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Before directing the workmen, however, to remove any of these stalagmitic accumulations, the Superintendents carefully examined them for inscriptions. Nevertheless, one inscription was overlooked—that already referred to as the only specimen of the kind within the Labyrinth ; and it was not until a portion of the largest boss was blasted off that it was found to have on it “G. Knight, June 1, 1836."

The upper surface of both the Cave-earth and the Breccia rose, with some irregularities, 38 inches from the mouth of the Labyrinth to its innermost extremity, giving a mean ascending gradient of about 1 in 17.

The total number of " finds” in this branch of the Cavern was 135, and the specimens they included were as follow :-

Lying on the surface.—Three portions of ribs and two other bones (No. 6780), the two latter having been cut with a sharp tool, perhaps by an existing butcher, and one bone of Bat in a heap of “ Pipes” of Stalactite, probably collected by man.

In the Granular Stalagmite.--One tooth of Lion.

In the Cave-earth.32 teeth of Hyæna, 7 of Bear, 6 of Fox, 3 of Horse, 2 of Rhinoceros, 3 plates of a molar of a young Mammoth, 1 of Lion, 1 of Ox, and 1 of Sheep (of doubtful position); several bones and portions of bone, including a tarsus of Bird, and two pieces of bone apparently charred; 1 coprolite; and 1 small chip of Int.

In the Crystalline Stalagmite.—6 teeth of Bear, of which 5 were in one and the same jaw.

In the Breccia.215 teeth of Bear, and a considerable number of bones, of which many are good specimens.

As in all other parts of the Cavern where he had made researches, Mr. MacEnery simply cast aside the material he dug up, without taking it to the exterior for final examination. The Superintendents took outside the Cavern the “broken ground” met with in the Labyrinth and examined it carefully by daylight, as in all previous cases of the kind. It yielded 17 teeth of Bear, 14 of Hyæna (three of them in pieces of jaws), 2 of the Gigantic Irish Deer (in part of a jaw), 1 of Deer, 1 of Horse, 1 of Sheep; bones and pieces of bone ; and part of a Crab's claw, no doubt quite recent.

The exploration of the Labyrinth, commenced on October 28, 1875, was completed on July 10, 1876, upwards of 8 months having been spent on it.

Matthews's Passage.—Having finished their researches in the Labyrinth, the Committee proceeded at once to explore the small branch leading from it to the Bear's Den, and termed, as already stated, Matthews's Passage, thus leaving the two other and adjacent small ramifications to be undertaken on some future occasion. To this course they were tempted partly on account of the severe and protracted labour which, from their very limited breadth and the character of their deposits, must attend the excavation of these branches, and partly by the wealth of osseous remains which, from Mr. MacEnery's description, they are likely to find in the Bear's Den.

Matthews's Passage consists of two Reaches: the first, opening out of the Labyrinth, extends for about 14 feet towards the south-east, where the second turns sharply towards east-north-east, and after a somewhat tortuous course for about 15 feet, enters the Bear's Den. Their height is from 9 to 10 feet almost everywhere (measuring, as usual, from the bottom of the excavation, which nowhere reaches the limestone floor), and they vary from 3.5 feet to 7 feet in width. The walls and roof, the latter especially, bear evident traces of the erosive action of a flowing stream, succeeded by the corrosion due, no doubt, to acidulated water, as the surfaces are much fretted. Holes, having the aspect of mouths of small watercourses, open out of the walls and roof in various places; and about midway in the Second Reach the roof rises into a small water-worn dome, from the apex of which a cylindrical flue ascends into the limestone, and, like the watercourses just mentioned, is quite empty.

There were but scanty traces of a Stalagmitic Floor in the First Reach, in which, however, the earlier explorers had here and there broken ground; but throughout the entire length of the Second Reach a Floor of Granular Stalagmite extended from wall to wall, varying from 10 to 24 inches in thickness; and at about 10 feet from its entrance there was also a portion of a Floor of Crystalline or old Stalagmite adhering to the left wall, whence it probably never extended to the opposite side. It was about 15 inches thick, below and almost in contact with the Granular Floor, but separated from it by a layer of Cave-earth about one inch thick.

The mechanical deposits in the First Reach were the usual thin layer of Cave-earth above, and the Breccia of unknown depth below; but in the Second Reach the space beneath the Stalagmitic Floor was mainly occupied with large loose masses of limestone, some of which required to be blasted more than once in order to remove them. The spaces between them were filled with Cave-earth or Breccia, with comparatively few specimens of any kind.

The upper surface of the Cave-earth was almost perfectly horizontal in the First Reach ; but in the Second there was a gradual and total ascent of 27 inches, giving a mean gradient of about 1 in 7 for that Reach.

