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CHAP.IV PBOPesBOB OF HixEBAioer AMD osoLoar in la^itbbsity college: {Late Professor in University College, London.) ek^ TORONTO : VRINTED BY LOVKLL AND GIBSON, YONGE STREET. 1860. f ll i yj ^ EXAMPLES •I THE APPLICATION OF TRIGtONOMETRY to CrgistaIl0grap|k CaMationis. Drawn up for the use of Students In the University of Toronto, By E. J. CHAPMAN, PBOIBSBOB OF HIKE ''.A.LOOY AND QEOLOOT IN VNIVBB81TT COLLEOB: (Late Professor in University College, London.) TORONTO : PRINTED BY LOVELL AND GIBSON, YONGE STREET. 1860. (> /- 'W'i^nfihlK :) I !"*i^ ^'' ' - ;■ ;^; ; <*,: iJ ' ADVERTISEMENT. The Application of Trigonometry to Crystallographic Calculations having been adopted by the University of Toronto, as one of the " Honor subjects " in the Department of Mineralogy and Geology, for students of the Fourth Year,* the accompanying Examples have been drawn up to convey a general idea of the principles involved in this application : more especially with a view to assist candidates who may not be able to attend the regular lectures given on the subject in University College. The Trimetric System has been selected, with regard to these Ex- amples, as the best adapted to exhibit the nature of Crystallographic Calculations in general. A knowledge of the common principle* of Crystallography, and of Plane and Spherical Trigonometry — so far, at least, as regards the working out of ordinary cases — is, of course, presupposed on the part of the student. As many persons, however, may find it desirable to take up this branch of inquiry, without being much versed in mathematical investigations, a few explanatory hints have been added here and there, and the calculations have been rendered as simple as the nature of the subject will admit. Abbreviated formulee have been thus avoided ; and the calculations are elucidated, moreover, by various original diagrams, designed expressly for this memoir. If these examples be thoroughly mastered, the solution of kindred questions in other systems of crystallization will occasion no difficulty. University College, Toronto, March, 1860. * The additional honor-subject in this Department, for University Students of the Fourth Tear, comprises the Geology of North America, with the Eock Formations, Fossils, and Economic Minerals of Canada considered in detail. The subject of Crystallography, it should be observed, has not only important bearings on Mineralogy proper, but on Chemistry also, oa well as on Optics, and other branches of Physical Science. ft> The writer's best thanks are due to Messrs. Lovbll and Oibsoit, for the care with which the symbols of the various crystal-forms described in the following pages, have been printed. The setting up of these symbols, with ordinary type, is a work involving mach ■kill and labour. EXAMPLES OV TH> Crgstallflgraplit gipplitatiotts OI TRIGONOMETRY. TRIMETRIC SYSTEM OF CRYSTALLIZATION. 1. In this System — the Rhombic System of Naumann> Prismatic System of Miller — the three axes are of unequal lenf^ths. They cross one another at right angles in the centre of the crystal. As shewn in figure 1, the vertical axis may be conveniently denoted by the symbol X ; the longer horizontal (the macrodiagonal, or right and left axis) by the symbol «; and the shorter horizontal (the brachydiagonal, or back and front axis) by the symbol x. 2. The forms of this system (see § 3,) comprise Basal, Polar, and Vertical Forms. The Basal form (B) consists of but two opposite planes, placed horizontally, one at each extremity of axis x. The Polar, or Pyramidal forms, are of three general kinds : — Front polars (= m p ) or " Macrodomes," between x and as , and consequently parallel to » ; Side polars {= m p) or " Brachydomes," between x and Xf And consequently parallel to x ; and intermediate polars, or polars properly so-called (Octahedrons, &c.) between x, j, and «. These latter are of two kinds, = m P, m P n : they cut all the axes. The vertical forms are also of three general kinds : J^ont verticals (== v) comprising merely two tangent planes,* one at each extremity of * In GryBtallography, a plane is Raid to he tangent to an axis, when the plane is placed rectangularly at the point of the axis referred to. In the Trimetric system, there