ON.

26

I place the latter quantities under the former: then the 6b quantities to be subtracted being 3a2b, I change their signs, and they become + 3 a-26; these I add, thus, 3a to + 4 a, and the sum is +7 a to put down; 26 to 8b to put down, by Art. 36.

3b

6b, and the sum is

Changing the signs of the lower quantities, they become -2x+7y; then put down, and + 7y added to down, by Art. 37.

2 x added to + 3 x, gives +x, to

ax take

-5y gives +2y, to put

in the upper; therefore I put the former down, with its proper sign, and the latter with its sign changed, according to the rule.

41. To subtract quantities under the vinculum.

RULE I. If the quantities under the vinculum are in all respects alike, change the sign of the coefficient of the lower vinculum, then add it to that of the upper, and to the result subjoin the common vinculum, &c.

II. But if they are not alike, put all the quantities down in succession, with the sign of the lower coefficient changed1.

15. From 14 a + x take

OPERATION.

14 √a + x

9/a + x

23 √a + x

The quantities under the vinculum being alike, I only change the sign of the lower coefficient, (-9 to +9,) then add it to the upper, and to the sum subjoin the common vinculum.

16. From 12.ax — 2) take 15.ar

OPERATION.

The quantities under the vinculum are alike; I change the sign of the 15 from + to −, then adding -15 to +12, the sum is 3, to which I subjoin the vinculum, &c.

3 az take 2 √a + x.

OPERATION.

3 a
2 va + x

3 va

18. From 5

Explanation.

The quantities under the vinculum being unlike, I only change the sign of the lower coefficient from+to-, and put all the quantities down in succession.

x − y take 2 √x — Y.

Diff. 3x-Y.

√1 + x take x + √1+x. Diff. 3x-9

1 The observations contained in the note on Art. 39. are applicable to this

rule, and therefore need not be repeated.

22. From a2

Diff. a2 - 2 ax

23. From 36

2 ax + x √x · 2x-2y

2y take 3 x x − 2 y + 1.

1.

4+ a + b take 12 x + 3. a + b2. Dif

ference 3b 16 + x + √a + b - 3.a + b2..

24. From a 3 xyz - y3

take ao + 3 .xyz — yo + 4 — a3.

Diff.6 xyz y3 + y2 — 4 — a'.

MULTIPLICATION.

To multiply simple quantities together.

42. When the factors have the same letters in each; RULE I. If the factors have like signs, (viz. both +, or both -,) make the sign of the product +; but if they have unlike signs, (viz. one and the other—,) make the sign of the pro

duct

II. Multiply both coefficients together, and the result will be the coefficient of the product.

III. Add the index of each letter in one factor, to the index of the same letter in the other, and the sum will be the index, which must be placed over that letter in the product.

IV. Place the sign, coefficient, and letters, as found above, in order, and it will be the product required ".

m The rule for the numeral coefficients is plain from the nature of arithmetical multiplication; that for the indices follows from the method of notation: thus, suppose a3 is to be multiplied into a2, the product will be a5, or a with the index 5, which rises from adding the indices 3 and 2 together; for a3=aaa, and a2 = aa, (Art. 20.) therefore a3 × a2=aaa × aa, or aaaaa (by Art. 20.) but this expression equals a5 (Art. 20.) therefore aa × a2=a5; or the multipli cation of like algebraic quantities is performed by adding the indices of like letters in both factors together, and placing the sum as an index over its letter in the product.

That like signs in the factors give plus in the product, and unlike signs in the factors give minus in the product, is thus explained.

1. When + a is to be multiplied into + b, it is implied, that + a is to be taken as many times as there are units in b; that is, as many a's are to be added together, as b contains units; and since the sum of any number of affirmative terms is affirmative, it follows, that a taken + b times is affirmative, or that + a x+b= + ab.

2. When quantities are to be multiplied together, it is indifferent in what

EXAMPLES.

1. Multiply 4a3 into 5a2.

Explanation.

The signs of both factors being +, (understood,) that of the product will be + (understood;) then 4 × 5 = 20 for the coefficient, and 2 + 3 = 5 for the index; wherefore 20 a 5 is the product required.

4 as 5 a2

20 as product

Or thus,

4 a3 × 5 a2 = 20 a3 product.

order they are placed, for a × bis the same as ba; therefore when a is to be multiplied into + b, or + b into ―a, this is the same as taking -ɑ as many times as there are units in +b; and since the sum of any number of negative terms is evidently negative, it follows that - a times+b, or + a times-bis negative, that is, − a x + b, or + a × —b, will each produce -ab.

3. When ―a is to be multiplied into — b, it is implied that - a is to be subtracted (not added) as often as there are units in b, because the sign - denotes subtraction; but subtracting a negative quantity, is the same as adding an affirmative one of equal value, (see the note on Art. 40.) consequently -a subtracted b times, is the same as + a added b times; that is, a x-b is the same as + ax + b, which produces +ab, as has been shewn in the former part of this note. Therefore multiplied by +, and each produces +; also + multiplied by -, or

duces

multiplied by -, multiplied by +, pro

The same may be shewn otherwise; thus, it is evident, that if a compound quantity equal to nothing, be multiplied by any quantity whatever, the product will be nothing. Now since a-a= =0, let this be multiplied by + b, the first term of the product will evidently be + ab, (by the former part of the note,) wherefore the second term of the product must be ab, otherwise the product of a-a=0 multiplied by + b, (or 0 multiplied by b,) will not be equal to nothing, which is absurd; wherefore if + ax + b + ab, then will ~ax+b=-ab.

Let now a-a=0 be multiplied by -b; the first term of the product being -ab, from what has just now been shewn, the second term must of course be +ab, to make the product =0: therefore a × −b= + ab.

That these conclusions are true, appears from their application; let 8-4 =4, be multiplied into 5-3=2.

40—44+12=52-44-8, but 4 X 2=8 also ;

Wherefore 8-4 × 5-3=4 × 2, or the product of the two compound quantities, (observing the above rule for the signs,) equals the product of their equivalent simple numbers found by common multiplication: which was to be shewn.

3. Multiply

Thus, - 2 ab2c × 3 a3bc = — 6 a4b5c2, the product.

4. Multiply

5 ar3 = 10 a2x5, product.

5. Multiply 3x2y into 8 x2y2.

Prod. 24 x4y3.

6. Multiply xyz into xyz. Prod.xyz3.

7. Multiply - 4 ab2 into — 3 a2b2. Prod. 12 a3ba.

43. When the factors consist of different letters.

RULE. Find the sign and coefficient of the product as before, and to the coefficient subjoin all the letters in both factors for the product".

8. Multiply 3ab into 5xy.

OPERATION.

Explanation.

The signs being unlike, that of the product will be -> then 3 X 5 15 the coefficient; then putting down the sign, coefficient, and all the letters, (in alphabetical order,) we have the product required,

3 xy into

3 xy × 7 az

44. When one of the factors is a compound quantity.

RULE I. Place the simple quantity under the left hand place of the compound one; or place both factors in a line, with the

This rule is evident from the nature of algebraic Notation, Art. 10. It is most convenient to place the letters alphabetically.