________________NOTE E.onphilo’sline.
________________
NOTE E.
onphilo’sline.
I am indebted to Professor Galbraithfor the following proof of the minimum property of Philo’sLine. It is due to the late Professor Mac Cullagh:—LetAC,CBbe two given lines,Ea fixed point,CDa perpendicular onAB; it is required to prove, ifAEis equal toDB, thatABis aminimum.
Dem.—ThroughEdrawEMparallel toBC; makeEN=EM; produceABuntilEP=AB.Through the pointsN,PdrawNT,RPeach parallel toAC, and throughPdrawPQparallel toBC. It is easy to see from the figure that the parallelogramQRis equal to theparallelogramMF, and is therefore given. ThroughPdrawSTperpendicular toEP. Now, sinceAE=DB,BPis equal toDB; thereforePS=CD. Again, sinceOP=AD,PTis equaltoCD; thereforePS=PT. HenceQRis the maximum parallelogram in the triangleSV T.
Again, if any other lineA′Bbe drawn throughE, and produced toP′, so thatEP′=AP′, thepointP′must fall outsideST, because the parallelogramQ′R′, corresponding toQR, will be equaltoMF, and therefore equal toQR. Hence the lineEP′is greater thanEP, orA′B′is greater thanAB. HenceABis a minimum.
________________NOTE F.onthetrisectionofanangle.
________________
NOTE F.
onthetrisectionofanangle.
The following mechanical method of trisecting an angle occurred to me several years ago. Apartfrom the interest belonging to the Problem, it is valuable to the student as a geometricalexercise:—
To trisect a given angleACB.
Sol.—ErectCDperpendicular toCA; bisect the angleBCDbyCG, and make theangleECIequal half a right angle; it is evident thatCIwill fall betweenCBandCA.Then, if we use a jointed ruler—that is two equal rulers connected by a pivot—and makeCBequal to the length of one of these rulers, and, withCas centre andCBas radius,describe the circleBAM, cuttingCIinI: atIdraw the tangentIG, cuttingCGinG.
Then, sinceICGis half a right angle, andCIGis right,IGCis half a right angle; thereforeICis equal toIG; butICequalCB; thereforeIG=CB—equal length of one of the two equal rulers.Hence, if the rulers be opened out at right angles, and placed so that the pivot will be atI,and one extremity atC, the other extremity atG; it is evident that the pointBwill bebetween the two rulers; then, while the extremity atCremains fixed, let the other bemade to traverse the lineGF, until the edge of the second ruler passes throughB: it isplain that the pivot moves along the circumference of the circle. LetCH,HF, be thepositions of the rulers when this happens; draw the lineCH; the angleACHis one-third ofACB.
Dem.—ProduceBCtoM. JoinHM. ErectBOat right angles toBM. Then, becauseCH=HF, the angleHCF=HFC, and the angleDCE=ECB(const.). Hence the angleHCD=HBC[I.xxxii.], and the right anglesACD,CBOare equal; therefore the angleACHisequal toHBO; that is [III.xxxii.], equal toHMB, or to half the angleHCB. HenceACHisone-third ofACB.
________________NOTE G.onthequadratureofthecircle.
________________
NOTE G.
onthequadratureofthecircle.
Modern mathematicians denote the ratio of the circumference of a circle to its diameter by thesymbolπ. Hence, ifrdenote the radius, the circumference will be 2πr; and, since the area of a circle[VI.xx.Ex.15] is equal to half the rectangle contained by the circumference and the radius, thearea will beπr2. Hence, if the area be known, the value ofπwill be known; and, conversely, if thevalue ofπbe known, the area is known. On this account the determination of the value ofπiscalled “the problem of the quadrature of the circle,” and is one of the most celebrated inMathematics. It is now known that the value ofπis incommensurable; that is, that it cannot beexpressed as the ratio of any two whole numbers, and therefore that it can be found onlyapproximately; but the approximation can be carried as far as we please, just as in extractingthe square root we may proceed to as many decimal places as may be required. Thesimplest approximate value ofπwas found by Archimedes, namely, 22 : 7. This value istolerably exact, and is the one used in ordinary calculations, except where great accuracy isrequired. The next to this in ascending order, viz. 355 : 113, found by Vieta, is correct to sixplaces of decimals. It differs very little from the ratio 3.1416 : 1, given in our elementarybooks.
