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^'fc^f

^V^E ClRci^

AND

STRAIGHT LINE,

l(i^^-

PART THIRD.

= ^

slip

iii,..

THE CIRCLE

AN'D

STEAIG-HT LIIsTE.

PART THIRD.

JOHN HARRIS.

MONTREAL:
JOHN LOVELL, ST. NICHOLAS STRKET.

MARCH, H74.

:«|*

Knteml nccording to Act of Parliament in the year one thousand f ipht hundred
and eevcnty-iour, by John Hakius, in the otTice of the Mini.-tir of Agriculture
and Statistics at Ottawa.

^^^ £i I'm

MoNTUKAL— John Lovbll, PniNXEn.

THE CIRCLE AND STRAIGHT LINE.

l*r()\ e <i!/ tilings ; hold fast that which ia gooi,'

St. P'nil.

PART THIRD.

THEORV OF UURVATl-RE.

Tht UUimatc-Siu(\ und the arc of incrcas'mg cnrvafin-r. —
If ii iiiiniln'f of intiTuiodiaie nres be described with a
proportionally increased radius, between 31. an<l )i.,
between h. and o., between o. and ^)., &c., &c., respec-
tively, it is evident that a continuous curved line drawn
through the terminal points oi' all those arcs, will form
a compounded arc of a peculiar character and which
fioin its forming a connection as it were between the
arc-length of the half-cpiadrant and the ultimate sine — i.e.,
the sine of the ultimate fraction of the half-quadrant —
possesses much interest. We will here briefly indicate
one mode in which the change in longitudinal magnitude
in conse(|uence of the elimination of curvature, as the
fractional arc is diminished by successive bisections, may
be investigated.

Taking the radius eqmd to 10. The length of the half-
quadrant equals the sine-length increased by addition of
' the sine length divided by the units of radius less one.'

^: = .s

K-i

We thus obtain the magnitudinal length of the arc
in units of radius. Now if this arc (half-quadrant) be
bisected and the lonijitudinal magnitude of the remain-
ingarc be duplicated, the half-quadrant is not reproduced

.*v

TIIK CIRCM-: AND STKAIOIIT LINE.

but nil arc is obtaiiuHl into whitli one half the curvature of
tlie half-(|uadraiit does not enter and in which the other
half of the uirvature beloni,'inf» to the half-(|Uadraiit is
divided over twice the quantity of linear extension.* The
second arc therefore, although it is a inagnitudinal dupli-
cation of the one-half of the first arc, is a lesser longitu-
(hnal magnitude than the lirst arc, because the niagnltu-
(Hiial (hiplication which has restored the direct linear
extension has not replaced the curvature.

The process may be thus conducted: —
^^ — The arc-length equals the sine-length increased by
S addition of the sine divided by the units in the
radius less one.

C . — The arc-length equals the sine-length increased by
J(» addition of the sine divided by the units in the
(duplicated) radius less one, multiplied by two.

C. — The arc-length equals the sine -length increased by
'32 addition of the sine divided by the units in the
(duplicated) radius less one, multiplied by two, multi-
plied by four.

(j, — The arc length equals the sine-length increased bj'
64 addition of the sine divided by ihe units in tho
(duplicated) radius let^s one, multiplied by two, multiplied
by four, multiplied by four. And so on, proceeding in like
manner.

The successive
divisors will
be therefore.

f 1st arc

( ^-1)

2 do

(2/?-l) X 2

3 do

(2/«?-l) X 2 X 4

4 do

(2 7e-l) X 2 X 4 X 4

5 do

(2 /?-l) X 2 X 4 X 4 X 4

, 6 do

(2 7?-l)x2x4x4x4X

*If the liaU'-(ina(Jrant, instead olbii^ected, were to be partially straiglit-
oiied into the same form as tlie lialt'-arc, it would exceed in lengtli
the diipulicated half-arc by the difference of the curvature, (i.e..
the difference between the curvature in the half-quadrant and in the
arc of 22 J degrees.)

THE CIKCLE -wsl) STHAKSHT LINK.

Taking the Bucocssive einc len^'ths from tlio talilo given at
pa^'e 54, Part yocoml, we obtain : —

707107

•785074

;{H

705306

•201413

152

7S03G1

-051339

008

7-841.37

•012897

2432

7-85082

•0032281

1)728

7-85319

•0008073

38'.) 12

7-85379

■f-

•0002018

155648

785393

-t-

•0000504

022592

7-85390

-f

•0000120

1 R-

l R-

•J

10 R-

04 R -

250 ye -

128

1024 R~

512

4090 R -

2048

10384 R-

8192

05530/?-

32768

= 7850744
= 7855080
= 7 •854951

= 7-8545!)7

- 7-854055

- 7-8539!i7
= 7853991
= 7-153984
= 7-853978

III tins tablo, tlie divisors in the first column E, result
li-()ui the last column D. Under A. are tiie sine lengths.
Under IS. are the <|Uotient8 of the division of the sine
lengths by the numbers under E, which (inotients adth-d
to the sine lengths yive the arc-lenirths in column C.

