REVIEWS. 
295 
curve in order to determine its area. Pascal seems to have regarded their 
relations as of sufficient difficulty to he selected for his 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 hook itself is divided into seven sections, dealing successively with 
the right cycloid, the epicycloid, and hypocycloid, trochoids with an 
appendix or elliptical hypotrochoids, the trisectrix, the spiral of Archi- 
medes, and the spiral track of a planet’s shadow in space, motion in 
cycloidal curves, epicyclics, equations to cycloidal curves, and the graphical 
use of cycloidal curves to determine the motion of planets and comets, and 
of matter projected from the sun. 
Regarding the treatment of these different topics it is impossible within 
the space of a notice such as this to give more than a few general indications, 
although the writer, contrary to the supposed practice of reviewers, admits to 
having read with interest the 250 pages of which the work consists from 
title-page to colophon. 
In the first place it is a refreshing study in geometry as opposed to 
analysis. There are many minds, especially those of the mechanical type, to 
which the former gives more objective reality than the latter ; and there are 
others to whom deltas and sigmas are a sealed language, hut who can nearly 
always puzzle out a construction and its consequent demonstration. 
Secondly, the subject, somewhat spinous from outside, really contains much 
interesting matter, and touches on practical points more often than at first 
appears. Examples of this are demonstrations that the area of the cycloid 
and its base is three times that of the generating circle ; that the evolute of 
the cycloid is an equal cycloid — resumed more fully in a later chapter — a 
property utilized in Huyghens’s ingenious though unsuccessful cycloidal cheeks 
for producing isochronism in pendulums; that the hypocycloid becomes a 
straight line when the diameter of the rolling circle is equal to the radius of 
the fixed circle, as seen in “sun and planet wheel” mechanisms; the descrip- 
tion of the cardioid or epicycloid traced by a point on the circumference of a 
circle rolling on an equal circle ; of the involute of the circle regarded as an 
epicycloid whose generating circle has infinite radius ; that the hypotrochoid 
becomes an ellipse when the diameter of the rolling circle is equal to the 
radius of the fixed ; the few lines about the trisectrix, and its manner of 
solving the much vexed problem of the trisection of an angle, and the proof 
of the identity of the epitrochoid with the spiral of Archimedes. 
The comments on planetary and lunar epicycles, with two beautiful plates 
in illustration, deserve special mention ; as also do those on which Lescar- 
bault’s observations of the supposed intra-Mercurial planet Vulcan turned. 
The note on Mr. Perigal’s geometric chuck, by which many of the illustra- 
tions have been furnished, will recall an old friend to many readers. The 
companion to the cycloid, better known as Roberval’s “ curve of sines,” has 
the deepest interest to students of acoustics, vibration, or .simple harmonic 
motion in any form ; and the graphical use of the cycloid opens a compara- 
tively fresh field of labour. 
Speaking generally, the writer has derived so much pleasure from a perusal 
of this little book that he confidently recommends it to his brother students. 
Mr. Proctor would confer a boon on many mechanics and physicists if he 