are three pidrs of these tangent planes:— B, placed horizontally at the ends of axis «; and V and T placed vertically at the extremities of » and «, respectively. a2 ■ I 6 THE APPLICATION OF TRIGONOMETRY axis », and consequently parallel to x and x ; tide verticals (= V) also of two planes, tangents at the extremities of axis i, and consequently parallel to x and x ; and intermediate verticals, or verticals properly BO'Called, between x and xt and consequently parallel to x. These latter are denoted by the symbols V, and Vn. In all of the above symbols, m stands for the length of the vertical axis X, and n for the length of the shorter horizontal axis «, compared with that of the longer horizontal axis x as unity. 3. The crystals of this system may be reduced to four simple types : Riffht Rectangular Prisms, made up of the basal planes and front and side verticals (fig. 2) ; Right Rhombic Prisms, made up of the basal and vertical planes (fig. 8) ; Rectangular octahedrons (base a rect- angle), made up of the front and side polars (fig. 4) ; and Rhombic Octahedrons (base a rhomb), made up of the polar planes properly so- called, or those which cut the three axes (fig. 5). The relations of these crystals to one another, are shewn by the lettering on the figures.* * The correspondence of the notation employed in our figures, with the symboli adopted by If aunuuin and Dana, is shewn in the following table : Chapman: Naumann: Dana: B = OP O V - ooPoo iT f =s ooPcx> iT V= Poo 1 p= Poo T ? = ?oo r P = P 1 The accompanying Table may also prove useful to the student. The sign 00, the usual sign of infinity, when attached to the symbol of an axis, denotes the form to be paraUel to that axis, or to cut it at the distance infinity : B = 6o 00 = X, OCX, oox V =s 00 00 = CO X, 00 X, X V = 00 00 = oox, X, oox V = 00 = oox, X, X P = 6o = X, ooxj z P = 00 = X, X, 00 X P = • = X. I, X The use of the symbols B, V, Ac., has this advantage : it enables us to tmniform the signi into word* with great readiness— the signs and the names of the forms which they represent being more or less alike. Thus B=B, or Base; YstFront V, or Front Vertietai Y^Sidt F» or Side Vertical; and so on with regard to the rest. TO CRY8TALLOGRAPHIC CALCULATIONS. 4. The map-diagram, figure 6, exhibits the relative positions of all the forms belonging to this system. The observer is supposed to be looking down at the top of the crystal, parallel to the common plane of the two horizontal axes, » and r. The vertical or prismatic forms — those parallel to the vertical axis — lie on the outside of the ellipse. Owing to want of space, the signs , „ ,„ have been substituted, respectively, for ^ Vn, Vn and m Pn ; and for ^ P ^ P ■^, mV\. Figure 6a, on the next plate, is a vertical projection, shewing the same forms. A portion only of the projection is given, the other parts (as will be seen by an inspection of the horizontal projection, fig. 6,) being merely repetitions of this. 5. In calculating the axial ratios, &c., of crystals, by means of trigonometry, we have in every case to subdivide the crystal (or that portion of it to which the calculations refer,) into one or more plane or spherical triangles— -from the known angles and sides of which, the other angles and sides may be deduced. 6. Pigure 7 represents ti crystal of pyrolusite, containing the basal form, B, and two vertical prisms. The axial relations of these latter have to be determined. The inclination of the two outer faces (V) in front = 93° 40°. Half this equals the angle A in fig. 8.* As the axes cross at right angles, C, in this figure, = 90°. Axis x is considered equal to unity. Hence, to determine axis x (or the side £,) we have the formula : R : Qoi A '.: a : b :. Log h =(log cot 46° 60')- 10 = 1.9721882 = log 0.9380. The inclination of the two inner faces over a front edge = 129® 46'. Half this, as before, = angle A' in fig. 8. And, consequently, Log h =(log cot 64» 53')- 10 = 1.6709774 = log 0.4C88. In one of these forms, therefore, the shorter axis compared to the longer axis as unity, is just twice as great as in the other. As the outer form is by far the more common of the two, we assume it to be the protaxial prism, and give it the simple symbol V. The inner form then becomes V^. See the remarks at the close of § 7. 