'^ >^. v^ ^ '¥ IMAGE EVALUATION TEST TARGET (MT-S) 1.0 I.I ^1^ l£^ ^ "'" 12.2 lit 2.0 L25 i 1.4 I 1.6 Photographic Sciences Corporation % // ^/ A signifie "A SUIVRE", le symbols y signifie "FIN". Mapa, plates, charts, etc.. may be filmed at different reduction ratios. Thoaa too large to be entirely included in one expoaura are filmed beginning in the upper left hand corner, left to right and top to bottom, as many framea aa required. The following diagrama illustrate tha method: Lea cartes, pienches, tableaux, etc., peuvent Atre fiimte A des taux de reduction diff Arents. Lorsque le document est trop grand pour Atre reproduit en un seul cllchA, ii est fiim6 d partir de I'angle supArieur gauche, de gauche ib droite, et de haut en bas, en prenant le nombre d'Images nicessaire. Las diagrammes suivants iliuatrant la mithoda. 1 2 3 1 2 3 4 5 6 CENTRIFUGAL FORCE AND GRAVITATION B7 ParU. 1. 2. 3. 4. 5. 6. 7. (K U K LOS) JOHS HARRIS. } Tho Attractive Force and Tangential Motion. The Planetary axis of rotation, and plane of the Ecliptic. The neighbouring Stellar System, and Aberration of Light. Tho Cometaiy Orbit of llevolution. PartH I. & 11. The Undulatory and other Theories of Light. Part III. Light, and other manifcHtations of Force. THE CmCLE AND STUAIGIIT LINE, 1. The Geometuioal Relationship uemonstbateu. 2. The Construction of the CiRci.t;. 3. Conclusion. -^ ^V^E ClHci^ AND STRAIGHT LINE, ^ tS^- THE CIRCLE AND STKAIGHT LIKE. \ hy JOHN HARRIS. MONTREAL: JOHN LOVELL, ST. NICHOLj^S STREET. JANUARY, 1374, if Kntenid according to Act of rarlinnipnt in the year ono thousand eight hundrod and 8ovouty-four, by Joii.s IIauims, in the ollico of the MinistiT of App-iculture and Statistics at Ottawa. HAi^il! r .••• 4 - r- ■' I-— MoMUKAL — Jon:< Lovkll, I'nixTEii. P R E F A C 1\ hundrpd riculture Deeming the suUject of this work, vi/ : tho sohition of that geometrical proljlein which dciuoiistratos the rehi- tion of the circle to the straight line, to he peculiarly of jmhlic importance, we will briefly state wliat has taken place in respect to it, and the present position of the matter. The discovery ot the solution was conmumicated hy letter, dated 2Uth Decendjer, ]^7(), accompanied with demonstration, &c., to the Astronomer Royal. It was afterwards found that the cast', as contained in tho papers thus presented in the lirst instance, was in a strictly geometrical sense imperfect and faulty ; tho solution and demonstration, however, were virtually c("i.- tained in them, and no objection whatever was made to their form. The papers were acknowledged, but, afler some con- siderable time, a decisive intinuition was received from Greenwich that the Astronomer Kuyal declined to examine the case. In the month of January, IS/O, the papers were [re- sented to the president of the lloyal Society with a formal request (claim) in writing to have the case judicially examined by that Society. The documents were returned with a note from the President, {(>. B. Airy, Es([..) decidedly refusing, as President, to receive the papers or to acknowledge the claitn for investigation. Before this refusal was received, considering that the subject if correctly understood would be found to have I 6 PRKFACE. ail especial interest for the public of a practical aii'l immediate character, four letters containing a general cxplrnation of the great importance of the subject with regard to the immediate interests of the public, were forwarded to the Editor of the "Times" (London) tor publication in that Journal ; and, when the letter of the President of the Royal Society, refusing to entertain the case, was received, a copy of that letter was also forwarded to the Editor of the " Times." About the same time that the case was presented for investigation to the Royal Society, a communication thereof was made to McGill College, also formally re- questing an investigation. Not long afterwards, however, the papers were returned witliout, as it would seem, having undergone examination. Cojiies of the several documents referred to will be found at the end of the Appendix to this book. ».*. INTRODUCTION. The subject of this work is that gvonietiical problem which recjuiresu straight line to be drawn equal in length to the given arc of a circle ; or, a circle to be described equal in length to a given straight line — accompanied, in either case, with demonstration that the conditions of the requisition have been mathematically fulfdled. This problem we have succeeded in solving accordinu to the strict rules of geometry, and the denionstratiou that we have so done, is herein presented to tiie public. We publish our solution with the dit,Jnct statement that it is essentially in strict accordance witii that scien • ti fie system known as Euchd's. We claim to have our demonstration admitted or disproved, and we challenge objection or axlverse argument on that system. The book known as 'Euclid's Elements of Geometry,' although possessing a high degree of completeness com- pared with other scienti*ie treatises, and including a con- siderable part of the subjects belonging to tiiat division of science, is a human production and imperfect — it is neither absolutely complete, nor absolutely conqn-c- hensive. However desirable it may be to retain the formal method in and by which Euclid taught his application ol the system, there can be no good reason why the method siiould be restricted to any particular number of prob- lems, or why it should not be extended to include cases of the same character as those treated in the ' Elements,' but which did not come under the consideration of Euclid. I\TR<»I)ICTI<>.V. Lt't US brictly «'.\imiine rlie otsmtiiil I'liiinu'trristicsor the scieiititic system tanu;lit l»y Euditl, — ami also tlu! ainmm'UUMit ot" Kiidid's lorinal nietliod. The first essential ot" tlio systnn is tliat then' shall he a siinplo (elniuMitary) l)asis . . . of which tlu» n-aliiy, actiialiry, or truth, is absolutfly certain. This fuutla- nuMital basis (or each such basis, because there may be an in(lefniit«; numbt'r) beins^ elementary ami manifestly real and true, requires merely to be delined or stated with precision ; and it is termeil, accordingly, — d ilrji/ntioit. A geometrical «lefmitiou may therefore be called the verbal «M|uivalent of a fact. • When the basis is not (element.iry) quite simple, but re(|uires only a brief explanation to satisfy the reason- able mind as to its un([Uestionable truth or reality . . it is called — (di (ixii>»i. (Strictly speaking an axiom is a proposition or thc*)- rem, of which the manifest truth becomes so readily a}>parent as to render formal demonstration umiecessary ; or in some instances, the delinite statement of the axiom may include its demonstration.) t The postulates of Euclid's ' elements ' are the rules of his systematic method, * It may he said to assort the existence of the fact which it defines. f Therefore it is in the ^ame case with the definition, and it may ho considered a compound definition ; or, the defi- nition may be considered an elementary axiom. In either ca>e tliere is the definition only of a reasonable recognition by tlie mind of an actual existence or reality. (Def) c. ij., ' A straight line is that which lies evenly between its points,' the mind at once recognizes and reasonably accepts ilxafacf, which is matiifest whether tho line be considered only a natural or oidy an ideal realit}'. (Axiom.) e. ^., ' The wliolo is greater than its part,' the comparison has already been made b^- the mind, and the inter-relation of iho two things^ thus defined, is at once recognized by the mind as actual, (/. r., as an ideal act.) INTKoDrCTloN. 