4 A-r ,/' 
 
 K U K L O S , 
 
 AN 
 
 EXPERIMENTAL INVESTIGATION 
 
 INTO THE 
 
 KHLATIOXSIUP OF CHRTAIN LINES. 
 
 BY JUIIN HARRIS. 
 
 PART FIRST. 
 
 MONTREAL : 
 
 PRINTFD BY JOHN LO\ELL. ST. NICHOLAS STREET. 
 
 1870. 
 
^ (j I^UO? H3 
 
 Euiered accorJine to Act of riulinmcut ...f Cnna.ln, in il,o vcar 1870. bv Joiiv TIabris, io llie Office of tbe 
 
 Minisicr of Agiiculiure. 
 
INTENDED ARRANGEMENT OF THE COMPLETE WORK. 
 
 Pakt Fikst Preface. 
 
 Preliinniary Argument. 
 
 Introiluction. 
 Part Second Nomenclature and Definitions. 
 
 Experimental Examination. 
 
 Theorems. 
 
 T\ fli . nil Tk 11 
 
 Oiiiismiiiix mill h'rri'f'i. 
 
 I'.i-f lit. Hh liiir 
 
 '■ •■ .'nil 
 
 • 2:}. i.'jth 
 " •• -jfitii 
 
 '• ;{•_'. Uottom '• 
 
 •• ;!:t. !»tii 
 
 in ilic Foot Niiii^ III paui^ ;il. t 
 
 .Sov('r;il ermrs dceiir in the lottery of ri'l'orcnci'. From tlio 2nd to the in\t line should re.'id as 
 lollows. ' Tiicn with coMtn; B, jmuI railius BlI, describe !in arc of 45 decrees, terminated by the line 
 BO. Il(i is till! tangent of that iiic and Ho is the tanjrent of an arc of 22^ dcj;recs described with 
 the same centre and radius; th;it is ...Ho = 41422, therefore (by prop. V) Up. ~ 41422 x § = 
 27(>l4tJ lll>. Now lit liic radius AB ^ 1... etc, etc. 
 
 .!■ FiiT. 
 
 , 
 
 re.i 
 
 d Fii:. 4. 
 
 " Fi^. 
 
 
 .i 
 
 Vi'j. .-.. 
 
 ■ Fijr. 
 
 
 
 Fi.u'. 11. 
 
 •• Pane 
 
 
 ■• 
 
 Pa!re21. 
 
 '• Dose ri be 
 
 BmC, 
 
 
 Describe the cjuadrarit Bm(\ 
 
 ' Detinite 
 
 Arc or (circle, 
 
 
 Detinite Arc of a Circle. 
 
^ !^ .0? h3 
 
 / 
 
 
 :!k. 
 
INTENDED ARRANGEMENT OF THE COMPLETE WORK. 
 
 Part Fikst Preface. 
 
 Prelimniary Argument. 
 
 Introcluction. 
 Part Second Nomenclature and Definitions. 
 
 Experimental E.xamination. 
 
 T]ieorcni8. 
 
 Part Tiiihd The Problem. 
 
 Part Fourth Concluding Chapter. 
 
 Part Fifth '• " 
 
 CONTENTS OF PART FIRST. 
 
 Taoe. 
 
 Preface 5 
 
 Preliminary argument 9 
 
 Introduction 35 
 
 Plate I containing Figures 1 to 7 inclusive 
 
 Do II " " 8 to 13 
 
 Do III •' " 14 to 18 " 
 
 Do IV •' " 19 to 24 " 
 
 Do V " " 2.5 to 26 " 
 
 Do VI " " 28 ". 
 
 Do VII " " 29 " 
 
 Do VIII " " 27 and 31 " 
 
 Do IX „ , 30 to 33 " 
 
 Do X " " 34 " 
 
Wi. must not unly search for an-I jToouro a groater nutiilier of cxi.criinc:its, l.ut also 
 iiitro.I.ico a com!.letdy (liff.Tcnt mctlio.l, or.lor an-l pro-ress of continuing an>l jirotnoting 
 experience. F.r va-uo an-l arbitrary experience i,s Us we have observe.!), mere groping in 
 the dark, an^l ratiier astonishes than histructs. r.i.t wlien experience shall proceed regularly 
 au.l nninternipteaiy hy a .{•■tenuinea rule, we may entertain brtter hopos of the Sciences. 
 
 FkAXCIS B.At'OX. 
 
P R E F A C K 
 
 The primniy and immediate snliject of the follinvin^ work is tlic Geometrical Quadrature 
 of the Circle, viz., the solution of that problem whicli recjuires a straight line to lie drawn 
 equal in length to the pivcn arc of a circle : or a straight line being given, rcciuires a definite arc 
 of a circle to be described equal in length to the given straight line ; accompanied in either 
 case with demonstration that the ref|uisitir>n has been fulfilled It is hereby asserted that the 
 following work will bo found to contain the satisfactory solution of this problem in both forms : 
 moaning by the word " satisfactory." that the reasoning a;.d demonstration will be conducted 
 strictly in accordance with the method laid down and pursued in Euclid's Geometry. 
 
 Ihc work will also have a more general purpose, but the explanation of this purpose will 
 be deferred until the concluding chapter ; because any importance or value that may be 
 attached to it, will in a great measure depend upon the reader's conviction that the above 
 assertion has been made good— that is to say, whether or not our claim of having solved the 
 problem is judged by him to have been substantiated. The work is published with full belief 
 on the part of the author, that the reasoning will prove to be essentially sound, and that the 
 substantial correctness of the demonstrations will be admitted by Mathematicians : but it 
 will not be surprising if such faults, as a want of mathematical coherence in arrange- 
 ment, and clumsiness in the use of geometrical expressions, should frequently manifest 
 themselves. In reference to these, no apology is offered, but some e.xiilanation will be proper. 
 Granting for the moment an assumption that the author has had the good fortune to find a 
 clue which enables him to contribute something of value and importance to the general stock 
 of mathematical knowledge— if such assumption is well founded, the cause of Science will be 
 better served by making the nature of the contribution at once known— that is. so soon as the 
 
subject c-aii l.e jiut intf> an intelli^'il'le shape— than l>y tlelayi'ig it for the purjinse of giving 
 more perfection and finish to the foiaiof the communication. If, on the other hand, such 
 sujiposition is a fallacy, the communication itself would not be worth the additional labour 
 best"we<l on improving its form. Some few additions to the usual nomenclature of Geometry 
 will be found carefully defined at the commencement of the work. These additions have 
 not been adopted without considering the inconvenience which a use of strange (un- 
 authorized) terms in Geometry is calculated to cause. In the present instance, however, 
 it is thought that the peculiar nature of the subject investigated v.ill be held to justify the 
 innovations ; and further, that the gain in clearness arising from their use, will show, perhaps, 
 that these expressions, or others equivalent to them, might be advantageously adopted by 
 geometricians generally. Again, to s<3me extent, the arrangement and method of conducting 
 the invesdsatlon will have reference to the more general purpose of the work, which purjose, 
 as already stated, will be exj'lained in the concluding chapter. The subject herein investigated, 
 besidct* having certain difficulties, properly belonging and peculiar to itself, is also surrouisded 
 by a soit of outside difficulty, rciuiring to be attended to and surmounted in the first instance. 
 The nature of this outride difficulty, and the mode in which we propose to deal witli it, we 
 will now endeavour to explain. 
 
 The (juestion of the <.,»ua'lratuR' of the Circle is looked at from a diflferent point of view, 
 and even the nature of the question is very differently estimated, by each of three distinct 
 parties. Firstly : — There is the mathematician, who knows that there is an actual and 
 imp.irtant problem having this title, which has hitherto remained, in a geometrical sense, 
 unsolved : he is acquainted with tne conditions of the problem and with the nature of the 
 difficulties to be overcome. Secondly : — There is the man, who, with a tape-line or some more 
 or less ingenious contrivance of the Uke kind, persists in measuring the Circle, unaware that 
 it has long since been matliematically measured. Thirdly : — There is the non-mathematical 
 public, whicli regarding the whole thing as an absurdity — something between a joke and an 
 exploded chimera — puts it on the same sLelf with the philosopher's stone and jierpetual 
 motion — whence one or the other is occasionally taken down to " point a moral " or " adorn a 
 tale,' and then shelved again. 
 
• Of these three iiartiea, it is the first, the mathematician, with wliich wo have to do ; hut 
 hero atamU the difficulty referred to — party number one lias been persistently tormented and 
 aggravated by party number two ; and the consenucnco now is that the mathematician, no less 
 Bcnaitive than tiie war horse, sniffs danger in the very breeze that wafts the name of a 
 new adventurer in connection with Quadrature of the Circle — the apparition of the man with 
 tlie tape line rises before his mind's eye growing desperate, he hardens his heart, and 
 at once resolves, like the deaf adder, let the iiew comer come in never so (luestionablo 
 a shape, to have nothing to say to iiim. A mathematician is, huwever, a luuuan Iteing, 
 and although as s\ich, subject to some slight extent to the weakness of tlie flesh, and other 
 imperfections pertaining to that unfortunate condition, ho is, nevertheless, bound by virtue of 
 his profession as a mathematician, to be very reasonable. A powerful outside influence, or 
 even lacking this, a bold assertion of. individual right, may yet gain the adventurer the much 
 coveted ojiportuuity of making heard his instructive voice. If on the one hand he is provided 
 with credentials from resitonsible parties, recognised by the mathematician as those having 
 authority, he may, by virtue of such credentials, at once demand a hearing: but otherwise, 
 and if not provided with any credentials, he may yet claim his right to the • trial by touch.' 
 In this case, however, he must be prepareil to abide by the coudiiions of the )rdeal. The 
 jiresenbed " modus, " as we understand it, is this — the adventurer must hand in a concise 
 statement of reasons why he should be permitted to unfold his tale, and why the mathematician 
 shiiuld be called upm to witness his performing the process in an orthodox and satisfactory 
 manner. This statement the mathematician is bound to look at, but it by no means f'l.JlDws 
 that he can bo got to see it ; and if he cannot, the adventurer must at once understand that 
 his views are too much in advance of the times to be made intelligible in the present obscure 
 state of Science ; he must therefore lock up the important secret in his breast or some other 
 secure place for at least a quarter of a century, after which he may take it out and try again : 
 but if the mathematician can be got to sec it, the adventurer has his will and may reckon to 
 hold on to the mathematician precisely as " ye ancient mariner held ye wedding guest," that 
 is to say, by his optical sensorium. 
 
8 
 
 Aa we are not proviJed with any crcJentuils wliatovor, we must go on our right o the 
 "trial by touch," and — accepting the conditions — will endeavour to meet the preliminary tliffi" 
 culty by a preliminary argument, put in as brief and concise a form as the circumstances of 
 the case permit. We must, however, beg the courteous reader kindly to take note, that tf^'ta 
 preliminary argument is not intended to stand as part of the systematic work, but is of tbe 
 nature of a dissertation, interp'olateJ for the express purpose stated above. 
 
PRELIMIXARY ARGUMENT. 
 
 PART FIRST. 
 
 Wo have Loon fortunate cnoui:li to find in one ol' the numbers of a Literary and Scientific 
 Journal, a brief review of the niitiakos made by persons who, at different times, have vainly attempted 
 to ' square the Circle;' aceonip-.uiiod with an explanation of the erroneous nature of the hypothesis 
 which has occasioned the-c attenipt=. and giving aho a concise definition of tliat prubiem which 
 forms the subject of tlie follv>wing work. The article alluded to may be found in the number of the 
 (London) Athcnrrnm J->urii-i>. publish.d on the 14tli October, 18(55, and is the lust of a series called 
 " A Budget of I'aradose-V from the fion of the eminent mathematical professor, De Morgan. We 
 give here the concluding part of the article, which appears well adapted to serve as an introduction to 
 our argument. •■X.iw, since dozens of methods, to which dozens more might be added at pleasure, 
 concur in giving one and the same result ; and since these methods arc declared by all who liave shown 
 knowledge of niathematics to be demonMrahd ; it is not asking too much of a person who has just a 
 little knowledge of the first eknients that he should learn more, and put his hand upon the error 
 before he intrudes his assertion of the tsistcncc of error upon those who have given more time and 
 attention to it than himself, and who are in po.ssession over and above many demonstrations, of many 
 consequences verifyinir each other, f^f which he can know nothing. This is all that is required. Let 
 any one square the Circle, and persuade his friends, if he and they please ; let him print, and let all 
 read who choose. But let Lim abstain from intruding liiniself upon those who have been satisfied 
 by existing demonstration, until ho is prepared to lay his finger on the point in which existing 
 
 demonstration is wrong. Let him also s.ay what this mysterious 3.14159 really is, wliich comes 
 
 in at every door and window, and down every chimney, calling itself the citcumferencc to a unit of 
 diameter. This most impudent and successful impostor holds false title deeds in his hand-^, and 
 invites examination ; surely those who can find out the rightful owner are equally able to detect the 
 
 forgery. All the quadrators are agreed that, be the right what it may, 3.14159 is wrong. It 
 
 would be well if they would put their heads together, and say what this wrong result really means. 
 
 A large number of opponents unite in declaring this result wrong, and all agree in two points : 
 first, in differing amongst themselves ; secondly, in declining to point out what that curious result 
 really is which the mathematical methods all agree in giving." 
 
