OF THE MOST IMPORITANT GEOMETRICAL SURFACES. DRIAWINGS IN DESCtIPTIVE GEOIIETIIY.!r5aiing at tle saale timnc as a iCatalogue of 1lbelris aerutck acrcorbing to tlc aforesaib projectiolrs 13 Y a~~~~~~~;I H.sT~ ca iensc S 0 >B E D I T I OiN FOR A M E RI C A 9r.'hat llalstal, I H. O FE S S OR IN' THE IT N IVER S I O rF I HA L L E. N E W Y ORK, H.. ( 0 E B E L E t., 3i43 B R O -A D W A Y. BOSTON: JOSEPIH M. W\TGfITMAN.-PItLADELPHIA:'A. HART.-BALTIMORE: N. HICKMAN.-CINCINNATI: R. ROOT.-MONTREAL: J. ARMOUR. PRIN Y. TEUBE, ANN S, N55 I'PRIN'I7I, TED BY G. B. TEIYBNER:, IT ANN ST'., N. ~J. medaL To put the value, of that acknowledgment in a I should exceed the limit P" It E T", A, C E. proper light I remark, that Sir DAVIDBREWSTER Was the to dwell ever so little on i blenis which were to be s, chairman of the Jury, and Sir JOYINHERSCHEL among its THr,, collection of mo dels by Mr. ENGEL, and the drawings the models. A few more words aboul which be is about to publish as a catalogue of his collection, The models No. 3-1.2 represent the five principal classes -ire r ally so very important for the study of Supe'ior Geometry of surfaces of the second order with their circular sections intended to serve as ar and Optics that I most willingly yield to the desire Of riubt li-nes and lines of curvature. For the construction of lection of models, they are c and ask in a few words for these interesting'teachiD O' Superior and De Mr. ENGELI these lines of curvature Mr. ENiGEL7 at first, made use only publications, the attention of friends of mathematics and of are -not very well acquaints( of -the projections designed by MONGE. In doing this be Geometry, will understand iiatural philosophy. found-which I Malce a point of, in order to give a notion It was the model of the wave surface, (Nr. I aDd 2) the ENGEL'Sm6thod of Projects of the accuracy of his graphical constructions-tbat the V first preparation of which presented the main difficult. For Perspective. y rectilinear diagonals of any square formed by arcs of lines the purpose of giving some notion of the surface with its Some time since* Mr. EC of curvature are equal to each other. This is quite new, 7 two sheets and singular points, it was thought sufficient till optical drawings which w for aught I know, at least in this sbape; we may derive it -now to represent its principal sections by means of wires. Geometers, even in foreig, from the well known theorem of TVORY ifwe add toit the Mr. ENGEL-was tbe first to succeed in modelling in wood remark Of CHASLES, that a curved line, perpendicularly in- therefore, that every friend the solid included between the two sheets; this model.. accept with no little pleas tersectinp,, a s stera of confucal surfaces meets them in corresy 7 properly dissected-permitting an exact inspection of the same. author. ponding points. The pr'ofit which Mr. ENGEL has derived shape of these sheets. The Jury of the London exhibition, from this circumstance will be mentioned in these explanations. first section, class Xth, (physical, chemical, and other in- PROFESS, The models Nr. 13-20 represent cones, combinations of struments) oil account of that model-which i's a master- byperbolical paraboloids, &c.; Nr. 21-27 several helicoids piece in its way*-bestowed upon Mr. ENGEL the prizeand screws; Nr. 28-30 three rectilinear oblique planes (not belonging to the family of surfaces just me tioned) The first model made byMR. ENGEL is inthe possession of Professor PLdCKEn at Bonn, it is mentioned with great praise [but Nr. 31 and 32 are two developable surfaces; Nr. 35-37 without telliDgthe artist's name] in the: "Einleitung in die h6here 21 refer to the theory of spherical curves and. their polar curves, Optik von BEER. A second exemplar is in the physical cabinet of the University of Berlin, and so fortb. ;CA-T-XtOG-UE 1 ~~~~~~~~6. The same with -ts lines of curvature, P. $12. 27. Screw of-fv~roeP$:o. 7. Hyperboloid of one sheet, st ewing two circular sections 28. Recttilineai obli'qu'e surface..8$0!to 12. OF DRAWINGS OF MODELS FOR'.T}-I:E: STUD)Y OF OPTICSP and some rectilinear generatrices, P. $8. 29..Another su!faee-of the.same-kiil (elliptieal wedge), AND TIM HIGHER% BRANCoHES.' OF GEOMJETRY,2.Ante.~~~~~~~~~~~~~~h smP. $1. 8 The same with its lines of curvature, P. $12. VV, $9. P $5... with the prices of these models made of wood (W.) or of plastereonmpositk, n (P.). 1 9. Elliptical Paraboloid, shewvino, two circular sections, 30. A thilrd surface of the same'kinlc!, (semi-ciretular wedge), P. $4. $ i10. The same with its lines of curvature, P. 6.evelop "The tacitels are to be h'ad by forwardi~}g'the amount and 75 cts. for packing, to H1. GOEBELER, 343 Broadway, New. Yok...| 11. Ityperbolical Pa,'aboloid (oblique plane), P. $6. 32.' Another:developable surface;' W:,$1. P.$6." Besides these models the undersigned has constantly-for sale a large 12. The same withx its lines of curvature, P. $8. Serpentine' body, P. 86 to 8. number of models and diagrams, made by himself, inttended chiefly for in-. - re.uction in Descriptive as well as i, the higher branches of Analytical 13. Right Elliptica'l Cone, with two circular sections, on 34. Amnular body, P,.- - G.ome.' wooden Support, P. $8. 3i. Spherica! Curve,'with its pl'ar curve, P. 85 to 6.'';......:':.''~ 14. The same with its lines of curvature, P. $10. 36. sphere with four main- circles, 1>...;1 to 2. 1. Fre-snel's Wave' Sur'face of Ligh t in double refracting' 1I. Double Right Circular Cone, with 3 sections, P. $8. 