9 usdistiiijfiiislied from tin' laws or rules ol'tlio pliiI(jso|ili\', which hist are to hr Irariit from their iij»[»lieali(jii aii*l illustration throiiirhoiit the work. In statin*? this, we are not making a tancil'iil (listini'tiuii, but iiulicaiing a (litlbri'neo which it is of great nioii ent to correctly ajipreciate. Kuclid'.s ' Ideal Philosojdiy,' or * Scientific System,' if perfected, would he a perfect system of human reasoninir, that is, to say, a syst«'m un(U'r which, its rules and r«'gi;- lations heing strictly observed, the gidance of reason could be obtained (in compou'>dii'g kiiowledgi) on all subjects to which human knowledge is permitted to ex- tend, so as to insure the attainment of certainty and truth ill all cases. Euclid's systematic method, as taught in the work known as the ' Elements of Geometry,' is an endeavour to ap[dy that system to one of the divisions of science, — namely, the science of Form and l\raguitude ; an en- deavour which, witb the exception of one grievous and calamitous mistake, must be considered as having be«'ii successful in an astonishing degree ; for, as a complete, comprehensive and important work in itself, the (sc- called) 'Elements of Geometry ' may be justly regardetl as the greatest nrodiict, and as tbe iiroudest monument, 1 pr< of the human intellect the world has yet to show. The postulates of the ' Elements ' are the arbitrary rules* framed to insure method and [»recision intlie ap[)li- cation of the system to the science of ' Form and IMag- ■'' Arhiti-ary. liuwevor, in a relative, not in an absolute scn.ense, in the same relative manner that the lan> of material Xatuic are arbiti-ary in a spiritual sense; in either ea>c thoro is a limitation and res- ti'iction which may be termed artitieial, hut . . the one is a contrivance of human reason . . the othe-, an arrani^emei t by IHvine reason. #••* 10 INTRODUCTIOX. iiitude.' * Fonnally, they give permission to conduct the necessary process of construction in a particuhir manner ; thereby strictly prohibiting other methods of conducting tiie process. Now the importance of such strict rules in science generally, and in each of the divi- sions of science, is not to be lightly esteemed. For the purpose of preserving order and distinctness, and of en- abling a number of persons to work harmoniously (in concert) on the same subject, some such rules may be considered practically indispensable; the indirect benefit of their influence, also, in training the mind to a due re- gard for law and system, in its endeavours to acquire increased knowledr^e by compounding the elements there- of, is un(]uestionably very great. It would be, however, superstitious, slavish, and quite unreasonable, for men to allow themselves collectively or individually to be absolutely bound and fettered by a particular set of rules, framed long since by other men, viz., by Euclid and his predecessors, as the most advan- m/"4 * The science of ' Form and Magnitude ' may be correctly termed an abstract science, because the expi-ession 'abstract ' is so U( IXrUODLCTIOX. ■ l pose of giviiii^ coui[»lrt(Miess to the I'g nv, so as to enuUlo the necessary coinbiuatioiis and comparisons to be made in an onlerly and systematic manner. The geometrician is now in a position to ol)tain the answer to the rc(inis:- tion. Tliis is at lirst briefly stated as a positive or decided coiichision. The demonstration tlien followj« ; in which the answer to the nMjiiisiti i as stated is justi- fied by the facts upon which it is based, and sliown thereby to be the oidy correct exphmation of the result. We have elsewhere stated that the primary and most important of the two-fold purj)ose which Euclid's work had in view was to teach and illustrate the philosophy — i.e., the sci(Mitific system of reasoning, and that the application of the philosophy in his treatise on the science of ' Form and ^Magnitude,' does not justify the inference that the philosophy has a peculiar connection with that one division of science, and that it is not, witli the requisite modifications, equally apphcable to the other divisions of science. * Xote (a). In writing thus it is to be understood that we are idealizing Euclid and considering him as the representative author of a work, which cannot reason- ably be supposed the production of one individual only ; it m{iy be that a large part of the ^Elements,' as well as tiie arrangement of the parts and the coherent completc- the conclusion that it contemplates accepting) for tho ap- jn'oval of reason. If reason approve*, the experiment suc- ceeds ; if reason disapproves, the cxporiniciit fails. It may be an exhibition experiment : the experimenter has himself accepted the result (he has repeatedly performed the experi- ment and has knowledge that reason a^jproves the arranged ea-^e), but ho exhibits the experiment. For what purpose '^ In order to demonstrate to the s[)eetators tho approval of reason ; or, more strictly speaking, to demonstrate tho legi- timacy of the arrangement and soundness of its conclusion, wliich conclusion is thus authorized and commended l>y reason for the mind's acceptance as knowledge. LAi. IXTnODlCTlOX. 13 to onaUle be made iii'trieiau ! 10(111 i. si- si tive or follows* ; I is justi- d shown le result, lid most d's work, •sopliy — that the i oil the itify the imectioii ot, with he other ood that 1 as the reasoi;- al only ; 3 well as Jinpletc- tho ap- lOllt SllC- It may s him.solf c oxpori- iirrangod )urpo:so '<* >rovaI of tho logl- icliision, iided hy iiess of the exposition as a whole, is attribiitalth' to Euclid himself, but, to some extent, at least, the book must be considered as the airangement of the work of his prede- cessors by Euclid. We do not assert that Euclid himself had, in arranfinw the work, a clear and distinct appreciation of the philo- sophy as a system of reasoning not peculiarly connected with that one division of science to which it is applied i I the * Ehnnents.' It is almost certain that he had not, and it is, humanly speaking, almost impossible that he «'Ould have had such a distinct appreciation of the fact. In the first place, the choice of the subject of the appli- cation was not originally his, the compound of the philosophy with that particular division of science was known as Geometry before his time, and was already r garded, not as a compound, but as a peculiar (mystical) division of knowledge; if Euclid had clearly perceived the fallacy of thus regarding it, some decisive indication that he did so would appear in the work. And, secondly, it must be remembered that most of the divisions of science, as we now recognize them, were at t'.iat time rpiite unknown, and, it may be said that, scientific knowledge was in a great measure comprised i 1 that one division wdiicli alone had been scientifically arranged. It was, therefore, evidently almost impossible f »r Euclid to clearly appreciate the general relationship of philoso])hy to all the divisions of science, because he was quite unac(|uainted with even the e:^istence of the major part of those divisions, and was consequently not in a position to appreciate the inter-relation of the divisions to each other as parts of '^ne general science. (Note h.) — There is, however, some apparent evidence that Ewdid did in a measure regard the exposition of the particular science primaribj as a means of illustrating and teaching the general philosophy; instances of proposi- tions treated in a very indirect and elaborate manner, at least, suggest such an explanation- Notable instances 14 INTKODUCTIOX. of this kind are the 1st prop, of Book iir, and the 2nd prop, of Book XII.)* The rules of philosophy are not in the same case with the postulates of Euclid's formal method. The laws of reason — that is, the authoritative rules by which know- ledge is to be compounded, and according to which com- pound knowledge* is to be accepted as sound or rejected as unsound, cannot be altered, and may not be tampered with in any degree, nor is it permissible to disregard them. An unintentional neglect of these rules through iijnorance or carelessness entails in all cases some degree of retributive punishment on the oflender, according to the circumstances and the greater or lesser importance of the subject on which the mind thus neglects the laws of its intellectual existence. A wilful disregard, contempt, or defiance, of these rules is an intellectual crime, and, it may be, if the subject be of great importance, and, espe- cially, if the mind be of great capacity and the acknow- ledged representative of many others, a crime of fright- ful magnitude, of which the consequences fall, not on\v on the individual ottender, but on an indelinite iHiniber (a vast number, periiaps) of other persons. It is our purpose to bring, almost immediately, t'lis sul)ject, namely, the 'law of intellectual existence' and the 'responsibihties belonging to knowledge,' particularly before the public. We will now proceed to tin? demonstration of the geometrical problem, in its several forms — that is to say, the several forms of the general relationship in form and magnitude between tlie straight-line and the perimeter of the circle. ^ Wc will in the Appendix make some examination of these props, relatively to the above suggestion. THE CIRCLE AND STRAIGHT LINE, ]. Befinition. — If a circle be applied upon a straight line,c.iKl the circle be then moved ir vii the straight line in such wise that each point in the circumference, successive to the point in tiie circumference which is first in contact with the straight line, be brought successively into con- tact with the straight line — to wit, with each similar successive point in the straight line, conunencing from the point in the straight hne which is first in contact with the circle, and if the circle be in such wise con- tinually moved until a given point in the circumference, at Fin'. 