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Article 17. Having the Dimensions of any of the parts of a Circle, to find
the Side of a Square, equal to the Circle

454

18. Having the Area of a Circle, to find the Diameter

455

19. Having the Area, to find the Circumference

455

20. Having the side of a Square, to find the Diameter of a Circle,

which shall be equal to the Square whose side is given 455

21. Having the side of a Square, to find the Circumference of a Cir-

cle equal to the given Square

456

22. Having the Diam. of a Circle, to find the Area of a Semicircle 150

23. Having the segment of a circle, to find the length of the Arch Line 156

24. Having the Chord avd Versed Sine of a Segment, to fiud the Di-

ameter of a Circle

457

25. To measure a Sector

457

26. To measure the Segment of a Circle

458

27. To measure an Ellipsis

460

Directions for applying Superficies to Surveying

460

Section II. Or Solids

461

Art. 28. To measure a Cube

461

29. To measure a Parallelopipedon

403

Having the side of a Square Solid, to find what Length will make

a Solid Foot

465

30. To measure a Cylinder

46.3

Having the Diameter of a Cylinder given, to find what length will

make a Solid Foot

466

'To find how much a round tree, which is equally thick, from end to

end, will hew to when made square

466

31. To measure a Prism

467

32. To measure a Pyramid or Cone

467

33. To measure the Frustum of a Pyramid or Cone

469

34. To measure a Sphere or Globe

472

35. To measure a Frustum or Segment of a Globe

473

36. To measure the middle Zone of a Globe

473

37. To measure a Spheroid

474

38. To measure the middle Frustum of a Spheroid

474

39. To measure a Segment, or Frustum of a Spheroid

474

40. To measure a Parabolick Conoid

474

41. To measure the lower Frustum of a Parabolick Conoid

475

42. To measure a Parabolick Spindle

475

43. To measure the middle Zone, or Frustum of a Parabolick Spindle 475

44. To measure a Cylindroid or Prismoid

475

45. To measure a solid Ring

476

46. To measure the Solidity of any irregular Body whose Dimensions

cannot be taken

475

Of the five Regular Bodies

477

47. To measure a Tetraedron

478

48. To measure an Octaedron

478

49. To measure a Dodecaedron

478

50. To measure an Eicosiedron

478
51. To gauge a Cask

479

62. To gauge a Mash Tub

480

53. Having the Difference of Diameters, Height and Content of a

Mash Tub, to find the Diameters at Top and Bottom

480

54. To ullage a Cask, lying on one side, by the Gauging Rod

487

55. To find a Ship’s Tonnage

481

The Proportions and Tonnage of Noah's Ark

482

Questions in Mensuration

489

Book Keeping by Single Entry

488

Pouble Entry

508

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OF THE CHARACTERS MADE USE OP IN THIS TREATISE. = The sign of equality : as 12 pence = 1 shilling, signifies that 12 pence are equal to one shilling; and, in general, that whatever precedes it is equal to what follows.

+ The sign of Addition: as 5+5=10, that is, 5 added to 5 is equal to 10. Read 5 plus 5, or 5 more 5 equal to 10.

The sign of Subtraction : as, 12—458, that is, 12 lessened by 4 is equal to 8, or 4 from 12 and 8 remains. Read 12 minus 4, or 12 less 4 equal to 8.

* The sign of Multiplication : as 6 X5530, that is, 6 multiplied by 5 is equal to 30. Read 6 iuto 5 equal to 30.

or 5)30( The sign of Division : as 30-5=6, that is, 30 divided by 5 is equal to 6. Read 30 by 5 equal to 6. 875

Numbers placed fraction wise, do likewise denote division, the numerator 25 or upper number being the dividend, and the denominator or lower number, the

875 divisor; thus, is the same as 875--25=35.

25 :::: The sign of proportion, thus, 2:4::8:16, that is, as 2 is to 4 so is 8 to 16.

Signifies Geometrical Progression. 9-2+6=13 Shews that the difference between 2 and 9 added to 6 is equal to 13. Read 9 minus 2 plus 6 equal to 13. And that the line above (called a Vinculum) connects all the numbers over which it is drawn.

12—3+4=5 Signifies that the sum of 3 and 4 taken from !2 leaves or is equal 73 Signifies the second power, or Square.

13 Signifies the third power, or Cube:

vjen signifies any power in al, as 6/2 = square of 6; and 501"=cube of 50, &c. thus m signifies either the square or cube, or any other power. V, or

Prefixed to any number or quantity, signifies that the square root of That number is required. It Likewise (as also the character for any other root) stands for the expression of the root of that number or quantity to which it is presised. As 136=6, and V 100+30=12, and 3612=6, &c.

j, or li Prefixed to any number, signifies that the cube root of that number is required, or expressed. As 121636, and 15134-216=9, &c. or 2167* = 6, &c. Siguities any root in general. As 36]

As 36)=square root, 216,3=cube V,

im
rool,. &c. Thus, signifies either the square root, cube root, or any oth
root whaterer.

abcd When several letters are set together, they are supposed to be multi-
plied into each other; as those in the margin are the same as a xoxoxd, and
represent the continual product of quantities or numbers.
Is the reciprocal of a, und is the reciprocal of --

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ita be the root, then a Xa=aa or az is the square of a, and axa Xa=aac
or 3 is the cute of a, &c.

.Note. The figure above is called the index of the power.

It is usual to write shillings at the left hand of a stroke, and pence at the ajght; thus, 1:3/4 is thirteen shillings and four pence.

