Digitized by the Internet Archive in 2010 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/generalinvestigaOOgaus Karl Friedrich Gauss General Investigations Curved Surfaces 1827 AND 1825 TRANSLATED WITH NOTES AND A BIBLIOGRAPHY BY JAMES CADDALL MOREHEAD, A.M., M.S., and ADAM MILLER HILTEBEITEL, A.M. J. S. K. FELLOWS IN MATHEMATICS IN PRINCETON UNIVERSITY THE PRINCETON UNIVERSITY LIBRARY 1902 Copyright, 1902, by The Princeton University Library BOSTON COLLEGE LVBR^^ ^ChStNV3T hill. MASS. 241441 C S. Robinson & Co., University Press Princeton, N. J. INTRODUCTION In 1827 Gauss presented to the Royal Society of Gottingen his important paper on the theory of surfaces, which seventy-three years afterward the eminent French geometer, who has done more than any one else to propagate these principles, charac- terizes as one of Gauss's chief titles to fame, and as still the most finished and use- ful introduction to the study of infinitesimal geometry \ This memoir may be called: General Investigations of Curved Surfaces, or the Paper of 1827, to distinguish it from the original draft written out in 1825, but not published until 1900. A list of the editions and translations of the Paper of 1827 follows. There are three editions in Latin, two translations into French, and two into German. The paper was origin- ally published in Latin under the title : I a. Disquisitiones generales circa superficies curvas auctore Carolo Friderico Gauss Societati regise oblatse D. 8. Octob. 1827, and was printed in : Commentationes societatis regise scientiarum Gottingensis recen- tiores, Commentationes classis mathematicse. Tom. VI. (ad a. 1823-1827). Gottingse, 1828, pages 99-146. This sixth volume is rare ; so much so, indeed, that the British Museum Catalogue indicates that it is missing in that collection. With the signatures changed, and the paging changed to pages 1-50, la also appears with the title page added : I b. Disquisitiones generales circa superficies curvas auctore Carolo Friderico Gauss. Gottingse. Typis Dieterichianis. 1828. II. In Monge's Application de I'analyse a la geometric, fifth edition, edited by Liouville, Paris, 1850, on pages 505-546, is a reprint, added by the Editor, in Latin under the title: Recherches sur la theorie generale des surfaces courbes; Par M. C.-F. Gauss. • G. Darboux, Bulletin des Sciences Math. Ser. 2, vol. 24, page 278, 1900, iv INTRODIJCTION" Ilia. A third Latin edition of this paper stands in: Gauss, Werke, Herausge- geben von der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, Vol. 4, Got- tingen, 1873, pages 217-258, witliout change of the title of the original paper (la). Ill 5. The same, without change, in Vol. 4 of Gauss, Werke, Zweiter Abdruck, Gottingen, 1880. IV. A French translation was made from Liouville's edition, II, by Captain Tiburce Abadie, ancien eleve de I'Ecole Poly technique, and appears in Nouvelles Annales de Mathematique, Vol. 11, Paris, 1852, pages 195-252, under the title : Recherches generales sur les surfaces courbes ; Par M. Gauss. This latter also appears under its own title. Vfl. Another French translation is : Recherches Generales sur les Surfaces Courbes. Par M. C.-F. Gauss, traduites en fran<5'ais, suivies de notes et d'<^tudes sur divers points de la Theorie des Surfaces et sur certaines classes de Courbes, par M. E. Roger, Paris, 1855. Yb. The same. Deuxieme Edition. Grenoble (or Paris), 1870 (or 1871), 160 pages. VI. A German translation is the first portion of the second part, namely, pages 198-232, of: Otto Boklen, Analytische Geometrie des Raumes, Zweite Auflage, Stutt- gart, 1884, under the title (on page 198) : Untersuchungen liber die allgemeine Theorie der krummen Flachen. Von C. F. Gauss. On the title page of the book the second part stands as : Disquisitiones generales circa superficies curvas von C. F. Gauss, ins Deutsche iibertragen mit Anwendungen und Zusiitzen .... VII «. A second German translation is No. 5 of Ostwald's Klassiker der exacten Wissenschaften : Allgemeine Flachentheorie (Disquisitiones generales circa superficies curvas) von Carl Friedrich Gauss, (1827). Deutsch herausgegeben von A. Wangerin. Leipzig, 1889. 62 pages. VII 3. The same. Zweite revidirte Auflage. Leipzig, 1900. 64 pages. The English translation of the Paper of 1827 here given is from a copy of the original paper, I a ; but in the preparation of the translation and the notes all the other editions, except V«, were at hand, and were used. The excellent edition of Professor Wangerin, VII, has been used throughout most freely for the text and notes, even when special notice of this is not made. It has been the endeavor of the translators to retain as far as possible the notation, the form and punctuation of the formulae, and the general style of the original papers. Some changes have been made in order to conform to more recent notations, and the most important of those are mentioned in the notes. INTKODUCTIOl^ T The second paper, the translation of which is here given, is the abstract (Anzeige) which Gauss presented in German to the Royal Society of Gottingen, and which was published in the Gottingische gelehrte Anzeigen. Stuck 177. Pages 1761-1768. 1827. November 5. It has been translated into English from pages 341-347 of the fourth volume of Gauss's Works. This abstract is in the nature of a note on the Paper of 1827, and is printed before the notes on that paper. Recently the eighth volume of Gauss's Works has appeared. This contains on pages 408-442 the paper which Gauss wrote out, but did not publish, in 1825. This paper may be called the New General Investigations of Curved Surfaces, or the Paper of 1825, to distinguish it from the Paper of 1827. The Paper of 1825 shows the manner in which many of the ideas were evolved, and while incomplete and in some cases inconsistent, nevertheless, when taken in connection with the Paper of 1827, shows the development of these ideas in the mind of Gauss. In both papers are found the method of the spherical representation, and, as types, the three important theorems : The measure of curvature is equal to the product of the reciprocals of the principal radii of curvature of the surface, The measure of curvature remains unchanged by a mere bending of the surface, The excess of the sum of the angles of a geodesic triangle is measured by the area of the corresponding triangle on the auxiliary sphere. But in the Paper of 1825 the first six sections, more than one-fifth of the whole paper, take up the consideration of theorems on curvature in a plane, as an introduction, before the ideas are used in space ; whereas the Paper of 1827 takes up these ideas for space only. Moreover, while Gauss introduces the geodesic polar coordinates in the Paper of 1825, in the Paper of 1827 he uses the general coordinates, ^?, q, thus introducing a new method, as well as employing the principles used by Monge and others. The publication of this translation has been made possible by the liberality of the Princeton Library Publishing Association and of the Alumni of the University who founded the Mathematical Seminary. H. D. Thompson. Mathematical Seminary, Princeton University Library, January 29, 1902. CONTENTS PAGE Gauss's Paper of 1827, General Investigations of Curved Surfaces ... 