In some weight-clocks the striking-train and tell comprise the driving-weight. The striking mechanism is released by pins projecting from the back of the little plates carrying the hoursigns. These pins trip a small lever as the train passes. Clocks drawn by a spring have the spring-barrel located in the lower part of the case A clock of this type in my possession has the general appearance of a hall clock of our grandfathers' days except for its diminutive size. It is eight inches high, three-fourths of an inch deep, and one and one-fourth inches wide. The case is beautifully made of dark wood. The upper part of it, enclosing the works, has glass front and sides, the cap over the balancewheel, as well as the front plate of the works, which are of brass, is open-work of graceful design and is gilded. Another clock of this type, also in my possession, is still more diminutive in size, being only three and three-fourths inches high, one-fourth inch deep, and three-fourths inch wide. It is made entirely of brass except the numerals, which are of silver, and is beautifully engraved and gilded. At the bottom of the case there is a small compartment closed by a hinged door. This contains the key. The numerals are fitted into a dovetail groove in the front of the case, and the hand is carried on a sliding piece attached in the manner before mentioned to the fusie chain. There are no divisions to indicate the fractions of the hour. Another interesting example of this type has a dial engraved with a series of logarithmic curves. On the faces of these clocks there are two rows of characters; when the dials are rectilinear, the characters are arranged in two vertical columns; when circular, in two concentric circles. These rows are some little distance apart, and the characters are unequally spaced. Each numeral is connected to its opposite one by a logarithmic curve. The space between the columns is divided into twelve equal parts by parallel vertical lines, each line having at its upper extremity the sign of a month. The space included between the intersections of one of these lines with two successive logarithmic curves, will indicate the length of the corresponding hour for the first day of the month which is indicated by that line. In this clock the index is borne on a cross-bar, which extends across the dial from one column to the other and is attached to the weightcord. The index is so affixed to this bar that it can be moved along its length, thus passing from one line to the other as the months elapse. When this kind of clock is provided with a circular dial, the logarithmic curves are laid out in the same manner and intersected by parallel concentric circles. The hand moves over the dial and is constructed so as to slide through its attachment to its arbor, thus being lengthened and shortened. Another clock of this type has a much more complicated structure. Its circular dial revolves and is furnished with movable hour-signs, which are arranged in concentric circular grooves on its face. A pin projecting from the posterior face of each opposite hour-sign enters the groove in a slotted arm which extends across the back of the dial. These arms are acted upon by an eccentric, which in its turn is driven by a train of wheels completing its cycle in a year. The action of this mechanism is such that the opposite ends of the arms and consequently the hoursigns are separated and approximated as the days and nights vary in length. It only remains to describe the clocks of the second class, viz., those in which the rate is made to vary in accordance with the seasons. None of these clocks, as far as I am aware, have the balance-wheel and hairspring, but they have its forerunner and immediate ancestor, the escapement of Huygens, which consists of a vertical staff suspended by a fine silk thread attached to its upper end. This staff is provided with lugs which engage the teeth of a crown-escapement wheel, and it bears a horizontal arm from which small weights are suspended like a scale-beam. The rate of the clock is regulated by the adjustment of these weights. In general form, these clocks are rectangular or cube-shaped. The gong is placed on top of the case. The dial is circular and revolves from right to left, the hand being stationary. The case is of brass and is usually highly ornamented. The variation of rate in these clocks is accomplished in two ways, viz., (1) entirely by the adjustment of the weights borne on the arm of the escapement, and (2) partly in the foregoing manner and partly by the mechanism itself; the latter form having a double escapement, which will be described later. The specimen of the former kind which I have is two and onehalf inches wide, two and one-half inches deep, and seven inches high over all. The case is of brass, and is beautifully ornamented by chasing, and the wheels, which are cut by hand, are very ac curately made. The characters are engraved