 B 
 
10 
 
 "Most of til' .luaJrators aro not aware that it has Ijeoii fa'ly 'l- iiiii-ti'atcJ tint no two numbers 
 wllat^oev.■r can ropre-cnt tho ratio of the ilianiotor to the oiivunit'crcnce with [.orfect aceiirapy. When, 
 therefore, w^- are toM that oiiher ^ to l'.'i or f'.-l t,, -Jdl i> tlio true rati", we know that it is no such 
 thiuL', without the neei'!':-ity of examination. Tlio [mint tliat is left open, as not fully ilemonstratcd 
 to be impossible, is the;,'eonietrieal ([Uaih-ature, tli; ileteniiination '^f the cireumfereiiee by the straight 
 lii.e and eirclo, used as in Kuelil. The general run of circlesquarer.s, hearing that the quadrature is 
 n^t pr>noMnc'd to be Jemons/idliri Ij impossibl\ imagine that the <tritliii>cticiil fjuadrature is opi'u to 
 tiieir ini;enuity. Before attempting the aiithmeticil problem, tliey ought to aequire knowledge 
 enough to read Lambert's demonstration (last given in Brewster's translation of Logendre's Geometry), 
 and if they can. i.i refute it. Probably some have begun in this way. and h.ve caught a Tartar, wlio 
 refuse I to let I hem go. I have never iieardofany one who, in [irodacing liis own demonstration, 
 has laid hi* finger on tl:e faulty part of Lambert's investigation. Tiiis is the answer to those who 
 think that tlie inatiiematicians treat the arltlmu'tieal squarers tcto liglitly, and tliat as some person may 
 sueceed at last, a'.l attempts jhoulJ be e.Kaminel. Those wlio have .so thought, not knowing that 
 there is demon.str.ition on the point, will jnob.ibly admit that a person who contradiets a tlieorein (jf 
 which the demon-tration has been acknowledged for a century by all who have alluded to it as read 
 by themselves, may reasonably be required to point out the error before he demands attention to his 
 own result.' 
 
 If the literary character of the Journal Irid admitted of such a mode of treatment. Prof. Do 
 Morgan might perhaps by the aid of a geonietrieal diai:raui, and, by explaining the relationship of 
 certain lines to the subject under consideration, have made the strength of his pisition still more 
 clearly apparent, and have put tho.se against whom his argument is directed yet more hopelessly iu 
 the wrong. As it is desirable to put the subject befon our readers in this form, we .shall venture, so 
 far as our ability will enable us. to take the place of the professor, and endeavour to supply such a 
 brief geometrical exposition by way of appendix to his st.itement. 
 
 The reader is referred to Fig. I. From the centre A with the radius AB describe the quadrant 
 BmC. from the point B draw the line BK at right angles to AB. Bisect the quadrant at the point 
 m. From A tlirough m draw A D intercepting the line BK at D. -Join AC and DC From m 
 draw the line m n at right angles to AB, and intercepting AB at n. Join m B. We have now the 
 square ACDB. The are Bm C. containing r>0 degrees, and the lesser are B)n, containing 45 degrees. 
 The secant AD, the shie m n, the o.-ine An, the chord Bm, and the tangent BD of the arc Bm. 
 We have also the three triangles Amn, Am B and ADB. of which Amn and ADB arc similar 
 triangles. Amn is part of ADB, and therefore ADB is greater than Amn, and the base of the 
 triangle AmB is intermediate between the base of the triangle Amn and the base of the triangle ADB. 
 
 To prevent confusion of lines, wo will now refer to Fig. 2, which is a repetition of Fig, 1, and 
 iu which similar letters denote similar lines. Bisect the arc Bm at the point 0, and from A through 
 draw Ap, intercepting BD at p. From the point 0, draw Or at right angles to AB, and inter- 
 cepting AB at r. Join OB. We have now again three triangles Aor, AoB, and ApB, of which 
 Aor and ApB are .similar triangles. ApB is greater than Aor, and the base of AoB is intermediate 
 between the base of Aor and the ba>e of ApB. D is manifest that if we bisect the fractional arc Bo, 
 b'st cut off, and draw the secant, the sine and the chord of the half of the fractional arc Bo, intercepted 
 between B and the point of bisection, that we -ball again have three triangles, and if the fractional 
 
11 
 
 Jirc, wliicli i.-i tlio Iiali' of Do is i\\< i bi-oc'cJ. tliroe more trian-'is, anrl finall}-. that so long as any 
 part uf lliu arc ISiiiC reniains. if'uc bis et tliat remaining [.art (howowr sii.all that part luay Le) we 
 ,*li:ill (il)taiii throe Iriai.irles, whi h will 1 c related to cieh other in the same manner as the three 
 tiiantilo- lirnt > htiin d, naiiuly, the trian.Ie formed by the peeant, the tangent and tlie radius of the 
 arc (however siii ill th'' arc may be) will br similar to and will be also greater than the trian-le formed 
 by the radius, tiie sine and the cosine of tlu same are ; and the base of tlic tliird triangle, formed by 
 the two radii and the chord of the same are, will b^ intermediate between the bases of the other two 
 triaiif;ks. It is also obvious that, as the pro?css of liiscction is carried further, aud the part of the arc 
 remaining becomes less, tlic lines forming the bases of the three triangles respectively, will continually 
 nppr.iaeh each other, and that wlien the part of the are remaining becomes extremely small, the three 
 lines will approximate very closely. Now, whether we take the whole arc of 4.") degrees, or any one 
 of the fractional arcs cut oflF by the process of bisection (however small that fractional arc may be), 
 it is manifest that the figure contained by the arc itself, together with the two radii, is k.ss than the 
 figure contained by the tangent, tog<'ther with the secant an 1 ra lius of that arc, and grcat'jr than the 
 figure contained by the chord, together with the two radii, ami th refore also greater than the figure, 
 contained )iy the sine, together with the radius and cosine of that are. It tlicrefl)re follcws that, 
 however sn?:dl the fractional are may be, the arc itself is less than the bas:' of the greatT triande, and 
 greater than the base of either of the other two triangles. ^Ve will now suppose a very small fractional 
 nrc remaining of only one degree * and leavitig out of consideration the intermediate triani.'le, 
 we will confine our attention to the arc it.self, and to the two lines, one of which (the tangent to the 
 arc), forma tlic base of the greater triangle, and the i.thur (the sine of \h>' arc) forms t!ie ba.sc of the 
 lesser triangle ; and since these two lines, one of which is greater and the other less than the arc. 
 approximate closely to each other, it is evident that, if we obtain the hn-jths of these two lines, we 
 shall also obtain a close approximation to the Kngth of the are, which is intermediate betwe n them. 
 
 The lengths of these lines, that is to say, the length of the tangont and of the sine of an are of 
 one degree, h ive been long since obtained ; and it is not to bo supposed that any one will call in 
 question the correctness of the mathematical nieasuremenf of these straight lines. The actual ler.gth 
 of these lines, taken from an authorised table (the radius beiuL' unity) are: Tangent, 0174r>.">; ."^inc, 
 017402. If we multiply the first of these '(Uantities by IMO degrees, we shall get the approximate 
 measurement of the cireuinference of the Circle, but the product so obtained will m'tntfrstlij be a little 
 mon than the actual length of the circumference ; if wo mul'iply the ,see'ind quantity by 3(10, the 
 product will manifestly be a little Ass than the actual length of the cireuuiferenee. To find the 
 approximate ratio of tiie circumference to a unit of diameter, the number so obtained must be divided 
 by two — beeau.se the diameter is twice the radius. Therefore, 
 
 .017455x300 = 0.283800 and C.2S3S00 ^^ ^^^g^ 
 
 2 
 
 .017452x30(1 = 0.282720 and C.282720 „,,,„,, 
 
 — = J. 141 do 
 
 Note. — It is evidcut that tlie [iroceis uf bisection, if commenced with an arc of 45 degrees, will result in a 
 sm.all arc eiilier gie.iter or U'ii tliiiu one degree ; the naUor iiiay iherefoa' suppose the prucuss to have coniiuenced 
 with an fire of 32 degrees. An arc of 45 degrees has ' e. n taken in Ihc first instauce fur the purfose of simpli- 
 fying the exphinition, but a little further on an arc of 52 degrees is substituted for the reasjn just referred to. 
 
12 
 
 So that we have .'i.UinO, a fiuaiitity which must inauiiVstly bo -reatcr tlian the actual leii'rth of tlic 
 circuuilerence, auJ wo have 3.1il3l3, a quantity whicli must manifestly bo less tliau tiio actual Icn-tb 
 of the circumforLuee. We have supposed the process of bisection to stop at a fractional arc of one 
 Jo!,'roo, but ihe reader, if he ] leases, can carry the process further, and cut d. wn the arc to a small 
 fraetion of on-; degree , he will still be able to obtain the leuL'tii of the tuni;ent and of the sine, and it 
 will be found that, however far the process may be carried, the produst of the tangent will never 
 become less than .'!.141o',),... nor will the product of the sine ever become greator than :5.14159...a 
 number which will be always intermediate between these products, prteisoly as the arc itself is inter- 
 mediate between the base of the greater and the base of the lesser trianule. '■<• 
 
 Now. although we hold in common with mathematicians that tlie process of which we have here 
 endeavoured to give a brief explanation, is geometrically sound, by which we mean that the .■easoning 
 is itself iueoutrovertiblo, and the facts upon which it is based indisputable, and that, therefore, the re- 
 sult must be certainly correct, we find, ueven!n.lcs.s, reason to believe that the result of that process 
 has not been hitherto wholly understood, or perhaps it would be better to say, that the meaning of the 
 result lias not beeu fully realized, and that certain imperfect assumptions arising in conse(|uence have 
 caused an obstacle to appear insurmountable, which obstacle can be ( vercome without any very great 
 dilEcuhy wheu the moaning of that result is fully and correctly understood. In order to bring at 
 once bct'jro the reader the precise nature of the ea.se, to which wo wi^h to direct his attention, we will 
 refer again to Fig. I. and Fig. II., and again suppose that the arc Bm has been bisected, the fractional 
 arc resulting therefrom also bisected, and so on until we again have a fractional arc oi'one degree only 
 remaining, 
 
 It appears manifest that this fractional are of one degree is the 4r)th part of the whole arc of 45 
 degrees, and in one sense such is undoubtedly the fact, .since it is a fraction of the wiiole arc, and 
 consists of one divisiimal unit out of the forty-five of which the entire arc consists. It will then 
 appear to lollow that if the fractional arc of one liegree is multiplied by 45, the product will equal the 
 entire arc of I'orty-fivo degrees ; and again in one sense such will be undoubtodly the fact; but wo 
 siy ihcre has been hitherto, directly or indirectly, an assumption that if the fractional arc of one 
 degree is increased to 45 times its magnitude, it will reconstruct the original arc, or, to put it in other 
 words, that if the fractional arc of one degree is magnified to 45 times its size, it will in its magnified 
 condition be similar as well as equal to the entire are of forty-five degrees. If .such assumption is 
 
 • We ar" aware tliut the latter part of this eximsition i?, in a scieiitilic sense, open to ot.jeclion ; inasmuch 
 as the measurcaient of these slriiijiht Um'.'S may be s-iiJ to be a ilorivative from ami a part of the same compulation 
 which gives the result 3.14159. . . .etc. . . .aail that therefore the fact of the part harmonizing with the whulc 
 does not furnish a proof of the correctness of the general result. Xevertheles?. addressed to the quadrators or 
 those who are sceptical as to the soundness of the mathematical result, the .irguraent as stated above is, we think, 
 leRilimaie ; many of the quadrators, fur instance, hive insisted on a result cuusijernhly greater than even the 
 product of the tangent, and others again have been equally ur.?ent iu favour of a quantity less than the product 
 of the sine, of ore degree. The Irigonometricul prnces', based . n the cnnlinucd hisoclion of an arc, may be said 
 to commence from tne vanishing-point at which the two lines, and the arc intermediate between ihem, terminate 
 or originate in common, and from that point to build up the meas irement of the entire circle. Formerly the 
 compiittttion was attended with the e.Tpendilure of very much time and labour, but more recently such com- 
 putations hare been greatly facilitated by the contrivance of very ingenious formula", and other improvements in 
 trigonometry 
 
13 
 
 subiuittcd to Geometry, its fallacy will become apparent. Prop. I. It is refjuireJ to dc.'cribe an arc» 
 similar atiJ ef|uivalent to a gi\cn a c rtilucJ tu half its magnitude. Fig. 3. Let tbe arc Bm, 
 deseribej as in Fig. I., he the ,i;;ven arc, it is required to reduce the arc Bm to half its magnitude 
 Bisect the radius AB at tbe p.'int L. With the centre L and the radius LB describe the quadrant 
 Bst. Bisect Bst at the point s. B.- shall be the rcijuirtd are Because the arc Bs is half the arc 
 Bst, and the arc Bm is half the arc .{mC, and the arcs Est and BmC arc both of them quadrants, 
 the arc Bs is similar to the arc Bm. Becau.sc the radius LB is half the radius AB, the circle of 
 whicli tbe arc Bst is a fractien, is half he maj.'nitude of the circle of wMch BniC is a fraction. But 
 Bs is half of Bst, and Bm u half of BmC, therefore Bs is half the magnitude of Bui, and Bs has been 
 shown to be a similar arc to the arc Bm. Wherefore, the arc Bs is similar and ec|uivalciit to the arc 
 Bm, reduced to half its mairnitude, which was required. It is unnecessary here to dcmoii.stratc 
 because it will be at oncj admitted, that the resulting arc may, by bisecting the radius LB, be 
 reduced to half its ma<:nitude. and that the process may be so continued until the arc is reduced to 
 any small ma^rnituJe rcjuirvd. We will now suppose u given arc of thirty-two degrees, and subject 
 it to the process of repeated bisection, as described in our appendix to Do Morgan's statement ; at the 
 fifth bisection we shall have a fractional arc remaining of one degree, which fractional arc we will call 
 m. We again suppose the same given arc of thirty-two degrees, and reduce it by the process shown 
 in Prop. L, to the thirty -second of its magnitude, and the resulting arc so obtained we wiil call n 
 Let us now consider and compare together these two small arcs, the fractional arc m, and the reduced 
 ;.rc n ; each of them is the one thirty-second par t of the same given arc, and they are necessarily equal 
 in length to each other, but they differ in form ; tlie radius, the secant, the tangent, and the chord 
 belonging to the one arc are neither equal, nor if taken together, are they similar to the radius, the 
 tangent, and the chord of the other. The two small arcs pos.'=ess distinct and dift'ereiit geometrical 
 properties ; they differing from each other, are, each uf them, the thirty-second part of the same thing. 
 But tills cannot be in the same sen^e. Manifestly if the small arc n, resulting from the prncchs of 
 rnluction, i.~ inerea.scd by rcTersinir that process to thirty-two times its magnitude, it will recinstract 
 the orii:inal arc, from which it was reduced. But then, to what does tbe small arc m belong in the 
 Buuie sense, and what wiii the arc m produce if .-iimilirly iucreased in magnitude. Tlie arc m, in the 
 eanie sense, is tfo. thlrty-sewnd pnrt uf nn arc of one degne. belonying to a circle thirfi/-two times the 
 mmpiitiide o/tht: circU to tchirh the original arc (from which it was cut off) bi lunged. And if the 
 tmall arc of one d^grtt m, u incnased to thirty-two times its vi'ignitiide by racrsing the jyrocrss of 
 reduction thoicn in Prop. I., it irill stdl remain an arc of only one degree, but it will have become one 
 degree of a circle thirty-ttco timet the mugnitude (f the circle to which the original arc belonged. 
 And the arc of one d' grif belonging to the circle of greater inagnifude will necessarily be equal in 
 length to the original giitn arc <f thirty-two degrees. Prop. II. — An arc (being thedctinitu fraction 
 of a circle of any magnitude ) and a straight line forming the tangent to the arc, being given ; it is 
 required to describe upon the same tangential straight lino, a second are, equal in length to the given 
 arc, and belonL'ing to a circle of double :Iie magnitude of the circle to which the given arc belongs ; 
 and it is further required to describe upon tlie same straight line a third arc, equal in length to each 
 of the other arcs, and belonging to a circle of double the magnitude of the circle to which the second 
 arc belongs. Fig, IS. — With the centre A and the radius AB, describe the quadrant BmC. Bisect 
 BmC at the p<3iut m. From theioint B draw tlie tangential straight line BT. Let the arc Bm of 45 
 
n 
 
 •Ifprrcos lio the irivon arc, .'iml lot BT be the ;;iv:ii -t 't:lit lino, it i> iv.|nirLd to iKsnilf tlio »( (;oni.l 
 ami lliiril arcs u;uju tlio .-traiL'lit lint- BT. 
 
