6. The investigation of such rules for calculation is one of the two objects of Algebra.

The other object, which is subservient to the former, is the discovery of the different operations which may be performed with the same given numbers, and shall produce the same ultimate numerical results as any given operations different from these shall produce, without regard to what those numbers may chance to be. Thus, if the square of the sum of a and b were sought in another form, it may be exhibited thus a × a + 2 × a × b + b × b. And the statement of this fact is thus written :

(a + b) x (a + b) : = (a × a) + (2 × a × b) + (b × b.) Shorter modes of writing it will be exhibited presently; but here the simple symbols used in the arithmetic have been alone employed, for the purpose of showing the nature of algebraic notation in its earliest forms, and to illustrate the objects for which it was devised.

7. The discovery of formulæ for the solution of questions constitutes the algebraical problem; and the discovery of formulæ of transformation, or of those which give equivalent results independently of the particular value of the quantities which enter into their composition, constitute the algebraical theorem.

8. The motives which gave rise to the use of alphabetic letters as symbols of number in preference to any other system of symbols, arbitrarily selected for the same purpose, are principally the following. First, As they have no numerical signification in themselves, they are subject to no ambiguity, having in reference to numbers no other signification than they are defined to have in the outset of each problem, or either defined, or understood from general practice, to have in each theorem. Secondly, Being familiar to the eye, the tongue, the hand, and the mind, that is, having a well-known form and name, they are easily read, written, spoken, remembered, and discriminated from one another, which could not be the case were they mere arbitrary marks, formed according to the caprice of each individual who used them, and always different, as in such case they must almost of necessity be, at each different time that the same person required to use them. Thirdly, The order in which the letters are arranged in the alphabet, facilitates the classification of them into groups much more easy to survey and comprehend in the expressions which arise from the performance of any assigned operations, and thereby renders the investigator much less likely to omit any of them by an imperfect enumeration, than if they were composed of marks that were used for that purpose only, and selected for each individual occasion from the various combinations that could be formed of such simple linear elements as the hand could readily trace, and the eye readily distinguish from all other combinations.

II. Definitions, Notation, and Fundamental Principles.

THE principal symbols which are employed to designate the operations of algebra and arithmetic, and the relations which subsist between quantities, are the following. Their object is to abbreviate.

I. 1. signifies addition, and is read plus. Thus 2 + 3 or a + b + c respectively signify that 3 is to be added to 2, and that b is to be added to a, and that then c is to be added to the sum of a and b.

A quantity to which the symbol + is prefixed, is called a positive or affirmative quantity.

2.- signifies subtraction, and is read minus. Thus, 3

- 1, or b

a, signify

respectively that I is to be subtracted from 3, and a from b. The number to be subtracted is always placed after the symbol.

A quantity to which the sign is prefixed is called a negative quantity *. 3. signifies the difference of the quantities between which it is placed; and is used either when it is not known or is not necessary to specify which is the greater of them. In this case a b, or b a, signify the same thing.

4. x is the symbol of multiplication, and is placed between the factors which are to be multiplied together. Sometimes a point. (placed at the lower part of the line, to distinguish it from the decimal point, which is placed at the upper part of the line,) is employed for the same purpose, and especially between the numerical factors, as 3. 5. 7, or 1. 2. 3. 4: and in the case of simple literal factors, the practice is now almost universal to drop all marks between the simple factors, and write them in consecutive juxta-position. Thus a x b b.c. x, or abca designate the same thing, viz. the continued product of the numbers which a, b, c, and x are put to represent ↑.

xc xx, or a .

When one of the factors is a number, it is called a coefficient: thus in 2 × a × bor 2ab, the 2 is called the coefficient of ab, and in 53xyz, 53 is called the coefficient of xyz. When no coefficient is written, 1 is understood to be meant, the quantity being taken once.

Also, in some cases where letters are put for numbers, the letters which represent given or known numbers are likewise called coefficients; as in 3 axz, 3a is called the coefficient of xz. In the case of a number being actually given, the coefficient is said to be a numeral coefficient; but when it is given in literal symbols, it is called a literal coefficient.

Moreover it may be remarked that cases of algebraical investigation sometimes present themselves in which even the symbols of the unknown quantities are conveniently considered as coefficients: but these will be pointed out when they arise.

Though, as is proved in the note, the order of the factors in multiplication, so

* Quantities affected with the signs+and+ or - and, are said to have like signs; and those affected with and +, or + and -, are said to have unlike signs.

It is manifest from the nature of addition and subtraction, that the disposition of the quantities as to order is immaterial; for a + b, or b + a, is the same thing, and a + bc, a - c + b, or -c+b+a, express the same quantity, only under a different arrangement, as to relative position.

It may be readily shown that it is immaterial in what order the factors are taken for the purpose of multiplication: that is, which is made the multiplicand and which the multiplier. For if a number of dots (or units) be placed horizontally equal in number to the units in the factor selected as the multiplicand, and this be repeated under this horizontal band till there are as many bands as there are units in the other factor: then the same number of dots considered as forming vertical columns will be constituted of as many times the number there is in the multiplier as there are units in the multiplicand, and representing therefore the result of a multiplicand with the order of the factors inverted. Thus if we take four times three, the dots will stand

and if we turn the column which is vertical into a horizontal position, it becomes

....

And in the same way it is shown of m and n as factors.

EXAMPLE.

To find four geometrical means between 3 and 96.

Here 3) 96 (32; the 5th root of which is 2, the ratio.

Then 3 x 2 = 6, and 6 x 2 = 12, and 12 x 2 = 24, and 24 x 2 = 48.
Or 96 ÷ 2 = 48, and 48 ÷ 2 = 24, and 24 ÷ 2 = 12, and 12 ÷ 2 = 6.
That is, 6, 12, 24, 48, are the four means between 3 and 96.