1 Gauss's Abstract of the Paper of 1827 ........ 45 Notes on the Paper of 1827 51 Gauss's Paper of 1825, New General Investigations of Curved Surfaces . 79 Notes on the Paper of 1825 Ill Bibliography of the General Theory of Surfaces ...... 115 DISQUISITIONES GENERALES CIRCA SUPERFICIES CURVAS AUCTORE CAROLO FRIDERICO GAUSS SOCIETATI REGIAE OBLATAE D. 8. OCTOB. 1827 COMMENTATIONES SOCIETATIS REGIAE SCIENTIARUM GOTTINGENSIS RECENTIORES. VOL. VI. GOTTINGAE MDCCCXXVIII GOTTINGAE TYPIS DIETERICHIANIS MDCCCXXVIII GENERAL INVESTIGATIONS OF CURVED SURFACES BY KARL FRIEDRICH GAUSS PRESENTED TO THE ROYAL SOCIETY, OCTOBER 8, 1827 Investigations, in which the directions of various straight lines in space are to be /j p 'a^f* .■i-oJ\ considered, attain a high degree of clearness and simplicity if we employ, as an auxil- ^' •'j' . ' iary, a sphere of unit radius described about an arbitrary centre, and suppose the ^a^- "-""^^ different points of the sphere to represent the directions of straight lines parallel to I p, ^ ^o the radii ending at these points. As the position of every point ia space is deter- mined by three coordinates, that is to say, the distances of the point from three mutually perpendicular fixed planes, it is necessary to consider, first of all, the directions of the axes perpendicular to these planes. The points on the sphere, which represent these directions, we shall denote by (1), (2), (3). The distance of any one of these points from either of the other two wiU be a quadrant; and we shall suppose that the direc- tions of the axes are those in which the corresponding coordinates increase. It wiU be advantageous to bring together here some propositions which are fre- quently used in questions of this kind. I. The angle between two intersecting straight lines is measured by the arc between the points on the sphere which correspond to the dii-ections of the lines. II. The orientation of any plane whatever can be represented by the great circle on the sphere, the plane of which is parallel to the given plane. 4 KARL FRIEDRICH GAUSS III. The angle between two planes is equal to the spherical angle between the great circles representing them, and, consequently, is also measured by the arc inter- cepted between the poles of these great circles. And, in like manner, the angle of inclina- tion of a straight line to a plane is measured by the arc drawn from the point which corresponds to the direction of the line, perpendicular to the great circle which repre- sents the orientation of the plane. IV. Letting x, y, s ; x', y' , s' denote the coordinates of two points, r the distance between them, and L the point on the sphere which represents the direction of the line drawn from the first point to the second, we shall have x'=^ X + r cos, {1)L y'—y + rcos (2)Z s'= s + r cos (3)Z » V. From this it follows at once that, generally, cos'' (l)i + cos' {2)L + cos' (3)Z = 1 and also, if L' denote any other point on the sphere, cos (1)Z . cos (1)X'+ cos {2)L . cos (2)X'+ cos (3)Z . cos (3)X'= cos LL'. VI. Theorem. If L, L', L", L'" denote four points on the sphere, and A the angle which the arcs LL', L"L"' make at their point of intersection, then we shall have cos LL" . cos L'L'"— cos LL'" . cos L'L"^ sin LL' . sin L"L'" . cos A Demonstration. Let A denote also the point of intersection itself, and set AL = t, AL'=t', AL"=^t", AL'"=t'" Then we shaU have -jv^ ^-^^.1 cos LL" = cos ^ . cos t" + sin t sin t" cos A cos L'L'" = cos f cos t'" + sin t' sin f" cos A ^,. j,, ■ 'tv^. cos LL'" = cos t cos t"'+ sin t sin t'" cos A cos L'L" — cos t' cos t" + sin t' sin t" cos A and consequently, cos LL" . cos L'L'"— cos LL'". cos L'L" = cos A (cos t cos t" sin t' sin f" + cos f cos f" sin t sin t" — cos t cos t'" sin f sin t" — cos f cos t" sin t sin t'") = cos A (cos t sin t' — sin t cos f) (cos t" sin t'" — sin t" cos t'") = cos A . sin {t'—t) . sin {t"'—i") = cosA . sin LL' .sin L"L'" GEKERAL ESTVESTIGATIOITS OF CURVED STJEFACES 5 But as there are for each great circle two branches going out from the point A, these two branches form at this point two angles whose sum is 180°. But our analysis shows that those branches are to be taken whose directions are in the sense from the point Z to Z', and from the point Z" to Z'"; and since great circles intersect in two points, it is clear that either of the two points can be chosen arbitrarily. Also, instead of the angle A, we can take the arc between the poles of the great circles of which the arcs Z Z', Z" Z'" are parts. But it is evident that those poles are to be chosen which are similarly placed with respect to these arcs; that is to say, when we go from Z to Z' and from Z" to Z'" , both of the two poles are to be on the right, or both on the left. VII. Let Z, Z', Z" be the three points on the sphere and set, for brevity, cos {1)Z = X, cos (2)Z = y, cos (3)Z = z cos (l)X' = z', cos {2)Z' = y', cos (3)Z' = z' cos (l)i/"= a;", cos {^)Z"=y", cos (3)X"= z" and also X y' s" + x' y" z -\- x" y z' — x y" z' — x' y z" — x" y' z =i ^ Let \ denote the pole of the great circle of which ZZ' is a part, this pole being the one that is placed in the same position with respect to this arc as the point (1) is with respect to the arc (2) (3). Then we shall have, by the preceding theorem, y z' —y' z = cos (1)\ . sin (2)(3) . sin ZZ' , or, because (2)(3) = 90°, y z' — y' z = cos (l)X. . sin ZZ', and similarly, z x' — z' X — COS (2)X . sin ZZ' xy' — x' y = COS (3)X . sin ZZ' Multiplying these equations by x", y", z" respectively, and adding, we obtain, by means of the second of the theorems deduced in V, A = cos X Z" . sin ZZ' Now there are three cases