It becomes evident that, by continuing the table, the
(|uantities in colunui B., which represent the curvature,
would beconie continually less, and the quantities under
A. and C. would approximate more closely. In reference
to column B. it may be observed that the third quotient
is rather greater than the fourth part of the :2nd, that
the 4th is a litt.e more than the fourth part of the ;ird,
the 5tli a very little more than the fourth part of the
4th and that, of the remaining quotients, each is the fourth
part of that preceding it, so far as the figures are carried ;
a greater number of decimals would show each of the suc-
cessive lesser quantities to slightly exceed the one-fourth
of the quantity preceding it, but the approximation to
the one-fourth becoming continually closer as the process
is carried further. To appreciate the meaning of this it
is necessary to remember that the chord of each arc,
which chord equals the sine of the next succeeding arc,
is the square root of the square of the versed sine added
t© the 8(iuare of the sine ; and that the versed sine be-

THE CmrLE AND STRAIGHT LINE.

loiigititf to ejicli arc is always greattM' than the ono-lialf of
that bolonijiiiu; to tlie arc precediiii? it ; but m the ter-
minal ratlins approaches the perpendicnlar (i.e., as the
cnrvature becomes more nearly eliminated), the versed
sine of each arc approximates more and more closely t«
the one-half ot that belonging to the arc preceding it.

The process thus snggestetl is put forward as worthy of
attentive consideration, and for the purpose of illustrating
the subject; but we wish it to be particularly observed that
(for the present) it stands by itself, nothing else is based
upon it ; it is not put forward as a demonstration, nor can
we (at present) vouch for the correctness of the process
or for the accuracy of the figures.

Thk akc of INCREASING CURVATL'RE — This arc is dis-
tinguished from the arc of the arc-length (that is from
an arc described with the length of the arc as a radius),
and illustrated in Figs. 21 and 2'2. In Fig. iil, XX
represents the arc described with the arc-length for a
radius, and S. n. o. p. the arc of increasing curvature.
In Fig. 22, the relation of the same two arcs to the
construction of the circle is illustrated on a larger scale.

{Note. — With these Figures compare Figs. 20 and 25)

The TANGENTIAL LINE B. T).

(Fig. 23 is a development of Fig. 12.)
Fig. 23. Bisect the radius A.B. in «., and with centre
a., and radius a.B. describe the quadrant B.K. ;
bisect the quadrant in m. and bisect the half-quadrant
B.m. in '«. ; through "«. draw a.n. intercepting the line
B.D. at n. Bisect also CD. in b., and w-ith radius b.D.
describe the quadrant D.K.; bisect D.K. in 31. and bisect
the half quadrant D.M. in 'iS^.; through 'N. draw h.N.
intercepting the line B.D. at N. And bisect the arc B.S.
in jj., and through j). draw A. p. With centre C. and
radius CD. describe D.A. bisected by the line R.T. in Z.,
bisect D.Z. in 'P. and through 'P. draw CP. intercepting
IID. at P.

*..

TIIK CIRCLE AND STRAIGHT LINK.

W*. now have a division of the (lefinitc line B.f). into
rortaiii known (|uantitivo niagnitnih-s. Ik'caiisc JJ.u. is
tilt* tangent of the are /?.'«., an<l JJ.p. is the tanj;«'nt of the
similar gn-ater arc //.';>. wliich is \\\ ce the niagnituch'
of li.ii., J!. p. is twiee tlie lenirtli of 7/. /<., and n.p. is
equal to Ji.n. At the o[)posite extremity tlie similiir
points D.X. and D.P. are similarly related, and X.J*. is
(Mpial to I).X., which iserpial to Ji.».
Also because ij.7 is re(|uivalent to) the ^ ne of ]>.)».,
B.fj. is the half of Il.W. which is (e(|Mivalent to) the
sin«» of the similar greater arc U.S.
Taking the radius A.Ii.- 10, the numerical values will

li.u. =-J'07H)7
n.p. -i'14:>l4
B.q. -7-07107^:.'

iAX-10-:2-0710()S

IKK,
D.X. ^-- 7i.p.

D.r.

•so7SG4:-i>.A'.

Having in mind the relative magnitudes of the primary
lines determined by trigonometry, these values present
themselves at once as evident; we may then obtain, from
and by means of these, the rpiantitive values of other
lines, and may also, thus (pumtitively, detenuine the
geometrical (magnitudinal) relation of other lines; that is
to sav — the inductive reasonini;, by means of which
further knowledge of the relation of the parts of the
circle to each other is to be obtained, may be based upon
the facts belonging to the science of 'Nundjer and Quan-
tity' in place of those belonging to ' Fokia and ^lagintude.
For example : —

Because B.n = 2-07i07, B.K=^o^ and B.W^l-OllOl
Therefore K.n.= W.D. = S.W. = S.T.^K.X. =2-92SiK5
And, 2'9289:J x 7-07107 = 20-7107....
Therefore, S.W. x B.W. - B.n. x B.D. That is—

the versed sine x the sine of the hulf-quadrant = the tangent
to the arc of 22^" x liie tangent to the half-quadrant.

f
¥

mi !

niK CIRCLE AXD STKAIGHT LINE.

Rerause K.p. = B.K. - B.p. = (o - 4- 14-? 146) ^ '857804
therefore K.p^ 10- {OK.p - (A'.//)* = '^•02893*= 8"57S04

And P.iK = -2 K.p = 1 -7 lo7if.

Hut the sqvuiie of'4-14-2136= 17-lo72

Therefore P.p. x \0^-{S.DY=iB.py^{D.P)% &c., &c.

