• Agterma Sina
R
13
16
ANS & CO
hobia adrian di kezd
A 546123

1
1837
SI
ARTES
LIBRARY
UNIVERSITY OF MICHIGAN
TAS
E PLURIBUS UNUM".
שננו
TUEBUF
SCIENTIA
OF THE
QUAERIS PENINSULAM AMOENAM
CIRCUMSPICE
DADAGDAGANURGARARAUCO
1900
_________
!
i

brg
QA
623
P96
7076
THE
GEOMETRY OF CYCLOIDS
LONDON: PRINTED BY
AND Co.,
NEW-STREET SQUARE
AND PARLIAMENT STREET
SPOTTISWOODE
FIG. 1.
PLATE 1.

C
B
Α΄
B
C
B
b
E
T
K
K
K
7:
A-
E
D'
Ε
D'
E'
d
FIG. 19.
B
B
THE EPICYCLOID.
P
p
P
A
A
B
THE RIGHT CYCLOID.
FIG. 45.
L
B
THE PROLAte Cycloid.
FIG. 46.
I
A
E
יח
a
A
C
+c
q
B
Z
THE CURTATE CYCLOID.
Q
I
'7'
L
P
T
A
E
Ρ
D
E
D
༤
E
d
FIG. 20.
B
Q
C
A
B
P
THE HYPOCYCLOID.
L
L
E
A
L
7
Q
B

B

ά
A
C
h

A TREATISE ON
THE CYCLOID
AND ALL FORMS OF
CYCLOIDAL CURVES
and on the Use of such Curves in dealing with the
MOTIONS OF PLANETS, COMETS, &c.
MATTER PROJECTED FROM THE SUN
Berat
AND OF
AUTHOR OF 'SATURN AND ITS SYSTEM
'THE UNIVERSE OF STARS'
STAR ATLAS'
(
RICHARD Ar PROCTOR, B.A.
=3
SCHOLAR OF ST JOHN'S COLLEGE, CAMBRIDGE
MATHEMATICAL SCHOLAR AND HON. FELLOW OF KING'S COLLEGE, LONDON
BY
thony
1
·
>
'THE MOON
THE SUN
TRANSITS OF VENUS
ESSAYS ON ASTRONOMY' 'THE GNOMONIC
LIBRARY STAR ATLAS' ETC.
>
WITH 161 ILLUSTRATIONS AND MANY EXAMPLES
FOR the USE of STUDENTS in UNIVERSITIES &c.
LONDON
LONGMANS, GREEN, AND CO.
1878
All rights reserved
}
}
•
198
10-4-37
Realasted
PREFACE.
THIS WORK deals primarily with the geometry of
cycloids, curves traced out by a point in a circle roll-
ing on a straight line, or on or within another circle,
and trochoids (or hoop-curves), curves traced out by a
point within or without a circle so rolling.
Although the invention of the cycloid is attributed
to Galileo, it is certain that the family of curves to
which the cycloid belongs had been known, and some
of the properties of such curves investigated, nearly
two thousand years before Galileo's time, if not earlier.
For ancient astronomers explained the motion of the
planets by supposing that each planet travels uniformly
round a circle whose centre travels uniformly round
another circle. By suitably selecting radii for such
circles, and velocities for the uniform motions in them,
every form of epicyclic curve can be obtained, including
the epicycloid and the hypocycloid. When the radius
of the fixed circle is indefinitely enlarged, or, in other
words, when the centre of the moving circle advances
vi
PREFACE.
uniformly in a straight line, the curve traced out by
the moving point becomes a trochoid, and may either
be a prolate, a right, or a curtate cycloid, according as
the velocity of the moving centre is greater, equal, or
less than the velocity of the point around that centre.
Lastly, if the radius of the moving circle is indefinitely
enlarged, so that a straight line is carried uniformly
round a centre while a point travels uniformly along
the line, the curve traced out becomes a spiral of the
family to which belong the spiral of Archimedes and
the involute of the circle.
It is of these curves, which are all included under
the general name epicyclical curves, that I treat in
the present volume, though the cycloid, epicycloid,
hypocycloid, and trochoid are more fully dealt with,
in their geometrical aspect, than the epitrochoidal and
spiral members of the epicyclic family.
Ancient geometers were not very successful in
their attempts to investigate any of these curves. It
is strange indeed to find a mathematician even of
Galileo's force so far foiled by the common cycloid as
to be reduced to the necessity of weighing paper
figures of the curve in order to determine its area.
Pascal dealt more successfully with this and other
problems. Yet he seems to have regarded their rela-
tions as of sufficient difficulty to be selected for his
PREFACE.
vii
famous challenge to mathematicians, to try whether a
priest who had long given up the study of mathematics
was not a match for mathematicians at their own
weapons. The argument, in so far as it was intended
to prove the soundness of Pascal's faith, was feeble
enough. But the failure, or partial failure, of many
who attacked his problems, is noteworthy.
We
find, for instance, that Roberval laboured for six
years over the quadrature of the cycloid, and only
succeeded at last in solving it by the comparatively
clumsy method indicated at p. 199, inventing a new
curve for the purpose. It will be seen that in the
present work this famous problem comes very early
(Prop. III., pp. 5, 6), and is made to depend on the
fundamental (and obvious) relation of the cycloidal
ordinates. The method-which so far as I know is a
new one-is extended to the epicycloid, hypocycloid,
trochoid, epitrochoid, and hypotrochoid. It will be
found that, in all, thirteen distinct methods of solving
the problem geometrically are either given in full or
indicated (seven of these methods being new so far as
I know), while seven independent methods are indi-
cated for determining the area of the epicycloid and
hypocycloid (of which five are new), besides one
method (see footnote, p. 50) derived from the properties
of the cycloid. After the first demonstration of the
viii
PREFACE.
area, however, those methods only are given in full
which involve other useful relations.
The position of the centre of gravity of the
cycloidal arc, and of the cycloidal area, has been fully
dealt with geometrically in Section I. (so far as I know,
for the first time). It seems to me that the treatment
of such problems by geometrical methods usefully in-
troduces the student to the use of analytical methods.
For instance, Prop. XIV. is a geometrical illustration
-in reality, so far as my own mathematical studies
were concerned, a geometrical anticipation*—of the
familiar relation
Su
dv
ገ dx = uv
dx
-So du
dx,
dx
of the Integral Calculus.
Most of the propositions in the first three sections
were established in the same manner as in this volume,
in notebooks which I drew up when at Cambridge;
* I may mention, as a circumstance in which some may perhaps
find encouragement and others a warning, that (owing chiefly to
my liking for geometrical studies) I knew very little of the Diffe-
rential Calculus, and scarcely anything of Astronomy, when I took my
degree. Possibly I owe to this circumstance no small share of the
pleasure derived from the study of these and other mathematical
subjects since. The hurried rush made at our universities over the
domain of mathematics has always seemed to me little calculated
to develope a taste for mathematics, though it may not invariably
destroy it when it already exists. The withdrawal of the mind
during three years from other subjects of greater importance,---
general literature, history, physical science, and so forth,-is still
more pernicious: yet it is practically forced on those who wish for
university distinctions, fellowships, and so forth.
PREFACE.
ix
but the proofs have been simplified and their arrange-
ment altogether modified more than once since then.
In fact anyone who compares the first two sections with
recent papers of mine on the Cycloid, Epicycloid, and
Hypocycloid, in the English Mechanic, will perceive
even that in the interval since those papers were written
the subject-matter has been entirely rearranged.
In defining epicycloids and hypocycloids I have
made a change by which an anomaly existing in the
former treatment of these curves has been removed.
The definitions hitherto used run as follows:
circles being in
The
S epicycloid
hypocycloid
is the curve traced out by a
point on the circumference of a circle which rolls with-
out sliding on a fixed circle in the same plane, the two
{
external
internal S
For this I substitute :-
contact.
The {epicycloid
hypocycloids is the curve traced out by a
point on the circumference of a circle which rolls with-
out sliding on a fixed circle in the same plane, the rolling
circle touching the
f outside
} of the fixed circle.
That the latter is the more correct definition is
proved by the fact that, while the former leads to an
altogether unsymmetrical classification of the resulting
X
PREFACE.
!
curves, the latter leads to a classification perfectly
symmetrical. According to the former every epicy-
cloid is a hypocycloid, but only some hypocycloids are
epicycloids; according to the latter no epicycloid is a
hypocycloid, and no hypocycloid is an epicycloid.
In the fourth section on motion in cycloidal curves
I have adopted a somewhat new method of arranging
the demonstrations to include cycloids, epicy cloids,
and hypocycloids. The proof that the cycloid is the
path of quickest descent is a geometrical presentation
of Bernouilli's analytical demonstration.
The section on Epicyclics was nearly complete
when my attention was directed to De Morgan's fine
article on Trochoidal Curves in the Penny Cyclopædia,
the only complete investigation of any part of my
subject (except a paper by Purkiss on the Cardioid)
of which I have thought it desirable to avail myself.
I rewrote portions of the section for the benefit of
those who may already have studied De Morgan's
essay, deeming it well in such cases to aim at
uniformity of definition, and, as far as possible, of treat-
ment. It will be observed, however, by those who
compare Section V. with De Morgan's essay, that
my treatment of the subject of epicyclics remains
entirely original, and that in some places I do not
adopt his views. For instance, I cannot agree with
PREFACE.
xi
him in regarding the angle of descent as negative under
any circumstances consistent with the definition of
the epicyclic itself. The radius vector indeed ad-
vances and retreats in certain cases; but in every
case it advances on the whole between any apocentre
and the next pericentre. De Morgan has also misin-
terpreted the figures on p. 187, as explained, p. 186.
In two respects this treatise has gained from
my study of De Morgan's essay. In the first place,
I had not originally intended to devote a section
to the equations of cycloidal curves. Secondly, and
chiefly, I was led, by the study of the very valuable
illustrations engraved by Mr. Henry Perigal for Prof.
De Morgan's article, to cancel all the drawings which
I had constructed to illustrate Section V., and to
apply to Mr. Perigal for permission to use his me-
chanically traced curves. A study of Plates II., III.,
and IV., and of other figures illustrating Section V., will
show how much the work has gained by the change.
For figs. 119 to 122, and two of those of Plate IV.,
also mechanically drawn, I am indebted to Mr. Boord.
I may add, to show the value of these illustrations,
Budget of Paradoxes,'
that Prof. De Morgan, in his
says that without Mr. Perigal's diagrams direct from
the lathe,' his article on Trochoidal Curves could not
have been made intelligible.' Yet even those cuts,
C
C
6
xii
PREFACE.
6
and many others added to them in this volume, will
give the reader but inadequate ideas of the immense
number, variety, and beauty of the sets of diagrams
published by Mr. Perigal himself, in his Contributions
to Kinematics.' In these the curves are shown white on
a black background, and hundreds of varieties at once
instructive and ornamental are presented for study
and comparison. Even for the mere patterns thus
formed, and apart from their mathematical interest,
these sets of diagrams possess great value. (See
further the note, pp. 193-195.)
The portions of Section V. relating to planetary
motions, and the concluding section relating to the
graphical use of cycloidal curves for determining the
motion of bodies in elliptical orbits under gravity and
of matter projected from the sun, will be useful, I
trust, to students of astronomy. In some respects
cycloidal curves are even more closely related to
astronomy than the conic sections. If planets and
comets travel approximately in ellipses about the sun,
and moons in ellipses about their primaries, the planets'
paths, relatively to our earth regarded as at rest, are
epicyclic curves; while the cycloid and its companion
curves supply an effective construction for dealing
with Kepler's famous problem relating to the motion
of a body in an ellipse round an orb in the focus
attracting according to the law of gravity.
PREFACE.
xiii
A treatise such as this is rather intended to afford
the means of solving such problems as may be suggested
to the student than of supplying examples. I have,
however, added a collection of about 150 examples.
All except those to which a name is appended are
original. They are, in fact, a selection from among
those which occurred to me as the work proceeded.
Many which I had intended to present as riders have
ultimately been worked into the text among the co-
rollaries and scholia. If these had been included as
examples, the total number would have amounted to
about 300; but it seemed to me better in their case
to indicate the nature of the proof.
LONDON: December, 1877.
On p. 59, line 11, for
,, 129,
17,
""
P.S.-As the last sheets are receiving their latest
corrections for press, I receive, through Mr. Boord's
kindness, the eight figures on p. 256. Of these, figs.
154, 158 represent orthoidal, figs. 155, 159 cuspidate,
and figs. 156, 160 centric epicyclics; while fig. 157 is
a transcentric, and fig. 161 a loop-touching epicyclic.
"
RICH. A. PROCTOR.
Errata.
Area ABD,' read 'Area OBD.'
'D,' read O.'
2
CONTENTS.
THE RIGHT CYCLOID.
SECTION I.
SECTION II.
THE EPICYCLOID AND HYPOCYCLOID
TROCHOIDS
Appendix to Section II.
The Straight Hypocycloid.
Useful General Proposition
The Four-pointed Hypocycloid
The Cardioid
The Bicuspid Epicycloid
The Involute of the Circle
Centre of Gravity of Epicycloidal and Hypocycloidal Arcs
and Areas
SECTION III.
Appendix to Section III.
Elliptical Hypotrochoids
The Trisectrix
The Spiral of Archimedes
Planet's Shadow in Space shown to be spiral
·
•
•
PAGE
1
40
8888
66
68
72
73
79
888888
80
85
92
124
125
127
133
xvi
CONTENTS.
EPICYCLICS
MOTION IN CYCLOIDAL CURVES
SECTION IV.
EXAMPLES.
SECTION V.
Appendix to Section V.
Right Trochoids regarded as Epicyclics
Spiral Epicyclics
Planetary and Lunar Epicyclics.
Forms of Epicyclic Curves
Forms of Right Trochoids
The Companion to the Cycloid
PLATE I.
PLATES II. AND III.
PLATES IV. and V.
PLATE VI.
EQUATIONS TO CYCLOIDAL CURVES
SECTION VI.
SECTION VII.
GRAPHICAL USE OF CYCLOIDAL CURVES TO DETERMINE
(1) the Motion of Planets and Comets
(1) the Motion of Matter projected from the Sun
PLATES.
""
•
"}
""
PAGK
135
148
167
168
169
182
195
197
201
frontispiece
to face each other between pp. 182, 183
192, 193
to face p. 209
209
216
234
THE
GEOMETRY OF CYCLOIDS.
SECTION I.
THE RIGHT CYCLOID.
NOTE. Any curve traced by a point on the circumfer-
ence of a circle which rolls without sliding upon either
a straight line or a circle in the same plane is called a
cycloid, but the term is usually limited to the right
cycloid, and will be so employed throughout this work.
DEFINITIONS.
The right cycloid is the curve traced by a point
on the circumference of a circle which rolls without
sliding upon a fixed straight line in the same plane.
The rolling circle is called the generating circle;
the point on the circumfererce the tracing point.
Similar terms are employed for all the curves dealt
with in this work.
Let AQB (fig. 1, Plate I.) be the rolling circle,
KL the fixed straight line. Let the centre of the
B
2
GEOMETRY OF CYCLOIDS.
rolling circle move along the line 'C c parallel to
KL through C the centre of AQB, in the direction
shown by the arrow. Then it is manifest that at
regular intervals the tracing point will (i.) coincide with
the line KL, as at D' and D (E'q'D' and Eqd being
corresponding positions of the generating circle), and
(ii.) will be at its greatest distance from KL, as at A
(AQB being the corresponding position of the genera-
ting circle), this distance being the diameter of AQB,
so that ACB the diameter through the tracing point is
at right angles to KL. It is clear also from the way
in which the curve is traced out that the parts AP'D'
and APD are similar and equal. Therefore ACB is
called the axis of the cycloidal curve; D'D is the
base; and A the vertex. The points D' and D are
called the cusps. The radius CA drawn to the tracing
point is called the tracing radius, the diameter through
the tracing point the tracing diameter. The radius of
the generating circle may be conveniently represented
by the symbol R. Where the tracing diameter coin-
cides with the axis, the generating circle is said to be
central, and AQB so placed is called the central gene-
rating circle. A diameter to the generating circle
parallel to ACB, that is perpendicular to D'D, is said
to be diametral. The line c' Ce is called the line of
centres.
The complete cycloid consists of an infinite number
of equal cycloidal arcs; but it is often convenient to
speak of the cycloidal arc D'AD as the cycloid.
It is clear that if D'E' and DE be drawn perpen-
THE RIGHT CYCLOID.
3
dicular to D'D, the semi-cycloidal arcs on either side
of D'E' and DE are symmetrical with respect to
these lines. Therefore D'E' and DE may conveniently
be called secondary axes.
A straight line E'AE through A parallel to D'D
manifestly touches the cycloid at A; for there is one
position, and one only, of the generating circle (be-
tween D'E' and DE) which brings the tracing point to
the distance AB from D'D. E'AE is called the
tangent at the vertex.
PROPOSITIONS.
PROP. I.-The base of the cycloid is equal to the
circumference of the generating circle.
This is manifest from the way in which the curve
is traced out; for every point of the generating circle
AQB (fig. 1) is brought successively into rolling con-
tact with the base D'D; so that necessarily
D'D circumference of the circle AQB.
Cor. 1. BD BD semicircular arc AQB.
Cor. 2. Drawing D'E' and DE square to D'D and
c' Cc parallel to D'D,
Area E'D = 2 area AD = 4 area CD
4 rect. under CB, BD
= 4 rect. under CB, arc AQB
4 times area of generating circle AQB.
=
B 2
4
GEOMETRY OF CYCLOIDS.
4
PROP. II.-If through P, a point on the cycloidal arc
APD (fig. 2), the straight line PQM be drawn
parallel to the base BD, cutting the central generating
circle in Q and meeting the axis AB in M; then
QParc AQ.
Let A'PB' be the position of the generating circle
when the tracing point is at P, C' its centre, A'C'B'
diametral, cutting MP in M'. Draw the tracing dia-
meter PC'p. Then MQ M'P; MM' QP; and
=
arc AQ =arc A'P. Now, since PC'p is the tracing
diameter, p is the point which had been at B when the
FIG. 2.
A
IM
C
B
S
Α'
R
QM'
C
B
P
9
P
E

=
tracing point was at A; hence the arc p B' BB',
for every point of p B' has been in rolling contact
with BB'. But
Arc pB'arc A'Parc AQ; and BB'= MM'=QP.
Wherefore,
QParc AQ.
/
Cor. 1. PMarc AQ+MQ.
Cor. 2. Since BD=arc AQB-arc AQ+arc QB,
BD>PM; wherefore the whole arc APD lies on the
left of DE, perpendicular to BD.
THE RIGHT CYCLOID.
5
10
Cor. 3. Let MP produced meet DE in m. Then
Pm-Mm-PM-arc AQB-arc AQ-MQ
=arc QB-MQ.
Cor. 4. Arc A'P=BB'; and arc PB'B'D.
Cor. 5. If through P', a point on the arc PD,
P'qQ' be drawn parallel to BD, meeting AQB in Q
and cutting A'PB' in q; then Q'P'arc A'q, and
QParc A'P; wherefore
S s = QP = A'P; and SR= arc A's;
wherefore Rs = arc s P = arc SQ.
qP' (=Q'P' — Q'q=Q'P' — QP) = arc A'q-arc A'P;
that is,
9 P'=arc P q.
Cor. 6. If through R, a point on the arc AP, s RS
parallel to BD meet the arcs AQB, A'PB' in S and s,
then
PROP. III. The area D'AD (fig. 1, Plate I.) between
the cycloid and its base is equal to three times the area
of the generating circle.
A, B, D, E, C, &c. (fig. 3), representing the same
points as in the preceding proposition; take CL=CL'
on AB, and draw LP 1, L'P'7' parallel to BD, cutting
the cycloid in P and P', and the central generating
circle in Q and Q', respectively. Complete the ele-
mentary rectangles PN, P'N', Lk, of equal width,
(PM=P'M'). Then
=
QP =arc AQ, and Q'P' arc AQ' =arc BQ;
therefore QP+Q'P' semicircle AQB Ll; and
the two rectangles NP and N'P' are together equal
=
6
GEOMETRY OF CYCLOIDS.
to the rectangle Lk.
gular elements as NP
limit area AQBDP =
(Prop. I. Cor. 2.)
Hence the area between the cycloid and its base
(= 2AQBDP+ circle AQB) = three times the area
of the generating circle. Q.E.D.
Another proof.-Let AP"D be a cycloidal arc
having A as cusp, D as vertex, and DE as axis. Let
N
A
K
C
B
P
m
T
Taking all such pairs of rectan-
and N'P', it follows that in the
rectangle CE = circle AQB.
FIG. 3.
N
N
P
H
盒
​I
=
1
M
D.
P
7
=
E

τ
k
A
IL cut AP"D in P" and be produced to meet the
circle AQB in Q". Then
τ'
=
LP are AQ+LQ; and LP" arc AQ - LQ
(Prop. II. Cor. 2). Wherefore
P"P=LP-LP"=2LQ=Q"Q; and the elementary
area Pm=the elementary area Q″N.
Taking all such elementary rectangles, we have in
the limit area AP"DP circle AQB rectangle CE.
Hence, taking these equals from the rectangle BE, it
follows that the equal areas ABDP" and APDE are
together equal to the rectangle CD, that is, to the
THE RIGHT CYCLOID.
7
Therefore AP"DB = the semicircle
circle AQB.
AQB; APDB = three times the semicircle AQB;
and the area between the cycloid and the base = three
times the generating circle.
Cor. 1. Rectangle Al-area AQP+ area BQ'P'D.
Cor. 2. Rectangle Cl=area QPP'Q'.
Cor. 3. If AE and BD be bisected in H and I,
and HI cut PQ and P'Q' in h and i; then if, as in
the figure, P and P' are on the same side of HI,
Ph+Pi Ph+P"h
P"P =
=
Q″Q=2LQ.
If P falls between AB and HI, as at p, then, com-
pleting the construction indicated by the dotted lines,
p'ï' — ph' — p″'h' —pk=p"p=gq=2 qj»
=
That is, if two points are taken on the cycloidal arc
equidistant from Cc, the sum or difference of the per-
pendiculars from these points upon HI will be equal
to the chord of the generating circle formed by either
perpendicular produced, according as the points on the
cycloid are on the same or on opposite sides of HI..
This relation will be found useful hereafter in deter-
mining the centre of gravity of the cycloidal area.
Cor. 4. When the tracing point is at P, the gene-
rating circle passes through P"; for its chord through
P parallel to AE=QQ″=PP”.
Cor. 5. Area AQ"Q area AP"P; and area
AQ"P" area AQP. The latter relation, established
independently (by showing that QP = Q″P”), leads
to a third demonstration of the area.
=
8
GEOMETRY OF CYCLOIDS.
:
PROP. IV.—If P (fig. 4) is a point on the cycloidal
arc APD, APB' the generating circle when the
tracing point is at P, A' C'B' diametral, then PB' is
the normal and A'P is the tangent to the cycloid at
the point P.
Since, when the tracing point is at P, the generating
circle A'PB' is turning round the point B', the direc-
tion of the motion of the tracing point at P must be
FIG. 4.
M
N
C
B
Α΄
Β΄
I
P
Ag
G
E

D
at right angles to B'P; wherefore PB' is the normal
and A'P is the tangent at the point P.
Another demonstration.-The objection may be
raised against the preceding proof, that, by the same
reasoning, B' would be proved to be the centre of curva-
ture at P, which is not the case. Although the objection
is not really valid, an independent proof may conve-
niently be added.
Take P'a point near to P, and draw PQM, P'Q'N
parallel to BD, cutting AQB in Q and Q', and P'Q'N
cutting A'PB' in q. Join PC'. Then q P' arc Pq
(Prop. II. Cor. 4), and ultimately PqP' is an isosceles
THE RIGHT CYCLOID.
9
triangle, whose equal sides Pq and qP' are respectively
perp. to the equal sides C'P and C'B' of the isosceles
triangle PC'B'; wherefore the third side PP' is perp.
to the third side PB'.* That is, PB' is the normal at
P, and therefore PA' the perp. to PB' is the tangent
at P.
Cor. 1. If Pn be drawn perp. to P'N, then the
figure PP'n is in the limit similar to the triangles
A´B´P, A'Pm, PB'm (m being the point in which
A'B' and PM intersect).
Cor. 2. If B'P cut P'N in 7, the triangle /P'P is
similar to the four triangles named in Cor. 1.
Cor. 3. Triangles Pql, Pq P' are similar respec-
tively to triangles PC'A' and PC'B' ; and 1q = q P'.
Cor. 4. AQ is parallel to the tangent at P.
Cor. 5. If AQ prod. meet P'N in r, QQ' ulti-
mately = Q'r.
SCHOL. A tangent may be drawn to the cycloid
from any point on the curve. For if we draw PQ
parallel to BD, the tangent PA' is parallel to AQ.
To draw a tangent from any point A' on the tangent
at vertex, we draw A'B' perp. to base, and the semi-
circle A'PB' on ADB' intersects APD in the point
P such that A'P is tangent to APD.
M
Saja.
*Thus, let the triangle Pq P' be turned in its own plane round
the point P till P coincides with PC'—that is, through one right
angle; the other sides q P' and PP' will also have been turned
through a right angle, therefore q P' will be parallel to C′ B', and
4 P' being equal to q P, P' will fall on B'P (for any parallel to C'B'
will cut off an isosceles triangle from B'PC'); hence B'PP' is the
angle through which PP' has been turned, and is therefore a right
angle.
10
GEOMETRY OF CYCLOIDS.
PROP. V.-If PQ (fig. 5), parallel to the base of cy-
cloid APD, and above the line of centres Cc, meets
the central generating circle in Q, and QN, PM are
perpendicular to Cc,
Area Ah QP+ rect. QM=rect. CF
(F being the point in which NQ produced meets the
tangent at the vertex AT).
If P'Q' be a parallel to the base below the line of centres
Q'L, P'M', perpendicular to Cc,
Area Ah Q'P'
rect. Q'M' = rect. CF'
(F' being the point in which LQ produced meets the
base BD).
Take p a point near to P, and let pn perp. to QN
cut arc AQQ' in q; join AQ and produce to meet pn
FIG. 5.
A
B
h
-

FƒT
So
n
N LKC
P
K M
P
m
R Mm'e
D.
P
p
D
SFT
in r; draw fq L, rK, pm perp. to Cc, and join C q.
Then in passing from P to p,area Ah QP+rect. QM
is increased by Ppm M and diminished by Q9LN, or in
the limit, increased by rect. Mp or Nr (since Qrp P
THE RIGHT CYCLOID.
11
is a parallelogram, Prop. IV. Cor. 4) and diminished by
rect. Ng; wherefore total increase rect. Lr. But
ng : qQ (=qr, Prop. IV. Cor. 5) :: qL : Cq(=NF),
rect. under nq, NF = rect. under qr, qL;
that is, rect. Nƒ= rect. Lr,
or incrt. of rect. CF incrt. of (area AhQP+ rect. QM).
But these areas start together from nothing, at A,
... rect. AhQP + rect. QM =rect. CF.
..
=
Cor. 1. Area AQC'RP=square CT=square CT',
TCT' being the tangent to AC'B at C' on the line of
centres.
Again, making a similar construction for the second
case (for convenience in figure Q'q' is so taken that
Qq' and q Q' are perp. to Cc), we have ultimately
decrement of area (Ah Q'P' - Q'M') = Lq' + P'm'
rect. Lyrect. n'K' (ultimately) = rect. Nr.
But since n'q': Q' d′ (= q'r') :: q′N : C q′ (
C q′ ( = Nƒ˜),
rect. under n'q, Nf' rect. under q'r', q' N;
=
that is,
rect. NF' = rect. Nr', or
decrt. of rect. CF = decrt. of area (AhQ'P'-QM').
But these areas begin together from the equal areas
AQC'R and square CT',
... area AhQ'P'
S
rect. Q'M' =rect. CF'.
Cor. 2. Area AC'BDR=rect. CBD c-generating
circle, so that we have here a new demonstration of
the area.
12
GEOMETRY OF CYCLOIDS.
:
PROP. VI.—lf from P a point on the cycloid APD
(fig. 6) PQ drawn parallel to the base, meets the
generating circle in Q,
arc AP = 2 chord AQ.
With the same construction as in Prop. IV., join
AQ and B'q; produce B'q to meet PP' in k; and
draw C'K perpendicular to B'P. Then ultimately,
FIG. 6. (Join Aʼq.)

M
L
2
B
T
Β΄
K
a
=
P
d b
R
qk is perpendicular to PP', and the triangle PqP' is
isosceles ;
... PP' 2P ultimately.
But PP' is ultimately the increment of the cycloidal
arc AP; and Pk is ultimately the increment of the
chord A'P (for A'q= A'k ultimately).
A´k ultimately). Hence the
increment of the cycloidal arc AP = twice the incre-
ment of the chord A'P or of the chord AQ. There-
fore, since the arc and chord begin together at A,
Arc AP = 2 chord AQ.
Cor. 1. Arc APD = 2 AB = 4R, and the entire
cycloidal arc from cusp to cusp = 4AB = 8R.
THE RIGHT CYCLOID.
13
Cor. 2. Since the square on AQ = rect. AB.AM,
sq. on st. line equal to arc AP = 4 rect. AB. AM,
and we have,
Arc AP 2/2R. ✓AM,
=
that is,
Arc AP
AM.
Cor. 3. Arc AP: arc PD :: AL: LB.
PROP. VII. PROB.-To divide the arc of a cycloid
into parts which shall be in any given ratio.
Let a straight line ab (fig. 6) be divided into any
parts in the points c and d: it is required to divide the
arc APRD in the same ratio.
Divide AB in L and I so that
AL: LI:7B: ac: cd: db.
With centre A and radius AL and Al, describe circular
arcs LQ, Ir, meeting the semicircle AQB in Q and r.
Through Q, r, draw QP, rR, parallel to BD. Then
2AQ 2AL; and arc AR = 2A1.
Arc AP
=
Therefore
Arc PR = 2L7; and similarly arc RD = 27B.
Therefore
Arc AP: arc PR :: arc RD :: AL:LI: IB :: ac:cd: db;
or the arc APD has been divided in the points P and
R in the required ratio.
Similarly may the arc APD be divided into any
number of parts, bearing to each other any given ratios.
14.
GEOMETRY OF CYCLOIDS.
PROP. VIII. With the construction of Prop. IV.
Area APB'B: sectorial area A'B' Ph
:: area PB'D: segment PFB' :: 3 : 1.
Let a P'b (fig. 7) be the position of the tracing
circle when the tracing point is at P' near to P, on the
A
B
Sp
Á a
Q M
FIG. 7. (Join a P'.)
K
IN'
c
h
L
h
GTA
E

D
side remote from A; a cb diametral.
Join b P', draw
P'ql parallel to BD meeting A'PB' in q and PB′ in 1,
join 7 B', which is parallel to b P', because qP' =
P q = B'b. Then ultimately P'q ql (Prop. IV.
Cor. 3), wherefore parallelogram qb = twice the tri-
angle 1q B' and trapezium P'b B' 3 times the
triangle 1q B': that is, ultimately (when the triangle
1 PP' vanishes compared with IP'b B'), the elementary
1
area
B′PP′ 6 = 3 times the elementary area PB'q
= 3 (area A'B'q h - area A'B'P h)
= 3 (area a bP' area A'B'P).
Thus the increment of the area ABB'P = 3 times the
THE RIGHT CYCLOID.
15
increment of the area A'B'P, and the decrement of
area PB'D = 3 times the decrement of the area PFB'.
But the areas ABB′P and A'B'P commence together,
and the areas PB'D and PFB' end together, as P
passes from A to D. Hence
ABB'P 3 times sectorial area A'B'P h.
3 times the segment PFB'
Area PB'D
and
=
Area APB'B: sectorial area A'B'P h
:: area PB'D: segment PFB' :: 3: 1.
Cor. 1. Area PFB'D = 2 segment PFB'. This
is easily proved independently. For any elementary
parallelograms ff" and FF' (having sides parallel to
BD), are manifestly equal; wherefore area q Fb F
parallelogram qb twice triangle B'ql (ulti-
mately) twice the decrement of segment b F'P'.
Cor. 2. Area AQBB'P (BQ straight) = 2 sec-
torial area AQB.
Cor. 3. Area Qs BDP = 2 seg. QsB+ par. PB
= 2 seg. QsBrect. BM'.
= =
SCHOL.-Prop. VIII. affords another proof of the
relation established in Prop. III. The first corollary,
established independently, gives another proof.
16
GEOMETRY OF CYCLOIDS.
2-2
Prop. IX.—With the same construction as in the pre
ceding propositions.
Area APA
segment A'hP.
Join PA', q A', and P'a. Then A'PP' is ulti-
mately a diameter of the. parallelogram A'aP'q, and
the ultimate triangle A'PP'a is equal to the triangle
A'PP'q, or in the limit to the triangle A'Pq. But
A'PP'a is the increment of the area APA', and A'Pq
is the increment of the segment A'h P. Since these
areas then begin together and have constantly equal
increments, they are constantly equal. Therefore
Area APA' segment A'hP.
Cor. 1. Draw PL, PM'M perp. to AE, AB respec-
tively, PM intersecting AB in M'. To each of the equal
areas APA' and A'h P add the equal triangles A'PL
and A'MP. Then the area APL = area A'h PM'
area AQM. This may be proved independently. For
drawing P'K, P'N' perp. to AE, A'B', we see that
A'PP' is ultimately a diameter of the rectangle N'K,
and therefore the rectangles PK and PN', being com-
plements to rectangles about the diameter, are equal:
or ultimately the increment of the area APL = incre-
ment of the area A'h PM'; wherefore, since these areas
begin together, area APL= area A'h PM' area AQM.
Cor. 2. Area AQP rect. ML-2 area AMQ.
Cor. 3. Area Qs BDP = circ. AQB area AQP
= circle AQB rect. ML + 2 area AMQ
= 2 (semicircle AQB + area AMQ) — rect. ML.
Gam
=
1
THE RIGHT CYCLOID.
17
=
Cor. 4. Area AA'h P 2 area AA'P=2 segment
A'h P. This may be proved independently, in the
same way as Cor. 1, Prop. VIII. Area A'a P'qh,
ultimately equal to the area A'a P'Ph, is shown to be
equal to the area of the parallelogram A'aP'q, that is,
to twice the area A'PP'a or A'PP'q (the ultimate in-
crements of AA'P, A' h P, respectively).
SCHOL.-Prop. IX. and Cor. 1 and 4 (established
independently) afford three new demonstrations of the
area of the cycloid. For they severally show that
area APDE = semicircle DQ'E, on DE as diameter;
and since BE twice the generating circle, the area
APDB 3 times the semicircle AQB.
It will be noticed that the area AEQ'DP = area
As BDP. This, which may easily be proved inde-
pendently, affords yet another proof of the area of the
cycloid. Thus let APD, AP'D (fig. 8) be cycloidal
FIG. 8.
P
A
B
a
jand
Q
Á α
B' b
P
E

arcs, placed as in Prop. III.; A'PB'P' and ap·b pº
adjacent positions of the tracing circle. Then, Prop.
III. Cor. 4, P'P and p'p are both parallel to BD.
Hence ultimately area A'a p P = area A'ap' P'; but
C
18
GEOMETRY OF CYCLOIDS.
=
these are the increments of the areas AA'P, and AA'P',
which commence together. Hence area AA'P area
AA'P', wherever P and P' may be. Wherefore
(taking P to D) area AEQ'DP = area AE qDP'
area AQBDP. Therefore the arc APD divides the
area AEQ'DBQ into two equal parts.
But area
AEQ'DBQ = area AEDB = twice the generating
circle. Hence area AQBDP area APDQ'E = the
generating circle; area APDB = 3 the semicircle
AQB; and area AEDP semicircle AQB.
PROP. X.-The radius of curvature at P (fig. 9) is
equal to twice the normal PB'.
With so much of the construction of fig. 7 as is con-
FIG. 9. (For O′, O read o, o'; and join o a'.)
A
A'
E
B
d
=

for
a"
น
To
e
D
tained in fig. 9, produce P'b, which is parallel to q B',
THE RIGHT CYCLOID.
19
to meet PB' produced in o'. Then since ultimately
IP' = 2lq; lo' ultimately =2lB'. So that if the
normals at the adjacent points P and P', intersect ulti-
mately (when P' moves up to P) in o (which, there-
fore, is the centre of curvature at P),
Rad. of curvature P o = 2 normal PB'.
Cor. The radius of curvature diminishes from the
vertex, where it has its maximum length, to the cusp,
where the radius vanishes or the curvature becomes
infinite.
PROP. XI. The evolute of the cycloid APD (fig. 9)
is an equal cycloid Do'd, having its vertex at D, and
its cusp d on AB produced to d so that Bd AB.
=
Complete the rectangle DB de, produce A'B' to
a', and join o a'. Then in the triangles A'B'P and
a'B'o the sides A'B', B'P, are equal to the sides a'B',
B'o, each to each, and enclose equal angles; therefore,
the triangles are equal in all respects, and the angle
a′o B' (― the angle B'PA') is a right angle. Hence a
circle described on B'a' as diameter will pass through
o. Again, in the equal circles A'B'P and a'B'o, the
angles A'B'P and a'B'o at the circumference are
equal. Therefore the arc o d = the arc PA'
= BB'
(Prop. II. Cor. 4) da'. Wherefore o is a point on
a cycloid having de for base, a cusp at d, and B'o d' as
tracing circle. Since de BD = arc B'o a', De is
the axis and D is the vertex of the evolute cycloid.
=
Cor. oP = 20 B′ = arc o D (Prop. VI.); so that,
© 2
20
GEOMETRY OF CYCLOIDS.
if a string coinciding with the arc do D and fastened
at d be unwrapped from this arc, its extremity will
always lie on the cycloid APD, which may, therefore, be
traced out in this way as the involute of the arc do D.
PROP. XII.—If APD (fig. 9) be a semi-cycloidal arc,
do Dits evolute, and o B'P the radius of curvature
at any point P on APD, cutting the base BD in
B', then the area APB'B = three times the area
d BB'o.
If P'o' be a contiguous radius of curvature cutting
BD in b, and P'l parallel to BD meet PB' in 7; then
in the limit ol 2 o B', and therefore the area of the
ultimate triangle ol P' 4 times the area of the ulti-
mate triangle o B'b; or ultimately the area B'l Pb
= 3
times the area o B'b. But these areas are the element-
ary increments of the areas APB'B and d BB'o, which
begin together from AB d. Wherefore the area APB'B
- 3 times the area d BB'o.
Cor. 1. Area ABD = 3 times area d BD = 3 times
area AED rect. BE=3 times the generating circle.
We have here another demonstration of the area.
Cor. 2. Area o B'D area B'DP = segm. Pq B'
(Prop. VIII.). This may be proved independently;
for triangle o B' b = triangle B'lq = (ultimately) tri-
angle B'Pq; but triangles o B'b, B'P q, are decre-
ments of area o B'D and segment Pq B' which end
together at D; .. o B'D = seg. Pq B'.
1
Hence, d B'D = generating circle. We have
here, then, yet another demonstration of the area.
THE RIGHT CYCLOID.
21
PROP. XIII.-If G (fig. 10) is the centre of gravity
of the cycloidal arc APD, then GK, perp. to AE
(the tangent at the vertex A) = AB.
A
Let PP' be an element of the arc APD and let
PM, P'N perp. to AB intersect the semicircle AQB
in Q and Q. Join AQ cutting MQ in n. Then
ultimately PP' is parallel and equal to n Q' (Prop. IV.).
M
2 3
IS
N
C
B
FIG. 10.
K
m
2
E

Now, representing the mass of element PP' by its
length, the moment of PP' about AE ultimately
= PP'. AN = n Q'. AN
= MN. AQ'
(since n Q': MN :: AQ': AN)
and may be represented therefore by the elementary
rectangle MN q'm, of which the side Ng AQ'.
Thus the moment of the arc APD about AE may
be represented by the area A q'b B obtained by draw-
ing the curve A q'b through all the points obtained as
q' was. But since square of Ng = square of AQ'
=rect. under AB, AM; Aqb is part of a parabola
=
22
GEOMETRY OF CYCLOIDS.
having A as vertex, AB as axis and parameter (focus
at S, such that ASAB). Therefore area AB b
AB. Bb; and moment of arc APD about AE
2 AB.Bb arc APD. B b
= arc APD. KG)
3
3
B b = AB.
MN. AQ.
KG =
Cor. 1. Moment of PP' about AE
Cor. 2. Still representing the mass of arc by its
length, that is, taking for unit of mass the mass of one
unit of length of the arc,
Moment of arc APD about AE = } (AB)².
Cor. 3. Momt. of AP about AE is represented by
area AM 7
AB. AM ÷ AM¹. AB¹.
do 202
9 =
AM
ep
or
PROP. XIV.-If G (fig. 11) is the centre of gravity of
the cycloidal arc APD, then GL perp. to the axis
AB = BD - AB.
A
With same construction as in Prop. XII.,
FIG. 11. (AQ' and NQ intersect in ".)
a'
N
M
L
B
a
P
22
=
k
G
=

To
E
H
momt. of PP' abt. AB=PN. PP'=2PN. inct. of AQ
THE RIGHT CYCLOID.
23
(Prop. VI.). Draw P a, P'a' parallel to AB and equal,
respectively, to AQ, AQ'; complete the rectangles Na,
Ma'; and produce a P to meet MP' in k. Also join
BQ and let NP, MP' prod. meet a cycloidal arc BE
having B as vertex and E as cusp in p and p'. Then,
rect. Ma ultimately exceeds rect. Na by
rect. under PN, (P'a'—P a) + rect. under a´P'. k P'.
That is,
inct. of rect. Na
PN. inct. of AQ + AQ'. k P'
momt. of PP' about AB+BQ'. MN
(since AQ': BQ' :: k P` : k P′)
momt. of PP' about AB+momt. of p p' about BD
(Prop. XIII. Cor. 1).
or
Wherefore, taking all increments from A, where rect.
Na has no area, to D, where N a=rect. AD, we have
2 rect. AD=momt. of arc APD about AB
+ 2 momt. of B p E about BD;
that is,
DE
arc APD. GL = 2 AB. BD – 2 are Bp E.
3
2AB. GL = 2AB. BD AB. DE;
... GL = BD – } AB.
1944942
-
Cor. Draw GH perp. to DE.
= BD = GL + GH. Therefore GH
;
Then GL + AB
AB.
24
GEOMETRY OF CYCLOIDS.
PROP. XV.-If G (fig. 11) is the centre of gravity of
the cycloidal arc APD, and GH, GJ be drawn perp.
to DE and BD, JH is a square, whose sides are
each equal to AB.
=
2
From Prop. XIII. EH = AB; .. DH AB.
≈ &
From Prop. XIV. Cor., GH AB. Therefore,
the rectangle JG is a square having each of its sides
= AB.
PROP. XVI.-Iƒ G' (fig. 12) is the centre of gravity
of the area APDE, then G'K perp. to AE=AB.
Take PP' an element of the arc APD; draw P'n
perp. to AE, and PQM, P'Q'N perp. to AB, inter-
½
A
Σ
N
C+
B
FIG. 12.
1
K
secting AQB in Q and Q. Complete rectangles Pn,
QN. Then from Prop. IX. Cor. 1,
rect. Pn = rect. QN.
Now momt. of element Pn about AE, ultimately
P' n. rect. P n
E

AN. rect. NQ
momt. of NQ about AE.
THE RIGHT CYCLOID.
25
Taking all such elements, we have
Momt. of area APDE about AE
AQB about AE.
That is, G'K. area APDE
But,
M
= K
K
PROP. XVII.—If G' (fig. 13) is the centre of gravity
of the area APDE, HI parallel to AB through H
the bisection of AE, and G'L perp. to HI, then
G'LAB AB: 3B1, or G'L= AB.
с
N
Take elements MN and M'N' equal to each other
and equidistant from A and B respectively; draw
M
B
area APDE = area AQB;
.. G'K =
q
FIG. 13.
H
L
P
AC. area AQB.
ra
2.
momt. of area
AC = AB.
ກ.
.
4
3π
n' E

R
MQP, NP', N'R, and M'q R' parallel to BD, meet-
ing APD in P, P', R and R' (Q and q being points on
circle AQB). Draw P'n and R'n' perp. to AE, and
complete the elementary rectangles Pn, Rn', QN and
q N'. These four rectangles are equal. Now, sum of
·
26
GEOMETRY OF CYCLOIDS.
moments of Pn, Rn' about HI
= Hn. rect. Pn + Hn' rect. R n'
(H n + H n') rect. QN
2QM. rect. QN (Prop. III. Cor. 3)
= 2 moment of rect. QN about AB.
[This relation holds whether Pn and Rn' lie on
the same side as in fig. 13 or on opposite sides of HI;
for in the latter case, the moments being in opposite
directions, their difference is the effective moment, and
instead of (H n' + Hn) rect. QN, we get (H n' — H n)
rect. QN; but when n' and n are on opposite sides of
HI, Hn' — H n = 2QM. Prop. III. Cor. 3.]
-
Wherefore taking all the elements such as MN,
M'N', from A and B to the centre C, we get
Momt. of area APDE about HI= 2 momt. of semi-
circle AQB about AB;
that is, LG'. area APDE 2 Cg. area AQB
(g being the centre of gravity of the semicircle AQB
and Cg perp. to AB). And since area APDE=area
AQB, LG′ = 2C
9.
But we know that
Cg AB AB: 3 arc AQB* (= 3BD);
: :
(or Cg
2AB
3π
wherefore LG: AB: 2 AB 3 BD :: AB: 3BI;
}
(or LG'₁ = 44B)
* If the reader is unfamiliar with this property, he may esta-
blish it thus:-First show that projection of any element of semi-
circle on tangent at the middle point of the arc has a moment about
THE RIGHT CYCLOID.
27
PROP. XVIII.—If G and G' (fig. 14) are the centres
of gravity of the areas APDB and APDE re-
spectively, O the centre of gravity of the rectangle
BE (that is the point in which HI, drawn as in last
proposition, and CC, the line of centres, bisect each
other), and GK, G'L are drawn perp. to HI, then
·AB=
4
OK=1AB=1AC; and GK=LG' =
2
Уп
A
C
M
F B
Since O is the centre of gravity of the rectangle
BE, that is, of the area APDB + the area APDE, the
FIG. 14.
H
L
P
3π
Cg: 2r: 2r: 3 arc of semicircle.
K
E
C´
D
8AC
16

That is to say Cg: 4r:: 1: 3″ :: 7 : 3πï, or
moments of APDB and APDE about COC' are
equal; that is,
diameter equal to the moment of the element; therefore moment
of semicircular arc, or π rad. x dist. of C.G from diameter = diameter
× rad.; that is distance of C.G from diam. = diameter ÷ ñ.
Now a
semicircular area may be supposed divided into an infinite number of
equal small triangles having centre for apex, and each triangle may
be supposed collected at its C.G. at a distance from centre
Hence C.G. of semicircular area lies at a dist. from diameter
2 diameter
rad.
=
28
GEOMETRY OF CYCLOIDS.
Similarly,
3 area APDE. OK
or
4
GK = 3LG':
LG'= AB
9TT
Cor. 1. Since LG': AB:: AB: 3 BI
(Prop. XVII.),
= area APDE. OL;
OK = {OL
OLAB =¿ AC.
1
2
GK: AB:: AB: 9 BI.
Cor. 2. G, O, and G' lie in a straight line, and
OG'=30G.
Cor. 3. Since moment of area AQBD about BD
= (moment of ABDP-moment of AQB) about BD
TAC2 3AC
4
(.3ACAC)
2
7. AC2; it follows
that the C.G. of area AQBD lies at a distance = AC
from BD.
SCHOL.-The position of G may be thus ob-
that is,
or
tained:-
Take OKAC. Also, take BM = AB ;
join MI, and let MF perp. to MI intersect DB pro-
duced in F: draw KG perp. to OI and equal to BF.
Then G is the centre of gravity of the area APDB.
For OK = AB; and
1 2
M
8 AC
9π
KG (=FB): BM ::BM : BI;
KG: AB::AB: BI:: AB: 3 BI,
1
3
KG : AB :: AB : 9 BI.
THE RIGHT CYCLOID.
29
PROP. XIX.-If from G (fig. 15), the centre of gravity
of semi-cycloidal arc APD, GL be drawn perp. to
AB, and Gl making with AB produced the angle
GIA = the angle ADB; then the surface gene-
rated by the revolution of the arc APD about the
axis AB is equal to eight times the rectangle having
sides equal to AB and Ll.
By Guldinus's First Property (see note following
this Proposition), the surface generated by the revolu-
FIG. 15.
A
B
7%
=
P
K
E

H
D
A
tion of APD about AB = rect. under straight lines
equal to APD and circumference of circle of radius
LG. But APD-2AB, and since GL is similar to
ABD, and BD the circumference of circle of
radius AB, it follows that Ll circumference of
circle of radius LG. Hence the surface produced by
the revolution of APD about AB
=rect. under 2 AB and 4 L 7
= 8 times the rectangle under AB and L 7.
30
GEOMETRY OF CYCLOIDS.
Cor. 1. In revolving round AB through half a right
angle, APD generates a surface equal to rectangle
under AB and L 1.
Cor. 2. Since GL = BD-AB (Prop. XIV.),
L1 = (BD —÷AB); and the surface generated by re-
volution of APD about AB 4AB (BD-AB) T
=8AC (T. AC-‡AC) π = π (π —‡) (AC)²,
2
= 8 (7) generating circle.
NOTE.—Guldinus's properties, usually demonstrated by the in-
tegral calculus, are essentially geometrical. His First Property
may be stated and established as follows:-
If a plane curve revolve through any angle a about an axis in its
own plane, the curve lying entirely on one side of the axis, the area
generated by the curve is equal to a rectangle having its adjacent sides
equal in length to the curve and to the are described by the centre of
gravity of the curve, in revolving about the axis through the angle a.
Let APB (fig. 16) be a curve lying in the same plane as OX, and
entirely on one side of OX, and let it revolve around OX through
FIG. 16.

