Εικόνες σελίδας
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Diam.

Squares.

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6 in. 36

37.515625 39.0625 40.640625 42.25 43.890625 45.5625

47.265625 7 in. 49

50.765625 52.5625 54.390625 56.25 58.140625 60.0625

62.015625 8 in. 64

66.015625 68.0625 70.140625 72.25 74.390625 76.5625

78.765625 9 in.

83.265625 85.5625 87.890625 90.25 92.640625 95.0625

97.515625 10 in. 100

102.515625 105.0625 107.640625 110.25 112.890625 115.5625

81

Cubes.

118.265625 11 in.' 121

123.765625
126.5625
129.390625
132.25
135.140625
138.0625

141.015625 12 in. 144

216 229.783203125 244.140625 259.083984375 274.625 290.775390625 307.546875 324.951171875 343 361.704078125 381.078125 401.130859375 421.875 443.322265625 465.484375 488.373046875 512 536.376953125 561.515625 587.427734375 614.125 641.619140625 669.921875 699.044921875 729 759.798828125 791.453125 823.974609375 857.375 891.666015625 926.859375 962.966796875 1000 1037.970703125 1076.890625 1116.771448375 1157.625 1199.462890625 1242.296875 1286.138671875 1331 1376.892578125 1423.828125 1471.818359375 1520.875 1571.009765625 1622.234375 1674.560546875 1728

113.0976 117.8590 122.7187 127.6765 132.7326 137.8867 143.1391 148.4896 153.9384 159.4852 165.1303 170.8735 176.7150 182.6545 188.6923 194.8282 201.0624 207.3946 213.8251 220.3537 226.9806 233.7055 240.5287 247.4500 254.4696 261.5872 268.8031 276.1171 283.5294 291.0397 298.6483 306.3550 314.1600 322.0630 330.0643 338.1637 346.3614 354.6571 363.0511 371.5432 380.1336 388.8220 397.6087 406.4935 415.4766 424.5576 433.7371 443.0146 452.3904

113.0976 120.3139 127.8320 135.6563 143.7936 152.2499 161.0315 170.1682 179.5948 189.3882 199.5325 210.0331 220.8937 232.1235 243.7276 255.7121 268.0832 280.8469 294.0095 307.5771 321.5553 335.9517 350.7710 366.0199 381.7017 397.8306 414.4048 431.4361 448.9215 466.8763 485.3035 504.2094 523.6000 543.4814 563.8603 584.7415 606.1318 628.0387 650.4666 673.4222 696.9116 720.9409 745.5004 770.6440 796.3301 822.5807 849.4035 876.7999 904.7808

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In calculating the preceding tables of circumferences, squares, cubes, areas, &c.; the following simple rules have been adopted,

1. The Circumferences.

The circumferences were obtained by adding į of 3.1416, or, .3927 constantly for the first table; to of 3.1416, or, .31416 for the second; and iz or .2618 for the third table; thus,

The circumference of a circle, whose diameter is

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2. The Squares. After the first square of each succeeding series was found by the common rule; twice the root of that square, plus 1, added to the square number obtained, gave the square

of the next number required; thus, The square of 1.1 = 1.21 And 1.1 X 2+1 3.2 +

4.41 the square of 2.1 2.1 X 2+1

5.2 +

9.61 the square of 3.1 3.1 X 2+1

7.2 + 16.81 the square of 4.1 &c.

Again, the square of 1.2

= 1.44 And, 1.2 X 2+1=

3.4 + 4.84

the square of 2.2 2.2 X 2+1

5.4 +

10.24 the square of 3.2 3.2 X 2+1

7.4 + 17.64 the square of 4.2, &c.

3. The Cubes.

In finding the cubes; the first of each series was also found in the usual form, then the root of the number so obtained being multiplied by 3, and by the root of the number required, and the product, plus 1, added to the former cube, gave the cube of the next number required; thus,

The cube of 1.1 1.331
And 1.1 X 3 X 2.1 + 1 = 7.93 +

9.261 the cube of 2.1
2.1 X 3 X 3.1 +1 20.53 +

29.791 the cube of 3.1 3.1 X 3 X 4.1 +1= 39.13 +

68.921 the cube of 4.1, &c.

the cube of 2.2

Or the cube of 1.2 = 1.728
And 1.2 X 3 X 2.2 +1

8.92 +

10.648
2.2 X 3 X 3.2 + 1 = 22.12 +

32.768
3.2 X 3 X 4.2 +1= 41.32 +

74.088

the cube of 3.2

the cube of 4.2, &c.

The Areas.

After the first area of each succeeding series was obtained by the common rule, the others were found by the following: namely,-a table of constant numbers was formed by multiplying -7854 by twice the fractional number contained in the diameter; thus,

x 2

.25 x .7854 .19635 x 2 =

.5 X.7854 .3927
x 2

.75 X.7854 .58905
x 2 = 1 X.7854 = .7854
2 = 1.25 X.7854 = .98175
= 1.5 x .7854 = 1.1781

1.75 x.7854 = 1.37445 for the first table.

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2 =

.1 x 2 .2 x .7854 = .15708
.2 x 2 = .4 x .7854 = .31416
.3 x 2 = .6 x .7854 .47124
.4

.8 X.7854 = .62832
.5 x 2 = 1 X.7854 = .7854
.6 x 2 = 1.2 x .7854 = .94248
.7 x 2 = 1.4 x .7854 = 1.09956

(2 = 1.6 x .7854 = 1.25664
.9 x 2 = 1.8 x .7854 = 1.41372 for the second.

X.7854

And, .0833 x 2 .1666 x .7854 .13084764

.1666 x 2 = .3333 x 7854 26177382
.25 x 2 = .5

.3927
.3333 x 2 .6666 x.7854 .52354764
.4166 x 2 .8333 X.7854 = .165447382
.5
x2 = 1

X.7854 = .7854
.5833 x 2 = 1.1666 x .7854 = .91624764
.6666 x 2 = 1.3333 X.7854 = 1.04717382
.75 x 2 = 1.5 X.7854 = 1.1781
.8333 x 2 = 1.6666 x .7854 = 1.30894764

.9166 X 2 = 1.8333 X.7854 = 1.43987382 for the third table, &c.

Then twice the whole numbers of the first circle, plus 1, and multiplied by .7854, produced a sum which, when added to the area and constant number of the fractional part, gave the area of the next in the series; thus, The area of a circle whose diameter is 15

.994021875 And 1x2+1=3x.7854= 2.3562 + the constant number .19635

3.546571875 the area of 2 2 X 2+1=5 X .7854= 3.927

+ .18635

7.7669921875 = the area of 3 3 X 2+1=7 x .7854 5.4978

+ .19635

13.364071875 = the area of 4f &c.

Again, the area of 14 = 1.2271875 To which add as above, 2.3562 And the constant number .3927

the area of 24

3.9760875 3.927 .3927

the area of 34

8.2957875 5.4978 .3927

14.1862875 = the area of 44, &c.

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