on the dial in two circles, the outer one being composed of the signs of the Chinese Zodiac, and the inner one, of the hour-signs. Below the dial, on the face of the clock, are two openings, through each of which may be seen an astrological character. These characters change once in twenty-four hours. The weight-cords run over spiked pulleys and have small counter-weights. The clock has a striking-train and a going-train. Another clock of this form in my possession is of more compli cated construction. It has two escapements, the horizontal arms of which are of different lengths. In this clock the variation of rate is accomplished partly by hand and partly by the automatic operation of the mechanism itself. One escapement remains idle during the day and the other during the night, the staff of one being lifted from its engagement with the escapement-wheel at the same time that the other is brought into gear. This is accomplished by two levers which lie directly below the ends of the vertical staves of the balance. The opposite ends of these levers are acted upon by two cams on the same arbor which cause one of them to rise and the other to fall at the proper moment. I have omitted to say anything of the fantastic astrological meanings of the various characters found on these clocks and of the intimate connection between the astronomy, astrology, and horology of the Japanese, and will only add that if they are children in imagination they are certainly giants in mechanical execution. In writing this article I have availed myself of the articles written by Emil James, Journal Science D'Horology, Vol. VIII.; Aneè and Thomas Eggleston, Ph.D., in the School of Mines Quarterly for July, 1892. SOME BIOLOGICAL NOTES ON AMBLYSTOMA TIGRINUM I. BY HENRY LESLIE OSBORN, PH.D., ST. PAUL. MINN. THERE is a salamander, most probably of the species named above, which is very common in this vicinity. In the autumn months, especially during September, it can be found abundantly in cellars or in damp, dark, or semi-dark places about buildings. I have often seen it on the railroad tracks imprisoned between the rails, and many specimens which had been run over and killed by the cars can be found at this season. Occasionally they are seen creeping about on the walks or in the grass, where they are frightened by man's approach and run actively away. They are familiarly called lizards, and the use of that word among the people of this vicinity can almost always be understood to refer to this animal. It lives in aquaria for an indefinite time, remaining on the bottom, and coming to the surface for renewal of air of the lungs rarely. 1. The markings of this salamander are vivid yellow spots upon a ground of brown-black upon the back, giving place to faint bluish ground and lighter color on the ventral surface. There is a very great deal of variation in the shape and distribution of the spots. In general, they are irregular, elongate figures of various sizes from very small rounded ones up to those of considerable size, whose length may be equal to half an inch. The directions of the long ones of the spots are not the same, while they are chiefly antero-posterior, some are oblique from behind, forward and inward, while others are oblique from behind, forward and outward. The patterns of the two sides are not "mated," they are entirely independent. Not only so, but there is a distinct line which separates them, and in the middle a black line often cuts directly through the spots, so that, while they meet, they do not match. This last-named condition is very noticeable in the tail, as shown in the accompanying figure. It is very conspicuous in many cases, but perhaps less noticeable in specimens not so largely spotted as the one used in making the figure. This absence of bilateral symmetry in the skin markings is a more or less general phenomenon in the coloration of animals; they rarely having their two sides perfect counterparts. It is in fact a case of a general law, applying to all bilateral organs, perfect bilateraling being a very rare phenomenon, due, on modern biological views, to the preponderance of growth in the cells of one organ over its homologue of the opposite side through the operation of any of the several causes which influence vitality Scale N of cells, e g., use, nutrition, disease, perhaps inheritance. But, in animal coloration, while perfect bilateraling of marking is unusual, and a certain independence of the opposite sides is usual, it is rarely carried so far as here. The markings of birds. etc., blend across the middle line, so, too, the blotches of snakes, frogs, and other familiar cases, and I have never seen an animal in which the independence of the color markings of the two sides is as pronounced as it is in this form. A fact of this kind would appear to have some important suggestions in relation to the ontogenetic history of the yellow color producing cells. If