in this field of enquiry, claiming to have propounded several simple and
exact methods, offers the following as sufficiently demonstrative:--
I construct my diagrams in the following way:--I draw two straight lines
at right angles, making O the right angle. From the point O, in the
direction OA, I mark off four equal parts together equal to OA, and from
O, in the direction of OB, I mark off three of such equal parts
together, equal to OB, and join AB. It is obvious, or rather
self-evident, that AOB is a right-angled triangle, of which the sides
that contain the right angle are in the ratio of 4 to 3, by
construction. With A as centre and AB as interval, I describe the circle
X, produce AO and BO to meet and terminate in the circumference of the
circle at the points G and C, and join AC, CG, and BG, producing the
quadrilateral ACGB. I bisect AG at F, and with O as centre, and OF as
interval, describe the circle Z. The line OF is the line that joins the
middle points of the diagonals in the quadrilateral ACGB; and it follows
that, {AG^2 + CB^2 + 4(OF^2)} = {AC^2 + CG^2 + BG^2 + AB^2.}
When AO = 4, we get the following equation:--
{5^2 + 6^2 + (4 × 1'5^2)} = {5^2 + sqrt(10^2) + sqrt(10^2) + 5^2,} or,
{25 + 36 + 9} = {25 + 10 + 10 + 25} = 70. From the points B and C, I
draw straight lines at right angles to AB and AC, and therefore
tangential to the circle X, to meet AG produced at D, and join BD and
CD, producing the quadrilateral ACDB. I bisect AD at E, and with O as
centre, and EO as interval describe the circle XY, and with E as centre,
and EA or ED as interval describe the circle Y.
Now, to square the circle, or, in other words, to get exactly equal in
superficial area to the circle X, I will show how to find it. From the
point G draw a straight line--say G _m_--perpendicular to ED, making G
_m_ equal GD. Produce GA to a point _n_, making G _n_ equal to 2AG - GD,
and join _n m_. The square on _n m_ will be the required square. (I have
indicated this square by dotted lines.) For example:--If AO = 4, then AG
= 5, and GD = 1'25; therefore {2 AG - GD} = {10 - 1'25} = 8'75 = Gn: and
Gm = 1'25; therefore, Gn^2 + Gm^2 = 3-1/8 (AB^2); that is, {8'75^2 +
1'25^2} = 3-1/8 (5^2), or, {76'5625 + 1'5625} = {3'125 × 25}; and this
equation=Area of the Circle X; and area of the square on _n m_ :: and it
follows, that the area of every circle, is equal to the area of a square
on the hypotenuse of a right-angled triangle, of which the sides that
contain the right angle are in the ratio of 7 to 1, and the sum of these
two sides equal to the diameter of the circle. In many ways I have
proved this fact, by practical or constructive geometry.
PLATE IV.
_Duplication of the Cube._--In his "Young Geometrician; or, Practical
Geometry without Compasses," 1865, Mr. Oliver Byrne's 40th Problem is as
follows:--
Let AB be the side of a given cube BD. It is required to find AC, the
side of another cube CE, so that the solid contents of the cube CE are
double the solid contents of the cube BD.
Ancient and modern mathematicians (says Mr. Byrne) have in vain
attempted to solve this problem geometrically, that is, by the ruler and
compasses only.
Let AB = BG = GR = RQ = QP = QO = OR = VZ. The length of the shortest
side of the lesser set square; a line of any other given length may be
applied. Draw OP and VR parallel to it; then apply the set squares in
close contact, the edge OV of OVT passing through the point O, while the
points of V and Z of ZSV fall exactly on the lines RV, RZ. Then draw the
line ZBC, cutting FA produced in C; then the cube on AC is double the
cube on AB.
PLATE V.
_Trisection of an Angle._--In his work entitled _Young Geometrician_,
1865, Mr. Oliver Byrne gives as the 39th Problem: To divide a given
angle BAC into three equal angles:--
The line A _m_ is made = _p q_, the least side of the lesser triangular
ruler; by (II) _p m_ is drawn parallel, and _m n_ perpendicular to AB.
Then both rulers are kept in motion, and at the same time in close
contact, as represented in the figure, until _p_ falls on the line _p
m_, and _n_ on the line _m n_; _r n_A passing through the angular point
A.
Then the angle DAB is one-third of the angle CAB. Mr. Byrne asserts that
this problem is not capable of solution by the straight line and circle.
Mathematicians have in vain attempted to solve it geometrically, that
is, by the ruler and compasses only.
PLATE VI.--FIGURE 1.