37. Spherical Triang]e wlthits syinmetrical and polar trimalon a wpoden. stand. The 3 axes, a, b, are to 16. Oblique Circular Cone, with four'sections, W. $10. angole. W. $8 P. $4.. each otber as -/3' /2:* 1. a==0 min. Wood $60, 17. Combination of a Sphere and a Right Elliptical Cone. 38. Combination of five Cubes, a crystaline form found P~laster;1 0. intersecting each other in two circles W. $6e somneimes in p yrites, P. $4. This model shews the t-wo sheets of the surface and is de- "n composable by meansof four central sections into two parts 18. Body, enclosed between two squares and four oblique 39. Samne body, each of the five cubes diferently coloured, in four different -ways.. planes' P $5. P~. $6 to 8 2. Inner part of the, Wave Surface convexly represented, of 19. Body, enclosed between one square and four oblique 40. Four Measuring-scales'on we e d, intended for the axenoW. $ 8, P. 8:3. planes, W. $6. mnetrical method of projection, W $3. 2az. Fresnel's WAav~e Surfalce in which the rcatio of tle axes is... 20. Parallelopiped, divided by an oblique plane into two un- 41. Cube,W. A 3 0.30. 1,53' 1,32'I1. a —I170min. W. 8100. P. 820.rb 1715 3: 1) 2:. a 7om. W.Wo. P $20.equal parts, Zinc 88. P. $6.42The I eqtlal parts, Z~nc S. P. E>6,. 142. Th1ree axis inltersectinlg each other perpen~dicullar, W7. 2b. Inner part of the Wave Surface convexly represented, Four Se-'ew-surfaces -with their nuts shewing the recti- | 00.30 W. $20. P. $5. linear generatrices in different positions. F21 a more detailel description of the suofaces enumerated 3. Triaxial Ellipsoid, shewing two circular sections, P. $3. | in thi 22. Obique t1elieoid, P. 8 6 to 9. i n t i c a l o u s eth"Epntos. 3a. The same divided into two parts by a circular central section. W. $l0. P. $4. 23. General Hlelicoid, P. 86 to 9. 4. The same with the lines of curvature, P. 810. 24. Developable ttelicoid, P. $6 to 9. 5. Hlyperboloid of two sheets, shewing two circular sections, 2 5. The same with its lines'of curvature, P. $8 to 10. oll wooden support, P. $8. 26. Screw of four grooves, P. $6 to 8. AXONOiMETiRICA.L PROJECTIONS AXONOiMET!RISC'H[ PROJ[CTIONEN PI OJECTIONS AXONOMUTI I.UES OF TH E'MOST I MPORTANT GEOM ETRICAL SURIF'AC[$ DElR WICHTIGCSTEN CEO M[ETR ISCHiEN FLACH[EN DES SURFACES CEOM ETRIQ.U[ES L[S PLUS IMPOR]TANT[S SERVING IN THE SAME TIME AS A CATALOGUE OF MODELS ZUGLEICH ALS CATAL0G EINER M.ODELLSAM'MLUNG VON KtORPERN SERVANT EN MEME TEMPS DE CATALOGUE D'UNE COLLECTION CARRIED OUT ACCORDING TO THIE AFORESAID PROJECTIONS DIE NACH DEN VORGENANTEN PROJECTIONEN AUSGEFUJHRT WORDEN $1ND DE SOLIDES DEOCEOMETRIE CONSTRUITS SUR CES PROJECTIONS vv~~~~~ BY VON PAR FERDINAND EINCEL. FERDINAND ENGEL. FERDINAND ENCEL.. I~~~~~~~~~ZV 1Y J-f X A PLA'SE M IT I FICU RETAFELN.Aty Et 2 W1X 1 S E@.$A '~~~~~~~~~~~~~~~~~~~~~~~- -R.-.-.......' j:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~".;,:-._~ ii, i~~~~~~~~~~~~~~~~~~~~~~~~~. x:,., —-— __:__~~~~~~~~~~~~~~~~~~~~~~~~UC —, —----— ~......." —---.........."/ <~~ ~.I i.':> ~~~~~~~~~~~~~~~~~~~~~.,.:i~~~~~~~~~~~~~~~~~~~~~~\ ~::~~~~~~~~-_~' —-':':-:> --— %~~~.....?i i ~ I"~, ~'~i'~ ~~~~~~~~~~~~~~~~~~;,, ~~~~~~~~~~~~~~~~~~-:;1..._~.. _~ ~_~..~:~'':,~~~~~~~~~~~~~~ —-i —-- W-:~~~~~~~~.. ~~~I>,... _ I~~'~ ~~~~~~ / i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i'~~Ai t:'~t I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,~ ~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~/ ~~~~~~~~~~~~~~~~~~~~, ~:. o: I, _~... i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i / I / ~.".~.. i i:.~- ~ -',.' ~...x/'/ / 5: ~ ~~ ~~~~ ~ J .?..' i!''.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.;i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \ I~~~~~~~~~~ i~~~~~~~~~~~~~~~~,i" ~:;,i~~~~~~~~~~~~~~~~~~ ~::" i,' I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" I:'~i ~\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _i~:~ !! ~1 i', 1.....:4~~~~~~~~~~~~~~~I;.i — ~~~~~~~ —:: —---.............. i,, ~ ~~~~-____.., I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" I' I i' \ 1" —- i"i!... ~~~~~~~~~~~i ~ ~ ~ ~ ~ ~ ~ ~ ~ l.' i:.... ~,i..............._\........ ii:~~~~~~~~~~~~~~~~~~~~~~~~~~~~...:I_.:t.~ — /;,.........i-.....:.......:....::-...:.~...-....:_.......i:./ —:::-:".... i\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_...:__... 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II, ii:~~~~~~~~~~~~~~~L ''~..i'!i i I t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I/'' I'I ti~ I /'"~~'~i ~~~~~ ~r! i'?.. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~!.' i'.~~~~~~~~~\.-;.. i',.'. _ ~~~~~ —— _~~~~~~~~~~~~~~.i.,i~. —— ~-c......... ii II~~~~~~~~~~~~~~~~~~~~~~~~~~~ /i~'""~ j ---- i i// ~~~~~~~~~~~~~~/1.-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —'._.-"'-i''~ ~~~~~~~~~~~~~~~~~J~.'T,.~ —~~: ~~~~~~~~~~~.~',,,/ i —-...........~.......;i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~?,........................ ~~-\1 —--- ~ ~ ~. ~ —'.I..i: _~~~~~~~~~~~~~~~~~ ~ _ \I' i: I-\i i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- ~~~~I - ii~~~~~~~~~........:-~i ~ l',.:?: ~ ~: i,, r'1 ~~~:'':-" "ii ~~~~~~~~~~~~~~~~ii., ii:: i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i.~.........~ —.... ~Ii i,......i~~~~~~~~~~~~~~~~~~~~~~~~~~~~;'h~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:,,: ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~` —-i i i ~~~~~~~~~~~~~~~~~~ i~~;. ]_~~~~~~~~~~~~~~~~~~~~~~~~~~~~.::~i-i, )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'. ~:i i` -1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "~~~ i -- — ~~~~~~~~~ —--— ~~J: — ~ " ___...._.....-_ _ _~_~ I~::___:...........'. __~................................."_:._ l~. ~._~ ~:~ ~_~:~ ~/....-' X / J. ~'~~~~~~~,',,,' ~ ~' ~ "~'..,,'- / /' / i ~ i — ~ ~ ~.'i /I`,/ " /~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~~~i f --- J.., —,, \l i /,. ~~.. ~: ~.. -- : h ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! 