Es wiederfährt uns in unserm Leben
(§ 15. 2. d.) manches Glück und
manches Unglück.

Es wiederfährt Manchem mehr
Ehre, als er verdient".

Der Vogel ist zum Fenster hinaus'. geflo'gen.

Die Freunde entzwei'ten sich.

Die Pflaume ist ein Steinobst.
Sie verließen sich darauf, daß er
sein Versprech'en halten würde.
Man soll wie eher in eine Sache
ein'willigen als bis man tissel'be
wohl überlegt hat.
Ist es nicht, als ob dieses Volk mich
zum Gotte mache? (Schiller.)

1. Dieses Jahr ist das Obst, sowie alle Früchte, wohl gerathen. 2. Dieser Baum trägt jedes Jahr sehr vieles Obst. 3. Ist alle Frucht Obst? 4. Nein, nicht alle, sondern nur solche, die (§ 63. 2.) an Bäumen wächst. 5. Dieser junge Mann verläßt sich zu viel auf seine Verwandten und zu wenig auf seine eigenen Fähigkeiten. 6. Er verläßt sich darauf, daß wir ihn die nächsten Woche besuchen. 7. Er verließ sich darauf, daß ihm Gott helfen werde. 8. Wer sich zu viel auf Andere verläßt, kann leicht 9. Ich halte (Sect. 69. II.) viel auf meine Freunde. getäuscht werben. 10. Er hält viel auf ein gemächliches Leben. 11. Dieser Mann halt seine Klugheit, weshalb er den Nath wohl. zu viel auf sich und meinenter Freunde verschmäht. 12. Nur unter dieser Bedingung kann ich darein willigen. 13. Ich willige varein, in so fern (Sect. 79. V.) 14. Er willigte varein, ohne mit allen es feine üblen Folgen hat. Schwierigkeiten bekannt zu sein. 15. Dieses Kind thut gerade, als ob es hier zu Hause wäre. 16. Der Matrose stellte sich, als ob er von Sinnen

wäre.

17. Er geberdet sich, als ob ihm das größte Unrecht widerfahren sei. 18. Dieser Mann stellt sich, als ob er beleidigt wäre. 19. Er stellt fich wie ein Kind von fünf Jahren. 20. Der Nachbar warf den Zudringlichen zur Thüre hinaus. 21 Der Knabe cilte zur Thüre hinaus, als ich dieselbe öffnete. 22. Zur Thüre hinaus, wer sich entzweit! (Göthe.) 23. Es hängt ganz von Umständen ab, ob ich schon nächstes Jahr nach Amerika reise oder nicht. 24. Es hängt sehr von Umständen ab, was er thur wird. 25. Ein so abhängiges Leben die Bauern in Deutschland führen, ein eben so unabhängiges führen sie in Amerika. 26. Ganz unabhängig vermag kein Mensch auf Erten zu werden.

1. Last year the fruit did not turn out well. 2. This tree yields fruit but seldom. 3. This young gentleman relies too much upon his abilities. 4. No, he does not rely too much upon his abilities, but he knows it is not well to be dependent upon those of others. 5. I rely upon you that you will visit me next week. 6. Do exactly as if you were at home. 8. This 7. The criminal acted as if he were out of his senses. man acts exactly as a child. 9. Where is your canary.bird? It is flown out of the window. 10. How can I assent to a thing which is against my inclination? 11. Whoever quarrels shall be expelled the house. 12. It depends upon circumstances, whether I shall go to my friends. 13. Every man strives to be independent. 14. Depend upon it that I shall not help you again.

SECTION LXXXII

"not to be in fault, or

Nichts or, nicht dafür können, signifies to blame," &c.; as, ich kann nichts dafür; it is not my fault, or I cannot help it; literally, I cannot, or can nothing therefore. Er kann nichts dafür, daß er so arm ist; he cannot help it, i. e. So also interrogatively; he is not to blame that he is so poor. as, kann die Welt etwas dafür, daß sich ein großer Weist in ein schlech tes Kleid versteckt? (Rabener.) Is the world t blame, that a great soul conceals itself in a plain dress? That is, Die Welt kann nichts dafür.

