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explanation in which the notion of rotation is freely but judiciously used. The Syllabus does not (like Euclid) limit the notion of an angle to one less than two right angles, but it does not explicitly recognize an angle greater than four right angles. Possibly, considering the difficulties of expression which the complete notion of an angle of unlimited magnitude involves, this limitation at the outset is wise. The Committee note with approval the use of the term conjugate for the two angles which, being contained by the same pair of lines drawn from a point, together make up four right angles.
They also approve the introduction of the term “identically equal” for figures which, differing only in respect of position, can be made to coincide with one another, while the term "equal” is reserved for such as are equal in area, but not necessarily in other respects.
The Syllabus divides the Axioms (as, indeed, Euclid did) into General Axioms (Euclid's kouvai ēvvocal), which find their fitting place in the Logical Introduction, and specially Geometrical Axioms (Euclid's airhuara), which are nearly those of Euclid—that about the equality of right angles being omitted, while that asserting that “two straight lines cannot enclose a space” is extended so as to assert coincidence beyond as well as between the two points which coincide.
The Postulates are those of Euclid's · Elements, with a modification in the third postulate, which admits of the direct transference of distances by the compasses, as before remarked.
The Theorems of Book I. are mainly those of Euclid I. 1-34, rearranged. The guiding principle of the rearrangement appears to have been the nearness or remoteness of the theorems from the possibility of proof by the direct application of the fundamental principle of superposition, the free use of this principle being indicated as desirable in many cases where Euclid prefers to keep it out of sight.
The discussion of the cases of identical equality of two triangles is rendered complete by the introduction of a theorem asserting the true conclusion from the equality of two sides and a non-included angle in each, namely, that the other non-included angles are either equal or supplementary, and that in the former case only are the triangles identically equal.
For the treatment of Parallels, Playfair's Axiom that “Two straight lines that intersect one another cannot both be parallel to the same straight line," has been substituted for Euclid's twelfth Axiom, and, in the opinion of the Committee, judiciously. It may, in fact, be regarded as merely an improved form of that axiom.
5. Book II. Areas. This book contains in thirteen Theorems the various theorems contained in Euclid between I. 35 and the end of Book II. Beyond noting the fact that it brings together more completely than in Euclid those theorems which are naturally related to one another, no comment is necessary which is not of the nature of that detailed criticism which the Committee do not think it their duty to offer.
6. Book III. The Circle. In this Book the sequence of Theorems differs materially from that of Euclid, those propositions being placed first which are fundamental in the sense that they follow directly from superposition. Other criticisms which might be offered on this part of the Syllabus are chiefly on points of detail on which the Committee think it unnecessary here to enter.
They would remark, however, with respect to the two modes of treatment of tangents in the Syllabus, that they would not recommend the second (depending on the notion of limits) in any case as a substitute for the first, however desirable it may be that it should be freely used by way of illustration and as leading up to the methods of Higher Geometry.
7. Books IV and V. Ratio and Proportion, and their application to Geometry
A theory of Proportion which shall be at once perfectly rigorous and complete is necessarily difficult. The Committee recognize with satisfaction that the Syllabus does not attempt to attain simplicity by any sacrifice of rigour, nor in Book IV. by any sacrifice of completeness. In Book IV. the theory is essentially that of Euclid in his famous, though (at the present day) little studied, Fifth Book: it is suggested, however, by an unusually full indication in this part of the Syllabus of the forms of demonstration recommended, that his theory may be presented in a form more easy to be grasped and applied be the adoption of the late Prof. de Morgan's notation, in which magnitudes are denoted by capital letters, instead of by straight lines, and their multiples by prefixing to the capitals small letters denoting integral numbers, instead of denoting them by longer lines. Opinions will probably differ as to the wisdom of retaining Euclid's treatment in any shape * ; but the Committee doubt whether any rival theory, which is equally rigorous and equally complete, would be more generally accepted.