1. a definite distance from the first point of contact, become in contact with the straight line — the circle is said to bo rolled upon the straight line from tlie point in the circumference first in contact with the straight line to the given point in the circumference. (Fig. 1.) Let A. B. C. be the circle applied upon the straight line B. E. F., and in contact with the straight line at A., and let B. be the given point in the circumfer- ence. If the circle ^ c' moved upon the straight line in the direction B. E. F., and be so moved that each successive point in the circumference between A. and B. become successively in contact Vk'ith each similar point successive in the straight line from the point A., and be so moved continually until the given point B. in the circumference n I I I 1^ , : t If I'. :1: lit ■I ; 'I J 10 THE CIRCLE AND STRAICHT LINE. of the circle be in contact witli the straiglit line — the circle A. B. C. is said to be rolled upon the straight line 7). E. F., from A. to B. on the circle, {a.) But if the given point be upon the straight line, at a definite distance from the first point of contact between tlie circle and the straightline, and the circle be moved, in such wise as before, until some point in the circumference be- come in contact with the given point on the straight line — the circle is said to be rolled upon the straight line from the first point of contact to the given point on the straight line. (Fig. 1.) Let F. be the given point on the straight line» and let tlie circle A. B. C. be continually moved from the point A., in the direction D. E. F., in the manner directed, until some point in the circumference be- come in contact with the point F. on the line — the circle,. A. B. Cis said to be rolled upon the straight line fiom. A. to F. on the line. (/>.) 2. Definition. — If a straight line be made to deviate from its evenness between the two extreme points thereof in such wise that the line be made to contain a circle or to be any jiart of the circumference of a circle — the straight hue is said to be bent or curved into the arc of a circle. Similarily if a curved line be extended until it lies evenly between its extreme points — the line is said to be unbent into a straight line. Postulates. — Let it be granted that : 1. A circle, or tliat any arc of a circle may be rolled upon a straight line. 2. A straight line may be bent into a curved line. 3. The arc of a circle, or other curved line may be unbent (extended) into a straight line. * * Should any cue be disposed to object that the operations for the pei'f'ormancc of which leave is here taken are of a mcehimieal nature, a little consideration mav conviuco them THE CIRCLE AND STRAIGHT LINE. 17 Axioms. — 1. The arc of a circle formod by bending a straight line into a curved line is equal in length to that straight line. 2. The straight line formed by unbending the arc of a circle into a straight line is equal in length to that arc. 3. If a circle be rolled upon a straight line from the point of contact in that straight line until a given point in the circumference become in contact with the straight line . . . the distance between the first point of contact and the last point of contact on the straight line, is equal to the distance between the first point of contact and the given point on the circumference of the circle. Construction, Plate 1 , Fig. 2. With the centre A, and radius A. B. describe the quadrant B. F. Bisect B. F. at M. Draw M. N., the sine of the arc B. M., at right angles to A. B., inter- cepting A. B. at N. From B. at right angles to A. B., draw B. E. of indefinite length, tangential to the arc B. M. Divide the arc M. F. into ten equal parts at the points of equal division a. h. c. d. e.f. g. h. i. From M. draw M. D. perpendicular to B. E. and intercepting B. E. at D. Scholium. — If the curved line, forming the arc B. M. cut oft' from the quadrant at M., is supposed to bo straightened upon the line B. E., the line B. M. will then throughout its entire length be applied upon the line B. E. {i.e., will coincide with a part of the straight line B. E.) and the point M. at the extremity of B. M. will that describing a circle with a centre and radius is quite as much so. But in fact the cii'cle is to be geometrically not mechanically rolled, and the reasoning is quite independent of any mechanical operation in both cases. It may be thu.s explained : — The ciz'cle is merely supposed to be rolled. Thr reasoning investigates and determines the alteration which would be occasioned in the relative positions of certain points if the circle were to be rolled. -ys,*. 18 THE CIRCLE AND STRAIGHT LINE. J . manifestly be in contact with (i.e. coincide with) a point in the line B. E., ut some place more distant than the point D. from B. Now the distance of the point of contact of M. from B. can be approximately detennined, and is approximately known. Let 0. indicate the unknown locality of the actual point of contact of M. when the curved line B. M. is straightened upon the line B. E. Again, if the arc B. M. in contact with the line B. E. at B. is supposed to be rolled from B. upon the line B. E.j until the point M. at the opposite extremity of the arc, become in contact with the line B. E., then, since each and every constituent (or component) part of the arc B.3I. is successively brought into contact with each similar constituent (or component) part of the line B. E., the point of contact must necessarily be the same point 0. touched by M. when the curved line forming the arc B. 31. is straightened upon B. E., and the straight line B. 0. {i.e. the straight line contained between the point B. and the unknown point which is indicated by 0.) must necessarily contain the same quantity of length contained in the arc B. M. Construction continued. — Suppose the arc B. M. to be straightened upon the line B. E. ; and let 0. indicate the point on the line B. E. which coincides with the point M. at the extremity of B. 31. when so straightened. From 0. perpendicular to B. E. draw 0. C. equal in length to A. B. With centre C. and radius C. 0. describe the quadrant 0. P. Bisect 0. P. at S. Through C. C. join S. 31. intersecting C. 0. at E., and through S. draw C. Q. intercepting B. E. at Q. Join C. P., P. Q. From S. perpendicular to B. E. draw S. T. Produce 31. S. through S. intercepting P. Q. ; and, from the point X. on the line so produced, taken at the distance S. X. equal to the distance 31. E. on the same line, draw the perpendicular X. Y. intercepting B. E. at Y. Divide the line E. S. into nine equal parts at the points of equal division h. r. d. c. f. y. h. i. THE CIKCLE AND STKAIOIIT LINE. 19 Examination by Hypothesis. (Kuclid. 2. xii.) On the as- sumption that M.a. on the line M. S. cqiiais M. a. on tho arc M. F. * Scholium. — Since, if the quadniiit B. F. be rolled upon the line B. E. until the point M. bisecting the quadrant arrive at 0. (which indicates the unknown actual poii't of contact), the point F. at the extremity of the (puidrant, must necessarily arrive at S., (0. »S^. represents what will then be the place of 3/. F.