7. In figure 9, a crystal of sulphate of lead is represented, com- * In this figure the side a represents half m ; and the side b, half x. The former equals 1 or unity, and the latter has to be determined. The temi-axet employed in these oonstruo* tions, bear, of course, to each other, the same relations as the entire aies. 8 THE APPLICATION OF TRIGONOMETRY i I' prising the base, a vertical prism, two front polars or macrodomes, and a side polar or brachydome. V : V = 103° 38'. B : P = 127° 46'. B ; iP = 157° 33'. B : ip = 140° 27'. To determine axis 5 (J being unity,) we have — ' — = 51° 49'. Then- (See Fig. 8.) ' 2 (Log cot 61° 49') - 10 = 1.8966719 = log 0.78645 = log x. To determine axis Xt we assume the side polar P to be a protazial form : this side polar being of almost constant occurrence, and often predominating in the crystals. B : p = 127° 45'. This, less 90° = the angle A in the diagram, fig. 10. Consequently (axis x being unity) : R : cot 37° 45' : : 1 : a? ; whence : >. Log X = (log cot 37° 45') -^ 10 = O.U^/^2 = log 1.29$i. Turning now to the two front polars, we find the inclination of the base on the one adjacent to it = 157° 33'. Deducting 90° from this, we get the angle A' in fig. 11. Then, t.. obtain the vertical axis x, we have I he formula : R : cot A' :: X : X. Log X (as already found,) = 1.G956719 Log cot 67" 33' - - = 9.6161614 Log B 9.5118233 = 10 1.5118233 = log 0.3250. This value being just one fourth that of oc in the protaxial form, the symbol of this front polar, or macrodome, becomes ^ P. The inclination of the base on the lower form = 140° 27'. De- ducting 90° from this, and proceeding as before, we obtain : iog :? - - - = r. 8956719 Log cot. 60" 27' = 9.9168765 Log R 9.8125484 = 10 1,8125484 = log 0-6418 TO CRYSTALLOGRAPHIC CALCVLATION8. 9 68, 49'. I This value being half that of a? in the protaxial form, the symbol becoaies i P. Uote. — The assumption of this protaxial form or starting-point, is, of course more or less arbitrary. Aa a general rule, we select a form of common occurrence, or one that predominates in the combinations ; or otherwise, one to which the cleavage planes are parallel. Any one form, however, being chosen, the axial ratios of all the other forms belonging to the substance, will bear some simple relation to it. Thus, if the lower front polar were assumed to bear the symbol P, the upper front polar (in which x = .3250) would be ^ P ; and the side polar (with x = 1.299) would be 2 P. In like manner, if the upper front polar were taken as a starting point (= P, id est, IP), the lower front polar would be 2P ; and the side polar, 4l*. 8. Fig. 12 represents a crystal of sulphur : a combination of three rhombic octahedrons or polars, each face cutting the three axes. The measured inclinations are as follows : r Over front edge... 106*' 38'. P on P< Over side edge .. 84° .58'. 1 I Over middle edge 143" 17'. Over front edge 127'' Over side edge ... 118° 10'. Over middle edge 90° 16'. {Over front edge... Over side edge . . . Over middle edge To calculate the axial ratios of those forms, we construct the sphe- rical triangle, figure 13, in which A = half the inclination over a front edge ; B = half the iuclitiation over a aide edge ; and C — 90°, or the meeting of two sections taken through the axes. A simple inspection of the figure will render this evident.* We first determine the side a opposite tho angle A. Here (with A and H given, and a required), A becomes the middle part, and fl and a the extremes dis- junct, or opposites. Hence : R cos .4 = sin if cos a. And, consequently, li cos A 1 P 142° 4'. 132° 44'. 62° 9'. cos a = sin B * The leas-experienced atudont is advised to fashion a solid triangle of this Iiind out of a piece of soft wood or chalk, and to mark upon its sides the outlines of the spherical tri« angles as givau in the text : Figs. 13, 14, and 16. 10 THE APPLICATION OF TRIGONOMETRY <■.■. * !i Then to determine the side b (with A and B), B becomes the middle part, and A and h the extremes disjunct or opposite*. Hence : R cos B =: am A cos b. And, consequently, R cos B cos b = sin A. With the two sides a and b, thus obtained, the axes x and £ are readily calculated. As » = unity, a? = cot a, or log a? = log cot a)— log B. An inspection of the figure will show this. Finally, axis S =,(tan i) x x ; or log » = (log tan b + log «) — log R. This understood, let us proceed, by way of example, to calculate the axial ratios of the three octahedrons in our crystal of sulphur, fig. 12. 