Saul's, is 256 acres from 1500, leaving 1244 to be divided equally; hence, 1244-3-4143 acres, Saul's share.

Then, 4143+72, Seth's; and 4143+72+112, Silas' share.

2. What is the difference between six dozen dozen, and half a dozen dozen ?

First, a dozen dozen is 144,÷2-72, a dozen.

Then, 144X6-72-792 difference, Ans.

3. What number, deducted from the 32d part of 3072, will leave the 96th part of the same?

First, 3072÷32-96; then, 3072÷96-32.
Then, 96-32-64, Ans.

4. There is a certain number, which being divided by 7, the quotient resulting multiplied by 3, that product divided by 5, from the quotient 20 being subtracted; and 30 added to the remainder, the half sum shall make 35. What is the number?

Thus, 35X2-30+20X5X7÷÷3-700, Ans.

5. How many trees, four feet apart each way, may grow on an acre of ground?

First, 1 acre 43560 feet, and four feet every way-16 feet; then, 43560-16=2722 trees, Ans.

6. A sheepfold was robbed three nights successively; the first night, half the sheep were stolen, and half a sheep more; the second night half the remainder were stolen, and half a sheep more; the last night, they took half of what were left, and half a sheep more, by which time they were reduced to 30. How many were there at first?

Thus, +30 of what were in the fold before any were taken the last night, therefore, 30,5+30,5-61 in the fold before any were taken the last night, and half of them were stolen and half a sheep more, consequently, 31 taken the third night, 62 the second, and 124 the first.

Then, 30+31+62+124—247, Ans.

7. What part of 33 is 2811?

First, 3342776, and 2811=2420; then, 2478=7, Ans. 8. Find two numbers that of one shall equal

other.

Thus, X-18, Ans.

9. If of 6 be 3 what will of 40 be?

of the

First, of 6 is 2, and of 40 is 10; then, if 2 is 3, 10 is 15. Ans. 15.

10. At what time, between twelve and one o'clock, do the hour and minute hands of a clock point in directions exactly opposite ?

The minute hand must gain 30 minutes on the hour hand, before they will point in opposite directions, and the minute hand, in moving 1 minute, gains 1 minute, because the motion of the hands are in the ratio of 11 to 1; consequently, 30X132 minutes past 12, Ans.

11. Seven-eighths of a certain number exceeds four-fifths of the same by 6; what is that number?

First, 40X7,÷8-35, and 40-35-5; 40×4,÷5=32,

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and 40-32-8; and 8-5-3. Then if 3:6::40:80, 7 of which exceeds by 6.

12. From 14 years, take 11 yrs. 11 mo. 11 w. 11 d. 11 h. 11 m. 11 s.

13. From 1 fur. take 39 rods, 4 yds. 2 ft. 5 inches.

14. What is the gross weight of a hogshead of tobacco, weighing neat 11 cwt. 1 qr.; tare 14 lbs. per. cwt.?

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First,112-14-98; then, if 98:112::11 cwt. qr.: 12 cwt. qrs. 12 lbs., Ans.

15. The births, in a certain town, were 475, and the proportion, 13 boys to 12 girls; what was the number of each?

Thus, 13-12:475:13:247 boys; then, 475-247-228 girls.

16. What sum of money will produce as much interest in 34 years as $210,15 can produce in 5 years?

First, 34 yrs. 39 months, and 55 yrs. 65 months.
Then, if 39:210, 15::65: $350,25, Ans.

17. Divide the number 360 into four such parts, which shall be to each other as 3, 4, 5, and 6. First, 3+4+5+6=18. Then, 18:360::3:60 And, 18:360::5:100 ":"6:120) ::6:120 Ans.

66. 66

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Ans.

18. C hired A and B to cut wood; A could cut a cord in 4 hours, and C in 6 hours; in what time could both cut a cord? *First, 6+4,÷2=5 of 12 hours; then,5:60:: 12:2 hours, 24 minutes, Ans.

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*The signifies the operation of the preceding question begins at that place.