1 r-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ J:?., I i — j - /,/'~ l! _ _ _ _ _ _ _ _: _ _ _ = _ _ _,'x~~~~~~~~~~~~~~~~~~~~ i~~~~~~~~~~~~~~~~~~~'.~ ~, i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:.-~~..~.',.....' 7-.'.........'................-...........`~~~ —1~~~ —-- _ -—,..;..~r —.......... ~~_.:~~ ~.....~~~~~~~~~~ /. ~ ~~~~~~~~~~~....-' ~~~~~~~~~~~~~~~~~~~~~~~~-~: __-'-... ~ —-~...............................................................~.... i~~~~~~~~~~~~~~~~~~~~~~~~~~_~,' ~. ~~~~cc ~ ~.~~. T~~t~,.... -~ ~ ~ ~ ~ ~ ~~~_.~ - __ -!... 1\~~~~~~~~~~~~~~~~~~~' ": — - - _::........'~~- -------... B'-..' I'",........-...... —'$ G~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-,,., ~i i?~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:.-~ —.'':': "''"~'::-' ->:"'"'~"-:'"''.-" \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' i.''''',/ -.' — i.'.~-~, -_i t-l-~i~ ~ ~ ~ ~ ~ ~ ~~~t',,..,. —,....,..., C:_i- /.,' I~ I.'-.. T~~~~~~' /L` t-j`-X~~~~~~~~~ 1 1X-;~~~~~~~`1:/-: /I ~l....~ —'i /i i,>"''x''.......,"' ~ III i i I/:////'1 ".,,,, \>..'..x x'% IIjI! t ~.'///...!................: -,' —-I~i....?r ~'.........................? -- - ~..................::-: ~-~: ~ i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ii: ii/i Ii i Ir --— c-~~~~~. Fig. 9, PI. II represents the wave-surface, the upper part Fig. 8 represents an ellipsoid divided into two halves by on the right hand and the lower one at the left hand are a circular section. withdrawn, to shew better the interior of the surface. The For the better understanding of the following we add first principal section consists of a circle A n Alnq which in- these remarkls: The name: "Line of curvature of a Sulface" closes the ellipse B s Bjs. The second principal section con- is given to a curved line along wihich the nolrmals ofthe sists of an ellipse A r Alrl which incloses the circle s COsle l. surfaice form a developable suiface. Every surface has two The third principal section consists of an ellipse n CnC,, systems of lines of culvature which intersect each other at E X P L AN A T I 0 NgSi, and of a circle B r Blrl, the diameter of which is equal to right angles. Among all the plane sections drawn through thle mean axis of the ellipsoid. These two curves meet in a normal to the surface, those which contain the tangents to four points, two of which are denoted by P and P 1 Wh7ile the lines of curvature, have the greatest or the least curTowvarde the end of these explanations, some details;vllLbeentered. n re *nto, reti n e. a r pe.s l _t-1 tangents dlrawTn thirough a point of a slulface are in vatulre, giving for the concavo-convex surfaces, a different into, respecting the particular proi- a-triea jection of some German autholrs —vhich has been made use of for gene ated in one single plane the tangrent-plane, they sign to the curvature of the sections, which ara in a different the drawings. form for tles oints P a cone of the second order. The dis- position to the tangent plane. The points of the sulfate, coveryfthlese singular points of the surface is due to Sir where all the normal sections have the same curvature are WILLIAM HAMILTON of Dublin. The straight line PP,, and called: its umnbilici. For the theory of the lines of curva1. FIreslel'e(s wavesp surface in biaxial crystals. a second straight line symmetrically situated with respect ture we are indebted to MONGE. (Plate II, fig. 9.) | to the former are called secondary optical axes.-Two sections hate been drawn through these axes and through the axis 4 Ellipsoi with is ines o crvare Draw throug1"f'tle centre 0 of a triaxial ellipsoid any ~ li~odwtlit iiso u~tl~ s S, 5 which are indicated in the figure. The points T and plane whatever, Z, which will intersect the surlftce in an el-. plane, whatever, E- wi'h will intersect the surface in an el- Q are the points of contact of one of the four tangents com- (Fig. 14, P1. III.) ipse; erect in 0 a perpendicular to E, take on elther side of, r r d t ei mon to the circle and the ellipse, the point Tbelongs to the O the lengths of the semi-axes of the ellipse; the resultin gr|; Figure 14 represents the two systems of the lines of curvao s aelliecircle. The plane drawn through T Q perpendicular to the four points will be four points of the wave-surface. Giving llture of the ellipsoid; the curve defg, for instance, belongs fou~ pitwilbfupotsothwe Ging principal section, touches the surface along a circle of which all possible directions to the transversal plane, you obtain to the first system, the curve our to the other. The prinT Q is a diameter. The figure shows these four circles of the complete surface. It consists of two sheets, which cor- a ctheipal sections AB AtBh and oB CBaCe, drawn tshrough the co ntact, the straight line a n is one of the two optical axes. respond to the najor and minor axes of the intersection- mean axis, belong to different systems As for tile third ellipses; and since among these ellipses there are only two 2. Figure 11 represents the Boy ionlsed by the sectionA CA, C, two ofitsarcs, n2'/ andnllnl,arethelimitwith two equal axes-that is to say: which are circles the interior second system. The four points z/n, 1/n two sheets will only have four points in, common. are the umbilici of the ellipsoid. The tangent-planes of the The wave sulface has three principal sections, each of We ill add, that PB, PB1 are arcs of aciile,and that surface in these points are paraleto the two systes of The~~~~~~~~~~ ~ twave suyaehs theeprniplsecin ec of which consists of a circle and ar, ellipse. Suppose, to prove PC, PC]. are elliptical arcs. circular sections; the