to be distinguished. First, when Z" lies on the great cii'cle of which the arc ZZ' is a part, we shall have Xl*"— 90°, and consequently, A = 0. If Z" does not lie on that great circle, the second case will be when Z" is on the same side as X ; the third case when they are on opposite sides. In the last two cases the points Z, Z', Z" will form a spherical triangle, and in the second case these points will lie in the same order as the points (1), (2), (3), and in the opposite order in the third case. 6 KARL FRIEDRICH GAUSS Denoting the angles of this triangle simply by L, L', L" and the perpendicular drawn on the sphere from the point L" to the side LL' by jj, we shall have sin^ = sin X . sin LL" ^ sin L' . sin L' L", and \Z"=90° +p, the upper sign being taken for the second case, the lower for the third. From this it follows that i A = sini. sinXX'. sinXZ" = sini'. sin ZX'. sini'i/" . » \^ n^- ^ = sin L" . sin L L" . sin L' L" \'' ;; '^- Ci^''' ZJ^ Moreover, it is evident that the first case can be regarded as contained in the second or third, and it is easily seen that the expression ± A represents six times the volume of the pyramid formed by the points L, L', L" and the centre of the sphere. Whence, finally, it is clear that the expression ± ^ A expresses generally the volume of any pyramid contained between the origin of coordinates and the three points whose coor- ■vj dinates are ^, y, s ; x', y' , z' ; x" , y", s" . 3. A curved surface is said to possess continuous curvature at one of its points A, if the directions of all the straight lines drawn from A to points of the surface at an infinitely small distance from A are deflected infinitely little from one and the same plane passing through A. This plane is said to iouch the surface at the point A. If this condition is not satisfied for any point, the continuity of the curvature is here interrupted, as happens, for example, at the vertex of a cone. The following investigations will be restricted to such surfaces, or to such parts of surfaces, as have the continuity of their curvature nowhere interrupted. We shall only observe now that the methods used to determine the position of the tangent plane lose their meaning at singular points, in which the continuity of the curvature is interrupted, and must lead to indeterminate solutions. 4. The orientation of the tangent plane is most conveniently studied by means of the direction of the straight line normal to the plane at the point A, which is also called the normal to the curved surface at the point A. We shall represent the direction of this normal by the point L on the auxiliary sphere, and we shall set cos (1)Z =X, cos (2)Z = Y, cos (3)X =Z; and denote the coordinates of the point A by x, y, z. Also let x + dx, y + dy, z + dz be the coordinates of another point A' on the curved surface ; ds its distance from A, GENEEAL INVESTIGATIONS OF CUEVED SUEFACES 7 which is infinitely small ; and finally, let X be the point on the sphere representing the direction of the element ilJ.'. Then we shall have dx = ds. cos {1)\, di/ — ds.Gos{2)\, ds = ds. cos {B)\ and, since X Z must be equal to 90°, X cos (1)X + Zcos (2)X + Z cos (3)X = By combining these equations we obtain Xdx+Ydi/+Zds = 0. There are two general methods for defining the nature of a curved surface. The Jirst uses the equation between the coordinates z, y, z, which we may suppose reduced to the form W = 0, where W will be a function of the indeterminates x, ij, z. Let the com- plete differential of the function W be dW^P dx-^ Qd>/ + Rdz and on the curved surface we shall have P dx+ Qdy^ Rds^^ and consequently, P cos (1)X + Q cos (2)X + R cos (3)X = Since this equation, as well as the one we have established above, must be true for the directions of all elements ds on the curved surface, we easily see that X, Y, Z must be proportional to P, Q, R respectively, and consequently, since X^+Y^+ Z^=^l, we shall have either _ P Q R X _ / / Tii I 7^2 i fw\' Y ,//D2_l_ /n2_L r>2\' Z V {P'+ Q' + R') V{P'+ Q'+ R') V{P'+ Q' + R') ^ ~ t/ (P^ + ^^ + R") ^ i/ (P^ + ^^ + R") ^ V {P^+ Q^^- R') The second method expresses the coordinates in the form of functions of two varia- bles, jt?, q. Suppose that differentiation of these functions gives dz =z a dp + a' dq dy =^h dp -\- b' dq dz =^c dp -\- c' dq 8 KAEL FREEDEICH GAUSS Substituting these values in the formula given above, we obtain {a X + b Y + c Z) dp + {a' X + b' Y + c' Z) dq =^ Since this equation must hold independently of the values of the differentials djo, dq, we evidently shall have aX+b Y+ c Z=^, a'X+b' Y+c' Z=0 From this we see that X, Y, Z will be proportioned to the quantities be' — cb', ca' — ac', aV — ba' Hence, on setting, for brevity, i- v'j^^ V (^{bc' — cb'f + {ca' — ac'Y + {aV — ba'Y) = A --• '' ''J^^^ u^^^ we shall have either : \ f^^P be' — eb' ea' — ae' ab' — ba' X= 7 3 Y 7 3 Z T AAA or __ cJ' — be' j^_(f(^' — (^C'' rr_^^' — ^^' ^~ A ' ^~ A ' ^~ A With these two general methods is associated a third, in which one of the coordinates, 2, say, is expressed in the form of a function of the other two, x, y. This method is evidently only a particular case either of the first method, or of the second. If we set dz zz^tdx -\- udy we shall have either X = —TT\ i rt i ~2\ ' Y =^ _ / 1 -\ i 75 i Tax ' Z = 1/ (1 + f + %"■) "■ V{\^f-\-v?) 1/ (1 + i!' + M^) V/ (1 + f + «') V {l^f^U^) l/ (1 + ^^ + M^) The two solutions found in the preceding article evidently refer to opposite points of the sphere, or to opposite du-ections, as one would expect, since the normal may be drawn toward either of the two sides of the curved surface. If we wish to distinguish between the two regions bordering upon the surface, and call one the exterior region and the other the interior region, we can then assign to each of the two normals its appropriate solution by aid of the theorem dcri\'ed in Art. 2 (VII), and at the same time establish a criterion for ilistinguishing the one region from the other. GENERAL rNTESTIGATIOITS OF CUEVED SURFACES 9 In the first method, such a criterion is to be drawn from the sign of the quantity W. Indeed, generally speaking, the curved surface divides those regions of space in which W keeps a positive value from those in which the value of W becomes negative. In fact, it is easily seen from this theorem that, if W takes a positive value toward the exterior region, and if the normal is supposed to be drawn outwardly, the first solution is to be