 Join BC, tlic clior.l of the (nudvunt BmC. I'rwlucc tin.- nidius BA, throu-h A, iiml iimkc J5AD 
 twice the length of BA. AVith the centre 1) and radius 1>B dc.scrihe the arc BiiE ; of any Icn-tii). 
 From the point D throu.di ui dr.iw ilic line Dn. iiiterecptinir tlic arc last descrihed at the point n, the 
 arc Bii so cut otf froi.i the arc BnE, shall 1 j the second arc rcf|inred. Because the radius DAB is 
 twice the length of the radius AB, the cird to which tlic arc BuK hclonps is douhle the magnitude 
 of the circle to which th arc BniC belongs. Fn lu the jKjint m and at right angles to the line Dm, 
 draw the chord Bin of the arc Bin. Bisect the arc Bm. at the point o. Join Ac perpendicular to the 
 chord Bm, j'in also uio tmd Bo, ami pp'iluce the line Bo through o, uutil it meets the line Dn, 
 at the point n. Bee:iuse o i.s the point of bisection of the arc Bni, the triangle Aom is similar and 
 equal to the triangle AnB and becau-'e the line Dn is pcrfK;ndicul,ir to the line Bni, and the line Ac 
 is" also perpendicular to the line Bm. and bec:r.i.-e the line DB is a production of the line AB, 
 therefore the triangle DuB is al-^o .simi'ar to th" triangle AoB. but DAB, the side of the greater 
 triangle, is twice the length of A13. the side of the lesser triangle AoB, therefore Bi'u, the base o: 
 the greater, is likewise twice the length of Bo, the base of the lesser, and coiise(iuently o is the jioint 
 of bisection of the line Bon. Since the triangle formed by the chord together with the two radii of 
 the Ks.ser are Bo, is similar to the tritiiigle formed by the chord, together with the two radii of 
 the greater arc Bn. therefore the arcs Bn and Bu form similar frictions of the cireirs to which they 
 respectively belong. But the arc Bo is part of the rjuadrant BmC, and the arc Bn is part of the 
 arc BiiE, and it has been 'linvn tliat the arc BnE bel' n-s to a circle of double the magnitude of the 
 circlet'! which the arc BmC belongs, therefore the arc Bn is twice the length of the arc Bo; and 
 ai'ain, the arc Bo is half the arc Bni, mil therefore the arc Bn is equal iu length to the arc Bm. 
 AVheref>re, tiie second are Bn, Uescrihud uf>on the given straight line, is equal in length to the 
 given arc Bm, and belongs to a circle twice the magnitude of the circle to which the given arc 
 bcFongs as required. But further, produce the r.adius BD, through D, making BDF twice tiie length 
 of BD. With the centre F, and theridiu- FB, de-cribe tlic arc BLG. and from F, througli the point 
 n, and it right angles to the chord Bon, draw tin- line FL, intercepting the arc last described at the 
 point L. The arc HL. so cut off from the arc BLG, .shall be the third arc required. Because 
 the radius FDB is twice the length of the radius DB, the circle to wliich the arc. BLG belongs is 
 double the magnitude of the circle to which the arc BnE belongs. Bisect the arc Bn at the point 
 p. Through o join Dp. perpendicular to the chord Bon, of the arc Bn. Join np and Bp, and 
 produce the line Bp through the point p, uutil it intercepts the line FL at the point L- liccauso 
 p is the point of bisection of the arc Bn, the triangle Dpn is similar and equal to the triangle 
 DpB. and because the line FL is perpendicular to tlie line Bod, and the line Dp is also perpendicular 
 to the line Bon, and because the line BDF is a production of the line BD, therefore the greater 
 triangle FLB is also similar to the lesser trianLde DpB, but FDB, the side of the grc iter triangle, is 
 twice the length of DB, the similar side of the lesser triangle DpB, therefore BpL, the base of the 
 greater, is al.«o twice the length of Bp, the bise of the lesser. .Since the triangle formed by the 
 chonl, together with the two radii of the le-ser jirc Bp, is sim'lar to th3 trimgle firmed by the 
 chord, together with the two radii of the greater arc BL, the arcs Bp and BL are therefore similar 
 
15 
 
 fractious of the ciicl'sto wLlIi they rc-poclivoly li«.!on;.'. IJut tlif arc Bi) is p;irt of the arc Bj.n, 
 and tlic ac I?L is p:rt ol" tlu; .-irc BLU ; a'l'l it has bcou shown that the arc BLG belnni^s to n 
 circlo Jou jIo tlie maj;iiitu'le of tlio circle to which Bpn bdo'ic;-*, thcrcfiro the arc BL is twice 
 •he lc'iij;tli of the arc Bj.. But the arc Bp is lialf tlic arc Bpn. Wherefore tlie third arc BL, described 
 upon tlie niven straiu'ht line, is equal in lenv'th to the sceond arc Bn, de-crilicd upon the same 
 straight line, and belongs to a circle twice the ninguitude of the circle to which the second arc belonc's, 
 as was required. 
 
 CoRnLLAiiV. — Because the chord Bm of the arc Bni is also the sine of the arc Bn, and becau.se 
 
 the chord Bn of the arc Bn, is also the sine of the arc Bl, it follows that If a nuadrant, or any 
 
 definite arc less than a quadrant, be given, and if a second arc be described upon the same straight 
 line, equal in length to the given arc, and belonging to a circle double the magnitude of the circle to 
 which the given arc belongs, — the sine of the second arc shall be the same line a^ the chord of the 
 given arc ; and if a third arc be described upon the same straight linn, equal in length to the .second 
 arc, and belonging to i circle double the magnitude of the circle to which llic second arc belongs, then 
 shall the sine of the third arc be the same line a.s the chord of the scconil arc ; and so on 
 
 It will be hereafter shown that if this process is continued, namely, if we continue to describe 
 arcs upon the same straight line, each of them equal in length to the preceding arc, and belonging to 
 a circle twice the magnitude of the circle to which the preceding arc belongs; — the eventual result 
 will be that we shall cbmpletely unbend the given arc, and project the given arc (so to .speak ) upon the 
 straiglit line. See Figures 24 and I>0. 
 
 If, however, it were neces.sary to actually double the length of the radius, and to describe the 
 successive arcs as specified in the foregoing proposition, a limit would be reached bef:)re the arc was 
 completely unbent, the radius becoming of such great length that it would be impracticable to proceed 
 further; we shall be able, however, to show that the estremitics of the successive radii ran be drawn, 
 and the point at the extremity of each re(iuired arc can be found, without neccssarir drawing the 
 entire radius, and without requiring to describe the arc itself. 
 
preld:inary argument. 
 
 PART SECOND. 
 
 Definition? and Postulates. 
 
 Let BAG be any anisic contained by two lines BA aud CA 
 With the centre A and radius AU, describe an arc lutereejitiug AC 
 at E. Bisect the arc BE at the point D, and j"in AD. 
 
 iKf. 1. — Where the exprcs-ion • bi.<i:ct the anijrle BAE at the 
 line AD,' is used ; the reader will understand the mcaninjz to he that 
 the above operation has been pcrfurnied, and the j^sitiou of the line 
 AO so obtained. The lino ..f biseclion (,.VD.) may be of in i tinitc 
 length, i.e.. the arc may bo described with any part or with any [to- 
 ductiou of the Hue AS. 
 
 Prf. 2. — Where the expression ' biiJeet the andu BAE at the point D,' is used; tlie point of 
 bisection of the aniile may bo a point at any distance from A, but must be somewhere upon tho 
 line Al\ or the production of AD. 
 
 Dtf. 3. — 'iVe term tin: point B, 'the original point,' the point D, 'the central point,' and the 
 p.iint E, 'the urniiniil puint,' nf tho arc BK. We term likewise tho line AB. 'the oriu:inal radius,' 
 the line AD, ' the central radius,' and the line AE, the terminal radius,' of the arc BE. 
 
 Ihf. \. — The point at which the radius intercepts the arc, we term ' the point at the terminal 
 extremity of the radius,' as the point B at the terminal extremity of the radius AB. 
 
 Def. 5. — An indefinitely small part of the radius terminated by the point at the terminal extrem- 
 ity of tho radius we call ' tho extremity of the radius,' as/"B. the extremity of the original radius AB. 
 
 PCSTULATES. 
 
 The demonstrated and est.iblished Elementary propositions and theorems of Geometry, we assume 
 to be admitted by the reader without reference. 
 
 The above dcfiuitioiis will be raaiic use of in 'the rre'iminary argument' enly in a few inslanees, aud 
 are here iuserit-d more particularly ua suggestions. 
 
; THE REQCJISITIOiV. 
 
 Pro?, a. (Analttioai,,) 
 
 An arc containing tlie dofinitc fraction of a circle, and a strai^lit line forming tlie tangent to the 
 arc being given, it is required to cut off from the straight line a part thereof equal in length to the 
 given arc. 
 
 Paop. B. (Stnthetical.) 
 
 It io required upon a given straight line to describe an arc such that the arc shall be equal in 
 length to the given straight line, and shall contain the definite fraction of a circle of definite magnitude 
 
 The demonstrations to these propositions, wliiyh it is one of the leading objects of tliis work to 
 furnish, will be properly duferrod until the ficts upon which the demonstrations must be necessarily 
 based, have been placed in some degree of order bcfire the reader. It will then appo.ir th it these 
 two propositions may be satisfactorily solved by api)]ying to the investigation a devclopcmcnt of that 
 process indicated in prop. II, namely, that of unbending an arc into a straight line, or the reverse, of 
 bending a straight line into the arc of a circle. We shall, however, be enabled to exhibit and apply to 
 the same investigation a different and distinct fundamental proces.*, namely, that of ' rolling an arc 
 geometrically upon a straight line.' We shall be able to demonstrate that the results obtained by 
 the several methods of the one process are strictly in agree.uent with each other, and also in agreement 
 with the result obtained by the other process, and we shall be able to show that the general result 
 obtaiiiud in either w.iy is strictly in agreement with the result of that process (of which one of the 
 methods was briefly oxplaintd in our appendix to Prof. De Morgan's statement) which, fur the sake 
 
 of distinction, we will term the " trigonometrical proees.s," and which gives the number 314159 
 
 as the ratio of a circle to a unit of diameter. But although proper to defer the formal demonstration 
 until we have at lesist endeavoured to establish those primary theorems which, if established, will enable 
 us, whether the quadrant is resolved into the straight line or the straight line is bent into the quad- 
 rant, to show in each successive stage of the process the important rclation.ship of the parts to each 
 other; yet we may hero direct the attention of the mathematician to the significance of some of tlie 
 most prominent lines in the figures belonging to the foregoing propositions ; we may, moreover, partio 
 ularly indicate some of the leading facts upon which our demonstration will be hereafter based,— 
 and we may also take advantage of the irregular character of this preliminary argument to call to 
 our aid and apply such indirect evidence as may assist us in establishing the actual relationship of the 
 lines in question to be in fuel that which wo have assigned to them. 
 
 B 
 
18 
 
 III oidOi- to y'w a Jibtinctlv uljo'ivf elm dor to tlii> lart il' cur ar^iiiiKut tlio loailint; 
 niiubitiuii miiy Le pii.-' iiicd ia the loUuwinj; t'lTin ; — 
 
 (iNTLUIiOU \ riVE rilnl'. X.) 
 
 An arc bi in" tlic ilciiiiitc fraclioii nf a circle', iiml .1 straight lino rnrniiiii; the tan'^'iMit In tin- arc 
 bcini: L'ivon ; it Is ip(iuiiTil to cut iff trum llic -tiai-iit line a pnrt tlK'not'eiiual in Icngtli to tlie given 
 
 arc. 
 
 Pi^r. 4 — I'^niui tlie ccntif A and with the radius AB ilcscribc iho i|Uiiilr:int HinC. Bisect the 
 (lUaJraiit BinC at tiic point lu. 'J'hnitij;ii 111 draw the (<■ tMlit AD, and draw also lliu tangent BD, of the 
 arc Bni. Lit the arc Bni, so cut off from the (|u:idr,int bo the given an — it is roijuirod to cut off 
 tVoni tlio straight line B[). a part thorouf 0((Ual in lonLith to the arc I'm. 
 
 Produce tl.c radius HA ihrcuuli .\. and make BAE tho produclion of BA, twice the longili of 
 BA. — 1 roni K through in dr^w El', interccpliiii,' tho line BD at f — Bisoot the radius BA at the 
 point e, Join Do, cuttin;,' Kt', at the point y. Throui;h y, ami at right angles to BD, draw <!jll, 
 iiiteieopting tho lio" BD. at tho point II. Shall Bll tho part so cut off from the straight line BD bo 
 the rc((uired line o(|ual in length to tho aie Bm ? 
 
