ALGEBRA IDENTIFIED WITH GEOMETRY; IN A SERIES OP FIVE TRACTS:I. EUCLID'S CONCEPTION OF RATIO AND PROPORTION. II. "CARNOT'S PRINCIPLE" FOR LIMITS. III. THE LAWS OF TENSORS, OR THE ALGEBRA OF PROPORTION. IV. THE LAWS OF CLINANTS, OR THE ALGEBRA OF SIMILAR TRIANGLES LYING UPON THE SAME PLANE. V. STIGMATIC GEOMETRY, OR THE CORRESPONDENCE OF POINTS IN A PLANE. BY ALEXANDER J. ELLIS, F.R.S. LONDON: (APRIL, 1874) C. F. HODGSON & SONS, GOUGH SQUARE, FLEET STREET. Price Five Shillings. With the Author's Compliments. ALGEBRA IDENTIFIED WITH GEOMETRY; THAT IS TO SAY, ORDINARY OR COMMUTATIVE ALGEBRA, INCLUDING INCOMMENSURABLES, NEGATIVES, AND IMAGINARIES, SHEWN TO BE A PURELY GEOMETRICAL (AND NOT A PURELY ARITHMETICAL) CALCULUS, AND THE HIGHER PLANE GEOMETRY OF DESCARTES, PLUECKER AND CHASLES SHEWN TO BE PARTICULAR RESULTS Or, ORDINARY COMMUTATIVE ALGEBRA, WHICH INCLUDES THE MIUCH MORE GENERAL PLANE GEOMETRY OF STIGMATICS; IN A SERIES OF ROUGH NOTES, FORMING FIVE TRACTS:I. EUCLID'S CONCEPTION OF RATIO AND PROPORTION, EXPLAINED IN A PROPER FORM FOR ELEMENTARY INSTRUCTION: II. "CARNOT'S PRINCIPLE" FOR LIMITS, REDUCED TO AN ELEMENTARY GEOMETRICAL FORIM: III. THE LAWS OF TENSORS, OR THE ALGEBRA OF PROPORTION, AN ORIGINAL CONCEPTION AND DEMONSTRATION, COMPLETING THE ALGEBRA OF THE GENERAL GEOMETRY OF M3AGNIIUDE OR OF RATIOS: IV. THE LAWS OF CLINANTS, OR THE ALGEBRA OF SIMILAR TRIANGLES LYING UPON THE SAME PLANE, AN ORIGINAL CONCEPTION, DEMONSTRATION, AND) EXPOSITION, COMPLETING THE ALGEBRA OF THE PLANE GEOMETRY OF DILRECTION, WITH EXAMPLES OF PROCESSES: V. STIGMATIC GEOMETRY, OR THE CORRESPONDENCE OF POINTS IN A PLANE, AN ORIGINAL CONCEPTION AND DEMONSTRATION, WITH EXAMPLES AND ILLUSTRATIONS, GIVING AN ELEMENTARY GEOMETRICAL CONSTRUCTION FOR ALL CASES OF SO-CALLED IMAGINARY POINTS, LINES, AND FIGURES, HITHERTO CONSIDERED AS EXCEPTIONAL, BUT NOW SHEWN TO BE THE USUAL CASES OF THIS HIGHER PLANE GEOMETRY, WHICH INCLUDES THE FORMIER REAL FIGURES AS RARE AND PARTICULAR OCCURRENCES. BY ALEXANDER J. ELLIS, B.A., F.R.S., F.S.A., F.C.P.S., F.C.P. MEMBER OF TITE MATHFI',MATICAL SOCIETY, FORMERLY SCHOLAR OF TRINITY COLLEGE, CAIBRIDGE, SIXTH WRANGLER IN 1837, TRANSLATOR OF MARTIN OHM'S "t GEIST DER MIATHEIATISCHEN \ANALYSIS,' PRESIDENT OF THE PHILOLOGICAL SOCIETY, AUTHOR OF " EARLY ENGLISH PRONUNCIATION." VWITH ONE PHOTO-LITHOGRAPHED TABLE OF FIGURES. LONDON: (APRIL, 1874) C. F. HODGSON & SONS, GOUGH SQUARE, FLEET STREET. LONDON: PRINTED BY C. F. HODGSON AND SONS, GOUGH SQUARE FLEET STREET. PRELIMINARY NOTICE. The Publication and Form of the Following Notes were thus conditioned. A Sub-committee appointed by the Association for the Improvement of Geometrical Teaching, laid before its members, at its last sitting, 13 January 1874, several schemes for the Treatment of Proportion, on which the members were requested to give their opinion not later than 31 Mlarch following. In writing my own opinion as one of the members of the Association, I found it impossible to complete my argument in favour of the retention of Euclid's method, and take into consideration several points suggested in those schemes, without communicating at least the bases of certain unpublished results of researches on which I had been engaged for many years. Hence it seemed necessary, as a matter of date, to give that communication a public form. But as all remarks upon the schemes submitted by the Sub-Committee were of course private and confidential, this publicity could not be attained without dividing my remarks into two parts. The first part, called Rough Notes on Proportion, has been sent round privately. The present, or second part, has been made fuller than would have been necessary for the purpose of my argument, although it is still very defective as an exposition of my theory, and is formally published, in order that it may be procurable in the usual way; but copies will be presented to all members of the Association for the Improvement of Geometrical Teaching, and of the Mathematical Society, to some other learned Societies, and to some Public Libraries, and to several other English and Foreign mathematicians. So far as copies remain on hand, I shall always be happy to present one to any Professor, Tutor, or Teacher of Mathematics, at any College or Public School, or any mathematical Author, English or Foreign, who favours me with a written request. Both parts have been written and printed under great pressure of other work, and at great speed (Appendix III.), leaving doubtless many marks of haste. for which indulgence is requested, as also for the figures, which had to be photolithographed from my own roughly executed drawings. My ambition is to present the Arsenal of Mathematics with a New Arm of Precision, of which the following pages contain the Specification, and a Sketch of its Action, Power, and Range. The sense in which the term " original" is used on my title page is duly explained hereafter (art. 25). Absolute originality is claimed for my Stigmatic Geometry alone (art. 35, and Appendix III.). But I have always felt it to be almost impossible that in the vast extent of mathematical literature some traces of a similar conception should not exist, although I feel confident that it can never have been worked out in the detail here indicated. I should therefore esteem it a great favour if any one into whose hands these pages may fall, would furnish me with an exact reference to any work or paper even of a later date than 1864, (when I first publicly stated the nature of my conception of Stigmatics,) which to his mind seems to have pointed in the same direction, or to have covered any part of the same ground.' ALEXANDER J. ELLIS. 29 April, 1874. 25, Argyll Road, Kensington, London, W. CONTENTS. Preliminary Notice, p. 3. I. EUCLID'S CONCEPTION OF RATIO AND PROPORrTION. 1. Nature of the Conception, p. 4. 2. Predagogical Exposition of the Conception. First Step, p. 7. 3. The same. Second Slep, p. 9. 4. The same. Third Step, p. 10. 5. The same. Fourth Step, p. 11. 6. Parallels, a Parenthesis, p. 12. 7. Paedagogical Exposition resumed. Fifth Step, p. 14. 8. The same. Sixth Step, p. 15. 9. Paedagogical Appendix to Proportion, p. 16. II. "CARNOT'S PRINCIPLE" FOR LIMITS. 10. " Carnot's Principle," p. 18. 11. Examples, p. 19. III. THE LAWS OF TENSORS, ORt THE ALGEBRA OF PROPORTION. 12. Proportion expressed by Tensors, p. 20. 13. Commutative Multiplication of Tensors, p. 21. 14. Division of Tensors, p. 21. 15. Associative Multiplication of Tensors, p. 21. 16. Results for Multiplication and Division of Tensors, p. 22. 17. Addition and Subtraction of Tensors, p. 22. 18. Distributive Character of Tensors, p. 24. 19. Applications of Tensors, p. 24. IV. THE LAWS OF CLI'NANTS, OR THE ALGEBRA OF SIMILAR TRIANGLES LYING ON THE SAME PLANE. 20. Data from the Geometry of Direction, p. 25. 21. Directionally Similar Triangles expressed by Cli'nants, p. 26. 22. Conml'utative and Asso'ciative Multiplication of Cli'nants, p. 27. 23. Division of Cli'nants, p. 27. 24. Addition and Subtraction and Distrib'utive Character of Cli'nants, p. 27. 25. On the Originality of these Conceptions of Tensors and Clinants, p. 28. 26. Subsidiary Cli'nants, p. 29. 27. Some Relations of Subsidiary Cli'nants, p. 31. 28. General Exponential Expressions, p. 31. 29. Solution of General Exponential Equations, p. 32. 30. Logoomntric and Binomial Series, p. 32. 31. General Goniometric Series, p. 33. 32. Completion of the Laws of Cli'nants, p. 33. 33. Geometrical Construction of Cli'nant Combinations, p. 34. 34. Applications of Cli'nants, p. 34. V. STIGMAT'IC GEOMETRY, OR THE CORRESPONDENCE OF POINTS IN A PLANE. 35. No previous complete representation of Algebra by Geometry, p. 39. 36. General Conception of Stigmat'ic Geometry, p. 40. 37. Integral Stigmat'ics, p. 43. 38. Pri'mals, or Carte'sian Straight Lines generalised, p. 44. 39. Intersections of Pri'mals, p. 45. 40. Dis'tals, or Plucker's Coordinates generalised, p. 48. 41. Trilat'erals, or Triangular Relations generalised, p. 49. 42. Pencil of Four Ray'als, or the Anharmon'ic Properties of Rays generalised, p. 50. 43. U'niqua'drals, or the Relations of Involu'tion and Homog'raphy generalised, p. 51. 44. In'vals, or Chasle'sian Involu'tion of Points generalised, p. 51. 45. Hom'mals,or Chasle'sian Homog'raphy of Points generalised, p. 53. 46. Ray-hommals and Ray-invals, or the Chas] e'sian Homograph'ic Relations of Rays, generalised, p. 55. 47. Transordination, or the Carte'sian Transformation of Coordinates and of Curves, generalised, p. 56. 48. Du'oqua'drals or Co'nals, or Con'ic Sections, generalised, p. 58. 49. Intersections of Duo'qua'drals by Pri'mals, p. 62. 50. Sym'metrals, or Conjugate Diameters generalised, p. 66. 51. Tangen'tals, Po'lals, Po'larals, Fo'cals, Confo'cal Centrals, and Curva-Cy'clals, or the Relations of Tangents, Poles, Polars, Foci, Confocal Conics, and Circle of Curvature, generalised, p. 67. 52. Parab'bals, p. 71. 53. Multindic'ials, or the meaning in Plane Geometry of Algebraical Equations with Several Inde. pendent Variables, p. 73. 54. Solid Stigmat'ics, p. 74. 55. Conclusion, p. 76. APPENDIX. I. On the Impossible in Geometry, p. 76. II. On the Imaginary in Geometry, p. 77. III. On the History of the Conception of Stigmat'ic Geometry, p. 81. ALGEBRA IDENTIFIED WITH GEOMETRY. I. EucLID's CONCEPTION OF RATIO AND PROPORTION. 1. Natzure of the Conception.-(i.) The Latin terms rati = calculation, and proportio = portioning forward, do not convey the force of the Greek Xooyoc and dvaXoyta, and have by their arithmetical character served to lead the mind astray. Of the second Greek term Cicero, to whom its Latinisation is due, says (UTimaeus, seu de Universo, cap. iv.): " Omnia duo ad cohaerendum tertium aliquid requirunt, et quasi nodum vinculumque desiderant. Sed vinculorum id est aptissimum atque pulcherrimum, quod ex se, atque de his, quae astringit, quam maxime uinum efficit. Id optime assequitur quae Graece cvaXoyia, Latine, (audendum est enim, quoniam haec primum a nobis novantur) comparatio proportiove diec potest." It is a pity that subsequent Latinists preferred Cicero's second proposal to his first. But Cicero was not thinking mathematically. The Greek term Xoyoc has its radical sense in collec-ting, or bringing together for the purpose of thought, and dvaXoyia was the comparison of such collections, by running them through from bottom to top (dU'a). This general conception must necessarily have influenced any Greek in applying the terms. Euclid meagrely defines Xooyo.thus, in two separate definitions, of which the second has not been usually construed as a development of the first. y'. AoyoC eric Soo feyeOGv o'iloyev4v i Kara 'rrXiKodrrrlTa rpoc arXXrcX 7otd arXlO',c. X'. Ao'yov 'Eetv rrpo' a`XXi1Xa peye'Or Xeyerat, t( vYrarrt lroXXarkXaOtaO'jieva a dXXXeiY V rrepeXeLv. (ii.) Now I first observe that Euclid does not define homogeneity, as he uses the term (Juoyevwv without any explanation, as if well understood, and hence that it is an error to suppose that in def. 4. he intended to define it, although of course that definition is incomprehensible unless the magnitudes compared are homogeneous. In modern language we may I think render the meaning of these definitions thus: "3. The term logos is used to express a certain standing towards one another in respect to size, of two homogeneous magnitudes. 4. Two magnitudes will be said to have a logos towards each other, when a multiple of either can be formed so as to exceed a multiple of the other." (iii.) The term mzutiple, of which much more in art. 3, is not, properly speaking, defined by Euclid. He first tells us that he intends to B 6 I. EUCLID9S CONCEPTPION [ART. 1. iii. —V. limit the ordinary word pepoc by using it as an aliquot part, which he defines by means of the unexplained term perpov, thus: a'. pepoC e7rn eye6os pey/eovc, rO ehUov\a cov ro eitovoC, orctv KaTraperpr TO feCIov. that is: "The less magnitude will be termed a imeros of the greater, when it measures the other without remainder (carad)." And then he observes as an additional remark (shewn by a e) meant to render this notion more complete, and also distinguish a multitude from an agg'regate: F3'. woXXa7rXacnlov e r7o telov rov e\ado((7ovoc, orav cKaratireTert(1r V7TT rtU iXEarrovos, "In this case (Ne) the greater magnitude will be a multiple of the less, when it is measured by the less without remainder (1carc)," which is only saying: "of course, then, any magnitude is a meros of any multiple of it." (iv.) Returning to the definitions in (ii.) The term oXe'otc rpoc is exactly rendered by our "standing towards." The use of "mutual relation" seems to be tautological, on account of the popular use of the word ratio, which the Germans have even translated by the same word, verhlbtniss, that they use for relation, just as in French our distinction of ratio and reason is lost in the single word raison. The use of 7rola before orX6r~c is precisely similar to our use of the word certain, meaning ' undefined, of some kind or other," and hence requiring future limitation, and in Plato's Greek constantly it is joined to nc, as 7roi( Trt. In def. 3. the only limitation regards size, which is expressed by 7rqN^XrWT-)C compared, as distinct from etyeOoc unctiompared magnitude. There isno notion of measuring out ctarcaierpetv, in 7rqXiKdrrqC, which is therefore not well rendered by quantuplicity or manifoldness, for which in literary Greek as in literary English there seems to have been no term. Now there are many ways in which two magnitudes may be compared in respect to size; 1) with regard to greater and less, the only method used in the previous books of Euclid, and by that very circumstance here excluded, 2) with regard to one measuring out the other, which was a particular case, already considered in def. 1. and 2.; 3) with regard to both being measurable out by a third magnitude, which Euclid wisely saw to be included in the next case; 4) with regard to successive multiples of one continually exceeding successive multiples of the other, and as a particular case one multiple of one being of the same size as the same or another multiple of the other. The object of def. 4., appears to me to have been the limitation of the roitd aXdec wpoc dXrXXXact, or certain sctaclding towards one another, to this last case, which is alone general and includes all the preceding. Observe that the article e points out UXeaxLc as the subject of the sentence. Euclid proceeds then to examine this conception, namely, that logos is the interdistribution of multiples. (v.) He begins by considering the possibilities that may occur. If we take two magnitudes A and B, and two others C and D, and compare any of their multiples nmA, nB, and qmO, nD with regard to greater, equal and less, we find that for each of the three cases of AmA being greater than, or equal to, or less than nB, qnC may be greater than, or equal to, or less than nD. There are therefore 9 cases to consider. Euclid already knew from the properties of parallel transversals cutting two intersecting straight lines, that it was possible that when mA > -- < nB, then mO might be > = < nD respectively, and therefore he begins by saying, dlef 5., that in that case the logos of A to B is the ART. 1. V.-2. i.] OF RATIO AND PROPORTION. 7 same as the logos of C to _D, ev rZ avrw), not ev rJ 'va, Xdyp, adding def. 6., "let then (I3) two pairs of magnitudes which have the same logos, as thus determined (included in le), be called analoga," rd be rov acvrov exovra IeyeOrl Xo'yov, cdvdXoya raX\eia0. He does not think it necessary to shew from the first, that if one and one only of the two, A or B, be altered in any way however slight, the logos will be changed. He proceeds to the cases in which the interdistribution of multiples is not the same for each pair of magnitudes considered. These he reduces to one. Suppose that when miA > nB, mC is not > nD; "in that case (rore) the first logos is said (Xdyerat) to be greater than the second," the metaphorical use of greater and less as applied to logos is justified by the ordinary use of the term greater and less applied to the multiples considered, rdre rod zrpWrov 7rpoq ro sevrepov peiova Xoyov xeCtv XEyerat, "7rep TO rpirov rpoc rd TerapcTov. This being settled, he is able to introduce the abstract term cnalogic for sameness of ratios. The word used is d/otAoTr]c, usually rendered similarity. It is evident from the ev T(J crVTo) Xo6y) in def. 5., that the Aristotelian racrorsc should have been used, but perhaps Euclid, if he was acquainted with the word (we know that he was no school-logician) possibly thought it barbarous. It remained for theologians to wrangle over 6poi0oovtos and o6poov'tos. Euclid at any rate did not invent dOPO-rI (which was never Greek), but contented himself with using dpoitoric. Perhaps logically considered two thoughts, just because they are tzo, are not the same, although indistinguishable except in point of time of entertainment. But the use of similarity has led to the use of eqzcality as applied to logoi, which Euclid did not contemplate, and this use of equality has led to bringing analogia under the axiom of " things which are equal to the same thing are equal to one another," which is a mere verbal quibble. What Euclid says is in English: "sameness of l6goi then (5e) is analogic," dvaXoyia 3S CTiv 4t TOV X6yw, OO/OTdrlc, the use of the 'j pointing out the subject of the sentence, and its absence the predicate, as before (iv.) (vi.) This appears to me Euclid's real conception, and it is a conception which places its author in the very first rank of thinkers, that is, among those who have discovered the one simple key to an apparently insoluble difficulty-in this case the passage from discontinuity to continuity.. It remains to shew how this conception can be imparted to learners, whose minds have been arithmetically cribbed, confined, and hence distorted from earliest childhood. Of course the Greek words logos, canalogia, here used to prevent ambiguity, will henceforth be discontinued. 2. Paedagogical Exposition of the Conception. First step.-(i.) In the following pages a method is suggested for leading pupils up to the conception of ratios of magnitudes, independently of commensurability, and to the mode of comparing them. No child who has not been taught arithmetic has any general conception on these points. Every child who has been so taught has a more or less incorrect conception. We have to furnish him with progressive experience to make him familiar with the geometrical conception, and understand how far the arithmetical conception is useful and where it makes default. It is not till after the modes of comparing magnitudes are understood that 8 I. EUCLID'S CONCEPTION [ART. 2. i. —viii. the term proportion should be introduced, as proportion is only one case of comparison. It will be understood that these are merely hints, and not even a detailed syllabus. (ii.) Arrange boys (or, for convenience, straws or sticks) in order of height (or length). Show how this can be done by marking their heights in any order against the same standard, because the terminal points of lengths which have the same origin arrange themselves in the order of the lengths of the lines. No statement is to be made of actual height or length in reference to a standard. (iii.) Arrange boys (or, for convenience, stones) in order of weight. Shew that this may be done by scales, but more conveniently by taking the stones at hazard and weighing them by a balanced lever with arms of unequal length, a fixed scale being attached to the shorter arm, and a small weight (another stone) hitched by a string over the longer, a mark being made on the longer arm where the balance is attained. Shew that these marks naturally arrange themselves in order, the mark for the heaviest being furthest from the fulcrum. No statement is to be made of actual weight in reference to a standard. This is an extremely important reduction of order of weights to order of lengths. Practically it leads to a mechanical mode of finding two straight lines which bear to each other the same ratio as any two weights, without any considerations of commensurability. But this reduction requires some mechanical knowledge and is not to be attempted at first. (iv.) Arrange stones by volume. Shew that this may be done by placing a large enough vessel f/dl of water within a larger one which drains into a glass cylinder outside of which a slip of paper is pasted vertically. On immersing any stone in any order carefully in the first vessel, the overflow is conducted through the second into the cylinder, and the height to which the water rises is to be marked on the paper. Empty the cylinder and fill the first vessel again. Immerse a second stone and proceed as before, and so on. The marks on the slip of paper arrange themselves naturally in the ascending order of the size of the stones. This will subsequently reduce ratios of any volumes to ratios of lengths without regard to commensurability. No reference to any standard volume is to be made. (v.) Arrange any number (4 or 5 are enough) of rectilinear areas (mixed, triangles and polygons) in order of magnitude. Shew that they may be all reduced to rectangles of the same altitude, and then that the bases may be arranged as in (ii.) (vi.) Arrange curvilinear, or amorphous, or mixed rectilinear and other areas, plane or other, in order of magnitude. Cut them out in "lead paper," which is sufficiently homogeneous, flexible and heavy, to convey the required notion, and treat the slips as weights (iii.). (vii.) Arrange curves or broken lines in order of length. Pass threads round them, and straighten them by tension, and apply (ii.). (viii.) The processes in (ii., v.) are strictly geometrical. The other processes require " idealising," and suggest geometrical problems, which the teacher should carefully explain have not been completely solved, but that in general we can by refined geometrical methods approach more nearly to the truth than by the rough physical methods here employed, when some of the magnitudes to be compared are very nearly -4 PT. 2., viii.-3. i.] OF RATIO AND PROPORTION. 9 the same; a difficulty which should be introduced in a second or third trial in every case. But shew also that the idealisation of those rough methods conclusively proves that we can always conceive a series of straight lines arranged in the order of magnitude of any series of magnitudes such as those already experimented on. (ix.) Then draw attention to the fact that we first compared straight lengths with one another, then weights with one another, then volumes, then areas, and then general lengths, but that we did not compare lengths with weights, &c., for we could not say of a length that it was either greater or less than a weight, although we were able to arrange lengths in the same order as weights. Hence lead to a conception of k7inds, and to the order of arrangemlents of things of the same kind independently of the par'ticular lkind. These are difficult abstractions, and must be treated cautiously. Terrible mistakes are made by children who have to grub them out unguided. But merely to tell them is pouring water on a duck's back-neither tale nor water is ever taken in. (x.) This completes the first step in the way of preparation, and the absence of all approach to arithmetic or commensurability is of the utmost importance for what follows. 3. Second step.-(i.) The next step includes the formation of multiples, and the point to be borne in mind by the teacher is that the child, through arithmetic, has been trained to consider "bags of stones,"-that is, separate discontinuous magnitudes artificially aggregated without losing their discontinuity,-and that he has to be led to comprehend an addition which results in absolute continuity, without a trace of the original individuality. This is best done by grouping quantities of liquids. Take a small glass vessel, with an external band marked on it, but not all round it (a short slip of paper is best); pour coloured water in till the top of the water is seen to coincide with the top of the band. Have ready a series of larger glass vessels of the same shape, cylinders of the same radius, which, to avoid arithmetical conceptions, are marked by the letters A, B, C, &c. Empty the small vessel into A. Fill it again and empty into B; fill it again and empty into B again. Fill it three more times and empty each time into C, and so on. Then place the vessels in the order of the height of the water. This will be also in order of the volumes and also of the weights of the water. Draw attention to the fact that the water in each vessel shews no trace of having been poured in by instalments, so that it is absolutely impossible to say in what manner it was poured in. But as the operation was witlnessed, it is 7Lownb that this continuity resulted from the discontinuous operation of adding equal instalments. These discontinlzuous instalments can be counted like anything else. A had 1, B had 2, 0 had 3, and so on. Hence the volumes of the water are called the first, second, third &c. ImzttTlple of the volume of water in the original smaller vessel, and the order of arrangement of these minltijples of volume is consequently the order of the arrangement of the scale of whole numbers, and this order m7?ust be the soame whatever be the size of the original small vessel, althoughl the multiple volLuzes themselves are diffrent. Moreover if any volumes are arranged in order of mag 10 I. EUCLIDIS CONCEPTION [ART. 3. i.-4. nitude it is easy to see,-not whether they have been formed by instalments, but-whether they can be formed by instalments, by simply emptying A into a new vessel, marking the height of the water, and throwing it away. Then pouring from B into this new vessel up to the line, emptying, seeing if the remainder will fill the new vessel up to the same line, and so on. (ii.) The points which should be gained are: 1) that multiples of magnitudes are simple continuous magnitudes; 2) that these can be arranged in order of magnitude; 3) that this order is constant, and is that of the numerical scale by which they are named; 4) that any magnitudes being arranged in order, it can be ascertained whether they are or are not multiples of the same magnitude, whenever subtraction is possible. (iii.) Next make the learner construct multiples of straight lines in the form of straight lines with no mark of division; multiples of rectilinear areas not being parallelograms, in the form of parallelograms of the same height with no mark of division; multiples of circular arcs in the form of circular arcs, also with no mark of division, but with a rough internal or external spiral which by the number of its coils shews the amount of revolution when exceeding a semi-revolution; and finally multiples of angles in the form of angles in the same way. (iv.) De 2Morgan said that Euc. vi. 33 fairly gave up Euclid's conception of angle, Euc. i., def. 8 to 12. But really this was given up in Euc. i. 13 and i. 32, especially in its corollaries. I think it advisable to retain the term angle for sums of angles not exceeding two right angles, and to use the term ro'tte for larger amounts. An extension of the term angle to any sums of less than four right angles does not meet the case of Euc. i. 32, cor. And it will be seen that for directional angles the limitation here proposed is important (art. 20. x.). Also it is clear that only in the case of such limitation can we dispense with the use of the subsidiary spirals. Angles themselves will then become rotates of less than a certain amount. The sums of angles (i. e. rotates) are always rotates, and may (exceptionally) be angles. Great trouble is at present experienced by learners from the sum of several angles exceeding even four right angles, and hence not being an angle at all, even when its meaning is extended as above. (v.) The next point is to shew that, knowing Euc. i. to iv., we cannot take multiples of curvilinear magnitudes; for example, we cannot draw a circle which shall be double the area of a given circle. Shew, however, that we can easily describe one much more or much less than double, and hence that it is only our want of geometry that prevents us from hitting the exact radius required. State that for this particular case we shall find a solution (art. 11. iii.); but that in general we are at present able only to form such multiples hypothetically in conception, and approximatively in practice. Thus we cannot with geometrical accuracy compare the length of a circular arc with its chord. The teacher should, however, shew why we know the arc to be greater, as this is a very important result. 4. Third step.-Take two series of multiples of two original magnitudes of the same kind, and, as they are all magnitudes, arrange them ARiT. 4. —5. iii.] OF RATIO AND PROPORTION. 11 in order of magnitude. Our preparation (art. 2. ii.-ix.) enables us to reduce this case to that of comparing multiples of length. Make the learner mark off lengths, as OA2, OA3, &c., and OB2, OB3, &c., which are multiples of OA and OB, by marking their terminations A2, &c., B2, &c. with short ticks on opposite sides of the same straight line. The point to be established is that, if one of the original lines be ever so slightly altered, some multiple of the altered line can be found which will exceed or fall short of the same multiple of the unaltered line by more than the other original line, and that consequently the order of the multiples of the two original lines, if enough of them are taken, will differ from the order of the multiples of one of the original lines, and of a line differing from the other. Hence, when two magnitudes are known, the order of their multiples is fixed and known. And it must be also seen that, conversely, if by any means the order of multiples is known, and also one of the original lines, the other is of fixed length, although we are not yet in a condition to find it. 5. Fourth step.-(i.) Shew that it is possible to alter the lengths of both the original lines of art. 4. in such a way that the order of the multiples of the two altered lines will be the same as that of the two original lines. (ii.) The first case is that of commensurability. If nm. OA = n. OB, (notation to be thoroughly explained as representing multiples, not aggregates,) then the multiples divide into groups of Am multiples of OA and n multiples of OB, and the order in which the multiples interlie will be the same in each group. This should be exemplified by a figure. The consequence is that when we know the order for the first group we know the order for ever-without any veiled application of the principle of limits. This conclusion is extremely important. (iii.) The second case is that of parallels. Let OAB, OCD be straight lines (the unconnected letters in the margin will show how any figures are to be constructed) A B drawn from a common origin O; and AC, BD B r parallel lines. Take OA,.= r. OA, OB,= s. OB, A and draw the lines A,.C,., BDs parallel to AC, C BD1; then 00,. = r.OC, and OD,= s.OD. D And as parallels do not intersect, the order of the multiples of OA, OB, determiined by the terminal points A,., B,, will be the same as the order of the multiples of OC, OD, determined by the terminal points C,., Ds. Here the geometrical property of parallels enables us to know with certainty that the order of multiples of OC, OD is the same as that of those of OA, OB, independently of the rlnumber of muZltiples compared, without any veiled application of limits, and also ir.dep2endently of como?,7enssucrability. This conclusion therefore holds for all those cases which have been shewn to be reducible to straight lines. It should be verified by examples of triangles and rectilinear areas generally. The converse must also be proved. Four arrangements are possible: 1) the lines are parallel, and the order the same; this we have seen to be the case, and it excludes 2), the lines are parallel, and the order is not the same; 3) but the lines may not be parallel, and yet, for all we know, the order may 12 I. EUCLID'S CONCEPTION [ART. 5. iii.-6. ii. be the same; but this is excluded by 4), when the lines are not parallel, the order not the same; because we know that when the order is not the same, the lines joining the extremities of the multiples must cross, which is impossible for parallels. There remain therefore only the first and fourth cases, and this proves the correctness of the conversion. (iv.) The third case is that of angles or rotates and their subtending arcs, which presents no difficulty. 6. Parallels, a Parenthtesis.-(i.) Here I interpose some parenthetical remarks suggested by the assumption that parallel lines never meet, in order to shew that the theory of parallels is not one of veiled limits, for, if it were, then indeed Euclid's conception would be one also, except in the case of commensurables. Now in modern geometry any system of parallels is said to have one and only one point in common, which is conveniently placed out of sight, at infinity. Townsend (Modern Geometry, 1863, p. 11, see also the citations in Appendix I.) says that the truth of this conclusion has been " long placed beyond all question by the simplest considerations of projection and perspective." I believe that it has been much longer rendered impossible by the elementary consideration that two straight lines cannot inclose a space. The conclusion (ibid. p. 12) that "the two opposite directions of every [straight] line, not itself at infinity, are to be regarded, not as reaching infinity at two different and opposite points, but as rnnlning into each other and meetivng at a single point at infinity," amounts to saying that diametrically opposite directions are the same. Again (ibid.), "every [straight] line not at infinity may be regarded as a circle of infinite radius whose centre is the point at infinity in the direction orthogonal to the line," i.e., the single point common to a system of parallel straight lines is the common centre of the concentric circles with which they coincide circumferentially, and which have no common circumferential point. The assumption that such circles have two imaginary points at infinity where they are touched by the two imaginary non-touchers (asymptotes) common to all concentric circles, is in the mere field of imaginaries, and will be disposed of hereafter (art 48. v.) The touching of curves by real non-touchers has more to be said in favour of it than the intersections of parallels, because asymptotes do constantly approach the curve, but no two points in two parallel lines are ever nearer each other than their common normal which itself never diminishes in length, so that the assumption that parallel lines intersect requires that an unchangeable length should discontinuously shrink into nothingness " at infinity " (which has no "at"). To my mind these are mere contradictions in terms which must lead, and I believe have led, to serious error. They are however nothing more than terminology, invented to make discontinuity continuous, and thus hide the real state of the case. Hence I hold that unless we assert that two straight lines will under certain circumstances inclose a space,-and thus give up the proof of the fundamental proposition, Euc. i. 4, in which case plane geometry will itself drift off to infinity,-we must consider that there are no veiled limits in the proof from parallels in art. 5. iii. ART. 6. ii.