M
====
A
X
B
**
an angle a to the position a p b. Then PP', an element of the arc
APB, generates a corical shred of constant breadth PP' and of area
ultimately PP'. arc Pp PP'. PM . a = a. moment of PP' about
OX. Taking all the elementary arcs of APB in this way, the sur-
face generated by the arc APB = a. moment of arc APB about OX
= a. GN. arc APB; (G being the centre of gravity of the arc APB,
and GN perp. to OX).
a, and the area of the
Or, if length of curve APB L, GN
surface generated A, then
A = L. ù . *
THE RIGHT CYCLOID.
31
If the axis intersect the curve, then the two portions of the
curve lying on either side of the axis must be separately dealt with.
It is easily seen that if the curve APB is not plane, or if (whether
plane or not) it is not in the same plane as OX, a similar property
may be established. Let the curve be carried once round OX, and
let a plane through OX intersect the surface thus generated in a
curve A'P'B' (any parts of A'P'B' through which more than one part
of APB may have passed being counted twice or thrice or so many
times as they may have been traversed in one circuit of APB). Let
L' be the length of A'P'B' (thus estimated); G' its centre of gravity
(correspondingly estimating the weight of its various parts), and ū'
the distance of G' from OX. Then the surface generated by the
revolution of APB round OX through the angle a =
L'. '. a (any
part of the generated surface traversed more than once by the
generating curve being counted as often as it has been so traversed).
Again, if APB so move as to generate a cylindrical surface either
right or oblique, and two planes through OX intersect the surface
thus generated, the portion of this surface intercepted between
those planes may be thus obtained :-through OX take a plane perp.
to the axis of the cylindrical surface and intersecting that surface
in a curve A'P'B' of length L' and having centre of gravity G'at
distance a' from OX; let the portion of a straight line through G'
parallel to the axis of the cylindrical surface, intercepted between
the boundary planes = h; then the surface intercepted L'. a'. h.
The proofs of this and the preceding extensions of Guldinus's
first property depend on the same principle as the proof of 'the pro-
perty itself given above. In fact, the student who has grasped the
principle of that proof will perceive the extensions to be little more
than corollaries.
-
It may be of use to note that the two extensions require two
lemmas. The first requires this lemma :-If an element of arc PP'
be projected orthogonally on a plane through OX and P into the
elementary arc Pp, then PP' and Pp in rotating through any angle
round OX generate equal surfaces. This is obvious, since they
generate equal elementary surfaces in rotating through an elemen-
tary angle round OX. The second extension requires this lemma :-
If two planes through OX cut two parallel lines Pp, P'p' in P, P'
and p,p′, the lines PP' and pp' being elementary, then two other
planes through OX near to these last cutting Pp and P'p' in R, R′
and r, r', such that PR=pr, intercept equal areas PRR'P' and prr'p'.
These areas are in fact ultimately parallelograms on equal bases and
between the same parallels.
32
GEOMETRY OF CYCLOIDS.
PROP. XX.-If from G (fig. 15), the centre of gravity
of the semi-cycloidal arc APD, GH be drawn perp. to
ED, and Gh making with ED produced the angle
Gh H= angle ABD, then the surface generated by
the revolution of the arc APD about ED as an
axis is equal to eight times the rectangle under AB
and Hh.
The demonstration is in all respects similar to
that of Prop. XIX.
Cor. 1. In revolving through half a right angle,
APD generates a surface equal to the rectangle under
AB and H h.
3
Cor. 2. Since GH = AB (Prop. XIV. Cor.),
TAB; and the surface generated by the revo-
Hh=
3
lution of APD about ED=8.AB.
(AB)2
32π
3
AB
3
(AC)² = 32. generating circle.
= 14
8π
3
PROP. XXI.—If from G (fig. 15), the centre of gravity
of the semi-cycloidal arc APD, GK be drawn perp.
to AE, and G k parallel to AD meet AE in k, then
the surface generated by the revolution of the arc
APD about AE as axis=eight times the rectangle
under AB and Kk.
The demonstration is similar to that of Prop. XIX.
Cor. 1. In revolving through half a right angle
THE RIGHT CYCLOID.
33
FRO
APD generates a surface equal to the rectangle under
AB and Kk.
Cor. 2. Since GK = AB (Prop. XIV.), K k =
AB; and the surface generated by the revolution of
6
APD about AE=8πAB. AB_4π AB2= (AC)²
6 3
generating circle.
16
3
•
APD about BD
PROP. XXIII.—If from G (fig. 15), the centre of
gravity of semi-cyclvidal arc APD, GJ be drawn
perp. to BD, and Gj parallel to AD to meet BD
produced in j, then the surface produced by the revo-
lution of the arc APD about BD as axis=eight times
the rectangle under AB and Jj.
The demonstration is similar to that of Prop. XIX.
Cor. 1. In revolving through half a right angle
APD generates a surface equal to the rectangle under
AB and Jj.
Cor. 2. Since GJ = AB (Prop. XV.), Jj =
16π
3
8πT
3
π
AB; and the surface generated by the revolution of
3
32π
(AB)²=== (AC)².
3
generating circle.
32
3
A
34
GEOMETRY OF CYCLOIDS.
PROP. XXIV.—If from G (fig. 17), the centre of
gravity of the cycloidal area APDB, GL be drawn
perp. to AB, and Gl making with AB produced the
angle GIA angle ADB, then the volume gene-
rated by the revolution of the area APDB around
the axis AB is equal to six times the volume of a
cylinder having the generating circle AQB for base
and height equal to Ll.
By Guldinus's Second Property (see note following
this proposition) the volume generated by the revolu-
3
4
A
B
FIG. 17.
a
H
た
​K
h
[[]]
tion of surface APD around AB= volume of a right
cylinder having APDB as base and height circum-
ference of circle of radius LG. But area APDB-
generating circle; and, as in Prop. XIX., L1=
circumference of circle with radius LG. Hence the
volume generated by the revolution of area APD
around AB is equal to (×4 times, or) six times the
volume of a cylinder having circle AQB as base and
height
= L %
Cor. 1. The volume generated by the revolution of
E

=
THE RIGHT CYCLOID.
35
APDB through one-third of two right angles about
AB is equal to a cylinder having circle AQB as base
and height = L 1.
Cor. 2. Since LG = OC
π
=3. AC_8 AC
9πT
(Prop. XVIII.)
L1 = (·
•G.AC-AC
π; and the sur-
face generated by the revolution of APDB about AB
= 67 (AC)² (= AC - 44C) × = (37³ – 8″) (AC)².
T
4
2 3
4AB
=
AL
Cor. 3. Since the rectangle BE in revolving
around AB generates a cylinder whose volume
= AB. 7. (BD)²=2AC . π (TAC)²=2π³. (AC)³, it
follows from Cor. 2 that the volume generated by
APDE in revolving around AB
元
​V = A. a. a.
16
8
3 8
= 2*³ (AC)³ — (37³ — 3- ) (AC)³= ('~' +3) (AC)².
—
2 3π
NOTE.--Guldinus's Second Property may be thus stated and es-
tablished:-
If a plane figure revolve through an angle a about an axis in its
own plane (the figure lying entirely on one side of the axis), the volume
of the solid generated by the figure is equal to that of a cylinder having
the figure for base and its height equal to the are described by the
centre of gravity of the surface in revolving through the angle a,
Let AQB (fig. 18) be a plane figure, and let it revolve through
an angle a about an axis OX in the same plane (AQB lying en-
tirely on one side of OX) to the position of aqb. Then PP', an ele-
ment of the figure's area, generates a ring of constant cross section
PP' and of volume ultimately PP'. Pp PP'. PM. α = a. moment
of PP' about OX. Taking all the elements of area of AQB in this
way, the volume generated by the surface AQB = a. moment of the
area AQB about OX = a. GN. area AQB, G being the centre of
gravity of the figure AQB, and GN perp. to OX.
A, GN = π, and the volume of the solid
Or if area of AQB
generated
V,
=
Ca
D 2
36
GEOMETRY OF CYCLOIDS.
PROP. XXV.-If from G (fig. 17), the centre of
gravity of the cycloidal area APDB, GH be drawn
perp. to BD and Gh parallel to AD to meet BD in
h, then the volume generated by the revolution of the
area APDB about BD as axis is equal to six times
the volume of a cylinder having the generating circle
AQB for base and height equal to Hh.
The demonstration is in all respects as in Prop.
XXIV.
Cor. 1. The volume generated by the revolution
of APDB through one-third of two right angles about
It is easily seen that if the figure AQB is not plane, or if,
whether plane or not, it is not in the same plane as OX, a similar
FIG. 18.

M
X
A
P
¿
B
property may be established. Let the figure AQB be carried
once round OX, and let a plane through OX intersect the surface
thus generated in a curve A'Q'B' (any parts of the plane figure
A'Q'B' through which more than one part of AQB may have passed
being counted twice or thrice, or so many times as they may have
been traversed in one circuit of AQB). Let A' be the area of
A'Q'B' (thus estimated), G' its centre of gravity (correspondingly
estimating the weight of its various parts), and 'the distance of
G' from OX. Then the volume generated by the revolution of AQB
round OX through the angle a = A'. a'. a (any part of the volume
generated which is traversed more than once by the generating
curve being counted as often as it is so traversed).
THE RIGHT CYCLOID.
37
BD is equal to a cylinder having the circle AQB as
base and height=H h.
Cor. 2. Since GH=&AC (Prop. XVIII.), Hh=
5πAC; and the volume generated by the revolution of
12
APDB about AB=T. (AC)². TAC=§T² (AC)³.
Cor. 3. Since the rectangle BE in revolving
around BD generates a cylinder whose volume =
.π (AB)²=πAC.4π (AC)² = 4π² (AC)³, it follows
from Cor. 2 that the volume generated by APDE in
revolving around BD
2
= 4π² (AC)³ — §π² (AC)³ = 3π² (AC)³.
Again, if AQB so move as to generate a cylindrical surface either
right or oblique, and two planes through OX intersect the surface
thus generated, the portion of the volume of this cylinder inter-
cepted between these planes may be thus obtained: -Through OX
take a plane perp. to the axis of the cylindrical surface, and inter-
secting that surface in a curve A'Q'B', enclosing a figure of area A',
and having its centre of gravity G' at a distance a' from OX; let
the portion of a straight line through G' parallel to the axis of the
cylindrical surface intercepted between these bounding planes
then the volume intercepted = A'. ā'. h.
h;
The proof of this and the preceding extension of Guldinus's
second property will be found to require the two following lemmas:
First, if an element of area PP' be projected orthogonally on a
plane through OX and P into the elementary area Pp', then PP'
and Pp' in rotating through any angle around OX generate equal
elementary solids. This is obvious, since they generate equal ele-
mentary solids in rotating through an elementary angle around OX.
Secondly, if two planes through OX cut a parallelopipedon of ele-
mentary cross section in the parallelograms PP' and pp', Pp and
P'p' being two opposite edges of the parallelopipedon, then two
other planes through OX near to these last, cutting Pp and P'p' in
R, R', and r, r', such that PR pr, intercept equal elementary solids,
P’RR'P' and prr'p'. These solids are, in fact, ultimately parallelo-
pipedons on equal bases and between the same parallel planes.
38
GEOMETRY OF CYCLOIDS.
PROP. XXVI.—If from G' (fig. 17), the centre of
gravity of the cycloidal area APDE, G'K be drawn
perp. to AE and G'k parallel to AD to meet AE in k,
then the volume generated by the revolution of the area
APDE about AE as axis is equal to twice the volume
of a cylinder having the generating circle AQB for
base and height equal to Kk.
The demonstration is as in Prop. XXIV., except
that the area APDE = a third only of the area
APDB.
Cor. 1. The volume generated by the revolution
of APDE through two right angles about AE = a
cylinder having circle AQB as base, and height equal
to K k.
Cor. 2. Since G'K=AC (Prop. XVI.), Kk =
ZAC; and the volume generated by the revolution of
APDE about AE = π (AC)².
(AC)². "AC="²(AC)³.
2
2
Cor. 3. Since the volume generated by the revolu-
tion of rectangle BE around AE=472 (AC)³ (see
Prop. XXV. Cor. 3), it follows from Cor. 2 that the
volume generated by APDB in revolving around AE
= 4″² (AC)³ — ~² (AC)³ = 77² (AC)³.
π2
2
2
THE RIGHT CYCLOID.
39
PROP. XXVII.-If from G' (fig. 17), the centre of
gravity of the cycloidal area APDE, G'J be drawn
perp. to DE and G'j parallel to AD to meet DE
in j. then the volume generated by the revolution of
the area APDE around DE as axis is equal to twice
the volume of a cylinder having the generating circle
AQB for base and height equal to Ïj.
The demonstration is as in Prop. XXIV., modified
as in Prop. XXVI.
Cor. 1. The volume generated by the revolution
of APDE through two right angles about AE = a
cylinder having circle AQB as base, and height equal
to Jj.
Cor. 2. Since G'J =
AE 8
2 3π
AC (Prop. XVII.)
8
4
= (¦) — 3,2) AC, J ;= (1,2 − 1 ) AC; and the volume
j (7
2
generated by the revolution of APDE around DE
T 8
.2
7
= π (AC)² ( 7 ² — § ) AC = ( ;² — 87 ) (AC)².
2 3
Cor. 3. Since the volume generated by the revo-
lution of the rectangle BE around DE=27³ (AC)³
(Prop. XXIV. Cor. 3), it follows from Cor. 2 that the
volume generated by APDB in revolving around DE
3
= 2π³ (AC)³ — (1² — 8,7) (AC)³ = (37³+87) (AC)².
—
2 3
40
GEOMETRY OF CYCLOIDS.
SECTION II.
THE EPICYCLOID AND HYPOCYCLOID.
DEFINITIONS.
The Epicycloid is the curve (as D'AD, fig. 19, Plate I.)
traced out by a point on the circumference of a circle
(as AQB) which rolls without sliding on a fixed
circle (as BDB') in the same plane, the rolling circle
touching the outside of the fixed circle.
The Hypocycloid is the curve (as D'AD, fig. 20, Plate
I.) traced out by a point on the circumference of a
circle (as AQB) which rolls without sliding on a
fixed circle (as BDB') in the same plane, the rolling
circle touching the inside of the fixed circle.
What follows applies to both figures unless special reference is
made to one only, and in every demonstration in this section two
figures are given, one illustrating a property of the epicycloid, the
other illustrating the same property of the hypocycloid, but the de-
monstration applying equally to either figure, unless special refer-
ence is made to one only. The student will do well to read each
proof twice, using first one figure, then the other. For convenience
the word 'cycloidal' throughout this section is to be understood to
signify either epicycloidal or hypocycloidal according to the figure
followed.
[NOTE.-It will be shown in Prop. I. of the pre-
THE EPICYCLOID AND HYPOCYCLOID.
41
sent section that if two circles AQB and AQ'B',
touching at B, touch a fixed circle BDB' at the ex-
tremities of a diameter BOB', then the same curve is
traced out by the point A on the circle AQB rolling
in contact with the circle BDB', as by the point A on
the circle AQ'B' rolling in contact with the same circle
BDB'. We may therefore, in what follows, limit our
attention to cases in which the centre O lies outside
the rolling circle. According to the definitions given
above, the curve traced out by A, fig. 19, is an epi-
cycloid whether AQB or AQ'B' is the rolling circle.
It may be well to mention that it has hitherto been
customary to regard the curve traced out by A on
AQB, fig. 19, as an epicycloid, and the same curve
traced out by A on AQ'B' as an external hypocycloid.
Instead of defining the hypocycloid as the curve ob-
tained when the rolling circle touches the outside of
the fixed circle, it has hitherto been usual to define it
as the curve obtained when either the convexity of
the rolling circle touches the concavity of the fixed
circle, or the concavity of the rolling circle touches the
convexity of the fixed circle. There is a manifest
want of symmetry in the resulting classification, see-
ing that while every epicycloid is thus regarded as an
external hypocycloid, no hypocycloid can be regarded
as an internal epicycloid. Moreover, an external hypo-
cycloid is in reality an anomaly, for the prefix 'hypo'
used in relation to a closed figure like the fixed circle
implies interiorness.]
Let BDB' (radius F) be the fixed circle, AQB
42
GEOMETRY OF CYCLOIDS.
(radius R) the rolling circle. If the centre of the
latter circle move in the direction shown by the arrow,
it is manifest that at regular intervals the tracing
point will coincide with the circumference BDB', as
at D', D, &c. (E'q'D' and EqD being the correspond-
ing positions of the rolling circle), while midway be-
tween two such coincidences the tracing point will be
at its greatest diametral distance from D'BD as at A
(AQB being the corresponding position of the rolling
circle), ACB the diameter through the tracing point
passing when produced through O, the diameter of the
fixed circle. It is clear also from the way in which
the curve is traced out that the parts AP'D and APD
are similar and equal. Wherefore AB is called the
axis of the cycloidal arc D'AD. The circular arc
D'BD is the base, A the vertex, and the points D'
and D are the cusps. It is convenient to call the
radius to the tracing point the tracing radius, and the
diameter through the tracing point the tracing diameter.
The tracing circle in the position AQB is called the
central generating circle; and straight lines passing
through the centres of both the fixed and rolling circles
are said to be diametral. The arc Ce is called the are
of centres, and the circle of which it is part the circle
of centres.
Let a circle EAE be described with centre O and
radius OA, and let OD' and OD (produced if neces-
sary) meet this circle in E and E'; then it is clear that
D'd' and Dd, the parts of the cycloidal curve on either
side of D'E' and DE, are symmetrical with regard to
THE EPICYCLOID AND HYPOCYCLOID.
43
these lines respectively, which are therefore secondary
Also E'AE touches the curve D'AD in A.
axes.
The complete curve, either of an epicycloid or of
a hypocycloid, consists of an infinite number of equal
cycloidal arcs, but when the radii F and R are com-
mensurable in length, the curve is re-entering, and
may be described as consisting of a finite number of
arcs.* Thus if R = F the rolling circle will make one
complete circuit of the fixed circle between each suc-
cessive coincidence of the tracing point with the fixed
circle; hence D and D' will coincide, and there will be
but one cusp. (No hypocycloid can be traced with
these radii.) If RF, each base as DD' will be
equal to half the circumference of the fixed circle, and
there will be but two cusps. Similarly if RF, 1F,
1F, &c., there will be 3, 4, 5, &c., cusps, respectively.
In these cases the complete cycloidal arc will consist of
a number of equal arcs, standing on equal parts of one
circuit of the fixed circle's circumference. Again, if
mR = nF, where n and m are integers prime to each
other, then m circumferences of the smaller circle
will be equal to n circumferences of the larger. Con-
sequently there will be m cusps in the complete cy-
cloidal curve, and the base of each cycloidal are will
be equal to one mth part of n circumferences of the
N
fixed circle, that is to theth part of the circumfer-
M.
*Theoretically it consists in that case of an infinite number of
arcs, occupying a finite number of positions, and consequently each
are coinciding with an infinite number of other ares belonging to
the curve.
44
GEOMETRY OF CYCLOIDS.
ence of this circle. Wherefore if n > m, the base is
greater than the circumference of the fixed circle,
but if n <m the base is less than this circumference.
If munity, that is Rn F, then the base of each
cycloidal arc = n times the circumference of the fixed
circle.
PROPOSITIONS.
PROP. I.-If a circle q'Dq (figs. 21 and 22), having
FIG. 21.

K
oc
U
B
radius Dc, roll in contact with a circle KDb, having
radius OD, c and O lying on the same side of D,
then the point D on q'Dq will trace out the same
curve as the point D on a circle QDQ having radius
DC equal to c 0 (measured in direction c O), rolling
in contact with the circle K D b.
Let b be the point in which the rolling circle q'Dq
THE EPICYCLOID AND HYPOCYCLOID.
45
touches KDb, when the tracing point is at P, c' being
the centre of q'D q (c', O, and b lying in the same
straight line). Through O draw OC', equal and
parallel to c P, meeting KDb in B'; and join PC'.
Then PC'O' is a parallelogram; PC'-c'O=DC;
also, since OC'-c'P-c'b', and OB'Ob, C'B' Oc
-
=DC. Hence a circle equal to QDQ', touching KD
in B′ (on the same side as QDQ), has its centre at C'
FIG. 22.

V
ջ
and passes through P.
Q
B
며
​Moreover, since arc P barc D b,
K
≤ Pcb (= ¿C′Ob= ¿PC′B′): ≤ DOb:: Ob: c'b,
... PC'B: 4 DOB':: Oc: Ob:: DC: OD.
Therefore arc PB'=arc DB', and P is a point on the
curve traced out by D on the circle QDQ' rolling in
contact with the circle KD b.
SCHOL.-It is manifest that when P arrives at the
vertex of the curve the rolling circles are placed (re-
latively to each other) as in figs. 19 and 20.
46
GEOMETRY OF CYCLOIDS.
PROP. II.— The base of the epicycloid or hypocycloid
is equal to the circumference of the generating circle.
This, as in the case of the cycloid, needs no demon-
stration.
Cor. 1. Arc D'B (figs. 19 and 20) =arc BD
-half the circumference of the generating circle.
Cor. 2. Arc Cc: arc BD: CO: BO.
Or for the epicy cloid,
arc Cc=
and for the hypocycloid,
arc C c
F+R
R
=
C =
F-R
R
arc BD
Cor. 3. Area E'AEDBD' – 2 AED'B
=
arc BD =
4CO
BO
4
•
4
= 4 rect. under AC, C c*
CO
=4
BO
F+R
R
F-R
R
F+R)
R
•
arc AQB,
(F-R)
R
arc AQB.
rect. under AC, BD
for the epicycloid
circle AQB
for the hypocycloid
circle AQB.
* The relation here employed is almost self-evident. It may be
thus demonstrated: Divide the area AEDB into a series of ele-
mentary areas by drawing radial lines from O: each element is in
the limit a trapezium whose area - rectangle under AB and half
the sum of those elementary arcs of AE and BD which form (in
the limit) the parallel sides of the trapezium. Therefore the area
AEDB rectangle under AB and half the sum of the arcs AE,BD
rectangle under AB and the arc Cc.
circle AQB
THE EPICYCLOID AND HYPOCYCLOID.
47
PROP. III.-If through P, a point on the epicycloidal
or hypocycloidal arc APD(figs. 23 and 24), the orc
PM be drawn concentric with the base BD, cutting the
central generating circle in Q and meeting the axis
AB in M, then arc QP: arc AQ:: OM: OB.
Let A'PB' be the position of the generating circle
when the tracing point is at P; C' its centre; A'C'B'O
diametral, cutting PM in M'. Draw the tracing dia-
FIG. 23.
FIG. 24.
A

M
K
B
Q
B
Α'
M
valtra dil de a háár as I
q
P
P
m
E
P
E
C'
A
པད➢
K
M
C
meter PC'b. Then it is manifest that arc QM =arc
M'P; arc MM' = arc QP; and arc AQ =arc A'P.
Now is the point which was at B when the tracing
point was at A; and since every point of the arc b B'
has been in rolling contact with BB', the arc b B' the
arc BB'. But arc b B'= arc A'P
=arc A'Parc AQ; and
arc MM² (= arc QP): arc BB'::OM : OB;
.. arc QP are AQ:: OM OB.
48
GEOMETRY OF CYCLOIDS.
Cor. 1. Arc MP
Cor. 2.
Let are MQ prod. meet OE(drawn as in figs. 19,20) in m
OM
then
OB ·
arc M m -
OM
OB ·
arc MP =
OM
OB
OM
OB
OM
OB
arc BD
OM
[But arc BQ > QK perp. to AB; .'. .arc BQ >
OB
ML, perp. to AB and meeting OQ produced in L (for
: :
OM OB > OM OK). But ML are MQ.
>
A
fortiori, then,
OM
OB
arc BQ> arc MQ.]
... since arc M m =
OM
OB ·
•
arc AQ+arc MQ.
arc AQ
arc AQ +
while
and P falls between OA and OE;
arc M m > arc MP,
that is, the whole arc APD lies between OA and OE.
Cor. 3. The arc P marc M m
are MP
OM
OB
padahal
OM
OB are AQ + are MQ,
ов
arc BQ.
W
.
*
•
arc AQB.
My
•
OM
ов arc BQ,
arc MQ.
Cor. 4. If through P', a point near P, arc P'pQ
be drawn concentric with the base BD, meeting
AQB in Q and cutting A'PB' in q, then in the limit
arc AQ arc MQ
* The part in [] fails for hypocycloid. Substitute the follow-
ing:---Let OQ produced meet arc BD in H, draw BF perp. to OH and
describe BFO. Then, arc BH=arc BF (of half rad. and double
Z at centre); but arc BF < arc BQ, .chd. BF <chd. BQ (BFQ
being a rt. angle) while seg. BF contains a larger angle than seg.
OB
BH >
OM
BQ'Q. Hence arc BQ > arc
> are MQ.
ΟΜ
OB
arc MQ; i.c.
•
arc BQ
THE EPICYCLOID AND HYPOCYCLOID,
49
OM
OB
and
(when P' is very near to P), arc P'Q':
arc PQ
OM
arc P'Q'-arc PQ (=qP')= OB (arc A'q — arc A'P)
ов
or, in the limit,
OM
OB are A'P;
•
:
=
OM
arc Pq;
OB
I P' arc Pq:: OM : OB.
arc A'q;
PROP. IV.—A, B, C, D, E, &c. (figs. 25 and 26, p. 51)
representing the same points as in the preceding propo-
sition, the area APDBQ = half the area ABDE;
or area APDBQ generating circle: OC: OB.
E
therefore,
Take CL=CL', on AB; and LK, L'K' equal ele-
ments of AB, both towards C. Draw LQ, Kq, K'q',
and L'Q' at right angles to AB to meet AQB; and
about O as centre describe arcs QP, qp, q'p', and
Q'P, meeting APD. Let Oq, produced if necessary,
meet QP in n; draw Qk perpendicular to Kq; join
C q, and draw Cm perpendicular to O q, produced if
necessary. Then ultimately the triangles Qkq and
q KC are similar, as are the triangles Q qn and q C m
(for Q4 C being ultimately a right angle, Qq n is ulti-
mately the complement of C q m and therefore equal
to q Cm). Hence the quadrilateral Q nqk is similar
to the quadrilateral qm C k, and
qn : Q k(=LK) :: Cm : Kq :: CO : 90
(triangles CO m and 9 OK being similar). Hence
50
GEOMETRY OF CYCLOIDS.
Area QP p q (ult.=rect. n q, QP): rect. LK, QP
:: CO: 90;
but,
:
rect. LK, QP rect. LK, Aq: QP: A q
:: 90: BO (Prop. II.);
area QP p q rect. LK, Ag:: CO: BO
:: Cc: BD;
... ex æq.
similarly, area Q'P'p'q': rect. L'K', A q' (or LK, Bq)
:: Cc: BD;
с
· ·. QPpq+Q'P'p'q' : rect. LK, Aq+Bq (or LK, BD)
:: Cc: BD;
wherefore
QPpq+Q'P'p'q' rect. LK, C c.
... summing all such elements between AE and BD,
Area APDBQ=rect. under AC, Cc area ABDE.
or, area APDBQ gen. :: OC: OB.
:
Cor. 1. Since, for epicycloid, C c =
F+R
F+R
F
O
AQB,
F gen. O
F+R
area APDBQ= F.AC. AQB=
and the area between epicycloidal arc and base
F+R
3 F+2R
(2. F +1) gen.
F gen. O
For the hypocycloid, area APDBQ =
F-R
•
F gen. ;
•
and the area between hypocycloidal arc and base
3 F-2 R
F . gen. O
Cor. 2. If AB is the axis of a cycloid (A the ver-
tex) and LQ produced meet this cycloid in R, then
Area AQP area AQR:: OC: OB.*
* This relation, which follows directly from the proportion on
the fifth line of this page, might have been employed to establish
the main proposition. I preferred, however, to give an independent
proof.
THE EPICYCLOID AND HYPOCYCLOID.
51
Cor. 3. Epicyc. area APDE
= (F+R)
gen. O
Hypocycloidal area APDE =
A
K
RX.
a
FIG. 25.
m
D
F+2R
F-2 R
E
APDBQ — AQB
2 F gen. 0.
P
2 F gen. O
P
FIG. 26.

ALK
B
Cor. 4. Area AQP + area BQ'P'D=rect. AL, C c ;
and,
area QQ'P'P
rect. under LC, C c.
PROP. V.—If P is a point on the epicycloidai or hy-
pocycloidal arc APD (figs. 27 and 28) A'PB' the
generating circle when the tracing point is at P,
A'C'B' diametral, then PB' is the normal and A'P
is the tangent at the point P.
Since, when the tracing point is at P, the generat-
ing circle A'PB' is turning round the point B', the
direction of the motion of the tracing point at P must
52
GEOMETRY OF CYCLOIDS.
be at right angles to PB';-wherefore PB' is the
normal and AP is the tangent at P.
Another Demonstration. (See p. 8.)
Take P'a point near to P and draw PQM, P'Q
concentric with BD; PQM meeting AB in M and
cutting AQB in Q; and P'Q'N cutting AQB and
FIG. 27. (Join PC', AQ.)
A
M
L
K
B
4.
P
R
R
G
FIG. 28. (Join AQ.)
di

["]
E
E
PnP
A
IM
JO
K
B
A'PB' in Q and q.
Join PC', PO, and let C's pa-
rallel to PO meet PB' (produced in case of epicycloid)
in s.
Then (Prop. III. Cor. 4)
arc q P' arc Pq:: PO B'O:: C's: C'B' (= C'P);
or the sides about the angles Pq P', PC's are propor-
tional; but these angles are ultimately equal, for P q
is ultimately perp. to C'P, and P'q to PO, that is to C's.
Therefore the triangles P q P' and PC's are ultimately
similar; and the third side PP' of one is perp. to the
THE EPICYCLOID AND HYPOCYCLOID.
53
third side P s of the other. That is PB' is the normal
at P, and therefore PA' perp. to PB' is the tangent at P.
Cor. 1. If PB' intersect Q'P' in l, and s C' pro-
duced meet PA' in k, the triangle PP'7 is ultimately
similar to the triangle s P k.
Cor. 2. If B'q be joined and produced to meet PP'
in n, then qn is ultimately perp. to PP'; wherefore if
C'N be drawn perp. to B'P, the figure P'q P'n l is ulti-
mately similar to the figure PC's Nk; whence
PP' Pn:: Ps: PN.
SCHOL. As in Schol. p. 9 (obviously modified), a
tangent may be drawn to APD from any point on
APD or AA'E.
djm
PROP. VI. With the same construction as in Prop. V.,
Arc AP chord AQ: 2CO: BO.
:
Since qn is ultimately perpendicular to PP', Pn
is ultimately equal to the excess of chord A'q over
chord A'P. Now from Cor. 2, Prop. V.,
PP' PnsP: NP: 2 s P : B'P
:
::2C0:B0::2CO : B0,
or, inct. of AP: inct. of ch. A'P (or AQ):: 2 CO: BO.
But arc AP and chord AQ begin together, wherefore
Arc AP chord AQ:: 2 CO : BO.
Cor. 1. Arc APD: AB:: 2 CO: BO.
Cor. 2. For the epicycloid,
Are APD = AB.
2 ( F + R) __ 4 R (F + R)
F
F
54
GEOMETRY OF CYCLOIDS.
For the hypocycloid,
Arc APD = AB.
2 (F-R) 4 R (F-R
F
F
Cor. 3. PP': Pn :: 2 CO: BO.
Cor. 4. PP'
n P' :: 2 CO: 2 CO-BO
:: 2 CO : 40.
Cor. 5. P n n P' :: BO : AO.
PROP. VII.—PROB. To divide the arc of an epicycloid
or a hypocycloid into parts which shall be in any
given ratio to each other.
Let a straight line ab (figs. 27 and 28) be divided
into any parts in the points c and d it is required to
divide the arc APD in the same ratio.
:
Divide AB in L and K, so that
Similarly
Therefore
AL LK KB:: ac: cd: db;
:
:
with centre A and radii AL and AK, describe circular
arcs LQ, Kr, cutting the semicircle AQB in Q and r ;
through which points draw the arcs QP, r P, concen-
tric with BD. Then
Arc AP chord AQ (= AL):: 2 CO BO.
:
Arc AR
Arc PR
AK:: 2 CO: BO;
LK :: 2 CO: BO.
Similarly Arc RD: KB:: 2 CO: BO,
therefore
Arc AP arc PR arc RD:: AL: LK : KB
accddb;
..
► •
THE EPICYCLOID AND HYPOCYCLOID.
55
or, the arc APD is divided into the points P and R in
the required manner.
Similarly may the arc APD be divided into four,
five, or any number of parts, bearing to each other any
given ratios.
PROP. VIII.—With same construction as in Prop. V.,
Area ABB'P (figs. 27 and 28): sectorial area A'B'P
:: area B'PD : segm. PFB' :: 2 CO + BO: BO.
Let b be the point of contact of tracing and fixed
circles, when tracing point is at P'; join b P', BQ, and
BQ'; and draw bi perpendicular to Ps. Then triangle
b B'i is similar to B'C'N, therefore to PC'N, and
therefore (Prop. V., Cor. 2) to Pqn; and B' b=Pq:
therefore P qn and b B'i are equal in all respects; and
Pn=bi. Now elementary area PP'b B' is ultimately
equal to trapezium Pib P,
-half rect. under Pi and (PP' + bi)
half rect. under PB' and (PP' + Pn) ultimately
and elementary area QBQ' is ultimately equal to tri-
angle PB' q
= half rect. under PB' and P n, ultimately.
‚·. area PP' b B': area QBQ' :: PP' + P n : Pn
:: 2 CO + BO : BO (Cor. 3, Prop. VI.).
Thus the increment of area ABB'P, or the decrement
of area B'PD, bears to the increment of area A'B'P,
or the decrement of area PFB', the constant ratio
56
GEOMETRY OF CYCLOIDS.
(2 CO+BO): BO. But the areas ABB'P and B'PD
commence together, and the areas A'B'P, PFB′ end
together, as P passes from A to D; hence
Area ABB'P: sectorial area A'B'P
:: area B'PD: segment PFB' :: 2 CO + BO : BO.
and
Cor. 1. Pn = bi;
PP' bi :: PP' : Pn :: 2 CO: BO.
:
Cor. 2. Area B'FPD: seg. B'FP :: 2 CO : BO.
This can be proved independently, in the same manner
as the corresponding relation for the cycloid, Cor. 1,
Prop. VIII., Cycloid.*
SCHOL.-The above affords a new demonstration
of the property proved in Prop. IV. Cor. 2 also, if
independently established, gives another proof of the
area.
* The proof may be effected in two ways, both analogous to the
proof for cycloid,-viz., either by making the sides of elements
such as ff' and FF' concentric with BD, or by making them perpen-
dicular to A'B'. In the former case we find the decrement of space
PFB'D P'q B'b, that is (ultimately) = P'n B'b, and the rest of the
proof is like the above. In the latter case we find the decrement of
PFB'D = a rect under C'e' (c' centre of F'P') and projection of
B'q on A'B'; and decrement PFB′ = triangle PB'b = rect. under
B'b and projection of B'q on A'B'; therefore
=
decrement of PFB'D: decrement of PFB' :: 2C'c': B′b;
area PFB'D: area PFB' :: 20'c': B'b: 2CO: BO.
that is,
THE EPICYCLOID AND HYPOCYCLOID.
57
PROP. IX.—If P(figs. 28 and 29) be a point on the
epicycloidal or hypocycloidal arc APD, ånd OP, OA,
OD be joined, and PM be drawn perp. to A'B', the dia-
metral of the generating circle A'PB' through P, then
Area APO : rect. OC (arc A'P+ PM) :: OA : 2 BO.
The area APO sector OBB' + AOB'P ± area
ABB'P (taking the upper sign for the epicycloid, and
the lower sign for the hypocycloid, throughout);
FIG. 30.
M
BA
M
therefore,
FIG. 29.

C
B
2 area APO = OB. arc BB' + OB'. PM
2 CO + BO
BO
(2 area A'B'P);
= OB arc A'P + OB. PM
+
2 CO+ BO
BO
(OBAC) arc A'P + (OB ± AC) PM
. (AC. arc A'P+AC. PM);
58
GEOMETRY OF CYCLOIDS.
+1
= (co
Now,
and
... ex æq.,
But
...
2 CO. AC
BO
CO±
•
arc A'P +
2 CO. AC)arc A'P
BO
2
+(CO±²CO.AC) PM;
BO±
:
2 CO. AC
BO
= CO
OA CO. (arc A'P + PM);
BO
BO
12 AC) (arc A'P + FM);
BO
therefore,
area APO : rect. OC (arc A'P + PM) :: AO: 2 BO.
Cor. 1. Area APDO: rect. OC, BD:: AO: 2 BO.
Cor. 2. Area DPO : rect. OC (arc B′P – PM)
:: AO: 2 BO.
Cor. 3. APDO: sect. OBD:: AO. CO: (BO)2.
Cor. 4. APDO: sect. OC c (figs. 25 and 26)
::sect. OA a: APDO :: AO : CO.
•
NOTE.-The above demonstration might have been readily made
geometrical in form as it is in substance; but it would have been
more cumbrous and not so easily followed. The student should,
however, note the following independent demonstration (which
occurred to me after the above had been corrected for press) :-
In figs. 27, 28, p. 52, let OP intersect P' in h; draw PH perp.
to sk and PM' perp. to A'B'. Then the ultimate increment of area
APO = rect. OP, h P'; while the corresponding increment of rect.
OC (arc A'P+ PM') = rect. OC, inct. of (arc A'P + PM'). Therefore,
former inct. latter inct.::OP.h P: OC, inct. of (arc A'P + PM').
h P': Pq :: s H: C'P
:
Pq inct. (arc A'P + PM') :: C'P: B'M'
h P' inct. (arc A'P+ PM'): :sH: B'M'
OP:
::s C': C'B'
OB'
. PM;
OP. h P': OB'. inct. (arc A'P + PM') : : 8 H . s C′ : B′M'. C'B'
::s P.SN: B'P. B'N (since C', P', H, P, N, lie on a O).
THE EPICYCLOID AND HYPOCYCLOID.
59
or,
Wherefore, increasing OB' in 2nd term to OC, and B'P in 4th to 8 P
(or both in the same ratio, since triangles & B'C, PB'O are similar),
OP.h P': OC. inct. (arc A'P + PM'): :s P. s N:s P. B'N
:: 8N B'N :: C'B' + B'O : B'O
¦
:: AO: BO;
inct. area APO: inct. rect. OC (arc A'P + PM')::AO: 2 BO
Area APO rect. OC (arc A'P + PM') :: AO : 2 BO.
Cors. 1, 2, 3, and 4, follow as before.
SCHOL.-We have here an independent demonstration of the
area of the epicycloid and hypocycloid, since
Area APDO = area ABD ± area APDB.
OBD
PROP. X.-With the same construction as in former
Propositions (figs. 31 and 32),
Area APA': segment A' h P :: AO : BO.
Let a P'B be the position of the tracing circle when
tracing point is at P' near to P; a cb O diametral.
Draw q P' concentric with BD and AE, join A' q, a P',
A'P'; also producing A'a to T and qP to t, draw P'T
and A't perp. to A'T and P t respectively.
Then A'PP'a, the increment of AA'P=rect.
under A'a, P'T ultimately; and A'P 4, the increment
of segment A'h P=rect. under P q, A't ultimately.
But ultimately the right-angled triangles A'tq and
P'T a are equal in all respects (since A'q= a P', and
angle A'qt angle at circumference on segment A'q
angle at circumference on segment a P′ angle
P'at) therefore P't A'T, and
increment of AA'P: increment of segment A'h P
:: A'a : Pq (= B′b) :: AO : BO;
60
GEOMETRY OF CYCLOIDS.
:
or since these areas begin together,
area AA'P: segment A'h P :: AO : BO.
Cor. Area AA'h P: seg. A'h P :: 2 CO : BO
( :: AO+BO : BO). This may readily be established
independently—by showing that ultimately
area A'ah' P’Ph: ▲A'P q:: 2 C'c : B′b :: 2 CO : BO.*
FIG. 31.
A
SOR
C
B
-X
Q
B
b
T
q
12
B'
P
T
X-
SCHOL. Since it follows that
area APDE:
!

E
B
ทา
A
R
FIG. 32.
gen. o :: AO : BO,
* A line from h', perp. to A'B', to meet A' P= C'c; and a line
from P, perp. to A'B', to meet A'q=Pq=B' b.
THE EPICYCLOID AND HYPOCYCLOID.
61
we have here another demonstration of the area of
APDE. Further, since.
1 gen. o: area ABDE:: CB, BD : 2. CB, arc C c'
:: BD:4Cc :: BO:4CO,
it follows, ex æquali, that
area APDE: area ABDE :: AO: 4 CO.
Yet again, from the corollary we see that
Area APDQ´E: generating circle :: 2 CO: BO
:: ½ area ABDE: generating circle,
... area APDQ'E = area ABDE,
which is the relation established in Prop. IV. If
established independently, as explained above, this
leads to another demonstration of the area.
NOTE. Arc APD divides the area AQBDQ'E
into two equal areas.
PROP. XI.—If PB'v (figs. 33, 34) is the radius of
curvature at P, and PB' the normal, then
Po: PB' :: 2 CO: AO.
With so much of the construction of figs. 27, 28
as is shown in figs. 33, 34, produce P'b to meet PB'
produced in o', then o is the limiting position of o' as P'
moves up to P. Now since PP' is ultimately parallel
to bi, therefore ultimately
Po B'o' PP' bi: 2 CO: BO
:: :
(Prop. VIII., Cor. 1), wherefore
Po: PB': 2 CO: 2 CO - BO:: 2 CO: CO + AC,
or ultimately Po: PB': 2 CO: AO.
62
GEOMETRY OF CYCLOIDS.
Cor. 1. For the epicycloid,
radius of curvature =
and for the hypocycloid,
radius of curvature =
A
B
a
FIG. 33.
C
O.
2 (F + R)
F+2R
2 (FR)
F-2R
E
•
E
NA
·
FIG. 34.
normal;
B
normal.