circular section kB k B has been it, a system El, of planes drawn through an axis A of the traced..3. riaxial Ellipsoid with two circular sectionsT ellipsoid; for all the ellipses, which will result, A will be the The lines of curvature of an ellipsoid call be constructed principal axis, whereas the second axis will be a diameter of (Fig. 7 and S, P1. II.) by the help of these four points, for-accolding to a most the ellipsoid, perpendicular to A. Construct the points of Fig. 7 represents the triaxial ellipsoid which has been remarkable theorem due to,;:hl. ICHAEL nROs TS of Dubthe wave, which belong to all the planes El,-and you will made use of in the construction of the wave-surface: the axes lin —they bear with respect to the lines of curvature, a part see, that they form a circle, the diameter of which is equal are to each other in the proportion /': /2: 1. Two dia- analogous to that of the foci with respect to an elipse; i. e.: to the axis A, and an ellipse, the principal axes of which meters, KK, and kq, of the ellipse A CAlC, are eq~al to the if we fix the extremities of a thread totwoumbilici which are equal to the two other axes of the ellipsoid. Tile circle mean axis BB], it follows, that the sections containing th are not diametrically opposed, and stretch it by means of will be exterior to the ellipse, if A be the greatest axis, it mean axis and one of these two diameters, are circles; the a style, thus bringing it close to the surface, the style, glidwll be interior to the ellipse, ifa be the least axis; and if only ones the planes of which pass through the centre of ing over the surface, willdescribeoneofitslinesofcurvature; A be the mean axis, the circle will intersect the ellipse at thesurface. It is known, that every section parallel to one but this is not the way by which the author determined four points. | of these two central sections is a circle. them. He has found that in every quadrilateral formed by two couples of lines of curvature, the Irect~ilinear diagonals are asyrmptotfs to the two hypelrbolas. An ellipsis described systems, tbita h igtlnso n sse nesc eqyual to each other. With the aid of this pr~oposition, which o 01 a al, b b, as axes, detelrmines with tbe- vertex rC a cone all the right lnso h te ytm hra h ih ie subsists not only for thle surfaces of the second order but which is asymptote. t6 the sutrface. The points nn, are the of one singl ste n o it cmn T aalso for the developable belicoid lize began. by determining umbili i a section drawn through no -npol 1 q-parallel to gnst h ie fcrauedvd h nlsbtwe the vertices of a selries of lines of. curvature. One may con- the tangent plane in n, is a cii-cle; both these circular see- t:e generatin ih ne t qu prs sider, indeed, the figure Cn' ef as a quadrilateral formed by tions have been traced in the figure. The diimensions of the Sem i-axelcftehp'onfla 4 eidsac two couples of lines of curvature; although two contigunous bvTperbola -A Z A, are: S Z=- 18.2, distance of the foci=- of the two foi e-as ~h lptc mnusides Cn1 and nle are in one single plane. I'L follows: that 47.4, those of the ellipsis A f i 1.3, B M == 40,7. (Unit section = an = 2(ni=1milmee. Ce ~fnl (these distances being measured in a straight I mrillimeter.) line,); therefore, after having arbitrarily determined a ver- 9o 9 Elliptitk a~~blo i aijBa4 I$BaIs~ios tex f on the principal section B CB, C,, the other vertex e 6, The same surfale with its linaes of curvatua$Pres wvill be found by the rehateion just indicated. The four vertices f e dg detelrmine the ellipse —projection of the curve Fi.1,P I.)The g~ene-,to ftes2~ae b oaleelpei r def.9 on the section -AB,AB —and the alre of the hyper- r yp fbo id oa set m Iw cfua logousS to tla ftehprood u h ircigcre bola-proj~ectio~n of the same curve on B CB, C1. After hav- scinoftshere ar~e two plaoa aigtesm etxadtesm ing consrce these t opoections we sball be able to axis; AB-A1i n oiio fteelpe deter~mine on the ellipsoid as many points of the lines of (Fig. 25, PI. VI.) The two pit n,aeuniii vr eto s curvature as we please. For the curves of the second system. rsraei nfnt n otnou,8m oeo allel to tbepne agnti n L i ice-( we make use of the curves already drawn of the first system, see in the Ii~le tecrlsec o hc asstrnl and f te prncial sctin CB01B whih blong tothe generating as for the byperboloid of two sheets, i. e.: by a a miiu and of the princial section GB CB which bmova n y, te at it is tue secod sytem forta~ig abitrtril an poit whtevr of movble ellipse, with this differe~nce nl ta i h secod sste; fo taingarbtrarly ny ointwhaeve of imaginary axis which is; common to the dfirecting byper~bothe principal section A, CA C7, as the vertex of a, curve of laI hthplbl, i hc hrn xsi iemnr10' 9 Sa & it t iuso o~aoe the second system, it is obuvious that one can deterinine, by ls nta yebl nwihtera xsi h io one, there exist two dliameters equal to the real axis AA, (i 4 1. the aid of our theorem, all its intersect ons with the curves ofteohr tRlos 7atyucndn hog hs of the first svstem. It is to avoid repetition, that we have The paraiee fteprbl, A sad oteae diarnet~~~~ers and through AA1 two circular sections A o Ann of25:6:43 entered on some details on this subject. W~e remark that of the ellips(hAA wocl~ulr seton A ~ln and A~ o A 721. Tbrough every p~oint of the surface you can in the hyperboloid of 6~ne sheet, and in the byperbolical pa- dlraw two right lines which lie altogether in the surfitce, geblc pabbod iBqa ln, raboloid, the existence of the genelrating~ straightt lines allows asfBg n B. vr opeo ili ie iie h of simplifications which the reader will find without diffl- surface into four reo-ions, so that a plain section