-Viii. ] OF RATIO AND PROPORTION. 13 (ii.) This leads me to consider the question of parallels as a subject to be taught to children. No one would dream of teaching them the bewilderments just mentioned; but some men whose opinions I respect, are inclined to make parallels a "reserved question," whereas it seems to me that no geometrical teaching, and especially none on proportion, is possible without making it elementarily exoteric. I must crave indulgence for briefly stating my own views on this subject, which, however crotchety in appearance, are the result of years of reflection tested by other years of application. (iii.) Bring the edges of two surfaces (pieces of paper may be used for illustration, but any surfaces will do) to touch in two points; observe whether there is any intermediate point, at which they are also in contact; turn the surfaces about the first two points like a door on its hinges, and observe if they still touch in that third point, throughout the movement. If they do, for all such third points observable, the edges intermediate to the points are straight lines. This is our only test of straightness. Some writers gain the second line by cutting off a bit of the first, which disguises without altering the principle. (iv.) Straight edges can slide one on the other, that is, can move so as always to have two points of the one coincident with two of the other, and hence coincide intermediately. (v.) But straight edges can also move one on the other so as to have one fixed point of one coincident with one fixed point of the other, and at least one fixed point of the one not coincident with any point of the other. In this case they can have only the one first mentioned point in each coincident. They then rotate. Here explain the generation of planes, plane rotation, circularity, angularity. (vi.) When straight edges have thus rotated they can be clamped, by a transversal having fixed points, one in common with each straight edge. They then form a biradial (Sir W. R. Hamilton's word, see also art 34. v.), of which the original straight lines are the arlms, the transversal not being further regarded. In this case the motion of one arm entails the motion of the other, and neither can rotate unless the other rotates also. (vii.) Now let one arm of a biradial slide on a given straight edge, the trace of the other arm having been marked in its original position. Then in every new position of this second arm there will exist a new straight line having at least one point not in common with the original trace, while, as it has not rotated, it can have no other point in common with the original trace, quite independently of length. None of these positions of the second arm therefore ever meet the original trace. The existence of parallels is therefore demonstrated without any veiled reference to limits. The experiment is best shewn to a single pupil by lines on tracing paper moved over lines on other paper, and to a class by lines drawn with gum and whiting on glass, and moved over the chalk lines on the black-board. It leads to the best practical method of drawing parallels by sliding one " set square" along another. A straight line thus moved is said to be translated. The advantage of early familiarity with the notions of rotation and translation is obvious. (viii.) The usual propositions as to equality of external and internal 14 I. EUCLID'S CONCEPTION [A RT. 6. "x.-7. i. angles, &c., in the case of parallels are now to be proved, but not their converse (Euc. i. 29). (ix.) The addition of angles which have not a common vertex is now to be shewn, by first sliding and then rotating, the sum of any number of rotations being independent of interposed slides or translations. In this way Euc. i. 32 may be immediately proved without using Euc. i. 29, for which purpose this proposition is mainly required: ABC (fig. 1) being a triangle, the rotation of a line A'D' originally lying over AD, by turning it about A as a pivot until it falls on AB, is the same as if this line were first slid till A' fell on C, and then rotated to fall on CB; were then slid along CB till A' pass from C to B, and A'D' falls on BF; were then rotated about B to BE, (the angle F1BE being shewn to be equal to CBA by merely continuing the line A'D' backwards to C' over C, and seeing that on rotation this A'C' comes to fall on BE,) and were then slid till A' falls on A, so that A'D' has rotated from AD to AB by the help of two rotations separated by intermediate slides. The exterior angle DAB is therefore equal to the two interior and opposite angles ACB, CBA, whenever two intersecting straight lines AB, CB are crossed by a transversal DAC. (x.) To prove Euc. i. 29, we have however still to prove Ax. 12, which lmay be made to depend on this principle: if a straight line BE (fig. 2) pass through a given point B and be translated in any manner till it again pass through B, it will wholly coincide with its former trace. For if it did not, it would have rotated, which is against the hypothesis. (xi.) Let AC, BD (fig. 2) be parallel lines, and angle ABE be less than angle ABD; to prove that AC, BE will meet. Take A'B'E' as a biradial over ABE; slide B'A' to fall on AH, so that B'E' falls on AF, and continue AF indefinitely both ways. AF necessarily cuts AC. Clamp B'E', now falling over AF, with A'G', falling over AG; slide A'G' along GAC. Then there is no point in the plane ABE over which B'E', which is attached to A'G', when sufficiently produced, will not pass. Hence it will pass over B. And then B'E', having been only translated, coincides with BE again. And as B'E', during the last translation, has never ceased to cut AC, BE also cuts AG. This seems to me a complete proof of this axiom on the data assumed, and the assumption of these data also appears to me more directly connected with the subject, and to make the point of this Axiom 12 more evident than any other. 7. Paedagogicca Exposition resumed. —Fifth step. (i.) The paedagogical introduction to proportion is now resumed. Having shewn that the order of multiples is constant when the originals are constant, and may be constant when the originals are both altered in certain ways, it becomes convenient to have a name for this order. Let the magnitudes be A and B, then the order in which the multiples of A are distributed among the multiples of B, (so that, given any imultiple of A, we know the two nearest multiples of B between which it lies,) is called the ratio of A to B, and is written A: B. Similarly B: A, or the ratio of B to A, means the order in which the multiples of B are distributed among those of A, (so that, given any multiple of B, we know the two nearest multiples of A between which it lies). ART. 7i. ii.'8. iii- OF RATIO AND PROPORTION.N 15 (ii.) If then the multiples of C are distributed among those of D in the same order as those of A among those of B, the ratio of C to D is the same as that of A to B. This is written A: B:: C': D, which I prefer reading " A to B same as C to D," omitting the word ratio, and using same as instead of equal to for the reasons in art. 1. v.; and I also prefer, at least paedagogically, not to use the old formula "as A is to B so is C to )," because of the marvellous ambiguity of the as and so, and because of the old false associations produced by the Rule of Three as usually taught. Of course in this case also B: A:: D: C. (iii.) The idealised elementary processes (art. 2. ii.-viii.) now lead us to infer that, given any two magnitudes of the same kind, we might always find (if our processes were accurate enough) two straight lines which would have the same ratio-i. e., whose multiples would have the same order of magnitude. Hence a ratio is always (conceptionally) expressible as that of two straight lines. (iv.) And this leads us to consider the case where A: B not:C: D, that is, where the multiples of A are not distributed in the same order among those of B, as those of C are among those of D. Two cases will arise: 1) Either some multiple of A is greater than some multiple of B, while the multiple of C corresponding to that of A is not greater than that of D corresponding to that of B. In this case, for brevity, the term greater is transferred from corresponding multiples to the orders of distribution of the multiples of A among those of B, and of C among those of D; and we say laconically, A: B > C: D, reading > as "greater than" (compare art. 1. v.). Stress should be laid on this abbreviation, because in the ratios there is no real greater or less. Numerous examples must be formed. 2) Or else there will be some multiple of A which is less than a multiple of B, while the multiple of C corresponding to that of A is not less than that of D corresponding to that of G. Here, in the same way, we write A: B < C: D, reading < as "less than," with the same warning as before. Numerous examples required. 8. Sixth step.-(i.) Up to this point there has not been a word of proportion. The word is used in common speech so ungeometrically, and has been so much perverted arithmetically, that I prefer reserving it for the Sixth step. (ii.) Stand before a mirror. Hold a book parallel to its surface. Advance and withdraw it, keeping the head steady. Observe the great apparent change of size in the image of the book, shewn by the amount of the surface of the mirror covered by it (easily marked off), whereas the shape remains unaltered. When this occurs, we say that all the dimensions in any one image are prop)ortiolnate to those in the other, or that they all alter proportionably or in prop)or1tion. Observe that the example is chosen so as to exclude commensurability, which would necessarily intrude in drawings made to different scales in the usual way. The shadow of a book cast on the wall from a single point of light (a candle) will serve as well; and better for a class. Examine what is meant by this. (iii.) The simplest figure to deal with is a triangle. Draw one con 16.1. ETTCTTJ)IS CONCEIPTION. [AR.. ii-9. ii. necting three points on the book, or cast the shadow of a set square on the wall. Observe that the sameness of s7hape depends on the sameness of angles between corresponding sides, and that the difference of size depends on the alteration of lengths. What is the law by which the lengths alter? This should enable us, when we know the length of one line in the original figure and that of the corresponding line in the altered figure, from the length of any line in the original figure to construct the length of the corresponding line in the other figure. Take two corresponding triangles. The sameness of angles allows of their superimposition so that any pair of corresponding vertices being brought together, the adjacent sides will lie on one another, and the opposite sides be pcrallel. We have the case of art. 5. iii. Hence if ABC, IA'B'C' be corresponding triangles of which AB, A'B' are the parallel sides, we have, by art. 7. ii., CA: CB:: C'A': C'B'. That is, j)roJortion (or the law of alteration of length in figures of the same shape and different size) consists in sameness of ratio between correspozndinzg le'ngtts. (iv.) Having thus arrived at an essentially geometrical view of proportion, exclusive of arithmetic and commensurability, it only remains to explain that figures which in popular language are said to be in proportion, are in geometry called sim1ilar; that their properties of size evidently depend on the sameness of the ratios of corresponding lengths; that the examination of the properties thus discoverable forms the principal part of geometry, and that it hence becomes important to discover all cases where this relation exists originally, and also what new relations of the same kind can be inferred from knowing one or more such relations. This then is the object of Euc. v. and vi., which would be made mutually illustrative if fused. In the elementary explanations, the main propositions, Euc. vi. 1. 2. 33, have already been proved. It would be of advantage to interpose Euc. vi. 3-17 between Euc. v. 16 and 17. The mode of treating the necessary propositions presents no difficulty whatever when this stage is reached, and I pass it over, to abridge this already too lengthy exposition, without which I felt that it was impossible to make my own views intelligible. 9. Paedacogical Appenldix to Proportioi.-(i.) After the general propositions on proportion in Euc. v., interspersed with some of their easiest and fundamental applications in Euc vi., have been thoroughly taught and understood in their real geometrical, as opposed to their arithmnetical, which is also the usual algebraical sense, the question arises: how can we proceed, when it is not geometrically possible to find, as suggested in art. 2. ii.-viii., two straight lines which bear to each other the came ratio as any two given homogeneous quantities, but it'is at the same time important to deal with that ratio? (ii.) This leads to the consideration of approximzate ratios. Of two ratios X: Y and X: Z where X, Y, Z are straight lines, that is nearer to the ratio A B, (where A and B are any homogeneous magnitudes,) for which the order of the multiples is the sa(me for the greater number of multiples of the greater term. When the number of multiples is very ART. 9. ii. —V.] OF RATIO AND PROPORTION. 17 great in both cases, and the ratio X: Y<A ' B, but X: Z > A B, there can be but a small difference between Yand Z, and the required line V, for which X: V:: A B, will be <Y and >Z. If then we can find successive values of Y and Z, nearer and nearer to each other, we shall obtain ratios which more and more nearly approximate to X: V. (iii.) When we require to find V for practical use, we may previously determine the amount of error, E, deemed sensible, and we lay down the principle that for szch p1ractical ends, if we can find Y and Z such that Y- Z, shall be < E, we shall have practicclly solved the problem, because the error will be insensible. (iv.) Now, conceptionally, we may suppose that by actual formation of multiples we obtain mA = nz3+D, where D is homogeneous with and <B, in which case, if amX = Y, and mXX = (z+ 1) Z, (whence, when X is given Y and Z can be found by parallels,) we shall have X: Y<A:B, and X: Z>A:, while Y-Z = m1 X, andhence ni (I + 1) may be made less than any line E, by simply increasing n. And this concepjtionally solves the practical problem. (v ) Of course the idea of discovering in and n by actually forming multiples, when mi and n are very large indeed, is practically illusory. Hence the usual process pursued for finding ma — n is to throw it into a continued fraction, and I particularly urge teachers to approximate to the values of v/2, N/3, &c. from two given lines in each case, (diagonal and side of the corresponding rectangles,) first by actually forming multiples, and secondly by actually forming continued fractions; and especially to shew that the diagonal and side of a square are incommensurable, both geometrically and arithmetically, to force on the learner the sensation, impossible to acquire without such actual trials, of the meaning, first, of approximation (with its practical uncertainty), and secondly, of incommensurability. An attempt to approximate to the ratio of the circumference to diameter of a circle by using strings of the length of both, is also very instructive. A gallipot, or tub head, or, better, a circular table, will give one or two places of decimals. Taking the best approximate commensurable ratio to be expressed by 355 1l 355 feet: 113 feet, and observing that = 3 +, it will be found extremely interesting to watch the hesitation about the 7, and see how it will wander from 5 to 8 or 9, according to circumstances, in different trials. To reach the Archimedean 7 is a triumph or a " fluke." Nothing is better adapted to make pupils feel the practical difficulty in the way of " squaring the circle," by such a simple process as "rolling a circle on a straight line and marking off the length." In a London draper's shop I learned that 11 metres are 12 yards, and I think the man who told me would have been puzzled had he been told that a yard is eleven-twelfths of a metre. In all comparisons of length we really use multiples. If we say that 1 yard is 0'9144 metres, we scarcely convey a notion to most people who would quite understand 10000 yards being 9144 metres. Similarly, for general intelligibility, I would back against any fractional statement such approximations as 8kilometres are 5 miles, 2 hectares are 5 acres, 5 kilogrammes are 18 II. t CARNOT'S PRINCIPLE. ' [ART. 9. Vi.-10O. i. 11 pounds avoirdupois, 200 grammes are 7 ounces av., 4 litres are 7 pints, which the draper's information led me to calculate. I may state, by the way, that we come within one unit of the truth up to 1000 times the French units of measure (for metres up to 11000) by adding 1 part in 400 to the yards and pounds, subtracting 6 parts in 1000 from the miles, and adding the same to pints, and subtracting 12 parts in 1000 from the acres. The calculation is much easier than for decimals, and the results furnish admirable materials for exercising pupils in approximating to ratios of magnitudes arithmetically. (vii.) Observe that if mA = nB +D, and we do not know the limit of the value of D, we can tell by the mere division mn- m =?' + proper fraction, that m'A lies between n'B and (i'+ 2) B, but that we cannot tell whether it lies between n'B and (n'+ 1) B, or between (n'+ 1) B and (n' + 2) B, however great In may be. If, then, we want to find, not V, but n'V within the limit E, we must find n'tmA = n" 'B D', where D'< B. This is important in settling the limits of error, or " the number of decimal places required." (viii.) But the processes of finding multiples, or throwing into a continued fraction, are alike illusory when certainty is required, as the suggested trials shew. Then arises the great problem of higher geometry: to find a series of terms (taken as geometrical magnitudes) continually diminishing, and connected by a lawm such that when a few are known any required number can be found, and such also that their (geometrical) sum continually approaches to the required limit, and may be made to differ from that limit by less than any assigned amount. The practical problem is then perfectly solved, but that practical problem gives birth to a theoretical problem. Suppose V to be the fixed limit toward which the series S converges, then V-S will be a magnitude (a straight line, see ii.) of continually diminishing size, which can be made less than any assignable magnitude, while at every moment V-(V- S) = S. Can we then neglect V-S, and deal with S as if it were V, not merely for a practical approximation, but for theoretical exactness? II. " CARNOT'S PRINCIPLE" FOR LIMITS. 10. " Carnot's Principle."-(i.) The only satisfactory answer which 1 have been able to find to the question just propounded, (and I have paid minute attention to the subject at various times for nearly 40 years,) is contained in Reflexions sur la ]ietaphc7ysique cld Calcul Icfinitesimal par CARNOT (3rd ed., Paris, 1839, pp. 254), which the name of the writer is enough to recommend to the careful study of all teachers. I wish here to state the principle in connection with the author's name, in that simple geometrical form which is suitable for learners, without any anticipation of the infinitesimal calculus. ART. 10. ii.-11. ii.] II. c CARNOT'S PRINCIPLE." 19 (ii.) Let A, B be two homogeneous magnitudes, of which we only know that they are invariable; and let X, Y be two other magnitudes homogeneous with A, B, of which we at first only know that they are constantly changing. And then in addition suppose that we know, by given geometrical or other relations, that through all the changes of X and Y, the following condition subsists: A- X = B - Y. What relation between A and B will such a condition allow us to infer? (iii.) First the given relation makes A-B- X-Y; and as A and B are invariable, A -B is invariable, and hence X-Y is invariable. Hence also we cannot have one of the two variables X and Y increasing and the other diminishing, because in that case the difference X- Y would necessarily vary. X and Y must both increase, or both diminish. (iv.) If X and Y both increase, X- Y may remain constantly equal to any unknown homogeneous magnitude whatever, as i, and then A-B = 1M, that is, some unknown. In this case, then, the relation A-X= B -Y leads to no result. (v.) But if X and Yboth decrease, and each can become less than any assignable homogeneous magnitude, the difference X- Y must also vary and become less and less, unless X = Y at all times. The condition that the difference should not vary, entails therefore the necessity that X = Y, and as a necessary consequence that A = B. (vi.) If, then, A-X=B -Y under these last circumstances (v.), we know with theoretical exactness, without any approximation at all, that A = B and X = Y. (vii,) If, then, our interest consists only in finding the relation between A and B, and we have no sort of interest at all in knowing that between X and Y, we may from the moment that the relation A - X = B- Y has been established, neglect the consideration of X and Y, and infer, with perfect exactness, that A = B. What we have neglected is not a decreasing magnitude, nor anything which affects the relation of A to B, but only something which affects the relation of X to Y, which we do not care about, but which we could at any time revert to if desired. (viii.) By neglecting variable infinitesimals, then, when seeking relations between invariable finites, we merely simplify the chain of argument, without impairing exactness. We do not neglect them as magnitudes so small that they are of no consequence (on the principle de mzinimis notn crlcat lex), but merely leave them out of consideration, because we do not happen to want to know anything about them. See the citations in Appendix I. 11. Examples.-(i.) It is important that the pupil should appreciate the working of this principle by applying it to the two main cases in elementary geometry; the expression of the ratios of the circumferences and areas of two circles. (ii.) Within any two circles describe, say, hexagons, Euc. iv. 15, as offering the least geometrical difficulty. These are similar polygons, and the ratio of their perimeters, P: P', is the same as that of the corresponding radii, B: B'. By bisecting the arcs subtended by the sides of these hexagons, and so on, we get other similar polygons, for all of which P: P':: '. Now if C, C' be the invariable circumferences of the circles, and D), D' the values of C-P, C'-P', which constantly diminish, and may be 20 III. THE LAWS OF TENSORS, OR THE [ART. 11. ii.-12. ii. made less than any assignable, the last proportion may be written C-D: C'-D':: R: R'; whence by Euc. vi. 16, rect. (C, R') -rect. (D, V') = rect. (C', )-~rect. (D', R). The second rectangles on each side may become less than any assignable, and hence, by "Carnot's principle," we have always, whether D, D' are large or small, rect. (C, R') = rect. (C', R), and rect. (D, R') = rect. (D', R), that is, both C: ':: R: ', and D: D'::: -R'. The last result was not wanted, (although it is useful to draw attention to it when D is large,) and hence, as soon as we had stated the proportion as C-D: C'-D'::R,: E', we might have inferred CG (':: R: R', which was all of the truth we wanted, although not the whole truth. (iii.) Apply the reduced process to find the ratio A: A' of the areas of the circles, Q: Q' being that of the areas of the similar polygons, which is the same as that of sq. on 1: sq. on B'. E and E' being the varying differences between the invariable areas of the circles and of the variable areas of the polygons, which differences may become less than any assignable areas, the constant proportion Q: Q':: sq. on i: sq. on R' can be expressed as A-E: A'-E':: sq. on l: sq. on R', and hence by " Carnot's principle " we infer A A':: sq. on R: sq. on R', and also, if required, E: E':: A: A'. This solves Art. 3. v. (iv.) It is obvious that the application of this principle in higher algebraical geometry, the differential calculus, &c., is impossible unless we assume that we can deal with incommensurable expressions by the ordinary laws of commutative algebra. I have never seen any attempt to prove the justifiability of this condition, which seems to be taken as an axiom. Yet it is evident that if the limit of a convergent series is incommensurable, we cannot, from conclusions drawn from the (commensurable) sum of any finite, or ever increasing (infinite) number of its terms, conclude an exact relation which depends solely on its limit, until we know what are the laws by which we may calculate with incommensurables. The ordinary algebraical proof that v/2. V/3 = 6, by " squaring each side," is absurd if we do not know the meaning of multiplying v/6 by /6 or of multiplying 1/2 by /3. To this question, then, the next Tract is devoted. III. THE LAWS OF TENSORS, OR THE ALGEBRA OF PROPORTION. 12. Proportion expressecl by Tensors.- (i.) Euc. v. and vi. are assumed. It is also assumed that no ratio is known till two straight lines have been found having that ratio. And only known ratios are here dealt with. (ii.) OI, fig. 3, is a straight line continued indefinitely beyond I, on which ART. 12. ii.-15. i.] ALGEBRA OF PROPORTION. 21 I is a fixed known point, 01 being the standard length to which all others are referred. All other points mentioned, as A, B, 0, &c., are supposed to lie on 01 and on the I side of 0. The length, position, and direction of OI are quite arbitrary. But for ulterior purposes I shall assume 0O to be horizontal, and drawn from right to left. (iii.) Given two points A and B, find a third point 0, so that 01: OA:: OB: 00. This is a perfectly simple known geometrical operation (art. 5. iii.), not in the slightest degree involving, but also not excluding, commensurability. As this is an operation performed on OB, through the instrumentality of 0A, (it is not necessary to consider the invariable 01,) I designate it by the small letter a, of which A is the capital, (a relation between forms of letters constantly observed,) and call it a tensor, (the name is borrowed from Sir W. R. Hamilton.) and I write OC = a. OB (read a. OB as "a ante OB",) to shew that OC is the result of performing the operation called the tensor a, upon the operand OB. Hence this equation has no other meaning than the original proportion 01: OA:: OB: OC. (iv.) Since 01: OA:: 01: OA, we must have OA = a. 01, and similarly OB = b. OI, 00 = c. 0f, so that the fundamental equation in (iii.) may be written c. 0I = a. (b. 01), observing or'der of letters. 13. Commutative MzcltipUlication of Tensors. (i.)-Now let ab be a symbol which has the same resultant meaning as c, but shews that c has been reached by the two operations a, b, performed in the order of the equation in (art. 12. iv.), then c = ab (read ab as "a ante b"), represents the effect of that equation without the use of 01, and defines what may be termed the vmultiplication of tensors, the order of the symbols being observed. It must be remembered that no knowledge of arithmetical operations is assumed. (ii.) But 01: OA:: OB: OC gives byEuc.v.16, OT: OB:: OA: OC; and hence, by precisely the same process as before, c= ba, and hence ab = ba, or the multi:plication of tensors is commuttative. (iii.) A product of tensors represents a compound ratio; compare art. 19. iii. 14. Division of Tensors.-Given any two of the three points A, B, 0, the third may be found by well-known tensor operations. Express them by writing a = - b = c, (read e, as "c super b). b- a b Thcen c ab Then. b = ab = c, a = = - b b b which are the two laws of division of tensors. 15. Associative lMultipliccation of Tensors.-(i.) Let OI: OM:: OA: OB.........................(1), and 01: OM:: 00: OD................ (2), whence OA: OB:: C: OD.....................(3), and also OA: O:: OB: OD..................... (4), and OB: OA:: OD: OC.........()........... ). 0 22 III. THE LAWS OF TENSORS, OR THE [ART. 15. i.-17. i. b Cd b d Then, by art. 14, m =-, n = -, and =...............(6) ac c a c and hence equation (6) becomes the tensor expression of the proportion (3); and as this involves (4) and (5), which may be similarly expressed, we find that b dci, c c c d a if -=-, then, - =, and a=... (7); a c b d' a b' c c whence follow various results, and among others, b dc b d c ac if =-, then -. c=-.cd, and -.b= b = d, a c a c a b a ac which is, in fact, the great law of association of tensors in multiplication, which may be put in the usual form thus: (ii.) Given m=ab, find n=bc; then = -, c = -, and b b ab. c = m. c =. -. = n. - n.a= bc. a, b b or ab. c = a. be, the usual form of this law, the extreme importance of which, and its occasional independence of the law of commutation, is well shewn by the laws of quaternions. 16. Results for M3uztitilication and Division of Tensors. —(i.) The following are immediate results of these laws: If p is any tensor, b pr b. pa pb. a pa a b. - --- -- b -= f, or E —, pb pb b p,b pb b c c a. - a.- d a c _ dc d ac ~~and b d b bd bdc' muipliion ad ivision a cocened, therefore, hold for (indi and b d b b \ d b bc and if then a bd C.bd, or ad= bc, &c. b d b d (ii.) The whole of the laws of commensurable fractions, so far as multiplication and division are concerned, therefore, hold for (indifferently, commensurable or incommensurable) tensors, and each equation represents a geometrical relation between points on the line 01, that is, between pure lengths. 17. Addition and Subtraction of Tensors.-(i.) To find the point G from A and B, by the addition of OA to OB in either order, as OA-+ OB = 00 = OB+ OA, is a geometrical commutative operation. It must now be shewn (v.), that under these circumstances we can find B from A and C, or A from ART. 17. i. —viii.] ALGEBRA OF PROPORTION. 23 B and C, so that OA = 00-OB, OB = OC —OA, and (00-OB) +OB = 00, (OA+ OB)-OB = OA. (ii.). Expressed by means of tensors, these equations give ac. 01+ b. 0I = c. 0I = b. 0I + a. 01, a. 0I = c. O1- b. 0,] b. OI = c. 0I- a. OI, (c. O. O-0)+b. I. I = c. OI, (a. OI+ b. OI)-b. 0 = a. OL (iii.) Taking then the symbol a+ b to mean c, as derived from OG, thus related to OA and OB, and similarly for c-b, we have a b = c = b+ a............................ (1), a = c-b, b = c-a, and (c-b)+b = c, (a C+b)-b= a. (iv.) From these definitions it follows that in c-b the c is always greater than b, that is, derived from a line O0 longer than OB. No meaning can be given to b -c, because OB- 00 has no meaning in the geometry of magnitude. The algebra of tensors has therefore, as regards subtraction, the same defect as the ordinary algebra of commensurables. The resultant difficulty of negatives and consequent imaginaries is overcome in the next section. (v.) To form OC- OB, we set off a line CA = OB from 0 towards 0, and find A such that OA = O- OB. Now although CA could not project beyond 0, the point A might lie on O itself, in which case B lies on 0, and we may shew this by writing 00 = 00- 00, wherever C may lie. Now from the point 00 we can form no tensor, because OI, 00 can have no ratio, as no multiple of 00 has any length (art. 1. ii.). But the operation indicated by C0-0C is that of reducing any length to a point, forming one of its extremities; and if we represent this by o, we may inquire what are the laws of o. (vi.) First, by the nature of o, whatever be C, o. -0 = 00, that is, o. (c. 01) = o. 0I, so that we may write o. c = o, whatever be c. This equation, wherever it arises, shews that c is indeterminate. It also shews that if any conditions require o. e = b, where b is not = o, those conditions are impossible. That is, there is no point 0 at all which will answer the condition that, when you measure from O to C and then back to 0, you should stop at a point B short of O. This is the one geometrical impossibility of simultaneous coincidence and separation (Appendix I.). C is not "at infinity," as usually stated. And with this the whole of the conceptions deprecated in art. 6. i. fall to the ground, to be replaced by intelligibilities. See the observations and citations in Appendix I. (vii.) Then from the proved associative character of geometrical addition and subtraction, namely, that (OA 4 OB) - o0 = OA~ (OB + OC), taking all the upper or all the lower signs, we find for tensors (aIb)~c = a (b +c), provided always that the subtractions be possible. And thence we obtain the further laws of o, ac- o = a, whatever be a. (viii.) Since i, o, answer to the arithmetical 1, 0, the laws of addition give i + i as 2, 2 + i as 3, and so on. But in these Tracts the geometrical c 2 24 Ill. THE LAWS OF TENSORS. [ART. 17. viii.