taken. Moreover, it will be easy to decide in any case whether the same rule for the sign of W is to hold throughout the entire surface, or whether for different parts there will be different rules. As long as the coefiicients P, Q, R have finite values and do not all vanish at the same time, the law of continuity will prevent any change. If we follow the second method, we can imagine two systems of curved lines on the curved surface, one system for which p is variable, q constant ; the other for which q is variable, jo constant. The respective positions of these lines with reference to the exte- rior region will decide which of the two solutions must be taken. In fact, whenever the three Unes, namely, the branch of the line of the former system going out from the point ^ as JO increases, the branch of the line of the latter system going out from the point .A as ^ increases, and the normal drawn toward the exterior region, are similarly placed as the X, y, z axes respectively from the origin of abscissas (e. ^., if, both for the former three lines and for the latter three, we can conceive the first directed to the left, the second to the right, and the third upward), the first solution is to be taken. But when- ever the relative position of the three lines is opposite to the relative position of the X, y, z axes, the second solution will hold. In the third method, it is to be seen whether, Avhen z receives a positive increment, x and y remaining constant, the point crosses toward the exterior or the interior region. In the former case, for the normal drawn outward, the first solution holds ; in the latter case, the second. 6. Just as each definite point on the curved surface is made to correspond to a definite point on the sphere, by the direction of the normal to the curved surface which is trans- ferred to the surface of the sphere, so also any line whatever, or any figure whatever, on the latter will be represented by a corresponding line or figure on the former. In the comparison of two figures corresponding to one another in this way, one of which will be as thQ map of the other, two important points are to be considered, one when quantity alone is considered, the other when, disregarding quantitative relations, position alone is considered. The first of these important points will be the basis of some ideas which it seems judicious to introduce into the theory of curved surfaces. Thus, to each part of a curved 1 .1% JX 10 KARL FRIEDEICH GAUSS surface inclosed within definite limits we assign a total or integral curvature, which is represented by the area of the figure on the sphere corresponding to it. From this integral curvature must be distinguished the somewhat more specific curvature which we shall caU the measure of curvature. The latter refers to a point of the surface, and shall denote the quotient obtained when the integral curvature of the surface element about a point is divided by the area of the element itself; and hence it denotes the ratio of the infinitely small areas which correspond to one another on the curved surface and on the sphere. The use of these innovations will be abundantly justified, as we hope, by what we shall explain below. As for the terminology, we have thought it especially desirable that aU ambiguity be avoided. For this reason we have not thought it advantageous to follow strictly the analogy of the terminology commonly adopted (though not approved by all) in the theory of plane curves, according to which the measure of curvature should be called simply curvature, but the total curvature, the amplitude. But why not be free in the choice of words, provided they are not meaningless and not liable to a misleading interpretation ? f-i/"^The position of a figure on the sphere can be either similar to the position of the corresponding figure on the curved surface, or opposite (inverse). The former is the case when two lines going out on the curved surface from the same point in different, but not opposite directions, are represented on the sphere by lines similarly placed, that is, when the map of the line to the right is also to the right ; the latter is the case when the con- trary holds. We shall distinguish these two cases by the positive or negative sign of the measure of curvature. But evidently this distinction can hold only when on each surface we choose a definite face on which we suppose the figure to lie. On the auxiliary sphere we shaU use always the exterior face, that is, that turned away from the centre ; on the curved surface also there may be taken for the exterior face the one already considered, or rather that face from which the normal is supposed to be drawn. For, evidently, there is no change in regard to the similitude of the figures, if on the curved surface both the figure and the normal be transferred to the opposite side, so long as the image itself is represented on the same side of the sphere. The positive or negative sign, which we assign to the measure of curvature accord- ing to the position of the infinitely small figure, we extend also to the integral curvature of a finite figure on the curved surface. However, if we wish to discuss the general case, some explanations will be necessary, which we can only touch here briefly. So long as the figure on the curved surface is such that to distinct points on itself there corres- pond distinct points on the sphere, the definition needs no further explanation. But whenever this condition is not satisfied, it will be necessary to take into account twice or several times certain parts of the figure on the sphere. Whence for a similar, or GENEEAL INVESTIGATIONS OF CURVED SURFACES 11 inverse position, may arise an accumulation of areas, or the areas may partially or wholly destroy each other. In such a case, the simplest way is to suppose the curved surface divided into parts, such that each part, considered separately, satisfies the above condition ; to assign to each of the parts its integral curvature, determining this magni- tude by the area of the corresponding figure on the sphere, and the sign by the posi- tion of this figure ; and, finally, to assign to the total figure the integral curvature arising from the addition of the integral curvatures which correspond to the single parts. So, generally, the integral curvature of a figure is equal to fJcdcr, da denoting the element of area of the figure, and h the measure of curvature at any point. The