second order and surfaces of rotation belong to the Liouville surfaces. Gauss established the theorem that the sum of the angles of a geodesic triangle (the sides are geodesic lines) is greater than, less than, or equal to, , according as the triangle lies in an elliptic, hyperbolic or parabolic region. The only surface that can contain an area of parabolic points is a developable surface (see 16), i.e., a surface developable upon a plane.

15. Representation of One Surface upon Another Surface. Conformal Representation. Applicability. In map drawing one has an instance of the depiction or representation of one surface upon another. To each point of one surface corresponds a definite point of the other surface. The character of the depiction is a matter of the law of relation of the corresponding points of the two surfaces. If the equations of a surface A are expressed in parameters u, v and those of surface B in parameters, u, v, any equations ug (u, v),

h(u, v) will give a law of correspondence of points, provided that to a pair of values of u, there corresponds a pair of values u1, vi, and conversely.

When each infinitely small triangle on the one surface is depicted in an infinitely small and similar triangle on the other surface, the depiction is said to be conformal. It follows that corresponding angles on A and B are equal; also that ds2 = kds, where ds is arc length on A measured from a point P, ds, the corresponding arc length on B and k a quantity depending on the position of P and independent of the direction of ds. The analytical side of conformal representation is completely resolved by recourse to thermal parameters. The arc element of A in the thermal parameters u, v is_ds2 = a(du3 + dv3); similarly, arc element of B in the thermal parameters un, V1 is ds1 = ẞ (dur + dv2). The relations

u1+ iv = f(u + iv), u1 — iv1 = f1(u — iv), where f and fi are arbitrary conjugate functions establish a conformal representation, since, by virtue of these relations, ds2 = kds1. Any two surfaces can, in general, be conformally represented upon each other in an infinity of ways. The functions ƒ and fi can be chosen to furnish the most advantageous conformal representation.

Two surfaces are applicable or developable upon each other if the corresponding infinitely small triangles are equal in all respects. This requires that corresponding arc elements shall be everywhere equal, namely, that u1 and vi shall be such functions of u and v as to transform the first member of the equation Edu+2Fidu1dv1 + G1dv1

= Edu+2Fdu dv + Gdv

into the second member. The letters with subscripts indicate the elements of the second surface. In general, this transformation cannot be made, and hence two arbitrarily given surfaces are, in general, not developed upon each other. It is obvious that all surfaces derived from a given surface by bending without stretching (see 8) are applicable upon each other. Hence the parameters of any one of them are expressable in the parameters of the original surface. All the surfaces may, accord

ingly, be assumed definite in the same parameters u, v, whence it follows that the fundamental magnitudes E, F, G will be identically_the same for the entire series of surfaces. The three magnitudes E, F, G and all functions formed from them and their partial derivatives are invariants of bending. Some important conclusions can immediately be drawn from these statements. We observe that the lefthand member of equation (7) is the Gauss measure of curvature and that the right-hand member is a function of E, F, G alone. conclude that the Gauss curvature does not change in any deformation of a surface by bending. One notes also that equation (P) depends only on E, F, G, whence the theorem that a geodesic curve remains a geodesic in the deformation by bending.

We

As earth-dwellers the most interesting depiction to us is that of a sphere upon a plane. The sphere is not developable upon a plane and, therefore, any depiction is bound to be a distorted image of the original. A conformal representation will at least preserve angles, and the picture and original will be similar in the corresponding infinitely small parts. The two bestknown examples of a conformal representation are the stereographic projection (Hipparchus, Ptolemy) and the projection of Mercator. Expressing the sphere of radius one in thermal parameters u1, V1:

=x

221

,y

=

201

=

̧u12+v12 — 1 u12 + v12 + 1 u12 + v12 + 1 ́u12+v12+12 and a plane in thermal parameters u, v: x = u, y=v; the stereographic projection is furnished by the relations u + iv = u1 + iv1, u — — iv — u1 — iv1, or simply u= U1, V=V1. For the Mercator correspondence one sets up the relations u1 + iv1 = eu+iv, u1 -iv1 = eu—iv, or u1 = eu cos v, V1eu sin v. In the stereographic projection the circles of the sphere are represented by circles (or straight lines) in the plane; in the Mercator map the meridians and parallels of latitude appear in the plane as a system of orthogonally intersecting right lines.