Matthews's Passage yielded a total of 49 “ finds," consisting of specimens which may be thus distributed :

In the Cave-earth.-26 teeth of Hyæna (some of them in portions of jaws), 2 of Bear, 1 of an immature Mammoth, 1 of Fox, and a considerable number of bones, many of them being more or less broken and a few of them gnawed.

In the Breccia.100 teeth of Bear and a large number of bones, including many good specimens. The richest “finds" were met with in a small narrow recess in the outer angle at the junction of the two Reaches, where the teeth and bones lay huddled confusedly together, suggesting that a rush of water had probably carried them to the spot they occupied.

No trace of man was detected in any part of this branch of the Cavern.

The exploration of Matthews's Passage, begun on 11th July, 1876, was completed on 31st August, having occupied about 7 weeks: and operations were commenced in the Bear's Den on 1st September.

In looking over the work accomplished, and the discoveries made, since the Eleventh Report was presented at Bristol in 1875, the following noteworthy facts present themselves :

1st. In their Eleventh Report the Committee sketched the distribution in the Cavern of the remains of the various species of Mammals which characterize the Cave-earth. Of this sketch the following is a brief summary : The Hyæna had been met with wherever the Cave-earth was found; the Hare had not been detected anywhere in the “ Western Division” of the Cavernthat most remote from the external entrances; the Badger, Wolf, and Ox had not been found beyond the “Charcoal Cave;” and relics of Horse, Rhinoceros, Deer, Fox, Elephant, and Lion had not appeared beyond the • Long Arcade.”

The discoveries which have since been made require that this sketch should be corrected in the following particulars :- Remains of Ox, Horse, Rhinoceros, Deer (?), Fox, Elephant, and Lion have all now been found beyond the Long Arcade, in one or more of the three branches of the Cavern explored since the Bristol Meeting. In all other particulars the distribution remains at present as sketched in 1875.

2nd. No tooth, or, so far as is at present known, other trace of Machairodus latidens has been met with since the last Report was drawn up. In short, the only evidence of the presence in the Cavern of this extinct species of Mammal which the Committee have detected during the continuous labour of almost twelve years, is the one solitary, but well-marked, incisor found 29th July, 1872-a fact well calculated to impress one with the unsatisfactory nature of merely negative evidence. It cannot be doubted that had this comparatively small specimen been overlooked, the palæontologists who, prior to its discovery, were sceptical respecting the occurrence of Machairodus in Kent's Holo, as stated by Mr. MacEnery, would have believed their scepticism to be strongly confirmed by the labours of your Committee, whilst the number of their followers would have been greatly increased.

3rd. As has been already stated, the Committee commenced the exploration of the Labyrinth on 28th October, 1875, and from that time to 31st August, 1876 (a period of upwards of ten months), they were occupied in it and in Matthews's Passage, both of which they completely explored; yet, during all that time, and in those two important branches of the Cavern, they found no trace whatever of prehistoric man. Had your Committee, on receiving their appointment from the British Association in 1864, commenced their researches in either of the branches just named (and such a course was by no means without its advocates), instead of beginning at the externál mouth of the Cavern and proceeding thence steadily through the successive chambers and galleries, there can be little or no doubt that Kent's Hole would have been pronounced to be utterly destitute of any evidence on the question of Human Antiquity, and but poorly furnished with the remains of extinct Mammalia. The work would probably have been closed without going further, to the great loss of Anthropology and Palæontology, as well as of popular education in these important branches of science.

Report of the Committee, consisting of Prof. SYLVESTER, Prof.

CAYLEY, Prof. Hirst, Rev. Prof. BARTHOLOMEW PRICE, Prof. H.J. S. SMITH, Dr. SPOTTISWOODE, Mr. R. B. HAYWARD, Dr. SALMON, Rev. Prof. R. TOWNSEND, Prof. FULLER, Prof. KELLAND, Mr. J.M. Wilson, Prof. HENRICI, Mr.J.W.L.GLAISHER, and Prof. CLIFFORD, appointed for the purpose of considering the possibility of Improving the Methods of Instruction in Elementary Geometry, and reappointed to consider the Syllabus drawn up by the Association for the Improvement of Geometrical Teaching, and to report thereon. Drawn up by Mr. HAYWARD. In a previous Report (Report for 1873, p. 459) the Committee recognized the fact that the main practical difficulty in effecting an improvement in the existing methods of teaching elementary geometry is that of reconciling the

claims of the teacher to greater freedom with the necessity of one fixed and definite standard for examination purposes. They also expressed their conviction that “no text-book that has yet been produced is fit to succeed Euclid in the position of authority ;” and that in the absence of such a text-book, whether the existence of a standard authority in the future such as Euclid has been in the past be regarded as desirable or not, it is important to secure “ the requisite degree of uniformity and no more by the publication of an authorized Syllabus” of propositions in a definite sequence, which should be regarded as a standard sequence for examination purposes, and subject to which alone any amount of variety in demonstration and general treatment of the subject should be admissible.