prin- cipal points concerning the geometric representation of this integral reduce to the fol- lowing. To the perimeter of the figure on the curved surface (under the restriction of Art. 3) will correspond always a closed line on the sphere. If the latter nowhere intersect itself, it will divide the whole surface of the sphere into two parts, one of which will correspond to the figure on the curved surface ; and its area (taken as positive or negative according as, with respect to its perimeter, its position is similar, or inverse, to the position of the figure on the curved surface) will represent the inte- gral curvature of the figure on the curved surface. But whenever this line intersects itself once or several times, it will give a complicated figure, to which, however, it is possible to assign a definite area as legitimately as in the case of a figure without nodes ; and this area, properly interpreted, wiU give always an exact value for the integral curvature. However, we must reserve for another occasion the more extended exposition of the theory of these figures viewed from this very general standpoint. 7. We shall now find a formula which will express the measure of curvature for • any point of a curved surface. Let dcr denote the area of an element of this surface ; p.f- then Zd(T will be the area of the projection of this element on the plane of the coor- dinates X, y ; and consequently, if c? S is the area of the corresponding element on the sphere, Z d'% wiU be the area of its projection on the same plane. The positive or negative sign of Z will, in fact, indicate that the position of the projection is similar or inverse to that of the projected element. Evidently these projections have the same ratio as to quantity and the same relation as to position as the elements themselves. Let us consider now a triangular element on the curved surface, and let us suppose that the coordinates of the three points which form its projection are X, y X + dx, y -^ dy X + S:r, y + 8y 12 KARL FRIEDRICH GAUSS The double area of this triangle wiU be expressed by the formula dx .hi/ — di/ . Sa; and this will be in a positive or negative form according as the position of the side from the first point to the third, with respect to the side from the first point to the second, is similar or opposite to the position of the ^-axis of coordinates with respect to the :r-axis of coordinates. In like manner, if the coordinates of the three points which form the projection of the corresponding element on the sphere, from the centre of the sphere as origin, are X, Y X+dX, Y+dY X+SX, Y+SY the double area of this projection will be expressed by dX.SY—dY.SX and the sign of this expression is determined in the same manner as above. Where- fore the measure of curvature at this point of the curved surface will be ^ dX.SY—dY.SX dx . hy — di/ . hx If now we suppose the nature of the curved surface to be defined according to the third method considered in Art. 4, X and Y will be in the form of functions of the quanti- ties X, y. We shall have, therefore, ^X dX dX — ^ — dx + -r- dy ox oy dx dy "^ dY dY 8F= -::—bx-r -ir-oy ox Oy '^ When these values have been substituted, the above expression becomes _dX BY dX ^dY dx dy dy dx GENERAL INVESTIGATIONS OF CURVED SURFACES 13 Setting, as above, and also cs , ds — — t, — = M dx dy dt z= Tdx + Udy, du = Udx + V dy „PI -(l-^ vc ; and hence we have from the formulae given above . -5 -"^ !• ^^'^ '' X=-tZ, Y^-uZ, {l-Vf^-u^)Z^=\ dX^—Zdt—tdZ dY^= — Zdu — u dZ {l + f+u^)dZ+Z{tdt + udu) = dZ=-Z'{tdt + udu) dX = -Z''{l + u^) dt + ZHu du dF= + Z' tu dt—Z' (1 + f) du g V" — = Z^i-{l + u^)T+tuUy ~-=Z^i-{l + u')U+tuV^ ^^Z^ituT-{l + f)U) ^=Z^ituU-{l + f)Vy Substituting these values in the above expression, it becomes k^Z'{T V— m) (1 + ^+ u^)=Z' [T V— U') TV—U' and so {1 + f+uY By a suitable choice of origin and axes of coordinates, we can easily make the values of the quantities t, u, U vanish for a definite point A. Indeed, the first two 14 KARL FRIEDEICH GAUSS conditions will be fulfilled at once if the tangent plane at this point be taken for the a:^-plane. If, further, the origin is placed at the point A itself, the expression for the coordinate s evidently takes the form s=i T°a? + U°xy + I FY+ fi where fl will be of higher degree than the second. Turning now the axes of x and p through an angle M such that 9 TT° tan2itf^ ^ .. T° F° it is easily seen that there must result an equation of the form z=k Ta^+ hVtf+ D. In this way the third condition is also satisfied. When this has been done, it is evi- dent that I. If the curved surface be cut by a plane passing through the normal itself and through the z-axis, a plane curve will be obtained, the radius of curvature of which at the point A wUl be equal to -„, the positive or negative sign indicating that the curve is concave or convex toward that region toward which the coordinates s are positive. II. In like manner -^^ will be the radius of curvature at the point A of the plane curve which is the intersection of the surface and the plane through the j/-axis and the s-axis. III. Setting x — r GOS
+V sin^ >) r^+ n
from which we see that if the section is made by a plane through the normal at A
and making an angle <^ with the a;-axis, we shall have a plane curve whose radius of
curvature at the point A wiU be
1
rcos^<^+Fsin^<^
IV. Therefore, whenever we have T—V, the radii of curvature in all the normal
planes will be equal. But if T and V are not equal, it is evident that, since for any
value whatever of the angle >, T cos^ (f)+ V sin^ <^ falls between T and V, the radii of
curvature in the principal sections considered in I. and II. refer to the extreme curva-
tures ; that is to say, the one to the maximum curvature, the other to the minimum,
GEKEEAL EsTVESTIGATIONS OF CUEVED SUKFACES 15
if T and F have the same sign. On the other hand, one has the greatest convex
curvature, the other the greatest concave curvature, if T and V have opposite signs.
These conclusions contain almost all that the illustrious Euler was the first to prove
on the curvature of curved surfaces.
V. The measure of curvature at the point A on the curved surface takes the
very simple form
k=TV,
whence we have the
Theorem. The measure of curvature at any point whatever of the surface is equal to a
fraction whose numerator is unity, and whose denominator is the product of the ttvo extreme
radii of curvature of the sections by normal planes.