drawn ~g 5,P.T. cultS5: ~ ~~~~~~~~~~~through the, point of the surface in question is a 2lyperbola or an ellipse, according to its situation in one or tile other Suppose Iw crbls%~Z, adS iutdi 51 Hfyperb~oloid off rtwo sheetso couple of these regions. Tile plane which contains all the two planes Iepniua oec thrac nsc ~oi tangents drawn through a point of the sur~face, passes also tion trttervlie oniead that th( xso n (Fig. 12, PI-III.) throug-l- those two Iright lines which cross each other there, of tbein is h rlnai o e x f e tl~- I This surface consists of two infinite and equal sheets, and cuts the surface as well as it touches it. one cf thes tv ues S o ntac-me i separated from one anotlier. Suppose two hyperbolas bav- "sc anr ~a t in ean aallt t is inggthe same vertices and the planes of which alre per~pendi- So Same surfacert witha its linesa of curvature. position, an( htisvre osaogteohrcre cular to each other; an ellipse with variable axes, situated it will describeaIye~oia n~blil faltesr (Fig. 26, PI. 111. )v etexlmnd hi s heolyon hepln in a plane perpendicular to those of the two hyperbolas and fcsw a the vertices of which are situated in these curves, gener~ates The model and the diag-ram. present besides the lines of sections of mvih aenvrcoe crei,eiss the surface in question; —A B A B, is ou e of the positions cur~vature a i reat -number of couples of gener~ating righlt lines. Likre the Iyebli foes~e thste ryryo of the variable ellipse. The right lines a S a,, b S b are the It will easily be remnarkzed Lhat these rigrht lines form two having ever, on asdtruhb w ih ie yn altogether in the surface; the right lines of each of the two hyperbola, and the section parallel to the side s W which is planes, the generatrices of whic h remain constantly parallel systems (see No. 8) are parallel to one plane. Thence fol- not marked with letters is a parabola. to the basis. Every ho lows a second way of generation for the paraboloid: It is with four equal sides. generated by a right line which, always remaining parallel 17, Combination of a sphere and a right elliptical to a plane, rests upon two oblique right lines; a property cone. 20, Parallelepiped divided by an oblique plane into which gave to this paraboloid the name of "Oblique plane". tw (Fig. 3, Pl. I.) t 12. Same surface with its lines of curvature. We have given one horizontal projection and two verti- (Fig. 19, Pl. IV.) c al. ones. The oblique plane passes frn the edge ad to the did(Fig. 16, PI. IV.) 6 The ellipse. AB CD is the base of the cone. Round the onal A C of the oppos The tangents of the lines of curvatur e are the bisectors of | point m" in the axis of the cone a sphere has been descr ibed T ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~The tnament of "He l inx"es gie of curvaue staretedbsctr ofTenam ofa l the angles between the right lines of the surface. which touches the two sides drawn from the vertex of the circular-based right cyli The proportion of the parameters of the two parabolas is 1 surface to the extremities of the basis. Their projection on Th~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ae proportiona of the paramtes of the toprboase ihse are prpetoalsurted 25., ~ 10. |the first vertical plane is the ellipse e" i" d ", on the second 2-,6:10. ~~~~~~~~~~~~~~~~~~~~~~~~~~~from the point, where theHlxmestebai.Tehgt vertical plane drawn through the axis CD theiT projections 13. Right elliptical cone with two circular sections, | are two right lines, whereas their horizontal projections are p of the helix He Right elliptical cone with two circular sections two ellipses. path of the helix. two ellipses. (Fig. 4, PI. I) We find by a similar method the circular sections in the 21 Right Helicoid, All the circular sections as hfi# and ahdie are perpen- other surfaces of the second order. (Fig 0 l I. dicular to the plane drawn through the vertex and the minor axis B Bi of the basis. I-So Body inclosed between two squares and four The surface is gener oblique planes. which, remaining parallel to the circular base of a right cy14. Same surface with its lines of curvature, (Fig. 20, P1.lV.) |inder, passes through i s axis and is supported by a helix which is traced on the surface of the cylinder. The curve (Fig. 5, PI. I.) In a right prism with-e square base the diagonals of the a 1, 2 3 4 5 6 7 b is the curv fcnato yidr icm The sides of the surface are the first system of lines.of cur- lateral faces have been di-awn. A right line A/B which- scribed to the helicoid, i vatufo, the curves of the second system are the intersections always remaining parallel to the base-moves along two of C5is half the path of tile firtoe-Iisradonaecd of the cone with concentric spheres described round the these diagonals Aa, Bb will describe a part of a hyperbolical cylinder which contains vertex S of the cone as their center. paraboloid, i. e. of an oblique plan. horizontal section of theciumciecyndrsaccld The same construction is to be used for the three other the generating eh'ele of which is equal to the base of the 15. Double right circular cone with three sections, faces. Every section parallel to the base is a square; the. gh ~ ~~~~~~~~~smallest section is that which passes through the center of scn yidr (Fig.39, PI IX) The surface being infinite, the diagram represents only the body. that part of it which is comprised between two planes perThe curve A~D A D is an ellipse, u G t is a parabola, endlar to the xis an A e hl H EH, a hyperbola. 1 9. Body inclosed between one square and four | first one. Same observainfrteolwngsfes oblique planes, 16. Oblique circular-based cone with four sections. 