they are separated early in their history and continue distinct, we should expect such a separation in their ultimate products. There are facts enough to indicate that in lower forms, such as annelids, the cells of the two sides of the body in many of the organ systems are separate from an early date, even as early as in the early segmentative stage of the egg (cf., E. B. Wilson, “The Cell Lineage of Nirus,” Journal of Morphology, vol. vi., p. 36, 1892). This supposition would not be out of accord with the fact that the independence of coloration is found in a lower rather than one of the higher animal groups and in a lower member of its group, for it is the characteristic of the higher forms to have more and more intimate relation of parts. The distribution of the color-spots I cannot as yet reduce to any law by study of adults, and I know of no observations in the embryology of Amblystoma which have been directed upon this point. There seem to be some faint suggestions of metamerism in the coloration of the area of the side walls of the body, especially between the limbs. The body wall in this region is marked on the ventral aspect and laterally by rings (Myotorms), which correspond with the attachments of the muscle fibres, and the color spots are rather noticeably located upon the rings rather than on the spaces between them. The rings look like somites of an annelid, and it would be interesting to know if they correspond with the segmentation of the vertebræ and nervous system. 2. The movements and locomotion of the salamander are very interesting to observe. They suggest an animal which is passing from the use of the back-bone and its curvatures as a mechanism for locomotion to the use of limbs. The locomotive movements are of two classes, the first are those performed under ordinary circumstances, the second those performed to escape from a pursuer, as when one attempts to seize the creature. The former are made by means of a combined use of the back bone, which is thrown into gentle curvatures, and the legs, which are the chief instruments in the act. The curvature of the back-bone is such as to throw the limb to be used forward further than it would be with the spine kept straight. The limbs are used in strict alternation, the right front leg and the left hind leg going forward together, and then backward together, while the spine has a convexity toward the right in the brachial region and toward the left in the sacral region. The creature, in water, when disturbed by one's hand generally either makes a disorderly scramble with the limbs, which has but little result, or it swims swiftly with a truly fish-like sinuation of the body, including the large post-anal region or "tail," which is much compressed and forms a very efficient organ of swimming. It has seemed to me that this swimming motion may be a case of physiological reversion. We know that the vertebral musculature is far more ancient phylogenetically than the limb musculature, and we may suppose that hence the power to control it nervously is far greater than that to control the more recently acquired limb musculature. It is a case of the tendency to fall back on the ancestral mode of action so long as the structure will permit, especially under circumstances in which the animal is under the influence of strong excitement, which would tend to weaken the more recently acquired powers and allow the ancient lines of habit to become dominant. This tendency can be discerned in many other cases; thus, for instance. I regard the case of the crayfish as precisely similar to the one just cited. It commonly moves by a walking motion, not using the flexion of the abdomen, but under excitement of escape it reverts to this ancestral action, and the familiar "crawfish" movement results. I do not think it is at all beyond the range of reason to include the tendency of people to lapse into a native language from an acquired one in moments of excitement under the same principle of physiological reversion. In this connection, I may speak of a specimen of Necturus, which I had for some time in an aquarium in the laboratory, in which the swimming movements were even more noticeable than in the salamander, a fact co-ordinated with its more piscine peculiarities in other respects. It is possible to discover in the movements a suggestion of the origin of limbs. The limbs are usually in a line, and the front right leg is thrown forward by the curvature of the body at the same time that the left hind leg is thrown forward by the curvature in its level. Limbs at these points, if at first mere stumps, would be of advantage by the hold they would give to the squirming body. Then elongation would increase the advantage. No loss of this function would be necessary, but a gain, if the limb acquired some independent motion, and this might be developed enough to render it capable of officiating as the sole locomotive organ. If such