2. Express in the quinary scale, 3824, 5861, and 3843. 3. Express in the septenary scale, 5163, 6842, and 4276. 4. Express in the quaternary scale, 3947, 5439, and 3854. 5. Express in the duodecimal scale, 6193, 8427, and 6958.

542. To change from any scale to the decimal scale. 1. Express 3432, in the decimal scale.

3432

5

19

5

98

5

492

EXPLANATION. Since each higher unit is equal to 5 of the next lower order, 3 units of the fourth order are equal to 15 of the third, and adding 4, the number of the third order given, we obtain 19, the number of the third order.

Proceeding in the same manner, until the number of units of the first order is obtained, the number in the decimal scale is 492.

Change the following to the decimal scale: 2. 5867,; 23123,; 34254; 523647.

3. 3432; 231t5; 41324,; 413426

4. 6735; 3819e12; 34514,; 268te12

543. Arithmetical processes in any scale.

The processes are performed in the same manner as in the decimal scale. The student must simply bear in mind each time the number of units of each order required to make one of the next higher order.

1. Add 31235, 41245, 32435, 42335.

2. Add 52437, 62317, 56347, 35437.
3. Add 43849, 52769, 83469, 74369.
4. Subtract 34562, from 624567.
5. Subtract 41375, from 73245g.
6. Multiply 3424 by 234.

PROOFS.

FUNDAMENTAL PROCESSES.

544. The proofs given under addition, subtraction, multiplication, and division are the most practical and reliable that can be given. A briefer method, however, has been discovered, which may be employed as a test of accuracy.

545. Method by casting out the nines.

It has been discovered that when the number of 9's in a number is found, the remainder is equal to the sum of the digits of the number, or the sum with the 9's omitted.

Thus, 743 700 ÷ 9+ 40 ÷ 9+3; and the remainders in each instance correspond with the digits which express the number. Hence the sum of the digits 7 + 4 + 3, or 14, or (with the 9 omitted) 5 is the number remaining after the 9's have been found.

546. The method of proof by casting out the 9's is based upon the presumption, that when the remainders in the results agree with the remainders in numbers, from which the results were obtained, the work is correct.

PROOF OF ADDITION.

547. 1. Prove that 893 + 296 + 452 + 368 = 2009.

EXPLANATION.-The 9's in the first addend are a certain number and 2 units remaining; in the second, a certain number and 8 units; in the third, a certain number and 2 units; in the fourth, a certain number and 8 units. The sum of the units remaining is 20, or casting out the 9's from that sum it is 2. The remainder after casting out the 9's in the sum 2009 is also 2. Hence, the work is probably correct.

It should be borne in mind, however, that this is not an accurate test of correctness, for the same excess of nines will be obtained in whatever order the figures are arranged.

PROOF OF SUBTRACTION.

548. Prove that 18945-93269169.

18945=0 EXPLANATION. - Casting out the 9's from the minuend 93262 there is 0 for a remainder. Casting out the 9's from the subtrahend there is a remainder of 2, 2 subtracted from

9619

= 7

a unit of the next higher order, or 9, leaves a remainder of 7. Casting out the 9's from the remainder there is also a remainder of 7. Hence the result is presumed to be correct.

PROOF OF MULTIPLICATION.

549. Prove that 718 x 28 = 20104.

718=7 EXPLANATION. - Casting out the 9's from the multi28 = 1 plicand the remainder is 7. Casting out the 9's from the multiplier, the remainder is 1. The product of these 20104 = 7 remainders is 7, and it is the same as the remainder after the 9's have been cast out from the product. Hence the work is probably correct.

PROOF OF DIVISION.

3

550. Prove that 8232 ÷ 21 = 392.

6

5 EXPLANATION. - Casting the 9's out of the divisor 21)8232 (392 and quotient, the remainders are 3 and 5 respectively. Their product is 15, from which, when the 9's are cast out, the remainder is 6. This number corresponds with the remainder in the dividend after the nines have been cast out of it; and the work is presumed to be correct, since the divisor multiplied by the quotient is equal to the dividend.

643)5926431 (9216 Rem. 543.

EXPLANATION. The product of the divisor by the quotient, plus the remainder is equal to the dividend. Casting the 9's out of the divisor the remainder is 4; casting them out of the quotient the remainder is 0. The product of 4 and 0 is 0; to which is added the excess of 9's in the remainder which is 3. The number remaining after casting out the 9's from the dividend is 3, therefore since the results agree, the work is presumed to be correct.

DIVISION BY FACTORS.

551. 1. What are the factors of 32? 25? 64? 96?

2. If a number is divided by 8, by what must the quotient be divided that the number may be divided by 16?

3. If a number is divided by 8 and the quotient by 6, by what is the number divided?

4. What factors may be used to divide a number by 36? 5. What factors may be used to divide a number by 48? 6. Divide 5683 by 32, using factors.

4 5683 21420 4 710

177

3

2

...

3+ (2 × 8) = 19 true Rem.

17712 Quotient.

EXPLANATION. -32 is equal to 4 x 2 x 4. Dividing 5683 by 4 gives a quotient of 1420 fours and 3 units remaining.

Dividing 1420 fours by 2 gives a quotient of 710 eights. Dividing 710 eights by 4 gives a quotient of 177 thirty-twos and 2 eights remainder.

The first partial remainder is 3 units, and the second, 2 eights, or 16; hence, the entire remainder is 3 + 16, or 19, and the quotient is 17712.

RULE. - Divide the dividend by one factor of the divisor, the quotient thus obtained by another factor, and so continue until all the factors have been used successively as divisors.

If there are remainders, multiply each remainder by all the preceding divisors except the one that produced it. The sum of these products will be the true remainder.

MEASUREMENT OF SOLIDS.

552. A surface such that a straight line joining any two points of it lies wholly in the surface is a Plane Surface.

553. A surface no part of which is a plane surface is a Curved Surface.

554. A plane figure bounded by straight lines is a Polygon.

555. The length of the lines that bound a figure is its Perimeter.

556. Anything that has length, breadth, and thickness is a Solid or Body.

The plane surfaces or planes which bound a solid are called its faces, and their intersections, its edges.

557. A solid whose two ends are equal polygons, parallel to each other, and whose sides are parallelograms, is a Prism. Prisms, from the form of their bases, are named triangular, quadrangular, pentagonal, etc.