_Perpetuum Mobile._ Desaguliers demonstrated the absurdity of attempting
to raise weights enclosed in a cellular wheel, simply by providing for
their approach in succession nearer to the centre on the ascending side,
while they should be projected further from the centre on the descending
side. He remarks:--
Those who think the velocity of the weight is the line it describes,
expect that that weight shall be overpoised, which describes the
shortest line, and therefore contrive machines to cause the ascending
weight to describe a shorter line than the descending weight.
For example, in the circle A B D _a_, the weights A and B being supposed
equal, it is imagined that, if by any contrivance whatever, whilst the
weight A describes the arc A _a_, the weight B is carried in any arc, as
B _b_, so as to come nearer the centre in its rising, than if it went up
the arc B D; the said weight shall be overpoised, and consequently, by a
number of such weights, a perpetual motion produced.
Now the velocity of any weight is _not_ the line which it describes in
general, but the height that it rises up to, or falls from, with respect
to its distance from the centre of the earth. So that when the weight
describes the arc A _a_, its velocity is the line A C, which shows the
perpendicular descent, and likewise the line B C denotes the velocity of
the weight B, or the height that it rises to, when it ascends in any of
the arcs B _b_, instead of the arc B D: so that, in this case, whether
the weight B, in its ascent be brought nearer the centre or not, it
loses no velocity, which it ought to do, in order to be raised up by the
weight A.
Indeed, if the weight at B, could by any means spring as it were, or be
lifted up to _x_, and move in the arc _x b_, the end would be answered,
because then the velocity would be diminished, and become _x_C.
FIGURE 2.
In "The Life, Times, and Scientific Labours of the Marquis of
Worcester," 1865, page 454, will be found a full account of the present
diagram, which is intended to illustrate as far as possible, an approach
to the probable construction of the wheel by the Marquis in the 56th
article of his memorable "Century of Inventions."
If any likely-looking method, could, more than another, render
hopelessness more hopeless, surely this mechanical demonstration must
prove most efficient for that purpose. For here, we actually produce a
wheel agreeing to the terms with which Desaguliers closes his
demonstration, when he suggests the only likely method to effect the end
proposed, namely, perpetual motion. We find the fallen weight is
absolutely "lifted up" as he desires, and "moves in the arc" he
describes, and yet although he declares that then "the end would be
answered"--it absolutely is _not_ answered in this instance.
It is not requisite to calculate throughout the effect of the Marquis's
entire number of 40 weights; four will suffice, taking the vertical and
horizontal spokes _a a a a_, showing two rings _a_ and _b_; one, _b_, 12
inches within the other, so that the wheel being, as the Marquis says,
14 feet diameter, the inner ring will be 12 feet diameter. Now let each
weight D be attached in the centre of a cord or chain _a_´, D, b´, 2
feet long, and then secure one end, as _a_´, so the extreme end of each
spoke _a_´, and the other end of the cord, as b´, to place on one lesser
ring, as at _b_, or 12 inches from each spoke.
We shall then find by admeasurement that the upper weight on the
vertical spoke is 7 feet from the centre, and the lower weight 6 feet;
while at the same time there appears to be a preponderance due to the
superior length of the horizontal arm A´; but against this latter we
have the rising weight _b_´D, 1 foot from the centre, which, added to
the 6 feet on the horizontal spoke, neutralizes the hoped-for effect,
and the wheel remains in _statu quo_.
THE END.
* * * * *
[Illustration: Plate 1.
LONDON. E. & F. N. SPON. 48, CHARING CROSS.]
* * * * *
[Illustration: Plate 2.
LONDON. E. & F. N. SPON. 48, CHARING CROSS.]
* * * * *
[Illustration: Plate 3.
LONDON. E. & F. N. SPON. 48, CHARING CROSS.]
* * * * *
[Illustration: Plate 4.
LONDON. E. & F. N. SPON. 48, CHARING CROSS.]
* * * * *
[Illustration: Plate 5.
LONDON. E. & F. N. SPON. 48, CHARING CROSS.]
* * * * *
[Illustration: Plate 6.
LONDON. E. & F. N. SPON. 48, CHARING CROSS.]
* * * * *
OTHER WORKS BY THE SAME AUTHOR.
One Volume 8vo., of 650 pages, illustrated with Steel Engravings of two
unpublished Portraits and 45 Wood Engravings, price 24_s_,
THE LIFE, TIMES, AND SCIENTIFIC LABOURS OF
EDWARD SOMERSET,
SIXTH EARL AND SECOND
MARQUIS OF WORCESTER,
To which is added,
A REPRINT OF HIS CENTURY OF INVENTIONS (1663),
WITH A COMMENTARY THEREON.
