RULE I. Multiply together the natural series, 1, 2, 3, &c. up to the number to be taken at a time.

II. Take a series of as many terms, decreasing by 1, from the number, out of which the choice is to be made, and find their continued product.

III. Divide this last product by the former, and the quotient will be the answer.

1. How many combinations may be made of 7 letters out of 12 ? 1X 2X 3X4 X5 X6 X 7 (the number to be taken at a time)= 5040. 12 x 11 x 10 x 9 x 8 x 7 x 6 (same number from 12)=3991680.

Then 5040)3991680(792, Ans.

2. A general was asked by his king what reward he should confer on him for his services; the general only required a penny for every file, of 10 men in a file, which he could make out of a company of 90 men: what did it amount to? Ans. £23836022841 7s. 1165d.

1134

3. A butcher agreed with a farmer for a dozen of sheep, at $2 a head, which were to be picked out of 2 dozen; but being long in choosing, the farmer told him that if he would give him a cent for every different dozen which might be chosen out of the two dozen, he should have the whole, which was agreed to: pray what did the sheep cost him?

Ans. $27041.56. THEORETICAL QUESTIONS. What is permutation? What the RULE ? What is combination ? the RULE ?

DUODECIMALS.

231. DUODECIMALS,* that is, TWELFTH PARTS, are chiefly used in measuring surfaces and solids.

232. In duodecimals, A FOOT is divided into TWELVE equal parts, called inches or primes, marked thus, (); and each of these again into TWELVE other equal parts, called seconds, ("). In like manner, each second is divided into twelve equal parts, called thirds, (""); each third also, into twelve equal parts, called fourths, ("'"'); and so on.

In this way of dividing A FOOT, is plain, that

1' inch or prime, is

1" second is

2 of 12

1"" third is

of 1⁄2 of 12.

1""" fourth is of 12 of 12 of 11⁄2

12

20736 ... .. .......

233. Duodecimals are added and subtracted the same as compound numbers, 12 OF A LESS denomination making 1 of a GREATER, as in the following

12 inches or primes 1 FOOT.

234. The marks, ' "' ""&c., are called the INDICES.

* Derived from the Latin duodecim, which signifies twelve.

MULTIPLICATION OF DUODECIMALS.

235. IN SUPERFICIES the LENGTH multiplied by the BREADTH gives the SUPERFICIAL CONTENTS.

15 9

1 5'

=

45

or 45"; 45"=

1. How many square feet in a board 15ft. 9′in. long, and 1ft. 9′in. wide? 9 inches or primes x 5' x 3' 9". We set down 9", and reserve the 3 to carry ....... 15ft x 5' = 15 x 5 = 15 or 75', and 3′ we had to carry, gives 78′ = 6 9" ft. 6 inches. We set down the 6', and carry the 6ft. to the left. Then, multiplying by the 1ft., and adding the two products together, we obtain 22ft. 3' 9", Answer.

6 6' 15 9/

22 3′ 9′′

Hence we see, (AND LET IT BE REMEMBERED,) that 236. The product of any two denominations will always be of THAT denomination denoted by the SUM OF THEIR INDICES. 2. How many cubic feet in a block 13ft. 9' long, 11 ft. 10' wide, and 9ft. 7' thick?

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NOTE. It will generally be found most convenient, to multiply FIRST by the feet of the multiplier, then by the inches, and so on.

13 9/

Length.
Breadth. 11 10/

236 a. In SOLIDS the LENGTH multiplied by the BREADTH, and that product by the THICKNESS, gives the SOLID CONTENTS.

From these examples we derive the following

237. RULE. Write down the denominations as compound numbers, and in multiplying, REMEMBER, to make the product of 6"" any two denominations TO BE OF THAT denomination denoted by the SUM OF THEIR INDICES.

EXAMPLES IN DUODECIMALS.

1. How many square feet in a board 17ft. 7' long, lft. 5′ wide?

Ans. 24ft. 10' 11". 2. How many solid feet in a stick of timber 12ft. 10' long, Ift. 7' wide, and 1ft. 9' thick? 3. How many feet in a load 6ft. 7' long, 3ft. 5′ high, and 3ft. 8' wide?

Ans. 35ft. 6' 8"′ 6′′

Ans. 82ft. 5' 8" 4′′.

4. How much wood in a load 11ft. 4in. long, 5ft. 9in. wide, and 3ft. 11in. Ans. 225ft. 2′ 10′′. high? B. How many solid feet in a block 15ft. 8' long, lft. 5' wide, and lft. 4' Ans. 29ft. 7' 1" 4"". thick? 6. What is the product of 371ft. 2′ 6′′ multiplied by 181ft. Ï' 9"? Ans. 67242ft. 10 1" 4" 6. 7. How many cords, &c., in a pile 176ft. in length, 3ft. 9' wide, and 4ft. Ans. 21 C. 117ft. 3' high? 8. What is the price of a marble slab, whose length is 5ft 7', and 1ft. 10, Ans. $10.23. at $1 per foot?

9. There is a house with three tiers of windows, 3 in a tier, the height of the first tier is 7ft. 10', of the second 6ft. 8, of the third 5ft. 4', and the

1

breadth of each is 3ft. 11'; what will the glazing come to at 14d. per foot? Ans. £13 11s. 104d. 10. If a house measures within the walls 52ft. 8 in length, and 30ft. 6' in breadth, and the roof be of a true pitch, or the rafters of the breadth of the building, what will the roofing come to, at 10s. 6d. per square? Ans. £12 12s. 11 d.