SPECIAL SURFACES.

16. Ruled Surfaces.-The continuous motion of a straight line through a simple infinity of position generates a ruled surface. When the consecutive generators intersect, that is, when the generating line is always tangent to one and the same space curve, the surface is a developable surface. In the contrary case the surface is called a skew surface. The director cone of the ruled surface is formed by drawing through an arbitrary point of space lines parallel to the generators of the ruled surface. The equations of the most general ruled surface are x = a1 + B1u, y = a2 + ẞ2u, z = a3 + ẞзu, where a, a, as; B1, B2, B3 are functions of v alone. The curves v constant are the rightline generators; the curves u constant are trajectories of the generators. The trajectory 10 is sometimes called the director curve. The important elements are: (1) The angle do between two consecutive generators g and g1; (2) the shortest distance dk between and g1; (3) the value of u corresponding to the point,

do

B1 B2 B3

da, da, das

u =

B1 B2 B3 B1 B2 B3 da, da, da, VB12 + B22 + B22' VB12 + B22 + B32 The locus of the central points of the generators is a curve called the line of striction. Its equation is the third of the group above. For a developable surface dk=0. The surface is cylindrical if B1 = B2 = B=0. The tangent plane to a skew surface at a point contains the generator through the point, and the plane rotates about the line as the point of contact moves on the line. The normals of the surface along a generator form a hyperbolic paraboloid. A ruled surface may undergo a deformation by bending so as to remain a ruled surface. The developable surfaces are so named because they are developable upon a plane. Their linear element ds admits of being thrown into the form of the ds of the plane.

17. Surfaces of Constant Gauss Curvature. -These surfaces are of three kinds: (1) Surfaces of constant positive curvature for which K

1

or is positive and constant at every point. RR

The sphere is the type of this class. (2) Surfaces of zero curvature for which K=0 at all points. These are the developable surfaces (see 16) of which the plane is the type. (3) Surfaces of constant negative curvature, K≤0, of which the pseudosphere is the type. The pseudosphere is the surface generated by rotating a tractrix about its asymptote. Minding showed (Crelle's Journal für Mathematik, 19, 1839') that all surfaces of the same constant curvature are developable upon each other, and in 3 of ways. The geometry of figures on a surface of constant positive curvature may, therefore, be studied on a sphere; that of developable surface on a plane; and, finally, the geometry of surfaces of constant negative curvature is identical with that of the pseudosphere. All surfaces of constant negative curvature are called pseudospherical surfaces. In employing a geodesic polar system of reference (see 14) the ds of pseudospherical surfaces with measure of curvature takes the form ds2 : du2 + R2 sin / dv2. A R2 R characteristic distinction between the geometries of the three kinds of surfaces of constant curyature is indicated by the number of geodesics that may be drawn through a point parallel to a given geodesic. On a pseudospherical surface two parallels may be drawn; on a developable surface only one parallel; on the surface of positive curvature, none.

น

18. Minimal Surfaces. They are defined to be the surfaces of mean curvature zero, H= (see 12). At every point of such a surface R1 = - R. Historically the theory of these surfaces had its origin in Lagrange's investigation of the surface of minimum area with prescribed boundary curve. It was ascertained that for such a surface R1 = R2, but

F and F are conjugate functions of the conjugate complex variables u and v. All minimal surfaces are contained in these formulas. When the function F is algebraic the surface is algebraic, and conversely. A minimal surface can be deformed by bending so as to remain a minimal surface. For this it is necessary and sufficient that one replace F (u) and F() by eai F(u) and eaiFi(v) respect. ively. All the surfaces corresponding to values of a (real constants) are developable upon each other as having the same ds. The only ruled minimal surface is the ordinary helicoidal surface with director plane. The only minimal surface of rotation is the catenoid, i.e., the surface generated by rotating a catenary about its base. These two surfaces are developable upon each other. The minimal surface is the only surface (aside from the sphere) whose spherical representation is conformal (see 12 and 15). For particulars as to these surfaces consult Schwarz, 'Gesammelte Mathematische Abhandlungen' (Vol. I, 1890); Darboux, 'Leçons sur la Theorie Générale des Surfaces' Vol. I, 1887).