A
C
B
d
Cor. 2. PB': B'o :: 2 CO-BO: BO :: AO: BO
:: F + 2 R : F for the epicycloid;
:: F-2 R: F for the hypocycloid.
SCHOL.-We see from Cor. 1 that when F = 2 R
the radius of curvature of the hypocycloid is infinite,
or the hypocycloid degenerates into a straight line.
See further the Appendix to this section, pp. 66 to 68.
THE EPICYCLOID AND HYPOCYCLOID.
63
you
PROP. XII.—The evolute of the epicycloid or hypocy-
clvid APD (figs. 33 and 34) is a similar epicycloid or
hypocycloid, do D, having its vertex at D, and its
cusp d so placed on OA (produced if necessary), that
dB: BA: OB: 0A;
or, which is the same thing, Od: OB :: OB : OA.
Join OD and describe the arc dae with O as
centre and radius Od. Produce A'B' to O, cutting (fig.
33) or meeting (fig. 34) de in a', and join o a'. Then
B'A': a'B' :: BA: dB:: AO: BO :: PB': B'o
(Prop. XI., Cor. 2);
that is, the sides about the equal angles o B'a'; PB'A'
are proportionals; therefore the triangles 'B'a',
PB'A' are similar, and the angle a'o B' (= the angle
B'PA') is a right angle. Hence a circle described
on B'a' as diameter will pass through o. Again the
angles A'B'P and a'B'o at the circumferences of the
circles A'B'P and a'B'o being equal,
arc o a: arc PA' (=arc BB') :: a'B': B'A':: OA: OB
:: OB: Od :: arc da' : arc BB'.
Therefore, arc o d'arc d a', and o is a point on an
epicycloid (fig. 33) or hypocycloid (fig. 34) having de
for base, its cusp at d and B'o a' as tracing circle.
Since de BD:: od: OB:: Bd: AB
:
:: arc B'o a': arc A'PB' (= BD);
de arc B'o a';
therefore
so that e D is the axis and D the vertex of the epi-
cycloid or hypocycloid do D.
64
GEOMETRY OF CYCLOIDS.
Cor. If c is the bisection of e D,
с
oP: 0 B': 2CO: AO: 2c0: DO;
therefore (Prop. VI.),
o Parco D.
If, then, a string coinciding with the arc do D and
fastened at d, be unwrapped from this arc, its extremity
will always lie on the arc APB, which may thus be
traced out as the involute of the arc do D.
SCHOL. A convenient construction for finding the
base, &c., of the evolute do D is indicated by the dotted
lines in the figures: thus, join AD, then Be parallel
to AD gives Oe (on OE, produced if necessary), the
radius of the base e d.
PROP. XIII.—If do D (figs. 33, 34) be the evolute of
the epicycloid or hypocycloid APD, and o B'P, the
radius of curvature at any point P on APD, cut the
base BD in B', then
area APB'B: area d BB'o
:: rect. under AO (AO + 2BO): square on BO.
If P'o' be a contiguous radius of curvature cutting
BD in b, and bi is drawn perp. to o B'P, then in the
limit
o Poi: 2CO: BO;
therefore
ult. area Po P' : ult. area o ib :: 4(CO)² : (BO)²,
whence, ultimately
area PB'b P: area o B'b: 4 (CO)² — (BO)² : (BO)²
::rect. (2CO - BO) (2CO + BO): sq. on BO
:: rect. AO (AO + 2BO) : sq. on BO.
THE EPICYCLOID AND HYPOCYCLOID.
65
But the areas PB'b P' and o B'b are the elementary
increments of the areas APB'B and d BB'o, which
begin together. Therefore,
area APB'B: area d BB'o
:: rect. under OA (AO + 2BO) : sq. on BO.
Cor. 1. Area APDB: area do DB
:: area PB'D: area o B'D
::rect. under OA (AO + 2BO): sq. on BO.
Cor. 2. Since
area do DB : area APDE :: (BO)² : (AO)²,
it follows (ex æq.) that
area APDB: area APDE :: AO (AO+2BO) : (AO)²
:: AO + 2BO: AO
:: (3F + 2R) : (F + 2R) for the epicycloid
:: (3F — 2R): (F2R) for the hypocycloid.
SCHOL.-It follows from Cor. 2 that
Area APDE: area ABDE :: AO : 2(AO + BO)
:: AO: 4CO,
which is one of the relations established in the scho-
lium on Prop. X. Hence we have in Prop. XIII.
another method of demonstrating the area of the epi-
cycloid and the hypocycloid.
F
66
GEOMETRY OF CYCLOIDS.
APPENDIX TO SECOND SECTION.
There are many forms, both of the epicycloid and
of the hypocycloid, which possess interesting proper-
ties. For the most part the general properties esta-
blished in the preceding section will suffice to enable the
student to deduce the properties of special forms of
these curves. For this reason, and also because of the
requirements of space, I shall only touch briefly here
on a few points in connection with the forms assumed
by epicycloids and hypocycloids for certain values of
the radii of the fixed and rolling circles. I do not
make set propositions of these points, but present them
in such sequence as appears most convenient and suit-
able.
THE STRAIGHT HYPOCYCLOID.
The hypocycloid becomes a straight line when the
diameter of the rolling circle is equal to the radius of
the fixed circle.
This in reality has been already demonstrated, be-
cause we have seen in the scholium to Prop. XI. that
the radius of curvature of the hypocycloid becomes
infinite when F 2R. Also the relation is involved.
in the demonstration of Prop. I. For when the two roll-
ing circles (figs. 21 and 22) are equal, each having its dia-
meter equal to the radius of the fixed circle, the curve
THE EPICYCLOID AND HYPOCYCLOID.
67
traced out by each must be a straight line. Thus,-
let BOB' (fig. 35) be the diameter of the fixed circle,
and its halves BO, OB', the diameters of the two equal
rolling circles; then by what is shown in Prop. I. of
this section the point O on BQO will trace out the
same curve as the point O on B'Q'O, but since the
circles BQO and B'Q'O are equal, this curve, regarded
0.
FIG. 35.
B

D
as traced out by O on BQO, must bear the same re-
lation in all respects to the axis OB that the same.
curve regarded as traced out by O on B'Q'O bears to
the axis OB', and the only line which can possibly
fulfil this condition is the diameter D'OD at right
angles to BOB'. This then must be the path traced
out by the point O in each case.
Let us proceed, however, to an independent de
monstration.
F 2
68
GEOMETRY OF CYCLOIDS.
When the circle OQB has rolled to the position
Opb (Ocb its diameter), let p be the point which had
been at B, so that drawing the diameter pc P, P is the
position of the tracing point. Then the arc pb is equal
to the arc Bb; and therefore, since F=2R, the angle
Bob is equal to half the angle bep, that is to the
angle bPp: but BOp and Ob P are alternate angles ;
wherefore bP is parallel to BO; and OP, which (OP¿
being a semicircle) is perpendicular to bP, is perpen-
dicular to BO. P therefore lies on the diameter D'OD
at right angles to BOB'; which was to be shown.
Cor. The point p lies on OB (the angles c Op and
c OB being each equal to half the angle b cp).
USEFUL GENERAL PROPOSITION.
The following property is worth noticing. It is
true of course for the cycloid also.
A diameter of the generating circle of an epicycloid
or hypocycloid constantly touches the epicycloid or
hypocycloid which would be generated by a circle of
half the diameter, alternate cusps of this epicycloid or
hypocycloid falling on successive cusps of the former.
It will suffice to demonstrate the property for the
epicycloid.
Let AQB (fig. 36) be the generating circle of an
epicycloid when the tracing point is at A, the vertex
of the epicycloid. When the circle has rolled to posi-
tion a Pb, let pc P be the position of the diameter
which had originally been in position ACB. Draw
b P' perpendicular to p P, and on cb describe the semi-
circle cP'b, having c' as its centre and passing through
THE EPICYCLOID AND HYPOCYCLOID.
69
}
P' because c P'b is a right angle. Then because the
angle P'c'b twice the angle P'cb, and c'b = half
cb, the arc P'b = arc pb =arc Bb. Wherefore P'
is a point on an epicycloid traced out by the rolling of
c Pb on BD, B being a cusp. D is the next cusp, be-
cause the base of the smaller epicyloid being equal to
the circumference of generating circle e P'b circum-
ference of semicircle AQB=BD. Also p P'c P is the
tangent at P' by what has been already shown respect-
ing the tangent to an epicycloidal arc.
The student will find it a useful exercise to prove
the property established in Prop. I. of the present
Fic. 36. (Draw in epicyeloid on base BD, touching cp in P'.)
A

B
a
D
:
section in the manner illustrated by figs. 37 and 38,
where APB is the arc traced out by point A on each
of the circles AQB, AQ'B'. The construction and
proof for the epicycloid (fig. 37) run as follows:
ABOB' being a common diameter of all three circles
at the beginning of the rolling motion, let P be the
position of the tracing point of the smaller rolling
circle when its centre is at c. Draw the diametral line
acb Of, and the diameter Pep. Join Pb and pro-
70
GEOMETRY OF CYCLOIDS.
20
duce to meet the circle BDB' in b', produce b’O to c',
taking Oc' R, so that b'e F + R, and join Pe;
=
then since
bPbb abbf: R: F:: Oc': Ob
Pc' is parallel to Ob, and the triangle b'e'P, like tri-
angle 'Ob, is isosceles (c'b' d'P). With centre d'
and radius c'P or c′b′ ( = F + R) describe the circle
b
p'
FIG. 37.
A
R
B
B
=

૧
b
a
D
'Pa'; produce Pc' to meet this circle in p'. Now,
arc Bb arc b p;
... angle pcb: angle BOb:: F: R;
=
but angle pcb 2 angle cb P = 2 angle Obb'
= angle l'Of
...angle b′Oƒ(= angle b'c'p') : angle BOb :: F: R;
and b'ép': U'′OB' :: F: F+R :: B′O: b'c'.
टोक
THE EPICYCLOID AND HYPOCYCLOID.
71
Whence it follows that arc b'p' = arc b'B'; and P is,
therefore, a point on the curve traced out by A (on
the circle AQ'B'), rolling so that its inside touches the
outside of the fixed circle BDB', ABOB' being ori-
ginally diametral. The same curve APB is traced
out, then, by the point A on each of the circles AQB
and AQ'B'.
के
FIG. 38.

p
Q
p
Cor. If we produce b'O to meet the circle b'P a' in
a', and join Pa', then a P and Pa' are in the same
straight line.
The construction and proof for the hypocycloid
(fig. 38) are similar, writing only - R for R.
The curve enveloped by a diameter of the gene-
rating circle of an epicycloid produced by the rolling
72
GEOMETRY OF CYCLOIDS.
of a circle larger than the fixed circle, and touching
this circle internally, will be an epicycloid if the radius
of the rolling circle exceeds the diameter of the fixed
circle; but if the rolling circle has a radius less than
the diameter of the fixed circle, the curve enveloped
by a diameter of the rolling circle will be a hypocycloid.
The proof for both cases is easily derived from the
demonstration in pp. 68, 69, the dotted line and circle.
of fig. 37 showing the nature of the construction.
The curve enveloped by a diameter of the gene-
rating circle of a hypocycloid is shown by reasoning
similar to that in pp. 68, 69, to be the hypocycloid
traced out by a generating circle of half the diameter,
alternate cusps of the smaller hypocycloid agreeing with
successive cusps of the larger. The dotted line and
circle in fig. 38 indicate the requisite construction when
the rolling circle has a diameter greater than F.
THE FOUR-POINTED HYPOCYCLOID.
It follows from the property indicated in the preced-
ing paragraph that the diameter OB of the rolling circle
BQO (fig. 35) constantly touches a hypocycloid having
four cusps, at B, D, B', and D'. As the extremities
p and P of the diameter lie always on BB' and D'D
respectively, we have in this result the solution of the
problem to determine the envelope of a finite straight
line pc P, whose extremities slide along the fixed straight
lines BOB' and DOD' at right angles to each other.'"
The direct proof is simple, however. Thus let p P be
C
THE EPICYCLOID AND HYPOCYCLOID.
22
the straight line in any position. Complete the rect-
angle Opb P, whose diagonals Ob and p P are equal
and bisect each other in c. With centre O and radius
Ob, describe the circle BbDB', and draw b P' perpen-
dicular to pP. Then a circle on cb, as diameter, passes
through P'. Let c' be the centre of this circle; then
c'b=10b: but ▲ bc'P' 2 ≤ b c P′ = 4 ≤ b OB;
therefore arc b P' arc pB. Hence P' is a point on
the hypocycloid traced out by circle b P' c rolling on
the inside of the circle BDB', the cusps lying at B,
D, B', and D'.*
=
THE CARDIOID.
The cardioid, or epicycloid traced by a point on
the circumference of a circle rolling on an equal circle,
has some interesting properties. Here, however, space
cannot be found for more than a few words about the
chief characteristics which distinguish this curve.
Let AQB (fig. 39) be the rolling circle, B b S the
fixed circle, A the tracing point when at the vertex,
so that ACBOS is diametral. Now let a Pb be another
*The four-pointed hypocycloid BDB'D' is interesting in many
respects. It bears the same relation to the evolute of the ellipse
that the circle bears to the ellipse. Its equation may readily be
obtained. Thus, let DOD' be axis of x, BOB' axis of y, and r, y
co-ordinates of P'; put Bob = 0; OB = a; then,
x = p P' sin - pb sin 0-a sin ³ 0;
0 =
PP' cos 0 bp cos2 0=a cos³ 0;
=
3
У
.. x²+y=a³, the required equation.
2
Come + y²
b2
The equation to the evolute of the ellipse
び
​#
(*) + (42) = 1; where a' = a —
44
>
and l'
*
= 1) is
V.
a²
b
74
GEOMETRY OF CYCLOIDS.
position of the rolling circle, a cb O diametral. Draw
the common tangent bm, meeting ABS in m; draw
also mpc P through c, the centre of circle a Pb; join
PS, cutting mb in n; bk perpendicular to AS; and
join b P, b S. Then, since cb = b 0, and 1 m is perp.
to c O, triangle cbm triangle Obm in all respects;
and arc bp arc b B. Wherefore, P is the position
of the tracing point; Pa is the tangent to the cardioid
FIG. 39. (Produce Pb to meet Ob BS in g; join pb, b' P′.)
A
ఆ
g
=

C
V
ሙ
B
R
Q
U
a
}
I
es
S
P
at P, Pb is the normal. The curve will manifestly
have the shape indicated in the figure, the only cusp
being at S, and the tracing point returning to A after
tracing the other half SP'A. AS divides the curve
symmetrically.
Note first that Pn=nS; or the cardioid is similar
to the curve obtained by drawing perpendiculars from
THE EPICYCLOID AND HYPOCYCLOID.
75
S (as Sn) to tangents at all points of a circle Bb S.
We might then obtain the cardioid P'APS, by draw-
ing a circle on AS as diameter, and from S letting
perpendiculars fall on tangents to this circle. This
property is expressed by saying that the pedal of a
circle with respect to a fixed point on its circumference
is a cardioid.
Secondly,
n Pb alt. Pbc4bPm =
= ▲
bSm; hence Sn-Sk. So that if we draw any line
Sn from S, and from b, in which the bisector of BS n
meets the circle on SB as diameter, draw bn per-
pendicular to Sn, the locus of n is a cardioid. [The
larger cardioid, P'APS, would be similarly described
by producing Sn and Sb, and from point in which
Sb meets circle on AS as diameter, letting fall perpen-
dicular on Sn (meeting S» in P).]
Or, thirdly, we may obtain a cardioid by taking
any finite line as SB, drawing Bb square to bisector
of any angle BS n, and from b drawing bn square to
Sn: the locus of n will be a cardioid.
Fourthly, draw circle OGD about S as centre cut-
ting Sn in e, and draw el perp. to SB; then Sn =
S k = SO+ O k = SD + S 1 (because Se is parallel and
equal to Ob)=DI. Thus the cardioid may be obtained
by drawing radii as Se to a fixed circle OGD, and on
Se, produced if necessary, taking n so that Sn = D l.
This is the usual definition of the cardioid.
Fifthly, let Pn S cut circle Bb S in f. Then pro-
ducing Pb to meet circle b BS in g, we have b P = bg,
and rectangle P b . P g (= 2P 7²) = rectangle Pƒ, PS
76
GEOMETRY OF CYCLOIDS.
•*
2 rect. Pf. Pn. Hence Pb² Pn. Pƒ, and Pb ƒ
is a right angle. Wherefore pbf is a straight line,
and (Pb bisecting angle p Pƒ) Pƒ=Pp= SB. Hence
the cardioid P'APS may be obtained by drawing
straight lines as Sf to circumference of circle BƒS,
and taking on Sf produced Pf= BS. (The cardioid
is therefore a limaçon.)*
=
Cor. If we draw s'S s tangent to circle BfS at S,
and take Ss Ss' = BS, then s, s' are points on the car-
dioid. We see that s's SA; and it is easily seen that if
P'SP is a straight line through S, PP'SA. For,
according to the definition just obtained, we should
have P' on a point on the curve if ƒ SP' = BS=ƒP;
therefore P'SP=SA. It may be well, however, to
show how this can be directly proved when the cardioid
is regarded as an epicycloid. For this purpose we have
only to notice that if on ab O produced we set centre
of generating circle as at c', then 'P', the arc of the
generating circle to tracing point P', must equal 'S,
wherefore P'S is parallel to c'O, and in same straight
line with PS. But since PSP' is parallel to cc' joining
centres of equal circles a P b, b'P'a', a P is parallel to
'P', and therefore PP'b'a=2ba = SA. This pro-
perty gives a method of tracing out the cardioid me-
chanically. For if there be a circular groove as BfS,
and we take a ruler of length SA (twice diameter of
groove), having a vertical pencil point at each extremity
* The limaçon is the curve obtained by drawing radii rectores to
a circle from a point on its circumference, and producing and re-
ducing all of them by a constant length.
THE EPICYCLOID AND HYPOCYCLOID.
77
and a point at its middle point moving in the groove,
while the rod itself always passes through S (either
through a small ring there or by having a projecting
point at S and a groove along the rod), the pencils at
the extremities of the rod will trace out the cardioid.
While one pencil moves over APs the other will move
over SP's', and while the former passes on from s to S,
the latter passes on from s' to A, completing the tracing
of the curve.
The evolute of the cardioid As Ss' is a cardioid
Sr O, having its vertex at S, cusp at d, on OB, such
that O d= OB, and linear dimensions equal to one-
third those of the cardioid A s s'.
S, the cusp of the cardioid, is also called the focus.
Since Pb is the normal at P and angle SP b = = angle
bPm, we perceive that if S be a point of light, and
the arc of the cardioid reflect the rays, P m will be the
course of the ray reflected from P. Hence the caustic
or envelope of the reflected rays will be the curve
constantly touched by the diameter Pp in the tracing
out of the cardioid. This curve, as shown at pp. 68,
69, will be the epicycloid traced out by a circle whose
diameter CB, and which has S as one of its cusps.
The other cusp will be at B, and the curve will have
the position shown by the dotted curve BRS and its
companion lobe in fig. 39.
Let us now determine how far the cardioid ranges
in distance from the diameter AS, and beyond ss'.
We note that (i.) when P (fig. 39) is at the greatest
possible distance from AS, the tangent Pa must be
78
GEOMETRY OF CYCLOIDS.
parallel to AS; and (ii.) when P' is at its greatest
distance from sSs, the tangent at P' must be parallel
to s s', and therefore P'b', the normal, must be parallel
to SA. Wherefore, since P'b' has been shown to be
parallel to Pa, we see that when P is at its greatest
distance from SA, P' is at its greatest distance from
ss'. Now, when Pa is parallel to AS, so also is pbf,
and as the arc bf arc Bb, the position of bf is at
once assigned: for if a chord bƒ (fig. 40) is parallel to
=
CL
IN
P n = 3 b n =
FIG. 40.
A

SO
On =
2
72
S
f
C
BS, arc B b = arc Sf, and since arc bf arc Bb
Sf, we have Bb=3rd the semi-circumference Bb S,
and the angle BSƒ= two-thirds of a right angle.
Sƒ= SO = SP'; and SP = 3SO. Also,
SO
3 3
2
2
; and Sm
S
a
SO; and P'n =
1/3
2
SO.
It follows from the parallelism of the tangent Pa
and the normal P'b', that when the cardioid is being
THE EPICYCLOID AND HYPOCYCLOID.
79
described by the continuous motion above indicated, one
end of the rod is always moving in a direction at right
angles to that of the other end of the rod. Thus the
tangents and normals at P and P' (fig. 39) intersect
on the circle which has PP' for its diameter. The
normals also intersect on the circle B g b' (at g), and the
tangents on the circle having centre O and radius OA.
Cor. The curve cuts s s' at equal angles, each equal
to half a right angle.
THE BICUSPID EPICYCLOID.
The epicycloid with two cusps (the dotted curve of
fig. 39, which, from its shape, we may call the nephroid)
presents also many interesting relations. I merely
indicate, however, in a few words the chief points
to be noticed at the outset of an inquiry into the re-
lations of the bicuspid epicycloid.
Let P (fig. 41) be a point on the epicycloidal arc
traced by the rolling of AQB on the circle DBD',
whose radius BO = AB.*
Let a Pb be position of rolling circle through P.
Draw common tangent bt, meeting OA in t; and join
ta, cutting a P in p. Then, since Obbu, angle
tOb= angle ta O, and arc pb=arc Bb; wherefore
pc P is a diameter of circle APb. Angle ca P =
compt. of cap compt. of t0b-angle bOD'. Hence
TO a is isosceles, and tb T is a straight line. Draw
* The curve has been omitted from fig. 41. The student should
trace it in pencil from the cusp D through A and P (touching PT)
to D'-forming a branch like either half of the dotted curve of fig. 39.
80
GEOMETRY OF CYCLOIDS.
bn perpendicular to OT, and join n P, b P, bp; then
triangle p b P= triangle On b in all respects, b P = b n,
and Pm = m n. Wherefore the bicuspid epicycloid
may be described thus: draw from any point b on
FIG. 41. (Join b p.)
D
A

ט
V
B
A
72
m
D'
T
circle DBD', bn perpendicular to fixed diameter,
DOD', and nm perpendicular to tangent at b; then
if nm is produced to P so that m P = mn the locus
of P is a bicuspid epicycloid.
THE INVOLUTE OF THE CIRCLE
REGARDED AS AN EPICYCLOID.
The curve traced by a point on a straight line which
rolls on a circle in the same plane may be regarded as
an epicycloid whose generating circle has an infinite
radius. The curve is the involute of the circle. Thus,
let DQB (fig. 42) be a circle, T'DT a tangent at D,
and let this tangent roll without sliding over the circle
DQB (DOB a diameter), the point D tracing out the
curve DP. Then when the tangent has the position
PB'p, having rolled over the arc DQB' once only, B'P
THE EPICYCLOID AND HYPOCYCLOID.
81
having been in contact with every point of the arc
B'QD is equal in length to this arc. Therefore the
point P lies on that involute of the circle DQB' which
commences at the point D. But T'DT may be re-
garded as part of a circle of infinite radius touching
the circle DQB' in D, and the arc DPR therefore as
an epicycloid. In fact this arc is the extreme case of
the epicycloid when the radius of the rolling circle is
indefinitely enlarged, precisely as the right cycloid is
the extreme case when the radius of the fixed circle is
indefinitely enlarged. The part of the curve near to
DQB manifestly has the shape shown in the figure, D
being the cusp. The branches of the curve extend
without limit outwards. It is obvious that if the line
B'P be produced to meet the next whorl of DPR (not
the curve Dp R), the portion of this line intercepted
between P and that whorl will be equal to the circum-
ference of the circle DQB. Again, if PB' produced
meet the branch Dp R in p, PB'p is also equal to the
circumference of DQB'; for B'Parc B'QD, and
B'parc B'B'D. The straight line r DR, perp. to
T'DT, passes through all the points of intersection of
the two branches, for the curve must necessarily be
symmetrical on either side of OD from the way in
which it is traced out. Qt, the tangent parallel to
OD, and equal to the quadrant QD, determines the
greatest range of the branch DtP above DT, for the
curve is perp. to Qt at t; also, if Qt be produced both
ways indefinitely, its intersections with the prolongation
of DtP above DT determine the greatest range of
G
82
GEOMETRY OF CYCLOIDS.
each successive whorl of that branch above DT, while
its intersections with the branch Dp R below DT
determine the greatest range of each whorl of that
branch below DT. Similarly of the tangent to DQB
parallel to Qt, and of the tangents perp. to DOB.
Many other relations of a similar kind exist which the
student will have no difficulty in discovering for him-
self. Both branches manifestly approach more and
FIG. 42.

T
Que
R
P
NA
O
B
Q
А
N
M M
T
more nearly to the circular form as their distance from
the centre increases; for from the manner of generation
the normals to the curve touch the circle DQB, and
for branches at an indefinitely great distance the di-
mensions of DQB are relatively evanescent, wherefore
the normal at any remote point of the curve is inclined
at an evanescent angle to the line joining that point
with O. Or, a whorl of the spiral may be regarded as
changing its distance from the fixed point O during one
THE EPICYCLOID AND HYPOCYCLOID.
83
complete circuit by a distance, as p' R', p'R", &c.
(these lines being diametral), equal to the circumfer-
ence of DQB, and this distance vanishes compared with
the radius vector of the spiral in its remote parts, so
that the radii vectores of a single whorl, though differ-
ing by a finite quantity and therefore not absolutely
equal, are yet in a ratio of equality; and in that sense
the whorl corresponds with the definition of a circle.
The circle DQB is the evolute of the curve RpDPR,
&c.; but we have seen (second section, Prop. XII.)
that the evolute of an epicycloid is a similar epicycloid:
hence we must regard the circle DQB as consisting of
an infinite number of infinitely close whorls, similar to
the remote whorls of the curve Rp DPR.
The rectification and quadrature of the epicycloid
in the preceding section manifestly fail for the involute
of the circle regarded as an epicycloid. But it is easy,
as follows, to compare the length of any arc DtP with
the corresponding arc DQB' of the fixed circle, and
the area DtPB'Q with the area of the sector DQB'O.
ARC OF THE INVOLUTE OF THE CIRCLE.
Let PP (fig. 42) be an elementary arc, PB', P'B"
the corresponding positions of the tracing tangent, then
since OB' is perp. to B'P and OB" to B'P', the angle
B'OB" = the angle PB'P', in the limit. Hence
:
Arc PP' arc B'B' :: B′P : OB' :: arc DQB': DO.
Now in Dr take Dd OD; and in DT take DM =
=
€ 2
84
GEOMETRY OF CYCLOIDS.
arc DQB', and MM' arc B'B". Complete the rect-
angles Dd NM, NM'. Also draw MK DM, perp.
to DM, and complete the rectangle KM'. Then if
we represent the arc B'B" by the area NM', the arc
PP' will be represented by the area KM', for
KM: NM' :: P'P: B'B".
But since KM DM, K lies on a straight line, DK,
bisecting the angle rDT; and every element of arc as
PP' has a corresponding representative element of
area, as KM', in the space KDM. Therefore the
length of the arc DtP is represented ultimately by
the area DMK; or
=
Arc DtP: arc DQB':: area DMK: area d M
DM. KM: DM.OD
or,
::
::
:: arc DQB': BD.
DM OD (since DM = KM)
arc DQB': OD
That is, the arc DtP is a third proportional to BD
and the arc DQB'.
This is the relation required. It may conveniently
be replaced by the following:
Cor. Rect. under arc DtP and BD=square on B'P,
(B'P)2
BD
Arc DtP=
AREA BETWEEN CIRCLE, ITS INVOLUTE, AND
THE NORMAL TO INVOLUTE.
Take Dn = OD and complete the rectangle » M.
Draw ML perp. to DM, cutting n N' parallel to DM
in N', and take L so that
THE EPICYCLOID AND HYPOCYCLOID.
85
ML: MN' (= Dn) :: (PB′)² (= DM²) : (OB′)².
Complete the rectangle LM'. Then by construction
Area N'M' = triangle OB'B" ultimately;
and ultimately
▲ B'PP': A OB'B" :: (PB')2: (OB')2
::rect. LM': rect. NM'.
Rect. LM' triangle B'PP'.
Therefore
Now from the construction L is a point on a parabola
DIL, having D as vertex and n as focus, or BD as
parameter. Hence, every elementary triangle as B'PP'
has a corresponding representative elementary rect-
angle LM'. Therefore
=
Area DtPB'Q = parabolic area DILM
rect. under DM. LM.
DM=arc DQB';
Now
and by property of parabola,
.. LM . BD=-(DM)² = (PB′)²;
or LM is a third proportional to BD and PB',
and therefore, as shown in last page,
LM = arc DtP,
... area DtPB'Q=3. rect. under arcs DQB' and DtP.
(B'P)3
Cor. Area DtPB'Q 13 OD
CENTRE OF GRAVITY OF EPICYCLOIDAL AND
HYPOCYCLOIDAL ARCS AND AREAS.
There is no simple geometrical method for de-
termining the position of the centre of gravity of an
86
GEOMETRY OF CYCLOIDS.
epicycloidal or hypocycloidal arc or area; and there-
fore, strictly speaking, these problems do not belong to
my subject. But it may be as well to indicate the
analytical method of solving them, which has not
hitherto, so far as I know, been discussed in any
mathematical treatise. I shall consider the case of
the epicycloid only. The solution for the hypocycloid
is similar, and the result only differs in the sign of R,
the radius of the rolling circle.
FIG. 43.
A

ROR
m
C
B
-X
Q
B b
모
​pand
R
P
E
B'
T
X
Q
E
T
A
77.
B
FIG. 44.
First, then, to determine the ordinates x, y of the
THE EPICYCLOID AND HYPOCYCLOID.
87
centre of gravity of the arc APD, fig. 43 (fig. 44 for
the hypocycloid), O being taken as origin, OX perp.
to OA as axis of x, and OA as axis of Y.
Let A'C'P=0; PC'q=d0. Then,
2R (F+R) Ө
COS.
3.2.do.
arc PP' =
F
Also, if P n is perp. to OA, then ultimately,
moment of arc PP' about OA = P ǹ. PP'
N
= {(F
(F + 2 R) sin
X
and similarly,
2 R(F+R)
F
= { (F+2R) cos
R
X
F+R
2
F
sin
Ө
0 + 2 R sin COS
2
2 R (F+R)
F
R
moment of arc PP' about OX=0n. PP'
F
COS
Ө
2
m¸ d◊, say;
Mx
do
◊ + 2 R sin
2 R (F+R)
F
2 R(F+R)
F
F-2 R
2 F
Ө
2
COS
På la
sin
Ө
R
0 + sin
2
(F+2R)
2 F
m, do, say.
We have to integrate these two expressions between
the limits =0, and ≈7, to obtain the moments of the
arc APD around the axes OA and OX.
Ө
Now Sm,de =S [F+2R in F+2R
2 F
d o
F+2R
F
R}
}
3 F+2R] do,
d◊,
да
1
88
:
GEOMETRY OF CYCLOIDS.
0
··S m₂ do = 2 F sin 2
Mx
π
Similarly
+
... x
X
+
S
π
+
0
My d o
F(F+R)
F-2 R
FR
3 F+2R
2 F(F+R)
F-2R
2 FR
3 F+2R
= {(F+2R) sin
2 R(F+R)
F
F+2R
F
sin
R
F
F+2R
4 F
0 +
(F+2R)
F
R2 sin²
sin 2
sin 2
F sin
sin
3 F+2R
2 F
2 R
3
Ө
F-2 R
2 F
and similarly
ÿ = M₂.
To determine X and Y, the coordinates of the centre.
of gravity of the area APDE, we have,—
Area of element A'P'a:
F+2R
F
π
3F+2R
4 F
F-2 R
4 F
and if g be the C. G. of this element, ultimately a
triangle, A′g=}A'P′ =²
2 R
3
sin
ultimately.
Also if gm is perp. to OA,
moment of element A'P'a about OA=gm. area A'P'a,
Ө
d o
Mr
arc APD
2
F+2R
2 F
"
π
Ө
sin COS
2
75
π = M,, say.
π= Mx, say.
π
== Mr;
Ө
R2 sin² do;
2
2
R². ax do, say;
F+2R
2 F
Ө
}
THE EPICYCLOID AND HYPOCYCLOID.
89
and similarly,
moment of element A'P'a about OX=0m. area A'P'a,
2 R
sin 02 sin
3
= { (F+2 R) cos
F+2R
F
+
R
F
+
..X
R2. sin² d o
2
(F+2R) R2
F
Now
Ө
So,de=f [2F-3R sin R
I
F
0 +
Similarly
Lay d0 = (2 F-3 K)
ㅠ
​(3 F +5 R)F
3 (F-R)
FR
3 (2F+R)
6 (F-R)
FR
(3 F+5R) F
sin2
and similarly Y=
3 F +5 R F-R
+
sin
Ө
6
F
6
F
R
2
- Ja, d9 = (2 F-3R) } sin² B --Fsin? F+R,
απ
R
2 F
2 F
F R
2 F
6(2 F+R)
F
2 R
sin
sin
π
2A,
a, dô, say.
sin2
2 F+R
2 F
Ay
sin
F+2R)
F
since area APDE
R2.
R
F
F-R
F
π
2 F+R
F
π
F+R F+R
2
F
R 2 F + Ro] do;
sin
F
F+-2 R
2 F
元
​1
7 =
sin
π = Ax, say.
F
Ar
area APDE
F+2R
2 F
2
sin
Ay, say.
3
}
2 A
π R²;
F+R
F
T
A
t
π
90
GEOMETRY OF CYCLOIDS.
It is easy to obtain in a similar manner X and Y',
the coordinates of the centre of gravity of the area
APDB, though the expressions are rather more cum-
brous. We take such elementary areas as PP' B' in
fig. 27 (fig. 28 for hypocycloid), and find,
R
Moment of element about OA= [(3 F+2R) sin
F
+
5 F +4 R
F
+
5 F +4 R
R
Moment of element about OX=
cos R
X'= (I
B3
B³,
Ө
2
sin
X
Y' = (B³
Ө
R cos
2
These expressions can be easily integrated. It will,
however, be more convenient to proceed as follows:
Moment of area ABDE about OA
F+2R
2 F
COS
ม
Moment of APDB about OA=B³-
Moment of APDB about OX=B³½
F+2R
R² Ax)
:).
F
= [(3
F+2R
2 F
R
= }} [(F+2R)³ - F³] sin2
2 F
Moment of area ABDE about OX
= } [(F+2R)³ — F³] sin RB,³, say.
Ꭱ
3
F
R
Ө
(3 F + 2 R) cos F
Ө
•] R² cos² 2 de.
Ө
dĉ.
?
R². Ay
»).
0]
-
= 14
A
R² cos do.
2
3
Bó, say.
F+2R
F
Ө
F+2R
F
3 F+2R
2 F
-
π R².
F+2R
3 F+2R
2 F
π R2.
F
SCHOL.-It should be noted that these solutions
might be presented geometrically, if it were worth
R² Ar.
R² A,,
THE EPICYCLOID AND HYPOCYCLOID.
91
while; but only at great length and with complicated
diagrams. The student will observe that all the rea-
soning in each demonstration, up to the point where the
integral calculus is employed, is manifestly capable of
being presented geometrically, the ratios dealt with
(including the trigonometrical ones) being those of
lines to lines, areas to areas, or solids to solids (in deal-
ing with moments of areas). Again, the only relations
derived from the integral calculus, are these—
S
a
10
J
sin a 0 d0 =
a
1
a
1
a
(1—cos a) = 2 sin²
2
sin a.
cos a 4 do
a
These (which are in effect one) are both capable of
easy geometrical demonstration, and are in fact de-
monstrated further on in the quadrature of the 'com-
panion to the cycloid.'
The student not familiar with the integral calculus,
will find no difficulty in proving by trigonometrical
series,* that the sum of the series whose general term is
sin
(r taking all integral values from 0 to n), is
a
ra
22
22
G
a
2 sin2 when n is indefinitely increased; and that the
comm
a
sum of the series whose general term is COS is sin a.
τα
N
>
N
These summations (or such as these) suffice for sum-
ming the elements dealt with in the above demon-
stration.
* See the chapter on the Summation of Trigonometrical Series
in Todhunter's Plane Trigonometry.'
(
92
GEOMETRY OF CYCLOIDS.
SECTION III.
TROCHOIDS.
NOTE. Any curve traced by a point, within or without
the circumference of a circle, which rolls without
sliding upon a straight line or circle in the same
plane, is a trochoid; but the term is usually limited
to the right trochoid, and will be so employed through-
out this section.
DEFINITIONS.
The right trochoid is the curve traced out by a
point either within or without the circumference of a
circle, which rolls without sliding upon a fixed straight
line in the same plane.
If the tracing point is within the circle, the trochoid
is called a prolate or inflected cycloid. The shape of
such a trochoid is shown in fig. 45, Plate I.
If the tracing point is outside the circle, the trochoid
is called a curtate or looped cycloid. The shape of
such a trochoid is shown in fig. 46, Plate I.
An epitrochoid is the curve traced out by a point
either within or without the circumference of a circle
which rolls without sliding on a fixed circle in the same
TROCHOIDS.
93
plane, the rolling circle touching the outside of the
fixed circle.
A hypotrochoid is the curve traced out by a point
either within or without the circumference of a circle
which rolls without sliding on a fixed circle in the
same plane, the rolling circle touching the inside of
the fixed circle.
It may readily be shown that every epitrochoid
can be traced out in two ways-viz., either by a point ·
within or without a circle which rolls in external con-
tact with a fixed circle, or by a point without or within
a circle which rolls in internal contact with a fixed
circle of smaller radius. Also every hypotrochoid can
be traced out either by a point within or without a
circle which rolls in internal contact with a fixed
circle of radius larger than rolling circle's diameter, or
by a point without or within a circle which rolls in
internal contact with a larger fixed circle, but of radius
not larger than rolling circle's diameter. Instead,
however, of giving a demonstration of these relations,
after the manner of Prop. I., Section II., I leave the
point for more general demonstration in Section V.
J
In what follows, reference is made to right trochoids,
unless special mention is made of epitrochoids and
hypotrochoids. Either fig. 45 or fig. 46 may be fol-
lowed. The reader is recommended to read the follow-
ing remarks twice over-once with each figure, and to
adopt the same plan with the demonstration of each of
the following propositions.
Let AQB (radius R) be the rolling circle, KL
94
GEOMETRY OF CYCLOIDS.
the fixed straight line. Let the distance of the tracing
point from the centre be r, so that the tracing point
lies on the circumference of the circle a qb, of radius
r, and concentric with AQB. This circle, aqb, is
called the tracing circle. Let D'D be the fixed straight
line, touching the circle AQB in B. Let the centre
of the rolling circle move along a line c' Cc, parallel
to D'D through C, the centre of AQB, in the direction.
shown by the arrow.
Draw ee and dd parallel to
c' Cc, and touching the tracing circle a qb. Then it
is manifest that at regular intervals the tracing point
will fall upon the straight lines e e and d' d. When
at a on the straight line e' e, the tracing point is turn-
ing around the centre of the rolling circle in the direc-
tion in which this centre is advancing, and is at its
greatest distance from the fixed straight line. When
at d' and d, the tracing point is turning round the
centre of the rolling circle in the opposite direction,
and is at its greatest distance from c'e on the side
towards which lies the fixed straight line KL. The
curve will manifestly be symmetrical on either side of
the diameter a Cb, perp. to KL. Therefore ab is
called the axis of the trochoidal curve: d' d is the base;
and a the vertex. The radius Ca, drawn to the
tracing point, may conveniently be called the tracing
radius. D'AD is called the generating base. The
rolling circle AQB is called the generating circle,
and when in the position AQB, is called the central
generating circle. The circle a qb is called the tracing
circle, and when in the position aqb, is called the
TROCHOIDS.
95
1
central tracing circle. The complete trochoid consists
of an infinite number of equal trochoidal arcs, but it is
often convenient to speak of a single trochoidal arc,
d'a d, as the trochoid.
It is clear that if D'c' E', Dc E, be drawn perp.
to the fixed straight line through d' and d, and inter-
secting e'a e in e' and e, respectively, the parts of the
trochoid on either side of d'e' and de are symmetrical
with respect to these lines. Therefore d'e' and de may
conveniently be called secondary axes.
The straight lines e'a e and dbd are tangents to
the trochoid at a, and at d' and d, respectively.
PROPOSITIONS.
PROP. I.-The base of the trochoid is equal to the
circumference of the generating circle (figs. 45, 46).
For dbd D'BD = circumference of the circle
AQB.
Cor. 1. d' b = b' d = half the circumference of the
generating circle.
Cor. 2. Area e d d'e' = 2 rect. a d = 4 rect. C d
7°
R
T
=4 rect. CD = 4 circle AQB.
R
Cor. 3. The base d'bd: circumference of the trac-
ing circle a qb: circumference AQB: circumference
a qb Rr.
::
Cor. 4. Area ed de 4 rect. under
R
R
= 4 rect. under Cb,
r
7°
arc aq b=4
Cb, bd
circle aqb.
96
GEOMETRY OF CYCLOIDS.
PROP. II.—If through p, a point on the trochoidal arc
apd (figs. 47, 48), the straight line pq M be
drawn parallel to the base bd, cutting the central
tracing circle in q, and meeting the axis AB in M;
then,
A
→
Let A'PB', a' p b' be the position of the generating
and tracing circles when the tracing point is at p,
FIG. 47.
M
C
น
B
·
A
M
N
♡
qp =
的
​A'
IM'
B'
a
R
r
a Mq
FIG. 48.
♡
19%
arc aq.

n
71
P
P
7
TAL

L
d
C' their common centre, A'C'B' diametral cutting
p M in M'.
Draw the diameter P p CB. Then it is
TROCHOIDS.
97
M'p;
MM' = q p ; and arc a q
manifest that M 9 - M'
arc a' p. Now ẞ is the point which was at B when the
tracing point was at a, and since every point of the
arc ẞ B' has been in rolling contact with BB', the arc
ß B′ = BB'.
B'
But arc ß B' = arc A'P =
r
Cor. 1. M
R
and BB'= MM'=qp; wherefore qp ==
r
Ip
11
R
R
r
Cor. 2. Since b d = AQB
R
R
ጥ
arc a' p =
arc a q + M
1.
R
7'
agb
R
arc a q″ + N q″
Ağa
για
(arc a g+ arc q b),
2
R
. arc qb necessarily > M q,
it follows that in the case of the prolate cycloid, where
R>r, and therefore
bd> Mp, and the whole arc apd lies on the same
side of de as a b.
y
But in the case of the curtate cycloid (fig. 48),
where R<r, there must be a point d' on a qb where
arc bq'
R
Nq" (drawn perp. to AB),
Ju
and if p" be the point in which N q" produced meets
the trochoid, then will p' fall on e d, for
R
p" N
7
arc aq;
arc aq.
arc a q' + arc b q″) = b d.
The part of the trochoid lying between p" and d mani-
H
98
GEOMETRY OF CYCLOIDS.
!
festly falls on the side of ed remote from a b; and as
the complete curve is symmetrical with respect to e d,
it follows that the curtate cycloid has a loop of the
form p'r dr'. It is also clear that the point p' lies
between D and e, since if L be the point in which BD
cuts the arc a qb, and CL cuts AQB in 7, the arc Bl
is less than BL. The point p' may lie nearer to e
than E does, however, and the arc dr' p' may intersect
a b. It is easily seen either from the mode of genera-
tion or from Cor. 1, that if the ratio r R be small,
the curve may cut ed a great number of times before
the tracing circle has been carried entirely past ed.
Observe that if C q' cuts AQ'B in point Q'
arc BQ' Nq".
=
Cor. 3. Let Mp produced (if necessary, in the
case of curtate cycloid) meet ed in m; then
pm = Mm p M
Ꭱ
r
R
arc b q
M 7.
r
For points of the arc p'rd (fig. 48) this relation still
holds, regarding lines drawn perp. to ed from the right
as negative.
arc a q b
Cor. 4. Arc a' p =
arc b'p
=
M
2
Ꭱ
r
R
•
R
p
arc a q M 9
b b', and
b' d.
Cor. 5. If from p' on pd, p' q' be drawn parallel to
b d to meet a' p b' in q',
q'p': arc pq': R: r.
TROCHOIDS.
99
The proof of this is similar to that of Prop. II., sec. 1,
cor. 5.
SCHOL. The reader will find no difficulty in
making the necessary modifications for the epitrochoid
and hypotrochoid, deducing properties bearing to
those established above the same relation which those
established in Prop. III., section 2, bear to the pro-
perties established in Prop. II., section 1.
PROP. III.-The area d' ad (figs. 45, 46) between
the trochoid and its base: area of the generating circle
:: (b C + b A) : b C :: 2 R + r : r.
This may be proved in either of two ways corre-
sponding in all respects with the two proofs of Prop.
III., section 1. In the first proof, we show that ele-
mentary rectangles qp, q'p' (figs. 49, 50) are equal
to elementary rectangle L7; whence areas a qp, q'b'dp',
together, are equal to rectangle L7; and the area
R
a qb dp to the rectangle C e == circle a qb. Whence
7
area d'ad (figs. 45, 46)
=
©
a qb +
2 R
y
O aqb,
or
O
area d'ad: agb: 2R+rr: be + b A: b C.
c
In the second proof, having drawn the inverted
trochoid a p'd, with ae as half base, and d e as axis,
we show that the elementary rectangles p'p and q'q
are equal, whence
area q'a qarea p'a p; and area a p'dp circle a gb.
-
H 2
100
GEOMETRY OF CYCLOIDS.
The equal areas ap'db and ap de are, therefore, each
(rect. be circle a bq)
=
R
(−1) circle abq;
therefore
R
the area a pdb = (²² + 4) circle a bq ;
2
d'ad = (2 R+r) circle a bq,
and
as before.
g
4 P\?\
C
r
a'
Ap
FIG. 49.

a
FIG. 50.
'a'
D d

SCHOL. The reader will find no difficulty in deal-
ing in like manner, so far as first proof is concerned,
with the area between the epitrochoid or hypotrochoid
and the base. The demonstration bears precisely the
same relation to that of Prop. IV., section 2, which the
above first proof bears to the first proof of Prop. III.,
section 1. We thus show that the area between the
TROCHOIDS.
101
generating semicircle a q b, the arc base b d (radius F)
and the trochoidal arc apd: generating circle a qb
:: 2 CO (bC + b A): b0. b C, that is, in the ratio
compounded of the ratios 2 CO : b O and (b C + b A)
: 6 C.
In all cases,—for cycloid, epicycloid, hypocycloid,
trochoid, epitrochoid, and hypotrochoid,-
area a qb dp in the trochoidal figures = area abde.
From Prop. II., section 1,
C
PROP. IV.—If the cycloid, a PD, and the trochoid,
a pd (figs. 49 and 50), have a common axis ab,
area a qbdp: area a qb DP :: R : r.
q P = arc aq;
but from Prop. II., of the present section,
R
gp= arc aq
2°
P :: R : ",
IP: r,
q
and elem. rectangle pq elem. rectangle q P:: R : r ;
whence
area a qp: area a q P
r.
:: area a 9
bdp: area a 9 b DP :: R : 7.
Cor. Area a qp area a q P :: R; r.
R:
SCHOL. A similar property can be readily esta-
blished for epicy cloids and epitrochoids, or for hypo-
cycloids and hypotrochoids, having a common axis.
In this case, qp, q P, and b d, are concentric arcs, and
in place of elementary rectangles we have elementary
102
GEOMETRY OF CYCLOIDS.
areas like Qp, q'P' of figs. 26 and 27; but the
ratios are the same, and we therefore still find
area a qp area a q P :: area a qb dp: area a q b DP
:: R: r.
:
PROP. V.—If p (figs. 51, 52) is a point in a trochoidal
arc, a' p b', the tracing circle when the tracing point
is at p, a' C'b' diametral, meeting the generating
base in B', then B'p is the normal at p; and if
Tat is the tangent to the tracing circle at a',
Tp,t p, tangents to the trochoid and tracing circle
respectively at p, then
Tt at :: R:
FIG. 51.
K
Mi
R: r.
FIG. 52.

T
p'
Since, when the tracing point is at p, the generating
circle is turning around the point B', the direction of
the tracing point's motion at p must be at right
angles to B'p, which is, therefore, the normal at p.
The tangent p T at p is therefore perp. to p B'. Also,
TROCHOIDS.
103
since pa is perp. to pb', and pt to C'p, triangle
Ta' is similar to p B'b', and pa't to pb'C'; there-
fore
P
Tt: at: B'C': C'b:: Rr.
Another Demonstration.
From p and p' (near p), on the trochoidal arc, draw
p M, p'M' perp. to a'b', p'M' cutting a'p b' in q. Then
q p′ : p q :: C′B′ · C′b′ (=C'p),
P
Prop. II., cor. 5, and since ultimately the sides qp',
qp are perp. to the sides C' B', C'p,
=
angle p q P' angle p C'B'.
Hence the triangle p q p' is ultimately similar to the
triangle p C'B', and p p', the third side of one, is ulti-
mately perp. to p B', the third side of the other.
Wherefore p B' is the normal at p. And, as in the
preceding proof, Tt: a't:: C'B' : C'b' :: R : r.
Cor. 1. Triangle pqp' being similar to triangle
p C'B',
Pp' pq: B'p: C'p.
:
Cor. 2. If pm be perp. to p'M', and p T cut or
meet a'b' in K, then pp' m is in the limit similar to
triangles p B'M, Kp M, KB' p.
Cor. 3. If B'p cut p' M' in l, the triangles lpm
and lp'p are similar to the four triangles named in cor.
1. Also, lpg is similar to Kp C', and q p p' to C'p B'.
Wherefore
lq qp' :: KC': C'B'::p N: NB'.
:
104
GEOMETRY OF CYCLOIDS.
Cor. 4. If p b' cut p' M'in k, kp p' is similar to
a' p B', and kp q to a' p C'. Wherefore
pq=qk, and kq q p' :: p q q pr: R.
:
:
Cor. 5. If in the case of the prolate cycloid, illus-
trated in fig. 51, the tracing point is at r, where the
tangent from B' meets the tracing circle a q b', then
the normal B'r has its greatest inclination to a'B',
and its least inclination to the base. It is manifest,
therefore, that r is a point of inflection. At the point
of the prolate cycloid corresponding to r', in which
B'p cuts the tracing circle, the tangent is parallel to
the tangent at p.
Cor. 6. If in the case of the curtate cycloid, illus-
trated in fig. 52, the tracing point is at r on the generat-
ing base, the normal B'r coincides with the generating
base. Therefore the curtate cycloid cuts the generat-
ing base at right angles.
Cor. 7. B'q produced to meet p p' in n is ultimately
perp. to p p', and if C'N is drawn perp. to p B', p q n
is similar to p C'N, and p'qn to B'C'N; and
Pp' pnp B': p N.
C
SCHOL. It is easy to prove that p B' is the normal
in the case of epitrochoid or hypotrochoid. We have
only to draw C's parallel to the line joining p with the
centre of the fixed circle, to meet p B',* and proceed
as in Prop. V., section 2. (In both figs. C's is drawn
for the case of the epitrochoid; C's', for the case of
My
The reader will note that, in fig. 51, C's' does not extend far
enough. It should be produced to meet p B'.
TROCHOIDS.
105
the hypotrochoid). If, in the former case, the straight
line joining p with O, the centre of the fixed circle, be
perp. to B'p, which can only happen when r> R (fig.
51), the tangent at p passes through O. This deter-
mines the position of the tangent from the centre to
the curtate epicy cloid corresponding to the direction of
the stationary point in the looped epitrochoid, regarded
as a planetary curve. It is well to note the construction
for determining this point. Produce C'b' (fig. 51) to
O, the centre of the fixed circle, and on B'O describe
a semicircle cutting a'p b' in r'; then B'r' is perp. to
r'O, and therefore a circle described about O as centre,
with radius Or', intersects the curtate epicycloid in
the point where the tangent passes through O. This
relation is demonstrated and dealt with under Prop. X.
Cor. 8. In the case of epitrochoids and hypotro-
choids the triangle p qp' is similar-not to p CB'—
but to p C's (the s accented throughout for hypotro-
choid);
p p' : p q :: p s : p C',
and pp': np::ps: PN.
Since then Np and n p are the same for the epi-
trochoid or hypotrochoid as for the right trochoid, with
the same generating and tracing circles (and, of course,
the same angle, p C'a', between tracing radius and
diametral), while
p B' B's: F: R,
and therefore p B′ : p s :: B′O : C'O (see figs. 28 and
B'
29), it follows that p p', regarded as an arc of an epitro-
106
GEOMETRY OF CYCLOIDS.
choid or hypotrochoid, bears to pp', regarded as an
arc of a trochoid (p q being the same for both), the ratio
sp : p B', or C′O : B′O, or F±R : R (the upper sign
for epitrochoid, the lower for hypotrochoid).
The student will find it a useful exercise to com-
plete the construction indicated in the scholium, noting
that the figs. 51 and 52 are correct for the cases there
considered, as well as for the case considered in the
text, except only that the lines p M and p'y M' must
be concentric with the generating base through B'—
that is, must have for centre the point O mentioned
in the scholium.
PROP. VI.—From a point p (figs. 53 and 54), on the
trochoid ap d, above the line of centres cc' C, let q p
be drawn parallel to c C to meet the central tracing
circle a cb in I, and qn, pm, perp. to c C′; then, if
the rectangle a c nf be completed,
area a hq p+rect. pn: rect. cf:: R: r.
And if from p' on a pd below c C, p'q' parallel to c C
meet a c′b in q' ; q'n', p'm' are drawn perp. to c C ;
and rect. n cbf' is completed, then
area a h c'q'p'-rect. p'n : rect. cf' :: R: r.
Let a PD be a semi-cycloid having a b as axis;
then it is easily seen that every element of either area
a hq p + p nor ahq'p'-p'n parallel to c C, bears to
the corresponding element for the case of cycloid a PD,
the ratio R r; and therefore the sum of all such ele-
TROCHOIDS.
107
ments of either area in case of trochoid : sum of all
such elements of either area in case of cycloid (¿.e.,
a
¿
"}'
FIG. 53.