B.P== B.K + K.P= o-8o 7804 = 2 WD^ 2Kn.

Now the tangent to the half of the bisected half-

(piadrant duplicated, = 8-28428= 2 B.p = 2 B.P. And

(2 B.p-B.W) = {K.n-2K.p.) = (8-28428- 7-07107) =

(2-92893-- 1-71572)= 1-21321, Ac, &c.

A fact of a primary cluiracter, which has not, we
believe, been as yet applied in the art of computation, or
otherwise utilized, is exhibited in the following: —

TiiKOKKM. — If the quadrant of a circle be bisected and
the secant to the half-quadrant be drawn through the
point of bisection; and if the radius be produced through
the centre of the circle and the produced radius be made
twice the length of the origintd radius — that is, equal to
the diameter of the circle — and a line be drawn joining
the extremity of the produced radius and the extremity
of the tangent ; then shall the sine of the arc cut off from
the quadrant by the line so drawn have to the tangent of
tiie half-quadrant the ratio of four to five.

Fig. 24. With centre A. and radius A.B. describe the
(|uadrant B.C., and bisect B.C. in the point S. Draw
the tangent B.B. and secant A.D. Join D.C. and A.C.
Produce the radius A.B. through A. and make E.B. the
production oi'A.B. double the length of ^l.D. Join U.D.
intersecting the quadrant at K. and bisecting the line
A.C. at g. Bisect the line CD. in the point d. and
Join B.il. intersecting the quadrant at the same point K.
From /iT. draw if. ^i^. perpendicular to Z?.Z). and intercept-
ing B.D. at Q. From Q. draw Q.II. parallel to A.D. and
intercepting A.B. at the })oint //. From H. draw H.P.
parallel to A.C. and equal in length to B.Q. Join P.K,

FIG. 2J^.

^>'< «rRAiOiri i,iM".

— ^H

csiftr^ -ifv? 'eue;th r,i' B.D., mux I-t.J>.
■m>u}^ B.d.D. ih simihir to
. v'iWiirU.'s B Jut 8i;ri k'J:.D
rli^M'ofore xhe l>n»» Ji.ti. is

,; ' > I • < V r- ■ ! I (. < ■ ."-

!roJ:

.<. -."U

/-'. at rv*.

/(./^

iV '.

f^'i

. k1**WI i' tU»1J«^

\. *

"^ .,1

-.ii* X

; f t.

THE CIRCLE AND STRAIGHT LINE.

Because E.B. is twice the length of B.D.y aiul B.D.
twice the length of d.I). the triangle B.d.D. is similar to
the triangle -fc'.D.^. But theang.es B.D.d. and E. B.D.
are both of them right angles, therefore the line B.d is
at right angles to the line E.D. Now since fl.Q. is
parallel toA.D. and H.P. is parallel to A.C. the triangle
H.P.Q., is similar to the triangle A. CD. And since the
line B.d. bisects CD., the same line B.d. which inter-
sects P.Q. at A', also bisects the line P.Q. in the point
K. [And the line B.K. (part of B.d.') has been shown to
be at right angles to the line K.D. (part of E.D.) tiiere-
fore the triangle K.D.Q. is similar to the triangle B.K.Q.
and K.Q. is the half of P.Q.I And P.Q. is (by the con-
struction,) equal to B.Q., therefore ^.^.is equal to twice
K.Q., and K.Q. is equal to twice D.Q.; therefore the line
B.Q. has the same ratio to the whole line B.D. as 4 : 5.
Draw K.e. the sine of the arc B.K. K.e. is manifestly-
equal to B.Q.

Problem. — Upon a given straight line it is required to
(Ascribe an arc containing 45 degrees, such that if the
tangent of the arc be divided into five equal parts, four
of those parts shall be together equal to the given straight
line.

Fig. 24. — Let B.Q. be the given straight line, it is
reciuired to describe the arc upon B.Q. From the
point Q., perpendicular to B.Q., draw Q.P., equal
in length to B.Q. ; and from B., perpendicular to B.Q.,
draw ^.77., also equal to B.Q. Join 77.^. Bisect P.Q.,
in the point K. and join B.K. Produce B.Q. through Q.
and produce B.H. t' -ough 77. (Through K., at right
angles to B.K, draw K.D. intercepting the production
of B.Q., at D. From 7). parallel to Q.H.* draw D.A.,
intercepting the production of B.H. at A. With centre ^.,

• Or, join B.P., and froni D. at right angles to B.P. draw JJ.A.,
intercepting the production oi' B.H. at A.

THE CIRCLE AND STliAlOHT LINE.

and radius A.B., describe the quadrant B.S.C., inter-
sectingD.^l. in the point S. JB.S. shall be the required arc.
The demonstration to this solution is substantially
contained in the demonstration to the theorem.)

Bisect K.Q. and, through/, the point of bisection, at right
angles to B.P., draw D.A. intercepting the production
of B.Q. at D,, and intercepting the production of B.II.
at A. With centre A, and radius A.B. describe the
quadrant B.S.C bisected by the line D.A. in the point
S. B.S. shall be the required arc.

Because B.Q. f. is a right angle and /. bisects ^•^.
QB. is equal to the one half of K.Q. But K. bisects
P.Q.. and P. Q. equals B.Q., therefore Q.B. is equal to
the one fourth part of BQ. Wherefore if B.B., the
tangent of the arc B.S., be divided into five equal parts,
four of those equal parts, shall together contain the given
straight line. Q.E.D.