 (I.NTERUiMiATIVK I'lli'l'. V.) 
 
 It is rcijuircd ui)ou a given straight lino to dcs( rilie an arc, sucii lliat the arc sliall be oi|ual in 
 len'.'th to the i;ivon straight lino, and .sli dl contain tlie definite fraction of a circle oi detinite magiiitudo. 
 
 Fig. J — .Let BII bo tho .:ivoii straight lino, it i.i required upon Bll to describe a definite arc ccjual 
 in Icngtli to BII. 
 
 From the pi int B, at one extremity ot the given straiglit line, dr.aw liie perpendicular 
 BK, equal in length to BII, and from II, at tho opposite cxtromily draw the perpendicular IDl, 
 also equal to Bll ; join K(.', and join also BG, Divide the lino BII into three 0(iual parts by 
 intercoption of the points R and S, so that the parts, BU, RS, and SII, .shall be equal each to eacl). 
 I'roui the centre K, with tho radius KB, describe tlie quadrant Bid. Bisect BIO at 1. From B, 
 tlirouah 1, draw B(), inlercoptiiig tho line OH at O. From 11 and parallel to BO, draw Up, inter- 
 cepting the line (ill at p. From S and parallel to Rp draw Sy, intercepting tlie line OH at y. 
 Produce the lino BK through K, anil produce the line Bll thrriugh H. Through tho point p, and at 
 right antrles to BO. draw tho line ApD, iuteroopting tho production of Bk at A, ami intercepting the 
 production of BII at D, With the centre A and the radiu.s AB, describe the arc Bin, intercepting 
 the line ApD. at m. Shall the arc Bm. containing 45 degrees, so described, be equal in length to tho 
 given straight line Bll ? 
 
 But, again, produce tho line BK through K, and produce the line BH through II ; and through 
 the point y, and at right angles to the line BO, draw the line Eyt intercepting the production of BK 
 at F, and intercepting the production of BH at f. With the centre E and the radius EB, dcBcribo 
 the arc Bn intercepting the line Ef at the point n. Shall the arc Bn containing 22A degrees, so 
 dc'cribed, be also equal in length to the straight line BII ? 
 
1» 
 
 It i.s evi k'lt that [.lop. Y is Ji'i'Cii.lent on iiro;>. X ; s • tliat [rop. V i.-* nvi.lily clouionfilr.iWc if the 
 quostinn subinittcil in prop, X ciin be answered in ilio iifliriuiitivo, tliiit is, if it can be sliown tliat tho 
 line IMI ((if pmp. X) is ('■|ii;il in li'MLflli to the iirc Mm. For, in lutli fijinres tlie I nc Kf U ;it ii;.'lit 
 aiijjles to tlip line Bni, and nl.-o in both fi^nres the jioint _v his 'ets the lin • pll. The arc Urn of Fig, 
 
 i.s rcconstrupted in Fi^;. fr'mi tlie straight line lUI, Sn'., kn. 
 
 It iii:iy hi! here ol served — 1st, That the length of the line lUI niu^l evidently iippro.xiuiate some- 
 what elosdy (if it be ii >i pijual) to the length of the arc IJni,— 2nd. That the line BlI has evidintl)- 
 «n important rclation>hip to the other parts of that gemral tlgure whieh is the guhj.-ct id' investigation. 
 The ijuestion whether the line IMI i/nm or iln,3 imt in /mi cipial the are I5m, we will snb:iiit to the 
 test of letn.'il exjH'riinent by applying the proces.i of unh<iuJinij tlf err itnil/, after the relatiuii-hip of 
 some of the other parts of the gencr i! 6gurc has been brii'fly investigated. 
 
 We propose now to co isidcr the si^nifieance and analytic d v.ilue tif the line Kf shown in Figs, 4, 
 and ■"), l!ut, in the first jilace, eon>i(lering one of tlio dilVienlties of this subject to consist in its ex- 
 treme coniplici tion, in such wise, thit a number of cases endeivour (so to speak") to crowd together 
 upon the ni:itheii>alician's attention, wc will submit the three following simple propositions with the 
 Cxpre->3 object of isolating that c ise whieh we wish here to bring under i)artieular eon-,ideration. 
 
 Prop, a ( Fig, S.) —Let mC be a line drawn fmni the p"int in, and let CA be a line drawn from 
 the point C at right angles to niC, ami let map be a lino driwn from the same point m, cutting the lino 
 CA at a, and together with mC containing an angle of -15 degrees. It is reijuired to draw a. line from 
 the piiint ni, which shall cut the line CA, anil shall together with the line mC, contain an angle of 
 22i degrees. With centre m and radius mC describe Cbd intercepting map at d, bisect Cbd and 
 through the point of bisection b draw mbo. — nibo is manil'estly the line re(|uired. 
 
 I'rop. b (Fig. VIII,) — Let mC, c\, and map be a repetition of the same lines (as prop, a). It 
 is rt(juired to bisect the line Ca, (cut off from CA by the line map) and through the jioint ul' bisection 
 to draw a lino which, if produc d, .shall intercept a production of the line Cm, through the point m, 
 and -hall together with the production of Cm contain an angle of 'S2\ degrees. L^'t f be the jniint of 
 bisection of the line Ca, From f draw a lino at right angles to map, and intercepting map at g. 
 From fC take fh equal to fg; and from the point h draw a line at right angles to IV and intercepting 
 map at n. From n through f draw nf'i|, nl'i) shall be the required line, Because Ig is perpendicular to 
 ng and bec.iuse fh is perpendicul.ir to nh, and fg and fh are equal, the triangle ngf is simil.ir and equal 
 to the triangle nlif Now ah, which is part of aC, is at right angles to hn, and nC is at right angles 
 to Cn» ; but pain, of whicli gn is a part, together with Cm, contains an angle of 45 degrees and (since 
 hn is parallel to Cm) gn together with Jin also contains an angle of 45 degrees; therefore (as fg and 
 fh arc equal) qfii together with hn eontain.s an angle of 22A degrees, consequently if qfn be produced 
 through n, and Cm be produced through ni, until the two lines so produced intercept each other, the 
 production of qfn will together with tiie production of Cm, contain an angle of 2'2\ degrees; where- 
 fore qfn is the line required, (Corollary — Ilencc it follows that if a straight line be drawn from h to 
 g, nfq will bisect that line ; and also if with the centre n and radius nh an arc bo described, terminat- 
 ing at the interception of the line iiga, nfq will bisect that arc,) (See I'rop, a). 
 
 Prop, c (Fig. IX,)— Let mC, CA and map be the sanie lines (as in prop, a) repeated. It is 
 required to draw a line from the point m whieh shall cut thj line Ca, through the point of bisection of 
 
20 
 
 tie line Ca (i. f. i>'vill bi!«cct Ch\ Kroin ii ilrnw nH ;it ri^ht aii>;lo(( to Ca arul I'^iuul in lon};tli to Cm, 
 pr Juce I'M tiirou^li a .mil m.iki i)iiH twiic tin- Uii^tli nf l>.i. J. in niB culling .ic at f.- ml H A\\\\\ 
 be the line n<iuircil. IKc.ium all i.-> muiil to uiC .ind <'u is at ri^:lit anj:li"!< to niC iind al.-o at ri^rlit 
 angle* to all, and kcauM- nit is part of nit'H, the lint.' aC biHicto tlif line rniB nmi tluirlori) tlio tri- 
 «nj:Ic niR' is ><iniiliir and ri|nal to tlio trianj;Io Mta, and tht; b.isc t'f ufliie cnn' imijual ti» the base I'li 
 ot tie ullier ; wlicrcfure tiic lino ml 15 liiwcin the line tiC as ri'c(uiri'd. 
 
 Eil«rinicntal examination of the i;oneral tiirure with Hjiopial nfcreiiee to the line Kt'. 
 
 Fi(5. 1 1.— .*"hi'W.« n ri'petitidM "f thr rndinnntary liu'nii-, ijie >4|uare ABlH.'. the (|iiailraiil hni*', 
 and the fecmt Ai>. I'rciin H, lhi()ii,i;h A, prodm-o the radins UA, and niaLc IJAK twice the Icnmh 
 of BA. Fnini K. throuu-h m, draw Kf, intiTee|ilinj,' I?D, at f; join UC, ciittini; Al) at S. 
 
 From BD. tlin'U.-h H. draw YSj;, at rii,'ht ■m^le'* ti BD. euttini' Kfat !>, an I intererpiin;^' AC at 
 g. Frum b, at ri^ht anj;les (>) AB, draw biF, cutiinj,' AD at u, and iul« rceptiiii.' AB at •!. With 
 centr- B and ra(liii> M.I, deseribi' the arc Jl> I', interc'iiiin;; BI>, at I'. Juin #11'. The le>.ser triangle 
 BI'J is similar to the greal<r trianj^le BDA, trnd nianil'.stly I'D i-'juals JA. But bi-raiisc KA ia 
 e^inal to AB. whieh is ei|Ual to ;*tn, KA is also eipial to Am, and since the les-sc-r triani;le bSni i.s simi- 
 lar to the jrreater trinnule KAni. b.'* » (|uals ,*'iii. From m draw w'/,. at ri;.'!it an;:lof. to i; Y, eiitlin^' .SY 
 at V, and iiiterci'ptinj: AB at l. Now, b^ cause AD i> fwrpendieular to CB. if a line int is drawn iVoin 
 m, int. rc< ptin.: AC at riuht anjrles, At mn.st e(|ual Si ; and since inS is cfjuul to )>.*«, and .lb ei|Ual to bv, 
 it follows tiiat AJ is efjual to mv ; and thcr<lnre tuv also cfjuals I'D. Produce Jb throiii;li b, until 
 Jbr intercepts CD at r. The li: o Ef bisects Jbr at the piint b . I'rop. bj. The line Ff cuts AC at 
 the joint e; from e draw eu, perpondicular to AD, iiml from m draw mt, pirpendicular to AC. ct 
 e<|Uals eu , that is. if with O'-ntre m, .ind radius mf, ^;n are tu is doeribid, the line mbK bi-oets that 
 arc (/'>/■'.//.;/•_// Pm^K h.) Fioni P, periioudieular to BI), draw I'l, euttini; AD at \V, and inlerecpt- 
 ing Jr at d. The liiic dJ is the base of the trimje P.Id, and du fpart of dJ ) i.s the base of the triani;!c 
 Wud ; th'/ iKiiiit b bistots du. .I^in ED. cuttint; A(.' at l', ami j'lin al.-o \Vb , Wb is parallel to Dg. 
 Bi'^ct dJ at p aii'l jiiiii Pp; Pp is also paralkl to \^^ ( betauje DAC, Wud and P.Jd, are .similar 
 triangle^ and consefpn-ntly Di.', M'b, ami Pp are similar lines.) Throuuh W, at ri^ht angles to CD, 
 draw Vo, interecptint: JP at n. The k>s.r tri.m-le PoW is similar to the irreater tri.oijile PJd, of 
 which it is a pap. and the tri.m;_'le DWV is miMilVstly similar and eijual to the trian^;le I'OW, 
 ED cuts dp at s. The Iriaiifxle DsP is siuijl:':- to the trianjilc DKB, (of whieh it in apart), but 
 DA bi-.-ets EB, and therefore DW .part (if DA ■ bi^eets xP.* 
 
 Fi;.;. 12 is a ri'[K;'.ition of the rudimentary Cj:ure. 
 
 With centre E, and radius EB, describe the arc Bn, intercepting; the line Ef at the point n. 
 Join I'n. With centre B, and radij- Bn, describe the ark TUK. Draw Kc, at ri-ht nicies to AB, 
 and join cT. at right anirl's to BD. euttini; the line \D at y The Ls^ieT trianule KTB is similar 
 to the prcater triangle ADB, of which it is a part. Throu<,'h y, at right angles to cT, draw Uyq ; 
 intercefftini: CD at T, and intercepting KT at rj. The lesser triangle DyT is similar to the greater 
 triangle DAB (of which it is a part), and the three triangles Dyl', DyT, and Tqy, are manifestly 
 similar and eijual, each to each. Join ED cutting cT at x. The triangle DxT is similar to tho 
 
 Vote.— The exjifrimentnl eiamination oftlie gcnenl figure will be currieJ somewtiat further at the coucln- 
 li'jn of the ai'peudii to tlip ireliminary org' meat. (.Seepage ). 
 
21 
 
 trion^'le DKR; but DA him'cts KB; tlnTofoic I>y bisects xT. (Bisect cK at n, and join uT ; aT is 
 inunifi^lly piir:illfl fn i.'I>. » 
 
 Sciliii.iiM. — Am tlu! circmnli'ii'iicc (pf a circlo liiis been iii.itlu'iimt!o;il!y iiiiriNUrrd, and tlio 
 niPiiMiircinciit (>f ccrtnin of tbi- liiics bt'luiij^iti;^ to fi;^'iirc,s X and XI is hIsh known, the data obtained 
 Irniii tiic jin 'Cecil II;.' < finiiiniitiiiii iiiiiy bo applied to iiscurtaiii ii/i/ifirnu'ifii- /ij tlio puint in tbe lino 
 IM>, at wbicli to cut (.J a part ibcrcid' ( iiu-asuriiig (Voin tlic cxtri.inity B,) c"iual in icii;^'tli to tbc arc 
 Bill. Tbc line BT iw tiic clioid of mi afc of 2'2\ dc^rceH and tiic loiii;tb nf wbicli ic!;itiv( ly to tlio 
 railiiiM KB is ibcrdiiro knnwii. I'D, ( Fij;. l<b) is cc|Uai to in\', but iijZ is tin flw, uud vZ is bulf 
 tbc tani;cnt of an aic nf 4r) degrees described with tbe radius AB, tiieref'oro I'D equals tbc leti^tli of 
 tlie sine mhnii balf tbe i(.nj,'tb of tbe tant;i'iit of tbc arc Bin (of •!') do;;ret!s). Tlio Iciejtb of BO, 
 tbe taiii^int of tlio arc Bin, inini'ii tbe bnutb of I'D -ives tlie leiigtb of BP, wbieb is greater tban 
 tlie fi|nivali;iit to tlie luatbematica! nicasureineiit of tbo aio Bin, and .-iiice tiic lonf^tb of tbc clmrd of 
 any arc is b'ss tban tlie len^'tb of tbe arc itself, it Ibllows tbat tbc point in tbc line BD at wbicb to 
 cut ' ff tberefroni ii r|iiantiiy of lenL'tb, (iiieasuriiif,' from B) enuivaleiit to tlio len^tli of tbc arc Bni, 
 (assuiniiiL' tbu inatbeinatical nica.-ureiiK nt correct) liiu^t necessarily fall between tbe point P and tbo 
 point T.—I'i- XII. 
 