Anʼgeben, to give, spe-
cify;
An'strengung, f. exer-
tion, effort, labour;
Bereit, ready;
Beruf', m. calling, vo-
cation;
Beruhigen, to quiet;
Bestimmen, to fix, de-
termine;

it;
Dafür, therefore, for

Dank, m. thanks, ac

knowledgment;
Grret'ten, to save, res-
cue, deliver;
Furcht, f. fear, dread;

EXERCISE 85.

Kern, m. kernel;
Kutscher, m. coachman;
gulation;
Ordnung,ƒ. order, re-
Schale, f. shell;
ling, weakly per-
Schwächling, m. weak-

son;

Umschlic'ßen, to in-
close, surround;
Un'ortentlich, disorder-
Umwerfen, to upset;
ly, irregular, con-
fused;
Unterlass'en, to leave
off, omit, fail;

Geld verloren habe.
Ich kann nichts dafür, daß ich mein

Diese Uhr geht vor (or, zu schnell),

und jene geht nach, (or, zu lang,
fam).

Hat man mein Zimmer in Ortnung
gebracht'?

In der Reihe seiner Schmeichler hat
er keinen wahren Freund.
Es giebt Viele, die glauben, daß in

ben meisten Fällen das Glück over
Unglück eines Menschen vom Zu ́-
fall abʼhänge.

Leben Sie wohl, mein Herr, und em
pfehlen Sie mich gütigst Ihrer
Frau Gemahlin.

worthiness, indig

nity; Teller, m.plate; Verter'ben, to spoil,

corrupt, destroy; Verzich'ten, (auf Etwas), to resign, i. e. as a privilege or a claim on anything; Vor'gehen, to go before,

go too fast;
Wagen, m. carriage;
Weisheit, f. wisdom;
Wesen, n. being, exis-

tence:

Zerbrechen, to break (in pieces).

lost my money. It is not my fault, that I have

This watch goes too fast and that (one) goes too slow.

Has my room been put in

order?

has not a true friend.

In the ranks of his flatterers he

There are many who believe, that in (the) most cases, the fortune or misfortune of a man depends on chance. member me kindly to your Farewell, sir, and please relady.

1. Sie können nichts dafür, daß Sie so unglücklich sind. 2. Er konnte nichts dafür, dieses Glas zerbrochen zu haben. 3. Ich kann nichts dafür geben, als meinen Dank. 4. Die Gründe dafür werde ich angeben, wenn es verlangt werden follte. 5. Können Sie mir sagen, (Sect. 83. I.) wie viel Uhr (Sect. 25. IX.) es ist? 6. Nein, tenn meine Uhr ist stehen geblieben. 7. Steht Ihre Uhr schon lange? 8. Ja, beinahe eine Stunde. 11. Leben Sie 9. Meine Uhr geht zu schnell, sie geht beinahe eine halbe Stunte vor. 10. Die Uhr meines Freundes geht fünf Minuten vor. Sie wohl, mein Herr! 13. Wann wollen wir zusammen Herrn N. wohl, und vergessen Sie nicht, mich bald wieder zu besuchen. 12. Leben besuchen? 14. Es hängt ganz von Ihnen ab (Sect. 81. I.), welche 3eit Sie dazu bestimmen wollen, ich bin zu jeder Zeit bereit, mitzugehen. 15. G8 hängt von Ihnen ab, diese Hamilie zu erretten oder zu verderben. 16. Der Nachbar arbeitet in feinem Garten und sucht denselben in Ortnung zu bringen. 17. Bei Aller Anstrengung bringt er diese Sache nicht in Ordnung. 18. Er suchte mich in tie Reihe seiner Kameraten zu bringen. 19. Es hält schwer (Sect. 46. II.), einen unordentlichen Menschen an Ortnung zu gewöhnen. 20. Nach vieler Mühe hat er die Rechnung in Ortnung gebracht. 21. Wer an dem Fuße eines steilen Berges stehen bleibt und aus Furcht ver Anstrengung denselben zu er, flimmen unterläßt und lieber auf die schöne Aussicht verzichtet, ter zeigt damit an, taß er ein Schwächling und eines solchen Genusses unwerth ist, und wer aus eigner Schuld in der Mitte seiner geistigen Ausbildung stehen bleibt, und den süßen Kern der Weisheit entbehren will, weil eine rauhe und harte Schale denselben umschließt, der zeigt ebenfalls nicht nur seine Unwärtigkeit, denselben zu genießen, an, sondern auch, wie wenig er den Beruf und die Pflicht des Menschen, als eines geistigen Wesens, erkannt hat.

