صور الصفحة
PDF
النشر الإلكتروني

THE KORAN. (Vol. II, p. 520.) Are there any lines for the memorization of the books or contents of the Korân similar to those of the Bible, published in Vol. II, p. 487 ? F. K. G.

The following is a metrical account of the verses, etc., of the Korân, taken from a very beautiful copy, once the property of the unfortunate Tippoo Sultan, but preserved in the public library at Cambridge, Eng. It was copied by Godfrey Higgins from a manuscript belonging to Prof. Lee, deposited with the Secretary of the Royal Society of Liter(See Higgins' Anacalypsis, Vol. II, p. 195.)

ature.

"The verses of the Koran, which are good and heart-delighting,
Are six thousand, six hundred, and sixty-six.

One thousand of it command, one thousand of it prohibit;

One thousand of it promise, one thousand of it threaten;

One thousand of it read in choice stories;

And know, one thousand of it to consist in instructive parables;

Five hundred of it in discussions on lawful and unlawful;

One hundred of it are prayers for morning and evening.
Know sixty-six abrogating and abrogated.

Of such an one, I have now told you the whole."

In the 6666, and 6600, exclusive of the abrogated part, Mr. Higgins thinks may be seen the remains of the cyclic system.

The preliminary discourse to George Sale's translation (p. 45) says there are seven principal editions, or ancient copies of that work: first two published at Medina, third at Mecca, fourth at Cufa, fifth at Basra, sixth in Syria, and the seventh the common or vulgate. The first makes the whole number of verses, 6,000; the second and fifth, 6,214; the third, 6,219; the fourth 6,236; the sixth, 6,226; the seventh, 6,225. Mr. Sale's edition contains 6,226. Mr. Sale says they all are said to contain the same number of words, namely, 77,639; and the same number of letters, namely, 323,015. Another authority says 114 chapters, 99,464 words, and 330,113 letters.

The Alexandrian Library which history inform us contained 700,000 manuscript volumes, is said to have been destroyed by Omar upon the following logic:

"Either these books are in conformity with the Korân, or they are not. If they are they are useless, and if not they are evil; in either event, therefore, let them be destroyed."

AN ARABIC PROVERB. The following is an Arabic proverb which was taken down from an Oriental: Men are four—

He who knows not, and knows not he knows not, is a fool. Shun him He who knows not, and knows he knows not, is simple. Teach him He who knows, and knows not he knows, is asleep. WAKE HIM He who knows, and knows he knows, is wise. FOLLOW HIM

"God's

GOD'S WOUNDS. (Vol. II, p. 460.) What is meant by Wounds?" (See Zounds, in "Webster's Dictionary."> J. J. J. "God's wounds" were the five wounds received by Christ on the PRIGGLES, San Francisco, Cal.

cross.

The Value of Pi ().

We re-publish the following article on computations, by request of several subscribers; also others who failed to secure a copy of the October number, 1884, containing it. The value of, in the last ten periods as previously printed, as stated, were taken from the work of William Shanks. Mr. Shanks afterwards re-computed his value, and communicated his later result to the Roya! Society of London, which was published in their Proceedings, Vol. XXI, 1873. This result changes the last 68 decimals of his former computation, and extends it to 44 more decimal places; or to 751 decimals in all. The following value is Mr. Shanks's last value. These facts were communicated by PROF. H. A. WOOD, School of Applied Science, Cleveland, Ohio. A second chapter on computations, including 63 pairs of "Amicable Numbers," furnished by Prof. Wood, will appear in a future issue of NOTES AND QUERIES.

THE VALUE Of Pi (≈ ). (Vol. II, p. 651.) To how many places of decimals has the value of, the ratio of the circumference of a circle to the diameter, been computed? Please give the decimals. H. A. W., New York City.