Several expeditious methods, depending on the higher mathematics, are known for calculatingthe value ofπ. The following is an outline of a very simple elementary method for determining thisimportant constant. It depends on a theorem which is at once inferred from VI., Ex.87, namely“Ifa,Adenote the reciprocals of the areas of any two polygons of the same number of sides inscribedand circumscribed to a circle;a′,A′the corresponding quantities for polygons of twice the number;a′is the geometric mean betweenaandA, andA′the arithmetic mean betweena′andA.”Hence, ifaandAbe known, we can, by the processes of finding arithmetic andgeometric means, finda′andA′. In like manner, froma′,A′we can finda′′,A′′relatedtoa′,A′; asa′,A′are toa,A. Therefore, proceeding in this manner until we arrive atvaluesa(n),A(n)that will agree in as many decimal places as there are in the degree ofaccuracy we wish to attain; and since the area of a circle is intermediate between thereciprocals ofa(n)andA(n), the area of the circle can be found to any required degree ofapproximation.
If for simplicity we take the radius of the circle to be unity, and commence with the inscribedand circumscribed squares, we have
a = .5, A =.25. a′ = .3535533, A′ =.3017766. ′′ ′′ a = .3264853, A =.3141315.
These numbers are found thus:a′is the geometric mean betweenaandA; that is, between.5and.25, andA′is the arithmetic mean betweena′andA, or between.3535533 and.25. Again,a′′isthe geometric mean betweena′andA′; andA′′the arithmetic mean betweena′′andA′. Proceedingin this manner, we finda(13)=.3183099;A(13)=.3183099. Hence the area of a circle radiusunity, correct to seven decimal places, is equal to the reciprocal of.3183099; that is, equalto 3.1415926; or the value ofπcorrect to seven places of decimals is 3.1415926. Thenumberπis of such fundamental importance in Geometry, that mathematicians havedevoted great attention to its calculation.Mr.Shanks, an English computer, carriedthe calculation to 707 places of decimals. The following are the first 36 figures of hisresult:—
3.141,592,653,589,793,238,462,643,383,279,502,884.
The result is here carried far beyond all the requirements of Mathematics. Ten decimals aresufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals wouldgive the circumference of the whole visible universe to a quantity imperceptible with the mostpowerful microscope.
CONCLUSION.
CONCLUSION.
In the foregoing Treatise we have given the Elementary Geometry of the Point, the Line, and the Circle, and figures formed by combinations of these. But it is important to the student to remark, that points and lines, instead of being distinct from, are limiting cases of, circles; and points and planes limiting cases of spheres. Thus, a circle whose radius diminishes to zero becomes a point. If, on the contrary, the circle be continually enlarged, it may have its curvature so much diminished, that any portion of its circumference may be made to differ in as small a degree as we please from a right line, and become one when the radius becomes infinite. This happens when the centre, but not the circumference, goes to infinity.
THE END.
THE END.
THIRD EDITION, Revised and Enlarged—3/6, cloth._____A SEQUELTO THEFIRST SIX BOOKS OF THE ELEMENTS OF EUCLID.BYJOHN CASEY, LL.D., F.R.S.,Fellow of the Royal University of Ireland; Vice-President, Royal Irish Academy;&c.&c.________________Dublin: Hodges, Figgis, & Co. London: Longmans, Green, & Co.________________EXTRACTS FROM CRITICAL NOTICES.“Nature,”April17, 1884.
THIRD EDITION, Revised and Enlarged—3/6, cloth.
_____
A SEQUEL
TO THE
FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID.
BY
JOHN CASEY, LL.D., F.R.S.,
Fellow of the Royal University of Ireland; Vice-President, Royal Irish Academy;&c.&c.
________________
Dublin: Hodges, Figgis, & Co. London: Longmans, Green, & Co.
________________
EXTRACTS FROM CRITICAL NOTICES.