At the same time it is clear that the measure of curvature is positive for con-
cavo-concave or convexo-convex surfaces (which distinction is not essential), but nega-
tive for concavo-convex surfaces. If the surface consists of parts of each kind, then
on the lines separating the two kinds the measure of curvature ought to vanish. Later
we shall make a detailed study of the nature of curved surfaces for which the meas-
ure of curvature everywhere vanishes.
9.
The general formula for the measure of curvature given at the end of Art. 7 is
the most simple of all, since it involves only five elements. We shall arrive at a
more complicated formula, indeed, one involving nine elements, if we wish to use the
first method of representing a curved surface. Keeping the notation of Art. 4, let us
set also
^!^=p, ^^=0' ^^-W
a'^^^„ an^^^,, a^^^„
5v
dy .dz ' dx.ds ^ ' dx.dy
SO that
dP = P' dx + R" dy + Q" ds
dQ = R"dx+ Q! dy +P"dz
dR = Q" dx + P"dy +R' ds
P
Now since t — ~^i we find through difi'erentiation
R'dt = -RdP + PdR = {PQ"—RP')dx + {PP"—RR")dy+{PR'—RQ")dz
16 KAKL FRIEDRICH GAUSS
or, eliminating dz by means of the equation
R^ dt={—R'' P' ^ 2 P RQ"—P^ R') dx +{PRP"+ QRQ"—PQR'—R^R")dy.
1 i • b '^
In like manner we obtain r
R^du = {PRP"+QRQ"—PQR'—R'R")dx+{—R'Q'+2 QRP"—(^R')dy
From this we conclude that
WT=-R''P'^-'lPRq"—P^R'
R^U=^PRP"-V QRQ"-PQR'-R'R"
R^V^-R^Q'+2 QRP"-Q^R'
Substituting these values in the formula of Art. 7, we obtain for the measure of curv-
ature k the following symmetric expression :
(P^+ Q^+R-'fk=P^{Q'R'-P"^)+Q\P'R'-Q"^) +R\P'Q'-R"^)
+ 2 QR{Q" R" -P' P") + 2 PR {P" R" - Q' Q") + 2PQ {P" Q" - R' R")
10.
r We obtain a still more complicated formula, indeed, one involving fifteen elements,
if we follow the second general method of defining the nature of a curved surface. It
is, however, very important that we develop this formula also. Retaining the nota-
tions of Art. 4, let us put also
■^ ^% ■ , a/~^' dp.dq f^' dq^ ^
and let us put, for brevity, « y,-)
Ic'-cV^A
ca' — ac'===B
aV-ba' = C
First we see that
Adx^Bdy+Cd3==^,
or
a2 = — Yfdx — jT dy.
O'^ -.
GENERAL INVESTIGATIONS OF CURVED SURFACES 17
Thus, inasmuch as z may be regarded as a function of x, y, we have
dx^'^~ G
d3_ ^_B_
dy-""- C
Then from the formulae
dx — adp-\- a' dq, dy — bdp + b'dq, (5 > " I ■
we have
Cdp^= V dx — a' dy
Cdq'=—b dx + a dy
Thence we obtain for the total differentials of ^, m
C^dt=lA^.-C^){b'dx-a'dy) + {0^-A^){bdx-ady)
\ dp dpi \ d q cql
C^du=^(B^-C^-p\{b'dx-a'dy) + (c^-^-B^){hdx-ady)
\ dp dp ' \ d q d q I
If now we substitute in these formulae
^-^-c'fi^by'-c^'-b'y
dA
-j^ = c'fi'^by"-c^"-b'y'
dB
-z — ='a' y + e a' — ay' —c a
d 7?
^-^a'y'+ea."-ay"-c' a'
dq ' '
~ = b' a+afi' -ba' -a' ^
^ = b'a'+a /8"- b a"- a' /8'
and if we note that the values of the differentials dt, du thus obtained must be equal,
independently of the differentials dx, dy, to the quantities T dx-VU dy, Udx+Vdy
respectively, we shall find, after some sufficiently obvious transformations,
C''T=aAb'^+/3Bb'^+yCb'^
-2 a' Abb'-2 13' Bbb'-2y' Cbb'
+ a"Ab' + ^"Bb''+y"Cb''
18 KARL FRIEDRICH GAUSS
C^ i7= -aAa'h'-^Ba'b'- y Ca'V
+ a' il {aV^ ha')+^' B [aV + ba') + y' C{ab'+ ba')
— a"Aab-^"Bab-y"Cab
C^V=aAa'^ + ^Ba'^+yCa'^
— 2a'Aaa'—2^'Baa'—'iy'Caa'
. +0." Aa^+ ^" Ba^+y" Ca""
Hence, if we put, for the sake of brevity,
Aa +BI3 +Cy =D (1)
Aa' +B/3' + Cy' =1)' (2)
Aa"+BI3"+Cy"=-D" (3)
we shall have
C T=D V- 2 D'bb' + D" b"-
C^ U=-Da'b' + D' {ab'+ ba') -D" ab
Cn^=D a'^-2 D' aa' + D" a"
From this we find, after the reckoning has been carried out,
C {T V- m) = {DD"-D'^) {ab'-ba'f = {DD"-D'^) C
and therefore the formula for the measure of curvature
_ DD"—D"'
11.
By means of the formula just found we are going to establish another, which may
be counted among the most productive theorems in the theory of curved surfaces.