^ because M. arrives at the point indicated by 0., and F. then, necessarily, arrives at S.), the point e. bisecting tiie arc M F. must also, neces- sarily, (when the arc B. M. has rolled through half its length) arrive at e. on the line 31. S., because the line B. 0. is manifestly equal to the line M. S. which is con- sequently equal to the arc M. F. ; it, therefore, follows that each of the divisional points a. h. c. d. e. f. g. h. i. on the arc M. F. must arrive successively at each of the divisional points a. h. c. d. e. f. g. h. i. on the line M. S. Again, if the quadrant be further rolled upon the line B. E. until the point F., at the extremity of the quadrant, be in contact with B. E., the point of contact must neces- sarily be the point Y. (because the arc 0. S. (or M. F.) is similar and equal to the arc B. M., and the movement of the one arc is similar and equal to the movement of tho other) and T. Y. is equal to 2). 0-, therefore, since B.. arrives at the point indicated by 0., and since F. (or 5'.) must then necessarily (if the rolling be continued) arrive at F., the line 0. F.is equal to the line M. S. which is equal to the line B. 0., which by the construction is the same length as the arc B. M. Verbal definition. — When a magnitude i'^ said to contain a certain number of lesser magnitudes, i: is defined that the greater magnitude is wholly compounded of tliut certain number of those lesser magnitudes ; similarly, when a lino (or a definite quantity of longitudinal space) is stated to contain a certain number of (equal) parts, tho line (or definite quantity of space), is defined ])y the statcTuent to be wholly compounded of that certain number of those parts. ♦ See Demonst. {a) Part Second. Pkop, a. — Theorkm. If an arc vontaining one-eighth of a circle, be applied upon a straight line, and, from the terminal extremity of the arc, a perpendicidar he drawn intercepting the straight line, and if from the arc one-tenth thereof he cut off, then, if the remaining arc (to wit, the arc containing nine- tenths of the ivhole arc,) he rolled upon the straight line, the point of contact shall he the same jwint on the straight line intercepted hy the perpendicular drawn from the terminal extremity of the tvhole arc. Fig. 1. Let the arc B. M., containing one-eighth of a circle and described with the radius A. B., be the arc applied upon the stiaight line B, E., and let M. D. be the perpendicular drawn from M. intercepting the straight line at D. And let the arc B. m., nine-tenths the length of B. M., be the arc cut oif from B. M. If the arc B. m. be [rolled upon the straight line until m. arrive at the straight line, the point of contact shall be the point Z>. intercepted by the perpendicular M. D. THE CIRCLE AND STRAIGHT LINE. 21 From the radius A. B. take K. B. nine-tenths the length of A. B., and with centre K. and radius K. B., describe the arc B. n. similar to the arc B. M., and equal in length to the arc B. m.* Draw M. N. the sine of the arc B. M., and n.p. the sine of the arc B. n. From n, at the extremity of the arc B. n. draw the perpendicular H. C. intercepting the straight line at C. Let the arc B. M. be rolled upon the straight line until M. become in contact, and let 0. indicate the point of contact. Let also the arc B. n. be rolled until n, become in contact on the straight line, and let d. indicate the point of contact. Because the radius K. B. is nine-tenths of the radius A. B., the sine n. p. is nine-tenths of the sine itf. JV., and the arc length of B. w., which is indicated by B. (?., is nine-tenths the arc length of B. M. which is indicated by B. O.J and 0. d., the difference of the sine and arc length of J?, w., is equal to nine-tenths of D. 0. the differ- ence of the sine and arc length of the arc B. M. Now if the arc B. M. be rolled upon the straight line (back again) from the point of contact (indicated by 0.) until the opposite extremity of the arc be in contact with the opposite extremity of the straight line at B., the point M. must again arrive at M. the extremity of the perpendicular M. 2)., as in the first position of the arc, and since the difference of the arc length and sine oi B.n. has the same propOi*tion to the difference of the arc length and sine of B. M. which the arc B. n. has to the arc B. M., therefore, if the arc B.n. be rolled upon the straight line (back again) from the point of contact indi- cated by d.j until the opposite extremity of the arc be in contact with B. at the opposite extremity of the straight * Circles are proportional to each other directly as their radii ; there- fore, since the radius K. B. is nine-tenths oi A. B. the arc B. n. is nine-tenths the arc B. M. But the arc B. m. is nine-tenths of £. M., therefore, the arc B, n. is equal to the arc B. m. ;f- oo THK CIRCLE AND 8TRAI0IIT LINK. It line, V. the point at the extremity of the perpendictihir n. C.J at which the extremity n. of the arc arrives, must be arrived at by the point n. from a horizontal distance 0. d. having? the same proportion to D. 0., which the sine n. p. has to the sine 31. K. Now thv) arc B. n. is equal to the arc B. M. diminished by one-tenth,* and, C. 0. is the difi'erence (C. d.) of the ^^ine and arc-length of B. n., and the difference (D. 0.) of the sine and arc- length of B. M. taken together . . . that is, the difference of the sine of the lesser arc and the arc-length of the greater arc. But, because the ratio of B. B. to B. C. is the same as the ratio of B. 0. to B. d, and the same as the ratio of d. 0. to c. d., the part G. D. of the whole distance C. 0. has necessarily the same proportion to the part B. 0. of the whole distance which the sine n. p. has to the sine M. N., and, therefore, the distance 0. d. is the same as the distance C. 2)., and the point indicated by d. is the same point D. at the extremity of the perpen- dicular M. D. Wherefore it is demonstrated that if an arc containing one-eighth of a circle, &c., &c. Q. E. D. Corollary. — Hence it follows, — because the point «., when the arc B. n. is rolled, becomes in contact on the straight line at the point intercepted by the perpendi- cular M. B., and, because the distance B. D. is (conse- quently^ nine-tenths of B. 0., — that the sine of an arc containing one-eighth of a circle is nine-tenths the arc length, t {i. e., the ratio of the length of the sine to the arc-length is the ratio of nine to ten. Note. — Having regard to the great importance of the subject, and as this theorem is fundamental, it will be now repeated in a somewhat different form :— • That is, B. M. is the arc B. n. increased by one-ninth of B. n. t Indepeiwient demonstration will be given of this proposition, but the corollary becomes apparent so soon as the point of contact, at which the extremity of the lesser arc arrives on the straight line, is determined. THE CIRCLE AND STRAIGHT LINK. 23 Scholium. — In considering,' the following cuse, since il several similar arcs of (litlerent nuignitudes be rolled on a straight line, the motion of the terminal point at the extremity of each arc is compounded of the vertical and oblique motion of each and all the component parts of that arc, it is requisite only in comparing the results of the motions of two or more of the similar arcs with each other, to take the straight horizontal longitudinal advance — that is, the horizontal motion — into consideration ; be- cause the result of the horizontal advance includes the result of the vertical motion of which it is in part com- pounded, and which must be necessarily proportional to the horizontal advance and, also, to the magnitude of the arc. Theorem A. {repeated.) Fig. 1. When the arc B. M. is rolled, M. advances to a point (t. e., some point) on the straight line, indicated by 0. When the arc B. w., which is similar to the arc B. M., and contains nine-tenths of the length contained by B. M.f is rolled upon the straight line, n. advances to a point (some point) indicated by d. Now the distances O. and d. from B. are so related to each other by the construction that B. d. contains nine-tenths the length contained by B. 0. Also the sine n. p. of the arc B. n. is nine-tenths the sine M. N. of the arc B. M. (2.) Thedistance D. 0., which is the horizontal advance of the point M., contains the difference between the sine and arc length of B. M. Now, if B. M. be re- duced by one-tenth part thereof, and the remaining arc, by diminishing the radius in the same proportion, is converted into a similar arc, containing one-tenth less length, the sine of the lesser arc B. n. thus described will have the same proportion to the sine of the greater arc B. 31., — to wit, the sine n.