Vote. The use of these characters must be perfectly understood by the pupil, as he may have occasion for them.

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NEW AND COMPLETE

SYSTEM OF ARITHMETICK.

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bers, and consists both in Theory and Practice. The Theory considers the nature and quality of numbers, and demonstrates the reason of practical operations. The Practice is that, which shews the method of working by numbers, so as to be most useful and expeditious for business, and is comprised under five principal or fundamental Rules, viz. Notation or NumeraTION, ADDITION, SUBTRACTION, MULTIPLICATION, and Division; the knowledge of which is so necessary, that scarcely any thing in life, aod nothing in trade can be done without it.

NUMERATION

1. TEACHES the different value of figures by their different places, and to read or write any sym or number by these ten characters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.-0 is called a cypher, and all the rest are called figures or digits.* The names and significations of these characters, and the origin or generation of the numbers they stand for, are as follow; 0 nothing ; 1 one, or a single thing calleri an unit; 1+1=2, (wo; 2+1=3, three ; 3+1=4, four; 4+1=5, five; 5+1=6, six; 6+1=7, seven ; 7+138, eight; 8+1=9, nine; 9+1=10. ten; which has no single character; and thus, by the continual addition of one, all numbers are generated.

2. The value of figures when alone, is called their simple value, and is invariable. Besides the simple valve, they have a local value, that is, a value which varies according to the place they stand

* 'These figures or digits were obtained from the Arabians, and were introduced into Europe in the ninth century. The Arabs probably derived the deci. mal notation from India. The sexagesimal division had previously been in gencral use in Europe. This mode of division is yet retained in a few cases, as in the division of time, where fixty minutes make an hour, fixty seconds a minute, &c. The figures are doubtless called digits from digitus, a finger, hecause conoting used to be performed on the fingers.

in when connected together. In a combination of figures, reckon ing from the right to the left, the figure in the first place represents its simple value; that in the second place, ten times its simple value, and so on ; each succeeding figure being ten lines the value of it in the place immediately preceding. There is no reason in the nature of numbers that their local value should vary according to this law. They might have been made to increase in 3, 4, 5, &c. fold, or in any other ratio. The tenfold increase is assumed because it is most convenient.

3. The values of the places are estimated according to their order: The first is denominated the place of units ; the second, tens ; the third, hundreds ; and so on, as in the table. Thus in the number—5293467; 7, in the first place signifies only seven ; 6, in the second place, signifies 6 tens, or sixty ; 4, in the third place, four lundred; 3, in the fourth place, three thousand ; 9, in the fifth place, ninety thousand ; 2, in the sixth place, two hundred th00 sand; 5, in the seventh place, is five millions; and the whole, taken together, is read thus ; five millions, two hundred and ninety three thousand, four hundred and sixty seven.

The process of Numeration may be more clearly seen by the following

TABLE,

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Six places of figures, beginning on the right, are called a perivni, and each successive six places another period. Each period is cosidered as divided into two half periods of three figures each. These are distinguished by the comma, and the point for a period. There is an obvious reason for this division into periods, for at the begin

ning of each period, there is a new denomination of units, of which , the lens, hundreds, thousands, &c. ar numerated as in the first period.

4. A cypher, though it is of no signification itself, yet, it pos. sesses a place, and, when set on the right hand of figures, in shola numbers, increases their value in the same tenfold proportion; thus, 9 signifies only nine ; but if a cypher is placed on its right band, thus, 30, it then becomes ninety ; and, if iwo cyphers be placed on its right, thus, 900, it is nine hundred, &c.

5. To enumerate any parcel of figures, observe the following Rule.

First, commit the words at the head of the table, viz. units, tens, hundreds, &c. to memory, then, to the simple value of each figure, join the name of its place, beginning at the left hand, and reading towards the right.-More particularly-1. Place a dot under the right hand figure of the 21, 4th, 6th, 3th, &c. half periods, and the figure over such dot will, universally, have the name of thousands. -2. Place the figures, 1, 2, 3, 4, &c. as indices over the 2d, 3d, 4th, &c. period. These indices will then shew the number of times the millions are increased.--The figure under 1, bearing the name of millions, that under 2, the name of billions (or millions of millions) that under 3, trillions.

EXAMPLE
Sextillions. Quintilli. Quatrill. Trillions. Billions. Millions. Units.

th. un.
tl.. un.
th. un.
th. un.
th. un.
th, un.

c.x.t.c.x.u. an

MMM Manns 6 5 4

1

3 913,208,000,341,620,057,219,356,809,379,120,406,129,763

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Note 1. Billions is substituted for millions of millions : Trillions, for millions of millions of millions; Quatrillions, for millions of millions of millions of millions, and so on.

These names of periods of figures, derived from the Latin nume. rals, may be continucl withont end. They are as follows, for twenty periods, viz. Units, Millions, Billions, Trillions, Quatrill100s, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, I'redecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillioni, Noveindecillions, Vigintillions.

THE APPLICATION.
Write down, in proper figures, the following numbers.
Fifteen.

15 Two hundred and seventy nine.

279 Three thousand four hundred and three. Thirty seven thousand, five hundred and fixty seven.

37567 Four hundred, one thousand and twenty cight. Nine millions, seventy two thousand and two hundred.

9072200 Fifty five millions, three hundred, nine thousand and nine. Light hundred millions, forty four thousand, and fifty five. i'wo thousand, five lundred' and forty three millions, four

2543431702 hundred and thirty one thousand, leven hundred and iwo.

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