1. It is not my fault that you have had the mishap. 2. You are not to blame that the servant has broken the plate. 3. He could not give me anything for it, except his thanks. 4. He could not help it, he only spoke the truth. 5. Is the coachman 7. Can you to blame, that the carriage was upset? 6. No, he is not to be blamed, for the horses could not be quieted.

SECTIO LXXXIII

tell me that time it is? 8. No, my watch goes too slow. 9. fm đà xát thư más Fun & Si nét pla de fo To fx the hour of my departure depends upon my parents 34 girl of is mine wine gelle tn Set 38 X. 15 Farewell, Madam; please do not forget, to remember me gut 1 Cum Brant waj uit ja nu net as the de to your parents 11 1: depende opon you, what time you will 11. £. maj folk iz Séry zé, a ja ga fx to it your friend; I shall always be ready to accompany you. 12 Fortune and misfortune. He and death, poverty and fat: tern auch im Schery fans was beinnig 12 gad riches, all depend on the will of God 13. Je gehe mat morem Armelii. 14 Ev nes bales se pa gries ,15 20 m til Ente ter Sutt 16 S Lange behen Sie zu gehen 17 lei eine Stante 15 Garn wie mera Slypaja; the Ba go 19. 34 bin bie in der F12be tel 8279 gemeet. 20. Funer we langen Svajiergang bekin. Su gemnát? 21 36 h fter car fade Sturze franeren gegangen 22 x large in and tem &arie grE se left £4, (literally, it understands itself, ie it is un. 23. 34 war trei viertel Sturen and emilia 24. Bac derstood, is self-evident, answers to our phrase, “of course," Sue writ won temielben entfernt? 25. 24 bu benite erre balle Start: or, as a matter of course." Ex: Gericht or cê ver west non demieiben entfernt geweien 20 34 15: rub soapine, i fit from feith tab is when Eltern geharden mat of course, 27. Der Gefangene meinte a matter of course, I must cher my parents. The word c un auf vieler, ever fet es in jezer Belt „nst naturally, is often used in the same manner; as, mathi mt fe fris, of course, it must be so

Bid werheben, (to understand one's self, with a signifies to be a judge of, to be skilled in. Ex: Gr 4 Ast he is skilled in everything.

or a

II. Bert is often answered in English by "gone, off," &c Ex: 3ft er schon lange fort? Has he already been gone long?

III. Es sei denn, raf=unless, except, &c, Ex: Der Vienic Lann nicht wahrhaft glücklich sein, es sei tenn, das er tugenthaft iei; man can not be truly happy, unless he be virtuous Wahrlich, mahrlich, ich faze dir: Es sei denn, ras Jemant von Neuem geboren were, fann er tas Reich Gettes nicht sehen:

Do you know how far you have

e sei man lange genug daß a ten warmen Schweiz ter Sense und te fuit: žuht kale entbiten måßen. 28 34 faz megen rik på to Lo offer tenn, daß mein Bruter bit ralin wieter gir gelat wine 24 3d fann heute unaniglich drejta. Di ef beentigen, es sei teaz, tai vé tez Nahmittag weniger geniet werke 30. Ge wit Niemand in to Stat singelaffen, es sei tean, tas er einen Voz Habe

I. Tell me if that is your own horse? 2 That farmer told me many things about agriculture. 3. I shall not go out today unless necessity compels me. 4. You will not enter the kingdom of heaven, unless you acknowledge the blessings of God 5. My brother went off yesterday and we have heard nothing of him. 6. It is self-evident that without nourishment man, animals, and plants cannot exist. 7. My knife is gone, and none of the children know where. 8. Our money is all gone. 9 I know very well how far I have to go in this matter 10. Where do you go to? 11. I am going to my brother. 12 How far have you to go? 13. Just to the park. 14 What distance have you to go? 15. About three quarters of a mile 16. He believed the time had now arrived to open his own path through life.