22. ob (Fig. 18, PI. IV.) (Fig. 40, PI. IX.) (Fig81P.VI) The right line EF is a parallel to the diagonal A C of The section A BA B is an ellipse, the angle A4 S A is the square A B CD, the right lines E P, F C are perpen- It differs from the preceding surface inasmuch, as the the greatest of all the angles formed by two sides of the dicular to the plane of the square. G, the middle of EF', movable right line forms with the axis a constant angle cone: gd~/~d, is the circumference of a circle the plane of has been joined to the points D and B by two right lines which is not a right one. The curve c'd'a'g'v' is the inwhich is not parallel to the base (see No. 13); t/ Bh is a which form with EA and F C the directrices of four oblique tersection of the surface with the superior horizontal plane. 23. General Helicoid. consecutive turns of the thread. The surface of these grooves 30. Oblique plane; semicircular wedge. of the screw is shaped in the diagcram, No. 206 according to (Fig. 17, P1. IV.) (Fig. 32, PI. VII.) the general helicoid. It is described by a right line of constant inclination The base is a semicircle; otherwise the same generation towards the basis of the cylinder and of constant distance 27. Screw of five grooves with its nut. as for No. 29. from the axis, while one of its points, the extremity of the I shortest distance, describes a helix. The horizontal projections of all the generating right lines touch a circle, the | Every section perpendicular to the axis is a regular penta- (Fig. 28, P. VI.) base of the cylinder onw hich thethelix is situated. Thle gon;the screw is generated by the motion of this pentagon, curve c'd" v' is a horizontal section of the surface; v'/_p'~ or/ the five vertices of which describe five helices situated on n"is a verticlal section drawn through the axis. cne ota isaectiondraw|the same cylinder. meets a sphere discribed with the raius a round the l ~~~~~~~~~~~~~~~~center so that one of its sides passes tharougch the centre of 24, Developable Hlellicoidl, 28, Oblique surfaece 24. Developable Elelicoid. l 2S, Oblique surface. ~~~~the sphere. Thleir intersection is a curve of double cur vature of which p I a is a part. The developable surface leresented (Fig. 33, P1. VII.) byFig. 218, is. t. (Fi. 3,Pl VI., | Fi. 1,Pl V)O by fig. 2 8, is the locus of the tangents to this curve. The A ruled surface is said to be developable if two generat- I iwg right lines, infinitely near to one another, are situated in whic o not meet d the angle between theirdii is rendered visible, the upper one being limited b the vertito be a right one as well as the angles between a b, AB cal plane b ca. one plane, which allows of extending its parts on a plane:. and the righlt line nmH which joins their centers. Describe without any tearing or creasing. The most simple developable surfaces ate the cylinders and cones in which the gen- abl sufacs ae te yiieles ad cne inwhih te gneon these tw~o lines a b and A B two semi-cir cumferences, 32 ~, Developable surface. el atots are paral lel or co n vergent. In all other d evel op abl e d ivid e them into equal parts, and from these points of division eratrs re aralelor onvrget. n al oherdevlopble draw perpendiculars to a b -and A B. We in this way get, on surfaces the right lines are tangents to a curve of double d each of these lines, a series of points unequally distant fiom The surface7represented by fig. 29 is the locus of the tancurvature called the edge of regression. The developable *eioid is t he lous ofthetangntsei one another. The surface represented by No. 21 will be g ents to the curve sS sis which is the intersection of two belicoid is the locus of the tangents to a helix the vertical *ot to vh * described by a straight line joining at first the middle M right cylinders. The base of one of them is the circle IAA projection of which is seen in t~' o" c'". The curve c' dl.q /V /n1: (partoftheinvoluteofacircle)isa horizontal section, the and the extremity b and meeting then successively every intheholizont (part of the involute of a circle) is a ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ i th horizontal pneadhaftebsecoftieothe, the curve mff pl o/n// belongs to a vertical section. drawn two points of division of the two right lines; b id and semi-circumfel through the axis. reesnsap through the axis. v in~~~~~~ A are two of its positionfs. It is very easily seen that it lepresents a palrtof the two sheets contained between four vetpasses twice over a b anid A B. Every plane section per- tical and one horizontal plane. The curves h b u and r bf are 25~ Same sulrfaco with its lines of curvature. penldicular to the axis M is an ellipse the vertices of whlich the i ntersections of the two sheets with one of the vertical lie in the triangles A mn B and a M b. The mean section is planes; the curye o/ —the prolongation of which has been (Fig. 38, P1. IX.) the circumference of a circle. dotted-is tha intersectio n of a horizontal plane; so it is We have represented only the inferior half of the surface. with the curve n k. 2~9, Oblique surface; ellipticfal wedge, The generating right lines are one of the two systems of lines of curvature (which they are in'all developable sur- (Fig33. Serpentine body. faces) the other system are plain sections perpendicular to The straight line a al is equal and parallel to the axis (Fig.3,1,ii.soeielpoeto) the axis. B B~ of the ellipse A`B, AB; and a B, a1B, are perpendicular to the plane of the ellipse. A movable straight l ine This body is generated by the motion of a circle, the plane 26, Screw of four grooves with its nat, supported by the ellipse, always remaining parlllel to the of which remains perpendicular to a helix while its centre (Fig. 34:,'P1. VIII.) (Fig. 34, Pl. VIII.) |plane mRA Ar will describe the surface ill question. Every| describe~ the. latter curve. To construct the outline of the plain section perpendicular to the axis m M is an ellipse. body spheres have been described round several points of A screw is said to have two, three, four grooves, if one,[ If AAl is greater than 1 BB one of these sections will be the helix with a radius equal to that of the generating circle. two, three other grooves have been inserted between two a circle. The envelope of the outline of these spheres gives the out 9 required. he fio'ur exhibitsseveral elipses whch are pr- I]7 reqlired. The ffioie exli~its several ellipses which ale pro- 1 37. cSp herical triangle with its symmetrical and projection has been given (seefig. 