-19. iii. operations i, o will be kept distinct from 1, 0. Thus geometry and arithmetic will be completely separated. The figures 2, 3, &c. represent certain arithmetical operations so that 20A = 01+ OA, 2a = a+ a, 2 2i 2i = - z+, -3a - 3a, &c. Again, a a.a, = a.ii., and so on. While o. OA = OA-OA, o. a = a-a, o. = i-z. 18. Distributive Character of Tensors.-(i.) Let A, B, C, D, P be such points that pa = b, pc = d, whence a = b, or 0: OC:: OB: OD, by Art. 15. i. (6.). Then Euc. v. 17 and 18, supposing OA > 00 for the lower sign, a 4-c b Lctd (OA 4 OC): 00:: (OB - OD): OD, or = c d ' whence b d = d. (a l c), or papec =p. (a=Lc), that is, tensors are distributive. (ii.) And taking, therefore, p. (a —a) to mean pa —pa, we have p.o = o, as well as o. = o (art. 17. vi.), or o is commutative with any tensor. 19. Applications of Tevsors.-(i.) This completes all the laws that need be adduced; powers with integral coefficents follow immediately, roots become intelligible as expressed either geometrically (for square roots, and for any roots by "Peaucellier's cell"), or as tensor limits to converging sums of tensors or fractions (art. 11. iv.). The whole of arithmetical algebra has been shewn to hold for tensor algebra, which also includes incommensurable algebra,. (ii.) The application to all the numerous cases for which merely quantitative geometry is used, is too evident to need explanation. And. if, when OM = AL', we agree to represent the tensor mz by AB in calculations, the whole of the usual algebra of quantitative geometry becomes exact, without any trace of limits. Moreover it would be possible to take proportion immediately after tththleory of parallels, and prove by tensors numerous propositions which, although usually (and very properly) otherwise proved, it will be a useful exercise to prove by tensors. (iii.) The whole of Euc. ii. admits of this treatment. The algebraical proots of these propositions, so mucch eschewed, now become strictly geometrical. Thus (AO+ CB)2 = A (;2 + Ci52 + 2A10. CB, with the conventional notation of (ii.), is a tensor relation. Referring each term to 0O, it becomes a relation of lengths only, and as such should be dravnal by the learner, that he may fully feel its meaning. Thus take 03 O, ON 1, O, Q so that in lengths 01: AC:: AC: OM, 01: CB:: CB: ON, 01: AC:: B(-: OP, and 01: AB:: AB: OQ. The meaning of the equation is that OQ = OMI+ ON+ 20P. Referred to the square on 01, it becomes the relation of rectangular areas, which is figured in Euc. ii. 4; for (fig. 4.) since 01: OM:: rect. (01, OF): rect. (03l, OP), we have rect. (01O, OP) = mi. rect. (OJ, OP). ART. 19. iii.-20. vi.] IV. THE LAWS OF CLINANTS. 25 Similarly, on taking OJ on OP of the length 01, so that rect. (OI, OJ) = square on OI, we have OJ: OP:: rect. (01, OJ): rect. (OI, OP), so that, on using p for the tensor of OP, rect. (Of, OP) -). square on OI, whence rect. (OiM, OP) = mp. square on Of = 01M. OP. square on OI, on using the ordinary notation. The proof is here conducted by proportion only, and quite independently of commensurability, so that the objections to the "algebraical proof"-really, commensurable proof-of Euc. ii. no longer hold. Referred to cube on OI, it becomes a relation of rectangular solids, having one constant dimension OJ. (iv.) The demonstration of all these relations flows at once from the laws of tensors. But there is no room for negatives or imaginaries in an algebra derived from the geometry of magnitude only. The laws of both are obtained at once from the geometry of direction, as follows. IV. THE LAWS OF CLlNANTS, OR TnHE ALGEBRA OF SIMILAR TRIANGLES LYING ON THE SAME PLANE. 20. Data from the Geometry of Dilection.-(i.) The following propositions are borrowed from the geometry of direction, as opposed to that of ratio, or magnitude only. See fig. 5. (ii.) AB, without further limitation, always represents the line AB, as respects both magnitude and direction, considered as the trace of the motion of a point along a straight line from, A to B, A being its iitial and B its.final point. When length is considered without direction, write len AB, and read " length of AB." (iii.) The equation AB=-OD implies that AB and CD are the opposite sides of the parallelogram ABDCA, the points lying in this order. (iv.) Directional Addition is defined by the equation AB+AC = AB+RBD = AD, or the sum of two adjacent sides of a parallelogram measured from their point of intersection is the included diagonal. And since in this case AC.+AB = AC+ CD = AD, directional addition (which is quite different in its results from quantitative or rational addition) is commutative. It is also easily shewn to be associative. (v.) Directional Subtraction is defined by the equation AB-AC = AB+CA = AB+BE = AE = CB, or the directional difference of two adjacent sides of a parallelogram is the transverse (non-included) diagonal, measured from the final extremity of the subtrahend to that of the minuend. (vi.) IOI', JOJ' (fig. 6 and 8) are diameters of a stancLcdard (or unit) circle, drawn at right angles. The position of centre O (taken as origin) and radius OI, (taken as the standaLrd of length and of orifginal direction,) and the direction of right angle I)OJ, (taken as standard of ang ular rotation,) are arbitrary, but once fixed remain throughout the problem, and deter 26 IV. THE LAWS OF CLINANTS, OR THE [ART. 20. Vi.-21. i. mine the plane IOJ on which all points are situate. The above lettering (founded on Sir W. R. Hamilton's) is assumed throughout. (vii.) M, N, P (fig. 6) being any points in the plane, (these words omitted in future,) and M', N', P' the points in which the straight (this word omitted in future) lines OM, ON, OP cut the standard circle, then the arc M'N' is measured from if' to N' through not more than a semicircle, so that its direction is not ambiguous except for a semicircle such as arc II', which may be either arc IJI' or arc IJ'I'. If the length of chord M'N' = length of chord I"N", and direction of arcs M'N', M"N" the same, then arc M'N' = arc MI"N"; but ch M'N' is not = ch 1M."N", unless they are coincident, and so of all similar cases. (viii.) Always, arc 'N' + arc N'P = arc MI'P', arc l'P' - N'P' = arc lM'P' + arc P'N'= arc 1'N', and the law of association also holds. All this is similar to, but different from (iv. v.) (ix.) The directional sum of any number of directed arcs is therefore a directed arc not exceeding a semicircle. If in be an integer, then (see fig. 7, where m=3) m.arclX = arcIV, determines IV unambiguously when IX is known; but there are m different directed arcs IX1, IX2... IX,, which satisfy the condition m. arc IX = arc IV when IV is known. And, assuming the power of finding an arc whose length bears any given ratio to that of a given arc, if m is an incommensurable tensor, the above equation admits of an infinite number of solutions. This is the source of the ambiguity of equations in all cases. (x.) The conclusions in vii., viii., ix., hold for any directed angle, or Z MON, fig. 6, subtended by the directed arc iM'N', and having OMl for its initial and ON for its final arm. Under these conditions no directed angle greater than the directional sum of two right angles in the same direction can occur (art. 3. iv.). Z 101 is called a null angle, IOI' a straight angle. If AB -- OM and CD = ON, by L (AB, CD) is meant z MON = Z M'ON'. When the amount of rotation in angles is alone considered, independently of the direction, the angles are said to be rationally equal. Shew this thus, amt MION, and read "amount of angle MON." (xi.) By directionally similar triangles are meant similar triangles in which the rationally equal angles are also directionally equal, so that their differences two and two = Z IOI. By conjugately similar triangles are meant similar triangles in which the rationally equal angles are directionally opposite, so that their sums two and two = Z 10. Any three separate points determine a triangle, whether they do or do not lie on the same straight line. The notation ABC A A'B'C' (read "ABO sim A'B'O' ") denotes the directional, and ABC V A'B'C' (read "ABC con-sim A'B'C' ") the conjugate similarity of the triangles ABC, A'B'C'. Sir W. R. Hamilton (Elements of Quaternions, p. 112, art. 118) uses the terms directly and inversely similar, and the notations A ABC c A'B'C', and A ABC a' A'B'C', for the present ABC A A'B'C' and ABC V A'B'C' respectively. 21. Directionally similar Triangles expressed by Clivants.-(i.) This being premised, let A, B, fig. 8, be any points, and determine C, so that IOA A BOC, which results from a simple geometrical construction. In this case also IOB A AOC. ART. 21. ii.-24. i.] ALGEBRA OF SIMILAR TRIANGLES. 27 (ii.) Then 00 is found by an operation on OB, determined by the point A, which operation will be called a clinant, a term introduced by myself in 1855, see Appendix III. It will be marked by the small letter a, corresponding to the large letter A, by which the point is noted. The tensor of OA (art. 12. iii.) will henceforth be marked Ta, as explained in art. 26. ii. It is sometimes convenient to use small Greek letters, a, /, /, A, e, for clinants. In such cases, when the corresponding capitals are the same in the Latin and Greek alphabets, I find it necessary to distinguish the latter by an apostrophe, which is not otherwise used, thus A', B', F, A, E', see fig. 33. The result is then written (read a.OB as "a ante OB"), 00 = a. OB, from IOA A BOO, and 00 = b. OA, from IOB A AOC. 22. Commutative and Associative Multiplication of Clinants.-(i.) From art. 21. ii., OA = a. OI, &c., fig. 8; and expressing c by ab in the first case, and ba in the second, we obtain, as in art. 13, c = ab = ba, or clinants are commutative in mnultiplication. (ii.) In this case' len 01 len O:: len: len 00, so that Tc = Ta. Tb, and ZIO = Z IOB + BOC=O += Z IOB + Z IOA= Z OA + Z IOB. (iii.) Then if A, B, C are any points, the proved association of tensors in multiplication, art. 15. ii., and of directed angles in directional addition, immediately establishes that a. be = ab. c, or that clinantts are associative in multiplication. 23. Division of Clinants.-(i.) When we have given A, C to find B (fig. 8), or B, C to find A, on the condition that c=ab, the geometrical operations are of the same kind as before, and will be represented by o C a c whence - a = ba = c, =, a a a which are the laws of division. b d (ii.) If then (fig, 9) we have in -, and m = -, A, B, C, D being a c b d different points and M determined as above, we have =, and a c 10O A AOB, and 101 A COD; whence AOB A COD, so that - = - represents the relation of directional similarity between these a c triangles. (iii.) All the relations found for tensors in art. 16. can now be proved for clinants, and in each case establish relations between the positions of points, or relations of directional similarity between triangles, and hence of directed angles and directed arcs. 24. Addition, Subtraction,, and Distributive Character of Clinants.(i.) The relations in art. 20, on putting OA+OB = C0, OC-OB = OA, 28 IV. THE LAWS OF CLINANTS, OR THE [ART. 24. i.-25. and properly defining a+b, c-b, after the model of art. 17, give a+b = c, c-b = a, so that the associative laws of directional addition make clinants associative in addition. Similarly the reduction of directional subtraction to directional addition (art. 20. v.) effects the same for clinants. (ii.) But the equation OA = 00- OB = BO +OC = BC, by art. 20. iv v., shews that BC = OA = a. I1= (c-b). 01, and thus gives us power to find the clinant of any finite directed line on a plane. (iii.) Now if (fig. 9) PQ= OA, PR= OB, PQ'= 00, P'R'= OD, and - -c, so that BOA A DOO, then will also RPQ A I'P'Q'. But b c under these circumstances a=q —p, b = r —p, c = q'-p', d = r'-p'. Hence the equation q — =- - -, means RPQ A R'P'Q', r-_P r -p and consequently gives a perfect algebraical representation of this geometrical relation, implying an equality of angle and proportionality of length, without any reference to commensurability or limits. (iv.) Let COD (fig. 10) be any triangle. Alter the lengths of the sides C0, OD by extending or contracting them to OC' and OD' in such a way that both C', D' lie on OC, OD, on the C and D side of O respectively. Then if len OC: len OC':: len OD: len OD':: len 01: len ON, we have, by art. 21. ii., c' = Tn. c and d' = Tn. d. (v.) Next suppose the whole triangle to be revolved about the point O, into the position C"OD", so that L C'O0" = L D'OD" Z ION, in which case also Z (Ci'D, C"D") = Z C'C00= Z ION, because C'D' can not have rotated differently from the arm OC' to which it is attached. Consequently c'" nc, d"= nd, and C"D" = n. CD = n. (cd-c). 01, as is well shewn in fig. 10, where C"E= CD, and ION A OC"ADOD" A EC"D". But C"D"- OD" — O"= ndc. 01-nc. OI. Consequently n. (d-c) = nd - nc, or clinants are distributive. 25. 0 the originality of these Conceptions of Tensors and Clinants.The slow and painful degrees by which I have at length arrived, after twenty years of thought and detailed work, at the above extremely simple fundamental laws and notation for tensors and clinants, and at the results to be subsequently sketched, may be seen by reference to Appendix III. Since the year 1855, when I first became acquainted with Sir W. R. Hamilton's Quaternions, I have as far as possible made use of his terminology and notation, which however I have been obliged to modify to suit my own objects. Although in some respects clinants may be regarded as colmplanrcar quaternions, and hence the theory of clinants may be brought under the theory of quaternions, the introduction of three dimensions, and all its complications, with its generally non-commutative algebra, was opposed to the object I had in view; and hence I have had to pursue a completely independent course, and in especial my term vector (art. 26. v.) has not precisely the same meaning as Sir W. R. Hamilton's, but only a correlative signification. The ART. 25.-26. vi.] ALGEBRA OF SIMILAR TRIANGLES. 29 two may be distinguished as clinant and quaternion vectors when needed. Before seeing Sir W. R. Hamilton's Quaternions, I had used czumbent and sistent for what are here termed scalar and vector. Some terms and expressions, and most of the algebra, are entirely my own, though the reader must carefully attribute to Sir WV. R. Hamilton whatever can be fairly traced to him, as I have had his magnificent labours constantly in mind and at hand. But notwithstanding his views, I believe that I may claim originality for the conceptions I have formed of tensors and clinants, as derived from pure geometrical proportion and similar triangles, and for my demonstrations of their laws. In particular I cannot recollect having seen elsewhere an approach to my proof of the associative character of tensors. And 1 know how gladly I should have availed myself of any such help, and how readily I should have acknowledged it. For general work I am of course deeply indebted to Augustus De Morgan and \Martin Ohm, and all the usual sources of information on the subject of imaginaries and complex numbers. See also art. 35. and Appendix II. 26. Suzbsidiary clinants. —(i.) 0, fig. 11, being any point, the biradial (art. 6. vi.) 100 determines the clinant c. Then the following subsidiary clinants can be readily formed. (ii.) Tensors. With centre 0 and radius C0 describe a circle cutting 0I, on the I side of 0, in T', then t' is the tensor of c, and is written t'= Tc. T2c means (Tc)2, see (xii.), and = T (c2) or Tc2. (iii.) Versors. Let the unit circle cut 00, on the C side of 0, in U', then u' is called the versor of c, and written '=- Uc. Observe that t'c= c, or c=Tc. Uc. (iv.) Scalars. Let a perpendicular from 0 cut 01, on either side of 0, in 8', then s' is called the scalar of c, and written s'= Sc. Scalars constitute the real orpossible, positive and negative, expressions of ordinary algebra, and are always represented by points on II' produced either way. (v.) Vectors and Jactors (my own term). From C let fall a perpendicular on JOJ' cutting it in V', and make V'OW' A JOI, so that W' falls on the I' side of 0, if V' falls on the J' side of O. Then v' is called the vector, and w' (which is always scalar) the jactor of c, and they are written v'= Vc, w'=W, in which case jw'= v or Vc =j Wc, the letter WT being used for jactor in preference to J, to shew its relation to V. Observe that c = s'+ v'= Sc+ Vc = Sc+j.We. As Sc,Wc are both scalars, the last is the usual form of imaginaries, which c represents when 0 does not lie on IO'. By the reduced jactor is meant Wrc = rWc2r7ri, where r is so chosen that TWVrc is not greater than ti. In fig. 11, Tw' being already < 7ri, W.c = We. (vi.) Conjuigates. Continue CS' to K' where S'K'= CS', then k' is called the conjugate of c, and is written k'= -Kc. Observe that 100 V 10K', and hence conjugates furnish the method of dealing with conjugately similar triangles. Observe that KKc = c, Kec = S — Vc, SKc = Sc, VKc = i'. Vc,.Kc Te T. = T = T. U,. c = T Uc. UKe = i, c - Kc = U2C, S2c = c + Kc, 2Vc = c —Kc. 30 IV. THE LAWS OF CLINANTS, OR THE [ART. 26. vii.-xi. (vii.) Reciprocals. Find R' so that COI A IOR', then r' is called the reciprocal of c, and written r'= Ac. Observe c. Rc = i, URc = UKc. (viii.) Angles, Am2plitudes, and Cissals (my own term). The Z 100 is called the angle of the clinant c, and written Z IOC = Z c. If we take length of OA'= to the rectified length of arc IU', and put A' on IOI', to the I side of 0, if Z c is in the same direction as /j, and to the I' side of 0 if Z c is in the same direction as Lj', then a' is called the amplitude of c, and written a'= Ac. Observe that Ac is always scalar, and that TAc never exceeds 7ri. Also, since Z (cc2a... c,) = L c1 + L c2 +... + Z c, never exceeds i', and hence TA (cic2... c,) cannot exceed uri, we have A ( c1.C..c..) = Acl +- Ac 2+... +Ac,, 2r7ri, where r is an integer so chosen that the tensor of this sum never exceeds tri. Define cos Z c, sin Z c, tan Z c, WTsc, Vsc by the equations Tc.cos Z c = Sc, Tc.sin Z c = Wc, tan Z c = TWc - Sc = Ws c=j'Vsc, and cos Ac, &c. by the usual scalar series, giving cos Ac = cos Z c, &c. The cissal of c is a term for Uc in the form cos Z c +j sin Z c, or cos Ac+j sin Ac, where it is expressed in terms of the Z c, or Ac, and it is so called because of Sir W. R. Hamilton's extremely convenient abbreviation cis / c or cis Ac. If for Ac we substitute Ac = Ac + 2n7ri, the result is called the nth amp2litude of c, and cis Axc = cis Ac, by the properties of the well known scalar series for cos A,c and sin Ac. (ix.) Logometers (De Morgan's term). The napierian logarithm of any tensor is a scalar, and the usual process is supposed to be known, and assumed to be executable geometrically by some arrangement like " Peaucellier's cell." It is easily shewn that series in which the sum of tensors of the terms form a converging series, converge to a definite clinant. It is worth while constructing several terms of such a series as i+x+2+..., where x = -j, and seeing what is meant by its continually approaching to y, where (i —j) y = i. All the laws of convergent series therefore hold for clinants. Represent the napierian logarithm of Tc by X Tc, and find the points L1, L2, L' so that 1i= XTc, 12 =jAc, and OL'= OL1+ OL,, then I' is called the logometer of c, and written 1'-= Lc. If we take 12l = jAc, and OL'n= OLj + OL-,,2, I, is called the nth logometer of c, and written 1I, = L,,c. Hence Lc=Loc, or the original logometer of c. Also, SL,,c = = X Tc. (x.) Metrands (my own term). Using the expression Ex for the X2 a3 well known series i + + + 1.23 +..., when + is any clinant, 1.2 1.2. w and having found E. Sc which will always be a tensor, set off a point M' so that Tm'= E. Sc and Am'"= Itc, then in' is called the nzetrand of c, and written mn'= JJc. This gives Mc = ESc. cis We, and hence ML,c = ESL,c. ecis WVL,c. But SL,^c = SLc = XTc, and hence ESLc = EXTc = Tc. And cis YL,,c = cis A,,c = cis Ac, so that MLc = Tc. cisAc = c. If Ac = o, and hence c = Tc, then Lc = Xc, and MLc c = c X = EcLc. (xi.) If (fig.12), c = e-n -, then the construction of the figure comm-n mgg -- nb pared with fig. 11, shews that - t'= =Tc, 2n — = u'= Tc,?. - n in-n i ART. 26 Ki.- 28.] ALGEBRA OF SIMILAR TRIANGLES. 31 mn-s", -v' f -w" m,//, --- s =Sc, --- = v _ Vc, — = w - - =, k= c, — m - n m - n m' -n m -n -r = r'= Rc, L NMP = Z I00 = Z c, and hence all these can be m- n obtained without a previous reduction to the form c, but Ac, Lc, Mc require that reduction, or its equivalent. (xii.) Observe that when any one of these signs T, U, S, V, V, K, R, &c., are employed they refer to all letters which follow until either a point (.) or a (+, -) sign intervenes. Thus Tab = T (ab), UaRb = U (a.Rb), and not Ua. Rb. But the point may be used thus U..aRb, if thought more distinct. Also that T"a, UZ"a, &c., mean (Ta)", (Ua)", &c., as in trigonometry cos' x commonly means (cos x)". Since T(Ta) = Ta, U(Ua) = U-a, &c., this is the most convenient notation. Thus also L'a- = (La)n, and LLa must be used for L (La), which is seldom required. 27. Some Relations of Subsidiary Clinants.-The hints given in art. 26. shew the relations of these important subsidiary clinants, excluding logometers and metrands, for single clinants. The following gives some of their relations for combined clinants: S(a+b) = Sa+Sb, V(a+b) = Va+Vb, K(a+b) = Ka+Kb; Tab = Ta. Tb, T (a-b) = Ta - Tb, or TaRb = Ta. RTb; Uab= lUa. Ulb, UaRb = TUa. RUb; Kab = Ka. Kb, Kaeb = Ka. RKb; Rab = Ra. b; Sab = Sa. Sb + Va. Vb,;Tb = (Va. Sb + Sa. Vb). The two last equations are found by putting ab = (Sa + Va).(Sb + Vb), and when divided by T. ab give the trigonometric formulae for cos(Aa+Ab) and sin (Aa+Ab), see art. 26. viii., being their most general independent proofs. SaRb = (Sa. Sb -Va. Vb). T2Rb, VaCtb = (Va. Sb-Sa. Vb). T2Rb. These two last equations are found by putting aRb = (Sa + Va). (Sb - Vb). T2Rb, and when divided by T.aRb give the usual trigonometric formulae for cos (Aa-Ab) and sin (Aa-Ab), being their most general independent proofs. S (Sa. Sb)= Sa. Sb, V(Sa.Sb) = o; S (Va.Vb) = a. Vb, V (Va.Vb)=o; S (Sa. Vb) = o, V(Sa. Vb) = Sa. Vb; T2(a + b) = T2a + T2b + 2T2b. SaRb = T2a + T2b + 2T2a. SbRa, which contain Euc. ii. 12, 13, and i. 47, of which they form independent proofs. On putting for SaRb, SbRa the above values, they also contain the whole trigonometric theory of the solution of triangles. 28. General Exponential Expressions.-The clinant power I define unambiguously by the equation a( = M. bLa, throwing all variety of values on the solution of exponential equations (art. 29), so that 'VTa has only its one tensor value. Then, m, n, p being integers, 32 IV. THE LAWS OF CLINANTS, OR THE [ART. 28-30. ab. ac = ab'c generally; (ab)c = ab. M2nrcj, (a)b = abc. Mlr21pbj, ac. b = (ab)~. M2m7rcj, where mn, 1, p depend on adjustment of amplitudes (art. 26. viii.); L,)IMa = a+2n7rj, where Wa +2n7r = TT,.a + 2m7r (see art. 26. v.); Lma+L,,,b = L1,ab, where A,,ma+A,,b = Apab; L,,Cab = bLa + 2nrj, where 1Wa + 2mr = W ab + 2mTr (see art. 26. v.); 3l (a+ b) = la.. b, M2n7rj = 1, Ila = ll (a + 2n7rj), 'IjWa = cis VWa, 3la = ESa. cis VWa, Ea = Haa. 29. Solution of General Exponential Equations. (i.) Given Xa = b = 2iaLx, then = i1(Lb. b a). lI(2r7rj. Ra). If a be a scalar integer, this gives the usual expressions. The equation b = IMIaLxp has the same solutions. (ii.) Given b = a x = MxLa; then =L,.b.RLa. But if b = 6 LexLa. Ji2p7rjx, then a = L,.b. RLpa, which is Martin Ohm's general solution. 30. Logometric and Binomial Series.-If (art. 26. x.) we put Ely=i+x, then y = L,,(i+x) = — 1+... in the usual way, n being determined so that the amplitudes should be the same on both sides, and Tx being < i. Also, taliing (i+a)b = iEbL (i+a), we find in the usual way (i+a)b 2r rj+i,b+ b (b —i) a2+ b (b-i) (b-2i) 3.. 1.2 1a..3 where the value of m has to be adjusted for amplitude, Ta being supposed to be small enough for convergence. The first series gives 2 3 (+j) = 2mrjrj I j + 13 - -. =2m -j+ 2 (1- + -.., i+ (- + -... where both series are convergent, the first = Xv/2, and the second lies between 1 and 1 —, so that mi, which has to be adjusted so that 3, the whole amplitudes must be < wi, will = 0; hence L(i+j) = i. X/2+j (l1 —+ -... But A (i+j) = ~-.i, T(i+j) = V/2.i, so that L(i+j) = iL(V/2. cis 1 r) = i. X 2 + I 7j, whence, on comparing, we have Leibnitz's well-known series 7,=4(1 i —... ART. 30-32. iii.] ALGEBRA OF SIMILAR TRIANGLES. 33 Similarly, from L 1 (i. V/3 +j) we find 1 1 Taking 2V3 = 3'4641016, and proceeding only as far as +21' 3% = 00000008, this series gives 7r = 3'1415911, or five places correct. 31. General Gonionmetric Series.-(i.) When x is any clinant, let Gx = 1 (Ej + Ej'), Zx = — j (Wa- -E'i), Px = ZX. Gxa, Qx =. Rzx, so that when x = Sy, these expressions become cos Sy, sin Sy, tan Sy, cot Sy, respectively, of which series they are the clinant generalisations. (ii.) Let G-x, Z-x, P2x, Q-x, (read, " G-invert of x " &c.,) be determined by the equations c~~=j'z [ +j V('-?2)], zx = j'L [ 1(i-x2) +jx], PX = 2' " [L (i + xj) -L ( ---xj) ] -Q = I j [L (i +j)-L (x-j)] Then GG'x = ZZax = LP'x = QQd x = a,, but the order of the symbols is important. In the scalar case I use cos Sy, sin S, ty, n, cot Sy for the inverts, in place of cos-'Sy &c., which are inappropriate whether cos 2 mean (cos x)2 or cos (cos x), because the signs cos, cos-1 would not be commutative. (iii.) Given Gx = c, then - r = 2nr + GCc. Given Zx = c, then either x = 2n-+Z'c, or (2n+l)7r —Zc. Given Px = c, then X = -7w + P'c. Given Qx -- c, then X = n Tr + Qc, whence all the scalar cases may be deduced, giving forms equivalent to those in Martin Ohm'sVersuch eines vollkommen consequenten Systems der Ylathematik, vol. 2, third edition, 1855, chap. viii. 32. Comnpletion of the Lawts of Clinants.-(i.) This completes the whole of the fundamental laws of clinants, which are shewn to be those of ordinary algebra, including imaginaries; and as each clinant expression can be perfectly constructed on the principles already given, by the elementary process of forming directionally or conjugately similar triangles, every one of these so-called itaygi;nary forms comes to be expressed by a, real point on the plane IOJ. (ii.) The fact that J = a"' +ax'l +...+pljx+ q, where all the coefficients are clinant, but m is an integer, can be expressed as a product (x-a,) (a -a29)... (x-a,,,, is proved by a process precisely similar to that in Sir W. R. Hamilton's Elements of Quaternions, Book II., Chap. ii., section 5, p. 265, which however admits of considerable simplification. (iii.) The solutions of quadratic, cubic, and biquadratic equations are conducted in the usual way, but with material simplifications, and the roots 34 IV. THE LAWS OF CLINANTS, OR THE [ART. 32. iii.-34. iii. are constructible by an elementary process which is rendered occasionally troublesome by the practical difficulty of drawing equal angles with sufficient exactness when the arms are greatly extended. But the process is always strictly geometrical, granted the power of dividing angles in any ratio, and of interposing any number of geometrical means. I have myself constructed every case, assuming the coefficients to be given by any points on a plane, with sufficient exactness to verify all the relations between the roots and the coefficients by geometric construction. 33. Geometrical Construction of Clinant Combinzations.-It would be beyond the purpose of these rough notes to enter upon details, but the following simple and frequently recurring cases should be noted: (i.) ax = c, make IOX A 1AO, so that X is the B of fig. 8. te-m a-b (ii.) -- - a —, make MNX A BOA, as in fig. 13. (iii.) x' = ab gives a =, so that XOA A BOX; and hence, x b fig. 14, if COD bisects Z AOB, and len 00 = len OD is a geometrical mean between len OA and len OB, the points C, D construct the values of x. Either of the lines 00, OD is called the mnean bisector of OA, OB, or of the biradial AOB. When 0, A, B are not collinear, A, C, B, D are concircular, for ab = 2 = i'. cc, whence a —. = i', and a-d b-c hence COAD+ Z DBC = Z i'. (iv.) 2 = a2 - b = (a + b) (a- b) = hiL, fig. 15, from A draw AH = OB, AK = BO, and construct OX', OX" as mean bisectors of OH, OK, by (iii.) (v.) y2 = a+ b2 = — b'2, if b' = jb,fig.15. Draw OB'of the len OB, making Z BOB' = Z j, 'A = AH' = OB', and find OY', OY" as mean bisectors of OH', OK', by (iii.) (vi.) ax+ bx+c = o, then 2ax = bi'~,/(b2-4ac). Find A', B', by (iii., iv.), so that a'2 = 2a. 2c, b'2 = b2-cb'2, then 2ax = bi'-b', and the two positions of X are constructed by (i.) See an example with various cases in art. 34. viii., figs. 14, 21, 22. These simple constructions suffice for all the cases considered in the next Tract. 34. Applications of Clinants.-(i.) In Sir W. R. Hamilton's Eleents of Quaterniols, Book I., chap. ii., and Book II., chap. ii., will be found a large quantity of geometry suitable for direct treatment by clinants, and this treatment will be found to introduce much simplification. To these I need merely refer. The following will suffice to shew the nature of direct clinant treatment of geometrical problems, and will give some results required in the next Tract. (ii.) If A, B, C are collinear, V a-b = o. a-c (iii.) (1) If the straight lines AB, CID, EF converge to a common point X, then a - 6 C - o e - / v-e c-v -- = 0a c —b c-d -f ART. 34. iii. iv.] ALGEBRA OF SIMILAR TRIANGLES. 35 a- b c-a a-b e - a whence V. V V - -- e-f c-d c-d' ec-f' which by forming the auxiliar points B', C', D', E' may be made to take the form a-b' c'-a a-d' e'-a c- b' a-e ____ ~ -- - or -,, e-f c-d c-d e-f a- d' a-c" or B'AD' A E'AC', as the condition of convergence. (2) If three straight lines known as perpendiculars to OA, OB, OC (fig. 16), converge to X, we have s-X= 8 - S i, a b c and since S s (. = 8 S + V V, by art. 27, c o ao c a c a cc a ax we have i= 8- +V V, c c a a a a and similarly = S + V V b b a and hence eliminating V —, we find -c \ b / b \ which, by forming the auxiliary points M, N, as in art. 26. xi., may be to. 5, a -m. \ a-in a-n reduced to ar(i —, - or -- c b c / c —m, b-n or AMl A ANB, as the condition of convergence. (iv.) Let (abed) - (a-b). (-d) (a- d). (c- b)' the letters being carefully written in this order. Then (abed) will be called the aural (aL-harmonic r-atio + al) of the four points A, B, C, D anywhere situate on a plane. (See Appendix III. for the principle of this terminology.) It is also convenient to shew the omission of tho terms involving any one of the points by the notations c-d C-n a-b a-b (.b) b (a.. cb) -- (ab.. ) — ) and (abc..) -. c-b a-cl a-dl' c-b Then (abed) = (bade) = (cdab) = (dcba), (abed). (acb) =i,, (abd). ( b) =. (adbe ) = ', and, since (a-b)(o-c) +(b-c) (-d)+(c —)(b-cl) = o, also (abed,) + (acbd) = i, (abcc) + B- (adbc) = i, and (acdb) = [(abcd) -i]. B. (abed), so that all the 24 possible anrals can be expressed in terms of one. If a'=- a + n, = b + '= c+= e, d'= dcZ+n, (abed) = (a'b'c'c'). If V. (tabcd) = o, then Z BAD + Z DCB = z i or Z i', and the four points ABCD are either on a straight line or on a circle. Putting (abed) = (a'b'c'c'), the first case only is the foundation of all Chasles's theories of homography and involution, and the first and second cases 36 IV. THE LAWS OF CLINANTS, OR THE fART. 31. iV. V. combineed form the basis of Mobius's Kreisvertandtschaft, or circularrelationship, all the results of which are much more simply written and obtained by means of clinants-as I have found by actual work. If (abcd) = i', the anral becomes a hacrmal (hcaron-onic-ratio+at), and the points lie harmonically on a straight line, or on a circle. For example, in fig. 29, (edfd') = (a'eaf) = i', and EDFD' is a straight line and A'EAF a circle. The troubles experienced by Chasles (Geom. Sup., chap. V.) arising from imaginary points in harmonic ratios, at once disappear, and the investigations are not only simplified but generalised, and, as will be seen, are capable of still further generalisation (art. 44). (v.) If A, B be any two points on a plane, the cannal (an-gle+ al) and tan-nal (tan-gent+ a) of the directed biradial AIB (written bir AIB), which gives not merely the Z AIB, but the length and direction of both arms IA, IB (compare art. 6. vi.), are written as follows, and express the following functions of a, b respectively: an AB = - tal A -- aa — i —ab' the letter I being always understood between A and B, and the order of the letters being important. For the use of these forms, see art. 39. iv. If Sa = Sb = Sc = Sd = o, and hence Va = a, Vb =b, V =, Vd = d, then Vs.an AB =- anA.= tan AB =j tan AIB, S. -an AB and ( abcc) -= sin AIB. sin CID sin AID. sin CIB The general expressions will be found to include the whole geometrical theory of inmaginary angles (as distinguished from the algebraical theory of art. 31). The following properties shew some analogies and solve some previously incomprehensible relations. They should be all constructed geometrically. Generally tal AB + tan BA = o. tal AB - tal AE - tal BE i- tal AE. tal BE' tal AB + tal BO + tal CA + tal AB. tal B. tan CA = o. And if a+a' = b+ b'-= c+c'= o, and tal AB = c, then tal AO = a, tal OA = a'. tal AC = b, tal BA = c' tal BC'= a, tal CA = b', tal B' = a, tal AB' = tal BA'. If tal AX = tal XB, then IX1 and IY2 are the medials of bir AIB, where a,, x2 are the roots of x2 - 2. (i + ab). B (a + b). x + i = o. In this case (x,ax2b) = i', or X1, X2 are harmonically situate with respect to A, B; see (iii.). And since,xx2 = i = it, X1, X2 are also harmonically situate with respect to I and I'; see art. 44. iii. X,, X2 are the points of intersection of the two figures XAX2B and XIX21J', of which one may be a straight line and the other a circle, or both circles. ART. 34. V.-viii.] ALGEBRA OF SIMILAR TRIANGLES. 