the pavement been only 6 yards wide instead of 12, then the number of square yards would have been 72, viz., 12 × 6=72 This figure will also serve to illustrate the relative proportions between the square foot and the square inch. If each of the small squares represent a square inch, the large square will represent a equare foot, being twelve times the length of the side of the small square; the figure will then represent a square foot of 144 square inches.

Obs. 3. For measuring, land surveyors use Gunter's chain, described above. 10 square chains or 100,000 square links are an acre, 640 acres are a square mile.

Obs. 4. A piece of land, 10 chains long and one broad, or five chains long and two broad, is an acre.

1. In 654 acres, how many square yards?

2. In 689 square yards, how many square inches?

3. In 64 acres, 3 roods, 3 poles, 15 yards, how many square feet?

Obs. 1. The contents of solid bodies, such as blocks of marble or stone, logs of timber, &c., are measured by this measure. Square measure has only to do with length and breadth; cubic measure with length, breadth, and thickness. The cubic foot is one foot broad, one foot long, and one foot thick, and contains 12 x 12 x 12 = 1728 cubic inches.

Obs. 2. The dimensions of bodies are taken by lineal measure, but the quantity of matter they contain is calculated by cubic measure.

1. In 63 cubic yards, how many cubic inches ?

2. In 189648826 cubic inches, how many cubic feet?

Obs. The year is divided into 12 calendar months; the number of days of which they are composed vary as on the following page :

120 + 123

December 31

+

122-365

The ancients had various modes of computing time. The Jews reckoned by moons; the Greeks had a similar mode of computation. The manner of computing time in this country previous to the year 1752 was by the Julian Calendar, that is, by the Roman Calendar, as improved by Julius Cæsar. Before his time, the Calendar consisted of 365 days, but finding that the sun performed his course in 355 days nearly, he gave 365 days to each year for three years, but to every fourth year 266 days, by adding one day to the month of February, making this month to consist for this year of 29 days, instead of 28. The year which consists of 366 days, is called Bissextile, or Leap-year, from the Latin word bissextus. Leap-year may be found by dividing the number, which indicates the year, by 4. If it divide without leaving a remainder, then it is Leap-year; if there be a remainder, then the remainder shows the number of years after Leap-year. Thus 1835 is three years after Leap-year, and 1836 4) 1835 is Leap-year, for

458 - 3

and

4) 1830
459

A further reformation of the Calendar was made by POPE GREGORY XIII. It was found by astronomers that the time occupied by the revolution of the earth round the sun was exactly 365 days 5 hours 48 min. 48 sec. Therefore the Julian year exceeds the true year by 11 min. 12 sec., and in the space of 400 years the Julian computation will fall behind the course of the sun and of the seasons 3 days 2 hours 40 min. Between the time of Julius Cæsar and the year 1582, when the Gregorian improvement in the Calendar was made, a period of about 1600 years had elapsed; consequently, in that time, during which the Julian Calendar bad been in use, the computation had fallen behind the true time 12 days 10 hours 40 mins. To rectify this discrepancy, the Gregorian or NEW STYLE was introduced. By this new mode of computing time, an advance of twelve days was made in the year at once, and for the future it was arranged to retrench three Leap-years in every 400 years. By this arrangement, every hundredth year, where the hundreds are not divisible by 4, (such as 17 hundred, 18 hundred, 19 hundred, which would otherwise have been accounted as Leap-years) are to be reckoned as common years of 365 days. Thus in 400 years, the difference between true time and apparent time is only two hours 40 min.

The NEW STYLE, or Gregorian computation, was adopted by England by act of parliament in 1752, and is now used by every country in Europe, except RUSSIA, which still retains the OLD STYLE.

1. In 56 years, how many seconds?

2. In 689432 seconds, how many days?

3. In 36 years, how many hours?

4. In 24 years, 11 months, and 18 days, how many hours?

224. QUESTIONS FOR EXAMINATION

UNDER CASE 5.

1. What are compound numbers ?

2. In what do compound and simple numbers differ? 3. Do the same principles of calculation apply to compound numbers as to simple numbers?

4. How are numbers of one denomination charged into another denomination?

5. Define the term Reduction, according to the common acceptance of the term, and also when used to indicate a process in Arithmetic?

6. What is the difference between Ascending and Descending Reduction?

7. To what purposes is Reduction applied?

8. Explain the principle of Reduction as described in paragraph 209.

9. Are the several Reduction Tables, tables of equivalents? 10. Which is the general weight of commerce?

11. For what purposes is Troy weight used?

12. What is the difference between the Avoirdupoise pound and the Troy pound?

13. For what purposes is Apothecaries' weight used?

14. What difference is there between the Troy pound and the Apothecaries' pound?

15. For what purposes is Striked Measure of Capacity used?

16. For what purposes is Heaped Measure used?

17. What is the standard measure of extension, and where is such standard preserved ?

18. What is Gunter's chain?

What is the Perambulator?