Let us introduce the following notation :
■>%'^>," aa'+bb'+cc'=F
•yl a'^+b'^^c'^=G
^^' V?4 (i o. +b ^ +cy =m (4)
\ ~^f a a' ^-b fi' +c y' =m' (5)
-BF .1^" a a"+b fi"^-c y" = m" (6)
Vi' '"^T a'a -^b' fi +c'y =n (7)
f^,_aJl>p^' a'a'^b'^'+c'y-=n' (8)
' u'a"+b' fi"+c'y"^n" (9)
A^-\- B'-V C^=EG—F^=^
GENERAL INVESTIGATIONS OF CURVED SURFACES 19
Let us eliminate from the equations 1, 4, 7 the quantities 13, y, which is done by
multiplying them by bc'—cb', h'C—c'B, c -5 — J C respectively and adding. In this
way we obtain
(A{hc'-cb') + a{b'C-c'B) + a'{cB-bC))a
= D{bc'—cb') + m{b' C- c'B) + n {cB - b C)
an equation which is easily transformed into
AD = ai^ + a{nF—mG) + a'{mF~nE)
Likewise the elimination of a, y or a, /8 from the same equations gives
BD=-^^ + b{nF-mG) + b' {niF — nE)
CD = y^+ c{nF — mG) + c' {mF — nE)
Multiplying these three equations by a", /8", y" respectively and adding, we obtain
DD"={aa"+ ^fi"+yy")^+m"{nF-mG)+n" (mF-nE) . . . (10)
If we treat the equations 2, 5, 8 in the same way, we obtain
AD'= a' A+ a (n'F-m'G) + a' [m'F-n'E)
BD'= ^'A+b {n'F- m' G) + b' {m'F-n'E)
CD' = y' £^+ c {n' F -m' G) + c' {m'F-n'E)
and after these equations are multiplied by a', /8', y' respectively, addition gives
D'^= (a''+ /8''+ r") A + m' {n'F-m' G) + n' {m'F-n'E)
A combination of this equation with equation (10) gives
DD"-D'^={aa"+^^"+yy"-a'^-fi'^-y'^)A
+ E{n'^-nn") +F{nm"- 2 m'n'+mn") + G{m'^-mm")
It is clear that we have
3^ '
dE
dE
dF
dF
dG
= 2m,
= 2m'
= m' +
n,
^ =m
" + n'
= 2<
dp
dq
' dp
, dE
dq
, dE
' dp
dF
, dG
tn =
dp
dF
, dE
m'
dq
,dG
m" =
dq
^dG
dp
n =
dp
dq
n'-
^ dp'
n" =
*a.
Moreover, it is easily shown that we shall have
aa" + l3l3" + yy"-a'^-(3''-y'^=j^~^ = -jy-j^
= _i g'-g 3'F d'G
" dq^ dp.dq ' dp^
'^yr
\...-s,,ir^^''^^"
k> -
%J
20
KARL FRIEDIilCH GAUSS
^'t
If we substitute these different expressions in the formula for the measure of curva-
ture derived at the end of the preceding article, we obtain the following formula, which
involves only the quantities U, F, G and their differential quotients of the first and
second orders :
dE SG__^dF_ dG . idG\
dq ■ ■ — ■'
dp dq dq
i{I!G-F'fk=Js(^
a£_a^ d_G
dq dq
fdE
dp
•dE
dp
dq
dE
dp dq \ dp ' /
dE dE
dp
dE
dq
dE
dp
dG dE
dp dp
dE
dq
,(m')-.i.o^nm-.^yS
d'E
d_G_
dp
d'^E
dp .dq dp
12.
Since we always have
d^+ df+ dz^ = Edp''+2 Edp .dq+G dq\
it is clear that
V{Ed2f+2Edp.dq+Gdq^)
is the general expression for the linear element on the curved surface. The analysis
developed in the preceding article thus shows us that for finding the measure of cur-
vature there is no need of finite formulae, which express the coordinates x, y, z as
functions of the indeterminates p, q ; but that the general expression for the magnitude
of any linear element is sufficient. Let us proceed to some applications of this very
important theorem.
Suppose that our surface can be developed upon another surface, curved or plane,
so that to each point of the former surface, determined by the coordinates x, y, z, will
correspond a definite point of the latter surface, whose coordinates are x', y' , s' . Evi-
dently a;', y' , z' can also be regarded as functions of the indeterminates p, q, and there-
fore for the element V {dx''^+ dy''^-\- dz''^) we shall have an expression of the form
\/{E'dp^+ 2E'dp.dq + G'dq^)
where E' , E' , G' also denote functions of p, q. But from the very notion of the devel-
opment of one surface upon another it is clear that the elements corresponding to one
another on the two surfaces are necessarily equal. Therefore we shall have identically
E^E', E^E', G^G'.
Thus the formula of the preceding article leads of itself to the remarkable
TuEOREM. If a curved surface is developed upon any other surface whatever, the
measure of curvature in each point remains unchanged.
GEISTEEAL INVESTIGATIONS OF CUKVED SURFACES
21
Also it is evident that any finite part whatever of the curved surface loill retain the
same integral curvature after development upon another surface.
Surfaces developable upon a plane constitute the particular case to which geom-
eters have heretofore restricted their attention. Our theory shows at once that the
measure of curvature at every point of such surfaces is equal to zero. Consequently,
if the nature of these surfaces is defined according to the thu-d method, we shall have
at every point
d'^Z d'^Z I d^z \''_
a criterion which, though indeed known a short time ago, has not, at least to our
knowledge, commonly been demonstrated with as much rigor as is desirable."^
13.
What we have explained in the preceding article is connected with a particular
method of studying surfaces, a very worthy method which may be thoroughly devel-
oped by geometers. When a surface is regarded, not as the boundary of a solid, but
as a flexible, though not extensible solid, one dimension of which is supposed to
vanish, then the properties of the surface depend in part upon the form to which we
can suppose it reduced, and in part are absolute and remain invariable, whatever may
be the form into which the surface is bent. To these latter properties, the study of
which opens to geometry a new and fertile field, belong the measure of curvature and
the integral curvature, in the sense which we have given to these expressions. To
these belong also the theory of shortest lines, and a great part of what we reserve to
be treated later. From this point of view, a plane surface and a surface developable
on a plane, e. g., cylindrical surfaces, conical surfaces, etc., are to be regarded as essen-
tially identical; and the generic method of defining in a general manner the nature of
the surfaces thus considered is always based upon the formula
V{Edp''-\- 2 Fdp .dq^a dq%
which connects the linear element with the two indeterminates p, q. But before fol-
lowing this study further, we must introduce the principles of the theory of shortest
lines on a given curved surface.