We have now to make known a discovery which will
fjicilitate the investigation of the structural characteristics
of the circle and which may become hereafter of much
utility and value in simplifying trigonometrical (cyclome-
trical) processes. We propose for the present to treat
it as a fundamental and, in a measure, independent fact,
belonging to the plan of the circle. The discovery is
stated in the following: —

Theorem. — That if the radius of the half-quadrant be
produced through the centre of the circle until it become
equal to the versed sine (of the half-((uadrant) multiplied
by ten, and with this radius an arc be described terminated
by a line drawn from the centre of the greater circle —
i.e., from the point at the (central) extremity of the
radius — through the point at the extremity of the half-
quadrant, the tangent to the arc last described (of greater
magnitude), cut off by the line so drawn from the tangent
of the half-quadrant, will be equal to the arc-length of
the half-quadrant.

FIG. 25.

■iwLiiiVt:;. !"i

r I

I ■ >

>.X.

mi 4.

■idil

- *f ■ u* It ■>

-V- i'*.-- T: f , --jiv ^v^ «

•^i^Jffc**

^if*^

. S*.*!

'' ''f^.

mt-*v

THE CIKCLE AND STHAIGHT LINE.

I' "'•»»■.

Fig. 2.5. — Let B.S. be the lmlf-(|iia(lrn!it of which
A.B. is the radius ; produce A.B. through A. and make
H.B. ecjual to ten times S.W. Froiu JL through S.
(haw li.X. intersecting B.D. in the point X. B.X.
shall be equal to the arc B.S.*

Bemonstmtion. (Quantitive). Th«' length of the sine, »S'.iSr.
the arc -length of B S., the length of the radius 11. B.
and of the versed sine S.W, are known,
we have to obtain tlie length of B.X.
The radius, A.B.= \0.
'* V. sine, S.W. =-2-9-2^}>:3.
*• radius, /?. R - 10 S. II'- iO-OS!):}.
7?..Y. = B.B - S. W = (20-2S9:]-i>-9-?S9:3) - 20'3G0:3.

Now the lesser triangle S. W.X. is similar to the greater
triangle 7?..Y.*S'.

Therefore W.X: X.S.: : S.W.: B.N.

W.X. : 7-07107 : : Q-'JQS'JS : 20'3(iO:3.
IF.X- •78507 B\MN.S. + W.X. --= B.X.
Therefore, 7-07107 + •7S567 = 7-So074.

And B.S. the arc - length also ec^uals 7-S5G74.
Wherefore the tangent B.X., of the arc of greater mag-
nitude is equal to the arc length B.S., of the half
quadrant. (Q.E.D.)

Corollary. Hence manifestly TF.jir= the difference of
the sine and arc-length of the half-quadrant.

The Ultimate tangent — We have objected to Legendre's
circumscribed polygon — which eventually almost coincides
with the sine — that it cannot be a simple continuous figm-e
but must be fragmentary and compound ; and we based
this conclusion primarily and directly on the manifest
necessity that a circumscribed polygon if continuous must
be greater than the circle which it surrounds. We propose
now to show that Legendre's circumscribed polygon is
in fact compounded of the minute ultimate tangents

■K Or :— From X, through S, draw X- Y.li., intercepting the
production of B.A., at B. B.B. shall be ten timea the length
o(S.W.

'l^'Sf-*-

TIIK ( IIMI.K AND STUAIfSHT I.IXE.

s»M>iUMt«' from oiU'li otiior hut arranjjpd together in the
form of a polvsroii. To show this we will consider the
question whether the tangent to an arc, if thaturc be an
indefinitely small fraction of the circle, is necessarily
greater than the arc itself. It is at present held that the
tangent is always greater than the arc by the evidence
of trigojiometry, thus : — Let A.B. be the arc, and A.d.

Fig. 2G.

the tangent. The arc is bisected ; and from the point ?,
e.B. is drawn perpendicular to U.B., intercepting R.B.
at B. Now the triangle d.e.B. is similar to the triangle
d.R.A., and the side d.c. is greater than the side B.c. of
the lesser triangle, in the same ratio that the side d.R. is
greater than the side A.R. of the greater triangle. Since
evidently so long as it is possible to assign any quantity
of magnitude to the remaining arc, it is possible to
imagine the same process to be performed, it is supposed
to follow that, however small a fraction of the circle an
arc may be, so long as there be an arc, the same reason-
ino: demonstrates the least tangent^belongins: thereto to
be greater than the arc, because it is thereby shown that
the outer tangent (as A.d.) is greater than the two

THK ClUCLK AND SrUAKlIIT I.IXi:,

intorior tnngeiits (as A.r. iiiul r.H.) taken together;
nnd it is nssuniod as niaiiifest that the two interior tangents
taken together are always greater tiian the are.

We submit that such conclusion is sliovvn to be erro-
neous by the following : Let A.B. be a given straight
line. At each extremity of the line draw a perpendicu-
lar A.B. and S-f. {t. being the same point as B.) Let

Fig. 27.

the same line A.B. be supposed to be curved* into tiie
form of an arc containing a very small fraction of the
circle.