D
f
FIG. 54.
cf or cf, as shown in Prop. V. sec. 1) :: R: r.
That is,
area a h
qp+rect. qm: rect. cf
area a h q'p' - rect. g'm': rect. c'f'
Cor. Area ac'bdrrect. cd=
::R: r.
Ꭱ
R
T
. circle a qb
(Prop. I., cor. 4). Thus we have here another de-
monstration of the area of trochoid.
108
GEOMETRY OF CYCLOIDS.
PROP. VII.-Let a (fig. 55) be the vertex of the
trochoidal are ap, a'p b' the tracing circle through p,
a' C′b' diametral, A' C'B' the corresponding diameter
of generating circle. Describe the quadrant A'PA"
having b' as centre and b'A' as radius; produce b'p to
meet A'PA" in P; and draw Pl perp. to b′A″.
Then, if b'B' = b'B', and B"PA", an elliptic
quadrant having b'B" and b'A' as semi-axes, inter-
sect Pl in P',
arc ap = twice the elliptic arc B'P′.
Let p'* be a point on the trochoid near p, and let
p'q parallel to the base meet a'p b' in q. Produce b'q
A
FIG. 55.

A"
to meet A'PA" in Q; draw QL perp. to b’A″, cut-
ting B'Q'A" in Q'. Join a'p, B'p, and draw b'n
parallel to B'p (dividing a'p in n, so that a'n: np
:: a'b' : b'B' :: A'B' · B'b'′). Join C'p, PQ, and P'Q'.
The secants PQ, P'Q' being ultimately tangents at
* p' does not lie on Pl.
TROCHOIDS.
109
P and P', meet ultimately when produced on b'A" ; let
them thus meet in T.
Then P'Q' PQ :: P'T : PT :: b'n b'a' (since tri-
angle a'b'p is similar to PT1, and a'p and Pl are
similarly divided in n and P' respectively) :: B'p: B'a'.
Also, PQ pq :: A'b' (=a'B'): a'b'
(because Pb'Q is an angle at centre of quadrant A'PA"
and at circumference of semicircle a'p b').* Where-
fore, ex æquali,
P'Q pq :: B'p a'b'. But
pp': p q :: B'p: C'p (Prop. V., cor.1):: 2 B'p: a'b';
therefore,
Pp' = 2 P'Q'.
But p p' and 2P'Q'are increments of arc ap and are
B''P' respectively, which arcs begin together.
Therefore,
arc a p=2 arc B'P'.
Cor. The arcs apd (figs. 45 and 46) = elliptic
are B'A'B', and arc d'ad circumference of an
ellipse having semi-axes b A, b B, that is, R+r and
R~r.
PROP. VIII.-If a'p b' (figs. 56 and 57) is the
position of the tracing circle through p, a'b' diame-
tral, a b the axis, and pb' be joined, then
area a pb'b: sect. area a bq (or a'b'ph)
area pb'd segment bs q (or b' fp)
:: 2 R+r: r.
Let a PD be a cycloid, having ab as axis, and let
Pp be parallel to bd; then area a qb B'P = 2 sec-
*
PQ
A'b'
= circ. meas. of p b'q=circ. meas. of p C'q = } PI_P_1
C'aa'b''
110
GEOMETRY OF CYCLOIDS.
torial area A'B'P. But every element of the area
a qb b'p parallel to base b d (as in Prop. III.): corre-
sponding element in case of cycloid :: Rr. Wherefore
area a qbb'p sectorial area ab q:: 2 R r, and area
ac bb' sectorial area a bq :: 2 R+r: r. Similarly
area pb'd segment b s q:: 2 R+r: r.
FIG. 56.

M
Z
a
n
♥
a'A'
B
b
Z
H
H2
C
D
area q s b d p =rect. bm, q P +
d
FIG. 57.
Cor. 1. Area pfb'd segment pfb' :: 2 R: r.
Cor. 2. Area a qbb'p: sectorial area abq:: 2 R: r.
Cor. 3. If p q produced meet a b in m,
2 R
segment b sq.
r
SCHOL. Two independent methods of demonstra-
ting the area of trochoids can be derived from the above
proposition, as in the case of cycloid. For, carrying p
to d, we have area a p db: circle a qb :: 2 R+r: r,
as in Prop. III.
The proof may be extended to epitrochoids and
hypotrochoids, and the following proportion esta-
blished:-
TROCHOIDS.
111
Area abb'p sectorial area a' b p
:: area b'p d seg. b sq
:: (2 CO+BO) (2 R + r): BO . r, where BO is the
radius of the base, and CO is the radius of the arc of
centres, or
:: (3 F±2R) (2 R+r): F. r
(where F is the radius of fixed circle), the upper sign
for epitrochoid, the lower for hypotrochoid.
PROP. IX.-To determine the area of the loop of the
curtate cycloid a p d, fig. 48.
By cor. 3, Prop. VIII., area q'p'r db, fig. 48,
(=rect. Nd+ loop r'r — area N bq")
½
2 R
=rect. b N, q'p" + seg. q'Lb;
+
... loop r r' =
.loop r'r = area Nb q″- rect. under b N, Nq'
2 R
seg. q'L b
T
2 R+r
r
4 R+2r
**
ľ
seg. q'L b-triangle b N q' ;
seg. q'L b-rect. N n.
PROP. X. With the same construction as in Proposi-
tion VIII., area a pha': segment a'hp::2 R: r.
:
Since area a q p
area AQP :: Rr :: area
aqpha' aq PHA' (PHA' being the arc of tracing
○ A'PB', for cycloid, not wholly shown in the figure);
:
112
GEOMETRY OF CYCLOIDS.
it follows that area apha' area a PHA':: R: r.
But area a PHA' = 2 segment A'HP or 2 segt. a'h p;
... area a pha' segt. a' hp :: 2 R: r.
Cor. 1. Area apdge circle e q'd:: 2 R r.
: :
ap de + circle a q b,
1224
Since a p d q'e
and rect. be: circle e q'd:: 4 R: r, it follows that
rect. be area ap de + circle a q b b::2
as in schol. to Prop. III., so that we have here a new
demonstration of the area.
2:1
Cor. 2. In the case of the prolate cycloid, fig. 57,
in which po' does not intersect the arc a p,
area apa' segment a'h
2 R-r: r.
P: 2 R
circle e q'd
Cor. 3. Proceeding to d, area ap de
:: 2 R-r: r, in case of prolate cycloid.
Cor. 4. In the case of the curtate cycloid, fig. 56,
pa' cuts the curve in some point k, between p and a'.
Here then
area a ka'
area kp: segment a'h p :: 2 R−r:r,
or passing to d,
area a re-semi-loop r p'a : circle eq'd :: 2 R−r : r.
SCHOL.-Another independent demonstration of
the area of trochoids is worthy of notice. Let us suppose
that the circle a qb, figs. 49 and 50, slides uniformly
between a e and b d to the position e Qd (e d diametral).
Let p'a'p be the position of the upper segment when
the circle passes through p'p (=q'q, so that the circle
reaches p' and simultaneously), and let a closely
į
}
TROCHOIDS.
113
adjacent segment, as in the figure, give the elementary
areas a'p and a'p". These are ultimately in a ratio of
equality, but they are the respective increments of the
areas a pa', a p'a' (or as actually drawn in the figure,
they are the elementary increments next before the
attainment of these areas a pa', a p'a'), and these
areas begin together. Hence
area a p a' = area a p''a' ;
and carrying the moving circle to its final position,
area a pd Qe area a p''d Q'e area a p db q',
whence the result of Prop. III. follows at once.
PROP. XI.-Let p o (figs. 58-62) be the radius of
curvature at p, on the trochoid; a'p b' the tracing
circle through p. Then, if a' C'b' meet the generating
base in B', and C'N be drawn perp. to p B',
pop B' :: p B': p N.
With so much of the construction of Prop. V. as
is indicated in fig. 58 (illustrating the prolate cycloid),
let p'L be the normal at p' (near p). Then
R
arc p q (Prop. II., Cor. 5) = B′ L.
7°
I p'
Join q B'. Now p'L, being parallel to 9 B', is not
parallel to p B', unless the point q falls on p B'; that
is, unless the tangent to the circle a'q b' passes through
B', the case illustrated by fig. 60. In this case the
radius of curvature is infinite, or p is a point of inflec-
I
114
GEOMETRY OF CYCLOIDS.
T
tion. In all other cases, p B' and p'L meet when pro-
duced,―towards B'L, when p'q has to be produced to
meet p B' (in 7), and towards pp' when p B' intersects
FIG. 58.
FIG. 59.
FIG. 61.

ά
-O
b
3
PO
N
B' L
H
FIG. 60.
T
K
D'
N
C
d
0 T
FIG. 62.
T
p'q (in 7) between p' and q, fig. 59.
Then in the limit
in o.
ດ.
H
K
ex
Let them meet
lo IB': lp' 1qpB': pN (Prop. V., cor. 3).
:
That is, ultimately,
opp B'p B' p N.
Cor. Rect. under o p, p N=square on p B'.
SCHOL.-The following construction is indicated for
determining the centre of curvature.
On B'p, pro-
duced if p is beyond N, otherwise not, take p H=p N,
nd on the tangent p KT at p take p T=p B'; then
-H
TROCHOIDS.
115
To perp. to HT will meet p B' produced in o, the
centre of curvature at p. For
op, pH=(p T)²,
that is,
op, p N=(p B')².
ор,
The student will find no difficulty in dealing with
the corresponding demonstration for the curtate cy-
cloid. Fig. 61 gives the construction for one general
case, p above the base; and for the case of a point on
the generating base where B' becomes the centre of
curvature (for the latter case r and r' are put for P
and p', while the letters H, T, and N are accented).
Fig. 62 gives the construction for a general case, p
below the base.
For the vertex, N coincides with C',p N=a'C' = r,
and p B' a'B' R+r.
= =
Therefore,
(R+ 2)²
r
radius of curvature at a =
both for prolate and curtate cycloids.
For the point d, N also coincides with C',p N=r
in absolute length, and must be regarded as negative
in case of prolate cycloid, because N falls outside p B'
beyond p, whereas in case of curtate cycloid N falls
on the same side of p as B', though beyond B'. Also
±p B' = (R − r). Therefore, rad. of curvature at d
(R − r)²
negative for prolate cycloid, and positive
"
2'
for curtate cycloid.
But it is to be noticed that in considering the
curvature in the case of the curtate cycloid as constantly
positive, regard is had to the intrinsic nature of the
curve. If the curvature is considered with reference
1 2
116
GEOMETRY OF CYCLOIDS.
f
to the base, there is a change of sign at the moment.
when N passes the point B', or where the curve cuts
the generating base-viz., at r.
.
At this point r,
(r B')²
= r B' ;
r B'
square on rad. (r B')² = (C′ r)² — (CB')² = r² — R².
radius of curvature =
ΟΙ
PROP. XII.—Let p o (figs. 63, 64) be the radius of cur-
vature at the point p of an epitrochoid or hypotrochoid;
a'pb' the tracing circle through p; and a'b' O dia-
metral, cutting generating base in B'. Draw C'N
perp. to p B'; and C's parallel to p 0 meeting p B'
(produced if necessary) in s. Then
pop B'ps: ps - NB'.
[Two illustrative cases only are dealt with (one of
a prolate epicycloid, one of a prolate hypocycloid). The
student will find no difficulty in modifying the demon-
stration and figure for other cases.]
Let p' be a point near p; p'L the normal at p'; p'q
concentric with generating base B'L, meeting a'p b' in
7. Draw
I
n perp. to pp'; qiin direction perp. to
a'b' to meet p p' in i, and Lh perp. to B'p. Then, as
in case of right trochoid,
R
qi= arc p q = B'L,
p
and triangle B'Lh is equal in all respects to triangle
qin. Also triangles pqn,pqi,pqp' are similar to
triangles p C'N,p C'B',p C's. (See Prop. V., Cors.
and Schol.) Now Lh is parallel to p'p; wherefore,
po: hopp : hL (= n i) :: ps: NB',
or ultimately pop B': vs: (ps - NB').
TROCHOIDS.
117
Cor. Since ps: C'O::p B': B'O, we see that
po : C′O :: (p B')²: (ps-NB') B'O
(p B')² : p B'. C'O— NB'. B′O.
See p. 166. At vertex, and at pt. on base, rad. of cur-
(R+r)² (F+R) R− r)² (F+R)
R² + r (F+R) R² —r (F+R)'
respectively, R being regarded as negative for hypo-
cycloid.
vature
K
FIG. 63.
N
"
and

FIG. 64.

K
SCHOL.-A construction similar to that for the
radius of curvature at points on right trochoids can
readily be obtained. Thus produce B'p to H (as in fig.
58), taking p H-ps-NB'; on the tangent p K take
p T, a mean proportional between p B' and ps; then
To perp. to TH will intersect p B' produced, in o, the
centre of curvature at p. For by the construction
po (ps - NB') = (p T)² = p B' . ps
.. pop B'ps: (ps NB').
"
At a point of inflection the radius of curvature
becomes infinite. Now p B' is always finite, and
į
118
GEOMETRY OF CYCLOIDS.
since ps p B':: C'O: B'O, ps is also necessarily
finite. Wherefore, the radius of curvature can only
become infinite by the vanishing of p s-NB', that is,
when
NB' ps, or Np B's,
or p must have such a position as is shown in figs. 65
and 66, for the epitrochoid and hypotrochoid respec-
tively. Wherefore,
NB' p B': ps p B':: C'O: B'O:: F±R: F
(upper sign for epitrochoid, lower for hypotrochoid),
FIG. 65.
FIG. 66.
- 30
=

N
d
d
or, drawing p I parallel to NC'—that is, perp. to B'N
-to meet C'O in I,
C'B' B'I:: C'O: B'O::FR: F.
Wherefore, the construction for determining points
of inflection is as follows:-Take I in C'O (figs. 65
and 66), so that
C'B': B'I':: C'O' : B'O::F±R: F
or B'I =
C'B'. B'O
C'O
R. F
FR
TROCHOIDS.
119
Then if the circle on IB' as diameter cuts the tracing
circle, as at p, a circle about centre O with radius Op
cuts the epitrochoid or hypotrochoid in its points of
inflection. If the circle on IB' as diameter does not cut
the tracing circle, there are no points of inflection.
Cor. C'B' C'I:: C'O: C'B',
}
R2
and
(C'B')²=C'I. C'O; that is, C'I = FR
If, in case of epitrochoid, I falls at b',-that is, if
C'B': B'b' :: C'O: B'O::FR: F,
the radius is infinite at the point d; but there is no
change of curvature: two points of inflection coincide,
and the curvature has the same sign on both sides of
the double point of inflection. In this case,
C'b' : C'B' :: C'B' : C'O::R: F + R
orr R R F + R.
→
:
This indicates the relation between r, R, and F, when
in the case of epitrochoid the curve just fails, at d, of
becoming concave towards the centre.
If, in case of hypotrochoid, I falls at a', that is, if
C'B' B'a':: C'O: B'O:: FR: F,
the radius is infinite at the vertex ɑ. Two points of
inflection coincide, the curvature having the same sign
on both sides of the double point of inflection. In
this case
C'a': C'B':: C'B': C'O:: R: FR
or r: R:: R: F R.
This indicates the relation between r, R, and F, when,
—
120
GEOMETRY OF CYCLOIDS.
:
in the case of the hypotrochoid, the curve just fails at
a of becoming concave towards the centre.
PROP. XIII.-If p (figs. 65 and 66) be a point of in-
flection of an epitrochoid or hypotrochoid, a'qp the
corresponding position of the generating circle ;
a' CO diametral, meeting the generating base in
B'; pz perp. to B'C' ; and k the centre of semi-
circle B'p I; then will
rect. C'B'.C'I± sq. on C'p = 2 rect. C'k, C'z
(the upper sign for epitrochoid, the lower for hypo-
trochoid).
We have
(C'p)² = (C'z)² + (p z)² = (C'z)² + (k I )² — (k z)²,
and for epitrochoid
C'B'. C'I (C'k)² — (k I)²
... C'B'. C'I + (C'p)² = (C'z)² + (C'k)² — (k z)²
= 2 C'k. C'z.
For hypotrochoid
C'B'. C'I = (I)² - (C'k)²
k
... C'B'. C'I (C'p)² = (k z)² — (C'z)² — (C'k)²
—
= 2 C'k. C'z.
=
SCHOL. This prop. may also be treated in the
manner adopted for the next-i.e., starting from the
relation (Ip)² + (B'p)² = (I B')², and taking triangles
I C'p and B'C'p.
TROCHOIDS.
121
PROP. XIV.-Let p (figs. 67, 68) be the point of the
loop of an epitrochoid or hypotrochoid where the
tangent to the curve passes through the centre of the
fixed circle; o'p b' the corresponding position of the
tracing circle; and a' C'B' diametral, meeting the gene-
rating circle in A' and B'; then, if p K is drawn perp.
to OC",
Rect. OA', C'K = sq. on C'b' + rect. O C', C'B',
for epitrochoid, and
=rect. OC. C'B'-sq. on C'b',
for hypotrochoid.
Since p B' is the normal at p, B'p O is a right
angle, and sq. on B'p+ sq. on p 0
O
sq. on B'O.
FIG. 67.

A
that is,
C
K
ܩ
72
C
PP
P
T
FIG. 68.
C
BAZ
៨
IK
Now (B'p)² = (C'p)² + (C'B')² — 2 C'B' . C'K
and (Op)² = (C'p)² + (C′O)² + 2 C'O . C'K
(lower sign for hypotrochoid)
.·. (B'p)² + (Op)² = 2 (C'p)² + (C'B')² + (C'O)²
— 2 (C'B' + C'O) C'K;
(B'O)² = 2 (C'p)² + (C'B')² + (C'O)²
2 OA'. C'K.
122
GEOMETRY OF CYCLOIDS.
Or, for epitrochoid,
—
2 OA'. C'K = 2 (C'b')² + (C'O)2 + (C'B')² - (B'O)²;
i.e. (Euc. II., 7) OA'. C'K = (C'b')² + OC' . C'B'.
For hypocycloid,
20A'. C'K'=(B′O)² — 2(C′b′)² — (C′O)²—(C'B')²;
i.e. (Euc. II., 4) OA'. C'K' OC'. C'B' —(C'b')².
SCHOL.-This prop. may also be treated in the
manner adopted for the preceding, bisecting KO in »,
and noting that rect. OC' . C'B'= ± [(C'n)² — (n B′)²],
upper sign for epitrochoid, lower for hypotrochoid.
Observe that C'K (regarded as positive or negative,
according as K lies on C'O, or C' on KO)
r² + R² + FR
F+2R
the upper sign for epicycloid, the lower for hypocy-
cloid.
72 (FR) R
F± 2 R
or =
This is the relation existing at a stationary point
in an epicycloidal planetary orbit.
PROP. XV.-If G (figs. 47 and 48) is the centre of
gravity of the trochoidal area d'a d,
b G : 3 R + 2 r :: r : 2 (2 R + r).
Since every elementary rectangle of the part of area
d'a d outside circle a q b, taken parallel to base : corre-
sponding element of part of cycloid having ab as axis
lying outside same circle a qb:: R: r, it follows that
the distance of CG of former areas from bd (along
TROCHOIDS.
123
b C, evidently) = distance of CG of latter areas from
b (along b C)=6C (Prop. XVIII., sec. 1st, cor. 3).
... Mom. of d'a d about b d
and
... bG..
R
2 circle a q b
r
2 R + r
r
area d'a d
O a q b
3 R + 2 r
2
bG.
3r
4
+ circle a qb.r
circle a qb
circle a qb =
2 R + r 3 R + 2 r
2
p
or b G : 3 R + 2 r :: r : 2 (2 R + r).
3 R + 2 r
Cor. b G
2 R + r
2 R+r
για
2 R + r
r
π
r
2
PROP. XVI.—The volume generated by the revolution
of a trochoid about its base is equal to that of a
cylinder having the circle a qb for base and height
equal to the circumference of a circle of radius
R+r; that is, this volume= r²(3R+2 r) π².
(area d'a d) 2π b G
r=aqb (3R+2r) π
= vol. of cylinder having circle a qb for base, and
height equal to circumference of a circle of radius
R+ r; or, vol. = r² (3 R + 2 r) «².
By Guldinus' 2nd prop., vol. =
3 R + 2 r
2 R + r
circle a q b
3 R + 27 circle aq b
r
2
124
GEOMETRY OF CYCLOIDS.
APPENDIX TO SECTION III.
ELLIPTICAL HYPOTRO CHOIDS.
The hypotrochoid becomes an ellipse when the diameter
of the rolling circle is equal to the radius of the fixed
circle.
Let BB'D (fig. 69) be the fixed circle, BQO the
rolling circle, when tracing point a is on the radius
FIG. 69. (Note that two lower a's are Greek.)

=
B
b
a
a
Q
с
7 N
M
d
BCO. We have already scen (p. 68) that when the
circle has rolled to position B'A'O, the tracing radius
has its extremity A' on OD perp. to OB, and B'A' is
perp. to OD (OC'B' being diametral). Take C'a' on
C'A', equal to Ca, then a' is the tracing point. Taking
съ Ca, describe arc b b'd about O as centre, cutting
OB' in b'. Then C'b' C'a', and... b'a' is parallel to
B'A' and perp. to OD, which let it meet in M, and
draw C'N perp. to B'A', bisecting b'a' in n. Then
Spanjem p
07
TROCIIOIDS.
125
a'M: a'n :: a' A' : a'C' :: a O: a C
... a'M : U'M ( = a'M + 2 a'n): a 0: Ob.
Wherefore a' is a point on an ellipse having O a
as semi-minor axis, and bb'd as auxiliary circle,-
i.e., having Od and O a (or R + r and Rr) as semi-
axes.
M
If r > R, or the tracing point is in CO produced,
as at a, it may be shown in like manner that when the
tracing radius has any other position C'A'a', the
tracing point a' lies on an ellipse having O 8 (DS = 0 a)
and O a as semi-axes, that is, having semi-axes equal
tor + R and r R, respectively.
SCHOL.—An ellipse with given semi-axes, a and b,
can be traced out equally by taking the radius of the
fixed circle equal to (a + b) or (a−b). In the former
case, the tracing radius = (a+b)− b = 1 (a−b); in
½
the latter the tracing radius = ( a − b ) + b = 1 (a+b).
THE TRISECTRIX.
When the radius of the rolling circle of an epitro-
choid is equal to that of the fixed circle, and r = 2 R,
the curve is called the trisectrix. The property of
trisecting angles from which it derives its name may
be thus established.
Let BDB' (fig. 70), centre O, be the fixed circle;
EQD, centre C, the rolling circle (ECDO diametral),
when the tracing radius is in the position CDO, or
(since CD=D0=R=1r) the tracing point is at O.
When the rolling circle is in position B'QA', A'C'B'O
126
GEOMETRY OF CYCLOIDS.
diametral, let C'Pp be the tracing radius, cutting
Then arc PB' = arc B'D; ... angle
OC'p
angle C'OC; and since C'p = OC, the tri-
angles OC'p and C'OC are equal in all respects.
B'QA' in P.
Thus,
and
... angle pOC = angle pC'C
right angle
right angle
right angle -
=
angle p OC' = angle CC'O
angle COC' = angle p C'O;
E
w Qu
P
—
R
[૨
3
angle OC'Cp C'O
angle COC' - angle p C'O
angle p C'O
angle OC p.
FIG. 70.

a
Wherefore, if Op produced meet in R a circle de-
scribed about C as centre, through O,
angle ROC +angle CRO = 2 angle p OC
= 2 right angles - 3 angle OCp;
TROCHOIDS.
127
h
but angle ROC + angle CRO
C
2 right angles — angle RCO;
... angle RCO 3 angle OC p.
=
Hence the trisectrix affords the following construction
for trisecting any given angle RCO. With centre C
and radius CO describe arc OR, cutting CR in R.
Join OR, cutting the loop OBC in p; then angle
RCO = 3 angle p CO, or Cp trisects the angle RCO.
SCHOL. Both the tricuspid epicycloid and the
tricuspid hypocycloid are trisectrices. See Exs. 91, 92.
THE SPIRAL OF ARCHIMEDES REGARDED AS AN
EPITROCHOID.
The curve traced out by a point retaining a fixed
position with respect to a straight line which rolls
without sliding on a circle, in the same plane as line
and point, may be regarded as an epitrochoid, whose
generating circle has an infinite radius.
Supposing the tracing point on R r, fig. 71, T'DT
the rolling straight line, it will easily be seen that if
this point is near D, the curve will resemble DPR,
only instead of a cusp near D there will be simply
strong curvature convex towards O, and two points of
inflexion, one on each side of Rr. When the point is
remote from D, the curve will be concave towards O
throughout. It is easily seen from the formula at page
119 (or it can be readily proved independently*) that
* For the independent geometrical proof, it is only necessary
to show that the tracing point recedes from Rr initially at the
128
GEOMETRY OF CYCLOIDS.
if the tracing point lies at d such that D d = DO, the
radius of curvature will be infinite at d, the two points
of inflexion coinciding there, for from the proportion
we have
R
T
R
r: RR: F+R,
R-r
стра
=
FIG. 71.
TL
RF: F+R.

B
♡~~~
M M'
Wherefore, since the ratio R: F+R is one of
equality when R is infinite,
T
R-r F; that is, d D DO.
*
When the tracing point is on DR there will be a loop.
We need not consider the various curves traced out
according to the varying position of the point d, either
same rate at which the point of contact between the generating line
and the fixed circle recedes from Rr; which is obvious, since D d
as it moves with the rolling tangent is constantly parallel to the
radius from 0 to the point of contact just named, and in its initial
motion the point D moves in direction Dr.
TROCHOIDS.
129
on Dr or on DR. There is, however, one case
which is historically interesting, and may therefore be
considered here, though briefly.
When the tracing point is at O, the curve traced
out becomes the spiral of Archimedes, a curve so called
because, though invented by Conon, it was first inves-
tigated by his friend Archimedes. It was defined as
the curve traversed by a point moving uniformly along a
straight line, which revolves uniformly around a centre.
So traced it is only perfect as a spiral when the moving
point is supposed first to approach the centre from an
infinite distance, and after reaching the centre to recede
along the prolongation of its former course to an in-
finite distance. Regarded as a trochoid, the complete
spiral (or rather the part near the centre) will be traced
out by supposing TDT' to roll first in one direction
from the position where the tracing point is at D, and
afterwards in the other direction.
The identity of this epitrochoid with the spiral
of Archimedes is easily demonstrated. Thus, let p
(fig. 72) be a point on the curve, B'P the corre-
sponding position of the rolling tangent, P p being the
position of the line which had been coincident with
OD, so that Pp is perp. to B'P, and B'P equal in
length to the arc DQB'. Then, since OB' is perp.
to B'P and equal to Pp, Op = B'P.
B'P. And Op is
parallel to B'P, the rolling line, whose direction has
changed through an angle measured by the arc DQB',
which is equal to B'P or Op. Hence the distance of
p from O is proportional to the angle through which
K
130
GEOMETRY OF CYCLOIDS.
Op has revolved from its initial direction OQ' (parallel
to DT'). Therefore p is a point on a spiral of Archi-
medes.
AREA OF THE SPIRAL OF ARCHIMEDES.
The area of the curve is thus determined :--Let p p'
be neighbouring positions of the tracing point; B'Py',
B'P'p' corresponding positions of the rolling tangent
T
k
B
FIG. 72.

d
B B
R
@
U
Pod
N
TMM'
WE ARE
IKI
with its perp. Then Op is equal and parallel to B'P;
Op' to B'P'. Wherefore, in the limit, area p Op'=
area PB'P'. Hence, increment of area O krp=incre-
ment of area D t PB'Q; and these areas begin together:
they are therefore equal. But PB' and P'B" are nor-
mals to D t P, the involute of the circle DQB;
TROCHOIDS.
131
therefore,
that is,
or
or
area D t PB'Q = ÷
1/48
or,
area Orp = }
ARC OF THE SPIRAL OF ARCHIMEDES.
The arc of this spiral may be thus determined.
Drawing DK (fig. 72), as in fig. 71, and representing
element of arc PP' by an element of area KM' (KM
= DM = B'P), let LM be so taken that element of
area LM' represents the increment of arc pp'. Now
the tangent at p is perp. to B'p, so that in the limit
(angle p Op' being equal to angle PB'P'),
pp': PP' :: B'p : B'P ;
(B'P)3
OD
3
(Op)³
OD
··· (pp')² : (PP′)² :: (B′p)² : (B′P)²
::(PO)²+(OB′)² : (B′P)²
(LM)² : (KM)² :: (DM)²+(OD)² : (DM)²
::(KM)²+(OD)² : (KM)²
... (LM² = (KM)²+(DO)²
(LM)² — (KM)² = (OD)²
(B'P) 2
2 OD
since arc DtP=
rect. DL
Arc Orp =
2 OD
; (see p. 85)
Wherefore L is a point on rectangular hyperbola d q'I,
having D d = OD as semi-axis, D as centre, and DK
as an asymptote; and
arc Orp arc DtP:: hyperh. area DdLM: ADKM.
DM+ML
::rect. DL+sq. on OD(log.
ML)
DO
+
(p. 84)
(loge
: sq. on DM.
DM2
2 OD ³
ML)
DM+ML
DO
K 2
132
GEOMETRY OF CYCLOIDS.
Cor. The loop cuts the axial line BO d in a point
r, such that Or Qt (the tangent drawn to DQB,
parallel to OD, meeting involute DtR in t)= arc DQ.
SCHOL. The curve, as it recedes from O, ap-
proaches more and more closely to the involute of the
circle Q'DQ, the curves being asymptotic. All that has
been said about the figure of the involute of the circle
at a great distance from O (pp. 82, 83), applies there-
fore to the spiral of Archimedes.
We have seen that the epicycloid, traced by the
point O, fig. 72, carried along with T'DT, as it rolls on
the fixed circle Q'DQ, is a spiral of Archimedes. To
prove the converse of this,—
Let a point start from O in direction OQ', tra-
velling uniformly with velocity v along radius OQ',
while this radius turns uniformly with angular velocity
w around O in direction Q'DQ. After a time t, let
the point be at p; then Op = vt and
=
angle Q'Oq (greater than 2 rt. angles) = wt.
Now if, with radius OQ F, we describe a circle
Q'DQB about O as centre, intersecting Op in q, then
arc Q'Dq Fwt;
and if F be such that F w = v
(in other words, if F be such that motion in a circle of
radius F, with angular velocity a round the centre,
gives linear velocity v), then arc Q'Dq (= F w t)
vt=Op. Wherefore, drawing OB' perp. to Op,
and completing the rectangle OB′P p,
=
and
B'P Op arc Q'D q = arc B'QD;
= OB' = F.
= =
=
Pp:
TROCHOIDS.
133
.. P is the position of the point D on tangent DT
after rolling round arc DQB' to tangent at B', and P p
is the position then taken up by DO. Hence as T'DT
rolls on the circle Q'DQ, the point O regarded as rigidly
attached to TOT, the tangent to circle Q'DQ of radius
F, at D, will trace out a spiral of Archimedes in which
the linear velocity of the moving point along the revolv-
ing radius is equal to F. angular velocity of the latter.
PROP.-The axis of a planet's shadow in space is a
spiral of Archimedes.
The spiral of Archimedes is interesting as the path
along which the centre of a planet's shadow (the
earth's for example) may be regarded as constantly
travelling outwards with the velocity of light.
This is easily seen if we suppose the earth and its
shadow momentarily reduced to rest, and, with the sun
as pole, imagine a radius vector carried from an initial
position coinciding with the earth and retrograding
through the various portions of the shadow. Let V be
the velocity of the earth in her orbit, D her distance from
V
the sun, and therefore her angular velocity about the
D
sun. Also let L be the velocity of light. Then if our
radius vector, carried back through an angle 0, corre-
sponding to the earth's motion in time t, is equal to r, we
V
D
L.D
have t =
= 0, or t= 0; and r = Lt:
Ꮎ .
D
V
V
Wherefore, since the radius vector varies as the vecto-
rial angle, the corresponding point of the shadow's axis
:
134
GEOMETRY OF CYCLOIDS.
(which was at the earth at time t before the epoch we
are considering) lies on a spiral of Archimedes. We
have in fact L, the velocity of light, for the velocity
along the radius vector (v in the preceding demonstra-
tion), when the angular velocity about the sun is taken
V
w
equal to the earth's angular velocity in her orbit, or D
(corresponding to o in preceding demonstration).
The radius F of the fixed circle by which this
tremendous spiral could be traced out, would therefore
V
L
D
L, or F =
V
be such that F
the radius of
D
the earth's orbit increased in the ratio in which the
velocity of light exceeds the velocity of the earth in
her orbit.
Thus
F 92,000,000 miles x
—
187,000
18.4
= 5,000,000 x 187,000 miles
(roughly)
935,000,000,000 miles.
[It is convenient to remember that the sun's dis-
tance is nearly equal to five million times the mean
distance traversed by the earth in one second.]
NOTE.--The student will find further information
respecting spiral epitrochoids in the examples on pp,
254-256. The solution of these examples presents no
difficulty.
135
SECTION IV.
MOTION IN CYCLOIDAL CURVES.
LEMMA.—When a body at rest at ▲ (fig. 73) is acted on by
an attractive force residing at C, and varying as the dis-
tance from the centre, the body will travel to C in the same
time whatever the distance CA; and if µ. CA is the measure
A
fro
of the accelerating force at A, time of fall to ▲ =
Α
Let AB, perp. to CA, represent the accelerating force at
A; join CB, then Mp perp. to CA, meeting CB in p, repre-
FIG. 73.
B'
b
♫ M
B
T
√ µ
μ
A
2.

sents the accelerating force at M; (vel.2 at M) is represented
by 2. Mµ BA* =2CAB (CA)²—(CM)²=rect, b A. QMY
(OM)²
2
CA²
CQ
(AQA' being a circle about C as centre). That is,
* Any elementary rectangle pm represents M m.accelerating
force at M; or since the force may be considered uniform throughout
the space Mm, p M represents half the increase of the square of the
velocity (by well-known relation in case of uniform force). Hence,
area p A represents (vel.² at M-vel,² at A) = į (vel.² at M).
136
GEOMETRY OF CYCLOIDS.
or,
Vel. at M is represented by
Vel. at M
QM
CQ
Therefore time of traversing m M
incrt. of time from beginning
•
m M =
where V is the velocity with which a particle would reach
C after traversing distance AC under the force at A con-
tinued constant.
•
But if Qg is a small element of arc at Q and qm perp.
to CA, then, ultimately,
QM
CQ
QM V
CQ
Qq.
•
1
V
time of fall from A to M =
Thus, time of fall to C=
1
•
•
ν μ
rect. b A ;
m M
vel. at M
པ་
or V = √μ. CA;
4
Hence,
But if μ CA is the measure of the accelerating effect of
the force at A, V²=2 µ CA. CA=µ (CA)²
2
2
the incrt. of arc AQ.
arc AQ
V
Q X
V
Thus, vel. at M=√μ. QM; and time of fall from A to M
1
arc. AQ
√μ. CA
circular measure of CQA.
ν μ
or,
circ. meas. of rt. angle=
2√ μ
and is therefore independent of the original distance CA.
"
SCHOL.-The general relation of this lemma may be re-
garded as obvious, seeing that a force varying as the dis-
tance from the centre is in this case a force varying as the
distance remaining to be traversed; and this relation holding
from the beginning, it follows that whether such distance be
MOTION IN CYCLOIDAL CURVES.
137
large or small, it will be traversed in the same time. The
general relation may be considered, in this aspect, as
follows:
Let C, fig. 74, be the centre of force, and let one particle
start from A, another from a, in the same straight lin~ CA.
Divide CA and Ca each into the same number of equal
elements, and let l, m, n, and L, M, N be the points of divi-
FIG. 74.
nza la
لب
NMLA
sion nearest to a and A, respectively. Then the force on the
particles starting from A and a may be regarded as severally
uniform while these particles traverse the spaces AL, a
respectively; hence these spaces being proportional to AC,
a C, that is to the uniform forces under which they are tra-
versed, will be traversed in equal times; and velocities pro-
portional to the forces, that is to ML and Im respectively,
will be generated in those times. Again, since the forces
acting on the particles at L and I are proportional to the
spaces LM, 7 m, and the velocities with which the particles
begin to traverse these spaces also proportional to LM, lm,
it follows that the times in LM, lm, will be equal; and the
total velocity acquired at the end of those times will still be
proportional to ML and ml, or to MN and m n, the spaces
next to be traversed. And so on continually. Hence the
particles will arrive at C simultaneously; and the velocities
with which they reach C will be proportional to AC and a c.
It is manifest, also, that if the particles during their
progress to C be resisted in a degree constantly proportional
to the velocity, the times of reaching C will still be equal.
5
138
GEOMETRY OF CYCLOIDS.
PROPOSITIONS.
PROP. I.-If A (fig. 75) be the vertex, A B the axis of an in
verted cycloid DPA, a particle let fall from a point F on
the arc APD (supposed perfectly smooth) will reach A in the
same time wherever F may be.
Let P be a point on the arc AF; draw PM perp. to AB
cutting the generating circle in Q and join AQ. Represent
FIG. 75. (Join AQ.)
D'
또
​B
K
=g.
M
నన
AQ
AB
H
p
g.
ར * 、 ༠༦༨,、……….…
the accelerating force of gravity by g. Then since the tan-
gent at P is parallel to AQ,
N
Accs. force at P along PP' : g :: AM : AQ :: AQ : AB ;
or, the accelerating force at P in direction of motion
q
D

arc AP
2 AB
Hence if the straight line ad arc AD, and we take
af=arc AF, and a parc AP, the acceleration of the particle
at P is the same as that of a particle moving from ƒ to a under
the action of a force varying as the distance from a, and
ap
ad
or at d to g. =g. The time of fall,
g.
equal at p to g· 2AB'
2AB
then, (by lemma, p. 135) is independent of the position of F.
MOTION IN CYCLOIDAL CURVES.
139
Since in this case the accelerating force at D = g =
arc APD, the µ of lemma, and time of fall from any
μ
g
4R
2R'
4R π
point of arc APD to A =
VR
2
g
The time of oscillation from rest to rest on either side
of A
/R
g
SCHOL. This proposition is easily established indepen-
dently. Thus take an elementary arc PP; draw ordinates
FHK, PQM, and P'Q' (Q,Q', on BQA); arcs Qn, Q'n' about
A as centre, to AH; and nq, n'q' perp. to AH, meeting
quadrantal arc HqN on AH in q, q'. Then, (vel.)2 along
PP = 2 g. KM
= 2π
=2g (AK-AM) = (AH²— AQ²)
2g
AB
.. vel. at P
=
√3
time along PP'
· 2 n n' ÷
and time along FPA
2g (nq); & PP=2(AQ—AQ)=2 nn';
AB
2AB
g
2AB
g
√20 (ng)=
AB
ㄒ​ㄧ​2
•
π
に
​2g
AB
(ng)=√√√2AB
!!!
√ R
>
g
2
(nq)²; or,
circ. meas, of q A q' ;
2AB II
as before.
A q
;
PROP. II.-A particle will pass in the same time to A along
a smooth epicycloidal arc APD (A the vertex, APBO dia-
metral,) under the action of a repulsive force at O varying
directly as the distance, from whatever point on APD the
particle starts.
Let the particle start from F. At P on the arc FA,
draw the tangent A'PT, and the normal PB'; then OB'A' cuts
140
GEOMETRY OF CYCLOIDS.
the generating circle through P diametrally in B'A', (B' on
the base BD); and OT perp. to A'T is parallel to B P. De-
scribe arc PQ about O as centre, to meet central generating
circle AQB, and join OP, AQ.
Then if the measure of accelerating force at A
µ. OA,
accelerating force at P. OP; and the accelerating
PT
force in direction PP/
μ PT
OP
καταστατικού μ
AP.
OB'
BA
μ. F2
4R(F+R)
Sa
μ OP.
2AQ.OC
(OB)2
2BA.OC
ов
Arc APD (Prop. VI., sec. 2).
μ.
•
FIG. 76. (Accent upper n and q.)

M
IB
R
B'
Therefore (applying lemma, p. 135, as in the case of
cycloid) the time in which particle reaches A
7
2
T
√4R(F+R)
F √ μ
✓R(F+R),
μ
The time of oscillation from rest to rest on either side of
A is twice this.
π
F
SCHOL.-This proposition may be proved independently
of the lemma, by a demonstration similar to that used for the
cycloid. The figure indicates the construction. We begin
1
MOTION IN CYCLOIDAL CURVES.
141
by showing that (Vel.)2 at M = μ [(OP)² - (OF)2]
µ —
= µ [(OM)² + (MQ)² — (OK)² — (KH)²]
=µ[MK (OM+OK)+AM (MK+KB) — KB(MK + AM)]
=
≈µ· 2MK (OA+OB)
G
Galant an
µ [(AH)² ~ (AQ)²)]
= μ
and
(ng)2 R
The rest follows directly, as in case of cycloid.
F+R
PROP. III.-A particle will pass in the same time to the ver-
tex of a smooth hypocycloidal are under the action of an
attractive force at the centre varying directly as the distance,
from whatever point on the arc the particle starts.
The construction and demonstration are in all respects
similar to those in the case of epicycloid, Prop. II.
/R (F-R)
μ
Time of motion from F to A
F+R
R
Since this gives μ F =
F+R
F
Magg
F
(Vel.)² at P = µ (ng)² (F=R).
SCHOL.-The time of oscillation in the epicycloid under
force above considered: time of an oscillation in cycloid
under gravity (the radii of generating circles being equal)
:: √/g (F+R) : F√µ
This follows directly from the values above determined
for the times of motion to A.
That the times of oscillation may be equal, we must have
F+R
(F+R) g = μ F2; or μ= g.
F2
g, it follows that the accele-
142
GEOMETRY OF CYCLOIDS.
rating force at A in the epicycloid must exceed the force of gra-
vity in the ratio OC : OB, in order that the oscillations may be
performed in the same time as in a cycloid of equal generating
circle, under gravity. The force in the epicycloid will equal
F2
gravity at a distance from 0 =
OK', obtained as in
F+R
fig. 76 by drawing BK' perp. to OC to meet semicircle on
OC as diameter in K'.
If we take μ F g, a cycloid in which the oscillations
under gravity will be the same as the oscillations in the epi-
cycloid must have a generating circle whose radius
OC. CB
ов
2
(CK')²
ов
Bb, obtained by draw-
ing semicircle Bk O, taking Bk CK', and drawing b
k
perp. to BO.
(F+R) R
F
Corresponding considerations and constructions apply in
the case of hypocycloid.
It is manifest (see scholium to lemma) that if the par-
ticle in its passage along the epicycloidal, hypocycloidal, or
cycloidal arc, be resisted in a degree constantly proportional
to the velocity, the periods of oscillation will still be isochro-
nous; the arc of oscillation, however, will no longer be sym-
metrical on either side of the axis, but will continually be
reduced, each complete arc of oscillation being less than the
arc last described.
A weight may be caused to oscillate in the arc of an
inverted cycloid in the manner indicated in fig. 77. Here
a A is a string swinging between two cycloidal cheeks a p D,
a p' D', a being a cusp, and DD, the common tangent at the
vertices D, D, being horizontal. The length of the string
a A being equal to twice the axis of a p D, or to the arc
a p D, the weight swings in the cycloidal arc DAD' (Prop. XI.
section 1). Such a pendulum would vibrate isochronously,
MOTION IN CYCLOIDAL CURVES.
143
if there were no friction and the string were weightless; but
in practice the cycloidal pendulum does not vibrate with
perfect isochronism.
An approach to isochronism is secured in the case of an
ordinary pendulum by having the arc of vibration small
compared with the length of the pendulum. In this case
the small circular arc described by the bob may be regarded
as coincident with a small portion of the cycloidal arc DAD'
(fig. 75) near to A, and the isochronism thence inferred. But
D
P
FIG. 77.
a

A
in reality the approach to isochronism in the case of a long
pendulum oscillating in a small arc, is best proved as a direct
consequence of the relation established in the lemma.
Thus, let ACA' (fig. 73) be the arc of oscillation of a pen-
dulum, whose length l is so great, compared with AA', that
ACA' may be regarded as straight. Then the accelerating
force in the direction of the bob's motion when at M
CM
g. sin. deflection from the vertical=g . ι very nearly,
or varies as CM. Hence the time of oscillation is very
nearly constant, whatever the range on either side of C, so
only that the arc of oscillation continues very small com-
pared with 7.
The accelerating force towards C at M being. CM,
144
GEOMETRY OF CYCLOIDS.
the time of an oscillation from rest to rest is π
g
Vel. at M=QM√√√√√√ (CA²—CM²).
D
D
P
and the
A pendulum may be made to swing in an epicycloidal
arc in the way shown in fig. 78, or in a hypocycloidal arc in
FIG. 78.
P
P
d
A
A
P'
the way shown in fig. 79 (Prop. XII. sect. 2); but of course
the oscillations will not be isochronous under gravity. In the
FIG. 79.
d
√√
g
g


p
case of the hypocycloid, if the plane of fig. 79 be supposed hori-
OF MOTION IN CYCLOIDAL CURVES.
145
zontal, P a smooth ring running on the arc DAD', and this
ring be connected with the centre of the fixed circle by an
exceedingly elastic string, very much stretched, the oscilla-
tions of the ring will be very nearly isochronous. For the
tension of a stretched elastic string is proportional to the
extension, and if when the ring is at A the string is
stretched to many times its original length, the extension
when the ring is at different parts of the arc DAD' is very
nearly proportional to the extended length. Suppose, for
instance, that when at A. the string were extended to 100
times its original length, then the extension would only be
less than the actual length by one 100th part.
If the circular arc DD' represent part of a great circle
of the earth's surface, DAD' a hypocycloidal tunnelling hav-
ing DD' as base, then, since the attraction at points below
the surface of the earth varies directly as the distance from
the centre, a body would oscillate in DAD in equal periods.
It would not, however, be possible to construct such a tun-
nelling, or to make its surface perfectly smooth.
PROP. IV. The path of quickest descent from D to any
point F not vertically below D, is a cycloidal arc through
F, having its cusp at D and its axis vertical.
The following is a modification of Bernouilli's original
demonstration.
The path of descent will necessarily be in the vertical
plane through D and F. Let it be DPF, and let PP' P" be
a small portion of this path, represented on a much enlarged
scale in fig. 80a.
Let P be a point on a horizontal line through P', and close
to P. Then since DPF is the path of quickest descent, the
L
146
GEOMETRY OF CYCLOIDS,
time of descent down the arc PP'P" is a minimum, and
from the nature of maxima and minima it follows that the
change in the time of fall resulting from altering the arc
PP'P" into the arc Pp'P" is evanescent, compared with the
total time of fall down PP'P". If this time were increased in
an appreciable ratio by passing from P' to a point p on one
side, it would be appreciably diminished by passing from P'
to a point on the other side of P', which is contrary to the
supposition that DPF is the arc of quickest descent. Now
regarding PP' and P'P" as straight lines, draw p'l perp. to
PP' and P'm perp. to P" p', so that ultimately Pl= Pp', and
D'
FIG. 80.
cos PP'p'
cos P'p'P"
B
•
A
d'
IQ
FIG. 80a.
P"m=P"P'. Therefore, if we suppose PP and Pp' traversed
Pl
with the uniform velocity V, then
V represents the ex-
cess of time in PP' over time in Pp'; and if we suppose
P' P" and p' P" traversed with the uniform velocity V',
p'm
then represents the defect of time in P' P" from
V
time in p'P".
along Pp' P", we must have V
¿