[Asterism] Thirty copies are printed on Large Paper, 1 vol. 4to. with
INDIA PROOFS before the Letters of the Portraits, price £3. 3_s_.
"A monument raised late, it is true, but not too late, to a great and
modest genius. A national biography which illustrates and elevates our
ideas of the past, and a contribution which the world will recognize to
the European history of Science."
_Dublin University Magazine_, September, 1865.
"A work which displays a high order of literary ability, careful
antiquarian research, much ingenuity, and withal thorough honesty of
purpose.
"[Lord Worcester], his life, told as Mr. Dircks has told it, is one of
much interest.
"Here we have an elaborate--although of course not a completely
exhaustive--account of his life; at any rate the most complete account
of him ever likely to be written--a work filled with abundant evidence
of the most painstaking research, a work written in a generous and
sympathising spirit, and with every attribute of conscientiousness."
_Engineering_, 5th January, 1866.
"The production of this volume is no common achievement; Mr. Dircks has
undertaken to write the life of a man about whom the public know very
little.
"He has, we think, collected some curious information, and established
the claim of the Marquis to be the first constructor of a steam-engine.
The reprint of the celebrated _Century of Inventions_ adds greatly to
the interest of the volume."--_The Spectator_, 14th September, 1867.
* * * * *
One Volume, 8vo., price 21_s_, only 100 copies printed,
WORCESTERIANA;
A COLLECTION OF
BIOGRAPHICAL AND OTHER NOTICES, ALPHABETICALLY ARRANGED,
RELATING TO EDWARD SOMERSET, SECOND
MARQUIS OF WORCESTER,
AND HIS IMMEDIATE FAMILY CONNECTIONS; WITH OCCASIONAL NOTES.
"The present volume is, as it were, a supplement. [To. Mr. Dircks's
_Life of the Marquis of Worcester_.] It contains what the French call
'pièces justificatives,' on which that biography was founded; and such
other materials connected with the history of Lord Worcester's family
and his invention of the steam-engine as will prevent, as far as
possible, a repetition of the gross errors hitherto promulgated on these
subjects."
_Notes and Queries_, February 3, 1866.
* * * * *
One Volume, post 8vo., with 130 wood engravings, price 10_s_ 6_d_,
PERPETUUM MOBILE;
OR,
HISTORY OF THE SEARCH FOR SELF-MOTIVE POWER
DURING THE 17TH, 18TH, AND 19TH CENTURIES,
WITH AN INTRODUCTORY ESSAY.
"The literature of this subject [Perpetual Motion] is very extensive,
but scattered mainly through Patent Records and ephemeral pamphlets. We
would especially refer the curious reader to a recent work by Mr.
Dircks, entitled _Perpetuum Mobile_, to which we have been indebted for
historical notices. It is extremely complete and interesting as a
history."
_Chambers's Encyclopædia_, Part 15, 1865.
"A very useful collection on the history of the attempts at perpetual
motion, that is, of obtaining the consequences of power without any
power to produce them."--_Professor De Morgan's_ Budget of Paradoxes,
No. 28.--_Athenæum_, July 15, 1865.
* * * * *
One Volume, post 8vo., with portrait, price 3_s_ 6_d_,
CONTRIBUTION TOWARDS A HISTORY OF
ELECTRO-METALLURGY,
ESTABLISHING THE ORIGIN OF THE ART.
"In his Introduction, Mr. Dircks has clearly stated the claims of
invention, and fairly discussed the only just grounds that can give
claim to priority of invention."--_The Mining Journal_, February 7,
1863.
"In the collection of chronological and other data for the history of
various branches and application of science, Mr. Dircks appears to be
indefatigable."--_The Electrician_, February 27.
"It is a useful and clear digest of evidence, and apparently impartially
put together."--_The Practical Mechanics' Journal_ (_Glasgow_), July.
* * * * *
One Volume, post 8vo., with two portraits, price 4_s_,
INVENTORS AND INVENTIONS,
IN THREE PARTS.
I. THE PHILOSOPHY OF INVENTION, considered strictly in relation to
Ingenious Contrivances tending to facilitate Scientific Operations,
to extend Manufacturing Skill, or to originate New Sources of
Industry.--II. THE RIGHTS AND WRONGS OF INVENTORS, Legally and
Politically Examined.--III. EARLY INVENTORS' INVENTORIES OF SECRET
INVENTIONS, employed from the 13th to the 17th Century, in
substitution of Letters Patent.