19. Concluding Remarks.—In addition to the special surfaces here described may be noted the surface of centres, i.e., the locus of the centres of curvature of a given surface. It consists of two sheets S1 and S2 corresponding to the two centres of curvature at every point. Also may be noted the W-surfaces or Weingarten surfaces, in which R and R2 are functionally related. A functional relation, ƒ(R1, R2) =Ő, defines an infinity of surfaces. In passing to the surface of centres of each individual of such a group, Weingarten showed that all the sheets S are developable upon each other and upon the same surface of rotation. The same theorem holds of course for the sheets S2. Surfaces of constant curvature, and minimal surfaces are special W-surfaces. Finally, there are certain analytical expressions constructed in the fundamental magnitudes, some arbitrary functions, and derivatives of the functions, which have the same value whatever the parameters employed. In other words they are quantities invariant with respect to a change of parameters. When the arbitrary functions are present they are called differential parameters. When the arbitrary functions are not present they are called differential invariants.

Mani

festly these magnitudes are connected with those geometrical properties that are essentially independent of the particular system of parameter reference. For example, Gauss curvature is obviously an invariant.

1

RiR2 Bibliography. Some of the principal treatises on the theory of surfaces are Darboux,

G., 'Leçons sur la Théorie Générale des Surfaces (4 vols., Paris 1887-96); Bianchi, L., 'Lezioni di G'ometria Differenziale' (Pisa, 2d ed., 1902, translated into German by Lukat, 'Differentialgeometrie,' Leipzig 1896); Raffy, L., Leçons sur les Applications Geométriques de l'Analyse' (Paris 1897); Scheffers, G., 'Anwendung der Differential-und integralrechnung auf Geometrie (2 vols., Leipzig 1902); Salmon, Geometry of Three Dimensions (Dublin 1882); Salmon-Fiedler, Analytische Geometrie des Raumes' (Leipzig 1880-98). JAMES MACLAY,

In

Professor of Mathematics, Columbia University. SURGEON IN THE ARMY AND NAVY OF THE UNITED STATES. The history of the connection of a regular staff of surgeons to the army dates back to the siege of Boston in 1775. At that time the Second Provincial Congress of Massachusetts Bay was in session and on 8 May 1775 appointed a committee "to examine such persons as are or may be recommended for surgeons with the army now forming in this colony." Of the 16 candidates examined only six were rejected. On 21 July 1775 Washington urged in a letter to the Colonial Congress that a "Hospital Department" be established, and on 27 July this department was created, having a director-general, chief physician and 20 surgeon's mates. April 1776 Congress passed an act that the board of surgeons be increased "not exceeding one surgeon and five mates to every 5,000 men, to be reduced when the army is reduced." Between 1784-89 no medical department was officially recognized by the government, but was in the latter year recognized and remained the same until 1821, when it took the form which it retained with no decided change until 1908 when the titles of all medical officers, excepting the surgeon-general of the army, became the military rank, major, captain, etc., followed by the words, Medical Corps, U. S. A. Applicants for the medical corps must be between 22 and 30 years of age, be citizens of the United States and graduates of a reputable medical school and have had at least one year's hospital training. After successful examination they are commissioned as first lieutenants. The Act of 1908 also established the Medical Reserve Corps, the members of which rank as first lieutenants and are appointed by the President after an examination. During the Civil War and the Spanish-American War the corps was increased to meet the emergencies, but on the cessation of hostilities was again placed upon a peace footing. Consult Lamphere, United States Government' (Philadelphia 1880); Farrow, 'Military Encyclopedia' (New York 1885); Hamersley, Army and Navy Register' (New York 1888) and other War and Navy Department records. See UNITED STATES, ARMY OF THE; UNITED STATES, NAVY OF THE.

