Remarks on Professor Johnson's Query

  Early  Journal  Content  on  JSTOR,  Free  to  Anyone  in  the  World   This  article  is  one  of  nearly  500,000  scholarly  works  digitized  ...

0 Downloads 14 Views
 

Early  Journal  Content  on  JSTOR,  Free  to  Anyone  in  the  World   This  article  is  one  of  nearly  500,000  scholarly  works  digitized  and  made  freely  available  to  everyone  in   the  world  by  JSTOR.     Known  as  the  Early  Journal  Content,  this  set  of  works  include  research  articles,  news,  letters,  and  other   writings  published  in  more  than  200  of  the  oldest  leading  academic  journals.  The  works  date  from  the   mid-­‐seventeenth  to  the  early  twentieth  centuries.      We  encourage  people  to  read  and  share  the  Early  Journal  Content  openly  and  to  tell  others  that  this   resource  exists.    People  may  post  this  content  online  or  redistribute  in  any  way  for  non-­‐commercial   purposes.   Read  more  about  Early  Journal  Content  at  http://about.jstor.org/participate-­‐jstor/individuals/early-­‐ journal-­‐content.                     JSTOR  is  a  digital  library  of  academic  journals,  books,  and  primary  source  objects.  JSTOR  helps  people   discover,  use,  and  build  upon  a  wide  range  of  content  through  a  powerful  research  and  teaching   platform,  and  preserves  this  content  for  future  generations.  JSTOR  is  part  of  ITHAKA,  a  not-­‐for-­‐profit   organization  that  also  includes  Ithaka  S+R  and  Portico.  For  more  information  about  JSTOR,  please   contact  [email protected]  

?89? NEW

RULE

FOR

CUBE

ROOT.

BY J. B. MOTT, WORTHINGTON, MINNESOTA. In the following example the ordinary method is pursued till the second figure ofthe root is found. We then find the triple product (t. p.) by placing the second figure of the root to the right of three times the preceding

part of the root and multiplying this by said second figure, and so on for all the triple products, finding one from the other. Thus: lst trial divisor + 1st t. p. = 1st complete divisor. lst complete divisor+lst t. p. + b2+2nd t. p. = 2nd complete divisor. = 3rd com. divisor, &c, 2nd complete divisor+2nd t. n.+c2+3rd tp. the to the under t. two each preceding divisor; b, c p. figures right placing the &c. of root. &c, being 2nd, 3rd figures Each complete divisor is used as a trial divisor for the next figure of the The constant left hand figures of any divisor, used as in simple divi? determine as many more figures of the root. will sion,

root.

Example.?To

find the cube root of 2 to fourteen places of decimals.

=3 First trial divisor ? t. = p. 32X2=J>4_ " com. divisor = 364

2(1.259921

8625 lst t. p. +4 + 2d t. p., 45025 2nd complete divisor, 2nd t. p+25 + 3d t. p., 218831 4721331 3rd complete divisor,

1_ )1000 728

3731211 3rd t. p.+81 + 4th t. p., 47586431"! 4th complete divisor, 34765144 t. p., 4th t. p.+81+5th 47621196244 5th complete divisor,

)272000 225125_ )46875000 42491979 )4383021000 4282778799

)100242201Q00 95242392488 79375561 5th t. p.+ 4 +6th t. p., 47621989989(51 6th complete divisor, )4999808512000 4762198998961 3779762 6th t. p.+ 0 + 7th t. p., 4762202778723 7th complete divisor, )237609513039 = .04989487 + . This united Dividing, we have 2376095 -s- 47622028 to the root above gives f2 = 1.25992104989487+. by R. J Adcock.?Mr. Query Jonson's paradox involves two questions, the value of u, and the value of du-i-dx = cos ax, when a = co. He says "now if a = co, u = 0 indepen? This I deny. 0 is used to represent both actual zero and dently of x". Remarks

on Professor

Johnson's

?90? infinitesimal

It cannot however be so used unless it can be quantities. In shown that no error will result from such use in the particular case. this case when a = oo, u = actual zero or an infinitesimal, that is u = 0 -r- a or some infinitesimal between l~-a and ? 1-j-a, these infinitesimals

depending for their values upon a and x. And the rate at which these infin? itesimals change their values is du~dx = cos ax, for all values of a and x. When a is infinite, that is greater than any assignable number, then a is indeterminately great, and by consequence indeterminate; cos ax is the co? sine of an are in a given circle, the are being as indeterminate as a, and therefore cos ax is indetermiate both in "form" and value, and from the

given conditions can no more be affirmed to be zero than any other value between +1 and?1. There being no preference or reason for the termina? tion of the are ax in one part of the circumference rather than another. Mathematics

having to deal with truth, like the "Scripture is of no private

interpretation". DIFFERENTIATION

OF

THE

LOGARITHM

OF A VARIABLE.

BY PROF. LABAN E. WARREN, COLBY UNIV., WATERVILLE,

ME.

To differentiate the logarithm of a variable, let y = ex; = d(\oge y). . ?. x = loge y; .-.dx = ex(edx-~~l), y + dy = ex+dx, dy = ex+dx?ex or dy eP* =

but dx =

1 +dx+

d^-+dfr+&^

or edx =

1 + dx;

or dy = exdx = ydx, or dx = dy-*-y, dy = ex(l+dx?1), d (loge 2/); . ?. d(logey) = dy~yf differential of Napierian logarithm. = md(\ogey); ^g10y = m(\ogey); .-. d(log10 y) . ?. d(logl0

SOLUTIONS

7/) =

OF

m-^,

differential of common log.

PROBLEMS

NUMBER

TWO.

of problems in No. 2 have been received as follows: Solutions From Prof. L. G. Barbour, 434; Prof. W. P. Casey, 430, 431, 433; G. E. Curtis, 429, 434; Geo. Eastwood, 431; Wm. Hoover, 429, 430; Prof. P. H. Philbrick, 429, 430, 431, 433, 434; Prof. E. B. Seitz, 430,431, 433, 435; Prof. J. Scheffer, 428, 430, 431, 433.