“Nature,”April17, 1884.
“We have noticed (‘Nature,’ vol.xxiv., p.52; vol.xxvi., p.219) two previous editions of thisbook, and are glad to find that our favourable opinion of it has been so convincinglyindorsed by teachers and students in general. The novelty of this edition is a Supplement ofAdditional Propositions and Exercises. This contains an elegant mode of obtaining thecircle tangential to three given circles by the methods of false positions, constructions fora quadrilateral, and a full account—for the first time in a text-book—of the Brocard,triplicate ratio, and (what the author proposes to call) the cosine circles. Dr.Casey hascollected together very many properties of these circles, and, as usual with him, hasadded several beautiful results of his own. He has done excellent service in introducing thecircles to the notice of English students.…We only need say we hope that this editionmay meet with as much acceptance as its predecessors, it deserves greater acceptance.”
The\MathematicalMagazine,” Erie, Pennsylvania.
The\MathematicalMagazine,” Erie, Pennsylvania.
“Dr.Casey, an eminent Professor of the Higher Mathematics and Mathematical Physics in theCatholic University of Ireland, has just brought out a second edition of his unique ‘Sequel tothe First Six Books of Euclid,’ in which he has contrived to arrange and to pack moregeometrical gems than have appeared in any single text-book since the days of the self-taughtThomas Simpson. ‘The principles of Modern Geometry contained in the work are, inthe present state of Science, indispensable in Pure and Applied Mathematics, and inMathematical Physics; and it is important that the student should become early acquainted withthem.’
“Eleven of the sixteen sections into which the work is divided exhibit most excellent specimensof geometrical reasoning and research. These will be found to furnish very neat models forsystematic methods of study. The other five sections contain 261 choice problems for solution. Herethe earnest student will find all that he needs to bring himself abreast with the amazingdevelopments that are being made almost daily in the vast regions of Pure and AppliedGeometry. On pp.152 and 153 there is an elegant solution of the celebrated Malfatti’sProblem.
“As our space is limited, we earnestly advise every lover of the ‘Bright Seraphic Truth’ andevery friend of the ‘Mathematical Magazine’ to procure this invaluable book without delay.”
The\Schoolmaster.”
The\Schoolmaster.”
“This book contains a large number of elementary geometrical propositions not given in Euclid,which are required by every student of Mathematics. Here are such propositions as thatthe threebisectors of the sides of a triangle are concurrent, needed in determining the position of the centreof gravity of a triangle; propositions in the circle needed in Practical Geometry and Mechanics;properties of the centres of similitudes, and the theories of inversion and reciprocations so useful incertain electrical questions. The proofs are always neat, and in many cases exceedingly elegant.”
The\EducationalTimes.”
The\EducationalTimes.”
“We have certainly seen nowhere so good an introduction to Modern Geometry, or so copious acollection of those elementary propositions not given by Euclid, but which are absolutelyindispensable for every student who intends to proceed to the study of the Higher Mathematics. Thestyle and general get up of the book are, in every way, worthy of the ‘Dublin University PressSeries,’ to which it belongs.”
The\SchoolGuardian.”
The\SchoolGuardian.”
“This book is a well-devised and useful work. It consists of propositions supplementary to thoseof the first six books of Euclid, and a series of carefully arranged exercises which follow each section.More than half the book is devoted to the Sixth Book of Euclid, the chapters on the ‘Theory ofInversion’ and on the ‘Poles and Polars’ being especially good. Its method skilfully combines themethods of old and modern Geometry; and a student well acquainted with its subject-matter wouldbe fairly equipped with the geometrical knowledge he would require for the study of any branch ofphysical science.”
The\PracticalTeacher.”
The\PracticalTeacher.”
“Professor Casey’s aim has been to collect within reasonable compass all those propositions ofModern Geometry to which reference is often made, but which are as yet embodied nowhere.…Wecan unreservedly give the highest praise to the matter of the book. In most cases the proofs areextraordinarily neat.…The notes to the Sixth Book are the most satisfactory. Feuerbach’s Theorem(the nine-points circle touches inscribed and escribed circles) is favoured with two or three proofs, allof which are elegant. Dr.Hart’s extension of it is extremely well proved.…We shall have givensufficient commendation to the book when we say, that the proofs of these (Malfatti’s Problem, andMiquel’s Theorem), and equally complex problems, which we used to shudder to attack, even bythe powerful weapons of analysis, are easily and triumphantly accomplished by PureGeometry.
