England, Scotland, Ireland, and France, (the three largest

in each,) was as follows:

Required the number of inhabitants contained in the three largest towns in each country.

Ans.

33. The Pyramids of Egypt are thought to have been built 337 years before the founding of Carthage, Carthage to have been founded 49 years before the destruction of Troy, and Troy to have been destroyed 431 years before Rome was founded; Carthage was destroyed 607 years after the founding of Rome, and 146 years before the commencement of the Christian era; the western empire of Rome ended in the year 476 of the Christian era, and 590 years before the Norman conquest; Constantinople was taken by the Turks 387 years after the Norman conquest, and 323 years before the American Revolution. How many years elapsed between the first and last of these events?

What is addition ?

Questions.

How is addition performed?

Repeat the rule of operation,

How is addition proved?

SUBTRACTION OF WHOLE NUMBERS.

18. Subtraction of whole numbers is the taking of a less number from a greater, and thereby showing the remainder or difference.

The number which is to be subtracted from the other is usually called the subtrahend; the number from which the subtraction is made, the minuend. If we have to subtract 346 from 579, it is plain that we may subtract the units, tens, and hundreds of the minuend; and that the sum of the remainders 233 is the remainder sought. And in such a case, it matters not whether we proceed from left to right, or from right to left. But if any digit of the minuend be less

than the digit in the corresponding column of the subtrabend, for instance, if we have to subtract 279 from 546, as we cannot subtract 9 units from 6 units, nor 7 tens from 4 tens, we may suppose the minuend resolved into the parts 16, 130, and 400; and then subtract the 9 units from 16, the 7 tens from 13 tens, and the 2 hundred from 4 hundred. And thus, when any digit of the minuend is less than the corresponding digit of the subtrahend, conceiving a unit prefixed to it, and performing the subtraction, when we proceed to the next column we have to conceive the next digit of the minuend less by 1, on account of the 1 which has been already borrowed from it. But it affords the same result, in practice, to conceive the next digit of the subtrahend increased by one, and the digit of the minuend unaltered: as it obviously gives the same remainder to subtract 8 from 14, as to subtract 7 from 13. And hence appears the reason of what is called carriage, in subtraction; and the reason of proceeding from right to left: though the result may be obtained by repeated subtractions from left to right. The carriage in subtraction may be accounted for on another principle; namely, that if the two numbers be equally increased, their difference will remain unvaried. Thus, in subtracting 19 from 56, when we take 9 from 16, we may conceive that we have added 10 to the minuend, and therefore must add 10 also to the subtrahend.

Besides the same attention to the arrangement of the numbers as is necessary in addition, the scholar ought to be exercised in performing the operation of subtraction whether the subtrahend be above or below the minuend.

The remainder found being the difference between the given numbers, or the numbers by which the minuend exceeds the subtrahend, it is plain that adding the remainder to the subtrahend must give a total equal to the minuend : or that subtracting the remainder from the minuend must give a remainder equal to the subtrahend. This affords two methods of proving subtraction. And, in addition, if we subtract any one of the numbers from the total, the remainder must be equal to the sum of all the other numbers. It is proper to observe, that the sign - (called minus, from the Latin word signifying less,) interposed between two numbers, denotes that the latter is to be subtracted from the former: thus, 25-4 (read 25 minus 4) denotes the remainder 21, subtracting 4 from 25.

--

Rule for the Subtraction of Whole Numbers.

19. Place the less number below the greater, with units under units, tens under tens, &c. as in addition. Beginning with the units, take, if possible, each figure in the lower line from the figure above it, and set down the remainder. But if any figure in the lower line be greater than the figure above it, add ten to the upper; then subtract as before, and carry one to the next figure in the lower line.

The reason of this rule is evident from the preceding article.

Methods of Proof.

1. Add the remainder and the less of the given numbers together if the sum be equal to the greater, the work is

correct.

2. Subtract the number found from the greater of the given numbers; if the remainder be equal to the less, the work is correct.

Examples in Subtraction of Whole Numbers.