37 Although tal II and tal 'I' are indeterminate, yet when A is neither I nor I', tal AA = o, tal AI = i', tal IA = i, tal AI'= i, tal AI. tal AI'= i'. If b = -a, or ab = i, R tal AB= o, and tal AB does not exist. In this case, if Sa = Sb = o, or A, B lie on OJ, Z AIB is a right angle. Hence, in the general case, I say that the biradial AIB is orth.a (dpO-oc- + c), a conception of considerable importance in stigmatic geometry. Bir X1IX2, formed by the medials of the bir AIB, is orthal, because xz2 = i. If an AY = an YB, then (y-i)2 = (a-i)(b-i); and y,, y, being the roots of this equation, IYI, IYE are the mean bisectors of the bir AIB (art. 33. iii.). If both A and B lie on OY, one of the means coincides in direction with one of the medials, but the lengths are different. (See art. 44. iii.) (vi.) If A, B, C be points in a circle of which 0 is the centre, T2a = T2b = T2c, or a. Ka = b. Kb = c. Kc, whence =K-, c a c c-b and ten b a-c K (b -c) a = 2a-c r —~, and then.K or - b, adh a K(a-c) b-c b b-c' that is z AOB = 2. Z ACB. This is a general and independent proof of Euc. iii. 20. 21. 22, in their proper statement for directed angles. See also art, 48. x. If D be a fourth point on the circle, it follows that U2 — c = U2 a — b-c b-d or U(acbd) = i or i', or V (acbd) = o; compare (iv.) (vii.) To find the points X, Y, Z on the unit circle so that Z IOX = 2. Z OXI, (the solution of which is evident,) Z OYI = 2. Z JOY, (which is Euc. iv. 10, with directed angles,) and Z IOZ = 3. Z OZI, (which is immediately constructible in Euclidean geometry from Euc. iv. 10). The properties of the unit circle give Tz=i, Kx=Rx, Ux=x, and similarly for y, z. The statement of the three problems in clinants is Ux = U2 -i, UY- = U2y, and U3Z- = UZ, x y z or U3 = U-(x i), U(y-i) = U3y, and U3(z-i) = U4z. Since U2 (x-i) = - i'. x, and similarly for y and E K(x-i) kx-i z, the first equation gives z3 = i'. x, or x2 =i', whence x,=j, x2 =j', the two triangles being OJ and TO', as is evident. The other two equations, on being squared (which introduces adventitious roots) and reduced, give y5= i' and z — i', and as one root of each is i', all the roots can be readily found. On calling them yY2, Y2, Y3 4, y, as the points Y1, Y2, Y3, Y4, Ys lie in order on the circumference of the unit circle in the direction IJT', we find that yl, Y3, y, and Z2 = y2, Z3 = yo, z4 = Y4 give the required solutions. (viii.) A, B, (fig. 17. 18. 19. 20.) being any points, find X and Y, so that IOX A ABX, and IOY V ABY. This is selected as giving rise to general simple equations, x-b and y-b = K a-b a-b D 38 IV. THE LAWS OF CLINANTS. [ART. 34, viii. ix,. (1) In no case can a-b = o, or A coincide with B. (2) If b=o, or B coincide with 0, then x=ax, or a=i, that is, A also coincides with I, and x is indeterminate; of course IOX A IOX whereverX maybe. But y=-aKy, or U2y=a, whence Ta=i, or OA is a radius on the unit circle, and any point Y on the line bisecting the the L IOA, fig. 17, makes IOY V ABY, that is VAOY. (3) Excluding these cases, make OC=BA, or c = a-b; first let c=i, or BA=0O-, then the first equation gives x-b=x, or b=o, (which reduces this to case 2). The second equation gives y-b=-Kiy, or y-K ij=b; and operating with K on each side, k y-y =Kb, so that, on adding, o=b + Kb, or Sb-ho, that is, B must lie on OJ, and 2 Fy= b, or Y lies anywhere on the line through D (where OD = OB) parallel to 01, fig. 18. (4) Take c = a-b as before, we have x-b = cx, or OXB A I07, fig. 19, or (i —c) x = b; or, making ld = i-c, we have dcx=b, that is, IOX A DO B, from either of which X is immediately constructed. (5) The second equation gives b+c. Kb y-b = c. Ky, whence Ky-Kb =- Kc. y, and y c= I c. Now if b + c. Kb = o, both b and Kb may = o, in which case / = o, and we have the evident similarity AOO A 100. But generally Kb +Kc. b = o gives cKc=i?, which leads to a mere identity o. y = o, from which notling can be determined. Reserving this case, and first putting b'=Kb, c'=Kc. fig. 20, and then cb'=- d, cc'= e, b+d = g, i —e = h, we shall have hy-=g, or IOY A flOG, and then IOY V ABY. (6.) For the reserved case, put y =, r + b/3+3', c - + /, where v, /3, 7 are scalars, and,', /3', ' are vectors. Since then IKy = n —'? &c., the two equations b + c. Kb = o, and y = b + c. -Ky, give, on equating the scalar and vector portions of each, / + /3 - y'/3= o, 3' + '/33- y/'= o, j3+r7- / '= 7, / +y'- 7y = /. Eliminating r) and 7' alternately from the two last equations, we obtain their value in the general case 5, frorm /3 (y+l)-I' + [(y-l)-y2' ] 7 = 0, /'-' (Y-/1) + [(y - (2 1)] j = 0; but applying the two first equations, the parts independent of r7, r7' each =o, and hence — =o, 1'=-o, and?/=o, which necessitates b=o, so that the reserved case only gives, as before, TOY A AOO. (ix.) Let A, B, C be any three points, and D a fourth, so that OD = CO. To find a point X so that the mean bisectors of XA, XB shall be equal to 00 and OD. This is selected as giving the general quadratic equation (- a)(x-b) _= c2 d2, or a2 - (a + b)x = c- ab. (1) If C0, OD are the mean bisectors of OA, OB, fig. 14, c2-ab = o, and a=o, or a + b, that is, X is at either of the two extremities of the diagonal OE, of the parallelogram of which OA, OB are adjacent sides; thus EF is clearly = OD. ART. 34. ix. —35.] V. STIGMATIC GEOMETRY. 39 (2) If c2-ab is not = o, then, fig. 21, putting 2m = a+b and 22 = a-b, on adding m2 to each side we have (am = )2 = 2 + c = p2b + c = 2 + or n,2, where Nr, N' are constructed as in art. 33. v., and x = xrn+,n = T'. or x = m-n- == m + n' = x'. The mean bisectors of X'A, X'B and X"A, X"B are X'E, X'E and X"F, X"F', which are = OC, OD respectively. (3) If p2+-c2= o, or OP, 00 are of the same length and at right angles to each other, fig. 22; then n = it' = o, and the two positions X', X" coalesce at M, so that there is only one position which satisfies the conditions. The mean bisectors of 3MA, 7fMB are IMG, 121'. (x.) To determine the points where a line perpendicular to OA cuts a circle with radius OB. As in iii. (2) we have in the line S. xPac = i, and as in (vi.) for the circle Tx = T2b. Then, by art. 26. vi., xRBa+KC. xRa = 2SxBc = 2i, and xKi: = T2b, whence eliminating K'v, we have = 2Cax- -ITja. T2b, and x = a -I- Uca /(T-a-T'2b); whence,. Ra = i 4 PTa. v/(T22- Tb). Unless then Ta, = < Tb, S. xIRt will not = i, and this is therefore the condition of possibility. There are no "imaginary " intersections. No "imagination" can make i = i4-k, where k is not = o, for this would lead to the impossibility of Appendix IT. A circle and straight line have therefore no "imaginary" intersections. This term applies only to a derived case, considered in art. 49. v. The meaning of this distinction is assigne in art. 36. v. When Tac=Tb, x=a, and there is only one point of intersection A. When Ta< Tb. x = a:kj. Ulc. V/(Tb-T'2a), wihich gives the two points determined by drawino Y'CA-X" perpendicular to OA, and making len AX' = len AX" = length of the perpepndicular of a right-angled triangle, of which the lengths of base and hypothenuse are the lengths of OA and OB respectively. V. STIGMA.TIC GEOMIETRY, OR THE CORERTSPONDENCE OF POINTS IN A PLANE. 35. No pr'evious compjlete represe1tatfion of Al7ebra by Geometry.Some of the results hitherto adduced have been already obtained (although less directly, and always by a more or less implied use of limits) from various geometrical " explanations " of "ilnaginaries," advanced with somne degree of he:titation, often on metaphysical grounds, and (except by Sir W. H. Hamilton) always by means of " complex numbers," or clinants of the form aSa-+j'Va, where Sa, TIVa were considered as the limits of convergent " possible " (that is, scalar) series. The class of problems embraced under the theory of Stigmatics have also been attacked with immense acuteness and wide success, in particular instances, but the occurrence of imaginaries have constantly baffled 40 V. STIGMATIC GEOMETRY, OR THE [ARET. 85. —36. i. the very lions of mathematical science, towards whom I feel but as the mouse that gnaws their net asunder by my clinant teeth. My firm belief is that there is not known to exist any intelligible, workable general theory but my own, nay, even any tenable, hypothetical particular explanation of the geometry of those imaginaries which constantly occur in the algebraical plane geometries of Descartes and Pliicker, or the higher plane geometry of Chasles; and that, until such a general theory has been furnished, there is no complete representation of geometry by algebra, or of algebra by geometry. The solution of this problem, the furnishing of one general theory which will embrace all cases of plane geometry from a single simple point of view, which shall never meet with any difficulties by the way from "imaginary" lines, "imaginary" angles, or " imaginary" figures; which shall make every step in every problem a pure piece of geometry (conceding the division of angles in any ratio and the interposition of any number of geometrical means between two extremes); which shall, in fact, iclentify Alyebr~a qiith Geom.etry,-this has been the ideal of my mathematical life, and I believe that it has at length beeli realised to the letter by means of my clinants and stigmatic geometry. Other labours have hitherto prevented me from sending it out in the form I have always wished to give it, with numerous illustrative and comparative diagrams; and I am now so far advanced in life that my power ever to do so becomes very problematical. The following brief notes, which contain my last unpublished notations and nomenclature, will enable any one of those distinguished mathematicians to whom they will be sent, if he finds time to scan them, to apply my theory far better than I could do it myself. Those who care to learn the history of the birth and growth of my conception of Stigmatic Geometry will find it in Appendix III. On the facts therein detailed, and on the citations from the works of eminent mathematician s in Appendix II., I distinctly claim originality for a conception, in forming which I have not obtained a scrap of help from the best writings of the best writers that I could consult. The mouse asserts her teeth. 36. General Goncertion of Stigmactic Geometry.-(i.) Let X and Y, fig. 23, be two points on a plane, connected by the clinant equation f (e, y) = o, which, so far as it can be solved, or so far as the properties of clinant equations are known, will enable us to construct the different positions of Y for every assumed position of X, (that is, with certainty so far as biquadratic equations extend,) and to deduce various relations between X and Y in all other cases. The continuous correspondence of the points X and Y, given by any such law, while X moves continuously over the plane, forms a stiygmat'ic. The point X, which moves independently, is called the inz'lex, and geometrically represents the independent variable x. The point Y, which is determined from X by the given law f (x, y), is called the stig',)a, and geometrically represents the dependent variable y. The pair of corresponding points, index and stigma, is termed a stig'm al, (stigmn-a 4- l; see an explanation of the origin of this nomenclature in Appendix III.,) and is written (XY), or (rly), or (x, y), according to convenience. The line OX is called the abscis'sc, the line XY the or'diCnate, and the line OY the ra'dius ART. 36. i. —iii.] CORRESPONDENCE OF POINTS. 41 of the stigmal (xy), and x, y-x, y are their clinants respectively. These three lines form the sides of the stigmal triangle OXY. To each index there may correspond several stigmata, in the same or different stigmatics. Stigmals with a common index are called co-stigmals, and their stigmata are called co-stigmata. (ii.) The points X, Y are said to be co-ordinated by the equation f (x, y) = o. If by simple geometrical constructions X', Y' can be determined from X. Y, so that X', Y' may be co-ordinated by a derived equation f (x', y') = o, then X, Y are said to be trans-orcinated to X', Y'; and the second stigmatic is said to be a transordination of the first. Such transordinations are frequently convenient for the purpose of simplifying the discovery of the points X, Y by means of the points X', Y'. The general theory is given in art. 47. Thus we may form subsidiary stigmatics having the same index X, but different stigmata U, V, by putting, as in fig. 23, 24, y-x=v, ju=v, y=x +v=x+jj, whereby the stigmatic equations become f (x, x+v) = o, / (x, x+ju) = 0, forming the connected ordinar and orthar stigmatics, which are related to the original stigmatic, stigmal for stigmal, as particular cases of transordinated stigmatics. If from the orthar stigmatics we select those particular stigmals for which both x and A are scalars (fig. 24), the stigmata of the corresponding stigmals form the real points of Cartesian plane geometry referred to reclangyldar co-ordinates, the Cartesian axes of the abscissae and ordinates being Of OJ; and all stigmata for which the one or the other or both of the points X, U do not lie on O.L or V does not lie on OJ, form the imaginary points of Cartesian plane geometry so referred. If (no figure) we make v = hAu', where h is any unit radius, y = x +7 h', and the new stigmatic is f(x, x + h1') = o, from which those stigmals (xy) for which x, u are scalar, have as their stigmata the real points of Cartesian plane geometry referred to the oblique co-ordinates of which 01, OH are the axes. For comparing stigmatic and Cartesian geometry it is convenient to have special names for these cases, which may be provided by the prefixes Cartesicca (abbreviated to car-,) and non-Cartesian, more briefly incar(in = negative+ Car-tesian). Thus carstig'mal, carstig'm2a, carin'dex, and so forth. Carstig'mata, are " real points;" not simply geometrical points, but points referred by ordinates to other points in the axis of the abscissae; incarstigmnata are "imaginary" points, that is, points which the former algebra indicated should be similarly referred, but which no one had been able to refer on the old theory, and hence merely " imagined" to be so referred, in order to preserve the old terminology. Rectangular co-ordinates will be assumed unless otherwise expressed, but the prefixes rec-, ob-, will distinguish the two cases. A carstigmatic is that part (if any) of a stigmatic for which the stigmals are carstigmals. A Cartesian stigmatic contains a carstigmatic, that is, some carstigmals, but also contains incarstigmals. (iii.) As any plane geometric curve whose properties are known may be treated as a carstigmatic, and expressed by f (x, x +jn) = o, with the condition that x, u are scalar; and as this can be immediately thrown into the general form f (x, y) = o, which will agree with the former 42 V. STIGMATIC GEOMETRY, OR THE [ART. 36. iii.-vi. as long as x, u are scalar, and which will also give all the relative positions of Y, when x is still scalar, but u not scalar, (that is, " imaginary,") or even when x is also not scalar,-it is evident that every result from any Cartesian form can be immediately included in its proper general clinant stigmatic, in which shape it is usually much easier to treat. " Imaginary" points can only thus arise in Cartesian Geometry; compare art. 34. x. If we further proceed to make the constants clinants, that is, refer them to any point on the plane, instead of those from which the scalar case was deduced, any such particular carstigmatic will suggest a still more general stigmatic, which is equally easy to treat, and is the only form which fully shews the geometrical relations. (iv.) Stigmatics are said to intersect in their commoni stigmals or stin'ctals (sti-gmals of in-tersection+al), of which the stigmata and indices are called stig'iinzs (stifz-ata + in-tersection) and indins (incd-ices +ii-tersection) respectively. The laws of such intersection are now precisely those in Pliicker's ThJeorie der algebriscl2en Curven (Bonn, 1839), the wq.7tole of which, transferred to stigmatic geometry, after the following theory of primals and quadrals is understood, may be interpreted as strictly geometrical. (v.) When the index moves on any path, the stigma moves on another path, corresponding point by point; these are the inz'it (indc-icis it-er) and stig''mod (ariyp-aroc 6;-oe). All incits which intersect in the index of a stinnal, have stigmods which intersect in its stigma. In carstigmatics the indit is a straight line, part or all of the Cartesian axis of abscissae, and the stigmod is that curve which was alone considered when Descartes founded his algebraical geometry, by referring any curve, point for point, to the axis of the abscissae by ordinates parallel to the ordinate axis. This reference was the egg from which the present stigmatic geometry was hatched. It was an addition to the ancient geometry, invented as a mere expedient for reducing it to algebraical computation, without any perception of the principle involved. It is evident from the preface to Chasles's Geome'trie S'aeprieure that he had not recognised this principle as identical with that of his own homographic geometry. But the fact of the identity of principle is shewn by the present inclusion of both as particular cases under Stigmatic Geometry, so that the method of working the two becomes indistinguishable. It will be seen, also, that the clinant stigmatic view is the only one which perfectly explains the principles of "signs " and "continuity." A carstigmod differs from a simple curve of the same form, by its impl.yizg a carindit, to which it is referred. The distinction is important. Thus when a simple straight line does not cut a simple circle, the line and circle have only to be considered as carstigmods, and Cartesian stigmatics are generated, which do intersect, although only in two in-carstin'nals. Compare art. 34. x. with art. 49. v. (vi.) From the theory of intersection, the analogous theories of contact (of any order) and asymptoticity may be immediately deduced. If f(x, y) = f (x, ). f2 (, y)+c = o, then f,(x,, y)=o, andf2 (,y) = o give stigmatics which have no stigmal in common with f (x, y) = o, but, as X recedes, have stigmata continually approaching to the co-stigmata in the original stigmatic, and are hence called its asymp'tals (asymp t-otes + al). ART. 36. vii.-37. iii.] CORRESPONDENCE OF POINTS. 43 (vii.) There is nothing in the form of the stigmatic equation f (x, y) = o to distinguish the index from the stigma. Either may be assumed as either, but the two stigmaties thus formed necessarily differ, unless the equation is symmetrical with regard to x and y, as in (s —x) (s —y) = (s-e)2, see art. 44. Given the direct stigmatic, with X as index, and Y as stigma, the inverse stigmatic, with Yas index and X as stigma, is the geometrical representative of the inversion of functions, which can be here only indicated. In this case one stigma may have many indices, giving con-in'dices and con-indi 'ial stigmals. (viii.) From the general conception of functions the meaning of clinant differential and clinant integral calculus, &c., is given. These are the only points which I have not yet worked out in detail. But the indications in Sir. W. R. Hamilton's ElemzenLts of Qalcternions, Book III. chap. ii., in Martin Ohm's Geist der DiWerential- ndc Intecrcal-Rechnng (Erlangen, 1846), in Casorati's Teoricac delle Flnzioni di VarCiabili Conmplessi (Pavia, 1868), in Hankel's Vorleszngen iiber die Coiplexen Zahlen und ih're Fnlctionen (Part I., Leipzig, 1867, Part II. will be the especial part when published), will suffice, with the present indications, to work out this part of the complete reconstruction of plane geometry. For the differential calculus, Taylor's theorem holds, and processes analogous to those for maxima and minima, and for tangents, immediately follow. 37. Integral Stiygmqatics -(i.) Henceforth attention will be confined to the integral stigayctic equations of the form M. (ay^+a'y-I+...) +- +-l (b+'y,- +...) +... =, where gn and n are integers and the other letters clinants. This is the fundamental form of equation assumed by Chasles in his Theory of Characteristics, (Comptes Rendics, 27 June, 1864, vol. 58, p. 1175), the whole of which theory (after primals are understood) may be incorporated in stigmatics, and applied to any points on a plane. (ii.) Dividing by y't, the sum of the terms not containing powers of y in the denominator is cax"'+bw' x +.., and if we put this = o, we shall obtain rn values of x, which, when substituted for x in the original equation, have no corresponding values of y. These point out m solitary ind-ices, having no corresponding stigmata. Similarly ayI' + a'y'l"+. =o gives n solitary stigincata, which have no corresponding indices. If we put x = y = z, we find an equation of mi+n dinensions in z; these give in +- doable poinbts Z, in which -the index coincides with the stigmata. When any one point is at once a solitary index and a solitary stigma, it is termed simply a solitary point. The above are called the peculiar points in a stigmatic. (iii.) Of this general form I shall give only the fundamental cases of primal (arts. 38. to 42.), uniiqtuadral (arts. 43. to 46.), and duoquaccdral (arts. 48. to 51.) stigmatics, but none will be treated with even a distant approach to detail. My second memoir on Plane Stigmatics, when the nomenclature is properly changed in accordance with that here used, and the notation altered by putting the present b-a- and (b —a) (d —c) for the ab and ab. cd there used, gives sufficient details to shew the power of the method; but it is impossible to abstract, much less to reproduce in the present improved form, the whole even of that memoir (itself a mere sketch) within the time and space at my command. 44 V. STIGMATIC GEOMETRY, OR THE [ART. 38. i. —iii. 38. Primals, or Cartesian Straight Lines generalised.-(i.) The simple stigmatic equation ax + b'y + c' = o, can, when b' is not = o, be reduced to the form y + (a-i). z = b = ac, which is the standard form of a primal stigmatic. There is no solitary index or stigma. C is the double point, B is the original point, that is, the stigma when the origin is taken as index. A is called the direction point, the triangle IOA A CXY (fig. 25) being the direction triangle. As it is necessary to become familiar with the geometrical relations of the primal, the reader should construct many figures with different positions of A, B, and hence C, beginning with cases where A and B lie on OJ, and C on OI, for which CB is the ordinary Cartesian line, as in fig. 34, and if X is chosen on 01, XY is parallel to OJ. But positions of X1 not on 01 should also be chosen, and the abscissa OX1 and ordinate X1Y, then give the imaginary Cartesian abscissa and ordinate of the imaginary point Y1. Fig. 25 gives a general case, and will indicate the method to be pursued. (ii.) Any two stigmals (xy), (x'y'),or (xy), (cc), or (xy), (ob), or (cc), (ob), will determine a primal, which may be written pri (xy, x'y'), &c. The direction point A and any stigmal (xy) or (cc) will also determine a primal, which may then be written pri (A, xy) or pri (A, cc) &c., the capital letter distinguishing the point. A primal is said to be drawn when a quadrilateral XYY'X'X has been constructed by joining the extremities of the ordinates XY, X'Y'. In drawing stigmatics generally it is convenient to guide the eye to the correspondences by making the stigmod YY' an unbroken line,- the indit curve XX' a broken line - - -, and the ordinates XY, X'Y' dotted lines.................. This will make the constant directional similarity, CXY A IOA, very evident in the primal. (iii.) The general form does not hold when b'= o, in (i.) In this case x=o, or x=m, and there is no direction point. The following eight peculiar cases occur so frequently that I have found it convenient to give them special names; they are here given in terms of both y and v = y-x, see art. 36. ii., for which the general equation becomes v + ax = b. Assume rm + M = o. direction original double NAME AND EQUATION. point. point. point. I. Ax'als and Parax'als. or'dinal, = o........................ none 0 0 or'dinal, x =o....m.................. none one M parordinal, an.none none 1 abscis'sal, y = x, v=o............ 0 all parabscis'sal, y =x + -, v =... 0 M none II. As'sals and Paras'sals. u'nal, y= o, v+x=o............... I 0 0 paru'nal, y= m, v+x=m......... I M M du'al, y 2zx, v- x=o............ ' 0 0 paradu'al, y = 2x +-m, v — = M M' ART. 38. iii.-39. i.] CORRESPONDENCE OF POINTS. 45 The name axal (ax-is +al) is given from the relation of these primals to the Cartesian axes, and the name assal (as-ymptote + at), because these primals are the asymptals of a cyclal (art. 48. v.), the so-called "imaginary asymptotes " of a circle. The prefix par-, or para-, denotes the sameness of the direction points, or para-llelism of the primals. If in the quadrilateral XYY'X' of (ii.) the two indices X, X' coalesce in X1, then pri (xy, xy') is a parordinal with constant index; but if the two stigmata Y, Y' coalesce in Y2, then pri (xy2, x'y2) is a parunal, with constant stigma. If the ordinate XY= ordinate X'Y', then pri (xy, xy') is a parabscissal with constant ordinate. If the line YY' joining two stigmata is always equal to double the line XX' joining the two corresponding indices, then the pri (xy, x'y') is a paradual. In fig. 33 pri (pe, ee) is a parunal, and pri (pe, of) a paradual; and in fig. 26, pri (mmn, mi) is a parunal, and pri (t't', ot) a paradual; in fig. 34, pri (oo, x'o) is a unal, and pri (oo, x'y0) a dual, and these two are there the asymptals of the cy'clal; see art. 48. v. (iv.) Given two stigmals (pq), (p'q') to find, fig. 25, the direction point A, original point B, and double point C. Make p- r = -q, then P- = i —a, or P'PR A OIA giving A, and - = a= -, p - P c -p p -p or CPQ A IOA A PPR giving C from A or from (pq), (p'q'), direct, and CPQ A COB giving B. (v.) If two stigmals (plq), (p'') are given, any other stigmal (.y) can be found without previously constructing A, B, or C, by putting the equation to the primal into the form X -- = Y-q,, or XPPI A YQQ', which also shews that every stiymod of a primal is similar to its own indit (compare the stigmod CQQ'YC with indit CPP'XC, fig. 25), and is the condition that three stigmals (xy), (p2), (p'q) should be coprimal, or lie on one primal. As this equation is satisfied by mn = ~ (p +p) and n = - (q + q'), (rmn) will be a stigmal on the pri (pq, p'q'). This stigmal (man) is called the middle stigmal between the stigmals (pq), (p'q'), and is said to bisect the chordal (pq, p'q'), bounded by the stigmals (_pq), (p'q'), or to be its bisec'tional. (vi.) It is evident that if we take aniy set of points in a plane, and, considering them as stigmata, refer two of them to any other two points as indices, we can by (v.) construct indices to all the other points so that they should lie on a primal. All points in a plane may therefore be considered as stigmata of a primal, of which two indices are determined arbitrarily, and may be chosen so as to satisfy certain conditions. In particular, the points thus regarded as stigmata may be themselves indices and stigmata of any stigmatic. In this way is formed the homma-primal, from the stigmatic called a hommal, in fig. 33; see art. 46. iv. Generally the new primal thus formed may be called a stigmatoprimal. The stigmals on these primals, which have former indices as their stigmata, may be distinguished as indi-stigmals (indi-cis + stigmal), and the others as stigmo-stigmals (stigms-at-o-s +stigmal). These terms save long periphrases in cases of frequent occurrence. 39. Intersections of Primals.-(i.) Let y +(a-i)x = b = ac, y + (a-i)=: b'=- a' 46 V. STIGMATIC GEOMETRY, OR THE [ART. 39. i.-iV. be two primals (for which a Cartesian case has been taken in fig. 26), it is easy to determine their stinnal (hk) from (a-a') Ih = b —b', or from 7-_ = -, that is, C0O' A A'OA. When merely two stigmals are h-c a given in each, it is generally most convenient to find A and C as in art. 38. iv., and apply this form. (ii.) If two pairs of co-stigmals are given, forming the primals (xp, xp'), (xq, x'q'), and (hk) be their stinnal, then P. = p - which shews q-q q-C ' that the stigmin K is the double point of the pri (pq, p'q'), from which property it may be immediately constructed as before, and then the indin 11 can be found from either primal. (iii.) A parordinal x=am has a constant index 11, and hence (mnn) its stinnal with pri (cc, xjp) is the stigmal of that primal for the index 1M, and is immediately found. A parabscissal y = x — 1 has a constant ordinate = OL, so that the index R of its stinnal (rs) with pri (A, ob) is found from ar = b - 1 = ', whence IO1 A AOL', or, from l=a (c -r), whence, on putting 1=c — ", wehave L"CRAAOI; and then the stigma S is constructed from R as an index in the primal. A parunal y=m has a constant stigma, which will therefore be that of the stinnal (m',m), the index of which I1M in the primal is immediately constructed from ClAiM A IOA. Aparadual y = 2x- t, of which T is the original and '2 the double point, where t+t'= o, intersects pri (A, ob) in (uv) where (a+-i) = b-t, or, (putting c = a+i, e= b-t,) where dul= e, that is, IOU A DOE, and then T'V= 2T'U. Observe that GUV A 1OA. The geometrical operation of finding the stinnal of two primals, especially in the four last named cases, must become extremely familiar to those who wish to construct figures in illustration of general stigmatics. The process is entirely disguised in ordinary Cartesian geometry. (iv.) If in (ii.) the directionpoints A, A' have been determined, we have --, which is the an'nzal of AIA' art. 34. v., and may be q-q a-, spoken of as the annal between the two primals, but continue to be written an AA', where A, A' are their direction points. Similarly tal AA' may be spoken of as the tannal of the annal between the two primals. a-a' = -(a-i) — -i (a — W) n Here w = tal AA' = i — = - i+ (a-i) +i(a' — When the pimals are given by two stigmals each, as pri (up, up') and pri (zcx, x'q'), then, since (p —') + (a- i)(x —')=o, and (cq —') + (a'-i)(x-x')=o, the second expression allows tanAA' to be expressed immediately in terms of the respective abscissae and ordinates and is often useful; see art. 48. x. It is seldom necessary actually to construct ws = tan AA'. In the Cartesian case of fig. 26, L TVIO = Z AIA', and W lies on OJ; the same construction holds for all primals representing Cartesian straight lines. But generally put a-a'- a,, aa'= a2, i-a,=a3, and w = a,. Ra3. The points A,,, A are omitted in the figure. By these expressions all cases where the sines and cosines and tangents of imaginary angles between real and imaginary lines, or two imaginary lines, occur, they may be treated with the greatest ease. ART. 39. V.-X.] CORRESPONDENCE OF POINTS. 47 (v.) Also, - - -i b —k (v.) Also, = a'-i = b'' or PJQ A AIl' A BKB', which q-k a -i b-k7c is a very useful property. (vi.) Para-primals, or parallel primals, have c = ca, or anAA'= i, tal AA'= o. Orthal primals (art. 34. v.) have aac = i, an AA' = i'. a = i'. Ra, tal AA' = none, or AO1 A IOA'. These generalise the conditions of parallelism and perpendicularity. Any parabscissal with direction point 0 is also said to be orthal to a parordinal which has no direction point, for the reason in (ix.) The direction point is the stigmin of the ordinal with a paraprimal through (ii). (vii.) The condition that three primals, having the direction points A, A', A" and original points B, B', B", should be co-stinnalC, or have a b - b' a - a' common stinnal, is b- =, or BB'B" A A'A'. b b a-aC (viii.) If in (vii.) we consider A as an index and B a stigma, and A', A" and B', B" as fixed points in the last equation, a primal results such that any other y + (c — ) x = b having any such pair of points A, B as direction point and original point, will have the same stinnal. Hence this is the equation to a pencil of raycls (1ray + -l) or system of primals with a common stinnal, or to their common stinnal itself. The primal of their direction points is then called a ray-prismal, with ray-indices and cray-stigycata. The direction points of any system of lines are the stigmins of pencils of rayals drawn through (ii) parallel to the primals in the system, to cut the ordinal; compare (vi.). For many purposes this is an important view of them to take. (ix.) If from the common stinlmal (hic) a pair of rayals be drawn having the direction points X', Y', and we substitute x', y' for X, y in the fundamental function f (x, y) = o, we determine relations, termed direction- or r'cy-stiyicltiCS, between pairs of rayals by means of those between pairs of direction points which act as index and stigma. Stigmals, of which index and stigma are direction points, may be called ^ay-stignmals, with r y-indices and rcay-stigmata, and the corresponding rayals may be termed indi-rayals and stizgmo-raycls, and the pair composed of an indi-rayal and stigmo-rayal referred to each other may be termed simply a crayacr. If we apply this transformation to the fundamental equation of art. 37. i., we shall have the results of Chasles's second lemma of Characteristics (Coli2tes Rendvus, 27 June, 1864, vol. 58, p. 1175), so that the whole of that theory becomes perfectly generalised in stigmatic geometry, and its imaginaries become geometrically intelligible. Observe that when the ray-index X' is solitary, that is, has no ray-stigma Y', the stigmo-rayal, having no direction point, is a parordinal through (h07), and hence still exists, so that a rayar pair is always complete. Similarly for the case of a solitary ray-stigma Y', in which case the indi-rayal, having no direction point, is also a parordinal through (idk). The double rayals are coincident, corresponding to coincident ray-index and ray-stigma. (x.) Thus, if we take aca= i as a direction-stigmatic, the corresponding rayals will be all orthal as long as either A or A' does not fall on 0, in which case the other does not exist, (vi.). If ca=c=i, or =', (in which case the primals are parassals art. 38. iii.), and we continue 48 V. STIGMATIC GEOMETRY, OR THE [ART. 39. X.-40. iv. to use the term orthal to express the relation of the rayals, we shall find that any parassal is orthal to itself (explaining the anomaly that either imaginary asymptote to a circle is perpendicular to itself). If a=0, or one rayal is parabscissal, A becomes solitary, and the corresponding rayal is parordinal; that is, retaining the term orthal, parabscissals and parordinals are mutually orthal (vi.), as in the usual Cartesian case of rectangular coordinates. 