“After showing what great results this book has accomplished in the minimum of space, it isalmost superfluous to say more. Our author is almost alone in the field, and for the present needscarcely fear rivals.”
The\Academy.”
The\Academy.”
“Dr. Casey is an accomplished geometer, and this little book is worthy of his reputation. It iswell adapted for use in the higher forms of our schools. It is a good introduction to the larger worksof Chasles, Salmon, and Townsend. It contains both a text and numerous examples.”
The\JournalofEducation.”
The\JournalofEducation.”
“Dr.Casey’s ‘Sequel to Euclid’ will be found a most valuable work to any student who hasthoroughly mastered Euclid, and imbibed a real taste for geometrical reasoning.…The highermethods of pure geometrical demonstration, which form by far the larger and more importantportion, are admirable; the propositions are for the most part extremely well given, and will amplyrepay a careful perusal to advanced students.”
PREFACE.
PREFACE.
Frequent applications having been made toDr. Caseyrequesting him to publish a ”Key” containing the Solutions of the Exercises in his ”Elements of Euclid,” but his professorial and other duties scarcely leaving him any time to devote to it, I undertook, under his direction, the task of preparing one. Every Solution was examined and approved of by him before writing it for publication, so that the work may be regarded as virtually his.
The Exercises are a joint selection made by him and the late lamented Professor Townsend,s.f.t.c.d., and form one of the finest collections ever published.
JOSEPH B. CASEY.
JOSEPH B. CASEY.
86, SouthCircular-road,December23, 1886.
Price 4/6, post free.]
THE FIRST SIX BOOKSOF THEELEMENTS OF EUCLID,With Copious Annotations and Numerous Exercises.BYJOHN CASEY, LL.D., F.R.S.,Fellow of the Royal University of Ireland; Vice-President, Royal Irish Academy;&c.&c.________________Dublin: Hodges, Figgis, & Co. London: Longmans, Green, & Co.________________OPINIONS OF THE WORK.
THE FIRST SIX BOOKS
OF THE
ELEMENTS OF EUCLID,
With Copious Annotations and Numerous Exercises.
BY
JOHN CASEY, LL.D., F.R.S.,
Fellow of the Royal University of Ireland; Vice-President, Royal Irish Academy;&c.&c.
________________
Dublin: Hodges, Figgis, & Co. London: Longmans, Green, & Co.
________________
OPINIONS OF THE WORK.
The following are a few of the Opinions received by Dr. Casey on this Work:—
“Teachers no longer need be at a loss when asked which of the numerous ‘Euclids’ theyrecommend to learners. Dr.Casey’s will, we presume, supersede all others.”—TheDublinEveningMail.
“Dr.Casey’s work is one of the best and most complete treatises on Elementary Geometry wehave seen. The annotations on the several propositions are specially valuable to students.”—TheNorthernWhig.
“His long and successful experience as a teacher has eminently qualified Dr.Casey for the taskwhich he has undertaken.…We can unhesitatingly say that this is the best edition of Euclid that hasbeen yet offered to the public.”—TheFreeman’sJournal.
From theRev.R.Townsend, F.T.C.D., &c.
From theRev.R.Townsend, F.T.C.D., &c.
“I have no doubt whatever of the general adoption of your work through all the schools ofIreland immediately, and of England also before very long.”
FromGeorgeFrancisFitzGerald, Esq., F.T.C.D.
FromGeorgeFrancisFitzGerald, Esq., F.T.C.D.
”Your work on Euclid seems admirable, and is a great improvement in most ways on itspredecessors. It is a great thing to call the attention of students to the innumerable variationsin statement and simple deductions from propositions.…I should have preferred somemodification of Euclid to a reproduction, but I suppose people cannot be got to agree toany.”