* If the cxpre.-^siuii * licmiiiig' or ' curvinji the line ' he ohjected to,
then, let a fractional arc equal in length to the given .straight line,
and bclongingtoa circle of any very great niagnituile, be applied upon
the straight line A.B. in suth \vi.se that the point at one extremity of
the straight Vu-.'! f^hall coincide with the ]ioint at one extremity ot the
arc, dec, &c.

THE CIRCLE AND STRAIGHT LINE.

It is manifest that the line being curved will be less
in direct length, measuring horizontally between the
perpendiculars, than when it was straight, and, although
the same line and of the same actual length as when
straight, it will not now extend quite so far from A.
as to reach the line S.t. Let any point h., less distant
than S.t. from A., by a very small quantity of space, be
the terminal extremity of the arc. It is evident that,
however small the distance of the point b. from the
line S.t., if a line be dravv-n from the point f. through h.
and the line so drawn be produced until it meets the
production ofA.Ii. througli B. then, if the line so drawn
may be correctly considered the secant of the arc, the
tangent is A.t. and is not greater than the arc. The
question, therefore, assumes the fonii : — whether the line
T.t. drawn through the point b. at the extremity of the
arc, is at right angles to the extremity of the arc. If it
be not at right angles, the tangent may be greater than
the arc ; because the point /.. at the extremity of the secant
might then fall beyond tiie distance B. from A. On this
question, however, we have the evidence of trigonome-
try as given in the polygon computation of Legendre
(already quoted) and others of the like kind, and which
evidence is decisive because it is shown thereby that
ultimately the tangent-length almost coincides with the
sine-length, and, since the sine-length is manifestly less
than the arc-length, it is evident that in fact the
extremity of the secant to the ultimate arc does not fall
beyond the arc-length on the tangential line, and, since
the ultimate tangent-length cannot be less than the arc-
length, it appears safe to conclude that the tangent-length
actually coincides with the arc-length of the ultimate arc.

Let D.E.A.F.B. (Fig. «) be four equal divisional parts
of the half-quadrant D.B. Fig. b. represents one of
those equal divisional parts on a larger scale, the straight
line being equal to A.e. Now if we apply four straight
lines each equal to A.e. as tangents to the divisions of

^:^.i^~

THE TANGENT.

Fig. a.

Fig. b.

Fig. c.

U'C.

\rts

Ight

Is oi

jii'i

iiiii^

,l,i;,:!!Tli;ii

lllllllllllllllill

Fig. d.

MIR

^:'^

i^i-^

THE CIRCLE AND STRAIGHT LINE.

the luilf-((uaclrant, as shown in Fig. a. it is evident that
they cannot meet each other at the extremities which are
not in contact with the circle. If they had sufficient
length to meet each other, angles would be formed at the
points E. and F., but, the length being insufficient, spaces
are left at those points. Supposing the four lines to be
again divided by bisection, and the eight half-lines to be
arranged as tangents to the circle, there will be four
spaces instead of two where the extremities of the
lines will not quite meet each other, but tliey will be
much more nearly in contact than before. If the lines
be in like manner repeatedly divided, there will be
eventually obtained a polygon, formed by the ultimate
tangents which will then be in contact, the extremities
of each with the extremities of those next thereto. "VVliat
are the conditions of that contact ? If we accepted the
dogma of Euclid, we should have to consider the contact
as complete and the polygon as a simple figure, and to
accept the sum of the ultimate tangent-lengths as the
measurement of the circle ; but we reject the dogma and
insist on a real circle containing area and, consequently,
containing breadth.

We have, a< 'ordingly, to enquire how the lines are
divided. Is the section oblique or perpendicular to the
length of the line f Now this is readily answered — because
the ultimate tangent-lines, derived from continued
bisection, are divisions of a straight line evenly divided
at right angles to its length. Therefore, by placing tlie
divisional lines of Fig. a. in the positions, relative to each
other, which they would occupy as tangents to the circle,
and considering the rectangular extremities of the one
with reference to the rectangular extremities of the next
the actual conditions of the contact between them will
become manifest. Fig. c.

The fragmentary tangent-lines compounding the
circumscribed) polygon are, tlierefore, in contact with
respect to the inner surface only, but, if a straight line be

■^■

'■r.

THE CIRCLE AND STRAIGHT LINE.

compounded of the fragmentary lines, the contact will
be complete — that is, perfect with respect to both sur-
faces — and the original straight line will be restored.

This case may be taken by itself. The fact of the
ultimate arc equal in length to its tangent, and of the
half-quadrant greater than the sum of all its ultimate
tangents, presents itself for acceptance. That it is indis-
putable and must be accepted seems to us evident. But,
what then : if there be some persons to whom this fact
appears irreconcilable with the facts (as now taught) of
trigonometry 1

In concluding this treatise it may be not out of place
to consider a little the meaning of the familiar statement
(phrase) that one fact cannot contradict another. Proba-
bly no one at the present time with any pretension to
scientitic knowledge or education will directly deny or
expi'css any doubt as to the truth and certainty of such
statement in its simple form ; but the question is as to
absolute certainty in the strictest and most comprehen-
sive sense ; and in the writings of men of scientific
reputation, at the present time, opinions are expressed,
theories propounded, and statements made, which, if
referred inferentially to their basis, make apparent that,
in the minds of the writers, no clear and reasonable
conviction has been arrived at that, as a scientific cer-
tainty, one fact may not or cannot contradict another ;
i.e., be otherwise than in harmony with other facts.
We call attention here to this circumstance because we
believe that, in many cases, if the writer distinctly
understood the nature of the conclusions involved or of
the reasonable inferences belonging as corollaries to those
theoretical opinions and statements, they would ht
reconsidered and, being found to be um'easonable, wouM
not be put forth. This circumstance, which must needs
be regarded as of the gravest importance, may in
innt ;it least, be a consequence of tlie increasing neglect