SURGEON-FISH, a fish of the family Teuthida, allied to the butterfly-fishes (q.v.) and distinguished by the presence of one or more erectile lancet-like spines on each side of the root of the tail which may inflict an ugly wound. Some 80 species are known, mostly of the genus Teuthis, scattered through the warm seas of the world and variously known as doctor-fish, lancet-fish, tangs and among the Spanish-speaking fishermen of the West Indies

as barberos and médicos. All are oblong, compressed, brownish or bluish fishes from 6 to 12 inches long, with narrow protruding incisorlike teeth. Several are good food, especially the ocean surgeon-fish (T. bahianus) found throughout the West Indies, and the blue tang (T. caruleus), found near Porto Rico.

SURGERY, History of General. The history of medicine, fascinating at all times, is most valuable to the surgeon when applied to the subject of general surgery. At the present time there is much in the literature of medicine that is very instructive and it becomes more and more so as we are put in possession of facts pertaining to prehistoric man and of those presented in the study of manuscripts antedating the Christian era.

Skulls of the prehistoric period have been discovered in caves and dwelling-places which show undoubted evidence that trephining had been performed on them and that healing of the bone had later taken place. The examination of these specimens is exceedingly interesting. Again, as we study the periods between this age and the beginning of the Christian era we note a great deal that leads to the conviction that surgery was understood in those times. The early embalmers, in their familiarity with the human body, must have acquired some knowledge of surgery, and this as far back as 1700 B.C.

While medicine as a profession was confined to the duties of the priest, surgery suffered, and was more and more neglected. Historians of that time have just reason to condemn Egyptian surgery. There are some very excellent books appearing at the present time giving a full description of what little was known, bestowing full credit upon those who practised surgery, or advocated it, and who were making some progress in that branch of medicine. The Egyptians were proud and fond of their work as scribes or writers, their learning in that direction leading them to advise their sons to take up what was then known on the subject of medicine; but very little seems to have been developed in the way of clinical observation, or, more particularly, in surgical procedure.

In the study of medicine by the Hindu one cannot but note that in their writings there is plain evidence that surgery had reached an advanced stage. "Surgery," says their great Susruta, "is the first and highest division of the healing art, least liable to fallacy, pure in itself, perpetual in its applicability, the worthy produce of heaven, the sure source of fame on earth." He also makes the very excellent observation that "he who knows but one branch of his art is like a bird with one wing." This writer was a very careful observer and beyond doubt did much to advance the art of surgery centuries before the birth of Christ. Some of the Hindu works and operations are yet spoken of by modern writers. They knew how to perform successfully the operation of developing a new nose by flaps taken from the forehead and were familiar with like procedures. The surgery of ancient India is worthy our most thorough and careful investigation. Great interest is being manifested in its study and there can be no doubt that good will result from the recent organization of the Charaka Club in

New York. This body of investigators bids fair to give us papers of great value in the elucidation of that period when Hindu surgery developed - and, it may be said, ceased. Why it disappeared so mysteriously has never been shown.

Through the study of manuscripts of succeeding centuries, and of other records of different peoples, one is greatly impressed with the difficult operations performed, now and then, by some predominating man, who, perhaps having more knowledge of the anatomy of the human body, and being somewhat bolder than his fellows, would perform an operation, leaving a report of possible success. Then for centuries this work would be forgotten, then 'revived, perhaps modified. Possibly to the operator it was a new operation (since he was not aware that it had been performed previously), yet to be developed into a line of work that was to lead to greater success. Thus we see reported very important advances in the line of surgical procedure from our earliest knowledge on.

In

The early Greek knowledge of surgery is not so apparent or abundant as that of later periods, yet Homeric medicine gives endorsement to the fact that there were surgeons capable of rendering aid in emergencies, such as the removal of foreign substances from the body, and who were able to control bleeding by the application of what were evidently understood to be drugs possessing some styptic power and to bind up and dress wounds. Even fractures and dislocations were treated. time of war these men were looked upon with reverence and the aid they were expected to render was highly valued. Of their real work there is little known before the Trojan War. The most observing student of cases in antiquity was Hippocrates (q.v.) born 460 B.C. He wrote on the treatment of articulations, luxations, fractures and also on the subject of the use of instruments, but his knowledge of anatomy must have been very meagre. The Greeks had great respect for their dead, which prevented dissection of the human body. They knew nothing of physiology and, therefore, anatomical structures, such as arteries, veins, nerves, tendons, ligaments and membranes were hopelessly confused. Hippocrates gave classifications not unlike those of the present day, "internal medicine" and "external or surgical medicine,” which were convenient, but not philosophical. The period of his life marked the transition from mythology to history. His doctrine and clinical observations were received with great respect. He advanced the science and art of surgery, but only a little later ignorance again reigned in the school which he made celebrated.