LESSONS IN ALGEBRA.-No. I

SECTION I.
INTRODUCTION.

to go in the matter? (how far ART. 1.-ALGEBRA is a general method of solving problems, and you are at liberty to go) How long a (pleasure) ridef investigating the relations of quantities by means of letters and

have you taken?

It is self-evident that a lazy
scholar can make no ad-
vancement.

This Italian is a judge of music

Mr. M. left (is off) this morning
for North America.

signs.

ILLUSTRATION.

PROB. 1.-Suppose that a man divided 72 pounds among his three sors in the following manner :-To A he gave a certain number of pounds; to B he gave three times as many as to a; and to che gave the remainder, which was half as many pounds as A and a received. How many pounds did he give to each?

1. To solve this problem arithmetically, the pupil would reason thus: A had a certain part, i. e., one share; B received three As far as he resides from here, times as much, or three shares; but c had half as much as a and and so great a distance as IB; hence he must have received two shares. By adding their have to walk, I, nevertheless, respective shares, the sum is six shares, which, by the conditions of visit him every day. the question, is equal to 72 pounds. If, then, 6 shares are equal Whither are you hastening so to 72 pounds, 1 share is equal to of 72, viz., 12 pounds, which rapidly? is A's share. E had three times as many, viz., 36 pounds, and c half as many pounds as both, viz., 24 pounds.

I am going to the dentist.
Well, be it (the thing) as it may
I shall not forgive him, unless
he ask my pardon.

2. Now, to solve the same problem by algebra, he would use letters and signs, thus:

Letz represent A's share; then by the conditions,

3 will represent B's share; and

4r2 will represent c's share.

Add together the several shares, or a's; thus, x+3x+2x=6xThen 6x=72, for the whole is equal to all its parts; and 1=12 pounds A's share; 3x=36 pounds B's share; and 2x=24 pounds

1. Der Dieb ist seines Verbrechens überführt werden und es versteht sich von selbst, daß er bestraft werden wirt. 2. Der Vater ist seit diesem Morgen fort und bis fezt noch nicht wieder zurückgekehrt. 3. Das Buch ist fort und keiner dieser Schüler will (§ 83. 8. Rem.) wissen, wo es hinc's share. gekommen ist. 4. Meine Neffen sind fortgegangen, ohne zu sagen wohin sie gehen würden. 5. Unser Obst ist alle. (Sect, 41. III.) 6. Auch noch so vieles Geld wird all, wenn man verschwenderisch ist. 7. Der türkische Kaiser Soliman II. fagte kurz vor seinem Lote: meine Kräfte

PROOF.-Add together the number of pounds received by each, and the sum will be equal to 72 pounds, the amount divided. In this algebraic solution it will be observed: First, that we represent the number of pounds which a received by z. Second, to obtain B's share, we must multiply a's share by 3. This multigk

cation is represented by two lines crossing each other like a capital X. Third, to find c's share, we must take half the sum of A's and B's share. This division is denoted by a line between two dots. Fourth, the addition of their respective shares is denoted by another cross formed by a horizontal and perpendicular line. Take another example:

PROB. 2.-A boy wishes to lay out 96 pence for peaches and oranges, and wants to get an equal number of each. He finds that he must give 2 pence for a peach, and 4 pence for an orange. How many can he buy of each?

is used; as

If the first is less than the other, the character ab, i. e., a is less than b. In both cases, the quantity towards which the character opens, is greater than the other.

16. A numeral figure is often prefixed to a letter. This is called a co-efficient. It shows how often the quantity expressed by the letter is to be taken. Thus 26 signifies twice b; and 95, 9 times b, or 9 multiplied into b.