"The author enters fully and effectually into the claims and grievances
of the inventor. He discusses the arguments for and against the concession
of patent right, and examines very ably leaders in the _Times_ on patent
monopoly; very clearly dissipating the sophism of the opponents of patent
right; also Sir William Armstrong's evidence regarding 'patent monopoly,'
&c., affording an interesting and useful publication from its many
excellences."--_The Scientific Review_, September 2, 1867.
"The second part of the volume discusses the right of inventors to a
property in their inventions, and thus raises the question of the patent
laws, and the twofold issue, whether it will be better to retain them
and reform them, or to sweep them away altogether. We are bound to admit
that he treats this topic in a fair spirit, and without any taint of
bigotry. Mr. Dircks is a man whose opinions are entitled to a hearing."
_The London Review_, September 21.
"Mr. Dircks treats the real problem and discusses the comparative merits
of the existing system, and the advantages which he, together with many
others, hopes would follow on the establishment of some judicial council
of inventions. The difficulties of the question are enormous, and no one
will think the less of them after having gone through this volume.
"The third part, or the lists of their inventions left by many great and
some ingenious persons, is interesting and curious."
_The Westminster Review_, October.
* * * * *
One Volume, post 8vo., price 3_s_ 6_d_,
A BIOGRAPHICAL MEMOIR OF
SAMUEL HARTLIB,
MILTON'S FAMILIAR FRIEND,
With Bibliographical Notices of Works published by him; and a reprint
of his Pamphlet entitled
"AN INVENTION OF ENGINES OF MOTION."
"Mr. Dircks's is the first careful attempt to make posterity his
(Hartlib's) friend."--_The Examiner_, 18th February, 1865.
"A scholar-like little monograph, giving all the information that can be
given about a man whose name occurs in the correspondence of almost
every eminent literary or scientific person of the time of the
Commonwealth."--_The Spectator_, 20th May.
* * * * *
One Volume, post 8vo., with engravings, price 2_s_,
THE GHOST!
AS PRODUCED IN THE SPECTRE DRAMA,
POPULARLY ILLUSTRATING THE MARVELLOUS OPTICAL ILLUSIONS
OBTAINED BY THE APPARATUS CALLED THE DIRCKSIAN
PHANTASMAGORIA.
"Mr. Dircks gives us the benefit of all his progressive discoveries in
the matter, from the paper first read at the British Association Meeting
at Leeds, in 1858, to the more recent improvements, with full
explanations of the machinery, apparatus, and processes adopted in these
ghost dramas, and further favours the public with a number of new
adaptations. As a curious description of these spectral illustrations,
the book is most interesting."--_The Technologist_, January, 1864.
"A volume explanatory of the uncommonly clever and scientific "spectral
illusion" which has of late fairly turned the public head."
_The Dublin Builder_, January 1.
"A few months ago all London was rushing off to see Professor Pepper's
Ghost, as it was called, but which it now appears was the property of
Mr. Dircks, and from which his good name was filched in a very
unhandsome manner. Here then he tells us all about it, how the spectre
was raised, and how we may ourselves at pleasure call spirits from the
vasty deep."--_The Bookseller_, February 29.
* * * * *
Transcriber's Notes
Page 36: changed "Sorbiere" to "Sorbière" (15. Samuel Sorbière visited
the works at Vauxhall)
Page 61: changed "Jupiper" to "Jupiter" (of cloves, &c. to Jupiter;)
Page 83: changed "BD^2" to "BG^2" ({AG^2 + CB^2 + 4(OF^2)} = {AC^2 +
CG^2 + BG^2 + AB^2.})
Page 83: changed "sqrt(10^2 5^2)" to "sqrt(10^2) + 5^2" ( ... = {5^2 +
sqrt(10^2) + sqrt(10^2) + 5^2,})
Page 84: changed closing parenthesis to closing braces ( ... 9} = {25 +
10 + 10 + 25} = 70)
Page 84: changed "tangental" to "tangential" (tangential to the circle
X)
Page 84: changed "Q" to "2" (making G _n_ equal to 2AG - GD)
Page 84: added missing opening parenthesis in "(I have indicated this
square by dotted lines.)"
Page 84: changed "+ 1'25}" to "× 25" ({76'5625 + 1'5625} = {3'125 × 25})
Page 84: changed "hypothenuse" to "hypotenuse" (the area of a square on
the hypotenuse)