1. The Lower Form. In this form (lettered P), A (see figure 13) = 53° 19' ; and B = 4&° 29'. Then : (Log cos 53° 19') + 10 =19.7762593 Log sin 42° 29' - = 9.8295454 9.9467139 = log cos a = log cos 27°48'. (seconds being neglected. The log cot of this latter value (27048') = 10.2779915. Deduct- ing 10 (or log R) from this, and seeking for the corresponding number, we obtain, for axis x, the value 1.897. Secondly : (Log cos 42''29') + 10 = 19.8677466 Log8in53n9 = 9.9041470 9.9636996 = log cos b = log cos 23''8'. The log tan of this angle (23»8') = 9.6306556. Adding the logarithmic value of x to this, and deducting 10 (or log RJ from the sum, we obtain (9.6306556 + 0.2779915) - 10 = 1.9086571 = log. J = log. 0.8103. 2. The Middle Form. In this form, A becomes 63*30' ; and B, 56"36'. Calculating from TO CRYSTALLOGRAPHIC CALCULATIONS. U these yalues, the axial ratio, as before,"' we obtain for axis x, 0.6326 ; and for axis x, 0.8102. Whilst, therefore, the latter axis remains of the same length as in the Lower Form, the vertical axis x, is only a third of the length of x in the form referred tr. As the Lower Form is always present in crystals of Trimetii-. phur, whilst the other forms are only occasionally present, we sc. i!C it for our start- ing point or " protaxial form," and attach to it the simple symbol F. The symbol of the Middle Form then becomes ^P. In the former, the axes x, x, x are as 1.897:1:0.8103; in the latter, as 0.6326:1:0.8102. 3. The Upper Form. In this form (see the angles at the commencement of § 8), A = 71°2' ; and 5 = 66»22'. (Log cos 71»2')+ 10=19.5119074 Log Bin 66'22' = 9.9619569 9.5499505 = log cos a = (neglecting se- conds) log cos 69''13'. Log cot 69'»13'-10=I.5792479alog «?=log 0.3795. Secondly : (Log cos 66-22') + 10=19.6030166 Log sin 71°2' = 9.9757570 9.6272596 =log cos 6 = log cos 64»55'9". Log tan 64«65'9" = 10.3296891 Log X = 1.5792479 Log R 9.9089370 = 10 1,9089370 = log ^ = log 0.8108. The axes x, x, x, therefore, are to one another, as 0.3795 : 1 : 0.8108. Comparing these values with the axial ratios exhibited by the pro- taxial form P, the symbol of the present form becomes ^ P. * The formulie, for more easy reference, are recapitulated here. Inclination over front edge = 2 A; over side edge - 2B. J S Log 008 a = (log cos ^, + 10)~ log sin B ; < Log X s log cot a, — 10. 2 i Log 008 & = (log cos B, + 10)— log sin A ; ' Log xss (log tan b, + log a?)— 10. 12 THE APPLICATION OF TRiOONOMETRY ; I 9. The relations of the axes in these tri-polar form? or octahe- drons, may also be calculated from the measured inclinations oyer a front and middle edge, or over a side and middle edge, as shown below. The student should work out these formulsB (from the angles given at the commencement of § 8) for the three sulphur forms : 1. Given the inclination over a front edge = 2 A, and the incli- nation over a middle edge « 2 D : required a and x ; x being unity. To obtain » , (see Fig. 14) : Log COB a=[(log cos A) + 10] —log sin D ; Log * = (log cot a)— 10. To obtain x, (see Fig. 14) : Log cos «?= [(log cos D) + 10] —log sin A ; Log «'=[(log tan. rf)4-log »]— 10. 2. Given the inclination over a side edge s 2 J9, and the inclina- tion over a middle edge = 2 D ; required x and x } x being unity. To obtain 34'. Then to determine a : 2 Sin 90° : i :: sin 46® 34': a Whence log a = log sin 46<> 34'— 10=f.86l0412. Secondly, to determine the vertical axis x, we deduct 90^, as before, from the inclination of the Base on the form in question. This (see the measurements, as given above) = 135'* 4'. The angle A conse- quently (as in figure 19) =45" 4'. Then : JR : cot A::a:x. Whence : Log X = [(log cot ^) +log o]— log R. Log cot 45" 4' = 9.9989893 Log a (as found above) ... = 1 .8610412 Log. S 9.8600305 = 10. 1.8600305 = log 0.7245. This is to the length of t)ie vertical axis in the protaxial forms, as 1.6 : 1.0. Consequently, the symbol becomes | P2. The index 2, in this form, is known at once, by the form being in the vertical zone of the prism V2. 