THEORETICAL QUESTIONS.

What are duodecimals? In what are they chiefly used? In duodecimals, how is a foot divided? How are duodecimals added or subtracted? What are the marks that distinguish the denominations, called? How do you find superficial contents? -solid contents? What is the RULE FOR DUODECIMALS?

INVOLUTION,

OR THE

RAISING OF POWERS.

238. INVOLUTION is multiplying any number into itself a certain number of times.

239. The PRODUCTS obtained are called POWERS.

240. The NUMBER ITSELF is called the ROOT or first

power.

241. If the ROOT or first power be multiplied into itself, the product is called the SQUARE or second power; if the square be multiplied by the first power, the product is called the CUBE or third power, &c.

25 is the square, or 2d power 125 is the cube, or 3d power

= 5%. = 53.

5 × 5 × 5 × 5625 is the biquadrate, or 4th power

54.

242. The little figure denoting the power is called the

243. It is evident, that a fraction is involved by involving

the NUMERATOR AND DENOMINATOR. (Art. 121.)

9. What is the square, cube, and 4th power of .5?

the 4th power? Ans. 1, 4, 286'

Ans. .25, .125, & .0625.

10. What is the square, cube, and 4th power of 1.5?

11. What is the 6th power of 2}?

Ans. 2.25, 3.375, & 5.0625.

244. A mixed number is best involved by first reducing it to an IMPROPER FRACTION. 2} = 3, and the 6th power of is 117643 = 1612 729

12. Involve 24 to the 6th power.

Ans. 531441 13. Involve 26, that is to say, involve 2 to the 6th power.

14. Involve 37?

68 ?

1312?

4096 = 1293057

79? 1213?

Ans. 64. 810?

Ans. 9878.

Ans. to last, 106993205379072.

The powers of the 9 digits, from the 1st power to the 5th, may be seen in the following

Sursolids or 5th powers | 1 | 32 | 243 | 1024 | 3125 | 7776 | 16807 | 32768 | 59049|

EVOLUTION,

OR THE

EXTRACTION OF ROOTS.

245. EVOLUTION is finding the ROOT of any given power or number.

246. The root we have seen, is that number, which being multiplied into itself continually, will produce the given power. 247. The SQUARE ROOT of any given number is a number, which being multiplied into itself, will produce that number, And the CUBE ROOT, is a number which being cubed, or involved to the 3d power, will produce the same given number.

The square root of 144 is 12, because 122 = 144.

The cube root of 512 is 8, because

83512, &c.

248. This character ✓ placed before any number, expresses the square root of that number.

Thus 25 expresses the square root of 25.

249. The same character is made to express any other root, by placing the index of the root above it.

Thus
And

27 expresses the cube root of 27.

625 expresses the 4th root of 625, &c.

250. It appears from the preceding table of powers, that the square of any root can have but TWICE as many figures as the root itself.*

Hence we may always tell how many figures there will be in the SQUARE ROOT of any number, by pointing it off from unit's place, into PERIODS of TWO FIGURES each.

251. It also appears, that the cube of any root can have, at most, but THREE TIMES as many figures as the root itself. +

Hence we may ascertain how many figures there will be in the OUBE ROOT of any number, by pointing it off from unit's place, into periods of three figures each.

252. Although there is no number which will not produce a perfect power, yet there are numbers of which exact roots can never be obtained.‡ Numbers, whose exact roots cannot be obtained, are called SURD NUMBERS, and those whose roots can be thus obtained, are called RATION

AL NUMBERS.

EXTRACTION OF THE SQUARE ROOT.

253. TO EXTRACT THE SQUARE ROOT is to find a number, which being multiplied into itself, will produce the GIVEN number.

Required the side of a square, that shall contain 625 square feet.

The contents of a square, as we have seen, (page 49,) is found by multiplying the length of one side INTO ITSELF. Hence, if we have the contents of any square given, we may always find the side by extracting the SQUARE ROOT.

This is done by a SORT OF TRIAL.

'1st. We must ascertain how many figures there will be in the root. This we can do, by pointing off the number from units, into periods of TWO FIGURES each; which, in this case, being done, we find that the root will consist of two figures, a ten and a unit.

2d. We seek for the first figure, that is, for the TENS of the root, and it is plain, that we must extract it from the left hand period 6 (hundred). The greatest square in 6 (hundred) we find, by trial, to be 4 (hundred), the root of which is 2 (tens=20,) we therefore set 2 (tens) in the root. The root, it will be recollected, is one side of a square. Let us, then, form a square, (A, Fig. I,) each side of which shall be supposed 2 tens,= 20ft., expressed by the root now obtained. The contents of this square is 20 X 20=400 ft., now disposed of, and which, consequently, is to be deducted from the whole number of feet, (625,) leaving 225ft. This deduction is most readily made by subtracting the square 4 (hundred), or the square of 2, (the figure in the root already found,) from the period 6 (hundred), and bringing down the next period by the side of the remainder, making 225 as before. 3d. The square A is now to be enlarged by the addition of the 225 remaining feet; and, in order that

Because the square of 9, the highest digit contains but two figures.

† Because the cube of 9, contains but THREE figures.

By the help of decimals, however, we can approximate towards the true to any degree of exactness required.