It may, however, be thought that this complete theory is one which the ordinary student can hardly be expected to master at an early stage of his mathematical studies, even though he may be well prepared for the study of the geometrical applications of the theory of Proportion. At the same time it is undesirable that the study of Similarity of Figures &c. should be commenced without some definite groundwork of demonstrated properties of Ratios and Proportions. The Syllabus suggests a mode of meeting this difficulty by prefixing to Book V. an indication of a method of treatment of the general doctrine of proportion, in which greater simplicity is obtained, not by the sacrifice of rigour, but by a certain sacrifice of completeness, in limiting the magnitudes considered to such as are commensurable.
The notion of Ratio may be regarded as an extension and generalization of the notion of quantuplicity, the simplest expression of which is contained in the question, “How many times does a magnitude A contain another magnitude B?” This question may be generalized so as to apply to any pair of commensurable magnitudes in two ways—the question taking the shape either “ How many times does A contain some aliquot part of B?” or else “ What multiples of A and B are equal to one another ?” The former leads to a treatment of proportion such as is usually given with more or less exactness in treatises on Arithmetic or Algebra, while the latter leads to a treatment similar in principle to Euclid's, but simplified by its limitation to commensurables. The Syllabus indicates a few of the more important general properties of proportion which ought to be proved by one or other of these methods, but leaves it open to the teacher to adopt whichever mode of treatment he may prefer.
In the Geometrical Applications of Proportion the Syllabus groups together
* Prof. Cayley is strongly of opinion that it ought to be retained,
all the theorems which directly depend on the definition of proportion, indicating that the demonstrations are to be adapted to the complete or to the partial theory according as the one or other has been studied. After these follow the usual standard theorems on Similar Figures, &c., on which it is unnocessary for the Committee to offer any comment.
The Association for the Improvement of Geometrical Teaching has not yet published any Syllabus of Solid Geometry. Should the present Syllabus of Plane Geometry be successful in leading to the establishment of a standard sequence of propositions in that subject, it is to be hoped that the Association will continue its labours in the field of Solid Geometry, where the Committee believe they are equally needed.
Results of a Comparison of the British- Association Units of Electrical
Resistance. By G. CHRYSTAL and S. A. SAUNDER *.
among the Reports.] Difficulties encountered.—The difficulties of the kind of measurement we had to make are confined almost entirely to the temperature determinations. Were it not for these a much higher degree of accuracy could be attained ; for while resistances comparable with the B.A. unit can be measured without difficulty to the 100,000th part, it is very difficult to determine the temperature of a wire imbedded in paraffin, as are the wires of the standards, nearer than the one tenth of a degree Centigrade, an error to which extent entails in some of the coils an error of .03 per cent. of resistance.
A mere comparison of the coils at the temperatures given on page 483 of the B.A. Report on Electrical Standards (1867)*: would hardly have been satisfactory, since it would have given no check on the accuracy of the observations and afforded no information as to the temperature value of a variation in resistanee, and conversely.
Object aimed at. The object aimed at in the experiments was to get the differences between the resistances of the several coils at some standard temperature, and also the coefficients of variation of resistanco with temperature in the neighbourhood of the standard temperature.
That it is inadmissible to apply to any given coil the variation-coefficient for its supposed material, as found by Matthiessen and others from experiments on naked wires, is abundantly evident. This appears very strikingly in the case of coils Nos. 2 and 3 (A and B in our subsequent numbering); and an examination of the results of Lenz, Arndtsen, Siemens, and others for platinum shows that within certain limits its behaviour is very uncertain. This arises no doubt from the presence of more or less iridium or other platinoids, a small admixture of which, without altering the value of platinum commercially, affects its electric resistance very considerably.
* In the spring of last year a series of experiments was made by one of the authors (G. Chrystal) with a view of comparing the different resistance-coils of the set of BritishAssociation units formerly deposited at Kew Observatory and now in the Cavendish Laboratory at Cambridge. In the month of October a final set of experiments was made, which was the work of both of us, sometimes working together and sometimes separately.
t Or Reprint, p. 146.
Preliminary experiments.--The preliminary experiments gave the differences between the coils and the variation-coefficients approximately.
The results appeared in some cases different from former measurements, so that it was thought better not to rely on these, but to make a more careful set of experiments on which to found the final comparison.