Therefore since time along PP'P" = time
Pl
V Ρι
√₁ = p'm
V
That is, the velocity at different points along
the arc of descent varies as the cosine of the angle at
which the arc is inclined to the horizon at these points. But
D-A-D
p'm
V
or
OF MOTION IN CYCLOIDAL CURVES.
147
this is a property of motion in an inverted cycloid. For if
DPFAD' is a cycloidal are having D and D as cusps, AB
as axis, and AB vertical, and PL is drawn perp. to AB,
cutting central generating circle in Q, then
(Vel.)² at P = 2 g. BL =
(BQ) 2
B = 4g R.
g
AB
29
•
BQ
(B9)
AB
i.e. Vel. at P=2g R. cos ABQ = 2√7 R. cos AQL,
vg
the required relation, since AQ is parallel to the tangent
at P.
2
Hence DPF is part of a cycloid having its cusp at D and
its axis vertical.
To describe the required arc, draw any cycloid Df d'
having D as cusp, its base D d' horizontal, and cutting DF
in ƒ; then D' so taken that
DD' : D ď' :: DF : Dƒ
is the base of the required cycloid through F. The axis BA,
bisecting DD' at right angles, bears to ba, the axis of D a d',
the ratio DF: Dƒ
SCHOL. The arc is not necessarily one of descent
throughout. If F' be the point to be reached, and the angle
of inclination of Df' to the horizon is less than the angle
b Da, the path from D to F' will include the vertex A, and
the particle will be ascending from A to F'.
The cycloid DAD' is the path of quickest motion from
D to D' at the same horizontal level as D.
L 2
148
GEOMETRY OF CYCLOIDS.
}
SECTION V.
EPICYCLICS.
DEF.- If a point travels uniformly round the circumference
of a circle, whose centre travels uniformly round the cir-
cumference of a fixed circle in the same plane, the curve
traced out by the moving point is called an epicyclic.
Let AQB (fig. 81) be the circle round which the tracing
point travels, CC'K the circle in which the centre C of the
moving circle AQB is carried, O the centre of the fixed circle
CC'K. Then the circle CC'K is called the deferent, AQB the
FIG. 81. (Join C'p.)
1

K
B
P
epicycle, O the centre, C the mean point, P the tracing point.
At the beginning of the motion let the tracing point be at A
in OC produced, or at its greatest possible distance from O.
When the centre is at C' let the tracing point be at P. Draw
the epicyclic radius C'a parallel to CA, and let OC produced
EPICYCLICS.
149
↑
meet the epicycle in A'; also let OA and OA' cut the
epicycle respectively in B and B'. Then C'a is the position
to which CA has been carried by the motion of the epicycle,
and a A'P is the arc over which the tracing point has tra-
velled, in the same time. The angle PC'a is called the epi-
cyclic angle, and the angle C'OC the deferential angle. Both
motions being uniform, the deferential angle bears a constant
ratio to the epicyclic angle. Call this ratio 1 : n; so that
1 is the ratio of the angular velocities of mean point
round centre, and of tracing point round mean point. If we
represent the radius of the deferent by D, and the radius of
the epicycle by E, the linear velocities of the motions just
mentioned are in the ratio D:n E.
ጎ
The deferential motion may be conveniently supposed to
take place in all cases in the same direction around 0,-that
indicated by the arrow on CC'. Such motion is called direct.
Angular motion in the reverse direction is called retrograde.
When the motion of the tracing point round the mean point is
direct, n is positive; we may for convenience say in this case
that the epicycle is direct, or that the curve is a direct epicy-
clic. When the motion of the tracing point round the mean
point is retrograde (as, for instance, if the tracing point had
moved over arc a q' P' while mean point moved over arc
CC'), n is negative, and we say the epicycle is retrograde, or
that the curve is a retrograde epicyclic.
The straight line joining the centre and the tracing point
in any position is called the radius vector. A point such as
A, where the tracing point is at its greatest distance (D+E)
from O, is called an apocentre. A point where the tracing
point is at its least distance (DE) from the centre is
called a pericentre. Taking an apocentre as A for starting
point, OA is called the initial line, and the angle between the
150
GEOMETRY OF CYCLOIDS.
radius vector and the initial line is called the vectorial angle.
This angle is estimated always in the same direction as the
deferential angle: so that if at the beginning the motion of
the tracing point round O was retrograde, the vectorial angle
would at first be negative.
Whatever value n may have, save 1 (in which case
the tracing point will manifestly move in the circle AA'),
the tracing point will pass alternately from apocentre on the
circle AA' to pericentre on the circle BB', thence to apocentre
on the circle AA', and so on continually. The angle between
an apocentral radius vector and the next pericentral radius
vector is called the angle of descent. It is manifestly equal to
the angle between a pericentral radius vector and the next
apocentral radius vector, called the angle of ascent.
PROP. I.—The angle of descent: two right angles :: n~1 : 1.
When n is positive and greater than 1, the epicyclic
angle PC' a (fig. 81) exceeds the deferential angle C'OC, or
A'C'a, by PC'A', or angle PC'A' (n-1) deferential angle.
But, at the first pericentre, angle PC'A'=2 right angles, and
the deferential angle is the angle of descent. Hence,
PROPOSITIONS.
01'
−
2 right angles = (n -
(n − 1) angle of descent,
the angle of descent: two right angles::n 1: 1.
When n is positive and less than 1, A'C' a exceeds the
epicyclic angle p C'a by p C'A', or angle p C'A' = (1 — n)
deferential angle; and proceeding as in the last case, we find
the angle of descent: two right angles :: 1 n: 1.
When n is negative, we have the epicyclic angle a C'P'
+angle A'C'a angle P'C'A', or (taking the absolute value
=
=
EPICYCLICS.
151
of n without regard to sign) angle P'C'A' (n+1) deferen-
tial angle. Wherefore (proceeding as before),
the angle of descent: two right angles: (n+1): 1.
But n being negative, the sum of the absolute values of 1
and n is the difference of their algebraic values, or n ~ 1.
Hence for all three cases,
~
angle of descent: two right angles n1: 1.
SCHOL.-The angle of descent is always positive. See
note, p. 185.
•
PROP. II.—The epicycle traced with deferential and epicyclic
radii D and E, respectively, and epicyclic vel. : deferen-
tial vel.::n: 1, can also be traced with deferential and
epicyclic radii E and D respectively, and epicyclic vel. : de-
ferential vel. :: 1 : n.
•
In fig. 81, complete the parallelogram PC'Oc'. Then
O c = C P = E and c'P OC' = D. Moreover c'OC
= ▲ PC'a, and c'P is parallel to OC'. Wherefore we see that
while the epicyclic curve is traced out by the motion already
described, the point c' travels in a circle of radius E about O
as centre, with the same velocity as P round C; while P
travels uniformly in a circle of radius E round c, and with
the same velocity as C' round O.
Therefore the same epicyclic curve is traced out with
deferent and epicycle of radii D, E, respectively, having
angular velocities as n 1, or by deferent and epicycle of
radii E, D, respectively, having angular velocities as 1 : n.
SCHOL. Thus the deferential and epicyclic radii, D and
E, can always be so taken that D is not less than E. When
D = E, the curve can still be regarded as traced in either of
two ways, viz., with epicyclic vel, to deferential vel. ::n: 1
or ::1: n. In this case all the pericentres fall at the centre.
152
GEOMETRY OF CYCLOIDS.
PROP. III.-Every epitrochoid is a direct epicyclic; and every
hypotrochoid is a retrograde epicyclic.
Let O be the centre of a fixed circle BB'D (fig. 82) on
which rolls the circle AQB; and let the tracing point be at
ron CA. Let the circle AQB roll uniformly to the position.
A'Q'B', C'p P being the position of the generating radius, p
the tracing point. Draw C'Q' parallel to OC. Then the
centre C of the rolling circle has travelled uniformly in
circle CC' about O as centre. Again ≤ Q'C'p = ≤ Q'C'A'
+ ▲ A'C'p=COC' (1+) (since are A'P = are B'B).
≤
Wherefore p is a point on an epicyclic arc, whose defer-
ent and epicycle have radii OC and Cr, or (R + F) and r
respectively, and whose epicyclic angle: deferential angle
::R+F:R. Or, by preceding proposition, we may have
rand RF for radii of deferent and epicycle respectively,
having R: R+F for ratio of epicyclic and deferential angles.
In this case n is greater than 1 and positive.
Next, fig. 83, let the circle AQB roll around instead of on
the circle BB'D. Then the above proof holds in all respects,
save that the angle Q'C'p now = 4. Q'C'A'
4. Q'C'A' — ▲ A'C'p, and
radius OC = R F instead of R + F. Thus in this case, the
epitrochoid gives an epicyclic curve having for deferential
and epicyclic radii (R-F) and r, respectively, and deferen-
tial angle: epicyclic angle:: R-F: R; or else, deferential
and epicyclic radii r and (R-F) and ratio of deferential and
epicyclic angles as R R-F.
:
In this case n is less than 1 and positive.
Next let O be the centre of a fixed circle BB'D, inside
which, figs. 84 and 85, rolls the circle AQB; and let the
* Or at r', on CA produced, in which case read p' for p through-
out the demonstration, for all four cases.
4
EPICYCLICS.
153
D
tracing point be at r. Then following the words of proof
for the case of epitrochoid with modifications corresponding
to the two figs. 84 and 85, the student will have no difficulty
in showing that the hypotrochoid, in the case illustrated by
each of these figures, may either have deferential and epi-
cyclic radii (F-R) and r, and deferential angle: epicyclic
angle :: F F - R: R; or epicyclic and deferential radii 7
FIG. 83.
B
a
R
M
P
D
M
B
FIG. 82.
σ
p
B
B
C
TA
D
M

B
M
P
B
D
B
FIG. 84.
FIG. 85.
and (FR), and deferential velocity epicyclic velocity
:: R: FR.
cases.
Since P has moved round C' in a direction contrary to
that in which C' has moved round O, n is negative in both
If F R> R or F> 2 R, n is >1; this is the
case illustrated by fig. 84. If F - R < R or F <2 R, the
case illustrated by fig. 85, n is < 1.
154
GEOMETRY OF CYCLOIDS.
SCHOL.-We may find in this proposition another reason
for regarding the curve traced out by a point on, or within,
or without a circle which rolls outside a fixed circle, but is
touched by that circle internally, as an epitrochoid, not as a
hypotrochoid, for this definition leads again (while the other
does not) to a symmetrical classification, giving epitrochoids.
as direct epicyclic curves, and hypotrochoids as retrograde
epicyclic curves.
PROP. IV. Every direct epicyclic is an epitrochoid; and
every retrograde epicyclic is a hypotrochoid.
Let p be a point on an epicyclic curve pp', OC ( = D)
the radius of deferent, Cp (= E) the radius of epicycle;.
FIG. 86.
T
N
Σ
P
1

17
k
B
n positive and > 1. Then the motion of p may be resolved
into two, one perp. to CO, the other perp. to Cp. Repre-
sent these by the straight lines p N, p M, taking p M=p C
; then the diameter p T of the pa-
со
ጎ
and therefore p N
rallelogram Np MT represents the motion of p in direction
and magnitude. Complete the parallelogram pCO c; take
PN'=p N; and draw N'B parallel to c O to meet OC in B.
EPICYCLICS.
155
Suppose the parallelogram NM turned (in its own plane)
round the point p through one right angle in the direction
shown by the curved arrow, making p M coincide with pC
and the parall. NM with parall. N'C. Then p B, the dia-
meter of the parallelogram N'C, is the normal at p.
Now, by the preceding proposition, if a circle DBB, having
centre at C and radius CB, roll on the fixed circle KBL having
centre at O and radius OB, the epitrochoid traced out by p,
at distance Cp from C, will be the epicyclic having Cp as
radius of epicycle, CO as radius of deferent, and epicyclic
ang. vel. deferential ang. vel. :: CO; CB :: n: 1. It will
therefore be the epicyclic p p'.
:
FIG. 87.
d
N
D
"P´
Z"
B
N
B
ד
个
​N
FIG. 88.

m
8
Thus the epicyclic pp' is an epitrochoid having
D
; R = CB
= ; and r =
F = BQ = D ( 1 - 11 )
=E.
ብ
N
We get precisely the same construction for the position
of the normal p B by interchanging the radii and the
angular velocities of deferent and epicycle, that is, taking
Oc as radius of deferent and cp as radius of epicycle. Let
p B and c O, produced (if necessary) intersect in b'. Then
с
156
GEOMETRY OF CYCLOIDS.
b'Ob'c::OB: cp::n-1: n; and by the preceding propo-
sition, if a circle dbb", with centre at c and radius cb', roll
outside but in internal contact with the circle k b'l having
centre at O and radius Ob', the epitrochoid traced out by p at
distance cp from c will be the epicyclic having cp as radius
of epicycle, c O as radius of deferent, and epicyclic ang. vel. :
deferential ang. vel. :: c0: c b' :: 1 : n.
с
It will therefore
be the epicyclic pp'. Therefore p p' is an epitrochoid having
F = b'0 = · D (n − 1); R = cb' = D.n; and r = E.
It will be found that the demonstration applies equally
to the case of the direct epicyclic where n < 1, illustrated in
fig. 87, only that N' lies on pc produced.
sponding epitrochoids have
J
The two corre-
D
(1) F = B0 = D D -
(1 − 1 ) ; R = CB =
N
N
(2) F=6'0=D (1 − n) ; R = cb =D n ; and r =E.
Moreover, it will be found that the demonstration
applies with slight (and obvious) alterations to the case of
T
M
FIG. 89.
P
Z
O
C
T M
N
FIG. 90.
"
B
b
; and r =E.

the retrograde epicyclic illustrated in fig. 88. (In the case
illustrated, n> 1: it is not necessary to illustrate sepa-
rately the case in which n < 1). We obtain for the two
corresponding hypotrochoids,-
EPICYCLICS.
157
D
(1) F = B0 = D ( 1 + 1 ) ; R = CB =
BÓ
N
(2) F = 60 = D (1 + n) ; R = cb' =Dn; and r = E.
SCHOL.-A number of cases resulting from varieties in
the position of p are illustrated by the dotted constructions,
and in figs. 89 and 90 (cases in which there is retrogression
about O, b lying between O and B). The reader will have
no difficulty either in understanding these, or in illustrat-
ing many other cases resulting from variations in the values
of D, E, and n.
; and r = E.
PROP. V.-The normal at any point p of an epitrochoid or
hypotrochoid passes through the point of contact B of the
fixed circle with the rolling circle when the tracing point is
at p.
"
The demonstration of the preceding proposition includes
the proof of this general proposition. The motion of p being
at the instant precisely the same as though the circle B were
rolling on the tangent to the fixed circle at B, it follows that
if Np (CB) represent the linear velocity of p in direc-
tion perp. to CO due to the advance of centre C of rolling
circle D BB p M C represents on the same scale the
p
linear velocity of p in direction perp. to Cp; wherefore p T,
the diameter of the parallelogram NM, represents the re-
sultant linear velocity of p; and as in the demonstration of
preceding proposition, if the parallelogram NM be rotated
round p in its own plane, through a right angle, in the direc-
tion indicated by the curved arrow, p T is brought to coin-
cidence with p B, which is therefore the normal at p.
158
GEOMETRY OF CYCLOIDS.
PROP. VI.—To determine the apocentral and pericentral
velocities in epicyclic curves.
From Prop. IV. fig. 86, we see that if the linear velocity of
p around C is represented by p C, that is, by E, the linear velo-
city of p is represented by p T, perp. to p B, in direction, and
D
by p T in magnitude, where CB (= ) represents the linear
22
velocity of C about 0.
Hence the velocity at an apocentre is represented on the
same scale by Ba, and the velocity at a pericentre by O b, a
and b being the points in which OC, produced if necessary,
meets the circle pp₁p3, a the remoter. That is, the linear
D
+E. On the same scale the linear
velocity at apocentre =
n
D
velocity of the mean centre =
D
n
n
lin. vel. at apocen. lin. vel. of mean cen. lin. vel. at pericen.
D D
E
+ E :
; and
n
n
:D+nE: D : D − n E;
n being positive in case of direct epicyclic and negative in
case of retrograde epicyclic.
Thus in the case of the direct epicyclic the motion at an
apocentre is always direct; while the motion at a pericentre is
direct, retrograde, or negative, according as D < or n E, or
as CB, fig 86, (= 2)>or<Cb. In the case of the
retrograde
epicyclic the motion at an apocentre is direct or retrograde
according as D> or <n E, or as CB
fig. 88; while the motion at a pericentre is always direct.
SCHOL.—If D=nE, there is a cusp at pericen. or apocen.
(=m) > or < Ca,
EPICY CLICS.
159
.. ܝ
PROP. VII. To determine the position of the points, if any,
where the motion of the radius vector becomes retrograde.
܂ܘ
It is manifest that if, as in the cases illustrated by figs.
86, 87, and 88, the point B lies outside the circle pP₁ P3, or
D > n E, the motion, direct both at apocentres and pericen-
tres, is direct throughout. For the motion to be retrograde
in part of the epicyclic, then, we require that D be <n E, or
CB < Ca. Since the direction at p is perp. to Bp, the mo-
FIG. 91.

e
K
FIG. 92.
A
"
C
K
tion will be directly towards or from centre if Bp is at right
angles to Op, for then Op will be the tangent at p. We
have then the relations presented in fig. 91 for direct epi-
cyclic, and in fig. 92 for retrograde epicyclic.
Op is the distance from O at which the epicyclic becomes
retrograde (for all smaller distances in case of direct epicyclic,
and for all greater distances in case of retrograde epicyclic).
Manifestly the distance Op is determined by describing a
semicircle on OB intersecting a'p b' in p. Now the angle
pC'a' (n-1) deferential angle (measured from apocen-
tral initial radius vector), say ≤ p C'a' = (n − 1) ø, and we
might proceed by the epicyclic method of treatment to
=
160
GEOMETRY OF CYCLOIDS.
determine ø geometrically. We have, however, already the
means of doing this, in the result of Prop. XIV., Sect. III.
Thus, draw p K perp. to C O; then
Cos p Ca
CK
Cp
P
and
(CA of Prop. XIV. sec. 3=
Cos (n - 1) 0 =
CK.OA'
Cp. OA
tan pOb
E² + D.
fore, p Od
Ø 1 + p O b'
It can easily be shown that
sin (n − 1),:
N
E(D. D)
1
=
n
D
n being negative in case of retrograde epicyclic.
Cor. If, be the value of y determined from this equation,
360°
the motion is retrograde from o
2 1
1 to 0 =
Φι·
I
SCHOL.-The angle is of course the angle which OC
makes with the initial line, and does not directly indicate
the arc of retrogradation, which is twice the angle pOd.
This, however, may be readily deduced in any given case. For
p K
Esin (-1)
tanp Ob
is known, and there-
ΚΟ
D+ Ecos (2-1)41
180°
· 1
is also known.
n
✓
(Ca)²+OC. CB
Cp (00+ 1/1)
P
D
D
N
√(D² - E²) (n2E2-12)
(1 + n) DE
n² E2-D2
D2-E2
D² + n E²
(1 + n) DE '
1
EPICYCLICS.
161
PROP. VIII.—To determine the tangential, transverse, and
radial velocities, and the angular velocity around the
centre at any point of an epicyclic curve.
Let p (figs. 86, 87, 88) be the position of the point on
the epicycle a p₁b. Join Op₁ and draw Bh perp. to Op₁.
Then when C₁ (= E) represents the linear velocity in the
epicycle, Op₁ represents the linear vel. at p₁ in magnitude,
but is at right angles to the direction of motion at p.
Hence Pi h represents the linear velocity perp. to the radius
vector, and Bh represents the linear velocity in the direction
of the radius vector, the direction of the motion in either
case being determined by conceiving p, C turned around p₁,
carrying with it p₁ B and p, h, in the plane of the figure,
through a right angle, to coincidence with the direction of
p's motion in the circle a p b. This includes all cases geo-
metrically, and the student will have no difficulty in effect-
ing the construction and deducing the proper directions for
the tangential, transverse, and radial velocities, for any given
values of D, E, and n, and for any given position of the
moving point. The angular velocities are determined by
the same construction. Thus in the case illustrated by fig. 86:
The tangential velocity of p, is represented by P B in
magnitude and is in advancing direction shown by arrow
at Pi∙
1
The transverse velocity of p, is represented by p₁h in
magnitude, and in direction by Bh.
The radial velocity of p, is represented by Bh in magni-
tude, and in direction by p, h.
The angular velocity of p, about 0: uniform angular velo-
mih
city of p₁ about C
Pi
Pr C
ph: Opp.
P₁ C
Opi
And similarly for all other cases.
:
M
162
GEOMETRY OF CYCLOIDS.
It is more convenient, however, where so many cases
arise, to obtain the analytical expressions for these quanti-
ties; for we know that by rightly considering the signs of the
values used and obtained, the same expression will be correct
for all possible cases. Let then the angle p₁ Ca (fig. 86)
(n-1); that is, let the deferential angle = ; let the
linear velocity of the mean point (C) be V, wherefore the
E
linear velocity of the moving point in the epicyclen V. D'
This is what we have represented linearly by p, C in figs.
86, 87, and 88, so that since p₁ C = E, we have to affect all
the above linear representations of velocity with the co-effi-
22 V
cient
:
D
=
Therefore, the tangential vel.
n V
DP B
n V
D √(P₁C)²+(CB)²+2p₁ C. CB cos p Ca.
n V
D
E²+
now, cos Вp₁ 0=
-
D2
+2
n²
n
E² +
V
√ D² + n² E² + 2 n DE cos (n − 1) p.
D
The transverse vel.
D2
n2
The radial vel.
n V
D
and transverse vel. (direct)
+
2 DE
22
cos (2-1) p
Pi B. cos B p₁ 0 :
(B p₁)² +(p, O²)—- (BO)²
2 p₁ B. p₁ O
P↓
Pi
DE
cos ( n − 1)p + E² + D² + 2 DE cos (n − 1)ø
-
2 P, O
; ..p₁ B.cos Bp₁ O
1
− (D –
V D² + n E² + (n + 1)DE cos (n—
D'
√ D² + E² + 2 DE cos (n − 1) ø
−1)p.
V
DPB. sin Bp¸0:
D.
n
EPICYCLICS.
163
now
sin BP10
sin p₁ OB
therefore,
D
P₁B sin Bp;0=B0 sin p,OB= (D − 1).
and radial vel. (towards centre)
=(n-1)V.
The angular velocity about O
BO
PIB ;
E sin (n-1)
√ D²+E²+2DE cos (n−1) p
transv. vel. V D²+n E²+(n+1) DE cos (n-1)
D' D² + E²+2 DE cos (n-1) p
rad. vect.
a B > b B
a
O
bo'
E sin (n−1).
PIO
The transverse vel. and the angular vel. about O vanish, if
D² + n E² + (n + 1) DE cos (n-1)=0,
the condition already obtained.
If v is the velocity in epicycle, v =
D
n E
or V-v
which value substituted for V in the above formulæ gives
formulæ enabling us to compare the various velocities with
the velocity in the epicycle.
SCHOL.-We see from the geometrical construction that
the radial velocity has its maximum value towards or from
the centre, when the moving point is at p or ps (figs. 86, &c.),
where a tangent from O meets the circle a p₁ b; for then Bh
or Bh has its greatest value. This also may be thus seen:
—Since the deferential motion gives no radial velocity, the
radial velocity will have a maximum value when the epicyclic
motion is directly towards or from the fixed centre,—that is,
at the points where a tangent from the fixed centre to the
epicycle meets this circle.
or as
VnE
D
Cor. The angular vel. at apocentre> or < angu-
lar vel. at pericentre, according as
a B > a O
b B = 10
:
M 2
164
GEOMETRY OF CYCLOIDS.
PROP. IX. To determine when epicyclic loops touch.
For this we must have Z p0d (figs. 93, 94)
descent; that is, see Schol. to Prop. VII.,
FIG. 93.
1 / 100-1 [ -1
COS
18-1 -
J
a
L
D² + n E²-
1 + pob' -
T
(1 + n) DE
+tan-1
FIG. 94.
10.
}}
ດ
K
180°
n — 1
[ √
N
180°
n-1
•
angle of

n2E2-D27
D2-E2
]
Now by Cor. to Prop. XII., Sect. III., C'I=
2 C'k. C'z C'B'. CI (C'p)2
(lower sign for retrograde epicyclic)
or (C'B C'I) C'z C'B'. C'I (C'p)²
±
C'B' . C'I ±(C'p)²
(C'B'± C'I) C'p
C' z
cos (n-1) o
C'p
= (1/2)² +
or'
PROP. X.-To determine the position of points of inflexion.
If p, figs. 95, 96, as in Prop. XIII., sec. 3, is a point of
inflexion, we have as in that proposition
÷D
360°
n - 1
Ala
D
no
EPICYCLICS
165
}
Cos (n-1) o
to be regarded as negative for retrograde epicyclic. Hence
D D
N
N2
FIG. 95.
C
D
360°
n-1
p
N
+
ย
+ E2
D
n2
9
tan pOb (figs. 95 and 96) =
E
n being negative in case of retrograde epicyclic.
Cor. If p2 be the value of 4 determined from this equa-
tion, there is a point of contrary flexure when ቀ Փշ
and
another when
FIG. 96.
p z
≈0
wherefore, if d be the pericentre
180°
P O d
22 — 1
It is easily shown that
sin (n-1) 42:
D² + n³ E²
n (1+ n) DE

$2.
SCHOL.-The angular range round O of the arc between
the points of flexure can be determined, as in case of arc of
retrogradation, see scholium to Prop. VII. We have
;
E sin (n-1) 2
D+Ecos (2-1) 2
P2 − p 0 b, is also known.
√ (n² E² — D²) (D2-24 E2)
n (1 + n) DE
"
166
GEOMETRY OF CYCLOIDS.
and
For the critical case where the points of inflexion coincide,
we have, from Cor. 1, cos (2-1) ₂ = −1 ;
D² + n 3 E2
2
= n(1 + n) DE
that is
(the same condition, both for direct and retrograde epicyclic,
due account being taken of the sign of n);
n (n² ED) E = (n² E D) D
D
(n E
D) (22² E D) = 0,.
which is satisfied, (i), if n = the condition (Schol. p. 158)
E'
ΟΙ
or'
✓
—
tan p 0 b' — √ (n² E² — D²) (D² — „ª E²)
= D (n² + n − 1) — n³ E²
for a cusp (at pericentre in case of direct epicyclic, and at apo-.
D
centre in case of retrograde epicyclic), and (ii), if n²= cor-
E'
responding to the case when this curve becomes straight at
pericentre both for direct and retrograde epicyclic. Com-
pare scholium to Prop. XII., Section III., from which the
relation between n², D, and E, can be directly obtained.
―――
PROP. XI.-To determine the radius of curvature, p, at a
point on epicyclic where deferential angle = 4.
ρ
From Cor. p. 117, noting value of p B' (as in p. 162);
that C'O=n. C'B'; and that BN=C'B cos p B'C'; while
p B cos p BC=B'C'+p C' cos (n-1), it is easily shown
that

2
[D²+n² E² + 2 n DE cos (n−1) ø
D²+n³ E² + n (n+1) DE cos (n-1)p
(D+n E)².
D+n² E
; at pericentre p
at apocentre, p=
(D—n E)²
D-n² E
EPICYCLICS.
167
APPENDIX TO SECTION V.
RIGHT TROCHOIDS REGARDED AS EPICYCLICS.
It is often convenient to regard right trochoids as
epicyclics. The radius of the deferent is in their case
infinite, the centre of the epicycle moving in a straight
line. It is necessary to substitute linear for angular velo-
cities, the value of n becoming infinite when the deferent
becomes a straight line. It is manifest that if the centre of
the rolling circle of a right trochoid moves with velocity v in
the line of centres, the tracing point moves with ve-
J'
locity v around the tracing circle; and conversely, it is
R
manifest that if a point moves with velocity m v round the
circumference of a circle of radius E, whose centre moves with
velocity v in a straight line in its own plane, the point will
trace out a right trochoid, having a tracing circle of radius E
and a generating circle of radius m E. We may put v= 1,
in which case m represents the velocity of the tracing point
round the circumference of the moving circle (m). It
is obvious also that if m 1 there is a loop; if m=1, a cusp;
if m<1 the curve is inflected. These cases correspond to
those of right trochoids in which r > R, r = R, and r < R.
Since right trochoids may be regarded as special cases of
epicyclic curves, it is not necessary to discuss them further
in their epicyclic character. It will be found easy to deduce
any required relation for right trochoids from the relations
above established for epicyclics, combined with the considera-
tions noted in the preceding paragraph. A single illustra-
tion will suffice to show how this may be effected.
M
168
GEOMETRY OF CYCLOIDS.
Suppose we wish to determine when the tracing point
ceases to advance in the looped trochoid. We have, from
Prop. VII., in case of epicyclic,
D² + n E²
(1 + n) DE
and if m represents the ratio of linear velocities in epicycle and
Also n is the angle swept out in
cos (n − 1) 41
Φι
D
deferent, n = m
E
•
epicycle, and when D becomes infinite is the same as (n−1)ø,
so that the angle 1 (the angle a CL of fig. 48) is deter-
mined by the equation
cos 91
D² + m DE
(E+ mD) D
when D is infinite.
The student will, however, find it a useful exercise to go
independently through the various propositions relating to
epicyclics, for the case in which the deferent is a straight.
line. The relations involved are simpler than those dealt
with in the present section. It is to be noticed that m
may always be regarded as positive, the same curve being
obtained for a negative value of m as for the same positive
value, if r remains unaltered.
1
M
SPIRAL EPICYCLICS.
When the radii of epicycle and deferent are both infinite
but (D-E) finite, the epicyclic becomes one of the system of
spirals of which the involute of the circle and the spiral of
Archimedes are special cases. We must of course suppose the
curve traced out on either side of the pericentre, since the
remoter parts of the curve pass off on each side to infinity.
Instead, however, of imagining a deferent of infinite radius
carrying an epicycle also of infinite radius, it is more con-
venient, in independent researches into these spirals by
epicyclic methods, to consider a deferent radius as revolving
EPICYCLICS.
169
uniformly round a fixed point, this radius bearing at its
extremity a straight line perp. to it in the plane of its own
motion, along which line a point moves with uniform
velocity. Let the length of the revolving radius=d,
velocity of its extremity 1, and velocity of moving point m.
Then if m= 1, the curve is the involute of the circle traced
out by the end of the revolving radius; if m> or < 1, the
curve is one of the system of spirals bearing the same relation
to the involute of the circle which the curtate and prolate epi-
cycloid respectively bear to the right epicycloid. If d= 0, the
infinite straight line revolves about a point in its own centre ;
and the curve traced out by the moving point is the spiral
of Archimedes, whatever the uniform angular velocity of the
revolving line, and whatever the uniform velocity of the
tracing point along the line. See also examples 131-133.
PLANETARY AND LUNAR EPICYCLES.
The ancient astronomers discovered that the paths in
which the planets travel with reference to the earth are
approximately epicyclic. It is easily shown that this follows
from the fact that the planets, as well as our earth, travel in
nearly circular paths about the sun as centre.
The general property is as follows :—
PROP. I.-Regarding the planets as travelling uniformly in
circles about the sun as centre, and in the same plane, the
path of any planet P (fig. 97) with reference to any other
planet, p, regarded as at rest, is the same as the path of p
with reference to P regarded as at rest, the corresponding
radii vectores lying in opposite directions; and each such
path is a direct epicyclic.
Let S be the sun, p and P two planets (p being the
inferior planet, and P the superior), in conjunction on the line
170
GEOMETRY OF CYCLOIDS.
Sp P. Let the planet p move to p', while P moves to P'.
Draw pQ and Pq parallel and equal to p' P'. Then, with
reference to the planet p, regarded as at rest, the planet P
has moved as if from P to Q; while considered with refer-
ence to P, regarded as at rest, the planet p has moved as if
from p to 9 and since p Q is equal and parallel to Pq, the
path of the outer planet with reference to the inner, regarded
FIG. 97.

S
Pa
MANGA
q h
S'
m
_M
as at rest, is the same as the path of the inner planet with
reference to the outer regarded as at rest,-each path being,
however, turned round through 180° with regard to the
other.
Join p'q, P'P, p' p, and P'Q. Draw S s' parallel to
p'q, and SS' parallel to P'Q, and join s'q, s'P, S'p, and
S'Q. Also draw s'm and S M parallel to SP, and complete
the parallelograms PMS'S, and pm 8'S.
Then, by construction, the figures S'p', p' Q, S'P, Sq,
q P', and s'P', are parallelograms. Wherefore p S'=p'S=
Sp; and SpS' LpSp'; S'M SP=SP'S'Q and
Z
< MS'Q= 4 PSP'; so that the relative motion of the outer
planet from P to Q around p may be regarded as effected
by the uniform motion of S to S' in a circle about p as centre
EPICYCLICS.
171
(corresponding to the real motion of p to p' around S
as centre), accompanied by the uniform motion of P (which,
if at rest, would have been carried to M), in a circle around
the moving S as centre to Q,-that is, through the arc M Q
P P'. Hence the motion of P with reference to p is that
of a direct epicyclic having D = Sp, ESP, and
N
N
Since
Ang. vel. of P round S
Ang. vel. of p round S
Similarly the relative motion of the inner planet from p
to q, around P, may be regarded as effected by the uniform
motion of S to s' around P as centre (corresponding to the
real motion of P to P' around S as centre), accompanied by
the uniform motion of p (which, if at rest, would have been
carried to m) in a circle around the moving S as centre to
q,—that is, through the arc m q = p p ⋅
Hence the motion
of
P with reference to P is that of a direct epicyclic having
D = SP, E =
Sp, and
•
Ang. vel. of p round S
Ang. vel. of P round S
SCHOL.—If the distances of the planets p and P from the
sun are r and R respectively, the epicyclic of either planet
about the other has DR, E=r, and
N
R\ 3
(
10:00
Ꭱ
(27) 9 >> R
2*
;
for the angular velocities of planets round the sun vary
inversely as the periods-that is, as the sesquiplicate power
of the mean distance.
D
or n > E'
the motion of one planet with reference to another is always
retrograde when the planets are nearest to each other;
therefore every planetary epicyclic is looped.
172
GEOMETRY OF CYCLOIDS
The arc of retrogradation of one planet with reference
to the other may be obtained as explained in scholium
to Prop. VII. of this section. The duration of the retrogra-
dation follows directly from the formula for determining
cos (n - 1), as in that proposition; for ₁ is the angle
swept out by the superior planet around the sun between the
time of inferior conjunction and first station. This formula,
with the values above given for D, E, and n, becomes
1
COS
sin
COS
P
P
R$ -r3
2.3
P-p
p
Φι
$1 = √1
Φι
or, putting P, p, for the respective periods of the planets,
P
R²r + R³ p2
Rr + Rr
Rr³ + R³r•
R} + ri
Rr
R - Rr + r
S
(R
R² +
Rr+
RY
(+1) 3
A
K
The arc of retrogradation,-
(
Ꭱ
2º¹³½ ) √R + r
Rå rå + r
2.
、/ (R + 2) ( R − 2 R³ r³ + r)
R Rir + r
2.2
Rr
√ Rr (R + r)
cos (n− 1)ø, √1 + cos (n − 1) ø1
Φι
Rr
M
=241 +2pOb' - 360°
R
Wherefore tan p Ob' (see fig. 91, and schol. p. 160)
r² ) √/R + r
r (Ri
R(RR + 2) — R} r}
rì r)
r (R − r) √/R+r
R(R+r)-R r (R + r)
; and

2°
R$ √ R + r
360° (p2
(p2-7) -
can be readily determined. Thus, the arc of retrogradation
EPICYCLICS.
173
=24pOb'
2 tan-1
y
P
P - P
p
R√R + r
P - P
; (
(180°
360° - cos-1
180°+cos-i
√ Rr
√ Rr - R-r
Rr
R−√Rr+rl
Apocentral distance = R + r;
Pericentral distance = R — r
p
P - P
This formula gives the arc of retrogradation. The angle
between pericentral and stationary radii vectores is half the
arc of retrogradation.
Thus the epicyclic path of a superior planet (period P)
with respect to an inferior planet (period p), or of latter
planet with respect to former, will have—
180°.
•
(1)
Angle of descent
The arc of retrogradation is determined by formula (1) above.
All the tables of planetary elements give R, r, P and p.
If one of the planets is the earth, the calculation is simpli-
fied, because the tables of elements give the distances of other
planets with the earth's mean distance as unity.
If a satellite be regarded as travelling uniformly in a
circle around its primary, while the primary travels uni-
formly in a circle in the same plane around the sun, the
path of the satellite is an epicyclic about the sun as fixed
centre.
All the satellites travel in the same direction round their
primaries as the primaries round the sun, except the satel
lites of Uranus, whose inclination is so great that their
motion does not approach the epicyclic character. The
174
GEOMETRY OF CYCLOIDS.
direction of the motion of Neptune's satellite, sometimes
given in tables of astronomical elements as retrograde, can-
not yet be regarded as determined. The inclination of
Saturn's satellites, seven of which travel nearly in the same
plane as the rings, is considerable; but these bodies may be
regarded as having paths of an epicyclic character. Our own
moon's path is but little inclined to the ecliptic, and the
paths of Jupiter's moons are still nearer the plane of their
planet's motion. The discussion of the actual motions of
these bodies belongs rather to astronomy than to our present
subject. We need consider here only some general relations.*
PROP. II.—To determine under what conditions a satellite,
travelling in a direct epicycle about the sun, will have its
motion (referred to the sun) looped, cusped, or direct
throughout, or partly convex towards the sun, or just fail-
ing of becoming convex at perihelion, or partly concave
towards the sun.
Let M be the sun's mass, m the primary's, R the dis-
tance of primary from the sun, the distance of satellite
from primary; also (though these values are only for con-
venience) let P be the primary's period, p the satellite's, and
assume that m is so small compared with M, and the satel-
lite's mass so small compared with m, that both the ratios
(M+m): M, and (m + satellite's mass): m may be regarded
throughout this inquiry as equal to unity.
We have first to obtain the means of comparing the
velocities in the primary and secondary orbits under any
* In a work on the 'Principles of Astronomy,' which I am at
present writing, the nature of the planetary and lunar epicycles
will be found fully treated of.
EPICYCLICS.
175
or
and
given conditions. The most convenient way of doing this is
perhaps as follows:-Let V, v, be the respective velocities
of bodies moving in circles around the sun, and round the
primary, at the same distance, R; and let v be the velocity
of the satellite at distance r. Then we know that
or as
V2
R
or as
2
V :: M: m,
R
V: v': √M: √m,
v': v VT: VR.
.. V:v: √/Mr: √m R,
V v
Rr
and
This is the ratio of the angular velocities of primary and
satellite in their respective orbits. It gives us
M µ³.
The path of the satellite will therefore be looped, cusped,
or direct throughout, according as
:: √/M 7³: √m R³.
7.3
3
n: 1(:: P:p): √m R³:
m R³ > R
✓ M23
R3
V
m R≥ Mr; or
m R² ≥ Mr²; or
<
And the path of the satellite will be partly convex towards
the sun, or just fail of becoming convex at perihelion, or be
partly concave towards the sun, according as
m R³ > R
M
3
m
M
7"
→
V ^|| V
m > r
MZR
!!
2.2
R2
Or'
V
v
m
M>
X
R
The student will find no difficulty in obtaining formula
for the range of the arc of retrogradation, if any, or of the
176
GEOMETRY OF CYCLOIDS.
arc of convexity towards the sun, if any, following the course
pursued at pp. 172, 173 (using in the latter case the formula
of p. 165), remembering that in this case D = R and E = r
as in the case of planetary motion, but that in
>
and n =
P
P
reducing the formula he must employ the relation
N
V
m R3
M 1.3'
I have not thought it necessary to occupy space here
with the reduction of these formulæ, because they are of no
special use. The path of our own moon has no points of
retrogradation or of flexure, and the position of such points
on the paths of Jupiter's moons, or Saturn's, is not a matter
of much moment.
We may pause a moment, however, to inquire into the
limits of distance at which, in the case of these planets and
our earth, convexity towards the sun, or retrogradation,
would occur.
M
In the case of our earth,
mi
322,700 = = (568)2 about;
and R = 92,000,000. Therefore a moon would travel in a
cusped epicycle, or come exactly to rest at perihelion, if (the
earth's whole mass being supposed collected at her centre)
92,000,000
the moon's distance from the earth's centre were
322,700
miles, or about 285 miles. That a moon should travel in a
path convex to the sun in perihelion, the distance should not
92,000,000
exceed
or about 162,000 miles. Thus the
568
moon's actual distance being 238,828 miles, her path is
entirely concave towards the sun.
M
In the case of Jupiter,
ՊՈՆ
****
= 1,046 = (323)² about; and
EPICYCLICS.
177
⠀
R=478,660,000 miles. Therefore a moon would travel in
a cusped epicycle, or come exactly to rest in perihelion, if its
478,660,000
1,046
distance from Jupiter's centre were
457,600 miles. Thus the two inner moons, whose distances
are 259,300 and 412,000 miles, have loops of retrogradation ;
whereas the two outermost, whose distances are 658,000 and
1,155,800 miles, have paths wholly direct. But all the
moons travel on paths convex towards the sun for a con-
siderable arc on either side of perihelion; since for the path
of a Jovian moon to just escape convexity towards the sun at
478,660,000
perihelion, its distance from Jupiter should be 321
miles, or about 14,804,000 miles; which far exceeds the
distance even of the outermost moon.
M
In the case of Saturn
ጎራ
cusped epicycle if its distance from Saturn were
3,510 (59)2 about, and
R = 877,570,000 miles. Hence a moon would travel in a
877,570,000
3,510
or about 250,700 miles. This is rather less than the distance
of his fourth satellite, Dione, 253,442 miles; and, owing to
the eccentricity of Saturn's orbit, it must at times happen
that Dione comes almost exactly to rest for an instant at a
cusp in epicyclic perihelion, or only has a motion perpendicular
for the moment to the path of Saturn. The three satellites
nearer to Saturn travelling at distances of 124,500, of 159,700,
and of 197,855 miles, have loops of retrogradation, as have all
the satellites composing the ring system. The other satellites,
having distances of 353,647, of 620,543, of 992,280, and of
2,384,253 miles respectively, have no loops; but their paths
are convex towards the sun for a considerable arc on either
N
=
"
or about
178
GEOMETRY OF CYCLOIDS.
side of epicyclic perihelion; since, for a satellite's path just to
escape convexity towards the sun, the satellite's distance
877,570,000
miles, or about 14,874,000 miles.
59
should be
PROP. III.-Regarding the planets as moving uniformly in
circles round the sun in the invariable plane, the projec-
tions of the paths of the planets in space upon a fixed
plane parallel to the invariable plane of the solar system
are right trochoids.
This follows directly from the fact that the sun is
advancing in a right line (appreciably, so far as ordinary
time-measures are concerned), with a velocity comparable
with the orbital velocities of the planets. His course being
inclined to the invariable plane, the actual path of each
planet is a skew helix, as shown in the last chapter of my
treatise on the sun.
PROP. IV.—To determine the tangential, transverse, and
radial velocities (linear) of a planet in its orbit relatively to
another planet, and its angular velocity about this planet
Let R be the distance, P the period, V the velocity of the
planet which is regarded as the centre of motion; the
distance, p the period, v the velocity of the other planet.
"
Then, in the formula for the tangential transverse, and
radial velocities in epicyclics, we have to put
R
P
• (1) ' = 1;
but it will be convenient to retain n, remembering its value.
We may also conveniently write = p, so that n = p-
2°
R
D=R; E =r; and n =
3
EPICYCLICS.
179
Moreover, with the units of distance and time in which R, r,
P, and p are expressed,
V
RɅ
V
Also is the angle swept out around the sun by the planet
of reference since the last conjunction of the sun and the
other planet, the conjunction being superior in the case of
an inferior planet.*
Thus the tangential velocity is equal to
R
{
R² +
V =
R\ 3
2 TR
P
2*
3
= (p− −1) V.
R² +
•
= V √] + p¬¹ + 2 pi cos (n − 1) p
= V
j² + 2 (R
The formula can obviously assume many forms, but per ·
haps this, which enables us at once to compare the tangential
velocity with V, the velocity of the planet of reference in
its orbit, is the most convenient.
The transverse velocity (direct)
© (1) ³
(+) =
√R² + r² + 2 R r cos (n − 1) p
P-P
p
COS
The radial velocity (towards centre)
V.
1 + p³ + (p¬³ + p) cos (n − 1)ø.
√1 + p² + 2 p cos (n
1) φ
R² + rå
x² + -R r cos (n - 1)}
22
Φ
−
r sin (n − 1)
√ R² + ¿² + 2 Rr cos (n − 1) p
0
(0-3 p) sin (n − 1) p
√1 + p² + 2 p cos (n − 1)ø
•
* The conjunction must be such that the sun is between the two
planets. It is a convenient aid to the memory, in distinguishing
between the superior and inferior conjunctions of inferior planets,
to notice that inferior conjunction is that kind of conjunction with
the sun which only inferior planets can enter into.
N 2
180
GEOMETRY OF CYCLOIDS.
The angular velocity of the planet about the planet of
reference
V
= R
ω.
3
p² + R² + (p˜³ +1) Rr cos (n − 1) ø
R² + 7² + 2 r cos (n 1)
Ξω
p² + 1 + (p + p) cos (n - 1)
1 + p² + 2p cos (n − 1) ø
V
putting == angular velocity of the planet of reference
ω
R
in its orbit.
Cor. 1. In conjunction (superior if moving planet is in-
ferior) = 0;
.. Angular velocity in superior conjunction
p³ + 1 + p˜ } +p
ω
1 + p² + 2
(1 + p) x (1 + p )
(1 + p)²
• (1+p+).
11
3
C
Cor. 2. Similarly since in opposition if the moving planet.
is superior, or in inferior conjunction if the moving planet is
inferior, (n-1) = 180°, angular velocity of a planet in op-
position or inferior conjunction
(n−1) ø
W
pi + 1 − p
2
1+p² - 2 p
(1 - p) - p- (1 - p)
(1 - p)²
-
ω 1 - pi
p
A

♥(=엄​)
ω
p
ω
P
√p + p
SCHOL.—All the above formulæ are susceptible of many
modifications depending on the relations subsisting between
the periods, distances, real velocities, and angular velocities
of the planets in their orbits. From Kepler's third law all
such modifications may be directly deduced.
EPICYCLICS.
181
PROP. V.--A planet transits the sun's disc at such a rate
that the sun's diameter S would be traversed in time t ;
assuming circular orbits and uniform motion, determine
the planet's distance from the sun.
*
sun's
Let the planet's distance = p, earth's distance being unity,
and let w be the earth's angular vel. about the sun =
angular vel. about earth. Then, if t' be the time in which
the sun in his annual course moves through a distance equal
to his own apparent diameter, wt' S, and the planet's
angular velocity about the earth when in inferior conjunction
or
Wherefore, the planet's retrograde gain on the sun (which
advances with angular velocity w)
a quadratic giving
or
ω
věto
ω
+w,
věto
ω
= ~( 1 + √p + p ) = &= ~ "
ω
>
√p + p
t
P+ √ == 1)
vē
ť'
t
√p = − } ± √ ³ ¢ + "' = ↓
√3t+t'
2
J
t t
$ ( + + + + √
G
t
→
ܒ
t t'
( ± √ 38 +8 - 1),
+
t'
- t
→→
3
t +
ť - t
1).
Magda
The lower sign must be taken, the upper giving a value of
℗ greater than unity.
Cor. Let us take the supposed case of Vulcan, whose
* This was the problem Lescarbault had to deal with in the case
of the supposed intra-Mercurial planet Vulcan. He failed for want
of such formulæ as are here given.
182
GEOMETRY OF CYCLOIDS.
rate of transit was such that the sun's diameter would have
been traversed in rather more than four hours. Since in
March (the time of the supposed discovery) the sun traversed
by his annual motion a space equal to his own apparent
diameter in rather more than 12 hours, we may say that
(with as near an approximation as an observation of this
kind—inexact at the best-merits) t' = 3 t. Thus
p = 1 / (2 − √✓ 3)
-
= (2 — 1·732) = (0·268) = 0·134.
½½
This is very near the estimated value of the imagined planet's
distance.
FORMS OF EPICYCLIC CURVES.
The relations discussed in the propositions of this section
enable us to determine the shape and general features of
epitrochoids or direct epicyclics and of hypotrochoids or re-
trograde epicyclics, for various values of D, E, and n. I
propose to consider these features, but briefly only, because
in reality their consideration belongs rather to the analytical
than to the geometrical discussion of our subject.
In the first place, since we obtain the same curve by
interchanging deferent and epicycle, and at the same time
interchanging the relative angular velocities of the motions.
in these circles, we shall obtain all possible varieties of epi-
cyclic curves by taking D as not less than E, so long as we
give to n all possible values from positive to negative in-
finity.
The whole curve lies, in every case, between circles of
radii D+E and D-E, the apocentres falling on the former
circle, the pericentres on the latter. When DE, the whole
curve lies within the apocentral circle; and all the pericentres
lie at the fixed centre.
FIG. 98.
FIG. 99.
PLATE II.