 Ni)TK.--Sonie ol" our readers tiiay like to liavc tbis applioitioii of tbe data in aritbiui tic d form. 
 \Ve take tbe radius AB - 1 . Ilelerriim to a inatbem itieal table we tind tbc sine of an arc of l\\ 
 dej:rces M !•.")<•>».'», and because tbe cbord of an arc of 2-i dcj;rec.s P((Uals tbe sine of an arc of 11^ 
 decrees, if tbo two arcs lie of equal len'_'tb (Corollary prop. 1 i and bee.iuse tbc r.iduis KB is twice 
 tbc lengtb of AB, tberelore "lOjO.SO x 4 'TMCJ J(.i .• tbe len^tb of tbe cbord of tl.c are BN (^of 2L'J 
 de;:reer-r Tbe sine (niZ) of tbe arc of 45 de.irroo.s- '707107. And balf tbc fanj;ent (vZ; of tbc same 
 arc^ Tmioimi. Ti.er.'fore PD 707107 ■• TiOdnO '•2071(»7, and BP - POOdO— ::O71()7--'7!:i2.S03. 
 Now tbc uiatb.'iuatieal nieasureiiieiit of tbc circunifereiu'c of a seniicirele to a unit vi radius — 
 
 3'1I1.")9 and coiLseijuently tbc arc of 45 degrees ^ — = "^^'jo!'. It tberelore follow-', if we 
 
 assume tbc correctness of tbese measurements, tbat tlii? requiicd point must f.!! b'tweeii tbe point P 
 and tbe point T. and i little nearer to tbe point T tban to tbe ["■iiit P. Tbe distance of tbc point from 
 the point P to tbe di.-tance from tbe point T being nearly in tlie proportiLn of 3.2. 
 
 Tbe particular si;.'niticaiico of tbe line in ((Ueslion in.iy be lure exeniplilied by -bowing tluit we 
 arc now in a position to answer, for instance, tbc following re(|Uisition. 
 
 ( Prop. 3 a.) — 1 'poll n given straigbt line it is reijuired to de.>cribe an arc containing '121 degrees, sueh 
 tbat tbc cbord of tbc arc ."-ball be Kjual in lengtb to tbc given .straigbt line. pig. 14. Let BT be tbe 
 given straigbt line, it is rc(|uired to describe tbc arc upon BT. Prom tbc point B at one extremity uf 
 tiic given straigbt line draw tbe perpendicular BK ei|U al in lengtb to BT, and fioiii T at tbe ojiposiio 
 extremity, draw tbe jicrpendieular Tc also cijual to BT. Join Kc. Witb centre K and radiu.s KB, dc.>eribe 
 tbe quadrant Bic. Join KT bisecting Blc at 1, and join al-o Be bisecting KT at tbc point r. Bisect 
 tbe line Kc at tbc point p. Join pT cutting Be at q. From q draw i|t at riglit angles to cT, and 
 intcrccptinp; cT at t, and from qt cut off qs equal to qr. From .s draw sy perpendicular to qt, and 
 intercepting KIT at the point y. Join sr, (and join also qy bisecting sr.) From B througb I draw 
 Blni at right angles to qy, and intercepting sy at m— produce BK and produce also BT indefinitely, and 
 
tlirou-li 1)1 (li-w a line paralkl I-:- KT. and pRidueiMt in both iliroction? iiitcrcopting th-i fcc-lacljon 
 ufBIs. at A. anJ iiitircoj.tin^' the j.roduction of IJT at D. Witli the centre A. atij tvllt* AB de- 
 scribe ail arc ii.terecptin-.' the liuo AmD at in. Produce the line BA throui;h A, mafcitt.' BAE twice- 
 the lini;tli of UA. From E driw Kn tliroujli the point ni. With centre E and radiiu LB descHbe 
 the arc Bn iiitoreoptinu' the lino E li proluced throii;_'h ni at the point n — Bu .shall be the r«ijiwi*darc. 
 Because Bni i> an arc, contiiniiig 4-") de-pTecs, and because the radius EAB is twict! the len^«li ^■f tbe 
 rolius A!', and Eiiin i- a pp-wjijctioa of Em tliruuyii lu. therefore the arc Bn i- an arc CJntjiLiing 22J 
 ile^'ree-i. 
 
 (Fig. 15 is a rejictition of tiiosv lines bcloDjring to Fig. 14 necessary for the democc'ii-i'ktn) 
 Join Bn. tlie chord of the an? Bu, and produce the line Hn through n, intercepting cT at W . Froiii 
 n at right angles to Bn draw a line, intercepting BD at c, and cutting WT at x. Join Bx. 
 
 With centre B and radius B."*. describe the arc .Sabd, intercepting BD at d. l>raw tho lin* Bajbl-eel- 
 ing the arc Sabd at a, the line Bn bLstxMing the hiif arc abd at the point b. and the line Bs !t-*rt:ng tbe 
 fiictional arc bd. Because Bu i; a production of I'p. and BT a production oi' Bd, ari'I l»eaFii<£ j!w linj 
 Bx bisects the arc bd. ther'.forens I? c>jual to Ts. But nx is perpendicular to Bu and Tx is ptirjiecai'fTilar 
 to BT. and the triangle Bxn is similar to the triangle Bxt. consc.|Uently, the ratio oF thi lisii Bo to 
 the li le Bx is equal to the ratio of the line BT to the line Bx, and two lines which have efjiajil ratios 
 to a third line are cfjuul to each other, therefore, the line Bn is equal to the BT. Wh-^rit' ft jLe arc 
 Bi!, containing 2'2\ degroi* h-..« bt-en de:<ribed upon the given straight liuo BT. an>l it hm been 
 demonstrated that the chord of the arc Bn is ajual in length to the given straight line. wSak-ii wss 
 required to be done. 
 
 For the purpose of further illustrating the relationship of the line Ef to the g'^QcriE figoiv. we 
 will suppose the s:ime requisition refit-ated. 
 
 (I'r'p. 3, b, fig. ICi). Lot BT be the given straight line. li i.- required to df.scriW ii>e arc 
 upon BT. From the centre B. with the radius BT, describe the quadrant TK, draw Kc ai light 
 angles to KB. Join Tc and j lin aL?o Be, biscctiug the arc TK at I. Bisect the arc IT ai d und 
 join Bd. Bisect the li lif arc dl at a. and biiect the half arc dT at n. Throujrh the piaijD and 
 a. and at right angles to Bd, draw dj, produce the line BK through K and the line BT iLroui:! 
 T, indefinitely, (see note) and produce tlie line na in both directions, intercepting the j ^lo'tskiD of 
 BK at E. and intercepting the pr.>]uction of BT at f With the centre E and the radias EB-ieKiibe 
 an arc intercepting the line Eanf at tlie p-Zmt n. The arc Bn shall be the required are. 
 
 Bisect the radius EB at the [-oiiit A. Witli centre A and radius AB describe tte qnsulriiiii BmC, 
 and bi.-ect the quadrant at ni. B-.rau.-H.- Bm is an arc of 45 degrees, and because the ridtos EB is 
 twice the len;_'th of the radiu- AB. and the line En is a production of the line Em iL^jJaifc nj, 
 therefore the arc Bn is an arc containing 22}, degrees (prop. 1). Again, because the line Bn is a 
 radius of the same are of whi^ h the line BT is also a radius. Bu is equal to BT. Bat iLe iioe Bn 
 is the chord of the arc Bn. Wherefore it has been demonstrated that the chord ot the art Ba b 
 c.:iual in length to the given straight line BT, and the arc Bn has been shown to contain 224 Jejrees, 
 and has been describetl upon the ■.'iven straight line. Q, E, D. 
 
 (Note, — The construction and demonstration to this prop, may be again varied thos. Jo'a nx, 
 cutting the line Bd at the pdnt m. From the point m, through the line Be, and at tvAt angles to 
 
23 
 
 Be ilniw !ii.\. intfrcoptin:; tlie iToJuution of BK at A, jiikI cuttiiii.' the line Ce at .S produce na 
 through a, iiitorcoptinir tlie proiluction nf I5K at K, ami li-oin S, thron.'h Kc, and at right angles to 
 Ki', draw a lini- iiitcrceptin;; naK at b. With the ecntro h and radius iiS, describe the '(uadrant Soc, 
 bisect Si-e at <>. : iid bi^x-t ^'o at the point p on the line uaH. With the centre K and r.idlus KB, 
 describe an arc. i itcrruptinj; the line naK at the point n. 
 
 Bicause the liue bo bisects the quadrant Soe, aud the lino bp bisects the arc So ; bp, tn.-i.tlier 
 with bS contains ai> anirle of 2-h degrei'S, but bp is a part of En, and bS is parallel to EB ; thfrefore 
 the line En, togiihcr with the line KB contains an angle of 22^ deirrccs, &c., kc. It may be also 
 nuted, that since uiS is a part of iiiA, and bp is a part of Em, and bS is parallel to EA, the triangle 
 bSin is similar to the triangle KAM. With centre 8 and radius .Sb, desciibe an arc intercepting the 
 pi'int m. sfincc Sb is equal to Sin, AE is "qual to AM. But AM is a radius of the arc BJF, ol 
 which arc AB is also a radius, therefore the radius EB is twice the length of the radius AB. 
 
 ( I'rop. 4 >. Upon a given straight line it is reijuired to describe an arc cont'iining 45 degrees, 
 fcuch that the differeucc between the sine of the arc and half the radius of the arc, shall be equ; 1 to 
 the diflerencc between the tangent of the arc and the given straight line. Fig . Let BF bo 
 the given stniight line. It is required to describe the arc upon Bl*. With centre B, and radius 
 Bl*, describe the quidrant I'il-J. With centre J aud radius JB, describe t!ie qua Ir.mt Bid. Join 
 Pd. Jd. and Bd. Bisect the quadrant Bid at I. Join Bl, and produce Bl through 1 indefinitely. 
 BLscct the given line Bl', at h, and with centre h, and radius hP. descrilx; the ark Po, inlcrcopting 
 the production of Bl at o. Bisect the arc Po it n. Produce BJ through J, and produce BP 
 through P, indelinittly. Through the point n, and at right angles to the production of Bl, draw the 
 line fuE. intercepting the pr duction of BP, at f, and i!iterc''pting tlu pr o luction '>f BJ at E. 
 Blse'Ct BE at tlie f«:iiut A. Willi centre A, and radius AB, describe the arc BmC* BuiC shall be 
 the re.|uired arc. i^The demonstration to this i'rop. is substantially cmitaii vd 't the preceding 
 examination) see page 
 
 EXPERIMEXTAI. Ex.^MISATKiN — it\illt!n>ial.) 
 
 The lino ED. Fig. 13 is a repetition (if the rudiment uy liguie. Join F-D. The line ED 
 bisects AC, at the point g, (Prop. c. ) The base of the triangle DAC is bisected by the line ED, at g, 
 therefore, since the lines CD and gY are parallel to the line I'^A, the three triangles EACl, Dl'd. and 
 gYD, are -iiuilar and equal, each to each. Bisect CD at i, and join Bl, cutting ED at k. Through k. 
 perp^'ndieular to BD. draw ckQ, intercepting BC at c, cutting AD at y, and iutcrcopting BD at 
 Q. From c. draw cll at right angles to AB, intercepting AB at II. Join IIQ. The triangle BeQ 
 is similar to the triangle BCD, of which it is a part, and Bi bisects CD ; therefore Bk < part of Bi) 
 bisects ctj. The triangle DkQ is similar to the triangle DEB, of which it is a part, and DA bisects 
 EB; therefore Dy (^part of DA) bLsects kQ. Through y, at right angles to kQ, draw Zyp, inter- 
 
 • Note. — Or oiherwise : — Tlirougb tlic point n, lU riglit tinglog lo the prdduetioo of Bl, driiw fnni, iiitercppting 
 the productioa of BP at f: and imerccpling the pi-oduction ef lil at ra. From ni, paralKl lo I'J, draw niA, 
 intercepting the proJuction of UJ ut A. Willi centre A, and ladiiii AH, ice. 
 
24 
 
 cepting CD at Z, an J iiiturcoptiii<; IIQ at p. The triangle ZyD is nianifustly similar and equal to 
 the trian^lo y [>(.}. (Beoause DQ is part of DB, and DZ part of DC, and because Qk is equal to Zp; 
 and Dk iutcrcopts the arc BniC at k, and Qp intercepts the same arc at p ; therefore the arc Bp is 
 equal to the arc Ck ; and the ark pin, equal to the arc km.) 
 