a history of the limb were true, the salamander is midway in the line of descent. In A LABORATORY OF PLANT DISEASES. BY C. W. WOODWORTH, BERKELEY, CAL. THERE has recently been equipped at the University of California a laboratory for the study of the subject of plant diseases in its broadest sense; and, as there are but few if any others where the whole subject is taught as a unit, it may be well to give an outline of the equipment for this class of work. We will not consider that part of the equipment for this work afforded by the grounds, orchard, nursery, gardens, and greenhouses of the agricultural department, but confine ourselves to the laboratory proper. The subject of plant diseases is now, and will continue to be, associated with that of entomology, so that the same equipment, to a considerable extent, serves for the two subjects. 3. The post-anal region of the salamander is piscine, while the anterior portions of the body are not, but are distinctly higher. This fact is more or less familiar in a general way and called by Professor Hyatt, who pointed it out many years ago, by the name "cephalization." This advance of the anterior part of the body of the salamander has left the "tail" to be in many respects not amphibian so much as piscine. Of course the term tail here means post-anal region of the body and the portion, roughly speaking, homologous with the post-anal region of the fish. the higher fishes this region has acquired a "tail," while the amphibia have not shared the acquisition of a structure supported by five rays, which does not belong to the ancient vertebrate stock. In this sense the tail of the salamander and its correlate, the post anal region of the fish, are not only similar in function, being organs of locomotion, but they are comparable in their anatomy. The back-bone is acentrum with bi-concave surfaces with twɔ equally developed arches, a neural arch containing the spinal cord, and an haemal arch containing a vein and an artery with oblique intervertebral muscles forming the back of the organ. In vertebrates above the urodela, with the loss of its locomotor function and the development of arms and legs, the post-anal region becomes of less and less importance, though not always disappearing; thus in many lizards it is large at its origin, as large as the body before it, and it has the peculiar power of autotony, as it has been called; that is, of breaking off in the hands of a captor, whereby the animal escapes capture. There is a gradual degeneration of the region among the higher vertebrates, with many varieties of direction and degree of development and occasional utilities in peculiar directions and the salamander stands at the bottom of this series. The laboratory-room is something over twenty by thirty feet, and is situated on the north side of the Experiment Station building. It is lighted by four windows, having an entirely unobstructed view, and so giving ample light for microscope work. A corner of the room is partitioned off for a private laboratory, and a closet is fitted with a ruby window, affording an opportunity for photo and blue-print work. The figure below will give a good idea of the arrangement of the room. 4. The death of the salamander is accompanied by a loss of powers of movement, which is first manifest in the last acquired (phylogenetically) of the powers, i.e., in the limbs, and finally in the vertebræ muscles. In specimens killed under the influence of chloroform, after all movement had ceased in the limbs, the sinuations of the back-bone continued for some time, and were the last movements observed to take place. REFLEX ACTION IN TURTLES. BY M J ELROD, ILLINOIS WESLEYAN UNIVERSITY, BLOOMINGTON, ILL. RECENTLY I had a number of map turtles (Malaclemys geographicus Le Sueur) for student work, and observed, what is to me, a remarkable instance of reflex muscular action, both in the head and limbs. In one specimen the head had been severed from the body fully an hour, when I observed the students amusing themselves by tapping the nose of the severed head, when almost as quickly as in life the jaws would open, and when a pencil or other hard object was thrust in would close upon it with seemingly as much viciousness as in life, continuing to hold for some time, gradually relaxing, when the experiment would be tried over again. This was the case not only with the one in question, but with a half-dozen others of the same lot. Taking a specimen with the head cut off and all the viscera cleared away, leaving the legs attached to the carapace, the legs manifested sensitiveness to a marked degree. In one specimen the four legs extended from the body almost straight; a very gentle touch with the point of a pencil on the tip of a claw caused that leg to be drawn within the shell, so to speak, as quickly as in life. This was done alternately with each foot to the first again, all giving the same results. Several other