36 anc ) and to the jections of the generating circle; the straight lines belolng ge a paP triangle.hat o 9~~~~~~~~~~~~~~~~~~eea projetoltafaxooeria triangleon to a right helicoid. Suppose (fig. 1, pl. I) y' o' (;/) c the base of cube situated (Fig.. 10 PI. IL.) |in the horizontal plane, turn this face about o' o" downward, till the perpendicular distance fiom the point c to the hori34, tnnu1lar biod, I Three diameters of a sphere A-"Al B/"B C//C, determine the vertices of two symmetrical triangles A B: C,. - of the cube and especially ot the three ed ges drawn through (Fig. 37, Pl. VIII. Isometrical projection.) - Blx B,1 C,,. The three great circles the planes of which are (wh ich ray be considered as three axes of orthooonal 01 (wih.a:b omi perpell(licllar to the three diametersdetermine eight triangles. coordinates) on a vertical p lan e draw n through o'o". To This body is generated by the motion of a circlet he center ITe angles ard sides of two of thesetriales are the suppl eTh. zive more dearness to the figuea, w thae g constuctted this of -which describes a second circle, while, its plane remains ments of the sides and angles of the given triangle; or projection in.0O to the rgt bv ota ~o om perpencticular to the directing circle. The construction of which comes to the same of the symmetrical triangle.' ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~a part of the horizon o y//aThvetcldgdrw the outline has been performed on the same principles as These two triangles are called the polar triangles of the through o' will remi erpndiuarotehrznin O -that of the preceding body. Several positions of the given triangle. The figure shows only one of them': 11 | t wll be slortenedt. 11 ~~~~~~~~~~~~~~~~~~~~~but it will be shor tened. Draw throulgh d a par allel to o" a generating circ-le have been indicated. ZIDe~at~l1 -llc~e llave been lllC El, ~11' and describe with o"/a an arc of a circ le about o", join their intersection and the point o"/by a right line: the angle a o" 35. Spherical curve with its polar curve, 8 35. Spherical curve with its p~~~~ola uv. 13" mbinatio~n or five cubes. ( z) will be the measulre of the revolution of the cube. Erect I < [ in o/ on o"/ (z) a perpendicular o" (x~) equal to the edge (Fig. 41, PI. IX,) 0/X/:t ng ~ / (Fig. -27, PL. VI.)afrsianl.Eetr (Fig. 27 Pl. VI.) [ (Fig. 41 Pl. IX ) | "~~~~~~~o/x": the angle z~:o": (x~). evidently, w ill be equal to the, I | ~~~~~~~~~~~~~~~~~~~~~~foresaid angle. Erect in 0 a perpendicular to the huorizon, The cube aI all all, a~ av avI avTM avT is pierced by four This spherical curve is the intersection of a sphere and of draw throug! (x) a, parallel to the latter line, and we shall two rirhlt cylinders. The bases of these cyinders hae a oter cubes C, E. Each of ts four dagoals coincides btain the projection of the vertical edge of the cube.with a diagonal of/B, 6', _D ai;J cl. The lines of intersectionTodtriehepjeinofheoiztaegeoy'ak diameter equal to the radius of the sphere; they touch each To determine the p oject of the faces of the first cube and of those of the four others other and the equator of the sphere, in the plane of which I on the ho rizon, at the right side from O a legth equal to they are described. In fig. 27 these two bases are seen in *form wit the edges ofcthe first t its extremgity a perpendicular to the horizon of the first species are equal to 45~, and those of the other the horizontal plane; the vertical projection of the intersec- 0 which goe s downwards, male —on o" ( o"/(y) equal to are 2 6 -83',1 9 ie. arc (tang —- 0, 5). ting curve is p" d//b"-. o(ay", draw tkirough (y) a paall el to the horizon which Suppose a system of planes drawn through the centre of meets the last mentioned perpendicular in y, and Oy will the sphere and through the tangents to the spherical curve be th e projection of the edge o:y'. Same construction for ald erect diameters perpendicul~ar to these planes' thiein- t... le second horizonltal edgfe o:~x' the projection of which Most of the bodies of which our' collection contains the s tersectinog curve of the spherical surface formed by these is Oz. C models have three axes perpendicular to one another. To diameters is thel polar curve to the given curve. We have make the understandi ng of the d iagrams wh ich represent If the edge of the cube is the u nit of length the three constructed the horizontal and the vertical projection of this them more easy, we may choose as plane of projetion a right lines ox, oy, oz may serve as units of length for the polar curve. | vertical plane wllich is llot pal allel to any one of theve rtical plane which is not parallel to any one of the axes.ae proectillnsolvte sriltines pr ow the xs It is evident, that the lengths of a line paralhl- to an axis We will solve now the inverse problem' stand to their projections in a constant ratio which, generally, The units of length L, M, 7 being given