To show this property to be general, we must observe, that the second term is equal to the first increased by the difference, and that the fourth is equal to the third increased by the difference; hence it follows, that the sum of the extremes, composed of the first and fourth terms, must be equal to the first increased by the third increased by the difference. Also, that the sum of the means, composed of the second and third terms, must be equal to the first increased by the difference increased by the third term; these two sums, being composed of the same parts, must consequently be equal.

We have gone on the supposition, that the second and fourth terms were greater than the first and third; but the contrary may be the case, as in the four numbers 8, 5, 15, 12; the second term will be equal to the first decreased by the difference, and the fourth will be equal to the third decreased by the difference. By changing the word increased into decreased, in the preceding reasoning, it will be proved that, in the present case, the sum of the extremes is equal to that of the means.

We shall not pursue this theory of equidifferent numbers further, because, at present, it can be of no use.

Questions for practice.

A and B have gained by trading $182. A put into stock $300 and B $400; what is each person's share of the profit? Ans. A $78 and B $104.

and that the name of arithmetical proportion was given to an assemblage of equidifferent numbers, which were not treated of till a much later period. These names are very exceptionable; the word proportion has a determinate meaning, which is not at all applicable to equidifferent numbers. Besides, the proportion called geometrical is not less arithmetical than that which exclusively possesses the latter M. Lagrange, in his Lectures at the Normal school, has proposed a more correct phraseology, and I have thought proper to follow his example.

name.

Equidifference, or the assemblage of four equidifferent numbers, or arithmetical proportion, is written thus; 2.7: 9.14.

Among English writers the following form is used;

Divide $120 between three persons, so that their shares shall be to each other as 1, 2, and 3, respectively.

Ans. $20, $40, and 60. Three persons make a joint stock. A puts in $185,66, B $93,50, and C $76,85; they trade and gain $222; what is each person's share of the gain?

Ans. A $114,171, B $60,5782431, and C 47,253877. Three merchants, A, B, and C, freight a ship with 340 tuns of wine; A loaded 110 tuns, B 97, and C the rest. In a storm the seamen were obliged to throw 85 tuns overboard; how much must each sustain of the loss?

Ans. A 27, B 241, and C 331. A ship worth $360 being entirely lost, of which belonged to A, to B, and the rest to C; what loss will each sustain, supposing $500 of her to be insured?

Ans. A $45, B $90, and C $225. A bankrupt is indebted to A $277,33, to B $305,17, to C $152, and to D $105. His estate is worth only $677,50; how must it be divided? Ans. A $223,813533, B $246,28

83957

C $122,663, and D $34,738. A and B, venturing equal sums of money, clear by joint trade. $154. By agreement A was to have 3 per cent. because he spent his time in the execution of the project, and B was to have only 5 per cent.; what was A allowed for his trouble?

Ans. 35,53.

Three graziers hired a piece of land for $60,50. A put in 5 sheep for 4 months, B put in 8 for 5 months, and C put in 9 for 6 months; how much must each pay of the rent?

Ans. A $11,25, B $20, and C $29,25. Two merchants enter into partnership for 18 months; A put into stock at first $200, and at the end of 8 months he put in $100 more; B put in at first $550, and at the end of 4 months took out $140. Now at the expiration of the time they find they have gained $526; what is each man's just share?

Ans. A's $192,951. B's $333,04111 A, with a capital of $1000, began trade January 1, 1776, and meeting with success in business he took in B capital of $1500 on the first of March following.

partner, with a

Three months

after that they admit C as a third partner, who brought into the stock $2800; and after trading together till the first of the next year, they find the gain, since A commenced business, to be $1776,50. How must this be divided among the partners? Ans. A's $457,46 466

Alligation.

B's 571,83

C's 747,198.

128. We shall not omit the rule of alligation, the object of which is to find the mean value of several things of the same kind, of several values; the follows examples will sufficiently illustrate it.

ing

A wine merchant bought several kinds of wine, namely; 130 bottles which cost him 10 cents each,

he afterwards mixed them together; it is required to ascertain the cost of a bottle of the mixture. It will be easily perceived, that we have only to find the whole cost of the mixture and the number of bottles it contains, and then to divide the first sum by the second, to obtain the price required.