specimens tested showed as much and as sudden movement, and one killed at 2 P. M., when touched at 11 A M. the day following, withdrew its feet instantly. While these observations are common for turtles, I have not observed such marked results in other species. The cloth bearing for the glass is treated with corrosive sublimate, and the paste and glue used are arseniated. These boxes are kept in cabinets, the glass doors of which are fitted with a rabbited groove on all four sides, thus making them also dust- and insect-proof. The collections kept in these cabinets are arranged in three series. Series one is the systematic collection, where the organisms producing injuries to plants are grouped in the ordinary order, beginning with mammals and ending with the higher plants. The second series is the "host" collection, where the various plants are taken up in an agricultural order, as, for instance, seed crops, fruit crops, etc., and the injuries to each particular crop illustrated. In the third series, the symptomatic collection, all diseases having a common symptom are brought together, thus all galls and distortions from whatever cause or on whatever plant are assembled and classified. Besides these there are the beginnings of a cryptogamic herb arium in drawers and a collection representing the materia medica of plant diseases. There are in the laboratory a sterilizer and all the other necessary apparatus for this class of bacteriological work. For microscopical and histological work there is also a good equipment including paraffin bath, microtome suitable for the highest grade of work, compound microscopes and accessories, and a very good outfit of reagents. All reagents, as far as possible, are kept in standard strengths, and the bottles marked to serve as graduates for dilution. Thus the chromic acid is made up in a large bottle into a 5 per cent solution. The 1 per cent solution is made by filling the bottle to contain it to a mark and adding water. Most of the chromic mixtures are made from the one per cent. The chromic-acetic killing mixture, for instance, is made, as is indicated on the label, from one-half per cent chromic acid to the first mark, 95 per cent alcohol to the second, and 10 per cent acetic acid to the neck. Mixtures liable to deteriorate are kept in small bottles, and such as the acid-alcohols for decolorizing are not kept mixed at all, but large homo vials are properly labelled and the mixtures made up as used. This sketch gives merely the present condition of the laboratory, it is expected that apparatus will be added from time to time as opportunity offers and as it is needed for the work in hand; indeed, there is considerable new apparatus at the present time being constructed for the laboratory. AN IMPORTANT COLLECTION OF MOLLUSCA. It is not generally known, even among specialists, that one of the most valuable and most instructively arranged collections of Mollusca in America, is that which Professor Henry A. Ward has brought together at Rochester, N.Y. This collection the writer has recently had an opportunity to examine, and it is believed that some account of it may be useful not only to specialists in Mollusk morphology, or conchologists desiring to see rare shells, but also to those who look upon a collection especially as an instrument of education for class or public use. The primary idea of Professor Ward's collection is to give the spectator not only a comprehensive but a comprehensible view of all phases of Mollusk life; and to this end a number of the more typically developed forms of each genus have been selected for exhibition. The practical advantage in limiting the number of species representing each genus will be readily admitted by those who have observed the effect, on the non-scientific observer, of the vast wilderness of similar species exhibited in some of the public museums of our large cities. A further purpose has been to procure the best specimens obtainable of each species represented, and to select not merely the rare and beautiful, but, before all, species and specimens which have a life history worth knowing, and can tell it themselves. The dry specimens of shells are contained in horizontal glazed cases disposed around the sides of two rooms,― in all, about 220 linear feet of cases. Wall-cases behind them contain alcoholic Mollusks, and drawers below hold additional species. The specimens are mounted upon light wooden tablets, appropriately colored, and made by gluing two pieces together, crossing the grain to prevent warping. Labels for families and higher groups are printed, and in most cases contain a concise statement of the fundamental characters of the group. The shellless forms, such as most Cephalopods and the Nudibranchiata are represented by Blatschka's beautiful models, now, alas! no longer obtainable. A few hasty notes upon some of the specimens may be of interest. Upon entering the outer room one