40. Dis'tals, or Pliicker's Coordinates generalised.-(i.) Let (xy) be any stigmal and (xp') its co-stigmal on the primal p'+ (a-i) z = b, (fig. 27 gives a Cartesian case,) then y —p = y+ (a-i) x-b, and y —p' is called the ordinar distal (dist-ance+ca), or simply the distal of the stigmal (xy) from pri (A, ob). It is evident that y -p' = o may be used as the equation to that primal. (ii.) Draw pri (T, xy) cutting pri (A, xp') in (xz p,); then, as (xipo) is the stinnal of these two primals, we have (by art. 39. iv.) 1l —Y _ t-i P -P' a - t-i t —i whence Y —1i =. (y -2) = t [y +(a-i) x-b]; t -a t - a and y-pi is called the general or T-distal of Yfrom the primal (A, ob), because T is the direction point of the primal which determines it. The usual or ordinar distal y-p' is determined by the intersection of the parordinal through (xy) with pri (A, ob). (iii.) It is evident that either y-p'= o or y-p1 = o may be taken as equations to the primal, and that the relations of the clinants y-p- ' or y-p- determine relations between PY or P1Y which are real distances measured directionally towards the arbitrary stigma Y from its co-stigma P' on the primal, or from the stigma PI of the stinnal of a known pri (T, xy) with the original pri (A, ob), and these relations of distances, directionally measured, determine and generalise a multitude of relations, hitherto most imperfectly noted even by Pliicker, who first drew attention to their value. The equations thus deduced are called distal equations. (iv.) Taking another primal (A', ob') intersecting the former, and determining the distals y-q' or y-ql as before, we may determine x and y from the corresponding values, I-i t —i / —P1= t (Y —p) = [ + (c[ — i) - b] = p, I - a t -a t' — i ( _,) -- y-q' = -,a (y-) = -_. [y + (a'-i)x —b'] = q. Finding from these equations the values of x, y in terms of p, q, and substituting them for x and y in f (x, y) = o, obtain first the distal equation 0 (y —p, y —1q) = o to the original stigmatic, and next 0 (p, q) = o as the equation to a subsidiary (or bi-primal) stigmatic, in which the relations of the original points X, Y, are determined by means of the subsidiary points P, Q, where OP, OQ represent the directional distances P Y, Q1 Y of the correspond ART. 40. iv.-41. iv.] CORRESPONDENCE OF POINTS. 49 ing stigmata PI, Q1 in two fixed primals (A, ob), (A', ob') from a movable stigma Y. The indices X1, X2 to the stigmrnata P1, Qi are found from the two known primals, and the index X to the stigma Y is known, because (xy) is the stinnal of the primals (x1p1, T), (xq1, T'). This may be called the bi-primal stigmatic, and is the basis of Plicker's Punct- Coordinaten. (v.) The equation to a ray-primal (art. 3.9. viii ) allows of establishing precisely similar transformations answering to Pliicker's Coordinaten gerader Linien, giving hi-stigimal stigmatics, in which the index and stigma relate to subsidiary points derived from the distals of two fixed stigmals from a movable primal, instead of the distals of a movable stigmal from two fixed primals. 41. Trilat'erals, or Triangular Relations generalised.-(Fig. 28 represents a Cartesian case.)-(i.) Let the three stigmals (i'v), (v'v), (w'w) be connected two and two by the primals (v'v, w'w), (z', w'wu), ('au, v'v), having the direction points T, '", T" respectively. These three primals form a trilateral of which the three stigmals in the above order are the apicals (apical stigmals) opposite to the laterals (lateral or side primals) in the above order. This is written tri (zuu, vv', ww'). (ii.) Let (~'Z) be a stigmal on the lateral opposite (~'q), then (art.39.iv.) -v_, t"- i,t -w t- i a - v (t-t (t- i) =, and — =, whence - u-Z t-t u- z t-t )- (t- i) u- n 7- V V-7i W_ - U, and generally --- ----- (t" — t) ( — V ) '(t-t(t -) (t )(ti) ( t) (t i) the symmetry of which is evident. These equations give all the relations of all " triangles real or imaginary." (iii.) The following particular cases for which the above assume inadmissible forms, with o in the denominator, are easily investigated independently. The three stigmals lie on one primal (W6q), (vvl1), (v'v), so that t =t' = t; the relation art. 38. v. must be used. The tri (z'v2, u', Iw') has the parordinal lateral (u',v, u'c) which has no direction point; but then (/t't2), (I'u) are co-stigmals and (w'w) the stinnal of primals (z'v2, w'w), (u'u, wiw), having the direction points T1, T' respectively, so that, by art. 39. iv., '6 tw = t- i If further, v- Wz t - as in fig. 28, pri (u'v2, w'w) is parabscissal, t =o, and --- i-t, and 2 — '. V-, IJ V — 2 Z (iv.) When the two last conditions are satisfied, we have an orthal trilateral. We may call its parabscissal lateral the ba'sal, and its parordinal lateral the perpendic'ulal, and the third lateral the hypothenz'sal. As we have shewn that tal T'O = t'= V2~-, we might invent a v.2-a si'nal (sin-e + al), cosi'nal (cosin-e+al) and cotan'nal (cotal-gent+al) of T'O, written sal T'O, cosal T'O, cotal T'O, defined thus, sal T'O = 2- c tTO V,_ i -W i V2 -- col = sa '_ - - cotal T'O =, - 'w-u t - q - t- i' 2- t 50 v. STIGMATIC GEOMETRY, OR THE [ART. 41. iv.-42. iv. from which, in the Cartesian case, by taking tensors, the usual formulae of trigonometry, as derived from the triangle only, in this case the triangle IOT', readily follow. For if t'= j, where p is scalar, T. sal T'IO=T. w _, and T. cosal T'IO= T. = PSjt /(p2 +i) t —i ( -2 + The former expressions, however, give what corresponds 1/(-P, +) to the sines, cosines, tangents, and cotangents of imaginary angles. Thus the direction triangle IOT' gives rise to a direction trilateral tri (oo, ii, of') which is clearly orthal. The imaginary trigonometrical functions in Cartesian and Pliickerian and hence also in Chaslesian geometry arose from applying the terminology of the simple triangle to this trilaterca, and the difficulties which hence arose are to be attributed to the omission to notice the directions of the sides of the triangle, that is, the direction points of the laterals of this trilateral. (v.) The condition that the primals given by the distal equations y-p= y —q y —= y o, (art. 40. iii.) and having the direction points t, t, t"' respectively, should be the laterals of this trilateral, and hence have no common stinnal, is (y —pI) * (t" —t') (Y - ') * (t- t") + (V — ). (-t-t) = e, pher a _ (t - t) (t- t") where e7 (" * (t" -t( I) t-)(vt-W).2) /_W.(vi-t ( V — t) (,- () -U )( - () = i- -- -V). i-t' ' i —I" (vi.) A multitude of propositions on the properties of the trilateral, deducible from these fundamental properties, are necessarily omitted. 42. Pencil of Four Rba/cls, or the Anhar'monic Properties of Rays generalised.-(i.) Let there be five rayals, having the common stinnal (he) and the direction points T, T1, T7, T73, T respectively, (a Cartesian case is shewn in fig. 31). Let a transversal primal be drawn parallel to the first primal, and intersecting the four last in the stinnals (1,y1), (x2a2), (x333) and (xY/4) respectively. (ii.) Then from tri (he, x/YI, X2y2) and tri (te, X^2y2, xay3) we find?/ -?!2 t1 t2 t3t- /-:Y 1 = t,-t. t1- 2 ~= ( t,,12t), art. 34. iv. 3-iY2 t1-t t3-ti That is, the anral of the direction points is expressed by the simple quotient of the differences of the clinants of the stigmins. (iii.) Similarly?1 t. ~ t = (t4tt3t), y3-/Y4 tl — t t3 - t and dividing the first of these results by the second, (/1!2y3y4) =) (t1i2 3t4), that is, whntever be the direction point of the transversal, the anral of the four stigmins, when they exist, is constant and equal to the anral of the direction points. And if there be only three stinnals, from the coincidence of T' with T4, we see by (ii ) that the aural, reducing to (y1?/2.3.), remains = (tjftt3t4). This constant aural of the direction points is called the anral of their four rayals. (iv.) This is a perfect generalisation of the fundamental property ART. 42. iV.-44. ii.] CORRESPONDENCE OF POINTS. 51 whence Chasles deduces the whole of his theory of anharmonic ratios, homography and involution (Ge'om. Sup., art. 13.; see also below, art. 45. vii.). But this generalisation has the advantage of including every case of " imaginary" rays, angles, and points of intersection. The deductions in this general case may be made in a manner precisely similar to his, using the same arguments, mnutatis mnutandcis. But the stigmatic calculus much facilitates the operation, as I have found by actually working out every proposition in the clinant form. (v.) The whole of homography &c. has also been worked out with distals, on the method of Plficker, taking (hk, rxp), (/, xq) to be two fixed rayals, and (/hk, xy), (7th, xy') two variable rayals determined by the equations (y-p)-e. (y-q)- o, g. (y'-p)-e. (y'-q) = o, where g is constant and e variable, which give y-P 8 -Y -1=, or (ypy'q) = Y — Y' —i which now becomes perfectly simple, because unperplexed by the "imaginaries" which are so plentifully strewn among Piiicker's demonstrations. 43. U'niqzci'drals, or the Relations of Involutio and Homograp7y generalised.-(i.) The general equation to quadrals is ax2 2xy 2 + Yy 2 + 2x+ 2e+y + = o, of which it is first convenient to consider the forms not involving x2 and y2, because they never give more than one value of y for each value of x. and conversely, whence the name zn'iquca'drals. These are (ii.) 2/ixy + 2 x + 2~y + 2 =o, in which x and y are symmetrically involved, giving an in'val (inv-olution + al), and (iii.) 2oy + 2Jx + 2ey +,2 = o, in which x and y are unsymmetrically involved, giving a hoin'mal (hornography + al). 44. ln'vals, or Chasle'sian Involution of Points generalised.-(i). From the general equation, art. 43. ii, determine the solitary index and solitary stigma, as in art. 37. ii. By dividing, out first by y and then by x, and putting = o the sum of the terms not containing y and x respectively in the denominator, we obtain 2/3-)x + 2 = o, /3y 2 = o, so that there is merely one solitarS point 8, where 2/3s + 2 = o. If e and fbe the roots of the equation 23fz2 + 47z + p = o, then s == (e+f), and E, F are the double points of the inval. These results give (,_-2)(s-y) = (s-e)2- (s-/)2, to which is adapted fig. 29, where AA', BB', CC', DD', GC', Hi', &c., are various ordinates. (ii.) To construct the stigmals, draw the characteristic circle, with centre S and radius SE or SF'. A being any index, to find the stigma A', draw ASE A ESA', by making Z ElSA' = Z ASE, and (B, B' being the intersections of SA, SA' with the char. cir.) BA' parallel to AB. The lengths of the corresponding SX, SY are thus always found, and it is then easy to separate >SX, SY by any angles from S.E. (iii.) From (i.) we find, on putting (aa'), (bb'), &c., for (xy), s- s-b a-b}' a- b' — a s —b' a — a — b if BA'= AN, so that if two stigmals s-b s-a' b- a' -n 52 V. STIGMATIC GEOMETRY, OR THE [ART. 44. iii. —vi. (ad), (bb') are known, the solitary point S is found by making ASB A B'AN, and then the double points E, F are found from SA, SA', as in art. 33. iii. Two stigmals being then sufficient to determine an inval, we may write it as inv (aa', bb'), which for the solitary point may be inv (aa', S). The true nature of the equations ab = i = i2 and (y-i)2 = (a-i) (b-i), art. 34. v., p. 37, is now evident. (iv.) From equations similar to those in (iii.) it is easy to shew that all the properties of Chasles's Involution hold strictly, of which the following need only be cited. First, 's-a s-y s-a s-b' s-c s-y s-c s-b' from - =, = - = s-x s-a s-b s-a s-x s-c s-b s- c wefn - s-.- y s-a s-b' s-c s-y s —c s-b' we ind _ --, - / --- a-x y-a' a-b b'-a" c-x y-c" c-b cwhence, eliminating s-a, s-c, s-y, s-b', we find (abcx) = (a'b'c'y), or any four indices have the same anral as their stigmata; and this would of course remain true if the former were drawn on a separate plane or different portion of the same plane from the latter. But this result is not characteristic of invals. Second, (abxy) = a'b'yx), or in any stigmal the index and stigma may be reversed. This result is characteristic, for on multiplying out we obtain the characteristic equation of invals, for which the planes cannot be separated. Third, (abs..) =(ab'.. s), as in (iii.) See art. 34. iv. Fourth, (efxy) = (efyx), whence (eyfx) = i', or any index and stigma form a harmal with the double points, and hence these four points will lie either on the same straight line or the same circle, as shewn in the figure. Hence also the construction: draw any circle of which EF is a chord, take any points A, A' upon it, so that Z ESA = z A'SE, then (aa') is a stigmal in the inval. In this case A and A' lie harmonically with respect to E, art. 34. iv. In the figure G is the centre of the circle containing A'BAF, which however is not drawn; but see fig. 14. If inv (ee, ff) and inv ('e',f'f'), have the common stigmal (ry), then (yexf) = (ye'f'), and hence (yy, xx) are the double points of inv (ef, ef'), whence (xy) may be constructed. This fails when the invals have a common solitary point, and in that case only they can have no stinnal. (v.) The equations of angles resulting from the above anrals also shew how the stig'mod varies for different straight lines or circles assumed as indits ~ thus the indit circle ABC has the sigmod circle A'B'C', but the indit circle SHDL, passing through X, has the stigmod straight line rI'D'L', S having no stigma. MTbius, in the papers cited in Appendix II., seems to have first treated the involution of points in a plane, but it will be found that his treatment is much more complicated, and that the present theory brings out all his results and many others with the greatest simplicity. (vi.) It may be observed that, in the old theories of involution of points on a straight line, when X Y lay as at D, D' on the same line as A, F, these last double points were called real, but when X, Y lay on a perpendicular to EF through 8, as at G, G', these double points, though remaining unchanged, were called "imaginary." By forming ART. 44. vi.-45. iii.] CORRESPONDENCE OF POINTS. 53 two inva-primals (art. 38. vi.), so taken that the carstigmod gives the line ESF in the first case, and the perpendicular to ESF in the second, it will be seen that E, F are carstigmata in the first, and incarstigmata in the second case. 'This is the meaning of the above confusing distinction, which could not be previously avoided. Again, until a Cartesian inva-primal had been formed, since the ordinates XY Ily on the same straigiht line, and not perpendicular to it, as in Cartesian geometry, the two cases were kept entirely separate. In uniquadrals XY was termed a segment, and in Cartesian geometry an ordinate. Until the stigmatic conception had been formed, it was impossible to perceive the real identity of the segments and the ordinates, as simply the straight lines connecting the indices with the stigmata, that is, shewing the pairs of corresponding points. The immense facilitation produced in the application of the homographic theories by the fusion of the Cartesian and Chaslesian geometries, will be strongly felt by every one who works out the cases in detail. 45. Homn'nals, or COhaslesicta Ilomogra.plhy of Points generalisec.(i.) To determine the solitary index S and solitary stigma Z' in the hommal, fig. 30, we find from art. 43. iii., first 2Js+2e = o, and then 2f3z'+ 2b = o, and for the double points E, F we have 2/3e" + (2b+2e) e + = o. These values easily reduce the general form of equation to (s -) (z'- ) = (s -C) (z - ). (ii.) From this, by a process like that in art. 44. iv., we find (abcz) = (a'b'c'y), which relation remains when the plane containing the indices is separated from that containing the stigmata. This enables us to determine the solitary index and stigma when three stigmals (ar'), (bb'), (cci) are known, because (abcs) = (aib'c'..), and (U bc..) (ab'/cY), that is to say, a- b c-s - a'-b' a-b a'- b' c'-z' c-b a-s c'-b' c- a c'-b a ' —z To construct the solitary points from these equations, 0'/- 1b' 9qv - b first construct TV from -- = -, or A'-B'C A T'BG; c'- b' c- ' and then S from C -- =- or CSA A TBA; a - s a - and Z' from c- = _,,, or CSA A A'Z'C'. a - s c - z (iii.) When S and Z' have been found from three stigmals, all other stigmals can be found from a subsidiary inval, thus: Suppose that the part of the plane containing the stigmata is slid over that containing the indices, by sliding Z'S over Z'S till Z' falls on S, and A' on A1, B' on B1, &c. Then z'- s = a'- co = - b'-b =...... = y-y,, and hence s-a1 = z-a',...... s-y = Z-y; and hence (s-x) (Z'-y) = (S —x)(s-y1) = (s-a) (s-a1) = (s-_m)2, when 31- is properly determined. Hence the subsidiary inval (s —) (s-yl) = (s-?-n)2 determines Y1 from XY, and then Y]Y= SZ' gives Yfirom Y1. Hence also a hormmal is merely an inval with its E 54 V. STIGMATIC GEOMETRY, OR THTAE [ART. 45.C5 iii,-vii. stigmod (or its indit) tl anslated in the same plane without rotation, that is, a transordinated inval. (iv.) There are now two easy constructiors to find the double points E and 'F. First select O so as to bisect SZ', whence s+ '= o, and find O' the stigma of 0 considered as an index, whence (s-c) ('- e) = (s —o) ('-o'), or e2 = i' so'= 'O', as shewn in the figure. Again, (SQ-1)2 = (s) = ( -e) ('- ) = (e-,s) (e +s) = e2 —s', or e2 = S+ (s - 1)2, which is constructed as in art. 33. v.; by drawing USV perpendicular to -lSM, and making US= SV, both of the length of SMll, so that S —u = j (s-m), S-V = - -s, which gives e2 = s2 — (s —q)2 = ztv. This shews that (uv), (z'o') lie on inv (ce, ff). (v.) It is convenient to call O (or common middle point of EF and SZ') the centre, EF the double ax,'s, SZ' the solitary axis, and JLN (where mn +? = 2s) the subsidiary axis of the hommal. For the hommal determined by three stigmals we may write hom (aa', bb', cc'), which for the solitary index and stigma may be written horn (aac', S..,..Z'). (vi.) The relative forms of the indit and stigmod are the same as for the inval (art. 44. v.), but the angular properties of the double points are peculiar to the homlmal. See fig. 30. First (eatbc) = (ec,'b'c'), hence if A, B2, C are collinear with each other and hence with S, in which case also A'JB'C' are collinear with each other and hence with Z'; then tan ABEG = tan A'EC', and tan AEA' = tan CEC'. Hence if two straight lines intersect at B, and are indefinitely produced each way, and then being clamped, are made to revolve, and to cut two given straight lines.PQS and P'Q'Z', they will intersect, the first in the indices and the other in the stigmata of a hommal, of which the solitary index S is in PQ, and solitary stigma Z' in P'Q', and E is one of the double points. In fig. 30, the lines PfQS, P'Q'Z' are so chosen as to make (p'), (qq') parts of the same honmmal as before. In any such case Z', S are easily found, by making one armL of the biradial parallel to PQ and P'Q' respectively, in which case the second arm cuts P'Q' and iP(2 in Z' and S respectively. F is then the fourth point of the parallelogram SEZ'F'. Also tan PFP' = tan QFQ', but they are not generally = tan PEP'. The same will be true if PQ, P'Q' coalesce in SZ', and then E, F' are the " imaginary " double points of the " real homography " on the line SZ'. This is a new demonstration of Chasles, Geol1. Stlp. art. 171, which it completes, shewing the nature of the points. But this property will be greatly generalised in art. 46. iii. By taking -E as the centre of a circle, there will now Ie no difficulty in explaining and completing the result in Ge'om. SLp. art. 664. (vii.) Observe that in applying the general property art. 42. iv. as Chasles has done to the construction of a homographic theory, we have from any stigmal (he), see fig. 31, a movable rayal cutting two primals which have the stinnal (kf). In this case the stigmins of the movable rayal on the first of the primals issuing from (kJ), taken as indices have their stigmata formed by the stigmins of the same rayals ART. 45. vii.-46. iv.] CORRESPONDENCE OF POINTS. 55 with the second primal, and the stigmals thus formed make a hommal, of which the stigmata E, F of the two stinnals (hle), (7f) are the double points. When the primals represent Cartesian straight lines (as in fig. 31), confining ourselves to the stigmods, we may say, if rays from E cut two rays issuing from F, the points of intersection form a hommnal, of which E and F are the double points, and of which the solitary index land stigma are found by drawing rays from E parallel first to one and then to the other of the rays issuing from F. This view will be found to shed a new light upon many of Chasles's investigations (especially Ge0om. Sup. chap. vi., &c.), but was of course impossible so long as the points in an homography were considered to lie necessarily on the same straight line. (viii.) Secondly, (/ftb)= (eJft''); thirdly, (efs) = (ef.. a'); fourthly, (esfji) =(e.. /t'); fifthly, (ea-s) = (ea'b'..); sixthly, (csa..) = (e.. a'); from all of which angular properties may be readily deduced. 46. RCay-hommals acnd RCayic-ivalCs, or the Chiaslesian Homograph7o Relations of Rays, gene'alised.-(i.) If the indices of a hommal are made direction points of the rayals emanating fiom a fixed stinnal (hk), and the stigmata of the same hommral are taken as the direction points of the rayals fiom anotherstinnal (mb), thus generatinga direction-homnmal, (art. 39. ix.), the rayals in these two pencils form a dozuble racyhoimial. If the two stinnals (hk), (m)?) are coincident, the result is a single ray-hoia)2mlcal. These rayals cut any primal in stigmals forming a homnma-primal. The stigmo-(or indi-)rayals corresponding to those direction stigmata (or indices), which have solitary indices (or stigmata) respectively, will be parordinal. (ii) If (a, a), (bb,), (cc2), (x1lJ2) be the stigmals on the direction hommal, and '1, Z2 the solitary points, then (si- Ia) (Ga - Y2) = (s - al) (2 - a), and (ab1cl,) = (a2b92C2J2), whence all properties may be deduced, (compare art. 39, ix. x.,) and the angular properties of the double points of honmmals duly generalised. (iii.) The following is the only case that can be noticed in this Tract. If from any stinnal there issue two rayals having their variable direction points X, Y2 so related that tal XI, Y2 is constant, so that, for example, _ -?/" = Ro, or xy, Y + m (l — y,)- i - o, these pair of primals will be the analogues of the various positions assumed by the revolving lines in art, 45. vi. Now in this case the direction points of the double rayals determinec by putting cl = y-, = e, =-f, give c =2 i =f i, so that they are, I', and the rayals are parassals (art. 38. iii.), that is, parallel to the asymptals of a cyclal, or, as used to be said, " they pass through the circular points at infinity" (!); and this will also be true when some pairs of rayals are Cartesian; and will also be true although these parassals among other rayals will of course be incarprinials. (iv.) Conversely, form a homma-primal from the indices and stigmata of a hommnal (ee, ff S..,..Z'), by assigning 2, Z" as the indices of X, Z', where (as), ('a') are carstigmals in fig. 33. Let E',,h be the indices of E, F, in which case (ee), (of) are necessarily incarstigmals in the figure. Then it is always possible to give new indices P, Q to 2 66 v. STIGMATIC GEOMETRY, OR THE [ART. 46. v.-47. ii. E and F, so that rayals from (pe), (qf) to (ee), (of) will be parassal, and in that case the tannal between any indi-rayal and stigmo-rayal will be constant. This condition gives e-e e-f. f-e - = — -=o, i —i' =2i, - = - I =i - f- =-2i, p —e. —P q-J' q-e and hence -EF = 2P@, TFE = 2 QE', and as D, E' are known, P and Q are determined. Let T be the direction point, and C the double point of the pri (as, ('z'), and let 2n- e f, 2v = e- p, and e —n = n-f = k ( —c). Then,n-c = (1-c) (i-t, f-c = (. (i- ), e-t = e. (i-t) whence e-p = n — = (k —t)(v-c), c-p- = (kt-i) (v - c), f-q = - e = ( k+t)(c- ), c-q = (7 +i) (c-,'). In the Cartesian case t, k7 are vectors. Hence C, N', P, Q are collinear, and EP, FQ perpendicular to CN', that is, (pe), (qf) are carstigmals. The extremely perplexing investigation of this whole question in Chasles, Geon. Sul. arts. 171, 172, 181 (especially see table of errata for p. 126 in this art.), 651, and Sect. Con. art. 293, will serve to shew the great simplification introduced by stigmatic geometry. But in the present Tract a mere indication must suffice. The whole subject has been carefully examined in detail. (v.) RCa?/-nvals result from similar considerations. Thus, i2= xSya is a rav-inval, of which all the rays are orthal (art. 39. vi.), the double rayals being parassals, and the rayals corresponding to the solitary index and solitary stigma, or for xI = o, y2 = none, y =- o, x, = none, being paraxals (art. 89. x.). As two invals have always a common stinnal (art. 44. iv.), any direction-inval, t2= =,Y2, will intersect i2=,xy2, and hence the corresponding ray-inval will always contain two orthal ravals. (vi.) A sheaf of parallel primals may be used in place of a pencil of rayals, provided their different original points be substituted for their common direction point. 47. Transordination, or tAhe Cartesian Transformation of Coordinates and of Curves, generaliseld.-(i.) The general nature and object of this operation is explained in art. 36. ii. The change is not perfect unless every single indi-stigmal (that is, every single stigmal in the first stigmatic) corresponds to one and only one stigmo-stigmal (that is, to one and only one stigmal in the second stigmatic). (ii) This cannot be effected except by assuming relations of the first order, such as x = b+ (ad-a), or x = \x'+uy+r,, which, changing the index without changing the stigma, produce indicial transordination, and are the foundation of the ordinary Cartesian change of coordination. The values of the constants are assumed so as to facilitate subsequent calculation. Similar changes have already been made. Thus the lhommal (s-x)(z' — y) = (s-m)2, on putting z' —y = s —y', becomes transordinated into the inval (s-x)(s-y') = (s —z)2. Again, from this last equation, on taking s-x = s-x'+ (y'-x'), we find (s-.x')-(yi' —x)2 = (s-_M)2, where 2x'= — +y' and is hence readily found. This however is a cyclal (art. 48. v.). ART. 47.. iii.-vVi.] CORRESPONDENCE OF POINTS. 57 (iii.) More generally, assume such a relation as ax - /3/ y = a' ' + 3'y' + ',y Xa + y' + y + =X'' +,'y' +', which on elimination give results of the form 7r.(y- x') = y+(t - i) - 6; K. (y-Y-) = y+ (t2-i)x -bb, and, on putting y-x'=p, y-y'= q, these lead at once to the distal transformation and biprimal coordination. (iv.) Still more generally, putting for brevity A = ax+a'y +a", B = 3x+/J3'y+ /", C = yx +y'y + ", and D=o for the result of eliminating x, y from the equations A=o, B=o, C=o, (that is, for the condition that the three corresponding primals have a common stinnal,) we may assume Cx'= A, Cy'= B. On determining the values of x, y in terms of ', y', they will be found to have a common denominator which will be a factor of the numerator when D=o, that is, when these primals have a common stinnal. Rejecting this case, the three primals form a trilateral such as (u'I, v'v, w'w) with the conditions (art. 41, v.). Then, taking P', Q', ' to be co-stigmata for index Xin these straight lines, and putting A =y-p'= p, B =y —q'=q, C =y —r'= r, we obtain a homogeneous distal equation between p, q2, r, or 7rp, q, pr, which is the foundation of tri-primal coordination. (v.) The primal (oo, xy), or y+(t —i) = o cuts the stigmatic f (X, y) = o in (xy). Eliminating z, we obtain 0 (y, t) = o, which is the foundation of polar coordination. (vi.) Taking a less perfect form of transordination, that is, one in which the condition (i.) is not perfectly satisfied, we may connect X with X', and Y with Y' by hommals, as X'++ X + x' + v = o, y +\'y+ X 'y'+ - O. In this case we shall occasionally have complete stigmals in. one answering to defective stigmals (that is, solitary indices, or solitary stigmata) in the other. It was probably the desire to avoid these relations of continuities to discontinuities, that the extraordinary assumptions mentioned in art. 6. i., and Appendix I., were introduced, by which the real nature of the solitary points was illogically distorted. Thus it was not seen, or, if seen, repudiated, that it was possible to have analogies which held for all but a definite number of cases. The attempt to conceal this important logical fact by a mere juggle of language, shews the danger of studying logic from simple arithmetic and geometry, of wvhich numerous instances could be cited besides those in Appendix I. The attempted passage from discontinuous arithmetic to continuous geometry (exceptingo only by Euclid's really " royal road"), like the attempted passage from discontinuous Cartesianism to some imagined continuity, has led to so much "stretching" of language, that the logical feeling of mathematicians, though dealing with "exact science," is in great danger of being entirely perverted. Thus Dean Peacock put forth his "permanence of equivalent forms," a logical fallacy long since exploded, but defended by him with great warmth and pertinacity. And "perspective projections," admirable as a piece of geometry, have landed us in the contradictions detailed in art. 6. i. and Appendix I. I have even heard these results defended by an excellent mathematician as " illogical, but convenient," as if want of logic, i. e. incorrect reasoning, were not the height of mathematical inconvenience. 58 V. STrGMATIC GEOMETRY, OR THE [ART. 47. vii.- 48. iii. (vii.) These hommal relations may be obtained from equations like x +by+c _ a +x + '+y a'x + b' + c' _ +' '+ i(y'f+' x -- b"y + c" a'x +I +"y'y + y" a + " + c" u/x + 1"y whence, on elimination, x, V, ', y' are obtained in similar forms, but then, on multiplying up, we find (xx'), (xy'), (yx'), (yy') given as stigmals on different hommals. In this case, by equating to o the denominators in the values of x, y, x', ' thus found, we obtain equations to primals in which (X/) and (xy') are stigmals, such that not one of the stigmals in either primal for the one stigmatic will have a corresponding stigmal in the other. Hence, relatively to each other, these stigmatics will have solitary indi-prirmals and solitary stiogmo-primals. In this way honmma-stigmatics are formed, which include the Cartesian case of homographic figures. And by proper changes of the constants these homma-stigmatics are brought into another relatioln whic may be called hom'olo-stigmatics, and include the Cartesian case of homologic figures. In consequence of the old " imnaginary" points, none of these relations are completely exhibited except in stigmatic geometry. 48. Du'ocqua'drals or Co'nals, or Conic Sections, generalised.-(i.) Duoquadrals are derived from such forms of the general quadral equation (art. 43. i.) as always give two stigmata Y, Y' for each index X. When they have any Carlesian portion, these stigmatics give as the carstigmods (paths described by the stigmata of the Cartesian portion), the well known conic sections, and are hence also called co'nals (con-ics+al), a name which may then be applied generally to all duoquadrals. (ii.) The extreme variety and the length of conal investigations preclude me from giving them in this Tract any even approximatively systematic form. I have myself carefully applied the present conception of stigmatic geometry, and the clinant calculus, to the treatment ofconals, by generalising the usual Cartesian methods, and also those in Plficker's System and.Entwiclcelungei, as well as those in Chasles's Sections Coqniques, in great detail, and have always found satisfactory results, easier calculation, and complete geometrical realisation. The previous explanations of primals and uniquadrals render any other result impossible, and I shall therefore content myself with giving a few notes as to some methods, and a few results, together with the nomenclature which I have found it convenient to adopt, and inviting mathematicians to test the stigmatic theory by minuter applications. Several of these are contained in my second memoir on Plane Stigmatics, but with my old notation and nomenclatture. If I may judge of the effect on others by that on myself, the continual explanation of formerly insuperable difficulties, the strictly geometrical meaning of calculations which seemed hopelessly analytical, and the absence of any difficulties in the assignment of positive and negotive, will render such a process a source of intense delight to the geometer. (iii.) When in the general quadral equation (art. 43. i.), P2-ay = o, but ce- fic is ot -= o, the stigmatic is a non-central, and by indicial transordination (retaining the stigmata, but altering the origin and indices) may be reduced to the form (y-x)2+4sx = o, which is here called a pyarab'bal (pacrab-ola+cl). When s is scalar and x is also ART. 48. iii. —v. ] CORRESPONDENCE OF POINTS. scalar, sx being tensor, y - is vector; or when S and X are both on the Iside of 0 on 01, then XY, X'Y' are parallel to OJ, or there is a Cartesian portion, and the carstigmod is a parabola. Y, Y' are constructed by art. 33. iii. Here 0 is the vertex, (oo) the vertical; S the focus, (ss) the focal. When 0, S are known, we may write par (0, S), or par (xy, S.) This case will not be further considered till art. 52., after the treatment of centrals (iv.) When nleither /2-ay-, nor. (y —/;e)2- (32 —cL) (e2-y ) are = o, the conal is central, and by indicial transordination can be reduced to the form g22+ e2(1/-x))2 =( e2g2, which embraces many cases according to the positions of E and G, as follows:-Generally let e +f = g + h = s+z = o, and 2 = e2 +g2, found as in art. 33. v. This may be called the central (ee, oo, og), or (E, O, G). There are no solitary points. E, F, in fig. 32, are the double or major points; G, II the original or m[i(. r points, and S, Z the foci of the central. (v.) Cy'clal (v;,,-oc+cal), E on 01, G on OJ, Te = T, e++g2= o, equation x —(y-x2) = e2. This may be called cyc (0, E). The equation gives (y-x)2 = -e2 = (a-e) (n+e) - (x —e) ( —f), which gives the contraction of Y, Y' from X immediately, and shews that Y, Y' lie harmonically with respect to E, F. When X is on I between _E and Fy, then XY, XY' are parallel to OJ, and the carstigmod or locus of Y, Y' is a circle of which 0 is the centre and EF the diameter (fig. 34). When the indit is L[MN, or X lies on the line 1M11, as at X1, M, X, on MN, the stigmod consists of two branches proceeding fiom Yj and Y. so that the circle is but an extremely small part of the cyclal. If OE had been taken on OJ at OG, so that g =je, we should have " — (y — ')2+ 2 =- o, whence (y-z)2 = x"+ g; hence when X is on 01, Yis always on Of; when X is on OJ, and Te < T~/, XY being parallel to 01, Y will describe the same characteristic circle as before, but every stigmal (xy) is non-cartesian. This is Chasles's 'imaginary" circle, more particularly referred to in art. 41. v. (2). Also since e2= y(2x-y), the primals, that is, the assals y =o anid 2x-y- -o are the asyip'tals (asympt-otes + at) of the cyclal; see art. 38. iii. These have no carstigmod. The nature of their asymptoticity is easily seen, for as.Xretreats in any direction, the angle EF diminishes, EX, EX become more nearly of the same length, Y approximates to 0, and Y' to a point Yo, where XYo = OX, while 0, Yo are the stiogmata of X in the assals. The asymptals of all concentric cyclals are parassal, and hence paraprimal. Since in the cycal 2x =?