19. For what purposes is Lineal Measure used? 20. For what purposes is Square Measure used? 21. Define a square?

22. What is an acre of land?

23. For what purposes is Cubic Measure used?

24. By what means did the Jews and Greeks compute time? 25. What was the mode of computing time in this country before the year 1752? What has been the mode of computing time since that period?

26. What is understood by the Julian Calendar ?

27. What is a Leap-year? How is Leap year found?

28. What is the Gregorian mode of computing time, and on what particular circumstance is the Gregorian computation founded?

29. In what space of time does the the earth revolve round the Sun?

30. What is understood by the New Style? What is understood by the Old Style?

31. What is the difference between true time and apparent time in 400 years, and how is that difference estimated?

32. What countries use the New Style of computation, and what countries use in the Old Style? When was the New Style introduced into this country?

85

CHAPTER VI.

COMPOUND ADDITION AND SUBTRACTION.

224. No rules in Arithmetic are more frequently used in the various transactions of business, than these; therefore, the student should practise them until he can perform operations in them with great facility and accuracy.

225. The student must be careful to place the different denominations of the question in proper order.

Set

£. s. d. 14 6 8 2 104

15

10 1 GI 12 13 3 25 15 111

78 0 45 48 10 94

29 9 63

4

of pence,

226. Arrange questions as in the margin, each denomination in regular columns, pounds under pounds, shillings under shillings, and pence under pence: the question being thus arranged, begin with the farthings, and say, 2 and 3 are 5, and I are 6, and 1 are 7, and 2 are 9,-the sum of the farthings being divided by 4, the number of farthings in a penny, gives 2 for a quotient, and a remainder of 1; the 1 being a farthing set it down under the column of farthings, and carry the 2 to column of pence, and say 2 and I are 3, and 3 are 6, and 6 are 12, and 8 are 20. down 0 on the off side, and carry 2 to the column of tens in the pence, and say 2 and I are 3, and 1 are 4, making 40 for the sum of the column which divide by 12, the number of pence in a shilling, and you have three for a quotient, and a remainder of 4, which 4 being pence set it down under the column of pence, and carry the three to the column of shillings, and say 3 and 5 are 8, and 3 are 11, and 1 are 12, and 2 are 14, and 6 are 20,-set down the 0 at the off side, and carry 2 to the column of tens in the shillings, and say 2 and 1 are 3 and 1 are 4, making 40 for the sum of the column of shillings, which divide by 20, the number of shillings in a pound, and you have 2 for a quotient without any remainder. You therefore set down 0 under the column of shillings, and carry the 2 to the column of pounds, and say, 2 and 5 are 7, and 2 are 9, and 5 are 14, and 4 are 18, set the 8 under the column of units in the pounds, and carry i to the column of tens, and say 1 and 2 are 3, and I are 4, and

12)40

34

2'0)4'0

2

4)9

2 1

1 are 5, and 1 are 6, and 1 are 7, which 7 place under the column of tens of the pounds, and you have the whole sum of the question, viz. £78 0s. 44d.

227. In the above example, the student will perceive that the question is so arranged as to enable the operator to add up each denomination separately, and find the amount of each. First, we add up the farthings, and having found their amount, we reduce them by the rule of Reduction to pence; the farthings we place in their proper place, and carry the pence to the column of pence; we then proceed in the same manner successively with the pence and shillings, and lastly add up the pounds, and write their amount under the column of pounds.

He will also observe, that in carrying from one denomination to another, we observe the notation of Compound Arithmetic; but so far as regards these denominations themselves, the notation of simple Arithmetic is observed.

228. In treating of these two rules, I have adopted the same plan as in Simple Addition and Subtraction, of making one rule prove the other. The student thereby learns two rules at once with the same ease he would learn one. He thereby understands his subject better, besides having the advantage of satisfying himself of the truth of his performances. See paragraphs 40 to 41.

229. Having added the three last terms of the question together, and placed their sum under the sum of all the terms, we proceed with the Subtraction thus: saying a halfpenny from a farthing we cannot, but borrow a penny; a halfpenny from a penny, and a half-penny remains, which, added to the farthing above, makes three farthings, which set down in its proper place, and add one to the nine for the penny borrowed at the farthings; then say 1 and 9 make ten; ten from four I cannot, but borrow a shilling, and say ten from twelve, and two, which, added to the four, make six; set down the six in its place, and add one to the shillings for what you borrowed at pence; then say eleven from 0 I cannot, but borrow one pound, and say eleven from twenty, and 9, set down the 9 in its place, and carry one to the pounds for what you borrow at the shillings; then say 9 from 8 I cannot, but borrow ten, 9 from 10, and 1 and 8 make 9; set down 9 in the place of units of pounds, and carry one to the place of tens for the ten borrowed in the place of units; then say five from seven, and 2, set down the 2 in its place, and the process of Subtraction is complete.

230. The remainder, observe, is equal to the sum of the two first terms.

231. In the same manner may all questions in these rules be performed; the principle is the same in all, they differing only in the Notation.