})' of the lesser arc will be nine-tenths the length of the sine M. N. The difference (D. 0.) between the arc length and sine of B. M. is to the difference (C. d.) between the arc length and sine of 24 THE CIRCLE AND STRAIGHT LINE. V.I B. n. in the proportion of ten to nine. But the one- tenth by which D. 0. exceeds C. d. is manifestly the increase belonging to the one-tenth of the arc B. M. by which that arc exceeds the arc B. n. Therefore, when the arc B. M. is diminished by one-tenth thereof, if the lesser arc, thence resulting, is rolled until the terminal point become in contact on the straight line, the advance of the terminal point must be less than D. 0. by one- tenth of D. 0- But the sine of the arc B. n. is one- tenth less than the sine of the similar arc B. M., there- fore the distance between 0. and d, must be less than the distance C. 0. by one-tenth of D. 0. and one-tenth of B. D. taken together ; that is by the distance D. 0. ; be- cause B. D. and D. 0. taken together equal B. 0., and D. 0. is one-tenth of B. 0. ; therefore B. d. is equal to B. 0. diminished by D. 0. ; but the distance B. 0. dimi- nished by the distance B. 0. is B. D., and therefore the THE CIRCLE AND STRAIGHT LINE. 36 when distance B. d., is the same as the distance B. D. Where- fore it is demonstrated that the point of con act indicated by d. on the straight line B. E., at which the terminal point of the arc B. n. arrives, is the point intercepted by the perpendicular M. I), drawn from the terminal point M. of the greater arc B. M. . d. : B. 0. ) therefore . d. : B. 0. 3 B. d. = B. D. (3.) Because : — B. C. : B. D. : : B and B. C. : ^. d. : : B That is :— The sine of B. n. is to the sine of ^. M. as the arc-length of B. n. is to the arc-length of B. M. The sine of B. n, is to the arc-length of B. n. as the arc-length of B. n. is to the arc-length of B. n. increased by one-ninth. Wherefore the arc-length of B. n. equals the sine of B.Jd, It immediately follows — because d. 0. is one-tenth of B. 0., that D. 0. (which is the same) is also the one-tenth of B. 0. (4.) Again ; let it be assumed to be possible that n. ■ ''vftnces to some point, d., less distant than D. from JB., then must the ratio of the distance B. 0., to the distance B. d,j be greater than the ratio of Jd. K, to n.p., because the point D. is, by the assumption, in advance of the point d., and the greater arc must advance from (its starting point) the point Z>. proportionally to the advance of the lesser arc (from its starting point) the point C. Now the distance B, 0. contains the distance B. C, together with the advance of the two arcs taken together — to wit, the advance of M. from D., together with the advance of n. from C, and together with the difference, if there be any, between d., the advanced place to which w. advances, and D. the place from which M. commences to (xlvance — that is, if d. be less distant than D. from a, C. 0. includes the diff. D. 0. together with i 26 THE CIRCLE AND STRAIGHT LINE. the difF. C. d. and together with the diff. d. D. ; (thus . . . C. 0. includes the advance of n. through C. d.j of M. C. d. + 0. d. and the distance d. D.) But B. through 9 C. and C. d. taken together equal B. d. which is equal to n. p. And M. N. is to n. p. as ten to nine- There- fore B. d. + C.d.+ C. d. is to B. C. + C. d, as ten to nine. 9 By the assumption B. 0. contains QB. G-^C. 0.) B. d. + C.d. + C. d. + d. B.J and therefore the ratio of B. 0. to 9 B. d. is gi-eater than the ratio of M. N. to «. p. by the distance d. D. Now, it is manifestly impossible that B. M. can increase by advancing in a greater ratio of proportion to the increase of B. n. than the proportion of the sine M. N. to the sine n. p.j because B. M. and B. n. are similar arcs ; therefore the distance C d. cannot be less than the distance C. D. By similar reasoning it may be shown that the distance C. d. cannot be greater than CD., because then the advance of the arc B. M. would be pro- portional to the advance of the arc B. n. in a ratio of proportion less than the ratio of the sine M. N., to the sine n. p., to suppose which would be absurd. Wherefore it is demonstrated that the point of contact of the lesser arc B. n. on the straight line B. E., indicated assump- tively by d., is the same point I), at the extremity of the perpendicular M. D. Q. E. D. (5.) Again — Fig. 3 (repeats the construction of Fig. 1.) Produce B. M. through M.., and make F. D. equal to A, B. With centre F. and radius F. D. describe the arc D. N. equal and similar to the arc B, M., produce the straight line E. B. through B. indefinitely, and upon the line so produced roll the arc B. N. from B. in the direction B. B. until the extremity N. of the arc becomes in contact upon the line at T, Now the distance B. T. is manifestly equal to the distance B. 0., THE CIRCLE AND STRAIGHT LINE. 27 because if the arc be rolled upon the line to the ex- tremity of the arc opposite to the point of contact, the advance in that direction must necessarily be equal to the advance in the opposite direction, if the arc be simi- larly rolled in the opposite direction ; but the other extremity of the arc — to wit, the extremity which is in contact with the line at D., must likewise advance (to *S'.) in the direction D. B., a horizontal distance D. h. equal to the distance B. T. ; manifestly the arc T. S. must be equal to the arc JJ. N. ('because both are the same arc in an altered position.) Also I), h,, which is equal to D. 0., has the same ratio to C. d. which M. N. has to n. p. — to wit, the ratio often to nine. And the arc S. T. has likewise the same ratio to the arc B, n., and the line B. T. the same ratio to the line C d. Therefore d. B. + B. T. (^ d. T.) = B. d. + d. 0. 'f 28 THE CIRCLE AND STRAIGHT LINE. 'k '■■{ 'i'm (= B. 0.) Now B. 0. contains B. C. + C, D. + D. 0., and d. T. contains B. T. + B. C. + 0. d., and since B. T. equals D. 0., C. d. must equal C. D. Wherefore it is demonstrated; &c. Note. — This last method is given as furnishing an inde- pendent means of testing and verifying the preceding demonstrations. Prop. B. — Theorem. That if a straight line he drawn from the radius of a circle at right angles to the raditiSf and the Une be drawn to the circumference of the circle, such that one-eighth part of the circle be contained between the point on the circum- ference intercepted by the straight line, and the point intercepted by the radius — then if the one eighth-part of the circle be divided into ten equal parts, the straight Une so draum shaU contain nine equal parts, each of them equal to each of the ten equal parts contained in the one- eighth part of the circle. Fig. 4. Let M. N. be the straight line drawn from the radius A. B. of the circle, of which circle the quadrant B. M. F. is bisected by the straight line in the point M. The straight line M. N. shall contain nine equal parts, each of them equal to each often equal parts con- tained in the half-quadrant B. M. From B. at right angles to A. B. draw the straight line of indefinite length, B. E. ; and, through M., pro- duce the straight line ^. M, indefinitely . From M. draw the pei*pendicular B. D., intercepting the line B. E. at D. Roll the arc B. M. upon the straight line from B. in the direction B. E., and from the point 0.,* where the ter- minal extremity, M., of the arc becomes in contact, draw the perpendicular 0. C. equal to A. B., intersecting the production of N. M. in the point i?. With centre G., * Theorem A. ' ., THE CIRCLE AND STRAIGHT LINE. and radius C. 0., describe the quadrant 0. P. bisected by the production of JV^. M. in the point S. Divide the line R. S.j cut off from the production of N. M. by the arc, into nine equal parts at the points of equal division h. c. d. e.f. g. h. i. Because the lines N. B., M T}., R. 0. are all of them perpendicular to the straight line