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NOTES AND QUERIES.

1. Problems in Division, in which all the figures in the divisor but the right-hand one are nines. Let it be required to divide 864835 by 996. Proceed as follows:

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864835 864 × 1000+ 835 = 864 (996 +4)+ 835,

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Hence 996 is contained in 864835, 864 times, and 3456 + 835 remainder. But

3456 = 3(996 + 4) + 456 = 3 × 996 + 3 × 4 + 456;

hence 3456 contains 996, 3 times, with 3 X 4 + 456 = 468 remainder. Hence, 864835 contains 996 864+ 3 times, with 835+468= 1303 remainder.

But

1303 = 1(996 + 4) + 303 = 1 × 996 + 4 + 303,

= 1 × 996 + 307;

that is, 1303 contains 996 once, with 307 remainder; and the final result is 868 as the quotient, and 307 remainder.

We can now condense the work into the following

OPERATION.

864 835 3456

867

12=

864 X 4
3 X 4
= quotient
remainder
4= 3 X 4
= quotient

1303

EXPLANATION.

Cut off as many figures from the right of the dividend as there are figures in the divisor. Multiply the remaining figures of the dividend, 864, by 4, the complement of the divisor, and set the product underneath the dividend. All the figures of this product over three, the number in the divisor, multiply by 4, and set the product underneath; and so continue to do until none fall to the left of the vertical line. Next add, and all at the left of the vertical line, as 867, will be quo

868 307

remainder

tient; and all to the right, as 1307, will be remainder. Treat this remainder in the same way, and so on; and the last additions will give the true quotient and remainder.-Prof. DEVOLSON WOOD, University of Michigan, Ann Arbor.

2. Derivative of a*.-Let u= a; then log ux log a. By algebra the Naperian log of u is

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The series in the parenthesis is converging for all positive values of u, and by summing an infinite number of terms the equation be

comes

4 du (11) = d x log a.

-LUCIUS BROWN, Fall River, Mass.

du dx

... = a log a.

3. Remarks upon a (supposed) new Instrument for the Mechanical Trisection of an Angle.-The Philosophical Magazine for April, 1860, contains an account, by Mr. TATE, of an instrument for the mechanical trisection of an angle, which is there described as new. The instrument is simple and elegant, and very convenient for the purpose intended. It will be seen, however, that the construction in question has long been known; for on turning to the 252d page of the second edition of the Traité Analytique des Sections Coniques, by DE L'HÔPITAL, first published in Paris in the year 1707, and republished in 1776, we

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find the description of an instrument designed for the same object, and which, in its essential parts, is precisely the same as that mentioned above. It is somewhat singular that this should thus have escaped notice, especially as the name of DE L'HÔPITAL is by no means an obscure one, and the problem of the trisection of an angle is one of so great historical celebrity. As the subject is one of some interest, we will give a brief description of the instrument employed by DE L'HÔPITAL, AX and AZ

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are two rulers united by a joint at A. CB, CD, CE, and CF are four others, all equal, and united together by a pin at C, so as to turn freely. CB and CD are likewise connected with AX and AZ by

joints at B and D in such a manner that AB = BC, and AD=DC, so that ABCD is always a parallelogram, the ends E and F of CE and CF sliding easily upon A X and AZ respectively. Through A and C draw A CY. The <ECY <AEC+<EA C. But

AEC=<EBC=<BAC+<BCA = 2<E A C.
.. ECY-34EA C

Similarly, FCY-3 <FA C. .. ECF-3<EAF.

Therefore, in order to trisect any angle, we have only to place the instrument in such a position that the sides of the rulers CE and CF coincide with the lines forming the given angle; in which case the straight lines drawn through C, parallel to AX and A Z, will divide it into three equal parts. DE L'HÔPITAL has also shown (as is indicated by the dotted lines in the figure) how, by the addition of four rods, each equal to those meeting at C, and connected to each other

in a similar manner at G, as also to CE, CF, A X, and AZ at the points E, F, H, and K respectively, any angle may be divided into five equal parts, and that the construction may be extended in the same way, so as to effect the division of an angle into any odd number of equal parts. ARTHUR W. WRIGHT, New Haven, Ct.

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method for approximating to a root of an equation of the form

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Now, in order to find an approximate value of this continued fraction, take a few of its terms, and, beginning with the last, find the square root of a + b, divide b by this root, then add a to the quotient, extract the square root again, and continue the same process upward. The values of the radicals, thus successively found (which we may denote by R, R, R", &c.), will be nearer and nearer approximations to the value of x. Considering all the radicals as positive, we shall thus find a positive root of the equation. It may then be depressed to a quadratic. The work may be somewhat abbreviated by the use of logarithms. In the following solution all the figures are retained.

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Here, after four approximations, a value of x is correctly found to four places of decimals. In the equation 23-3x=1, we shall in the same manner find, after four approximations, x = 1.8793, which is also correct to four places of decimals.

An equation of the form 3-ax = -b reduces to the form

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In this case, if we consider each radical as negative, the second term under each radical will be additive; and the process will be exactly the same as before, and the approximations as rapid. Thus in the equation x3-7x=-7, the fourth approximation gives x3.048, which is correct to every figure.

This method, which is new to the writer, applies to the irreducible case of CARDAN, of which the foregoing equations are examples. It serves to find a value of x "to any required degree of correctness." It will be observed that it is entirely independent of any trial.

It is evident that the approximations will be most rapid when a3> b2; a condition which the irreducible case always involves. — Prof. E. W. EVANS, Marietta College, Ohio.

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