Peter Barlow, in his" New Mathematical and Philosophical Dictionary" says, under CIRCLE, that Vieta by means of the inscribed and circumscribed polygons of 393,216 sides carried the ratio to 10 decimal places; Van Coulen, by the same process, carried it to 36 places Abraham Sharp extended it to 72 places; Machin went to 100 places; De Lagny extended it to 128 places. Dr. William Rutherford of the Royal Military Academy, Woolwich, in a "Paper on Determining the Value of," sent to the Royal Society of London, calculated its value to 441 decimal places. William Shanks of London coöperated with Rutherford in verifying the 441 decimals, and extended the same to 607 places. These results are published in a royal octavo volume by William Shanks entitled "Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 places of decimals," 1853. The value of the Base of Napier's Logarithms, the Modulus of the Common System, the 72 1st power of 2, and several other numbers are also given in the same work. We print some of them here as they are of frequent use and reference.

=

VALUE OF TO 721 DECIMAL PLACES.

3.1415926 5358979 3238462 6433832 7950288 4197169 3993751 0582097 4944592 3078164 0628620 8998628 0348253 4211706 7982148 0865132 8230664 7093844 6095505 8223172 5359408 1284811 1745028 4102701 9385211 0555964 4622948 9549303 8196442 8810975 6659334 4612847 5648233 7867831 6527120 1909145 6485669 2346034 8610454 3266482 1339360 7260249 1412737 2458700 6606315 5881748 8152092 0962829 2540917 1536436 7892590 3600113 3053054 8820466 5213841 4695194 1511609 4330572 7036575 9591953 0921861 1738193 2611793 1051185 4807446 2379834 7495673 5188575 2724891 2279381 8301194 9129833 6733624 4193664 3086021 3950160 9244807 2366886 1995110 0892023 8377021 3141694 1190298 8582544 6816397 9990465 9700081 7002963 1237738 1342084 1307914 5118398 0570985 (721 decimals totalize 2866.)

It may be stated here that there is a variation in the 113th decimal by several mathematicians. J. E. Montucla, Oliver Byrne, William Rutherford, and William Shanks make it an "8, designated above by an Italic 8, in the third column, third line, first figure. Peter Barlow, Thomas F. De Lagny, Olinthus Gregory, Edmund Halley, Charles Hutton, and several other authors make the 113th decimal a "7." The works of Benjamin Greenleaf, Uriah Parke, and several others vary in several other decimals.

SPECULATIONS.

Here are 721 decimals; there is one interesting feature that attracted the attention of Prof. Augustus DeMorgan. It might be expected that in so many figures, the nine digits and the cipher would occur each about the same number of times, that is, each about 61 times. But the figures stand as follows:

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

"One digit," Prof. De Morgan says, "is treated with an unfairness

[ocr errors]

that is incredible as an accident; and that number is the mystic number seven!"

Now, if all the digits were equally likely to appear, and 651 draw. ings were to be made, it is 48 to 1 against the number of 7's being as distant from the probable average (say 65) as 47 on one side or 83 on the other. There is probably some reason why the digit 7 is thus deprived of its representation in the number. Yet, in twice the number of decimal places 7 might receive a proper representation. Here is a field of speculation in which two branches of inquiries might unite. The value of to 36 places of decimals which was carried out by Ludolph Van Coeulen agrees with the first 36 decimals of the above, although it is erroneously printed in several works, and also on page 386 of this magazine, our attention being called to the error by a subscriber (Mrs. E. D. Slenker, Snowville, Va.) On Van Coulen's tombstone undoubtedly it is as follows, and correct to 36 decimals : 3,141592653589793238462643383279502884+

NAPERIAN BASE TO 205 DECIMAL PLACES.

=2.718281 828459 045235 360287 471352 662497 757247 093699 959574 966967 627724 076630 353547 594571 382178 525166 427427 466391 932003 059921 817413 596629 043572 900334 295260 595630 738132 328627 943490 763233 829880 748207 076730 493949 2+ (205 decimals totalize 958.)