FIG. 100.
FIG. 102.
FIG. 104.
FIG. 101.


FIG. 103.


FIG. 105.


FIG. 106.
FIG. 108.
FIG. 110.
FIG. 112.
'
PLATE III.
FIG. 107.


FIG. 109.


FIG. 111.


FIG. 113.
&


EPICYCLICS.
183
If n be infinite, whether positive or negative, we may
consider the deferential velocity zero, and that of the epicy-
clic finite, giving for the curve the direct epicycle itself if n
is positive, and the retrograde epicycle itself if n is negative.
When n is very great, we obtain such a curve as is
shown in fig. 98, Plate II. (p. 184) if n is positive, and such
a curve as in fig. 99, if n is negative.
As n diminishes the angle of descent increases, the loops
separate and we obtain such forms as are shown in figs. 100
and 101, for n positive or negative respectively.
With the further reduction of n, the loops become
smaller, the point of intersection approaching the pericentre
when n is positive, the apocentre when n is negative, until
D
finally, when n= the loops disappear and we have peri-
E'
central cusps as in figs. 102 and 104, or apocentral cusps as
in figs. 103 and 105, according as n is positive or negative.
In the former case the curve is the epicycloid, in the latter
the hypocycloid.
As n diminishes from towards unity the cusps disap-
D
E
pear and we have points of inflexion on either side of the
pericentres if n is positive, or of the apocentres if n is nega-
tive, as shown respectively in figs. 106 and 107, Plate III.
As n further diminishes the points of inflexion draw
further apart for a while in case of direct epicyclic, and after-
D
wards approach until n² when they coincide again at
E'
the pericentres, the curve being entirely concave towards the
centre for all smaller values of n. In the case of the retro-
grade epicyclic, the points of inflexion draw apart on either
side of the apocentres, and continue so to do till they meet
points of inflexion advancing from next apocentres on either
181
GEOMETRY OF CYCLOIDS.
have when 122
side; so that in this case, as in that of direct epicyclic, we
D
two points of inflexion coinciding at
E
the pericentres. These two cases are illustrated in figs. 114
and 115. The former is a direct epicyclic; n=5; and
D: E:: 25: 1; (apocentral dist. : pericentral dist. :: D+E
D-E 13: 12. The latter is a retrograde epicyclic;
::
n=-3; and D: E:: 9:1; (apocentral dist. : pericentral
dist. D+ED-E:: 5:4). Compare figs. 118, 121, 154,
158.
As n continues to decrease from the value
FIG. 114.
ос
In diminishing from the value
√
angle of descent continually increases if n is positive and we
have curves of the form shown in fig. 108.
FIG. 115.
D
E'
E
the


" passe; through
the value unity. When n + 1 the curve is a circle hav-
ing the fixed point as centre, and having for radius whatever
distance the tracing point may have from that centre ini-
tially; the radius vector therefore always lies in value
between D + E and D-E.
As n continuing positive diminishes in absolute value from
1 to 0, the angle of descent which had become infinite dimi-
nishes, remaining positive.* The curve continues concave
* De Morgan says, 'becomes very great and negative.' This is
correct on his assumption that the angle of descent is to be re-
EPICYCLICS.
185
towards the centre, resembling the appearance it had had
before n reached the value unity. As n approaches the value
0, however, the angle of descent becomes less and less, until
when n=0) it becomes 180°, the curve being now a circle hav-
ing radius D and centre at distance E from the fixed centre.
Thus, if the tracing point is initially at A, fig. 81, p. 148,
the centre is at c, but if the tracing point is initially at P,
the centre is at c, (Oc being parallel to C P).
As a diminishes in absolute value from-
K
////
D
E
to -1,
the angle of descent increases till it is equal to 90°, the
curve, always concave towards the fixed centre, forming a
series of arcs more and more approaching the elliptical form,
as in fig. 109, till when n = -1 the curve is the elliptical
hypocycloid, see p. 124. We see that the equality of the
diameters of the fixed and rolling circles is equivalent to the
condition n = 1 for retrograde epicyclic. The semi-axes
are (D+ E) and (D-E).
Lastly as n, still negative, diminishes from -1 towards
0, the curve at first resembles in appearance that obtained
before n reached the value -1, but the angle of descent
gradually increases, until at length, when » = 0, it is 180°
and the curve becomes the circle already described.
garded as positive when the radius of the epicycle gains in direc-
tion on the radius of the deferent, and negative when the radius of
the deferent gains in direction on the radius of the epicycle. There
is no occasion, however, to make this assumption, which is alto-
gether arbitrary. If we consider the actual motion of the tracing
point coming alternately at apocentre and at pericentre upon the
deferential radius, which constantly advances whatever the value of n
positive or negative (except + 1 only), we must consider the angle
of descent as always positive. We arrive at the same conclusion
also if we consider that the radius vector advances on the whole be-
tween apocentre and following pericentre, for all epicy clics, direct
or retrograde.
186
GEOMETRY OF CYCLOIDS.
The varieties of form assumed by epicyclics according to
the varying values of n, D, and E, are practically infinite.
It will be noticed that in all the illustrative figures, n is a
commensurable number, so that the curve re-enters into itself.
Of course, no complete figure of an epicycle in which n is not
a commensurable number could be drawn.
Certain special cases may here be touched on briefly.
When D = E, the direct epicyclic assumes such forms as
are shown in figs. 110, 112, the retrograde epicyclic such
forms as are shown in figs. 111 and 113. The distinction
between the two classes of epicyclics in these cases is re-
cognised by noting that the angle of descent, which must be
positive, can only be made so by tracing the curves in figs.
110 and 112 the direct way, and by tracing those in figs.
111 and 113 the reverse way.
A distinction must be noted between direct and retrograde
epicyclics, when D is nearly equal to E, and n approaches the
D
value which is nearly equal to unity. For the direct epi-
E'
cyclic, the angle of descent, 180° ÷ (n−1), becomes very
great, and we have a curve which passes from apocentre to
pericentre through a number of revolutions, before beginning
to ascend again by as many revolutions to the next peri-
centre.* On the other hand, in the case of the retrograde
epicyclic, when D is very nearly equal to E, the angle of
descent 180° (n + 1) approaches in value to 90°, or the
angle between successive apocentres approaches in value to
two right angles, so that the curve has such a form as is
shown farther on in fig. 119.
We have followed the effects of changes in the value of
* Prof. De Morgan strangely enough takes figs. 116 and 117 as
illustrating this case. But in both these figs. n=1; in fig. 117,
D= E. In neither is E very nearly equal to D.
EPICYCLICS.
187
}
n, where D and E are supposed to remain unchanged through-
out. The number of apocentres and pericentres depends, as
we have already seen, on the value of n. It will be a useful
exercise for the student to examine the effect of varying the
value of E, keeping D and n constant, or (which amounts
FIG. 116.

really to the same thing) to examine the effect of varying the
E keeping n constant. Since the angle of descent
value of
D'
is equal to 180° ÷ (n − 1) if n is positive, and to 180° ÷
FIG. 117.

(n+1) if n is negative, changing the value of
E
D
will not give
all the curves having any given number m of apocentres or
pericentres (for each revolution of the deferent). For this
purpose it is necessary to assume first n= (m + 1), giving
all the direct epicyclics having m apocentres and m peri-
188
GEOMETRY OF CYCLOIDS,
centres, and secondly n-(m-1) giving all the retrograde
epicyclics having m apocentres and m pericentres, for each
revolution of the deferent. (Of course, m is not necessarily
a whole number.)
Suppose we take n=, so that the angle of descent
(=180°÷) is equal to ths of two right angles. Then if
E> D we have such a curve as is shown in fig. 116. As
E diminishes until E=D, the loops turn into cusps as
5
II
FIG. 118.

T21
FIG. 119.

shown in fig. 117; as E diminishes still further until E
D
D (that is n²
E
form shown in fig. 118. Again, take n= 4. Then
>
the curve assumes the orthoidal
K
EPICYCLICS.
189
when E is nearly equal to D the curve has such a form
as is shown in fig. 119, merging into the cuspidate form
as in fig. 120, when E 3D; and into the orthoidal (or
straightened) form, as in fig. 121, when ED (or
=
FIG. 120.

122
= 1). For further illustrations see p. 256.
E
If we compare fig. 98 with fig. 122, we perceive that in
the former the loop between two successive whorls overlaps
FIG. 121.

two preceding loops, while in the latter each loop overlaps
but one preceding loop. A number of varieties arise in this
way. The determination of the condition under which any
given preceding loop may be just touched is not difficult;
190
GEOMETRY OF CYCLOIDS.
but in no case does the condition lead to a formula giving n
directly in terms of D and E. The simplest of these cases is
dealt with in Prop. IX. of this section. (See fig. 160, p. 256.)
Figs. 123 and 124 illustrate eight-looped epicyclics direct
and retrograde. By noting the different proportions between
FIG. 122.

their respective loops, and by comparing fig. 123 with fig.
100, a ten-looped direct epicyclic, and fig. 124 with fig. 101, a
ten-looped retrograde epicyclic, the student will recognise the
effect of varying conditions on the figures of epicyclics. (In
FIG. 123.

A
fig, 100, n = 11; in fig. 101, n - 9; in fig. 123, n = 9,
and in fig. 124, n = - 7).
It is a useful exercise to take a series of epicyclics and
determine the value of D, E, and n, from the figure of the
Suppose, for instance, the curve shown in fig. 125,
curve.
EPICYCLICS.
191
is given for examination. This closely resembles fig. 108 in
appearance; but in reality fig. 125 is a retrograde, whereas
fig. 108 is a direct epicyclic. The character of the curve in
this respect is determined by tracing it directly from any
apocentre and noting that the next apocentre falls behind
FIG. 124.

the one from which we started. The values of D and E are
determined at once from the dimensions of the ring within
which the curve lies,-its outer radius being D + E, its
inner D E.
The value of n is conveniently determined
FIG. 125.

by noting the angle between two neighbouring apocentres
(indicated best by the intersections of the curve next within
the apocentres, for from the symmetry of the curve all inter-
sections lie of necessity either on apocentral radii vectores
or on these produced). This angle one-tenth of 360°, so
192
GEOMETRY OF CYCLOIDS.
9
ΤΟ
ths of 180°; or n + 1.
that the angle of descent is
Thus in absolute value n =
1.
, but n is negative.
9
In like manner we find that in fig. 126, n = }
In each of the figs. 127, 128, and 129, n = 2, since
there is only one apocentre. In fig. 127, the trisectrix,
FIG. 126.
FIG. 127.
ö
DE; in fig. 128, the cardioid, D= 2 E; in fig. 129,
D = 3 E.
Figs. 130 and 131, Plate IV., illustrate some of the pleasing
combinations of curves which may be obtained by the use of
the geometric chuck, the instrument with which all the curves
of the present part of this section have been drawn. In
FIG. 129.
Fir. 28.
fig. 130 we have two direct cpicyclics, (D – E) of the outer
being equal to (D + E) of the inner. It will be found that
for the outer n = 7, while for the inner = 15. In fig. 131
we have four direct epicycles, having (D + E) constant, but
ratio D E different in each. It will be found that there
:
вит




FIG. 130.
Fig. 134.
#:1::29: 7.
MERCURY.
COUNTRIE
n: 1:21.
D: E5 2.
FIG. 136. MARS.
PLATE IV.
D E:3: 2.
Dod
"
FIG. 131.


FIG. 135. VENUS.


#: 1:13: 8.
Fig. 137. JUNO.
n: 1:13: 3.
APPROXIMATE FORMS OF
D: E::10: 7.


D: E::8: 3.
Fig. 138. JUPITER.
e
FIG. 132.
x00000
ė
# : 1 :: 12 : 1.
Fig. 140.
n: 1:85: 1.
D: E::5: 1.
URANUS,
$0.00
PLATE T.
FIG. 133.


FIG. 139. SATURN.


O
2000
n: 1:59:2.
D: E:: 19: 1.
THE PLANETARY EPICYCLICS.
000
Fig. 141. NEPTUNE.
#:1::217: 1.
D: E::19: 2.


DE: 36: 1.
EPICYCLICS.
193
GL
are 5
apocentres in each circuit; whence (n 1)=
3
1. 360 = 67, and n = 68. The inner part of the figure
is a retrograde epicyclic having 5 apocentral distances in
each circuit; whence in absolute value (n + 1) = 67½½, and
N
661/2.
Figs. 132, 133, Plate V., are further examples for the
student.
•
The remaining eight figures of Plates IV. and V., for
which I am indebted to Mr. Perigal, present the approxi-
mate figures of the epicyclics traversed by the planets, with
reference to the earth regarded as fixed Of course the real
curves of the planetary orbits with reference to the earth
do not return into themselves as these do, the value of n not
being in any case represented by a commensurable ratio.
Moreover, the orbits of the earth and planets around the
sun are not in reality circles described with uniform velocity,
but ellipses around the sun as a focus of each and described
according to the law of areas called Kepler's second law.
Therefore figs. 134-141 must be regarded only as repre-
sentative types of the various epicyclics to which the plane-
tary geocentric paths approximate more or less closely. In
the case of Mars, I may remark that either of the ratios
158 or 32 17 would have given a more satisfactory
approximation to the planet's epicyclic path around the
earth. It so chances that I have taken occasion during the
opposition-approach of Mars in 1877 to draw the true geo-
centric path of Mars around the earth for the last forty
years and for the next fifty, taking into account the eccen-
tricity and ellipticity of the paths, and the varying motion
of the earth and Mars in their real orbits around the sun.
The resulting curve, though presenting the epicyclic cha-
racter, yet falls far short of any of the curves of Plates IV.
0
194
GEOMETRY OF CYCLOIDS.
and V. in symmetry of appearance. The loops are markedly
unequal, a relation corresponding of course to the observed
inequality of the arcs of retrogradation traversed by Mars
at different oppositions.
NOTE.—Mr. H. Perigal, to whom I am indebted for all the illus-
trations of this part of the present work (except figs. 118-121, 132,
133, and 154-161, engraved by Mr. L. W. Boord, with a similar
instrument), gives the following account of the geometric chuck:—
'The geometric chuck, a modification of Suardi's geometric pen,
was constructed by J. H. Ibbetson, more than half a century ago, as
an adjunct to the amateur's turning-lathe. It is admirably adapted
for the purposes of ornamental turning; but is still more valuable
as a means of investigating the curves produced by compound cir-
cular motion. In its simplest form it generates bicircloid curves,
so called from their being the resultants of two circular movements.
This is effected by a stop-wheel at the back of the instrument giving
motion to a chuck in front, which rotates on its centre, while that
centre is carried round with the rest of the instrument and the train
of wheels which imparts the required ratio of angular velocity to the
two movements. A sliding piece gives the radial adjustment, which
determines the phases of the curve dependent upon the radial-ratio.
By the simple geometric chuck a double motion is given to a
plane on which the resultant curve is delineated by a fixed point;
but it may act as a geometric pen when it is made to carry the
tracing point with a double circular motion, so as to delineate the
curve on a fixed plane surface. The curves thus produced being
reciprocals, all the curves generated by the geometric chuck may be
produced by the geometric pen, and vice versâ, by making the angu-
lar velocity of the one reciprocal to that of the other. For instance,
the ellipse may be generated by the geometric chuck with velocity-
ratio 1: 2' (see, however, remarks following this extract), and
by the geometric pen with velocity-ratio 21, the movements
of both being inverse, that is, in contrary directions.
'The accompanying curves were turned in the lathe with the geo-
metric chuck (by myself, many years ago), of sufficient depth to
enable casts to be taken from them in type metal, so as to print the
curves as black lines on a white ground. These curves are therefore
veritable autotypes of motion.'
Mr. Perigal has invented, also, an ingenious instrument, called
the kinescope (sold by Messrs. R. & J. Beck, of Cornhill), by which
all forms of epicyclics can be ocularly illustrated. A bright bead
=
==
EPICYCLICS.
195
is set revolving with great rapidity about a centre, itself revolving
rapidly about a fixed centre, and by simple adjustment, any velo-
city-ratio can be given to the two motions, and thus any epicyclic
traced out. The motions are so rapid that, owing to the persist-
ence of luminous images on the retina, the whole curve is visible as
if formed of bright wire.
He has also turned hundreds of epicyclics (or bicircloids, as he
prefers to call them) with the geometric chuck. There is one point
to be noticed, however, in his published figures of these curves. The
velocity-ratio mentioned beside the figures is not the ratio n : 1 of
this section, but (n−1): 1, i.e., he signifies by the velocity-ratio, not
the ratio of the actual angular velocity of the tracing radius in the
epicycle to the angular velocity of the deferent radius, but the ratio
of the angular gain of the tracing radius from the deferent to the an-
gular velocity of the deferent. This may be called the mechanical
ratio, as distinguished from the mathematical ratio; for a mecha-
nician would naturally regard the radius C'A' of the epicycle PA'P'
(fig. 81) as at rest, and therefore measure the motion of the tracing
radius C'P' from C'A', whereas in the mathematical way of viewing
the motions, C'a is regarded as the radius at rest, and the motion of
CP is therefore measured from C'a. The point is not one of any im-
portance, because no question of facts turns upon it; but it is neces-
sary to note it, as the student who has become accustomed to regard
the velocity-ratios as they are dealt with in the present section (and
usually in mathematical treatises on epicyclic motion), might other-
wise be perplexed by the numerical values appended to Mr. Perigal's
diagrams. These values, be it noticed, are those actually required in
using the geometric chuck or the kinescope; for in all adjustments
the epicycle is in mechanical connection with the deferent.
K
FORMS OF RIGHT TROCHOIDS.
Right trochoids may be regarded as epicyclics having the
radius of deferent infinite, the centre of the epicycle travel-
ling in a straight line. A good idea of the form of trochoids
may be obtained by regarding them as pictures of screw-
shaped wires (like fine corkscrews), viewed in particular
directions. This may be shown as follows :—
If a point move uniformly round a circle whose centre
advances uniformly in a straight line perpendicular to the
0 2
196
GEOMETRY OF CYCLOIDS.
plane of the circle, the point will describe a right helix, the
convolutions of which will lie closer together, relatively to
the span of each, as the motion of the point in the circle is
more rapid relatively to the motion of the circle's centre.
Now if any plane figure be projected on a plane at right
angles to its own, by parallel lines inclined half a right
angle to each plane (or perpendicular to one of the two planes
bisecting the plane angle between them), the projection of
the figure is manifestly similar and equal to the figure itself.
Therefore if the circle and the point tracing out the helix just
described be projected on a plane parallel to the axis of the
helix, by lines making with this plane and the plane of
the circle an angle equal to half a right angle, the circle will
be projected into a circle whose centre advances uniformly
in the plane of projection in a right line. The projection of
the tracing point will be a point travelling uniformly round
this circle; and therefore the projection of the helix will be a
right trochoid. We may say then that every helix viewed
at an angle of 45° to its axis is seen as a trochoid,—or rather
that portion of the helix which is so viewed from a distant
point appears as a trochoid. When the tracing point of a
helix moves at the same rate as the centre of the circle, the
helix viewed at an angle of 45° to its axis appears as a
right cycloid. Thus a helicoid or corkscrew wire having a
slant of 45° and viewed from a great distance at the same
slant (so that the line of sight coincides with the direction
of the helix where touched, at one side, by a plane through
the remote point of view), appears as a cycloid.
The helix is projected into other curves if the line of sight
is inclined to the axis at an angle less or greater than 45°.
In this case the projected curve is that generated by a point.
travelling round an ellipse in such a way that the eccen-
tric angle increases uniformly while the centre of the ellipse
EPICYCLIC'S.
197
advances uniformly,-in the direction of the minor axis if the
angle of inclination exceeds half a right angle, and of the
major axis if the angle of inclination is less than half a
right angle.
A set of such curves, obtained from a helix of inclination
45°, are shown in fig. 144, plate VI., A bio T' being a semi-
cycloid, and Ab, T, A bg T', &c., other projections of the
10
9
8
same portion of the helix by lines inclined to the plane of
projection at an angle exceeding a right angle, A b T' being
the orthogonal projection of this portion of the helix.
Such curves, and varieties of them resulting when the
helix is skewed (the centre of the circle advancing in a
direction not perpendicular to the plane of the circle), possess
interesting properties; but they do not belong to our subject,
not being trochoidal. Moreover, for their thorough investi-
gation much more space would be required than can here be
spared. But one of these curves, the orthogonal projection
AbT' (fig. 144, Plate VI.) of a helix of inclination 45°,
must be briefly mentioned here, because associated histori-
cally as well as geometrically with the right cycloid.
THE COMPANION TO THE CYCLOID.
This curve, called also 'Roberval's Curve of Sines,' may
be obtained as follows:-
Let AB (fig. 142) be a fixed diameter of a circle AQB,
and through any point Q on AQB draw MQp perp. to ACB
and equal to the arc AQ; the locus of this point p is the
companion to the cycloid APD having AB as axis.
If CO c, the line of centres of semicycloid APD, be
bisected in O, the curve passes through O, because CO
quadrant AQC'.
Drawing pm, Q, n, perp. to CO c, we have mo
198
GEOMETRY OF CYCLOIDS.
CO-Cm=AC' — AQ=arc QC'; pm: 0m :: Qn: arc QC'
sin QCC circ. meas. of QCC'. Hence the part A p 0 of
the companion to the cycloid is a curve of sines.
Produce Qn to meet
AC'B in Q', draw M Q'p' paralle
23
M
N
M
B
78
Ta
him
T
FIG. 142.
x
P
22.
ת)

K
to BD to meet the curve Ap D in p' and AB in M', and draw
p'm' perp. to CO c. Then
M'p' = AC Q', and OC = AC'
arc C'Q' = arc C'Q
= 0 m ;
= n Q' = n Q =pm.
And
C m'
.. Om'
p' m'
Therefore the part Op'D of the curve bears precisely the
same relation to the line Oc, which the part Ap O bears to
OC. Thus the entire curve is a curve of sines.
Area Ap OC = area Op' Dc; wherefore, adding CODB,
area AODB = rect. CD rect. BE circle AQB.
It is also obvious that the same curve D p' Op A will be ob-
tained by taking Ec' D as the generating semicircle, and
drawing m'q'p': =arc q'D, m q p arc q q' D; so that the
figure ED p Op A is in all respects equal to the figure
BA p Op D.
EPICYCLICS.
199
Since MQP
arc AQ + MQ ; and M p = arc AQ,
MQ = p P;
so that an elementary rectangle QN elementary rectangle
p L of same breadth; whence it follows that area Ap D P
= semicircle AQB: for we may regard pL and NQ as
elementary rectangles of these areas respectively, and the
equality of every such pair of elements involves the equality
of the areas. Since
area AODB=circle AQB; and area A p DP=1 circle AQB;
.. Area APDB = 3 circle AQB;
and
2 area APDB = 3 circle AQB:
this is Roberval's demonstration of the area of the cycloid.
Draw 8r parallel and near to Qp, and ksh, CT, rl
perp. to OC; then
Ꮣ Ꮯ A 8 ; mC = AQ ; .. m l — Q 8; and
ml: nh: Q 8 : nh:: CQ (hk): Qn (ult. rl)
(=
..rect. m l.rl=rect. nh.hk; that is, rect. rm=rect. nk;
or inct. of area A pm C-inct. of rect. A n. But these areas
begin together. Hence area Apm C=rect. A n ; also
Area AOC rect. CT; and area pm 0= rect. n T.
Representing angles by their circular measure :---
Om
pm=r sin QC′
r sin
yo
r
0
Se
and rect. n Tr² (1
·(1-
sin x dx = 1 - cos x;
and similarly, since pmr cos
therefore, the proof that area pm0rect. n T, may be re-
garded as a geometrical demonstration of the relation
AQ
T
↑ COS
C m
gr
os0 m);
до
COS-
the proof
200
GEOMETRY OF CYCLOIDS.
that area A pm C = 'rect. A n may be regarded as a geome-
trical demonstration of the relation
0
I a
cos x d x = sin x.
It will easily be seen that for points on Op'D,
Area AO p' M' rect. M'm' rect. A n, or Bn,
S
leading again to the relation.
=
area AODB = rect. B c.
}
201
SECTION VI.
EQUATIONS TO CYCLOIDAL CURVES.
Although, properly speaking, the discussion of the equa-
tions to cycloidal curves belongs to the analytical treatment
of our subject, it may be well, for convenience of reference,
to indicate here the equations to trochoids (including the cy-
cloid), epicyclics, and the system of spirals which may be re-
garded as epitrochoidal (see p. 127, et seq.). For the sake of
convenience and brevity I follow the epicyclic method of
considering all these curves.
Let the centre of a circle a qb (figs. 45, 46, Plate I.), of
radius e, travel with velocity 1 along a straight line C c in its
own plane, while a point travels with velocity m round the
circumference of the circle. Take the straight line Cc for
axis of x, Ca for axis of y, and let the point start from a, in
direction a qb. When it has described an angle mo about C,
the centre has advanced a distance e p along Cc, and there-
fore, if x and y are the coordinates of the tracing point,
x = e 4 + e sin m ø,
y = e cos m p.
(1)
If we remove the origin to b, the centre of the base, taking
b d as axis of a and b a as axis of y, the equations are,
x
x=ep + e sin m 4,
y = e + e cos m p.
(2)
If we remove the origin to a, the vertex, taking a e as
axis of x and a b as axis of y, the equations are
x = e o + e sin m p,
(3)
y = c e cos m p.
Kalpagtatan
202
GEOMETRY OF CYCLOIDS.
If we remove the origin to c', taking c' C as axis of x,
and c' d' as axis of y, the tracing point starting from d in
the same direction as before, the equations are
y = e cos m 4.
x=e&
еф e sin o,
(4)
If in this case we remove the origin to e', taking e'e as
axis of x and e' d' as axis of y, the equations are
Car
y = e + e cos m p.
x=e8
еф e sin o,
(5)
And lastly, if we remove the origin to d', taking d'd as
axis of x and d' e' as axis of y, we have the equations
x = e p
e sin o,
y = e e cos m 0.
FIG. 143. (Join C'p.)
Barbar
K
T
M
(6)

=
If m 1, these equations represent the right cycloid; if
m < 1, they represent the prolate cycloid; and if m > 1, they
represent the curtate cycloid.
rent=
For epicyclics, take O (fig. 143), the centre of fixed circle
as origin, OA through an apocentre A as axis of x, and a
perp. to OA through O as axis of y. Put OC, radius of defe-
d; CA, radius of epicycle=e (using italics as more
convenient in equations than capitals); COC′ = p, and
angle a C'P=nø. Then, if x and y are the co-ordinates of P
(7)
x=d cos 4+ e cos n p,
y=d sin +e sin n p.
Φ
EQUATIONS TO CYCLOIDAL CURVES.
203
If OC, instead of passing through an apocentre when pro-
duced, intersects the curve in a pericentre at B, the equations
are
y=d sin o
x=d cos o e cos n o,
e sin n o. (8)
For a retrograde epicyclic, angle a C'P=n4, and the
equations (A being an apocentre) are
A
x=d cos + e cos n o, y=d sin p e sin np. (9)
If B is a pericentre of retrograde epicyclic, the equations
are
―
x=d cosp e cos no, y=d sin +e sin n o. (10)
But all these equations are derivable from form (7) ;—(8)
by rotating the axis through the angle of descent,
and
;
1
N
(9) and (10) from (7) and (8) respectively by changing the
sign of n. So that equations (7) may be used as the equa-
tions for the epicyclic in rectangular coordinates, without
loss of generality.
d
e
When, in (7) and (10), n = the equations are those of
the epicycloid and hypocycloid respectively, when an axis coin-
d
cides with the axis x; if, in equations (8) and (9), n=~
e'
equations are those of the epicycloid and hypocycloid, respec-
tively, when a cusp falls on axis of x. It will be remembered
that if F is radius of fixed circle and R radius of rolling circle,
d=R+F, and e=R; R being regarded as negative in case
of hypocycloid.
W
Τ
From (7) we get
2¹² + y² = r² = d² + e² + 2 d e cos (n − 1)ø,
and tan 0 =
the
(11)
d cos + e cos n o
d sin o + e sin n o
;
(12)
which are the polar equations to the curve, O being the pole
201
GEOMETRY OF CYCLOIDS.
and OA, though an apocentre, the initial line. [Equation
(11) is obviously derivable at once from the triangle OC'P.]
For the epicyclic spirals, suppose OC', fig. 143-ƒ, and that
a tangent at C to circle CK, carrying with it the perp. BCA,
rolls over the arc CK, uniformly, till it is in contact at C,
the angle C'OC being p. Then if AC g, and x and y are
the rectangular coordinates of the point to which A has
been carried, it is obvious (since CA in its new position is
parallel to OC) that (taking projections on axes of x and y)
x=(f+g) cos &+fp sino; y=(f+g) sin o-ƒ cos; (13)
the equations to the epicyclic spiral traced by A. The spiral
traced by B obviously has for its equations
x=(ƒ−g)cosp+ƒpsinp; y=(ƒ—9) sin p—ƒ¢ cos p. (14)
From (13) we get
x² + y² = r² = (ƒ + g)² + ƒ² q2;
p2
2
(f+g) sin of cos o
= (f+g) cos +ƒo sin
(15)
&
ƒ¢= √ r² — (ƒ²+g²); tan 0
the polar equations to these spirals. See also Ex. 133, p. 253.
If g=0, or the tracing point is on the tangent, equations
(13) become
y=fsin of cos;
&
x = ƒcos & +ƒo sin o,
(16)
the equations to the involute of a circle. The polar equation
to this curve is (from 15),
ftan
giving
and
tan 0 =
√72 - ƒ2
f2
f
√ p² — ƒ”²
f
√72 - ƒ2
+ √ p² -ƒ2
ftan
If gf, equations (13) become
x=fsino; y = ƒ cos ;
- & o
x² + y² = ƒ² 42; or r=fo;
tan 0 =
cot &;
or 0 = 0
M
or
π
(17)
EQUATIONS TO CYCLOIDAL CURVES.
205
whence
r = ƒ 0 + ƒ√ ;
=ƒ0 f
2
(18)
the polar equation to the spiral of Archimedes, with OD, fig.
72, p. 130, as initial line. If OQ be taken as initial line, the
equation is
r =ƒ0.
(19)
All the pairs of equations in rectangular coordinates can
readily, by eliminating p, be reduced to a single equation
between x and y. Thus (1) becomes
e
(
m
the general equation to the right trochoid.
From equation (11)
1
12 — 1
P =
x =
COS
cos-l
+ √e² - 1²;
2
x² + 3/2 - 172 +2
2 de
(20)
which combined with either of equations (7) gives the general
equation to the epicyc'ic in rectangular coordinates. To
obtain this general equation in a symmetrical form, note that
from (7)
y cos o x sin o = e sin (n − 1)ø.
(21)
However, in nearly all analytical investigations of the
properties of these curves, it is more convenient to use the
pair of equations (1) for trochoids, (7) for epicyclics, and
(13) for epicyclic spirals, or the polar equations (11) and (12)
for epicyclics, and (15) for epicyclic spirals.
The only use I propose to make, here, of the equations to
these curves, is to obtain the general equations to the evo-
lutes of trochoids, epicyclics, and epicyclic spirals. These
general equations, though they may be deduced from rela-
tions established geometrically in the text, are more con-
veniently dealt with analytically.
We have in equations (1), (7), and (13), x and y expressed
as functions of a third variable ; wherefore
206
GEOMETRY OF CYCLOIDS.
Also,
d x
ἀφ
and the equation to the evolute is derived from the two
equations
Jus
η
d2 y
do 2
wherefore
d x
(2/2)²
and if we put
XC
y +
J
W
where and ŋ are coordinates of the point in the evolute
corresponding to the point x, y, on the curve.
In the case of trochoids, we obtain from (1)
d² y d x
•dq² do
+
2
2 V2
{ (dx)² + (dy)² } }
=e+me cos m p;
Y
d x
d² x dy
2
2
d q² do d q² do
ρ
d²
to the evolute are
2
d x
dys
23{(c)² + (D)")
(1/7/20
d o
do
Φ

d² y d x
do² do
ψ 14
T
2
2
x x
da { (da)² + (1 %)²
)"}
d² y d x
dø² do
ἀφ
(2%
d 2
m² e cosmo;
d² x d y
dy² do
φ
dy
d o
e² (1 + 2 m cos m 4 + m²).
d² x dy
do² dp
d² x dy
2
dø² do
12x
d q²
2
me sin mo ;
(1 + 2 m cos m p + m²)
m² (cos m 4 + m)
me sin mo;
e² m² (cos m 4 + m) ;
e
(1 + 2 m cos m o + m²);
m² (cos mp + m)
;
k, the equations
E = eo + e (1 + km) sin m p,
n = ek + e (1 + km) cos m 4.
(22)
EQUATIONS TO CYCLOIDAL CURVES. 207
Φ
1+hm
If we put
may be written
I x
ἀφ
&= = e(1 + km) p' + e (1 + k m) sin m'p',
ทุ ek + e (1 + km) cos m'p';
and
from which we see that the evolute of the trochoid may be
regarded as traced by an epicycle of variable radius e (1 + km),
in which the tracing point moves with velocity bearing the
variable ratio m' to the velocity of the epicycle's centre,
while the deferent straight line shifts parallel to the axis of
x so that its distance from this axis is constantly equal to
ek on the negative side of the axis of
y.
If
1 (or curve (1) becomes the cycloid), k
and equations (22) become
(23)
६ = =e& e sin o; n= 2 e — e cos mp;
showing that the evolute is an equal and similar cycloid,
with parallel base, removed a distance 2 e, or one diameter of
the tracing circle, from the base of the involute cycloid
towards the negative side of the axis of y (that is from the
concavity of the involute), and having vertices coincident
with the cusps of the involute cycloid.
From equations (7) we obtain
=p' and m (1+km)=m', these equations
-
-d sin p—n e sin no;
-
ď²y dx
d q²
2
d x 2
dy
•· ( 1 ) ² + ( 1 )
y ) = d² + n² e² + 2 n d e cos (n−1) 4,
Φ
Φ
№² y
d 1 42
d2x
2
d q3
p
d y = d cos q+ne cos n 9 ;
d o
d sin o
2,
M
n² e sin no,
d cos o
n² e cos no
dy =d²+n³ e² + (n² + n) decos (n − 1)
Y 4
d² x
do do² dy
2
208
GEOMETRY OF CYCLOIDS.
wherefore, p =
and
and if we put
we obtain for the equations to the evolute
n = d sin & + e sin no
}
{d² + n² e² + 2 n de cos (n − 1)
d² + n³ e² + (n² + n) d e cos (n − 1)p
d² + n² e² + 2 n d e cos (n−1) p
d²+n³ e² + (n² +n) d e cos (n − 1) ø
E = d cos o + e cos n o - k (d cos o + n e cos n 4),
or
¿ = d (1−k) cos
and η d (1−k) sin
US
77
S
2 — 1
n+1
17
-1
n+ 1
_______
whence we see that the evolute may be regarded as traced
by an epicycle of variable radius e (1−n k) carried on a de-
ferent also of variable radius d (1−k).
It is easily seen (see p. 117, and figs. 63, 64), that
C'B'
k
C'O'
+e(1−nk) cos no
+ e (1 − n k) sin n o
2
1+2
•
(
When d = ne, so that the involute epicyclic is the epi-
cycloid or the hypocycloid (according as n is positive or ne-
gative), k reduces to
and the equations of the evolute
become
‚d cos -
Makka
•
k (d sin o + ne sin n ø)
}
ps
ps - NB'
d sin -
3
5.).
n-1
n+1
n — 1
n+1
=k,
e cos no
e sin no
; (24)
; (25)
which (we see from 8) are the equations of an epicycloid or
hypocycloid (according as n is positive or negative), whose
deferential and epicyclic radii (and in fact whose linear pro-
portions) bear to those of the involute the ratio (n−1):
(n+1), and whose vertices touch the cusps of the involute
epicycloid or hypocycloid. If n is positive the ratio (n−1)
: (n+1) is the same as (-e): (d+e), or F: (F+ 2 R), as
in Section II. If n is negative the ratio (n-1): (n+1) is
the same as (d+e): (d—e), or F: (F-2 R), as in Section II.
PLATE VI.
A
M
Ma
Mi
S
Se
2
Sa
S+
S&
Se
S4
SA
Sa
ΑΙ
Bo
Ba B
BSBF
92
91
M
فن
P2
by by by Bra
be
FIG. 144. CONSTRUCTION FOR MEASURING MOTION IN ELLIPTIC ORBITS UNDER GRAVITY.
T
C'

Τ
209
SECTION VII.
GRAPHICAL USE OF CYCLOIDAL CURVES.
GRAPHICAL USE OF THE CYCLOID AND ITS COMPANION TO
DETERMINE THE MOTION OF PLANETS AND COMETS.
[From the Monthly Notices of the Astronomical Society for April
1873.]
The student of astronomy often has occasion to deter-
mine approximately the motion of bodies, as double stars,
comets, meteor systems, and so on,-in orbits of considerable
eccentricity. The following graphical method for solving
such problems in a simple yet accurate manner is, so far as I
know, a new one. * By its means a diagram such as fig. 144,
Plate VI., having, once for all, been carefully inked in on
good drawing card, the motion of a body in an orbit of any
eccentricity can be determined by a pencilled construction of
great simplicity, which can be completed (including the
construction of the ellipse) in a second or two.
Let APA', fig. 145, be an elliptical orbit of which ACA'
is the major axis, C the centre, S being the centre of force,
so that A is the aphelion, and A' the perihelion. Let H be
*New as a method of construction, though the principle on
which it depends is of course not new. The curve Ap T' (fig. 145), for
instance, is an orthogonal projection of a particular prolate cycloid
which, as Newton long since showed, if accurately drawn, gives the
means of determining the motion in the ellipse APA'. But, as he
remarks, this prolate cycloid cannot readily be drawn; whereas the
curve A p T can be very readily drawn.
P
210
GEOMETRY OF CYCLOIDS.
half the periodic time, and T the time in which the body
moves from A to P.
On AA' describe the auxiliary semicircle Ab A'.
Then
A
M
CS
AU QM: AQA'
AC
Now if Am1' be a cycloid having AA' as its diameter,
then
C
S
A
T: H area ASP area ABA'
we have
•
:: (ACQ + SCQ); area A b A'
:: AC . AQ + ČS . QM ; AC. AQA'
P
:: AQ +
Ordinate M m = AQ + QM.
FIG. 145.
B
b
IP
•

·
17
And if we take M
I
AQ, we have q a point on A q T',
the companion to the cycloid. The line qm is then equal to
QM; and if we take a point p on mQ such that
I
SC
SC
AC IM
AC·• QM
1
T
GRAPHICAL USE OF CYCLOIDAL CURVES. 211
Mp = AQ +
wherefore
SC
AC
QM; and A'T′ =
AQA';
TH:: Mp: A'T'
Thus we may represent the time in traversing the arc AP
by the ordinate Mp to a curve Ap T', obtained by dividing
all such lines as qm (joining the cycloid and its companion,
and parallel to A'T') so that qp: q m as SC: AC.
10
Accordingly, if we construct such a diagram as is shown
in fig. 144, plate VI., in which AT' is a semi-cycloidal arc
and Ab T' its companion, while intermediate curves are
drawn dividing all such lines as b b₁, into ten or any other
convenient number of equal parts, the curves through the
successive points b, b₁, ba, &c., to bio, give us the time-ordi-
nates for bodies moving in ellipses having A and A'as apses,
and their centres of force respectively at C, S1, S2, S3,
S₁, and A'.
In the plate the semi-ellipses corresponding to these posi-
tions of the centre of force are drawn in, and it will be
manifest that any ellipse intermediate to those shown can be
pencilled in at once, with sufficient accuracy. Ellipses within
AB,A' have their focus of force between S, and A', and are
exceptionally eccentric.* It is easy to construct such an
ellipse, however, in the manner indicated for the semi-ellipse
AB,A'. For the radial lines and the parallels to AT
through their extremities are supposed to be inked in; and
(taking the case of ellipse AB,A') we have only to draw the
semicircle a B, a', and parallels to AA' through the points
where the radial lines intersect this semicircle, to obtain by
* It is manifest that when the centre of force is at A' we have
the case of a body projected directly from a centre of force, and the
time-curve becomes the cycloid A b₁T. Thus the above lines give a
geometrical demonstration of the relation established analytically
in the paper which follows.
+
P 2
212
GEOMETRY OF CYCLOIDS.
the intersections of these parallels with the parallels to AT
a sufficient number of points on the semi-ellipse.
The illustrative diagram has been specially constructed
for the use of those who may have occasion to employ the
method, and will be found sufficiently accurate for all ordi-
nary purposes. Before proceeding, however, to show how
the method is applied in special cases, I shall describe how
such a diagram should be constructed :-
First the semicircle ABA' must be drawn, and the lines
AT, A'T' perp. to AA'. Then CA' must be divided into
ten equal parts (and when the figure is large, a plotting scale
for hundredths, &c., should be drawn). Next A'T and AT
must be each taken equal to 3·1416 where CA' is the unit.
Join TT'. Now AT and A'T' represent, as time-ordinates,
the half-period of any body moving in an ellipse having AA'
as major axis. Each must now be divided into the same
number of equal parts, and it is convenient to have eighteen
such parts. (So that in the illustrative case of our Earth,
three divisions represent a month.) Next the semicircle.
ABA' must be divided into eighteen equal parts. Through
the points of division on the semicircle, parallels to AT and
A'T' are to be drawn,* and the points of division along AT and
A'T' are to be joined by parallels to AA' and TT'. Then the
curve Ab T', the 'companion to the cycloid,' runs through
the points of intersection of the first parallel to AT and the
first to AA', the second parallel to AT and the second to
AA', the third parallel to these lines, the fourth, and so on.
We have now only to take b bio equal to CB; q₁ P₁ equal to
M₁ P₁; q2 P₂ equal to M, P₂; and so on, to obtain the re-
quired points on the cycloid A b₁0 T'; and the equidivision
10
2 2
* Practically it is convenient to draw another semicircle on TT,
divide its circumference into eighteen parts, and join the correspond-
ing points of division on the two semicircles.
1
GRAPHICAL USE OF CYCLOIDAL CURVES. 213
of all such lines as b bio, q1 P1, 42 P₂ (into ten parts in the
illustrative diagram) gives us the required points on the
interme liate curves.
Next let us take some instances of the application of the
diagram.
I. Suppose we wish to divide a semi-e lipse of given
eccentricity into any given number of parts traversed in
equal times, and let the eccentricity be, and 18 the given
number of parts * :—
5
Then S, is the centre of force; AB,A' the semi-ellipse;
and Ab, T' the time-curve. The dots along A b,T' give
the intersection of the time-curve with the time-ordinates
parallel to AA'; and therefore parallels to AT, though these
dots (not drawn in the figure, to avoid confusion) indi-
cate by their intersection with the semi-ellipse ABA, the
points of division required.
II. Suppose we wish to know how far the November
meteors travel from perihelion in the course of one quarter
of their period, that is, one half the time from perihelion to
aphelion :-
The curve AB,A, is almost exactly of the same eccen-
tricity as the orbit of the November meteors. To avoid
additional lines and curves, let us take it as exactly right.
Then Ab, T' is the time-curve. For the quarter period
from the perihelion (or aphelion), we take of course the
middle vertical line, which intersects Ab,T in c. This
point by a coincidence is almost exactly on a parallel to AT,
and this parallel meets the semi-ellipse AB, A' in n, the re-
quired point on the orbit. In other words, the journey of
the November meteors from A to n occupies the same time
as their journey from n to A', S, being the position of the
9
* This selection is made solely to avoid the addition of lines and
curves not necessary to the completeness of the diagram.
214
GEOMETRY OF CYCLOIDS.
Sun, and the Earth's distance from the Sun approximately
equal to A'S,.
III. Suppose we require, in like manner, the quarter-
period positions in different orbits, all having AA as major
axis, but their centres of force variously placed along CA'.
We get any number of points, n, 1, k, precisely as n was
obtained; m, of course, is on the parallel through Co; and
we obtain, in fine, the curve mnlk B, which resembles, but
is not, an elliptic quadrant.
IV. Suppose we require to know in what time the half
orbit from aphelion or perihelion is described in orbits of
different eccentricity. The required information is manifestly
indicated by the intersection of CC' with the time-curves, in
b, b₁, b₂, &c. Thus in the circle, AB is described in the time
represented by Cb; in the semi-ellipse AB, A, AB, is
described in the time represented by Cb3, and B,A' in the
time represented by b,C; and so on for the other semi-
ellipses.
V. Suppose we require to determine approximately the
'equation of the centre' for a body when at any given point
of its orbit of known eccentricity. Take the case of Mars,
whose eccentricity being nearly 1o, his path is fairly repre-
sented by the ellipse next within ABA', and his time-curve
by Ab, T. Then the equation of the centre, when Mars is
at his mean distance, is represented by bb,; when Mars is
at P, (not on the circle, but on the curve just within), the
equation of his centre is represented by q₁ r₁; and so on.
Many other uses and interpretations of the time-curves
will suggest themselves readily to those who are likely to use
the diagram.
After the above method had been briefly described, Pro-
fessor Adams, who was in the chair, mentioned a method
GRAPHICAL USE OF CYCLOIDAL CURVES. 215
6
(devised by himself many years since) by which the same
results can be obtained from the companion to the cycloid'
or curve of sines.' Professor Adams's method may be thus
exhibited :—Let a ba' be the y-positive half of one wave of
the curve of sines,' C its diameter: AbA', a semicircle with
radius C. Let ABA', fig. 146, be a half-ellipse having its
↳
focus at S. Then the time in any are AP of this ellipse may
he thus determined. Join b S, produce the ordinate PM to Q
on circle ABA', draw Qq parallel to a a', and qp parallel to
bS; then ap represents the time in traversing AP, where
a a' represents the half period. And vice versâ, if we require
+
A
M
Trc. 146.