The co-efficient may be either a whole number or a fraction. Thus b is two-thirds of b. When the co-efficient is not expressed, 1 is always to be understood. Thus a is the same as la, i. e., once a, or one times.

Let x denote the number of each. Now, since the price of one peach is 2 pence, the price of a peaches will be x X 2 pence, or 2o 17. The co-efficient may also be a letter, as well as a figure. In pence. For the same reason, x x 4, or 4x pence, will denote the the quantity mb, m may be considered the co-efficient of b; price of oranges. Then will 2x-4x, that is, 6x, be equal to because b is to be taken as many times as there are units in m. 96 pence by the conditions of that question, and 1x is equal to of If m stands for 6, then mb is six times b. In 3abc, 3 may be con96 pence, viz., 16 pence, which is the number he bought of each. sidered as the co-efficient of abc; 3a the co-efficient of be; or 3ab 2. QUANTITIES in algebra are generally expressed by letters, as the co-efficient of c. in the preceding problems. Thus b may be put for 2 or 15, or any other number which we may wish to express. It must not be inferred, however, that the letter used has no determinate value. Its value is fixed for the occasion or problem on which it is employed; and remains unaltered throughout the solution of that problem. But on a different occasion, or in another problem, the same letter may be put for any other number. Thus, in prob. 1, x was put for A's share of the money. Its value was 12 pounds, and remained fixed through the operation. In prob. 2, was put for the number of each kind of fruit. Its value was 16, and it remained so through the calculation.

18. A simple quantity is either a single letter or number, or several letters connected together without the signs and Thus a, ab, abd, and 8b, are each of them simple quantities. 19. A compound quantity consists of a number of simple quan. tities connected by the sign+ or -. Thus a+b, d−y, b-d+ 3h, are each compound quantities. The members of which each is composed are called terms.

3. By the term quantity, we mean anything that can be multi-A plied, divided, or measured. Thus, a line, weight, time, number, &c., are called quantities.

4. The first letters of the alphabet, a, b, c, &c., are used to express known quantities; and the list, z, y, x, &c, letters, those which are unknown.

5. Known quantities are those whose values are given, or may be easily inferred from the conditions of the problem under con

sideration.

6. Unknowr quantities are those whose values are not given but required.

7. Sometimes, however, the given quantities, instead of being expressed by letters, are given in figures.

8. Besides letters and figures, it will also be seen that we use certain signs or characters in algebra to indicate the relations of the quantities, or the operations which are to be performed with them, instead of writing out these relations and operations in words. Among these is the sign of addition (+), subtraction (−), equality (=), &c.

9. Addition is represented by two lines (+), one horizontal, the other perpendicular, forming a cross, which is called plus. It signifies "more," or "added to.' Thus a signifies that is to be added to a. It is read a plus b, or a added to b, or a and b. 10. Subtraction is represented by a short horizontal line (-) which is called minus. Thus, a-b signifies that b is to be "sub. tracted" from a; and is read a minus b, or a less b.

11. The sign + is prefixed to quantities which are considered as positive or affirmative; and the sign, to those which are supposed to be negative. For the nature of this distinction, see Arts.

36 and 37.

12. The sign is generally omitted before the first or leading quantity, unless it is negative; then it must always be written. When no sign is prefixed to a quantity, is always understood. Thas a + is the same as +a+v. 13. Sometimes both and (the latter being put under the former, +) are prefixed to the same letter. The sign is then said to be ambiguous. Thus a+b signifies, that in certain cases, comprehended in a general solution, b is to be added to a, and in other cases subtracted from it.

Observ. When all the signs are plus, or all minus, they are said to be alike; when some are plus and others minus, they are called unlike.

14. The equality of two quantities, or sets of quantities, is expressed by two parallel lines =. Thus a+b=d, signifies that a and together are equal to d. So 8+4=16-410+2=7+ 2 + 3.

15. When the first of the two quantities compared is greater than the other, the character > is placed between them. Thus a> signifies that a is greater than b.

