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BABBAGE'S CALCULATING MACHINE. (Vol. VI, p. 268.) The answer to this question can best be given in Babbage's own words taken from his “ Ninth Bridgewater Treatise," pp. 43-48, Philadelphia edition, 1841. After giving a description of the calculating engine and his services to the English Government, and delays in its perfection, he says

“Let the figures thus seen be the series of natural numbers, 1, 2, 3. 4, 5, 6, &c., each of which exceeds its immediate antecedent by unity. Now, reader, let me ask how long you will have counted before you are firmly convinced that the engine, supposing its adjustments remain unaltered, will continue whilst its motion is maintained, to produce the same series of natural numbers ? Some minds perhaps are so constituted, that after passing the first 100 terms, they will be satisfied that they are acquainted with the law. After seeing 500 terms, few will doubt; and after the 50,00oth term, the propensity to believe that the succeeding term will be 50,001, will be almost irresistible. That term will be 50,00 1 ; the 5,000,oooth and the 50,000,oooth term will still appear in their expected order ; and one unbroken chain of natural numbers will pass before your eyes, from i up to 100,000,000

True to the vast induction which has been made, the next succeeaing term will be 100,0000,001; but after that the next number presented by the rim of the wheel, instead of beeing 100,000,002, is 100,010,002. The whole series from commencement being thus :

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The law which seemed at first to govern this series at 100,000,002nd term failed. That term is larger than we expected by 10,000. The next term is larger than was anticipated by 30,000, and the excess of each term, above what we expected, forms the following table ;

10,000 30,000 60,000 100,000 150,000

being, in fact, the series of triangular numbers, each inultiplied by 10,000. The numbers 1, 3, 6, 10, 15, 21, 28, &c., are formed by add. ing the successive terms of the series of natural numbers, thus :

I = I

10, &c.

1+2=3 1+2+3 :6

1+2+3+4 They are called triangular numbers, because a number of points corresponding to any term can always be placed to form a triangle.

If we continue to observe, we shall discover another law then coming into action, which also is dependent, but in a different manner, on triangular numbers. This will continue through about 1430 terms, when a new law is again introduced, which extends over about 950 terms; and this, too, like all its predecessors, fails, and gives place to other laws, which appear at different intervals.

Now it must be remarked, that the law that each number presented by the engine is greater by unity than the preceding number, which law the observer had deduced from an induction of 100,000,000 instances, was not the true law that regulated its action ; and that the occurrence of the number 100,010,002 at the 100,000,002d term, was as necessary a consequence of the original adjustment, and might have been as fully foreknown at the commencement, as was the regular succession of any one of the intermediate numbers to its immediate antecedent. The same remark applies to the next apparent deviation of the new law, which was founded on an induction of 2761 terms, and to all the succeeding laws; with this limitation only, that whilst their consecutive introduction at various definite intervals is a necessary consequence of the mechanical structure of the engine, our knowledge of analysis does not yet enable us to predict the periods at which the more distant laws will be introduced. *

The engine we have been considering is but a very small portion (about 15 figures) of a much larger one, which was preparing, and is partly executed; it was intended, when completed, that it should have

presented at once to the eye about 130 figures. In that more extended form which recent simplifications have enabled me to give to machinery constructed for the purpose of making calculations, it will be possible, by certain adjustments, to set the engine so that it shall produce the series of natural numbers in regular order, from unity up to a number expressed by over 1,000 places of figures. At the end of that term, another and different law shall regulate the succeeding terms; this law shall continue in operation perhaps for a number of terms, expressed perhaps by unity, followed by 1,000 zeros, or 101000 ;* at which period a third law shall be introduced, and like its predecessors, govern the figures produced by the engine during a third of those enormous periods. This change of laws might continue without limit. Each individual law being destined to govern for millions of ages the calculations of the engine, and then give way to its successor to pursue a like career.