Now, the 130 bottles at 10 cents cost 1300 cents

5737 divided by 463 gives 12,39 cents, the price of a bottle of the mixture.

The preceding rule is also used for finding a mean of different results, given by experiment or observation, which do not agree with each other. If for instance, it were required to know the distance between two points considerably removed from each other, and it had been measured; whatever care might have been used in doing this, there would always be a

little uncertainty in the result, on account of the errors inevitably committed by the manner of placing the measures one after the other.

We will suppose that the operation has been repeated several times, in order to obtain the distance exactly, and that twice it has been found 3794yds. 2ft., that three other measurements have given 3795yds. Ift., and a third 3793yds. As these numbers are not alike, it is evident that sum raust be wrong, and perhaps all. To obtain the means of diminishing the error, we reason thus; if the true distance had been obtained by each measurement, the sum of the results would be equal to six times that distance, and it is plain that this would also be the case, if some of the results obtained were too little, and others too great, so that the increase, produced by the addition of the excesses, should make up for what the less measurements wanted of the true value. We should then, in this last case, obtain the true value by dividing the sum of the results by the number of them.

This case is too peculiar to occur frequently, but it almost always happens, that the errors on one side destroy a part of those on the other, and the remainder, being equally divided among the results, becomes smaller according as the number of results is greater.

According to these considerations we shall proceed as follows; yds. ft. We take twice 3794 2

ft.

or

7589 1

Dividing 22768yds. 1ft. by 6, we find the mean value of the required distance to be 3794yds. 2ft.

129. Questions sometimes occur, which are to be solved by a method, the reverse to that above given. It may be required, for example, to find what quantity of two different ingredients it would take to make a mixture of a certain value. It is evident, that if the value of the mixture required exceeds that of one of the ingredients just as much as it falls short of that of the other, we should take equal quantities of each to make the compound. Arith.

16

So also, if the value of the mixture exceeded that of one twice as much as it fell short of that of the other, the proportion of the ingredients would be as one half to one. To mix wine at $2 per gallon with wine at $3, so that the compound shall be worth $2,50, we should take equal quantities, or quantities in the proportion of 1 to 1. If the mixture were required to be worth $2,663, the quantities would be in the proportion of to 1, or of

1

1

663 to 331; and, generally, the nearer the mixture rate is to that of one of the ingredients, the greater must be the quantity of this ingredient with respect to the other, and the reverse; hence, To find the proportion of two ingredients of a given value, necessary to constitute a compound of a required value, make the difference between the value of each ingredient and that of the compound the denominator of a fraction, whose numerator is one, and these fractions will express the proportion required; and being reduced to a common denominator, the numerators will express the same proportion, or show what quantity of each ingredient is to be taken to make the required compound.

When the compound is limited to a certain quantity, the proportion of the ingredients, corresponding to it, may be found by saying; as the whole quantity, found as above, is to the quantity required, so is each part, as obtained by the rule, to the required quantity of each.

Let it be required, for example, to mix wine at 5s. per gallon and 8s. per gallon, in such quantities that there may be 60 gallons worth 6s. per gallon. The difference between 6s. and 5s. is 1, and between 6s. and 8s. is 2, giving for the required quantities the ratio of to, or 2 to 1; thus, taking r equal to the quantity at 5s. and y equal to the quantity at 8s. we have these proportions; 3: 60 :: 2 : x, and 3 : 60 :: 1:y, giving, for the answer, 40 gallons at 5s. and 20 gallons at 8s. per gallon.

Also, when one of the ingredients is limited, we may say; as the quantity of the ingredient found as above, is to the required quantity of the same, so is the quantity of the other ingredient to the proportional part required.

For example, I would know how many gallons of water at Os. per gallon, I must mix with thirty gallons of wine at 6s. per