11. It frequently happens that, in a given crystal, only certain angles can be obtained by measurement : the other angles have then to be calculated. The following example will afford the student a general idea of the method of procedure in cases of this kind. Given, in the crystal represented by figure 21, the inclination of V on V in front=102' 22' ; and the inclination of P on V=126° .58' : required, the respective inclinations of P on P, over the front, side, and middle edges. I ;! 16 THE APPLICATION OF TRIGONOMKTRY 1). To obtain the inclination of P : P over a mtUfpolar edge. In the triangle constructed for this purpose, figure 22, C = 90* ; J yfi*»>A^ inclination of V : V. A, on the other hand, = half the required^ a^ ^,^-* D = the inclination of (P : V) — 90* angle. and a = half the front \ Then, to determine A from D and a ; A becomes the middle part, and D and a the extremes disjunct or appoaites. Whence : i2 cos ^ = sin D cos a. And, therefore, . sin D cos a. cos A = 5 Log sin 36°58' = 9.7791276 Log cos 5ini' =9.7971501 Log A 19.5762776 = 10 9.5762776 = Log. cos. 67''5n9". Twice this, or 135°42'3S" equals the required inclination. 2). To obtain the inclination of P : P over a side polar edge. In the constructed triangle, figure 23, C (as before) = 90°; D, again = the inclination of (P on V) —90° ; and b = half the inclination of V:V over a side vertical edge, or 180°— ^Q^lO.'. B equals half the required angle. "With B, b, and D, B becomes the middle part, and b and D the extremes disjunct or opposites. Whence : B cos B = sin D cos b ; and, consequently, sin D cos b cos B = R Log sin D = log sin 36°58'= 9.7791275 Log cos b = log cos 38°4';'= 9.8916242 Logi? 19.6707517 = 10. 9.6707517 = log cos 62°'324". Twice this, or 124°6'48" = the required inclina- tion over the side polar edge. TO CRYSTALL06RAPBIC CALCULATIONS. If 8). To obtain the inclination of P : P over a middle edge. Here no calculation is required. / n inspection of the three pre- ceding figures will show plainly thai the angle in question = D x 2 ; or, in other words, that the given angle (P : V) — 90° = one half the inclination of P : P over a middle edge. To render this still more clear, the vertical section, figure 24, is added. The required inclina- tion, consequently = 73°.56' 12. Finally, in bringing these examples to a close, we give, in the following formulae, the means of calculating the more important angles of Trimetric forms, from the given ratios of the axes. A (in figure 25) = half the angle over the front polar edge of an octahedron. B = half the angle over a side polar edge. D = half the angle over the middle edge ; D + 90° = »» P : V. a = half the inclinatiou of a front polar over the summit, a + 90° = w p : Base. h = half this inclinatiou over a middle e4ge. b + 90° = mv'.V. c and d =, respectively, the same angles in a side polar or brachy- dome. c + 90° = m v : Base ; «? + 90° = w P : v . e = half the front angle of a vertical prism, V or V w ; and / =: the half side-angle of the same, e + 90° = inclination of a V plane on V ; / + 90° = inclination of a V plane on V. Log tan a = (log H) + 10 — log a?. Log cot b = (log «) -+- 10 — log X. Log cot c = (log x) -f 10. Log tan d = (log x) + 10. Log cot e = (log ») 4- 10. Log tan/ = (log *) + 10. Log cot d = (log cot c + log sin a) — 10. Log cot B = (log cot a + log sin c) — 10. Log tan D = (log cot C+ log si"/) — 10. j&ajamp/e.— Required, in a crystal of Topaz, the inclination of the Base on the side polar 4 ¥. In the protaxial form, x = 0.4831 : see § 10, in which the axial ratios of Topaz are worked out. In 4 p, consequently, x must equal 1.9324. If THE APPLICATION OF TRIGONOMETRY To determine the inclination demanded, we require the angle c in figure 25. ■ ■ ' . x\ , Log cot y^^. ^ /'f^S EfJThapniai/ . nleJ ?'uUer.v.2f( -j:ke lith Vi/t-SjILTcvntK. \ 4'« i u.r> /;./ 6.' % 0. v-^-/2„:i->/' /iy// NffY <-" B \ \ p- .--■-'- y r /.^^ l-i V N --_ _ — ^'' AV- f^ Fig. /O. fn^ n. TH.J. r.' 'a}> v<^n ' Jpi' . '}lil,>j-i,:£e'.'i-JLi>J,irA^Vu-/MaicJb>v>'cn r^L£t^ 7///. /:i /Vy. /;. /;^. /'/ ' fJ.O . it^^^iar- dei- //'. ^l p r V r tig. UZ. n^. 2/*. EJC^az mwi . oijc- Tuiisj-ssBeKcJceJLizh. l^JctWojuJhrfyitn , f- ... 1 t EJ. riq u: //y 21 Fv^. ^fK 11^.29. EJCf:apTr.cc':. iM F:i.'-^,rir'B(nckf,.J,ith. l'.ct-7It)UZ ToronTy?,