Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

same side of the sun, they are in Perigo; when 'on dif ferent sides, in Apogeo: What is the difference of their distances in both these positions? Answ. 118,000,000

miles.

2. 20. If the mean distance between the earth and sun be 81 millions of miles, and between the earth and moon 240 thousand, how far are these two luminaries asunder in an eclipse of the sun, when the moon is lineally between the earth and sun? And in another of the moon, when the earth is in a line between her and him?

Answ.

In an eclipse of the sun, 80,760,000 miles.
In an eclipse of the moon, 81,240,000 miles.

CHAP. IV.

MULTIPLICATION.

1. A Number is said to multiply a number, when a num

ber is produced which contains the multiplied number, as often as the multiplying number contains unity. 2. The multiplied number is called the Multiplicand. 3. The multiplying number, the Multiplier. 4. And the number produced, the Product.

THE TABLE.

8 |

9 |10||r1| 12} 18 | 20 22 24 27 | 30| 331 36 36 | 40 441 48| 45|5v|55| 60 54|60|66| 72

63

| 2 3 4|5| 6| 7 | 21 4 6 8 | 10 | 12 | 14 | 16 | 3 6912 15 | 18 | 21 | 24 | | 8 | 12 | 16 | 20 | 24 | 28 | 32 | | 10 | 15 | 20 | 25 | 30 35 40 6 | 12 | 18 | 24 30|36| 42 | 48 7 14 21 28 | 35 42 | 49 | 56 8. 1024 32 | 40 | 48 | 56 | 64 | 9|S|27| 30 | 45 | 54 | 63 |72 | 10 | 20 | 30 | 40 | 50: 60 | 70 | 89 | 90 | 100|110|120| 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110121|132| 12|24|36| 48 | 60 72 | 84 | 96 | 108 | 120|132|144| Rule.

701 771 84

72 ||

80 88 96

81 |

901 99/108

Rule.

Multiply the first figure of the multiplicand by the gi ven multiplier, if the product be less than ten, set it down in the same place with the tuultiplied figure; but if the product be above ten (or tens) set down the overplus only, and reserve the ten or tens (in mind,) then by the same multiplier multiply the next figure of the multiplicand, and to the product add the ten or tens reserved, and proceed in the very same manner, until all the figures of the multiplicand are multiplied.

[1] 256745

Examples.
[2] 785403

[3] 27540098

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

1. Multiply the multiplicand by the first or units figuré of the multiplier, and subscribe the product (as last ;) in like manner multiply by the second figure of the mul tiplier, and so successively by every figure one by one, whereby there will be as many products as there are sig nificant figures in the multiplier.

2. Place these products in order under one another, so that the first figure stand directly underneath, or in the same place with the multiplying figures; that is, the first figure of the second product must stand under the second figure of the first product, the first figure of the third, under the second figure of the second, and the third figure of the first, &c.

3. Add all these products together, and their sum is the product sought.

C 2

Examples.

[blocks in formation]

When the multiplier hath cyphers intermixed with the significant figures.

Rule.

1. Multiply first by the first significant figure, and by every other successively, (omitting the cyphers) so that there will be as many particular products as significant figures in the multiplier.

2. As before, let the first figure of every particular product be put in the same place with the multiplying gure.

3. Add these products together, &e. as in the last.

Examples.

[20] 570684

[21] 8504593

304

709

[blocks in formation]

When the multiplicand, or multiplier, or both have cyphers in the lowest places.

Multiply by the other figures, as before taught, neglecting the cyphers till the product be found; and then put all the cyphers, both at the end of the multiplicand and multiplier, to the right hand of the product.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small]

When the multiplier is such a number that any two figures (in the table,) being multiplied together, will pro

duce it.

Rule.

Multiply the given number by one of these figures, and that product by the other, which will give the desired product.

C 3

Examples.

[blocks in formation]

Multiplication was shewn to be a compendious method of adding a number to itself, a determinate number of times, and from this consideration the universal Multiplication Table was constructed: In like manner we may construct a table of any multiplicand whatever, 2. e. find the product thereof by each of the single figures by Addition only, and this method is very useful in case of large multipliers.

Let the given multiplicand be the number 987654321.

Construction.

1987654321 Add the given multiplicand to itself, 21975308642 then we have the product thereof, mul32962962963 tiplied by 2. Again, add the said pro43950617284 duct or sum, to the given multiplicand, 54938271606 and the sum is the product thereof by 65925925926 3; and so adding every new product or 7 6913580247 sum to the given number, we may get 87901234568 the products to the said numbers multi98888888889 plied by all the single figures.

109876543210

2. Put each figure in the same (horizontal) row with the product it produceth, in a column to the left hand, as in the margin, and then the table is constructed.

3. A number is multiplied by 10 if a cypher be put before it whence we get a most easy proof of the truth or error of the operation; for, if the product of the given number by 9 be added to the said given number, we get the product thereof by 10, which, if it comes out the said given number with a cypher annexed, proves every proluct right, otherwise some error is contracted.

THE USE.

The use is evident, for let the given multiplier be 122156789. 987654321

« ΠροηγούμενηΣυνέχεια »