B. E,, the line M. N. is equal to the line B. D., and M. M. is equal to 0. D. Now it has been demonstrated (Prop. A.) that if the arc B. M. be rolled upon the straight line B. E. until the terminal extremity M. of the arc becomes in contact on the line, the magnitude B. 0., cut oiF from the line B. E. by the point of contact 0. is greater than the mag- nitude B. D., cut off by the perpendicular M. 2>., in the ratio of ten to nine. But the arc-length M. F. is equal to the arc-length B. Jlf., and R. 8. is equal to B. D., and M. R. is equal to D. 0., therefore M. S. is equal to the arc- length ofM. F. And since M. R. is one-tenth of M. S. and R. S. is divided into nine equal parts, M.R. is equal to one ef those equal parts. But the line M. S. is equal to the arc-length of M. F., therefore, M. F. contains ten equal parts, each of them equal to each of the equal parts con- tained in the line R. S. Now the line R. S. is equal to the line M. N., and the arc M. F, is equal to the arc B. M. Wherefore it is demonstrated that if a straight line drawn from the radius of a circle, &c. q.e.d. Corollary. — Hence — since if the line, drawn from the radius, intercepting the circle on the one side, be drawn from the circle through the radius until it intercepts the circle on the other side of the radius, the part of the line on the one side of the radius is necessarily equal to the part of the line on the other side, and the fraction of the circle, contained between the point of interception of the line on the one side and the extremity of the radius, is equal to the similar fraction of the circle cut off between the extremity of the radius and the point of interception Ill 80 TIIK CIRCLE AND STRAIGHT LINE. of the line on the other side — it follows that, if a straight line be drawn from a circle througli the radius, at right angles to the radius, until the line intercept the circle at the opposite side of the radius, and the line be so drawn that the part of the circle next the extremity of the radius be the one-fourth part of the circle, then — if the straight line so drawn be divided into nine equal parts, the fourth part of the circle shall contain ter equal parts each of them equal to each of the nine equal parts con- tained by the line, and the whole circle shall contain forty such ecjual parts. Prop. C— Problem. Itcquisition. — It is required upon a given straight line to describe an arc, such that the arc siudl be equal in length to the given straight line, and shall contain a definite fraction of a circle of definite magnitude. DcfinHion. — Fig. 5. Let B. 0. be the given straight line : it is required upon B. 0., to describe a definite arc, equal in length to B. 0. Construction. — Divide the line B. 0. into ten equal parts at the points of equal division a. h. c. d. e.f. (j. h. i. From the point B., at the extremity of the straight line, draw the perpendicidar of indefinite length B. fi., and from the point /., on the straight line B. 0., draw the perpendicular of indefinite length i. H. On the line B. c. take B, d\, equal to B. ?'., and join i. d\ From B. at right angles to /.f?'., draw the line of indefinite length /?.7:i'., and on the line B. E. take B. C. equal to B. i. Prodiu'e B. 0. inde- finitely through 0. and, through the point C, draw a line at right angles to B. E., intercepting B. c. at A. and intercepting the production of ^. 0. at D. With centred, and radius A. B, describe the quadrant B. M. F. bisected at M. by the perpendicular i. //. From the point 31., and at right angles to A. B., draw 31. N., intercepting A. B. at ^V. THE CIRCLE AND STUAIGHT LINE. 31 Eesult. — The arc B. d. described with the definite radius A. B. shall be the required arc, equal in length to the given straight line B. 0. Demonstration. — Because the arc B. 31. F. is a quad- rant, and the point ilf. bisects the quadrant, tlierefore the arc B. M. contains the eighth part of a circle. Nov M. N. is the sine of the arc B. M. and M. N. is manifestly equal to B.i. on the line B. 0.; but B. i. con- tains nine of the ten equal divisional parts of tlie line B. 0., and it has been demonstrated ( Prop, b. ) that, if the sine of an arc, containing the eighth part of a circle, be divided into nine equal parts, the arc contains ten equal parts, each of them equal to each of the nine equal parts contained by the sine. Therefore B. M is equal to B. 0. Wherefore the arc B, M. containing the eighth part of a circle has been described with tlie definite radius A. B. upon the given straight line B. 0., and it has been shown to be equal in iiiigth to the given straight line, as was required to be done. Corollary 1. — To describe itpon the given straight line a quadrant equal in length to the given straight line. (Fig. 6^) Bisect the vudius A. B. at the point c^., and from a. at right angles to A. B draw the line of inde- finite length a. g. With centre a. and radius a. B. describe the quadrant B. c. Now because a. B. is the one half of yl. B., the quad- rant B. c. is the fourth part of a circle of half the magni- tude of the circle of which the arc B. M. (Fig. 5.) is the one eighth part ; consequently the quadrant B. c. is equal in length to the arc B. M. ; and it has been demonstrated that the arc B. M. is equal in lengtli to the line B. 0., wiiich is the given straight line. Wherefore the quad- rant B, c. is also equal to the given straight line. 32 TKE CIRCLE AND STRAIGHT LINE. iW )■■!{ 1 Corollary 2. — To describe upon the given straight line an arc containing the sixteenth part of a circle such that the arc shall he equal in length to the given straight line. {FW. 6'^) Produce A. B. through J.., and from the production take R. B. twice the length of A. B. From B. through F. draw B. F. G. of indefinite length. Witii centre B. and radius B. B. describe the arc B. P., inter- cepting B. F. G. at P. Bisect the arc B. P. at Q. Now, since the arc B. Q. is the one half of the arc B. P., which contains the eighth part of the circle ; and, because the radius B. B. is twice the length of the radius A. B. ; the arc B. Q. contains the sixteenth part of a circle twice the magnitude of the circle of which the arc B, M. con- tains the eighth part. Therefore the arc B. Q. is equal in length to the arc B. M., and it has been shown that the arc B. M. is equal in length to the given straight line B. 0. Wherefore the arc B. Q., which contains the six- teenth part of a circle, is also equal to the given straight line. {Note.^ — It would be more strictly methodical to give these two corollaries as separate propositions, formally supported by demonstration that the magnitudes of circles are in the same ratio one to another as the ratios one to another of their respective radii. But the theorem to which such demonstration belongs is one of the funda- mental facts upon which the science of trigonometry is indirectly based. It seems, the:' 3fore, preferable to avoid complicating the immediate subject, and to give these secondary demonstrations in the less formal but more concise manner ir which they are here presented. This same Note has reference, also, to the corollaiy of Prop. A. which in strictness requires formal definition of the sine of an arc ; but the approved trigonometrical nomeu- elature of the lines belonging to the circle is quite definite, and we ma}^ for the present purpose, assume the definition to be contained in the expression. THE CIRCLE AND STRAIGHT LINE. 33 Prop. D. — Problem. acquisition. — Through ii given point in tlie circumfer- ence of a given circle it is required to draw a atraiglit line tangential to the circle, which shall be equal in length to the given circle. Definition. — Fig. 6. — Let B. E. G. H. be the given circle, and let B. be the given point, — it is required to draw a straight line through the point B. which shall be equal in length to the circle B. E. G. II. Construction. — Through the point B. draw the hue d. B. a. of indefinite length, and from B. perpendicular to d. B. a. draw the diameter B. G. ; bisect B. G. at C, the centre of the circle ; and, through C. at right angles to B. G., draw the diameter E. H. Bisect the quadrant B. E. at M. ; and, at right angles to B. C, draw the line M. 0., intersecting B. C. SitN., and intercepting