 TiiKOttEM. — If the quadrant of any circle be bisected and the secant to tlic half-arc be drawn through 
 tiip pfiint of bisect i iii, and the tangent to the half-arc be drawn touching the arc at tlie original point thereof. 
 And if the original radius of the (juadrant (see drfinitions) be produced through the centre, and the radius 
 so produced be made twice the length of the original radius, and a line bo drawn joining tlie point at the 
 extremity of tlie radius so produced, and the cxtron)ity of the tangent to the half-arc ; then shall the 
 sine of the are cut off Iron) the quadrant by the line so drawn have the same ratio to the tangent of 
 the half-arc, as the ratio which four has to five ( / c 4 : 5j, Fig. 17. Let the arc Bm(J, described with 
 
 the centre A, and radius AB, be the quadrant. Hi>ect BniC at the point m througli ni draw tlie 
 
 secant AD, an I the tangent BU to the half are Bni. Produce the radius AB through A, and make 
 EB twice the length of AB. Join ED, catting the quadrant at the point k. Join CD, and bisect CD 
 at the point d, Join Bd, cutting the quadrant, and intersecting the line ED at the same point k. 
 Tlirnugh k draw cQ perpendicular to Bl), an(l intcn-cpting BD at Q. From Q draw Qll parallel to 
 the secant AD, and iiitereoptinir AB at H. From II, at right angles to AB. draw He, intercepting 
 cQ. at c. Because EB is twice the length of BD, and BD twice the length of dD, the triangle BdD 
 is similar to the triangle EDIJ. But the angles BDd and EBD are both of them right angles, 
 theref'iro the line Bd is at right anjles to the line ED. Now since HQ is parallel to AD, and Ho 
 parallel to AC, the line Bd bisects cQ at the point k...and the line Bk (part of Bd) has been shown to 
 be at right angles to the line kD (part of ED) therefore tlie triangle kDQ is similar to the triangle 
 BkQ : and as kQ is equal tu the half of BD, DQ is also necessarily equal to the half of kQ. There- 
 fore BQ is equal to twice kQ. and kQ is equal to twice J'Q; whereiore the line BQ has the same 
 r:.tio to tlie whole line BD as 4 5. Draw ko, the sine of the arc Bk. ke is manifestly equal to BQ. 
 
 (Prop. 5 ) Upon a given .straight line it is required to describe an arc containing 45 degrees, 
 siic'i that if the tangrnt of ihe are bo divided into five equal parts, four of these paits shall be together 
 equal to the given straight line. 
 
 Fig. 17. — Let BQ be the given straight line, it is required to describe the arc upon BQ. 
 
 From the point Q, perpendicular to BQ, dr.-iw Qe, ecjual in length to BQ ; and from B, perpendicular 
 to BQ, draw BH, also equal to BQ. Join Be and IIQ. Bisect cQ at the point k, and join Bk. 
 Produce BQ through Q, and produce BII through H. Through k, at right angles to Bk, draw kD, 
 intercepting the production of BQ, at D. From D, pir.dlel to QH, draw DA, intercepting the 
 production of BU at A. With centre A. and radius AB, djscribe the arc Bm, interoepting AD at 
 m. Bm shall be the required arc. The demonstration to this solution is substantially contained in 
 the demonstration to the Theorem. 
 
 \\'e will now submit a proposition having a more general purpose but which may, at the same 
 time, serve to exemplify the relationship of the line ED to the general figure. 
 
 Prop. 5. Let a straight line A be so related to a second straight line B, that if A be divided 
 into five equal parts, four of such divisional parts taken together shall equal B, (that is, let A : B : ; 5 : 4) 
 
26 
 
 and let the second line B bo so related to a third lino C, that the ratio of B to C shall be the same as 
 the ratio of A to B, (that is, let B:C': ;A;B) and again let the ratio of C to a fourth line D be the 
 same as the ratio of B to C and so on, (that is, let A B : : C : D ; : D ; E : : E : F, &c., &c., : ; Y : Z : : 5 ; 4.) 
 It is required from a given straiglit line equal in length to A to cut off a part equal in length to B, 
 and from the par. so cut off equal to B to cut off a part equal to C, and from the part equal to C, to 
 cut off a part cqu'il to D, and so on ; and it is required to demonstrate that the process may be so 
 continued until the part last remaining shall be of less length than can be divided. 
 
 Fig. 22. Let the straight Hue AZ, drawn from the point 3, he the given straight line. It is 
 required to divide AZ ia the manner stated above. From the point Z, and perpendicular to ZA, draw 
 the line Za, equal in length to ZA. With the centre a, and radius aZ, describe the quadrant ZmC ; 
 join aC and AC ; join also Aa, bisecting the are at the point m. Bisect AC at the point d. Bisect 
 also Ca at g. Join Zd, and from g, at right angles to ZD, drav? gA, cutting ZD at the point K. 
 From K, perpendicular to AZ, draw KB, intercepting AZ at B. The part BZ, so cut off from AZ, 
 shall have the required ratio to the whole line AZ (that is, BZ shall be to AZ; :4:5). Bisect Ad at 
 the point e, and join Zo. Because AC is eqtial to AZ and the point d bisects AC, the half of AZ 
 is equal to AD, and because the point c bisects Ad, the half of the half of AZ is equal to Ao, there- 
 fore if AZ is divided into four equal parts, each one of those parts is equal to Ae. But the lesser 
 triangle BZK is similar to the greater triangle AZD, of which it is a part, and therefore the half of 
 BZ is equal to BK, and the line BZE, which bisects AD, also (manifestly) bisects BK at J, there- 
 fore if the line BK is divided into four equal parts, each of these parts is equal to BJ. And again, 
 the lesser triangle ABJ is similar to the greater triangle AZa, of which it is a part, and since AZ is 
 equal to aZ, A13 is also equal to BJ. But it has been shown that if BK is divided into four equal 
 parts, BJ is equal to each one of those parts, therefore AB is also equal to each one of the four divi- 
 sional parts of BZ, therefore the whole line ZA contains five equal divisional parts, of which parts AB 
 is one. Wherefore the line BZ, cut off from AZ, contains four equal divisional parts, each of which is 
 equal to each of the five equal divisional parts contained by the line AZ. Again from B draw Bb, 
 parallel to Aa, and intercepting Za at b. With the centre b and radius bz, describe the quadrant znc 
 bisected at n by the line Bb. Join cb and cB. Bisect cb at the point g. and join Bg. From 1, 
 the point at which Bg cuts the quadrant znd, draw IC, perpendicular to ZB, and intercepting ZB at 
 C. The fractional line ZC shall have the required ratio to the line ZB, from which it is cut off. From 
 the point Z through K and also through 1 join ZD. Because the lesser triangle CZl is a part of the 
 greater triangle BZK, and similar to BZK, and because BK, the base of the greater, is equal to half 
 of BZ, therefore CI, the base of the lesser is equal to half of CZ. But the line ZE bisects BK, and 
 therefore also bisects CI at G, consequently, if ZC is divided into four equal parts, CG shall be equal 
 to each one of those parts, and since the lesser triangle BCG is similar to the greater triangle BZb, 
 and BZ is equal to bZ; BC is also equal to CG. Wherefore CZ, cut off from BZ, contains four equal 
 divisional parts, each one of which is equal to each of the five equal divisional parts contained by the 
 line BZ. And again, if from C, a line parallel to Bb be drawn iuterccpting aZ at c, and with the 
 centi'e o, and radius cZ, a third quadrant be described, and if, from C, a liue parallel to Bg be drawn 
 and if from the point where the line so drawn cuts the quadrant a line be drawn perpendicular to CZ, 
 and intercepting CZ at D, it follows from a parity of reasoning that DZ, the part so cut off from CZ 
 
 s 
 
26 
 
 shall have tlio same ratio to CZ, as CZ has to BZ, ami as BZ has to AZ. And that, in like manner, 
 a part may be cut off from the lino PZ, having the same ratio to D/ that DZ has to CZ. And a^'aiii, 
 from the part last cut off another similar fractional part may be cut off, and that the process may bo 
 so continued in like manner so long as any such part of the line A'^ be remaining that it be still pos- 
 sible to describe a quadrant and to draw linos parallel to the linos Aa and A'i. Wherefore the line 
 AZhas boon divided Ac, iSio., Q. E. D. Scholium. Tiie chord ZK of the arc ZK, cut off from the 
 quadrant ZniU by the line G A, includes the chords of ail and each of the lesser similar arcs Again, if 
 the line AG be produced through md the line Za be produced through a, the production of Za, 
 intercepted by the production "f -hall be twice the length of Zu, Similarly of the lines B(J and 
 
 Zb, and so on"'-'. Hence... Coiiiiary. The linos belonging to an arc which contains any definite fraction 
 of a circle have the same relationship to each other and to the arc itself, as the similar lines belonging 
 to all other arcs which contain similar fractions of other circles, have to each other and to the arcs 
 respectively to which they belong ; and any line bolnnging to the one arc has the same ratio to the 
 similar line belonging to the other arc, as the one arc which contains the definite fraction of a circle 
 has to the other arc. which contains the similar fraction of any other circle, however groat or however 
 small the magnitude of the circles may be to which the arcs respectively belong. t 
 
 Kx.VMI.VATlOX CONTINUED. 
 
 The lines BD, BC, and Bm. Fig. 25. Join Bm. On the line BD, take Ba, equal in length to 
 BS, and from the point a. perpendicular to BD draw am. Because BS is equal to Ba, and Sm is 
 perpendicular to BS, and am perpendicular to Ba— the triangle BmS is similar and equal to the 
 triangle Bma, and consequently the line Sm is equal to the line am. But the line BS is a part of 
 BC, and il:o line Ba i'^ a part of BD. Again, the line Ef cuts the line B J at tlie point e, and 
 the line Ef is perpendicular to the line Bm, therefore the triangle Bme is similar and equal to the 
 
 * We woiu'i suggest tbat tbis proposition may be found of utility in applied Geometry, {i.e. for the 
 jiurposcs of tlie mccbanical draughtsman, &c. It is, however, to its application in quanliiive arithmetic that 
 ivo wish here to invite the reader's attention. With this view we repent the lignre wiiliont describing the arcs, 
 nt Fig. 23 wliieh may tlais i)e con^lructi'd. Draw the iipiare nc.Vz. Bisect ac, at g ; and bisect c.\, at 
 d. Join Aa, gz. dz, and .-Vg. Through the point where gA intersects dz, draw a line perpeudiculur to zA, inter- 
 cepting zA at li. From li, draw lib, parallel lo Aa. I'roduce llie line i erpendicular to z.\, from li, tlirough the 
 point of intersection of gA, and dz. From the point b, (where 15b intercepts az) draw be, at right angles to az, 
 meeting the line produced from B, at the point c. be is bisected by the line gz. From the point of bisection, 
 join gB parallel to gA ; and through the point of bisection, perpendicular to zA, draw cC ; and so on. The 
 division of the line zA will then have the ratios, each to the others, shown in the prop. And evidently the 
 
 other lines rad'ating from the point z will be similarly divided (i.'-. the lines Z\ zc, zg, kc. The line gz bisects 
 each of the line.i perpendicular to Az : and so on. When the process of u 'no ding an arc has been presently 
 ex|ilniued, it will appear that the method here shown in conjunction with thai process may furnish the means 
 of obtaining a very gre.i' number of various lines having definite ratios, each to the others, and to a given line 
 of known length. 
 
 t This may seem much like proving a truism since every one knows, for instance, that the tangent of an arc 
 of 45 degrees, is equal to the radius whether the figure be drawn on n large or ft small scale | but, if we are 
 correct in supposing that no direct demonstration of the general law has been given— the omission in a Geom- 
 etrical (Cyclometrical) S'.-nse, is not, as we opine, justifiable. 
 
27 
 
 triangle Biuf, and the line mc U eriual to the line mf. Ilcnco, by induction, •,'onoi'ally ; if from any 
 point on the line Bui (or production of Bm) a lins bo drawn to tlio lino BC. at riglit anu'los to BU, 
 and from the saiuo p'jiut on tho line Bui — a line be drawn to the line BD at ri/lit an^'lcs to BD, tho 
 two Hues must h • necessarily eijual to c ich other. And also if a line bo drawn at any point from the 
 line BC to t!ie lit e BH, throuirh the lino Bin, and at ri;^lit an^lo.s to B.n, tlio lino sj drawn ^•lulll bu 
 bisected by the lii-C Bin. or production of Bni.* 
 
 r>F,VF.r.OI'F,MEXT (1F THE PROCESS OF UNUENDIN(i AN ARC. 
 
 Fiir. -4. which is a repetition of the original figure, .shows the arc Bin unbent through tiivcc successive 
 duplicate ni.i;_'nitude.«, by rei»eatcd duplications of the radius, ns previously shown and explained in ]irop. 
 II. In this fii^ure the entire r.adius of each arc is drawn, and each sueces.sivo arc is described. Wo sliall 
 now explain three methods, differing somewhat from each other, by either of which the points at the 
 terminal extremity of eacli successive arc can be found, and the extremities of the successive radii, 
 properly pertaining to the arcs, can be drawn without actually describing the arcs, and without drawing 
 the entire radii. 
 
 MExnon the First.— Fi(i.2G. Through the point xa, and porpondiculir to the lino BD, draw 
 the line ma, of indefinite length, with centre m and radius ma, describe an arc, intercepting AD at a. 
 Bisect the arc aa at R, and joiv Bin. Juin Bm, and with centre B and radius (Bm, or any pan of 
 Bm, as) Bb, describe an arc intercepting BD at b. Bisect the arc bb at c. Produce the line Hm 
 through ni, and ppKluce the the lino Be tlirough c, until the two lines so produced intercept each other at 
 the pr>int n. Rn .sliall be the extremity of the first duplicated radius, and the point n shall be the point 
 at the terminal estremity of the first arc of duplicated magnitude. Because am is perpendicular to BD, 
 and consoi|Uently parallel to the radius AB, and because am, which is part of the secant AD, contains 
 together with am an angle of 45 degrees, tliercfore tlie line Rn which is a production of Rm contains, 
 together with Am. an angle of 22i degrees. Bisect the original arc of 45 degrees Bm, at the 
 point c. Now the line Bn inter-sects the same point e, and tiie point e also bisects tlie line Bn, and since 
 it is manifest that if the radius Ae, of the arc of 22A degrees Be, be drawn the line Ae shall be parallel to 
 the line Rn, and becau-i-e the line Bn i.! twice the length of Be, and is intercepted by Rn, drawn 
 through the point ni, therefore the line Bn is the chord of an arc, erjual in length to the arc Bm, 
 and belonging to a circle double the magnitude of the circle to which the arc Biu belongs, and the 
 point n, at the terminal extremity oftlie chord of the arc, must be necessarily the point at tl;e terminal 
 extremity of the arc itself if the arc bo actually described. Througli the point n, and perpendicular 
 to BD, draw na of indefinite length. Produce nR through R, and with centre n and radius na describe 
 the arc aa. Bi>cct aa at S and join ?n. With centre B, and with (any part of the line Bn, as) Bb 
 describe the arc bb, intercepting the line BD. Bisect the arc bb atd, and join Bd, Produce the lino 
 Sn through n, and produce the line Bd, through d, until the two lines, so proiluecd, intercept each 
 other at the poiut o. The line Sno shall be the extremity of the second duplicated radius, and the 
 
 • The figures belonging to tlie Exi'eriniental Exdrai lation, and a\?o, a part of tbc figures belonging to the 
 props. deJuctil from tlie Eijil. Exns. are conibincj in Fig. 25 drawn on a larger scale, with the object of defining 
 more clearly the relative positions of the lines under consideration. 
 
poirt shall be the terminal point of the second arc of duplicate magnitude. Bisect the arc Be of 22J 
 degrees at the point f, and join Af. Af is necessarily parallel to So. The line Bo cuts the arc 
 at the point f, and the part Bf, cut off from Bo by the point f, is one-fourth the length of the whole lino 
 Bo. Wherefore So is the extremity of the second duplicated radius, and Bo is the chord of an arc 
 equal in length to the arc Bm, and belonging to a circle four times the magnitude of the circle to 
 which tlie arc Bm belongs, and the point o must be necessarily the terminal point of the second aro 
 of duplicate magnitude if the arc be actually described. By continuing the process in like mannen 
 and by a parity of reasoning, it will appear that the lines Tp and Uq arc the extremities of the radii' 
 and the points p and q the terminal points belonging to the successive arcs of duplicate magnitude. 
 