MODULUS OF COMMON SYSTEM TO 205 DECIMAL PLACES.

M.434294 481903 251827 651128 918916 605082 294397 005803 666566 114454 084295 210320 561389 388912 264709 669534 911420 043393 805647 056134 312230 230604 429277 441521 725473 726681 842901 672329 470756 458650 612932 297550 246842 915649 9± (205 decimals totalize 865.)

2 RAISED TO THE 721ST POWER.

11 031304 526203 974597 457456 414861 827591 216226 218170 224705 794538 792432 397774 848431 640257 320003 887617 175667 569787 102671 861633 294128 382337 464639 166223 001902 133228 245297 232354 359845 986844 033174 623155 170927 185464 197384 241152. (218 figures totalize 929.)

SQUARE ROOT OF 2 CARRIED TO 486 DECIMAL PLACES. √21.414213 562373 095048 801688 724209 698078 569671 875376

948073 176679 737990 732478 462107 038850 387534 327641 572735 013846 230912 297024 924836 055850 737212 644121 497099 935831 413222 665927 505592 755799 950501 152782 060571 470109 559971 605970 274534 596862 014728 517418 640889 198609 552329 230484 308714 321450 839762 603627

( 23 )

995251 407989 687253 396546 331667 408283 959041 684760 297667 684273 862638 670905 164606 038203 518674 278823 457716 756598 936147 683830 428020 835398 973351 758630 743182 214425 593909 415560 306506 208077 018188 034610 (486 decimals totalize 2111.)

622246 + (This number was computed and verified by Mr. J. M. Boorman of New York, and is found in The Mathematical Magazine (published by Artemas Martin, M. A., Erie, Pa.,) No. 10, page 164. Mr. Boorman computed the square root of 2 to 34 more decimal places, or 520 in all, which is only 87 less than the extent to which the received value of was computed by Mr. Shanks.)

[ocr errors]

In a work entitled "The Square Root of Two, or the Common Measure of the Side and Diagonal of the Square; also, the Square Root of Two, by Division alone, to 144 Decimal Places,' by William A. Myers, 1874, the square root of 2 coincides with the above to the 96th decimal place inclusive; from the 97th to the 144th inclusive, he gives the following figures, which are probably wrong:

563643 977195 724018 929160 771077 122365 330384 600627

integer 1 with first 333 decimalS OF THE √ 2, SQuared. The integer with the first 333 decimals of the 2, squared produces the following, which contains 667 figures, and Mr. Boorman says, is "undoubtedly the largest square number ever computed : " 1.999999 (and 324 more 9's inserted here, in 54 periods, six in each,) 997849 553453 840534 947811 584454 819326 925014 318295 914544 801818 976523 919014 733545 342539 155429 965387 461306 426495 155193 487390 836452 559388 759965 846607 768752 313845 826516 448345 426858 870864 082136 258680 639474 906311 646204 546628 773418 189094 922517 782767 761054 697553 522368 472093 420695 554590 621177 140096 690265 912807 512270 189629 675625 207498 095649 555856 (667 figures totalize 4575).

Mr. Boorman also computed the square root of 3 to 246 decimal ⚫ places, and verified the first 98 decimals. The figures are as follows: SQUARE ROOT OF 3 TO 246 DECIMAL PLACES.

√3 =

=1.732050 807568 877293 527446 341505 872366 942805 253810 380628 055806 979451 933016 908800 037081 146186 757248 575675 626141 415406 703029 969945 094998 952478 993520 846889 105764 348475 097760 422180 593969 224053 405731 716104 909309 807129 140548 504914 094494 944077 202209 398943 ± (247 figures totalize 1118).

ASHER B. EVANS'S VALUE OF x AND y FOR x2-940751y2=1. In 1860, Prof. Asher B. Evans, of Lockport, N. Y., solved the equa

« السابقةمتابعة »