14.
The nature of a curved line in space is generally given in such a way that the
coordinates x, y, z corresponding to the diflferent points of it are given in the form of
functions of a single variable, which we shall call w. The length of such a line from
^ fw-^,r\(\[^
y
/
4/\ »^v«*i-^.^j^-
A ^.^-^
22 KARL FRIEDRICH GAUSS
an arbitrary initial point to the point whose coordinates are x, y, z, is expressed ))y
the integral \ iV"-^
f'-Mthm-m) ^-^''^/'
If we suppose that the position of the line undei'goes an infinitely small variation, so
that the coordinates of the different points receive the variations hx, Sy, 8 s, the varia-
tion of the whole length becomes
'dx .dSx +dy .dSt/ + ds .dSs
r
Vidx'+df+ds')
which expression we can change into the form
dx .8x-\-d7/ .hy -\-ds . 8z
vJdJ+dfTdJ)
J V^-^Vidx'+df+dz') '^^^•^V{dx'+df+d^) '^^^•'^ Vidx'+df+dzy
We know that, in case the line is to be the shortest between its end points, all that
stands under the integral sign must vanish. Since the line must Ue on the given
surface, whose nature is defined by the equation
Pdx + Qdy+Rds-=0,
the variations Sx, By, 8s also must satisfy the equation
PSx + Q8y+RSs = 0,
and from this it follows at once, according to well-known rules, that the differentials
d _ , , n .1 1 J n I 7 „2 \ > d _ , I J ^1, I 7.9 I T72V' d-
V{d:^-^dy'\d^Y "^ V {dx"^ dy''+ ds^Y 'W {d:^^ dy''-^ ds^)
must be proportional to the quantities P, Q, R respectively. Let c?;- be the element
of the curved line ; X the point on the sphere representing the direction of this ele-
ment ; L the point on the sphere representing the direction of the normal to the curved
surface ; finally, let ^, -q, t, be the coordinates of the point \, and X, Y, Z be those of
the point L with reference to the centre of the sphere. We shall then have
dx~$dr, dy = r}dr, ds='l,dr
from which we see that the above differentials become d^, d-q, dt,. And since the
quantities P, Q, R are proportional to X, Y, Z, the character of shortest lines is
expressed by the equations
dl^drj ^dC ^ >
X" Y ~ Z o"
GENERAL INVESTIGATIONS OF CUEVED SITRFACES 23
Moreover, it is easily seen that
is equal to the small arc on the sphere which measures the angle between the direc-
tions of the tangents at the beginning and at the end of the element dr, and is thus
equal to — ? if p denotes the radius of curvature of the shortest line at this point.
Thus we shall have
pd^=Xdr, pdrj = Ydr, pdt,^Zdr
15.
Suppose that an infinite number of shortest lines go out from a given point A
on the curved surface, and suppose that we distinguish these lines from one another
by the angle that the first element of each of them makes with the first element of
one of them which we take for the first. Let j> be that angle, or, more generally, a
function of that angle, and ?• the length of such a shortest line from the point A to
the point whose coordinates are z, y, z. Since to definite values of the variables r, <^
there correspond definite points of the surface, the coordinates x, y, z can be regarded
as functions of r, ^. We shall retain for the notation A, L, ^, rj, ^, X, Y, Z the same
meaning as in the preceding article, this notation referring to any point whatever on
any one of the shortest lines.
All the shortest lines that are of the same length r wUl end on another line
whose length, measured from an arbitrary initial point, we shall denote by v. Thus v
can be regarded as a- function of the indeterminates r, = Y^^{YdX-XdY) + ^_^, \CY-'qX^Z+ iXYZ\
-j^lCX+rtXYZ-^Y'Z}
+ dUy^X-iY).
?v
,,x.-P
102
Since, further.
KARL FRIEDRICH GAUSS
= 7)Z{i-z') + cyz\
rjXYZ-iY'Z = -^X^Z-i:XZ'-^Y^Z
=^-^Z{l~Z^)-iXZ\
our whole expression becomes
d^ = i-z^ {YdX-XdY)
+ iXY-'nZ)d^ + {^Z-iX)dri + {'qX~^Y)dl
15.
The formula just found is true in general, whatever be the nature of the curve.
But if this be a shortest line, then it is clear that the last three terms destroy each
other, and consequently
dj>==- ^ 5^, {XdY- YdX).
But we see at once that
^^^^ {XdY-YdX)
is nothing but the area of the part of the auxiliary sphere, which is formed between
the element of the line L, the two great circles drawn through its extremities and
(Z)
W
(J) (Z)
PP'
(5)
(J) (2)
(S)
PP'
(7)
(3), and the element thus intercepted on the great circle through (1) and (2). This
surface is considered positive, if L and (3) lie on the same side of (1) (2), and if the
NEW GENERAL INTESTIGATIOiSrS OF CURAT^D SURFACES [1826] 103
direction from P to P' is the same as that from (2) to (1) ; negative, if the contrary
of one of these conditions hold ; positive again, if the contrary of both conditions be
true. In other words, the surface is considered positive if we go around the circum-
ference of the figure LL'P'P in the same sense as (1) (2) (3); negative, if we go
in the contrary sense.
If we consider now a finite part of the line from L to L' and denote by <^, <^'
the values of the angles at the two extremities, then we have
<^' = ) + Area iX'P'P,
the sign of the area being taken as explained.
Now let us assume further that, from the origin upon the curved surface, infinitely
many other shortest lines go out, and denote by A that indefinite angle which the
first element, moving counter-clockwise, makes with the first element of the first line ;
and through the other extremities of the different curved lines let a curved line be drawn,
concerning which, first of all, we leave it undecided whether it be a shortest line or
not. If we suppose also that those indefinite values, which
for the first line were (^, <^' , be denoted by \/», i/;' for each of
these lines, then i/*' — i// is capable of being represented in
the same manner on the auxiliary sphere by the space
LL\P\P. Since evidently \\i =