 Method the Second. 
 
 Fig. 27. Through the point m and at right angles to BD, draw the line amb. With centre m, and 
 radius ma (of indefinite length), describe the arc ai, intercepting AD at i. Bisect ai at R. Join Rm 
 and produce Rm, through m (indefinitely). With centre m and radius mb, (of indefinite length) 
 describe the arc bb, intercepting the production of Rm at b. Bisect the arc bb at c, join cm and from 
 the point where cm cuts RD, draw a line perpendicular to BD and intercepting the production of Rm 
 at the point n. Rn shall be the extremity of the first duplicated radius, and n shall bfi the terminal 
 point of the first arc of duplicate magnitude. Join Bm. Bm is perpendicular to Rm. Join Bn. 
 Bn cuts the arc Bm at me point of bisection e. Therefore Rn is the same lino and n the same point 
 previously obtained by the first method and shown in Fig. 26. Through the point n, and perpendicu- 
 lar to BD, draw the line anb. produce nR, through R, and with centre n and radius na, of indefinite 
 length) describe the arc ai, intercepting the production of nR at L Bisect aA at 8. Join Sn and 
 produce Sn, through n indefinitely. With centre n and radius nb (of indefinite length) describe the 
 arc bb, intercepting the production of Sn at b. Bisect the arc bb at d. Join dn, and from the point 
 where dn cuts the line BD dr.-iw a line perpendicular to BD intercepting the production of Sn, at tho 
 point ; So, shall be the extremity of the .second duplicated radius and o, the terminal point of the second 
 arc of duplicated magnitude. Join Bn. Bn is perpendicular to So. Join Bo. Bo cuts the half-arc Be, 
 at the point of bisection/. Therefore So is the same line, ando the same point previously obtained by 
 the first method, and shown in Fig. 26. 
 
 By continuing the same method and by a parity of reasoning, the remaining points and lines 
 may be obtained, namely the points p q &c., and the lines Tp and Uq, &c., and will be found to 
 coincide with the points and lines denoted by the same letters in Fig. 26 and previously determined 
 by the first method. 
 
 Now taking either of the figures belonging to these two methods if a line be drawn from the 
 point n, to the line BD, jtrpendicular to BD, the line so drawn shall be equal to mn. And if a line 
 be drawn from the point o to BD perpendicular to BD, the line so drawn shall be equal to n o. 
 And a line from p to BD perpendicular to BD equal to op, and so on. (See experimental exami- 
 nation, page 26). 
 
 For taking Fig. 24 let nb, oc and pd be the lines drawn as directed, and join also mb, no and 
 od, the lesser triangle mnb is manifestly similar to the greater triangle nEF, and again noc, similar to 
 oFG, and so on, but EF is equal to En, and therefore rm is equal to nb, also FG is equal to Po, and 
 
29 
 
 therefore on \a equal to oc and so on. Or again taking Fig. produce Bm through m and with 
 centre B and radius Bn describe an are in both directions intercepting the production of Bni at J. 
 uud intercepting BD at K; then because Be bisects bob, Bn bisects JnK. nm is the sine of the half- 
 arc Jn, and D) (perpendicular to BD) the sine of the half arc nK. But the sines of equal and 
 similar arcs art equal to each other, therefore nm is equal to nb. Similarly if an arc is described with 
 the centre B and radius Bo intercepting the production of Bn at one extremity, and intercepting the 
 line BD at the other extremity, then since the production of Bn is a production of Be, and BD a pro- 
 duction of Bb, the arc described with the radius Bo (which is a production of Bd) shall be manifestly 
 bisected at the point o, consequently no, must equal the line drawn from o, to the line BD perpendi- 
 cular to BD, and so also, by a parity of reasoning, of the other lines drawn from the successive points 
 p, q, r, etc., etc. 
 
 Method the Thiud. 
 
 Fig. 28 (a,) represents a part of the original figure on an enlarged scale. mD is part of the secant 
 AD belonging to the arc Bm out off at the point m, bD is part of the tangent BD (belonging to the 
 arc Bm) cut off by interception of the line mb drawn from m, and perpendicular to BD. Bisect the 
 angle Dmb at the line mf (sec definitions). Bisect the angle fmb at the point g. From g, and 
 perpendicular to bD, draw a line intercepting the line mf at the point n. mn shall be the extremity 
 of the first duplicated radius, and n shall be the point at the terminal extremity of the first arc of 
 duplicate magnitude. Bisect the angle fng at the line nh. Bisect the angle hng at the point i and 
 from i perpendicular to bD, draw a line, intercepting the line nh at the point o. no, shall be the 
 extremity of the second duplicated radius, and o shall be the point at the terminal extremity of the 
 second arc of duplicate magnitude. Bisect the angle hoi at the line ok. Bisect the angle koi, at the 
 point 1, and from 1 draw a line perpendicular to bD intercepting the line ok at the point p. The line 
 op shall be the extremity of the third duplicated radius and p shall be the point at the terminal ex- 
 tremity of the third arc of duplicate magnitude. Similarly by proceeding in the same manner the 
 remaining lines and points may be obtained, namely, the lines pq, qr, &e., and the points q. r. &c., 
 coinciding with the lines and points denoted by the same letters in Figs. 2G and 27 and determined by 
 the two previous methods. 
 
 The two following theorems may serve, perhaps, to assist the reader in examining the several cases 
 eliminated in the development of the process. Theorem (a). If, in any right-angled triangle, one 
 of the two sides together containing the right angle be bisected, and from the point of bisection a line 
 be drawn to the vertex of the opposite angle, then all lines drawn between the other two sides of the 
 triangle, if drawn perpendicular to that side which is at right angles to the side from which the first 
 line was drawn, shall be bisected by the line first drawn. Fig. 20. Let ABC be the triangle, and 
 let AB be bisected at e. Join eC, and between AC and BC draw any line Gf, perpendicular to BC. 
 Then, because the triangle CGf is similar to the greater triangle CAB, and because Ce bisects AB, 
 therefors Ce bisects Gf. 
 
 Theorem (b). If, in any right-angled triangle, one of the acute angles be bisected, (see definitions) 
 and from the point in the side of the triangle intercepted by the line of bisection, a line be 
 drawn to the opposite side, and the line be drawn perpendicular to that side from which it is drawn, 
 then shall the line so drawn be equal in length to that part of the opposite side cut off by interception of 
 
33 
 
 til.' liii.' >n ,lr:.\vii, iinil coiituinoil \c:\\mi tlir vrtcx m 'li.' ;.rut.> iiuplo, wliich w;is liisrctod, mv\ tl.o 
 p,.iiit i.t iiiici'.i ption I f ihu iiio so ilniwn,- Im-. 2 1 . i.ct A UC l>o tlic riglit-miL'li'il triiinulc, iiml let tho 
 nciito uicie VMi Iha bisoitta iii tlio line /.■•. !• miij tlif |K.iiit o. dniw vl\ ]ior|ionilicul!ir In BC, .•nid 
 iiit.rc.jMinj!; AC ut 1'. A' n^mU lA. U\>:';t tlv line AC at ir, iinJ jnin Qf lu'ipondicular to AC, tho 
 tri;iii:.'lc ul'o M siiiiiliir to llii; tiiaii,i:Io ;.'I'A. Now tlu' two A'h* L't' iniil u'C «>f ilio ono iuc c'i(u;il to tlio 
 two sides <.'!' and l'A of tlio ollior, tin'ivloro tlio tliivd sides of and t'A arc also ofjual. Or a^rain, let 
 ABe (snmc fiauro), he the riuht-angled trianL'lc, and let the aniilo AeB he hisccled at the line eh, and 
 from h, draw lim, perpendieiilar to AB, and intercoiiting Ae at ni, hm ei|n:ils me. Similarly, if tho 
 an^'le eAB he bi.seetcd ut the line Ai, and from i, the line i n, jierpendicular to Bo be drawn, inter- 
 ocptinL' Ae at n, tlic line i n, shall necessarily ef|Ual the line nA. 
 
 We nr<y here furnish an illustration of the practical application of the unbem'iut,' process, by submit- 
 ting to experiment the iiucstiun stated in the inteno;4ative prop.X.l page 18). In order, however, todefino 
 more precisely the nature of the question, it wil! bo desirable, in tlie first place, to esiimine with some 
 attention tlie form jriven tliereto in prop. Y— n.'imely. ' Upon a given straight lino it is roiiuireJ to 
 describe an arc of 45 dcfirecs, such that the are shall be equal in length to the given straight lino.' 
 Evidently, if we can bend the straight line itself into the required arc, we .shall bo able to fullill tho 
 recin'sition. Let AB (Fig. lli). be the given straight line. One extremity of the lino at tlie point 
 B, we will suppose to be list d. From tlie point A, ut the opposite extremity, draw the line (of inde- 
 finite length) CA, perpendicular toAB;and we will snppo.sc the lino CA to be also fixed. Wo 
 now comincncc to bend the line AB upward.s into the n quired are; as wo proceed, tho point A will 
 manifestly he separated and recede from the perpmdieular CA. And it is likewise manifest tliat 
 wlien tho lino AB has been cmnpletely bent into the arc of 45 degrees, tiie secant of tho arc will cross 
 the line C\ at an angle of 45 degrees, and will cut the line CA nmneuhcix at a point xnme distance 
 from the point A, (at the extremity of tho fixed lino CA). We will call the point at whieli the 
 secant (if ihe are of 45 degrees so cuts the line CA. (x). Now suppose the arc of 45 degrees to be 
 be unbent into an arc of l.'2A degrees, and it is again manifest that the .secant of the arc of 22i 
 degrees will cross the line CA at some point between the point s and the point A. Wo will call tho 
 point at which the secant of tlie arc so cuts the line CA (y i. One result of the experiment to which 
 we are about to submit the question must necessarily show eitlior that the point y will histct the 
 line Vx, (under tlu conditions sot forth above) or that tho point y will not bisect the lino Ax (under 
 those conlitions). Assuming for tho innnent tliat tho result of tho experiment is affirmative, it will 
 then follow that if tlic two arcs be each of thorn equi! in Icngtli to the line AB, and if tho secants 
 cross the line CA at au'^dos respectively of 4.'> and 22i degrees, the lino xy must be equal to the line 
 Ay. on th^ li'i ' C\. But if any otiier line as D 1, nnro dis^mt than C.\. I'roni the point B ; or Ff, loss 
 dist.nt than CV from B, be ilrawn, also perpcndieular to AB, at any very small distance from CA; and 
 if the points at which the socuits of the arcs cut tho lines so drawn be again called respectively x and 
 y. then (gr.inting the assumption.) it will bo evident that the point y cannot llKixt xd on the lino 
 Dd. Nor can y hisn-t xf on the lino Ff because (".'ranting the assumption) xy must be necessarily 
 graiti'v than yd on the line Dd, ainl Xy must be h ss than yf on tho lino Ff. It may be ob.served 
 that the line GIT, as drawn in the fi;;ure to prop. Y, (Fig. 5) corresponds to the line here denom- 
 inated CA, and that therefjro the assumption now proposed, as well as the question stated in proj 
 X, will be tested by the experiment. 
 