Erasistratus (about 300 B.C.) was a close observer as an anatomist and made use of his knowledge of the valves of the heart. He discovered the lymph vessels, described the epiglottis, successfully removed the spleen and performed other remarkable operations.

Aretæus was one of the most brilliant men of the second century before the Christian era. He recognized surgical affections of the brain and described the Syriac or Egyptian ulcer, tetanus and anal fistula.

Galen (q.v.), who died about 201 A.D., must have had considerable knowledge of anatomy.

History indicates that he was a vivisector, and to him is due a clear classification of the muscles, which is followed at the present day, as the "flexors and extensors." He came very near discovering the circulation of the blood, and divided the body into the "cranial, thoracic and abdominal cavities," whose proper viscera and envelopes he described. This was during a period when encouragement was given to the study of anatomy. The works of one Oribasius (q.v.), who died about 400, were based on the writings of those who had preceded him, but had a distinct importance of their own. He showed much originality in the treatment of hydrocephalus, advised paracentesis of thorax and abdomen, removal of vesical calculi, treatment of aneurism, excision of hypertrophied mammæ in men, etc. Antyllus who flourished about this period was one of the most distinguished and original surgeons of antiquity. He was the first to describe the extraction of small cataracts and is perhaps best known to the surgical world of to-day by his exceedingly bold plan of opening aneurisms, so successfully imitated by the late James Symes.

It is to be remembered that during the Greek period Galen and his followers dissected animals and occasionally a corpse on the field of battle.

In the 6th and 7th centuries the Arabians gave more encouragement to dissection and demonstrated that surgery required a knowledge of anatomy. One of the most celebrated Arabian physicians was Rhazes (q.v.), who died about 932. He compiled from all authors some 37 books on medicine and surgery.

Albucasis (936-1013) in one of his writings gives a most detailed account of necessary instruments and in speaking of their proper use and application to surgery, he emphasizes the fact that surgeons should be versed in the science of anatomy.

In visiting the various museums in Europe at the present time, especially Naples, one is greatly impressed with the variety of ancient surgical instruments that have been recovered from the ruins of Pompeii and other former surgical centres.

Avenzoar (q.v.), who died about 1169, wrote one of the most remarkable treatises on renal diseases, especially in reference to the treatment of calculus and further surgical intervention.

From the 9th to the 13th century the Jews and the Christian clergy shared the honors of the healing art, and during this time references are not infrequently made to the work of the barber-surgeon (see Barber). Lithotomy (q.v.) seems to have been developed in this period and it is noted (1022) that Henry II, Holy Roman emperor, was cut for stone by Saint Benedict himself.

In the 13th century Rolger of Palermo was evidently one of the most distinguished pioneers in modern surgery. He was the first to use the term "seton." His pupil, Roland, wrote a treatise on surgery, which became very famous and was mentioned by Guy de Chauliac, "restorer of French surgery in the 14th century. The latter was probably one of the most famous surgeons of that time. He opened the abdomen for dropsy and operated for the radical cure of hernia and for cataract.

The history of the school of Salerno in the 13th century indicates that practitioners of surgery had to devote a certain time to the study of anatomy, were obliged to pass an examination by the faculty of the university and were licensed by the royal hands. Surgeons recognized the importance of nausea, vomiting and hemorrhage from the ears, in injuries to the head. They used the trephine (q.v.) in treating fractures of the skull and treated hernia cerebri by pressure and caustics. Ligatures were used in wounds of the carotid arteries and jugular veins. The surgeons also treated wounds of the abdomen. Lithotomy was described with care and compound fractures were treated with splints.

The first important work on minor surgery appeared during the 14th century. It was written by Lanfranc, but although it grew into a second and larger treatise, surgery soon after this began to decline. The barber-surgeons of this time seemingly commanded considerable attention, although it is evident that the importance of this body of men has been greatly exaggerated.