OF HARMONICAL PROPORTION.

THERE is also a third kind of proportion, called Harmonical or Musical, which being but of rare occurrence in questions purely arithmetical, a very short account of it may here suffice. It will however be again noticed both in algebra and in geometry, but especially in the latter.

Musical Proportion is when, of three numbers, the first has the same proportion to the third, as the difference between the first and second has to the dif ference between the second and third.

When four numbers are in musical proportion, then the first has the same ratio to the fourth, as the difference between the first and second has to the difference between the third and fourth.

When numbers are in musical progression, their reciprocals are in arithmetical progression; and the converse, that is, when numbers are in arithmetical progression, their reciprocals are in musical progression.

=

So in these musicals 6, 8, 12, their reciprocals,,, are in arithmetical progression; for + į = 1/2 ; and + = } ; that is, the sum of the extremes is equal to double the mean, which is the property of arithmeticals. The method of finding a series of numbers in musical proportion and progression is best expressed by algebraic methods and symbols.

FELLOWSHIP OR PARTNERSHIP.

FELLOWSHIP is the rule by which any sum or quantity may be divided into any number of parts which shall be in any given proportion to one another.

By this rule are adjusted the gains or losses or charges of partners in company; or the effects of bankrupts, or legacies in case of a deficiency of assets or effects; or the shares of prizes; or the numbers of men to form certain detachments; or the division of waste lands among a number of proprietors.

Fellowship is either Single or Double. It is single, when the shares or portions are to be proportioned each to one given number only; as when the stocks of partners are all employed for the same time and double, when each portion is to be proportional to two or more numbers; as when the stocks of partners are employed for different times.

83

SINGLE FELLOWSHIP.

GENERAL RULE.

ADD together the numbers that denote the proportion of the shares: then

say,

As the sum of the said proportional numbers,

is to the whole sum to be parted or divided,
so is each of the several proportional numbers,
to the corresponding share or part.

Or, As the whole stock, is to the whole gain or loss,

so is each man's particular stock,

to his particular share of gain or loss.

TO PROVE THE WORK. Add all the shares or parts together, and the sun will be equal to the whole number to be shared, when the work is right.

EXAMPLES.

1. To divide the number 240 into three such parts, as shall be in proportion to each other as the three numbers, 1, 2, and 3.

2. Three persons, A, B, C. freighted a ship with 340 tuns of wine; of which A loaded 110 tuns, B 97, and C the rest in a storm the seamen were obliged to throw overboard 85 tuns; how much must each person sustain of the loss?

Here 11097

207 tuns, loaded by A and B ;

therefore 340 207 133 tuns, loaded by C.

3. Two merchants, C and D, made a stock of 1207; of which C contributed 757, and D the rest: by trading they gained 301; what must each have of it? Ans. C 187 15s, and D 117 5s. 4. Three merchants, E, F, G, make a stock of 7001; of which E contributed 1237, F 3587, and G the rest: by trading they gain 125/ 10s; what must each have of it? Ans. E must have 227 18 Od 23.q.

5. A General imposing a contribution * of 700l on four villages, to be paid in proportion to the number of inhabitants contained in each; the first containing 250, the 2d 350, the 3d 400, and the 4th 500 persons; what part must each village pay? Ans. the first to pay 1167 13s 4d.

* Contribution is a tax paid by provinces, towns, or villages, to excuse them from being plundered. It is paid in provisions or in meney, and sometimes in both.

6. A piece of ground, consisting of 37 ac 2 ro 14 ps, is to be divided among three persons, L, M, and N, in proportion to their estates: now if L's estate be worth 5007 a year, M's 3207, and N's 757; what quantity of land must each one have? Ans. L must have 20 ac 3 ro 3919 pls.

7. A person is indebted to O 577 15s, to P 1081 3s 8d, to Q 227 10d, and to R 737; but at his decease, his effects are found to be worth no more than 1707 14s; how must it be divided among his creditors?

belonged to S,

to T,

8. A ship, worth 9007, being entirely lost, of which and the rest to V; what loss will each sustain, supposing 5401 of her were insured? Ans. S will lose 451, T 90l, and V 2251.

9. Four persons, W, X, Y, and Z, spend among them 25s, and agree that W shall pay of it, X, Y 4, and Z; that is, their shares are to be in proportion as, , 3, 4, and what are their shares? Ans. W must pay 9s 8d 3449.

10. A detachment, consisting of 5 companies, being sent into garrison, in which the duty required 76 men a day; what number of men must be furnished by each company, in proportion to their strength; the 1st consisting of 54 men, the 2d of 51 men, the 3d of 48 men, the 4th of 39, and the 5th of 36 men? Ans., the 1st must furnish 18, the 2d 17, the 3d 16, the 4th 13, and the 5th 12 men

DOUBLE FELLOWSHIP.

DOUBLE FELLOWSHIP, as has been said, is concerned in cases in which the stocks of partners are employed or continued for different times.

RULE. Multiply each person's stock by the time of its continuance; then divide the quantity, as in Single Fellowship, into shares, in proportion to these products, by saying,

As the total sum of all the said products,

Is to the whole gain or loss, or quantity to be parted,

So is each particular product,

To the corresponding share of the gain or loss.

* Questions of this nature frequently occurring in military service, General Haviland, an officer of great merit, contrived an ingenious instrument, for more expeditiously resolving them; which is distinguished by the name of the inventor, being called a Haviland.

The proof of this rule is as follows: When the times are equal, the shares of the gain or loss are evidently as the stocks, as in Single Fellowship; and when the stocks are equal, the shares are as the times; therefore, when neither are equal, the shares must be as their products.