31 
 
 Kxpcrliiii'i t. — V\<i. 28 (c; niD is a pnrt of the socnnt AD, belonf;inj; to the arc Bm of tin' miiiinal 
 fiirnro, cut nil' u. tlio point m. bl) is a p:irt of tin' tiiM'.n'ii( BD, cut off by Interception of tlic line 
 iiib, drawn perpendicular to HD, from tlio point ui. Bisect the line nib nt the point L, and j lin 1>L. 
 Bisect tho an^le Dnib iit llic line nif (sec definitions) cuttinj; the line DL at the point y. From 
 tiic line ml), throu.di the point y, and ])i;rp "idicul ir to tht; line bl), dr nv the lin? xyll int.reeptln,' 
 bl) nt the point II. The line xyll so drawn corresponds to tiio line OH in the fi_'urc belonjin^ to 
 prop. X. (Fi^. 5) Bisect tho ftn'.^lo fnib, at tho point <,', and from tho point >,', perpcndicihir to of 
 the lino bd, draw jrn, interceptini; the line nif at the point n. n, is the point at the terminal extremity 
 the first arc of duplicate nia;j:nitude (see IJrd method of process) jiisect the an;,do fng, at the line 
 nil. Bisect the Hngle hnjr, at tiie point i, and from i, pcrpemiicular to bd, draw io, intercepting 
 nil, at 0. o is the point at the tcrmin;d estromity of the second arc of duplio.ite magnitude. Bisect 
 tho ani^le hoi, at the line ok. Bisect the air^Ie koi at the point 1, and from 1, porpendicuhir to bD, 
 draw Ip, intercepting; ok, at p. \\ is the point at the terminal extremity of the third arc of duplicate 
 magnitude. But it now becomes evident lliat the point p, fills bt'yond tho line xyll, and th'-roforc 
 (because any arc is greater than its chord) the line BlI cat off from BD by interception of the line 
 Gil in the figure belonging to prop. X, mUit be /rsn than the arc Bm, consci|uently the ((ucstioa 
 submitted in that prop, must be an^^wered in the neLritive; and it necessarily follows that ihs arc con- 
 structed by prop. V is greater than the given line BlI, upon which it was described. And also since 
 the pointy bisects the line .xyll, the assumption proposed at page 30 cannot bo affirmed ; for it is now seen 
 to be evidently impossible in the case supposed that the point y c ui bisect xA on the line CA.* t 
 
 'I'here remains a very iui[>ortant cjnsideration in the applicition of the '' unbending process " to 
 the analysis of the general figure, to which wo have to recjuest the reader's particular attention. It 
 has been shown (page 27t that, if wiili the centre B, and radius BC (or any part or production of BC) 
 iin arc is described, terminated by interception of tho line BD, (or the cij livalent part ur production of 
 BD) a production of (or part of) the line Bm shall be the central radius of that arc. Similarly if with 
 tlie radius Bm, an are terminated by the line BD is describe 1, the line B:i shall contain the central 
 radius, and so on with respect tci the lines Bn, Bo, Bp, etc. Now -^ince the arc Bn ol'duplicite magni- 
 tude is, if described, eipial in length to the are Bm, it is evident that ii' a part of the line Bit be taken 
 Kpial to the chord Bm, of the arc Bm, anl if an are similar and O'pi.d to the arc Bm be described upon 
 the line BD, such that the part Bl), so taken, be the chord of that arc, then if the last arc so described 
 be unbent, tho first arc of duplicate magnitude, eiiuid in length thereto, will also bo terminated by the 
 point n. That is, the figure may be invci-ted as shown at Fig. 28 (d,) and the process may be verified 
 by projecting the arc (so to speak ) upon the upper, as well as upon tho lower straight line. And it is 
 evident that this uictho 1 may be also adopted with any of the intermediate lines, namely. — we may 
 
 • See also Fig. 28 (a) on ciilargeil scale, in whicli tlic point y U ixljo indicated. 
 
 t Tlio actual qiianlitive value of the line lill, cut oft" from BD, by interception of Gil, in the interrogative 
 prop. X. ; may be (letunuiQcd by a simple trigonometrical computation ; thus . . . Lei Bit, Ciiual I. Then, with 
 centre B, and radius BH, describe an arc of 4J degress Icrmiuated by tlie proJuctiou of the line liui ; Ho is the 
 tangent of that arc, and lip is the tangent ot nn arc of 22} degrees, described with the same centre and radius ; 
 
 that is Hp... ='41422, therefore (by prop. V) Up = '--i-- X 2 = '270140 = IID. Now, let tlio radius 
 
 AB = I. Tlicn since AB = BD ; and AB:IJH:: 1'27014G; 1 ... .therefore 1: length of BlI:: r27GUO: 1 . ...and 
 
 _L*^-_— '"B3t31...,=the length of BH, taking the length of AB=1. 
 1'270146 ^ .Bo 
 
32 
 
 commcnco witli tlio line Bn, anil unbend t'.c arc ihcrcrroi) npm t'lo liiu- BD, nnJ then invert the pro- 
 cess, and comnienco with tlie.,,uivalcnt part of BD, an I unbena tlio arc upon the production of Bn, 
 and so on with the intermediate linos, niini'ly, tlio lines llo, Bp, etc. Ajiain, also, sinco the lino BC 
 is the central radius (or a part of the central radius) of an arc described with the radius BA (or with 
 a production, or a part thereof^ and terniinaied by th.; radius BD, it is evident that the half of tho 
 quadrant, contained between the line? BA and BC, may bo divided similarly to the half of the quadrant 
 contained between the lines BC and BD. Hence, for the purpose of analysis, wc may divide any quad- 
 rant described between BA and BD into as many segments as may be desired, and apply tlic process 
 dircc'ly and also iaicrstli/ to each segment. 
 
 THE REQT'ISITION. 
 
 Prop. A.— An arc containing the definite fraction of a circle, and a straij^ht line forming tho 
 tangent to the arc being given, it is required to cut off from the straight lino a part thereof, equal in 
 length to the given arc. 
 
 Fig. G (is a repetition of the original Dgure.) Let Bm be the given arc; and BD the tangent 
 to the arc Bm, be the given straight line ; it is requiri.d to cut off from BD, a part thereof, equal 
 in length to Bm. 
 
 Jcjin Bm and produce Bm, through ui, indefinitely. Through ni, perpendicular to BD, draw 
 amb. Bisect the angle Ania, at the lino Sni, and bisect the angle niBb, at the line Be, fSco defini- 
 tions.) Produce Sm through ni, and produce Be, through e, until the two lines so produced intercept 
 each other at tho point n. Through n, perpendicular to BD, draw dng. Bisect the angle Snd at 
 the line Tn, cutting the line AD at i. From i, through the production of Bm, and at right angles to 
 Bm, draw ik, intercepting am at k, and cutting the production of Bm at 1. With centre B, and 
 radius Bl, describe the arc h, intercepting the line BD at the point - B^ shall be tho required 
 line. 
 
 Prop. B. — It is required upon a given straight line to describe an arc such that tho arc shall bo 
 equal in length to the given straight line, and shall contain the definite fraction of a circle of definite 
 magnitude. 
 
 Fig. 7. Lot Bt be the given straight line, it is required to describe the arc upon Bt. From 
 B, draw BP, perpendicular to Bt, and equal in length to Bt, and from j^, draw ""c, perpendicular to 
 B~, and equal in length to Bt. With centre F, and radius ^B, describe the quadrant Be. Join 
 the chord Be, and bisect the quadrant at N. Join BN, and produce BN, through N, indefinitely. 
 With centre B, and radius B", describe the arc ^, intercepting the production of BN at 1. Bisect 
 the line ct at r, and join Br. Through 1, at right angles to Bl, draw a line intercej)ting Br at K 
 From K draw Kb perpendicular to B-, and, cutting the production of BN at the point m. Produce BF 
 through F, and produce Bt through t. Through the point m, and through tho lim Be, at right 
 angles to Be, draw a line intercepting the production of BF at A, and intercepting tho production oi 
 B" at D. With centre A and radius AB, describe BmC. BmC shall be the required urc. 
 
88 
 
 Wo liavo iilicaily stutoij that it it not our purim-c to furnish in tliis place formal Jomnnstrations 
 to the aliHve propusitious. If tho ' ]ir(»co!«.-i of uiibcndiu^' nn inv,' lu'rcin t'xplniiiuil, is reco^iii/uil as 
 tlio souml c'XjHi itiiiii iif II p!oiiii'trii'al /'"•/, tho rca^lor is furnithcJ in tlmt' proces.s ' with nn infallible 
 mcan.M of ti'Mtin: tiiu correct noH:* of tlii" nlalinnship ns<.Hii:nt<il to tiio lines in ipiestion. It is proper, 
 however, lliat Wt shouM here inclieale the direct connexion between the 'process of uubi.'ndinu' an arc' 
 iind the H(ilution» to tliesc priipnsitidiis : gucli tliitt if tho former bo reo<>^'nized and admitted tho 
 rceo,'niiion of lln' latter must then IIiUdw as a necessary 00Mse(|iienCi'. With this object in view, wo 
 will now on<lcavnur to state the more ;;(Mieral form of the sulution to pmp. A as a theorem. 
 
 'fiieorem 1\ . — If, in any i|uadrantor any defuiito nrc or circle less tlian a (|Uadrant, the terininnl 
 radius tie produced throu:;h tlie |)oint at the ti'rminal extremity of that are ( i.e. if the scc.mt to that 
 are ho drawn) (see definitions); and if the ehurd of llic nrc bo also produced througli the point at the 
 terminal extremity of the arc, and if from tlie point nt tlie original extremity of the arc a second arc, 
 e((Ual in len;;th to llie fir.-t arc and iielongiui; to a circle four times the ma^'iiitudc of the circle t'l which 
 tho tirst arc beloni,'s, be described , ami if, from tlie point where the terminal radius cif the soeond arc 
 cuts the production of the terminal radius ol'the tirst are, a line be <lrawn, iutereeptin;^ the production 
 of the chord of the tirst arc at ri^dit anvdes to the production of th:it chord, then shall the part cut 
 ort' from the production of the ehyrd of ihe first .'irc by iiiterceiiiion of tlie line so ilrnwn be cnual 
 in lenj^th to each of the arcs. 
 
 This theorem will evidently carry with it tlie corollary, that, if with the [lart so cut otf from the 
 production of the chord if the !iiv* arc as u radius and with the original point of the tirst arc i. which 
 is, also, the original point of the chord, and also of the secmd arc,) as a centre, an arc be described 
 intercepting the tangent ot' the first arc drawi' from the oriixinal point of the first arc ; then, if the 
 taiigeut to thf last are be drawn from the original p'dnt of the last arc, (' i.e. t'roni the point on the 
 proilnction of the chord of the first arc intercepted by the line drawn at right angles to the production 
 of ihiit chord.) the tangent so drawn shall cut the production of the terminal radius of the first arc 
 at the same point intersected by the terminal radius of the soeoiid arc. 
 
 The reader may, therefore, tost the solution to proji. A oxpi'rimontally ly [nodueiiig the chord 
 of each successive are of duplicate migiiiiude ; and by ap]>lyiiig tho theorem, may cut oti' from each 
 produced chord a line eijual in length to e.acli of the ares. The " process" may be then inverted as 
 already explained, by taking ri /> irl »/ l/i- tinjcnt belonging to the first arc eqitnl to the chord of the 
 i5an>e arc, and projoeting the nnbei.t are upon the upper line: oaeli <^tage of the process may be again 
 tested by applying the theorem. •■■ 
 
 Fig. 31, drawn on alarger.sc.de, is intoiided to illustrate the theonin, and may also serve to 
 assist the reader in examining tho .solution to pri ps. A i^ B. 
 
 The dotted line in Fig. 2S (,bi, shows the arc - - described according to prop. A. 
 
 Fig. oD shows a semicircle unbent upon the straight line Dd by drawing the radii and describing 
 the successive ares of duplicate magnitude. 
 
 Fig. 32 shows a semicircle unbent upon the straight line Dd, by drawing only the chords of the 
 
 • According to the tlieort-m, the arc ~~ may be described with the line (if tijuiil lenglli on iiiiy one of the 
 jirodaced cliords a^ n. radiii.s, and willi (Ac foiiit B as a centre ; and the arc ~~ shii'l tUi'u cut all the other pro- 
 duced eliords al lh'> terminal pouts ef tl>c lines ofeijual length. 
 
 E 
 
34 
 
 • 
 
 roccwsivc orc''. In each ViiX. the prooi*,^ in ciiiiiiilitcil n^'t'orlin^: to the- iilmvo .■Miluiinn tn I'riip. A. 
 The 8cvornl arcr< in eitlicr of tho fi;,'uri's arc oijual in Ini^th cadi to i-ich, ami carli ol'thr nrcn Ik i'(|iial 
 in lent'th to tho ntrni;.'hf lino "". . If thi' radius AB- 1 . . the «trai'.'ht line "- .'! 1 il.'i!! \e. 
 
 Fif.'. '-'f' kIiiiwb th • rchifive ((uiuititive values al the lim.- Ht^ III', H" \l\\ and UT. ihe lenu'lh 
 of the line BI> bviiii: taken aM IdO. 
 
 Vij!. .'Jt refire.-'ents four seniieirdes unbent upon a "trai^ilif line (--> intcriiiedi ili' lictween tiieni ; 
 the ctrai lit line btinj!; tangential to all the arcii. ami tuuehinj.' each are at the jioint tiTniinatin;: thu 
 central radiu.- of eaeh. The ra'liu-< EH, brloni.'ini; to tho two ^Toater wini circles is thu duplicate of 
 the radiu- CB, beloni;in^' to lie' lesser seniieireles. Tlio proee.-s of unbending.' tlie are is eomple'ed 
 by application of- prop. A.' Each of the greater arcs is equiil in Icni'th to the line II II, and each of 
 the lesser ares is equal to the line tt. The circle "tt niay be termed the circle for arc) of decomposi- 
 tion, and cuts all the produced chords of ccjual iiiif,'th (productinn- of Hni. Bn, Bo, ttc, beliinj;inj! to 
 the les.^er figxirc) acondiuf: tn ' tiieorem iv.' The letters D and f, on the line nil, denote the terminal 
 piiints of the lines El> and Ef. belon^'ini: tn the ji's.-er fi:_'.irc. 
 
 Wc now venture to assume that our pr. li'iiinury case is .•jufficicntly established and proceed aee.ird- 
 inulv to the introduction. 
 
L\TIK)T)rCTTOX, 
 
 What li a inagnitudc ? A m igiiituile is doiiotf.! in Euclicr^ Geometry by a straight line 
 of (li'finite len;^tli ; aii'l a lino i.s defineil as length without hreadth. Conse'iuontly, if a given 
 straight lino la hiseeted, one hall' the lino so eut olV j, half the magnitude of the given line, 
 and if the half line is douhled, ur increased to twice itd size, the re<inlt will he a lino equal and 
 .iimilar to the given line ; in other words, the original line will ho roconstructcd. But what 
 will be the result, if. in jilace of a straight lino of dofniite length, a curved line of definite 
 length is substituted, and if the curved line is bisected and the magnitude of the one-half is 
 doubled ? If, for instance, an arc containing A') degrees of a circle is bisected and if the one- 
 half (/. >■. the arc of 22i degrees) is increased to twice its size : will the original arc be recon- 
 structed? Supposing that an orange is cut through the centre and evenly divided into two 
 equal and similar jiarts, and one of those two jiarts is then magnified to twice its size, the part 
 so magnified will, in its magnified condition, f'c eiputl in sizt to the whole orange, but it will 
 evidently not /■< «i'w/7<^- thereto in/onn it will still remain a half orange, but of increased size. 
 
 What then will happen if any definite arc of a circle is bisected, and if the magnitude of 
 one-half the arc so cut oft' is doubleil ? The reader can easily submit the ijucstion to experi- 
 ment and ascertain the actual result. Upon the correct answer to fliis iiue3tion,'the following 
 work will be primarily based. 
 

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