It has been supposed that ten turns of the handle of the calculating engine might be made in a minute, or about 526,000,000 in a century. As in this case, each turn would make a calculation, after a million of centuries, and only the 15th place of figures would have then been reached."

[10000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000]

We print this stupendous number so one can see what 101000 represents ; a number of 333 periods, and in Henkle's method of enumeration would be Ten Tertio-Trigillions-Trecentillions. This numbers has never yet been equaled in computations when composed of digits, and is not quite one-half longer than William Shanks's computation for the value of . (See Vol. V, p. 120.)

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From any point A, ia an indefinite straight line A C, draw a perpendicular A B equal to the given diameter. Then set off A C equal to three times A B. Join B C, and from D, a point in the line A C, equal to twice A B, draw the perpendicular D H. Make D M equal to H C, and A N equal to s of jy of A B.

Then N M is nearly equal to the circumference of a circle, whose diameter is A B. Because, By construction, A N= f of 7o of A B

A DE 2 A B DM=HC= N(A B? + A C2)=}BC If A B=l; then of + 2 + 1 V 10 == 3.141592+, correct to the sixth decimal place.

T. P. STOWELL. CURIOUS PROPERTIES OF 2.61803399+ (Vol. V, p. 206.) I will add a few more curious properties to those already given of the number 2.61803399+

The reciprocal of .61803399+ is 1.61803399+

If a series of fractions be made thus, }, }, , 183, , j}, 34, etc., in which every numerator equals the preceding denominator, and each denominator equals the sum of the preceding numerator and denom. inator, then the farther we proceed with the series the nearer the fractions become to equal .61803399+ They are alternately a little greater and a little less; but the difference grows less at each successive step.

The cosine of the angle that the sides vi the Great Pyramid make with the plane of the base is .61803399+

T. S. BARRETT.

-O

or

Questions and Answers.
TRANSLATION OF PROVERBS XXII, 6. (Vol. V. p. 78.) Is this

Train up a child in the way he should go,"
Enoch hath been made into a boy, according to his path.

FIDES. Proverbs xxii, 6, reads in Hebrew thus : HHaNouCH L’NaGHAR GHaL Peel DaRCHVou, GaM CHeel IaZKeeIN LouA IOSVOOR MeeMMeNoH. The literal meaning of which is :

“ Initiate a youth after the manner of his way, (then) also if he gets to be old he will not deviate from it."

The proposed rendering is impossible ; because, (1st), the second part of the verse would not suit to it at all ; (2d), there is no verb in this passage to be rendered “hath been made"; (3d), therefore, HHaNouCH must be a verb, and not a noun; and (4th), the L in the word L’NagHaR does not mean always “to,” “into,” representing the dative case, but is frequently the sign letter for the accusative (=objective) case, and here is the objective case, NaHaR, after the verb HhaNouCH.

The reading of the English Common Version is insipid. Thousands of youths have been trained the way they should go, i. e. aright yet when they became old they left it. The author evidently inveighs against the ready-made, wholesale education of the schools, and he would rather have a youth educated according to his individual capacity into knowledge and wisdom, then he would not turn from them when he becomes older, and thinks for himself.

The HHANVouCH (English “Enoch ”) of Genesis v, 18, 19, 20 21, 22, 23, and 24, was very likely an initiate into divine wisdom His grandfather was M'HaLeL-AL, =“Praiser of God,” but his father was IeReD="a goer down,” an indifferent,” or “apostate,” and the traditional divine wisdom threatened to become extinct with him. But HHN VouCH, his son, became initiated, and begat his longest-lived son, whose name was MTHOOVIHeLaHH, which I would render “ Marking-Sender,” i. e. one who disseminated certain marks and signs, and it may be he who taught Noah the science of geometry, without which he could not have built the ark. In that ark I have shown* from Genesis vi, 16, how in it was involved the knowledge of the

" The Measures and Weights mentioned in the Bible, in connection with Structures, Worship and Narratives.”—International Standard, Vol. III, No. 2, pp. 101-100, 1885,

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