/+y', we can eliminate x from the equation e2 = x-(y-x)2 = 2y-_2/ = (y+y'/) y-_y2 = y'. Hence the pairs of co-stigmata form an inval of which 0 is the solitary point; B, F1 are the double points. The stigmods of Y and Y' for a given indit are therefore related as the indit and stigmod of an inval. There are really always two branches, which are disguised ii the Cartesian case, because they are then two semicircles united at their extremities by the double points E and F. This gives an easy way of finding Xfromn Y, and shews that though each index has two stigmata, each stigma has but one index, which is also apparent from the original equation being only of one dimension in y. We have already found that (,-y)2 = ( - e) (n;-f), which also shews that if we form an inval of 60 V. STIGMATIC GEOMETRY, OR THE [ART. 48. v.-vViii. mwhich X is the solitary point, and (ef) a variable stigmal, each stigmal determines a new circle having the common stigmals (xy), (xy') with each of the others. Compare art. 49. v. (1). (vi.) _E' qlier'bacl (efqui-lateral or e('i-a-ngular + hy-perb-ola + al), E and G alre coincident and both lie on 01, (no figure), e9+.q2 2e2 = s2, equation r+ /-t;-)( = - e2, whence (y/-x) = e' —2 = i'. (x-e) (x-f), so that lr, Y' in the equiperbal are found by turning YXY' in the cycial thlrouoo a rioht anole. This is the foundation of Poncelet's supplemental circle. Wihen X is on I, beyond Eand F, then XY, XY' will be paraiiel to OJ, and the carstigmod, or the locus of Y, Y', is an equilateral hyperbola, where the two branches are visibly separated. Also, since e-= [ +.j(y — x)]. [a -j(y — )], the asymptals are xz+j(y/-) -o, and — J (1/ -') = o, which Icave a Cartesian part, and their carstigmods wiNill be the loci of'P and Q, the extremities of PXQ, the YXY' of the asymptals to the cyclal, turned through a right angle about X. See the more generll case of the hyperbai, in (viii.) (vii.) liyi 'sal (elicps-e+ai-), E on 01, G on 0, T7< Te. In this case (nlo figure) let /, so that q 2 = o, I'. 7ie is a tensor, and K- lies upon Of. The equation becoimes e2(y- )2-' 1s2c2 '-c2, 0, xx aence e(y-') = ) 1. ( e2-) = 7'. (x-e) ( —t; elence AY, XY' are imlmediately found, by formining XcU, the imean bisector of XE, XF, as in the cycial, and altering its lenoth so that len XU': len XY:: len OE ien O(.. When XA is on l between E and F, thenl XY, XY' are parallel to )T, and the carstigmod or the locus of IY Y is an ellipse, of which Li' is thei 'ajcmor axis, aid GI- the mizinor ax;is, and S, Z tle j Aci. Also, since cl '= [ i:-e(il —a)]. [l7^x+e(y —a)], the primals tk - e (1/ -- ) =, J:x -ce - e(y- ) = o, will be the asymptals of the ellipsal, and will have no carstigm'od. The ellipsal includes the cyclal as a particular case. If in fig. 32, OE', 0j' (not 0E, OG) are taken as the semi-nmajor and semii-minor axes; S, Zwill be foci, and (1itl) a carstignlal in the characteristic ellipse. (viii.) /ijjlerbcal (o/-i:,l '-ola+ cl), E and G both on 01, so that e'. t2y iS a, tensor; no particular relatio- is needed between len OE and len OG, s2 = e'+2. The equation remains q/-'a+ (yyx) = e g2, whlence c2 (y —a2) = y2 (e2 —2) = i'. Y'( - e) (: -f), andl hence YXY' is found by turning the corresponding line of the ellipsal, for which g_ 7;2, through a rioht angle. Hence Poncelet's supplemental ellipses and hyperbolas. Wihen X is on I, beyond El', then XY, XY' are parallel to OJ, and the carstigmood or locus of Y, Y' is an hyperbola, of which EF is the qnzijor, and GC the minoor, or " iminaiary," axis. It has been usual to represent the minor axis by a line perpendicular to _E, and call it imaginary. In fact (og), (oh,), which are the stinnals of the ordinal with the hyperbal, are incarstigmais, and both points G, I lie on the line EE. If, in figure 32, OE" is taken as the semireal axis, and S the focus of the flat hyperbola there (very indifferently indicated rather than) drawn, OG" will be the minor semi-axis, (oy") being the stinnal of the ordinal with the hyperbal, determined by making y" s — e"2. The primal (oo, og") through (oo) will be the ordinal, and have OGC for its carsiigmod, and OG is parallel to the carstigmnod of the tangental at E". If len OG2 = len OG", OGC is the line usually drawn as the "imaginary" semi-minor axis. Similarly, ART. 48. viii. —x.] CORRESPONDENCE OF POINTS. 61 OE being any semi-diameter, OKYis usually drawn as the "imaginary" conjugate semi-ciameter, being parallel to the tangent at E, whereas it is only the carstigmod of the symmetral (art. 50.) to the diametral, of which OE is the carstigmod, and the proper stinnal (gg ) of that primal with the curve is found by turning OKi through a right angle to OG, and drawing GG1 perpendicular to OG'. We shall find in art. 50. iii. (3) that s2 = e,2+ g,12 = e//2_ 2 = e2+ 2 = e_2-2. Since e~g2 = [fgx —je (y+~x). [gx —je (y —)], the asymptals of the hyperbal are cyx +je (y-x) - o, and c;x-je (y —x) = o, and have a carstigmod, which will be found by turning the YXY' of the asymptals of the ellipsal through a right angle. Thus, in fig. 32, OL is an asymptote to the flat hyperbola on the right, where F'" = OG2. (ix.) I-/iperel (hyuper-bola+el-lipse, the final-ta omitted for euphony),.E and G lie anywhlere on the plane. This is the general case, to which all properties of centrals belong, The equations have the same forms as in (viii.) Given X (fig. 32), join XE, XF, make XF=l=FX, draw XU the mean bisector of XE, XE1, and revolve XU through L UXY= Z EOG, altering its lenglth so that len XU: len XY:: len OE: len OG. When X lies on EF between E and F, as at X, this construction gives Y as at YI, Y', on an ellipse of which Oi, OG are conjugate semidiameters. But if X lie beyond Z, F, as at X2, the same construction gives Y as at Y., Y. on a con focal hyperbola passing through E (the same as that described in viii.). From this circumstance is derived the name hiyperel, which thus becomes synonymous with the general central quadral. If the ordinate X~IY2 be revolved through a right angle to X2Y3, its termination will lie on one of Poncelet's supplementary hyperbolas, which is however quite useless in this case, as the stigmod is sufficiently clear in itself. The equations to the asymptals are the same as before; but if we put them into the proper cistal form (art. 40.), using (x/y'), (;my') for the costigmals in the asymptals, with (xy) in the central, they become y -p' = y-x +j'. Re. gx, y-c '= y -x -j.. Re. cx, whence (yl-) (y - q) = y2, or the mean bisectors of P'Y, Q'Y = OG and GO, as in fig. 32, where pri (oo, xp') and pri (oo, xc') are the asymptals. Now 2 (y —) = y —y', hence y'- '- (y-o')-2 (y-x) = i'. (y-p'), or Q'Y'= YP', a well known property in the hyperbola, but seldom directionally stated. (In the ordinary hyperbola, the parallelogram FPYQ'Y' becomes a straight line.) Also if y-pl = 7, (y- p'), yJ-ql = 7. (y-q'/), we have (y-p1) (y-(/,) -= 7r. 2. Hence the above property holds for the stigmins of any transversal drawn through (xy) and cutting both the central and the asymptals. Also if y —p = p, y-q~- = q, pq = 7r2. I2, or (pq) is the stig'mal of an inval depending on the direction of the transversal. And so on for the generalisation of all other properties deduced in Pluicker's System, p. 91. (x.) The unreduced duoquadral equations to the cyclal takes one of the forms 2xy -y2 + 2s"x + 2e' + 4' = o, or x- (_ X-)2 + 2p' + 2,'. (y - ) + ' = o. 62 V. STIGMATIC GEOMETRY) OR THE [ART. 48. x.-xi. If T, T' be the direction points of two intersecting rayals (Ib, xy), (Ed, xy), proceeding from fixed stigmals (/ib), (Ed), then (b-y) + (t-i) (/-x) = o, and (d-y)+(t'- i) ( —x) = o. Hence the condition tal TT'= f, giving +/ ( -i) - (, -i) is easily reduced to an equation in w and y, which on multiplying out will be found to be one of these two general forms of the cyclal. This generalises a portion of art. 31. v., and admits of the complete application of ray-hommals in the same way as Chasles uses the homographic properties of rays in a circle. This shews also that three stigmals, forming a trilateral (aa, /3b, yc) determine a cyclal. To construct it from them, it is necessary to find the axis, that is, the stigmals of the centre, and the major points. On drawing orthals through the middle stigmals of two of the laterals, their stinnal is the stigmal of which the centre is the stigma. Transordinate so as to make the central stigmal (oo), then (x'y) being one of the transordinated stigmals, draw X'Y' so that 2x'= I+y', and find E, F as double points of the inval (oo, yy'). On making this construction first in a Cartesian case, carefully marking the indices, its nature will be quite clear. A cyclal thus given may be noted as cyc (aa, l3b, yc). (xi.) For conals generally, if from (/rn), (va) rayals be drawn intersecting in fixed stinnals (,ca), (/3b), (yc), and a variable stinnal (xy), and the direction points of the rayals from (anm) be A,, B1, C, X1 and from (n1,) be A2, B2, 02, Y2 respectively, then we may find al-i, a2-i, b —i, b —i, &c., in the same way as in (x.), whence we can form a,-b, = (al —i) —(bl-i), and so on. Then if the movable rayals form a ray-hommal with the fixed rayals, we have (abclxl) = (a2,h2cy2). Substituting the values of ca,-b, &c., thus found, we obtain as the locus of (xy) a general quadral, of which it is easy to investigate the particular cases. Also if there be four fixed stigmals (ac), (3ib), (yc), (dc), whence rayals are drawn to a movable stinnal (xy), and Al, B1, C1, C D be their variable direction points; the condition (caLbcd1) = X, reduced as before, gives a general quadral. In the latter case, (abdc) is also constant; hence X = p (abed), where p. is a constant, or the anral of the rayals, now called chordals (chord + al) of the quadral, divided by the anral of the stigmata of the fixed stigmals is constant. These contain stigmatic generalisations of Chasles's fundamental propositions, Sections Cooiqu's, arts. 8. and 4. respectively. They can also be deduced in other ways. The deduction in Chasles is made from perspective projections of a circle; but this is inapplicable stigmatically when the centre of projection is not in the same plane as the curve. Hence it is not possible to pass in that way from the properties of general stigmals of a circle (non-Cartesian as well as Cartesian, "imaginary" as well as "real" points) by such projections. For the same reason it will be necessary to establish a stigmatic theory of contact before the corresponding generalisation of the fundamental proposition of tangents can be undertaken. That proposition is proved in art. 51. iv. After these chief propositions have been proved, the whole of the d3monstrations in Chasles's Sections Coniques can be adapted stigmatically by mere alteration of terminology. ART. 49. i. —V.] CORRESPONDENCE OF POINTS. 49. Intersections of Duoqciacials by Pgrimals.-(i.) The intersections of a hyperel e2(y_-)2 +g-2 = eqg2 bya primal y —xtx= b give at once (y2 +e2t2)x2- 2bte'2x = (2-_b2). e2...................(1), whence (2 + 2t2) = bte2~e,(+ e - ).................. (2), which is constructed by putting et I=, g2 + qi2 i= 2, t 2_ b2 = r2, b2 = nm', gr = -' ', whence q.X1 = eg e /', nx = -eii,' - er. In particular cases this construction may be greatly simplified. (ii.) There is no intersection, if g2+e2t2 = o and b = o; for the equation (1) in (i.) then reduces to o = e2y2, an impossibility. In this case, the primal is an asymptal, as already found. (iii.) If g2+e2t2 = o, but b not = o, the eqution (1) in (i.) reduces to (2-_b2) + 2tbx = o, giving only one value of x, or aparasymiptal cuts the hyperel in one stigmal only. (iv.) If b does not = o, but y2+e2t2 = b2, then there is also only one value of x, produced however not by the reduction of the equation (1) in (i.) to a simple form, but to a complete square. This makes the primal a tangental at (!/!), and on determining t from this condition, and from the equations to the primal and the hyperel, we find te2(y- x) = g2x, so that (xI/y) being any other stigmal on the tangental, its equation is e2(/ —x). (yl —x)+g2X. 2' = e22. Hence tangentals to a central can be drawn through any stigmal, except the centre stigmal (oo). The whole theory of the tangentkal and polar can now be deduced; see arts. 50. and 51. (v.) For the particular case of the cyclal proceed thus, fig. 34, where the lettering must first be understood in a general, not a Cartesian, sense. Primal y-xx+tx = b = ct; cyclal a2- (y-x)2 =e2 whence t~ /(b2 +e2- e2t'2) b F v/(b2 + e2- e2t2) t2- i t+ i Put (X1y1), (X2 Y2) for the two values of the stinnals. The orthal from (oo) on the primal is t (y-x) + x = o, and if its stinnal with the primal be (mrn), and with the cyclal be (di), (V'd'), b bt t —i 2 we have nt = — t =+ t) = t+ — ' e, whence 2n =!/I + Y2i, 2,mn = x, + x2, Y 1 y/2 = (c2, so that ('rnm) is the middle stigmal of chordal (x11Y, T2y2), and Y1, Y2 lie harmonically with respect to D, D'. Also in- = ~ (b2~+2 —e2t2) 2,-y - /(2+e2-e2t2) in bt ' b ' ( b')-l —e2-t 2 2 2 i -- t and (n y) = - e- t= ~, + e2. -- == 2-d2 = ('a-c) (n-cl'). (t+i)2 =2% i+t (1) First particular case. The primal and cyclal are Cartesian, e Se, b = Vb, t = Vt, or Ve = Sb = St - o; e2= T2e, 2= i'. T2b, t2 =i'. Tt, b- + e2- et2 = i. Tb + T-e + Te. Tt = (i + ( T) i T-e-T'n, sinne Y-]2b (i+ 2t). T2u. If then Te > Tn, or the line GB cuts the circle (this case is not drawn in the figure), 7 (b2- e-te'') = i, and hence VJfx= 0, and S = 0, Vv f 64 V. STIGMATIC GEOMETRY, OR THE [ART. 49. V. or X310 is a straight line, and ONY a right angle. This corresponds to the case of art. 34. x. But if Te < ThA, as in fig. 34, U(b2 + e- e2 ) =i, and hence S- = O, t — = 0, si n or OJiX is a right angle, (and hence Xl1 fX2 a straight line perpendicular to 031,) and ONY or Y1NYQO is a straight line. In this case T' (n-y) = Tn —T2e. Hence set off NZ' or NS of the same length as OE, and with centre Z' and radius of the same length as ON describe a circle which will cut OIN in Y1, Y2. It is easily seen that this construction is the same as that for finding the double points in the hommal resulting from the intersections with CB of rays from K, L, the extremities of the diameter parallel to 0, B passing through any points in the circle. Thus the tangents KZ', LS determine the solitary stigma and index, and the rays KD, LD two other points, (drawn but not lettered in the figure,) whence Y1, Y. are found. Chasles's definition of the imaginary points of intersection corresponds to their being the double points thus obtained. Then X1, X2 are found by making CX1Y1 A CY.2Y A COB. It is well to verify by construction that (x1 l1), (xzY/) are really stigmals belonging to the cyclal. If X1Y1 and X2Yq are produced to the same length backwards, they will fall on other parts of the stigmod corresponding to the indit X1X2. This is seen to be a two-branched curve in the figure. The stigmods described by two different stigmata for any indit are necessarily so; but the two branches of the carstigmod in this case, as mentioned in art. 48. v., coalesce and form the circle, whereby, as so frequently happens in Cartesian geometry, the real relations are completely disgcuised. Observe that since (it-)2 =- (, -d)( - d'), if we were to suppose e, and hence d, d', to vary, (dd') will become the stigma on an inval of which N is the solitary point and Y1, Y0 the double points. This would give a series of cyclals having the common chordal (z1y1, xa 2) on the primal, of which CB is the carstigmod, and hence being the only part hitherto recognisable, was used to represent that chordal and called the radical axis. Since (art. 50. ii.) the symmetrals of a cyclal are orthal, no generality is lost by considering this chordal to be the ordinal, and taking the origin 0 at N, and the equation to the cyclal as (c —z)2 —(y_-)2 = ( c-I)2 = (c-7)2, so that C is its centre, and Ilt its axis. Let (oe), (of) be the stinnals of the ordinal with this cyclal, then the inval becomes e2=f2= h7, and all the general cyclals which the ordinal intersects in (oe), (of) will be found from their axis lK, which forms an ordinate in this inval. This at once generalises and simplifies the investigation of the properties of this commoU chordal. (2) Next suppose the primal to be Cartesian, but the cyclal to be 2(y —)2 = g2, where g =je, and is hence a vector. This may be distinguished as the vec-cyclal, and corresponds to Chasles's "imaginary circle," (see below, p. 78, col. 1, at bottom,) which here becomes a geometrical reality; see art. 48. v. In this case, b+2 g2_t2t = i'. T - T2g —T T. Tt, and hence S v/(b2+g2-g2t2) = o in all cases. Hence we have as before S m x- =o, V - = o; but T2 (.-y) = Tn2 +Tg. 93n?' ART. 49. V. vi.] CORRESPONDENCE OF POINTS. 65 Hence make len NY' = len Y"N = len OZ', and the stigmins Y', Y' are determined. Then find X', X" from CX'Y' A CX"Y" A COB. The figure shews that X"Y" is a mean bisector of X"'G, X"H, and hence that (x"y") is a stigmal in the vec-cyclal as well as in the carprimal. This will suffice to initiate the very interesting relations of this case. (vi.) Carnot's Transversals for conals may be considered thus: —(1) Let two primals through any stigmal (xy) cut tihe conal whose equation is p (r, y) = o, in (f'ly,), (x2y2) and (x'y'), (r' y") respectively. Then if h be the coefficient of y2 in p( (x, y) and cl, c, be coefficients depending on thle direction points of the primals (put the equation in the distal form, and apply art. 40. iv.), each of the following expressions represents ) (x, y), and we have consequently,C. (y-yl) * (y-y2) =. (y-y)). (y-y')............ (1). If the second primal is tangental, the second side becomes 2. (y-y)2............................ (2). If the second primal is a parasymptal, it cuts the conal in one stigmal only, and the second side becomes t'3. (y-y')........................ (3). If the second primal be an asymptal, it does not cut the conal at all, and (Y-yYl). (Y-y2) = 4........................ (4). (2) If two primals be drawn intersecting each other in (xy) and the conal in (xy1i), (x2y/2) and (x'y'), (x"y") respectively. And two others parallel to the former respectively and intersecting each other in (siq) and the conal in (a nI), (422), and (E'"'), (4"r) respectively, then K (y.i-) (,Y2 -) = K" (Y- l) (Y"- Y), and K (ql~-?) ()2-i) = ('-?) (y/- ) so that, on eliminating C, '', (yl-y) (Qy2-y) = (/- ) (l/' -?) (3) Let the laterals (/3b, yc), (yc, aa), (aa, f/b), of the tri (aa, fb, yc) intersect the conal in (Xi), (X'Z'), in (fin), (''), and in (vn), (v''), respectively, and let hC1, 2,, K'3 be the coefficients due to their direction points respectively, then K1. (c-z). (c-I') =. (-n). (c-.(- 1),. (a-). ( = 3 (a-)n') = i. ( a-n'), K3 (b-n). (b-n) = K (b-). (b-)0; whence eliminating q, K2, C3 we have (c-I)(c-l ') (a-n) (a-n') (b —n) (b-) -n (c-r7) (c-rn') (a-n ) (a- n) (b-1) (b-') and this expression holds for non-Cartesian as well as for Cartesian intersections. Thus, in fig. 34, the laterals (oo, cc), (cc, ob), (ob, oo) of the Cartesian trilateral, (oo, cc, ob) intersect the Cartesian cyclal in (ee), (ff), in (X1yl), (X2 y2), and in (og, ohl), respectively, and hence gh. - 0)(e/-c 0!f. (/-b) (Y2,-b) i. ef (y - c) ( - ) (g -b) (h- b) 66 V. STIGMATIC GEOMETRY, OR THE [ART. 49. vi.-50. iii. The position of the triangle then shews that U (: — - -)= i, (y-c) ( 2-c) and hence z CY1B = Z BY2C, which on account of the perpendicularity of Y1Yv on BeG is easily verified, and shews a real geometrical relation of the "imaginary points" Y1, Y2. In a similar way all the other transversal relations may be generalised. 50. Synmmeirals, or Conjugcae Diameters generatlised.-(i.) A primal, drawn through the stigmnal of which the centre of a central is the stigma, cutting the central in two known stinnals, is called a diiaettali, and those stinnals its term1inals. The major and minor axals (ee, ff), (og, oh) of a central, which in this form are the abscissal and ordinal, are such diametrals, of which the stigmals just named are the terminals. The central expressed as y2_q+e2 (y-)2= e2y2 has then this property, that for any value of x the two values of y —x are equal and opposite. The equations to these principal diametrals are x =o and y-x = o. (ii.) Now, transordinate indicially (art. 47. ii.), putting x = ax'+ b (y —x')................................(1), whence y - = (i-a) '+ (i-b). (y-x')..................(2). Then, putting alternately '= o, oy —'= o, for the equations to new diametrals, they give, in the old coordination, x=by, x=ay respectively. If 1/, 1T be the two direction points of these primals, then b (i-t,) = i and a (i-t,) - i. Putting these values for a and b, and then the resulting values for x and y — in the equation to the central, and reducing, we find it (2 +e2 t ) 2 (y2 +2tlt2).'- ) + - (2 + e2 t ) (y_-x') i - 12 i - ti = e2..... (3). This therefore will have the same form as before, if g2+e2tt2 = o. Hence the pairs of diametrals satisfying this condition form a rayinval, and the two rayals in each rayar pair (art. 39. ix.) may be called synm.cetrals (con-jugate, con- represented by symz-, and dia-met/ral). The double rayals are determined by cg2+ 2t = o, but these are not diametrals, for, putting t= tj = tI, this condition reduces equation (3) to o = e2J2, which is clearly impossible. But these double rays are the asymptals (see art. 48. viii.), and, calling their direction points A1, A2, we have tlt2= a2 = a 2, which gives an easy construction, when the asymptals and one symmetral is known, to find the other symmetral. In the cyclal, since e2 +g2 = o, we have t1t2 = i, or the synmmetrals of any pair in the cyclal are orthal. (iii.) Let (2c, u'c'), (vd, v'd') be two symmetrals expressed by their terminals, having the direction points T1, T2. Let a primal fiom ('nc) orthal to (vd, v'c') cut the latter in (l2'222) having the direction point P1 so that t2pl =?i. Then g2,ta2+e2(c-u))2 = e2y2, gy2v2~+e(d-v)3 e=y2, =. -c, rt2= v-ct, i'.2- 2,-, (c-uz)(<l-v) t1 = u —C, Vt2- == V-c, Z - 2 2= ( U)( —) e?uV ART. 50. iii.-51. ii.] CORRESPONDENCE OF POINTS. 67 Substituting from the third and fourth in the first and second, and reducing by the fifth of these equations, g2t e = (c-i)2. (g+ t2 e2), 2t2 2 = vI. (/t-t Iy2),. (c-u) +v. (c —v) = o........................ (1), + e2, (c- )2+ (d-V)2 =................. (2), c2 d2 = e2+ pg - 82........................... (,3), v. (c - it)- u (d - v) = - e................... (4), c —a_ = l —i.- tl —, by art. 41. ii., c tl —i p)-1 -_ JRf2- t1-t2 _ d v (c-u)-ut (d-v) tl —i I' t-t2 o (d —7)2-v ' whence (c - ). (2v - c) = eg..................... (5). These are generalisations of mostly well known properties, but (3) was I believe never noticed till my second memoir on Plane Stigmatics (14 June, 1866), though it gives a very neat and useful construction by art. 33. v. for finding the focus from any pair of symmetrals of which the terminals are known, or the terminal of a second symmetral from the foci and one symmetral. Compare especially Chasles, Sect. Con., art. 205, and observe that that article applies only to the ellipse and to the case of "real" or Cartesian symmetrals, whereas the present equation applies generally. The reduction of these to the usual tensor relations in the Cartesian case of either ellipse or hyperbola presents no difficulty. (iv.) Putting for t1, t2 the values in (iii.), we have for the transordination in (ii.), X - = -_. + V* ( ) i - t —t, (c t, to -,+ d-v Y-'w = — 1- ' + t,7 —~ ' / - -) - -- - (y —x')' tl-i t2-) C d' and then substituting in the equation to the central and reducing by (iii.), we find cz'-+ 2 (y -x)2 = c2cF, so that the central referred to symmetrals has always the same form. 51. Tcngeitcdls, Polals, Polarals, Focals, Co'nfocal Ce1 ntrals, anid Curvacyclcts, or the Relatio72s of Tca17gents, Poles, Polars, Foci, Cojlfocal Convics, and Circle of Cauvctlnre, generalised.-(i.) Notation as in art. 50. If To be the direction point of the tangential to a central at (mc), and TI, T2 those of the diametral (oo, tic) and its symmetral, it appears by the equation to the tangental in art. 49. iv. that T to =. - = i' -c/ by art. 50. iii., = t, by art. 50. ii. e2' c - 32 -' The tangental is consequently parallel to the symmetral. (ii.) If the double point of the tangental at (zc) be JW, it appears by the equation in art. 49. iv. that Zw- = e2, or U, TW are harmonically situate with respect to E, F. As the stigma C does not appear, the co-stigmal (nzc') will have a tangental with the same double point. On 68 V. STIGMATIC GEOMIETRY, OR THE [ART. 51. ii.-V. account of art. 49. iv. the same is true for the co-stigmals of any index, when the central is referred to symmetrals. Hence, to draw two tangentals to a central through a given stigmal (w'w). first draw a diametral (oo, ic) through that stigmal, then its symmetral (oo,vd), and then determine the terminals (?,c), ('d) of both. Find X'so that x'. w = c2, and taking X'as the index of a stigmal referred to the symmetrals as axals, find its stigmata Yi, Y2, and then find the indices X1, X2 of these stigmata referred to the old axals. The two tangents referred to the symmetrals are (WVIc,,'y1) and (nzv, x,'y), and referred to the old axals are (w'w, my1), (al-c, 2:f/_o9). (iii ) The primal (x1y1, 2,/2) or cotact-c(chordal is the polaral (polar + at) of the stigmal (iv'w) in reference to the central, and this stigma is the poltl (pol-e+al) of that chordal. The properties of these stigmals and primals depend upon the inval equation 'z'w = c2 by which they were determined in (ii.). (iv.) " If through four fixed stigmals in a central there be drawn any four tangentals, intersecting any fifth tangental, and also four chordals meeting in any fifth stigmal of the central, the anral of the four stigmins of the four first with tlhe fifth tangental will he equal to the anral of the direction points of the four chordals." This is the stigmatic expression of Chasles's fundamental property (Sections Colniqtues, art. 2.) referred to in art. 48. xii. The following is the demonstration I gave in 1866, in my second memoir on Plane Stigmatics, art. 110, reduced to the present terminology. The aural of the four chordals remains unaltered, whatever be the fifth stigmal to which they are drawn (art. 48. xi.); hence it is sufficient to prove the proposition for any particular position of the fifth stigmal. Assume it to be the contact stigomal of the fifth tangental with the central, and through the stinnals of the four tangentals with the fifth, draw four rayals to the stigmal of which the centre of the central is the stigma. These will be symmetrals to the diametrals which are parallel to the four chordals (as they are all contact chordals), and their direction points will have the same anral as the direction points of these diametrals (on account of the inval, art. 50. ii.), and hence as the aural of the direction points of the four chordals. But the direction points of the four rayals have also the same anral as the four stigmins of the four tangentals with the fifth, throngh which the rayals were drawn (art. 42. iii.). Hence the proposition is established in all its generality for all central quadrals, Cartesian or non-Cartesian, and consequently all deductions made from it, by adapting the reasoning in Chasles's Sections Coniques to the stingmatic generalisations, must also be necessarily correct. For non-central quadrals, see art. 52. xii. (v.) If B be the original point of the tangental, and T its direction point, then, by art. 49. iv., g2+e22 = b2. Hence, if tangentals be parallel to the asymptals of a cyclal, that is, be parassal, so that t2 = i, we have b2 =- 2 + e2 =s2= z2. Hence all such tangentals contain the stigmals (os) or (oz). In this case then the equation to the tangental at (xy) reduces to y = s orz, and 2x -y = s or z. Now the double points in both cases are (ss) or (zz). Consequently there are four primals (ss, os), (zz, os), (zz, oz), (zz, o.s), having either ART. 51. V.-vii.] CORRESPONDENCE OF POINTS. 69 S or Z as the dnoble point, and also either S or Z as the original point, which possess the property of being at once parassal and tangental to the central. These two points, S, Z, are known as thel oci', and the four stigmals (ss), (os), (zz), (oz), may be termed the focals. By confusing foci with focals (i.e., stigmata with stigmals, as usual in Cartesian geometry), Piicker (Systema, p. 106, 1. 6) recognises four Brelnpzt cte or foci in a central; two real, lying on the major axis,-these are the focals (ss) and (zz); and two imaginary, lying on the minor axis,-these ame the focals (os), (oz). This results from his definition of focus, which is really only that of focal. Salmon (Coiic., 3rd ed. p. 233, 4th ed. p. 242) also says that the two imaginary points, meaning the two stigmals (os), (oz), " may be considered as imaginary foci of the curve." He also speaks of a quadrilateral, corresponcing to that stigmoatic quadrilateral of which the four are the four tangentals just named. Chasles (iSections Coniqifes, art. 294) speaks of this qualdrillteral, but recognises as foci two only of its apicals (ss), (zz), as will be found only translating his language stigmatically. His words are: " Les foyers d'une coniqcle dont les detx sonzmets reels du quadrilatere imaginaiire circonscrit i la courbe, et dent les points du concours des cotes opposes sent les deux points imaginair-es situes a l'infini sur un cercle." Points, which are either indices or stigmata, should be kept distinct from stigmalcs, which consist of stigmata referred to indices. If we use foci for the points, there are but two iii a central, determined by s~-2 = e2 — + = c-2 +d2 but there are four focals, which, referred to the principal axals, are (ss), (zz), (os), (oz), the first two on the abscissal and the second two on the ordinal. In fig. 32, S' is so taken that ss' = e2, hence the ordinal through (s's') is the contact-chordal for tangentals from (ss). Consequently (s's), which is a stigmal in the parunal through (ss), must be the stigmal of contact. It is readily seen by actual construction that (s's) is a stigmal in the central. If for any indit through S' we find the corresponding stigmod for the central, and also for the parunal, the latter would remain the point S, and hence the fact of contact would not appear to the eye. But on turning all the ordinates through a right angle, we obtair supplementary figures in which the contact is visible. For illustration this is shewn in fig. 32 for the car-ellipsal e'2y —y_)2-9'2 = ey'22, in the tangental from (zz), of which the contact-chordal is the parordinal (z'z', z'zi), where zz' —= e2. The ordinates turned through a right angle generate one of Poncelet's supplementary hyperbolas, and the tangent to this from z represents tIhe stigmod of the actual tangental, and is seen also to be a tangental from (zz). It must be remembered that this arrangement in the figure does not represent the actual state of things, but merely serves to make it clearer to the eye by separating points which would have otherwise coalesced, or have lain on the same straight line. (vi.) " If pairs of rayals be drawn from any focal of a central to the corresponding stinnals of a movable tangental and two fixed tangentals, the tannal of the direction points of the rayals will be constant." This is a generalisation of Chasles (Sectio;ns Goniqlaes, art. 293), and applies to all four focals; the demonstration follows from art. 43. iii. (vii.) "The sum of the tannals of the direction points bet.vesen the rayals drawn from any stigmal in a central to the two focals (ss), (zz), F 70 V. STIGMATIC GEOMETRY, OR THE [ART. 51. Viii.-ix. or of those drawn to the two focals (os), (oz), and the normal (or orthal to the tangental at the point) is null." This is a generalisation of the property whence the foci received their name. The existence of this property for both pairs of stigmals (s,), (zz) and (os), (oz), justifies therefore the application of the term focal to all four. Let N, be the direction point of the normal (that is, the orthal to the tangental) at (xy), and S1, Z1; S2, Z2, the direction points of the rayals from (xy) to (ss), (zz), (os), (oz) respectively. Then, art. 49. iv., 0(if —. e2. x/ -X- - _. g while s - z — s- - _ - (y-s) _ s- (y-) - -(/-z) _ + (/- x) 82 - --, ~ S2- = X Z2 = X X X. x an, (Y-X) n,-a Hence tal SNY = -- = s. (y-) __ = tal NiZ,, i —s<ii 1 2 i- z- ih and tal S - - = - tal N1Z2. ' —S1l1l e2 — n1 Z2 (viii.) The equations s2 = e2 +g2 = e2 + 2 = e2 + g2, fig. 32, point to a series of conals with a common centre 0 and common foci S, Z. These are called confocal centrals. If we put e2= X2, g2= ( - aX)2, these equations reduce to s2 =- -+ (y_ - )2, which gives an equiperbal (art. 48. vi.) whence, given S, Z, the whole system can be found. If we assume any pair of values of e, y, to give a standard hyperel, then by art. 50. iii. (3), another pair, as c, d, will give terminals of symmetrals, which must be referred to indices by being taken as clinants of stigmata in the hyperel determined by the other. To find the stinnals of two confocal hyperels (ee, oo, oy) and (e'e', oo, oq'), 22 + e2(y-,)2 = e29 g' + 2e - X2(- = e/g/2 where 2 = e2 + g2 = e2 + g2, fig. 32. These equations give s2a2 = e2e2, s2(y-a)2 = g2g2. If then ', T' be the direction points of the tangentals to these hyperels at (xy), we have t = -. 