Approximate coefficient of “ Flat Coil" and Middle Coils.--The variationcoefficient of the “Flat Coil” was taken from the preliminary experiments, This was given by a fairly good series of experiments; and a first approximation was considered sufficient, since the coil during the final experiments never varied in temperature more than two degrees, being always bathed in the tap-water. A similar remark applies to the middle coils. The coils used for middle coils were 29 and 43 (F and G) when neither of these was being measured, in which case 2 and 3 (A and B) were used. The coefficients of these coils, so far as required for small temperature-corrections, were taken from the preliminary experiments.
Method of experimenting. The method used in the final experiments was as follows:
First. All the coils (the flat coil, the two middle coils, and the coil to be compared with the flat coil) were bathed in a stream of tap-water, the temperature of which was carefully taken by means of a Casella's thermometer (lent us by Mr. Gordon), reading to tenths of a degree Centigrade and easily estimable to hundredths. After the temperature of the stream had been constant for twenty minutes or so, the difference between the coil to be compared and the flat coil was found.
Secondly. Another series of experiments was made in which the flat coil and the middle coils were kept at the temperature of the tap as before; but the remaining coil was raised by careful nursing, which lasted two hours or more, to the temperature (or to one of the temperatures) at which, according to the B.A. Report, it is correct.
Lastly. The coils were compared with each other at the standard temperatures, the middle coils being kept at the temperature of the tap-water.
Variation-coefficients, how found.—The first two sets were used to give the variation-coefficients, being peculiarly fitted to do so, because in them the temperature of the flat coil did not alter much in comparison with the alteration in the coil compared with it.
Differences between the coils, how found.—Then using the low-temperature experiments the differences of resistance between the respective coils and the flat coil (all at 10° C.) were found.
Control experiments, how used. From this, of course, the difference between any two coils at any temperatures could be calculated. This was done for the old standard temperatures, and the results compared with the results of direct experiment obtained from our third set of experiments. This gave a test of the accuracy of our work; and it is on this mainly that we rely in claiming to have stated the temperatures at which the coils are equal within 0°1 C. in all cases.
Degree of accuracy.--The degree of accuracy of resistance varies, of course, for the different coils. For the platinum units 0°1 C. corresponds to a variation of .03 per cent. resistance, for the platinum-silver to about .002 per cent.
In the B.A. Report, 1865 (p. 303)*, all the coils are stated to be accurate at the temperatures indicated within :01 per cent. This corresponds to about one thirtieth of a degree Centigrade for the platinum units. It is not stated
* Reprint, p. 137
how this degree of accuracy was attained. Some such statement was perhaps necessary, considering the difficulty of controlling the temperature of an inaccessible wire, even within 1° Centigrade.
Arrangement &c. of apparatus.—The instruments used in these experiments for resistance measurements were the Wheatstone's bridge and Thomson's galvanometer belonging to the Association. The arrangements in the low-temperature experiments were as in the annexed figure. At one corner of a large
table is the bridge A B (see B.A. Report, 1864, p. 353*); by means of mercurycups at Dand'G, are inserted the flat coil and the coil being compared with it; at E and F are similarly inserted the middle coils, which were always two of the units, as small and as nearly equal in temperature-variation as possible. X, Y, Z are three earthenware jars in which the coils are placed ; these stand in a trough, V W, provided with a waste-pipe going to the sink. The jars were kept constantly overflowing by means of a feed-pipe fitted with an offset for each. The temperature in all three jars was carefully observed, and it was found that after the tap had been turned on for fifteen minutes or so the temperature in all three in general became constant, and remained so within a tenth of a degree for a long time. Now and then irregularities occurred, which caused the rejection of the results concerned. Thin wires go from E and from the contact-block C to the galvanometer at the other end of the table. The last adjustments of the balance were made by observing the spot on the galvanometer-scale with the telescope from where the observer sits. The battery-circuit terminates at H and J, and is made and broken by means of a treadle worked by the observer's foot. A small Leclanché's cell was found sufficient to indicate a deviation from balance of a tenth of a millimetre on the bridge-scale. Since the contact of the block-piece could not be relied on within less than this, no higher batterypower was ever used.
Thermoelectric disturbances.—To avoid thermoelectric currents, owing to the junction of copper with brass at the block, the button of the block-piece was never touched by the fingers, but always by means of two pieces of wood, which were exchanged now and again to prevent heating. It was
* Reprint, p. 119.