3
IB
C P
is
a'
the position of the moving body after any time from the apse.
say aphe'ion, then take ap to represent the time, where a a
is the half period, ACA' the major axis, S the centre of force;
join Sb, draw pq parallel to $7, 9 Q parallel to AA', and
QP perpendicular to AA gives P the point required.
It will be manifest that in principle my method is iden-
tical with this, for in my figure the time is represented by
Mp, where M q (fig. 145) is equal to the arc AQ, and 9 p is
equal to QM reduced in the ratio of CS to CA. Now a P
fig. 146 is the projection of a q and 9p; and the projection of
aq is equal to the arc AQ (see p. 200), while the projection
in
216
GEOMETRY OF CYCLOIDS.
of
q P
is equal to QM reduced in the proportion of CS
to AC.
Although Professor Adams's construction has the advan-
tage of requiring but a single curve, yet for the particular
purpose described my construction is more convenient. We
see from the fig. 146 that to give the relation between the
times and positions in the case of the ellipse Ap A', we
require a series of parallels to bC, aa' and bS; and the
parallels to bS only serve for this one case. Therefore we
could not construct a reference figure for many cases, without
having many series of parallels and a very confusing result.
In my construction we have, instead, many curves, but a
result which is not confusing because each curve is distinct
from the rest.
GRAPHICAL USE OF THE CYCLOID TO MEASURE THE MOTION
OF MATTER PROJECTED FROM THE SUN.
[From the Monthly Notices of the Astronomical Society for
December 1871.]
Whatever opinion we may form as to the way in which
the matter of certain solar prominences is propelled from
beneath the photosphere, there can be little question that
such propulsion really takes place. It seems clear indeed
that some prominences, more especially those seen in the
Sun's polar and equatorial regions, are formed-or rather
make their appearance-in the upper regions of the solar
atmosphere, and even assume the appearance of eruption-
prominences by an extension downwards, somewhat as a
waterspout simulates the appearance of an uprushing column
of water though really formed by a descending movement.
But it is certain that other prominences are really phenomena
of eruption.
GRAPHICAL USE OF CYCLOIDAL CURVES. 217
4
In the case of any matter thus erupted, we shall clearly
obtain an inferior limit for the value of the initial velocity
of outrush, if we assume that the apparent height reached by
the matter is the real limit of its upward motion (that is,
that there is no foreshortening), and that the solar atmosphere
exercises no appreciable influence in retarding the motion.
The latter supposition is, however, wholly untenable under
the circumstances, while the former must in nearly all cases
be erroneous; and I only make these suppositions in order
to simplify the subject, noting that their effect is to reduce
the estimated velocity of outrush to its lowest limiting
value.
We are to deal then, for the present, with the case of
matter flung vertically upwards from the sun's surface and
subject only to the influence of solar gravity; I propose to
consider the time of flight between certain observed levels,
not the mere vertical distance attained by the erupted
matter; and (as I wish to deal with cases where a great
distance from the sun has been attained) it will be necessary
to take into account the different actions of the solar gravity
at different distances. Zöllner, in dealing with prominences
of moderate height, has regarded the solar gravity as con-
stant; but this is evidently not admissible when we come to
deal with matter hurled to a height of 200,000 miles, since
at that height solar gravity is reduced to less than one-half
the value it has at the surface of the sun.
It is easy to obtain the required formula; and though it
is doubtless contained in all treatises on Dynamics, it will
be as well to run through the work in this place.
In re-
ducing the formula I have noticed a neat geometrical illus-
tration (and a partial proof) which I do not remember to
have seen in that form in any book. It not only presents
in a striking manner the varying rate at which a body
218
GEOMETRY OF CYCLOIDS.
falls towards a centre attracting according to the law of
nature, but it supplies a means whereby the time of flight
between any given distances may be readily obtained from a
simple construction.
K
Let C, fig. 147, be the centre of a globe ABD, of radius
R, and attracting according to the law of nature; let y be the
accelerating force of gravity at the surface of the globe. Then
the attraction exerted at a unit of distance, if the whole
mass of the globe were collected at a point, would be g R2.
T
g
Q
וד
FIG. 147.
25 20
a
1/5
12x
d t2
1.5
10
Let a particle falling from rest at E
time t; and let AE = H, and CP = x.
of motion is
Y
R2
H
Illustrating the motion of a body descending from rest towards a globe
attracting according to the law of nature.
x:2
F
1

M
f
reach the point P i
Then the equation
GRAPHICAL USE OF CYCLOIDAL CURVES. 219
giving
Thus
so that, since the particle starts from rest at a distance
(R+ H) from C, we have
R
2g
R N D
d t
A x
Integrating, we have
R
Z
༡
(da) ²
For convenience write D for (R + H); then we have
2
(da) ² =
(1 - 11)
금​)
t
/29
D
•
0
t
=v2
2 g R2
R+H
= v² = 2 g R²
Dx
2 g R2
X
=
VD x
x2
+ C.
2 g R2 / D
x² +
D
2
MAN, K
+C;
2² (D = x).
X
But when t=0, x = D; so that C
hence we have
√3/1/1 = √ Dx - 202
✓
t
D
x2
D
·D cos-2 (D-24).
COS
Dπ
cos-1
(1)
+ C.
(2 x D), (2)
D
(where D is equal to the radius of the globe added to the
height from which the particle is let fall).
Equation (1) gives the velocity acquired in falling (from
rest) from a height H to a distance x from the centre, and
(2) gives the time of falling to that distance. The geo-
metrical illustration to which I have referred, relates to
the deduction of (2) from (1). We see from (1) that at the
point P
220
GEOMETRY OF CYCLOIDS.
so that
27
211 R2 (D
D
Bisect CE in F, and describe the semicircle CDE; then if
DE is a tangent to the circle DAB, and if DM is drawn
perpendicular to CE,
·292 =
Hence, from (a),
so that
J the vel.
at P
{
CM =
(11)
But if close by G, either on the tangent GH or on the arc
GE, we take G' and draw G'P' perpendicular to CE, and
Gn perpendicular to GP, we have
-(D=2).
(CD)2
CE
PE
r = √29. UM PP.
VCP
23
CM
GG' + Gn GF + FP
PP
GP
}:{
27
•
√2g. CM
R2
D
CP
PE
CP
✓CP. PE
PP'
GG + Gn
J velocity acquired in falling through
space CM, under const. accel. force y
}:
[ elem. space
PP'
Therefore the falling particle traverses the space PP' in the
same time that a particle travelling with the velocity acquired
in falling through space CM under constant accelerating
force g, would traverse the space (GG' + Gn). It follows
that the time in falling from E to P is the same as would be
occupied by a particle in traversing (arc EG + GP) with the
velocity acquired in falling through the space CM under a
constant accelerating force g. In other words,
sum of elementary
spaces GG and Gn
}
GRAPHICAL USE OF CYCLOIDAL CURVES. 221
or
PG arc GE
v2y. CM
Ꭱ .
·
J
D
t = √PE. PC + CF arc GE
√(D − x) x +
D
可
​COS
-1
(2 x D).
D
as before.
The relation here considered affords a very convenient
construction for determining the time of descent in any given
case. For, if PG be produced to Q so that GQ = arc GE,
Q lies on a semi-cycloid KQC, having CE as diameter; and
the relative time of flight from E to any point in AE is at
once indicated by drawing through the point an ordinate
parallel to CK. The actual time of flight in any given case
can also be readily indicated. For let T be the time in
which LC would be described with the velocity acquired in
falling through a distance equal to LC under accelerating
force y, and on LM describe the semicircle Lm M; then
clearly Cm ( CL. CM) will be the space described in
time T with the velocity acquired in falling through the
space CM under accelerating force g; and we have only to
divide Cm into parts corresponding to the known time-
interval T, and to measure off distances equal to these parts
on PQ to find the time of traversing PQ with this uniform
velocity, i.e., the time in which the particle falls from E to P.
The division in the figure illustrates such measurements in
the case of the sun, the value of T being taken as 18 minutes.
Moreover it is not necessary to construct a cycloid for
each case. One carefully constructed cycloid will serve for
all cases, the radius CA being made the geometrical variable.
As an instance of this method of construction, I will take
Professor Young's remarkable observation of a solar out-
(mai
)))
GEOMETRY OF CYCLOIDS.
burst, premising that I only give the construction as an illus-
tration, and that a proper calculation follows.
Fr. 148.

Ul
Vi
a
12h 55m.
Fig. 149.
"
14.5m.
}

GRAPHICAL USE OF CYCLOIDAL CURVES. 223
On September 7, 1871, Professor Young saw wisps of
hydrogen carried in ten minutes from a height of 100,000 miles
to a height exceeding 200,000 miles from the sun's surface.
A full account of his observations is given in the second and
FIG. 150.

(I
1.40m
third editions of my treatise on the sun. Figs. 148, 149, 150,
and 151, with the times noted, indicate the progress of the
changes. I assumed in what follows that there was no fore-
FIG. 151.
"

1.55".
shortening. The height, 100.000 miles (upper part of cloud
in fig. 148), was determined by estimation; but the ultimate
height reached by the hydrogen wisps (that is, the elevation
224
GEOMETRY OF CYCLOIDS.
at which they vanished as by a gradual dissolution) results
from the mean of three carefully executed and closely ac-
cordant measures. This mean was 7 49", corresponding to
a height of 210,000 miles (highest filaments in fig. 149). We
may safely take 100,000 miles as the vertical range actually
traversed, and 200,000 miles as the extreme limit attained.
We need not inquire whether the hydrogen wisps were
themselves projected from the photosphere,-most probably
they were not,--but if not, yet beyond question there was
propelled from the sun some matter which by its own motion
caused the hydrogen to traverse the above-mentioned range
in the time named, or caused the hydrogen already at those
heights to glow with intense lustre. We shall be under-
rating the velocity of expulsion, in regarding this matter
as something solid propelled through a non-resisting me-
dium, and attaining an extreme range of 200,000 miles.
What follows will show whether this supposition is ad-
missible.
Now g for the sun, with a mile as the unit of length and
a second for the unit of time, is 0.169, and R for the sun is
425,000. Thus the velocity acquired in traversing R under
uniform force y,
√2y. R
= √338 × 425
=379, very nearly.
(This is also the velocity acquired under the sun's actual
attraction by a body moving from an infinite distance to the
sun's surface.)
And a distance 425,000 would be traversed with this
velocity in 18m 40° (T).
Let KQE, fig. 152, be our semi-cycloid (available for
GRAPHICAL USE OF CYCLOIDAL CURVES. 225
many successive constructions if these be only pencilled), and
CDE half the generating circle.
Then the following is our construction :-Divide EC into
6 equal portions, and let EP, PA be two of these parts, so
that EA represents 200,000 miles and CA 425,000 miles
(the sun's radius). Describe the semicircle ADL about the
centre Cand draw DM perpendicular to EC; describe the half
circle M m L. Then m C represents T where the ordinate PQ
represents the time of falling from E to P.
FIG. 152.

K
Q
35 20
m
a
15
15 10
5
A
M
L
Illustrating the construction for determining time of descent of a particle from
rest towards a globe attracting according to the law of nature,
T 18h 50m, and PQ (carefully measured) is found to
correspond to about twenty-six minutes.
Thus a body propelled upwards from A to E would
traverse the distance PE in twenty-six minutes. But the
hydrogen wisps watched by Professor Young traversed the
distance represented by PE in ten minutes. Hence either
E was not the true limit of their upward motion, or they
226
GEOMETRY OF CYCLOIDS.
were retarded by the resistance of the solar atmosphere.
Of course if their actual flight was to any extent fore-
shortened, we should only the more obviously be forced to
adopt one or other of these conclusions.
But now let us suppose that the former is the correct
solution; and let us inquire what change in the estimated
limit of the uprush will give ten minutes as the time of
moving (without resistance) from a height of 100,000 to a
height of 200,000 miles. Here we shall find the advantage
к
FIG. 153.

CL
M
13 10
G
p'
A
F
f
M
с
Illustrating the construction for determining time of descent between given levels
when a body descends from rest at a given height towards a globe attracting accord-
ing to the law of nature.
of the constructive method; for to test the matter by calcu-
lation would be a long process, whereas each construction
can be completed in a few minutes.
Let us try 375,000 miles as the vertical range. This
gives CE 800,000 miles, and our construction assumes the
appearance shown in fig. 153. We have AC-425,000 miles;
GRAPHICAL USE OF CYCLOIDAL CURVES. 227
AP PP'100,000 miles; and Q 1 or (PQ-P'Q') to repre-
sent the time of flight from P to P'.
The semicircles ADL, M m L, give us m C to represent
T or 18m 50s; and QL carefully measured is found to corre-
spond to rather less than ten minutes. It is, however, near
enough for our purpose.
It appears, then, that if we set aside the probability, or
rather the certainty, that the sun's atmosphere exerts a
retarding influence, we must infer that the matter projected
from the sun reached a height of 375,000 miles, or there-
abouts. This implies an initial velocity of about 265 miles
per second.*
But it will be well to make an exact calculation,-not
that any very great nicety of calculation is really required,
but in order to illustrate the method to be employed in such
cases, as well as to confirm the accuracy of the above con-
structions.
In equation (2) put 2 g R = 379; R = 425,000;
D = 625,000; and x = 525,000; values corresponding to
Professor Young's observations. It thus becomes—
x=
125 (379) t
(379) t(100,000) (525,000)
+ 312,500 cos-1
(10505625
The value is of course deduced directly from (1), p. 219; but it
is worthy of notice that it can be deduced at once from fig. 153, by
drawing A a parallel to KC, and mf parallel to a E; then Cƒrepre-
sents the required velocity, CL representing 379 miles per second.
A similar construction will give the velocity at P, P', &c. Applied
to fig. 147, it gives Cf to represent the velocity at A, C' to represent
the velocity at P; mf and mf being parallel to a E and GE re-
spectively. Applied to the case dealt with in fig. 152, we get Cf t
represent the velocity at A, where E is the limit of flight: Cƒi
found to be rather more than 5 of CL; so that the velocity at A is
rather more than 210 miles
per second.
Q 2
228
GEOMETRY OF CYCLOIDS.
or
379 √/17 . t = 250,000 √21 + 1,562,500 cos−1
1562-7 t = 1,145,100 1,285,800
+
2,430,900,
t = 1,556ª = 25m 56s.
This then is the time which would have been occupied in
the flight of matter from a height of 100,000 to a height of
200,000 miles, if the latter height had been the limit of
vertical propulsion in a non-resisting medium.
In order to deduce the time of flight t between the same
levels, for the case where the total vertical range is 375,000
miles, we have, putting t, for the time of fall to 200,000
miles above the sun's surface, and t₂ for the time of fall to
100,000 miles, the equation,
125 (379) = √(175,000) (625,000)
ti
800
+(400,000) cos-1
N
425
800
(379) t₂ = √(275,000) (525,000)
t2
+ (400,000) cos-1
(12580),
giving (since t₂ — t₁ = t')
V
−
125 (379) t' = 25,000 { ✓11 × 21 − √7 × 25 }
800
(18) - C
16
161,066,
25
(105-80),
cos-1
+ 400,000 { cos-1
276.25 t' 49,250 + 111,816
t' = 583$ = 9m 43s.
I had
This is very near to Professor Young's ten minutes.
found that an extreme height of 400,000 miles gave 9m 24s
for the time of flight between vertical altitudes 100,000
9
(1/2)}
GRAPHICAL USE OF CYCLOIDAL CURVES. 229
miles and 200,000 miles. It will be found that a height
of 360,000 miles gives 9m 58s, which is sufficiently near to
Professor Young's time.
Now to attain a height of 360,000 miles a projectile from
the sun's surface must have an initial velocity
✓
= 1/2gR.
360,000
785,000
379
√
72
157
257 miles per second.
The eruptive velocity, then, at the sun's surface, cannot
possibly have been less than this. When we consider, how-
ever, that the observed uprushing matter was vaporous,
and not very greatly compressed (for otherwise the spectrum
of the hydrogen would have been continuous and the
spectroscope would have given no indications of the phe-
nomenon), we cannot but believe that the resisting action of
the solar atmosphere must have enormously reduced the
velocity of uprush before a height of 100,000 miles was
attained, as well as during the observed motion to the
height of 200,000 miles. It would be safer indeed to assume
that the initial velocity was a considerable multiple of the
above-mentioned velocity, than only in excess of it in some
moderate proportion. Those who are acquainted with the
action of our own atmosphere on the flight of cannon-balls
(whereby the range becomes a mere fraction of that due to
the velocity of propulsion), will be ready to admit that hy-
drogen rushing through 100,000 miles even of a rare atmo-
sphere, with a velocity so great as to leave a residue sufficient
to carry the hydrogen 100,000 miles in the next ten minutes,
must have been propelled from the sun's surface with a
velocity many times exceeding 257 miles per second, the
result calculated for an unresisted projectile. Nor need we
wonder that the spectroscope supplies no evidence of such
230
GEOMETRY OF CYCLOIDS.
velocities, since if motions so rapid exist, others of all
degrees of rapidity down to such comparatively moderate
velocities as twenty or thirty miles per second also exist,
and the spectral lines of the hydrogen so moving would
be too greatly widened to be discerned.
Now the point to be specially noticed is, that supposing
matter more condensed than the upflung hydrogen to be
propelled from the sun during these eruptions, such matter
would retain a much larger proportion of the velocity origi-
nally imparted. Setting the velocity of outrush, in the case
we have been considering, at only twice the amount deduced
on the hypothesis of no resistance (and it is incredible that
the proportion can be so small), we have a velocity of pro-
jection of more than 500 miles per second; and if the more
condensed erupted matter retained but that portion of its
velocity corresponding to three-fourths of this initial velocity
(which may fairly be admitted when we are supposing the
hydrogen to retain the portion corresponding to so much as
half of the initial velocity), then such more condensed
erupted matter would pass away from the sun's rule never to
return.
The question may suggest itself, however, whether the
eruption witnessed by Professor Young might not have been
a wholly exceptional phenomenon, and so the inference
respecting the possible extrusion of matter from the sun's
globe be admissible only as relating to occasions few and
far between. On this point I would remark, in the first
place, that an eruption very much less noteworthy would
fairly authorise the inference that matter had been ejected
from the sun. I can scarcely conceive that the eruptions
witnessed quite frequently by Respighi, Secchi, and Young
-such eruptions as suffice to carry hydrogen 80,000 or
100,000 miles from the sun's surface-can be accounted for
GRAPHICAL USE OF CYCLOIDAL CURVES. 231
without admitting a velocity of outrush exceeding consider-
ably the 379 miles per second necessary for the actual rejec-
tion of matter from the sun. But apart from this it should
be remembered that we only see those prominences which
happen to lie round the rim of the sun's visible disk, and
that thus many mighty eruptions must escape our notice
even though we could keep a continual watch upon the
whole circle of the sierra and prominences (which unfortu-
nately is very far from being the case).
It is worthy of notice that the great outrush witnessed
by Professor Young was not accompanied by any marked
signs of magnetic disturbance. Five hours later, however, a
magnetic storm began suddenly, which lasted for more than
a day; and on the evening of September 7, there was a dis-
play of aurora borealis. Whether the occurrence of these
signs of magnetic disturbance was associated with the
appearance (on the visible half of the sun) of the great spot
which was approaching or crossing the eastern limb at the
time of Young's observation, cannot at present be deter-
mined.
I would remark, however, that so far as is yet known
the disturbance of terrestrial magnetism by solar influences
would appear to depend on the condition of the photosphere,
and therefore to be only associated with the occurrence of
great eruptions in so far as these affect the condition of the
photosphere. In this case an eruption occurring close by the
limb could not be expected to exercise any great influence on
the earth's magnetism; and if the scene of the eruption were
beyond the limb, however slightly, we could not expect any
magnetic disturbance at all, though the observed phenomena
of eruption might be extremely magnificent.
In this connection I venture to quote from a letter
232
GEOMETRY OF CYCLOIDS.
addressed to me by Sir J. Herschel in March 1871 (a few
weeks only before his lamented decease). The letter bears
throughout on the subject of this paper, and therefore I
quote more than relates to the association between terrestrial
magnetism and disturbances of the solar photosphere.
C
After referring to Mr. Brothers' photograph of the corona
(remarking that the corona is certainly extra-atmospheric
and ultra-lunar'), Sir John Herschel proceeds thus :-
100
'I can very well conceive great outbursts of vaporous
matter from below the photosphere, and can admit at least
the possibility of such vapour being tossed up to very great
heights; but I am hardly yet exalted to such a point as to
conceive a positive ejection of erupted particles with a
velocity of two or three hundred miles per second. But
now the great question of all arises what is the photo-
sphere? what are those intensely radiant things-scales,
flakes, or whatever else they be-which really do give out
all (or at least ths of) the total light and heat of the
sun? and if the prominences, &c., be eruptive, why does not
the eruptive force scatter upwards and outwards this lu-
minous matter? .. Through the kindness of the Kew
observers I have had heliographs of the two great outburst-
ing spots which I think I mentioned to you as having been
non-existent on the 9th, and in full development on the 10th,
both [being] large and conspicuous, and including an area of
disturbance at least 2' (54,000 miles) across. They were both
nearly absorbed, or in rapid process of absorption, on the
11th. In my own mind I had set it down as pretty certain
that the outbreak must have taken place very suddenly at
somewhere about the intervening midnight. Well, now!
The magnet's declination curves at Kew have been sent me,
and, lo! while they had been going on as smoothly as
•
GRAPHICAL USE OF CYCLOIDAL CURVES. 233
possible on the 6th, 7th, 8th, and 9th, and up to 11 P.M. on
the latter day (9th), suddenly a great downward jerk in the
curve, forming a gap as far as 3 A.M. on the 10th. Then
comparative tranquillity till 11 A.M., and then (corresponding
to the re-absorption of the spots) a furious and convulsive
state of disturbance extending over the 11th and the greater
part of the 12th. I wonder whether anything was shot out
of those holes on that occasion! and, if so, what is going on
in the inside of the sun?'
8
234
GEOMETRY OF CYCLOIDS.
!
EXAMPLES.
All the examples which have no name appended to them are
original, except four or five familiar ones (as 125, 126, &c.), the
authors of which are not known.
1. A chord of a cycloid parallel to the base is equal in
length to the perimeter of the uppermost of the two seg-
ments into which the chord divides the generating circle.
from
2. A'PB' is the generating circle through P on the cy-
cloidal arc APD; A'B' diametral; and equal arcs P q and
Pq' are taken on A'PB'. Show that straight lines drawn
q and q', parallel to the base, to meet APD, are equal.
3. AQB is a semicircle on diameter AB; and from Q, QL is
drawn perp. to AB, and produced to P, so that QP= arc
AQ. Show that the locus of P is a cycloid having a cusp at
A, and AB as secondary axis.
4. If B'P (fig. 4, p. 8), the normal at P, be produced to
meet AA' produced, in G, then PB'. PG = A'P².
5. If the tangent A'P (fig. 4, p. 8), produced, meet the
tangent at D in T, show that A'T: A B :: arc PB': PB.
6. Show that the rectangle under PG (fig. 4, p. 8) and
the diameter of curvature at P = (arc AP)2.
7. Show that the chord in which the tangent at P (fig. 4,
p. 8) intersects the circle on B'G as diameter, is equal to the
arc AP.
8. PC'p is the tracing diameter of P on the cycloidal arc
EXAMPLES.
235
D'APD. If p P', parallel to the base, meet the arc D'A
in P, show that the tangents and normals at P and P' form
a rectangle.
9. An equilateral triangle AQC is described on AC
(fig. 4, p. 8) as side; show that QP, parallel to the base of the
cycloid, bisects the arc APD in P.
10. If through C, CP be drawn parallel to the base, to
meet the cycloid in P, show that (arc AD) 2 = 2 (arc AP)2.
11. If there are two cycloids APD and AP"D placed as
in fig. 3, p. 6, and the straight line drawn from any point P
in one to a point Q in the other, P and Q lying on different
sides of Cc, is equal to the diameter of the generating circle,
show that the circle on PQ as diameter touches BD and AE.
12. When the angle BAQ (fig. 4, p. 8) is equal to two
thirds of a right angle, then in the limit when P' moves up
to P,
14. In fig.
R, show that
2 n g = 2 7 n.
2 n q
PP' 2 MN, and q P' =
13. When the angle BAQ = one-third of a right angle,
then in the limit
PP' = q P' = 2n q = 3 l n.
៖
3, p. 6, if arc AQB intersect arc AP'D in
area AQRP"
= area BRD.
15. APD, AP'D are two equal semi-cycloids placed as in
fig. 8, p. 17; show that every generating circle A'PB' divides
the area APDP' into three parts, which are equal each to each
to the three parts into which the area of the circle A'PB' is
divided by the arcs APD, AP'D.
16. In the same case, if two generating circles P'RA'PB'
and p'r a pb cut APD in R, P and r, p, respectively, and
AP'D in P', p', show that
area P'R r p' = difference of areas RA'P, r a p.
236
GEOMETRY OF CYCLOIDS.
17. In fig. 5, p. 10, Area RD c = area AQC'T.
18. In fig. 11, p. 22,
Area AQB p" - area E p'' D = generating circle.
19. If in fig. 5, p. 10, RJ is drawn perp. to BD, and a
quadrant AIC about T as centre, show that
area RJD = area AQC’I.
20. If CQP parallel to base BD cut the central genera-
ting circle in Q and meet the cycloid in P, show that the area
AQP is equal to the triangle ABQ.
21. A semi-cycloid having BA as axis, B as vertex, cuts
the semi-cycloid APD (A vertex, AB axis, and D cusp) in P,
and AQB is the central generating circle, Q lying on the
same side of AB as P; show that the area AQBP is equal to
the square inscribed in the circle AQB.
22. The normal at any point of a cycloidal arc divides
the area of a generating circle through the point, and the area
of the cycloid, in the same ratio.
23. In Example 20, show that
(arc AP)²= (arc APD)2.
24. If a cycloidal arc DAD' is divided into any two parts
in P, and PB' is the normal at P (B' on the base), show that
arc DP. arc PD' — 4 (PB') ²
2
1
2
25. D is the cusp of a cycloid APD, C' the centre of the
tracing circle PKB' through P. If DC' cut the tracing
circle PKB' in K, and DP = 2 arc PK, show that DP
touches the tracing circle at P.
26. If APD is a semi-cycloid, having axis AB and base
BD; AP'D the quadrant of an ellipse having semi-axes AB,
BD; and AP"D the arc of a parabola, having AB as axis,
show that
area APDP area AP'DB: area AP/DB:: 9: 3π: 8.
EXAMPLES.
237
27. With the same construction, the radii of curvature of
the three curves at A are in the ratio 16: 22: π².
28. On the generating circle AQB the arc AQ = } cir-
cumference is taken, and through Q a straight line parallel
to the base is drawn, cutting the cycloid in the point P;
show that the radius of curvature at P is equal to the
axis AB.
29. The axis AB of a cycloid APD is divided into four
equal parts in the points D, C, and E, through which straight
lines are drawn parallel to the base, meeting the cycloid in the
points P1, P2, and P3; if the radii of curvature at A, P1, P2,
and P3, are respectively equal to p₁, P2, P3, and p₁, show that
P12: P22 P32: P2: 4:3: 2: 1.
P1,
4
30. OI (fig. 14, p. 27) is produced to a point J, such that
I J = 2 OK, and on OJ as base a cycloid is described; show
that radius of curvature at vertex of this cycloid
= LG'.
31. If a cycloid roll on the tangent at the vertex, the
locus of the centre of curvature at the point of contact is a
semicircle of radius 4 R.
32. If a cycloidal arc be regarded as made up of a great
number of very small straight rods jointed at their extremities,
and each such rod has its normal (terminated on the base of
the cycloid) rigidly attached to it, show that if the arc bet
drawn into a straight line, the extremities of the normals
will lie in a semi-ellipse, whose major axis 8 R, and minor
= 4 R.
axis
33. PB' and P'B' are the normals at two points P, P',
close together on a cycloidal arc, and PQ parallel to the base
BD' meets the central generating circle in Q; show that if
PP' is of given length, B'B" varies inversely as the chord
BQ.
34. From different points of a cycloidal arc, whose axis is
238
GEOMETRY OF CYCLOIDS.
vertical, particles are let fall down the normals through those
points; show that they will reach the base simultaneously in
R
time 2
g
If they still continue to fall along the normals pro-
duced, they will reach the evolute simultaneously in time.
2 R
2
V/
g
35. If the distance of P on semi-cycloidal arc APD (fig.
10, p. 21) from base BD 2 AB, show that
3 moment of PD about AE = 14 moment of AC about AE.
36. In same case, if PM parallel to BD meet AB in M,
show that
moment of PD about AE= (AB); [(AB)
}
37. Show that the moment of arc AP (fig.
about AB
(AM); }.
10, p. 21)
= 2 (NQ+arc AQ) AQ- AB (AB - BM).
38. If equal rolling circles on the same fixed circle
trace out an epicycloid and hypocycloid having coincident
cusps, the points of contact of the rolling circles with the
fixed circles coinciding throughout the motion, show that
the tangents through the simultaneous positions of the tracing
point intersect on the simultaneous common tangent to the
three circles.
39. A tangent at a point P on an epicycloidal arc APD is
parallel to AB the axis, and a circular arc PQ about O as
centre intersects the central generating circle in Q; show
that
Arc AQ arc BQ:: F: 2 R.
40. Two tangents P'T, PT to the same epicycloidal arc
D'P'APD intersect in T at right angles, and through P' and
EXAMPLES.
239
P circular arcs P'Q' and PQ are drawn around Q as centre
to meet the central generating circle in Q and Q, neither arc
cutting this circle; show that
are Q'AQ a semicircle :: F: F+2R.
41. If the rolling circle by which an epicycloid is traced
out travel uniformly round the fixed circle, the angular ve-
locity of the point of contact about centre of fixed circle being
w, show that the directions of the normal of the tangent also
F+2R
2 R
change uniformly with angular velocity
42. On the same assumption, the direction of the tracing
F+R
radius changes uniformly with angular velocity
R
ω.
W.
W.
43. If the rolling circle by which a hypocycloid is traced
out travel uniformly round the fixed circle, the angular
velocity of the point of contact about centre of fixed circle
being w, show that the direction of the normal and of the
tangent also change uniformly with angular velocity
F-2 R
2 R
44. On the same assumption the direction of the tracing
F - R
radius changes uniformly with angular velocity
R
W.
45. A is the vertex of a hypocycloidal arc APDP', D the
cusp, P' a point on the next arc; and the tangent at P' is
parallel to the axis AB. If a circular arc P'Q around O as
centre intersect the remoter half of the central generating
circle in Q, show that
Arc ABQ arc BQ:: F: 2 R.
46. Two tangents P'T, PT to the same hypocycloidal arc
D'P'APD, the base D'D less than a quadrant, intersect in T
at right angles; and through P' and P circular arcs P'Q'and
240
GEOMETRY OF CYCLOIDS.
PQ are drawn around O as centre to meet (without cutting)
the central generating circle in Q' and Q; show that
Arc Q'AQ: a semicircle :: F: F-2 R.
47. AQ, QB are quadrants of the central generating
circle of an epicycloid or a hypocycloid, and the circular arc
QB about O as centre meets APD in P; show that
Area APQ triangleABQ:: CO; BO.
48. In last example, show that (arc AP)² = ½ (arc APD)².
49. At any point B' in the base of an epicycloid DAD'
a tangent PB'P' is drawn to the fixed circle, meeting the
epicycloid in P and P; show that
PB' < arc DB', and P'B' < arc D'B'.
50. With the same construction, show that PB'P' has its
greatest value when B' is at B, the foot of the axis AB.
51. At P, a point on the epicycloid DAD', a tangent
PKD' is drawn cutting the fixed circle in K and K', and the
normal PB'b' cutting the fixed circle in B' and b' (B' on the
base DBD'); show that
PK . PK': (PB')2 :: F +R: R:: (Pb)²; PK . PK'.
52. With the same construction if OM be drawn perp.
to PKP', show that
OM PB Pb': F+2R: 2R: 2 (F+ R).
•
53. If tangent at P to epicycloid DAD' touches the
fixed circle, and PB'b' the normal at P meets the fixed circle
in B' and b' (B' on the base DBD'), show that
PB' (F+2R) = 2 R2; and Pb' (F+2R) = 2R (F+ R).
54. If tangent at P to epicycloid DAD' touches the fixed
circle and cuts the rolling circle in A', then
(A'P)2 (2R)2:: (F + R) (F+3R): (F+2R)2
EXAMPLES.
241
55. In figs. 21 and 22 (pp. 44, 45) the points P, B', b, lie
in a straight line.
56. In figs. 21 and 22, the tangent to DP at P cuts
Oc' produced in a point a such that ba 2b c'.
57. At D the cusp of an epicycloid D'AD (fig. 19, fron-
tispiece) a tangent Dt to the fixed circle DBD' meets D'AD
in t, and from t another tangent K is drawn meeting the
fixed circle in K; show that Dt is always less than the arc
DBK if the radius of the rolling circle is finite.
58. ACB is the axis of an epicycloid DAD'; D, D′ its
cusps; CQ, Oq radii of central generating circle and fixed
circle respectively, perp. to ABO and on same side of it.
If C'q cut Qq parallel to CO in K, and a straight line d K d'
り
​I
through K parallel to O q is the generating base of a prolate
eyeloid having AQB as central generating circle, show that
the area between the epicycloid DAD' and its base DD' is
equal to the area between the prolate cycloid d A d'aud its
base d'd'.
59. ACB is the axis of a hypocycloid DAD ; D, D' its
casps; CQ, Oq radii of central generating circle and fixed
circle perp. to BAO and on the same side of it. If Ca cut
Qy parallel to CO in K, and a straight line d K d' through
K parallel to Oq is the generating basis of a curtate cycloid
having AQB as central generating circle, show that the
area between the hypocycloid DAD' and its base DD' is
equal to the area between the curtate cycloid d A d' and its
base d' d'.
60. The area between the cardioid and its base is equal
to five times the area of the fixed circle.
61. The area between the cardioid and a circle concentric
with the fixed circle, touching the cardioid at the vertex, is
equal to three times the area of the fixed circle.
R
242
GEOMETRY OF CYCLOIDS.
I
62. The area of a circle touching the cardioid at the
vertex and concentric with the base, is divided into three
equal parts by the arc of the cardioid and the axis produced
to meet the circle.
63. Area A o P (fig. 39, p. 74) 3R (6k+ arc Bb).
64. If 0 = ▲ BO b (fig. 39, p. 74)
Area PSA R² (30 + 4 sin 0 + 1 sin 2 0).
65. The area between one arc of the tricuspid epicycloid
and the base is equal to 3 times the area of the generating
circle.
66. A complete focal chord is drawn to a cardioid.
Show that the lesser of the two segments into which the
focus divides the chord, is equal to the portion intercepted
between the fixed circle and the tracing circle through the
extremity of the longer segment.
67. A circle is described on the axial focal chord as
diameter, show that the segments of a complete focal chord
intercepted between the curve and this circle are equal.
(Purkiss.)
68. Lines perp. to focal radii vectores through their ex-
tremities have a circle for envelope. (Purkiss.)
69. From S, the focus of cardioid, a perp. SQ to a com-
plete focal chord PSP', is drawn, meeting the fixed circle in
Q; show that SQ is a mean proportional between SP and
SP'.
70. If SP be any focal radius vector of a cardioid whose
vertex is A, and the bisector of the angle PSA meet the
circle on SA in Q, SQ will be a mean proportional between
SP and SA. (Purkiss.)
71. PSP' is a complete focal chord of a cardioid; SQAQ'
a circle on SA as diameter; SQ, SQ' bisectors of the angles
EXAMPLES.
243
PSA, P'SA respectively; and Sq perp. to PSP' meets circle
SQA in 9 ; show that
q
SQ : Sq :: SB ; SQ'.
72. The pedal of a cardioid with respect to the focus is
also the locus of the vertex of a parabola which is confocal
with the cardioid and touches the circle on SA as diameter.
(Purkiss.)
The demonstration of this will be more easily effected by taking
for the cardioid the locus of n, fig. 39 (see p. 75). From a draw
ny a parallel to bf, then Sy, perp. to ny, gives y a point on the
pedal of this cardioid with respect to S. It can readily be shown
that a parabola having S as focus and y as vertex touches the
circle BS in b.
73. From a fixed point A any arc AQ is taken and bi-
sected in Q'. If P is a point on the chord QQ' such that
QP = 2 Q'P, show that the locus of P is a cardioid.
74. If rays diverge from a point on the circumference of
a circle and be reflected at the circumference, the caustic will
be a cardioid. (Coddington's 'Optics,' or Parkinson's 'Optics,'
Art. 72, which see.)
If Sb, fig. 39, p. 74, represent path of a ray, to circle B b S, re-
flected ray bg is in the line Pbg, normal to the caustic APS, and
therefore the envelope of the reflected rays is the evolute of the
cardioid APS, or is a cardioid having its vertex at S, SO diametral
and linear dimensions one third those of APS. This, however, is
not a direct proof. The preceding proposition will be found to
supply a direct proof. For if from A two rays proceed to neighbour-
ing points Q, q, and thence respectively after reflection to neigh-
bouring points Qʻ and q', arc Q' q′ = 2 arc Q4 ; and the point of in-
tersection of QQ' and q q' therefore lies on QQ' (equal to AQ), at a
point ultimately equal to one-third of the distance QQ' from Q.
-
75. A series of parallel rays are incident on a reflecting
semicircular mirror and in the plane of the semicircle; show
that the caustic curve is one half (from vertex to vertex) of
R 2
244
GEOMETRY OF CYCLOIDS.
}
a bicuspid epicycloid or nephroid. (Coddington's 'Optics,' or
Parkinson's Optics,' Art. 71, which see.)
76. A series of rays are incident on the concave side
of a reflecting cycloidal mirror to whose axis they are
parallel and in whose plane they lie; show that the caustic
curve consists of two equal cycloids each having one half of
the base of the cycloidal mirror for base, and the axis of this
larger cycloid as the tangent at their cusp of contact.
77. The linear dimensions of the evolute of the bicuspid
epicycloid (or nephroid) are those of the curve itself.
78. The area between one arc of the nephroid and the
base is equal to four times the generating circle.
:
79. The evolute of a nephroid is drawn, the evolute of
this evolute, the evolute of this second evolute, and so on
continually show that the sum of all the areas between
all the evolute nephroids, and their respective base-circles,
are together equal to one-third of the area between the
original nephroid and its base-circle.
80. If in the epicycloid m R = n F, show that the linear
dimensions of the evolute are to those of the epicycloid as
m: m + 2n.
81. If m R = n F, area between an arc of epicycloid and
3 m + 2 n
(3 m + 2 n) n²
its base
fixed O.
.gen.
m
M3
82. If PB'o Q is the diameter of curvature at the point
P of an epicycloid, o the centre of curvature, B' a point of the
base, then
*
Area of epicycloid
area of gen. :: QB': B'o.
83. If the arc of an epicycloid, from cusp to cusp = a,
and m R = n F, show that a + arc of evolute from cusp to
cusp + arc of evolute's evolute from cusp to cusp, and so on
ad infinitum,
(m + 2n) a
2 n
EXAMPLES.
245
84. If the area between an epicycloid and its base = A,
and m R = n F, show that A+ area between an arc of the
evolute and its base + area between an arc of the evolute's
evolute and its base, and so on ad infinitum,
(m + 2n)² A²
2
±n (m + n)
85. If in the hypocyloid m R= n F, show that the linear
dimensions of the evolute are to those of the bypocycloid as
m: m-2n.
F
Interpret this result when R =
2
86. If m R = n F, area between an arc of hypocycloid
3 m 2 n
(3 m −2 n)n²
and its base
fixed O.
gen.
M3
M
87. If PB'o Q is the diameter of curvature at the point
P of a hypocycloid, o the centre of curvature, B' a point on
the base,
QB: B'o::3CF-2R: F.
88. If the arc of a hypocycloid from cusp to cusp=a, and
m R = n F, show that a + arc of hypocycloid of which the
given hypocycloid is the evolute + arc of hypocycloid of
which this hypocycloid is the evolute, and so on ad infinitum,
ጎ
2 n
A
89. If the area between a hypocycloid and its base A,
and m R = n F, show that A + the area between one arc of
the hypocycloid of which the given hypocycloid is the evolute,
and its base the area between one arc of the hypocycloid
of which this hypocycloid is the evolute and its base, and
so on ad infinitum,
m²A
4 n(m—n)'
246
GEOMETRY OF CYCLOIDS.
"
90. D'AD is an arc of a tricuspid epicycloid, from cusp
to cusp, ACB the axis, AQB the central generating circle, C
its centre, OBCA diametral; show that an angle may be tri-
sected by the following construction :-Let ACQ be the
angle to be trisected. Join QB, QO; about O as centre
describe arc QP meeting D'AD in P (on AD): join PO;
make the angle OPB equal to the angle OQB, and towards
the same side, PB' meeting the base D'BD in B'; and join
B'O. Then the angle BOB' is equal to one-third of the
angle ACQ.
91. D'AD is an arc of a tricuspid hypocycloid from
cusp to cusp; ACB the axis; AQB the central generating
circle, C its centre, OACB diametral. Show that an angle
may be trisected by the following construction. Let ACQ
be the angle to be trisected. Join QB, QO; about O as
centre describe arc QP meeting D AD in P (on AD); join
PO and make the angle OPB' equal to the angle OQB, and
towards the same side, PB' meeting the base D BD in B';
and join BO. Then the angle BOB' is equal to one-third of
the angle ACQ.
92. D'AD is an arc of an epicycloid from cusp to cusp ;
ACB the axis; AQB the central generating circle, C its
centre; OBCA diametral. A radius CQ is drawn to AQB;
and BQ, OQ are joined. About O as centre the arc QP is
drawn meeting D'AD in P (on AD); PO is joined, and the
angle OPB is made equal to the angle CQB and towards the
same side, PB' meeting the base D'BD in B'. If OB' is
joined, show that
R
F. angle ACQ,
angle BOB'
so that, by means of a suitable epicycloid, an angle may be
divided in any required ratio.
93. D'AD is an arc of a hypocycloid from cusp to
EXAMPLES.
247
C
cusp; ACB the axis; AQP the central generating circle,
Cits centre; OACB diametral. From C a radius CQ is
drawn to AQB; and BQ, OQ are joined. About O as centre
the arc QP is drawn meeting D'AD in P (on AD); PO is
joined; and the angle OPB' is made equal to the angle
OQB, and towards the same side, PB' meeting the base
D'BD in B'. If OB' is joined, show that
angle BOB =
R
F
angle ACQ,
so that by means of a suitable hypocycloid an angle may be
divided in any required ratio.
94. If PC p is the tracing diameter at P on an epicycloid
or hypocycloid APD (vertex at A), o the centre of curvature
at P, show that op produced meets the tangent at P in a
point T such that TP is equal to the arc AP.
95. If an epicycloid roll upon the tangent at the vertex,
show that the locus of the centre of curvature at the point
of contact is a semi-ellipse having semi-axes
4 R (F+R)
K
R) and
4 R2 / F+R
FF+2R •
and
96. If a hypocycloid roll upon the tangent at the vertex,
show that the locus of the centre of curvature at the point of
contact is a semi-ellipse having semi-axes
4 R²/F Ꭱ
F F-2R
97. An arc DAD of the bicuspid epicycloid, or nephroid,
has its axis AB coincident in position with A b, the axis of a
cycloid whose vertex is at A; but AB = Ab. If the
nephroid and the cycloid roll on T'AT, the common tangent
at A, in such sort that they simultaneously touch the same
point on TT, show that the centre of curvature of the
nephroid at the point of contact will trace out the same
curve as the foot of normal to the cycloid at the point of
4 R (FR)
F
248
GEOMETRY OF CYCLOIDS.
contact (the foot of normal being understood to mean the
intersection of the normal with the base).
98. If a quadricuspid hypocycloid (radius of fixed circle
F) is orthogonally projected on a plane through two opposite
cusps, in such sort that the distance 2 F between the other
two cusps is projected into distance 2f, show that the pro
jected curve is the evolute of an ellipse having axes equal to
Fƒ2
F²ƒ
and
F2 -ƒ2 F2 -ƒ'2'
99. Show that the arc of the projected curve in 98, from
cusp to cusp,
2
F² + Fƒ +ƒ²
F+ƒ
100. ACA', BCB' are the major and minor axes of an
ellipse, C its centre; and a Ba' B' is a similar ellipse having
BCB as major axis; if the ellipse ABA'B' is orthogonally
projected into a circle, show that the evolute of a Ba'B' will
be projected into a quadricuspid hypocycloid, and determine
its dimensions.
101. With the same construction, show (independently)
that the portion of the projection of any normal of a Ba B,
intercepted between the projections of AA' and BB', is of
constant length. (This will be found to follow readily from
Propos. X. and XIV. of Drew's 'Conics,' chapter ii.)
NOTE.-This proposition, demonstrated geometrically, combined
with what is shown at pp. 72, 73, affords a geometrical demon-
stration of the nature of the evolute to the ellipse. See next
problem.
102. Let ACA', BCB be the major and minor axes of an
ellipse, b C b' the orthogonal projection of BCB on a plane
through ACA', so situated that bb': BB' :: BB : AA.
From B draw BL perp. to AB to meet A'C in L; and about
EXAMPLES.
249
C in the p'ane Ab A, describe a circle with radius LA cutting
CA, CA, C' b, and C'b', in K, K', k, and k, respectively. Draw
a four-pointed hypocycloid, having cusps at K, k', K', and h.
Then a plane perpendicular to the plane Ab A'b', through
any tangent to the hypocycloid K k'K'k, will intersect the
plane ABA'B' in a normal to the ellipse ABA'B', and a
right hypocycloidal cylinder on K 'K'k as base, will inter-
sect ABA'B' in the evolute of this ellipse.
103. Two straight lines intersect at right angles in a
plane perpendicular to the sun's rays, one of the lines being
horizontal. If the extremities of a finite straight line slide
along the fixed straight lines, and the shadow of all three
lines be projected on a horizontal plane, show that the
envelope of the projection of the sliding line is the evolute of
an ellipse. Determine the position and dimensions of this
ellipse.
If the sun's altitude is a, and the length of the sliding line 7,
then taking for axis of the shadow of the horizontal fixed line,
the equation to the envelope is x²+ y² sin a = 1²; and the
equation to the involute ellipse is a² cos¹ a + y² sin² a cos¹ a=
a = 1'.
104. At P a point on the hypocycloid DPAD' the tan-
gent KPK' is drawn, meeting the fixed circle in K and K',
and the normal b'PB' meeting the fixed circle in b′ and B'
(B' on the base DBD'); show that
KP. PK' (PB')2: F-R: R:: (P): KP. PK'.
:
105. With the same construction, OM is drawn perp. to
KPK'; show that
OM : PB': Pb' :: F-2 R : 2 R : 2 (F− R).
106. If the tangent to the cardioid at P touches the fixed
circle, and cuts the rolling circle in A', and the normal at
P cuts the fixed circle in B' and b', then
250
GEOMETRY OF CYCLOIDS.
4 R
PB' = R; PÚ
107. In the trochoid, if R b', the normal at p, meets the
generating base in B', and the tangent at p meets the tangent
at vertex in T, a'b' being diametral to tracing circle; show
that triangle TB'p' is similar to triangle a'b'p.
108. With same construction
3
; and A'P=
▲ TB'a' = ▲ b'p B' = ¿ Tpa'.
R
109. In fig. 48, triangle Cbq"=
110. In fig. 48, p. 96, show that
(r 2R)
loop p' rdr =2
4√2
3
2"
R.
-
sector b C q".
-R)
arc a b NLq"+2
rect. N n.
за
Jo
111. Show that the result obtained in the last example
agrees with that obtained in Prop. IX., Section III.
112. If in Q'q', fig. 48, produced, a point X is taken such
that (CX)2 =rect. a B a C, and a circular arc XY (less than
semicircle) with C as centre and CX as radius cuts a
duced in Y, show that
b
pro-
loop p'r d = 2 segment XY rect. N n.
113. In fig. 48, p'' x is drawn parallel to q'b to meet the
base b d in y; show that
area y d r p': seg. q'Lb:: a B ; a C.
114. From B (fig. 45, frontispiece) a straight line Bq q'
is drawn cutting the central tracing circle in q and q', and
straight lines qp and q'p' parallel to the base meet the arc
a d in p and p'; show that the tangent at p is parallel to the
tangent at p'.
115. P and P' are two points on an epitrochoid or hypo-
trochoid, C and C' the corresponding positions of the centre
of generating circle, O the fixed centre, OA, OB the apo
central and pericentral distances. If OP. OP' = OA . OB,
EXAMPLES.
251
show that the tangents at P and P' make equal angles with
OC and OC' respectively.
116. A cycloid on base BD (fig. 45, frontispiece) has its
cusps at B and D; show that it touches the prolate cycloid
a pd at a point of inflexion.
117. A series of prolate cycloids have the same line of
centres, their axes in the same straight line, and their bases
equal. Show that their envelope is a pair of arcs of a cycloid
having its base equal to half the base of each prolate cycloid
of the system, and the line of their axes as a secondary axis.
118. If the normals at p and q, two points on a prolate
cycloid a p q d, are parallel, and meet the generating base in
b' and b' respectively, then p and p' being the radii of cur-
vature at p and q respectively,
p: p' :: (p b')² : (9 b'')².
119. If p is the radius of curvature at the point where a
curtate cycloid cuts the generating base, and μ is a mean
proportional between the radii of curvature at the vertex
and at d on the base, show that p²
μη.
120. Show that that involute of the central gene-
rating circle of a cycloid which has its cusp at the vertex
passes through the cusps of the cycloid.
121. That involute of any generating circle of a cycloid,
which has its cusp at the tracing point, passes through the
cusps of the cycloid.
122. The sum of the two nearest arcs of the involute of
the circle, cut off by any tangent to the circle, is least when
the tangent touches the circle at the farther extremity of
the diameter through the cusp of the involute.
123. If the rolling straight line by which the involute of
a circle of radius ƒ is traced out has rolled over an arc a from
the cusp, show that the arc traced out
f
= 1/2 a².
252
GEOMETRY OF CYCLOIDS.
124. If the rolling straight line by which a spiral of
Archimedes is traced out, has rolled over an arc a from first
position, when the extremity of perp. carried with it was
at the centre of the fixed circle (radius f), show that
f
= 1/2 { ~ a √I + u² + log (a + √1 + a²) } .
arc traced out=
125. All involutes of circles are similar.
126. All spirals of Archimedes are similar.
127. If a straight line carrying a perp. of length d roll on
a circle of radius f, and another straight line carrying a perp.
of length D (on same side with reference to centre of fixed
circle) roll on a circle of radius F, show that the curves
traced out by the extremities of these perps. will be similar
Pp:: Fƒ
Ff.
if
128. In the spiral of Archimedes the subtangent is equal
to that arc of a circle whose radius is the radius vector,
which is subtended by the spiral angle. (Frost's Newton').
The subtangent is the portion of a perp. to radius vector,
through pole, intercepted between pole and tangent at extremity of
radius vector. What is required to be shown in this example is
that if p'p (fig. 72, p. 130), produced, meet B′O produced in Z, OZ
is equal to the arc corresponding to DQB' in a circle of radius Op.
129. Establish the following construction for determining
the centre of curvature at point p (fig. 72, p. 130) of a spiral
of Archimedes. Draw radius OB' to fixed circle, perp. to
Op; join p B'; and draw OL perp. to p B'. Then if B'L is
divided in o so that
.
B'o: oL:: B'p : B'L,
o is the centre of curvature at p.
130. From this construction (established geometrically)
show that, taking the usual polar equation to the spiral of
Archimedes, viz., r = a 0,
p =
a(1 + (42) 3}
2+02
EXAMPLES.
253
0 =
131. A straight line turns uniformly in a plane round a
fixed point, while the foot of a perpendicular of length l
moves uniformly along the revolving line; show that the
other end of this perpendicular will trace out one of the
spirals described at pp. 128, 129.
132. If the angular velocity in preceding problem is w,
the linear velocity of the foot of perpendicular v, and l
v
the perpendicular lying on the side towards which the revolv-
ing line is advancing, show that the other extremity of the
perpendicular will describe the involute of the circle.
133. If DT, fig. 42, p. 82, rolls on the circle DQB of radius
a, and a point initially on DO and distant b from D is carried
with DT to trace out a spiral in the manner described at
pp. 128, 129, show that the polar equation to the spiral, OQ
being taken as initial line, and the rolling taking place in the
usual positive direction, is
/p2
(a - b)2
a
+ tan-1
3
(a² p² + b²);
९२
L
(a - b)
(a - b)²
2
ز + ab + b²°
j.2 — (α,
134. Show that the construction given in Example 129
for determining the centre of curvature at a point on the
spiral of Archimedes is applicable to all the spirals of Ex-
amples 131 and 133.
135. In the case of one of these spirals, putting the arc
over which the rolling line has passed from its initial
position =4, show that
W
136. The locus of the foot of perpendicular from a point
on a cycloid upon the diametral of the generating circle
through the point is the companion to the cycloid.
254
GEOMETRY OF CYCLOIDS.
137. From D, the cusp of an inverted cycloid, and P, a
point near D, two particles roll down the smooth arc to the
vertex; show that in the limit the path of either relatively
to the other is a semicircle.
138. A particle is projected with given velocity from the
vertex of a cycloid whose axis is vertical, and vertex upper-
most; find where it will leave the curve, and the latus
rectum of its future parabolic path.-(Tait and Steele's
'Dynamics.')
139. A particle falling from rest at a point in an in-
verted cycloid has its velocity suddenly annihilated when it
has passed over half its vertical height above the lowest
point; then proceeds, again losing its velocity when half-
way down from its last position of no velocity, and so on
continually. Show that it will be at 1th of its original
22n
height above the vertex after n times the time it would have
taken to fall to the vertex undisturbed.—(Tait and Steele's
'Dynamics.')
140. If a curve of any form is rolling upon another
curve in the same plane, and p is a point on the curve
traced by any given point carried with the rolling curve and
in the same plane with it, b the point of contact of the fixed
and rolling curves, show that the following relation exists.
between P1, P2, the radii of curvature of the fixed and rolling
curves at b, and p, the radius of curvature of the traced
P3
curve at p (putting pbn and the angle between pb and
the normal of fixed curve at b A),
{n (p₁ + P₂) - P1 P2 cos 0} P3 = n² (p₁ + p₂).
141. A tube of uniform cross section, small compared with
its length, is bent into the form of a cycloid, its open ends
EXAMPLES.
255
and
lying at the cusps, and this cycloid is placed with its axis
vertical and its vertex downwards. Equal quantities of
fluids of specific gravity, and σ, are poured in at the two
cusps, the quantity of each being such as would fill a length
a of the tube (a being the length of the cycloid's axis, so that
4a is the length of the tube). If the fluids do not mix and
the distance of the upper levels of the fluids from the vertex
(measured along the cycloidal arc) be x₁, x, respectively,
show that
4x,(σ₁ + σ₂) = a(σ₁ + 30₂),
4x₂(0₁ + σ₂) = a(302 +03).
142. If in problem 141 an equal quantity of a third fluid
of specific gravity 3 is poured in upon the free surface of
the second fluid (sp. gr. σ2), and x1, x2, are the respective
distances of the free surfaces of the first and third fluids
from the vertex (measured along the cycloidal arc), show
that
4x₁(~1 +0₂+03) = a(1 + 3 0₂+ 503),
(T
4X2(1 + σ₂+03) = a(5 σ, +32 +03).
and
Under what condition will either the first or third fluid run
over?
0
o
143. If n fluids are poured in, as in Ex. 141, the specific
gravities of 1st, 2nd, 3rd, &c., to the nth, being 1, 2, 3,
&c., to on, respectively, the arcs occupied by the respective
fluids being 1, 2, 3, . . . „, and no fluid overflowing; and if
x is the distance of the free surface of the first fluid from the
vertex (measured along the cycloidal arc), show that
4x(014 + 0212 + 03/3 + ... + 0,1) = 0,1,² + 0₂(12² + 2₁₂)
21/2)
+ 03(13²+271/3+22/3) + . . .
2
+ o₁(?„² + 27, 7, +224 + ... + 2n_Vn).
ก
250
GEOMETRY OF CYCLOIDS.
Ex. 144. Fig. 154.
Ex. 145. FIG. 155.
Ex. 146. FIG. 156.
Ex. 147. FIG. 157.
Ex. 148. Fig. 158.