20. If there are two terms in a compound quantity, it is called a binomial. Thus a+b and a-b are binomials. The latter is also called a residual quantity, because it expresses the difference of two quantities, or the remainder, after one is taken from the other. compound quantity consisting of three terms, is sometimes called a trinomial; one of four terms, a quadrinomial, &c. 21. When the several members of a compound quantity are to be subjected to the same operation, they must be connected by a line (—) called a vinculum, or by a parenthesis (). Thus a-bc, or a (b+c), shows that the sum of b and c is to be subtracted from a." But a-be signifies that b and c is to be subtracted from a, while e is to be added.

22. A single letter, or a number of letters, representing any quantities with their relations, is called an algebraic expression, or formula. Thus a+b+3d is an algebraic expression.

23. Multiplication is usually denoted by two oblique lines and 6 X 3 is 6 times 3, or 6 into 3. Sometimes a point is used to crossing each other, thus X. Thus ab is a multiplied into b; indicate multiplication. Thus a. b is the same as a X b. But the sign of multiplication is more commonly omitted, between simple quantities; and the letters are connected together in the form of a word or syllable. Thus ab is the same as a.b or axb. And is to be multiplied, a vinculum or parenthesis is used, as in the case bede is the same as bXcXdXe. When a compound quantity of subtraction. Thus the sum of a and b multiplied into the sum of c and d, is a+b+c+d, or (a+b) x (c+d). (6+2) X 5 is 8X 5, or 40. But 6+2 x 5 is 6+10, or 16. When the marks of parenthesis are used, the sign of multiplication is frequently omitted. Thus (x+y) (x-y) is (x + y) × (x−y).

And

24. When two or more quantities are multiplied together, each of them is called a factor. In the product ab, a is a factor, and so is b. In the product aX(a+m), a is one of the factors, and a+m the other. Hence every co-efficient may be considered as a factor. (Art. 17.) In the product 3y, 3 is a factor as well as y.

25. A quantity is said to be resolved into factors, when any factors are taken, which, being multiplied together, will produce the given quantity. Thus 3ab may be resolved into the two factors 3a and b, because 3axb is 3ab. And 5amn may be resolved into the three factors 5a, and m, and n. be resolved into the two factors 2x24, or 3x16, or 4X12, or 6x8: or into the three factors 2×3×8, or 4×6×2, &c.

And 48 may

26. Division is expressed in two ways: 1st. By a horizontal line between two dots, which shows that the quantity preceding it is to be divided by that which follows. Thus, ac, is a divided by c.

2nd. Division is more commonly expressed in the form of a fraction, putting the dividend in the place of the numerator, and the divisor in that of the denominator. Thus - is a divided by b. b

a

27. When four quantities are proportional, the proportion is expressed by points, in the same manner as in the Rule of Proportion in arithmetic. Thus a:b::e:d signifies that a has to b, the

beginning. In doing this the sign X must not be omitted between
the numbers, as it generally is between factors expressed by letters.
Thus if a stands for 3, and 6 for 4, the product ab is not 34, but
3X4, i. e. 12. Suppose a=3; b=4; c=2; d=6; m=8;
and n=10.
Find the values of the following algebraic expressions.
EXAMPLE.-11.

same ratio which e has to d. And ab: cd::a+m: b+n, means
that ab is to cd, as the sum of a and m, to the sum of b and n.
28. Algebraic quantities are said to be alike, when they are
expressed by the same letters, and are of the same power; and
unlike, when the letters are different, or when the same letter is
raised to different powers. Thus ab, 3ab, ab, and 6ab, are
like quantities, because the letters are the same in each, although
the signs and co-efficients are different. But 3a, 3y, 3bx, are
unlike quantities, because the letters are unlike, although there is
no difference in the signs and co-efficients. So x, xx, and xxx, 92. Ans.
are unlike quantities, because they are different powers of the
same quantity. (They are usually written x, x and x3.) And
universally if any quantity is repeated as a factor a number of
times in one instance, and a different number of times in another,
the products will be unlike quantities; thus, cc, ecce, and e, are
unlike quantities. But if the same quantity is repeated as a
factor the same number of times in each instance, the products are
like quantities. Thus, aaa, aaa, aa1 and aaa, are like quantities.
29. One quantity is said to be a multiple of another, when the
former contains the latter a certain number of times without a
remainder. Thns 10a is a multiple of 2a; and 24 is a multiple
of 6.