2, t' = a-. p,'2 p -a y- e2' yp-a so that tt'= i, or the tangentals are orthal. This stinnal is very nearly the (X.i/,2) of fig. 32. If in the same figure we take the Cartesian ellipsal ('e', oo, of/), and the confocal Cartesian hyperbal (e'e", oo, og"), their stigmin is E, and the perpendicularity of the carstigmods of the two Cartesian tangentals at E is evident. (ix.) The theory of transversals in art. 49. vi. is sufficient to determine the curva-cyclal (curva-ture+cyclol) to any conal whatever. Let (aa), (cc'a'), (t3'b') be three stigmals in a central. (The reader should draw a Cartesian case; there was no room for the figures.) Draw the chordal (aa, a'a'), and through ('b') draw an orthal to this chordal, cutting it in (X/), and also cutting the central again in (f3b), and the cyclal drawn through the three first stigmals, in (3d). Take 2,u = /3 + f3', 2- b ~ b+b', and through (/mn) draw a primal parallel to the chordal (act, a'c'), and cutting the central in (yc), (y'c'). Let (w'o) be the stigmal of which the centre of the cyclal is the stigma, and draw the symmetrals ((w~,1 pj), (/ww, q'/), parallel to the chordals (aa, a'a') and (3b, 3'b'), so ART. 51. ix.-52. iii.] CORRESPONDENCE OF POINTS. 71 that ((,o-p)2+ (w - q)2 = o, because, being orthal, they are symmetrals in a cvclal, art. 50. iii. Then by transversals, in the cyclal -i - ) = -- = '.................. (1), (a - ) (d- l) ( —2)2 in the central (b -) (b'-l) (b-)(b' -n) i' (b —)2 in the central, — =, — _ -, 72 (a - ) (a'-i) (c- n) (c - n) ( C-) ) ( -.2). b-1- (b - 13) and by division b- = -(-......... (3) cd-i (c - r) (c - n)z) This holds for all circles. Now take the circle which is the limit as A, A', B' approach L. The tangental at (Xi) will be the limit of the chordal (act, ac'a'), and since the normal to it in the cyclal will be a diametral, (w'w) will lie on (X\, EcU), and d- I = 2 (w- 1). Also b -I = 2 (b - m) = i'. 2 (l-m7). Hence the last equation becomes I -, (c-r ) (c '- )....................... (4), i-n a new expression, giving an easy construction for the axis of the curvacyclal at (Xl) in the general case by making UMCT A O'ML and L&2 UfM. For the general form of the usual expression for centrals, from (oo) draw an orthal to the tangental cutting it in (pr), and, parallel to the tangental, a diametral to the conal cutting the latter in (vn), then w - 1 -= n2. Br. Make VON A NOR, and L2 = OV. 52. Pacab'bals.-(i.) There is no figure. If the reader will draw an ordinary Cartesian parabola with vertex 0, focus S, parameter OE = 408, directing point D, when DO= OS, axis OE, ordinate XY, he will probably experience no difficulty. (ii.) Putting 4s=e, the general equation to the parabbal (art. 48. iii.) is (y —x)'+e = o. To construct Y, join XO, draw OF-= EO, and make XY equal to the mean bisector of OF, OX. If X is on OE, the stigmod is the usual parabola. As long as X is on any straight line through 0, as OX1, the ordinates remain parallel to each other and len XY = len Xi Y, where X1Y1 is the Cartesian ordinate at X1 and len OXl = len OX. Hence the locus of Y is again an ordinary parabola, with "diameter" OX, and tangent at 0 parallel to XY. If the index X move on OF, away from 8', then XY, XY' lie on OF, and one of the stigmata will encroach on OS, but never farther than S. If these ordinates be turned through a right angle, the result is an ordinary parabola with focus D and axis OD. If X fall on 8, (y-s)2+4s2 = o, and len YY' = len Of. If X fall on D, (y —d)2 = 4s2, and if 2cd = s +s', then (ds), (ds') are the two stigmins of the directrix cd-x = o with the parabbal. In all cases (d —x) = (S + )2 = (s- )2 +4S = (S- )2- (y —)2, which is the generalisation of the property whence the directrix was named, giving in the Cartesian case, len SY = len DX. (iii.) To determine the stinnals of the primal y - x + t = b with the parabbal (y -)2+4sx = o, we find t2x' — 2tbz+ 2 - 4dx.......................... (1), 72 V. STIGMATIC GEOMETRY, OR THE [ART. 52. iii.-viii. whence tx = bt + 2d -- 2 v/(btc + d2)..................... (2). In the general case this is best constructed as in (viii.). (iv.) If t = o, or the primal is parabscissal, (i.) becomes b2 = 4cx, and hence there is only one stinnal. Such prinals are termed pCaraxials (jpni-a+axi-s + I) in preference to dicaietrals, a term applicable to central quadrals only. There is no asymptal. (v.) If bt=s, there will be only one stinnal by the reduction of (1) to a complete square. in this case t (y-x) = 2s gives T, the direction point of the tangental at (Xy), of which, if (x1yI) be any stigmal upon it, the equation is (y,- xn) (y-x) + 2s (I + x1) = o. If N, P be the double and original points of tangental at (xy), i +x = o, ) = ( (y —x) = S. t, s.p, n =. lt = s. R't = p2. Rs, p2 = sm = i. sx. If T' be the direction point of pri (ss, op), then r=p. s=Rit, or rt=i, so that this primal is orthal to the tangental. Also, since s (s-J) = s [s-s —(y -_)~ = S (S +p2.Rs-2p) = (s-p)2, SP is the mean bisector of SY, SO. These generalise known properties. The value of N being independent of Y, two tangentals can be drawn from (icn), and the ordinal (xy, xcy') will be the contact chordal. (vi.) Transordinate indicially; assuming x=u + c (x'-v) + b (y-Cx'). The equations to the new axals found by putting y=x', and x'= v alternately, are x = I + c (y- v), c = + b (y -v), which intersect in (uo). Substituting in (y-. )2+t4sx = o, and assuming (v — t)2+ 4ss' = o, a = i, (v -- - 2s) b = v- t, in which case (wV) is a stigmal on the parabbal, and the new axals are a paraxial and a tangental at (wv), we find (y- ')2+4 (s-v). (z'-v) = o, an equation of precisely the same form. as before. To find Y from X', draw VZ= 4S'V, and take X'Y equal to mean bisector of VX', VZ. (vii.) Let (x"y") be a stigmal referred to the axals in (vi.), and let 2v =- x+ x', then (y"'-x")2= '. 4 (s-v). ('"- v) = i'. 4 (s-v)(v-x') = i'. (y-x'), and hence these ordinates are of equal length and at right angles, so that (x"y") can be constructed from (x'y). (viii.) To determine intersections of pri (acc, ob) with the parabbal, see (iii.). Draw tangental (icn, op) parallel to (aa, ob), touching parabbal at (icv). It is determined by by s = as, = a, + = o, v- - = 2p; see (v.). Through (nv) draw a paraxial, cutting pri (aa, oh) in (w'x") and find (x"y,) and (X" y.2) as (x"y") was found in (vii.). In the Cartesian case YYA is perpendicular to AB. Then ("yl), ('"y2) are the stinnals referred to the paraxial and tangental as axes, and Y, Y2 are the required stigmins. To these the indices X, X,, referred to the old axes may now be found from the primal. But since w-u =- * -- v, (v-u-2s)(x-zt) = (v-u)(yj-x'), we find on substituting in (y,-x") + 4 (s -v) (x"'-v), that (x,-Vw)2 + 4t (u —zv) = o, so that (wx,), (wau2) are stigmals on a parabbal of which (un) is the vertical, and (oo) the focal. In the Cartesian case the same equations shew that if Y, TV be drawn perpendicular to the carordinate lVX", then w, -x"- x-,-w = w-x2, which give X1 and 'X immediately. ART. 52. ix.-53. i.] CORRESPONDENCE OF POINTS. 73 (ix.) For tangentals from (hk). Through (7k) draw a paraxial cutting parabbal in (zv), take '2v = xt+ k, and find (x"yl), ("Ya2), as in (viii.), then (kk, "y1),(kk, x"/!2) are the tangeuntals referred to the paraxial and tangental at its extremity, and (hk, x,1f1), (h7, 2/y2) the same referred to old axes, and (x1y1, x,1/2), that is (ac, oh) in the chordal of contact, or polaral of the polar (hk). The paraxial through the stinnal of the tangentals cuts the chordal of contact at its middle stigmal. (x.) For focal. If in iv. (2) we put bt = s, for tangental, andc make t = i or i', we obtain as the equations to the parassal tangeutals (see art. 51. v.) y = s, and 2x-y = s, or the prials (ss, os), andl (ss, oc). There is therefore only one focal (ss) where these two tarngeutals intersect. The stigmals of contact are respectively (ds), (ds') where s+s' = 2d, and hence (compare ii.) the contact chordal is the directrix. (xi.) If N2 be the direction point of the normal or orthal on tangental at (xy), and S1' of the pri (ss, xy) from the focal, then 2sn-l = y-x, (s —x) S1 =- y-x, whence tal S'lN -= h1 = tan NV10, which is the generalisation of the property that gave its name to the focus; see art. 51. vii. (xii.) To demonstrate (art. 51. iv.)for parabbals, proceed thus. Fromn any stigmal on a parabbal draw chordals to four other stigmals on it, and draw tangentals at all the five stigmals, and through the stinnals of the last four tangentals with the fifth draw paraxials (having therefore the same aural as the stigmins of these tangentals), these will pass through the middle stigmals in the four chordals of contact, and hence have the same anral as the original points of four paraxials drawn from the first four stigmals of contact (art. 46. vi.). But this last aural is equal to the aural of the four chordals, which is again equal to the anral of four chorclals drawn from the same four stignmals to any other stigmal. (xiii.) The aural of the stigmins of four tangentals with a fifth is equal to the aural of the direction points of these four talgerntals; see Chasles Sec. Con. art. 58, where, as the tangentals have no colmmon stinnal, he has been obliged to invent a new name, not here required. Let the four stiglmals of contact be (cCr), (/3b), (7c), (C d), and the four stinnals (c'C'), (f3b'), (/y'),c ('1'), and the four direction points of the tangentals at the four first stigmals be Al, B1, C1, D1; and the original points of the paraxials be A', B", C", D". Then, by (v.), 2a" =- a-ca = 2s. Ruca, 22b"= b —3 -- 2s. R]b1, &c., hence (a'b'c'd') = (Ca b " d) (= ( -C1 — R) 1 — 1)- = ( d). ('lw- Rd,) (nRe -b -l) 53. lMultiSndiciacs, or th e meaninLg in Planze Geometry Of AAlebraical Eqlatioins with severeal Independlent Vlriables.-(i.) In stating the general conception in art. 36. i., only one index, X, was mentioned, for clearness. But it is evident that in the equation f(:cl, c2,... x y) =-, the points X1, ~i,... X, may be assumed as indices respectively, and the resulting values of y determined, giving stigmata of which each one corresponds to many indices. Such stigmatics are distinguished as Tzult-indiicials. Hence there is no need to proceed beyond plane geometry for the perfect treatment of the relations of all such equations as are now referred to real geometry of three dimensions or ima 74 V. STIGMATIC GEOMETRY. [AIZT. 63. L-54.4~ iii. ginary geometries of n dimensions. As long as commutative algebra only is used, the stigmatic conception, with the algebra of clinants, allows of every result being clearly and distinctly considered as the algebraical expression of a geometrical relation of points on a plane. (ii.) But multindicials as well as sol-indicials (having one index) may be treated in the manner which originally suggested itself to me (Appendix III.) by assuming O1,, 02, OZ3... O,, 011', as unit radii, and determining a point IS, by the condition r = X-S1+ x242,... + ^,,+ yr. This is what is in fact done in Cartesian geometry, in the form r = x +ys, only scalar values of x and y being then admissible, whereas clinant values give the complete generalisation. We have thus derived stigmatics, of which the most general form would be r' = F [fi(X1, X2 **.., ) 41, A (X1, X2 X 7, y)......] Some of these I investigated in my original papers of 1855 and 1850, (see Appendix III.,) and the results are sometimes very curious. 54. Solid Stigmatics.-(i.) The Cartesian solid geometry results from a species of the derived stigmatics just mentioned, 0I, OJ, OK being three unit radii (here supposed to be rectangular) of a unit sphere, and R the point that we wish to investigate; on assuming Oi = x. 01+ y. OJ+ z. OK, any equation f(x, y, z) = o, will, for any given values for x, y, determine values of z. If the given values of x, y, and the determined values of z, be all scalar, the point R can be drawn. But if they be not scalar the conception is insufficient to determine i, until it is supplemented in various ways, and hence the custom of supposing R to become an " imaginary point," the fact being that no provision had been made for this case. (ii.) Among such provisions as might be suggested, the following would always give a position for E, which would agree with that now assigned so far as the Cartesian case is concerned. Suppose OIJ to be the clinant plane, but suppose it also to be movable, and that it can be placed so as to make 01, OJ coincide with OI, OK, or with OK, 01 respectively. This amounts to saying, allow OJ, OK on the plane JO~K, and OK, 01 on the plane KOI to flanction as 01, OJ on the plane Of. In this case,,. 01 gives a line OX1 on the plane 1OJ; y. OJ gives a line OYi on the plane JOK; and z. OK gives a line OZ1 on the plane KOI, with perfect certainty and distinctness; and then, as before, OR = 0-, + 0 Y +OZI, by the usual operations of directional addition of directed lines in space, I~ being the summit opposite to 0 of the parallelopipedon of which OXi, OYi, OZ1 are adjacent sides. This is only one out of numerous possibilities. It is clearly not a general conception. It is merely one of those geometric contrivances ad hoc, useful enough as illustrations, but not suitable for universal adoption, like Poncelet's supplementary ellipses and hyperbolas, all very well in their way, but needing no farther notice in a Tract on principles. (iii.) Clinant or purely commutative algebra is not adapted for the purposes of solid geometry, which involves non-commutative operations, when the plane on which the similar triangles are to be constructed, is constantly movable. The required instrument is furnished by quaternions, but the resultant stigmatic geometry differs from the former, ART. 54. iii-55.] CONCLUSION. owing to the variability of plane. In clinants, two points, 0 and I, could be considered fixed, and one only, X, being variable, could pass into any point of the plane, and hence determine any triangle on that plane. Now it might also pass into any point in space, but in doing so it would determine triangles only on such planes as intersect in Of. To complete the geometry of space, the standard line must be itself movable, but its origins may be fixed, and the length of its initial limit may be unchanged. Let then 01 be a unit radius in the same unit circle as before, so that OM. = n. O1, and Tm = i, where in is a clinant. OM may be called the (unit) base, llM the base point. Let X be any point in space, which may be called the vertex. Then MfOX0 will be any triangle on, or parallel to, any plane in space; and if OA be any line parallel to the plane of lOX, it is possible to construct A OB A MOX, and thus determine B. The operation thus performed is called a quaternion, and may be represented by,,,, the subscript letter referring to the clinant m, so that OB = x,,,. OA. This is the operation, differently conceived, of which Sir VW. R. Hamilton has investigated the laws, and we see that cliCnants are qulaternions with a constant base point and constant plane of rotation,, or for which,, always = xi = x on the plane IOJ. Now assume the laws of quaternions as established by Sir W. R. Hamilton, and let y,, be some other quaternion, and let (b (xl, y,,) = o. Then, so far as this equation can be solved, (which is not very far, for Sir W. R. Hamilton only solved the equation of the first degree completely,) the assumption of any two points Ml, X, forming a qZin (qn-aternion in-dex) will determine two other points N, Y, forming a qcgzas (qnca-ternion s-tigma). The relation then is not one between two points, index and stigma, forming a stigmal, but between two pairs of points, quin and quas, forming a qual (qc.-aternion stigmal), and hence partakes of the character of the relation between an indistigmal and a stigmo-stigmal in the case of a transordinated stigmatic, (art. 47. i.) This bare statement of the conception must here suffice. Solid stignmatics, and the correspondence of points lying in different planes, lie beyond the scope of this Tract, although the geometry here developed allows of such correspondence being expressed in various particular cases, by the aid of conventions similar to those in (ii.) and those indicated in the first case of art. 44. iv. CONCLUSION. 55. Such is my Stigmatic Geometry. The sketch is rough, and bare of detail, but the outline is, I trust, sufficiently firm and true for Mathematicians to recognise the main features of my Theory, and to justify my own confidence that Clinants and Stigmatics are a New Power in Mathematical Analysis, a New Instrument for Geometrical Investigation, and a New Form of Life for Algebra. 76 THE IMPOSSIBLE IN GEOMETRY. [APP. I. APPENDIX I. On the Impossible in Geometry. SEE Art. 6.i., 10. viii., 17. vi.,47.vi. The point a l'infini," whereas in G. S. 719 he followinrg is a revision of a note originally s peaks of the " double contact imaginaire appended to the private reprint of the Ab- a l'ijfimi " of two concentric circles, which stract of my -second Memoir on Plane are particular cases of concentric ellipses, Stigmatics. so that circles which, according to the first AXIO.si. It is imnpossible for two poeints to citation, have no point at infinity, have also be, at the same time, separate ctad coincident. twi o such points, according to the second.] Ahthough most of the following pro- 5. " The product of nothiny by infinity positions are employed without hesitation I a/y be fniteC;" Salbon, Conics, 4th edit. by eminent mathematicians, they appa- art. 67; [i. e., perpetual doing of nothing rently involve this impossibility. may produce something; or, an infinite 1. Anlz infites'i.ialedistance isno distance; agregotion of points, each having no [i.e., infinitesimal separationis coincidence; length, may produce some length; or, noni. e., as finite length is an aggregate of entity infinitely repeated may produce an infinitesimal lengths, each of which is, on entity]. Townsecnd, MI. G(. art. 13. this theory, a coincidence or single point, 6. Ifniity is a singyle straigqlt line havevery finite length is a single point, i. e. a ing an icndteirmiate direction. [Salmon's point, having no dimensions, may be re- equation to this line, 0. x - 0. y + C = 0, garded as a circle, having two dimensions, shews that its existence assumes that C is or a sphere, ha-ving three dimensions; as- both = and not = 0, unless proposition sumptions constantly made]. 5 holds. Townsend's proof of the equiva2. Wlhen the rateio of the distances of two lent proposition involves a threefold applipoints from a third, which? is snot midway cation of proposition 2.] loncelet, Probetween them, is one of egqality, the third jections Perspectives, p. 53, art. 107. point is at an infinite distance ftioi the other Chasles, G. S. 503, 651, &c. Salioni two, and con,'ersely; [i. e. the separation of Conics, 4th edit. art. 67. Townsend, the two first points is equivalent to a co- MI. G. art. 136, 150. incidence, or else lengths differing by a 7. Relations of noeviqn points whiZch hold finite length may have a ratio of equality]. as they appuroach a litit, hold also at the Chlsles, GcometrieSupcrieure, 15, etpassim. limizit. [The limit is a condition not Tozinseia, 3Modern Geometry, art. 15. reached: hence this asserts that what is 3. Paacllel straiyht lines moeet at one single not reached is reached (i. e., that separation point atifi)iity/; [i. e., two points, connected is coincidence), or that relations of existby an invariable straight line, and moving ence hold for non-existence, under which each upon a straight line, and hence never i last form the proposition is continually approaching, meet.] Poncelet, Proj. Persp. i applied, as wlhen four points reduce to p. 52, art. 103. Chl,,sl&s, G. S. passiil. t three, or two points to one.] C'hasles, G. S. Tow.nsend, Mi. G. art. 16. [The inter- i15. ToiCnse2nd, A1. G. art. 19, and examples. section of two concentric circles is liable 8. Varciabl, elationes of mozing points to a similar obiection.] waich approach a fixed relation as tze 4. 1 striarighit 1/6e hais a single point at points ap)propi'h a ftixecd limit, cassiome that ialinity; [i. e., if two poiiits move in oppo- fi ed relation whcn the points s'rech the site directions unpon a straii;ht line, they lbiiit. [This is liable to the same will meet at one point at infinity; i. e., a i objection of reaching a limit. The procontinually]) increasing separation promotes position is, however, almost universally coincidence, or else a straight line is an appllied in the theory of limits, except as enclosed curve, e. g. a circle with an in- lidl down by Carnot. See Tract II. finite radius; hence diametrically opposite above. It generally assumes that what is directions are the same]. Steiier, Geo- true for separation is true for coincidence.] metrische Gestalten, p. 2, note. Chaslces, These contradictions are avoided in the G. S. 20. ownsendc, M. G. art. 17. above Tracts. The operation of annihila[Similar objections apply to the single tion (o) has been distinguished from those point at infinity of a parabola, and the two of chainge (a, b, c, &c.). Incommensurables (and no more) points at infinity of the are treated independently of limits. The hyperbola, C'/hases, Sections Coniques, 13, essence of a limit is held to consist in its where he says that the ellipse "' n'a uenci joint approachability and unattainability. APP. II.] TIE IMAGINARY IN GEOMETRY. 77 APPENDIX II. On the Inaginacry in Geometry. This is a revision of a note added to the la courbe.... J'en dirai tout autant de private reprint of the Abstract of my second l'admission des iatyginaires; c'ost parce Alemoir on Plane Stigmatics, and contains que dans ces figures, la chose repr6sentee extracts from the works of eminent mathe- perdc soe exis'tece pr6cisement dans les maticians, shewing the condition of the memes limites ou l'expression algebrique problem of the geometrical signification of correspondante devient imaginaire, qu'il est imaginaries prior to the presentation of that possible et permis d'adopter, dans tous les M[emoir to the Royal Society:- cas, cette expression pour la dlfinition PLCI(EIt, Dr. Julius, Professor at Bonn, rigoureuse et representation exacte de cette Foreign M\ember of the Rloyal Society. chose; et ainsi s'etablit une continuite Aliay2 tscih -geoesctrische Ent;zickehligen, indcfinie, tantot absolue et tanto't fictive 2 vols. 4to, Essen, 1828-1831 (Analy- entre tous les etats d'un mtme systemo tico-geometrical Developments), vol. i. geom6trique." [The cli/c/nt expression p. 61: ".... so bedeutet die Gleichung remaining perfectly continuous, there is durchaus aicts, oder wenn wir lieber no longer any need for such a metaphysical wollen, eiincts imaegiindren Lreis,".. basis for "imaginaries."] " nwenn vorhin von imacgiCiiren Ir'eisen die M. CIASLEas. Trait, de G-eometrie SupeRede war, so ist cliess qzr cine Spraece, die rieure, 8vo, Paris, 18o2. Preface, p. xi.: den analytischen Formen angepasst ist." " is iwagyinaires, eln Geonisetrie ptre, prd(.... in this case the equation means sentent de gcraves dWf/celtes: souvent l'on ne not/idy at all, or if we prefer it, an sait comment les definir ni les introduire imotaginzary circle,... when we spoke of dans le raisonnement; et, d'autre part, les i)maginary circles, it was only a -angynage 61ements d'une demonstration peuvent accommodated to the analytical forms.) disparaitre quand quelques parties d'une Vol. ii. p. 64-: "Setzen wir tan qp = v/(- 1,) figure deviennent imaginaires. Ces diffiso kommt tan (0q + 4) == 1/(-l)".... cultes n'existent pas en analyse, ou les "4Um das Paradoxe dieser Behauptung von I imaginaires se manifestent et se caracterider analytischen Seite fortzuraumen (voPn sent par les racines d'une 6quation du geeoanetrischer Beelee(tentrg ir(ia gar keine Redfe second degre', dont les coefficients souls, et seen)..." (If we put tan q = /(-l), non les racines elles-mcmes, entrent dans we have tan (p + l) = /(-l)...... les relations que l'on considere." [When To clear up the paradoxical character of the coefficients are themselves imaginary, this assertion from the analytical side, this is of no assistance. If we start from tizere can be no queestion at all of a g/co- geometrical lines considered in the CarzetricalJ meanin......) [The geometrical tesian manner, we obtain equations with meaning is directly derived from art. 34. scalar coefficients; the imaginaries occur in v., in conjunction with art. 39. iv. above.] pairs only, and their product is scalar. J. V. PoXCELET. Traiti des Proprietes But if we start with stigmatics, we obtain Projectives des Figures, 4to, Paris, 1822, equations with clicsant coefficients, which p. 28: "En general on pourrait designer may be of the form Sa +j ta, and their par 1'adjectif istgioaire tout objet qui imaginaries do not necessarily occur in d'absolu et reel qu'il etait dans une certaine pairs, and their product is not generally figure, serait devenu entierement impossible scalar.] " Nos theories donnent lieu aussi oni iicostr (cttible dans la fig-ure corrdlative. 'a certaines equations du second degre, qui.... Car, de menime qu'on a dleja en permettent d'introduire, naturellement et Geometrie des noms pour exprimer les dans un sons parfaitement dotermine, los divers modes d'existence qu'on veut com- imaginaires dans loes speculations geoparer.... il taut aussi en avoir pour metriques, parce que ces objets imaginaires, exprimer ceux de la non-existence." points, lignes ou quantites, n'entrent pas IDEM. Applications d'Analyse et de eux-mimes explicitement dans le raisonneGeometrie, 2 vols. 8vo, Paris, 1862-1864, ment," [the next citation will shew that vol. ii. p. 321, Sur le principe de con- this statement admits of exception, even in tinuit' (originally written in 1819): "La Chasles's own method; in point of fact, the neccessite de son admission [de la rgle -des "imaginaries" are a substantial and explicit signes] derive ainsi de la volonte d'etablir part of the complete theory,] "mais s'y la continutit eitre les diverses regions de trouvent reprisentes par des Jelmesnts ton THE IMAGINARY IN GEOMETRY. [APP. II. jours reels, qui peuvent servir a les determiner." [This is at most only correct for the very limited class of imaginaries which M3. Chasles considers.] Page 64: "La plupart des relations entre deux couples de points en rapport harmonique, quo nous avons ddmontrees dans l'hypothese de quatre points r6els, ne sont plus applicables dans le cas ou deux de ces points sont imaginaires, comme il peut arriver; c'est-a-dire que ces relations peuvent ne plus avoir de sens explicite; les segments qui y entrent sont en quelque sort desagrege's, et ne representent que des quan.titis ilmaginaires, lesqueelles ne soot rien par elles-nmze)es, consid6eres isolement...... Mais si 1'on admet que 1'on puisse faire sur les quantites imaginaires les memes opbrations d'addition, multiplication, etc., que sur les quantites reelles," [a renunciation of pure geometry,] "principe pratique en algebre, alors on deduira de chacune de ces equations une relation ou les deux points.... n'entreront que par leurs deux 61ements.... Alors les segments qui entrent dans ces relations... doivent etre consideres comme des synyboles," [a second renunciation of pure geometry,] " anz mnoyen desqnels on fait alllsion act cas oi6 les points seraient reels, et qui, combines entre eux... conduisent hc des relations o n'entrent que les elements des deux points. De sorte que la relation symbolique primitive n'est, na fond, qu'une expression de cette relation entre des elements toujours reels. II serae done persmis cl'en)ployer ces relations synmboliqees, ou, en d'autres termes, de raisonner sur des points imaginaires, comme on le ferait dans le cas analogue ou ces points seraient reels. On peut determiner la position de deux droites issues d'un point donne, par celles des deux points ou ces droites rencontrent une droite fixe. Et quand ces deux points seront imaginaires, on dira que les deux droites sont elles-memes imaginaires." Page 546: "Nous ferons en Geometrie pure ce que l'on fait en G6ometrie analytique: nous admettrons que, soit dans des formules, soit dans des constructions" [concerning circles] "qui n'impliquent que le carre durayon d'un cercle, ce carre devienne negatif; et nozus dirons que le cercle est ilcyaginaire. II ny a point, bien enteCndu, d cre eele imainair'e; et ce nmot 'est quzze fiction qui sert a rattacher les resultats obtenus a un autre cas de la question generale, dans lequel la presence d'un cercle procure une image visible et une notion parfaitement claire des proprietes de la figure." Chasles's imaginary circle is the vec-cyclal, described suprct, art. 49. v. (2). G. SALMON, D.D., F.R.S. A Treatise on Conic Sections, 4th edit., 8vo, London, 1863, p. 79: "When the distance of the line from the centre [of the circle] is greater than the radius, the line, geomzetrically considered, does not meet the circle; yet we have seen that analysis furnishes definite imaginary values for the co-ordinates of intersection. Instead, then, of saying that the line meets the circle in no points, we shall say that it meets it in two inmaginary points, just as we do not say that the corresponding quadratic has no roots, but that it has two imaginary roots. By an inmayincary point we meanc nzothinqg more than ao point, oze or both of whose co-ordinates are inlaginary." [The meaning of an imaginary co-ordiniate is not explained.] " It is a plr ely analytical conception, whlich we o zot attempt to reprsesent geoometrically-just as when we find imaginary values for the roots of an equation, we do not try to attach an arithmetical meaning to our result." [The imaginary points (or stigmata of stigmals, for which the abscissa is not on 01, and ordinate not parallel to OJ, that is, for which the coordinates are imaginary) in this case are given geometrically in art. 49. v. That the piece of " imagination" which supplied these "airy nothings" with a "name," although it could not give them a " local habitation," involved a pure impossibility, is shewn in art. 34. x. R. TowNSEND, M.A., F.R.S. Chapters on the Modern Geometry of the Point, Line, and Circle, 2 vols. 8vo, Dublin, 1863-65, vol. i., p. 16: "In the language of modern geometry every two points, lines, or other similar elements of, or connected with, any compound figure, which with change of relative position among the constituents of the figure pass or are liable to pass, as above described, from separation, through coincidezce, to siomultaeous disapcearance, or conversely, are termed cooetiogesnt, as distinguished from permanent elements of the figure, and are said to be real or iizayinary, according as they hcap)en to be apparent or non-arpp2arent to sense or conception. Geoemeters of course hcave aot, zor do they profess to have, Cany conceptiom of the natutre of contin/gent eleiments in their imagyinary state, blt they find it preferable, on the grounds both of convenience and accuracy, to regard acnd speak of themw as im.aygiCnary rcatcher thanc as 0non-existent in that state: in the transition from the real to the imaginary state, and conversely, contingent elements pass invariably through coinciden ce,through which, as above described, they always change state together." APP. II.] THE IMAGINARY IN GeOMETRY. Sir WILLIAM ROWAN HAMILTON. Ele- z vector, then the two stigmatics will be ments of Quaternions, 8vo, London, Cartesian, and have n Cartesian points of 1866 (written 1865), p. 90, note: "It is intersection, which will furnish the nvalues to be observed that no intepretation is of y; and will also have n (n- 1) non-scalar here proposed for inmaginary intersections of points of intersection, whence are derived this kin;d, such as those of a sphere with a Sir ~W. R. H.'s imaginaries; but if we right line, which is wholly external there- combine their abscissae and ordinates, to." LThese are perfectly similar to the they will only yield the same n values of cases in art. 34. x.] "The language of y as were given by the scalar points. modern geometry requires that such In this, as in all other cases, the imagiimaginary intersections should be spoken nary points arise from passing unof, and even that they should be cnume- consciously from a merely geometrical rated-exactly as the language of algebra curve, to a stigmatic in which each point requires that we should count what are of that curve is referred to a point in called the imaginary roots of an equation. another curve according to a definite law. But it would be an error to confound geo- The unconsciousn ess arises from the second metrical imaginaries, of this sort, with curve being always the wholly auxiliary those square roots of negatives for which axis 01, and the mode of reference being it will soon be seen that the calculus of the drawing of ordinates perpendicular to quaternions supplies, from the outset, a it, so that the second curve is overlooked. definite and real interpretation." Page If to these we add the investigations in 218: ".... where /- 1 is the old and Peacock's Algebra, first (1830) and second ordinary ihmaginary symbol of algebra, and (1842-45) editions, and in De Morgan's is not invested here witlh any sort of geo- Trigonomnetry and _Doable Alqebra (1849), mnetrical intepj)retation. We merely express introducing a conception equivalent to a thus the fact of calculation, that.... the eclinant, and the papers of M6bius (Berichte formula.... when treated by the rules of fiber die Verhandlungen der k. Sachsischen quaternions, conducts to the quadratic Gesellschaft derhWissenschaften zu Leipzig, equation.... which has no real root,- 16 Oct. 1852, 5 Feb. 1853, 14 Nov. 1853, the reason being that the right line.... and 21 Feb. 1857), in which he has sucis, in the present case, wholly external to ceeded in applying the last-named geomethe sphere, and therefore does not really trical explanation of imaginary expressions intersect it at all, although, for the sake of to the involution and homography of points generalization of language, we nmay agree to on a plane, but has not approached the say, as asoial, that the line intersects the subject of imaginary intersections, lines, spthere in two im;aginary points." Page angles, &c., we shall obtain a fair view of 277: "The equation.... in complanar the state of the problem prior to the conquaternions" [=clinants] "of the ath ception of Stigmatics, which was first introdegree, with real" [clinant] "coefficients, duced bynamein the writer's paper on "Cliwhile it admits of only n real quaternion" nant Geometry," ' Proceedings of the Royal [clinant] "roots, is symbolically satisfied Society,' 26 Feb. 1863, vol. xii. p. 442. also by n (on-1) imaginary quaternion The impossibility of any satisfactory roots." Page 278: "Imaginary roots of representation of imaginaries in Cartesian this sort are sometimes useful, or rather Geometry was looked upon by Auguste necessary, in calculations respecting ideal Comte as so indisputably settled, that he intersections, and ideal contacts, in geo- regarded it as a philosophic principle, and metry." [These imaginaries of Sir WV. thus referred toit in his latest volume,which R. H., belonging to solid geometry, are not gives a connected view of mathematics. It considered in these Tracts; but by sup- is necessary to remember that he was a posing the unknown in the clinant equa- professional mathematician, who thus detion of n dimensions to be y, then putting scribes himself in a philosophical work on y = x + z, and developing, we obtain an geometry, from which I have derived equation which, since x is arbitrary, may much assistance:-" Traite Elementaire be split into two in various ways, each pair de Geomntrie a deux et a trois dimensions, of equations generally furnishing az2 values contenant toutes les th6ories generales de of x, z, and hence determining n2 stinnals geometrie accessibles a l'analyse ordinaire. of the various pairs of stigmatics; but in Par M. Auguste Coenote, ancien 6elve de every case these 9n2 stinnals only furnish l'Ecole polytechnique, r6eptiteur d'analyse ns values of x + z, that is, of y, and these et de mecanique rationelle a cette Ecole, coincide with the direct solutions of the et examinateur des candidats qui s'y destiequation. If we take the two equations to nent, auteur du Systenm de Philosophie be such as would occur if x were scalar and Positive. Paris, AMars, 1843." 