THE AREAS OF THE

Height. Area Seg. Height. Area Seg. Height. Area Seg.

289 188140324220404359 253590 290 189047 325 221340360 254550 291 189955326 222277 361 2 255510 292190864327 223215 362 256471 293191775 •328224154363 257433 294192684 329 225093364 258395 295193596330 226033365 259357 296194509331 226974366 260320 297195422 332 227915 332 227915 367 261284 298196337333 228858368 262248 299197252 334 229801369 263213

265144

300 198168 335 230745 370 264178 301 199085336 231689 371 *302 200003 337 232634 372 26211 303 200922 338 233580 373 267078 304 201841339234526 *374268045 305202761340235473 340 235473 375 269013 306 203683 341 236421 376269982 307 204605 342 237369 377 270951 308 205527 343 238318 378 271920 309 206451 344 239268 344 239268379 272890 310 207376345240218 380 273861 311

208301346241169381274832 312 209227347 242121 382 275803 313 210154 348 243074 383 276775 314 211082 349 244026384 277748 315 212011 350244980 385 278721 316 212940 351 245934 386 279694 317 213871 352 246889 387 280668 318214802 353 247845388 281642 319 215733354 248801389 282617 320216666 355249757 *390283592

321 217599356 259715391

322

284568

285544

218533357 251673 357 251673392 323219408358252631393286521

SEGMENTS OF A CIRCLE.

Height. Area Seg. Height. Area Seg. Height. Area Seg.

394 287498 430 322928466 358725 288476431323918467 359723

395

396 289453 432 324909468360721

403

405

325900469 361719 326892 470 362717 327882 471 363715 328874 472 364713

*397290432 433 398291411 434 399 292390 435 400 293369 436 401 294349 437 329866473 365712 402 295330 438 330858 474 366710 296311439331850475367709 404297292 440332843476368708 405298273 441 333836477 369707 299255442334829478 370706 407300238443335822 479 371705 408 301220444 336816480372704 409 302203 445 337810 481 *373703 410303187 446 338804 482 374702 411304171 447 339798 483 375702 412 305155 448340793 484 376702 •484376702 413 306140449 341787485377701 414 307125 450 342782 486 378701 415 308110451 343777 487 379700 416309095 452 344772 488 380700 31008453345768489381699 418311068 454 346764 490 382699 419 312054 455 347759491383699 420 313041456348755492384699 421314029 457349752 493 385699 422 315016458350748 494 386699 4233:6004 459 351745495 387699 424 316992460352742 +96 388699. 425 317981 461 353739 497 389699 426 318970 462 354736498 390699 427319959463355732499 391699 428320948464356730500392699

417

429321938465357727

In this table the diameter of the circle is 1, and the whole area 785398; the figures in the columns of heights are the height of the fegment, or the verfed fine of half its arc; and those in the columns of area fegments are the areas of the circular fegments, whofe heights stand on the left hand in the column of heights.

The ufe of this table is to find the area of any fegment of a circle, by the following rule:

RULE. Divide the height of the given fegment by the diameter of the circle, and the quotient is the height of the fegment: and oppofite this height, in the table, is the tabular area, which is to be multiplied by the square of the diameter, and the product will be the area of the fegment,

EXAMPLE. What is the area of a fegment of a circle, whofe height is 5 inches, and the diameter 30 inches?

30)5.000(.166 Tab. height

085544 Tab. fegment

900 Square of the diameter.

Anfwer. 76.989600

30

30

900

Note. In dividing the given height by the diameter, if the quotient does not terminate in three places of decimals, then the area for that fractional part ought to be proportioned for as follows:

Having found the tabular area anfwering to the first three decimals of the quotient, take the difference between it and the next following tabular area; which difference multiplicd by the fractional part remaining, the product will be the correspondent proportional part, to be added to the first tabular area.

Thus in the laft example the quotient is .166, and there is a remainder of, the tabular area of the fegment is

397

SECT. IV.

OF INSTRUMENTAL ARITHMETIC.

Inftrumental arithmetic is the method of calculation by inftruments made for that purpofe, for a quicker dispatch of bufinefs, and for the help of those who are deficient in common arithmetic. The most common inftruments are, the carpenter's rule, Coggeshall's fliding rule, Gunter's line, the gauging rule, and the diagonal rod.

1. The carpenter's rule confifts of two equal pieces, each a foot in length, connected together by a joint: one fide of this rule is divided into inches, and eighths of an inch; on the fame fide are also several plane fcales, divided into twelfth parts by diagonal lines, for planning dimensions in feet and inches.

2. On the other fide of this rule, on one piece, are set all the principal lines which are on Coggeshall's fliding rule: namely, four lines marked A B C, and D respectively; the two middle ones being on a flider, which runs in a groove. Thefe

These four lines are logarithmic lines; the three marked A, B, C, are all equal; and are called double lines, because the numbers fet upon them run from 1 to 10, twice: the lowest line D is a fingle line, the numbers running from 4 to 40. It is called the girt line, from its ufe in cafting up the contents of trees, and timber. This line alfo may be used for gauging; as upon it, at 17.15, is marked W. G. and at 18.95, A. G. for the wine and ale gauge points.

Upon the other part of this fide is a table of pounds, fhillings, and pence, showing the value of a load, or 50 cubic feet of timber, at all prices from fixpence to two fhillings a foot.

In the use of thofe lines it must be ftrictly observed, that when i at the beginning of a line is accounted unity, then the 1 in the middle of the line will ftand for 10, and the 10 at the end of the line will stand for 100: and when the 1 at the beginning of the line ftands for 10, the 10 in the middle of the line will ftand for 100, and the 10 at the end of the line 1000, &c. and all the intermediate divisions are altered in proportion.

The edge of the rule is divided decimally, each foot being divided into an hundred equal parts: by this measure dimenfions are taken in feet, and decimals of a foot, which is by much the best way.

Gunter's line is a line of figures, exactly the fame as the three fingle lines on the carpenter's rule, and therefore needs no further deféription. Its ufe is to multiply, or divide numbers, to perform the rule of three direct, &c. It was formerly set by itself on the carpenter's rule.

The figures 1, 2, 3, 4, &c. fometimes ftand fimply for themselves, at other times they fignify 10, 20, 30, &c. again at other times, 100, 200, &c. or 1000, 2000, &c.

4. The gauging rule ferves to compute the contents of cafks, or any other veffels, after the dimenfions have been taken. It is a fquare rule, with logarithmic lines on the fides, having three fliders running in grooves in three of the fides,

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