30. One quantity is said to be a measure of another, when the former is contained in the latter any number of times, without a remainder. Thus 36 is a measure of 156; and 7 is a measure of 35.

31. The value of an expression, is the number or quantity for which the expression stands. Thus the value of 3+4 is 7; of 3X4 is 12 of 10 is 2.

32 The RECIPROCAL of a quantity, is the quotient arising from dividing A UNIT by that quantity. The reciprocal of a is

a

; the

33. What is the algebraic expression for the following statement, in which the letters a, b, c, &c., may be supposed to represent any given quantities?

EXAMPLE. The product of a, b, and c, divided by the difference of c and d, is equal to the sum of b and c added to 15 times h.

EXERCISES.

abc c-d

Ans. =b+c+15h.

2. The product of the difference of a and h into the sum of b, c and d, is equal to 37 times m, added to the quotient of divided by the sum of h and b.

3. The sum of a and b, is to the quotient of b divided by c, as the product of a into c, is to 12 times h.

4. The sum of a, b and c, divided by six times their product, is equal to four times their sum diminished by d.

5. The quotient of 6 divided by the sum of a and b, is equal to 7 times d, diminished by the quotient of b, divided by 36. 31. What will the following expressions become, when words are substituted for the signs?

EXAMPLE.-12.

ad

с

+a+mn=3×6

+3+8×10=9+3+80=

2

POSITIVE AND NEGATIVE QUANTITIES.

36. A POSITIVE of AFFIRMATIVE quantity is one which is to be added, and has the sign + prefixed to it. (Art. 11.) 37. A NEGATIVE quantity is one which is required to be SUBTRACTED, and has the sign-prefixed to it.

When several quantities enter into a calculation, it is frequently necessary that some of them should be added together, while others are subtracted.

If, for instance, the profits of trade are the subject of calculation, and the gain is considered positive, the loss will be negative; because the latter must be subtracted from the former, to determine the clear profit. If the sums of a book account are brought into an algebraic process, the debit and the credit are distinguished by opposite signs.

38. The terms positive and negative, as used in the mathematics, are merely relative. They imply that there is, either in the nature of the quantities, or in their circumstances, or in the purposes which they are to answer in calculation, some such opposition as requires that one should be subtracted from the other. But this opposition is not that of existence and non-existence, nor of one thing greater than nothing, and another less than nothing. For in many cases either of the signs may be, indifferently and at pleasure, applied to the very same quantity; that is, the two characters may change places. In determining the progress of a ship, for instance, her easting may be marked +, and her westing-; or the westing may be +, and the easting - All that is necessary is, that the two signs be prefixed to the quantities, in such a manner as to show which are to be added, and which subtracted. In different processes they may be differently applied. On one occasion, a downward motion may be called positive, and on another occasion negative.

39. In every algebraic calculation, some one of the quantities must be fixed upon to be considered positive. All other quantities which will increase this must be positive also. But those which will tend to diminish it, must be negative. In a mercantile concern, if the stock is supposed to be positive, the profits will be positive; for they increase the stock; they are to be added to it. But the losses will be negative; for they diminish the stock; they are to be subtracted from it.

40. A negative quantity is frequently greater than the positive one with which it is connected. But how, it may be asked, can the former be subtracted from the latter? The greater is certainly not contained in the less: how then can it be taken out of it? The answer to this is, that the greater may be supposed first to exhaust the less, and then to leave a remainder equal to the difference between the two. If a man has in his possession 1,000 pounds and has contracted a debt of 1,500; the latter subtracted from the former, not only exhausts the whole of it, but leaves a balance of 500 against him. In common language, he is 500 pounds worse than nothing.

41. In this way, it frequently happens, in the course of an