From the 15th century through the 19th surgery developed more and more as a science. In England, Thomas Linacre of Canterbury (1460-1524) was one of the earliest writers. Jerome Fabricius (1537-1619) was also a noted writer and authority on the practice of surgery during this period. Benivieni during the same period was, according to Malgaigne, the first to impress upon the profession the importance of searching in the cadaver for the concealed cause of disease. His observations on gall-stones and conveyance of syphilis from the mother to the foetus were original.

Notwithstanding the progress in surgical science during the 15th and 16th centuries, the practice of surgery was largely abandoned to a class of ignorant barbers and bone-setters. Most of these operators traveled from city to city, individual practitioners limiting themselves to the operation for stone or for hernia. This condition of affairs, together with the prejudice against dissection, was most unfavorable for the profession of surgery.

France at this time presented the only special college for the instruction of surgeons. To the 16th century belongs the career of that most wonderful surgeon, Ambroise Paré. He was an original thinker, had the courage of his convictions and did away with the use of the cautery and boiling oil in amputations, using ligatures to control hemorrhage, the latter being the most important advance until the introduction of ether in 1846 (see ANESTHETICS). At the beginning of the 17th century surgery reached a higher social and intellectual plane than it had heretofore occupied. Amphitheatres for dissection developed in many European cities, together with hospitals and dispensaries in connection with the various schools. The term "inflammation" was then introduced and from that time until the present day has been a subject for continuous investigation. From this period may be dated the beginning of consultation work between expert practitioners; and clinical teaching and the presentation of surgical cases then advanced their claims. Surgical history from Valsalva on presents the names of many who became eminent operators,

and in their writings did much to advance the art of surgery. Notably was this so among the Italians, who in their plastic surgery developed the Italian method for construction of

a new nose.

In France, Morel (1674) invented the tourniquet, Denis performed the first transfusion of blood in man and other French surgeons became very expert in the operation for lithotomy. Mareschal (1658-1736) had a record of eight lithotomies performed in half an hour. He was one of the founders of the French Academy of Surgery.

In Holland, Rau (1658-1719) taught practical surgery upon the cadaver.

Wiseman (1625-86) was the first to develop English surgery. He was also the first to do external urethrotomy for relief of stricture. At this time in France alone was instruction in surgery well regulated, as it was the only country which possessed a proper surgical college.

In the 18th century hospitals began to multiply in Germany, benefiting general surgery to a great extent. Brasdor (1721-76) developed the method of distal ligation of aneurisms, while Sabatier (1732-1811) wrote a treatise on operations and recommended resection of the head of the humerus. DeSault and Chopart did much in developing operative surgery. In Italy Scarpa (1752-1832) advanced our knowledge of hernia and aneurisms. Spanish surgeons did little to improve the science and art of surgery. In England, Cheselden (1688-1752) did much in advancing the knowledge of pathology and general surgery. White, of Manchester, devised a method of reducing dislocations of the humerus with the foot in the axilla. A well-defined operation for excision of the joints was also first practised in England. The investigation of pathology and diagnosis in France at this time had much to do in the 19th century in developing the "new Vienna School." Percival Pott (q.v.) did much in elaborating and classifying diseases of the joints and especially spinal diseases. John Hunter (q.v.) was the most famous English surgeon of his day. He belonged to a family which in many ways assisted the development of pathological anatomy and surgical technique. At the close of the 18th century, Benjamin Bell was the first to make use of drainage by means of tubes of lead or silver. In France, Bichat (q.v.), although not generally so understood, became a forceful lecturer on surgery and did much to bring hospital-gangrene under observation and control. The Dutch during the 18th century developed some eminent surgeons, and it is interesting to note how dextrous they became in the use of instruments. At the same time their knowledge of anatomy enabled them to present some very able papers on the subjects of hernia and dislocation. Sandifort, of Leyden, first described a downward dislocation of the femur.

In reference to the surgery of our own country in the 18th century, one of the most interesting works, which was of great service to American surgeons during the Revolution, was that of John Jones, Plain, Practically Precise Remarks on the Treatment of Wounds and Fractures.'

The 19th century witnessed great advances and from 1838 was one continuous chapter of investigation toward the development of the