Lancaster daily intelligencer. [volume] (Lancaster, Pa.) 1864-1928, 1886-10-23

imufely tue, let the arc beever so small We
take the liberty, however, 10 reassert that II
the arc be taken small the expression IL up-
proriwdteiV correct and II the arc be inn.
uitcly small the expression is rtgidlV true.
But, by fair or foul means the professor Is de-
termined TO make his assertion good by
another resort 10 numbers, aud we are sorry
to say he is Just o. reckless here IA he was
In demoliShing the nfst proportion.
After trying to show how 1t Is proper
to proceed in such cases he says: let US try
this equation = r. arc r sin upon radius
Of iO with an arc of l., since 1t does not make
The area oi sector of l. Ia radius Of
500 500 "'''" - '''' '"'"' " WV
The area OF 4 sin-5.snle .0SiS, (reslly
.OsI2), or about lilo OF the value OF ~ arc.
Please understand clearly. Prot kershuer
says that the AREA OF circular sector OF
10 times the area of the triangle repre-
senied 10y radius, sin and GO sin, l., Now,
please go on iuSt a little farther, dear proies-
sor, and you will be sole TO prove 10 the
world that the area OF a circle 10 times the
area OF an inscribed polygon Of lSo sides
He says this result suggests a correction
required to y are r. sin.
Remember that the tables OF natural sines-
cosins, &c, are developed upon n basis OF
unity and when you assume radius OF 10 II
is necessary to multiply the given values OF
the functions by 10. Doing this you Will
discover that the equation = Z arc -'s sin is
all right SO manifest a blunder 10 be COM
mitted twice IN the course of this criticism
suggests TO me the importance of a little cor
rectluH somewhere else
Lastly the professor says, that even II it
were not for the errors already pointed out,
the area II il not constant He SAYS II is an
element OF the circle ThOsG radius tS PS.
There is no account taken of the fact he says
that r varies, which we reply, is indeed the
uerV fact taken into account l can do no
better here than to refer our critic to an ele-
mentary proposition on central forces in
Loomis' treatise on astronomy, p. l29, after
understanding which, ! am sure that he will
find that his last exception is no weightier
than the least OF the others.
In order to anticipate a question into which
our critic has not inquired, it seems proper
to state why the demonstration OF Th. VIII Is
confined 10 the movement Of the body in the
| difterential arc AP. The object is TO de-
| termine the iUw of force prevailing al
| the point \ which will cause a particle
| at A TO which \ centrifugal impulse at night
| angles with ALB is impaired, TO describe an
arc of a circle. 1t is shown by the demon
| strstiou that the law which will satisfy this
condition is the law OF grsvitation ITSELF
Now, if the differentisl arc caused 10y gravtta.
tIOn is circular 1t can of course not be ellpti-
cal 1t Will be shown 10y means OF other
demonstrations that the curvature COM
meuced At A is maintained during the entire
revolution that the curvature caused by
gIavitation is diiTGyS circular and that the
| distance from ; TO b, Cfrom the central force
| to the centre OF the orbit in other words,
| the eccentricity OF the orbit, depends upon
the ratio OF centrifugal 10 centripetal force,
The investigation of these points is the sub
ieCt OF other demonstrations and this Is why
.Th. VIII is confined TO the ditfereutisl arc,
Having announced that Newton gives
similar demonstration by which he arrive.
at a different conclusion, and having offered
| 10 point out his error, l will now presen
| the following, which is a correct copy of hi!
| demonstration as found in the WORK pub
lished by Perciyal Frost
PROF VII, PROBLEM IL.
a toiy moves in the circumference of
circle, TO ind the law of the centripetal force
| tending to any given point in the plane Of th
| circle.