80 THE IMAGINARY IN GEOMETRY. [APP. II. AUGUSTE COMTE, Synthese Subjective negativevalues of the symbols,whichwould ou Systeme Universel des Conceptions exclude all parts not contained within the propres a l'6tat Normal de l'Humanite, angular wedge 10J. Indeed, the introductome premier, contenant le Systeme de tion of the negative case was as great an Logique Positive, ou Traite de Philoso- innovationinCartesian, as the firtherintrophie Math6matique. Paris, Novembre duction of theimaginarycase is in Stigratic 1856, pp. 345-7. Speaking of Cartesian Geometry.] " L'appreciation philosophique geometry, he says:-" Il fact jfiiaoleueat des meilleurs modes propres a combler regatrder 'oomission gyeomctrique des solutions cette lacune peut directement confirmer imaginaircs cooz;re 1plus Ztile que nuisible un tel jugement, en faisant sp6cialement a lae constitution de la philosophie Czathe- sentir que la plenitude d'interpretation aotiqzqe. A cet egard, on doit d'abord tendrait a porter la confusion dans les reconnaitre les consequences g6enrales de tableaux correspondants. Instituee aussi cette lacine spontaune, d'apres laquelle la simplement que possible, la repr6sentation peinture des equations resto plus ou moins geometrique des solutions imaginaires incomplete en un cas quelconque, et peut consisterait 'a les construire en ecartant souvent devenir insuffisante," [it aliways is le facteur constant qui les rend ordinaireso if only scalar values of x and y are mlent telles, sauf 'a marquer distinctement admitted]. "Nous devons ainsi concevoir les points ainsi produits. Nous pourrions des equations, t deux ou trois variables, alors combiner, envers les memes dimenoui la representation, sans produire aucune sions, l'hyperbole et l'ellipse, de maniere ligne ou surface, se trouvera bornee ia ' rendre chacune de ces lignes propre a quelques points isoles, qui ne pourront representer les solutions imaaginaires de jamais caracteriser leur source algebrique. l'autre 6quation." [This refers to PonceConvenablement formdces, les equations let's "supplementary" figures, already peuvent meme devenir entierement de- mentioned.] " Etendue a la cissoide, et pourvues d'interp6tation concrete, si toutes successivement a tons les cas suflisamment les solutions y soit imaginaires, du moins favorables, cette peinture instituerait des envers l'une des variables." [In stigmatic accouplements plus contraires que congeometry such cases can only occur by formes t l'ensemble des comparaisons limiting the symbols, as in using Sx or geometriques. Assimilee algebriquement VT only for x.]~ "Une telle lacune peut, l'ellipse, l'hyperbole s'en l6oigne geomereciproquemente, alterer linstitution al- triquemn nt par ses proprietes principales, g6brique des figures, en suscitant des qui doivent la faire finalement classer modifications abstraites qui n'auront pas parmi les courbes susceptibles d'equation d'equivalent gdomdtrique. 3Mieux ap- binome envers deux asymptotes." [In the preciee, cette influence consiste ta sur- stigmatic case, the two figures are indischarger ou priver les equations de facteurs tinguishable, and are treated as one under incapable de peinture, en taut que d6- the name of Central.] "Mieme dans les pourvus de solutions reelles. A ces cas les plus favorables, la peinture ces accidcets il faut toutefois attribuor une solutions imaginaires" [after Poncelet's action plus salutaire que nuisible, parce fashion, the only one with which Comlte qu'ils peuvent quelquefois expliquer la was acquainted] " pourrait done troubler diversit6 des equations rectilignes d'une la goeometrie gen6rale, en y suscitant des meme figure diversement formulee. Nous rapprochements vicieux. On doit d'ailleurs devons generalment reoarder l'omission reconnaitre que cette representation resgeomlntrique de solutions imaginaires terait ainsi bornee a certains modes de comme plus propre a periectionner qu'a l'equation propre a chaque figure, et troubler la subordination de l'abstrait au deviendrait confuse envers ses types les concret. II importe davantage de pouvoir plus etendus. Si, par exemple, on conainsi noter algebriquement la discontinuite sidere la conjugaison ci-dessus indiqude partielle des lignes et des surfaces que de entre l'ellipse et 1'hyperbole, on recoi - representer g6omitriquemlent des equa- nait qu'elle ne convient qu'i leurs plus tions on solutions essentiellements inutiles." simples 6quations, de maniere a ne pouvoir [Now first, till they are known, hown can nettement s'adapter ' lear situation quelthe solutions be decllied " essentially use- conque." [This objection does not apply less"? And, seconldly, whatever advantage to my theory.] "A plus forte raison, un is gained by not representing them remains tel mode serait habituellement inapplicable in stigmatic geomntry by simply limliting au delt du second degre, sauf envers des cas thevalue of the symbols, just as in Cartesian de plus en plus exceptionnels." [T'his is geometry wemight still furthermutilatthe the e reverse of correct as respects stigmarepresentation of equations by excluding all tics.] " Belativement aux precedes plus APP. III.] HISTORY OF STIGMATICS. 81 generaux qui furent directement destines equations quelconque au lieu de la subora peindre les solutions imaginaires, ils sent donner a sa destination principale, comme trop indirect et trop compliques pour 61iment necessaire de la constitution devenir jamais admissibles." [But stig- propre a la geom6trie ggenerale." [But matic geometry is more direct and less the chief ground on which I base the complicated.] "Tel est le jugement claims of stigmatic geometry to attention fnal qui convient a des speculations is that it is a simple geometrical idea d6pourvues de direction philosophique, oui carried out by a calculus based on the l'on oublie le but, essentiellement geome- simplest geometrical relations -those of trique de l'institution cartdsienne. Elles similar triangles. Hence Comte's "final manifestent une tendance absolue a judgment" is altogether premature and indevelopper isolement la peinture des applicable.] APPENDIX III. 0,z the History of Stigmatic Geometry. TIE indulgence of the reader is requested as a freshman Dean Peacock's Algebra for the following personal record of the 1830 (ed.), which first shewed me that various steps by which I have arrived at the " imaginary" and " impossible" of the general conceptions sketched above. other writers might become geometrically About fifty years ago my father taught visible and possible. But I owe most at me Euclid after Playfair, and Algebra this period to discussions with my late from the "Elements of Algebra, by I friend Duncan Farquharson Gregory (fifth Leonard Euler,translated from the French; wrangler in 1837) from whom I derived with the notes of MI. Bernouilli, &c., and the germ of the conception of operation, the additions of 1I. de la Grange, 3rd ed., and not quantity, as the real meaning of by the Rev. John Hewlett, B.D., F.A.S., algebraical expressions. About this time &c., to which is prefixed a Memoir of the also I became acquainted with the works Life and Character of Euler, by the late of Martin Ohm of Berlin, (Versutch eiaes Francis Horner, Esq., M.P.," London, I vollkomnein conseqzenten Systems der Jattlte1822, pp. xxx. 593. Although I fear I did miatik, "An Attempt at a perfectly connot profit properly by such a book, it was sistent System of Mlathematics," in nine something to have begun Algebra under volumes, vol 1 and 2, second edition 1833, theguidance of a mathematician like Euler. third edition 1853; vol. 9, 1852; -Der At present, his exposition of infinity, surds, Geist der meathezmatischeln A4nalysis, meld inhr negatives, imaginaries, geometrical ratio, Verhailtniss zirr Schule, "The Spirit of &c., appears very defective. " Since all Mathematical Analysis, and its relation to numbers which it is positive to conceive, a logical system," pp. 159, Berlin, 1842, are either greater or less than 0, or are translated and published by me in 1843; 0 itself, it is evident that we cannot rank 2)er Geist der Differential- umd Integralthe square root of a negative number l echingy, nebst einer nezen uind griidamong possible numbers, and we must licheren Theorie der bestimnmten Interale," therefore say it is an impossible quantity. " The Spirit of the Differential and InIn this manner we are led to the idea of tegral Calculus, with a new and more numbers which from their very nature are Fundamental Theory of Definite Inteimpossible; and therefore they are usually grals," Erlangen, 1846, which, together called iimatginary quantities, because they with many others of his nearly thirty exist merely in the imagination..... Of volumes, I also translated, though I did such numbers we may truly assert that they not publish them),-and these occupied my are neither nothing, nor greater than thoughts for many years after I had taken nothing, nor less than nothing, which my degree. Of course I mention no books necessarily constitutes them imaginary or of regular routine, to which belong Laimpossible."-Ibid. p. 43. EFrom this con- grange, and Lacroix, and Newton, &c., &c. fusion to the clinant conception the reach But as yet I had hit upon no scheme for was long.) About forty years ago I studied solving those difficulties of incommensur 82 HISTORY OF STIGMATICS. [APP. III. ables and imaginaries, which even Ohm seemed ratherto evade thanto meet. In 1846 my thoughts were turned in another direction, and dropping mathematics altogether, I worked so hard at practical phonology that in 1849 I was completely prostrated, and remained for some three years incapable of head work. As I was recovering, I again recurred to my mathematical researches, and especially amused myself (I was not fit for real study) with Augustus' De Morgan's Trigonometry and Double Algcebra, 1849. At last, at five o'clock in the morning of Palm Sunday, 20 March 1853, while residing at Redland, Bristol, I awoke suddenly with a conception of the application of algebra as a measure of quantity,-the germ of the algebra of proportion. This thought kept working in my brain for some months, till during a walk at Scarborough, on Saturday 13 August of the same year, I noted in my journal that I had " very satisfactorily arranged the whole subject in my own mind," but that as it had "helped to tire my head," I should "not even write it down." The next morning I awoke at six o'clock with the thought clear upon me, and I noted, "I have made a real discovery in mathematics, and have discovered the real theory of analytical geometry, shewing that Descartes's is only its simplest case." As it is not always possible to trace an original conception to the moment of germination, I trust that those who have had the patience to look through these pages, will not take it amiss, if I here continue the quotation from my journal, shewing the rough-hewn form of this conception; and, considering what an immense amount of labour was still required "to shape its ends," its very comprehensive nature. Regarding this statement as an historical document, I do not change a word, but I should mention that the Roman i represents J^(-1), equivalent to my present j. "I will just register it here, in order to secure the date, and the germ of the idea, in case I should work it out further. "Let z be a unit line, then if r, p be both real numbers, rePi represents a line of the length r z, and inclined to z at p radial angles, (27r radial angles = 4 right angles). This I have established long ago by a true process, but I have improved my method a little lately. "Iff (x) = 0, then the values of x are necessarily of the form repi, and hence xz will represent a straight line on a plane; or its final terminal points. All the values, therefore, represent a series of disjointed points. " Iff (x, y) =0, and any value x' of the form rePi be put for x, an expression for y, = y', of the same form reCP will result. If then we take any function of x', y' as F (x', y') the result will also be of the same form reti, and therefore F (x', y'). a will represent a set of lines, or their terminal points. This is the most general case of geometry of two dimensions. To reduce it to Descartes's case, F (:', y') = x' + y'. As x' is of the form repl, where r, p are independent, it can have an infinite number of values. Restrict them to those in which p =nr, therefore x' is of the form + r, or is possible. Putting these values, we get a series of values for y' which are possible or imaginary, and the result corresponds to a series of points as before. This is Descartes's case with imaginary values of y, perfectly explained. But to come to Descartes's case precisely, take F (x', y') = x' + y'. i, where x', y' are both of the form + r, then putting x for x' in the equation, f (x, y) =0, we must only take the possible values of y which correspond to it, by the restriction already named. Substituting these in x' + y'. i, we determine a series of points by drawing two lines at right angles to each other, and marking the point of intersection. This is the case of rectangular co-ordinates. Oblique co-ordinates are easily shewn by taking F (x', y') = x'. e + y'. e5i, where x', y' are possible. Now this is beautiful, and must be worked out well." The result of this determination was a paper which I read before Section A of the British Association at Glasgow. (On a more General Theory of Analytical Geometry, including the Cartesian as a particular case, -Rep2orts of Sections, 1855.) Through reading this paper I became personally acquainted with Sir William Rowan Hamilton, and heard of his Lectures on Quaternions (Dublin, 1853), and also of Michel Chasles's Trait de G eomletrie Superieure (Paris, 1852), which, with De Morgan's writings, henceforward became my constant study. In 1858 I became acquainted with Julius Pliicker's works (Analytisch - Geometrische Entwielehlncgen, " Developments of Analytical Geometry," Essen, Part 1, 1828; Part 2, 1831; System der analyzischen Geomnetrie autf naee Betrachtlngsys isesen gegriiedet, " System of Analytical Geometry founded on new methods," Berlin, 1835; 'Theorie der algebraischen Cureen, " Theory of Algebraical Curves," Bonn, 1839). To these APP. III.] HISTORY OF STIGMATICS. 83 works I had been led by a mere reference I exhibited and explained some of the results at the foot of a page in Salmon's Treatise of this theory to the British Association, on Ifigher Plane Curves (Dublin, 1852), by means of large diagrams, shewing the when I discovered how inadequately his geometrical meaning of "imaginaries," in very original views had been introduced involution, homography, &c. (On Plane to English readers, and how much they Stigmatics, Reports of Sections, 1866.) had suffberd in the transit. The imme- Since 1866 I have been deeply engaged diate consequences were three papers with my treatise on GEarly English lProwhich I read before the Royal Society (On n unciatzon, which has so severely taxed the Laws of Operation and Systematisa- my strength that I have several times tion of Mathematics, 20th May 1859; On broken down for months together, but Scalar and Clinant Algebraical Co-ordinate during part of 1871 I was able to revert to Geometry, 22nd May 1860; On an Appli- my Stigmatic Geometry, and lay its founcation of the Theory of Scalar and Clinant dations firmly by establishing the laws of Radical Loci, 14th March and 20th June its operations (Tensors and Clinants) upon 1861; abstracts are given, and the last is the Euclidean theories of Proportion and printed at full in the "Proceedings" for Similar Triangles, and to work out the those dates). In all of these the above complete geometrical conception of algebra given original conception was worked out here presented. in considerable detail. But, finding that From time to time, also, I revised my this conception was incomplete, as it failed stigmatic nomenclature. Professor H. J. to give a proper direct explanation of those Smith had pointed out to me at Nottingrelations between pairs of points on a ham that my terms " stigmatic point, stigstraight line, and pairs of rays having a matic line, stigmatic circle," for the precommon point of issue, which form the sent " stigmal, primal, cyclal," were subjects of Chasles's Geomntrie Satperieenre, misleading, as the latter were not proI applied my conception of clinants, ex- perly speaking points, lines, and circles at pressed in a geometrical form, imitating all. I had hoped that the prefixed qualiChasles's notation, to obtain an expla- fication "stigmatic" would have prevented nation of the "imaginary" cases which, any confusion, and I was unwilling to give as is shown in Appendix II., caused him up the old Cartesian confusion, arising from so much difficulty. The result of my first neglecting the index and regarding the work, which was necessarily incomplete, stignma only, for the particular directions and indeed contained some positively erro- of the abscissa and ordinate in the old neous views on imaginary tangents, was algebraical geometry. But slowly the another paper read before the Royal necessity of inventing a new terminology Society, (On Clinant Geometry, as a means became evident, and the difficulties of of expressing the General Relations of forming one which should be brief, unPoints in a Plane, realizing Imaginaries, ambiguous, euphonic, and suggestive of reconciling Ordinary Algebra with Plane the old Cartesian and Chaslesian usages, Geometry, and extending the Theories of were extremely great. It was some time Anharmonic Ratios, 26 Feb. 1863, ab- before I could reconcile my philological stracted in the "Proceedings,") in which prejudices to the necessities of the case; I gave geometrical constructions for the but finally I was led to the conclusion that imaginary intersections of real straight the science of the nineteenth century was lines with real circles, and introduced the not to be bound by the rude habits of comterm stigyiatic. But the first clear state- pounding words which grew up among ment of the new conception implied by the old Aryans, who in pre-historic times this word, was given in a short paper read originated the mother of our European before the British Association at Bath, (On tongues. The example of chloroform (terStigmatics, ljeports of Sections, 1864,) and chlor-ide o-f foers-yle), chloral (chlor-ine + this fructified into two very long papers, al -um) and many other chemical names, and almost books, presented by me to the Royal in mathematics Sir XV. R. Hamilton's Society, (Introductory Memoir on Plane cis 0 (c-osine of 0 + i-maginary ante s-ine Stigmatics, 6 April 1865; Second 3Me- of 0, see art. 26. viii.) led me to see that moir on Plane Stigmatics, 7 June 1866; modern science requires an adaptation of both abstracted in the "Proceedings," the the North American Indian incoiporative last at considerable length; but the actual system of speech. No scientific word can state of my conception at that time can be possibly convey its compound meaning by appreciated only by those who consult the the meaning of the simple constituents of original memoirs preserved in the archives its name (compare astronomny and astroof the Royal Society). At Nottingham I logy). This is especially true in mathe 84 HISTORY OF STIGMATICS. [APP. III. matics. It is there absolutely necessary that the conception should be explained at length, and then d(ocActed by a name. For memorial and historical purposes the name should have reference to some salient points in that conception, but it is suficient that this reference should be made by cbbrevivations of the names of those salient points, and these abbreviations may be either initial letters (as the usual G.c.M. and L.C.M. for g-reatest c-ommon i-easure and I-east c-onmmon m-ultiple), or distinctive syllables (as the usual tan, log, &c., for tae-gent, log-arithm, &c.) Again, for the usual language of science, words should be employed -which could be readily adapted to our inflectional system and converted into any European language. Hence they should have a Latin or (;reek basis. This is the origin of my present nomenclature, explained as it is introduced, of which the clharacteristic is the Latin termination -al (a remnant of gener-cl-isation), which has the advantage of foiring both substantives and adjectives, and admlitting of many additions. This -al (except in a very few w ords, as lateral, nor)a ol) becomes distinctive of the stiomatic theory, while the preceding letters in each word allude to the particular historical case which has been generalised. I hope, therefore, that my philological friends will hold me guiltless of murdering English, and consider that I have rather endeavoured to s-ystematise the application of our old-world Eastern speech to modern science, by taking a hint fiom the new-world WVestern tongues. This nomenclature is made public for the first time in these Tracts. Simultaneously with this improvement proceeded another, namely, the complete adoption of an algebraical form for my geometrical conception, consequent on my having established that all the laws of commutative algebra held for the geometry of proportion and similar triangles, so that, in fact, it was impossible to form any algebraical expression nwhich had not a geometrical meanling. The converse is not true. It is only that part of pIcano geometry which depends on the relations of similar triangles that can be brought under the laws of commutative algebra. Solid geometry requires additional conceptions, (as the directional addition of spherical arcs, which is n2ot commutative,) and hence requires another algebra (quzatcrnions), embracing that of plane geo metry (cliants) as a particular case, while relations of length and opposition of direction of a straight line (tensors and scealars) belong to both algebras. There may be also I think there are) parts of plane geometry depending on higher principles, which will develop now algebras, with new- " inmaginaries," not here considered. Hence I have termed the results of my long investigation, "Algebra identified with Geoimetry," and not " Geometry identified with Algebra," and in my title have especially shewn swZcat part of Geometry can be considered as identified with Alebra. Those who read these pages are particularly requested to note this distinction. On reviening those Appendices, and especially I. and II., it will be seen that difficulties in aloebra arose fromo its foundation on arithmetic, essentially discontinuous, and its application to geometry, essentially continuous. All these dificuties vanish as soon as a purely geometrical basis is given to algebra, as in imy theory of Clilnants, by shewing it to be, in every one of its phases, nothiing but the calculus of the operation of describing triangles directionally similar to the variable triangles determined by a fixed base and a vertex continuously moving over a fixed plane. But to apply this principle to geometry, it was necessary to shew that the theories of functions, and of both coordinate and homographic geometry, were all particular cases of the correspondence of two such movable points. T'his is the first aim of my Stigmatic Geoometry. When such a conception has once been hfilnly grasped, mathenlatical analysis can only be regarded as the expression of certain definite and simple geometrical operations which can be always performed. That there should be purely analytical forms in ordinary algebra, without any geometrical significance whatever,-a proposition magain and agmain insisted on, by the most eminent mathematicians, in the citations of Appendix II.,-becomes henceforth inconceivable. This is my own view of my own work. As the last dates in this history, let me note that the first page of these Tracts was written on Friday, 6 lMarch 1874; that an explanation of the principle involved was given before the hMathematical Society on Thursday, 9 April 1874; anld that this last page was finally corrected for press on Wednesday 29 April 1874. Pinited by C. F. Hodgson and Sons, Gough Square, Fleet Street. rsfim., to ^uac, j Eujg..__~. br, ~p/ -A\ lR avg~. a6,jx 24.$-h23U2i- w d~ ^/ I/,^ /. A^A -,9 " ^ T\^. \.y~ ntified Mwith Geojnetry, cr a, fi y Me Ad% a. --- -- ----C- - -- L - - —C - ---- — p~ ----. I --- I-blll —l —~^ CL---I11 --- U -^1 - C _ ~ __ I r..13 Tpl2..355. (y f A \ i - 4 —. q.?. o-, r, e 6. -x.:,Y. (10. arl-.24 ii-,r,3.w x. -A l i ) )c~6ar3^ I /^ ly) 1-,ff~ i<3) -V"~ A< /~; /, ^ / s -_, /, * ^p/,-;^p ^-y~~~~~~rvx7~ _y~,J 4-.-a / ^ ^V~, ^ ^~~~~~~~ -A-^M / ^ '; <"~~ ~ \6' ->iC\-~ f^>< $d f z ^^^: /:;^ ^.^x" A^."' / ^; J,,> i~r ty ^ _.^ s. ~ ** *\ 3B/; ' I OW I I ^r I I L. ---0 -<x-r,6.3, Ar. 724( -ae,36., "Ita. 16 Oll.- —, / IT — I 't A\ J~YJ!?v,f/ I \1 I I I I I IJ ""It IP` 381:~~~~J~~~~" a i3'i~~~~~~~~~~,~ - -- - - S~. 5. rr4~t~z~ I a )! j I"' I #A' IJC fitZ,, 2\atlo, A ~ ~~~~~ / -r:c i I 1.: /: vS I,;.I 4/ ": 'ls~ ^.'V J~L_________- ____ ____~~( Y.4% * 1 R ki JC I^ II VI -flqyff. 'v..4.i1x.i.iv, 4I. / /4 jiq"3. - p.V., ',, -.20 h h,0__ I L. __ ~ _ _ I _ __ __ I i WO1T7O-L/H. F,,'-RANCIS, TOOK'S COU^T, CHANCE /t> L-ANE. BY THE SAME AUTHOR. ON EARLY ENGLISH PRONUNCIATION, WITH ESPECIAL REFERENCE TO SHAKSPERE AND CHAUCER, containing an Investigation of the Correspondence of Writing with Speech in England, from the Anglosaxon Period to the Present Day, preceded by a Systematic Notation of All Spoken Sounds by means of the Ordinary Printing Types, including a Re-arrangement of PROF. F. J. CHILD'S Memoirs on the Language of CHAUCER and GOWEE, and Reprints of the Rare Tracts by SASLESBURY on English, 1547, and Welsh, 1567, and by BARCLEY on French, 1521. 8vo. PARtT I. An Investigation of the Pronunciation ot English during the xivth, xvIth, xvIIth, and xvIIIth Centuries. 1 Feb. 1869. pp. 416, price 10s. PARTI II. An Investigation of the Pronunciltion of English during the xIIIth and Previous Centuries, of Anglosaxon, Old Norse, Modern Icelandic, and Gothic, with Illustrations, and Alphabetic Arrangements of the Successive Values of Letters and Expression of Sounds. 1 Aug 1869. pp. 417-632, price 10s. PART III. Illustrations of the Pronunciation of English during the xIvth, xvIth, xvIIth, and xvIIth Centuries, and of the Pronunciation of Welsh and French, and English Pronunciation of Latin, in the xvIth Century. CHAUCER (specilens, with introduction and new critical text of Prologue), GOWER (specimen, with text printed for the first time from three MSS.), WYCLIFFE (specimen),SPENSER (specimens aspronounced by GILL, with an examination of SPENSER'S, SIDNEY'S, TENNYSON'S and MOORE's Rhymes), SHAKSPERE (specimen, with an examination of his Puns and Rhymes). Reprints of SALESBURY and BARCLEY. Specimens from HART (from the original MS. in the British Museum, written 1551, now first printed, and from the edition of 1569), BULLOKAR 1580, GILL 1621, BUTLER 1632, PRONOUNCING VOCABULARY of the xvIth Century, from contemporary sources. 13 Feb 1871. pp. 633-966, price 10s. PART IV. Illustrations of the Pronunciation of English during the xvIIth, xvIIIth and xixth Centuries. Pronouncing Vocabularies of the xviith and xvIIIth Centuries from contemporary sources. Specimens and Words from HODGES 1643, PRICE andWILKINS 1668, COOPER 1685. DRYDEN, (specimen; with analysis of Rhymes of DRYDEN and his contemporaries). Specimens and Words from LEDIARD 1725, FRANKLIN 1768, KENRICK 1773, BUCHANAN 1769, STEELE 1775, WEBSTER 1789. Eduzcated English Pronunciation, in the xIxth Century. Critical Examination of M. BELL, S. S. HALDEMAN, B. H. SMART. Proposed New Orthographies. Unstudied Pronunciation. American and Irish Pronunciation. Natural English Pronunciation. GILL on English Dialects, 1621. Preliminary Dissertations on Dialectal Vowel and Consonant Relations. PRINCE L. L. BONAPARTE'S Vowel Identifications in 45 European languages, his complete Consonant system, his dialectal specimens, his Classification of English Dialects in the xIxth Century (incorporating MR. C. C. RoBINSON'S Yorkshire, MR. T. HALLAM'S Derbyshire, and Miss JACKSON'S Shropshire) and Dr. JAS. A. H. MURRAY'S Classification of Scotch Dialects. Comparative Specimen of Dialectal Pronunciation in more than 80 versions 'from living authorities, all named. Comprarative Vocabularies of Dialectal Pronunciations. Incidentally,abstracts of Schmeller's Bavarian Dialectal Pronuciation and Winklers Low German Dialecticon reduced to one system of writing, as well as the ancient and modern Indian Pronunciation of Sanscrit, and explanations of the Ancient Indian and of various other phonologic systems, together with various modern peculiarities of Pronunciation in numerous languages. [It is hoped that this Part will be ready in November or December 1874. It.will contain between 500 and 600 pages, of which 350 are already in type.] PART V. Supplement, Retrospect, and Conclusion. Additional information respecting English in xIIth to xIvth Centuries. ROBERT OF BRUNNE'S Chronicle. Examination of investigations and criticisms which have appeared since the commencement of this work. Digested Review of the Results obtained, anldof the work remaining to be done. Indices. tlhis Part may appear in 1877 or 1878, if the w —itlr's life and health permit. Many collections have already been made for it.] OPINIONS OF THE PRESS. "It is not saying too much to characterize this as a work which will make an epoch in English Philology, for no one can henceforth write an English Grammar with any scientific pretensions until he has mastered the contents of this book." [Es ist gewiss nicht zuviel gesagt, wenn ich es fiir ein geradezu epochemachendes im Gebiete der englischen Sprachforschung erklare; denn es wird fortan keiner eile irgend auf Wissenschaftlichkeit Anspruch machende englische Grammatik schreiben konnen ohne sich des Inhalts dieses Buchs bemiachtigt zu haben].-Review of Part I, in the Allgemneine Zeitung, for 1 August, 1869, p. 3291. " It is saying little, to say that Mr. Ellis has surpassed all predecessors in the same field. We believe that he is the first who has really endeavoured to collect everything that can throw light on the history of English pronunciation, and to treat the whole subject with scientific precision and thoroughness. In the collection of his material he has used exemplary diligence, sparing no pains to make it complete and exhaustive; and in the discussion of it he has shewn a fairness of mind a freedom from prejudice, a simple love of truth, not less exemplary."-Review of Parts I. and II. in the North American Review for April, 1870, pp. 420-437. Reprinted in Prof. James Hadley's Essays Philological and Critical, New York, 1873, pp. 240-262. See also reviews of Parts I. and 1I. in Athenceum, 4 June 1870, p. 737; Academy, 1 June 1871, p, 294; and Saturday Review, 1 July 1871, p. 18, and 8 July 1871, p. 55, (two long notices). Of Part III. in Academy, 15 June 1871, p. 320, and Atheneeum, 23 Sept 1871, p. 392. Of Parts I. II. and III. in Pall Mall Gazette, 19 Aug 1871, p. 12, and in WVestminster Review, No. 80, Oct 1871, p. 565. Although incomplete, this work has become a book of reference for students of Chaucer, Shakspere and Early English generally in England, America, and Germany. Published for the Philological Society by ASHER & Co., London and Berlin; and for the Early English Text Society, and the Chaucer Society, by TRUEBNER & Co., Ludgate Hill, London, from whom single Parts can always be obtained at the prices above mentioned,