the circle and bisecting the quadrant B. H. at 0. Divide the chord M. 0. of the quadrant M. B. 0. into, eighteen equal parts. On the line d. B. a. take, from B. in the direction -B. f/., B.f. containing ten equal parts, each of them equal to each of the eighteen equal parts contained in the chord M. 0. (Theor. B.,) and on the same line d. B. a., from B. in the opposite direction, take B. c. containing likewise ten equal parts, each of them equal to each of the eighteen equal parts contained in the chord M. 0. Increase B. f. through /. on the line B. d., and take B. B. equal to four times the length of B. /., and on the same line in the opposite direction increase B. e. through e., and take B. A- equfil to B. D. Bcsult. — The straight line D. B. A. contained between the points D. and A. shall be the required straight line equal in length to the circle B. E. G. II. Demonstration. — Because the point M. bisects the quadrant B. M. E., and because the sine M. N, of the 94 THE CIRCLE AND STRAIGHT LINE. arc B. M. is divided into nine equal parts, therefore B.f., part of the line d, B. «., containing ten equal parts each of them equal to each of the nine equal parts contained in the sine M. N., is equal in length to the arc B. M. (Prop. B.) But the part B. e. of the same line d. B. a. i s equal to the part B. /., and is similarly related to the arc B. 0., which is similar and equal to the arc B, M. ; therefore/ B. c, the two parts of the line taken together, is equal to the quadrant M. B. 0. containing the two arcs taken together. Now the line I). B. A. is (by the construction) equal to the part thereof /. B. e.,, taken four times together, and the circle B. E. G. H. is equal to the quadrant M. B. 0. taken four times together ; therefore the line D. B. A. is eqiml to the circle B. E. G. H. Wherefore the line D. B. A., drawn through the given point B. in the circumference of the circle, and tangen- tial to the circle, is equal in length to *^^he circle B. E. G. H. Q. E, D. Corollary. — Hence it follows that, if the chord of a quadrant be divided into (9) eighteen equal parts, a straight line containing (10) twenty equal parts, each of them equal to each of the (9) eighteen equal parts contained in the chord, is equal in length to the quadrant. Wherefore if a square be inscribed in a circle the ratio of the inscribed square to the circle is the ratio of nine to ten. 'it I I refore B. /., 1 parts each 8 contained arc B. M. ine d. B. a. ated to the arc B. M. ; 3n together, ng the two . is (by the '. e.j taken H. is equal s together; ircle B. E. h the given md tftngen- rcle B. E. E.D. chord of a al parts, a rts, each of jqual parts e quadrant, the ratio of I of nine to af ^ i. TPIE APPENDIX, (a) The question to be here briefly noticed ia, whether 'Euclid's Elements/ having regard to the apparent intention of the author (or authors) of that work, should be considered a treatise on one peculiar division of knowledge (one science) only — that is to say, a treatise on the inter-relation of the subjects of the science and of the laws which govern and belong to that peculiar (so-called) science of geometry ; or whether the work should be considered, in regard to its purpose, a practical treatise on applied reasoning — teaching, by illustration, the correct mode of building-up (compounding) science from its elements. We will briefly examine two important propositions of those belonging to the * Elements ' as to any evidence they may aftbrd in this respect. The first prop, of the third book : ' To find the centre of a circle.' It is at once evident that this prop., because of the use made by Euclid of the circle, must be con- sidered, if regarded as one of the Elements of a peculiar science (geometrv), as a prop, of great relative import- ance. The plan of the book, assuming the purpose of that plan to be a treatise on the science only, would call for a solution of this prop, in such a form as to constitute it a primary or fundamental prop., upon which secondary propositions and corollaries could be based and shown to be directly dependant. One or more definitions or axioms might obviously be provided to furnish any necessary support for a concise and comprehensive solu- tion. At least, in regard to such supposed purpose, a direct character for the solution would suggest itself as almost imperative. As it stands, the character of the solution is indirect, and may be termed negative ; it gets constructively at the answer to the requisition by a very simple operation, but, as a reasonable and sub- jective proceeding, this operation is only supported by % 36 APPENDIX. w i i -If m the negative results of an exhaustive process. The question naturally suggests itself whether the construc- tion has been, in the first place, merely the result of a fortunate guess. If it is based upon and Las been com- pounded from elements previously verified, why is it not shown to be so ? As it stands, we cannot take the result, and by inverting the order of the reasoning, get at the construction as subjective. No reason for per- forming the (given) constructive operation is apparent. Again, the corollary is stated to be manifest ; but, it is certainly not manifest as a corollary to the negative solution of the prop. If the corollary is manifest, it is manifest independently (in itself), and the solution of the prop, might be very well based upon the corollary. By exhibiting the fact stated in the corollary, in the fonn of a demonstrated theorem, the problem (Prop, i.) might be directly and positively solved in a few lines, and Prop. III. (Theorem) might become unnecessary. Now, if we suppose the primary purpose of the work was to illustrate the applied method, and that the parti- cular science (chosen for the purpose of illustration) was considered quite subordinate to that purpose, it is much easier to understand how such a form of solution might suggest itself as desirable in this and in some other similar instances. The negative exhaustive process illustrated in this prop, is undoubtedly in some cases of much value, and it may be considered, in some sense, a distinct kind of reasoning ; hence, having regard to the primary purpose, it might be thought desirable to exhibit it as illustrating a proposition which from its fundamental character and importance would be more likely to attract attention and interest the mind. We cannot consider this as furnishing an altogether satisfactory explanation of the fonn here given to the solution, but it seems to be much more intelligible when so considered, viz., as primarily intended to illustrate the applied method of reasoning. APPENDIX. 37 The second proposition of the 12tli book is a similar and still more striking instance, which appears to strengthen and confirm the explanation just suggested in the case of Prop. i. Book in. This proposition (Prop. ii. Book XII.) ' Circles are to one another as the squares of their diameters/ is also evidently a proposition of u primary and important character because defining the nature of a fundamental relation between circles, and therefore here again we should expect a direct, positive, and simple treatment of the case, and, on the contrary, we again find the indirect negative treatment exhibited in a long solution of considerable complexity ; the only characteristic difference between the treatment in this case and that of Prop. i. Book in., is that in this, the construcrion, like the solution to which it belongs, is complex and only indirectly related to the proposition, whereas, in the former instance, the construction is simple and directly related to its proposition. But the particu- lar mode of the reasoning, for the illustration of which this proposition appears to have been selected, is in itself of a very refined and instructive character. In the kindred science of ' Number and Quantity,' calculations of great importance are derived from and based upon the quantitive equivalent of this proposition : for example — ' Legendre's Numerical and Trigonometrical Proposi- tions ' concerning the quantitive relations of polygons and the perimeters of circles.