7854

Ex. 1. From 7854 take 4513.

4513

Set the numbers as in the margin, and pro

ceed thus: 3 from 4 and 1 remains, 1 from 5 Rem. 3341 and 4 remain, 5 from 8 and 3 remain, 4 from 7

and 3 remain; the remainder therefore is 3341. Proof 7854 To prove the work, to the less of the given numbers add the remainder, and the sum will be 7854, the greater; or, as in the second method, subtract the remainder from the greater number, and the result will be 4513, the Rem. 3341

7854

4513

less.

Proof 4513 1831.

From 3712

Take 1831

Ex. 2. Required the difference of 3712 and In this example, proceed thus: 1 from 2 and 1 remains, 3 from 11 and 8 remain, carry 1 to 8, and then 9 from 17 and 8 remain, carry 1, and then 2 from 3 and 1 remains. The differ- Rem. 1881.

ence therefore is 1881, and the operation would be proved in the same manner as before.

83

57

26

Ex. 3. Required the difference of 83 and 57. Here, as 7 cannot be taken from 3, we consider 83 as 70 and 13; and, subtracting 7 from 13, and 5 from 7, we find the difference to be 26. In this simple and natural method, the values of the given numbers undergo no change; and, with only one exception, it might be employed with as much facility as the common method, the next figure in the upper line being always diminished by a unit, when one would be carried to the figure below it, in the common method. The exception is the case in which the next figure in the upper line is a cipher; in this case, the common method is considerably preferable; and, as in practice that method is in no way inferior, it is universally preferred.

Whenever the figure in the lower line is greater than the corresponding figure in the upper line, the operation of sub. traction may also be performed in the following manner :subtract the less from the greater, and take the remainder from ten; set down the result; then take the next figure in the lower line from the corresponding figure in the upper line diminished by unity, and proceed as before. For instance, in the above example, as 7 cannot be taken from 3, subtract 3 from 7, and there remain 4, and 4 from 10 and 6 remain: now 8 diminished by 1 is 7, and 5 from 7 and 2 remain. By this method, we can perform the operation of subtraction without the aid of addition, and also without being under the necessity of carrying one, as in the common method.

Ex. 13. 1010101-101010

Ex. 17. 44444-35555

14. 100000001000101

15. 75634-12345

16. 9999-8888

18. 12222-3333 19. 303030-30303 20. 95000000-240000

21. Take four thousand four hundred and four from four millions.

22. Required the difference between six millions and six thousand.

23. Subtract nineteen millions and ninety-nine from one billion.

24. La Place, the celebrated French mathematician and philosopher was born in 1749; required his age in 1826.

25. The height of Mont Blanc. the highest mountain in Europe, is 15680 feet, and the height of Chimborazo, the highest mountain in America, is 21427 feet: how much is the latter higher than the former?

26. Required the difference between the population of New-York and that of New-Jersey ? Ans. 1095237.

27. How long is it since James Monroe was elected to the Presidency, which event took place in 1817? Ans. 28. Guns were first used in 1380: how many years since that time till the year 1827 ? Ans. 447 years.

20. How many inhabitants are there in Africa more than in America? Ans. 30000000. 30. George Washington was elected President of the United States in the year 1789: how many years have elapsed since?

Ans.

31. How long is it since the peace of Utrecht, which was concluded in the year 1713?

Ans. 32. La Fayette arrived in America in the spring of the year 1777. How many years have elapsed since? Ans.

33. The following are the years of the Christian era in which the under-mentioned events happened: required the number of years from each till the year 1827.. Commencement of the Hegira, or era of the flight of Mahomet, 622; the Arabic, or modern notation in Arithmetic, introduced from Arabia into Europe by the Saracens, 991; first Cru sade, 1096; Magna Charta signed by king John, 1215; linen first made in England, 1253; Termination of the Crusade, 1291; spectacles invented by a monk of Pisa, 1299; gunpowder first used in Europe, 1330; Algebra introduced into Europe from Arabia, 1412; printing invented, 1440; Constantinople taken by the Turks, 1453; America discovered