sees suspended from the ceiling a life-size model of the gigantic Squid (Architeuthis) of the North Atlantic, its suckered tentacular arms thirty feet in length. The actual existence of such monsters almost makes us forgive old Denys de Montfort for his picture of a "Poulpe Colossal" dragging down a full-rigged ship! The first horizontal cases contain shells of the Paper Nautilus; then several species of the Chambered Nautilus. A specimen of the animal of the latter (Nautilus pompilius) in its shell is one of a very few in America; though the shells are not uncommon, this remnant of a Palæozoic and Mesozoic race is rarely found in the flesh. The pelagic Pteropods are arranged after the Cephalopods, and then the air-breathing Gastropods. The latter series begins with carnivorous forms, the worm-eating genus Testacella, in which the shell is degenerate, owing to its subterranean habits, standing first,' followed by the Floridian Glandina, which has a well-developed shell, and subsists largely upon snails, swallowing them whole and digesting the soft parts out of the shells at leisure. Following these are the Achatinas of Africa, largest of land snails. The striped, oval shells are 8 or 9 inches long. With them are specimens of their eggs, hitherto, I believe, undescribed. They are about the size of a sparrow's egg, oval, with calcareous shell, and of a bright sulphur-yellow color; the only case known to me of a land snail having colored eggs. In an adjacent case are the South American Bulimi, Tomigerus and Anostoma, having upturned apertures. An Amazonian Indian who collected them said to Professor Ward, "God laughed when he made these shells." The numerous families of marine gastropods are represented by characteristic specimens, among them a good number of species which, to my knowledge, are not in any other American museum. The families Volutida, Conidee, and Muricida may be mentioned as affording valuable material. An example of Xenophora conchyliophora carried a load of rounded pebbles soldered to his shell instead of the usual disguise of shells and shell-fragments, obviously showing the character of the sea-bottom he lived upon, and an ability to adapt himself to unusual circumstances. In the Turbinida we examined the unique type of Astralium Wardii Baker, and incline to consider it a form of A. Japonicum Dkr. It will be of interest to conchologists to learn that the hitherto unknown operculum of A. modestum Rve. of Japan is represented by several specimens, and that it proves to be of the same abnormal type as that of the Mediterranean species, A. rugosum, the form and the position of the nucleus being the same in both. The operculum of A. modestum, however, is pure white, while that of the other species is scarlet. The series of Lamellibranchiata is of equally great extent. But further enumeration would be tedious. We may confidently state that those interested in science-education or in animal lifehistory and structure will find, as the writer has done, that this collection is full of most valuable suggestions and material, and will well repay a visit to Rochester. 1 The writer has recently shown that the South African genus Aerope is more highly specialized than any other carnivorous land-snail, and it should therefore be given first place in the series. 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It is invaluable to those who use it in soliciting information or seeking new positions. The name and address of applicants should be given in full, so that answers will go direct to them. The "Exchange" column is likewise open. NON-EUCLIDEAN GEOMETRY. BY G. A. MILLER, PH.D., EUREKA COLLEGE, EUREKA, ILL. EUCLID'S elementary geometry was written about three centuries before the Christian era. We must conclude that it was much superior to all preceding works on this subject. Proclus, who wrote a commentary on Euclid's Elements in the fifth century of our era, represents it such, and his statements are corroborated by the facts that all similar works of Euclid's predecessors have ceased to exist, and, if any elementary geometry was written by a Greek after Euclid, there is no mention made of this anywhere.1 The facts that Euclid's Elements are still used as a text-book especially in England -- and that the works used in its place are generally based upon it, are perhaps still stronger evidences of its excellence. No geometry can be written without making some assumptions with respect to the space with which it deals. These are generally of such a nature as to commend themselves to our full confidence by their mere mention, and are commonly called axioms. It is the duty of the geometer to demonstrate properties and relations of magnitudes by non-contradictory statements which rest ultimately upon these axioms. It is evident that the axioms should be as few and as clear as possible. Upon essentially different axioms essentially different geometries may be