I :i

^"•■?h

THE CIRCLE AND STRAIGHT LINE.

of mjithematical reasoning — by which we mean, strictly
lawful and (therefore) correct reasoning — as a necessary
element of educational training. A consequence of the
want of such training is a non-recognition by the mind
of the necessity of a primary basis, either securely
established or undoubtinTly believed in, upon which
the theory or statement must be actually or piesiimably
based. Practically the effect displays its'?irin the opinion
now more or less openly expressed by scientific teachers
that Theology is not an absolutely necessary and funda-
mental part of Science, but that, on the contrary. Science
and Theology, if not actually antagonistic, should be kept
quite distinct and separate.

This separation of Science from Theology appears to
be concurred in, and even supported by very many of
those who teach that the fundamental basis of Science
must be found in Theology. They, nevertheless, accept
the title of theologians, as distinguished from, and as
in some degree antagonistic to that of, men of Science.
Moreover the dislike of the man of Science to Theology,
is perhaps, more than reciprocated by the theologian
whose belief in the primary truth of Theology is often-
times a belief fearful and apprehensive lest the progress
of Science should discover and make known some adverse
truth irreconcikble with the truth of Theology.

That this view of the case is adopted by all or by
nearly all scientific men or theologians we are not sup-
posing. The exceptions on both sides are probably in
the aggregate very many, but the question is, taken
all together, what is the concludion or state of feeling
now entertained and held by the cducateil public gener-
ally ; and in what direction are the doctrines now taught
guiding those who are to succeed us ?

Assumingthe existing state of things to be sul)stantially
as we have suggested, the apprehensive belief of the
theologian which fears the facts and discoveries of
science is not a belief in a scientific sense ; such a fear

<r -

THE CIRCLE AND STRAIGHT LINE.

evidently includes a doubt as to whether one fact may
not contradict and be irreconcilable with another or with
some other facts. Nor can it help matters to say that
the truth of theology is different from the truth of science,
because, if it be meant thereby that the truth of the one
can be otherwise than in perfect harmony with the truth
of the other, the entertaining or admiting in any degree
such a supposition is the making over Theology to Meta-
physics which is expressly the antagonist of sound Theolo-
gy as being the essential basis of all sound Science. How
stands the case on the other side ! If the (piestion was
put : what is the especial principle contended for and
upheld by modern science 1 the answer most generally
concurred in would be — 'intellectual freedom,' the right
of each one to exercise his intellect in his own way,
without let or hindrance. What does that mean ? It
means inteUectual laicJessncss ; intellectual anarchy ; the
ria;ht of each one in matters which concern others as
well as himself iu the most important and momentous
sense, to do as seemeth good in his own eyes ; the demand
of men, who recognise the imperious necessity of laws
and submission to those laws in matters of comparatively
little importance, to be free from the restraint of all law
in matters of the greatest importance. It means the claim
of some men to be allowed to impede sound education,
to pollute things sacred, to contaminate all true know-
ledge, to teach the most pernicious doctrines, and to
deceive and confound those who look to them for
instruction.

Against this claim of so-called intellectual freedom,
whether it be iiesitatingly expressed, merely as a dislike
to mixing up theology with science, or boldly and defi-
antly asserted as a determination to be intellectually
unrestrained by any laws or rules, we distinctly protest
and earnestly caution those whom we may in any degree
influence. It may be said — let public discussion and
progressive education put right what is wrong, we

. f-

THE CIRCLE AND STRAIGHT LINE.

cannot expect to hi /e science quite perfect at any time ;
that which is true is stronger than that which is false —
increased knowledge and discussion may be relied upon
to eliminate error. No : such reasoning is not itself based
on fact ; and such a course is not to be relied on to elimi-
nate error ; on the contrary if allowed to proceed, its effect
would be to vitiate education and to deteriorate knowledge
n:Dre and more, until, at length, the education would be-
come an altogether evil education, and the knowledge an
essentially unsound and destructive knowledge. Truth is
J^tronger, much stronger, than falsehood, but a mixture of
that which is true and that which is false is not truth.
Neither is it fact tliat error can be eliminated and got rid of
by discussion in which sound and unsound knowledge are
mingled together, in which the reasoning is not strictly
lawful and not properly based on that which is true and
certain. Such discussion is the weapon with which Un-
truth, wearing the mask of Science, can best succeed in
destroying that which is true, and confounding those who
love the truth. A li nnan science wliich does not distinctly
recognize the prinuuy truths of Theology as its ultimate
basis, is not based on reality ; it has not and cannot have
any actual and secure foundation. If the Science of
England is not so based, no matter what seeming progress
may for a time be made, whenever the trial comes it will
be as the house built ou the shifting sand ; and, if not
destroyed by sudden catastrophe, will eventually become
a ruin, together with the civilization wliich rests upon it.

f;r*-:

:r ^;^

APPENDIX.