Ex. 149. FIG. 159.


Ex. 150. FIG. 160.
Ex. 151. FIG. 161.
Spottiswoode & Co., Printers, New-street Square, London.
Interpret figs. 154 161 in the way explained in pp. 191-193.




!
GENERAL LISTS
LISTS OF NEW WORKS
FEBRUARY 1881.
MESSRS. LONGMANS, GREEN & CO.
PATERNOSTER ROW, LONDON.
J
PUBLISHED BY
HISTORY, POLITICS, HISTORICAL MEMOIRS &c.
Armitage's Childhood of the English Nation. Fcp. 8vo. 2s. 6d.
Arnold's Lectures on Modern History. 8vo. 7s. 6d.
2 vols. 8vo. 23s.
Bagehot's Literary Studies, edited by Hutton.
Browning's Modern France, 1814-1879. Fep. Svo. 1s.
Buckle's History of Civilisation. 3 vols. crown 8vo. 24s.
Chesney's Waterloo Lectures. 8vo. 10s. 6d.
Epochs of Ancient History :-
Beesly's Gracchi, Marius, and Sulla, 2s. 6d.
Capes's Age of the Antonines, 2s. 6d.
Early Roman Empire, 2s. 6d.
02000
Cox's Athenian Empire, 2s. 6d.
Greeks and Persians, 2s. 6d.
Curteis's Rise of the Macedonian Empire, 2s. 6d.
Ihne's Rome to its Capture by the Gauls, 2s. 6d.
Merivale's Roman Triumvirates, 2s. 6d.
Sankey's Spartan and Theban Supremacies, 2s. 6d.
Smith's Rome and Carthage, the Punic Wars, 2s. 6d.
Epochs of English History, complete in One Volume. Fcp. 8vo. 5s.
Creighton's Shilling History of England (Introductory Volume).
Fcp. 8vo. 1s.
Browning's Modern England, 1820-1875, 9d.
Epochs of Modern History :-
gd.
Cordery's Struggle against Absolute Monarchy, 1603–1688, 9d.
Creighton's (Mrs.) England a Continental Power, 1066–1216,
Creighton's (Rev. M.) Tudors and the Reformation, 1485-1603, 9d.
Rowley's Rise of the People, 1215–1485, 9d.
Rowley's Settlement of the Constitution, 1688-1778, 9d.
Tancock's England during the American & European Wars,
1778-1820, 9d.
York-Powell's Early England to the Conquest, 1s.
Eldorado
Church's Beginning of the Middle Ages, 2s. 6d.
Cox's Crusades, 2s. 6d.
Creighton's Age of Elizabeth, 2s. 6d.
Gairdner's Houses of Lancaster and York, 2s. 6d.
Gardiner's Puritan Revolution, 2s. 6d.
Thirty Years' War, 2s. 6d.
London, LONGMANS & CO.
2
General Lists of New Works.
¡
į
Epochs of Modern History-continued.
—
Hale's Fall of the Stuarts, 2s. 6d.
Johnson's Normans in Europe, 2s. 6d.
Seebolim's Protestant Revolution, 2s. bd.
Stubbs's Early Plantagenets, 2s. 6d.
Warburton's Edward III., 2s. 6d.
Froude's English in Ireland in the 18th Century. 3 vols. crown 8vo. 18s.
History of England. 12 vols. 8vo. £8. 18s. 12 vols. crown 8vo. 728.
Julius Cæsar, a Sketch. 8vo. 16s.
Longman's Frederick the Great and the Seven Years' War, 2s. 6d.
Ludlow's War of American Independence, 2s. 6d.
Morris's Age of Queen Anne, 2s. 6d.
Gardiner's England under Buckingham and Charles I., 1624-1628. 2 vols. 8vo. 24s.
Personal Government of Charles I., 1628-1637. 2 vols. 8vo. 24s.
Greville's Journal of the Reigns of George IV. & William IV. 3 vols. 8vo. 36s.
Hayward's Selected Essays. 2 vols. crown 8vo. 12s.
Ihne's History of Rome. 3 vols. 8vo. 45s.
8vo. 36s.
Lecky's History of England. Vols. I. & II. 1700-1760.
European Morals. 2 vols. crown 8vo. 163.
Rationalism in Europe. 2 vols. crown 8vo. 16s.
Lewes's History of Philosophy. 2 vols. 8vo. 32s.
Longman's Lectures on the History of England. 8vo. 15s.
Life and Times of Edward III. 2 vols. 8vo. 28s.
Macaulay's Complete Works. Library Edition. 8 vols. 8vo. £5. 5s.
A
History of England:-
Student's Edition, 2 vols. cr. 8vo. 12s.
People's Edition. 4 vols. cr. 8vo. 16s.
Macaulay's Critical and Historical Essays.
Student's Edition. 1 vol. cr. 8vo. 63.
People's Edition. 2 vols. cr. 8vo. 8s.
May's Constitutional History of England, 1760-1870. 3 vols. crown 8vo. 18s.
Democracy in Europe. 2 vols. 8vo. 32s.
Merivale's Fall of the Roman Republic. 12mo. 7s. 6d.
Cabinet Edition. 16 vols. crown 8vo. £4. 16s.
-
Cabinet Edition. 8 vols. post 8vo. 483.
Library Edition. 5 vols. 8vo. £4.
General History of Rome, B.C. 753-A.D. 476. Crown 8vo. 7s. 6d.
History of the Romans under the Empire. 8 vols. post 8vo. 48s.
Minto (Lord) in India from 1807 to 1814. Post 8vo. 12s.
Rawlinson's Seventh Great Oriental Monarchy-The Sassanians. 8vo. 28s.
Russia Before and After the War, translated by E. F. Taylor. 8vo. 14s.
Russia and England from 1876 to 1880. By 0. K. Svo. 14s.
Seebohm's Oxford Reformers--Colet, Erasmus, & More. 8vo. 14s.
!
I ca
Cheap Edition. Crown 8vo. 3s. 6d.
Cabinet Edition. 4 vols. post 8vo. 24s.
Library Edition. 3 vols. 8vo. 36s.
Sewell's Popular History of France to the Death of Louis XIV. Crown 8vo. 7s. 6d.
Short's History of the Church of England.
Smith's Carthage and the Carthaginians.
Taylor's Manual of the History of India.
Todd's Parliamentary Government in England. 2 vols. 8vo. 37s.
Crown 8vo. 7s. 6d.
Crown 8vo. 10s. 6d.
Crown 8vo. 7s. 6d.
the British Colonies.
8vo. 21s.
S
Crown 8vo. 2s. 6d.
Trench's Realities of Irish Life.
Trevelyan's Early History of Charles James Fox. 8vo. 18s.
Walpole's History of England, 1815-1841. Vols. I. & II. 8vo. 36s. Vol. III. 18s.
Webb's Civil War in Herefordshire. 2 vols. 8vo. Illustrations, 42s.
London, LONGMANS & CO.
#
General Lists of New Works.
3
CO
Memoirs of Anna Jameson, by Gerardine Macpherson. 8vo. 12s. 6d.
Mendelssohn's Letters. Translated by Lady Wallace. 2 vols, cr. 8vo, ös, each,
Mill's (John Stuart) Autobiography. 8vo. 7s. 6d.
Missionary Secretariat of Henry Venn, B.D. Sve. Portrait. 18s.
Newman's Apologia pro Vita Suâ. Crown Svo. 6s.
Nohl's Life of Mozart. Translated by Lady Wallace. 2 vols. crown 8vo. 21s.
Overton's Life &c. of William Law. Svo. 15s.
BIOGRAPHICAL WORKS.
Bagchot's Biographical Studies. 1 vol. 8vo. 12s.
Burke's Vicissitudes of Families. 2 vols. crown 8vo. 21s.
Cates's Dictionary of General Biography. Medium 8vo. 28s.
Gleig's Life of the Duke of Wellington. Crown 8vo. 63.
Jerrold's Life of Napoleon III. Vols. I. to III. 8vo. price 18s. each.
Lecky's Leaders of Public Opinion in Ireland. Crown 8vo. 7s. 6d.
Life (The) and Letters of Lord Macaulay. By his Nephew, G. Otto Trevelyan,
M.P. Cabinet Edition, 2 vols. post 8vo. 12s. Library Edition, 2 vols. 8vo. 36s.
Marshman's Memoirs of Havelock. Crown 8vo. 3s. 62.
Spedding's Letters and Life of Francis Bacon. 7 vols. 8vo. £4. 4s.
Stephen's Essays in Ecclesiastical Biography. Crown 8vo. 7s. 6d.
Amos's View of the Science of Jurisprudence. 8vo. 18s.
padd
MENTAL AND POLITICAL PHILOSOPHY.
Fifty Years of the English Constitution, 1830-1880. Crown Svo. 10s. 6d.
Primer of the English Constitution. Crown 8vo. 68.
Bacon's Essays, with Annotations by Whately. 8vo. 10s. 6d.
Works, edited by Spedding. 7 vols. 8vo. 73s. 6d.
Bagehot's Economic Studies, edited by Hutton. 8vo. 10s. 6d.
Bain's Logic, Deductive and Inductive. Crown 8vo. 10s. 6d.
PART I. Deduction, 4s.
Bolland & Lang's Aristotle's Politics.
Brassey's Foreign Work and English Wages. 8vo. 10s. 6d.
Comte's System of Positive Polity, or Treatise upon Sociology. 4 vols. 8vo, £1,
Congreve's Politics of Aristotle; Greek Text, English Notes. 8vo. 18s.
PART II. Induction, 6s. 6d.
Crown 8vo. 7s. 6d.
Grant's Ethics of Aristotle; Greek Text, English Notes. 2 vols. 8vo. 32s.
Griffith's A B C of Philosophy. Crown 8vo. 5s.
Hillebrand's Lectures on German Thought. Crown Svo. 78. 6d.
Hodgson's Philosophy of Reflection. 2 vols. 8vo. 21s.
Kalisch's Path and Goal. 8vo. 12s. 6d.
Pod
Lewis on Authority in Matters of Opinion. 8vo. 14s.
Leslie's Essays in Political and Moral Philosophy. 8vo. 10s. 6d.
Macaulay's Speeches corrected by Himself. Crown 8vo. 3s. 6d.
Macleod's Economical Philosophy. Vol. I. 8vo. 15s. Vol. II. Part I. 12s.
Mill on Representative Government. Crown 8vo. 2s.
Compati
- Liberty. Post Svo. 7s. 6d. Crown 8vo. 1s. 4d.
Mill's Analysis of the Phenomena of the Human Mind. 2 vols. 8vo. 28.
Dissertations and Discussions. 4 vols. 8vo. 47s.
Essays on Unsettled Questions of Political Economy. Svo. 6s, 6d.
Examination of Hamilton's Philosophy. 8vo. 16s.
London, LONGMANS & CO.
4
General Lists of New Works.
Mill's Logic, Ratiocinative and Inductive. 2 vols. 8vo. 25s.
Principles of Political Economy. 2 vols. 8vo. 30s. 1 vol. crown 8vo. 5s.
Subjection of Women. Crown 8vo. 63.
Utilitarianism. 8vo. 5s.
Müller's (Max) Chips from a German Workshop. 4 vols. 8vo. 368.
Hibbert Lectures on Origin and Growth of Religion. 8vo. 10s. 6d.
Selected Essays on Language, Mythology, and Religion. 2 vols.
Plantag
crown 8vo. 16s.
Sandars's Institutes of Justinian, with English Notes. 8vo. 188.
Swinbourne's Picture Logic. Post 8vo. 53.
Thomson's Qutline of Necessary Laws of Thought. Crown 8vo. 63.
Tocqueville's Democracy in America, translated by Reeve. 2 vols. crown 8vo. 16s.
Twiss's Law of Nations, 8vo. in Time of Peace, 123. in Time of War, 21s.
Whately's Elements of Logic. 8vo. 10s. 6d. Crown 8vo. 48. 6d.
8vo. 10s. 6d. Crown 8vo. 4s. 6d.
English Synonymes. Fcp. 8vo. 3s.
Williams's Nicomachean Ethics of Aristotle translated. Crown 8vo. 7s. 6d.
Zeller's Socrates and the Socratic Schools. Crown 8vo. 10s. 6d.
Rhetoric.
Stoics, Epicureans, and Sceptics. Crown 8vo. 15s.
Plato and the Older Academy. Crown 8vo. 18s.
Pre-Socratic Schools. 2 vols. crown 8vo. 30s.
▬▬▬▬▬▬▬▬▬▬
MISCELLANEOUS AND CRITICAL WORKS.
Arnold's (Dr. Thomas) Miscellaneous Works.
8vo. 7s. 6d.
(T.) Manual of English Literature. Crown 8vo. 73. 6d.
English Authors, Poetry and Prose Specimens.
Bain's Emotions and the Will. 8vo. 15s.
Mental and Moral Science. Crown 8vo. 10s. 6d.
Senses and the Intellect. 8vo. 158.
Becker's Charicles and Gallus, by Metcalfe. Post 8vo. 78. 6d. cach.
Blackley's German and English Dictionary. Post 8vo. 7s. 6d.
Conington's Miscellaneous Writings. 2 vols. 8vo. 28s.
Contanseau's Practical French & English Dictionary.
Pocket French and English Dictionary.
Davison's Thousand Thoughts from Various Authors.
Farrar's Language and Languages. Crown 8vo. 6s.
Froude's Short Studies on Great Subjects. 3 vols. crown 8vo. 183.
German Home Life, reprinted from Fraser's Magazine. Crown 8vo. Bs.
Gibson's Cavalier's Note-Book, Small 4to. 14s.
Greville's (Lady Violet) Faiths and Fashions. Crown 8vo. 7s. 6d.
Hume's Essays, edited by Green & Grose. 2 vols. 8vo. 28s.
Post 8vo. 7s. 6d.
Square 18mo. 3s. 6d.
Crown 8vo. 7s. 6d.
Treatise on Human Nature, edited by Green & Grose. 2 vols. 8vo. 28s.
Latham's Handbook of the English Language. Crown 8vo. 6s.
English Dictionary. 1 vol, medium 8vo. 14s. 4 vols. 4to. £7.
Liddell & Scott's Greek-English Lexicon. Crown 4to. 36s.
Abridged Greek-English Lexicon. Square 12mo. 7s. 6d.
Longman's Pocket German and English Dictionary. 18mo. 58.
Macaulay's Miscellaneous Writings. 2 vols. 8vo. 21s. 1 vol. crown 8vo. 48.
London, LONGMANS & CO.
General Lists of New Works.
5
10
Macaulay's Miscellaneous Writings and Speeches. Crown 8vo. 6s.
Macaulay's Miscellaneous Writings, Speeches,
Cabinet Edition. 4 vols, crown Svo. 24s.
Mabaffy's Classical Greek Literature. Crown Svo. Vol. I. the Poets, 7s. GA.
Vol. II. the Prose Writers, 7s. 6d.
Müller's (Max) Lectures on the Science of Language. 2 vols. crown 8vo. 163.
Rich's Dictionary of Roman and Greek Antiquities. Crown 8vo. 7s. 6d.
Rogers's Eclipse of Faith. Fcp. 8vo. 5s.
Defence of the Eclipse of Faith Fcp. 8vo. 3s. 6d.
Roget's Thesaurus of English Words and Phrases. Crown 8vo. 10s. 62.
Savile's Apparitions, a Narrative of Facts. Crown Svo. 5s.
Selections from the Writings of Lord Macaulay. Crown 8vo. 6s.
The Essays and Contributions of A. K. H. B. Crown Svo.
Autumn Holidays of a Country Parson. 3s. 6d.
Changed Aspects of Unchanged Truths. 3s. 6d.
Common-place Philosopher in Town and Country.
Counsel and Comfort spoken from a City Pulpit.
Critical Essays of a Country Parson. 3s. 6d.
Graver Thoughts of a Country Parson. Three Series, 3s. 6d. each.
Landscapes, Churches, and Moralities. 3s. 6d.
Lays of Ancient Rome, &c.
3s. 6d.
3s. 6d.
Lessons of Middle Age. 3s. 6d.
Leisure Hours in Town. 3s. 6d.
Present-day Thoughts. 3s. 6d.
Recreations of a Country Parson. Three Series, 3s. 6d. each.
Seaside Musings on Sundays and Week-Days. 3. 6d.
Sunday Afternoons in the Parish Church of a University City. 35. 6ð,
White & Riddle's Large Latin-English Dictionary. 4to. 21s.
White's College Latin-English Dictionary. Royal 8vo. 12s.
Junior Student's Lat.-Eng. and Eug.-Lat. Dictionary. Square 12mo. 12s.
Dictionary, 5s. 6d.
Separately (The Latin-English Dictionary, 7s. 6d.
Wit and Wisdom of the Rev. Sydney Smith, 16mo. 3s. 6d.
Yonge's English-Greek Lexicon. Square 12mo. Ss. 6d. 4to. 21s.
ASTRONOMY, METEOROLOGY, GEOGRAPHY &c.
Freeman's Historical Geography of Europe. Svo. 31s. 6d.
Herschel's Outlines of Astronomy. Square crown Svo. 12s.
Smith's Air and Rain. 8vo. 24s.
The Public Schools Atlas of Ancient Geography. Imperial 8vo. 7s. 6d.
Atlas of Modern Geography. Imperial 8vo. 5s.
NATURAL HISTORY & POPULAR SCIENCE.
Arnott's Elements of Physics or Natural Philosophy. Crown 8vo. 12s. 6d.
Braude's Dictionary of Science, Literature, and Art. 3 vols. medium 8vo. 63s.
London, LONGMANS & CO.
Keith Johnston's Dictionary of Geography, or General Gazetteer. 8vo. 42s.
Neison's Work on the Moon. Medium 8vo. 31s. 6d.
Proctor's Essays on Astronomy. 8vo. 12s. Proctor's Moon. Crown 8vo. 10s. 6d.
Larger Star Atlas. Folio, 15s. or Maps only, 12s. 6d.
New Star Atlas. Crown 8vo. 5s. Orbs Around Us. Crown Svo. 7s. 6d.
Other Worlds than Ours. Crown 8vo. 10s. 6d.
Saturn and its System. 8vo. 14s. Proctor's Sun. Crown 8vo. 14s.
Universe of Stars. Svo. 10s. 6d.
:
!
6
General Lists of New Works.
Buckton's Town and Window Gardening. Crown 8vo. 2s.
Decaisne and Le Maout's General System of Botany. Imperial 8vo. 31s. 6d.
Dixon's Rural Bird Life. Crown 8vo. Illustrations, 7s. 6d.
Ganot's Elementary Treatise on Physics, by Atkinson. Large crown 8vo. 15s.
Crown 8vo. 7s. 6d.
Crown 8vo. 6s.
8vo. 15s.
Polar World. 8vo. 10s. 6d.
8vo. 10s. 6d.
Natural Philosophy, by Atkinson.
Goodeve's Elements of Mechanism.
Grove's Correlation of Physical Forces.
Hartwig's Aerial World. 8vo. 10s. Gd.
Sea and its Living Wonders.
Subterranean World. 8vo. 10s. 6d. Tropical World. 8vo. 10s. 6d.
Haughton's Six Lectures on Physical Geography. 8vo. 15s.
Heer's Primeval World of Switzerland. 2 vols. 8vo. 16s.
Helmholtz's Lectures on Scientific Subjects. 2 vols. cr. 8vo. 7s. 6d. each.
Helmholtz on the Sensations of Tone, by Ellis. 8vo. 36s.
Hullah's Lectures on the History of Modern Music.
8vo. 8s. 6d.
Transition Period of Musical History. 8vo. 10s. 6d.
Keller's Lake Dwellings of Switzerland, by Lee. 2 vols. royal 8vo. 42s.
Lloyd's Treatise on Magnetism. 8vo. 10s. 6d.
on the Wave-Theory of Light. 870. 10s. 6d.
Loudon's Encyclopædia of Plants. 8vo. 42s.
Lubbock on the Origin of Civilisation & Primitive Condition of Man. 8vo. 183.
Macalister's Zoology and Morphology of Vertebrate Animals. Svo. 10s. 6d.
Nicols' Puzzle of Life. Crown 8vo. 3s. 6d.
J
Owen's Comparative Anatomy and Physiology of the Vertebrate Animals, 3 vols.
870. 73s. 6d.
Proctor's Light Science for Leisure Hours. 2 vols. crown 8vo. 7s. 6d. each.
Rivers's Orchard House. Sixteenth Edition. Crown 8vo, 5s.
S
Rose Amateur's Guide. Fcp. 8vo. 4s. 6d.
Stanley's Familiar History of British Birds. Crown 8vo. 6s.
Text-Books of Science, Mechanical and Physical.
Abney's Photography, 3s. 6d.
Anderson's (Sir John) Strength of Materials, 3s. 6d.
Armstrong's Organic Chemistry, 3s. 6d.
Ball's Astronomy, 6s.
Barry's Railway Appliances, 3s. 6d. Bloxam's Metals, 3s. 6d.
Goodeve's Principles of Mechanics, 3s. bd.
Gore's Electro-Metallurgy, 65.
Griffin's Algebra and Trigonometry, 3s. 6d.
Jenkin's Electricity and Magnetism, 3s. 6d.
Maxwell's Theory of Heat, 3s. 6d.
Merrifield's Technical Arithmetic and Mensuration, 3s. 6d.
Miller's Inorganic Chemistry, 3s. 6d.
Preece & Sivewright's Telegraphy, 3s. 6d.
Rutley's Study of Rocks, 4s. 6d.
Shelley's Workshop Appliances, 3s. 6d.
Thome's Structural and Physiological Botany, 6s.
Thorpe's Quantitative Chemical Analysis, 4s. 6d.
Thorpe & Muir's Qualitative Analysis, 3s. 6d.
Tilden's Chemical Philosophy, 3s. 6d.
Unwin's Machine Design, 3s. 6d.
Watson's Plane and Solid Geometry, 3s. 6d.
Tyndall on Sound. New Edition in the press.
London, LONGMANS & CO.
General Lists of New Works.
7
(repea
Tyndall's Contributions to Molecular Physics. 8vo. 168.
Fragments of Science. 2 vols. post 8vo. 16s.
Heat a Mode of Motion. Crown 8vo. 12s.
Notes on Electrical Phenomena. Crown 8vo. 1s. sewed, 1s. 6d. cloth.
Notes of Lectures on Light. Crown 8vo. 1s. sewed, 1s. 6d. cloth.
Lectures on Light delivered in America. Crown 8vo. 7s. 6d.
Lessons in Electricity. Crown 8vo. 2s. 6d.
Von Cotta on Rocks, by Lawrence. Post 8vo. 14s.
Woodward's Geology of England and Wales. Crown 8vo. 14s.
Wood's Bible Animals. With 112 Vignettes. 8vo. 143.
Homes Without Hands. 8vo. 14s. Insects Abroad. 8vo. 14s.
Insects at Home. With 700 Illustrations. 8vo. 14s.
Out of Doors. Crown 8vo. 7s. 6d. Strange Dwellings. Crown 8vo. 7s. 6d.
CHEMISTRY &
PHYSIOLOGY.
Buckton's Health in the House, Lectures on Elementary Physiology. Cr. 8vo. 2s.
Crookes's Select Methods in Chemical Analysis. Crown 8vo. 12s. 6d.
Kingzett's Animal Chemistry. 8vo. 18s.
History, Products and Processes of the Alkali Trade. 8vo. 12s.
Miller's Elements of Chemistry, Theoretical and Practical. 3 vols. 8vo. Part I.
Chemical Physics, 16s. Part II. Inorganic Chemistry, 24s. Part III. Organic
Chemistry, Section I. price 31s. 6d.
Reynolds's Experimental Chemistry, Part I.
Thudichum's Annals of Chemical Medicine.
Tilden's Practical Chemistry. Fcp. 8vo. 1s. 6d.
Watts's Dictionary of Chemistry. 7 vols, medium 8vo. £10. 16s. 6d.
Third Supplementary Volume, in Two Parts. PART I. 365.
Fcp. 8vo. 1s. 6d.
Vol. I. 8vo. 14s.
THE FINE ARTS &
Doyle's Fairyland; Pictures from the Elf-World. Folio, 15s.
Dresser's Arts and Art Industries of Japan.
Jameson's Sacred and Legendary Art. 6 vols, square crown 8vo.
Legends of the Madonna. 1 vol. 218.
Monastic Orders. 1 vol. 21s.
Saints and Martyrs. 2 vols. 31s. 6d.
Saviour. Completed by Lady Eastlake. 2 vols. 42s.
Longman's Three Cathedrals Dedicated to St. Paul. Square crown 8vo. 21s.
Macaulay's Lays of Ancient Rome. Illustrated by Scharf. Fcp. 4to. 21s. imp.
16mo. 10s. 6d.
ILLUSTRATED EDITIONS.
K
Illustrated by Weguelin. Crown Svo. 6s.
Macfarren's Lectures on Harmony. 8vo. 12s.
Moore's Irish Melodies. With 161 Plates by D. Maclise, R.A. Super-royal 8vo. 21s.
Lalla Rookh, illustrated by Tenniel. Square crown 8vo. 10s. 6d.
Perry on Greek and Roman Sculpture. 8vo.
[In preparation.
Bourne's Catechism of the Steam Engine. Fcp. 8vo. 6s.
[In preparation.

THE USEFUL ARTS, MANUFACTURES &c.
Examples of Steam, Air, and Gas Engines. 4to. 70s.
London, LONGMANS & CO.
8
General Lists of New Works.
Bourne's Handbook of the Steam Engine. Fcp. 8vo. 9s.
Recent Improvements in the Steam Engine. Fcp. 8vo. 6s.
Treatise on the Steam Engine. 4to. 42s.
Brassey's Shipbuilding for War. 2 vols. 8vo.
Cresy's Encyclopædia of Civil Engineering. 8vo. 25s.
Culley's Handbook of Practical Telegraphy. 8vo. 16s.
Eastlake's Household Taste in Furniture, &c.
Fairbairn's Useful Information for Engineers.
S
Ma
Square crown 8vo. 14s.
3 vols. crown 8vo. 31s. 6d.
Applications of Cast and Wrought Iron. 8vo. 16s.
Mills and Millwork. 1 vol. 8vo. 25s.
Gwilt's Encyclopædia of Architecture. 8vo. 52s. 6d.
Hobson's Amateur Mechanic's Practical Handbook. Crown 8vo. 2s. 6d.
Hoskold's Engineer's Valuing Assistant. 8vo. 31s. 6d.
Kerl's Metallurgy, adapted by Crookes and Röhrig. 3 vols. 8vo. £4. 193.
Loudon's Encyclopædia of Agriculture. 8vo. 21s.
Gardening. 8vo. 21s.
Ma
Mitchell's Manual of Practical Assaying. 8vo. 31s. 6d.
Northcott's Lathes and Turning. 8vo. 18s.
Payen's Industrial Chemistry Edited by B. H. Paul, Ph.D. 8vo. 42s.
Piesse's Art of Perfumery. Fourth Edition. Square crown 8vo. 21s.
Stoney's Theory of Strains in Girders. Royal 8vo. 86s.
Ure's Dictionary of Arts, Manufactures, & Mines. 4 vols. medium 8vo. £7. 78.
Ville on Artificial Manures. By Crookes. 8vo. 21s.
RELIGIOUS & MORAL WORKS.
Abbey & Overton's English Church in the Eighteenth Century. 2 vols. 8vo. 36s.
Arnold's (Rev. Dr. Thomas) Sermons. 6 vols. crown 8vo. 5s. each.
Bishop Jeremy Taylor's Entire Works. With Life by Bishop Heber. Edited by
the Rev. C. P. Eden. 10 vols. 8vo. £5. 5s.
Boultbee's Commentary on the 39 Articles. Crown 8vo. 6s.
History of the Church of England, Pre-Reformation Period. Svo. 15s.
Browne's (Bishop) Exposition of the 39 Articles. 8vo. 16s.
Bunsen's Angel-Messiah of Buddhists, &c. 8vo. 10s 6d.
Colenso's Lectures on the Pentateuch and the Moabite Stone. 8vo. 12s.
Colenso on the Pentateuch and Book of Joshua. Crown 8vo. 6s.
PART VII. completion of the larger Work. 8vo. 24s.
Conder's Handbook of the Bible. Post 8vo. 7s. 6d.
Conybeare & Howson's Life and Letters of St. Paul :-
P
»
Library Edition, with all the Original Illustrations, Maps, Landscapes on
Steel, Woodcuts, &c. 2 vols. 4to. 42s.
Intermediate Edition, with a Selection of Maps, Plates, and Woodcuts.
2 vols. square crown 8vo. 21s.
Student's Edition, revised and condensed, with 46 Illustrations and Maps.
1 vol. crown 8vo. 7s. 6d.
Ellicott's (Bishop) Commentary on St. Paul's Epistles. 8vo. Galatians, 8s. 6d.
Pastoral Epistles, 10s. 6d. Philippians, Colossians, and
Philemon, 108. 6d. Thessalonians, 7s. 6d.
Ephesians, 8s. 6d.
Ellicott's Lectures on the Life of our Lord. 8vo. 12s.
Ewald's History of Israel, translated by Carpenter. 5 vols. 8vo. 633.
London, LONGMANS & CO.
:
General Lists of New Works.
9
Ewald's Antiquities of Israel, translated by Solly. 8vo. 12s. 6d.
Gospel (The) for the Nineteenth Century. 4th Edition. 8vo. 10s. 6d.
Hopkins's Christ the Consoler. Fcp. 8vo. 2s. 6d.
Jukes's Types of Genesis. Crown 8vo. 7s. 6d.
Second Death and the Restitution of all Things. Crown 8vo. 3s. 6d.
Kalisch's Bible Studies. PART I. the Prophecies of Balaam. 8vo. 10s. 6d.
PART II. the Book of Jonah. 8vo. 10s. 6d.
Historical and Critical Commentary on the Old Testament; with a
New Translation. Vol. I. Genesis, 8vo. 18s. or adapted for the General
Reader, 123. Vol. II. Exodus, 15s. or adapted for the General Reader, 12s.
Vol. III. Leviticus, Part I. 15s. or adapted for the General Reader, 8s.
Vol. IV. Leviticus, Part II. 15s. or adapted for the General Reader, 8s.
Lyra Germanica: Hymns translated by Miss Winkworth. Fcp. 8vo. 55.
Martineau's Endeavours after the Christian Life. Crown 8vo. 7s, bå.
Hymns of Praise and Prayer. Crown 8vo. 4s. 6d. 32mo. 1s. 6d.
Sermons, Hours of Thought on Sacred Things. 2 vols. 73. 6d. each.
Mill's Three Essays on Religion. 8vo. 10s. 6d.
Missionary Secretariat of Henry Venn, B.D. 8vo.
Monsell's Spiritual Songs for Sundays and Holidays.
Müller's (Max) Lectures on the Science of Religion.
Newman's Apologia pro Vitâ Suâ. Crown 8vo. 6s.
Passing Thoughts on Religion. By Miss Sewell. Fcp. 8vo. 3s. 6d.
Sewell's (Miss) Preparation for the Holy Communion. 32mo. 3s.
Private Devotions for Young Persons.
Smith's Voyage and Shipwreck of St. Paul. Crown Svo. 7s. 6d.
Supernatural Religion. Complete Edition. 3 vols. 8vo. 36s.
Thoughts for the Age. By Miss Sewell. Fcp. Svo. 3s. 6d.
Whately's Lessons on the Christian Evidences. 18mo. 6d.
White's Four Gospels in Greek, with Greek-English Lexicon. 32mo. 5s.
Portrait. 18s.
Fcp. 8vo. 5s. 18mo. 2s.
Crown 8vo. 10s. 6d.
TRAVELS, VOYAGES, &C.
Baker's Rifle and Hound in Ceylon. Crown Svo. 7s. 6d.
Eight Years in Ceylon. Crown Svo. 7s. 6d.
Ball's Alpine Guide. 3 vols. post 8vo. with Maps and Illustrations :-I. Western
Alps, 6s. 6d. II. Central Alps, 7s. 6d. III. Eastern Alps, 10s. 6d.
Ball on Alpine Travelling, and on the Geology of the Alps, 1s.
Brassey's Sunshine and Storm in the East.
Svo. 21s.
·
Voyage in the Yacht Sunbeam.' Cr. 8vo. 7s. 6d. School Edition, 2s.
Edwards's (A. B.) Thousand Miles up the Nile. Imperial Svo. 42s.
Hassall's San Remo and the Western Riviera. Crown 8vo. 10s. 6d.
Macnamara's Medical Geography of India. 8vo. 21s.
Miller's Wintering in the Riviera. Post 8vo. Illustrations, 7s. 6d.
Packe's Guide to the Pyrenees, for Mountaineers. Crown Svo. 7s. 6d.
Rigby's Letters from France, &c. in 1789. Crown 8vo. 10s. 6d.
Shore's Flight of the 'Lapwing', Sketches in China and Japan. Svo. 158.
The Alpine Club Map of Switzerland. In Four Sheets. 42s.
Tozer's Turkish Armenia and Eastern Asia Minor. Svo. 16s.
Weld's Sacred Palmlands. Crown Svo. 10s. 6d.
London, LONGMANS & CO.
10
General Lists of New Works.
WORKS OF FICTION.
Blues and Buffs. By Arthur Mills. Crown 8vo. 6s.
Buried Alive, Ten Years of Penal Servitude in Siberia. Crown 8vo. 10s. 6d.
Crookit Meg (The). By Shirley. Crown 8vo. 6s.
Endymion. By the Right Hon. the Earl of Beaconsfield, K.G. 3 vols. post 8vo.
31s. 6d.
Hawthorne's (J.) Yellow-Cap and other Fairy Stories. Crown 8vo. 6s.
Cabinet Edition of Stories and Tales by Miss Sewell :-
Amy Herbert, 2s. 6d.
Cleve Hall, 2s. 6d.
Ivors, 2s. 6d.
Katharine Ashton, 2s. 6d.
Laneton Parsonage, 28. 6d.
Margaret Percival, 3s. 6d.
Ursula, 38. 6d.
The Earl's Daughter, 2s. 6d.
Experience of Life, 2s. 6d.
Gertrude, 2s. 6d.
Novels and Tales by the Right Hon. the Earl of Beaconsfield, K.G. Cabinet
Edition, Ten Volumes, crown 8vo. price £3.
Lothair, 6s.
Coningsby, 63.
Sybil, 68.
Tancred, 63.
Venetia, 6s.
The Modern Novelist's Library. Each Work
complete in itself, price 23. boards, or 2s. 6d. cloth :-
By the Earl of Beaconsfield, K.G.
Lothair.
Coningsby.
Sybil.
Tancred.
Venetia.
Henrietta Temple.
Contarini Fleming.
Alroy, Ixion, &c.
The Young Duke, &c.
Vivian Grey.
Henrietta Temple, 6s.
Contarini Fleming, 63.
Alroy, Ixion, &c. 6s.
The Young Duke, &c. 63.
Vivian Grey, 6s.
By Anthony Trollope.
Barchester Towers.
The Warden.
C
By the Author of the Rose Garden.'
Unawares.
in crown 8vo. A Single Volume,
By Major Whyte-Melville.
Digby Grand.
General Bounce.
Kate Coventry.
The Gladiators.
Good for Nothing.
Holmby House.
The Interpreter.
The Queen's Maries.
C
By the Author of the Atelier du Lys.'
Mademoiselle Mori.
The Atelier du Lys.
By Various Writers.
Atherstone Priory.
The Burgomaster's Family.
Elsa and her Vulture.
The Six Sisters of the Valleys.
Lord Beaconsfield's Novels and Tales. 10 vols. cloth extra, gilt edges, 30s.
Whispers from Fairy Land. By the Right Hon. Lord Brabourne. With Nine
Illustrations. Crown 8vo. 3s. 6d.
Higgledy-Piggledy; or, Stories for Everybody and Everybody's Children. By
the Right Hon. Lord Brabourne. With Nine Illustrations from Designs by
R. Doyle. Crown 8vo. 3s. 6d.
POETRY & THE DRAMA.
Bailey's Festus, a Poem. Crown 8vo. 12s. 6d.
Bowdler's Family Shakspeare. Medium 8vo. 14s. 6 vols. fcp. 8vo. 218.
Cayley's Iliad of Homer, Homometrically translated. 8vo. 12s. 6d.
Conington's Eneid of Virgil, translated into English Verse. Crown 8vo. 9s.
London, LONGMANS & CO.
General Lists of New Works.
11
Goethe's Faust, translated by Birds. Large crown 8vo. 12s. 6d.
translated by Webb. 8vo. 12s. 6d.
edited by Selss. Crown 8vo. 5s.
Ingelow's Poems. New Edition. 2 vols. fcp. 8vo. 12s.
Macaulay's Lays of Ancient Rome, with Ivry and the Armada. 16mo. 3s. 6d.
Ormsby's Poem of the Cid. Translated. Post 8vo. 5s.
Medium 8vo. 14s.
Southey's Poetical Works.
RURAL SPORTS, HORSE & CATTLE MANAGEMENT &c.
Blaine's Encyclopædia of Rural Sports. 8vo. 21s.
Francis's Treatise on Fishing in all its Branches. Post 8vo. 15s.
Horses and Roads. By Free-Lance. Crown 8vo, 68.
Miles's Horse's Foot, and How to Keep it Sound. Imperial 8vo. 12s. 6d.
Plain Treatise on Horse-Shoeing. Post 8vo. 2s. 6d.
Stables and Stable-Fittings. Imperial 8vo. 15s.
Remarks on Horses' Teeth. Post 8vo. 1s. 6d.
Nevile's Horses and Riding. Crown Svo, 6s.
Ronalds's Fly-Fisher's Entomology. Svo. 14s.
Steel's Diseases of the Ox, being a Manual of Bovine Pathology. 8vo. 15s.
Stonehenge's Dog in Health and Disease. Square crown 8vo. 7s. 6d.
Greyhound. Square crown 8vo. 15s.
Youatt's Work on the Dog. 8vo. 6s.
Horse. 8vo. 7s. 6d.
Wilcocks's Sea-Fisherman. Post 8vo. 12s. 6d.
WORKS OF UTILITY & GENERAL INFORMATION.
Acton's Modern Cookery for Private Families. Fcp. 8vo. 6s.
Black's Practical Treatise on Brewing. 8vo. 10s. 6d.
Buckton's Food and Home Cookery. Crown 8vo. 2s.
Bull on the Maternal Management of Children. Fcp. 8vo. 2s. 6d.
Bull's Hints to Mothers on the Management of their Health during the Period of
Pregnancy and in the Lying-in Room. Fcp. 8vo. 2s. 6d.
Campbell-Walker's Correct Card, or How to Play at Whist. Fcp. 8vo. 2s. 6d.
Edwards on the Ventilation of Dwelling-Houses. Royal Svo. 10s. 6d.
Johnson's (W. & J. H.) Patentee's Manual. Fourth Edition.
Longman's Chess Openings. Fcp. 8vo. 2s. 6d.
Macleod's Economics for Beginners. Small crown 8vo. 2s. 6d.
Elements of Economics. Small crown 8vo.
8vo. 10s. 6d.
[In the press.
Theory and Practice of Banking. 2 vols. 8vo. 26s.
Elements of Banking. Fourth Edition. Crown Svo. 5s,
M'Culloch's Dictionary of Commerce and Commercial Navigation. Svo. 638.
London, LONGMANS & CO.
12
General Lists of New Works.
Maunder's Biographical Treasury. Fcp. 8vo. 6s.
Historical Treasury. Fcp. 8vo. 68.
Scientific and Literary Treasury. Fcp. 8vo. 6s.
Treasury of Bible Knowledge, edited by Ayre.
Treasury of Botany, edited by Lindley & Moore.
Treasury of Geography. Fcp. 8vo. 63.
Treasury of Knowledge and Library of Reference. Fcp. 8vo. 6s.
Treasury of Natural History. Fcp. 8vo. 63.
Pereira's Materia Medica, by Bentley and Redwood. 8vo. 258.
Pewtner's Comprehensive Specifier; Building-Artificers' Work. Crown 8vo. 61.
Pole's Theory of the Modern Scientific Game of Whist. Fcp. 8vo. 2s. 6d.
Scott's Farm Valuer. Crown 8vo. 5s.
Rents and Purchases. Crown 8vo. 6s.
Smith's Handbook for Midwives. Crown 8vo. 53.
The Cabinet Lawyer, a Popular Digest of the Laws of England. Fcp. 8vo. 93.
West on the Diseases of Infancy and Childhood. 8vo. 18s.
Wilson on Banking Reform. 8vo. 7s. 6d.
on the Resources of Modern Countries 2 vols. 8vo. 24s.
M
Fcp. 8vo. 6s.
Two Parts, 128.
MUSICAL WORKS BY JOHN HULLAH, LL.D.
Hullah's Method of Teaching Singing. Crown 8vo. 2s. 6d.
Exercises and Figures in the same. Crown 8vo. 1s. or 2 Parts, 6d. each.
Large Sheets, containing the 'Exercises and Figures in Hullah's Method,' in
Parcels of Eight, price 6s. each.
Chromatic Scale, with the Inflected Syllables, on Large Sheet. 1s. 6d.
Card of Chromatic Scale.
1d.
Exercises for the Cultivation of the Voice. For Soprano or Tenor, 2s. 6d.
Grammar of Musical Harmony. Royal 8vo. 2 Parts, each 1s. 6d.
Exercises to Grammar of Musical Harmony. 1s.
Grammar of Counterpoint. Part I. super-royal 8vo. 2s. 6d.
Willem's Manual of Singing. Parts I. & II. 2s. 6d. ; or together, 5s.
Exercises and Figures contained in Parts I. and II. of Wilhem's Manual. Books
I. & II. each 8d.
Large Sheets, Nos. 1 to 8, containing the Figures in Part I. of Wilhem's Manual,
in a Parcel, 68.
Large Sheets, Nos. 9 to 40, containing the Exercises in Part I. of Wilhem's
Manual, in Four Parcels of Eight Nos. each, per Parcel, 6s.
Large Sheets, Nos. 41 to 52, containing the Figures in Part II. in a Parcel, 9s.
Hymns for the Young, set to Music. Royal 8vo. 8d.
Infant School Songs. 6d.
Notation, the Musical Alphabet. Crown 8vo. 6d.
Old English Songs for Schools, Harmonised. 6d.
Rudiments of Musical Grammar. Royal 8vo. 8s.
School Songs for 2 and 3 Voices. 2 Books, 8vo. each 6d.
char
London, LONGMANS & CO.
Spottiswoode & Co. Printers, New-street Square, London.

UNIVERSITY OF MICHIGAN
i
:
#
3 9015 06387 2918
DATE DUE
0035
Call Number
CA
1623
pal
Volume
C
Copy
Please Check
Undergra
Grad
U
Sul M. NG
PROCTOR RA
Author
mends
Buclack
Title
لأنان
11.250+
4
Zip Code