* It may be also useful to * We may remark here the confusion liable to arise from using the same expression ' geometricar which is applied to the propositions of the science of Form aud Magnitude, for those of the science of Number and Quantity. To these last some of Legendre's propositions belong exclusively, whilst others are of a hybrid character, J; is true these sciences have a common boundary where they approach each other very nearly, but it is a stumbling-block in the way of the student to find them thus mixed together without indication as to their distinct and different characters. 38 APPENDIX. point out that this is one of the propositions in which Euclid defines, by illustration, the use of the hypothesis. In this instance, the hypothesis is assumed to be true, and is then made to temporarily occupy the place of, and serve as a fact, so that combination can be carried forward and the results tested by comparison. The terms in which this proposition is stated require particular notice, in regard to the sense in which the expression * square of the diameter ' is used 5 for, the very same expression is used in the kindred science of ' Number and Quantity,' and is used therein in an essentially different sense. Since, therefore, we are here, almost on the border land which connects these two sciences, there is much danger of the expression used in the sense belonging to the one science being mistaken for the same expression used in the sense belonging to the other science. The proposition under consideration is thus stated in the Elements of Euclid^ ' Circles are to each other as the squares of their diame- ters.' The expression 'square of the diameter' here means (as used by Euclid) a square of which the diameter of the circle is one of the four equal sides. The state- ment would therefore have the same (equivalent) mean- ing if the word 'square' be left out and we read simply ' circles are to each other as their diameters.' To de- monstrate the one statement is to demonstrate the other ; because if two magnitudes be proportional one to the other, equimultiples of those magnitudes are propor- tional in the same ratio ; and, if equimultiples of those magnititudes be taken from them respectively, the re- mainders are likewise proportional in the same latio. But in the applied science of ' Number and Quantity,' the same expression, * square of the diameter,' has a dif- ferent meaning. To correctly appreciate the nature of the difference it is nece»<-':iry to observe that all ' Number ' or ' Quantity ' is relative. It has, always, reference to a standard of '^:- APPENDIX. 39 comparison * (an apparent exception to this is tiie number one, or unity, but here the number is itself equivalent to its standard of comparison). In trigo- nometry ' square of the diameter ' means ' a quantity of magnitude (diameter) taken as many times as there are units contained in that quantity of magnitude (diameter).' An example will at once mai^e the charac- ter of this difference quite obvious. Let the two diameters be proportional one to the othei- in the ratio of 'four^ to ' Uvo' — then the square of theiv diameters in the sense intended by Euclid will he—fmir taken four times, and two t, ken four times ) that is, sixteen to eighth magnitudes proportional in the same ratio as before. But in the numerical or quantitive sense we get— four multiplied hy four gives sixteen (4x4 = 16); and — two multiplied by tivo gives four (2X2 = 4). The numerical proportion becomes then fore 10 : 4, instead of 16 : 8 ; — the ratio of the proportion is no longer the same as before. We have recently made public a notice of very grave errors in astronomical science, certain of which appear to have arisen from a fundamental misapprehension (non- appreciation) as to the relation of the semi-diameter (radial-distance) to the circumference of the circle — e. 7 . the doctrine of the (supposed) law of equable areas. The question may suggest itself whether the use of the expression ' square of the diameter ' with the two different meanings undistinguished, may not have assisted to prepare the deceptive foundation for that super- structure of unsound knowledge which has been since built upon it. * Which is commonly termed a ' unit. ' C( Tc tai or 'th giv fini be wh COK stri tha ciai 1 beii ent mei as t reqi mac dchi of t adra I re righ stric nam that and, mys j)ubl the I oft), Tl lege. !-am< COPIES OF THE DOCUMEXTS REFERRED TO IX THE I'REFACE. To G. SlR,- MoNTREAL, 20th Docerabor, 1872. B. Airy, Esq., President of the Royal Society, London. —I have the honor herewith to send you papers con- taining the solution to that geometrical problem (commonly or vulgarly known as ' Squaring the Circle '), which requires ' that a straight line shall be drawn equal in length to a given arc of a circle of definite magnitude,' or ' that a de- finite arc belonging to a circle of definite magnitude shall be described equal in length to a given straight line,' and which requires that the demonstration (proving that every condition of the requisition has been fulfilled,) shall be in strict accordance with the system of reasoning laid down in that w^ork generally known, and recognized by mathemati- cians, as the ' Elements of Euclid,' or ' Euclid's Geometry.' The papers are submitted to you for the purpose of their being examined by those whose mathematical knowledge entitlcr. them to express an authoritative judgment as to the merits of the case thus laid before the Royal Society ; and, as the subject is one of great public importance, I have to request that as soon as the necessary examination has been made, (i.e., within reasonable time and without unnecessary delay,) the correctness of m}'- demonstration and the result of that demonstration shall be publicly acknowledged and admitted as estp'ilished fact ; or, otherwise, if disputed— then I require, which I submit that I have an unquestionable right to do, that the objection or objections shall be made strictly in accordance with the same system of reasoning — namely, that laid down and illustrated by Euclid, — in order that I may have the opportunity to meet such objections and, if I am able, to disprove their validity, and so to put myself in a position to insist upon my demonstration being publicly acknowledged as sound and true. Yours respectfully,. JOHN IIAifRLS. (XoTE. — The above coi)y is taken from a I'ough draft of the letter, and may be possibly not a s'l-ictly accurate copy of the letter forwarded to the President of the R. S.) The letter accompanying the papers sent to IMcGill Col- lege, was of the same teiioi-, and in foi'ni substantially the ^ame, as the foregoing. IvOVAL OnSEIlVATOUY, (JiiEENwrcn, London, S.E., February -1, 1873. Sir, I have to acknowledge your letter of Decern ber 20, atUlresf^ed to me as President of the Royal Society, (which has reached me through the hands of John Marshall, Esq.,) inclosing papers which are STippo.sed to contain an investi- gation on trustworthy principles of the problem of ''Squar- ing the Circle." Yoii are perhajjs aware that the Foreign Acatlemies, in general, by express statutes, refuse to receive communica- tions on this subject. I do not know that the Royal Society of London is jirevented by statute from receiving ihcm, but I know that its practice is unvarying ; and, so far as my private judgment is concerned, I ajiprove of that practice. I must, therefore, decline to present the papers to the Royal Society. I cannot myself give any time to their examination; nor should I think it right to force them on the attention of others. I have therefore thought it best at once to return them entire to Mr. Marshall. I am. Si J', Your obedient servant, (Signed,) G. B. AIRY. Jno. Uakris, Esq • •f-t !■ :'# MoNTiiEAL, 21st December, 1872. To Mb. John Harris, Sir, I herewith return your papers relating to the problem known as Squaring the Circle. I have not received from any of the professors in McOill College any observations on the subject. Your obedient servant, (Signed,) CHAS. D. DAY. KNwrcn, vy i, 1873. oecinbor 20, icty, (which ■shall, Esq.,) n an investi- i of ''Squar- Jademios, in communicrt- ayal Socict}- g ill cm, but > far as my at practice, o the Royal nation; nor attention of ce to return S. AIRY. ior, 1872. ing to the lot received bservations ^nW D. BAY. %, V ■^^1 mm ?sft