As it was understood that the Association for the Improvement of Geometrical Teaching was engaged in the task of drawing up such a Syllabus, no further action was taken by the Committee until the present year, when, the Syllabus having been completed and published, they have proceeded to consider the same in accordance with the instructions contained in the resolution reappointing the Committee.

The Committee have not considered it to be their duty to examine the Syllabus in minute detail, but rather to report on its general character and its fitness as a basis for an authorized standard sequence of propositions.

The Committee have no hesitation in stating at the outset, as the result of their consideration of the Syllabus as a whole, that it appears to have been drawn up with such care, and with such regard to the essential conditions of the problem, as to render it highly desirable that it should be considered in detail by authorized representatives of the Universities and the other great examining bodies of the United Kingdom with a view to its adoption, subject to any modifications which such detailed consideration may show to be necessary, as the standard for examinations in Elementary Geometry.

It may be well to observe that the adoption of this or some such standard Syllabus would not necessitate the abandonment of the • Elements of Euclid' as a text-book by such teachers as still preferred it to any other, as it would at the utmost involve only such supplementary teaching as is contained in the notes appended to many of the editions of Euclid now in use; while it would greatly relieve that large and increasing body of teachers, who demand greater frecdom in the treatment of geometry than under existing conditions they can venture to adopt.

Having thus expressed their opinion of the general merits of the Syllabus as a whole, the Committee have only further to add a few remarks on its more important features, which may serve to call attention to those points in which it differs from Euclid, and which give it a claim on the consideration of all who are interested in the improvement of instruction in Geometry.

1. Geometrical Constructions. It has been found, in the experience of many who have taught Geometry to young beginners, that the attainment of a firm grasp of its fundamental conceptions and methods is much facilitated by a series of exercises in constructions made with the ruler and compasses, such exercises being given either as preliminary to, or simultaneously with, the study of the earlier parts of Theoretical Geometry. A judicious selection of such exercises is prefixed to the Syllabus ; and the Committee remark with approval that here, as well as in the Postulates of Book I., the use of the compasses for direct transference of distances is formally admitted.

2. Logical Introduction. The Syllabus is further prefaced with an introduction, in which are collected together and formulated the most important logical relations of the several propositions logically associated with a given proposition, namely its converse, its obverse (sometimes called its opposite), and its contrapositive. It is distinctly stated, in a note prefixed to this introduction, that it is not intended that a study of the abstract logical relations contained in it should precede the study of Geometry, but that the introduction should be referred to from time to time as instances of the applications of its principles arise, until the student obtains such a grasp of the principles and rules as to be able to apply them without difficulty. With this understanding the Committee regard the proposed logical introduction as a valuable feature of the Syllabus.

3. Separation of Theorems and ProblemsLoci. Throughout the Syllabus, the Problems, instead of being interspersed among the Theorems, are collected together in separate sections at the end of each Book. This may be regarded as equivalent to the assertion of the principle that, while Problems are from their very nature dependent for the form, and even the possibility, of their solution on the arbitrary limitation of the instruments allowed to be used, Theorems being truths involving no arbitrary element ought to be exhibited in a form and sequence independent of such limitations. In other words, constructions may be rightly assumed in the demonstrations of theorems, whether or not they have been shown previously to be capable of being effected by ruler and compasses, provided only they can be seen from the nature of the case, or be proved, to be possible. For instance, the existence of the third part of an angle being regarded as axiomatic, the impossibility of trisecting an angle with ruler and compasses only ought to form no obstacle to the proof of a theorem for which the trisection of an angle is required. It should be remembered that the acceptance of the principle here asserted by no means necessitates in teaching that separation of Theorems from Problems which seems desirable in a syllabus. It is probable that most teachers would prefer to introduce problems, not as a separate section of geometry, but rather in connexion with the theorems with which they are essentially related. The Syllabus in this respect leaves complete freedom to the teacher.

The early introduction of the notion of a Locus and its use in the solution of problems by the intersection of Loci the Committee regard with favour; and they observe with satisfaction that the Syllabus rightly insists on the demonstration of two theorems (a theorem, and either its converse or its obverse) as necessary for the complete establishment of a locus, a point which is too often neglected in the investigation of loci.

4. Book I. The Straight Line. The Definitions are substantially those of Euclid. An attempt to give a real definition of a straight line (Euclid's is only verbal) is to be commended, though the wording is difficult, and would for a beginner require detailed and familiar explanation.

The definition of an angle is another of the elementary difficulties of Geometry. The Syllabus in a note asserts that “an angle is a simple concept incapable of definition, properly so called," but enters into a somewhat detailed

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