for the pro36*~ Sphere with four great circles. will be a different one for every other axis. If these three jections of the straight lines parallel to the three axes, find (Fig. 6, Pl. I.) raiios are eqtual, the position of the axes in respect to the |he angles between the projections of file axes' plane ot projection is the same as that of three convergent Suppose r to be the unit of length (tie edge of the cube) Through the diameter Pp have been di-awn'three greatl edges of a cube iin respect to the plane'perpendicular to the a, y to be the angles between the plane of projection and circles of the sphere, the great circle A B C1) is perpendi- diagonal which passes through the vertex formed by the |he axes (three convergent edges o f the cube): thenyou cular to the diameter.Pp. edges. To this special projection the name of isometrical will obtain' 10 L — = r cos a, l = r cos, N = r cos, dius m n the lengtlls m' = L, m y/ M, m z/ =., erect are equal to mx/ and to the projection of m on m n, i.e. Whence the perpendiculars x/p, y q, zPs: then the angles p m x', to m/ v. L2 +M2 +_N2 = r2 (cos a2 +cos 32 +cos y2) qm y', s mz' will be equal to the inclinations a, 3, y. With the help of these three light lines the tliangle XO v but These being found, the problem of determining thle angles (fig. 2b) has betn constructed in which the angle in O is the CS+cos a2 +cos 2 COS =- 2| between the projections of the axes, the units of the scales angle between the projections of the two axes. Thee position thus of reduction being given, is to be solved in tihe following of the projection of the third axis will be found by dlawing = 7 f (L2+M2+N2) I manner: Suppose through p-that extremnitv of the edge O Z perpendicular to Xv'. Indeed, the third axis being a formula which allows of a very simple construction. TWe of the cube which is the nearer one to the plane of projection perpendicular to the plane of the projected tliangle, its prohave executed this construction supposing: L = -, M ~ =,, — a plane parallel to the latter. It is obvious. that this new jection must be perpendicular to the tlace of this plane; but N= 8 (see fig. 2, p..) These same numbers are adopted plane will determine on the second axis a segment equal to Xv' is parallel to this trace; OZ, therefore, must be pertoo for the scales (fig. 42, pl. IX). The scale designated by y n v. The rectangular triangle between the first axis and pendicular to Xv. A refers to the unity of length, while X, Y, Z are scales of the seglnent of the second one, will have for its plrojection The plroblemn w e have just solved is a special case of anreduction for the three axes. To come baclk to our problem: a second triangle one side of which is equal to the known other well known problem, viz of the reduction of an angle Decribe with the radius r a circle about m, take on the ra- hypotenuse of the triangle in space, while the two other sides to the horizon. I have great pleasure in offeringf my testimony to the extlraordinary beauty and valtie of MR. ENGEL S Collection of Geometrical and Optical Models. It is not always easy for a student to obtain clear and accurate conceptions of the relations of space from diagrams alone, and there are plrobably few teachelrs who have not often felt the want of a complete and well executed series of models. MR. ENGEL'S beautiful collection will be found to be a most desirable addition to the usual method of instruction in Geometly and Optics, and I earnestly recommend it to the attention of teachers and students. WOLCOTT GIBBS, -Prof. Chemistry and Physics in the Free Academy in New York Juze 26th, 1855. Beifolgend erlaube ich mir die mir guiitigst mitgetheilten Zeichnungen nebst Mat. ESGEL de cette ville, duquel la rare perfection avec laquelle il Text nrit besonderem Danke zu remitftiren. Ich weiss in delr That nicht, ob exScute toute solte de dessiun g&ometique s et la preision vl-ainlent sulpleieh Ihre "schijien Modelle" h6her sehatzen soil oder'"die Zeichnungen," und nante avec laquelle il constrluit en relief les objets si varies sur lesquels s'exerce glaube, dass beide in Verbilndung mit eiuander beim Unterricht erst recht la haute Geometlie, ont valu depuis hngtemps dans ce pays-ci 1a vive aoprofruchtbar sein werden. Vollenden die erstelrn die dem Anfn0'er oft; schwie- bation de tous les plrofesseurs qui onG et6 a mgme d'appleier!es glands rige Vorstellung der Flcehen und Linien, um die es sich handelt, so zeigen die secours que les dessins et les modules de Mi. ENGEL foulnisseut a l'enseigneletztern die grapllische Darstellung in einer Vollkommenheit, die niehts zu ment, etalnt sur le point de s'expatrier, je me plais a exprinel la haute estime wunschen iibrig lisst und in vielen Fallen mehr leistet als die k6rperliche que j'ai conque pour son beau talent. Puisse son talent aussi rare qu'il est Construction. utile, avoir aussi ailleuls tout le succes qu'il me'ite. Der Text ist zum Verst/ndniss dessen, was gegeben ist, ausreichend. BEaLINI le 7 AoAt 1854. Fur den Gebrauch, der in dem Bereich meiner Wirksamkeit von den Blittern G. LEJEUNE, DIRICIILET, gemacht werden wird, hatte ich an manclhen Stellen ein Eingehen auf die J~emzbre de l'Alcad~m. ie de Berlina, a~ssocik Entstehungsweise der Figuren, eine Angabe der Constructionlen gewiUinscht. de'lstitte de Frace. Dadurch hitten die Blitter zugleich als eine Aufgaben-Sammluug gedient. lndessen verkenne ich nicht, dass die Arbeit, wie sie volliegt, fiir das gl6ssere Publikum besser passt. Ich wiinsche, dass das letztele recht gross sein m6ge, und kann ich hierzu etwas beitragen, so wird es mir zum Vergniigen gereichen. BERLIN, 14. Juli 1854. DRUCKENMULLER, Director des Konigl. Gewerbeinstitittes.