Figs. 27, 28, 29, 30, are additional illustrations of the
complementary arc of absolute curvature, of the arc of
increasing curvature, and of the increasing sine-length.

It may be observed that in some of these figures the
same letter of reference, 3£., indicates the arc length and
the ultimate sine-length. On the scale of these figures
the two points fall so closely together as to appear almost
identical, the difierence being only about three parts in
eight thousand, and an attempt to show them separately
in the same figure would confuse the lines and letters.

It is supposed that, with the explanations which have
preceded them, the general puipose of these last figiu'es
will be sufficiently apparent without further remark.

In reference, however, to Figs. 21 and 22, where the
arc of increasing curvature appears as described with the
radius a.S. it is desirable to point out more particularly
the circumstance (already stated) that the curve is not
an arc described with a uniform radius and a fixed centre.
The curve may be considered as compounded of fractional
arcs of which the centres are not co-incident, or as be-
longing to an ellipse. The same remark would apply to the
arc of curvature. Fig. 25, if described from the same
centre «. with the increased radius a.X. For example (Fig.
22) — If the distance of a. from B. in Fig. 22, be made
two tenths of B.D., and B.X-Ba (-7-S.54- 2) = .'rS.54
be taken as the radius, and the arc be described from X.
as the original point, the arc will then fall inside X,S.^
because the distance of S. from a. is o'SoG, consequently

;..%• \:

■^

JE'

5^-

Al'PKXDlX.

the radius will fall short by the dillereiice of about 2 parts
in six tiiousaud. Similarly, in Fig. •J-'), (taking the
point a. at the same distance, B.u. as before,) U.X. -
J5.«. = 7-8-5G74-2 = 5-So074 ; tlie arc now falls beyond
X.S., because the distance a.S. is as before 5"S-5GO,
and the radius is now 5*S.5G74.

But, if we bisect the line S.X., (theextrennty of the line
B.Y.X., Fig. 2-'5,) and from the point of bisection a' right
angles to S.X. we draw a line intercepting the line B.D.,
the point of interception will be b. at a distance froP' B. =
2-0U4o2.

Therefore B.X. - B.h. = 7-8.5()74 - 2004-32 = ryS52'2-i. .
which is also the exact distance of the point S. at the ex-
tremity of the half quadrant B.S., from the point h. on the
line B.D. Wherefore, in tliis case, we lind the arc X.S.
is the true arc of a circle described from the centre h.
with the radius Z;.X. = o-8-522:J. . .

We have brougiit under consideration four ;ii"cs,
namely : —

(1) The arc of the arc-length X. X.

(2) The CO' plementaryarc(or, arcof curvature) /S.X
(S) The arc of the idtimate sine-length X. X.

(4) The arc of increasing curvature *S^. X.

Tlie first and third, ir.dicated in tiie Figmvs by the
same letter X. X., because too close to each other to be
clearly distinguishable, are both of tiiem arcs of a circle
described from the same centre B. Tlie radius of the
one (1) equalling 7^-3G74...&c. ; and the radius of the
other (3) equalling 7-8-5:39S...&c.

The second and fourth arcs we wish now again to com-
pare and contrast, in order to distinguish clearly between
them. The fourth is not the arc of a circle, but a curve
of a compound character; as already mentioned, it may
be considered as compounded of (ultimate) fractional
curves differing (very slightly) from each other, or (pre-
fei'ably) as a compound arc described from a centre, the
relative position of .vhich is constantly varying, the posi-

■^

APPENDIX. 25

tion and variation thereof being definite relatively to each
uUimate fraction of the arc described.

The following table exhibits the comparison through-
out the greater part of the curve : —

A. are the sine-lengths of the successive arcs — obtained
by the duplication of the successive fractions of the
repeatedly bisected arc — by the terminal extremities ef
which * the compound arc of increasing curvature ' i&
formed*

J?, are the sine-lengths of the equivalent succesaive

«rca (corresnonding to t>i"i» of coknnny),) — resulting

from the continual ' unbending ' or ' straightening ' of

the primaiy arc — by the terminal extremities of which

* the complementary arc of absolute curvature ' is

ibrmed.

A. B.

/ " — — > /■ * — \

?_ 7071068 7-071068

^ 7-653668 7-65517

£ 7-803613 7-80606

-^ 7-841371 7-84405

-^ 7-850828 7-85356

128

~ 7-853193 • 7-85539

256

~ 7-853785 785655

~ 7-853937 7-85670

— 7-853969 7-85673

2048

Finally 7-85398.. .&c. 7-85674.. .&c.

* These sine lengths agree with those of the tabl« at page
55 of Appendix, Part Second.

y^>

QC,

APrFNOIX.