established. Among the axioms of Euclid there is at least one which is not axiomatic. This is the axiom of parallels, which reads as follows: "If a straight line meet two straight lines so as to make the two interior angles on the same side of it taken together less than two right-angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right-angles." All the popular text-books on elementary geometry employ this axiom either in this form or in some shorter form, such as, "Through a point without a line only one line can be drawn parallel to the given line." Many efforts have been made to demonstrate this axiom. Since it does not depend upon more elementary axioms, such attempts must be futile. If we assume it to be true, it follows directly that the sum of the three angles of a plane triangle is two rightangles; and, conversely, if we should assume that the sum of the internal angles of a plane triangle is two right-angles, this axiom would follow.3 As the geometers who do not adopt all the axioms of Euclid deny this, non-Euclidean geometry is sometimes defined as the geometry which does not assume that the sum of the three angles of a plane triangle is two right-angles. A more satisfactory defiCantor's Vorlesungen über Geschichte der Mathematik, Vol. I., p. £24. 2 Encyclopædia Britannica, Vol. VIII., p. 657. 3 Frischauf's Absolute Geometrie, pp. 14, 15. nition is, non-Euclidean geometry is a geometry which assumes other properties of space in place of the following properties of Euclidean space: The sum of the three angles of a plane triangle is two rightangles, space is an infinite continuity of three dimensions, and rigid bodies may be moved in every way in space without change of form. Just one hundred years ago (1792) the famous mathematician Gauss began the study of a geometry free from the first of these assumptions. He did not publish the results of his study. We may infer something in regard to them from his letters It was not until 1840 that a geometry was published in which Euclid's axiom of parallels was replaced by another, and the sum of the angles of a plane finite triangle was thus shown to be less than two right angles. The work was written by a Russian mathematician named Lobatschewsky. It contains only sixty-one pages and bears the title "Geometrische Untersuchungen zur Theorie der Parallellinien." He began his treatment of parallels by observations, in substance, as follows: Given a fixed line (L) and a fixed point (A) not on this line. The lines through A lying in the plane determined by A and L may be divided with respect to L into two classes (1) those intersecting L, and (2) those not intersecting L. The assumption that the second class consists of the single line which is at rightangles with the perpendicular from A to L is the foundation of a great part of the ordinary geometry and plane trigonometry. While the assumption that the second class consists of more than one line leads to a newer geometry, whose results are also free from contradictions." This newer geometry was called nonEuclidean geometry by Gauss, imaginary geometry by Lobatschewsky, and absolute geometry by Johann Bolyai." It is certainly of interest to learn what some of the foremost mathematicians have said with respect to this geometry. Professor Sylvester said in regard to Lobatschewsky's work :— "In quaternions the example has been given of algebra released from the yoke of the commutative principle of multiplication— an emancipation somewhat akin to Lobatschewsky's of geometry from Euclid's noted empirical axiom." "What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatschewsky to Euclid." Something of the nature of this geometry may be inferred from a few of its theorems which differ from the corresponding theo rems of the ordinary geometry. In addition to the important theorem that the sum of the internal angles of a plane finite triangle is less than two right-angles, it is proved that if we have given a line (L) and a perpendicular (B) to L, the parallels to L through points on B will make angles with B varying from to 0; so that we can draw through B a parallel to L making any given angle with B." The locus of a point at a constant distance from a straight line is a curved line." The areas of two plane triangles are to each other in the ratio of the excesses of two right-angles over the sums of their angles.' We proceed now to some observations on the second property of Euclidean space mentioned above, viz., that space is an infinite continuity of three dimensions. We shall not take up the question of the infinitude of space nor Riemann's distinction between • Briefwechsel zwischen Gauss und Schumacher, especially Vol. II., pp. 268-271. 5 Lobatschewsky's Theorie der Parallellinien, Art. 22. • Frischauf's Absolute Geometrie, Art. 13. 7 Lobatschewsky's Theorie der Parallellinien, Art. 23. Frischauf's Absolute Ge, metrie, p. 18. 9 Frischauf's Absolute Geometrie, p. 50. |