TTereiii we obsorvo (l)}i quite iiulepeiideiit verification
of tiie fact, already a)ni>ly demonstrated, that 7'85G74 is
the actual arc-length of the half-quadrant U.S.; and
moreover we observe (-J) a very interesting definite rela-
tionship established between tlie arc X.S. (Fig. 2ij) and
the half quadrant B.S. / namely the magnitudinal ratio of
the circles to which they reBp'"'<''''^'y belong is . . as
X.S. : B.S. : : radius X.h, 5-8')2:>:3 : radius A.B., 10-0000,
(i.e., a half quachant described with the lesser radius
would have that ratio to the arc B.S.,)

Note. — Referring to Fig. 25, the following will indi-
cate the elements of the computation : —

Bisecting the line S.X. and drawing the line d.b. in-
tercepting B.D. at h. we obtain X.d. as the base of a
triangle similar to the triangle S.X. W. ; therefore

Now since S. W. = 2-92S9;j S.X. = 3-03247
And X.(l = S.X.-^2^ I -51023 5, therefore
-7S5G74 : 3-03247 : : 1-51G235 : 5-85223 .,
which last number is the length of the radius h.X.
7S-r/u74 5352i2 ^ 200452 - 13. 0,
But 7-07107- 2-00452 = 500655

And V5-0G655' + 2 •92893' = 5-85223..,
that is, the square of ' the sine of the half-quadrant
less B.h.,^ added to the square of the verscU sine S.W.
gives the square of the radius h.X.

In like manner the sine-lengths of the successive equi-
valent arcs of decreasing curvature (column B.) may be
verified.

) r

i'?^.

■;-.f

■MM

'^%Jf

ti^.

A'.

M| 1,^1, ,„,

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.•^.

':*#■

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ILLUSTRATION TO
Fig. 20 {R.)

t^fi

-M*-"---"

ffc

Supplementary Illustrations.

Analytifal Jvxaniiiuitioii of tlio <niaili':int and ot'tlic arc
of inci'casiiiir (.•urvaturc.

1— Fi.i;-. 20 (R.) The raiUui A.B. as erjualli)i'j 10.

The radius A. "a. A.N. :. A.M. ; A.B. Conf^equcntly
A. 'a . A.. VI :: A.X, : A.B. And the are--, bines, tangenti^,
and their equal divisions, and oilier related lines, belonging
to each of these radii, are in the same ratios respectively to
the .similar lines belonging to each of the other radii ; that
is,... the arc 'a.l>. ; N.II. :: M.e. ; B.S., the sine 'c.b. ;
d.H. :: a e. : >s..S., and so on. (Observe that the radius of
the arc X.JI. - the cosine of B.S.)

A.B. --= 1(»00
A.d. -= o'OU

A.N. - 7-OT107.
A. a. — o-5o5535.

As ('.'a. : 'a.d. :: d.N. : N.B., and therefore, inversely,
as2-9289;5 : 2-0710T :: l-4644()5 ; 1 035535.

And as b.e. : e.ll. :; U.S. : 8.D. i.i' , (inversely,)
as4-l-4214 ; 2-!:t2803 :: 2-07107 : 1-4(U4G5.

Conse(,uenl!y 4-14214 : 2'!t2sn3 :: 2-02893 : 2-07107;
&c., &c.

2— Fig. 20 {11.) Thr radius A.B. - 10.

A.e. - e.g. = B.V. K.B. = K.S. = S.D. = 4-14214.
A T. - B.Q. - C.Jl. = D.Z. = 2 B.V. = 8-28428.

N.K.= K.ll.= X.B. (thover.sd.sinoofB,.S.)= 2-92893.
Z.Y. = Px.V. = &c., &(;. . = 414214.

B.R. = A.Z. = D.T. = C.Q. = B.e. = 1000 - 414214.
= 5-8578(3 (= Cosine of half-fiuadrantdescrilied with A. g.)

Because the triangle A.X. K., is a part of the similar
triangle A. B.V., N.Jv. : B.V. :: A.X. : A.B. Therefore.—

2-92893 : 4142U :: 7-07107 : 10-000.
2 92893 : 7-07107 :: 4-14:' .4 : 10000.
2-92893 X 10000 ^ 4-14.14 x 707107.

... V

M^-

.^.

The quadrant e.y. ( = g.Z.) : B.C. :: Ac, : A.B.

The arc c.I. ( = g.K.) : B.S. :: A.o. : A.B. :: 1-U214 :
1000.

But tlic proportion c.y. (or g.z.) : B.S. :: 8'28428 :
10-900, is not strictly correct, because e.y., and g.Z., arc
quadrants, and contain twice the proportionate amount of
curvatui'c contained in the half-rpiadrant B.S. (See appendix
to Part Third.)

Draw g.W. the tangent of the half-quadrant g.K. g.W. =
B.V. the tangent of the arc B.U. of 22i doij-rees.

Scholium. — Observe the relationship... e.g. : A.B. ::
4'14214: : 10. And the tangent to the half-quadrant, of which
e.g. is the radius, equals the tangent to the arc of 22^ degrees
of which A.B. is the radius; but the sine (X.K.) of g.K. :
the sine of B.U. :: A.K. : A.U.

Some of Vi.e moi-e immediate relations of these (quantitivo
magnitudes) numbers may be noted, — as for instance : —

U-U2U-10 - 414214.

70710V X 2 - 1414214
7-07 107'' = 50-0000

4-14214 X 2 = 8-28428.

7-07107 - 5000 = 2-07107 (= half the tangent of-j^)

14-14214- = 200-00 1-414214^ = 2-000.
8-28428 X 2-07107 = 17-15732.

4-14214* = 17.15732.
8-28428 X 7-07107 ^ 58-5786.
2-92893 X 2 X 10 = 58-5786.

2-92893' = 8-5786.
7-653667* - 58-5786.

And so on. (The last number represents the duplicated
sine of-j^)

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